Subgroups — #28318856

Order 13568.

970 conjugacy classes of subgroups.

The subgroup lattice has 780 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅₃⋊#435	C₂	1	1	yes	maximal, characteristic
2	#20291169	C₂	1	1	yes	maximal, characteristic
3	#20290516	C₂	1	1	yes	maximal, characteristic, complemented
4	#20290509	C₂	1	1	yes	maximal, characteristic, complemented
5	#20290504	C₂	1	1	yes	maximal, characteristic, complemented
6	#20290458	C₂	1	1	yes	maximal, characteristic, complemented
7	#20290455	C₂	1	1	yes	maximal, characteristic, complemented
8	#20290455	C₂	1	1	yes	maximal, characteristic, complemented
9	#20290454	C₂	1	1	yes	maximal, characteristic, complemented
10	#20290446	C₂	1	1	yes	maximal, characteristic, complemented
11	#20290443	C₂	1	1	yes	maximal, characteristic, complemented
12	#20290433	C₂	1	1	yes	maximal, characteristic, complemented
13	#20290383	C₂	1	1	yes	maximal, characteristic, complemented
14	#20290377	C₂	1	1	yes	maximal, characteristic, complemented
15	#436×D₅₃	C₂	1	1	yes	maximal, characteristic
16	C₂×D₅₃⋊M₁₆	C₂²	1	1	yes	characteristic
17	C₂²×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×#13133296	C₂²	1	1	yes	characteristic
19	C₂²×C₄×#6157	C₂²	1	1	yes	characteristic
20	#16811655	C₄	1	2	yes	complemented
21	#16811654	C₄	1	2	yes	complemented
22	#16811653	C₄	1	2	yes	complemented
23	#16811652	C₄	1	2	yes	complemented
24	D₅₃⋊(C₄⋊C₈)	C₂²	1	1	yes	characteristic
25	D₅₃⋊(C₂²⋊C₈)	C₂²	1	1	yes	characteristic
26	D₅₃⋊(C₂²⋊C₈)	C₂²	1	1	yes	characteristic
27	#16811582	C₂²	1	1	yes	characteristic
28	C₂×#13133271	C₂²	1	1	yes	characteristic
29	C₂×#13133271	C₂²	1	1	yes	characteristic
30	C₂×#13133271	C₂²	1	1	yes	characteristic
31	C₂×#13133271	C₂²	1	1	yes	characteristic
32	C₂×C₈×#6157	C₂²	1	1	yes	characteristic
33	C₂×C₈×#6157	C₂²	1	1	yes	characteristic
34	#16811573	C₄	1	2	yes	—
35	#16811572	C₄	1	2	yes	—
36	#16811571	C₄	1	2	yes	—
37	#16811570	C₄	1	2	yes	—
38	C₄×C₄⋊#6157	C₂²	1	1	yes	characteristic
39	C₄×C₂²⋊#6157	C₂²	1	1	yes	characteristic
40	C₄×C₂²⋊#6157	C₂²	1	1	yes	characteristic
41	#16811559	C₂²	1	1	yes	characteristic
42	(C₂×#6157)⋊C₈	C₂²	1	1	yes	characteristic
43	(C₂×#6157)⋊C₈	C₂²	1	1	yes	characteristic
44	(C₂×#6157)⋊C₈	C₂²	1	1	yes	characteristic
45	(C₂×#6157)⋊C₈	C₂²	1	1	yes	characteristic
46	(C₂×#6157)⋊C₈	C₂²	1	1	yes	characteristic
47	(C₂×#6157)⋊C₈	C₂²	1	1	yes	characteristic
48	(C₂×#6157)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×#6157)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×#6157)⋊C₈	C₂²	1	1	yes	characteristic
51	(C₂×#6157)⋊C₈	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×D₅₃	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×D₅₃	C₂²	1	1	yes	characteristic
54	C₄×D₄×D₅₃	C₂²	1	1	yes	characteristic
55	#16810702	C₄	1	1	yes	characteristic, complemented
56	#16810701	C₄	1	1	yes	characteristic, complemented
57	#16810700	C₄	1	1	yes	characteristic, complemented
58	#16810699	C₄	1	1	yes	characteristic, complemented
59	#16810699	C₄	1	1	yes	characteristic, complemented
60	#16810698	C₄	1	1	yes	characteristic, complemented
61	#16810698	C₄	1	1	yes	characteristic, complemented
62	C₄⋊C₈×D₅₃	C₂²	1	1	yes	characteristic
63	C₂²⋊C₈×D₅₃	C₂²	1	1	yes	characteristic
64	C₂²⋊C₈×D₅₃	C₂²	1	1	yes	characteristic
65	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
66	C₅₃×#436	C₄	1	1	yes	characteristic, complemented
