Subgroups — #27371795

Order 13568.

706 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285302	C₂	1	1	yes	maximal, characteristic
2	#20285300	C₂	1	2	yes	maximal
3	#20285299	C₂	1	1	yes	maximal, characteristic
4	#20285224	C₂	1	2	yes	maximal
5	#20274332	C₂	1	1	yes	maximal, characteristic, complemented
6	#20274331	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274323	C₂	1	2	yes	maximal, complemented
8	#20274312	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274310	C₂	1	2	yes	maximal, complemented
10	C₈²×D₅₃	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#1707	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811178	C₂²	1	2	yes	—
13	#16811178	C₂²	1	2	yes	—
14	#16811028	C₂²	1	1	yes	characteristic
15	#16811026	C₂²	1	2	yes	—
16	#16811026	C₂²	1	2	yes	—
17	#16811026	C₂²	1	2	yes	—
18	#16811023	—	2	4	no	—
19	#16811021	C₂²	1	2	yes	—
20	#16811003	C₂²	1	1	yes	characteristic
21	#16811002	C₂²	1	2	yes	—
22	#16810975	C₂²	1	2	yes	—
23	#16810974	C₂²	1	2	yes	—
24	#16810973	C₂²	1	2	yes	—
25	#16810865	—	2	2	no	—
26	#16810863	—	2	4	no	—
27	#16810861	—	2	2	no	—
28	Q₁₆×Dic₅₃	—	2	2	no	—
29	#16810717	—	2	2	no	—
30	#16810715	—	2	4	no	—
31	#16810658	—	2	4	no	—
32	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
35	C₈×#88322	C₂²	1	1	yes	characteristic
36	C₈×#88322	C₂²	1	1	yes	characteristic
37	C₈×#88322	C₂²	1	1	yes	characteristic
38	C₅₃×#500	C₂²	1	1	yes	characteristic
39	C₅₃×#498	C₂²	1	2	yes	—
40	C₅₃×#494	C₂²	1	1	yes	characteristic
41	C₅₃×#487	C₂²	1	2	yes	—
42	C₈²×C₅₃	C₂²	1	1	yes	characteristic
43	C₂×Dic₂₁₂	—	2	2	no	—
44	C₂×#11719925	—	2	4	no	—
45	C₂×C₈×D₅₃	D₄	1	2	yes	—
46	C₂×C₈×D₅₃	D₄	1	2	yes	—
47	C₂×C₈×D₅₃	D₄	1	2	yes	—
48	#13133217	C₂³	1	2	yes	—
49	#13133217	C₂³	1	2	yes	—
50	#13133208	—	2	2	no	—
51	#13133208	—	2	2	no	—
52	#13133197	—	2	2	no	—
53	#13133197	—	2	2	no	—
54	#13133193	—	2	2	no	—
55	#13133193	—	2	2	no	—
56	Q₈×Dic₅₃	—	2	2	no	—
57	Q₈×Dic₅₃	—	2	2	no	—
58	#13133182	—	2	2	no	—
59	#13133182	—	2	2	no	—
60	#13133175	—	2	2	no	—
61	#13133175	—	2	2	no	—
62	C₄²×D₅₃	C₂³	1	1	yes	characteristic
63	#13133151	—	2	2	no	—
64	#13133150	—	2	4	no	—
65	#13133150	—	2	4	no	—
66	#13133150	—	2	4	no	—
67	#13133149	—	2	4	no	—
68	#13133142	—	2	4	no	—
69	#13133141	—	2	4	no	—
70	#13133140	—	2	4	no	—
71	C₄×#88322	C₂³	1	1	yes	characteristic
72	C₄×#88322	C₂³	1	1	yes	characteristic
73	C₄×#88322	C₂³	1	1	yes	characteristic
74	C₈×Dic₅₃	D₄	1	2	yes	—
75	C₈×Dic₅₃	D₄	1	2	yes	—
76	C₈×Dic₅₃	D₄	1	2	yes	—
