Subgroups — #27355269

Order 13568.

600 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20284780	C₂	1	1	yes	maximal, characteristic
2	#20283836	C₂	1	1	yes	maximal, characteristic
3	#20273610	C₂	1	2	yes	maximal, complemented
4	#20273586	C₂	1	2	yes	maximal, complemented
5	#20273570	C₂	1	2	yes	maximal, complemented
6	C₈×#11719924	C₂	1	2	yes	maximal, complemented
7	#20273503	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273170	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273095	C₂	1	1	yes	maximal, characteristic
10	#20272999	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2986	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#11719945	C₂²	1	1	yes	characteristic
13	#16810961	C₄	1	2	yes	—
14	#16810905	C₄	1	1	yes	characteristic
15	#16810900	C₄	1	1	yes	characteristic
16	#16810830	C₄	1	1	yes	characteristic
17	#16810825	C₄	1	1	yes	characteristic
18	#16810783	C₄	1	2	yes	—
19	#16810753	C₂²	1	2	yes	—
20	C₄×#11719925	C₂²	1	2	yes	—
21	C₄×#11719924	C₂²	1	2	yes	—
22	#16810697	—	2	2	no	—
23	C₈×D₂₁₂	C₂²	1	2	yes	—
24	#16810679	—	2	1	no	—
25	#16810676	C₂²	1	1	yes	characteristic
26	#16810647	C₂²	1	1	yes	characteristic
27	C₄⋊C₈×D₅₃	—	2	1	no	—
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	C₅₃×#533	C₂²	1	1	yes	characteristic
40	C₅₃×#446	—	2	1	no	—
41	C₅₃×#446	C₂²	1	2	yes	—
42	C₈×Q₈×C₅₃	—	2	1	no	—
43	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
44	C₅₃×#433	—	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	C₂×#11719925	C₂×C₄	1	2	yes	—
49	C₂×#11719924	C₂×C₄	1	2	yes	—
50	C₂×#11719917	—	2	1	no	—
51	C₂²×#88322	—	2	1	no	—
52	C₂×C₈×D₅₃	—	2	2	no	—
53	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
54	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
55	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
56	C₄×D₂₁₂	C₂³	1	1	yes	characteristic
57	C₄×C₂²⋊D₅₃	—	2	1	no	—
58	C₄×C₂²⋊D₅₃	—	2	1	no	—
59	#13133176	C₂×C₄	1	1	yes	characteristic
60	#13133174	—	2	1	no	—
61	C₄²×D₅₃	—	2	1	no	—
62	#13133150	C₂×C₄	1	2	yes	—
63	#13133148	C₂×C₄	1	2	yes	—
64	C₄⋊#88322	—	2	1	no	—
65	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
66	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
67	C₄⋊#88322	C₂³	1	2	yes	—
68	#13133146	—	2	2	no	—
69	#13133141	C₂×C₄	1	2	yes	—
70	#13133141	C₂×C₄	1	2	yes	—
71	#13133140	C₂×C₄	1	2	yes	—
72	#13133137	C₂×C₄	1	2	yes	—
73	C₂²⋊#88322	C₂×C₄	1	2	yes	—
74	#13133129	—	2	2	no	—
75	#13133127	—	2	2	no	—
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	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
81	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
82	#125×C₅₃	—	2	1	no	—
