Group #31797721 — subgroups

Subgroups — #31797721

Order 17536.

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

normal subgroups only

This group has 394 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	Q₈×#11764047	C₂	1	2	yes	maximal, complemented
2	#22578438	C₂	1	6	yes	maximal, complemented
3	#22578420	C₂	1	3	yes	maximal, complemented
4	#22578406	C₂	1	3	yes	maximal, complemented
5	#41×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555519	C₂²	1	6	yes	—
7	#17984155	C₂²	1	3	yes	—
8	#17984155	C₂²	1	1	yes	characteristic
9	C₄⋊#11764047	C₂²	1	6	yes	—
10	C₂²⋊#11764047	C₂²	1	6	yes	—
11	C₄×#11764047	C₂²	1	6	yes	—
12	Q₈×Dic₁₃₇	C₄	1	1	yes	characteristic
13	Q₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
14	D₄×Dic₁₃₇	C₄	1	3	yes	—
15	D₄×Dic₁₃₇	C₂²	1	3	yes	—
16	#17984047	C₄	1	3	yes	—
17	C₂×C₄×Dic₁₃₇	C₂²	1	3	yes	—
18	C₂×#41×C₁₃₇	C₄	1	1	yes	characteristic
19	#14555519	—	4	6	no	—
20	C₂×#11764047	C₂³	1	2	yes	—
21	C₂×#11764047	C₂³	1	6	yes	—
22	C₂²×Dic₁₃₇	C₂×C₄	1	3	yes	—
23	C₂²×Dic₁₃₇	C₂³	1	3	yes	—
24	C₄⋊Dic₁₃₇	C₂×C₄	1	3	yes	—
25	C₄⋊Dic₁₃₇	C₂×C₄	1	3	yes	—
26	C₂²⋊Dic₁₃₇	C₂×C₄	1	6	yes	—
27	C₄×Dic₁₃₇	C₂×C₄	1	3	yes	—
28	C₄×Dic₁₃₇	C₂×C₄	1	3	yes	—
29	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
30	C₄×Dic₁₃₇	C₂³	1	3	yes	—
31	#41×C₁₃₇	—	2	4	no	—
32	C₂×Q₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
33	C₂×D₄×C₁₃₇	C₂×C₄	1	3	yes	—
34	C₂²×C₄×C₁₃₇	C₂×C₄	1	3	yes	—
35	#11764047	—	2	6	no	—
36	#11764047	—	2	2	no	—
37	C₂×Dic₁₃₇	—	2	6	no	—
38	C₂×Dic₁₃₇	—	2	3	no	—
39	C₂×Dic₁₃₇	—	2	3	no	—
40	C₂×Dic₁₃₇	C₂²×C₄	1	3	yes	—
41	C₂×Dic₁₃₇	C₂²×C₄	1	3	yes	—
42	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₁₃₇	C₂⁴	1	1	yes	characteristic
44	C₂³×C₁₃₇	C₂²×C₄	1	3	yes	—
45	Q₈×C₁₃₇	M₁₆	1	4	yes	complemented
46	D₄×C₁₃₇	—	2	6	no	—
47	C₂×C₄×C₁₃₇	—	2	6	no	—
48	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₃₇	C₂²×C₄	1	3	yes	—
50	C₂²×C₁₃₇	—	2	3	no	—
51	C₂²×C₁₃₇	—	2	3	no	—
52	C₂²×C₁₃₇	—	2	3	no	—
53	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₁₃₇	—	2	1	no	—
55	C₄×C₁₃₇	C₂×M₁₆	1	6	yes	—
56	Dic₁₃₇	—	2	3	no	—
57	Dic₁₃₇	—	2	3	no	—
58	Dic₁₃₇	#131	1	2	yes	—
59	Dic₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
60	Dic₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
61	C₂×C₁₃₇	—	4	3	no	—
62	C₂×C₁₃₇	#520	1	1	yes	characteristic
63	C₂×C₁₃₇	C₂²×M₁₆	1	1	yes	characteristic
64	C₂×C₁₃₇	C₂×#131	1	1	yes	characteristic
65	C₁₃₇	#2991	1	1	yes	characteristic, complemented, Sylow
66	#2991	—	137	1	no	maximal, Sylow
67	C₄×#41	—	137	1	no	—
68	C₈×Q₈	—	137	2	no	—
69	#436	—	137	6	no	—
70	#431	—	137	3	no	—
71	#423	—	137	3	no	—
72	C₂×#41	#10587739	1	1	yes	characteristic
73	C₂×M₁₆	—	137	6	no	—
74	C₄×Q₈	—	137	1	no	—
75	C₄×Q₈	—	137	1	no	—
76	C₄×D₄	—	137	3	no	—
77	C₄×D₄	—	137	3	no	—
78	#117	—	137	3	no	—
79	C₂×C₄²	—	137	3	no	—
80	C₄⋊C₈	—	137	6	no	—
81	C₄⋊C₈	—	137	3	no	—
82	C₄⋊C₈	—	137	1	no	—
83	C₂²⋊C₈	—	137	6	no	—
84	C₄×C₈	—	137	6	no	—
85	#41	—	2	4	no	—
86	C₂×Q₈	C₂×#10587739	1	1	yes	characteristic
87	C₂×D₄	C₂×#10587739	1	3	yes	—
88	C₂²×C₄	—	137	3	no	—
89	C₂²×C₄	—	137	3	no	—
90	C₂²×C₄	C₂×#10587739	1	3	yes	—
91	M₁₆	—	548	6	no	—
92	C₂×C₈	—	137	2	no	—
93	C₂×C₈	—	137	6	no	—
94	C₄⋊C₄	—	137	3	no	—
95	C₄⋊C₄	—	137	3	no	—
96	C₂²⋊C₄	—	137	6	no	—
97	C₄²	—	137	3	no	—
98	C₄²	—	137	3	no	—
99	C₄²	—	137	3	no	—
100	C₄²	—	137	1	no	—
101	C₂³	C₂²×#10587739	1	3	yes	—
102	Q₈	#14555519	1	4	yes	complemented
103	D₄	—	2	6	no	—
104	C₂×C₄	—	2	6	no	—
105	C₂×C₄	—	274	3	no	—
106	C₂×C₄	—	274	3	no	—
107	C₂×C₄	—	274	6	no	—
108	C₂×C₄	—	137	3	no	—
109	C₂×C₄	—	137	3	no	—
110	C₂×C₄	—	137	1	no	—
111	C₂×C₄	—	137	1	no	—
112	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
113	C₂×C₄	C₂²×#10587739	1	3	yes	—
114	C₈	—	274	6	no	—
115	C₈	—	274	2	no	—
116	C₂²	—	2	3	no	—
117	C₂²	—	2	3	no	—
118	C₂²	—	2	3	no	—
119	C₂²	C₂³×#10587739	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	274	3	no	—
122	C₄	—	274	3	no	—
123	C₄	—	137	1	no	—
124	C₄	—	137	1	no	—
125	C₄	—	137	2	no	—
126	C₄	C₂×#14555519	1	6	yes	—
127	C₂	—	4	3	no	—
128	C₂	#22578472	1	1	yes	characteristic
129	C₂	C₂²×#14555519	1	1	yes	characteristic
130	C₂	D₁₃₇⋊#131	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#11764047