67	C₂³×#6157	C₂×C₄	1	2	yes	—
68	C₂×#11719936	C₂×C₄	1	2	yes	—
69	D₅₃⋊M₁₆	—	4	1	no	—
70	D₅₃⋊M₁₆	—	4	1	no	—
71	C₂×#11719940	C₂×C₄	1	2	yes	—
72	C₂×D₅₃⋊C₈	—	4	1	no	—
73	C₂×D₅₃⋊C₈	—	4	1	no	—
74	C₂×D₅₃⋊C₈	C₂×C₄	1	2	yes	complemented
75	C₂×D₅₃⋊C₈	C₂³	1	1	yes	characteristic
76	C₂×D₅₃⋊C₈	C₂³	1	1	yes	characteristic
77	C₂×D₅₃⋊C₈	C₂³	1	1	yes	characteristic
78	C₂×D₅₃⋊C₈	C₂³	1	1	yes	characteristic
79	C₂²×#88324	C₂×C₄	1	2	yes	complemented
80	#13133296	—	4	1	no	—
81	#13133296	—	4	1	no	—
82	C₂×C₄⋊#6157	C₂×C₄	1	2	yes	—
83	C₂×C₄⋊#6157	C₂×C₄	1	2	yes	—
84	C₂×C₂²⋊#6157	C₂×C₄	1	2	yes	—
85	C₂×C₂²⋊#6157	C₂×C₄	1	2	yes	—
86	C₂×C₂²⋊#6157	C₂×C₄	1	2	yes	—
87	C₂×C₄×#6157	—	4	1	no	—
88	C₂×C₄×#6157	—	4	1	no	—
89	C₂×C₄×#6157	C₂×C₄	1	2	yes	—
90	C₂×C₄×#6157	C₂³	1	1	yes	characteristic
91	C₂×C₄×#6157	C₂³	1	1	yes	characteristic
92	C₂×C₄×#6157	C₂³	1	1	yes	characteristic
93	C₂×C₄×#6157	C₂³	1	1	yes	characteristic
94	#13133284	C₂×C₄	1	2	yes	complemented
95	C₄⋊#88324	C₂×C₄	1	2	yes	complemented
96	#13133274	C₂×C₄	1	2	yes	—
97	#13133274	C₂×C₄	1	2	yes	complemented
98	C₂²⋊#88324	C₂×C₄	1	2	yes	—
99	C₂²⋊#88324	C₂×C₄	1	2	yes	complemented
100	#13133272	C₂×C₄	1	2	yes	complemented
101	#13133272	C₂×C₄	1	2	yes	complemented
102	#13133271	—	4	1	no	—
103	#13133271	—	4	1	no	—
104	#13133271	—	4	1	no	—
105	#13133271	—	4	1	no	—
106	C₈×#6157	—	4	1	no	—
107	C₈×#6157	—	4	1	no	—
108	(C₂×#6157)⋊C₄	C₂×C₄	1	2	yes	—
109	(C₂×#6157)⋊C₄	C₂×C₄	1	2	yes	—
110	(C₂×#6157)⋊C₄	C₂×C₄	1	2	yes	—
111	(C₂×#6157)⋊C₄	C₂×C₄	1	2	yes	—
112	(C₂×#6157)⋊C₄	C₂×C₄	1	2	yes	—
113	C₂×D₄×D₅₃	C₂×C₄	1	1	yes	characteristic
114	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
115	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
116	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
117	C₂×#11719917	C₂×C₄	1	1	yes	characteristic, complemented
118	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
119	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
120	C₂×#11719912	C₂×C₄	1	1	yes	characteristic, complemented
121	C₂×#11719912	C₂×C₄	1	1	yes	characteristic, complemented
122	M₁₆×D₅₃	—	4	1	no	—
123	M₁₆×D₅₃	—	4	1	no	—
124	M₁₆×D₅₃	—	4	1	no	—
125	M₁₆×D₅₃	—	4	1	no	—
126	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
127	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
128	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
129	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
130	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
131	D₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
132	#13133179	C₂×C₄	1	1	yes	characteristic
133	C₄×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
134	C₄×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
135	#13133177	C₂×C₄	1	1	yes	characteristic
136	#13133177	C₂×C₄	1	1	yes	characteristic
137	C₄⋊C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
138	C₂²⋊C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
139	C₂²⋊C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
140	C₄²×D₅₃	C₂³	1	1	yes	characteristic
141	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic, complemented
142	#13133146	C₂×C₄	1	1	yes	characteristic, complemented
143	#13133146	C₂×C₄	1	1	yes	characteristic, complemented
144	C₂²⋊#88322	C₂×C₄	1	1	yes	characteristic
145	C₂²⋊#88322	C₂×C₄	1	1	yes	characteristic, complemented
146	#13133129	C₂×C₄	1	1	yes	characteristic
