Subgroups — #27366529

Order 13568.

726 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285294	C₂	1	1	yes	maximal, characteristic, complemented
2	#20284170	C₂	1	2	yes	maximal, complemented
3	#20274159	C₂	1	2	yes	maximal, complemented
4	#20274152	C₂	1	2	yes	maximal, complemented
5	#20274109	C₂	1	1	yes	maximal, characteristic, complemented
6	#20274104	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274075	C₂	1	1	yes	maximal, characteristic, complemented
8	#20274070	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16810376	C₂	1	1	yes	maximal, characteristic, complemented
10	#20273362	C₂	1	2	yes	maximal, complemented
11	C₈×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133237	—	2	2	no	—
13	#16811173	C₂²	1	2	yes	—
14	#16811028	C₂²	1	1	yes	characteristic
15	#16811000	C₂²	1	1	yes	characteristic
16	#16810970	C₂²	1	2	yes	—
17	#16810859	—	2	1	no	—
18	#16810859	—	2	1	no	—
19	#16810845	—	2	2	no	—
20	#16810838	—	2	2	no	—
21	#16810833	C₂²	1	2	yes	—
22	#16810828	C₂²	1	2	yes	—
23	#16810823	C₂²	1	2	yes	—
24	#16810818	C₂²	1	2	yes	—
25	C₄×D₄₂₄	—	2	1	no	—
26	C₄×D₄₂₄	—	2	1	no	—
27	C₂×C₄⋊#88322	C₂²	1	2	yes	—
28	#16810617	—	2	2	no	—
29	#16810376	C₂²	1	1	yes	characteristic
30	#16810376	C₂²	1	1	yes	characteristic
31	#16810376	C₂²	1	2	yes	—
32	#16810368	C₂²	1	2	yes	—
33	#16810368	C₂²	1	2	yes	—
34	#16810366	C₂²	1	2	yes	—
35	#16810366	C₂²	1	2	yes	—
36	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
37	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₅₃	—	2	1	no	—
40	C₈×D₄×C₅₃	—	2	1	no	—
41	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
42	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
43	C₅₃×#431	—	2	1	no	—
44	C₅₃×#431	C₂²	1	1	yes	characteristic
45	C₂×C₄×C₈×C₅₃	—	2	1	no	—
46	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
47	C₂×#11719945	—	2	2	no	—
48	#13133237	—	4	8	no	—
49	C₂×#11719925	—	2	2	no	—
50	C₂×#11719923	—	2	2	no	—
51	C₂×#11719924	—	2	2	no	—
52	C₂×D₄₂₄	—	2	1	no	—
53	C₂×D₄₂₄	—	2	1	no	—
54	C₂×#11719922	—	2	2	no	—
55	C₂²×#88322	—	2	2	no	—
56	#13133217	C₂³	1	2	yes	—
57	#13133214	C₂³	1	2	yes	—
58	#13133205	—	2	2	no	—
59	#13133191	—	2	2	no	—
60	#13133191	—	2	2	no	—
61	C₄×D₂₁₂	—	2	2	no	—
62	C₄×D₂₁₂	—	2	2	no	—
63	#13133151	—	2	1	no	—
64	#13133151	—	2	1	no	—
65	C₄⋊#88322	—	2	2	no	—
66	C₄⋊#88322	C₂³	1	2	yes	—
67	C₄⋊#88322	C₂³	1	2	yes	—
68	#13133140	—	2	2	no	—
69	#13133140	—	2	2	no	—
70	#13133139	—	2	2	no	—
71	#13133139	—	2	2	no	—
72	#13133137	—	2	2	no	—
73	#13133137	—	2	2	no	—
74	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
75	C₂²×C₈×C₅₃	—	2	1	no	—
76	C₂²×C₈×C₅₃	—	2	1	no	—
77	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
78	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
79	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
80	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
81	C₄×D₄×C₅₃	—	2	1	no	—
82	C₄×D₄×C₅₃	—	2	1	no	—
83	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
84	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