77	C₂×Q₁₆×C₅₃	—	2	2	no	—
78	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
79	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
80	#107×C₅₃	—	2	2	no	—
81	#106×C₅₃	—	2	4	no	—
82	#103×C₅₃	—	2	4	no	—
83	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₂×Q₈×C₅₃	—	2	2	no	—
87	C₂×Q₈×C₅₃	—	2	2	no	—
88	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂×Dic₁₀₆	—	2	2	no	—
92	C₂×Dic₁₀₆	—	2	2	no	—
93	Q₁₆×C₅₃	—	4	4	no	—
94	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
97	C₄⋊C₄×C₅₃	—	2	2	no	—
98	C₄⋊C₄×C₅₃	—	2	2	no	—
99	C₄⋊C₄×C₅₃	—	2	2	no	—
100	C₄⋊C₄×C₅₃	—	2	2	no	—
101	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
102	#11719925	—	4	8	no	—
103	#11719921	—	4	2	no	—
104	#11719921	—	4	2	no	—
105	C₄⋊Dic₅₃	—	2	2	no	—
106	C₄⋊Dic₅₃	—	2	2	no	—
107	C₄⋊Dic₅₃	—	2	2	no	—
108	C₄⋊Dic₅₃	—	2	2	no	—
109	C₄⋊Dic₅₃	—	4	2	no	—
110	C₄⋊Dic₅₃	—	4	2	no	—
111	#11719919	—	4	2	no	—
112	#11719919	—	4	2	no	—
113	C₄×Dic₅₃	—	4	2	no	—
114	C₄×Dic₅₃	—	4	2	no	—
115	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
116	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
117	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
118	C₂×#88322	C₂×D₄	1	2	yes	—
119	C₂×#88322	C₂×D₄	1	2	yes	—
120	C₂×#88322	C₂×D₄	1	2	yes	—
121	Dic₂₁₂	—	4	4	no	—
122	C₈×D₅₃	—	2	4	no	—
123	C₈×D₅₃	—	2	4	no	—
124	C₈×D₅₃	Q₁₆	1	8	yes	complemented
125	C₂²×D₅₃	#127	1	1	yes	characteristic
126	Q₈×C₅₃	—	4	4	no	—
127	Q₈×C₅₃	—	4	4	no	—
128	C₂×C₄×C₅₃	—	4	2	no	—
129	C₂×C₄×C₅₃	—	4	2	no	—
130	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×Dic₅₃	—	4	2	no	—
134	C₂×Dic₅₃	—	4	2	no	—
135	C₂×Dic₅₃	#127	1	1	yes	characteristic
136	C₂×Dic₅₃	#127	1	1	yes	characteristic
137	C₂×Dic₅₃	#127	1	1	yes	characteristic
138	C₄×D₅₃	—	2	2	no	—
139	C₄×D₅₃	—	2	2	no	—
140	C₄×D₅₃	C₂×Q₁₆	1	4	yes	—
141	Dic₁₀₆	—	4	4	no	—
142	Dic₁₀₆	—	4	4	no	—
143	C₈×C₅₃	—	2	2	no	—
144	C₈×C₅₃	C₂×Q₁₆	1	4	yes	—
145	C₈×C₅₃	#135	1	4	yes	—
146	#88322	—	2	2	no	—
147	#88322	C₂×Q₁₆	1	4	yes	—
148	#88322	#135	1	4	yes	—
149	#19604	—	53	1	no	maximal, Sylow
150	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
151	C₂×D₅₃	—	2	1	no	—
152	C₂×D₅₃	—	2	1	no	—
153	C₂×D₅₃	#494	1	2	yes	—
154	C₄×C₅₃	—	8	2	no	—
155	C₄×C₅₃	—	8	2	no	—
156	C₄×C₅₃	C₂²×Q₁₆	1	2	yes	—
157	C₄×C₅₃	C₂×#135	1	2	yes	—
158	C₄×C₅₃	C₂×#135	1	2	yes	—
159	Dic₅₃	—	2	1	no	—
160	Dic₅₃	—	2	1	no	—
161	Dic₅₃	—	8	2	no	—
162	Dic₅₃	—	8	2	no	—
163	Dic₅₃	#494	1	2	yes	—
164	#3170	—	53	1	no	—
165	#3170	—	53	1	no	—
166	#3170	—	53	2	no	—
