Subgroups — #27536693

Order 13568.

822 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811031	C₂	1	2	yes	maximal
2	C₂×#16811030	C₂	1	2	yes	maximal
3	#20283091	C₂	1	1	yes	maximal, characteristic
4	#20283077	C₂	1	1	yes	maximal, characteristic
5	#20274882	C₂	1	2	yes	maximal
6	#20274880	C₂	1	2	yes	maximal
7	#20274879	C₂	1	1	yes	maximal, characteristic
8	#20274877	C₂	1	1	yes	maximal, characteristic
9	#20274877	C₂	1	1	yes	maximal, characteristic
10	C₄×#13133127	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#1846	C₂	1	1	yes	maximal, characteristic
12	C₂×#13133216	C₂²	1	2	yes	—
13	C₂×#13133208	C₂²	1	2	yes	—
14	#16811031	—	2	8	no	—
15	#16811030	—	2	8	no	—
16	C₂×#13133151	C₂²	1	2	yes	—
17	C₂×#13133150	C₂²	1	2	yes	—
18	C₂×#13133150	C₂²	1	2	yes	—
19	C₂×#13133149	C₂²	1	2	yes	—
20	C₂×#13133148	C₂²	1	2	yes	—
21	C₂×#13133148	C₂²	1	2	yes	—
22	C₂×#13133127	C₂²	1	1	yes	characteristic
23	C₂×#13133127	C₂²	1	1	yes	characteristic
24	C₂×#13133127	C₂²	1	1	yes	characteristic
25	C₂×#13133127	C₂²	1	1	yes	characteristic
26	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
27	#16810527	—	2	4	no	—
28	#16810520	—	2	4	no	—
29	#16810492	C₂²	1	2	yes	—
30	#16810489	C₂²	1	1	yes	characteristic
31	#16810489	C₂²	1	1	yes	characteristic
32	#16810488	C₂²	1	2	yes	—
33	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
34	C₂×#107×C₅₃	C₂²	1	2	yes	—
35	C₂×#106×C₅₃	C₂²	1	2	yes	—
36	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#382	C₂²	1	1	yes	characteristic
38	C₅₃×#382	C₂²	1	1	yes	characteristic
39	C₂²×Dic₁₀₆	—	2	2	no	—
40	C₂²×#88322	C₂³	1	1	yes	characteristic
41	C₂²×#88322	C₂³	1	1	yes	characteristic
42	#13133216	—	2	4	no	—
43	#13133216	—	2	4	no	—
44	#13133208	—	2	4	no	—
45	#13133208	—	2	4	no	—
46	C₂×C₄⋊Dic₅₃	—	2	2	no	—
47	C₂×C₄⋊Dic₅₃	C₂³	1	2	yes	—
48	C₂×C₄⋊Dic₅₃	C₂³	1	2	yes	—
49	C₂×#11719919	—	2	2	no	—
50	C₂×#11719919	—	2	4	no	—
51	C₂×#11719919	—	2	2	no	—
52	C₂×C₄×Dic₅₃	D₄	1	4	yes	—
53	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
54	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
55	#13133151	—	2	4	no	—
56	#13133150	—	2	4	no	—
57	#13133150	—	2	4	no	—
58	#13133149	—	2	4	no	—
59	#13133148	—	2	4	no	—
60	#13133148	—	2	4	no	—
61	#13133127	D₄	1	4	yes	—
62	#13133127	D₄	1	4	yes	—
63	#13133127	Q₈	1	4	yes	complemented
64	#13133127	Q₈	1	4	yes	complemented
65	C₄×#88322	Q₈	1	4	yes	complemented
66	#13133124	—	2	2	no	—
67	#13133124	—	2	2	no	—
68	#13133124	—	2	4	no	—
69	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
73	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
74	#107×C₅₃	—	2	4	no	—
75	#106×C₅₃	—	2	4	no	—
76	C₄×C₈×C₅₃	Q₈	1	4	yes	complemented