85	#117×C₅₃	—	2	1	no	—
86	#117×C₅₃	D₄	1	1	yes	characteristic
87	C₂×C₄²×C₅₃	—	2	1	no	—
88	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
89	C₄⋊C₈×C₅₃	—	2	1	no	—
90	C₄⋊C₈×C₅₃	—	2	1	no	—
91	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
92	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
93	C₂²⋊C₈×C₅₃	—	2	1	no	—
94	C₂²⋊C₈×C₅₃	—	2	1	no	—
95	C₂²⋊C₈×C₅₃	—	2	1	no	—
96	C₄×C₈×C₅₃	—	2	1	no	—
97	C₄×C₈×C₅₃	—	2	1	no	—
98	C₄×C₈×C₅₃	—	2	1	no	—
99	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
100	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
101	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
102	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
103	#41×C₅₃	—	2	2	no	—
104	#41×C₅₃	D₈	1	4	yes	complemented
105	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂×D₄×C₅₃	—	2	1	no	—
107	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	—	2	1	no	—
109	C₂²×C₄×C₅₃	—	2	1	no	—
110	C₂²×C₄×C₅₃	—	2	1	no	—
111	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
114	C₂×C₂²⋊D₅₃	—	4	2	no	—
115	#11719945	—	4	4	no	—
116	#11719945	—	4	4	no	—
117	C₂×D₂₁₂	—	2	2	no	—
118	C₂×D₂₁₂	—	2	2	no	—
119	C₂×D₂₁₂	—	2	2	no	—
120	C₂×C₄×D₅₃	—	4	2	no	—
121	C₂×Dic₁₀₆	—	2	2	no	—
122	C₂×C₈×C₅₃	—	2	2	no	—
123	C₂×C₈×C₅₃	—	2	1	no	—
124	C₂×C₈×C₅₃	—	2	1	no	—
125	C₂×C₈×C₅₃	—	4	1	no	—
126	C₂×C₈×C₅₃	—	4	2	no	—
127	C₂×C₈×C₅₃	—	4	2	no	—
128	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
130	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
132	C₄⋊C₄×C₅₃	—	2	1	no	—
133	C₄⋊C₄×C₅₃	—	2	1	no	—
134	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
135	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
136	C₂²⋊C₄×C₅₃	—	2	1	no	—
137	C₂²⋊C₄×C₅₃	—	2	1	no	—
138	C₂²⋊C₄×C₅₃	—	2	1	no	—
139	C₄²×C₅₃	—	2	1	no	—
140	C₄²×C₅₃	—	2	1	no	—
141	C₄²×C₅₃	—	2	1	no	—
142	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
143	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
144	C₄²×C₅₃	#41	1	1	yes	characteristic
145	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
146	#11719925	—	4	4	no	—
147	#11719924	—	4	4	no	—
148	#11719923	—	4	4	no	—
149	#11719922	—	4	4	no	—
150	#11719921	—	4	2	no	—
151	#11719921	—	4	2	no	—
152	C₄⋊Dic₅₃	—	2	2	no	—
153	C₄⋊Dic₅₃	—	2	2	no	—
154	C₂×#88322	—	2	2	no	—
155	C₂×#88322	—	2	2	no	—
156	C₂×#88322	—	4	4	no	—
157	D₄₂₄	—	4	2	no	—
158	D₄₂₄	—	4	2	no	—
159	C₂³×C₅₃	—	2	1	no	—
160	C₂³×C₅₃	#121	1	1	yes	characteristic
161	C₂²×D₅₃	—	4	2	no	—
162	Q₈×C₅₃	—	2	1	no	—
163	Q₈×C₅₃	C₂×D₈	1	2	yes	—
164	D₄×C₅₃	—	2	4	no	—
165	D₄×C₅₃	—	2	1	no	—
166	D₄×C₅₃	C₂×D₈	1	2	yes	—
167	C₂×C₄×C₅₃	—	2	1	no	—
168	C₂×C₄×C₅₃	—	2	1	no	—
169	C₂×C₄×C₅₃	—	2	1	no	—
170	C₂×C₄×C₅₃	—	2	4	no	—
171	C₂×C₄×C₅₃	—	2	1	no	—
172	C₂×C₄×C₅₃	—	2	1	no	—
173	C₂×C₄×C₅₃	—	4	1	no	—
174	C₂×C₄×C₅₃	—	4	2	no	—
175	C₂×C₄×C₅₃	—	4	1	no	—
176	C₂×C₄×C₅₃	—	4	1	no	—
177	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
178	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
179	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
