Group #31781591 — subgroups

Subgroups — #31781591

Order 17536.

332 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 141 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×#17984088	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984013	C₂	1	4	yes	maximal
3	#22577273	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₇₄	C₂²	1	2	yes	—
6	#17984088	—	2	2	no	—
7	#17984088	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₁₃₇	—	2	2	no	—
9	C₂×C₄⋊Dic₁₃₇	—	2	2	no	—
10	C₂×C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
11	#17984013	—	2	8	no	—
12	C₂²×C₈×C₁₃₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₃₇	—	2	2	no	—
15	C₂×Dic₂₇₄	—	2	2	no	—
16	C₂×Dic₂₇₄	—	2	4	no	—
17	C₂×Dic₂₇₄	—	2	4	no	—
18	C₂×Dic₂₇₄	C₂×C₄	1	4	yes	—
19	C₂²×Dic₁₃₇	—	2	2	no	—
20	C₂²×Dic₁₃₇	—	2	2	no	—
21	C₄⋊Dic₁₃₇	—	2	8	no	—
22	C₄⋊Dic₁₃₇	—	2	4	no	—
23	C₄⋊Dic₁₃₇	—	4	4	no	—
24	C₂²×C₄×C₁₃₇	D₄	1	2	yes	—
25	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₃₇	—	2	2	no	—
27	C₂×C₈×C₁₃₇	—	2	2	no	—
28	C₂×C₈×C₁₃₇	D₄	1	4	yes	—
29	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
30	C₄²×C₁₃₇	D₄	1	2	yes	—
31	Dic₂₇₄	—	2	8	no	—
32	Dic₂₇₄	—	4	4	no	—
33	C₂×Dic₁₃₇	—	2	4	no	—
34	C₂×Dic₁₃₇	—	4	2	no	—
35	C₂×Dic₁₃₇	—	4	2	no	—
36	C₂×Dic₁₃₇	—	4	2	no	—
37	C₂×Dic₁₃₇	—	4	4	no	—
38	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	4	yes	—
40	C₂×C₄×C₁₃₇	SD₁₆	1	4	yes	—
41	C₂×C₄×C₁₃₇	Q₁₆	1	4	yes	—
42	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₃₇	#41	1	2	yes	—
46	C₈×C₁₃₇	—	2	4	no	—
47	C₂²×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₁₃₇	#124	1	1	yes	characteristic
49	C₂²×C₁₃₇	#127	1	1	yes	characteristic
50	C₂²×C₁₃₇	C₂×SD₁₆	1	2	yes	—
51	C₂²×C₁₃₇	C₂×Q₁₆	1	2	yes	—
52	C₄×C₁₃₇	#103	1	8	yes	complemented
53	C₄×C₁₃₇	C₄×D₄	1	2	yes	—
54	C₄×C₁₃₇	#121	1	2	yes	—
55	Dic₁₃₇	—	4	4	no	—
56	Dic₁₃₇	—	8	2	no	—
57	C₂×C₁₃₇	#390	1	1	yes	characteristic
58	C₂×C₁₃₇	C₂×#103	1	2	yes	—
