Subgroups — #27278337

Order 13568.

602 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20277615	C₂	1	1	yes	maximal, characteristic
2	#20275479	C₂	1	1	yes	maximal, characteristic, complemented
3	#20274842	C₂	1	1	yes	maximal, characteristic, complemented
4	#20274716	C₂	1	1	yes	maximal, characteristic
5	C₂×#16810350	C₂	1	1	yes	maximal, characteristic
6	#20272554	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1296	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
9	C₂×#13133179	—	2	1	no	—
10	C₂×C₄⋊C₄×D₅₃	—	2	1	no	—
11	C₂×#13133149	—	2	1	no	—
12	C₂×#13133139	—	2	1	no	—
13	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
14	#16810519	—	2	1	no	—
15	#16810494	—	2	2	no	—
16	#16810493	C₂²	1	1	yes	characteristic
17	#16810480	—	2	1	no	—
18	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
19	#16810386	—	2	1	no	—
20	#16810386	—	2	1	no	—
21	#16810385	—	2	2	no	—
22	#16810350	C₄	1	4	yes	complemented
23	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#384	C₂²	1	1	yes	characteristic
26	C₂²×D₂₁₂	D₄	1	1	yes	characteristic
27	C₂²×C₄×D₅₃	—	2	1	no	—
28	C₂²×C₄×D₅₃	—	2	2	no	—
29	C₂²×#88322	—	2	1	no	—
30	C₄×D₂₁₂	—	2	1	no	—
31	C₄×D₂₁₂	—	2	1	no	—
32	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
33	C₄×D₂₁₂	Q₈	1	2	yes	—
34	#13133179	—	4	2	no	—
35	#13133179	—	4	2	no	—
36	C₂×C₄⋊Dic₅₃	—	2	1	no	—
37	C₂×C₄⋊Dic₅₃	—	2	1	no	—
38	C₂×C₄⋊Dic₅₃	—	2	1	no	—
39	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
40	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
41	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
42	C₂×#11719919	—	2	2	no	—
43	C₄⋊C₄×D₅₃	—	4	2	no	—
44	C₄⋊C₄×D₅₃	—	4	2	no	—
45	C₂×#11719921	—	2	2	no	—
46	C₂×#11719921	—	2	1	no	—
47	C₂×C₄×Dic₅₃	—	2	1	no	—
48	#13133149	—	4	2	no	—
49	#13133139	—	4	2	no	—
50	C₄×#88322	—	2	1	no	—
51	C₄×#88322	C₂×C₄	1	2	yes	complemented
52	#13133124	—	2	2	no	—
53	#13133124	—	2	1	no	—
54	C₂²×C₈×C₅₃	—	2	1	no	—
55	C₂²×C₈×C₅₃	—	2	1	no	—
56	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
57	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
58	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
59	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₅₃	—	2	1	no	—
61	C₄⋊C₈×C₅₃	—	2	1	no	—
62	C₄⋊C₈×C₅₃	—	2	1	no	—
63	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	complemented
64	C₂³×D₅₃	—	2	1	no	—
65	C₂²×C₄×C₅₃	—	2	2	no	—
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₅₃	—	2	1	no	—
70	C₂²×Dic₅₃	—	2	1	no	—
71	C₂²×Dic₅₃	—	2	1	no	—
72	C₂²×Dic₅₃	—	4	1	no	—
73	C₂×D₂₁₂	—	2	1	no	—
74	C₂×D₂₁₂	—	2	1	no	—
75	C₂×D₂₁₂	—	2	1	no	—
76	C₂×D₂₁₂	C₄⋊C₄	1	2	yes	—
77	C₂×D₂₁₂	SD₁₆	1	2	yes	—