180	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
181	C₂×C₄×C₅₃	#135	1	2	yes	—
182	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
185	C₂²⋊D₅₃	—	8	4	no	—
186	C₂×Dic₅₃	—	4	2	no	—
187	D₂₁₂	—	4	2	no	—
188	D₂₁₂	—	4	2	no	—
189	D₂₁₂	—	4	2	no	—
190	D₂₁₂	—	4	2	no	—
191	D₂₁₂	—	8	2	no	—
192	C₄×D₅₃	—	8	4	no	—
193	Dic₁₀₆	—	4	2	no	—
194	Dic₁₀₆	—	4	2	no	—
195	C₈×C₅₃	—	2	1	no	—
196	C₈×C₅₃	—	2	1	no	—
197	C₈×C₅₃	—	2	1	no	—
198	C₈×C₅₃	—	4	1	no	—
199	C₈×C₅₃	—	4	1	no	—
200	C₈×C₅₃	#135	1	2	yes	complemented
201	#88322	—	4	2	no	—
202	#88322	—	4	2	no	—
203	#19125	—	53	1	no	maximal, Sylow
204	C₂²×C₅₃	—	2	1	no	—
205	C₂²×C₅₃	—	2	1	no	—
206	C₂²×C₅₃	—	2	2	no	—
207	C₂²×C₅₃	—	4	1	no	—
208	C₂²×C₅₃	—	4	1	no	—
209	C₂²×C₅₃	#466	1	2	yes	—
210	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
211	C₂×D₅₃	—	8	2	no	—
212	C₂×D₅₃	—	8	2	no	—
213	C₂×D₅₃	—	8	2	no	—
214	C₄×C₅₃	—	2	1	no	—
215	C₄×C₅₃	—	2	2	no	—
216	C₄×C₅₃	—	2	1	no	—
217	C₄×C₅₃	—	2	1	no	—
218	C₄×C₅₃	—	4	1	no	—
219	C₄×C₅₃	—	4	1	no	—
220	C₄×C₅₃	#466	1	2	yes	—
221	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
222	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
223	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
224	C₄×C₅₃	#575	1	1	yes	characteristic
225	Dic₅₃	—	8	2	no	—
226	#3169	—	53	1	no	—
227	#3035	—	53	2	no	—
228	C₈×#41	D₅₃	1	1	yes	characteristic, complemented
229	#1682	—	53	2	no	—
230	#1681	—	53	2	no	—
231	#1674	—	53	1	no	—
232	#1673	—	53	1	no	—
233	#1668	—	53	1	no	—
234	#1667	—	53	1	no	—
235	C₂×#335	—	53	1	no	—
236	#1540	—	53	2	no	—
237	C₂×C₅₃	—	2	2	no	—
238	C₂×C₅₃	—	4	2	no	—
239	C₂×C₅₃	#3035	1	1	yes	characteristic
240	C₂×C₅₃	C₂×#466	1	1	yes	characteristic
241	C₂×C₅₃	#3067	1	1	yes	characteristic
242	D₅₃	—	16	2	no	—
243	C₂×#135	—	106	2	no	—
244	#532	—	53	2	no	—
245	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
246	#500	—	53	1	no	—
247	#493	—	53	1	no	—
248	#486	—	53	2	no	—
249	#466	—	106	1	no	—
250	#466	—	106	1	no	—
251	#464	—	106	2	no	—
252	#463	—	106	2	no	—
253	#462	—	53	2	no	—
254	#461	—	53	2	no	—
255	#460	—	53	2	no	—
256	#459	—	53	2	no	—
257	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
258	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
259	C₄×D₈	—	106	1	no	—
260	C₄×D₈	—	106	1	no	—
261	C₈×D₄	—	2	1	no	—
262	C₈×D₄	—	2	1	no	—
263	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
264	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
265	#431	—	2	1	no	—
266	#431	C₂×D₅₃	1	1	yes	characteristic
267	C₂×C₄⋊C₈	—	53	2	no	—
268	#416	—	106	2	no	—
269	C₂×C₄×C₈	—	2	1	no	—
270	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
271	#335	—	53	1	no	—
272	#335	—	53	1	no	—
273	#335	—	53	2	no	—
274	#332	—	53	2	no	—
275	#332	—	53	2	no	—
276	#331	—	53	2	no	—
277	#331	—	53	2	no	—
278	C₅₃	#19125	1	1	yes	characteristic, complemented, Sylow
279	C₂×#41	—	106	2	no	—
280	C₂×#41	D₂₁₂	1	1	yes	characteristic
281	#135	—	212	8	no	—
282	C₂×Q₁₆	—	106	2	no	—
283	C₂×SD₁₆	—	106	2	no	—
284	C₂×SD₁₆	—	106	2	no	—
285	C₂×D₈	—	106	2	no	—
286	C₂×D₈	—	106	1	no	—
