Group #31789729 — subgroups

Subgroups — #31789729

Order 17536.

348 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 189 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	#22577960	C₂	1	1	yes	maximal, characteristic
2	#22577531	C₂	1	8	yes	maximal, complemented
3	#22577429	C₂	1	2	yes	maximal
4	#22577427	C₂	1	1	yes	maximal, characteristic
5	#22577358	C₂	1	2	yes	maximal
6	C₂×#97×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#14555494	C₂²	1	2	yes	—
8	#17984088	C₄	1	2	yes	—
9	#17984080	C₄	1	2	yes	—
10	#17984067	C₄	1	4	yes	—
11	C₄×Dic₂₇₄	C₂²	1	4	yes	—
12	#17984047	C₂²	1	2	yes	—
13	C₄⋊#11764040	C₂²	1	4	yes	—
14	#17984017	C₂²	1	8	yes	—
15	#17983998	C₂²	1	4	yes	—
16	C₈×Dic₁₃₇	C₂²	1	4	yes	—
17	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
20	#97×C₁₃₇	C₂²	1	4	yes	—
21	C₂×Dic₂₇₄	C₂×C₄	1	4	yes	—
22	#14555494	—	2	4	no	—
23	C₂×#11764040	C₂³	1	4	yes	—
24	C₄⋊Dic₁₃₇	C₂×C₄	1	4	yes	—
25	C₄⋊Dic₁₃₇	C₂×C₄	1	4	yes	—
26	#14555489	C₂×C₄	1	8	yes	—
27	C₂²⋊Dic₁₃₇	C₂×C₄	1	4	yes	—
28	C₄×Dic₁₃₇	C₂³	1	4	yes	—
29	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₁₃₇	—	2	2	no	—
32	C₂×C₈×C₁₃₇	Q₈	1	4	yes	complemented
33	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
34	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
35	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₁₃₇	C₂³	1	2	yes	—
38	Dic₂₇₄	—	4	4	no	—
39	C₂×Dic₁₃₇	C₂²×C₄	1	8	yes	—
40	#11764040	—	2	4	no	—
41	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₃₇	—	4	2	no	—
43	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₁₃₇	#41	1	2	yes	—
48	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
49	C₈×C₁₃₇	—	2	2	no	—
50	C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
51	C₂²×C₁₃₇	—	2	2	no	—
52	C₂²×C₁₃₇	C₄×Q₈	1	2	yes	—
53	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₁₃₇	—	2	2	no	—
55	C₄×C₁₃₇	—	2	2	no	—
56	C₄×C₁₃₇	C₄×Q₈	1	2	yes	—
57	C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
58	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
59	Dic₁₃₇	—	2	8	no	—
60	C₂×C₁₃₇	—	2	2	no	—
61	C₂×C₁₃₇	C₂×C₄×Q₈	1	1	yes	characteristic
62	C₂×C₁₃₇	#568	1	2	yes	—
63	C₁₃₇	#2961	1	1	yes	characteristic, complemented, Sylow
64	#2961	—	137	1	no	maximal, Sylow
65	#533	—	137	1	no	—
66	#446	—	137	8	no	—
67	#424	—	137	2	no	—
68	#423	—	137	1	no	—
69	#405	—	137	2	no	—
70	C₂×#97	D₁₃₇	1	1	yes	characteristic
71	C₂×M₁₆	—	137	2	no	—
72	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
73	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
74	C₄⋊Q₈	—	137	2	no	—
75	#125	—	137	2	no	—
76	C₂²⋊Q₈	—	137	4	no	—
77	C₄×Q₈	—	137	4	no	—
78	#117	—	137	2	no	—
79	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
80	C₄⋊C₈	—	137	8	no	—
81	C₄⋊C₈	—	137	4	no	—
82	#97	—	137	4	no	—
83	#97	C₂×D₁₃₇	1	4	yes	—
84	C₄×C₈	—	137	4	no	—
85	C₂×Q₈	—	137	4	no	—
86	C₂²×C₄	C₄×D₁₃₇	1	2	yes	—
87	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
88	M₁₆	—	274	4	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	137	4	no	—
91	C₂×C₈	Dic₂₇₄	1	4	yes	complemented
92	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
93	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
94	C₄⋊C₄	—	137	8	no	—
95	C₄⋊C₄	—	137	4	no	—
96	C₄⋊C₄	—	137	4	no	—
97	C₂²⋊C₄	—	137	4	no	—
98	C₄²	—	137	4	no	—
99	C₄²	C₄×D₁₃₇	1	1	yes	characteristic
100	C₄²	C₄×D₁₃₇	1	1	yes	characteristic
101	C₄²	C₂²×D₁₃₇	1	2	yes	—
102	C₂³	C₂×C₄×D₁₃₇	1	1	yes	characteristic
103	Q₈	—	548	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	137	8	no	—
106	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
107	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
108	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
109	C₂×C₄	C₂×Dic₂₇₄	1	2	yes	—
110	C₂×C₄	#14555510	1	2	yes	—
111	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	—	274	4	no	—
114	C₈	C₂×Dic₂₇₄	1	4	yes	—
115	C₂²	—	2	2	no	—
116	C₂²	C₄×Dic₂₇₄	1	2	yes	—
117	C₂²	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	274	8	no	—
121	C₄	C₄×Dic₂₇₄	1	2	yes	—
122	C₄	C₂²×Dic₂₇₄	1	1	yes	characteristic
123	C₄	C₂×#14555510	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	C₂×C₄×Dic₂₇₄	1	1	yes	characteristic
126	C₂	#22578169	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22577960