59	C₂×C₁₃₇	#487	1	2	yes	—
60	C₂×C₁₃₇	#492	1	1	yes	characteristic
61	C₂×C₁₃₇	#494	1	1	yes	characteristic
62	C₁₃₇	#1956	1	1	yes	characteristic, complemented, Sylow
63	#1956	—	137	1	no	maximal, Sylow
64	C₂×C₄⋊Q₈	—	137	1	no	—
65	C₂×#103	—	137	4	no	—
66	C₂×C₄×C₈	D₁₃₇	1	1	yes	characteristic
67	#384	—	137	1	no	—
68	C₂²×Q₈	—	137	2	no	—
69	C₂²×C₈	C₂×D₁₃₇	1	2	yes	—
70	C₄⋊Q₈	—	274	2	no	—
71	C₄⋊Q₈	—	137	4	no	—
72	C₂×C₄⋊C₄	—	274	2	no	—
73	C₂×C₄⋊C₄	—	274	2	no	—
74	C₂×C₄⋊C₄	—	137	2	no	—
75	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
76	#103	—	274	8	no	—
77	C₄×C₈	—	2	2	no	—
78	C₂×Q₈	—	274	2	no	—
79	C₂×Q₈	—	274	4	no	—
80	C₂×Q₈	—	274	4	no	—
81	C₂×Q₈	—	137	4	no	—
82	C₂²×C₄	—	274	2	no	—
83	C₂²×C₄	—	274	2	no	—
84	C₂²×C₄	D₅₄₈	1	2	yes	—
85	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	D₅₄₈	1	4	yes	—
89	C₄⋊C₄	—	274	8	no	—
90	C₄⋊C₄	—	274	4	no	—
91	C₄⋊C₄	—	548	4	no	—
92	C₄²	C₄×D₁₃₇	1	2	yes	—
93	C₄²	C₂²⋊D₁₃₇	1	2	yes	—
94	C₂³	C₂×D₅₄₈	1	1	yes	characteristic
95	Q₈	—	274	8	no	—
96	Q₈	—	548	4	no	—
97	C₂×C₄	—	274	4	no	—
98	C₂×C₄	—	548	2	no	—
99	C₂×C₄	—	548	2	no	—
100	C₂×C₄	—	548	2	no	—
101	C₂×C₄	—	548	4	no	—
102	C₂×C₄	#14555487	1	4	yes	—
103	C₂×C₄	#14555499	1	4	yes	—
104	C₂×C₄	Dic₅₄₈	1	4	yes	—
105	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
107	C₂×C₄	C₂×D₅₄₈	1	2	yes	—
108	C₂×C₄	#14555510	1	2	yes	—
109	C₈	—	2	4	no	—
110	C₂²	C₂×#14555487	1	1	yes	characteristic
111	C₂²	#17984076	1	1	yes	characteristic
112	C₂²	#17984085	1	1	yes	characteristic
113	C₂²	C₂×#14555499	1	2	yes	—
114	C₂²	C₂×Dic₅₄₈	1	2	yes	—
115	C₄	—	548	4	no	—
116	C₄	—	1096	2	no	—
117	C₄	#17984013	1	8	yes	complemented
118	C₄	C₄×D₅₄₈	1	2	yes	—
119	C₄	#17984062	1	2	yes	—
120	C₂	#22577305	1	1	yes	characteristic
121	C₂	C₂×#17984013	1	2	yes	—
122	C₂	#22577754	1	2	yes	—
123	C₂	#22577776	1	1	yes	characteristic
124	C₂	#22577783	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984088

