Subgroups — #27539618

Order 13568.

482 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285228	C₂	1	1	yes	maximal, characteristic
2	#499×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#20277000	C₂	1	2	yes	maximal, complemented
4	#20276999	C₂	1	2	yes	maximal, complemented
5	#20276994	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276991	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276991	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276502	C₂	1	2	yes	maximal
9	#20276500	C₂	1	2	yes	maximal
10	C₄×#13133156	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#2253	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811179	—	2	1	no	—
13	#125×D₅₃	—	2	1	no	—
14	C₂×#13133156	C₂²	1	1	yes	characteristic
15	C₂×#13133156	C₂²	1	1	yes	characteristic
16	#16811027	C₂²	1	2	yes	—
17	#16811025	C₂²	1	1	yes	characteristic
18	#16811025	C₂²	1	1	yes	characteristic
19	#16811024	C₂²	1	2	yes	—
20	#16811023	—	2	1	no	—
21	#16811022	—	2	1	no	—
22	#16811022	—	2	1	no	—
23	#16811020	—	2	1	no	—
24	#16810660	C₂²	1	2	yes	—
25	#16810658	—	2	1	no	—
26	#107×D₅₃	C₂²	1	2	yes	—
27	#106×D₅₃	—	2	1	no	—
28	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
29	#16810451	C₂²	1	2	yes	—
30	#16810450	C₂²	1	2	yes	—
31	#16810450	C₂²	1	2	yes	—
32	#16810449	C₂²	1	2	yes	—
33	#16810448	C₂²	1	2	yes	—
34	#16810448	C₂²	1	2	yes	—
35	#16810398	C₂²	1	1	yes	characteristic
36	#16810398	C₂²	1	1	yes	characteristic
37	C₄×#11719908	C₂²	1	1	yes	characteristic
38	C₅₃×#499	C₂²	1	1	yes	characteristic
39	C₅₃×#499	C₂²	1	1	yes	characteristic
40	C₅₃×#367	C₂²	1	2	yes	—
41	C₅₃×#366	C₂²	1	2	yes	—
42	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
43	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
44	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
45	#13133216	—	2	2	no	—
46	#13133209	—	2	1	no	—
47	#13133209	—	2	1	no	—
48	#13133208	—	2	2	no	—
49	#13133207	—	4	2	no	—
50	#13133195	—	4	2	no	—
51	#13133182	—	4	2	no	—
52	#13133175	—	2	2	no	—
53	C₄⋊C₄×D₅₃	—	2	2	no	—
54	C₄⋊C₄×D₅₃	—	4	2	no	—
55	C₄²×D₅₃	D₄	1	1	yes	characteristic
56	#13133156	Q₈	1	4	yes	complemented
57	#13133156	Q₈	1	4	yes	complemented
58	C₂×#11719908	C₂³	1	1	yes	characteristic
59	C₂×#11719908	C₂³	1	1	yes	characteristic
60	#13133151	C₂³	1	2	yes	—
61	#13133151	C₂³	1	2	yes	—
62	#13133150	—	2	2	no	—
63	#13133150	—	2	2	no	—
64	#13133149	—	2	1	no	—
65	#13133149	—	2	1	no	—
66	#13133148	—	2	2	no	—
67	#13133148	—	2	2	no	—
68	C₄×#88322	D₄	1	1	yes	characteristic
69	C₈×Dic₅₃	D₄	1	1	yes	characteristic
70	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
71	#125×C₅₃	—	2	1	no	—
72	#125×C₅₃	—	2	1	no	—
73	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
75	#107×C₅₃	C₂³	1	2	yes	—
76	#107×C₅₃	C₂³	1	2	yes	—
77	#106×C₅₃	—	2	1	no	—
78	#106×C₅₃	—	2	1	no	—
79	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
80	C₂×C₄×D₅₃	—	4	2	no	—
81	C₂×C₄×D₅₃	D₈	1	2	yes	—
82	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂×Dic₁₀₆	—	4	2	no	—
84	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
86	C₄⋊C₄×C₅₃	—	2	2	no	—
87	C₄⋊C₄×C₅₃	—	2	2	no	—
88	C₄⋊C₄×C₅₃	—	4	2	no	—
89	C₄⋊C₄×C₅₃	—	4	2	no	—
90	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
