Group #31791342 — subgroups

Subgroups — #31791342

Order 17536.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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×#17984080	C₂	1	1	yes	maximal, characteristic
2	#22577809	C₂	1	4	yes	maximal, complemented
3	#22577802	C₂	1	4	yes	maximal, complemented
4	#22577446	C₂	1	4	yes	maximal
5	C₄×#14555494	C₂	1	1	yes	maximal, characteristic
6	#533×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#14555494	C₂²	1	2	yes	—
8	#17984080	—	2	2	no	—
9	#17984080	C₂²	1	2	yes	—
10	#17984080	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
12	C₂×#14555489	—	2	2	no	—
13	#17984021	C₂²	1	8	yes	—
14	#17984019	C₂²	1	8	yes	—
15	#17983999	C₂²	1	2	yes	—
16	C₄×#11764040	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
18	#125×C₁₃₇	C₂²	1	2	yes	—
19	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
20	C₄×Q₈×C₁₃₇	—	2	2	no	—
21	#117×C₁₃₇	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
23	C₂²×Dic₁₃₇	—	2	2	no	—
24	#14555494	Q₈	1	8	yes	complemented
25	C₂×#11764040	C₂³	1	4	yes	—
26	C₄⋊Dic₁₃₇	—	2	2	no	—
27	C₄⋊Dic₁₃₇	C₂³	1	4	yes	—
28	#14555489	—	2	4	no	—
29	#14555489	—	2	4	no	—
30	C₂×Q₈×C₁₃₇	—	2	2	no	—
31	C₂²×C₄×C₁₃₇	D₄	1	2	yes	—
32	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₁₃₇	—	2	4	no	—
34	C₄⋊C₄×C₁₃₇	—	2	2	no	—
35	C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
36	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
37	C₄²×C₁₃₇	—	2	2	no	—
38	C₄²×C₁₃₇	D₄	1	2	yes	—
39	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
40	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
41	C₂×Dic₁₃₇	—	2	4	no	—
42	C₂×Dic₁₃₇	—	4	4	no	—
43	#11764040	C₂×Q₈	1	8	yes	—
44	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
45	Q₈×C₁₃₇	—	4	4	no	—
46	C₂×C₄×C₁₃₇	—	2	4	no	—
47	C₂×C₄×C₁₃₇	—	4	2	no	—
48	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
52	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
53	C₂²×C₁₃₇	—	2	2	no	—
54	C₂²×C₁₃₇	C₄⋊Q₈	1	2	yes	—
55	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
56	C₄×C₁₃₇	—	2	2	no	—
57	C₄×C₁₃₇	—	2	2	no	—
58	C₄×C₁₃₇	—	4	4	no	—
59	C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
60	C₄×C₁₃₇	C₂²×Q₈	1	2	yes	—
61	Dic₁₃₇	—	4	4	no	—
62	C₂×C₁₃₇	—	2	2	no	—
63	C₂×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₁₃₇	#575	1	2	yes	—
65	C₁₃₇	#3166	1	1	yes	characteristic, complemented, Sylow
66	#3166	—	137	1	no	maximal, Sylow
67	#533	D₁₃₇	1	1	yes	characteristic
68	C₂×#125	—	137	1	no	—
69	#501	—	137	4	no	—
70	#499	—	137	4	no	—
71	#428	—	137	4	no	—
72	C₄×M₁₆	—	137	1	no	—
73	C₂×M₁₆	—	137	2	no	—
74	C₄⋊Q₈	C₂×D₁₃₇	1	2	yes	—
75	#125	—	274	2	no	—
76	#125	—	137	2	no	—
77	#125	—	137	2	no	—
78	#125	C₂×D₁₃₇	1	2	yes	—
79	C₂²⋊Q₈	—	2	2	no	—
80	C₄×Q₈	—	2	2	no	—
81	#117	C₂×D₁₃₇	1	2	yes	—
82	C₂×C₄⋊C₄	—	274	2	no	—
83	C₂×C₄⋊C₄	—	137	2	no	—
84	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
85	#107	—	137	8	no	—
86	#106	—	137	8	no	—
87	#97	—	137	2	no	—
88	C₄×C₈	—	137	2	no	—
89	C₂×Q₈	—	2	2	no	—
90	C₂²×C₄	—	274	2	no	—
91	C₂²×C₄	C₂²⋊D₁₃₇	1	2	yes	—
92	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
93	M₁₆	—	137	8	no	—
94	C₂×C₈	—	137	4	no	—
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	274	4	no	—
98	C₄⋊C₄	—	274	4	no	—
99	C₄⋊C₄	—	274	2	no	—
100	C₄⋊C₄	—	137	4	no	—
101	C₄⋊C₄	C₂²×D₁₃₇	1	4	yes	—
102	C₂²⋊C₄	—	2	2	no	—
103	C₄²	—	2	2	no	—
104	C₄²	C₂²⋊D₁₃₇	1	2	yes	—
105	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
106	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
107	C₂³	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
108	Q₈	—	4	4	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	274	4	no	—
112	C₂×C₄	—	548	4	no	—
113	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	2	yes	—
114	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	2	yes	—
115	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
116	C₂×C₄	Q₈×D₁₃₇	1	4	yes	—
117	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
118	C₈	—	137	8	no	—
119	C₂²	—	2	2	no	—
120	C₂²	#17984086	1	2	yes	—
121	C₂²	C₂²×C₂²⋊D₁₃₇	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	4	4	no	—
125	C₄	—	548	4	no	—
126	C₄	#17984086	1	2	yes	—
127	C₄	C₂×Q₈×D₁₃₇	1	2	yes	—
128	C₂	—	2	2	no	—
129	C₂	C₂×#17984086	1	1	yes	characteristic
130	C₂	#22578201	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984080

