Subgroups — #28319497

Order 13568.

810 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291193	C₂	1	2	yes	maximal
2	#20290652	C₂	1	1	yes	maximal, characteristic, complemented
3	#20290651	C₂	1	1	yes	maximal, characteristic, complemented
4	#20290638	C₂	1	2	yes	maximal, complemented
5	#20290593	C₂	1	2	yes	maximal
6	#20290592	C₂	1	2	yes	maximal
7	#20290529	C₂	1	2	yes	maximal, complemented
8	C₄×#13133286	C₂	1	2	yes	maximal, complemented
9	#405×D₅₃	C₂	1	1	yes	maximal, characteristic
10	C₂×#13133286	C₂²	1	2	yes	—
11	C₂×#13133286	C₂²	1	2	yes	—
12	C₂×#13133286	C₂²	1	2	yes	—
13	C₂×#13133285	C₂²	1	2	yes	—
14	#16811573	—	2	4	no	—
15	#16811572	—	2	4	no	—
16	#16811569	C₂²	1	2	yes	—
17	#16811567	C₂²	1	2	yes	—
18	#16811565	C₂²	1	4	yes	—
19	C₄×C₄⋊#6157	C₂²	1	4	yes	—
20	#16811497	C₂²	1	4	yes	—
21	#16811497	C₂²	1	4	yes	—
22	C₂×M₁₆×D₅₃	C₂²	1	1	yes	characteristic
23	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
24	#117×D₅₃	C₂²	1	1	yes	characteristic
25	#16810581	C₄	1	1	yes	characteristic, complemented
26	#16810580	C₄	1	1	yes	characteristic, complemented
27	#16810580	C₄	1	1	yes	characteristic, complemented
28	#16810579	C₄	1	1	yes	characteristic, complemented
29	#16810578	C₄	1	1	yes	characteristic, complemented
30	#16810577	C₄	1	1	yes	characteristic, complemented
31	#16810577	C₄	1	1	yes	characteristic, complemented
32	#97×D₅₃	C₂²	1	1	yes	characteristic
33	#97×D₅₃	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
35	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
36	C₅₃×#405	C₄	1	1	yes	characteristic, complemented
37	C₂×C₄⋊#6157	—	2	4	no	—
38	C₂×C₄⋊#6157	C₂³	1	4	yes	—
39	C₂×C₄⋊#6157	C₂³	1	4	yes	—
40	C₂×C₂²⋊#6157	—	2	4	no	—
41	C₂×C₄×#6157	—	2	4	no	—
42	#13133286	—	2	4	no	—
43	#13133286	—	2	4	no	—
44	#13133286	—	2	4	no	—
45	#13133285	—	2	4	no	—
46	(C₂×#6157)⋊C₄	—	2	4	no	—
47	(C₂×#6157)⋊C₄	—	2	4	no	—
48	(C₂×#6157)⋊C₄	—	2	4	no	—
49	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
50	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
51	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
52	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
53	M₁₆×D₅₃	—	2	2	no	—
54	M₁₆×D₅₃	—	2	2	no	—
55	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
56	C₂×C₈×D₅₃	—	2	2	no	—
57	C₂×C₈×D₅₃	—	2	2	no	—
58	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
61	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
62	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
63	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
64	#13133176	C₂×C₄	1	1	yes	characteristic
65	#13133175	Q₈	1	2	yes	—
66	#13133174	C₂×C₄	1	1	yes	characteristic
67	#13133174	C₂×C₄	1	1	yes	characteristic
68	#13133173	Q₈	1	2	yes	—
69	C₄⋊C₄×D₅₃	D₄	1	2	yes	—
70	C₂²⋊C₄×D₅₃	D₄	1	2	yes	—
71	C₄²×D₅₃	C₂³	1	1	yes	characteristic
72	C₄²×D₅₃	C₂³	1	1	yes	characteristic
73	#13133128	C₂×C₄	1	1	yes	characteristic
74	#13133128	C₂×C₄	1	1	yes	characteristic
75	#13133127	C₂×C₄	1	1	yes	characteristic