83	C₂²⋊Q₈×C₅₃	—	2	1	no	—
84	C₂²⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
85	C₄×Q₈×C₅₃	—	2	1	no	—
86	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
87	#117×C₅₃	—	2	1	no	—
88	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
89	C₄⋊C₈×C₅₃	—	2	2	no	—
90	C₄⋊C₈×C₅₃	—	2	2	no	—
91	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
92	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
93	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
95	C₂²⋊C₈×C₅₃	—	2	1	no	—
96	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
97	#97×C₅₃	—	2	2	no	—
98	C₄×C₈×C₅₃	—	2	2	no	—
99	C₄×C₈×C₅₃	—	2	1	no	—
100	C₄×C₈×C₅₃	C₂³	1	2	yes	—
101	C₂×Q₈×C₅₃	—	2	1	no	—
102	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
106	C₂×C₂²⋊D₅₃	—	2	1	no	—
107	#11719945	—	2	2	no	—
108	#11719945	—	4	1	no	—
109	C₂×D₂₁₂	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×D₅₃	—	2	1	no	—
111	C₂×C₄×D₅₃	—	2	1	no	—
112	C₂×C₄×D₅₃	—	2	1	no	—
113	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₅₃	—	2	1	no	—
115	C₂×C₈×C₅₃	—	2	1	no	—
116	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
117	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
118	C₄⋊C₄×C₅₃	—	2	1	no	—
119	C₄⋊C₄×C₅₃	—	2	1	no	—
120	C₄⋊C₄×C₅₃	—	2	1	no	—
121	C₄⋊C₄×C₅₃	—	2	2	no	—
122	C₄⋊C₄×C₅₃	—	2	2	no	—
123	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
124	C₂²⋊C₄×C₅₃	—	2	2	no	—
125	C₄²×C₅₃	—	2	2	no	—
126	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
127	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₅₃	#41	1	1	yes	characteristic
129	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
130	C₂²⋊Dic₅₃	—	2	1	no	—
131	#11719925	—	4	2	no	—
132	#11719924	—	2	4	no	—
133	#11719921	—	2	1	no	—
134	#11719921	—	2	1	no	—
135	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
136	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
137	#11719919	—	2	1	no	—
138	#11719919	—	2	1	no	—
139	C₄×Dic₅₃	—	2	1	no	—
140	C₄×Dic₅₃	—	2	1	no	—
141	C₄×Dic₅₃	—	2	1	no	—
142	#11719917	—	8	1	no	—
143	C₂×#88322	—	2	4	no	—
144	C₂×#88322	—	2	1	no	—
145	C₂×#88322	—	2	1	no	—
146	C₂×#88322	—	2	2	no	—
147	C₂×#88322	C₂²×C₄	1	2	yes	—
148	C₂×#88322	C₂²×C₄	1	2	yes	—
149	C₈×D₅₃	—	8	2	no	—
150	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
151	C₂²×D₅₃	—	2	1	no	—
152	Q₈×C₅₃	—	2	2	no	—
153	Q₈×C₅₃	—	2	2	no	—
154	Q₈×C₅₃	—	4	2	no	—
155	C₂×C₄×C₅₃	—	2	1	no	—
156	C₂×C₄×C₅₃	—	2	1	no	—
157	C₂×C₄×C₅₃	—	2	1	no	—
158	C₂×C₄×C₅₃	—	2	1	no	—
159	C₂×C₄×C₅₃	—	2	2	no	—
160	C₂×C₄×C₅₃	—	2	2	no	—
161	C₂×C₄×C₅₃	—	2	2	no	—
162	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