C₂

yes

maximal, characteristic

#22577531

C₂

yes

maximal, complemented

#22577429

C₂

yes

maximal

#22577427

C₂

yes

maximal, characteristic

#22577358

C₂

yes

maximal

C₂×#97×C₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555494

C₂²

yes

—

#17984088

C₄

yes

—

#17984080

C₄

yes

—

#17984067

C₄

yes

—

C₄×Dic₂₇₄

C₂²

yes

—

#17984047

C₂²

yes

—

C₄⋊#11764040

C₂²

yes

—

#17984017

C₂²

yes

—

#17983998

C₂²

yes

—

C₈×Dic₁₃₇

C₂²

yes

—

C₂²×C₈×C₁₃₇

C₂²

yes

characteristic

C₂²×C₈×C₁₃₇

C₂²

yes

characteristic

C₂×C₄²×C₁₃₇

C₂²

yes

characteristic

#97×C₁₃₇

C₂²

yes

—

C₂×Dic₂₇₄

C₂×C₄

yes

—

#14555494

—

C₂×#11764040

C₂³

yes

—

C₄⋊Dic₁₃₇

C₂×C₄

yes

—

C₄⋊Dic₁₃₇

C₂×C₄

yes

—

#14555489

C₂×C₄

yes

—

C₂²⋊Dic₁₃₇

C₂×C₄

yes

—

C₄×Dic₁₃₇

C₂³

yes

—

C₂²×C₄×C₁₃₇

C₂×C₄

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

Q₈

yes

complemented

C₂×C₈×C₁₃₇

C₂³

yes

—

C₂×C₈×C₁₃₇

C₂³

yes

—

C₄²×C₁₃₇

C₂×C₄

yes

characteristic

C₄²×C₁₃₇

C₂×C₄

yes

characteristic

C₄²×C₁₃₇

C₂³

yes

—

Dic₂₇₄

—

C₂×Dic₁₃₇

C₂²×C₄

yes

—

#11764040

—

C₂³×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×Q₈

yes

—

C₂×C₄×C₁₃₇

#41

yes

—

C₂×C₄×C₁₃₇

C₂⁴

yes

characteristic

C₈×C₁₃₇

—

C₈×C₁₃₇

C₂×Q₈

yes

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₄×Q₈

yes

—

C₂²×C₁₃₇

C₂³×C₄

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₄×Q₈

yes

—

C₄×C₁₃₇

C₂²×Q₈

yes

characteristic

C₄×C₁₃₇

C₂×#41

yes

characteristic

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

C₂×C₄×Q₈

yes

characteristic

C₂×C₁₃₇

#568

yes

—

C₁₃₇

#2961

yes

characteristic, complemented, Sylow

#2961

—

137

maximal, Sylow

#533

—

137

—

#446

—

137

—

#424

—

137

—

#423

—

137

—

#405

—

137

—

C₂×#97

D₁₃₇

yes

characteristic

C₂×M₁₆

—

137

—

C₂²×C₈

C₂×D₁₃₇

yes

characteristic

C₂²×C₈

C₂×D₁₃₇

yes

characteristic

C₄⋊Q₈

—

137

—

#125

—

137

—

C₂²⋊Q₈

—

137

—

C₄×Q₈

—

137

—

#117

—

137

—

C₂×C₄²

C₂×D₁₃₇

yes

characteristic

C₄⋊C₈

—

137

—

C₄⋊C₈

—

137

—

#97

—

137

—

#97

C₂×D₁₃₇

yes

—

C₄×C₈

—