147	#13133129	C₂×C₄	1	1	yes	characteristic
148	#13133129	C₂×C₄	1	1	yes	characteristic, complemented
149	#13133129	C₂×C₄	1	1	yes	characteristic, complemented
150	C₄×#88322	C₂×C₄	1	1	yes	characteristic, complemented
151	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic, complemented
152	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic, complemented
153	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
154	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
155	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
156	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
157	C₂²⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
158	C₂²⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
159	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
160	C₂³×D₅₃	C₂²×C₄	1	1	yes	characteristic
161	C₂³×D₅₃	C₂²×C₄	1	1	yes	characteristic
162	C₂²×#6157	—	4	2	no	—
163	C₂²×#6157	—	4	2	no	—
164	C₂²×#6157	—	4	2	no	—
165	C₂²×#6157	C₂²×C₄	1	2	yes	—
166	C₂²×#6157	C₂²×C₄	1	2	yes	—
167	C₂²×#6157	C₂²×C₄	1	2	yes	—
168	C₂²×#6157	C₂²×C₄	1	2	yes	—
169	C₂×D₄×C₅₃	C₄²	1	1	yes	characteristic
170	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
171	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
172	C₂×C₂²⋊D₅₃	C₄²	1	1	yes	characteristic
173	C₂×C₂²⋊D₅₃	C₄²	1	1	yes	characteristic
174	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
175	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
176	D₄×D₅₃	—	4	2	no	—
177	C₂×D₂₁₂	C₄²	1	1	yes	characteristic
178	C₂×C₄×D₅₃	—	2	1	no	—
179	C₂×C₄×D₅₃	—	2	1	no	—
180	C₂×C₄×D₅₃	—	2	1	no	—
181	C₂×C₄×D₅₃	—	2	1	no	—
182	C₂×C₄×D₅₃	—	2	1	no	—
183	C₂×C₄×D₅₃	—	2	1	no	—
184	C₂×C₄×D₅₃	—	2	1	no	—
185	C₂×C₄×D₅₃	—	2	1	no	—
186	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
187	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
188	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
189	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
190	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
191	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
192	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
193	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
194	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
195	C₂²⋊#6157	—	4	2	no	—
196	C₂²⋊#6157	—	4	2	no	—
197	C₂²⋊#6157	—	4	2	no	—
198	#11719940	—	4	2	no	—
199	C₂×#88324	—	4	2	no	—
200	C₂×#88324	C₂²×C₄	1	2	yes	—
201	C₂×#88324	C₂²×C₄	1	2	yes	—
202	C₂×#88324	C₂²×C₄	1	2	yes	—
203	C₂×#88324	C₂²×C₄	1	2	yes	—
204	C₄⋊#6157	—	4	2	no	—
205	C₄⋊#6157	—	4	2	no	—
206	C₄×#6157	—	2	1	no	—
207	C₄×#6157	—	2	1	no	—
208	C₄×#6157	—	2	1	no	—
209	C₄×#6157	—	2	1	no	—
210	C₄×#6157	—	2	1	no	—
211	C₄×#6157	—	2	1	no	—
212	C₄×#6157	—	2	1	no	—
213	C₄×#6157	—	2	1	no	—
214	C₄×#6157	—	4	2	no	—
215	#11719936	—	4	2	no	—
216	D₅₃⋊C₈	—	2	1	no	—
217	D₅₃⋊C₈	—	2	1	no	—
218	D₅₃⋊C₈	—	2	1	no	—
219	D₅₃⋊C₈	—	2	1	no	—
220	D₅₃⋊C₈	—	2	1	no	—
221	D₅₃⋊C₈	—	2	1	no	—
222	D₅₃⋊C₈	—	2	1	no	—
223	D₅₃⋊C₈	—	2	1	no	—
224	D₅₃⋊C₈	—	4	2	no	—
225	M₁₆×C₅₃	—	4	1	no	—
226	M₁₆×C₅₃	—	4	1	no	—
227	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
228	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
229	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
230	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
231	C₄⋊C₄×C₅₃	C₄²	1	1	yes	characteristic
232	C₂²⋊C₄×C₅₃	C₄²	1	1	yes	characteristic