165	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
166	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
168	C₂²⋊D₅₃	—	4	2	no	—
169	C₂×Dic₅₃	—	2	1	no	—
170	C₂×Dic₅₃	—	2	1	no	—
171	C₂×Dic₅₃	—	2	1	no	—
172	D₂₁₂	—	2	1	no	—
173	D₂₁₂	#131	1	2	yes	—
174	C₄×D₅₃	—	4	2	no	—
175	C₄×D₅₃	—	4	1	no	—
176	C₄×D₅₃	—	4	1	no	—
177	C₄×D₅₃	—	4	2	no	—
178	Dic₁₀₆	—	2	1	no	—
179	Dic₁₀₆	—	2	1	no	—
180	C₈×C₅₃	—	2	2	no	—
181	C₈×C₅₃	—	2	2	no	—
182	C₈×C₅₃	—	4	1	no	—
183	C₈×C₅₃	—	4	1	no	—
184	#88322	—	2	2	no	—
185	#88322	—	2	2	no	—
186	#88322	—	2	2	no	—
187	#88322	—	2	2	no	—
188	#88322	—	4	2	no	—
189	#18016	—	53	1	no	maximal, Sylow
190	C₂²×C₅₃	—	2	1	no	—
191	C₂²×C₅₃	—	2	1	no	—
192	C₂²×C₅₃	—	2	1	no	—
193	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
194	C₂×D₅₃	—	2	2	no	—
195	C₂×D₅₃	—	4	1	no	—
196	C₂×D₅₃	—	4	1	no	—
197	C₄×C₅₃	—	2	1	no	—
198	C₄×C₅₃	—	2	1	no	—
199	C₄×C₅₃	—	2	1	no	—
200	C₄×C₅₃	—	4	2	no	—
201	C₄×C₅₃	—	4	2	no	—
202	C₄×C₅₃	#443	1	2	yes	—
203	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
204	C₄×C₅₃	#568	1	1	yes	characteristic
205	C₄×C₅₃	#577	1	1	yes	characteristic
206	C₄×C₅₃	#578	1	1	yes	characteristic
207	Dic₅₃	—	4	1	no	—
208	Dic₅₃	—	4	1	no	—
209	Dic₅₃	—	4	1	no	—
210	#3107	—	53	1	no	—
211	#2990	—	53	1	no	—
212	#2986	D₅₃	1	1	yes	characteristic, complemented
213	#1591	—	53	2	no	—
214	#1587	—	53	2	no	—
215	#1585	—	53	2	no	—
216	C₈×SD₁₆	—	53	2	no	—
217	#1574	—	53	1	no	—
218	#1493	—	53	1	no	—
219	#1476	—	53	1	no	—
220	#1450	—	53	1	no	—
221	C₂×C₅₃	—	4	1	no	—
222	C₂×C₅₃	#2933	1	1	yes	characteristic
223	C₂×C₅₃	#2947	1	1	yes	characteristic
224	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
225	D₅₃	—	4	2	no	—
226	#533	C₂×D₅₃	1	1	yes	characteristic
227	C₄×#41	—	53	1	no	—
228	#485	—	53	2	no	—
229	#475	—	53	1	no	—
230	#474	—	53	1	no	—
231	#462	—	53	1	no	—
232	#461	—	53	1	no	—
233	#452	—	53	2	no	—
234	#446	—	2	1	no	—
235	#446	—	53	2	no	—
236	#446	C₂×D₅₃	1	2	yes	—
237	C₈×Q₈	—	2	1	no	—
238	C₈×Q₈	C₂×D₅₃	1	2	yes	—
239	C₄×Q₁₆	—	53	2	no	—
240	C₄×SD₁₆	—	53	2	no	—
241	#435	—	106	2	no	—
242	C₈×D₄	—	53	2	no	—
243	#433	—	2	2	no	—
244	#432	—	106	1	no	—
245	#431	—	53	1	no	—
246	#431	C₂×D₅₃	1	1	yes	characteristic
247	#431	C₂×D₅₃	1	1	yes	characteristic
248	#423	—	53	1	no	—
249	C₂×C₄⋊C₈	—	106	1	no	—
250	C₂×C₄×C₈	—	53	1	no	—
251	#335	—	53	2	no	—
252	#334	—	53	2	no	—
253	Q₈⋊C₈	—	53	2	no	—
254	Q₈⋊C₈	—	53	2	no	—
255	Q₈⋊C₈	—	53	1	no	—
256	Q₈⋊C₈	—	53	1	no	—
257	D₄⋊C₈	—	53	1	no	—
258	D₄⋊C₈	—	53	1	no	—
259	#322	—	53	2	no	—
260	C₈²	—	53	2	no	—
261	C₅₃	#18016	1	1	yes	characteristic, complemented, Sylow
262	C₂×#41	—	53	1	no	—
263	C₂×Q₁₆	—	53	2	no	—
