Group #30667572 — subgroups

Subgroups — #30667572

Order 16768.

260 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16862186	C₂	1	2	yes	maximal, complemented
2	C₂²⋊(C₁₃₁⋊C₁₆)	C₂	1	4	yes	maximal, complemented
3	C₂³×C₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#16862186	—	2	4	no	—
5	C₂×C₁₃₁⋊C₁₆	C₂²	1	4	yes	complemented
6	C₂³×C₄×C₁₃₁	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₁₃₁	—	2	4	no	—
8	C₂²×C₈×C₁₃₁	C₄	1	1	yes	characteristic
9	C₂²×C₈×C₁₃₁	C₄	1	2	yes	—
10	C₂²×C₈×C₁₃₁	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
12	C₁₃₁⋊C₁₆	—	2	4	no	—
13	C₂⁴×C₁₃₁	C₈	1	1	yes	characteristic
14	C₂²×C₄×C₁₃₁	—	2	2	no	—
15	C₂²×C₄×C₁₃₁	—	2	2	no	—
16	C₂²×C₄×C₁₃₁	C₈	1	1	yes	characteristic
17	C₂²×C₄×C₁₃₁	C₈	1	2	yes	—
18	C₂²×C₄×C₁₃₁	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₁₃₁	—	2	2	no	—
21	C₂×C₈×C₁₃₁	—	2	4	no	—
22	C₂×C₈×C₁₃₁	—	2	4	no	—
23	C₂×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₁₃₁	D₄	1	4	yes	—
26	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
27	C₂³×C₁₃₁	—	2	2	no	—
28	C₂³×C₁₃₁	—	2	4	no	—
29	C₂³×C₁₃₁	C₂×C₈	1	1	yes	characteristic
30	C₂³×C₁₃₁	C₂×C₈	1	2	yes	—
31	C₂×C₄×C₁₃₁	—	2	2	no	—
32	C₂×C₄×C₁₃₁	—	2	4	no	—
33	C₂×C₄×C₁₃₁	—	2	2	no	—
34	C₂×C₄×C₁₃₁	—	2	2	no	—
35	C₂×C₄×C₁₃₁	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₁₃₁	C₂×C₈	1	2	yes	—
37	C₂×C₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
38	C₂×C₄×C₁₃₁	M₁₆	1	2	yes	—
39	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
40	C₈×C₁₃₁	—	2	2	no	—
41	C₈×C₁₃₁	C₂²⋊C₄	1	2	yes	—
42	C₈×C₁₃₁	C₂×D₄	1	2	yes	—
43	C₂²×C₁₃₁	—	2	8	no	—
44	C₂²×C₁₃₁	—	2	2	no	—
45	C₂²×C₁₃₁	—	2	2	no	—
46	C₂²×C₁₃₁	—	2	2	no	—
47	C₂²×C₁₃₁	C₂²⋊C₈	1	2	yes	—
48	C₂²×C₁₃₁	M₃₂	1	4	yes	complemented
49	C₂²×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
50	C₄×C₁₃₁	—	2	2	no	—
51	C₄×C₁₃₁	C₂²⋊C₈	1	2	yes	—
52	C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₁₃₁	C₂×M₁₆	1	1	yes	characteristic
54	C₂×C₁₃₁	—	2	2	no	—
55	C₂×C₁₃₁	—	2	4	no	—
56	C₂×C₁₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
57	C₂×C₁₃₁	C₂×M₃₂	1	2	yes	—
58	C₁₃₁	#2112	1	1	yes	characteristic, complemented, Sylow
59	#2112	—	131	1	no	maximal, Sylow
60	C₂³×C₈	D₁₃₁	1	1	yes	characteristic
61	C₂×M₃₂	—	131	2	no	—
62	C₂²⋊C₁₆	—	131	4	no	—
63	C₂³×C₄	Dic₁₃₁	1	1	yes	characteristic
64	C₂²×C₈	—	2	4	no	—
65	C₂²×C₈	Dic₁₃₁	1	1	yes	characteristic
66	C₂²×C₈	Dic₁₃₁	1	2	yes	—
67	C₂²×C₈	C₂×D₁₃₁	1	2	yes	—
68	C₂²×C₈	C₂×D₁₃₁	1	1	yes	characteristic
69	M₃₂	—	262	4	no	—
70	C₂×C₁₆	—	131	4	no	—
71	C₂⁴	C₁₃₁⋊C₈	1	1	yes	characteristic
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	2	2	no	—
74	C₂²×C₄	C₁₃₁⋊C₈	1	1	yes	characteristic
75	C₂²×C₄	C₁₃₁⋊C₈	1	2	yes	—
76	C₂²×C₄	C₂×Dic₁₃₁	1	2	yes	—
77	C₂²×C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	4	no	—
80	C₂×C₈	—	2	4	no	—
81	C₂×C₈	C₂×Dic₁₃₁	1	2	yes	—
82	C₂×C₈	C₂×Dic₁₃₁	1	1	yes	characteristic
83	C₂×C₈	C₂²⋊D₁₃₁	1	4	yes	—
84	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
85	C₁₆	—	262	4	no	—
86	C₂³	—	2	2	no	—
87	C₂³	—	2	4	no	—
88	C₂³	C₂×C₁₃₁⋊C₈	1	1	yes	characteristic
89	C₂³	C₂×C₁₃₁⋊C₈	1	2	yes	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	C₂²⋊Dic₁₃₁	1	2	yes	—
95	C₂×C₄	C₂×C₁₃₁⋊C₈	1	2	yes	—
96	C₂×C₄	C₂×C₁₃₁⋊C₈	1	1	yes	characteristic
97	C₂×C₄	#14523497	1	2	yes	—
98	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	C₂²⋊Dic₁₃₁	1	2	yes	—
101	C₈	C₂×C₂²⋊D₁₃₁	1	2	yes	—
102	C₂²	—	2	8	no	—
103	C₂²	—	2	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	—	2	2	no	—
106	C₂²	C₂²⋊(C₁₃₁⋊C₈)	1	2	yes	—
107	C₂²	#16862186	1	4	yes	complemented
108	C₂²	C₂²×C₁₃₁⋊C₈	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	C₂²⋊(C₁₃₁⋊C₈)	1	2	yes	—
111	C₄	C₂×C₂²⋊Dic₁₃₁	1	1	yes	characteristic
112	C₄	C₂×#14523497	1	1	yes	characteristic
113	C₂	—	2	2	no	—
114	C₂	—	2	4	no	—
115	C₂	C₂×C₂²⋊(C₁₃₁⋊C₈)	1	1	yes	characteristic
116	C₂	C₂×#16862186	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16862186

