Group #31780912 — subgroups

Subgroups — #31780912

Order 17536.

172 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 81 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×#17984024	C₂	1	1	yes	maximal, characteristic
2	#22577358	C₂	1	1	yes	maximal, characteristic
3	#22577173	C₂	1	2	yes	maximal
4	#22577159	C₂	1	2	yes	maximal
5	#427×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555494	C₂²	1	2	yes	—
7	C₂×#14555494	C₂²	1	1	yes	characteristic
8	#17984047	C₂²	1	1	yes	characteristic
9	#17984024	—	2	1	no	—
10	#17984024	—	2	1	no	—
11	#17983998	C₄	1	2	yes	complemented
12	C₈×Dic₁₃₇	C₄	1	2	yes	complemented
13	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
14	#117×C₁₃₇	C₂²	1	2	yes	—
15	#107×C₁₃₇	—	2	1	no	—
16	#106×C₁₃₇	—	2	1	no	—
17	#14555494	—	2	2	no	—
18	#14555494	—	2	1	no	—
19	#14555494	—	2	1	no	—
20	#14555494	—	2	2	no	—
21	C₂×#11764040	—	2	2	no	—
22	C₂×#11764040	C₂×C₄	1	2	yes	—
23	C₄⋊Dic₁₃₇	—	2	1	no	—
24	C₂²⋊Dic₁₃₇	—	2	1	no	—
25	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
30	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
31	C₂×C₈×C₁₃₇	Q₈	1	1	yes	characteristic
32	C₂×C₈×C₁₃₇	Q₈	1	1	yes	characteristic
33	C₄⋊C₄×C₁₃₇	—	2	2	no	—
34	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
35	C₄²×C₁₃₇	—	2	2	no	—
36	C₂×Dic₁₃₇	—	2	2	no	—
37	#11764040	—	2	2	no	—
38	#11764040	—	4	2	no	—
39	C₂³×C₁₃₇	#41	1	1	yes	characteristic
40	C₂×C₄×C₁₃₇	—	4	2	no	—
41	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₁₃₇	#41	1	1	yes	characteristic
45	C₂×C₄×C₁₃₇	#41	1	1	yes	characteristic
46	C₂×C₄×C₁₃₇	#41	1	1	yes	characteristic
47	C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
48	C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
49	C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
50	C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₁₃₇	—	4	1	no	—
52	C₂²×C₁₃₇	#117	1	1	yes	characteristic
53	C₂²×C₁₃₇	#124	1	1	yes	characteristic
54	C₂²×C₁₃₇	#125	1	1	yes	characteristic
55	C₄×C₁₃₇	—	8	2	no	—
56	C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₄×C₁₃₇	#117	1	1	yes	characteristic
58	C₄×C₁₃₇	#124	1	1	yes	characteristic
59	C₄×C₁₃₇	#125	1	1	yes	characteristic
60	Dic₁₃₇	—	4	2	no	—
61	C₂×C₁₃₇	—	2	1	no	—
62	C₂×C₁₃₇	—	2	1	no	—
63	C₂×C₁₃₇	—	2	1	no	—
64	C₂×C₁₃₇	#382	1	1	yes	characteristic
65	C₁₃₇	#1833	1	1	yes	characteristic, complemented, Sylow
66	#1833	—	137	1	no	maximal, Sylow
67	C₂×#108	—	137	1	no	—
68	#427	D₁₃₇	1	1	yes	characteristic
69	#405	—	137	1	no	—
70	#344	—	137	2	no	—
71	H(3,ℤ/4)	—	137	2	no	—
72	C₂×M₁₆	—	137	2	no	—
73	C₂×M₁₆	—	137	1	no	—
74	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
75	#117	—	137	1	no	—
76	#117	C₂×D₁₃₇	1	2	yes	—
77	#108	—	274	1	no	—
78	#108	—	274	1	no	—
79	#107	—	2	1	no	—
80	#106	—	2	1	no	—
81	#97	—	137	2	no	—
82	C₄×C₈	—	137	2	no	—
83	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
84	M₁₆	—	274	1	no	—
85	M₁₆	—	274	1	no	—
86	M₁₆	—	274	2	no	—
87	M₁₆	—	274	2	no	—
88	C₂×C₈	—	274	2	no	—
89	C₂×C₈	—	137	2	no	—
90	C₂×C₈	C₄×D₁₃₇	1	1	yes	characteristic
91	C₂×C₈	C₄×D₁₃₇	1	1	yes	characteristic
92	C₂×C₈	C₂²⋊D₁₃₇	1	1	yes	characteristic
93	C₂×C₈	C₂²⋊D₁₃₇	1	1	yes	characteristic
94	C₂×C₈	Dic₂₇₄	1	1	yes	characteristic
95	C₂×C₈	Dic₂₇₄	1	1	yes	characteristic
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	274	1	no	—
98	C₂²⋊C₄	—	2	2	no	—
99	C₂²⋊C₄	—	274	1	no	—
100	C₄²	—	2	2	no	—
101	C₄²	—	137	2	no	—
102	C₂³	#14555509	1	1	yes	characteristic
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	274	2	no	—
105	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
107	C₂×C₄	C₂×Dic₂₇₄	1	1	yes	characteristic
108	C₂×C₄	#14555509	1	1	yes	characteristic
109	C₂×C₄	#14555511	1	1	yes	characteristic
110	C₂×C₄	#14555511	1	1	yes	characteristic
111	C₈	—	274	2	no	—
112	C₈	—	548	2	no	—
113	C₈	#14555489	1	1	yes	characteristic
114	C₈	#14555489	1	1	yes	characteristic
115	C₈	#14555489	1	1	yes	characteristic
116	C₈	#14555489	1	1	yes	characteristic
117	C₂²	—	4	1	no	—
118	C₂²	#17984046	1	1	yes	characteristic
119	C₂²	#17984075	1	1	yes	characteristic
120	C₂²	#17984079	1	1	yes	characteristic
121	C₄	—	8	2	no	—
122	C₄	—	548	2	no	—
123	C₄	C₂×#14555489	1	1	yes	characteristic
124	C₄	#17984046	1	1	yes	characteristic
125	C₄	#17984077	1	1	yes	characteristic
126	C₄	#17984079	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#22577268	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984024

