Subgroups — #28318632

Order 13568.

838 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅₃⋊#446	C₂	1	1	yes	maximal, characteristic
2	#20291174	C₂	1	1	yes	maximal, characteristic
3	#20290516	C₂	1	1	yes	maximal, characteristic
4	#20290516	C₂	1	1	yes	maximal, characteristic
5	#20290513	C₂	1	1	yes	maximal, characteristic
6	#20290513	C₂	1	1	yes	maximal, characteristic
7	#20290512	C₂	1	1	yes	maximal, characteristic
8	#20290509	C₂	1	1	yes	maximal, characteristic
9	#20290504	C₂	1	1	yes	maximal, characteristic
10	#20290504	C₂	1	1	yes	maximal, characteristic
11	#20290502	C₂	1	1	yes	maximal, characteristic
12	#20290373	C₂	1	1	yes	maximal, characteristic
13	#20290371	C₂	1	1	yes	maximal, characteristic
14	#97×#6157	C₂	1	1	yes	maximal, characteristic
15	#446×D₅₃	C₂	1	1	yes	maximal, characteristic
16	#16811675	C₄	1	2	yes	complemented
17	#16811675	C₄	1	2	yes	complemented
18	#16811674	C₄	1	2	yes	complemented
19	#16811673	C₄	1	2	yes	complemented
20	D₅₃⋊(C₄⋊C₈)	C₂²	1	1	yes	characteristic
21	D₅₃⋊(C₄⋊C₈)	C₂²	1	1	yes	characteristic
22	D₅₃⋊(C₄⋊C₈)	C₂²	1	1	yes	characteristic
23	#16811582	C₂²	1	1	yes	characteristic
24	#16811582	C₂²	1	1	yes	characteristic
25	C₂×#13133271	C₂²	1	1	yes	characteristic
26	C₂×#13133271	C₂²	1	1	yes	characteristic
27	C₂×#13133271	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	C₄×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₈×#6157	C₂²	1	1	yes	characteristic
33	C₂×C₈×#6157	C₂²	1	1	yes	characteristic
34	C₂×C₈×#6157	C₂²	1	1	yes	characteristic
35	C₂×C₈×#6157	C₂²	1	1	yes	characteristic
36	(C₂×#6157)⋊Q₈	C₄	1	2	yes	—
37	#16811574	C₄	1	2	yes	—
38	#16811572	C₄	1	2	yes	—
39	#16811572	C₄	1	2	yes	—
40	C₄×C₄⋊#6157	C₂²	1	1	yes	characteristic
41	C₄×C₄⋊#6157	C₂²	1	1	yes	characteristic
42	C₄×C₄⋊#6157	C₂²	1	1	yes	characteristic
43	#16811559	C₂²	1	1	yes	characteristic
44	#16811559	C₂²	1	1	yes	characteristic
45	C₄²×#6157	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	C₄×Q₈×D₅₃	C₂²	1	1	yes	characteristic
53	#16810754	C₄	1	1	yes	characteristic, complemented
54	#16810753	C₄	1	1	yes	characteristic, complemented
55	#16810752	C₄	1	1	yes	characteristic, complemented
56	#16810752	C₄	1	1	yes	characteristic, complemented
57	#16810751	C₄	1	1	yes	characteristic, complemented
58	#16810751	C₄	1	1	yes	characteristic, complemented
59	#16810750	C₄	1	1	yes	characteristic, complemented
60	C₄⋊C₈×D₅₃	C₂²	1	1	yes	characteristic
61	C₄⋊C₈×D₅₃	C₂²	1	1	yes	characteristic
62	C₄⋊C₈×D₅₃	C₂²	1	1	yes	characteristic
63	#97×D₅₃	C₂²	1	1	yes	characteristic
64	#97×D₅₃	C₂²	1	1	yes	characteristic
65	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
66	C₅₃×#446	C₄	1	1	yes	characteristic, complemented
