Group #31791292 — subgroups

Subgroups — C₂×#22577804 #31791292

Order 17536.

360 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 165 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×#17984079	C₂	1	2	yes	maximal
2	#22577804	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17984022	C₂	1	1	yes	maximal, characteristic
4	C₂×#17984019	C₂	1	2	yes	maximal
5	C₂×C₈×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#107×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11764040	C₂²	1	1	yes	characteristic
8	#17984079	—	2	4	no	—
9	#17984079	C₂²	1	8	yes	—
10	C₂×C₄⋊Dic₁₃₇	—	2	2	no	—
11	C₂×C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
12	C₂×#14555489	—	2	2	no	—
13	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
14	#17984022	C₂²	1	4	yes	—
15	#17984019	C₂²	1	8	yes	—
16	C₈×Dic₁₃₇	C₂²	1	4	yes	—
17	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
19	#107×C₁₃₇	C₂²	1	4	yes	—
20	C₂²×Dic₁₃₇	—	2	2	no	—
21	C₂²×Dic₁₃₇	D₄	1	1	yes	characteristic
22	C₂²×Dic₁₃₇	D₄	1	1	yes	characteristic
23	C₂×#11764040	Q₈	1	4	yes	—
24	C₂×#11764040	C₂³	1	2	yes	—
25	C₄⋊Dic₁₃₇	—	2	8	no	—
26	C₄⋊Dic₁₃₇	—	2	2	no	—
27	C₄⋊Dic₁₃₇	C₂³	1	4	yes	—
28	#14555489	—	2	8	no	—
29	C₄×Dic₁₃₇	D₄	1	2	yes	—
30	C₄×Dic₁₃₇	C₂³	1	2	yes	—
31	C₂²×C₄×C₁₃₇	—	2	2	no	—
32	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₁₃₇	Q₈	1	4	yes	—
34	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
35	C₄⋊C₄×C₁₃₇	—	2	2	no	—
36	C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
37	C₂×Dic₁₃₇	—	2	2	no	—
38	C₂×Dic₁₃₇	—	2	2	no	—
39	C₂×Dic₁₃₇	—	2	4	no	—
40	C₂×Dic₁₃₇	—	4	4	no	—
41	C₂×Dic₁₃₇	C₂×D₄	1	2	yes	—
42	C₂×Dic₁₃₇	C₂×D₄	1	2	yes	—
43	#11764040	C₂×Q₈	1	4	yes	—
44	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₃₇	—	2	4	no	—
46	C₂×C₄×C₁₃₇	—	4	4	no	—
47	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
51	C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
52	C₂²×C₁₃₇	C₄⋊Q₈	1	2	yes	—
53	C₂²×C₁₃₇	#135	1	2	yes	—
54	C₂²×C₁₃₇	#135	1	2	yes	—
55	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
56	C₄×C₁₃₇	—	4	4	no	—
57	C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
58	C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
59	C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
60	Dic₁₃₇	—	2	2	no	—
61	Dic₁₃₇	—	2	2	no	—
62	Dic₁₃₇	—	4	4	no	—
63	C₂×C₁₃₇	#499	1	4	yes	complemented
64	C₂×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₁₃₇	C₂×#135	1	1	yes	characteristic
66	C₂×C₁₃₇	C₂×#135	1	1	yes	characteristic
67	C₁₃₇	C₂×#499	1	1	yes	characteristic, complemented, Sylow
68	C₂×#499	—	137	1	no	maximal, Sylow
69	C₂×#125	—	137	2	no	—
70	#499	—	137	8	no	—
71	C₂×#107	—	137	1	no	—
72	C₂×#107	D₁₃₇	1	1	yes	characteristic
73	C₂×#106	—	137	2	no	—
74	C₂×C₄×C₈	—	137	1	no	—
75	C₂²×C₈	—	137	1	no	—
76	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
77	#125	—	274	4	no	—
78	#125	—	137	8	no	—
79	C₂×C₄⋊C₄	—	274	2	no	—
80	C₂×C₄⋊C₄	—	274	2	no	—
81	C₂×C₄⋊C₄	—	137	2	no	—
82	C₂×C₄⋊C₄	C₂×D₁₃₇	1	2	yes	—
83	C₂×C₄²	—	137	1	no	—
84	#107	—	137	4	no	—
85	#107	C₂×D₁₃₇	1	4	yes	—
86	#106	—	137	8	no	—
87	C₄×C₈	—	137	4	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	—	274	2	no	—
90	C₂²×C₄	—	137	1	no	—
91	C₂²×C₄	—	137	1	no	—
92	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
93	C₂×C₈	—	137	4	no	—
94	C₂×C₈	—	137	2	no	—
95	C₂×C₈	Dic₂₇₄	1	4	yes	—
96	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	274	8	no	—
99	C₄⋊C₄	—	274	8	no	—
100	C₄⋊C₄	—	274	2	no	—
101	C₄⋊C₄	—	137	4	no	—
102	C₄⋊C₄	C₂²×D₁₃₇	1	4	yes	—
103	C₄²	—	137	2	no	—
104	C₄²	—	137	2	no	—
105	C₂³	D₄×D₁₃₇	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	4	no	—
108	C₂×C₄	—	274	2	no	—
109	C₂×C₄	—	274	2	no	—
110	C₂×C₄	—	274	4	no	—
111	C₂×C₄	—	548	4	no	—
112	C₂×C₄	—	137	2	no	—
113	C₂×C₄	—	137	2	no	—
114	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
115	C₂×C₄	Q₈×D₁₃₇	1	2	yes	—
116	C₂×C₄	C₂×Dic₂₇₄	1	2	yes	—
117	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
118	C₈	—	137	4	no	—
119	C₈	C₂×Dic₂₇₄	1	4	yes	—
120	C₂²	#17984087	1	2	yes	—
121	C₂²	#17984107	1	2	yes	—
122	C₂²	#17984111	1	2	yes	—
123	C₂²	C₂×D₄×D₁₃₇	1	1	yes	characteristic
124	C₄	—	4	4	no	—
125	C₄	—	274	2	no	—
126	C₄	—	274	2	no	—
127	C₄	—	548	4	no	—
128	C₄	#17984087	1	2	yes	—
129	C₄	C₂×Q₈×D₁₃₇	1	1	yes	characteristic
130	C₄	C₂²×Dic₂₇₄	1	1	yes	characteristic
131	C₂	#22577804	1	4	yes	direct product factor
132	C₂	C₂×#17984087	1	1	yes	characteristic
133	C₂	C₂×#17984107	1	1	yes	characteristic
134	C₂	C₂×#17984111	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984079

