Subgroups — #27353541

Order 13568.

702 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283823	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133248	C₂	1	1	yes	maximal, characteristic
3	#20273532	C₂	1	2	yes	maximal, complemented
4	#20273525	C₂	1	2	yes	maximal, complemented
5	C₈×#11719925	C₂	1	2	yes	maximal, complemented
6	C₈×#11719924	C₂	1	2	yes	maximal, complemented
7	#20273468	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273095	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₈⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
10	#20272996	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2982	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133248	C₄	1	1	yes	characteristic
13	C₄×#11719945	C₂²	1	1	yes	characteristic
14	C₈×Dic₁₀₆	C₂²	1	2	yes	—
15	#16810728	C₄	1	2	yes	—
16	#16810721	C₄	1	2	yes	—
17	C₄×#11719925	C₂²	1	2	yes	—
18	C₄×#11719924	C₂²	1	2	yes	—
19	C₈×D₂₁₂	C₂²	1	2	yes	—
20	#16810686	—	2	2	no	—
21	#16810675	—	2	1	no	—
22	#16810663	C₄	1	1	yes	characteristic
23	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×D₅₃	—	2	1	no	—
25	#16810632	C₄	1	1	yes	characteristic
26	#16810622	C₄	1	1	yes	characteristic
27	C₄×#11719917	C₂²	1	1	yes	characteristic
28	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
29	#16810375	C₂²	1	2	yes	—
30	#16810373	C₂²	1	2	yes	—
31	Q₈⋊#88322	C₂²	1	2	yes	—
32	Q₈⋊#88322	C₂²	1	2	yes	—
33	#16810353	C₂²	1	1	yes	characteristic
34	#16810353	C₂²	1	1	yes	characteristic
35	#16810350	C₂²	1	1	yes	characteristic
36	#16810350	C₂²	1	1	yes	characteristic
37	#16810345	C₂²	1	2	yes	—
38	C₈×#88322	C₂²	1	2	yes	—
39	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
40	C₈×Q₈×C₅₃	—	2	1	no	—
41	C₈×Q₈×C₅₃	—	2	1	no	—
42	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
43	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
44	C₅₃×#431	—	2	2	no	—
45	C₅₃×#431	C₂²	1	1	yes	characteristic
46	C₅₃×#431	C₂²	1	1	yes	characteristic
47	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
48	#13133248	C₈	1	8	yes	complemented
49	C₂×#11719925	C₂×C₄	1	2	yes	—
50	C₂×#11719924	C₂×C₄	1	2	yes	—
51	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
52	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
53	C₂²×#88322	—	2	1	no	—
54	C₂²×#88322	—	2	1	no	—
55	C₂×C₈×D₅₃	—	2	2	no	—
56	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
57	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
58	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
59	C₄×D₂₁₂	C₂³	1	1	yes	characteristic
60	C₄×C₂²⋊D₅₃	—	2	1	no	—
61	C₄×C₂²⋊D₅₃	—	2	1	no	—
62	#13133176	C₂×C₄	1	1	yes	characteristic
63	#13133174	—	2	1	no	—
64	C₄²×D₅₃	—	2	1	no	—
65	#13133150	C₂×C₄	1	2	yes	—
66	#13133148	C₂×C₄	1	2	yes	—
67	C₄⋊#88322	—	2	1	no	—
68	C₄⋊#88322	C₂³	1	2	yes	—
69	#13133146	—	2	2	no	—
70	#13133141	C₂×C₄	1	2	yes	—
71	#13133141	C₂×C₄	1	2	yes	—
72	#13133140	C₂×C₄	1	2	yes	—
73	#13133137	C₂×C₄	1	2	yes	—
74	#13133129	—	2	2	no	—
75	#13133128	C₂×C₄	1	2	yes	—
