Subgroups — #27271278

Order 13568.

262 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274875	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133151	—	2	1	no	—
5	C₂×#13133151	C₄	1	2	yes	—
6	C₂×C₄⋊#88322	—	2	1	no	—
7	#16810493	—	2	1	no	—
8	#16810376	—	2	1	no	—
9	#16810376	C₄	1	2	yes	complemented
10	C₂²×C₁₆×C₅₃	—	2	1	no	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×C₄⋊C₁₆	—	2	1	no	—
13	C₅₃×C₄⋊C₁₆	C₄	1	2	yes	complemented
14	C₂²×#88322	—	4	1	no	—
15	C₂×C₄⋊Dic₅₃	—	2	2	no	—
16	C₂×C₄⋊Dic₅₃	—	4	1	no	—
17	C₂×C₄⋊Dic₅₃	—	4	1	no	—
18	#13133151	—	4	1	no	—
19	#13133151	—	4	1	no	—
20	#13133151	—	4	2	no	—
21	C₄⋊#88322	—	2	2	no	—
22	C₄⋊#88322	—	4	1	no	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₃	—	2	2	no	—
27	C₂×C₁₆×C₅₃	—	4	1	no	—
28	C₂×C₁₆×C₅₃	—	4	1	no	—
29	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
36	C₂²×Dic₅₃	—	4	2	no	—
37	C₂×C₈×C₅₃	—	2	1	no	—
38	C₂×C₈×C₅₃	—	2	1	no	—
39	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
44	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
45	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
47	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
50	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
51	C₄⋊Dic₅₃	—	4	2	no	—
52	C₄⋊Dic₅₃	—	4	2	no	—
53	C₄⋊Dic₅₃	—	8	2	no	—
54	C₄⋊Dic₅₃	—	8	1	no	—
55	C₄⋊Dic₅₃	—	8	1	no	—
56	C₂×#88322	—	4	2	no	—
57	C₂×#88322	—	8	1	no	—
58	C₂×#88322	—	8	1	no	—
59	C₁₆×C₅₃	—	4	2	no	—
60	C₂³×C₅₃	#102	1	1	yes	characteristic
61	C₂×C₄×C₅₃	—	2	1	no	—
62	C₂×C₄×C₅₃	—	2	1	no	—
63	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
68	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
70	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
71	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
73	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
74	C₂×C₄×C₅₃	SD₃₂	1	1	yes	characteristic
75	C₂×C₄×C₅₃	SD₃₂	1	1	yes	characteristic
76	C₂×C₄×C₅₃	Q₃₂	1	1	yes	characteristic
77	C₂×Dic₅₃	—	8	2	no	—
78	C₂×Dic₅₃	—	8	2	no	—
79	C₂×Dic₅₃	—	8	2	no	—
80	C₈×C₅₃	—	2	1	no	—
81	C₈×C₅₃	—	2	1	no	—
82	C₈×C₅₃	#102	1	1	yes	characteristic
83	C₈×C₅₃	#103	1	1	yes	characteristic
84	C₈×C₅₃	#106	1	1	yes	characteristic
85	C₈×C₅₃	#107	1	1	yes	characteristic
86	#88322	—	8	2	no	—
87	#7274	—	53	1	no	maximal, Sylow
88	C₂²×C₅₃	#331	1	1	yes	characteristic
89	C₂²×C₅₃	#332	1	1	yes	characteristic
90	C₂²×C₅₃	#340	1	1	yes	characteristic
91	C₂²×C₅₃	#357	1	1	yes	characteristic
92	C₂²×C₅₃	#358	1	1	yes	characteristic
93	C₂²×C₅₃	#361	1	1	yes	characteristic
94	C₂²×C₅₃	#362	1	1	yes	characteristic
95	C₄×C₅₃	—	2	1	no	—
96	C₄×C₅₃	—	2	1	no	—
97	C₄×C₅₃	#332	1	1	yes	characteristic
98	C₄×C₅₃	#333	1	1	yes	characteristic
99	C₄×C₅₃	#340	1	1	yes	characteristic
100	C₄×C₅₃	#343	1	1	yes	characteristic
101	C₄×C₅₃	#357	1	1	yes	characteristic
102	C₄×C₅₃	#358	1	1	yes	characteristic
103	C₄×C₅₃	#366	1	1	yes	characteristic
104	C₄×C₅₃	#367	1	1	yes	characteristic
105	Dic₅₃	—	16	2	no	—
106	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic
107	#1845	—	53	1	no	—
108	C₂×#335	—	53	1	no	—
109	C₂×C₅₃	#1296	1	1	yes	characteristic
110	C₂×C₅₃	#1361	1	1	yes	characteristic
111	C₂×C₅₃	#1362	1	1	yes	characteristic
112	C₂×C₅₃	#1364	1	1	yes	characteristic
113	C₂×C₅₃	#1365	1	1	yes	characteristic
114	C₂×C₅₃	#1380	1	1	yes	characteristic
115	C₂×C₅₃	#1387	1	1	yes	characteristic
116	C₂²×C₁₆	—	2	1	no	—
117	C₂×#107	—	53	2	no	—
118	C₂×#107	—	106	1	no	—
119	C₂×C₄⋊C₈	—	106	1	no	—
120	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
121	#384	—	106	1	no	—
122	C₄⋊C₁₆	—	2	1	no	—
123	C₄⋊C₁₆	Dic₅₃	1	2	yes	complemented
124	#335	—	53	2	no	—
125	#335	—	106	1	no	—
126	C₅₃	#7274	1	1	yes	characteristic, complemented, Sylow
127	C₂²×C₈	—	212	1	no	—
128	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
129	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
130	C₂×C₄⋊C₄	—	106	2	no	—
131	C₂×C₄⋊C₄	—	212	1	no	—
132	C₂×C₄⋊C₄	—	212	1	no	—
