Subgroups — #27368003

Order 13568.

716 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285143	C₂	1	1	yes	maximal, characteristic, complemented
2	#20284866	C₂	1	2	yes	maximal, complemented
3	#20274085	C₂	1	1	yes	maximal, characteristic, complemented
4	#20274080	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274075	C₂	1	1	yes	maximal, characteristic, complemented
6	#20274070	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274063	C₂	1	2	yes	maximal, complemented
8	#20274056	C₂	1	2	yes	maximal, complemented
9	#20273478	C₂	1	1	yes	maximal, characteristic, complemented
10	#20273418	C₂	1	2	yes	maximal, complemented
11	C₈×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133239	—	2	1	no	—
13	C₂×#13133239	—	2	1	no	—
14	#16811173	C₂²	1	2	yes	—
15	#16811007	—	2	1	no	—
16	#16811007	—	2	1	no	—
17	#16811003	C₂²	1	1	yes	characteristic
18	#16811000	C₂²	1	1	yes	characteristic
19	#16810996	C₂²	1	2	yes	—
20	#16810947	—	2	2	no	—
21	#16810908	C₂²	1	2	yes	—
22	#16810903	C₂²	1	2	yes	—
23	#16810823	C₂²	1	2	yes	—
24	#16810818	C₂²	1	2	yes	—
25	#16810768	—	2	2	no	—
26	#16810733	—	2	2	no	—
27	#16810721	—	2	2	no	—
28	#16810680	C₂²	1	2	yes	—
29	#16810376	C₂²	1	1	yes	characteristic
30	#16810376	C₂²	1	1	yes	characteristic
31	#16810374	C₂²	1	2	yes	—
32	#16810370	C₂²	1	2	yes	—
33	#16810370	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	#13133239	—	4	4	no	—
49	#13133239	—	4	4	no	—
50	C₂×Dic₂₁₂	—	2	1	no	—
51	C₂×Dic₂₁₂	—	2	1	no	—
52	C₂×#11719913	—	2	2	no	—
53	C₂×#11719913	—	2	2	no	—
54	C₂×#11719924	—	2	2	no	—
55	C₂×D₄₂₄	—	2	1	no	—
56	C₂×D₄₂₄	—	2	1	no	—
57	C₂×#11719922	—	2	2	no	—
58	#13133217	C₂³	1	2	yes	—
59	#13133214	C₂³	1	2	yes	—
60	#13133205	—	2	2	no	—
61	#13133191	—	2	2	no	—
62	#13133191	—	2	2	no	—
63	C₄×D₂₁₂	—	2	2	no	—
64	C₄×D₂₁₂	—	2	2	no	—
65	#13133150	—	2	2	no	—
66	#13133148	—	2	2	no	—
67	C₄⋊#88322	C₂³	1	2	yes	—
68	C₄⋊#88322	C₂³	1	2	yes	—
69	#13133141	—	2	2	no	—
70	#13133138	—	2	2	no	—
71	#13133137	—	2	2	no	—
72	#13133137	—	2	2	no	—
73	C₂²⋊#88322	—	2	2	no	—
74	#13133128	—	2	2	no	—
75	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
76	C₂²×C₈×C₅₃	—	2	1	no	—
77	C₂²×C₈×C₅₃	—	2	1	no	—
78	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
79	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
80	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
81	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
82	C₄×D₄×C₅₃	—	2	1	no	—
83	C₄×D₄×C₅₃	—	2	1	no	—
84	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
85	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
86	#117×C₅₃	—	2	1	no	—
87	#117×C₅₃	D₄	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	1	no	—
91	C₄⋊C₈×C₅₃	—	2	1	no	—
92	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
93	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₂²⋊C₈×C₅₃	—	2	1	no	—
95	C₂²⋊C₈×C₅₃	—	2	1	no	—