76	C₄×#88322	—	2	1	no	—
77	C₄×#88322	C₂³	1	1	yes	characteristic
78	C₄×#88322	C₂³	1	1	yes	characteristic
79	C₄×#88322	C₂³	1	2	yes	—
80	C₈×Dic₅₃	—	2	2	no	—
81	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
82	C₄×Q₈×C₅₃	—	2	1	no	—
83	C₄×Q₈×C₅₃	—	2	1	no	—
84	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
85	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
86	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
87	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
88	C₂×C₄²×C₅₃	—	2	1	no	—
89	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
90	C₄⋊C₈×C₅₃	—	2	2	no	—
91	C₄⋊C₈×C₅₃	—	2	2	no	—
92	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
93	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
94	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
95	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
96	C₂²⋊C₈×C₅₃	—	2	1	no	—
97	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
98	C₄×C₈×C₅₃	—	2	2	no	—
99	C₄×C₈×C₅₃	—	2	2	no	—
100	C₄×C₈×C₅₃	—	2	1	no	—
101	C₄×C₈×C₅₃	C₂³	1	2	yes	—
102	C₂×Q₈×C₅₃	—	2	1	no	—
103	C₂×Q₈×C₅₃	—	2	2	no	—
104	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	complemented
105	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
106	C₂²×C₄×C₅₃	—	2	1	no	—
107	C₂²×C₄×C₅₃	—	2	1	no	—
108	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
111	C₂×C₂²⋊D₅₃	—	2	1	no	—
112	#11719945	—	2	2	no	—
113	#11719945	C₂×C₈	1	4	yes	—
114	C₂×D₂₁₂	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×D₅₃	—	2	1	no	—
116	C₂×C₄×D₅₃	—	2	1	no	—
117	C₂×C₄×D₅₃	—	2	1	no	—
118	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₈×C₅₃	—	2	1	no	—
120	C₂×C₈×C₅₃	—	2	1	no	—
121	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
122	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
123	C₄⋊C₄×C₅₃	—	2	2	no	—
124	C₄⋊C₄×C₅₃	—	2	1	no	—
125	C₄⋊C₄×C₅₃	—	2	2	no	—
126	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
127	C₄²×C₅₃	—	2	2	no	—
128	C₄²×C₅₃	—	2	2	no	—
129	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₅₃	#41	1	1	yes	characteristic
132	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
133	C₂²⋊Dic₅₃	—	2	1	no	—
134	#11719925	C₂×C₈	1	8	yes	complemented
135	#11719924	C₂×C₈	1	8	yes	complemented
136	#11719921	—	2	1	no	—
137	#11719921	—	2	1	no	—
138	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
139	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
140	#11719919	—	2	1	no	—
141	#11719919	—	2	1	no	—
142	C₄×Dic₅₃	—	2	1	no	—
143	C₄×Dic₅₃	—	2	1	no	—
144	C₄×Dic₅₃	—	2	1	no	—
145	#11719917	—	2	2	no	—
146	#11719917	C₂×C₈	1	4	yes	complemented
147	C₂×#88322	—	2	1	no	—
148	C₂×#88322	—	2	2	no	—
149	C₂×#88322	—	2	1	no	—
150	C₂×#88322	—	4	2	no	—
151	C₂×#88322	—	8	1	no	—
152	C₂×#88322	C₂²×C₄	1	2	yes	—
153	C₂×#88322	C₂²×C₄	1	2	yes	—
154	C₈×D₅₃	—	8	2	no	—
155	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
156	C₂²×D₅₃	—	2	1	no	—
157	Q₈×C₅₃	—	2	2	no	—
158	Q₈×C₅₃	—	2	4	no	—
159	Q₈×C₅₃	C₂²×C₈	1	4	yes	—
160	C₂×C₄×C₅₃	—	2	2	no	—