77	#95×C₅₃	—	2	2	no	—
78	#95×C₅₃	—	2	2	no	—
79	C₂²×C₄×C₅₃	—	2	2	no	—
80	C₂²×C₄×C₅₃	—	2	2	no	—
81	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
83	C₂²×Dic₅₃	—	2	2	no	—
84	C₂²×Dic₅₃	—	2	2	no	—
85	C₂²×Dic₅₃	C₂×D₄	1	2	yes	—
86	C₂²×Dic₅₃	C₂×D₄	1	2	yes	—
87	C₂×Dic₁₀₆	—	4	8	no	—
88	C₂×Dic₁₀₆	—	4	4	no	—
89	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
92	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
93	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
94	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
95	C₄⋊C₄×C₅₃	—	2	2	no	—
96	C₄⋊C₄×C₅₃	—	2	2	no	—
97	C₄⋊C₄×C₅₃	—	2	2	no	—
98	C₄⋊C₄×C₅₃	—	2	2	no	—
99	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
100	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
101	C₄⋊Dic₅₃	—	2	2	no	—
102	C₄⋊Dic₅₃	—	2	2	no	—
103	C₄⋊Dic₅₃	—	2	2	no	—
104	C₄⋊Dic₅₃	—	2	2	no	—
105	C₄⋊Dic₅₃	—	4	4	no	—
106	#11719919	—	4	8	no	—
107	#11719919	—	4	4	no	—
108	#11719919	—	4	4	no	—
109	C₄×Dic₅₃	—	4	4	no	—
110	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
111	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
112	C₄×Dic₅₃	C₂×Q₈	1	2	yes	—
113	C₄×Dic₅₃	C₂×Q₈	1	2	yes	—
114	C₂×#88322	C₂×D₄	1	2	yes	—
115	C₂×#88322	C₂×D₄	1	2	yes	—
116	C₂×#88322	C₂×Q₈	1	2	yes	—
117	C₂×#88322	C₂×Q₈	1	2	yes	—
118	C₂×#88322	C₂×Q₈	1	2	yes	—
119	C₂×#88322	C₂×Q₈	1	2	yes	—
120	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	—	2	4	no	—
122	C₂×C₄×C₅₃	—	4	2	no	—
123	C₂×C₄×C₅₃	—	4	2	no	—
124	C₂×C₄×C₅₃	—	4	4	no	—
125	C₂×C₄×C₅₃	—	4	4	no	—
126	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
127	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
133	C₂×Dic₅₃	—	2	4	no	—
134	C₂×Dic₅₃	—	2	4	no	—
135	C₂×Dic₅₃	—	4	2	no	—
136	C₂×Dic₅₃	—	4	2	no	—
137	C₂×Dic₅₃	—	4	4	no	—
138	C₂×Dic₅₃	—	4	4	no	—
139	C₂×Dic₅₃	#127	1	4	yes	—
140	C₂×Dic₅₃	C₄⋊Q₈	1	4	yes	—
141	Dic₁₀₆	—	8	8	no	—
142	C₈×C₅₃	C₄⋊Q₈	1	4	yes	—
143	C₈×C₅₃	C₄⋊Q₈	1	4	yes	—
144	#88322	C₄⋊Q₈	1	4	yes	—
145	#88322	C₄⋊Q₈	1	4	yes	—
146	#33317	—	53	1	no	maximal, Sylow
147	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
148	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
149	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
150	C₂²×C₅₃	#575	1	4	yes	—
151	C₄×C₅₃	—	2	4	no	—
152	C₄×C₅₃	—	8	2	no	—
153	C₄×C₅₃	—	8	2	no	—
154	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
155	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
156	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
157	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
158	Dic₅₃	—	2	4	no	—
159	Dic₅₃	—	2	4	no	—
160	Dic₅₃	—	8	2	no	—
161	Dic₅₃	—	8	2	no	—
162	C₂×#501	—	53	2	no	—