67	C₂×D₅₃⋊C₈	C₂³	1	1	yes	characteristic
68	C₂×D₅₃⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×D₅₃⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×D₅₃⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊#6157	C₂×C₄	1	2	yes	—
72	C₂×C₄⋊#6157	C₂×C₄	1	2	yes	—
73	C₂×C₄⋊#6157	C₂×C₄	1	2	yes	—
74	C₂×C₄×#6157	C₂×C₄	1	2	yes	—
75	C₂×C₄×#6157	C₂×C₄	1	2	yes	—
76	C₂×C₄×#6157	C₂³	1	1	yes	characteristic
77	C₂×C₄×#6157	C₂³	1	1	yes	characteristic
78	C₂×C₄×#6157	C₂³	1	1	yes	characteristic
79	C₂×C₄×#6157	C₂³	1	1	yes	characteristic
80	#13133284	C₂×C₄	1	2	yes	—
81	#13133284	C₂×C₄	1	2	yes	—
82	#13133284	C₂×C₄	1	2	yes	—
83	C₄⋊#88324	C₂×C₄	1	2	yes	—
84	C₄⋊#88324	C₂×C₄	1	2	yes	—
85	C₄⋊#88324	C₂×C₄	1	2	yes	—
86	#13133272	C₂×C₄	1	2	yes	—
87	#13133272	C₂×C₄	1	2	yes	—
88	#13133272	C₂×C₄	1	2	yes	—
89	#13133272	C₂×C₄	1	2	yes	—
90	#13133271	—	2	2	no	—
91	#13133271	—	2	2	no	—
92	#13133271	—	2	2	no	—
93	#13133271	—	2	2	no	—
94	#13133271	—	2	2	no	—
95	#13133271	—	2	2	no	—
96	C₄×#88324	C₂×C₄	1	2	yes	—
97	C₄×#88324	C₂×C₄	1	2	yes	—
98	C₈×#6157	—	2	2	no	—
99	C₈×#6157	—	2	2	no	—
100	C₈×#6157	—	2	2	no	—
101	C₈×#6157	Q₈	1	4	yes	complemented
102	(C₂×#6157)⋊C₄	C₂×C₄	1	2	yes	—
103	(C₂×#6157)⋊C₄	C₂×C₄	1	2	yes	—
104	(C₂×#6157)⋊C₄	C₂×C₄	1	2	yes	—
105	(C₂×#6157)⋊C₄	C₂×C₄	1	2	yes	—
106	(C₂×#6157)⋊C₄	C₂×C₄	1	2	yes	—
107	(C₂×#6157)⋊C₄	C₂×C₄	1	2	yes	—
108	(C₂×#6157)⋊C₄	C₂×C₄	1	2	yes	—
109	C₂×Q₈×D₅₃	C₂×C₄	1	1	yes	characteristic
110	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
111	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
112	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
113	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
114	Q₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
115	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
116	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
117	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
118	#13133182	C₂×C₄	1	1	yes	characteristic
119	#13133182	C₂×C₄	1	1	yes	characteristic
120	#13133182	C₂×C₄	1	1	yes	characteristic
121	C₄⋊C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
122	C₄⋊C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
123	C₄⋊C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
124	C₄²×D₅₃	C₂³	1	1	yes	characteristic
125	C₄²×D₅₃	C₂³	1	1	yes	characteristic
126	C₄²×D₅₃	C₂³	1	1	yes	characteristic
127	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
128	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
129	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
130	#13133146	C₂×C₄	1	1	yes	characteristic
131	#13133146	C₂×C₄	1	1	yes	characteristic