161	C₂×C₄×C₅₃	—	2	4	no	—
162	C₂×C₄×C₅₃	—	2	1	no	—
163	C₂×C₄×C₅₃	—	2	1	no	—
164	C₂×C₄×C₅₃	—	2	2	no	—
165	C₂×C₄×C₅₃	—	4	1	no	—
166	C₂×C₄×C₅₃	—	8	1	no	—
167	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
168	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
169	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
170	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
171	C₂×C₄×C₅₃	#131	1	2	yes	—
172	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
173	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
175	C₂²⋊D₅₃	—	2	4	no	—
176	C₂×Dic₅₃	—	2	1	no	—
177	C₂×Dic₅₃	—	2	1	no	—
178	C₂×Dic₅₃	—	2	1	no	—
179	D₂₁₂	—	2	1	no	—
180	D₂₁₂	C₂²×C₈	1	2	yes	—
181	C₄×D₅₃	—	2	4	no	—
182	C₄×D₅₃	—	4	1	no	—
183	C₄×D₅₃	—	4	1	no	—
184	C₄×D₅₃	—	8	1	no	—
185	Dic₁₀₆	—	2	1	no	—
186	Dic₁₀₆	C₂²×C₈	1	2	yes	—
187	C₈×C₅₃	—	2	2	no	—
188	C₈×C₅₃	—	2	2	no	—
189	C₈×C₅₃	—	4	1	no	—
190	C₈×C₅₃	—	4	1	no	—
191	#88322	—	2	2	no	—
192	#88322	—	2	2	no	—
193	#88322	—	4	1	no	—
194	#88322	—	4	2	no	—
195	#88322	C₂²×C₈	1	4	yes	—
196	#17853	—	53	1	no	maximal, Sylow
197	C₂²×C₅₃	—	2	1	no	—
198	C₂²×C₅₃	—	2	1	no	—
199	C₂²×C₅₃	C₈×D₄	1	2	yes	—
200	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
201	C₂×D₅₃	—	2	2	no	—
202	C₂×D₅₃	—	4	1	no	—
203	C₂×D₅₃	—	4	1	no	—
204	C₄×C₅₃	—	2	4	no	—
205	C₄×C₅₃	—	2	1	no	—
206	C₄×C₅₃	—	2	1	no	—
207	C₄×C₅₃	—	2	1	no	—
208	C₄×C₅₃	—	4	2	no	—
209	C₄×C₅₃	C₈×D₄	1	2	yes	—
210	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
211	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
212	C₄×C₅₃	#577	1	1	yes	characteristic
213	C₄×C₅₃	#578	1	1	yes	characteristic
214	Dic₅₃	—	2	2	no	—
215	Dic₅₃	—	4	1	no	—
216	Dic₅₃	—	4	1	no	—
217	#2988	—	53	1	no	—
218	#2982	D₅₃	1	1	yes	characteristic, complemented
219	C₄×#137	—	53	1	no	—
220	#1579	—	53	2	no	—
221	SD₁₆⋊C₈	—	53	2	no	—
222	C₈×Q₁₆	—	53	2	no	—
223	C₈×SD₁₆	—	53	2	no	—
224	#1565	—	53	1	no	—
225	#1476	—	53	1	no	—
226	C₂×Q₈⋊C₈	—	53	1	no	—
227	C₈×M₁₆	—	53	1	no	—
228	C₂×C₅₃	—	2	2	no	—
229	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
230	C₂×C₅₃	#2947	1	1	yes	characteristic
231	C₂×C₅₃	#2956	1	1	yes	characteristic
232	D₅₃	—	4	2	no	—
233	C₂×#137	—	53	1	no	—
234	C₄×#41	—	53	1	no	—
235	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
236	C₈×Q₈	—	2	1	no	—
237	C₈×Q₈	—	2	1	no	—
238	C₈×Q₈	—	53	2	no	—
239	C₈×Q₈	C₂×D₅₃	1	2	yes	—
240	C₈×Q₈	C₂×D₅₃	1	2	yes	—
241	#441	—	53	2	no	—
242	#440	—	53	2	no	—
243	C₄×Q₁₆	—	53	2	no	—
244	C₄×SD₁₆	—	53	2	no	—
245	C₈×D₄	—	53	2	no	—
246	C₈×D₄	—	106	2	no	—
247	#431	—	2	2	no	—
248	#431	—	106	1	no	—
249	#431	C₂×D₅₃	1	1	yes	characteristic
250	#431	C₂×D₅₃	1	1	yes	characteristic
251	#428	—	53	1	no	—
252	C₂×C₄⋊C₈	—	53	1	no	—
253	C₂×C₄⋊C₈	—	106	1	no	—
254	#419	—	53	1	no	—
255	#417	—	53	1	no	—
256	C₄×M₁₆	—	53	1	no	—
257	C₂×C₄×C₈	—	53	1	no	—
258	#335	—	53	2	no	—
259	#334	—	53	2	no	—
260	Q₈⋊C₈	—	53	2	no	—
261	Q₈⋊C₈	—	53	2	no	—
262	Q₈⋊C₈	—	53	1	no	—
263	Q₈⋊C₈	—	53	1	no	—
264	D₄⋊C₈	—	53	1	no	—