96	C₂²⋊C₈×C₅₃	—	2	1	no	—
97	C₄×C₈×C₅₃	—	2	1	no	—
98	C₄×C₈×C₅₃	—	2	1	no	—
99	C₄×C₈×C₅₃	—	2	1	no	—
100	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
101	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
102	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
103	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
104	#41×C₅₃	—	2	2	no	—
105	#41×C₅₃	—	2	2	no	—
106	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₂×D₄×C₅₃	—	2	1	no	—
108	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₅₃	—	2	1	no	—
110	C₂²×C₄×C₅₃	—	2	1	no	—
111	C₂²×C₄×C₅₃	—	2	1	no	—
112	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
115	C₂×C₂²⋊D₅₃	—	4	2	no	—
116	#11719945	—	4	4	no	—
117	#11719945	—	4	4	no	—
118	C₂×D₂₁₂	—	2	2	no	—
119	C₂×D₂₁₂	—	2	2	no	—
120	C₂×D₂₁₂	—	2	2	no	—
121	C₂×C₄×D₅₃	—	4	2	no	—
122	C₂×Dic₁₀₆	—	2	2	no	—
123	C₂×C₈×C₅₃	—	2	2	no	—
124	C₂×C₈×C₅₃	—	2	2	no	—
125	C₂×C₈×C₅₃	—	2	1	no	—
126	C₂×C₈×C₅₃	—	2	1	no	—
127	C₂×C₈×C₅₃	—	4	2	no	—
128	C₂×C₈×C₅₃	—	4	2	no	—
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	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
133	C₄⋊C₄×C₅₃	—	2	1	no	—
134	C₄⋊C₄×C₅₃	—	2	1	no	—
135	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
136	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
137	C₂²⋊C₄×C₅₃	—	2	1	no	—
138	C₂²⋊C₄×C₅₃	—	2	1	no	—
139	C₂²⋊C₄×C₅₃	—	2	1	no	—
140	C₄²×C₅₃	—	2	1	no	—
141	C₄²×C₅₃	—	2	1	no	—
142	C₄²×C₅₃	—	2	1	no	—
143	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
144	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
145	C₄²×C₅₃	#41	1	1	yes	characteristic
146	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
147	#11719924	—	4	4	no	—
148	#11719922	—	4	4	no	—
149	#11719921	—	4	2	no	—
150	#11719921	—	4	2	no	—
151	C₄⋊Dic₅₃	—	2	2	no	—
152	C₄⋊Dic₅₃	—	2	2	no	—
153	C₂×#88322	—	2	2	no	—
154	C₂×#88322	—	2	2	no	—
155	Dic₂₁₂	—	4	2	no	—
156	Dic₂₁₂	—	4	2	no	—
157	D₄₂₄	—	4	2	no	—
158	D₄₂₄	—	4	2	no	—
159	#11719913	—	4	4	no	—
160	#11719913	—	4	4	no	—
161	C₂³×C₅₃	—	2	1	no	—
162	C₂³×C₅₃	#121	1	1	yes	characteristic
163	C₂²×D₅₃	—	4	2	no	—
164	Q₈×C₅₃	—	2	1	no	—
165	Q₈×C₅₃	#135	1	2	yes	complemented
166	D₄×C₅₃	—	2	4	no	—
167	D₄×C₅₃	—	2	1	no	—
168	D₄×C₅₃	#135	1	2	yes	complemented
169	C₂×C₄×C₅₃	—	2	4	no	—
170	C₂×C₄×C₅₃	—	2	1	no	—
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	2	no	—
179	C₂×C₄×C₅₃	—	4	1	no	—
180	C₂×C₄×C₅₃	—	4	1	no	—
181	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
182	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
183	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
184	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
187	C₂²⋊D₅₃	—	8	4	no	—
188	C₂×Dic₅₃	—	4	2	no	—
189	D₂₁₂	—	4	2	no	—
190	D₂₁₂	—	4	2	no	—
191	D₂₁₂	—	4	2	no	—
192	D₂₁₂	—	4	2	no	—
193	D₂₁₂	—	8	2	no	—
194	C₄×D₅₃	—	8	4	no	—
195	Dic₁₀₆	—	4	2	no	—
196	Dic₁₀₆	—	4	2	no	—
197	C₈×C₅₃	—	2	1	no	—