137

—

C₂×Q₈

—

137

—

C₂²×C₄

C₄×D₁₃₇

yes

—

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

M₁₆

—

274

—

C₂×C₈

—

C₂×C₈

—

137

—

C₂×C₈

Dic₂₇₄

yes

complemented

C₂×C₈

C₂²×D₁₃₇

yes

—

C₂×C₈

C₂²×D₁₃₇

yes

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₂²⋊C₄

—

137

—

C₄²

—

137

—

C₄²

C₄×D₁₃₇

yes

characteristic

100

C₄²

C₄×D₁₃₇

yes

characteristic

101

C₄²

C₂²×D₁₃₇

yes

—

102

C₂³

C₂×C₄×D₁₃₇

yes

characteristic

103

Q₈

—

548

—

104

C₂×C₄

—

105

C₂×C₄

—

137

—

106

C₂×C₄

C₂×C₄×D₁₃₇

yes

—

107

C₂×C₄

C₂×C₄×D₁₃₇

yes

—

108

C₂×C₄

C₂×C₄×D₁₃₇

yes

characteristic

109

C₂×C₄

C₂×Dic₂₇₄

yes

—

110

C₂×C₄

#14555510

yes

—

111

C₂×C₄

C₂³×D₁₃₇

yes

characteristic

112

C₈

—

113

C₈

—

274

—

114

C₈

C₂×Dic₂₇₄

yes

—

115

C₂²

—

116

C₂²

C₄×Dic₂₇₄

yes

—

117

C₂²

C₂²×C₄×D₁₃₇

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

274

—

121

C₄

C₄×Dic₂₇₄

yes

—

122

C₄

C₂²×Dic₂₇₄

yes

characteristic

123

C₄

C₂×#14555510

yes

characteristic

124

C₂

—

125

C₂

C₂×C₄×Dic₂₇₄

yes

characteristic

126

C₂

#22578169

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂³×C₄
Fitting	C₂×#97×C₁₃₇	C₂
prime radical	C₂³×C₁₃₇	C₂²×C₄
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	C₂×#97	D₁₃₇
hypocenter	C₁₃₇	#2961
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₃₇

commutator, socle

C₂²×C₁₃₇

C₂³×C₄

Fitting

C₂×#97×C₁₃₇

C₂

prime radical

C₂³×C₁₃₇

C₂²×C₄

cosocle

C₂×C₄×C₁₃₇

C₂⁴

hypercenter

C₂×#97

D₁₃₇

C₁₃₇

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂×C₄ ◁ C₂×#97

N	H ≅ G/N	type
2. #22577531	C₂	semidirect
32. C₂×C₈×C₁₃₇	Q₈	semidirect
63. C₁₃₇	#2961	semidirect
91. C₂×C₈	Dic₂₇₄	semidirect

H ≅ G/N

type

2. #22577531

C₂

semidirect

32. C₂×C₈×C₁₃₇

Q₈

semidirect

63. C₁₃₇

#2961

semidirect

91. C₂×C₈

Dic₂₇₄

semidirect

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

subgroup	index
#22577960	2
#22577531	2
#22577429	2
#22577427	2
#22577358	2
C₂×#97×C₁₃₇	2
#2961	137

subgroup

index

C₂×#97×C₁₃₇

137

Subgroups — #31789729

Conjugacy classes of subgroups

Complements

Maximal subgroups