167	#3160	—	53	2	no	—
168	#1713	—	53	1	no	—
169	#1713	—	53	1	no	—
170	#1710	—	53	2	no	—
171	#1707	—	53	2	no	—
172	#1707	—	53	1	no	—
173	#1707	D₅₃	1	1	yes	characteristic, complemented
174	C₂×C₈²	—	53	1	no	—
175	C₂×C₅₃	C₂×#494	1	1	yes	characteristic
176	C₂×C₅₃	#3147	1	1	yes	characteristic
177	C₂×C₅₃	#3147	1	1	yes	characteristic
178	D₅₃	—	2	2	no	—
179	#533	—	53	2	no	—
180	#533	—	53	2	no	—
181	#500	—	53	2	no	—
182	#500	—	53	2	no	—
183	#500	—	53	2	no	—
184	#500	—	53	1	no	—
185	#500	C₂×D₅₃	1	1	yes	characteristic
186	#499	—	106	4	no	—
187	#498	—	53	2	no	—
188	#498	C₂×D₅₃	1	2	yes	—
189	#494	—	53	2	no	—
190	#494	—	53	1	no	—
191	#494	C₂×D₅₃	1	1	yes	characteristic
192	#487	—	53	2	no	—
193	#487	—	53	2	no	—
194	#487	—	53	2	no	—
195	#487	C₂×D₅₃	1	2	yes	—
196	#467	—	106	4	no	—
197	#467	—	106	2	no	—
198	#467	—	106	2	no	—
199	C₄×Q₁₆	—	106	4	no	—
200	C₄×Q₁₆	—	106	2	no	—
201	C₄×Q₁₆	—	106	2	no	—
202	#427	—	106	4	no	—
203	C₂×C₄×C₈	—	53	1	no	—
204	C₂×C₄×C₈	—	53	1	no	—
205	C₂×C₄×C₈	—	53	1	no	—
206	C₈²	—	53	1	no	—
207	C₈²	—	53	1	no	—
208	C₈²	—	53	1	no	—
209	C₈²	C₂×D₅₃	1	1	yes	characteristic
210	C₅₃	#19604	1	1	yes	characteristic, complemented, Sylow
211	C₂×Q₁₆	—	2	2	no	—
212	C₂×Q₁₆	—	106	4	no	—
213	C₂×Q₁₆	—	106	2	no	—
214	C₂²×C₈	—	53	2	no	—
215	C₂²×C₈	—	53	2	no	—
216	C₂²×C₈	—	53	2	no	—
217	C₄⋊Q₈	—	53	2	no	—
218	C₄⋊Q₈	—	53	2	no	—
219	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
220	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
221	#125	—	106	2	no	—
222	#125	—	106	2	no	—
223	C₂²⋊Q₈	—	106	2	no	—
224	C₂²⋊Q₈	—	106	2	no	—
225	C₂²⋊Q₈	—	106	2	no	—
226	C₂²⋊Q₈	—	106	2	no	—
227	C₄×Q₈	—	106	2	no	—
228	C₄×Q₈	—	106	2	no	—
229	C₄×Q₈	—	106	2	no	—
230	C₄×Q₈	—	106	2	no	—
231	#117	—	106	2	no	—
232	#117	—	106	2	no	—
233	C₂×C₄²	—	53	1	no	—
234	#107	—	2	2	no	—
235	#107	—	106	4	no	—
236	#107	—	106	4	no	—
237	#107	—	106	4	no	—
238	#107	—	106	2	no	—
239	#106	—	2	4	no	—
240	#106	—	106	4	no	—
241	#103	—	2	4	no	—
242	#103	—	106	4	no	—
243	#103	—	106	4	no	—
244	#103	—	106	4	no	—
245	C₄×C₈	—	53	2	no	—
246	C₄×C₈	—	53	2	no	—
247	C₄×C₈	—	53	2	no	—
248	C₄×C₈	—	53	1	no	—
249	C₄×C₈	—	53	1	no	—
250	C₄×C₈	—	53	1	no	—
251	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
252	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
253	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	—	106	2	no	—
257	C₂×Q₈	—	106	2	no	—
258	C₂²×C₄	—	53	1	no	—
259	C₂²×C₄	—	53	1	no	—
260	C₂²×C₄	—	53	1	no	—
261	Q₁₆	—	4	4	no	—