C₂

yes

maximal, characteristic

C₂×#17984013

C₂

yes

maximal

#22577273

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₁₃₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₇₄

C₂²

yes

—

#17984088

—

#17984088

C₄

yes

complemented

C₂×C₄⋊Dic₁₃₇

—

C₂×C₄⋊Dic₁₃₇

—

C₂×C₄⋊Dic₁₃₇

C₂²

yes

—

#17984013

—

C₂²×C₈×C₁₃₇

C₂²

yes

—

C₂×C₄²×C₁₃₇

C₂²

yes

characteristic

C₄×C₈×C₁₃₇

—

C₂×Dic₂₇₄

—

C₂×Dic₂₇₄

—

C₂×Dic₂₇₄

—

C₂×Dic₂₇₄

C₂×C₄

yes

—

C₂²×Dic₁₃₇

—

C₂²×Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₂²×C₄×C₁₃₇

D₄

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

D₄

yes

—

C₄²×C₁₃₇

C₂×C₄

yes

—

C₄²×C₁₃₇

D₄

yes

—

Dic₂₇₄

—

Dic₂₇₄

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂³×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃₇

SD₁₆

yes

—

C₂×C₄×C₁₃₇

Q₁₆

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂×C₄×C₁₃₇

#41

yes

—

C₈×C₁₃₇

—

C₂²×C₁₃₇

C₂×C₂²⋊C₄

yes

C₂²×C₁₃₇

yes

C₂²×C₁₃₇

yes

C₂²×C₁₃₇

C₂×SD₁₆

yes

—

C₂²×C₁₃₇

C₂×Q₁₆

yes

—

C₄×C₁₃₇

#103

yes

complemented

C₄×C₁₃₇

C₄×D₄

yes

—

C₄×C₁₃₇

#121

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

#390

yes

characteristic

C₂×C₁₃₇

C₂×#103

yes

—

C₂×C₁₃₇

#487

yes

—

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₁₃₇

yes

characteristic, complemented, Sylow

#1956

—

137

maximal, Sylow

C₂×C₄⋊Q₈

—

137

—

C₂×#103

—

137

—

C₂×C₄×C₈

D₁₃₇

yes

characteristic

#384

—

137

—

C₂²×Q₈

—

137

—

C₂²×C₈

C₂×D₁₃₇

yes

—

C₄⋊Q₈

—

274

—

C₄⋊Q₈

—

137

—

C₂×C₄⋊C₄

—

274

—

C₂×C₄⋊C₄

—

274

—

C₂×C₄⋊C₄

—

137

—

C₂×C₄²

C₂×D₁₃₇

yes

characteristic

#103

—

274

—

C₄×C₈

—

C₂×Q₈

—

274

—

C₂×Q₈

—

274

—

C₂×Q₈

—

274

—

C₂×Q₈

—

137

—

C₂²×C₄

—

274

—

C₂²×C₄

—

274

—

C₂²×C₄

D₅₄₈

yes

—

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₅₄₈

yes

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

548

—

C₄²

C₄×D₁₃₇

yes

—

C₄²

C₂²⋊D₁₃₇

yes

—

C₂³

C₂×D₅₄₈

yes

characteristic

Q₈

—

274

—

Q₈

—

548

—

C₂×C₄

—

274

—

C₂×C₄

—

548

—

C₂×C₄

—

548

—

100

C₂×C₄

—

548

—

101

C₂×C₄

—

548

—

102

C₂×C₄

#14555487

yes

—

103

C₂×C₄

#14555499

yes

—

104

C₂×C₄

Dic₅₄₈

yes

—

105

C₂×C₄

C₂×C₄×D₁₃₇

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

characteristic

107

C₂×C₄

C₂×D₅₄₈

yes

—

108

C₂×C₄

#14555510

yes

—

109

C₈

—

110

C₂²

C₂×#14555487

yes

characteristic

111

C₂²

#17984076

yes

characteristic

112

C₂²

#17984085

yes

characteristic

113

C₂²

C₂×#14555499

yes

—

114

C₂²

C₂×Dic₅₄₈

yes

—

115

C₄

—

548

—

116

C₄

—

1096

—

117

C₄

#17984013

yes

complemented

118

C₄

C₄×D₅₄₈

yes

—

119

C₄

#17984062

yes

—

120

C₂

#22577305

yes

characteristic

121

C₂

C₂×#17984013

yes

—

122

C₂

#22577754

yes

—

123

C₂

#22577776

yes

characteristic

124

C₂

#22577783

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₄₈
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₇
Fitting	C₂×C₄×C₈×C₁₃₇	C₂
socle, prime radical	C₂³×C₁₃₇	C₂×D₄
cosocle	C₂²×C₄×C₁₃₇	C₂³
hypercenter	C₂×C₄×C₈	D₁₃₇
hypocenter	C₁₃₇	#1956
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₅₄₈

commutator

C₂×C₄×C₁₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₇

Fitting

C₂×C₄×C₈×C₁₃₇

C₂

socle, prime radical

C₂³×C₁₃₇

C₂×D₄

cosocle

C₂²×C₄×C₁₃₇

C₂³

hypercenter

C₂×C₄×C₈

D₁₃₇

hypocenter

C₁₃₇

#1956

LCS

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
7. #17984088	C₄	semidirect
52. C₄×C₁₃₇	#103	semidirect
62. C₁₃₇	#1956	semidirect
117. C₄	#17984013	semidirect

H ≅ G/N

type

7. #17984088

C₄

semidirect

52. C₄×C₁₃₇

#103

semidirect

62. C₁₃₇

#1956

semidirect

117. C₄

#17984013

semidirect

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

subgroup	index
C₂×#17984088	2
C₂×#17984013	2
#22577273	2
C₂×C₄×C₈×C₁₃₇	2
#1956	137

subgroup

index

C₂×C₄×C₈×C₁₃₇

137

Subgroups — #31781591

Conjugacy classes of subgroups

Complements

Maximal subgroups