233	C₂²⋊C₄×C₅₃	C₄²	1	1	yes	characteristic
234	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
235	C₂²⋊Dic₅₃	C₄²	1	1	yes	characteristic
236	C₂²⋊Dic₅₃	C₄²	1	1	yes	characteristic
237	#11719921	C₄²	1	1	yes	characteristic
238	#11719921	C₄²	1	1	yes	characteristic
239	#11719921	C₄²	1	1	yes	characteristic
240	#11719921	C₄²	1	1	yes	characteristic
241	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
242	#11719919	C₄²	1	1	yes	characteristic
243	#11719919	C₄²	1	1	yes	characteristic
244	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
245	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
246	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
247	#11719917	—	4	1	no	—
248	#11719917	—	4	1	no	—
249	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
250	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
251	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
252	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
253	#11719912	—	4	1	no	—
254	#11719912	—	4	1	no	—
255	#11719912	—	4	1	no	—
256	#11719912	—	4	1	no	—
257	C₈×D₅₃	—	2	1	no	—
258	C₈×D₅₃	—	2	1	no	—
259	C₈×D₅₃	—	2	1	no	—
260	C₈×D₅₃	—	2	1	no	—
261	C₈×D₅₃	—	2	1	no	—
262	C₈×D₅₃	—	2	1	no	—
263	C₈×D₅₃	—	2	1	no	—
264	C₈×D₅₃	—	2	1	no	—
265	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
266	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
267	C₂²×D₅₃	—	2	1	no	—
268	C₂²×D₅₃	—	2	1	no	—
269	C₂²×D₅₃	—	2	1	no	—
270	C₂²×D₅₃	—	2	1	no	—
271	C₂²×D₅₃	—	2	2	no	—
272	C₂²×D₅₃	—	2	2	no	—
273	C₂²×D₅₃	—	2	2	no	—
274	C₂²×D₅₃	—	2	2	no	—
275	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
276	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
277	C₂²×D₅₃	C₂³×C₄	1	1	yes	characteristic
278	C₂×#6157	—	2	2	no	—
279	C₂×#6157	—	2	2	no	—
280	C₂×#6157	—	2	2	no	—
281	C₂×#6157	—	2	2	no	—
282	C₂×#6157	—	2	2	no	—
283	C₂×#6157	—	2	2	no	—
284	C₂×#6157	—	2	2	no	—
285	C₂×#6157	—	2	2	no	—
286	C₂×#6157	—	2	2	no	—
287	C₂×#6157	—	2	2	no	—
288	C₂×#6157	—	2	2	no	—
289	C₂×#6157	—	2	2	no	—
290	C₂×#6157	—	4	8	no	—
291	D₄×C₅₃	—	4	1	no	—
292	C₂×C₄×C₅₃	—	2	1	no	—
293	C₂×C₄×C₅₃	—	2	1	no	—
294	C₂×C₄×C₅₃	—	2	1	no	—
295	C₂×C₄×C₅₃	—	2	1	no	—
296	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
297	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
298	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
299	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
300	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
301	C₂²⋊D₅₃	—	4	1	no	—
302	C₂²⋊D₅₃	—	4	1	no	—
303	C₂×Dic₅₃	—	2	1	no	—
304	C₂×Dic₅₃	—	2	1	no	—
305	C₂×Dic₅₃	—	2	1	no	—
306	C₂×Dic₅₃	—	2	1	no	—
307	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
308	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
309	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
310	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
311	C₂×Dic₅₃	C₂³×C₄	1	1	yes	characteristic
312	D₂₁₂	—	4	1	no	—
313	C₄×D₅₃	—	2	1	no	—
314	C₄×D₅₃	—	2	1	no	—
315	C₄×D₅₃	—	2	1	no	—
316	C₄×D₅₃	—	2	1	no	—
317	C₄×D₅₃	—	2	2	no	—
318	C₄×D₅₃	—	2	2	no	—
319	C₄×D₅₃	—	2	1	no	—
320	C₄×D₅₃	—	2	1	no	—
321	C₄×D₅₃	—	2	1	no	—
322	C₄×D₅₃	—	2	1	no	—
323	C₄×D₅₃	—	2	1	no	—
324	C₄×D₅₃	—	2	1	no	—
325	C₄×D₅₃	—	2	1	no	—
326	C₄×D₅₃	—	2	1	no	—
327	C₄×D₅₃	H(5,𝔽₂)	1	1	yes	characteristic
328	C₄×D₅₃	H(5,𝔽₂)	1	1	yes	characteristic
329	C₄×D₅₃	H(5,𝔽₂)	1	1	yes	characteristic
330	C₄×D₅₃	2₋¹⁺⁴	1	1	yes	characteristic
