Group #30675050 — subgroups

Subgroups — #30675050

Order 16768.

364 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 221 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	#22553443	C₂	1	6	yes	maximal, complemented
2	Q₈×C₁₃₁⋊C₈	C₂	1	2	yes	maximal, complemented
3	#22553369	C₂	1	3	yes	maximal
4	#22553365	C₂	1	3	yes	maximal
5	C₂×C₄×Q₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
6	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	6	yes	—
7	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	6	yes	—
8	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	3	yes	—
10	#16862157	C₂²	1	6	yes	—
11	C₄×C₁₃₁⋊C₈	C₂²	1	6	yes	—
12	C₂²×Q₈×C₁₃₁	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₁₃₁	C₄	1	3	yes	—
14	C₄×Q₈×C₁₃₁	C₄	1	1	yes	characteristic
15	C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₄×Q₈×C₁₃₁	C₂²	1	3	yes	—
17	C₂×C₄⋊C₄×C₁₃₁	C₄	1	3	yes	—
18	C₂×C₄²×C₁₃₁	C₂²	1	3	yes	—
19	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
20	C₂×C₁₃₁⋊C₈	C₂³	1	6	yes	—
21	C₂×Q₈×C₁₃₁	—	2	4	no	—
22	C₂×Q₈×C₁₃₁	C₂×C₄	1	3	yes	—
23	C₂×Q₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₃₁	C₂×C₄	1	3	yes	—
25	C₂²×C₄×C₁₃₁	C₂×C₄	1	3	yes	—
26	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₁₃₁	C₂×C₄	1	6	yes	—
28	C₄⋊C₄×C₁₃₁	C₂×C₄	1	3	yes	—
29	C₄⋊C₄×C₁₃₁	C₂×C₄	1	3	yes	—
30	C₄²×C₁₃₁	C₂×C₄	1	3	yes	—
31	C₄²×C₁₃₁	C₂×C₄	1	3	yes	—
32	C₄²×C₁₃₁	C₂³	1	3	yes	—
33	C₄²×C₁₃₁	C₂³	1	3	yes	—
34	C₁₃₁⋊C₈	—	2	2	no	—
35	C₁₃₁⋊C₈	—	2	6	no	—
36	C₂³×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
37	Q₈×C₁₃₁	—	2	6	no	—
38	Q₈×C₁₃₁	M₁₆	1	4	yes	—
39	C₂×C₄×C₁₃₁	—	2	6	no	—
40	C₂×C₄×C₁₃₁	—	2	6	no	—
41	C₂×C₄×C₁₃₁	—	2	2	no	—
42	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₃₁	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₁₃₁	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₁₃₁	C₂²×C₄	1	3	yes	—
46	C₂×C₄×C₁₃₁	C₂²×C₄	1	3	yes	—
47	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
48	C₂²×C₁₃₁	—	2	1	no	—
49	C₂²×C₁₃₁	—	2	1	no	—
50	C₂²×C₁₃₁	#131	1	2	yes	—
51	C₂²×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
52	C₄×C₁₃₁	—	2	1	no	—
53	C₄×C₁₃₁	—	2	3	no	—
54	C₄×C₁₃₁	—	2	3	no	—
55	C₄×C₁₃₁	—	2	3	no	—
56	C₄×C₁₃₁	C₂×M₁₆	1	6	yes	—
57	C₄×C₁₃₁	2₋¹⁺⁴	1	2	yes	—
58	C₂×C₁₃₁	—	2	2	no	—
59	C₂×C₁₃₁	#519	1	1	yes	characteristic
60	C₂×C₁₃₁	C₂²×M₁₆	1	1	yes	characteristic
61	C₂×C₁₃₁	C₂×#131	1	1	yes	characteristic
62	C₁₃₁	#2984	1	1	yes	characteristic, complemented, Sylow
63	#2984	—	131	1	no	maximal, Sylow
64	C₂×C₄×Q₈	D₁₃₁	1	1	yes	characteristic
65	#446	—	131	6	no	—
66	C₈×Q₈	—	131	2	no	—
67	#432	—	131	3	no	—
68	#431	—	131	3	no	—
69	C₂²×Q₈	Dic₁₃₁	1	1	yes	characteristic
70	C₄×Q₈	Dic₁₃₁	1	3	yes	—
71	C₄×Q₈	Dic₁₃₁	1	1	yes	characteristic
72	C₄×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
73	C₄×Q₈	C₂×D₁₃₁	1	3	yes	—
74	C₂×C₄⋊C₄	Dic₁₃₁	1	3	yes	—
75	C₂×C₄²	C₂×D₁₃₁	1	3	yes	—
76	C₄⋊C₈	—	131	6	no	—
77	C₄⋊C₈	—	131	6	no	—
78	C₂²⋊C₈	—	131	1	no	—
79	C₂²⋊C₈	—	131	3	no	—
80	#97	—	131	6	no	—
81	C₄×C₈	—	131	6	no	—
82	C₂×Q₈	—	2	4	no	—
83	C₂×Q₈	C₂×Dic₁₃₁	1	3	yes	—
84	C₂×Q₈	C₂×Dic₁₃₁	1	1	yes	characteristic
85	C₂²×C₄	C₂×Dic₁₃₁	1	3	yes	—
86	C₂²×C₄	C₂×Dic₁₃₁	1	3	yes	—
87	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
88	C₂×C₈	—	131	2	no	—
89	C₂×C₈	—	131	6	no	—
90	C₄⋊C₄	C₂×Dic₁₃₁	1	6	yes	—
91	C₄⋊C₄	C₂×Dic₁₃₁	1	3	yes	—
92	C₄⋊C₄	C₂×Dic₁₃₁	1	3	yes	—
93	C₄²	C₂×Dic₁₃₁	1	3	yes	—
94	C₄²	C₂×Dic₁₃₁	1	3	yes	—
95	C₄²	C₂²×D₁₃₁	1	3	yes	—
96	C₄²	C₂²×D₁₃₁	1	3	yes	—
97	C₂³	C₂²×Dic₁₃₁	1	1	yes	characteristic
98	Q₈	—	2	6	no	—
99	Q₈	#14523497	1	4	yes	—
100	C₂×C₄	—	2	6	no	—
101	C₂×C₄	—	2	6	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
104	C₂×C₄	C₂²×Dic₁₃₁	1	3	yes	—
105	C₂×C₄	C₂²×Dic₁₃₁	1	3	yes	—
106	C₂×C₄	C₂²×Dic₁₃₁	1	3	yes	—
107	C₂×C₄	C₂²×Dic₁₃₁	1	3	yes	—
108	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
109	C₈	—	262	2	no	—
110	C₈	—	262	6	no	—
111	C₂²	—	2	1	no	—
112	C₂²	—	2	1	no	—
113	C₂²	#16862253	1	2	yes	—
114	C₂²	C₂³×Dic₁₃₁	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	3	no	—
117	C₄	—	2	3	no	—
118	C₄	—	2	3	no	—
119	C₄	C₂×#14523497	1	6	yes	—
120	C₄	#16862294	1	2	yes	—
121	C₂	—	2	2	no	—
122	C₂	#22553790	1	1	yes	characteristic
123	C₂	C₂²×#14523497	1	1	yes	characteristic
124	C₂	C₂×#16862253	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22553443