198	C₈×C₅₃	—	2	1	no	—
199	C₈×C₅₃	—	4	1	no	—
200	C₈×C₅₃	—	4	1	no	—
201	C₈×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
202	C₈×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
203	#88322	—	4	2	no	—
204	#88322	—	4	2	no	—
205	#19269	—	53	1	no	maximal, Sylow
206	C₂²×C₅₃	—	2	1	no	—
207	C₂²×C₅₃	—	2	1	no	—
208	C₂²×C₅₃	—	2	2	no	—
209	C₂²×C₅₃	—	2	1	no	—
210	C₂²×C₅₃	—	4	1	no	—
211	C₂²×C₅₃	—	4	1	no	—
212	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
213	C₂×D₅₃	—	8	2	no	—
214	C₂×D₅₃	—	8	2	no	—
215	C₂×D₅₃	—	8	2	no	—
216	C₄×C₅₃	—	2	1	no	—
217	C₄×C₅₃	—	2	1	no	—
218	C₄×C₅₃	—	2	2	no	—
219	C₄×C₅₃	—	2	1	no	—
220	C₄×C₅₃	—	2	1	no	—
221	C₄×C₅₃	—	4	1	no	—
222	C₄×C₅₃	—	4	1	no	—
223	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
224	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
225	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
226	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
227	Dic₅₃	—	8	2	no	—
228	#3147	—	53	1	no	—
229	#3115	—	53	2	no	—
230	C₈×#41	D₅₃	1	1	yes	characteristic, complemented
231	#1670	—	53	1	no	—
232	#1669	—	53	1	no	—
233	#1668	—	53	1	no	—
234	#1667	—	53	1	no	—
235	#1666	—	53	2	no	—
236	#1665	—	53	2	no	—
237	#1568	—	53	1	no	—
238	#1553	—	53	2	no	—
239	C₂×C₅₃	—	4	1	no	—
240	C₂×C₅₃	—	4	2	no	—
241	C₂×C₅₃	#3038	1	1	yes	characteristic
242	C₂×C₅₃	#3050	1	1	yes	characteristic
243	C₂×C₅₃	#3067	1	1	yes	characteristic
244	D₅₃	—	16	2	no	—
245	C₂×#135	—	106	1	no	—
246	C₂×#135	—	106	1	no	—
247	#532	—	53	2	no	—
248	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
249	#495	—	106	1	no	—
250	#495	—	106	1	no	—
251	#494	—	53	1	no	—
252	#493	—	53	1	no	—
253	#492	—	53	2	no	—
254	#482	—	106	2	no	—
255	#475	—	53	2	no	—
256	#474	—	53	2	no	—
257	#460	—	53	2	no	—
258	#459	—	53	2	no	—
259	#449	—	106	2	no	—
260	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
261	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
262	#442	—	106	2	no	—
263	#440	—	106	2	no	—
264	C₈×D₄	—	2	1	no	—
265	C₈×D₄	—	2	1	no	—
266	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
267	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
268	#432	—	53	2	no	—
269	#431	—	2	1	no	—
270	#431	C₂×D₅₃	1	1	yes	characteristic
271	C₂×C₄×C₈	—	2	1	no	—
272	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
273	#335	—	53	1	no	—
274	#335	—	53	1	no	—
275	#334	—	53	2	no	—
276	#332	—	53	2	no	—
277	#332	—	53	2	no	—
278	#331	—	53	2	no	—
279	#331	—	53	2	no	—
280	C₅₃	#19269	1	1	yes	characteristic, complemented, Sylow
281	C₂×#41	—	106	2	no	—
282	C₂×#41	D₂₁₂	1	1	yes	characteristic
283	#135	—	212	4	no	—
284	#135	—	212	4	no	—
285	C₂×Q₁₆	—	106	1	no	—
286	C₂×Q₁₆	—	106	1	no	—
287	C₂×SD₁₆	—	106	2	no	—
288	C₂×SD₁₆	—	106	2	no	—
289	C₂×SD₁₆	—	106	2	no	—
290	C₂×D₈	—	106	2	no	—
291	C₂×D₈	—	106	1	no	—
292	C₂×D₈	—	106	1	no	—
293	C₂²×C₈	—	2	1	no	—
294	C₂²×C₈	—	2	1	no	—