C₂

yes

maximal, complemented

#22578438

C₂

yes

maximal, complemented

#22578420

C₂

yes

maximal, complemented

#22578406

C₂

yes

maximal, complemented

#41×Dic₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555519

C₂²

yes

—

#17984155

C₂²

yes

—

#17984155

C₂²

yes

characteristic

C₄⋊#11764047

C₂²

yes

—

C₂²⋊#11764047

C₂²

yes

—

C₄×#11764047

C₂²

yes

—

Q₈×Dic₁₃₇

C₄

yes

characteristic

Q₈×Dic₁₃₇

C₂²

yes

characteristic

D₄×Dic₁₃₇

C₄

yes

—

D₄×Dic₁₃₇

C₂²

yes

—

#17984047

C₄

yes

—

C₂×C₄×Dic₁₃₇

C₂²

yes

—

C₂×#41×C₁₃₇

C₄

yes

characteristic

#14555519

—

C₂×#11764047

C₂³

yes

—

C₂×#11764047

C₂³

yes

—

C₂²×Dic₁₃₇

C₂×C₄

yes

—

C₂²×Dic₁₃₇

C₂³

yes

—

C₄⋊Dic₁₃₇

C₂×C₄

yes

—

C₄⋊Dic₁₃₇

C₂×C₄

yes

—

C₂²⋊Dic₁₃₇

C₂×C₄

yes

—

C₄×Dic₁₃₇

C₂×C₄

yes

—

C₄×Dic₁₃₇

C₂×C₄

yes

—

C₄×Dic₁₃₇

C₂³

yes

characteristic

C₄×Dic₁₃₇

C₂³

yes

—

#41×C₁₃₇

—

C₂×Q₈×C₁₃₇

C₂×C₄

yes

characteristic

C₂×D₄×C₁₃₇

C₂×C₄

yes

—

C₂²×C₄×C₁₃₇

C₂×C₄

yes

—

#11764047

—

#11764047

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

C₂²×C₄

yes

—

C₂×Dic₁₃₇

C₂²×C₄

yes

—

C₂×Dic₁₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₁₃₇

C₂⁴

yes

characteristic

C₂³×C₁₃₇

C₂²×C₄

yes

—

Q₈×C₁₃₇

M₁₆

yes

complemented

D₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂²×C₄

yes

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂³×C₄

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×M₁₆

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

#131

yes

—

Dic₁₃₇

H(5,𝔽₂)