331	#88324	—	2	2	no	—
332	#88324	—	2	2	no	—
333	#88324	—	2	2	no	—
334	#88324	—	2	2	no	—
335	C₈×C₅₃	—	2	1	no	—
336	C₈×C₅₃	—	2	1	no	—
337	C₈×C₅₃	—	2	1	no	—
338	C₈×C₅₃	—	2	1	no	—
339	#88322	—	2	1	no	—
340	#88322	—	2	1	no	—
341	#88322	—	2	1	no	—
342	#88322	—	2	1	no	—
343	#20564	—	53	1	no	maximal, Sylow
344	C₂²×C₅₃	—	2	1	no	—
345	C₂²×C₅₃	—	2	1	no	—
346	C₂²×C₅₃	—	2	1	no	—
347	C₂²×C₅₃	—	2	1	no	—
348	C₂²×C₅₃	—	2	1	no	—
349	C₂²×C₅₃	—	2	1	no	—
350	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
351	C₂×D₅₃	—	2	1	no	—
352	C₂×D₅₃	—	2	1	no	—
353	C₂×D₅₃	—	2	1	no	—
354	C₂×D₅₃	—	2	1	no	—
355	C₂×D₅₃	—	2	1	no	—
356	C₂×D₅₃	—	2	1	no	—
357	C₂×D₅₃	—	4	2	no	—
358	C₂×D₅₃	—	4	2	no	—
359	C₂×D₅₃	—	4	2	no	—
360	C₂×D₅₃	—	4	2	no	—
361	C₂×D₅₃	#518	1	1	yes	characteristic
362	C₂×D₅₃	#520	1	1	yes	characteristic
363	C₂×D₅₃	#568	1	2	yes	—
364	C₂×D₅₃	#568	1	2	yes	—
365	#6157	—	4	2	no	—
366	#6157	—	4	2	no	—
367	#6157	—	4	2	no	—
368	#6157	—	4	2	no	—
369	C₄×C₅₃	—	2	1	no	—
370	C₄×C₅₃	—	2	1	no	—
371	C₄×C₅₃	—	2	1	no	—
372	C₄×C₅₃	—	2	1	no	—
373	C₄×C₅₃	#518	1	1	yes	characteristic
374	C₄×C₅₃	#520	1	1	yes	characteristic
375	Dic₅₃	—	2	1	no	—
376	Dic₅₃	—	2	1	no	—
377	Dic₅₃	—	2	1	no	—
378	Dic₅₃	—	2	1	no	—
379	Dic₅₃	#518	1	1	yes	characteristic
380	Dic₅₃	#520	1	1	yes	characteristic
381	C₂×#436	—	53	1	no	—
382	C₂×#435	—	53	1	no	—
383	#2290	—	53	1	no	—
384	#1771	—	53	1	no	—
385	#1770	—	53	1	no	—
386	#1768	—	53	1	no	—
387	#1753	—	53	1	no	—
388	#1753	—	53	1	no	—
389	#1753	—	53	1	no	—
390	#1752	—	53	1	no	—
391	#1750	—	53	1	no	—
392	#1749	—	53	1	no	—
393	#1746	—	53	1	no	—
394	#1729	—	53	1	no	—
395	#1728	—	53	1	no	—
396	C₂×C₅₃	—	4	1	no	—
397	C₂×C₅₃	—	4	1	no	—
398	C₂×C₅₃	#2275	1	1	yes	characteristic
399	C₂×C₅₃	#2873	1	1	yes	characteristic
400	C₂×C₅₃	#2875	1	1	yes	characteristic
401	D₅₃	—	4	1	no	—
402	D₅₃	—	4	1	no	—
403	D₅₃	#2978	1	2	yes	—
404	D₅₃	#2995	1	2	yes	—
405	C₂²×M₁₆	—	53	1	no	—
406	C₂²×M₁₆	—	53	1	no	—
407	C₂²×M₁₆	—	53	1	no	—
408	C₂³×C₈	—	53	1	no	—
409	C₂×C₄×D₄	—	53	1	no	—
410	C₂×#117	—	53	1	no	—
411	C₂²×C₄²	—	53	1	no	—
412	#436	—	53	1	no	—
413	#436	—	53	1	no	—
414	#436	—	53	1	no	—
415	#436	—	53	1	no	—
416	#436	—	53	1	no	—
417	#436	—	53	1	no	—
418	#436	—	53	1	no	—
419	#436	#6157	1	1	yes	characteristic, complemented
420	#435	—	53	2	no	—
421	#435	—	53	2	no	—
422	#435	—	53	2	no	—
423	#435	—	53	2	no	—
424	C₂×C₄⋊C₈	—	53	1	no	—
425	C₂×C₄⋊C₈	—	53	1	no	—
426	C₂×C₂²⋊C₈	—	53	1	no	—
427	C₂×C₂²⋊C₈	—	53	1	no	—
428	C₂×C₂²⋊C₈	—	53	1	no	—
429	C₂×C₂²⋊C₈	—	53	1	no	—
430	C₂×#97	—	53	1	no	—
431	C₂×#97	—	53	1	no	—
432	C₂×#97	—	53	1	no	—
433	C₂×#97	—	53	1	no	—
434	C₂×#97	—	53	1	no	—
435	C₂×C₄×C₈	—	53	1	no	—
436	C₂×C₄×C₈	—	53	1	no	—
437	C₂×C₄×C₈	—	53	1	no	—
438	#388	—	53	2	no	—
439	#387	—	53	2	no	—
440	#386	—	53	2	no	—
441	#385	—	53	2	no	—
442	C₄×C₄⋊C₄	—	53	1	no	—
443	C₄×C₂²⋊C₄	—	53	1	no	—
444	C₄×C₂²⋊C₄	—	53	1	no	—
445	#376	—	53	1	no	—
446	#336	—	53	1	no	—
447	#336	—	53	1	no	—
448	#336	—	53	1	no	—
449	#336	—	53	1	no	—
450	#336	—	53	1	no	—
451	#336	—	53	1	no	—
452	#336	—	53	1	no	—
453	#336	—	53	1	no	—
454	#336	—	53	1	no	—
455	#336	—	53	1	no	—
456	C₅₃	#20564	1	1	yes	characteristic, complemented, Sylow
457	C₂²×D₄	—	53	1	no	—
458	C₂³×C₄	—	53	1	no	—
459	C₂³×C₄	—	53	1	no	—
460	C₂³×C₄	—	53	2	no	—
461	C₂×M₁₆	—	53	2	no	—
462	C₂×M₁₆	—	53	2	no	—
463	C₂×M₁₆	—	53	1	no	—
464	C₂×M₁₆	—	53	1	no	—