133	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	4	1	no	—
136	C₂×C₁₆	—	4	1	no	—
137	#107	—	212	1	no	—
138	#107	—	212	1	no	—
139	#107	—	212	2	no	—
140	C₄⋊C₈	—	106	2	no	—
141	C₄⋊C₈	—	212	1	no	—
142	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
143	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
144	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
145	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
146	C₂²×C₄	—	212	2	no	—
147	C₂²×C₄	#11719913	1	1	yes	characteristic
148	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
149	C₂²×C₄	#11719921	1	1	yes	characteristic
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	424	1	no	—
153	C₂×C₈	—	424	1	no	—
154	C₂×C₈	—	212	2	no	—
155	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
156	C₂×C₈	#11719919	1	1	yes	characteristic
157	C₂×C₈	#11719919	1	1	yes	characteristic
158	C₂×C₈	#11719922	1	1	yes	characteristic
159	C₂×C₈	#11719923	1	1	yes	characteristic
160	C₂×C₈	#11719924	1	1	yes	characteristic
161	C₂×C₈	#11719925	1	1	yes	characteristic
162	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
163	C₄⋊C₄	—	424	2	no	—
164	C₄⋊C₄	—	424	1	no	—
165	C₄⋊C₄	—	424	1	no	—
166	C₄⋊C₄	—	212	2	no	—
167	C₄⋊C₄	—	212	2	no	—
168	C₄²	#11719913	1	1	yes	characteristic
169	C₄²	Dic₂₁₂	1	1	yes	characteristic
170	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
171	C₄²	#11719921	1	1	yes	characteristic
172	C₁₆	—	4	2	no	—
173	C₂³	#13133139	1	1	yes	characteristic
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	424	2	no	—
177	C₂×C₄	—	424	2	no	—
178	C₂×C₄	—	424	2	no	—
179	C₂×C₄	#13133124	1	1	yes	characteristic
180	C₂×C₄	#13133133	1	1	yes	characteristic
181	C₂×C₄	#13133135	1	1	yes	characteristic
182	C₂×C₄	#13133137	1	1	yes	characteristic
183	C₂×C₄	#13133139	1	1	yes	characteristic
184	C₂×C₄	#13133140	1	1	yes	characteristic
185	C₂×C₄	#13133142	1	1	yes	characteristic
186	C₂×C₄	#13133142	1	1	yes	characteristic
187	C₂×C₄	#13133149	1	1	yes	characteristic
188	C₂×C₄	#13133151	1	1	yes	characteristic
189	C₂×C₄	D₈₄₈	1	1	yes	characteristic
190	C₂×C₄	#13133162	1	1	yes	characteristic
191	C₂×C₄	#13133162	1	1	yes	characteristic
192	C₂×C₄	Dic₄₂₄	1	1	yes	characteristic
193	C₈	—	2	1	no	—
194	C₈	—	2	1	no	—
195	C₈	—	424	2	no	—
196	C₈	#13133138	1	1	yes	characteristic
197	C₈	#13133141	1	1	yes	characteristic
198	C₈	#13133148	1	1	yes	characteristic
199	C₈	#13133150	1	1	yes	characteristic
200	C₂²	#16810366	1	1	yes	characteristic
201	C₂²	#16810368	1	1	yes	characteristic
202	C₂²	#16810386	1	1	yes	characteristic
203	C₂²	#16810425	1	1	yes	characteristic
204	C₂²	#16810428	1	1	yes	characteristic
205	C₂²	#16810437	1	1	yes	characteristic
206	C₂²	#16810440	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	848	2	no	—
210	C₄	#16810370	1	1	yes	characteristic
211	C₄	#16810371	1	1	yes	characteristic
212	C₄	#16810385	1	1	yes	characteristic
213	C₄	#16810391	1	1	yes	characteristic
214	C₄	#16810425	1	1	yes	characteristic
215	C₄	#16810428	1	1	yes	characteristic
216	C₄	#16810449	1	1	yes	characteristic
217	C₄	#16810451	1	1	yes	characteristic
218	C₂	#20272599	1	1	yes	characteristic
219	C₂	#20272762	1	1	yes	characteristic
220	C₂	#20272765	1	1	yes	characteristic
221	C₂	#20272769	1	1	yes	characteristic
222	C₂	#20272772	1	1	yes	characteristic
223	C₂	#20272807	1	1	yes	characteristic
224	C₂	#20272824	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133139
commutator	C₂×C₈×C₅₃	C₄²
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₅₃	#102
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#7274
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
9. #16810376	C₄	semidirect
13. C₅₃×C₄⋊C₁₆	C₄	semidirect
123. C₄⋊C₁₆	Dic₅₃	semidirect
126. C₅₃	#7274	semidirect

Maximal subgroups

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

subgroup	index
#20274875	2
C₂×#16810376	2
C₂×C₅₃×C₄⋊C₁₆	2
#7274	53