264	C₂×SD₁₆	—	53	2	no	—
265	C₂×M₁₆	—	106	1	no	—
266	C₂²×C₈	—	106	1	no	—
267	C₂²×C₈	—	106	2	no	—
268	C₄⋊Q₈	C₄×D₅₃	1	1	yes	characteristic
269	C₄⋊Q₈	C₄×D₅₃	1	1	yes	characteristic
270	#125	—	2	1	no	—
271	C₂²⋊Q₈	—	2	1	no	—
272	C₂²⋊Q₈	C₄×D₅₃	1	2	yes	—
273	C₄×Q₈	—	2	1	no	—
274	C₄×Q₈	—	53	1	no	—
275	C₄×Q₈	—	53	1	no	—
276	C₄×Q₈	C₂²×D₅₃	1	2	yes	—
277	C₄×D₄	—	53	1	no	—
278	C₄×D₄	—	53	1	no	—
279	C₄×D₄	—	106	1	no	—
280	C₄×D₄	—	106	1	no	—
281	#117	—	2	1	no	—
282	#117	—	53	1	no	—
283	#117	—	106	1	no	—
284	C₂×C₄²	—	106	1	no	—
285	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
286	#107	—	53	2	no	—
287	#106	—	53	2	no	—
288	C₄⋊C₈	—	2	2	no	—
289	C₄⋊C₈	—	2	2	no	—
290	C₄⋊C₈	—	53	1	no	—
291	C₄⋊C₈	—	53	1	no	—
292	C₄⋊C₈	—	53	2	no	—
293	C₄⋊C₈	—	106	2	no	—
294	C₄⋊C₈	—	106	1	no	—
295	C₄⋊C₈	C₂²⋊D₅₃	1	1	yes	characteristic
296	C₄⋊C₈	C₂²⋊D₅₃	1	1	yes	characteristic
297	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
298	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
299	#103	—	53	2	no	—
300	#103	—	53	2	no	—
301	#103	—	53	2	no	—
302	#102	—	53	2	no	—
303	C₂²⋊C₈	—	2	1	no	—
304	C₂²⋊C₈	—	53	2	no	—
305	C₂²⋊C₈	—	106	2	no	—
306	C₂²⋊C₈	C₂²⋊D₅₃	1	2	yes	—
307	#97	—	2	2	no	—
308	#97	—	106	2	no	—
309	C₄×C₈	—	2	2	no	—
310	C₄×C₈	—	2	1	no	—
311	C₄×C₈	—	53	2	no	—
312	C₄×C₈	—	53	1	no	—
313	C₄×C₈	—	53	1	no	—
314	C₄×C₈	—	106	1	no	—
315	C₄×C₈	C₂²×D₅₃	1	2	yes	—
316	#41	—	106	2	no	—
317	#41	—	212	1	no	—
318	C₂×Q₈	—	2	1	no	—
319	C₂×Q₈	—	53	1	no	—
320	C₂×Q₈	C₂×C₄×D₅₃	1	2	yes	—
321	C₂×D₄	—	53	1	no	—
322	C₂×D₄	—	106	1	no	—
323	C₂²×C₄	—	106	1	no	—
324	C₂²×C₄	—	106	1	no	—
325	C₂²×C₄	—	106	1	no	—
326	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
327	C₂²×C₄	#11719945	1	1	yes	characteristic
328	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
329	Q₁₆	—	212	2	no	—
330	SD₁₆	—	106	4	no	—
331	M₁₆	—	424	1	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	1	no	—
334	C₂×C₈	—	53	2	no	—
335	C₂×C₈	—	53	2	no	—
336	C₂×C₈	—	106	4	no	—
337	C₂×C₈	—	106	1	no	—
338	C₂×C₈	—	106	1	no	—
339	C₂×C₈	—	106	2	no	—
340	C₂×C₈	—	424	2	no	—
341	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
342	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
343	C₄⋊C₄	—	2	1	no	—
344	C₄⋊C₄	—	2	1	no	—
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	53	1	no	—
349	C₄⋊C₄	—	53	1	no	—
350	C₄⋊C₄	—	106	1	no	—
351	C₄⋊C₄	—	106	1	no	—
352	C₄⋊C₄	C₂×C₄×D₅₃	1	2	yes	—
353	C₂²⋊C₄	—	2	2	no	—
354	C₂²⋊C₄	—	106	1	no	—
355	C₂²⋊C₄	—	106	1	no	—
356	C₂²⋊C₄	—	106	1	no	—
357	C₄²	—	2	2	no	—
358	C₄²	—	106	1	no	—
359	C₄²	—	106	1	no	—
360	C₄²	—	106	1	no	—
361	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