287	C₂×D₈	—	106	1	no	—
288	C₂²×C₈	—	2	1	no	—
289	C₂²×C₈	—	2	1	no	—
290	C₂²×C₈	—	106	2	no	—
291	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
292	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
293	C₄⋊Q₈	—	53	2	no	—
294	#127	—	53	2	no	—
295	#124	—	106	2	no	—
296	#121	—	106	2	no	—
297	#121	—	106	2	no	—
298	C₄×Q₈	D₂₁₂	1	1	yes	characteristic
299	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
300	C₄×D₄	—	2	1	no	—
301	C₄×D₄	—	2	1	no	—
302	C₄×D₄	—	106	2	no	—
303	C₄×D₄	—	106	2	no	—
304	C₄×D₄	D₂₁₂	1	1	yes	characteristic
305	C₄×D₄	C₂²×D₅₃	1	1	yes	characteristic
306	#117	—	2	1	no	—
307	#117	D₂₁₂	1	1	yes	characteristic
308	C₂×C₄²	—	2	1	no	—
309	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
310	#107	—	106	1	no	—
311	#107	—	106	1	no	—
312	C₄⋊C₈	—	2	1	no	—
313	C₄⋊C₈	—	2	1	no	—
314	C₄⋊C₈	—	53	2	no	—
315	C₄⋊C₈	—	53	2	no	—
316	C₄⋊C₈	—	106	2	no	—
317	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
318	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
319	#103	—	106	2	no	—
320	#103	—	106	2	no	—
321	#102	—	106	2	no	—
322	#102	—	106	2	no	—
323	#102	—	106	2	no	—
324	#102	—	106	2	no	—
325	C₂²⋊C₈	—	2	1	no	—
326	C₂²⋊C₈	—	2	1	no	—
327	C₂²⋊C₈	—	2	1	no	—
328	C₄×C₈	—	2	1	no	—
329	C₄×C₈	—	2	1	no	—
330	C₄×C₈	—	2	1	no	—
331	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
332	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
333	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
334	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
335	#41	—	2	2	no	—
336	#41	—	212	4	no	—
337	#41	—	212	4	no	—
338	#41	D₄₂₄	1	4	yes	complemented
339	C₂×Q₈	—	106	2	no	—
340	C₂×Q₈	C₂×D₂₁₂	1	1	yes	characteristic
341	C₂×D₄	—	2	1	no	—
342	C₂×D₄	—	106	2	no	—
343	C₂×D₄	—	106	2	no	—
344	C₂×D₄	—	106	2	no	—
345	C₂×D₄	—	212	2	no	—
346	C₂×D₄	C₂×D₂₁₂	1	1	yes	characteristic
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	2	1	no	—
350	C₂²×C₄	—	212	2	no	—
351	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
352	C₂²×C₄	#11719945	1	1	yes	characteristic
353	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
354	Q₁₆	—	212	4	no	—
355	SD₁₆	—	212	4	no	—
356	SD₁₆	—	212	4	no	—
357	D₈	—	212	2	no	—
358	D₈	—	212	4	no	—
359	D₈	—	212	2	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	1	no	—
362	C₂×C₈	—	2	1	no	—
363	C₂×C₈	—	4	1	no	—
364	C₂×C₈	—	4	2	no	—
365	C₂×C₈	—	4	2	no	—
366	C₂×C₈	—	106	2	no	—
367	C₂×C₈	—	106	2	no	—
368	C₂×C₈	—	212	4	no	—
369	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
370	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
371	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
372	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
373	C₄⋊C₄	—	2	1	no	—
374	C₄⋊C₄	—	2	1	no	—
375	C₄⋊C₄	—	106	2	no	—
376	C₄⋊C₄	—	106	2	no	—
377	C₄⋊C₄	C₂×D₂₁₂	1	1	yes	characteristic
378	C₄⋊C₄	C₂×D₂₁₂	1	1	yes	characteristic
379	C₂²⋊C₄	—	2	1	no	—
380	C₂²⋊C₄	—	2	1	no	—
381	C₂²⋊C₄	—	2	1	no	—
382	C₂²⋊C₄	—	212	2	no	—
383	C₂²⋊C₄	—	212	2	no	—
384	C₄²	—	2	1	no	—
385	C₄²	—	2	1	no	—
386	C₄²	—	2	1	no	—
