Group #31797691 — subgroups

Subgroups — #31797691

Order 17536.

364 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 166 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	#22578453	C₂	1	6	yes	maximal, complemented
2	Q₈×#11764047	C₂	1	2	yes	maximal, complemented
3	#22578425	C₂	1	3	yes	maximal
4	#22578422	C₂	1	3	yes	maximal
5	C₄×Q₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#11764047	C₂²	1	6	yes	—
7	C₄⋊#11764047	C₂²	1	6	yes	—
8	#17984145	C₂²	1	1	yes	characteristic
9	#17984145	C₂²	1	3	yes	—
10	#17984143	C₂²	1	6	yes	—
11	C₄×#11764047	C₂²	1	6	yes	—
12	C₂×Q₈×D₁₃₇	C₄	1	1	yes	characteristic
13	Q₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
14	C₄×Dic₂₇₄	C₄	1	3	yes	—
15	#17984053	C₂²	1	3	yes	—
16	C₄⋊C₄×D₁₃₇	C₄	1	3	yes	—
17	C₄²×D₁₃₇	C₂²	1	3	yes	—
18	C₄×Q₈×C₁₃₇	C₄	1	1	yes	characteristic
19	C₂×#11764047	C₂³	1	2	yes	—
20	C₂×#11764047	C₂³	1	6	yes	—
21	C₂×Dic₂₇₄	C₂×C₄	1	3	yes	—
22	Q₈×D₁₃₇	—	2	4	no	—
23	C₂×C₄×D₁₃₇	C₂×C₄	1	3	yes	—
24	C₂×C₄×D₁₃₇	C₂×C₄	1	3	yes	—
25	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
26	C₄⋊Dic₁₃₇	C₂×C₄	1	3	yes	—
27	#14555489	C₂×C₄	1	6	yes	—
28	C₄×Dic₁₃₇	C₂×C₄	1	3	yes	—
29	C₄×Dic₁₃₇	C₂³	1	3	yes	—
30	C₄×Dic₁₃₇	C₂³	1	3	yes	—
31	C₂×Q₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
32	C₄⋊C₄×C₁₃₇	C₂×C₄	1	3	yes	—
33	C₄²×C₁₃₇	C₂×C₄	1	3	yes	—
34	#11764047	—	2	2	no	—
35	#11764047	—	2	6	no	—
36	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
37	Dic₂₇₄	—	2	6	no	—
38	C₂×Dic₁₃₇	C₂²×C₄	1	3	yes	—
39	C₂×Dic₁₃₇	C₂²×C₄	1	3	yes	—
40	C₂×Dic₁₃₇	C₂⁴	1	1	yes	characteristic
41	C₄×D₁₃₇	—	2	6	no	—
42	C₄×D₁₃₇	—	2	6	no	—
43	C₄×D₁₃₇	—	2	2	no	—
44	Q₈×C₁₃₇	M₁₆	1	4	yes	—
45	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₃₇	C₂²×C₄	1	3	yes	—
47	C₂×C₄×C₁₃₇	C₂²×C₄	1	3	yes	—
48	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
49	C₂×D₁₃₇	—	2	1	no	—
50	C₂×D₁₃₇	—	2	1	no	—
51	C₂×D₁₃₇	#131	1	2	yes	—
52	C₄×C₁₃₇	—	2	1	no	—
53	C₄×C₁₃₇	—	2	3	no	—