262	Q₁₆	—	212	8	no	—
263	Q₁₆	—	212	4	no	—
264	C₂×C₈	—	53	2	no	—
265	C₂×C₈	—	53	2	no	—
266	C₂×C₈	—	53	8	no	—
267	C₂×C₈	—	53	2	no	—
268	C₂×C₈	—	106	4	no	—
269	C₂×C₈	—	106	4	no	—
270	C₂×C₈	D₄×D₅₃	1	2	yes	—
271	C₂×C₈	D₄×D₅₃	1	2	yes	—
272	C₂×C₈	D₄×D₅₃	1	2	yes	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	106	2	no	—
278	C₄⋊C₄	—	106	2	no	—
279	C₄⋊C₄	—	106	2	no	—
280	C₄⋊C₄	—	106	2	no	—
281	C₄⋊C₄	—	212	2	no	—
282	C₄⋊C₄	—	212	2	no	—
283	C₄⋊C₄	—	212	2	no	—
284	C₄⋊C₄	—	212	2	no	—
285	C₂²⋊C₄	—	212	2	no	—
286	C₂²⋊C₄	—	212	2	no	—
287	C₄²	—	53	1	no	—
288	C₄²	—	53	1	no	—
289	C₄²	—	53	1	no	—
290	C₄²	—	212	2	no	—
291	C₄²	—	212	2	no	—
292	C₄²	C₂³×D₅₃	1	1	yes	characteristic
293	C₂³	—	53	1	no	—
294	Q₈	—	4	4	no	—
295	Q₈	—	4	4	no	—
296	Q₈	—	212	4	no	—
297	Q₈	—	212	4	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	53	1	no	—
301	C₂×C₄	—	53	4	no	—
302	C₂×C₄	—	53	1	no	—
303	C₂×C₄	—	53	1	no	—
304	C₂×C₄	—	106	2	no	—
305	C₂×C₄	—	106	2	no	—
306	C₂×C₄	—	212	2	no	—
307	C₂×C₄	—	212	2	no	—
308	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
310	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	53	4	no	—
313	C₈	—	53	4	no	—
314	C₈	—	106	2	no	—
315	C₈	Q₁₆×D₅₃	1	4	yes	—
316	C₈	#13133238	1	4	yes	—
317	C₂²	—	53	2	no	—
318	C₂²	—	106	1	no	—
319	C₂²	—	106	1	no	—
320	C₂²	#127×D₅₃	1	1	yes	characteristic
321	C₄	—	8	2	no	—
322	C₄	—	8	2	no	—
323	C₄	—	53	2	no	—
324	C₄	—	106	1	no	—
325	C₄	—	106	1	no	—
326	C₄	—	424	2	no	—
327	C₄	—	424	2	no	—
328	C₄	C₂×Q₁₆×D₅₃	1	2	yes	—
329	C₄	#135×D₅₃	1	2	yes	—
330	C₄	C₂×#13133238	1	2	yes	—
331	C₂	—	106	2	no	—
332	C₂	#494×D₅₃	1	1	yes	characteristic
333	C₂	#20285138	1	1	yes	characteristic
334	C₂	#20285140	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1707	C₂
socle	C₂²×C₅₃	C₂×#127
prime radical	C₂²×D₅₃	#127
hypercenter	#1707	D₅₃
hypocenter	C₅₃	#19604
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1707

Complements

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

N	H ≅ G/N	type
5. #20274332	C₂	semidirect
6. #20274331	C₂	semidirect
7. #20274323	C₂	semidirect
8. #20274312	C₂	semidirect
9. #20274310	C₂	semidirect
11. C₅₃×#1707	C₂	semidirect
124. C₈×D₅₃	Q₁₆	semidirect
173. #1707	D₅₃	semidirect
210. C₅₃	#19604	semidirect

Maximal subgroups

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

subgroup	index
#20285302	2
#20285300	2
#20285299	2
#20285224	2
#20274332	2
#20274331	2
#20274323	2
#20274312	2
#20274310	2
C₈²×D₅₃	2
C₅₃×#1707	2
#19604	53