132	#13133146	C₂×C₄	1	1	yes	characteristic
133	#13133146	C₂×C₄	1	1	yes	characteristic
134	#13133146	C₂×C₄	1	1	yes	characteristic
135	#13133146	C₂×C₄	1	1	yes	characteristic
136	#13133128	C₂×C₄	1	1	yes	characteristic
137	#13133128	C₂×C₄	1	1	yes	characteristic
138	#13133127	C₂×C₄	1	1	yes	characteristic
139	#13133127	C₂×C₄	1	1	yes	characteristic
140	#13133127	C₂×C₄	1	1	yes	characteristic
141	#13133127	C₂×C₄	1	1	yes	characteristic
142	C₄×#88322	C₂×C₄	1	1	yes	characteristic
143	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
144	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
145	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
146	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
147	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
148	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
149	#97×C₅₃	C₂×C₄	1	1	yes	characteristic
150	#97×C₅₃	C₂×C₄	1	1	yes	characteristic
151	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
152	C₂²×#6157	C₂²×C₄	1	2	yes	—
153	C₂²×#6157	C₂²×C₄	1	2	yes	—
154	C₂²×#6157	C₂²×C₄	1	2	yes	—
155	C₂²×#6157	C₂²×C₄	1	2	yes	—
156	C₂×Q₈×C₅₃	C₄²	1	1	yes	characteristic
157	Q₈×D₅₃	—	4	2	no	—
158	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
159	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
160	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
161	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
162	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
163	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
164	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
165	C₂×Dic₁₀₆	C₄²	1	1	yes	characteristic
166	C₂×Dic₁₀₆	C₄²	1	1	yes	characteristic
167	C₂×Dic₁₀₆	C₄²	1	1	yes	characteristic
168	C₂×#88324	C₂²×C₄	1	2	yes	—
169	C₂×#88324	C₂²×C₄	1	2	yes	—
170	C₂×#88324	C₂²×C₄	1	2	yes	—
171	C₂×#88324	C₂²×C₄	1	2	yes	—
172	C₄⋊#6157	—	4	2	no	—
173	C₄⋊#6157	—	4	2	no	—
174	C₄⋊#6157	—	4	2	no	—
175	C₄×#6157	—	2	1	no	—
176	C₄×#6157	—	2	1	no	—
177	C₄×#6157	—	2	1	no	—
178	C₄×#6157	—	2	1	no	—
179	C₄×#6157	—	2	1	no	—
180	C₄×#6157	—	2	1	no	—
181	C₄×#6157	—	4	2	no	—
182	C₄×#6157	—	4	2	no	—
183	C₄×#6157	C₂×Q₈	1	2	yes	—
184	C₄×#6157	#41	1	2	yes	—
185	D₅₃⋊C₈	—	2	1	no	—
186	D₅₃⋊C₈	—	2	1	no	—
187	D₅₃⋊C₈	—	2	1	no	—
188	D₅₃⋊C₈	—	2	1	no	—
189	D₅₃⋊C₈	—	2	1	no	—
190	D₅₃⋊C₈	—	2	1	no	—
191	D₅₃⋊C₈	C₂×Q₈	1	2	yes	—
192	D₅₃⋊C₈	#41	1	2	yes	—
193	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
194	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
195	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
196	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
197	C₄⋊C₄×C₅₃	C₄²	1	1	yes	characteristic