C₂

yes

maximal, characteristic

#22577809

C₂

yes

maximal, complemented

#22577802

C₂

yes

maximal, complemented

#22577446

C₂

yes

maximal

C₄×#14555494

C₂

yes

maximal, characteristic

#533×C₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555494

C₂²

yes

—

#17984080

—

#17984080

C₂²

yes

—

#17984080

C₂²

yes

—

C₂×C₄⋊Dic₁₃₇

C₂²

yes

—

C₂×#14555489

—

#17984021

C₂²

yes

—

#17984019

C₂²

yes

—

#17983999

C₂²

yes

—

C₄×#11764040

C₂²

yes

—

C₄⋊Q₈×C₁₃₇

C₂²

yes

—

#125×C₁₃₇

C₂²

yes

—

C₂²⋊Q₈×C₁₃₇

—

C₄×Q₈×C₁₃₇

—

#117×C₁₃₇

C₂²

yes

—

C₂×C₄²×C₁₃₇

C₂²

yes

characteristic

C₂²×Dic₁₃₇

—

#14555494

Q₈

yes

complemented

C₂×#11764040

C₂³

yes

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

C₂³

yes

—

#14555489

—

#14555489

—

C₂×Q₈×C₁₃₇

—

C₂²×C₄×C₁₃₇

D₄

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

C₂³

yes

—

C₂²⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

—

C₄²×C₁₃₇

D₄

yes

—

C₄²×C₁₃₇

C₂³

yes

characteristic

C₄²×C₁₃₇

C₂³

yes

characteristic

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

#11764040

C₂×Q₈

yes

—

C₂³×C₁₃₇

C₂×D₄

yes

characteristic

Q₈×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×Q₈

yes

—

C₂×C₄×C₁₃₇

C₂⁴

yes

characteristic

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₄⋊Q₈

yes

—

C₂²×C₁₃₇

C₂²×D₄

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₄⋊Q₈

yes

—

C₄×C₁₃₇

C₂²×Q₈

yes

—

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₃₇

#575

yes

—

C₁₃₇

#3166

yes

characteristic, complemented, Sylow

#3166

—

137

maximal, Sylow

#533

D₁₃₇

yes

characteristic

C₂×#125

—

137

—

#501

—

137

—

#499

—

137

—

#428

—

137

—

C₄×M₁₆

—

137

—

C₂×M₁₆

—

137

—

C₄⋊Q₈

C₂×D₁₃₇

yes

—

#125

—

274

—

#125

—

137

—

#125

—

137

—

#125

C₂×D₁₃₇

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₁₃₇

yes

—

C₂×C₄⋊C₄

—

274

—

C₂×C₄⋊C₄

—

137

—

C₂×C₄²

C₂×D₁₃₇

yes

characteristic

#107

—

137

—

#106

—

137

—

#97

—

137

—

C₄×C₈

—

137

—

C₂×Q₈

—

C₂²×C₄

—

274

—

C₂²×C₄

C₂²⋊D₁₃₇

yes

—

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

M₁₆

—

137

—

C₂×C₈

—

137

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

100

C₄⋊C₄

—

137

—

101

C₄⋊C₄

C₂²×D₁₃₇

yes

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

C₂²⋊D₁₃₇

yes

—

105

C₄²

C₂²×D₁₃₇

yes

characteristic

106

C₄²

C₂²×D₁₃₇

yes

characteristic

107

C₂³

C₂×C₂²⋊D₁₃₇

yes

characteristic

108

Q₈

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

274

—

112

C₂×C₄

—

548

—

113

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

—

114

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

—

115

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

characteristic

116

C₂×C₄

Q₈×D₁₃₇

yes

—

117

C₂×C₄

C₂³×D₁₃₇

yes

characteristic

118

C₈

—

137

—

119

C₂²

—

120

C₂²

#17984086

yes

—

121

C₂²

C₂²×C₂²⋊D₁₃₇

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

548

—

126

C₄

#17984086

yes

—

127

C₄

C₂×Q₈×D₁₃₇

yes

—

128

C₂

—

129

C₂

C₂×#17984086

yes

characteristic

130

C₂

#22578201

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₃₇
commutator, cosocle	C₂×C₄×C₁₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	#533×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂²×D₄
prime radical	C₂³×C₁₃₇	C₂×D₄
hypercenter	#533	D₁₃₇
hypocenter	C₁₃₇	#3166
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₃₇

commutator, cosocle

C₂×C₄×C₁₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₇

Fitting

#533×C₁₃₇

C₂

socle

C₂²×C₁₃₇

C₂²×D₄

prime radical

C₂³×C₁₃₇

C₂×D₄

D₁₃₇

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #22577809	C₂	semidirect
3. #22577802	C₂	semidirect
24. #14555494	Q₈	semidirect
65. C₁₃₇	#3166	semidirect

H ≅ G/N

type

2. #22577809

C₂

semidirect

3. #22577802

C₂

semidirect

24. #14555494

Q₈

semidirect

65. C₁₃₇

#3166

semidirect

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

subgroup	index
C₂×#17984080	2
#22577809	2
#22577802	2
#22577446	2
C₄×#14555494	2
#533×C₁₃₇	2
#3166	137

subgroup

index

#533×C₁₃₇

137

Subgroups — #31791342

Conjugacy classes of subgroups

Complements

Maximal subgroups