C₂

yes

maximal, characteristic

#22577358

C₂

yes

maximal, characteristic

#22577173

C₂

yes

maximal

#22577159

C₂

yes

maximal

#427×C₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555494

C₂²

yes

—

C₂×#14555494

C₂²

yes

characteristic

#17984047

C₂²

yes

—

—

C₄

yes

complemented

C₈×Dic₁₃₇

C₄

yes

complemented

C₂²×C₈×C₁₃₇

C₂²

yes

characteristic

#117×C₁₃₇

C₂²

yes

—

#107×C₁₃₇

—

#106×C₁₃₇

—

—

—

—

—

—

C₂×C₄

yes

—

C₄⋊Dic₁₃₇

—

C₂²⋊Dic₁₃₇

—

C₄×Dic₁₃₇

C₂×C₄

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

C₂×C₈×C₁₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₇

D₄

yes

characteristic

C₂×C₈×C₁₃₇

D₄

yes

characteristic

C₂×C₈×C₁₃₇

Q₈

yes

characteristic

C₂×C₈×C₁₃₇

Q₈

yes

characteristic

C₄⋊C₄×C₁₃₇

—

C₂²⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

—

C₂×Dic₁₃₇

—

#11764040

—

#11764040

—

C₂³×C₁₃₇

#41

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×Q₈

yes

C₂×C₄×C₁₃₇

yes

C₂×C₄×C₁₃₇

yes

C₂×C₄×C₁₃₇

yes

C₈×C₁₃₇

C₄⋊C₄

yes

characteristic

C₈×C₁₃₇

C₄⋊C₄

yes

characteristic

C₈×C₁₃₇

C₄⋊C₄

yes

characteristic

C₈×C₁₃₇

C₄⋊C₄

yes

characteristic

C₂²×C₁₃₇

—

C₂²×C₁₃₇

yes

C₂²×C₁₃₇

yes

C₂²×C₁₃₇

yes

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×C₄⋊C₄

yes

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

#382

yes

characteristic

C₁₃₇

#1833

yes

characteristic, complemented, Sylow

#1833

—

137

maximal, Sylow

C₂×#108

—

137

—

#427

D₁₃₇

yes

characteristic

#405

—

137

—

#344

—

137

—

H(3,ℤ/4)