198	C₄⋊C₄×C₅₃	C₄²	1	1	yes	characteristic
199	C₄⋊C₄×C₅₃	C₄²	1	1	yes	characteristic
200	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
201	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
202	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
203	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
204	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
205	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
206	#11719919	C₄²	1	1	yes	characteristic
207	#11719919	C₄²	1	1	yes	characteristic
208	#11719919	C₄²	1	1	yes	characteristic
209	#11719919	C₄²	1	1	yes	characteristic
210	#11719919	C₄²	1	1	yes	characteristic
211	#11719919	C₄²	1	1	yes	characteristic
212	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
213	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
214	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
215	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
216	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
217	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
218	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
219	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
220	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
221	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
222	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
223	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
224	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
225	C₈×D₅₃	—	2	1	no	—
226	C₈×D₅₃	—	2	1	no	—
227	C₈×D₅₃	—	2	1	no	—
228	C₈×D₅₃	—	2	1	no	—
229	C₈×D₅₃	—	2	1	no	—
230	C₈×D₅₃	—	2	1	no	—
231	C₈×D₅₃	C₂×Q₈	1	2	yes	—
232	C₈×D₅₃	#41	1	2	yes	—
233	C₂²×D₅₃	C₂³×C₄	1	1	yes	characteristic
234	C₂×#6157	—	2	2	no	—
235	C₂×#6157	—	2	2	no	—
236	C₂×#6157	—	2	2	no	—
237	C₂×#6157	—	2	2	no	—
238	C₂×#6157	—	2	2	no	—
239	C₂×#6157	—	2	2	no	—
240	C₂×#6157	—	2	2	no	—
241	C₂×#6157	—	2	2	no	—
242	C₂×#6157	—	2	2	no	—
243	C₂×#6157	—	2	2	no	—
244	C₂×#6157	—	2	2	no	—
245	C₂×#6157	C₄×Q₈	1	4	yes	—
246	Q₈×C₅₃	—	4	1	no	—
247	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
248	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
249	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
250	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
251	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
252	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
253	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
254	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
255	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
256	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
257	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