265	D₄⋊C₈	—	53	1	no	—
266	#322	—	53	2	no	—
267	C₈²	—	53	2	no	—
268	C₅₃	#17853	1	1	yes	characteristic, complemented, Sylow
269	C₂×#41	—	53	1	no	—
270	C₂²×Q₈	C₄×D₅₃	1	1	yes	characteristic
271	#137	—	53	8	no	—
272	C₂×Q₁₆	—	53	2	no	—
273	C₂×SD₁₆	—	53	2	no	—
274	C₂×M₁₆	—	53	1	no	—
275	C₂×M₁₆	—	53	1	no	—
276	C₂²×C₈	—	106	1	no	—
277	C₂²×C₈	—	106	1	no	—
278	C₂²×C₈	—	106	2	no	—
279	C₄×Q₈	—	2	1	no	—
280	C₄×Q₈	—	2	1	no	—
281	C₄×Q₈	—	53	1	no	—
282	C₄×Q₈	—	53	1	no	—
283	C₄×Q₈	C₄×D₅₃	1	2	yes	—
284	C₄×Q₈	C₂²×D₅₃	1	2	yes	—
285	C₄×D₄	—	53	1	no	—
286	C₄×D₄	—	53	1	no	—
287	C₄×D₄	—	106	1	no	—
288	C₄×D₄	—	106	1	no	—
289	#117	—	53	1	no	—
290	#117	—	106	1	no	—
291	C₂×C₄⋊C₄	—	2	1	no	—
292	C₂×C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
293	C₂×C₄²	—	2	1	no	—
294	C₂×C₄²	—	106	1	no	—
295	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
296	#107	—	53	2	no	—
297	#106	—	53	2	no	—
298	C₄⋊C₈	—	2	2	no	—
299	C₄⋊C₈	—	2	2	no	—
300	C₄⋊C₈	—	53	2	no	—
301	C₄⋊C₈	—	106	2	no	—
302	C₄⋊C₈	—	106	1	no	—
303	C₄⋊C₈	C₂²⋊D₅₃	1	1	yes	characteristic
304	C₄⋊C₈	C₂²⋊D₅₃	1	1	yes	characteristic
305	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
306	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
307	#103	—	53	2	no	—
308	#103	—	53	2	no	—
309	#103	—	53	2	no	—
310	#102	—	53	2	no	—
311	C₂²⋊C₈	—	2	1	no	—
312	C₂²⋊C₈	—	106	2	no	—
313	C₂²⋊C₈	C₂²⋊D₅₃	1	2	yes	—
314	#97	—	53	2	no	—
315	C₄×C₈	—	2	2	no	—
316	C₄×C₈	—	2	2	no	—
317	C₄×C₈	—	2	1	no	—
318	C₄×C₈	—	53	2	no	—
319	C₄×C₈	—	53	1	no	—
320	C₄×C₈	—	53	1	no	—
321	C₄×C₈	—	106	2	no	—
322	C₄×C₈	—	106	1	no	—
323	C₄×C₈	C₂²×D₅₃	1	2	yes	—
324	#41	—	53	4	no	—
325	#41	—	106	2	no	—
326	C₂×Q₈	—	2	1	no	—
327	C₂×Q₈	—	2	2	no	—
328	C₂×Q₈	—	53	1	no	—
329	C₂×Q₈	C₈×D₅₃	1	4	yes	complemented
330	C₂×Q₈	C₂×C₄×D₅₃	1	2	yes	—
331	C₂×D₄	—	53	1	no	—
332	C₂×D₄	—	106	1	no	—
333	C₂²×C₄	—	2	1	no	—
334	C₂²×C₄	—	2	1	no	—
335	C₂²×C₄	—	106	1	no	—
336	C₂²×C₄	—	106	1	no	—
337	C₂²×C₄	—	106	1	no	—
338	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
339	C₂²×C₄	#11719945	1	1	yes	characteristic
340	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
341	Q₁₆	—	53	8	no	—
342	SD₁₆	—	53	8	no	—
343	M₁₆	—	53	4	no	—
344	M₁₆	—	106	2	no	—
345	C₂×C₈	—	2	1	no	—
346	C₂×C₈	—	2	1	no	—
347	C₂×C₈	—	53	2	no	—
348	C₂×C₈	—	53	2	no	—
349	C₂×C₈	—	106	1	no	—
350	C₂×C₈	—	106	2	no	—
351	C₂×C₈	—	106	1	no	—
352	C₂×C₈	—	424	2	no	—
353	C₂×C₈	—	424	1	no	—
354	C₂×C₈	—	212	2	no	—
355	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
356	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
357	C₄⋊C₄	—	2	2	no	—
358	C₄⋊C₄	—	2	1	no	—
359	C₄⋊C₄	—	2	2	no	—
360	C₄⋊C₄	—	53	1	no	—
361	C₄⋊C₄	—	53	1	no	—
362	C₄⋊C₄	—	106	1	no	—
363	C₄⋊C₄	—	106	1	no	—
364	C₄⋊C₄	C₂×C₄×D₅₃	1	2	yes	—
365	C₂²⋊C₄	—	106	1	no	—
366	C₂²⋊C₄	—	106	1	no	—
367	C₂²⋊C₄	—	106	1	no	—
368	C₄²	—	2	2	no	—
369	C₄²	—	2	2	no	—
370	C₄²	—	106	1	no	—
371	C₄²	—	106	1	no	—
372	C₄²	—	106	1	no	—
373	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
