Subgroups — #27293107

Order 13568.

306 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
2	C₈×#11719917	C₂	1	1	yes	maximal, characteristic
3	#20272634	C₂	1	2	yes	maximal, complemented
4	C₁₆×#88322	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2162	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₄	1	2	yes	—
7	#16810676	C₄	1	1	yes	characteristic
8	C₄×#11719917	C₄	1	1	yes	characteristic
9	C₄×#11719908	C₄	1	2	yes	—
10	C₄×#11719908	C₂²	1	1	yes	characteristic
11	C₄×#11719908	C₂²	1	1	yes	characteristic
12	#16810345	C₄	1	2	yes	—
13	C₈×#88322	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×C₄⋊C₁₆	C₄	1	2	yes	—
16	C₅₃×C₂²⋊C₁₆	C₄	1	2	yes	—
17	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
18	C₂×#11719917	C₈	1	2	yes	—
19	C₂×#11719908	—	2	4	no	—
20	C₂×#11719908	C₂×C₄	1	2	yes	—
21	C₂×#11719908	C₂×C₄	1	2	yes	—
22	C₄⋊#88322	C₈	1	2	yes	—
23	C₂²⋊#88322	C₈	1	2	yes	—
24	#13133128	C₈	1	2	yes	—
25	C₄×#88322	C₂×C₄	1	2	yes	—
26	C₄×#88322	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
38	C₂×C₈×C₅₃	—	2	2	no	—
39	C₂×C₈×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₄²×C₅₃	C₄²	1	1	yes	characteristic
45	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
46	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
47	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	#11719917	C₁₆	1	8	yes	complemented
49	C₂×#88322	C₂×C₈	1	4	yes	—
50	C₂×#88322	C₂×C₈	1	4	yes	—
51	C₁₆×C₅₃	—	2	4	no	—
52	#11719908	—	2	2	no	—
53	#11719908	M₁₆	1	4	yes	complemented
54	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
55	C₂×C₄×C₅₃	—	2	1	no	—
56	C₂×C₄×C₅₃	—	2	1	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×C₁₆	1	4	yes	—
62	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
65	C₈×C₅₃	—	2	1	no	—
66	C₈×C₅₃	—	2	1	no	—
67	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
68	C₈×C₅₃	#131	1	2	yes	—
69	#88322	—	2	4	no	—
70	#88322	C₂×C₁₆	1	8	yes	complemented
71	C₁₆×M₁₆	—	53	1	no	maximal, Sylow
72	C₂²×C₅₃	—	2	1	no	—
73	C₂²×C₅₃	—	2	1	no	—
74	C₂²×C₅₃	C₄×C₁₆	1	2	yes	—
75	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
76	C₄×C₅₃	—	2	1	no	—
77	C₄×C₅₃	—	2	1	no	—
78	C₄×C₅₃	C₄×C₁₆	1	2	yes	—
79	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
80	C₄×C₅₃	#405	1	1	yes	characteristic
81	C₄×C₅₃	C₂²×C₁₆	1	2	yes	—
82	C₄×C₅₃	#504	1	2	yes	—
83	#2162	D₅₃	1	1	yes	characteristic
84	C₂×C₄×C₁₆	—	53	1	no	—
85	C₈×M₁₆	—	53	1	no	—
86	#1312	—	53	2	no	—
87	C₈×C₁₆	—	53	2	no	—
88	C₂×C₅₃	—	2	2	no	—
89	C₂×C₅₃	C₈×M₁₆	1	1	yes	characteristic
90	C₂×C₅₃	C₂×C₄×C₁₆	1	1	yes	characteristic
91	C₂×C₅₃	#2108	1	1	yes	characteristic
92	C₂²×C₁₆	—	53	2	no	—
93	#431	—	53	1	no	—
94	C₄×M₁₆	—	53	1	no	—
95	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
96	C₄⋊C₁₆	Dic₅₃	1	2	yes	—
97	C₂²⋊C₁₆	Dic₅₃	1	2	yes	—
98	C₄×C₁₆	—	53	2	no	—
99	C₄×C₁₆	—	53	1	no	—
100	C₄×C₁₆	—	53	1	no	—
101	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
102	#322	—	53	2	no	—
103	C₈²	—	53	2	no	—
104	C₅₃	C₁₆×M₁₆	1	1	yes	characteristic, complemented, Sylow
105	C₂×M₁₆	—	53	2	no	—
106	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
107	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
108	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
109	C₂×C₁₆	—	53	2	no	—
110	C₂×C₁₆	—	53	2	no	—
111	C₂×C₁₆	—	106	4	no	—
112	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
113	C₄⋊C₈	—	53	2	no	—
114	C₂²⋊C₈	—	53	2	no	—
115	#97	—	53	2	no	—
116	C₄×C₈	—	53	2	no	—
117	C₄×C₈	—	53	2	no	—
118	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
119	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
121	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
122	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
123	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
124	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
125	M₁₆	—	53	8	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	53	4	no	—
129	C₂×C₈	—	53	4	no	—
130	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
131	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
132	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
133	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
134	C₄²	C₈×D₅₃	1	1	yes	characteristic
135	C₄²	C₈×D₅₃	1	1	yes	characteristic
136	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
137	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
138	C₁₆	—	2	4	no	—
139	C₁₆	—	53	4	no	—
140	C₁₆	—	106	2	no	—
141	C₂³	C₈×Dic₅₃	1	1	yes	characteristic
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
146	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
147	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
148	C₂×C₄	C₁₆×D₅₃	1	4	yes	—
149	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
150	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
151	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₈	—	53	8	no	—
155	C₈	—	106	4	no	—
156	C₈	M₁₆×D₅₃	1	2	yes	—
157	C₈	#13133224	1	2	yes	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	C₁₆×Dic₅₃	1	2	yes	—
161	C₂²	C₂×C₈×Dic₅₃	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	C₁₆×Dic₅₃	1	2	yes	—
165	C₄	M₁₆×Dic₅₃	1	1	yes	characteristic
166	C₄	#16810579	1	1	yes	characteristic
167	C₄	C₂×C₁₆×D₅₃	1	2	yes	—
168	C₄	#16811041	1	2	yes	—
169	C₂	—	2	2	no	—
170	C₂	#20272996	1	1	yes	characteristic
171	C₂	C₂×C₁₆×Dic₅₃	1	1	yes	characteristic
172	C₂	#20276314	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₅₃
commutator	C₂×C₅₃	C₂×C₄×C₁₆
Fitting	C₅₃×#2162	C₂
socle	C₂²×C₅₃	C₂×C₄×C₈
prime radical	C₂³×C₅₃	C₄×C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#2162	D₅₃
hypocenter	C₅₃	C₁₆×M₁₆
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
3. #20272634	C₂	semidirect
4. C₁₆×#88322	C₂	semidirect
48. #11719917	C₁₆	semidirect
53. #11719908	M₁₆	semidirect
70. #88322	C₂×C₁₆	semidirect
104. C₅₃	C₁₆×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11719908	2
C₈×#11719917	2
#20272634	2
C₁₆×#88322	2
C₅₃×#2162	2
C₁₆×M₁₆	53