258	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
259	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
260	C₂×Dic₅₃	C₂³×C₄	1	1	yes	characteristic
261	C₄×D₅₃	—	2	2	no	—
262	C₄×D₅₃	—	2	2	no	—
263	C₄×D₅₃	—	2	2	no	—
264	C₄×D₅₃	—	2	2	no	—
265	C₄×D₅₃	—	2	2	no	—
266	C₄×D₅₃	—	2	2	no	—
267	C₄×D₅₃	C₂²×Q₈	1	1	yes	characteristic
268	C₄×D₅₃	C₂×#41	1	1	yes	characteristic
269	C₄×D₅₃	C₂×#41	1	1	yes	characteristic
270	C₄×D₅₃	C₂×#41	1	1	yes	characteristic
271	Dic₁₀₆	—	4	1	no	—
272	Dic₁₀₆	—	4	1	no	—
273	Dic₁₀₆	—	4	1	no	—
274	#88324	—	2	2	no	—
275	#88324	—	2	2	no	—
276	#88324	—	2	2	no	—
277	#88324	C₄×Q₈	1	4	yes	—
278	C₈×C₅₃	—	2	1	no	—
279	C₈×C₅₃	—	2	1	no	—
280	C₈×C₅₃	—	2	1	no	—
281	C₈×C₅₃	C₄×Q₈	1	2	yes	—
282	#88322	—	2	1	no	—
283	#88322	—	2	1	no	—
284	#88322	—	2	1	no	—
285	#88322	C₄×Q₈	1	2	yes	—
286	#20518	—	53	1	no	maximal, Sylow
287	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
288	C₂×D₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
289	C₂×D₅₃	C₄×#41	1	1	yes	characteristic
290	C₂×D₅₃	#568	1	2	yes	—
291	C₂×D₅₃	#568	1	2	yes	—
292	#6157	—	2	4	no	—
293	#6157	—	4	2	no	—
294	#6157	—	4	2	no	—
295	#6157	—	4	2	no	—
296	C₄×C₅₃	—	2	1	no	—
297	C₄×C₅₃	—	2	1	no	—
298	C₄×C₅₃	—	2	1	no	—
299	C₄×C₅₃	—	2	1	no	—
300	C₄×C₅₃	—	2	1	no	—
301	C₄×C₅₃	—	2	1	no	—
302	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
303	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
304	Dic₅₃	—	2	1	no	—
305	Dic₅₃	—	2	1	no	—
306	Dic₅₃	—	2	1	no	—
307	Dic₅₃	—	2	1	no	—
308	Dic₅₃	—	2	1	no	—
309	Dic₅₃	—	2	1	no	—
310	Dic₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
311	Dic₅₃	C₄×#41	1	1	yes	characteristic
312	C₂×#446	—	53	1	no	—
313	C₂×#446	—	53	1	no	—
314	#2295	—	53	1	no	—
315	#1771	—	53	1	no	—
316	#1771	—	53	1	no	—
317	#1771	—	53	1	no	—
318	#1771	—	53	1	no	—
319	#1770	—	53	1	no	—
320	#1770	—	53	1	no	—
321	#1768	—	53	1	no	—
322	#1768	—	53	1	no	—
323	#1767	—	53	1	no	—
324	#1726	—	53	1	no	—
325	#1726	—	53	1	no	—
326	C₄×#97	—	53	1	no	—
327	C₂×C₅₃	C₄²×Q₈	1	1	yes	characteristic
328	C₂×C₅₃	#2873	1	1	yes	characteristic
329	C₂×C₅₃	#2875	1	1	yes	characteristic
330	D₅₃	#2961	1	2	yes	—
331	D₅₃	#2967	1	2	yes	—
332	C₂×C₄×Q₈	—	53	1	no	—
333	#446	—	53	2	no	—
334	#446	—	53	2	no	—
335	#446	—	53	2	no	—
336	#446	—	53	2	no	—
337	#446	—	53	1	no	—
338	#446	—	53	1	no	—
339	#446	—	53	1	no	—
340	#446	—	53	1	no	—
341	#446	—	53	1	no	—
342	#446	—	53	1	no	—
343	#446	—	53	1	no	—
344	#446	#6157	1	1	yes	characteristic, complemented
345	C₂×C₄⋊C₈	—	53	1	no	—
346	C₂×C₄⋊C₈	—	53	1	no	—
347	C₂×C₄⋊C₈	—	53	1	no	—
348	C₂×C₄⋊C₈	—	53	1	no	—
349	C₂×C₄⋊C₈	—	53	1	no	—
350	C₂×C₄⋊C₈	—	53	1	no	—
351	C₂×#97	—	53	1	no	—