465	C₂×M₁₆	—	53	1	no	—
466	C₂×M₁₆	—	53	1	no	—
467	C₂×M₁₆	—	53	1	no	—
468	C₂×M₁₆	—	53	1	no	—
469	C₂×M₁₆	—	212	1	no	—
470	C₂×M₁₆	—	212	1	no	—
471	C₂×M₁₆	—	212	1	no	—
472	C₂×M₁₆	—	212	1	no	—
473	C₂×M₁₆	—	212	1	no	—
474	C₂×M₁₆	—	212	1	no	—
475	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
476	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic, complemented
477	C₂²×C₈	—	53	1	no	—
478	C₂²×C₈	—	53	1	no	—
479	C₂²×C₈	—	53	1	no	—
480	C₂²×C₈	—	53	1	no	—
481	C₂²×C₈	—	53	2	no	—
482	C₂²×C₈	—	53	2	no	—
483	C₂²×C₈	—	53	1	no	—
484	C₂²×C₈	—	53	1	no	—
485	C₂²×C₈	—	53	1	no	—
486	C₂²×C₈	—	53	1	no	—
487	C₂²×C₈	—	212	1	no	—
488	C₂²×C₈	—	212	1	no	—
489	C₄×D₄	—	53	1	no	—
490	C₄×D₄	—	53	1	no	—
491	C₄×D₄	—	53	1	no	—
492	C₄×D₄	—	53	1	no	—
493	C₄×D₄	—	53	1	no	—
494	C₄×D₄	—	53	1	no	—
495	C₄×D₄	—	53	1	no	—
496	C₄×D₄	C₂×#6157	1	1	yes	characteristic
497	#117	—	212	1	no	—
498	#117	—	212	1	no	—
499	C₂×C₄⋊C₄	—	53	1	no	—
500	C₂×C₄⋊C₄	—	53	2	no	—
501	C₂×C₄⋊C₄	—	53	2	no	—
502	C₂×C₂²⋊C₄	—	53	1	no	—
503	C₂×C₂²⋊C₄	—	53	1	no	—
504	C₂×C₂²⋊C₄	—	53	2	no	—
505	C₂×C₂²⋊C₄	—	53	2	no	—
506	C₂×C₂²⋊C₄	—	53	2	no	—
507	C₂×C₄²	—	53	1	no	—
508	C₂×C₄²	—	53	1	no	—
509	C₂×C₄²	—	53	1	no	—
510	C₂×C₄²	—	53	1	no	—
511	C₂×C₄²	—	53	1	no	—
512	C₂×C₄²	—	53	2	no	—
513	C₂×C₄²	—	212	1	no	—
514	C₂×C₄²	—	212	1	no	—
515	C₄⋊C₈	—	53	2	no	—
516	C₄⋊C₈	—	53	2	no	—
517	C₄⋊C₈	—	53	1	no	—
518	C₄⋊C₈	—	53	1	no	—
519	C₄⋊C₈	—	53	1	no	—
520	C₄⋊C₈	C₂×#6157	1	1	yes	characteristic, complemented
521	C₂²⋊C₈	—	53	2	no	—
522	C₂²⋊C₈	—	53	2	no	—
523	C₂²⋊C₈	—	53	2	no	—
524	C₂²⋊C₈	—	53	2	no	—
525	C₂²⋊C₈	—	53	1	no	—
526	C₂²⋊C₈	—	53	1	no	—
527	C₂²⋊C₈	—	53	1	no	—
528	C₂²⋊C₈	—	53	1	no	—
529	C₂²⋊C₈	—	53	1	no	—
530	C₂²⋊C₈	—	53	1	no	—
531	C₂²⋊C₈	C₂×#6157	1	1	yes	characteristic
532	C₂²⋊C₈	C₂×#6157	1	1	yes	characteristic, complemented
533	#97	—	53	2	no	—
534	#97	—	53	2	no	—
535	#97	—	212	1	no	—
536	#97	—	212	1	no	—
537	#97	—	212	1	no	—
538	#97	—	212	1	no	—
539	C₄×C₈	—	53	1	no	—
540	C₄×C₈	—	53	1	no	—
541	C₄×C₈	—	53	1	no	—
542	C₄×C₈	—	212	1	no	—
543	C₄×C₈	—	212	1	no	—
544	C₄×C₈	C₂×#6157	1	1	yes	characteristic, complemented
545	#95	—	53	2	no	—
546	#95	—	53	2	no	—
547	#95	—	53	2	no	—
548	#95	—	53	2	no	—
549	#95	—	53	2	no	—
550	C₂⁴	—	53	1	no	—
551	C₂⁴	—	53	1	no	—
552	C₂×D₄	—	53	1	no	—
553	C₂×D₄	—	53	1	no	—
554	C₂×D₄	—	53	1	no	—
555	C₂×D₄	—	212	2	no	—
556	C₂×D₄	C₄×#6157	1	1	yes	characteristic
557	C₂²×C₄	—	53	1	no	—
558	C₂²×C₄	—	53	1	no	—
559	C₂²×C₄	—	53	1	no	—
560	C₂²×C₄	—	53	1	no	—
561	C₂²×C₄	—	53	1	no	—
562	C₂²×C₄	—	53	1	no	—
563	C₂²×C₄	—	53	1	no	—
564	C₂²×C₄	—	53	1	no	—
565	C₂²×C₄	—	53	1	no	—
566	C₂²×C₄	—	53	1	no	—
567	C₂²×C₄	—	53	1	no	—
568	C₂²×C₄	—	53	2	no	—
569	C₂²×C₄	—	53	2	no	—
570	C₂²×C₄	—	53	2	no	—
571	C₂²×C₄	—	53	2	no	—
572	C₂²×C₄	—	106	1	no	—
573	C₂²×C₄	—	106	1	no	—
574	C₂²×C₄	—	106	1	no	—
575	C₂²×C₄	—	106	1	no	—
576	C₂²×C₄	—	106	1	no	—
577	C₂²×C₄	—	106	1	no	—
578	C₂²×C₄	—	106	1	no	—
579	C₂²×C₄	—	106	1	no	—
580	C₂²×C₄	—	212	2	no	—
581	C₂²×C₄	—	212	2	no	—
582	C₂²×C₄	—	212	2	no	—
583	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
584	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
585	M₁₆	—	4	1	no	—
586	M₁₆	—	4	1	no	—
587	M₁₆	—	212	1	no	—
588	M₁₆	—	212	1	no	—
589	M₁₆	—	212	1	no	—
590	M₁₆	—	212	1	no	—
591	M₁₆	—	212	1	no	—
592	M₁₆	—	212	1	no	—
593	M₁₆	—	212	2	no	—
594	M₁₆	—	212	2	no	—
595	C₂×C₈	—	53	2	no	—