295	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
296	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
297	C₄⋊Q₈	—	53	2	no	—
298	#127	—	53	2	no	—
299	#124	—	106	2	no	—
300	#121	—	106	2	no	—
301	#121	—	106	2	no	—
302	C₄×Q₈	D₂₁₂	1	1	yes	characteristic
303	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
304	C₄×D₄	—	2	1	no	—
305	C₄×D₄	—	2	1	no	—
306	C₄×D₄	—	106	2	no	—
307	C₄×D₄	—	106	2	no	—
308	C₄×D₄	D₂₁₂	1	1	yes	characteristic
309	C₄×D₄	C₂²×D₅₃	1	1	yes	characteristic
310	#117	—	2	1	no	—
311	#117	D₂₁₂	1	1	yes	characteristic
312	C₂×C₄²	—	2	1	no	—
313	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
314	#107	—	106	2	no	—
315	#106	—	106	2	no	—
316	C₄⋊C₈	—	2	1	no	—
317	C₄⋊C₈	—	2	1	no	—
318	C₄⋊C₈	—	53	2	no	—
319	C₄⋊C₈	—	53	2	no	—
320	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
321	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
322	#103	—	106	2	no	—
323	#102	—	106	2	no	—
324	#102	—	106	2	no	—
325	#102	—	106	2	no	—
326	C₂²⋊C₈	—	2	1	no	—
327	C₂²⋊C₈	—	2	1	no	—
328	C₂²⋊C₈	—	2	1	no	—
329	C₂²⋊C₈	—	106	2	no	—
330	#97	—	106	2	no	—
331	C₄×C₈	—	2	1	no	—
332	C₄×C₈	—	2	1	no	—
333	C₄×C₈	—	2	1	no	—
334	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
335	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
336	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
337	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
338	#41	—	2	2	no	—
339	#41	—	2	2	no	—
340	#41	—	212	4	no	—
341	#41	—	212	4	no	—
342	C₂×Q₈	—	106	2	no	—
343	C₂×Q₈	C₂×D₂₁₂	1	1	yes	characteristic
344	C₂×D₄	—	2	1	no	—
345	C₂×D₄	—	106	2	no	—
346	C₂×D₄	—	106	2	no	—
347	C₂×D₄	—	106	2	no	—
348	C₂×D₄	—	212	2	no	—
349	C₂×D₄	C₂×D₂₁₂	1	1	yes	characteristic
350	C₂²×C₄	—	2	1	no	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	212	2	no	—
354	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
355	C₂²×C₄	#11719945	1	1	yes	characteristic
356	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
357	Q₁₆	—	212	2	no	—
358	Q₁₆	—	212	2	no	—
359	SD₁₆	—	212	4	no	—
360	SD₁₆	—	212	4	no	—
361	SD₁₆	—	212	4	no	—
362	D₈	—	212	2	no	—
363	D₈	—	212	4	no	—
364	D₈	—	212	2	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	2	1	no	—
368	C₂×C₈	—	2	1	no	—
369	C₂×C₈	—	4	2	no	—
370	C₂×C₈	—	4	2	no	—
371	C₂×C₈	—	106	2	no	—
372	C₂×C₈	—	106	2	no	—
373	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
374	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
375	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
376	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
377	C₄⋊C₄	—	2	1	no	—
378	C₄⋊C₄	—	2	1	no	—
379	C₄⋊C₄	—	106	2	no	—
380	C₄⋊C₄	—	106	2	no	—
381	C₄⋊C₄	C₂×D₂₁₂	1	1	yes	characteristic
382	C₄⋊C₄	C₂×D₂₁₂	1	1	yes	characteristic
383	C₂²⋊C₄	—	2	1	no	—
384	C₂²⋊C₄	—	2	1	no	—
385	C₂²⋊C₄	—	2	1	no	—
386	C₂²⋊C₄	—	212	2	no	—
387	C₂²⋊C₄	—	212	2	no	—
388	C₄²	—	2	1	no	—
389	C₄²	—	2	1	no	—
