Subgroups — #27537683

Order 13568.

626 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×C₂²⋊D₅₃	C₂	1	1	yes	maximal, characteristic
2	#20276544	C₂	1	2	yes	maximal, complemented
3	#20276543	C₂	1	2	yes	maximal, complemented
4	#20276540	C₂	1	2	yes	maximal, complemented
5	C₁₆×C₂²⋊D₅₃	C₂	1	2	yes	maximal, complemented
6	#20276515	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276511	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276326	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276324	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²⋊C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₂²⋊D₅₃	C₄	1	1	yes	characteristic
14	C₂×#13133157	C₂²	1	2	yes	complemented
15	C₂×#13133156	C₂²	1	2	yes	complemented
16	C₂×C₁₆×D₅₃	C₂²	1	2	yes	complemented
17	C₈×C₂²⋊D₅₃	C₄	1	1	yes	characteristic
18	C₈×C₂²⋊D₅₃	C₄	1	1	yes	characteristic
19	C₈×C₂²⋊D₅₃	C₄	1	1	yes	characteristic
20	C₈×C₂²⋊D₅₃	C₄	1	1	yes	characteristic
21	C₈×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
22	C₈×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
23	C₈×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
24	C₈×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
25	C₂×#13133146	C₄	1	1	yes	characteristic
26	C₂×C₂²⋊#88322	C₄	1	1	yes	characteristic
27	C₂×#13133129	C₄	1	1	yes	characteristic
28	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
29	#16810441	C₂²	1	2	yes	complemented
30	#16810441	C₂²	1	2	yes	complemented
31	C₂²⋊#11719908	C₂²	1	1	yes	characteristic
32	C₂²⋊#11719908	C₂²	1	1	yes	characteristic
33	C₂²⋊#11719908	C₂²	1	2	yes	complemented
34	#16810402	C₂²	1	2	yes	complemented
35	#16810402	C₂²	1	2	yes	complemented
36	#16810398	C₂²	1	2	yes	complemented
37	C₁₆×Dic₅₃	C₂²	1	2	yes	complemented
38	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	complemented
40	C₅₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
41	C₅₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
42	C₅₃×C₂²⋊C₁₆	C₂²	1	2	yes	complemented
43	C₂²×C₂²⋊D₅₃	C₈	1	1	yes	characteristic
44	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
46	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×D₅₃	—	4	2	no	—
48	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
51	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
52	C₄×C₂²⋊D₅₃	C₈	1	1	yes	characteristic
53	C₄×C₂²⋊D₅₃	C₈	1	1	yes	characteristic
54	C₄×C₂²⋊D₅₃	C₈	1	1	yes	characteristic
55	C₄×C₂²⋊D₅₃	C₈	1	1	yes	characteristic
56	C₄×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
57	C₄×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
59	C₄×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×#11719919	C₈	1	1	yes	characteristic
61	C₂×C₂²⋊Dic₅₃	C₈	1	1	yes	characteristic
62	C₂×#11719921	C₈	1	1	yes	characteristic
63	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
64	#13133157	—	4	2	no	—
65	#13133156	—	4	2	no	—
66	C₂×#11719908	C₂³	1	2	yes	—
