Subgroups — #27275228

Order 13568.

494 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133128	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810345	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
5	#20272542	C₂	1	1	yes	maximal, characteristic
6	#20272542	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊#88322	C₄	1	2	yes	—
9	C₂×#13133128	C₄	1	2	yes	—
10	C₂×#13133128	C₄	1	2	yes	—
11	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
14	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
15	#16810378	C₄	1	2	yes	—
16	#16810345	—	2	2	no	—
17	C₈×#88322	—	2	2	no	—
18	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₄³×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#336	C₄	1	2	yes	—
23	C₂²×#88322	C₂×C₄	1	2	yes	—
24	C₂²×#88322	C₂×C₄	1	2	yes	—
25	C₂²×#88322	C₂×C₄	1	2	yes	—
26	C₂²×#88322	C₂×C₄	1	2	yes	—
27	C₄⋊#88322	—	2	4	no	—
28	#13133128	—	2	4	no	—
29	#13133128	C₈	1	8	yes	complemented
30	C₄×#88322	—	2	1	no	—
31	C₄×#88322	—	2	1	no	—
32	C₄×#88322	—	2	1	no	—
33	C₄×#88322	—	2	1	no	—
34	C₄×#88322	—	2	1	no	—
35	C₄×#88322	—	2	1	no	—
36	C₄×#88322	D₄	1	2	yes	—
37	C₄×#88322	Q₈	1	2	yes	—
38	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₅₃	—	2	4	no	—
48	C₄×C₈×C₅₃	—	2	1	no	—
49	C₄×C₈×C₅₃	—	2	1	no	—
50	C₄×C₈×C₅₃	—	2	1	no	—
51	C₄×C₈×C₅₃	—	2	1	no	—
52	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₂×C₈×C₅₃	—	2	2	no	—
64	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
65	C₄²×C₅₃	—	2	2	no	—
66	C₄²×C₅₃	—	2	2	no	—
67	C₄²×C₅₃	—	2	2	no	—
68	C₄²×C₅₃	—	2	2	no	—
69	C₄²×C₅₃	—	2	2	no	—
70	C₄²×C₅₃	C₂×C₈	1	4	yes	—
71	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
73	C₄²×C₅₃	#41	1	1	yes	characteristic
74	C₄²×C₅₃	#41	1	1	yes	characteristic
75	C₂×#88322	—	2	2	no	—
76	C₂×#88322	—	2	2	no	—
77	C₂×#88322	—	2	2	no	—
78	C₂×#88322	—	2	2	no	—
79	C₂×#88322	—	2	2	no	—
80	C₂×#88322	—	2	2	no	—
81	C₂×#88322	—	2	2	no	—
82	C₂×#88322	C₄⋊C₄	1	4	yes	—
83	C₂×#88322	C₂×C₈	1	4	yes	—
84	C₂×#88322	C₂×C₈	1	4	yes	—
85	C₂×#88322	M₁₆	1	4	yes	—
86	C₂×#88322	M₁₆	1	4	yes	—
87	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	2	1	no	—
96	C₂×C₄×C₅₃	—	2	1	no	—
97	C₂×C₄×C₅₃	—	2	1	no	—
98	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
99	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
100	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
108	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
109	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
110	C₂×C₄×C₅₃	#131	1	2	yes	—
111	C₂×C₄×C₅₃	#131	1	2	yes	—
112	C₈×C₅₃	—	2	4	no	—
113	C₈×C₅₃	—	4	2	no	—
114	#88322	—	2	4	no	—
115	#88322	—	2	4	no	—
116	#88322	—	4	2	no	—
117	#88322	C₄⋊C₈	1	8	yes	complemented
118	#8262	—	53	1	no	maximal, Sylow
119	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
121	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
122	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
123	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
124	C₂²×C₅₃	#405	1	1	yes	characteristic
125	C₂²×C₅₃	#405	1	1	yes	characteristic
126	C₄×C₅₃	—	2	2	no	—
127	C₄×C₅₃	—	2	2	no	—
128	C₄×C₅₃	—	4	1	no	—
129	C₄×C₅₃	—	4	1	no	—
130	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
131	C₄×C₅₃	#431	1	2	yes	—
132	C₄×C₅₃	#435	1	2	yes	—
133	C₄×C₅₃	#435	1	2	yes	—
134	C₄×C₅₃	#446	1	2	yes	—
135	C₄×C₅₃	#446	1	2	yes	—
136	C₄×C₄⋊C₈	—	53	1	no	—
137	C₄×C₄⋊C₈	D₅₃	1	1	yes	characteristic
138	C₄×#97	—	53	1	no	—
139	C₂×#322	—	53	1	no	—
140	C₂×C₈²	—	53	1	no	—
141	#1273	—	53	1	no	—
142	#1273	—	53	1	no	—
143	C₂×C₅₃	C₈×M₁₆	1	1	yes	characteristic
144	C₂×C₅₃	C₈×M₁₆	1	1	yes	characteristic
145	C₂×C₅₃	#1453	1	1	yes	characteristic
146	C₂×C₅₃	#1455	1	1	yes	characteristic
147	C₂×C₅₃	C₄×C₄⋊C₈	1	1	yes	characteristic
148	C₂×C₅₃	#1771	1	1	yes	characteristic
149	C₂×C₅₃	#1771	1	1	yes	characteristic
150	C₂×C₄⋊C₈	—	53	2	no	—
151	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
152	C₂×#97	—	53	2	no	—
153	C₂×#97	—	53	2	no	—
154	C₂×C₄×C₈	—	53	1	no	—
155	C₂×C₄×C₈	—	53	1	no	—
156	C₂×C₄×C₈	—	53	1	no	—
157	C₂×C₄×C₈	—	53	1	no	—
158	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