78	C₂×D₂₁₂	SD₁₆	1	2	yes	—
79	C₂×C₄×D₅₃	—	2	2	no	—
80	C₂×C₄×D₅₃	—	2	2	no	—
81	C₂×C₄×D₅₃	—	4	1	no	—
82	C₂×C₄×D₅₃	—	4	1	no	—
83	C₂×C₄×D₅₃	—	4	1	no	—
84	C₂×C₄×D₅₃	—	4	1	no	—
85	C₂×C₄×D₅₃	—	4	4	no	—
86	C₂×C₄×D₅₃	—	4	4	no	—
87	C₂×C₄×D₅₃	—	8	2	no	—
88	C₂×C₈×C₅₃	—	2	2	no	—
89	C₂×C₈×C₅₃	—	4	1	no	—
90	C₂×C₈×C₅₃	—	4	1	no	—
91	C₂×C₈×C₅₃	—	4	1	no	—
92	C₂×C₈×C₅₃	—	4	1	no	—
93	C₂×C₈×C₅₃	—	4	1	no	—
94	C₄⋊C₄×C₅₃	—	4	2	no	—
95	C₄⋊C₄×C₅₃	—	4	1	no	—
96	C₄⋊C₄×C₅₃	—	4	1	no	—
97	C₄⋊C₄×C₅₃	—	4	1	no	—
98	C₄⋊C₄×C₅₃	—	4	1	no	—
99	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
101	C₄²×C₅₃	#41	1	1	yes	characteristic
102	C₄²×C₅₃	#41	1	1	yes	characteristic
103	#11719921	—	2	2	no	—
104	#11719921	—	2	2	no	—
105	#11719921	—	4	4	no	—
106	C₄⋊Dic₅₃	—	2	1	no	—
107	C₄⋊Dic₅₃	—	2	1	no	—
108	C₄⋊Dic₅₃	—	2	1	no	—
109	C₄⋊Dic₅₃	—	2	1	no	—
110	C₄⋊Dic₅₃	—	4	2	no	—
111	C₄⋊Dic₅₃	—	4	1	no	—
112	C₄⋊Dic₅₃	—	4	1	no	—
113	C₄⋊Dic₅₃	—	4	1	no	—
114	C₄⋊Dic₅₃	—	4	1	no	—
115	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
116	C₄⋊Dic₅₃	D₈	1	2	yes	—
117	#11719919	—	4	4	no	—
118	C₄×Dic₅₃	—	4	1	no	—
119	C₄×Dic₅₃	—	4	1	no	—
120	C₂×#88322	—	2	2	no	—
121	C₂×#88322	—	4	1	no	—
122	C₂×#88322	—	4	1	no	—
123	C₂³×C₅₃	#120	1	1	yes	characteristic
124	C₂²×D₅₃	—	2	2	no	—
125	C₂²×D₅₃	—	2	2	no	—
126	C₂²×D₅₃	—	2	2	no	—
127	C₂²×D₅₃	—	4	1	no	—
128	C₂²×D₅₃	—	4	1	no	—
129	C₂²×D₅₃	—	4	1	no	—
130	C₂²×D₅₃	—	4	1	no	—
131	C₂×C₄×C₅₃	—	2	1	no	—
132	C₂×C₄×C₅₃	—	2	1	no	—
133	C₂×C₄×C₅₃	—	4	2	no	—
134	C₂×C₄×C₅₃	—	4	2	no	—
135	C₂×C₄×C₅₃	—	4	2	no	—
136	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
141	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
145	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
146	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
149	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
150	C₂×Dic₅₃	—	2	2	no	—
151	C₂×Dic₅₃	—	2	2	no	—
152	C₂×Dic₅₃	—	4	1	no	—
153	C₂×Dic₅₃	—	4	1	no	—
154	C₂×Dic₅₃	—	4	1	no	—
155	C₂×Dic₅₃	—	4	1	no	—
156	C₂×Dic₅₃	—	4	1	no	—
157	C₂×Dic₅₃	—	4	1	no	—
158	C₂×Dic₅₃	—	4	1	no	—
159	C₂×Dic₅₃	—	8	1	no	—
160	C₂×Dic₅₃	—	8	1	no	—
161	C₂×Dic₅₃	—	8	1	no	—
162	D₂₁₂	—	2	2	no	—
163	D₂₁₂	—	2	2	no	—
164	D₂₁₂	—	2	2	no	—
165	D₂₁₂	#106	1	4	yes	complemented
166	C₄×D₅₃	—	4	2	no	—
167	C₄×D₅₃	—	4	2	no	—
168	C₄×D₅₃	—	4	2	no	—
169	C₄×D₅₃	—	4	2	no	—
170	C₄×D₅₃	—	8	8	no	—
171	C₈×C₅₃	—	4	2	no	—
172	C₈×C₅₃	—	8	1	no	—
173	#88322	—	4	2	no	—
174	#8894	—	53	1	no	maximal, Sylow
175	C₂²×C₅₃	#385	1	1	yes	characteristic
176	C₂²×C₅₃	#447	1	1	yes	characteristic
177	C₂²×C₅₃	#448	1	1	yes	characteristic
178	C₂²×C₅₃	#450	1	1	yes	characteristic
179	C₂²×C₅₃	#452	1	1	yes	characteristic
