Group #31789427 — subgroups

Subgroups — #31789427

Order 17536.

392 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 213 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	#22577464	C₂	1	8	yes	maximal, complemented
2	#22577427	C₂	1	2	yes	maximal, complemented
3	#22577367	C₂	1	2	yes	maximal
4	C₄×#14555494	C₂	1	2	yes	maximal, complemented
5	C₂×#117×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555494	C₂²	1	4	yes	—
7	C₄⋊#11764040	C₂²	1	8	yes	complemented
8	C₂²⋊#11764040	C₂²	1	8	yes	—
9	#17983999	C₂²	1	4	yes	complemented
10	C₄×#11764040	C₂²	1	4	yes	complemented
11	C₂³×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
12	#117×C₁₃₇	C₄	1	4	yes	—
13	#117×C₁₃₇	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₁₃₇	C₄	1	2	yes	—
15	C₂×C₂²⋊C₄×C₁₃₇	C₄	1	2	yes	—
16	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
18	#14555494	—	2	8	no	—
19	C₂×#11764040	C₂³	1	8	yes	—
20	C₂⁴×C₁₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₃₇	—	2	4	no	—
22	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
27	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₃₇	C₂×C₄	1	8	yes	—
29	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	8	yes	—
30	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
31	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
32	C₄²×C₁₃₇	C₂³	1	2	yes	—
33	C₄²×C₁₃₇	C₂³	1	2	yes	—
34	#11764040	—	2	8	no	—
35	C₂³×C₁₃₇	—	2	2	no	—
36	C₂³×C₁₃₇	—	4	2	no	—
37	C₂³×C₁₃₇	C₂²×C₄	1	2	yes	—
38	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₃₇	—	2	2	no	—
40	C₂×C₄×C₁₃₇	—	2	4	no	—
41	C₂×C₄×C₁₃₇	—	2	4	no	—
42	C₂×C₄×C₁₃₇	—	4	2	no	—
43	C₂×C₄×C₁₃₇	—	4	2	no	—
44	C₂×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
45	C₂×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
46	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₃₇	#41	1	4	yes	—
49	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
50	C₂²×C₁₃₇	—	2	2	no	—
51	C₂²×C₁₃₇	—	2	2	no	—
52	C₂²×C₁₃₇	—	2	4	no	—
53	C₂²×C₁₃₇	—	4	4	no	—
54	C₂²×C₁₃₇	#117	1	2	yes	—
55	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
56	C₄×C₁₃₇	—	2	2	no	—
57	C₄×C₁₃₇	—	2	4	no	—