352	C₂×#97	—	53	1	no	—
353	C₂×#97	—	53	1	no	—
354	C₂×#97	—	53	1	no	—
355	C₂×#97	—	53	1	no	—
356	C₂×#97	—	53	1	no	—
357	C₂×#97	—	53	1	no	—
358	C₂×#97	—	53	1	no	—
359	C₂×#97	—	53	1	no	—
360	C₂×#97	—	53	1	no	—
361	C₂×C₄×C₈	—	53	1	no	—
362	C₂×C₄×C₈	—	53	1	no	—
363	C₂×C₄×C₈	—	53	1	no	—
364	C₂×C₄×C₈	—	53	1	no	—
365	C₂×C₄×C₈	—	53	1	no	—
366	C₂×C₄×C₈	—	53	1	no	—
367	#391	—	53	2	no	—
368	#389	—	53	2	no	—
369	#387	—	53	2	no	—
370	#387	—	53	2	no	—
371	C₄×C₄⋊C₄	—	53	1	no	—
372	C₄×C₄⋊C₄	—	53	1	no	—
373	C₄×C₄⋊C₄	—	53	1	no	—
374	#376	—	53	1	no	—
375	#376	—	53	1	no	—
376	C₄³	—	53	1	no	—
377	#336	—	53	1	no	—
378	#336	—	53	1	no	—
379	#336	—	53	1	no	—
380	#336	—	53	1	no	—
381	#336	—	53	1	no	—
382	#336	—	53	1	no	—
383	C₅₃	#20518	1	1	yes	characteristic, complemented, Sylow
384	C₂²×Q₈	—	53	1	no	—
385	C₂²×C₈	—	53	1	no	—
386	C₂²×C₈	—	53	1	no	—
387	C₂²×C₈	—	53	1	no	—
388	C₂²×C₈	—	53	1	no	—
389	C₂²×C₈	—	53	1	no	—
390	C₂²×C₈	—	53	1	no	—
391	C₂²×C₈	—	53	1	no	—
392	C₂²×C₈	—	53	1	no	—
393	C₄×Q₈	—	53	1	no	—
394	C₄×Q₈	—	53	1	no	—
395	C₄×Q₈	—	53	1	no	—
396	C₄×Q₈	—	53	1	no	—
397	C₄×Q₈	—	53	1	no	—
398	C₄×Q₈	—	53	1	no	—
399	C₄×Q₈	—	53	1	no	—
400	C₄×Q₈	C₂×#6157	1	1	yes	characteristic
401	C₂×C₄⋊C₄	—	53	1	no	—
402	C₂×C₄⋊C₄	—	53	1	no	—
403	C₂×C₄⋊C₄	—	53	1	no	—
404	C₂×C₄⋊C₄	—	53	2	no	—
405	C₂×C₄⋊C₄	—	53	2	no	—
406	C₂×C₄⋊C₄	—	53	2	no	—
407	C₂×C₄²	—	53	1	no	—
408	C₂×C₄²	—	53	1	no	—
409	C₂×C₄²	—	53	1	no	—
410	C₂×C₄²	—	53	1	no	—
411	C₂×C₄²	—	53	1	no	—
412	C₂×C₄²	—	53	1	no	—
413	C₂×C₄²	—	53	1	no	—
414	C₂×C₄²	—	53	2	no	—
415	C₂×C₄²	—	53	2	no	—
416	C₄⋊C₈	—	53	2	no	—
417	C₄⋊C₈	—	53	2	no	—
418	C₄⋊C₈	—	53	2	no	—
419	C₄⋊C₈	—	53	2	no	—
420	C₄⋊C₈	—	53	2	no	—
421	C₄⋊C₈	—	53	2	no	—
422	C₄⋊C₈	—	53	1	no	—
423	C₄⋊C₈	—	53	1	no	—
424	C₄⋊C₈	—	53	1	no	—
425	C₄⋊C₈	—	53	1	no	—
426	C₄⋊C₈	—	53	1	no	—
427	C₄⋊C₈	—	53	1	no	—
428	C₄⋊C₈	—	53	1	no	—
429	C₄⋊C₈	—	53	1	no	—
430	C₄⋊C₈	—	53	1	no	—
431	C₄⋊C₈	C₂×#6157	1	1	yes	characteristic
432	C₄⋊C₈	C₂×#6157	1	1	yes	characteristic
433	C₄⋊C₈	C₂×#6157	1	1	yes	characteristic
434	#97	—	53	2	no	—
435	#97	—	53	2	no	—
436	#97	—	53	2	no	—
437	#97	—	53	2	no	—
438	#97	—	53	1	no	—
439	#97	—	53	1	no	—
440	#97	—	53	1	no	—
441	#97	—	53	1	no	—
442	#97	—	53	1	no	—
443	#97	—	53	1	no	—
444	#97	—	106	2	no	—
445	#97	—	106	2	no	—
446	#97	—	106	2	no	—
447	#97	—	106	2	no	—
448	#97	—	106	2	no	—
449	#97	—	106	2	no	—
450	#97	C₂×#6157	1	1	yes	characteristic
451	#97	C₂×#6157	1	1	yes	characteristic
452	C₄×C₈	—	53	2	no	—
453	C₄×C₈	—	53	2	no	—
454	C₄×C₈	—	53	4	no	—
455	C₄×C₈	—	53	1	no	—
456	C₄×C₈	—	53	1	no	—
457	C₄×C₈	—	53	1	no	—
458	C₄×C₈	—	106	2	no	—
459	C₄×C₈	—	106	2	no	—
460	C₄×C₈	—	106	2	no	—
461	C₄×C₈	C₂×#6157	1	1	yes	characteristic