596	C₂×C₈	—	53	2	no	—
597	C₂×C₈	—	53	2	no	—
598	C₂×C₈	—	53	2	no	—
599	C₂×C₈	—	53	1	no	—
600	C₂×C₈	—	53	1	no	—
601	C₂×C₈	—	53	1	no	—
602	C₂×C₈	—	53	1	no	—
603	C₂×C₈	—	106	1	no	—
604	C₂×C₈	—	106	1	no	—
605	C₂×C₈	—	106	1	no	—
606	C₂×C₈	—	106	1	no	—
607	C₂×C₈	—	106	1	no	—
608	C₂×C₈	—	106	1	no	—
609	C₂×C₈	—	106	1	no	—
610	C₂×C₈	—	106	1	no	—
611	C₂×C₈	—	106	1	no	—
612	C₂×C₈	—	106	1	no	—
613	C₂×C₈	—	106	1	no	—
614	C₂×C₈	—	106	1	no	—
615	C₂×C₈	—	106	1	no	—
616	C₂×C₈	—	106	1	no	—
617	C₂×C₈	—	106	1	no	—
618	C₂×C₈	—	106	1	no	—
619	C₂×C₈	—	212	2	no	—
620	C₂×C₈	—	212	2	no	—
621	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
622	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
623	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
624	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
625	C₄⋊C₄	—	53	1	no	—
626	C₄⋊C₄	—	53	1	no	—
627	C₄⋊C₄	—	53	1	no	—
628	C₄⋊C₄	—	212	2	no	—
629	C₄⋊C₄	—	212	2	no	—
630	C₄⋊C₄	C₄×#6157	1	1	yes	characteristic
631	C₂²⋊C₄	—	53	1	no	—
632	C₂²⋊C₄	—	53	1	no	—
633	C₂²⋊C₄	—	53	1	no	—
634	C₂²⋊C₄	—	53	1	no	—
635	C₂²⋊C₄	—	53	1	no	—
636	C₂²⋊C₄	—	53	1	no	—
637	C₂²⋊C₄	—	212	2	no	—
638	C₂²⋊C₄	—	212	2	no	—
639	C₂²⋊C₄	—	212	2	no	—
640	C₂²⋊C₄	C₄×#6157	1	1	yes	characteristic
641	C₂²⋊C₄	C₄×#6157	1	1	yes	characteristic
642	C₄²	—	53	1	no	—
643	C₄²	—	53	1	no	—
644	C₄²	—	53	1	no	—
645	C₄²	—	106	1	no	—
646	C₄²	—	106	1	no	—
647	C₄²	—	106	1	no	—
648	C₄²	—	106	1	no	—
649	C₄²	—	106	1	no	—
650	C₄²	—	106	1	no	—
651	C₄²	—	106	1	no	—
652	C₄²	—	106	1	no	—
653	C₄²	—	212	2	no	—
654	C₄²	C₂²×#6157	1	1	yes	characteristic
655	C₂³	—	53	1	no	—
656	C₂³	—	53	1	no	—
657	C₂³	—	53	1	no	—
658	C₂³	—	106	1	no	—
659	C₂³	—	106	2	no	—
660	C₂³	—	106	2	no	—
661	C₂³	—	106	1	no	—
662	C₂³	—	106	1	no	—
663	C₂³	—	106	1	no	—
664	C₂³	—	106	2	no	—
665	C₂³	—	106	2	no	—
666	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
667	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
668	D₄	—	4	1	no	—
669	D₄	—	212	1	no	—
670	D₄	—	212	1	no	—
671	D₄	—	212	1	no	—
672	C₂×C₄	—	2	1	no	—
673	C₂×C₄	—	2	1	no	—
674	C₂×C₄	—	2	1	no	—
675	C₂×C₄	—	2	1	no	—
676	C₂×C₄	—	53	1	no	—
677	C₂×C₄	—	53	1	no	—
678	C₂×C₄	—	53	1	no	—
679	C₂×C₄	—	53	1	no	—
680	C₂×C₄	—	53	1	no	—
681	C₂×C₄	—	53	1	no	—
682	C₂×C₄	—	53	1	no	—
683	C₂×C₄	—	53	1	no	—
684	C₂×C₄	—	53	1	no	—
685	C₂×C₄	—	106	1	no	—
686	C₂×C₄	—	106	1	no	—
687	C₂×C₄	—	106	1	no	—
688	C₂×C₄	—	106	1	no	—
689	C₂×C₄	—	106	1	no	—
690	C₂×C₄	—	106	1	no	—
691	C₂×C₄	—	106	2	no	—
692	C₂×C₄	—	106	1	no	—
693	C₂×C₄	—	106	1	no	—
694	C₂×C₄	—	106	2	no	—
695	C₂×C₄	—	106	1	no	—
696	C₂×C₄	—	106	1	no	—
697	C₂×C₄	—	106	1	no	—
698	C₂×C₄	—	106	1	no	—
699	C₂×C₄	—	106	1	no	—
700	C₂×C₄	—	106	1	no	—
701	C₂×C₄	—	106	1	no	—
702	C₂×C₄	—	106	1	no	—
703	C₂×C₄	—	106	2	no	—
704	C₂×C₄	—	106	2	no	—
705	C₂×C₄	—	106	2	no	—
706	C₂×C₄	—	106	2	no	—
707	C₂×C₄	—	106	2	no	—
708	C₂×C₄	—	106	2	no	—
709	C₂×C₄	—	106	2	no	—
710	C₂×C₄	—	106	2	no	—
711	C₂×C₄	—	106	2	no	—
712	C₂×C₄	—	106	2	no	—
713	C₂×C₄	—	106	2	no	—
714	C₂×C₄	—	106	2	no	—
715	C₂×C₄	—	212	8	no	—
716	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
717	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
718	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
719	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
720	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
721	C₈	—	2	1	no	—
722	C₈	—	2	1	no	—