C₂

yes

maximal

#22577804

C₂

yes

maximal, direct product factor

C₂×#17984022

C₂

yes

maximal, characteristic

C₂×#17984019

C₂

yes

maximal

C₂×C₈×Dic₁₃₇

C₂

yes

maximal, characteristic

C₂×#107×C₁₃₇

C₂

yes

maximal, characteristic

C₂²×#11764040

C₂²

yes

characteristic

#17984079

—

#17984079

C₂²

yes

—

C₂×C₄⋊Dic₁₃₇

—

C₂×C₄⋊Dic₁₃₇

C₂²

yes

—

C₂×#14555489

—

C₂×C₄×Dic₁₃₇

C₂²

yes

characteristic

#17984022

C₂²

yes

—

#17984019

C₂²

yes

—

C₈×Dic₁₃₇

C₂²

yes

—

C₂²×C₈×C₁₃₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₃₇

C₂²

yes

—

#107×C₁₃₇

C₂²

yes

—

C₂²×Dic₁₃₇

—

C₂²×Dic₁₃₇

D₄

yes

characteristic

C₂²×Dic₁₃₇

D₄

yes

characteristic

C₂×#11764040

Q₈

yes

—

C₂×#11764040

C₂³

yes

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

C₂³

yes

—

#14555489

—

C₄×Dic₁₃₇

D₄

yes

—

C₄×Dic₁₃₇

C₂³

yes

—

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

C₂×C₈×C₁₃₇

Q₈

yes

—

C₂×C₈×C₁₃₇

C₂³

yes

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

C₂³

yes

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

C₂×D₄

yes

—

C₂×Dic₁₃₇

C₂×D₄

yes

—

#11764040

C₂×Q₈

yes

—

C₂³×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×Q₈

yes

—

C₂×C₄×C₁₃₇

C₂×Q₈

yes

—

C₂×C₄×C₁₃₇

C₂⁴

yes

characteristic

C₈×C₁₃₇

C₂×Q₈

yes

—

C₂²×C₁₃₇

C₄⋊Q₈

yes

—

C₂²×C₁₃₇

#135

yes

—

C₂²×C₁₃₇

#135

yes

—

C₂²×C₁₃₇

C₂²×D₄

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

C₄⋊Q₈

yes

—

C₄×C₁₃₇

C₂²×Q₈

yes

characteristic

C₄×C₁₃₇

C₂²×Q₈

yes

characteristic

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

#499

yes

complemented

C₂×C₁₃₇

C₂×C₄⋊Q₈

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₁₃₇

yes

characteristic, complemented, Sylow

C₂×#499

—

137

maximal, Sylow

C₂×#125

—

137

—

#499

—

137

—

C₂×#107

—

137

—

C₂×#107

D₁₃₇

yes

characteristic

C₂×#106

—

137

—

C₂×C₄×C₈

—

137

—

C₂²×C₈

—

137

—

C₂²×C₈

C₂×D₁₃₇

yes

characteristic

#125

—

274

—

#125

—

137

—

C₂×C₄⋊C₄

—

274

—

C₂×C₄⋊C₄

—

274

—

C₂×C₄⋊C₄

—

137

—

C₂×C₄⋊C₄

C₂×D₁₃₇

yes

—

C₂×C₄²

—

137

—

#107

—

137

—

#107

C₂×D₁₃₇

yes

—

#106

—

137

—

C₄×C₈

—

137

—

C₂²×C₄

—

C₂²×C₄

—

274

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

C₂×C₈

—