163	C₂×#501	—	53	2	no	—
164	#2862	—	53	1	no	—
165	#2859	—	53	1	no	—
166	#1847	—	53	2	no	—
167	#1847	—	53	2	no	—
168	#1846	—	53	1	no	—
169	#1846	—	53	1	no	—
170	#1846	—	53	1	no	—
171	#1846	D₅₃	1	1	yes	characteristic
172	C₄×#97	—	53	1	no	—
173	C₂×C₅₃	#2868	1	1	yes	characteristic
174	C₂×C₅₃	#3153	1	2	yes	—
175	C₂×C₅₃	#3162	1	2	yes	—
176	C₂×C₅₃	#3166	1	2	yes	—
177	C₂×C₄⋊Q₈	—	53	2	no	—
178	C₂×#125	—	53	2	no	—
179	#501	—	106	8	no	—
180	#501	—	106	8	no	—
181	C₂×#107	—	53	2	no	—
182	C₂×#107	—	53	2	no	—
183	C₂×#107	—	53	2	no	—
184	C₂×#107	C₂×D₅₃	1	2	yes	—
185	C₂×#106	—	53	2	no	—
186	C₂×#106	—	53	2	no	—
187	C₂×#106	—	53	2	no	—
188	C₂×#106	C₂×D₅₃	1	2	yes	—
189	C₂×#97	—	53	1	no	—
190	C₂×#97	—	53	1	no	—
191	C₂×#97	—	53	1	no	—
192	C₂×#97	—	53	1	no	—
193	C₂×C₄×C₈	—	53	1	no	—
194	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
195	#391	—	106	4	no	—
196	#389	—	106	4	no	—
197	#384	—	53	2	no	—
198	#382	—	53	2	no	—
199	#382	—	53	1	no	—
200	#382	—	53	1	no	—
201	#382	C₂×D₅₃	1	1	yes	characteristic
202	#382	C₂×D₅₃	1	1	yes	characteristic
203	C₄³	—	53	1	no	—
204	C₅₃	#33317	1	1	yes	characteristic, complemented, Sylow
205	C₂²×Q₈	—	106	2	no	—
206	C₂²×C₈	—	53	1	no	—
207	C₂²×C₈	—	53	1	no	—
208	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
209	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
210	C₄⋊Q₈	—	106	4	no	—
211	C₄⋊Q₈	—	106	4	no	—
212	#125	—	106	4	no	—
213	#125	—	106	4	no	—
214	C₂×C₄⋊C₄	—	53	2	no	—
215	C₂×C₄⋊C₄	—	53	2	no	—
216	C₂×C₄⋊C₄	—	106	2	no	—
217	C₂×C₄⋊C₄	—	106	2	no	—
218	C₂×C₄⋊C₄	—	106	2	no	—
219	C₂×C₄⋊C₄	—	106	4	no	—
220	C₂×C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
221	C₂×C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
222	C₂×C₄²	—	53	1	no	—
223	C₂×C₄²	—	53	1	no	—
224	C₂×C₄²	—	53	4	no	—
225	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
226	#107	—	2	4	no	—
227	#107	—	106	4	no	—
228	#107	—	106	4	no	—
229	#107	—	106	4	no	—
230	#106	—	2	4	no	—
231	#106	—	106	4	no	—
232	#106	—	106	4	no	—
233	#106	—	106	4	no	—
234	#97	—	53	4	no	—
235	#97	—	53	4	no	—
236	#97	—	53	4	no	—
237	#97	—	53	4	no	—
238	C₄×C₈	—	53	4	no	—
239	C₄×C₈	Dic₁₀₆	1	4	yes	complemented
240	#95	—	2	2	no	—
241	#95	—	2	2	no	—
242	#95	—	106	4	no	—
243	#95	—	106	2	no	—
244	#95	—	106	2	no	—
245	C₂×Q₈	—	212	4	no	—
246	C₂×Q₈	—	212	8	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	53	2	no	—
250	C₂²×C₄	—	53	2	no	—
251	C₂²×C₄	—	106	2	no	—
252	C₂²×C₄	—	106	2	no	—
253	C₂²×C₄	D₄×D₅₃	1	2	yes	—
254	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
255	C₂×C₈	—	53	2	no	—
256	C₂×C₈	—	53	2	no	—
257	C₂×C₈	—	53	2	no	—
258	C₂×C₈	—	53	2	no	—
259	C₂×C₈	—	53	2	no	—