180	C₂²×C₅₃	#454	1	1	yes	characteristic
181	C₂²×C₅₃	#455	1	1	yes	characteristic
182	C₂×D₅₃	—	2	4	no	—
183	C₂×D₅₃	—	4	2	no	—
184	C₂×D₅₃	—	4	2	no	—
185	C₂×D₅₃	—	4	2	no	—
186	C₂×D₅₃	—	4	2	no	—
187	C₂×D₅₃	—	4	2	no	—
188	C₂×D₅₃	—	4	2	no	—
189	C₄×C₅₃	—	2	1	no	—
190	C₄×C₅₃	—	2	1	no	—
191	C₄×C₅₃	—	8	2	no	—
192	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
193	C₄×C₅₃	#428	1	1	yes	characteristic
194	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
195	C₄×C₅₃	#440	1	1	yes	characteristic
196	C₄×C₅₃	#474	1	1	yes	characteristic
197	C₄×C₅₃	#478	1	1	yes	characteristic
198	C₄×C₅₃	#481	1	1	yes	characteristic
199	C₄×C₅₃	#483	1	1	yes	characteristic
200	Dic₅₃	—	4	2	no	—
201	Dic₅₃	—	4	2	no	—
202	Dic₅₃	—	8	1	no	—
203	Dic₅₃	—	16	1	no	—
204	D₄×C₄⋊C₄	—	53	1	no	—
205	#1948	—	53	1	no	—
206	#1834	—	53	1	no	—
207	#1802	—	53	1	no	—
208	C₂×D₄⋊C₈	—	53	1	no	—
209	#1296	D₅₃	1	1	yes	characteristic, complemented
210	#1279	—	53	1	no	—
211	C₂×C₅₃	#1627	1	1	yes	characteristic
212	C₂×C₅₃	#1635	1	1	yes	characteristic
213	C₂×C₅₃	#1639	1	1	yes	characteristic
214	C₂×C₅₃	#1643	1	1	yes	characteristic
215	C₂×C₅₃	#1853	1	1	yes	characteristic
216	C₂×C₅₃	#1860	1	1	yes	characteristic
217	C₂×C₅₃	#1862	1	1	yes	characteristic
218	D₅₃	—	4	4	no	—
219	C₂×C₄×D₄	—	53	1	no	—
220	C₂×C₄×D₄	—	106	1	no	—
221	C₂²×C₄⋊C₄	—	106	1	no	—
222	C₂×#106	—	106	1	no	—
223	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
224	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
225	C₂×#102	—	106	1	no	—
226	C₂×C₄×C₈	—	53	1	no	—
227	#389	—	106	1	no	—
228	#385	—	106	2	no	—
229	#384	—	53	1	no	—
230	#384	C₂×D₅₃	1	1	yes	characteristic
231	#380	—	106	1	no	—
232	C₄×C₄⋊C₄	—	53	1	no	—
233	#340	—	106	1	no	—
234	#340	—	106	1	no	—
235	#340	—	106	2	no	—
236	D₄⋊C₈	—	53	4	no	—
237	C₅₃	#8894	1	1	yes	characteristic, complemented, Sylow
238	C₂²×D₄	—	53	1	no	—
239	C₂³×C₄	—	106	2	no	—
240	C₂³×C₄	—	106	1	no	—
241	C₂²×C₈	—	2	1	no	—
242	C₂²×C₈	—	2	1	no	—
243	C₂²×C₈	—	106	1	no	—
244	C₄×D₄	—	53	2	no	—
245	C₄×D₄	—	53	2	no	—
246	C₄×D₄	—	106	1	no	—
247	C₄×D₄	—	106	1	no	—
248	C₄×D₄	—	212	2	no	—
249	C₄×D₄	—	212	2	no	—
250	C₂×C₄⋊C₄	—	2	1	no	—
251	C₂×C₄⋊C₄	—	2	1	no	—
252	C₂×C₄⋊C₄	—	53	1	no	—
253	C₂×C₄⋊C₄	—	53	1	no	—
254	C₂×C₄⋊C₄	—	53	1	no	—
255	C₂×C₄⋊C₄	—	106	2	no	—
256	C₂×C₄⋊C₄	—	106	1	no	—
257	C₂×C₄⋊C₄	—	106	1	no	—
258	C₂×C₄⋊C₄	—	106	1	no	—
259	C₂×C₄⋊C₄	—	212	2	no	—
260	C₂×C₄⋊C₄	—	212	2	no	—
261	C₂×C₄⋊C₄	D₂₁₂	1	2	yes	—
262	C₂×C₂²⋊C₄	—	106	1	no	—
263	C₂×C₂²⋊C₄	—	106	2	no	—
264	C₂×C₄²	—	106	1	no	—
265	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
266	#106	—	212	2	no	—
267	C₄⋊C₈	—	2	1	no	—