67	C₂×#11719908	C₂³	1	2	yes	—
68	C₁₆×D₅₃	—	4	2	no	—
69	#13133146	C₂×C₄	1	1	yes	characteristic
70	#13133146	C₂×C₄	1	1	yes	characteristic
71	#13133146	C₂×C₄	1	1	yes	characteristic
72	#13133146	C₂×C₄	1	1	yes	characteristic
73	C₂²⋊#88322	C₂×C₄	1	1	yes	characteristic
74	C₂²⋊#88322	C₂×C₄	1	1	yes	characteristic
75	C₂²⋊#88322	C₂×C₄	1	1	yes	characteristic
76	C₂²⋊#88322	C₂×C₄	1	1	yes	characteristic
77	#13133129	C₂×C₄	1	1	yes	characteristic
78	#13133129	C₂×C₄	1	1	yes	characteristic
79	#13133129	C₂×C₄	1	1	yes	characteristic
80	#13133129	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
82	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
83	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
84	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
85	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
86	C₂²×C₈×C₅₃	—	4	2	no	—
87	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
88	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
89	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
90	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
92	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
93	C₂×C₁₆×C₅₃	—	4	2	no	—
94	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
95	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
96	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
97	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
98	C₂²×C₄×C₅₃	—	4	1	no	—
99	C₂²×C₄×C₅₃	—	4	1	no	—
100	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
103	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₂²⋊D₅₃	—	2	4	no	—
107	C₂×C₂²⋊D₅₃	C₂×C₈	1	1	yes	characteristic
108	C₂×C₂²⋊D₅₃	C₂×C₈	1	1	yes	characteristic
109	C₂×C₂²⋊D₅₃	C₂×C₈	1	1	yes	characteristic
110	C₂×C₂²⋊D₅₃	C₂×C₈	1	1	yes	characteristic
111	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
112	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
113	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
114	C₂×C₄×D₅₃	—	4	1	no	—
115	C₂×C₄×D₅₃	—	4	1	no	—
116	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
117	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
118	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₈×C₅₃	—	2	2	no	—
121	C₂×C₈×C₅₃	—	2	2	no	—
122	C₂×C₈×C₅₃	—	2	2	no	—
123	C₂×C₈×C₅₃	—	4	1	no	—
124	C₂×C₈×C₅₃	—	4	1	no	—
125	C₂×C₈×C₅₃	—	4	1	no	—
126	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
130	C₂²⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
131	C₂²⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
132	C₂²⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
133	C₂²⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
134	#11719921	C₂×C₈	1	1	yes	characteristic
135	#11719921	C₂×C₈	1	1	yes	characteristic
136	#11719921	C₂×C₈	1	1	yes	characteristic
137	#11719921	C₂×C₈	1	1	yes	characteristic
138	#11719919	C₂×C₈	1	1	yes	characteristic
139	#11719919	C₂×C₈	1	1	yes	characteristic