54	C₄×C₁₃₇	C₂×M₁₆	1	6	yes	—
55	Dic₁₃₇	—	2	3	no	—
56	Dic₁₃₇	—	2	3	no	—
57	Dic₁₃₇	2₋¹⁺⁴	1	2	yes	—
58	C₂×C₁₃₇	#519	1	1	yes	characteristic
59	C₂×C₁₃₇	C₂²×M₁₆	1	1	yes	characteristic
60	C₂×C₁₃₇	C₂×#131	1	1	yes	characteristic
61	D₁₃₇	—	2	2	no	—
62	C₁₃₇	#2984	1	1	yes	characteristic, complemented, Sylow
63	#2984	—	137	1	no	maximal, Sylow
64	C₂×C₄×Q₈	—	137	1	no	—
65	#446	—	137	6	no	—
66	C₈×Q₈	—	137	2	no	—
67	#432	—	137	3	no	—
68	#431	—	137	3	no	—
69	C₂²×Q₈	—	137	1	no	—
70	C₄×Q₈	—	137	1	no	—
71	C₄×Q₈	—	137	3	no	—
72	C₄×Q₈	—	137	3	no	—
73	C₄×Q₈	#10587739	1	1	yes	characteristic
74	C₂×C₄⋊C₄	—	137	3	no	—
75	C₂×C₄²	—	137	3	no	—
76	C₄⋊C₈	—	137	6	no	—
77	C₄⋊C₈	—	137	6	no	—
78	C₂²⋊C₈	—	137	1	no	—
79	C₂²⋊C₈	—	137	3	no	—
80	#97	—	137	6	no	—
81	C₄×C₈	—	137	6	no	—
82	C₂×Q₈	—	274	4	no	—
83	C₂×Q₈	—	137	3	no	—
84	C₂×Q₈	C₂×#10587739	1	1	yes	characteristic
85	C₂²×C₄	—	137	1	no	—
86	C₂²×C₄	—	137	3	no	—
87	C₂²×C₄	—	137	3	no	—
88	C₂×C₈	—	137	2	no	—
89	C₂×C₈	—	137	6	no	—
90	C₄⋊C₄	—	137	6	no	—
91	C₄⋊C₄	—	137	3	no	—
92	C₄⋊C₄	C₂×#10587739	1	3	yes	—
93	C₄²	—	137	3	no	—
94	C₄²	—	137	3	no	—
95	C₄²	—	137	3	no	—
96	C₄²	C₂×#10587739	1	3	yes	—
97	C₂³	—	137	1	no	—
98	Q₈	—	274	6	no	—
99	Q₈	#14555520	1	4	yes	—
100	C₂×C₄	—	274	6	no	—
101	C₂×C₄	—	274	2	no	—
102	C₂×C₄	—	274	6	no	—
103	C₂×C₄	—	137	3	no	—
104	C₂×C₄	—	137	3	no	—
105	C₂×C₄	—	137	1	no	—
106	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
107	C₂×C₄	C₂²×#10587739	1	3	yes	—
108	C₂×C₄	C₂²×#10587739	1	3	yes	—
109	C₈	—	274	2	no	—
110	C₈	—	274	6	no	—
111	C₂²	—	274	1	no	—
112	C₂²	—	274	1	no	—
113	C₂²	—	137	2	no	—
114	C₂²	C₂³×#10587739	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	3	no	—
117	C₄	—	274	3	no	—
118	C₄	—	274	3	no	—
119	C₄	—	137	2	no	—
120	C₄	C₂×#14555520	1	6	yes	—
121	C₂	—	274	2	no	—
122	C₂	#22578471	1	1	yes	characteristic
123	C₂	C₂²×#14555520	1	1	yes	characteristic
124	C₂	D₁₃₇⋊#131	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22578453