462	#95	—	53	2	no	—
463	#95	—	53	2	no	—
464	#95	—	53	2	no	—
465	#95	—	53	2	no	—
466	#95	—	53	2	no	—
467	#95	—	53	2	no	—
468	#95	—	53	2	no	—
469	C₂×Q₈	—	53	1	no	—
470	C₂×Q₈	—	53	1	no	—
471	C₂×Q₈	—	53	1	no	—
472	C₂×Q₈	—	212	2	no	—
473	C₂×Q₈	C₄×#6157	1	1	yes	characteristic
474	C₂²×C₄	—	53	1	no	—
475	C₂²×C₄	—	53	2	no	—
476	C₂²×C₄	—	53	1	no	—
477	C₂²×C₄	—	53	1	no	—
478	C₂²×C₄	—	53	1	no	—
479	C₂²×C₄	—	53	1	no	—
480	C₂²×C₄	—	53	1	no	—
481	C₂²×C₄	—	53	1	no	—
482	C₂²×C₄	—	53	2	no	—
483	C₂²×C₄	—	53	2	no	—
484	C₂²×C₄	—	53	2	no	—
485	C₂×C₈	—	53	2	no	—
486	C₂×C₈	—	53	2	no	—
487	C₂×C₈	—	53	2	no	—
488	C₂×C₈	—	53	1	no	—
489	C₂×C₈	—	53	2	no	—
490	C₂×C₈	—	53	2	no	—
491	C₂×C₈	—	53	2	no	—
492	C₂×C₈	—	53	2	no	—
493	C₂×C₈	—	53	2	no	—
494	C₂×C₈	—	53	1	no	—
495	C₂×C₈	—	53	1	no	—
496	C₂×C₈	—	53	1	no	—
497	C₂×C₈	—	106	1	no	—
498	C₂×C₈	—	106	1	no	—
499	C₂×C₈	—	106	1	no	—
500	C₂×C₈	—	106	1	no	—
501	C₂×C₈	—	106	1	no	—
502	C₂×C₈	—	106	1	no	—
503	C₂×C₈	—	106	1	no	—
504	C₂×C₈	—	106	1	no	—
505	C₂×C₈	—	106	1	no	—
506	C₂×C₈	—	106	1	no	—
507	C₂×C₈	—	106	1	no	—
508	C₂×C₈	—	106	1	no	—
509	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
510	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
511	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
512	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
513	C₄⋊C₄	—	53	1	no	—
514	C₄⋊C₄	—	53	1	no	—
515	C₄⋊C₄	—	53	1	no	—
516	C₄⋊C₄	—	53	1	no	—
517	C₄⋊C₄	—	53	1	no	—
518	C₄⋊C₄	—	53	1	no	—
519	C₄⋊C₄	—	53	1	no	—
520	C₄⋊C₄	—	53	1	no	—
521	C₄⋊C₄	—	53	1	no	—
522	C₄⋊C₄	—	212	2	no	—
523	C₄⋊C₄	—	212	2	no	—
524	C₄⋊C₄	—	212	2	no	—
525	C₄⋊C₄	C₄×#6157	1	1	yes	characteristic
526	C₄⋊C₄	C₄×#6157	1	1	yes	characteristic
527	C₄⋊C₄	C₄×#6157	1	1	yes	characteristic
528	C₄²	—	53	2	no	—
529	C₄²	—	53	2	no	—
530	C₄²	—	53	1	no	—
531	C₄²	—	53	1	no	—
532	C₄²	—	53	1	no	—
533	C₄²	—	53	1	no	—
534	C₄²	—	53	1	no	—
535	C₄²	—	53	1	no	—
536	C₄²	—	53	1	no	—
537	C₄²	—	53	1	no	—
538	C₄²	—	53	1	no	—
539	C₄²	—	106	1	no	—
540	C₄²	—	106	1	no	—
541	C₄²	—	106	1	no	—
542	C₄²	—	106	1	no	—
543	C₄²	—	106	1	no	—
544	C₄²	—	106	1	no	—
545	C₄²	—	212	2	no	—
546	C₄²	—	212	2	no	—
547	C₄²	C₂²×#6157	1	1	yes	characteristic
548	C₄²	C₂²×#6157	1	1	yes	characteristic
549	C₄²	C₂²×#6157	1	1	yes	characteristic
550	C₂³	—	53	1	no	—
551	Q₈	—	4	1	no	—
552	Q₈	—	212	1	no	—
553	Q₈	—	212	1	no	—
554	Q₈	—	212	1	no	—
555	C₂×C₄	—	53	1	no	—
556	C₂×C₄	—	53	1	no	—
557	C₂×C₄	—	53	1	no	—
558	C₂×C₄	—	53	1	no	—
559	C₂×C₄	—	53	1	no	—
560	C₂×C₄	—	53	4	no	—
561	C₂×C₄	—	53	1	no	—
562	C₂×C₄	—	53	1	no	—
563	C₂×C₄	—	53	1	no	—
564	C₂×C₄	—	53	1	no	—
565	C₂×C₄	—	53	1	no	—
566	C₂×C₄	—	53	1	no	—
567	C₂×C₄	—	106	2	no	—
