Subgroups — #27274674

Order 13568.

494 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810702	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133133	C₂	1	1	yes	maximal, characteristic
3	#20273035	C₂	1	1	yes	maximal, characteristic
4	C₈×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272566	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272564	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1451	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11719912	C₄	1	2	yes	—
9	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
10	#16810702	—	2	1	no	—
11	#16810702	—	2	1	no	—
12	#16810702	—	2	1	no	—
13	#16810702	—	2	1	no	—
14	#16810676	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
16	C₂×#13133133	C₄	1	2	yes	—
17	C₂×#13133129	C₄	1	2	yes	—
18	C₄×#11719917	C₂²	1	1	yes	characteristic
19	#16810391	C₄	1	2	yes	—
20	#16810346	C₄	1	4	yes	—
21	#16810345	—	2	1	no	—
22	C₈×#88322	—	2	1	no	—
23	C₅₃×#431	C₂²	1	1	yes	characteristic
24	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#322	—	2	1	no	—
27	C₅₃×#322	—	2	1	no	—
28	C₂²×D₂₁₂	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×D₅₃	C₂×C₄	1	2	yes	—
30	C₂×#11719917	C₂×C₄	1	2	yes	—
31	C₂×#11719912	—	2	2	no	—
32	C₂×#11719912	—	2	2	no	—
33	C₂×#11719912	—	4	4	no	—
34	C₂²×#88322	C₂×C₄	1	2	yes	—
35	C₄×D₂₁₂	—	2	1	no	—
36	C₄×D₂₁₂	—	2	1	no	—
37	C₄×D₂₁₂	—	2	1	no	—
38	C₄×D₂₁₂	—	2	1	no	—
39	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×#11719921	C₂×C₄	1	2	yes	—
41	C₄⋊#88322	—	2	1	no	—
42	C₄⋊#88322	—	2	1	no	—
43	C₄⋊#88322	—	2	1	no	—
44	#13133133	—	2	4	no	—
45	C₂²⋊#88322	C₂×C₄	1	2	yes	—
46	#13133129	—	2	2	no	—
47	#13133129	—	2	2	no	—
48	#13133128	—	2	1	no	—
49	C₄×#88322	—	2	1	no	—
50	C₄×#88322	—	2	1	no	—
51	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₅₃	—	2	1	no	—
56	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
57	#97×C₅₃	—	2	1	no	—
58	C₄×C₈×C₅₃	—	2	1	no	—
59	C₄×C₈×C₅₃	—	2	1	no	—
60	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
62	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
63	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
64	C₂³×D₅₃	C₄²	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×Dic₅₃	C₄²	1	2	yes	—
69	C₂×D₂₁₂	—	2	1	no	—
70	C₂×D₂₁₂	—	2	1	no	—
71	C₂×D₂₁₂	—	2	2	no	—
72	C₂×D₂₁₂	—	2	2	no	—
73	C₂×C₄×D₅₃	—	2	2	no	—
74	C₂×C₄×D₅₃	—	2	2	no	—
75	C₂×C₄×D₅₃	—	4	2	no	—
76	C₂×C₄×D₅₃	—	4	2	no	—
77	M₁₆×C₅₃	—	2	2	no	—
78	M₁₆×C₅₃	—	2	2	no	—
79	C₂×C₈×C₅₃	—	2	2	no	—
80	C₂×C₈×C₅₃	—	2	2	no	—
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	—	4	1	no	—
83	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₅₃	#41	1	1	yes	characteristic
90	C₄²×C₅₃	#41	1	1	yes	characteristic
91	#11719921	—	2	2	no	—
92	#11719921	—	2	2	no	—
93	C₄⋊Dic₅₃	—	2	1	no	—
94	C₄⋊Dic₅₃	—	2	1	no	—
95	#11719917	—	2	2	no	—
96	#11719917	M₁₆	1	4	yes	complemented
97	C₂×#88322	—	2	2	no	—
98	C₂×#88322	—	2	2	no	—
99	C₂×#88322	—	2	2	no	—
100	C₂×#88322	—	4	2	no	—
101	#11719912	—	4	4	no	—
102	#11719912	—	4	4	no	—
103	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
104	C₂²×D₅₃	—	2	2	no	—
105	C₂²×D₅₃	—	2	2	no	—
106	C₂²×D₅₃	—	2	2	no	—
107	C₂²×D₅₃	—	4	2	no	—
108	C₂²×D₅₃	—	4	2	no	—
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	—	2	1	no	—
111	C₂×C₄×C₅₃	—	4	1	no	—