76	#13133127	C₂×C₄	1	1	yes	characteristic
77	#13133127	C₂×C₄	1	1	yes	characteristic
78	#13133127	C₂×C₄	1	1	yes	characteristic
79	C₄×#88322	C₂×C₄	1	1	yes	characteristic
80	C₄×#88322	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	C₂×C₄	1	1	yes	characteristic
84	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
85	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
86	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
87	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
88	#97×C₅₃	C₂×C₄	1	1	yes	characteristic
89	#97×C₅₃	C₂×C₄	1	1	yes	characteristic
90	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
91	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
92	C₂³×D₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×#6157	—	2	4	no	—
94	C₂²×#6157	—	2	4	no	—
95	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
97	C₂×C₄×D₅₃	—	2	1	no	—
98	C₂×C₄×D₅₃	—	2	1	no	—
99	C₂×C₄×D₅₃	—	2	1	no	—
100	C₂×C₄×D₅₃	—	2	1	no	—
101	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
103	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
104	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
106	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
107	C₂²⋊#6157	—	8	4	no	—
108	C₄⋊#6157	—	2	4	no	—
109	C₄⋊#6157	—	2	4	no	—
110	C₄⋊#6157	—	2	4	no	—
111	C₄⋊#6157	—	2	4	no	—
112	C₄⋊#6157	—	8	4	no	—
113	C₄×#6157	—	8	4	no	—
114	M₁₆×C₅₃	—	2	2	no	—
115	C₂×C₈×C₅₃	—	2	2	no	—
116	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
121	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
122	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
123	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₂²⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
125	#11719921	C₄⋊C₄	1	2	yes	—
126	#11719921	C₄⋊C₄	1	2	yes	—
127	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
128	#11719919	C₄⋊C₄	1	2	yes	—
129	#11719919	C₄⋊C₄	1	2	yes	—
130	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
131	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
132	C₄×Dic₅₃	C₂×Q₈	1	2	yes	—
133	C₄×Dic₅₃	C₂×Q₈	1	2	yes	—
134	#11719917	—	2	2	no	—
135	C₂×#88322	—	2	2	no	—
136	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
137	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
138	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
139	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
140	#11719912	—	2	2	no	—
141	#11719912	—	2	2	no	—
142	C₈×D₅₃	—	2	1	no	—
143	C₈×D₅₃	—	2	1	no	—
144	C₈×D₅₃	—	2	1	no	—
145	C₈×D₅₃	—	2	1	no	—
146	C₈×D₅₃	—	2	2	no	—
147	C₈×D₅₃	—	2	2	no	—
148	C₈×D₅₃	#41	1	2	yes	—
149	C₈×D₅₃	#41	1	2	yes	—
150	C₈×D₅₃	#41	1	2	yes	—
151	C₈×D₅₃	#41	1	2	yes	—
152	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂²×D₅₃	—	2	1	no	—
154	C₂²×D₅₃	—	2	1	no	—
155	C₂²×D₅₃	—	2	2	no	—
156	C₂²×D₅₃	—	2	2	no	—
157	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₂²×D₅₃	C₂²×D₄	1	1	yes	characteristic
159	C₂×#6157	—	4	4	no	—
160	C₂×#6157	—	4	4	no	—