260	C₂×C₈	—	53	2	no	—
261	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
262	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
263	C₂×C₈	D₄×D₅₃	1	2	yes	—
264	C₂×C₈	D₄×D₅₃	1	2	yes	—
265	C₂×C₈	Q₈×D₅₃	1	2	yes	—
266	C₂×C₈	Q₈×D₅₃	1	2	yes	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	106	2	no	—
272	C₄⋊C₄	—	106	2	no	—
273	C₄⋊C₄	—	106	2	no	—
274	C₄⋊C₄	—	106	2	no	—
275	C₄⋊C₄	—	212	4	no	—
276	C₄⋊C₄	—	212	4	no	—
277	C₄⋊C₄	—	212	4	no	—
278	C₄⋊C₄	—	212	8	no	—
279	C₄²	—	53	2	no	—
280	C₄²	—	53	2	no	—
281	C₄²	—	53	2	no	—
282	C₄²	—	53	2	no	—
283	C₄²	—	212	4	no	—
284	C₄²	C₂×Dic₁₀₆	1	2	yes	—
285	C₄²	Q₈×D₅₃	1	2	yes	—
286	C₂³	C₂×D₄×D₅₃	1	1	yes	characteristic
287	Q₈	—	424	8	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	53	4	no	—
294	C₂×C₄	—	53	4	no	—
295	C₂×C₄	—	106	4	no	—
296	C₂×C₄	—	106	4	no	—
297	C₂×C₄	—	212	2	no	—
298	C₂×C₄	—	212	2	no	—
299	C₂×C₄	—	212	4	no	—
300	C₂×C₄	—	212	4	no	—
301	C₂×C₄	#13133216	1	4	yes	—
302	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
303	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
304	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
305	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
306	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
307	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
308	C₈	—	53	4	no	—
309	C₈	—	53	4	no	—
310	C₈	#13133216	1	4	yes	—
311	C₈	#13133216	1	4	yes	—
312	C₂²	#127×D₅₃	1	1	yes	characteristic
313	C₂²	C₄⋊Q₈×D₅₃	1	1	yes	characteristic
314	C₂²	C₂×#13133216	1	1	yes	characteristic
315	C₂²	#16811420	1	4	yes	—
316	C₄	—	2	4	no	—
317	C₄	—	8	2	no	—
318	C₄	—	8	2	no	—
319	C₄	—	106	4	no	—
320	C₄	—	106	4	no	—
321	C₄	—	424	2	no	—
322	C₄	—	424	2	no	—
323	C₄	C₄⋊Q₈×D₅₃	1	1	yes	characteristic
324	C₄	C₄⋊Q₈×D₅₃	1	1	yes	characteristic
325	C₄	C₂×#13133216	1	1	yes	characteristic
326	C₄	C₂×#13133216	1	1	yes	characteristic
327	C₂	#20283112	1	1	yes	characteristic
328	C₂	#20285176	1	2	yes	—
329	C₂	#20285238	1	2	yes	—
330	C₂	#20285263	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1846	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
hypercenter	#1846	D₅₃
hypocenter	C₅₃	#33317
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1846

Complements

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

N	H ≅ G/N	type
63. #13133127	Q₈	semidirect
64. #13133127	Q₈	semidirect
65. C₄×#88322	Q₈	semidirect
76. C₄×C₈×C₅₃	Q₈	semidirect
204. C₅₃	#33317	semidirect
239. C₄×C₈	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811031	2
C₂×#16811030	2
#20283091	2
#20283077	2
#20274882	2
#20274880	2
#20274879	2
#20274877	2
C₄×#13133127	2
C₅₃×#1846	2
#33317	53