C₂

yes

maximal, complemented

Q₈×C₁₃₁⋊C₈

C₂

yes

maximal, complemented

#22553369

C₂

yes

maximal

#22553365

C₂

yes

maximal

C₂×C₄×Q₈×C₁₃₁

C₂

yes

maximal, characteristic

C₄⋊(C₁₃₁⋊C₈)

C₂²

yes

—

C₄⋊(C₁₃₁⋊C₈)

C₂²

yes

—

C₂²⋊(C₁₃₁⋊C₈)

C₂²

yes

characteristic

C₂²⋊(C₁₃₁⋊C₈)

C₂²

yes

—

#16862157

C₂²

yes

—

C₄×C₁₃₁⋊C₈

C₂²

yes

—

C₂²×Q₈×C₁₃₁

C₄

yes

characteristic

C₄×Q₈×C₁₃₁

C₄

yes

—

C₄×Q₈×C₁₃₁

C₄

yes

characteristic

C₄×Q₈×C₁₃₁

C₂²

yes

characteristic

C₄×Q₈×C₁₃₁

C₂²

yes

—

C₂×C₄⋊C₄×C₁₃₁

C₄

yes

—

C₂×C₄²×C₁₃₁

C₂²

yes

—

C₂×C₁₃₁⋊C₈

C₂³

yes

—

C₂×C₁₃₁⋊C₈

C₂³

yes

—

C₂×Q₈×C₁₃₁

—

C₂×Q₈×C₁₃₁

C₂×C₄

yes

—

C₂×Q₈×C₁₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₁

C₂×C₄

yes

—

C₂²×C₄×C₁₃₁

C₂×C₄

yes

—

C₂²×C₄×C₁₃₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₁

C₂×C₄

yes

—

C₄⋊C₄×C₁₃₁

C₂×C₄

yes

—

C₄⋊C₄×C₁₃₁

C₂×C₄

yes

—

C₄²×C₁₃₁

C₂×C₄

yes

—

C₄²×C₁₃₁

C₂×C₄

yes

—

C₄²×C₁₃₁

C₂³

yes

—

C₄²×C₁₃₁

C₂³

yes

—

C₁₃₁⋊C₈

—

C₁₃₁⋊C₈

—

C₂³×C₁₃₁

C₂²×C₄

yes

characteristic

Q₈×C₁₃₁

—

Q₈×C₁₃₁

M₁₆

yes

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂²×C₄

yes

—

C₂×C₄×C₁₃₁

C₂²×C₄

yes

—

C₂×C₄×C₁₃₁

C₂²×C₄

yes

—

C₂×C₄×C₁₃₁

C₂²×C₄

yes

—

C₂×C₄×C₁₃₁

C₂⁴

yes

characteristic

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

#131

yes

—

C₂²×C₁₃₁

C₂³×C₄

yes

characteristic

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

C₂×M₁₆

yes

—

C₄×C₁₃₁

2₋¹⁺⁴

yes

—

C₂×C₁₃₁

—

C₂×C₁₃₁

#519

yes

characteristic

C₂×C₁₃₁

C₂²×M₁₆

yes

C₂×C₁₃₁

yes

C₁₃₁

yes

characteristic, complemented, Sylow

#2984

—

131

maximal, Sylow

C₂×C₄×Q₈

D₁₃₁

yes

characteristic

#446

—

131

—

C₈×Q₈

—

131

—

#432

—

131

—

#431

—

131

—

C₂²×Q₈

Dic₁₃₁

yes

characteristic

C₄×Q₈

Dic₁₃₁

yes

—

C₄×Q₈

Dic₁₃₁

yes

characteristic

C₄×Q₈

C₂×D₁₃₁

yes

characteristic

C₄×Q₈

C₂×D₁₃₁

yes

—

C₂×C₄⋊C₄

Dic₁₃₁

yes

—

C₂×C₄²

C₂×D₁₃₁

yes

—

C₄⋊C₈

—

131

—

C₄⋊C₈

—

131

—

C₂²⋊C₈

—

131

—

C₂²⋊C₈

—

131

—

#97

—

131

—