yes

characteristic

Dic₁₃₇

2₋¹⁺⁴

yes

characteristic

C₂×C₁₃₇

—

C₂×C₁₃₇

#520

yes

characteristic

C₂×C₁₃₇

C₂²×M₁₆

yes

C₂×C₁₃₇

yes

C₁₃₇

yes

characteristic, complemented, Sylow

#2991

—

137

maximal, Sylow

C₄×#41

—

137

—

C₈×Q₈

—

137

—

#436

—

137

—

#431

—

137

—

#423

—

137

—

C₂×#41

#10587739

yes

characteristic

C₂×M₁₆

—

137

—

C₄×Q₈

—

137

—

C₄×Q₈

—

137

—

C₄×D₄

—

137

—

C₄×D₄

—

137

—

#117

—

137

—

C₂×C₄²

—

137

—

C₄⋊C₈

—

137

—

C₄⋊C₈

—

137

—

C₄⋊C₈

—

137

—

C₂²⋊C₈

—

137

—

C₄×C₈

—

137

—

#41

—

C₂×Q₈

C₂×#10587739

yes

characteristic

C₂×D₄

C₂×#10587739

yes

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

C₂×#10587739

yes

—

M₁₆

—

548

—

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₂²⋊C₄

—

137

—

C₄²

—

137

—

C₄²

—

137

—

C₄²

—

137

—

100

C₄²

—

137

—

101

C₂³

C₂²×#10587739

yes

—

102

Q₈

#14555519

yes

complemented

103

D₄

—

104

C₂×C₄

—

105

C₂×C₄

—

274

—

106

C₂×C₄

—

274

—

107

C₂×C₄

—

274

—

108

C₂×C₄

—

137

—

109

C₂×C₄

—

137

—

110

C₂×C₄

—

137

—

111

C₂×C₄

—

137

—

112

C₂×C₄

C₂²×#10587739

yes

characteristic

113

C₂×C₄

C₂²×#10587739

yes

—

114

C₈

—

274

—

115

C₈

—

274

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

C₂³×#10587739

yes

characteristic

120

C₄

—

121

C₄

—

274

—

122

C₄

—

274

—

123

C₄

—

137

—

124

C₄

—

137

—

125

C₄

—

137

—

126

C₄

C₂×#14555519

yes

—

127

C₂

—

128

C₂

#22578472

yes

characteristic

129

C₂

C₂²×#14555519

yes

characteristic

130

C₂

D₁₃₇⋊#131

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10587739
commutator, socle	C₂²×C₁₃₇	C₂³×C₄
Fitting, prime radical	C₂×#41×C₁₃₇	C₄
cosocle	C₂×Dic₁₃₇	C₂⁴
hypercenter	C₂×#41	#10587739
hypocenter	C₁₃₇	#2991
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×#41

subgroup

G/H

center, Frattini

C₂²

C₂³×#10587739

commutator, socle

C₂²×C₁₃₇

C₂³×C₄

Fitting, prime radical

C₂×#41×C₁₃₇

C₄

cosocle

C₂×Dic₁₃₇

C₂⁴

C₂×#41

C₁₃₇

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂² ◁ C₂×#41

N	H ≅ G/N	type
1. Q₈×#11764047	C₂	semidirect
2. #22578438	C₂	semidirect
3. #22578420	C₂	semidirect
4. #22578406	C₂	semidirect
45. Q₈×C₁₃₇	M₁₆	semidirect
65. C₁₃₇	#2991	semidirect
102. Q₈	#14555519	semidirect

H ≅ G/N

type

1. Q₈×#11764047

C₂

semidirect

2. #22578438

C₂

semidirect

3. #22578420

C₂

semidirect

4. #22578406

C₂

semidirect

45. Q₈×C₁₃₇

M₁₆

semidirect

65. C₁₃₇

#2991

semidirect

102. Q₈

#14555519

semidirect

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

subgroup	index
Q₈×#11764047	2
#22578438	2
#22578420	2
#22578406	2
#41×Dic₁₃₇	2
#2991	137

subgroup

index

#41×Dic₁₃₇

137

Subgroups — #31797721

Conjugacy classes of subgroups

Complements

Maximal subgroups