161	C₂×#6157	—	4	4	no	—
162	C₂×#6157	—	4	4	no	—
163	C₂×#6157	—	4	4	no	—
164	C₂×#6157	—	4	4	no	—
165	C₂×C₄×C₅₃	—	2	1	no	—
166	C₂×C₄×C₅₃	—	2	1	no	—
167	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
169	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
170	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
171	C₂×Dic₅₃	—	2	1	no	—
172	C₂×Dic₅₃	—	2	1	no	—
173	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
175	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
176	C₂×Dic₅₃	C₂²×Q₈	1	1	yes	characteristic
177	C₄×D₅₃	—	2	1	no	—
178	C₄×D₅₃	—	2	1	no	—
179	C₄×D₅₃	—	2	2	no	—
180	C₄×D₅₃	—	2	2	no	—
181	C₄×D₅₃	—	2	2	no	—
182	C₄×D₅₃	—	2	2	no	—
183	C₄×D₅₃	—	2	1	no	—
184	C₄×D₅₃	—	2	1	no	—
185	C₄×D₅₃	—	2	1	no	—
186	C₄×D₅₃	—	2	1	no	—
187	C₄×D₅₃	C₂×#41	1	1	yes	characteristic
188	C₄×D₅₃	C₂×#41	1	1	yes	characteristic
189	C₄×D₅₃	C₂×#41	1	1	yes	characteristic
190	C₄×D₅₃	C₂×#41	1	1	yes	characteristic
191	C₈×C₅₃	—	2	1	no	—
192	C₈×C₅₃	—	2	1	no	—
193	C₈×C₅₃	#117	1	2	yes	—
194	C₈×C₅₃	#117	1	2	yes	—
195	#88322	—	2	1	no	—
196	#88322	—	2	1	no	—
197	#88322	#117	1	2	yes	—
198	#88322	#117	1	2	yes	—
199	#20821	—	53	1	no	maximal, Sylow
200	C₂²×C₅₃	—	2	2	no	—
201	C₂²×C₅₃	—	2	1	no	—
202	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
203	C₂×D₅₃	—	2	2	no	—
204	C₂×D₅₃	—	2	1	no	—
205	C₂×D₅₃	—	4	2	no	—
206	C₂×D₅₃	—	4	1	no	—
207	C₂×D₅₃	—	4	1	no	—
208	C₂×D₅₃	#532	1	1	yes	characteristic
209	C₂×D₅₃	#532	1	1	yes	characteristic
210	C₂×D₅₃	#576	1	2	yes	—
211	C₂×D₅₃	#578	1	2	yes	—
212	#6157	—	8	4	no	—
213	#6157	—	8	4	no	—
214	C₄×C₅₃	—	2	2	no	—
215	C₄×C₅₃	—	2	2	no	—
216	C₄×C₅₃	—	2	1	no	—
217	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
218	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
219	Dic₅₃	—	2	2	no	—
220	Dic₅₃	—	2	2	no	—
221	Dic₅₃	—	2	1	no	—
222	Dic₅₃	#533	1	1	yes	characteristic
223	Dic₅₃	#533	1	1	yes	characteristic
224	C₂×#405	—	53	1	no	—
225	#2314	—	53	2	no	—
226	#1846	—	53	1	no	—
227	#1845	—	53	1	no	—
228	#1836	—	53	2	no	—
229	#1809	—	53	2	no	—
230	#1808	—	53	2	no	—
231	#1776	—	53	2	no	—
232	C₄×#107	—	53	2	no	—
233	C₂×C₅₃	—	4	1	no	—
234	C₂×C₅₃	#2285	1	1	yes	characteristic
235	C₂×C₅₃	#2913	1	2	yes	—
236	D₅₃	—	4	1	no	—
237	D₅₃	#3175	1	2	yes	—
238	D₅₃	#3176	1	1	yes	characteristic
239	D₅₃	#3177	1	1	yes	characteristic
240	C₂²×M₁₆	—	53	1	no	—
241	C₂³×C₈	—	53	1	no	—
242	C₂×#117	—	53	1	no	—
243	C₂×#107	—	53	2	no	—
244	C₂×#107	—	53	2	no	—
245	C₂×#107	—	53	2	no	—
246	C₂×#106	—	53	2	no	—
247	#405	—	53	1	no	—
248	#405	—	53	1	no	—
249	#405	—	53	1	no	—
250	#405	—	53	1	no	—
251	#405	—	53	1	no	—
252	#405	—	53	1	no	—
253	#405	—	53	1	no	—
254	#405	#6157	1	1	yes	characteristic, complemented
255	C₂×#97	—	53	1	no	—
256	C₂×#97	—	53	1	no	—
257	C₂×C₄×C₈	—	53	1	no	—
258	C₂×C₄×C₈	—	53	1	no	—
259	#388	—	106	4	no	—
260	#387	—	106	4	no	—
261	#384	—	53	2	no	—
262	#382	—	53	2	no	—
263	#380	—	53	4	no	—
264	C₄×C₄⋊C₄	—	53	4	no	—
265	#340	—	53	4	no	—
266	#340	—	53	4	no	—
