Subgroups — #27282758

Order 13568.

370 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275474	C₂	1	2	yes	maximal
2	#20274872	C₂	1	1	yes	maximal, characteristic
3	#20274582	C₂	1	1	yes	maximal, characteristic
4	#20272599	C₂	1	2	yes	maximal
5	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133151	—	2	2	no	—
7	C₂×#13133149	—	2	1	no	—
8	C₂×#13133149	—	2	1	no	—
9	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
10	C₂×#13133127	—	2	1	no	—
11	#16810493	C₂²	1	2	yes	—
12	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
13	#16810386	—	2	2	no	—
14	#16810377	—	2	1	no	—
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#335	C₄	1	4	yes	complemented
18	C₂²×#88322	—	2	1	no	—
19	C₂×C₄⋊Dic₅₃	—	2	2	no	—
20	C₂×C₄⋊Dic₅₃	—	2	2	no	—
21	C₂×C₄⋊Dic₅₃	—	2	2	no	—
22	C₂×C₄⋊Dic₅₃	D₄	1	2	yes	—
23	C₂×C₄×Dic₅₃	—	2	1	no	—
24	#13133151	—	4	4	no	—
25	#13133149	—	4	2	no	—
26	#13133149	—	4	2	no	—
27	C₄⋊#88322	—	2	1	no	—
28	C₄⋊#88322	Q₈	1	2	yes	—
29	#13133127	—	4	2	no	—
30	#13133124	—	2	1	no	—
31	C₂²×C₈×C₅₃	—	2	2	no	—
32	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₅₃	—	2	2	no	—
36	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
37	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₅₃	Q₈	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
42	C₂²×Dic₅₃	—	2	2	no	—
43	C₂²×Dic₅₃	—	4	2	no	—
44	C₂×C₈×C₅₃	—	2	1	no	—
45	C₂×C₈×C₅₃	—	2	1	no	—
46	C₂×C₈×C₅₃	—	2	4	no	—
47	C₂×C₈×C₅₃	—	4	2	no	—
48	C₂×C₈×C₅₃	—	4	2	no	—
49	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
51	C₂×C₈×C₅₃	D₈	1	2	yes	—
52	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
53	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
55	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₅₃	#41	1	1	yes	characteristic
57	C₄⋊Dic₅₃	—	2	2	no	—
58	C₄⋊Dic₅₃	—	2	2	no	—
59	C₄⋊Dic₅₃	—	4	2	no	—
60	C₄⋊Dic₅₃	—	4	2	no	—
61	C₄⋊Dic₅₃	—	4	2	no	—
62	C₄⋊Dic₅₃	—	4	2	no	—
63	C₄⋊Dic₅₃	—	8	2	no	—
64	C₄×Dic₅₃	—	4	1	no	—
65	C₄×Dic₅₃	—	4	1	no	—
66	C₂×#88322	—	2	2	no	—
67	C₂×#88322	—	4	1	no	—
68	C₂×#88322	—	4	1	no	—
69	C₂³×C₅₃	#121	1	1	yes	characteristic
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#137	1	2	yes	—
82	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
83	C₂×Dic₅₃	—	4	2	no	—
84	C₂×Dic₅₃	—	4	2	no	—
85	C₂×Dic₅₃	—	4	2	no	—
86	C₂×Dic₅₃	—	8	2	no	—
87	C₂×Dic₅₃	—	8	2	no	—
88	C₂×Dic₅₃	—	8	2	no	—
89	C₈×C₅₃	—	2	2	no	—
90	C₈×C₅₃	—	4	4	no	—
91	C₈×C₅₃	#107	1	4	yes	complemented
92	#88322	—	4	2	no	—
93	#9617	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	#389	1	1	yes	characteristic
95	C₂²×C₅₃	#459	1	1	yes	characteristic
96	C₂²×C₅₃	#462	1	1	yes	characteristic
97	C₂²×C₅₃	#468	1	2	yes	—
98	C₂²×C₅₃	#471	1	2	yes	—
99	C₄×C₅₃	—	2	1	no	—
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
102	C₄×C₅₃	#428	1	1	yes	characteristic
103	C₄×C₅₃	#441	1	1	yes	characteristic
104	C₄×C₅₃	#442	1	1	yes	characteristic
105	C₄×C₅₃	#474	1	1	yes	characteristic
106	C₄×C₅₃	#477	1	1	yes	characteristic
107	C₄×C₅₃	#501	1	2	yes	—
108	Dic₅₃	—	8	2	no	—
109	Dic₅₃	—	16	2	no	—
110	#1947	—	53	2	no	—
111	#1844	—	53	1	no	—
112	#1770	—	53	1	no	—
113	C₂×#335	D₅₃	1	1	yes	characteristic
114	#1296	—	53	2	no	—
115	C₂×C₅₃	#1685	1	2	yes	—
116	C₂×C₅₃	#1692	1	2	yes	—
117	C₂×C₅₃	#1943	1	1	yes	characteristic
118	C₂×C₅₃	#1947	1	1	yes	characteristic
119	C₂×C₅₃	#1951	1	1	yes	characteristic
120	C₂×#107	—	106	2	no	—
121	C₂×#106	—	106	1	no	—
122	C₂×#106	—	106	1	no	—
123	C₂×C₄⋊C₈	—	53	1	no	—
124	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
125	C₂×#97	—	106	1	no	—
126	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
127	#384	—	53	2	no	—
128	C₄×C₄⋊C₄	—	53	1	no	—
129	#340	—	106	2	no	—
130	#336	—	106	1	no	—
131	#335	Dic₅₃	1	4	yes	complemented
132	C₅₃	#9617	1	1	yes	characteristic, complemented, Sylow