—

137

—

C₂×M₁₆

—

137

—

C₂×M₁₆

—

137

—

C₂²×C₈

C₂×D₁₃₇

yes

characteristic

#117

—

137

—

#117

C₂×D₁₃₇

yes

—

#108

—

274

—

#108

—

274

—

#107

—

#106

—

#97

—

137

—

C₄×C₈

—

137

—

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

M₁₆

—

274

—

M₁₆

—

274

—

M₁₆

—

274

—

M₁₆

—

274

—

C₂×C₈

—

274

—

C₂×C₈

—

137

—

C₂×C₈

C₄×D₁₃₇

yes

characteristic

C₂×C₈

C₄×D₁₃₇

yes

characteristic

C₂×C₈

C₂²⋊D₁₃₇

yes

characteristic

C₂×C₈

C₂²⋊D₁₃₇

yes

characteristic

C₂×C₈

Dic₂₇₄

yes

characteristic

C₂×C₈

Dic₂₇₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

274

—

C₂²⋊C₄

—

C₂²⋊C₄

—

274

—

100

C₄²

—

101

C₄²

—

137

—

102

C₂³

#14555509

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

274

—

105

C₂×C₄

C₂×C₄×D₁₃₇

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

characteristic

107

C₂×C₄

C₂×Dic₂₇₄

yes

characteristic

108

C₂×C₄

#14555509

yes

characteristic

109

C₂×C₄

#14555511

yes

characteristic

110

C₂×C₄

#14555511

yes

characteristic

111

C₈

—

274

—

112

C₈

—

548

—

113

C₈

#14555489

yes

characteristic

114

C₈

#14555489

yes

characteristic

115

C₈

#14555489

yes

characteristic

116

C₈

#14555489

yes

characteristic

117

C₂²

—

118

C₂²

#17984046

yes

characteristic

119

C₂²

#17984075

yes

characteristic

120

C₂²

#17984079

yes

characteristic

121

C₄

—

122

C₄

—

548

—

123

C₄

C₂×#14555489

yes

characteristic

124

C₄

#17984046

yes

characteristic

125

C₄

#17984077

yes

characteristic

126

C₄

#17984079

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#22577268

yes

characteristic

	subgroup	G/H
center	C₄	#17984046
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₇
Fitting	#427×C₁₃₇	C₂
socle	C₂×C₁₃₇	#382
prime radical	C₂³×C₁₃₇	#41
cosocle	C₂²×C₄×C₁₃₇	C₂³
hypercenter	#427	D₁₃₇
hypocenter	C₁₃₇	#1833
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₄

#17984046

commutator

C₂×C₄×C₁₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₇

Fitting

#427×C₁₃₇

C₂

C₂×C₁₃₇

C₂³×C₁₃₇

C₂²×C₄×C₁₃₇

C₂³

D₁₃₇

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
11. #17983998	C₄	semidirect
12. C₈×Dic₁₃₇	C₄	semidirect
65. C₁₃₇	#1833	semidirect

H ≅ G/N

type

11. #17983998

C₄

semidirect

12. C₈×Dic₁₃₇

C₄

semidirect

65. C₁₃₇

#1833

semidirect

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

subgroup	index
C₂×#17984024	2
#22577358	2
#22577173	2
#22577159	2
#427×C₁₃₇	2
#1833	137

subgroup

index

#427×C₁₃₇

137

Subgroups — #31780912

Conjugacy classes of subgroups

Complements

Maximal subgroups