C₄×C₈

—

131

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₃₁

yes

—

C₂×Q₈

C₂×Dic₁₃₁

yes

characteristic

C₂²×C₄

C₂×Dic₁₃₁

yes

—

C₂²×C₄

C₂×Dic₁₃₁

yes

—

C₂²×C₄

C₂²×D₁₃₁

yes

characteristic

C₂×C₈

—

131

—

C₂×C₈

—

131

—

C₄⋊C₄

C₂×Dic₁₃₁

yes

—

C₄⋊C₄

C₂×Dic₁₃₁

yes

—

C₄⋊C₄

C₂×Dic₁₃₁

yes

—

C₄²

C₂×Dic₁₃₁

yes

—

C₄²

C₂×Dic₁₃₁

yes

—

C₄²

C₂²×D₁₃₁

yes

—

C₄²

C₂²×D₁₃₁

yes

—

C₂³

C₂²×Dic₁₃₁

yes

characteristic

Q₈

—

Q₈

#14523497

yes

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂²×Dic₁₃₁

yes

characteristic

104

C₂×C₄

C₂²×Dic₁₃₁

yes

—

105

C₂×C₄

C₂²×Dic₁₃₁

yes

—

106

C₂×C₄

C₂²×Dic₁₃₁

yes

—

107

C₂×C₄

C₂²×Dic₁₃₁

yes

—

108

C₂×C₄

C₂³×D₁₃₁

yes

characteristic

109

C₈

—

262

—

110

C₈

—

262

—

111

C₂²

—

112

C₂²

—

113

C₂²

#16862253

yes

—

114

C₂²

C₂³×Dic₁₃₁

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

C₂×#14523497

yes

—

120

C₄

#16862294

yes

—

121

C₂

—

122

C₂

#22553790

yes

characteristic

123

C₂

C₂²×#14523497

yes

characteristic

124

C₂

C₂×#16862253

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₁
commutator, socle	C₂²×C₁₃₁	C₂³×C₄
Fitting	C₂×C₄×Q₈×C₁₃₁	C₂
prime radical	C₂³×C₁₃₁	C₂²×C₄
cosocle	C₂×C₄×C₁₃₁	C₂⁴
hypercenter	C₂×C₄×Q₈	D₁₃₁
hypocenter	C₁₃₁	#2984
LCS	G ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₃₁

commutator, socle

C₂²×C₁₃₁

C₂³×C₄

Fitting

C₂×C₄×Q₈×C₁₃₁

C₂

prime radical

C₂³×C₁₃₁

C₂²×C₄

cosocle

C₂×C₄×C₁₃₁

C₂⁴

hypercenter

C₂×C₄×Q₈

D₁₃₁

C₁₃₁

G ▷ C₂²×C₁₃₁ ▷ C₁₃₁

C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
1. #22553443	C₂	semidirect
2. Q₈×C₁₃₁⋊C₈	C₂	semidirect
62. C₁₃₁	#2984	semidirect

H ≅ G/N

type

1. #22553443

C₂

semidirect

2. Q₈×C₁₃₁⋊C₈

C₂

semidirect

62. C₁₃₁

#2984

semidirect

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

subgroup	index
#22553443	2
Q₈×C₁₃₁⋊C₈	2
#22553369	2
#22553365	2
C₂×C₄×Q₈×C₁₃₁	2
#2984	131

subgroup

index

Q₈×C₁₃₁⋊C₈

C₂×C₄×Q₈×C₁₃₁

131

Subgroups — #30675050

Conjugacy classes of subgroups

Complements

Maximal subgroups