362	C₄²	#11719945	1	1	yes	characteristic
363	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
364	C₄²	C₂³×D₅₃	1	1	yes	characteristic
365	C₂³	—	106	1	no	—
366	C₂³	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
367	Q₈	—	2	2	no	—
368	Q₈	—	2	2	no	—
369	Q₈	—	4	2	no	—
370	Q₈	—	106	1	no	—
371	Q₈	—	106	1	no	—
372	D₄	—	53	2	no	—
373	D₄	—	106	1	no	—
374	D₄	—	212	2	no	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	106	1	no	—
383	C₂×C₄	—	106	1	no	—
384	C₂×C₄	—	106	1	no	—
385	C₂×C₄	—	212	2	no	—
386	C₂×C₄	—	212	1	no	—
387	C₂×C₄	—	212	1	no	—
388	C₂×C₄	—	212	2	no	—
389	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
390	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
391	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
392	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
393	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
394	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
395	C₈	—	2	2	no	—
396	C₈	—	2	2	no	—
397	C₈	—	4	1	no	—
398	C₈	—	4	1	no	—
399	C₈	—	106	2	no	—
400	C₈	—	106	2	no	—
401	C₈	—	106	2	no	—
402	C₈	—	106	2	no	—
403	C₈	—	212	2	no	—
404	C₂²	—	2	1	no	—
405	C₂²	—	2	1	no	—
406	C₂²	—	2	1	no	—
407	C₂²	—	106	2	no	—
408	C₂²	—	212	1	no	—
409	C₂²	—	212	1	no	—
410	C₂²	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
411	C₄	—	2	1	no	—
412	C₄	—	2	1	no	—
413	C₄	—	2	1	no	—
414	C₄	—	4	2	no	—
415	C₄	—	4	2	no	—
416	C₄	—	212	1	no	—
417	C₄	—	212	1	no	—
418	C₄	—	212	1	no	—
419	C₄	#16810738	1	2	yes	—
420	C₄	#131×D₅₃	1	1	yes	characteristic
421	C₄	#16811388	1	1	yes	characteristic
422	C₄	#16811431	1	1	yes	characteristic
423	C₄	#16811438	1	1	yes	characteristic
424	C₂	—	4	1	no	—
425	C₂	—	212	2	no	—
426	C₂	#20283437	1	1	yes	characteristic
427	C₂	#20283558	1	1	yes	characteristic
428	C₂	C₂×#16810738	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#2986	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#2986	D₅₃
hypocenter	C₅₃	#18016
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #2986

Complements

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

N	H ≅ G/N	type
3. #20273610	C₂	semidirect
4. #20273586	C₂	semidirect
5. #20273570	C₂	semidirect
6. C₈×#11719924	C₂	semidirect
7. #20273503	C₂	semidirect
8. #20273170	C₂	semidirect
10. #20272999	C₂	semidirect
11. C₅₃×#2986	C₂	semidirect
212. #2986	D₅₃	semidirect
261. C₅₃	#18016	semidirect

Maximal subgroups

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

subgroup	index
#20284780	2
#20283836	2
#20273610	2
#20273586	2
#20273570	2
C₈×#11719924	2
#20273503	2
#20273170	2
#20273095	2
#20272999	2
C₅₃×#2986	2
#18016	53