112	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
119	C₂×C₄×C₅₃	#131	1	2	yes	—
120	C₂×Dic₅₃	—	2	2	no	—
121	C₂×Dic₅₃	—	4	2	no	—
122	D₂₁₂	—	4	2	no	—
123	D₂₁₂	—	4	2	no	—
124	C₄×D₅₃	—	4	4	no	—
125	C₄×D₅₃	—	4	2	no	—
126	C₄×D₅₃	—	4	2	no	—
127	C₈×C₅₃	—	2	1	no	—
128	C₈×C₅₃	—	2	1	no	—
129	C₈×C₅₃	—	2	1	no	—
130	C₈×C₅₃	—	4	2	no	—
131	C₈×C₅₃	—	4	2	no	—
132	C₈×C₅₃	#100	1	2	yes	complemented
133	#88322	—	2	4	no	—
134	#88322	—	4	2	no	—
135	#88322	—	4	2	no	—
136	#8166	—	53	1	no	maximal, Sylow
137	C₂²×C₅₃	—	2	1	no	—
138	C₂²×C₅₃	—	2	1	no	—
139	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
140	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
141	C₂²×C₅₃	#405	1	1	yes	characteristic
142	C₂×D₅₃	—	4	2	no	—
143	C₂×D₅₃	—	4	2	no	—
144	C₂×D₅₃	—	4	2	no	—
145	C₂×D₅₃	—	8	2	no	—
146	C₂×D₅₃	—	8	2	no	—
147	C₄×C₅₃	—	2	1	no	—
148	C₄×C₅₃	—	2	1	no	—
149	C₄×C₅₃	—	2	1	no	—
150	C₄×C₅₃	C₂×#100	1	1	yes	characteristic
151	C₄×C₅₃	#413	1	1	yes	characteristic
152	C₄×C₅₃	#435	1	2	yes	—
153	C₄×C₅₃	#436	1	2	yes	—
154	Dic₅₃	—	4	2	no	—
155	C₂×#436	—	53	1	no	—
156	C₄×#100	—	53	1	no	—
157	#1460	—	53	1	no	—
158	#1451	D₅₃	1	1	yes	characteristic, complemented
159	C₈×M₁₆	—	53	1	no	—
160	#1283	—	53	1	no	—
161	#1283	—	53	1	no	—
162	C₂×C₅₃	—	2	1	no	—
163	C₂×C₅₃	—	2	1	no	—
164	C₂×C₅₃	#1753	1	1	yes	characteristic
165	C₂×C₅₃	C₄×#100	1	1	yes	characteristic
166	C₂×C₅₃	#1757	1	1	yes	characteristic
167	D₅₃	—	8	2	no	—
168	C₂²×M₁₆	—	53	2	no	—
169	C₂×C₄×D₄	—	53	1	no	—
170	#436	—	106	1	no	—
171	#436	—	106	1	no	—
172	#436	—	106	1	no	—
173	#436	—	106	1	no	—
174	#431	—	53	1	no	—
175	#431	C₂×D₅₃	1	1	yes	characteristic
176	C₂×C₄⋊C₈	—	53	1	no	—
177	C₂×#100	—	53	2	no	—
178	C₂×C₂²⋊C₈	—	53	2	no	—
179	C₄×M₁₆	—	53	1	no	—
180	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
181	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
182	#343	—	53	2	no	—
183	#323	—	53	4	no	—
184	#322	—	2	1	no	—
185	#322	—	2	1	no	—
186	#322	—	106	1	no	—
187	C₈²	—	106	1	no	—
188	C₅₃	#8166	1	1	yes	characteristic, complemented, Sylow
189	C₂²×D₄	—	53	1	no	—
190	C₂³×C₄	—	53	2	no	—
191	C₂×M₁₆	—	53	2	no	—
192	C₂×M₁₆	—	106	2	no	—
193	C₂×M₁₆	—	106	2	no	—
194	C₂×M₁₆	—	212	4	no	—
195	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
196	C₂²×C₈	—	53	2	no	—
197	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
198	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
199	C₄×D₄	—	106	1	no	—
200	C₄×D₄	—	106	1	no	—
201	C₄×D₄	—	106	1	no	—
202	C₄×D₄	—	106	1	no	—
203	C₂×C₄⋊C₄	—	53	1	no	—
204	C₂×C₂²⋊C₄	—	53	2	no	—
205	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
206	C₄⋊C₈	—	2	1	no	—
207	C₄⋊C₈	—	106	1	no	—
208	C₄⋊C₈	—	106	1	no	—
209	C₄⋊C₈	—	106	1	no	—
210	#100	—	106	4	no	—
211	C₂²⋊C₈	—	53	2	no	—
212	C₂²⋊C₈	—	106	2	no	—
213	C₂²⋊C₈	—	106	2	no	—
214	C₂²⋊C₈	C₄×D₅₃	1	2	yes	—
215	#97	—	2	1	no	—
216	#97	—	106	1	no	—
217	C₄×C₈	—	2	1	no	—
218	C₄×C₈	—	2	1	no	—
219	C₄×C₈	—	106	1	no	—
220	C₄×C₈	—	106	1	no	—
221	C₄×C₈	D₂₁₂	1	1	yes	characteristic
222	C₄×C₈	D₂₁₂	1	1	yes	characteristic
223	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
224	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
225	C₂⁴	—	53	2	no	—
226	C₂×D₄	—	106	1	no	—
227	C₂×D₄	—	106	1	no	—
228	C₂×D₄	—	106	2	no	—
229	C₂×D₄	—	106	2	no	—
230	C₂²×C₄	—	53	2	no	—
231	C₂²×C₄	—	106	2	no	—
232	C₂²×C₄	—	106	2	no	—