140	#11719919	C₂×C₈	1	1	yes	characteristic
141	#11719919	C₂×C₈	1	1	yes	characteristic
142	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
143	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
144	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
145	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
146	C₂×#88322	—	4	1	no	—
147	C₂×#88322	—	4	1	no	—
148	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
149	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
150	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
151	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
152	C₈×D₅₃	—	2	2	no	—
153	C₈×D₅₃	—	2	2	no	—
154	C₈×D₅₃	—	4	1	no	—
155	C₈×D₅₃	—	4	1	no	—
156	C₁₆×C₅₃	—	2	2	no	—
157	C₁₆×C₅₃	—	2	2	no	—
158	#11719908	—	2	2	no	—
159	#11719908	—	2	2	no	—
160	C₂³×C₅₃	—	2	2	no	—
161	C₂³×C₅₃	—	2	2	no	—
162	C₂³×C₅₃	—	4	1	no	—
163	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
164	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
165	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
166	C₂²×D₅₃	—	2	2	no	—
167	C₂²×D₅₃	—	2	2	no	—
168	C₂²×D₅₃	—	4	1	no	—
169	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
170	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
171	C₂×C₄×C₅₃	—	2	1	no	—
172	C₂×C₄×C₅₃	—	2	1	no	—
173	C₂×C₄×C₅₃	—	2	1	no	—
174	C₂×C₄×C₅₃	—	2	1	no	—
175	C₂×C₄×C₅₃	—	2	1	no	—
176	C₂×C₄×C₅₃	—	2	1	no	—
177	C₂×C₄×C₅₃	—	4	1	no	—
178	C₂×C₄×C₅₃	—	4	1	no	—
179	C₂×C₄×C₅₃	—	4	1	no	—
180	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
181	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
182	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
183	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
184	C₂²⋊D₅₃	—	2	2	no	—
185	C₂²⋊D₅₃	—	2	2	no	—
186	C₂²⋊D₅₃	—	2	2	no	—
187	C₂²⋊D₅₃	M₃₂	1	4	yes	complemented
188	C₂×Dic₅₃	—	2	2	no	—
189	C₂×Dic₅₃	—	2	2	no	—
190	C₂×Dic₅₃	—	4	1	no	—
191	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
192	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
193	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
194	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
195	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
196	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
197	C₄×D₅₃	—	2	1	no	—
198	C₄×D₅₃	—	2	1	no	—
199	C₄×D₅₃	—	2	1	no	—
200	C₄×D₅₃	—	2	1	no	—
201	C₄×D₅₃	—	4	1	no	—
202	C₄×D₅₃	—	4	1	no	—
203	C₈×C₅₃	—	2	1	no	—
204	C₈×C₅₃	—	2	1	no	—
205	C₈×C₅₃	—	2	1	no	—
206	C₈×C₅₃	H(5,𝔽₂)	1	2	yes	—
207	#88322	—	2	1	no	—
208	#88322	—	2	1	no	—
209	#88322	—	2	1	no	—
210	#88322	—	2	1	no	—
211	#33488	—	53	1	no	maximal, Sylow
212	C₂²×C₅₃	—	2	1	no	—
213	C₂²×C₅₃	—	2	1	no	—
214	C₂²×C₅₃	—	2	1	no	—
215	C₂²×C₅₃	—	2	1	no	—
216	C₂²×C₅₃	—	2	1	no	—
217	C₂²×C₅₃	—	2	1	no	—
218	C₂²×C₅₃	—	2	1	no	—
219	C₂²×C₅₃	—	4	4	no	—
220	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
221	C₂²×C₅₃	#504	1	2	yes	—
222	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