C₂

yes

maximal, complemented

C₂²⋊(C₁₃₁⋊C₁₆)

C₂

yes

maximal, complemented

C₂³×C₈×C₁₃₁

C₂

yes

maximal, characteristic

#16862186

—

C₂×C₁₃₁⋊C₁₆

C₂²

yes

complemented

C₂³×C₄×C₁₃₁

C₄

yes

characteristic

C₂²×C₈×C₁₃₁

—

C₂²×C₈×C₁₃₁

C₄

yes

characteristic

C₂²×C₈×C₁₃₁

C₄

yes

—

C₂²×C₈×C₁₃₁

C₂²

yes

—

C₂²×C₈×C₁₃₁

C₂²

yes

characteristic

C₁₃₁⋊C₁₆

—

C₂⁴×C₁₃₁

C₈

yes

characteristic

C₂²×C₄×C₁₃₁

—

C₂²×C₄×C₁₃₁

—

C₂²×C₄×C₁₃₁

C₈

yes

characteristic

C₂²×C₄×C₁₃₁

C₈

yes

—

C₂²×C₄×C₁₃₁

C₂×C₄

yes

—

C₂²×C₄×C₁₃₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

C₂×C₄

yes

—

C₂×C₈×C₁₃₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₁

D₄

yes

—

C₂×C₈×C₁₃₁

C₂³

yes

characteristic

C₂³×C₁₃₁

—

C₂³×C₁₃₁

—

C₂³×C₁₃₁

C₂×C₈

yes

characteristic

C₂³×C₁₃₁

C₂×C₈

yes

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃₁

C₂×C₈

yes

—

C₂×C₄×C₁₃₁

C₂×C₈

yes

characteristic

C₂×C₄×C₁₃₁

M₁₆

yes

—

C₂×C₄×C₁₃₁

C₂²×C₄

yes

characteristic

C₈×C₁₃₁

—

C₈×C₁₃₁

C₂²⋊C₄

yes

—

C₈×C₁₃₁

C₂×D₄

yes

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

C₂²⋊C₈

yes

—

C₂²×C₁₃₁

M₃₂

yes

complemented

C₂²×C₁₃₁

C₂²×C₈

yes

characteristic

C₄×C₁₃₁

—

C₄×C₁₃₁

C₂²⋊C₈

yes

—

C₄×C₁₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₃₁

C₂×M₁₆

yes

characteristic

C₂×C₁₃₁

—

C₂×C₁₃₁

—

C₂×C₁₃₁

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₁₃₁

C₂×M₃₂

yes

—

C₁₃₁

#2112

yes

characteristic, complemented, Sylow

#2112

—

131

maximal, Sylow

C₂³×C₈

D₁₃₁

yes

characteristic

C₂×M₃₂

—

131

—

C₂²⋊C₁₆

—

131

—

C₂³×C₄

Dic₁₃₁

yes

characteristic

C₂²×C₈

—

C₂²×C₈

Dic₁₃₁

yes

characteristic

C₂²×C₈

Dic₁₃₁

yes

—

C₂²×C₈

C₂×D₁₃₁

yes

—

C₂²×C₈

C₂×D₁₃₁

yes

characteristic

M₃₂

—

262

—

C₂×C₁₆

—

131

—

C₂⁴

C₁₃₁⋊C₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₁₃₁⋊C₈

yes

characteristic

C₂²×C₄

C₁₃₁⋊C₈

yes

—

C₂²×C₄

C₂×Dic₁₃₁

yes

—

C₂²×C₄

C₂×Dic₁₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₁₃₁