390	C₄²	—	2	1	no	—
391	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
392	C₄²	#11719945	1	1	yes	characteristic
393	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
394	C₄²	C₂³×D₅₃	1	1	yes	characteristic
395	C₂³	—	2	1	no	—
396	C₂³	—	212	2	no	—
397	C₂³	#13133191	1	1	yes	characteristic
398	Q₈	—	2	1	no	—
399	Q₈	—	212	2	no	—
400	Q₈	—	212	2	no	—
401	Q₈	#13133239	1	2	yes	complemented
402	D₄	—	2	4	no	—
403	D₄	—	2	1	no	—
404	D₄	—	424	2	no	—
405	D₄	—	424	4	no	—
406	D₄	—	212	2	no	—
407	D₄	—	212	2	no	—
408	D₄	—	212	2	no	—
409	D₄	—	212	2	no	—
410	D₄	#13133239	1	2	yes	complemented
411	C₂×C₄	—	2	4	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	4	1	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	4	1	no	—
423	C₂×C₄	—	424	4	no	—
424	C₂×C₄	—	212	2	no	—
425	C₂×C₄	#13133191	1	1	yes	characteristic
426	C₂×C₄	#13133191	1	1	yes	characteristic
427	C₂×C₄	#13133191	1	1	yes	characteristic
428	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
429	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
430	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
431	C₈	—	2	1	no	—
432	C₈	—	2	1	no	—
433	C₈	—	4	1	no	—
434	C₈	—	4	1	no	—
435	C₈	—	212	2	no	—
436	C₈	—	212	2	no	—
437	C₈	#13133242	1	2	yes	—
438	C₈	#13133242	1	2	yes	—
439	C₂²	—	2	1	no	—
440	C₂²	—	2	1	no	—
441	C₂²	—	2	1	no	—
442	C₂²	—	2	2	no	—
443	C₂²	—	4	1	no	—
444	C₂²	—	4	1	no	—
445	C₂²	—	424	2	no	—
446	C₂²	—	424	2	no	—
447	C₂²	—	424	2	no	—
448	C₂²	C₂×#13133191	1	1	yes	characteristic
449	C₄	—	2	1	no	—
450	C₄	—	2	1	no	—
451	C₄	—	2	1	no	—
452	C₄	—	2	1	no	—
453	C₄	—	2	2	no	—
454	C₄	—	4	1	no	—
455	C₄	—	4	1	no	—
456	C₄	—	424	2	no	—
457	C₄	C₂×#13133239	1	1	yes	characteristic
458	C₄	C₂×#13133239	1	1	yes	characteristic
459	C₄	C₂×#13133242	1	1	yes	characteristic
460	C₄	C₂×#13133242	1	1	yes	characteristic
461	C₂	—	4	1	no	—
462	C₂	—	4	2	no	—
463	C₂	—	848	2	no	—
464	C₂	#20284195	1	1	yes	characteristic
465	C₂	#20284296	1	1	yes	characteristic
466	C₂	#20284435	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#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	#19269
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
1. #20285143	C₂	semidirect
2. #20284866	C₂	semidirect
3. #20274085	C₂	semidirect
4. #20274080	C₂	semidirect
5. #20274075	C₂	semidirect
6. #20274070	C₂	semidirect
7. #20274063	C₂	semidirect
8. #20274056	C₂	semidirect
9. #20273478	C₂	semidirect
10. #20273418	C₂	semidirect
11. C₈×#41×C₅₃	C₂	semidirect
165. Q₈×C₅₃	#135	semidirect
168. D₄×C₅₃	#135	semidirect
230. C₈×#41	D₅₃	semidirect
280. C₅₃	#19269	semidirect
401. Q₈	#13133239	semidirect
410. D₄	#13133239	semidirect

Maximal subgroups

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

subgroup	index
#20285143	2
#20284866	2
#20274085	2
#20274080	2
#20274075	2
#20274070	2
#20274063	2
#20274056	2
#20273478	2
#20273418	2
C₈×#41×C₅₃	2
#19269	53