268	C₄⋊C₈	—	2	1	no	—
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	C₄×D₅₃	1	2	yes	complemented
271	#102	—	212	2	no	—
272	C₄×C₈	—	53	2	no	—
273	C₄×C₈	—	106	1	no	—
274	#95	—	106	2	no	—
275	#95	—	106	1	no	—
276	C₂⁴	—	106	1	no	—
277	C₂×D₄	—	53	2	no	—
278	C₂×D₄	—	53	2	no	—
279	C₂×D₄	—	53	2	no	—
280	C₂×D₄	—	106	1	no	—
281	C₂×D₄	—	106	1	no	—
282	C₂×D₄	—	106	1	no	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	—	106	1	no	—
285	C₂²×C₄	—	106	1	no	—
286	C₂²×C₄	—	106	2	no	—
287	C₂²×C₄	—	106	2	no	—
288	C₂²×C₄	—	106	1	no	—
289	C₂²×C₄	—	424	2	no	—
290	C₂²×C₄	—	212	4	no	—
291	C₂²×C₄	—	212	4	no	—
292	C₂²×C₄	—	212	1	no	—
293	C₂²×C₄	—	212	1	no	—
294	C₂²×C₄	—	212	1	no	—
295	C₂²×C₄	—	212	1	no	—
296	C₂²×C₄	—	212	1	no	—
297	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
298	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
299	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	4	1	no	—
302	C₂×C₈	—	4	1	no	—
303	C₂×C₈	—	4	1	no	—
304	C₂×C₈	—	4	1	no	—
305	C₂×C₈	—	4	1	no	—
306	C₂×C₈	—	106	2	no	—
307	C₂×C₈	—	212	1	no	—
308	C₂×C₈	—	212	1	no	—
309	C₄⋊C₄	—	4	2	no	—
310	C₄⋊C₄	—	4	1	no	—
311	C₄⋊C₄	—	4	1	no	—
312	C₄⋊C₄	—	4	1	no	—
313	C₄⋊C₄	—	4	1	no	—
314	C₄⋊C₄	—	53	2	no	—
315	C₄⋊C₄	—	53	2	no	—
316	C₄⋊C₄	—	106	1	no	—
317	C₄⋊C₄	—	106	1	no	—
318	C₄⋊C₄	—	106	1	no	—
319	C₄⋊C₄	—	106	1	no	—
320	C₄⋊C₄	—	212	2	no	—
321	C₄⋊C₄	—	212	4	no	—
322	C₄⋊C₄	—	212	1	no	—
323	C₄⋊C₄	—	212	1	no	—
324	C₄⋊C₄	—	212	1	no	—
325	C₄⋊C₄	—	212	1	no	—
326	C₂²⋊C₄	—	106	2	no	—
327	C₂²⋊C₄	—	106	2	no	—
328	C₂²⋊C₄	—	212	4	no	—
329	C₄²	—	212	1	no	—
330	C₄²	—	212	1	no	—
331	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
332	C₄²	#11719945	1	1	yes	characteristic
333	C₄²	#11719947	1	1	yes	characteristic
334	C₄²	Q₈×D₅₃	1	1	yes	characteristic
335	C₂³	—	106	2	no	—
336	C₂³	—	106	2	no	—
337	C₂³	—	106	2	no	—
338	C₂³	—	212	1	no	—
339	C₂³	—	212	1	no	—
340	C₂³	—	212	1	no	—
341	C₂³	—	212	1	no	—
342	C₂³	#13133187	1	1	yes	characteristic
343	D₄	—	53	4	no	—
344	D₄	—	106	2	no	—
345	D₄	—	106	2	no	—
346	D₄	—	106	2	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	106	2	no	—
353	C₂×C₄	—	106	2	no	—
354	C₂×C₄	—	424	8	no	—
355	C₂×C₄	—	424	1	no	—
356	C₂×C₄	—	424	1	no	—
357	C₂×C₄	—	424	1	no	—
358	C₂×C₄	—	212	1	no	—
359	C₂×C₄	—	212	1	no	—
360	C₂×C₄	—	212	1	no	—
361	C₂×C₄	—	212	1	no	—
362	C₂×C₄	—	212	1	no	—
363	C₂×C₄	—	212	1	no	—
364	C₂×C₄	—	212	1	no	—
365	C₂×C₄	—	212	2	no	—
366	C₂×C₄	—	212	2	no	—
367	C₂×C₄	—	212	2	no	—
368	C₂×C₄	—	212	2	no	—
369	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
370	C₂×C₄	#13133177	1	1	yes	characteristic
371	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