58	C₄×C₁₃₇	—	2	4	no	—
59	C₄×C₁₃₇	#117	1	2	yes	—
60	C₄×C₁₃₇	C₂×#41	1	2	yes	—
61	C₂×C₁₃₇	—	2	2	no	—
62	C₂×C₁₃₇	—	4	2	no	—
63	C₂×C₁₃₇	C₂×#117	1	1	yes	characteristic
64	C₂×C₁₃₇	#568	1	2	yes	—
65	C₁₃₇	#2921	1	1	yes	characteristic, complemented, Sylow
66	#2921	—	137	1	no	maximal, Sylow
67	C₂×#117	D₁₃₇	1	1	yes	characteristic
68	#433	—	137	8	no	—
69	#423	—	137	2	no	—
70	#407	—	137	2	no	—
71	C₄×M₁₆	—	137	2	no	—
72	C₂³×C₄	C₂×D₁₃₇	1	1	yes	characteristic
73	C₂×M₁₆	—	137	4	no	—
74	#117	Dic₁₃₇	1	4	yes	—
75	#117	C₂×D₁₃₇	1	4	yes	—
76	C₂×C₄⋊C₄	Dic₁₃₇	1	2	yes	—
77	C₂×C₂²⋊C₄	Dic₁₃₇	1	2	yes	—
78	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
79	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
80	C₄⋊C₈	—	137	8	no	—
81	C₂²⋊C₈	—	137	8	no	—
82	#97	—	137	4	no	—
83	C₄×C₈	—	137	4	no	—
84	C₂⁴	C₂×Dic₁₃₇	1	1	yes	characteristic
85	C₂²×C₄	—	2	4	no	—
86	C₂²×C₄	C₂×Dic₁₃₇	1	1	yes	characteristic
87	C₂²×C₄	C₂×Dic₁₃₇	1	2	yes	—
88	C₂²×C₄	C₂×Dic₁₃₇	1	2	yes	—
89	C₂²×C₄	C₂×Dic₁₃₇	1	2	yes	—
90	C₂²×C₄	C₂²×D₁₃₇	1	2	yes	—
91	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
92	M₁₆	—	274	8	no	—
93	C₂×C₈	—	137	8	no	—
94	C₄⋊C₄	C₂×Dic₁₃₇	1	8	yes	—
95	C₂²⋊C₄	C₂×Dic₁₃₇	1	8	yes	—
96	C₄²	C₂×Dic₁₃₇	1	2	yes	—
97	C₄²	C₂×Dic₁₃₇	1	2	yes	—
98	C₄²	C₂²×D₁₃₇	1	2	yes	—
99	C₄²	C₂²×D₁₃₇	1	2	yes	—
100	C₂³	—	2	2	no	—
101	C₂³	—	4	2	no	—
102	C₂³	C₂²×Dic₁₃₇	1	2	yes	—
103	C₂³	C₂²×Dic₁₃₇	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	C₂²×Dic₁₃₇	1	4	yes	—
110	C₂×C₄	C₂²×Dic₁₃₇	1	4	yes	—
111	C₂×C₄	C₂²×Dic₁₃₇	1	2	yes	—
112	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
113	C₂×C₄	#14555510	1	4	yes	—
114	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
115	C₈	—	274	8	no	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	4	no	—
119	C₂²	—	4	4	no	—
120	C₂²	#17984047	1	2	yes	—
121	C₂²	C₂³×Dic₁₃₇	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	4	no	—
124	C₄	—	2	4	no	—
125	C₄	#17984047	1	2	yes	—
126	C₄	C₂×#14555510	1	2	yes	—
127	C₂	—	2	2	no	—
128	C₂	—	4	2	no	—
129	C₂	C₂×#17984047	1	1	yes	characteristic
130	C₂	#22578170	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22577464