137

—

C₂×C₈

—

137

—

C₂×C₈

Dic₂₇₄

yes

—

C₂×C₈

C₂²×D₁₃₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

100

C₄⋊C₄

—

274

—

101

C₄⋊C₄

—

137

—

102

C₄⋊C₄

C₂²×D₁₃₇

yes

—

103

C₄²

—

137

—

104

C₄²

—

137

—

105

C₂³

D₄×D₁₃₇

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

274

—

109

C₂×C₄

—

274

—

110

C₂×C₄

—

274

—

111

C₂×C₄

—

548

—

112

C₂×C₄

—

137

—

113

C₂×C₄

—

137

—

114

C₂×C₄

D₄×D₁₃₇

yes

characteristic

115

C₂×C₄

Q₈×D₁₃₇

yes

—

116

C₂×C₄

C₂×Dic₂₇₄

yes

—

117

C₂×C₄

C₂³×D₁₃₇

yes

characteristic

118

C₈

—

137

—

119

C₈

C₂×Dic₂₇₄

yes

—

120

C₂²

#17984087

yes

—

121

C₂²

#17984107

yes

—

122

C₂²

#17984111

yes

—

123

C₂²

C₂×D₄×D₁₃₇

yes

characteristic

124

C₄

—

125

C₄

—

274

—

126

C₄

—

274

—

127

C₄

—

548

—

128

C₄

#17984087

yes

—

129

C₄

C₂×Q₈×D₁₃₇

yes

characteristic

130

C₄

C₂²×Dic₂₇₄

yes

characteristic

131

C₂

#22577804

yes

direct product factor

132

C₂

C₂×#17984087

yes

characteristic

133

C₂

C₂×#17984107

yes

characteristic

134

C₂

C₂×#17984111

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₁₃₇
commutator, cosocle	C₂×C₄×C₁₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	C₂×#107×C₁₃₇	C₂
socle, prime radical	C₂³×C₁₃₇	C₂×D₄
hypercenter	C₂×#107	D₁₃₇
hypocenter	C₁₃₇	C₂×#499
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₁₃₇

commutator, cosocle

C₂×C₄×C₁₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₇

Fitting

C₂×#107×C₁₃₇

C₂

socle, prime radical

C₂³×C₁₃₇

C₂×D₄

D₁₃₇

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
2. #22577804	C₂	direct (N × H)
63. C₂×C₁₃₇	#499	semidirect
67. C₁₃₇	C₂×#499	semidirect
131. C₂	#22577804	direct (N × H)

H ≅ G/N

type

2. #22577804

C₂

direct (N × H)

63. C₂×C₁₃₇

#499

semidirect

67. C₁₃₇

C₂×#499

semidirect

131. C₂

#22577804

direct (N × H)

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

subgroup	index
C₂×#17984079	2
#22577804	2
C₂×#17984022	2
C₂×#17984019	2
C₂×C₈×Dic₁₃₇	2
C₂×#107×C₁₃₇	2
C₂×#499	137

subgroup

index

C₂×C₈×Dic₁₃₇

C₂×#107×C₁₃₇

C₂×#499

137

Subgroups — C₂×#22577804 #31791292

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22577804 #31791292

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22577804 #31791292