223	C₂×D₅₃	—	2	1	no	—
224	C₂×D₅₃	—	2	1	no	—
225	C₂×D₅₃	—	2	1	no	—
226	C₂×D₅₃	—	2	1	no	—
227	C₂×D₅₃	—	2	1	no	—
228	C₂×D₅₃	—	4	4	no	—
229	C₂×D₅₃	C₂×M₃₂	1	2	yes	—
230	C₄×C₅₃	—	2	1	no	—
231	C₄×C₅₃	—	2	1	no	—
232	C₄×C₅₃	—	2	1	no	—
233	C₄×C₅₃	#518	1	1	yes	characteristic
234	C₄×C₅₃	#568	1	1	yes	characteristic
235	Dic₅₃	—	2	1	no	—
236	Dic₅₃	—	2	1	no	—
237	Dic₅₃	—	2	1	no	—
238	Dic₅₃	—	2	1	no	—
239	Dic₅₃	—	2	1	no	—
240	Dic₅₃	C₂×M₃₂	1	2	yes	—
241	C₂×C₈×D₄	—	53	1	no	—
242	#2169	—	53	2	no	—
243	#2168	—	53	2	no	—
244	#2168	—	53	2	no	—
245	D₄×C₁₆	—	53	2	no	—
246	#2163	—	53	1	no	—
247	#2162	—	53	1	no	—
248	#2112	—	53	1	no	—
249	#2112	—	53	1	no	—
250	C₂×C₂²⋊C₁₆	—	53	1	no	—
251	C₂×C₂²⋊C₁₆	D₅₃	1	1	yes	characteristic, complemented
252	C₂×C₅₃	—	2	2	no	—
253	C₂×C₅₃	—	2	2	no	—
254	C₂×C₅₃	—	4	1	no	—
255	C₂×C₅₃	#2972	1	1	yes	characteristic
256	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
257	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
258	D₅₃	—	2	2	no	—
259	D₅₃	—	4	1	no	—
260	C₂³×C₈	—	53	1	no	—
261	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
262	C₂×C₄×D₄	—	53	1	no	—
263	C₂×M₃₂	—	53	2	no	—
264	C₂×M₃₂	—	53	2	no	—
265	C₂²×C₁₆	—	53	2	no	—
266	C₂²×C₁₆	C₂×D₅₃	1	2	yes	complemented
267	C₈×D₄	—	53	1	no	—
268	C₈×D₄	—	53	1	no	—
269	C₈×D₄	—	53	1	no	—
270	C₈×D₄	—	53	1	no	—
271	C₈×D₄	—	53	1	no	—
272	C₈×D₄	—	53	1	no	—
273	C₈×D₄	—	53	1	no	—
274	C₈×D₄	—	53	1	no	—
275	C₂×C₄⋊C₈	—	53	1	no	—
276	C₂×C₂²⋊C₈	—	53	1	no	—
277	C₂×C₂²⋊C₈	—	53	1	no	—
278	C₂×C₄×C₈	—	53	1	no	—
279	C₄⋊C₁₆	—	53	2	no	—
280	C₄⋊C₁₆	—	53	2	no	—
281	C₂²⋊C₁₆	—	53	1	no	—
282	C₂²⋊C₁₆	—	53	1	no	—
283	C₂²⋊C₁₆	—	53	2	no	—
284	C₂²⋊C₁₆	—	53	2	no	—
285	C₂²⋊C₁₆	—	53	2	no	—
286	C₂²⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
287	C₂²⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
288	C₂²⋊C₁₆	C₂×D₅₃	1	2	yes	complemented
289	#346	—	53	2	no	—
290	C₄×C₁₆	—	53	2	no	—
291	C₅₃	#33488	1	1	yes	characteristic, complemented, Sylow
292	C₂²×D₄	—	53	1	no	—
293	C₂³×C₄	—	53	1	no	—
294	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
295	C₂²×C₈	—	4	2	no	—
296	C₂²×C₈	—	53	1	no	—
297	C₂²×C₈	—	53	1	no	—
298	C₂²×C₈	—	53	1	no	—
299	C₂²×C₈	—	53	1	no	—
300	C₂²×C₈	—	53	1	no	—
301	C₂²×C₈	—	53	1	no	—
302	C₂²×C₈	—	212	2	no	—
303	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
304	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
305	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
306	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
307	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
308	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
309	C₄×D₄	—	53	1	no	—
310	C₄×D₄	—	53	1	no	—
311	C₄×D₄	—	53	1	no	—
312	C₄×D₄	—	53	1	no	—
313	C₄×D₄	—	53	1	no	—
314	C₄×D₄	—	53	1	no	—
315	C₄×D₄	—	53	1	no	—
316	C₄×D₄	—	53	1	no	—
317	C₂×C₄⋊C₄	—	53	1	no	—
318	C₂×C₂²⋊C₄	—	53	1	no	—
319	C₂×C₂²⋊C₄	—	53	1	no	—
320	C₂×C₄²	—	53	1	no	—