91	#11719921	—	4	2	no	—
92	C₄⋊Dic₅₃	—	2	2	no	—
93	C₄⋊Dic₅₃	—	2	2	no	—
94	C₄⋊Dic₅₃	—	4	2	no	—
95	#11719919	—	4	2	no	—
96	#11719919	—	4	2	no	—
97	#11719919	—	4	2	no	—
98	#11719919	—	4	2	no	—
99	#11719919	—	4	2	no	—
100	C₄×Dic₅₃	—	4	2	no	—
101	C₄×Dic₅₃	D₈	1	2	yes	—
102	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
103	C₂×#88322	C₂×D₄	1	1	yes	characteristic
104	C₂×#88322	C₂×D₄	1	1	yes	characteristic
105	C₈×D₅₃	Q₁₆	1	4	yes	—
106	C₈×D₅₃	C₂×Q₈	1	2	yes	—
107	C₈×D₅₃	C₂×Q₈	1	2	yes	—
108	C₁₆×C₅₃	C₂×Q₈	1	2	yes	—
109	C₁₆×C₅₃	C₂×Q₈	1	2	yes	—
110	#11719908	C₂×Q₈	1	2	yes	—
111	#11719908	C₂×Q₈	1	2	yes	—
112	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
113	C₂×C₄×C₅₃	—	4	2	no	—
114	C₂×C₄×C₅₃	—	4	2	no	—
115	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×Dic₅₃	—	4	2	no	—
118	C₂×Dic₅₃	—	4	2	no	—
119	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
120	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
121	C₄×D₅₃	—	4	2	no	—
122	C₄×D₅₃	—	8	4	no	—
123	C₄×D₅₃	C₄⋊Q₈	1	2	yes	—
124	C₄×D₅₃	C₂×Q₁₆	1	2	yes	—
125	Dic₁₀₆	—	8	4	no	—
126	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
127	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
128	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
129	#88322	C₄⋊Q₈	1	2	yes	—
130	#88322	C₂×Q₁₆	1	2	yes	—
131	#33732	—	53	1	no	maximal, Sylow
132	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
133	C₂×D₅₃	—	2	2	no	—
134	C₂×D₅₃	#500	1	2	yes	—
135	C₄×C₅₃	—	2	2	no	—
136	C₄×C₅₃	—	8	2	no	—
137	C₄×C₅₃	—	8	2	no	—
138	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
139	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
140	Dic₅₃	—	2	2	no	—
141	Dic₅₃	—	8	2	no	—
142	Dic₅₃	—	8	2	no	—
143	Dic₅₃	#500	1	2	yes	—
144	#3160	—	53	1	no	—
145	C₂×#499	—	53	1	no	—
146	#2255	—	53	2	no	—
147	#2255	—	53	2	no	—
148	#2253	—	53	1	no	—
149	#2253	—	53	1	no	—
150	#2253	—	53	1	no	—
151	#2253	D₅₃	1	1	yes	characteristic, complemented
152	#2159	—	53	2	no	—
153	#2159	—	53	2	no	—
154	C₄×M₃₂	—	53	1	no	—
155	C₂×C₅₃	C₂×#500	1	1	yes	characteristic
156	C₂×C₅₃	#3414	1	2	yes	—
157	D₅₃	—	2	2	no	—
158	#533	—	106	1	no	—
159	C₂×#125	—	106	1	no	—
160	C₂×M₃₂	—	53	1	no	—
161	C₂×M₃₂	—	53	1	no	—
162	#500	—	53	2	no	—
163	#499	—	53	2	no	—
164	#499	—	53	1	no	—
165	#499	—	53	1	no	—
166	#499	—	106	1	no	—
167	#499	—	106	1	no	—
168	#499	—	106	1	no	—
169	#499	C₂×D₅₃	1	1	yes	characteristic
170	#499	C₂×D₅₃	1	1	yes	characteristic
171	#498	—	106	1	no	—
172	#427	—	53	2	no	—
173	#427	—	106	1	no	—
174	C₂×#107	—	53	2	no	—
175	C₂×#106	—	106	1	no	—
176	C₂×C₄×C₈	—	53	1	no	—
177	#367	—	53	2	no	—
178	#367	—	53	2	no	—
179	#367	—	53	2	no	—
180	#367	C₂×D₅₃	1	2	yes	—
181	#366	—	53	2	no	—
182	#366	—	53	2	no	—
183	#366	—	53	2	no	—
184	#366	C₂×D₅₃	1	2	yes	—
185	#346	—	53	1	no	—
186	#346	—	53	1	no	—
187	C₄×C₁₆	—	53	1	no	—
188	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
189	C₅₃	#33732	1	1	yes	characteristic, complemented, Sylow
190	C₂²×C₈	—	53	1	no	—
191	C₂²×C₈	—	53	1	no	—
192	C₄⋊Q₈	—	106	2	no	—
193	#125	—	2	1	no	—
194	#125	—	2	1	no	—
195	#125	—	106	2	no	—
196	#125	—	106	1	no	—
197	#125	—	106	1	no	—
198	#125	—	212	2	no	—
199	C₂²⋊Q₈	—	212	2	no	—
200	C₄×Q₈	—	212	2	no	—