yes

—

C₂×C₈

C₂×Dic₁₃₁

yes

characteristic

C₂×C₈

C₂²⋊D₁₃₁

yes

—

C₂×C₈

C₂²×D₁₃₁

yes

characteristic

C₁₆

—

262

—

C₂³

—

C₂³

—

C₂³

C₂×C₁₃₁⋊C₈

yes

characteristic

C₂³

C₂×C₁₃₁⋊C₈

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₁₃₁

yes

—

C₂×C₄

C₂×C₁₃₁⋊C₈

yes

—

C₂×C₄

C₂×C₁₃₁⋊C₈

yes

characteristic

C₂×C₄

#14523497

yes

—

C₂×C₄

C₂²×Dic₁₃₁

yes

characteristic

C₈

—

100

C₈

C₂²⋊Dic₁₃₁

yes

—

101

C₈

C₂×C₂²⋊D₁₃₁

yes

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

C₂²⋊(C₁₃₁⋊C₈)

yes

—

107

C₂²

#16862186

yes

complemented

108

C₂²

C₂²×C₁₃₁⋊C₈

yes

characteristic

109

C₄

—

110

C₄

C₂²⋊(C₁₃₁⋊C₈)

yes

—

111

C₄

C₂×C₂²⋊Dic₁₃₁

yes

characteristic

112

C₄

C₂×#14523497

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

C₂×C₂²⋊(C₁₃₁⋊C₈)

yes

characteristic

116

C₂

C₂×#16862186

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₃₁
commutator, socle	C₂²×C₁₃₁	C₂²×C₈
Fitting	C₂³×C₈×C₁₃₁	C₂
prime radical	C₂⁴×C₁₃₁	C₈
cosocle	C₂×C₈×C₁₃₁	C₂³
hypercenter	C₂³×C₈	D₁₃₁
hypocenter	C₁₃₁	#2112
LCS	G ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₃₁

commutator, socle

C₂²×C₁₃₁

C₂²×C₈

Fitting

C₂³×C₈×C₁₃₁

C₂

prime radical

C₂⁴×C₁₃₁

C₈

cosocle

C₂×C₈×C₁₃₁

C₂³

hypercenter

C₂³×C₈

D₁₃₁

C₁₃₁

G ▷ C₂²×C₁₃₁ ▷ C₁₃₁

C₁ ◁ C₂×C₈ ◁ C₂³×C₈

N	H ≅ G/N	type
1. C₂×#16862186	C₂	semidirect
2. C₂²⋊(C₁₃₁⋊C₁₆)	C₂	semidirect
5. C₂×C₁₃₁⋊C₁₆	C₂²	semidirect
48. C₂²×C₁₃₁	M₃₂	semidirect
58. C₁₃₁	#2112	semidirect
107. C₂²	#16862186	semidirect

H ≅ G/N

type

1. C₂×#16862186

C₂

semidirect

2. C₂²⋊(C₁₃₁⋊C₁₆)

C₂

semidirect

5. C₂×C₁₃₁⋊C₁₆

C₂²

semidirect

48. C₂²×C₁₃₁

M₃₂

semidirect

58. C₁₃₁

#2112

semidirect

107. C₂²

#16862186

semidirect

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

subgroup	index
C₂×#16862186	2
C₂²⋊(C₁₃₁⋊C₁₆)	2
C₂³×C₈×C₁₃₁	2
#2112	131

subgroup

index

C₂×#16862186

C₂²⋊(C₁₃₁⋊C₁₆)

C₂³×C₈×C₁₃₁

#2112

131

Subgroups — #30667572

Conjugacy classes of subgroups

Complements

Maximal subgroups