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	—	106	1	no	—
135	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
136	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
137	C₂×C₄⋊C₄	—	53	2	no	—
138	C₂×C₄⋊C₄	—	106	2	no	—
139	C₂×C₄⋊C₄	—	106	2	no	—
140	C₂×C₄⋊C₄	—	106	2	no	—
141	C₂×C₄²	—	106	1	no	—
142	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
143	#107	—	212	4	no	—
144	#106	—	212	2	no	—
145	#106	—	212	2	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	53	2	no	—
148	C₄⋊C₈	—	106	1	no	—
149	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
150	#97	—	212	2	no	—
151	C₄×C₈	Dic₁₀₆	1	2	yes	—
152	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
153	#95	—	106	1	no	—
154	C₂²×C₄	—	106	2	no	—
155	C₂²×C₄	—	212	2	no	—
156	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
157	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
158	C₂²×C₄	#11719949	1	1	yes	characteristic
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	—	106	2	no	—
165	C₂×C₈	—	212	1	no	—
166	C₂×C₈	—	212	1	no	—
167	C₂×C₈	D₄₂₄	1	2	yes	—
168	C₂×C₈	Dic₂₁₂	1	2	yes	—
169	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
170	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
171	C₄⋊C₄	—	106	2	no	—
172	C₄⋊C₄	—	106	2	no	—
173	C₄⋊C₄	—	424	2	no	—
174	C₄⋊C₄	—	212	2	no	—
175	C₄⋊C₄	—	212	2	no	—
176	C₄⋊C₄	—	212	2	no	—
177	C₄⋊C₄	—	212	2	no	—
178	C₄²	—	212	1	no	—
179	C₄²	—	212	1	no	—
180	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
181	C₄²	#11719947	1	1	yes	characteristic
182	C₄²	Q₈×D₅₃	1	1	yes	characteristic
183	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
184	C₂³	#13133192	1	1	yes	characteristic
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	424	2	no	—
187	C₂×C₄	—	424	2	no	—
188	C₂×C₄	—	424	2	no	—
189	C₂×C₄	—	212	2	no	—
190	C₂×C₄	—	212	2	no	—
191	C₂×C₄	—	212	2	no	—
192	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
193	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
194	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
195	C₂×C₄	#13133197	1	1	yes	characteristic
196	C₂×C₄	#13133208	1	1	yes	characteristic
197	C₂×C₄	#13133216	1	1	yes	characteristic
198	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
199	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
200	C₂×C₄	#13133241	1	2	yes	—
201	C₂×C₄	#13133245	1	1	yes	characteristic
202	C₂×C₄	#13133249	1	2	yes	—
203	C₂×C₄	#13133250	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	4	4	no	—
206	C₈	—	212	2	no	—
207	C₈	#13133151	1	4	yes	complemented
208	C₂²	#16810522	1	1	yes	characteristic
209	C₂²	#16810819	1	1	yes	characteristic
210	C₂²	#16810834	1	1	yes	characteristic
211	C₂²	#16810870	1	2	yes	—
212	C₂²	#16810887	1	2	yes	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	848	2	no	—
216	C₄	—	424	2	no	—
217	C₄	C₂×#13133151	1	1	yes	characteristic
218	C₄	#16810665	1	1	yes	characteristic
219	C₄	#16810730	1	1	yes	characteristic
220	C₄	#16810735	1	1	yes	characteristic
221	C₄	#16810904	1	1	yes	characteristic
222	C₄	#16810919	1	1	yes	characteristic
223	C₄	#16811031	1	2	yes	—
224	C₂	#20274181	1	2	yes	—
225	C₂	#20274222	1	2	yes	—
226	C₂	#20275454	1	1	yes	characteristic
227	C₂	#20275474	1	1	yes	characteristic
228	C₂	#20275496	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133192
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#335	C₂
socle, prime radical	C₂³×C₅₃	#121
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#335	D₅₃
hypocenter	C₅₃	#9617
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
17. C₅₃×#335	C₄	semidirect
91. C₈×C₅₃	#107	semidirect
131. #335	Dic₅₃	semidirect
132. C₅₃	#9617	semidirect
207. C₈	#13133151	semidirect

Maximal subgroups

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

subgroup	index
#20275474	2
#20274872	2
#20274582	2
#20272599	2
C₂×C₅₃×#335	2
#9617	53