387	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
388	C₄²	#11719945	1	1	yes	characteristic
389	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
390	C₄²	C₂³×D₅₃	1	1	yes	characteristic
391	C₂³	—	2	1	no	—
392	C₂³	—	212	2	no	—
393	C₂³	#13133191	1	1	yes	characteristic
394	Q₈	—	2	1	no	—
395	Q₈	—	212	2	no	—
396	Q₈	—	212	2	no	—
397	Q₈	C₂×D₄₂₄	1	2	yes	—
398	D₄	—	2	4	no	—
399	D₄	—	2	1	no	—
400	D₄	—	424	2	no	—
401	D₄	—	424	4	no	—
402	D₄	—	212	2	no	—
403	D₄	—	212	2	no	—
404	D₄	—	212	2	no	—
405	D₄	—	212	2	no	—
406	D₄	C₂×D₄₂₄	1	2	yes	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	4	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	1	no	—
417	C₂×C₄	—	424	4	no	—
418	C₂×C₄	—	212	2	no	—
419	C₂×C₄	#13133191	1	1	yes	characteristic
420	C₂×C₄	#13133191	1	1	yes	characteristic
421	C₂×C₄	#13133191	1	1	yes	characteristic
422	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
423	C₂×C₄	#13133239	1	2	yes	—
424	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
425	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
426	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
427	C₈	—	2	1	no	—
428	C₈	—	2	1	no	—
429	C₈	—	2	1	no	—
430	C₈	—	4	1	no	—
431	C₈	—	4	1	no	—
432	C₈	—	212	2	no	—
433	C₈	—	212	2	no	—
434	C₈	#13133237	1	2	yes	complemented
435	C₂²	—	2	1	no	—
436	C₂²	—	2	1	no	—
437	C₂²	—	2	2	no	—
438	C₂²	—	4	1	no	—
439	C₂²	—	4	1	no	—
440	C₂²	—	424	2	no	—
441	C₂²	—	424	2	no	—
442	C₂²	—	424	2	no	—
443	C₂²	#16810859	1	2	yes	—
444	C₂²	C₂×#13133191	1	1	yes	characteristic
445	C₄	—	2	1	no	—
446	C₄	—	2	1	no	—
447	C₄	—	2	1	no	—
448	C₄	—	2	2	no	—
449	C₄	—	4	1	no	—
450	C₄	—	4	1	no	—
451	C₄	—	424	2	no	—
452	C₄	#16810859	1	2	yes	—
453	C₄	C₂²×D₄₂₄	1	1	yes	characteristic
454	C₄	C₂×#13133237	1	1	yes	characteristic
455	C₄	C₂×#13133239	1	1	yes	characteristic
456	C₄	#16811421	1	1	yes	characteristic
457	C₂	—	2	2	no	—
458	C₂	—	4	2	no	—
459	C₂	—	848	2	no	—
460	C₂	#20284170	1	1	yes	characteristic
461	C₂	C₂×#16810859	1	1	yes	characteristic
462	C₂	#20284435	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133191
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₈×#41×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#121
hypercenter	C₈×#41	D₅₃
hypocenter	C₅₃	#19125
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
1. #20285294	C₂	semidirect
2. #20284170	C₂	semidirect
3. #20274159	C₂	semidirect
4. #20274152	C₂	semidirect
5. #20274109	C₂	semidirect
6. #20274104	C₂	semidirect
7. #20274075	C₂	semidirect
8. #20274070	C₂	semidirect
9. C₂×#16810376	C₂	semidirect
10. #20273362	C₂	semidirect
11. C₈×#41×C₅₃	C₂	semidirect
104. #41×C₅₃	D₈	semidirect
200. C₈×C₅₃	#135	semidirect
228. C₈×#41	D₅₃	semidirect
278. C₅₃	#19125	semidirect
338. #41	D₄₂₄	semidirect
434. C₈	#13133237	semidirect

Maximal subgroups

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

subgroup	index
#20285294	2
#20284170	2
#20274159	2
#20274152	2
#20274109	2
#20274104	2
#20274075	2
#20274070	2
C₂×#16810376	2
#20273362	2
C₈×#41×C₅₃	2
#19125	53