267	C₅₃	#20821	1	1	yes	characteristic, complemented, Sylow
268	C₂³×C₄	—	53	1	no	—
269	C₂×M₁₆	—	53	1	no	—
270	C₂×M₁₆	—	53	1	no	—
271	C₂×M₁₆	—	53	1	no	—
272	C₂×M₁₆	—	106	2	no	—
273	C₂×M₁₆	—	106	2	no	—
274	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
275	C₂²×C₈	—	53	1	no	—
276	C₂²×C₈	—	53	1	no	—
277	C₂²×C₈	—	53	1	no	—
278	C₂²×C₈	—	53	1	no	—
279	C₂²×C₈	—	53	1	no	—
280	C₂²×C₈	—	53	1	no	—
281	C₂²×C₈	—	53	1	no	—
282	C₂²×C₈	—	106	2	no	—
283	C₂²×C₈	—	106	2	no	—
284	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
285	#117	—	53	1	no	—
286	#117	—	53	2	no	—
287	#117	—	53	2	no	—
288	#117	—	53	1	no	—
289	#117	—	53	1	no	—
290	#117	C₂×#6157	1	1	yes	characteristic
291	C₂×C₄⋊C₄	—	53	2	no	—
292	C₂×C₄⋊C₄	—	53	4	no	—
293	C₂×C₄⋊C₄	—	53	4	no	—
294	C₂×C₄⋊C₄	—	106	4	no	—
295	C₂×C₂²⋊C₄	—	53	2	no	—
296	C₂×C₂²⋊C₄	—	106	4	no	—
297	C₂×C₄²	—	53	1	no	—
298	C₂×C₄²	—	53	1	no	—
299	C₂×C₄²	—	106	4	no	—
300	#107	—	106	4	no	—
301	#107	—	106	4	no	—
302	#107	—	106	4	no	—
303	#106	—	106	4	no	—
304	#97	—	53	1	no	—
305	#97	—	53	1	no	—
306	#97	—	53	1	no	—
307	#97	—	53	1	no	—
308	#97	—	53	1	no	—
309	#97	—	53	1	no	—
310	#97	C₂×#6157	1	1	yes	characteristic
311	#97	C₂×#6157	1	1	yes	characteristic
312	C₄×C₈	—	53	1	no	—
313	C₄×C₈	—	53	1	no	—
314	C₄×C₈	—	53	1	no	—
315	C₄×C₈	—	53	1	no	—
316	C₄×C₈	—	53	1	no	—
317	C₄×C₈	—	53	1	no	—
318	C₄×C₈	C₂×#6157	1	1	yes	characteristic
319	C₄×C₈	C₂×#6157	1	1	yes	characteristic
320	#95	—	106	4	no	—
321	#95	—	106	4	no	—
322	#95	—	106	4	no	—
323	C₂⁴	—	53	1	no	—
324	C₂²×C₄	—	53	1	no	—
325	C₂²×C₄	—	53	1	no	—
326	C₂²×C₄	—	53	1	no	—
327	C₂²×C₄	—	53	2	no	—
328	C₂²×C₄	—	53	2	no	—
329	C₂²×C₄	—	53	1	no	—
330	C₂²×C₄	—	53	1	no	—
331	C₂²×C₄	—	106	1	no	—
332	C₂²×C₄	—	106	1	no	—
333	C₂²×C₄	—	106	1	no	—
334	C₂²×C₄	—	106	1	no	—
335	C₂²×C₄	—	106	4	no	—
336	C₂²×C₄	—	106	4	no	—
337	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
338	M₁₆	—	2	2	no	—
339	M₁₆	—	106	2	no	—
340	M₁₆	—	106	2	no	—
341	M₁₆	—	106	2	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	—	53	1	no	—
344	C₂×C₈	—	53	1	no	—
345	C₂×C₈	—	53	2	no	—
346	C₂×C₈	—	53	2	no	—
347	C₂×C₈	—	53	2	no	—
348	C₂×C₈	—	53	2	no	—
349	C₂×C₈	—	53	1	no	—
350	C₂×C₈	—	53	1	no	—
351	C₂×C₈	—	106	2	no	—
352	C₂×C₈	—	106	1	no	—
353	C₂×C₈	—	106	1	no	—
354	C₂×C₈	—	106	1	no	—
355	C₂×C₈	—	106	1	no	—
356	C₂×C₈	—	106	2	no	—
357	C₂×C₈	—	106	2	no	—
358	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
359	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
360	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
361	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
362	C₄⋊C₄	—	53	2	no	—
363	C₄⋊C₄	—	53	2	no	—
364	C₄⋊C₄	—	53	2	no	—
365	C₄⋊C₄	—	106	4	no	—
366	C₄⋊C₄	—	106	4	no	—
367	C₄⋊C₄	—	106	4	no	—
368	C₄⋊C₄	—	106	4	no	—
369	C₄⋊C₄	—	424	4	no	—
370	C₄⋊C₄	C₄⋊#6157	1	2	yes	—
371	C₂²⋊C₄	—	53	2	no	—
372	C₂²⋊C₄	—	53	2	no	—
373	C₂²⋊C₄	—	53	2	no	—
374	C₂²⋊C₄	—	424	4	no	—
375	C₂²⋊C₄	C₄⋊#6157	1	2	yes	—
376	C₄²	—	53	2	no	—