201	#117	—	106	2	no	—
202	C₂×C₄⋊C₄	—	106	2	no	—
203	C₂×C₄⋊C₄	—	212	2	no	—
204	C₂×C₄²	—	53	1	no	—
205	M₃₂	—	53	4	no	—
206	M₃₂	—	53	4	no	—
207	C₂×C₁₆	—	53	1	no	—
208	C₂×C₁₆	—	53	1	no	—
209	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
210	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
211	#107	—	53	2	no	—
212	#107	—	53	2	no	—
213	#107	—	106	2	no	—
214	#107	—	106	2	no	—
215	#107	C₂²×D₅₃	1	2	yes	—
216	#107	C₂²×D₅₃	1	2	yes	—
217	#106	—	2	1	no	—
218	#106	—	2	1	no	—
219	#106	—	106	2	no	—
220	#106	—	106	2	no	—
221	#106	—	106	1	no	—
222	#106	—	106	1	no	—
223	C₄×C₈	—	53	1	no	—
224	C₄×C₈	—	53	1	no	—
225	C₄×C₈	—	53	1	no	—
226	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
227	C₂×Q₈	—	212	2	no	—
228	C₂²×C₄	—	53	1	no	—
229	C₂²×C₄	—	53	2	no	—
230	C₂²×C₄	—	212	2	no	—
231	C₂×C₈	—	53	1	no	—
232	C₂×C₈	—	53	1	no	—
233	C₂×C₈	—	53	4	no	—
234	C₂×C₈	—	53	2	no	—
235	C₂×C₈	—	53	2	no	—
236	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
237	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	106	2	no	—
243	C₄⋊C₄	—	106	2	no	—
244	C₄⋊C₄	—	212	2	no	—
245	C₄⋊C₄	—	212	2	no	—
246	C₄⋊C₄	—	212	2	no	—
247	C₄⋊C₄	—	212	2	no	—
248	C₄⋊C₄	—	212	2	no	—
249	C₄⋊C₄	—	212	2	no	—
250	C₂²⋊C₄	—	212	2	no	—
251	C₄²	—	53	2	no	—
252	C₄²	—	53	1	no	—
253	C₄²	—	212	2	no	—
254	C₄²	D₄×D₅₃	1	1	yes	characteristic
255	C₁₆	—	53	2	no	—
256	C₁₆	—	53	2	no	—
257	C₁₆	Q₈×D₅₃	1	2	yes	—
258	C₁₆	Q₈×D₅₃	1	2	yes	—
259	C₂³	—	53	1	no	—
260	Q₈	—	424	4	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	53	1	no	—
264	C₂×C₄	—	53	2	no	—
265	C₂×C₄	—	53	2	no	—
266	C₂×C₄	—	53	2	no	—
267	C₂×C₄	—	424	4	no	—
268	C₂×C₄	—	212	2	no	—
269	C₂×C₄	—	212	2	no	—
270	C₂×C₄	—	212	2	no	—
271	C₂×C₄	D₈×D₅₃	1	2	yes	—
272	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
273	C₈	—	53	2	no	—
274	C₈	—	53	2	no	—
275	C₈	Q₁₆×D₅₃	1	2	yes	—
276	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
277	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
278	C₂²	—	53	2	no	—
279	C₂²	—	106	2	no	—
280	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	8	2	no	—
283	C₄	—	8	2	no	—
284	C₄	—	53	2	no	—
285	C₄	—	106	2	no	—
286	C₄	—	424	2	no	—
287	C₄	—	424	2	no	—
288	C₄	C₄⋊Q₈×D₅₃	1	1	yes	characteristic
289	C₄	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
290	C₂	—	106	2	no	—
291	C₂	#500×D₅₃	1	1	yes	characteristic
292	C₂	#20287848	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2253	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂²×D₅₃	C₂×D₈
hypercenter	#2253	D₅₃
hypocenter	C₅₃	#33732
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2253

Complements

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

N	H ≅ G/N	type
3. #20277000	C₂	semidirect
4. #20276999	C₂	semidirect
5. #20276994	C₂	semidirect
6. #20276991	C₂	semidirect
7. #20276991	C₂	semidirect
11. C₅₃×#2253	C₂	semidirect
56. #13133156	Q₈	semidirect
57. #13133156	Q₈	semidirect
151. #2253	D₅₃	semidirect
189. C₅₃	#33732	semidirect

Maximal subgroups

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

subgroup	index
#20285228	2
#499×D₅₃	2
#20277000	2
#20276999	2
#20276994	2
#20276991	2
#20276502	2
#20276500	2
C₄×#13133156	2
C₅₃×#2253	2
#33732	53