C₂

yes

maximal, complemented

Q₈×#11764047

C₂

yes

maximal, complemented

#22578425

C₂

yes

maximal

#22578422

C₂

yes

maximal

C₄×Q₈×D₁₃₇

C₂

yes

maximal, characteristic

C₄⋊#11764047

C₂²

yes

—

C₄⋊#11764047

C₂²

yes

—

#17984145

C₂²

yes

characteristic

#17984145

C₂²

yes

—

#17984143

C₂²

yes

—

C₄×#11764047

C₂²

yes

—

C₂×Q₈×D₁₃₇

C₄

yes

characteristic

Q₈×Dic₁₃₇

C₂²

yes

characteristic

C₄×Dic₂₇₄

C₄

yes

—

#17984053

C₂²

yes

—

C₄⋊C₄×D₁₃₇

C₄

yes

—

C₄²×D₁₃₇

C₂²

yes

—

C₄×Q₈×C₁₃₇

C₄

yes

characteristic

C₂×#11764047

C₂³

yes

—

C₂×#11764047

C₂³

yes

—

C₂×Dic₂₇₄

C₂×C₄

yes

—

Q₈×D₁₃₇

—

C₂×C₄×D₁₃₇

C₂×C₄

yes

—

C₂×C₄×D₁₃₇

C₂×C₄

yes

—

C₂×C₄×D₁₃₇

C₂³

yes

characteristic

C₄⋊Dic₁₃₇

C₂×C₄

yes

—

#14555489

C₂×C₄

yes

—

C₄×Dic₁₃₇

C₂×C₄

yes

—

C₄×Dic₁₃₇

C₂³

yes

—

C₄×Dic₁₃₇

C₂³

yes

—

C₂×Q₈×C₁₃₇

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₃₇

C₂×C₄

yes

—

C₄²×C₁₃₇

C₂×C₄

yes

—

#11764047

—

#11764047

—

C₂²×D₁₃₇

C₂²×C₄

yes

characteristic

Dic₂₇₄

—

C₂×Dic₁₃₇

C₂²×C₄

yes

—

C₂×Dic₁₃₇

C₂²×C₄

yes

—

C₂×Dic₁₃₇

C₂⁴

yes

characteristic

C₄×D₁₃₇

—

C₄×D₁₃₇

—

C₄×D₁₃₇

—

Q₈×C₁₃₇

M₁₆

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂²×C₄

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

—

C₂²×C₁₃₇

C₂³×C₄

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

#131

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×M₁₆

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

2₋¹⁺⁴

yes

—

C₂×C₁₃₇

#519

yes

characteristic

C₂×C₁₃₇

C₂²×M₁₆

yes

characteristic

C₂×C₁₃₇

C₂×#131

yes

characteristic

D₁₃₇

—

C₁₃₇

#2984

yes

characteristic, complemented, Sylow

#2984

—

137

maximal, Sylow

C₂×C₄×Q₈

—

137

—

#446

—

137

—

C₈×Q₈

—

137

—

#432

—

137

—

#431

—

137

—

C₂²×Q₈

—

137

—

C₄×Q₈

—

137

—

C₄×Q₈

—

137

—

C₄×Q₈

—

137

—

C₄×Q₈

#10587739

yes

characteristic

C₂×C₄⋊C₄

—

137

—

C₂×C₄²

—

137

—

C₄⋊C₈

—

137

—

C₄⋊C₈

—

137

—

C₂²⋊C₈

—

137

—

C₂²⋊C₈

—

137

—

#97

—

137

—

C₄×C₈

—

137

—

C₂×Q₈

—

274

—

C₂×Q₈

—

137

—

C₂×Q₈

C₂×#10587739

yes

characteristic

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

C₂×#10587739

yes

—

C₄²

—

137

—

C₄²

—

137

—

C₄²

—

137

—

C₄²

C₂×#10587739

yes

—

C₂³

—

137

—

Q₈

—

274

—

Q₈

#14555520

yes

—

100

C₂×C₄

—

274

—

101

C₂×C₄

—

274

—

102

C₂×C₄

—

274

—

103

C₂×C₄

—

137

—

104

C₂×C₄

—

137

—

105

C₂×C₄

—

137

—

106

C₂×C₄

C₂²×#10587739

yes

characteristic

107

C₂×C₄

C₂²×#10587739

yes

—

108

C₂×C₄

C₂²×#10587739

yes

—

109

C₈

—

274

—

110

C₈

—

274

—

111

C₂²

—

274

—

112

C₂²

—

274

—

113

C₂²

—

137

—

114

C₂²

C₂³×#10587739

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

274

—

118

C₄

—

274

—

119

C₄

—

137

—

120

C₄

C₂×#14555520

yes

—

121

C₂

—

274

—

122

C₂

#22578471

yes

characteristic

123

C₂

C₂²×#14555520

yes

characteristic

124

C₂

D₁₃₇⋊#131

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10587739
commutator, socle	C₂²×C₁₃₇	C₂³×C₄
Fitting	C₄×Q₈×C₁₃₇	C₄
prime radical	C₂²×D₁₃₇	C₂²×C₄
cosocle	C₂×Dic₁₃₇	C₂⁴
hypercenter	C₄×Q₈	#10587739
hypocenter	C₁₃₇	#2984
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#10587739

commutator, socle

C₂²×C₁₃₇

C₂³×C₄

Fitting

C₄×Q₈×C₁₃₇

C₄

prime radical

C₂²×D₁₃₇

C₂²×C₄

cosocle

C₂×Dic₁₃₇

C₂⁴

C₄×Q₈

C₁₃₇

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. #22578453	C₂	semidirect
2. Q₈×#11764047	C₂	semidirect
62. C₁₃₇	#2984	semidirect

H ≅ G/N

type

1. #22578453

C₂

semidirect

2. Q₈×#11764047

C₂

semidirect

62. C₁₃₇

#2984

semidirect

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

subgroup	index
#22578453	2
Q₈×#11764047	2
#22578425	2
#22578422	2
C₄×Q₈×D₁₃₇	2
#2984	137

subgroup

index

C₄×Q₈×D₁₃₇

137

Subgroups — #31797691

Conjugacy classes of subgroups

Complements

Maximal subgroups