372	C₂×C₄	#13133195	1	1	yes	characteristic
373	C₂×C₄	#13133197	1	1	yes	characteristic
374	C₂×C₄	#13133199	1	1	yes	characteristic
375	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
376	C₂×C₄	SD₁₆×D₅₃	1	1	yes	characteristic
377	C₂×C₄	SD₁₆×D₅₃	1	1	yes	characteristic
378	C₂×C₄	#13133238	1	1	yes	characteristic
379	C₂×C₄	#13133241	1	1	yes	characteristic
380	C₂×C₄	#13133244	1	1	yes	characteristic
381	C₂×C₄	#13133249	1	1	yes	characteristic
382	C₂×C₄	#13133249	1	1	yes	characteristic
383	C₈	—	4	2	no	—
384	C₈	—	8	1	no	—
385	C₈	—	212	2	no	—
386	C₂²	—	106	4	no	—
387	C₂²	—	212	2	no	—
388	C₂²	—	212	2	no	—
389	C₂²	—	212	2	no	—
390	C₂²	—	212	2	no	—
391	C₂²	—	212	2	no	—
392	C₂²	—	212	2	no	—
393	C₂²	#16810497	1	1	yes	characteristic
394	C₂²	#16810758	1	1	yes	characteristic
395	C₂²	#16810763	1	1	yes	characteristic
396	C₂²	#16810775	1	1	yes	characteristic
397	C₂²	#16810788	1	1	yes	characteristic
398	C₂²	#16810798	1	1	yes	characteristic
399	C₂²	#16810803	1	1	yes	characteristic
400	C₄	—	2	1	no	—
401	C₄	—	2	1	no	—
402	C₄	—	8	2	no	—
403	C₄	—	848	1	no	—
404	C₄	—	424	1	no	—
405	C₄	—	212	2	no	—
406	C₄	—	212	2	no	—
407	C₄	#106×D₅₃	1	1	yes	characteristic
408	C₄	#16810664	1	1	yes	characteristic
409	C₄	#16810703	1	1	yes	characteristic
410	C₄	#16810726	1	1	yes	characteristic
411	C₄	#16810902	1	1	yes	characteristic
412	C₄	#16810922	1	1	yes	characteristic
413	C₄	#16810940	1	1	yes	characteristic
414	C₄	#16810952	1	1	yes	characteristic
415	C₂	—	212	4	no	—
416	C₂	#20273819	1	1	yes	characteristic
417	C₂	#20273867	1	1	yes	characteristic
418	C₂	#20273895	1	1	yes	characteristic
419	C₂	#20273923	1	1	yes	characteristic
420	C₂	#20274909	1	1	yes	characteristic
421	C₂	#20274951	1	1	yes	characteristic
422	C₂	#20274963	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133187
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1296	C₂
socle	C₂³×C₅₃	#120
prime radical	C₂²×D₂₁₂	D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1296	D₅₃
hypocenter	C₅₃	#8894
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

Complements

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

N	H ≅ G/N	type
2. #20275479	C₂	semidirect
3. #20274842	C₂	semidirect
6. #20272554	C₂	semidirect
7. C₅₃×#1296	C₂	semidirect
22. #16810350	C₄	semidirect
51. C₄×#88322	C₂×C₄	semidirect
63. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
165. D₂₁₂	#106	semidirect
209. #1296	D₅₃	semidirect
237. C₅₃	#8894	semidirect
270. C₄⋊C₈	C₄×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#20277615	2
#20275479	2
#20274842	2
#20274716	2
C₂×#16810350	2
#20272554	2
C₅₃×#1296	2
#8894	53