374	C₄²	#11719945	1	1	yes	characteristic
375	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
376	C₄²	C₂³×D₅₃	1	1	yes	characteristic
377	C₂³	—	106	1	no	—
378	C₂³	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
379	Q₈	—	2	2	no	—
380	Q₈	—	2	4	no	—
381	Q₈	—	53	2	no	—
382	Q₈	—	106	1	no	—
383	Q₈	C₂×C₈×D₅₃	1	4	yes	—
384	D₄	—	53	2	no	—
385	D₄	—	106	4	no	—
386	D₄	—	106	1	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	4	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	4	1	no	—
393	C₂×C₄	—	8	1	no	—
394	C₂×C₄	—	106	1	no	—
395	C₂×C₄	—	106	4	no	—
396	C₂×C₄	—	106	1	no	—
397	C₂×C₄	—	106	1	no	—
398	C₂×C₄	—	424	1	no	—
399	C₂×C₄	—	212	1	no	—
400	C₂×C₄	—	212	1	no	—
401	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
402	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
403	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
404	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
405	C₂×C₄	#13133223	1	2	yes	—
406	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
407	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
408	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
409	C₈	—	2	2	no	—
410	C₈	—	2	2	no	—
411	C₈	—	4	1	no	—
412	C₈	—	4	1	no	—
413	C₈	—	53	4	no	—
414	C₈	—	106	2	no	—
415	C₈	—	106	2	no	—
416	C₈	—	212	1	no	—
417	C₈	—	212	2	no	—
418	C₂²	—	2	1	no	—
419	C₂²	—	2	1	no	—
420	C₂²	—	106	2	no	—
421	C₂²	—	212	1	no	—
422	C₂²	—	212	1	no	—
423	C₂²	C₈×C₂²⋊D₅₃	1	2	yes	—
424	C₂²	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
425	C₄	—	2	1	no	—
426	C₄	—	2	1	no	—
427	C₄	—	2	1	no	—
428	C₄	—	2	4	no	—
429	C₄	—	4	2	no	—
430	C₄	—	106	2	no	—
431	C₄	—	212	1	no	—
432	C₄	—	212	1	no	—
433	C₄	C₈×C₂²⋊D₅₃	1	2	yes	—
434	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
435	C₄	C₂×#13133223	1	1	yes	characteristic
436	C₄	#16811431	1	1	yes	characteristic
437	C₄	#16811438	1	1	yes	characteristic
438	C₂	—	2	2	no	—
439	C₂	—	212	2	no	—
440	C₂	C₂×C₈×C₂²⋊D₅₃	1	1	yes	characteristic
441	C₂	#20283558	1	1	yes	characteristic
442	C₂	#20283622	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#2982	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#2982	D₅₃
hypocenter	C₅₃	#17853
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #2982

Complements

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

N	H ≅ G/N	type
3. #20273532	C₂	semidirect
4. #20273525	C₂	semidirect
5. C₈×#11719925	C₂	semidirect
6. C₈×#11719924	C₂	semidirect
7. #20273468	C₂	semidirect
9. C₂×Q₈⋊#88322	C₂	semidirect
10. #20272996	C₂	semidirect
11. C₅₃×#2982	C₂	semidirect
48. #13133248	C₈	semidirect
104. C₂×Q₈×C₅₃	C₂×C₈	semidirect
134. #11719925	C₂×C₈	semidirect
135. #11719924	C₂×C₈	semidirect
146. #11719917	C₂×C₈	semidirect
218. #2982	D₅₃	semidirect
268. C₅₃	#17853	semidirect
329. C₂×Q₈	C₈×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#20283823	2
C₄×#13133248	2
#20273532	2
#20273525	2
C₈×#11719925	2
C₈×#11719924	2
#20273468	2
#20273095	2
C₂×Q₈⋊#88322	2
#20272996	2
C₅₃×#2982	2
#17853	53