568	C₂×C₄	—	106	2	no	—
569	C₂×C₄	—	106	2	no	—
570	C₂×C₄	—	106	2	no	—
571	C₂×C₄	—	106	2	no	—
572	C₂×C₄	—	106	2	no	—
573	C₂×C₄	—	106	2	no	—
574	C₂×C₄	—	106	2	no	—
575	C₂×C₄	—	106	2	no	—
576	C₂×C₄	—	106	2	no	—
577	C₂×C₄	—	106	2	no	—
578	C₂×C₄	—	106	2	no	—
579	C₂×C₄	—	106	2	no	—
580	C₂×C₄	—	106	2	no	—
581	C₂×C₄	—	106	2	no	—
582	C₂×C₄	—	106	2	no	—
583	C₂×C₄	—	106	2	no	—
584	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
585	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
586	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
587	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
588	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
589	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
590	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
591	C₈	—	2	1	no	—
592	C₈	—	2	1	no	—
593	C₈	—	2	1	no	—
594	C₈	—	53	4	no	—
595	C₈	—	53	2	no	—
596	C₈	—	106	2	no	—
597	C₈	—	106	2	no	—
598	C₈	—	106	2	no	—
599	C₈	—	106	1	no	—
600	C₈	—	106	1	no	—
601	C₈	—	106	1	no	—
602	C₈	Q₈×#6157	1	2	yes	—
603	C₂²	—	53	1	no	—
604	C₂²	—	53	1	no	—
605	C₂²	—	53	2	no	—
606	C₂²	—	53	2	no	—
607	C₂²	C₂²×C₄×#6157	1	1	yes	characteristic
608	C₄	—	2	1	no	—
609	C₄	—	2	1	no	—
610	C₄	—	2	1	no	—
611	C₄	—	2	1	no	—
612	C₄	—	2	1	no	—
613	C₄	—	2	1	no	—
614	C₄	—	53	1	no	—
615	C₄	—	53	1	no	—
616	C₄	—	106	4	no	—
617	C₄	—	106	1	no	—
618	C₄	—	106	1	no	—
619	C₄	—	106	1	no	—
620	C₄	—	106	1	no	—
621	C₄	—	106	1	no	—
622	C₄	—	106	1	no	—
623	C₄	—	212	2	no	—
624	C₄	—	212	2	no	—
625	C₄	—	212	2	no	—
626	C₄	C₂×Q₈×#6157	1	1	yes	characteristic
627	C₄	#41×#6157	1	1	yes	characteristic
628	C₂	—	53	2	no	—
629	C₂	—	53	2	no	—
630	C₂	C₄×Q₈×#6157	1	1	yes	characteristic
631	C₂	#20291278	1	1	yes	characteristic
632	C₂	#20291285	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂²×C₄²
Fitting	C₅₃×#446	C₄
prime radical	C₂²×D₅₃	C₂³×C₄
cosocle	C₂×C₄×D₅₃	C₂⁴
hypercenter	#446	#6157
hypocenter	C₅₃	#20518
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
16. #16811675	C₄	semidirect
17. #16811675	C₄	semidirect
18. #16811674	C₄	semidirect
19. #16811673	C₄	semidirect
53. #16810754	C₄	semidirect
54. #16810753	C₄	semidirect
55. #16810752	C₄	semidirect
56. #16810752	C₄	semidirect
57. #16810751	C₄	semidirect
58. #16810751	C₄	semidirect
59. #16810750	C₄	semidirect
66. C₅₃×#446	C₄	semidirect
101. C₈×#6157	Q₈	semidirect
344. #446	#6157	semidirect
383. C₅₃	#20518	semidirect

Maximal subgroups

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

subgroup	index
D₅₃⋊#446	2
#20291174	2
#20290516	2
#20290513	2
#20290512	2
#20290509	2
#20290504	2
#20290502	2
#20290373	2
#20290371	2
#97×#6157	2
#446×D₅₃	2
#20518	53