160	C₄³	C₂×D₅₃	1	1	yes	characteristic
161	#336	—	53	2	no	—
162	#336	Dic₅₃	1	2	yes	—
163	#322	—	106	2	no	—
164	C₈²	—	106	2	no	—
165	C₅₃	#8262	1	1	yes	characteristic, complemented, Sylow
166	C₂²×C₈	—	53	2	no	—
167	C₂²×C₈	—	53	2	no	—
168	C₂²×C₈	—	53	2	no	—
169	C₂²×C₈	—	53	2	no	—
170	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
171	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
172	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
173	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
174	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
175	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
176	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
177	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
178	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
179	C₄⋊C₈	—	2	4	no	—
180	C₄⋊C₈	—	106	4	no	—
181	#97	—	53	8	no	—
182	#97	—	106	4	no	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	—	2	1	no	—
185	C₄×C₈	—	2	1	no	—
186	C₄×C₈	—	2	1	no	—
187	C₄×C₈	—	53	2	no	—
188	C₄×C₈	—	53	2	no	—
189	C₄×C₈	—	106	1	no	—
190	C₄×C₈	—	106	1	no	—
191	C₄×C₈	—	106	1	no	—
192	C₄×C₈	—	106	1	no	—
193	C₄×C₈	—	106	1	no	—
194	C₄×C₈	—	106	1	no	—
195	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
196	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
197	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
198	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
201	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	53	4	no	—
208	C₂×C₈	—	53	4	no	—
209	C₂×C₈	—	53	4	no	—
210	C₂×C₈	—	53	4	no	—
211	C₂×C₈	—	53	4	no	—
212	C₂×C₈	—	106	2	no	—
213	C₂×C₈	—	106	2	no	—
214	C₂×C₈	—	106	2	no	—
215	C₂×C₈	—	106	2	no	—
216	C₂×C₈	—	106	2	no	—
217	C₂×C₈	—	106	2	no	—
218	C₂×C₈	—	106	2	no	—
219	C₂×C₈	#11719917	1	4	yes	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	2	no	—
225	C₄²	C₈×D₅₃	1	4	yes	—
226	C₄²	D₄×D₅₃	1	1	yes	characteristic
227	C₄²	#11719947	1	1	yes	characteristic
228	C₄²	Q₈×D₅₃	1	1	yes	characteristic
229	C₄²	#11719949	1	1	yes	characteristic
230	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
242	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
243	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
244	C₂×C₄	#13133175	1	1	yes	characteristic
245	C₂×C₄	#13133179	1	1	yes	characteristic
246	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
247	C₂×C₄	#13133182	1	1	yes	characteristic
248	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
249	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
250	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
251	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
252	C₂×C₄	C₂×#11719917	1	2	yes	—
253	C₂×C₄	#13133223	1	2	yes	—
254	C₂×C₄	#13133224	1	2	yes	—
255	C₈	—	2	4	no	—
256	C₈	—	4	2	no	—
257	C₈	—	53	8	no	—
258	C₈	—	106	4	no	—
259	C₈	—	106	4	no	—
260	C₈	—	212	2	no	—
261	C₂²	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
262	C₂²	C₂×C₈×Dic₅₃	1	1	yes	characteristic
263	C₂²	M₁₆×Dic₅₃	1	1	yes	characteristic
264	C₂²	M₁₆×Dic₅₃	1	1	yes	characteristic
265	C₂²	C₄×#11719917	1	1	yes	characteristic
266	C₂²	#16810578	1	1	yes	characteristic
267	C₂²	#16810579	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	2	2	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	1	no	—
272	C₄	C₄⋊C₈×D₅₃	1	2	yes	—
273	C₄	#16810675	1	2	yes	—
274	C₄	#16810693	1	2	yes	—
275	C₄	#16810695	1	2	yes	—
276	C₄	#16810752	1	2	yes	—
277	C₄	#16810754	1	2	yes	—
278	C₂	C₈×#11719917	1	1	yes	characteristic
279	C₂	#20272996	1	1	yes	characteristic
280	C₂	#20273010	1	1	yes	characteristic
281	C₂	#20273018	1	1	yes	characteristic
282	C₂	C₄⋊C₈×Dic₅₃	1	1	yes	characteristic
283	C₂	#20274590	1	1	yes	characteristic
284	C₂	#20274592	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₄×C₄⋊C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#8262
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
29. #13133128	C₈	semidirect
117. #88322	C₄⋊C₈	semidirect
165. C₅₃	#8262	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#88322	2
C₄×#13133128	2
C₂×#16810345	2
C₂×C₈×#88322	2
#20272542	2
C₄×C₄⋊C₈×C₅₃	2
#8262	53