321	M₃₂	—	212	2	no	—
322	M₃₂	—	212	2	no	—
323	C₂×C₁₆	—	4	2	no	—
324	C₂×C₁₆	—	53	2	no	—
325	C₂×C₁₆	—	53	2	no	—
326	C₂×C₁₆	—	212	2	no	—
327	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
328	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
329	C₄⋊C₈	—	53	1	no	—
330	C₄⋊C₈	—	53	1	no	—
331	C₄⋊C₈	—	53	1	no	—
332	C₄⋊C₈	—	53	1	no	—
333	C₂²⋊C₈	—	53	1	no	—
334	C₂²⋊C₈	—	53	1	no	—
335	C₂²⋊C₈	—	53	1	no	—
336	C₂²⋊C₈	—	53	1	no	—
337	C₂²⋊C₈	—	53	1	no	—
338	C₂²⋊C₈	—	53	1	no	—
339	C₂²⋊C₈	—	53	1	no	—
340	C₂²⋊C₈	—	53	1	no	—
341	C₄×C₈	—	53	1	no	—
342	C₄×C₈	—	53	1	no	—
343	C₄×C₈	—	53	1	no	—
344	C₄×C₈	—	53	1	no	—
345	C₂⁴	—	53	1	no	—
346	C₂⁴	C₈×D₅₃	1	1	yes	characteristic
347	C₂×D₄	—	53	1	no	—
348	C₂×D₄	—	53	1	no	—
349	C₂×D₄	—	53	1	no	—
350	C₂×D₄	—	53	1	no	—
351	C₂×D₄	—	106	4	no	—
352	C₂²×C₄	—	4	1	no	—
353	C₂²×C₄	—	4	1	no	—
354	C₂²×C₄	—	53	1	no	—
355	C₂²×C₄	—	53	1	no	—
356	C₂²×C₄	—	53	1	no	—
357	C₂²×C₄	—	53	1	no	—
358	C₂²×C₄	—	53	1	no	—
359	C₂²×C₄	—	53	1	no	—
360	C₂²×C₄	—	53	1	no	—
361	C₂²×C₄	—	212	1	no	—
362	C₂²×C₄	—	212	1	no	—
363	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
364	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
365	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
366	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
367	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
368	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	—	2	2	no	—
372	C₂×C₈	—	4	1	no	—
373	C₂×C₈	—	4	1	no	—
374	C₂×C₈	—	4	1	no	—
375	C₂×C₈	—	53	1	no	—
376	C₂×C₈	—	53	1	no	—
377	C₂×C₈	—	53	1	no	—
378	C₂×C₈	—	53	1	no	—
379	C₂×C₈	—	106	2	no	—
380	C₂×C₈	—	106	2	no	—
381	C₂×C₈	—	212	1	no	—
382	C₂×C₈	—	212	1	no	—
383	C₂×C₈	—	212	1	no	—
384	C₂×C₈	—	212	1	no	—
385	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
386	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
387	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
388	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
389	C₄⋊C₄	—	53	1	no	—
390	C₄⋊C₄	—	53	1	no	—
391	C₄⋊C₄	—	53	1	no	—
392	C₄⋊C₄	—	53	1	no	—
393	C₂²⋊C₄	—	53	1	no	—
394	C₂²⋊C₄	—	53	1	no	—
395	C₂²⋊C₄	—	53	1	no	—
396	C₂²⋊C₄	—	53	1	no	—
397	C₂²⋊C₄	—	53	1	no	—
398	C₂²⋊C₄	—	53	1	no	—
399	C₂²⋊C₄	—	53	1	no	—
400	C₂²⋊C₄	—	53	1	no	—
401	C₄²	—	53	1	no	—
402	C₄²	—	53	1	no	—
403	C₄²	—	53	1	no	—
404	C₄²	—	53	1	no	—
405	C₁₆	—	2	2	no	—
406	C₁₆	—	2	2	no	—
407	C₁₆	—	106	2	no	—
408	C₁₆	—	106	2	no	—
409	C₂³	—	2	2	no	—
410	C₂³	—	2	2	no	—
411	C₂³	—	4	1	no	—
412	C₂³	—	53	1	no	—
413	C₂³	—	53	1	no	—
414	C₂³	—	106	2	no	—
415	C₂³	—	106	2	no	—
416	C₂³	—	212	1	no	—
417	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
418	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
419	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
420	D₄	—	53	4	no	—
421	D₄	—	106	2	no	—
422	D₄	—	106	2	no	—
423	D₄	—	106	2	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	4	1	no	—