233	C₂²×C₄	—	212	2	no	—
234	C₂²×C₄	—	212	2	no	—
235	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
238	M₁₆	—	2	2	no	—
239	M₁₆	—	2	2	no	—
240	M₁₆	—	53	4	no	—
241	M₁₆	—	106	2	no	—
242	M₁₆	—	212	4	no	—
243	M₁₆	—	212	4	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	106	2	no	—
249	C₂×C₈	—	106	2	no	—
250	C₂×C₈	—	106	2	no	—
251	C₂×C₈	—	212	2	no	—
252	C₂×C₈	#11719921	1	1	yes	characteristic
253	C₂×C₈	#11719921	1	1	yes	characteristic
254	C₂×C₈	#11719921	1	1	yes	characteristic
255	C₂×C₈	#11719921	1	1	yes	characteristic
256	C₄⋊C₄	—	106	1	no	—
257	C₄⋊C₄	—	106	1	no	—
258	C₂²⋊C₄	—	106	2	no	—
259	C₂²⋊C₄	—	106	2	no	—
260	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
261	C₄²	#11719945	1	1	yes	characteristic
262	C₄²	#11719945	1	1	yes	characteristic
263	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
264	C₂³	—	106	2	no	—
265	C₂³	—	106	2	no	—
266	C₂³	—	106	2	no	—
267	C₂³	—	212	2	no	—
268	C₂³	—	212	2	no	—
269	C₂³	C₄²×D₅₃	1	1	yes	characteristic
270	D₄	—	212	2	no	—
271	D₄	—	212	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	106	2	no	—
276	C₂×C₄	—	212	4	no	—
277	C₂×C₄	—	212	2	no	—
278	C₂×C₄	—	212	2	no	—
279	C₂×C₄	—	212	2	no	—
280	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
281	C₂×C₄	#13133174	1	1	yes	characteristic
282	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
283	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
284	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
285	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
286	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
287	C₂×C₄	#13133225	1	2	yes	—
288	C₈	—	2	1	no	—
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	4	2	no	—
292	C₈	—	4	2	no	—
293	C₈	—	106	4	no	—
294	C₈	—	212	2	no	—
295	C₈	—	212	2	no	—
296	C₈	#13133133	1	2	yes	complemented
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	424	2	no	—
300	C₂²	—	424	2	no	—
301	C₂²	—	212	2	no	—
302	C₂²	—	212	2	no	—
303	C₂²	—	212	2	no	—
304	C₂²	C₄×#11719921	1	1	yes	characteristic
305	C₂²	#16810575	1	1	yes	characteristic
306	C₂²	#16810580	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	212	2	no	—
311	C₄	C₂×#13133133	1	1	yes	characteristic
312	C₄	#16810605	1	1	yes	characteristic
313	C₄	#16810696	1	2	yes	—
314	C₄	#16810699	1	2	yes	—
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	424	2	no	—
318	C₂	#20274500	1	1	yes	characteristic
319	C₂	C₄×#13133133	1	1	yes	characteristic
320	C₂	#20274516	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11719921
commutator	C₂³×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1451	C₂
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	C₂²×D₂₁₂	C₂×C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1451	D₅₃
hypocenter	C₅₃	#8166
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1451

Complements

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

N	H ≅ G/N	type
4. C₈×#11719917	C₂	semidirect
5. #20272566	C₂	semidirect
6. #20272564	C₂	semidirect
7. C₅₃×#1451	C₂	semidirect
96. #11719917	M₁₆	semidirect
132. C₈×C₅₃	#100	semidirect
158. #1451	D₅₃	semidirect
188. C₅₃	#8166	semidirect
296. C₈	#13133133	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810702	2
C₄×#13133133	2
#20273035	2
C₈×#11719917	2
#20272566	2
#20272564	2
C₅₃×#1451	2
#8166	53