723	C₈	—	2	1	no	—
724	C₈	—	2	1	no	—
725	C₈	—	106	2	no	—
726	C₈	—	106	2	no	—
727	C₈	—	106	2	no	—
728	C₈	—	106	2	no	—
729	C₈	—	106	1	no	—
730	C₈	—	106	1	no	—
731	C₈	—	106	1	no	—
732	C₈	—	106	1	no	—
733	C₂²	—	2	1	no	—
734	C₂²	—	2	1	no	—
735	C₂²	—	2	1	no	—
736	C₂²	—	2	1	no	—
737	C₂²	—	2	1	no	—
738	C₂²	—	2	1	no	—
739	C₂²	—	53	1	no	—
740	C₂²	—	53	1	no	—
741	C₂²	—	53	2	no	—
742	C₂²	—	53	2	no	—
743	C₂²	—	106	1	no	—
744	C₂²	—	106	1	no	—
745	C₂²	—	106	1	no	—
746	C₂²	—	106	1	no	—
747	C₂²	—	106	1	no	—
748	C₂²	—	106	1	no	—
749	C₂²	—	212	2	no	—
750	C₂²	—	212	2	no	—
751	C₂²	—	212	2	no	—
752	C₂²	—	212	2	no	—
753	C₂²	C₂²×C₄×#6157	1	1	yes	characteristic
754	C₄	—	2	1	no	—
755	C₄	—	2	1	no	—
756	C₄	—	2	1	no	—
757	C₄	—	2	1	no	—
758	C₄	—	53	1	no	—
759	C₄	—	53	1	no	—
760	C₄	—	106	1	no	—
761	C₄	—	106	1	no	—
762	C₄	—	106	1	no	—
763	C₄	—	106	1	no	—
764	C₄	—	212	2	no	—
765	C₄	—	212	2	no	—
766	C₄	—	212	2	no	—
767	C₄	—	212	2	no	—
768	C₄	#16811690	1	1	yes	characteristic
769	C₄	#16811692	1	1	yes	characteristic
770	C₂	—	4	1	no	—
771	C₂	—	4	1	no	—
772	C₂	—	53	2	no	—
773	C₂	—	53	2	no	—
774	C₂	—	212	1	no	—
775	C₂	—	212	1	no	—
776	C₂	#20291154	1	1	yes	characteristic
777	C₂	#20291278	1	1	yes	characteristic
778	C₂	#20291285	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂²×C₄²
Fitting	C₅₃×#436	C₄
prime radical	C₂×D₄×D₅₃	C₂×C₄
cosocle	C₂×C₄×D₅₃	C₂⁴
hypercenter	#436	#6157
hypocenter	C₅₃	#20564
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #20290516	C₂	semidirect
4. #20290509	C₂	semidirect
5. #20290504	C₂	semidirect
6. #20290458	C₂	semidirect
7. #20290455	C₂	semidirect
8. #20290455	C₂	semidirect
9. #20290454	C₂	semidirect
10. #20290446	C₂	semidirect
11. #20290443	C₂	semidirect
12. #20290433	C₂	semidirect
13. #20290383	C₂	semidirect
14. #20290377	C₂	semidirect
20. #16811655	C₄	semidirect
21. #16811654	C₄	semidirect
22. #16811653	C₄	semidirect
23. #16811652	C₄	semidirect
55. #16810702	C₄	semidirect
56. #16810701	C₄	semidirect
57. #16810700	C₄	semidirect
58. #16810699	C₄	semidirect
59. #16810699	C₄	semidirect
60. #16810698	C₄	semidirect
61. #16810698	C₄	semidirect
66. C₅₃×#436	C₄	semidirect
74. C₂×D₅₃⋊C₈	C₂×C₄	semidirect
79. C₂²×#88324	C₂×C₄	semidirect
94. #13133284	C₂×C₄	semidirect
95. C₄⋊#88324	C₂×C₄	semidirect
97. #13133274	C₂×C₄	semidirect
99. C₂²⋊#88324	C₂×C₄	semidirect
100. #13133272	C₂×C₄	semidirect
101. #13133272	C₂×C₄	semidirect
117. C₂×#11719917	C₂×C₄	semidirect
120. C₂×#11719912	C₂×C₄	semidirect
121. C₂×#11719912	C₂×C₄	semidirect
141. C₄⋊#88322	C₂×C₄	semidirect
142. #13133146	C₂×C₄	semidirect
143. #13133146	C₂×C₄	semidirect
145. C₂²⋊#88322	C₂×C₄	semidirect
148. #13133129	C₂×C₄	semidirect
149. #13133129	C₂×C₄	semidirect
150. C₄×#88322	C₂×C₄	semidirect
151. C₈×Dic₅₃	C₂×C₄	semidirect
152. C₈×Dic₅₃	C₂×C₄	semidirect
154. C₂×M₁₆×C₅₃	C₂×C₄	semidirect
156. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
158. C₂²⋊C₈×C₅₃	C₂×C₄	semidirect
159. C₄×C₈×C₅₃	C₂×C₄	semidirect
419. #436	#6157	semidirect
456. C₅₃	#20564	semidirect
476. C₂×M₁₆	C₂×#6157	semidirect
520. C₄⋊C₈	C₂×#6157	semidirect
532. C₂²⋊C₈	C₂×#6157	semidirect
544. C₄×C₈	C₂×#6157	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
D₅₃⋊#435	2
#20291169	2
#20290516	2
#20290509	2
#20290504	2
#20290458	2
#20290455	2
#20290454	2
#20290446	2
#20290443	2
#20290433	2
#20290383	2
#20290377	2
#436×D₅₃	2
#20564	53