431	C₂×C₄	—	4	1	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	53	1	no	—
434	C₂×C₄	—	53	1	no	—
435	C₂×C₄	—	53	1	no	—
436	C₂×C₄	—	53	1	no	—
437	C₂×C₄	—	53	1	no	—
438	C₂×C₄	—	53	1	no	—
439	C₂×C₄	—	106	2	no	—
440	C₂×C₄	—	106	2	no	—
441	C₂×C₄	—	106	1	no	—
442	C₂×C₄	—	106	1	no	—
443	C₂×C₄	—	106	1	no	—
444	C₂×C₄	—	106	1	no	—
445	C₂×C₄	—	212	1	no	—
446	C₂×C₄	—	212	1	no	—
447	C₂×C₄	—	212	1	no	—
448	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
449	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
450	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
451	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
452	C₈	—	2	1	no	—
453	C₈	—	2	1	no	—
454	C₈	—	2	1	no	—
455	C₈	—	106	1	no	—
456	C₈	—	106	1	no	—
457	C₈	—	106	1	no	—
458	C₈	—	106	1	no	—
459	C₈	#13133262	1	2	yes	—
460	C₂²	—	2	1	no	—
461	C₂²	—	2	1	no	—
462	C₂²	—	2	1	no	—
463	C₂²	—	2	1	no	—
464	C₂²	—	2	1	no	—
465	C₂²	—	2	1	no	—
466	C₂²	—	2	1	no	—
467	C₂²	—	4	4	no	—
468	C₂²	—	53	2	no	—
469	C₂²	—	106	1	no	—
470	C₂²	—	106	1	no	—
471	C₂²	—	106	1	no	—
472	C₂²	—	106	1	no	—
473	C₂²	—	106	1	no	—
474	C₂²	—	212	4	no	—
475	C₂²	M₃₂×D₅₃	1	2	yes	—
476	C₂²	#16811039	1	2	yes	—
477	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
478	C₄	—	2	1	no	—
479	C₄	—	2	1	no	—
480	C₄	—	2	1	no	—
481	C₄	—	53	2	no	—
482	C₄	—	106	1	no	—
483	C₄	—	106	1	no	—
484	C₄	—	106	1	no	—
485	C₄	—	106	1	no	—
486	C₄	—	106	1	no	—
487	C₄	#16811096	1	1	yes	characteristic
488	C₄	#16811389	1	1	yes	characteristic
489	C₂	—	2	2	no	—
490	C₂	—	2	2	no	—
491	C₂	—	4	1	no	—
492	C₂	—	106	2	no	—
493	C₂	—	212	1	no	—
494	C₂	#20283737	1	1	yes	characteristic
495	C₂	C₂×M₃₂×D₅₃	1	1	yes	characteristic
496	C₂	C₂×#16811039	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₂×C₅₃×C₂²⋊C₁₆	C₂
prime radical	C₂²×C₂²⋊D₅₃	C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×C₂²⋊C₁₆	D₅₃
hypocenter	C₅₃	#33488
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #20276544	C₂	semidirect
3. #20276543	C₂	semidirect
4. #20276540	C₂	semidirect
5. C₁₆×C₂²⋊D₅₃	C₂	semidirect
6. #20276515	C₂	semidirect
7. #20276511	C₂	semidirect
8. #20276326	C₂	semidirect
9. #20276324	C₂	semidirect
10. C₂²⋊C₁₆×D₅₃	C₂	semidirect
11. C₂×C₅₃×C₂²⋊C₁₆	C₂	semidirect
14. C₂×#13133157	C₂²	semidirect
15. C₂×#13133156	C₂²	semidirect
16. C₂×C₁₆×D₅₃	C₂²	semidirect
29. #16810441	C₂²	semidirect
30. #16810441	C₂²	semidirect
33. C₂²⋊#11719908	C₂²	semidirect
34. #16810402	C₂²	semidirect
35. #16810402	C₂²	semidirect
36. #16810398	C₂²	semidirect
37. C₁₆×Dic₅₃	C₂²	semidirect
39. C₂²×C₁₆×C₅₃	C₂²	semidirect
42. C₅₃×C₂²⋊C₁₆	C₂²	semidirect
187. C₂²⋊D₅₃	M₃₂	semidirect
251. C₂×C₂²⋊C₁₆	D₅₃	semidirect
266. C₂²×C₁₆	C₂×D₅₃	semidirect
288. C₂²⋊C₁₆	C₂×D₅₃	semidirect
291. C₅₃	#33488	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×C₂²⋊D₅₃	2
#20276544	2
#20276543	2
#20276540	2
C₁₆×C₂²⋊D₅₃	2
#20276515	2
#20276511	2
#20276326	2
#20276324	2
C₂²⋊C₁₆×D₅₃	2
C₂×C₅₃×C₂²⋊C₁₆	2
#33488	53