C₂

yes

maximal, complemented

#22577427

C₂

yes

maximal, complemented

#22577367

C₂

yes

maximal

C₄×#14555494

C₂

yes

maximal, complemented

C₂×#117×C₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555494

C₂²

yes

—

C₄⋊#11764040

C₂²

yes

complemented

C₂²⋊#11764040

C₂²

yes

—

#17983999

C₂²

yes

complemented

C₄×#11764040

C₂²

yes

complemented

C₂³×C₄×C₁₃₇

C₂²

yes

characteristic

#117×C₁₃₇

C₄

yes

—

#117×C₁₃₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₃₇

C₄

yes

—

C₂×C₂²⋊C₄×C₁₃₇

C₄

yes

—

C₂×C₄²×C₁₃₇

C₂²

yes

characteristic

C₂×C₄²×C₁₃₇

C₂²

yes

characteristic

#14555494

—

C₂×#11764040

C₂³

yes

—

C₂⁴×C₁₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₇

C₂×C₄

yes

—

C₂²×C₄×C₁₃₇

C₂×C₄

yes

—

C₂²×C₄×C₁₃₇

C₂×C₄

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₃₇

C₂×C₄

yes

—

C₄²×C₁₃₇

C₂×C₄

yes

—

C₄²×C₁₃₇

C₂×C₄

yes

—

C₄²×C₁₃₇

C₂³

yes

—

C₄²×C₁₃₇

C₂³

yes

—

#11764040

—

C₂³×C₁₃₇

—

C₂³×C₁₃₇

—

C₂³×C₁₃₇

C₂²×C₄

yes

—

C₂³×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

#41

yes

—

C₂×C₄×C₁₃₇

C₂⁴

yes

characteristic

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

#117

yes

—

C₂²×C₁₃₇

C₂³×C₄

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#117

yes

—

C₄×C₁₃₇

C₂×#41

yes

—

C₂×C₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

C₂×#117

yes

characteristic

C₂×C₁₃₇

#568

yes

—

C₁₃₇

#2921

yes

characteristic, complemented, Sylow

#2921

—

137

maximal, Sylow

C₂×#117

D₁₃₇

yes

characteristic

#433

—

137

—

#423

—

137

—

#407

—

137

—

C₄×M₁₆

—

137

—

C₂³×C₄

C₂×D₁₃₇

yes

characteristic

C₂×M₁₆

—

137

—

#117

Dic₁₃₇

yes

—

#117

C₂×D₁₃₇

yes

—

C₂×C₄⋊C₄

Dic₁₃₇

yes

—

C₂×C₂²⋊C₄

Dic₁₃₇

yes

—

C₂×C₄²

C₂×D₁₃₇

yes

characteristic

C₂×C₄²

C₂×D₁₃₇

yes

characteristic

C₄⋊C₈

—

137

—

C₂²⋊C₈

—

137

—

#97

—

137

—

C₄×C₈

—

137

—

C₂⁴

C₂×Dic₁₃₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×Dic₁₃₇

yes

characteristic

C₂²×C₄

C₂×Dic₁₃₇

yes

—

C₂²×C₄

C₂×Dic₁₃₇

yes

—

C₂²×C₄

C₂×Dic₁₃₇

yes

—

C₂²×C₄

C₂²×D₁₃₇

yes

—

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

M₁₆

—

274

—

C₂×C₈

—

137

—

C₄⋊C₄

C₂×Dic₁₃₇

yes

—

C₂²⋊C₄

C₂×Dic₁₃₇

yes

—

C₄²

C₂×Dic₁₃₇

yes

—

C₄²

C₂×Dic₁₃₇

yes

—

C₄²

C₂²×D₁₃₇

yes

—

C₄²

C₂²×D₁₃₇

yes

—

100

C₂³

—

101

C₂³

—

102

C₂³

C₂²×Dic₁₃₇

yes

—

103

C₂³

C₂²×Dic₁₃₇

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂²×Dic₁₃₇

yes

—

110

C₂×C₄

C₂²×Dic₁₃₇

yes

—

111

C₂×C₄

C₂²×Dic₁₃₇

yes

—

112

C₂×C₄

C₂²×Dic₁₃₇

yes

characteristic

113

C₂×C₄

#14555510

yes

—

114

C₂×C₄

C₂³×D₁₃₇

yes

characteristic

115

C₈

—

274

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

#17984047

yes

—

121

C₂²

C₂³×Dic₁₃₇

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#17984047

yes

—

126

C₄

C₂×#14555510

yes

—

127

C₂

—

128

C₂

—

129

C₂

C₂×#17984047

yes

characteristic

130

C₂

#22578170

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂³×C₄
Fitting	C₂×#117×C₁₃₇	C₂
prime radical	C₂⁴×C₁₃₇	C₂×C₄
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	C₂×#117	D₁₃₇
hypocenter	C₁₃₇	#2921
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₃₇

commutator, socle

C₂²×C₁₃₇

C₂³×C₄

Fitting

C₂×#117×C₁₃₇

C₂

prime radical

C₂⁴×C₁₃₇

C₂×C₄

cosocle

C₂×C₄×C₁₃₇

C₂⁴

D₁₃₇

C₁₃₇

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂×C₄ ◁ C₂×#117

N	H ≅ G/N	type
1. #22577464	C₂	semidirect
2. #22577427	C₂	semidirect
4. C₄×#14555494	C₂	semidirect
7. C₄⋊#11764040	C₂²	semidirect
9. #17983999	C₂²	semidirect
10. C₄×#11764040	C₂²	semidirect
65. C₁₃₇	#2921	semidirect

H ≅ G/N

type

1. #22577464

C₂

semidirect

2. #22577427

C₂

semidirect

4. C₄×#14555494

C₂

semidirect

7. C₄⋊#11764040

C₂²

semidirect

9. #17983999

C₂²

semidirect

10. C₄×#11764040

C₂²

semidirect

65. C₁₃₇

#2921

semidirect

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

subgroup	index
#22577464	2
#22577427	2
#22577367	2
C₄×#14555494	2
C₂×#117×C₁₃₇	2
#2921	137

subgroup

index

C₂×#117×C₁₃₇

137

Subgroups — #31789427

Conjugacy classes of subgroups

Complements

Maximal subgroups