377	C₄²	—	53	2	no	—
378	C₄²	—	53	1	no	—
379	C₄²	—	53	1	no	—
380	C₄²	—	424	4	no	—
381	C₄²	C₂²×#6157	1	1	yes	characteristic
382	C₄²	C₂²×#6157	1	1	yes	characteristic
383	C₂³	—	53	1	no	—
384	C₂³	—	53	1	no	—
385	C₂³	—	106	2	no	—
386	C₂³	—	106	1	no	—
387	C₂³	—	106	1	no	—
388	C₂³	—	106	2	no	—
389	C₂³	C₂×C₄⋊#6157	1	1	yes	characteristic
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	53	1	no	—
393	C₂×C₄	—	53	1	no	—
394	C₂×C₄	—	53	1	no	—
395	C₂×C₄	—	53	1	no	—
396	C₂×C₄	—	53	1	no	—
397	C₂×C₄	—	53	2	no	—
398	C₂×C₄	—	53	2	no	—
399	C₂×C₄	—	53	1	no	—
400	C₂×C₄	—	106	1	no	—
401	C₂×C₄	—	106	1	no	—
402	C₂×C₄	—	106	2	no	—
403	C₂×C₄	—	106	2	no	—
404	C₂×C₄	—	106	2	no	—
405	C₂×C₄	—	106	2	no	—
406	C₂×C₄	—	106	1	no	—
407	C₂×C₄	—	106	1	no	—
408	C₂×C₄	—	106	1	no	—
409	C₂×C₄	—	106	1	no	—
410	C₂×C₄	—	106	1	no	—
411	C₂×C₄	—	106	1	no	—
412	C₂×C₄	—	212	4	no	—
413	C₂×C₄	—	212	4	no	—
414	C₂×C₄	—	212	4	no	—
415	C₂×C₄	—	212	4	no	—
416	C₂×C₄	—	212	4	no	—
417	C₂×C₄	—	212	4	no	—
418	C₂×C₄	C₂×C₄⋊#6157	1	2	yes	—
419	C₂×C₄	C₂×C₄⋊#6157	1	2	yes	—
420	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
421	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
422	C₈	—	2	1	no	—
423	C₈	—	2	1	no	—
424	C₈	—	53	2	no	—
425	C₈	—	53	2	no	—
426	C₈	—	106	1	no	—
427	C₈	—	106	1	no	—
428	C₈	#13133296	1	2	yes	—
429	C₈	#13133296	1	2	yes	—
430	C₂²	—	2	2	no	—
431	C₂²	—	2	1	no	—
432	C₂²	—	53	2	no	—
433	C₂²	—	53	2	no	—
434	C₂²	—	53	1	no	—
435	C₂²	—	53	1	no	—
436	C₂²	—	106	2	no	—
437	C₂²	—	106	1	no	—
438	C₂²	—	212	1	no	—
439	C₂²	—	212	2	no	—
440	C₂²	—	212	1	no	—
441	C₂²	C₂²×C₄⋊#6157	1	1	yes	characteristic
442	C₄	—	2	2	no	—
443	C₄	—	2	2	no	—
444	C₄	—	2	1	no	—
445	C₄	—	53	1	no	—
446	C₄	—	53	1	no	—
447	C₄	—	106	2	no	—
448	C₄	—	106	2	no	—
449	C₄	—	106	1	no	—
450	C₄	—	424	4	no	—
451	C₄	—	424	4	no	—
452	C₄	C₂×#13133296	1	1	yes	characteristic
453	C₄	C₂×#13133296	1	1	yes	characteristic
454	C₂	—	4	1	no	—
455	C₂	—	53	1	no	—
456	C₂	—	53	2	no	—
457	C₂	—	53	1	no	—
458	C₂	—	212	1	no	—
459	C₂	#20291164	1	1	yes	characteristic
460	C₂	#20291370	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₅₃×#405	C₄
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₅₃	C₂×D₄
cosocle	C₂×C₄×D₅₃	C₂⁴
hypercenter	#405	#6157
hypocenter	C₅₃	#20821
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
2. #20290652	C₂	semidirect
3. #20290651	C₂	semidirect
4. #20290638	C₂	semidirect
7. #20290529	C₂	semidirect
8. C₄×#13133286	C₂	semidirect
25. #16810581	C₄	semidirect
26. #16810580	C₄	semidirect
27. #16810580	C₄	semidirect
28. #16810579	C₄	semidirect
29. #16810578	C₄	semidirect
30. #16810577	C₄	semidirect
31. #16810577	C₄	semidirect
36. C₅₃×#405	C₄	semidirect
254. #405	#6157	semidirect
267. C₅₃	#20821	semidirect

Maximal subgroups

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

subgroup	index
#20291193	2
#20290652	2
#20290651	2
#20290638	2
#20290593	2
#20290592	2
#20290529	2
C₄×#13133286	2
#405×D₅₃	2
#20821	53