Group #31792243 — subgroups

Subgroups — #31792243

Order 17536.

322 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 117 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	#22578015	C₂	1	2	yes	maximal
2	#22577711	C₂	1	4	yes	maximal, complemented
3	#22577661	C₂	1	2	yes	maximal
4	#22577644	C₂	1	2	yes	maximal
5	#22577614	C₂	1	4	yes	maximal, complemented
6	#424×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₅₄₈	C₂²	1	2	yes	—
8	C₂×Dic₅₄₈	C₂²	1	2	yes	—
9	#17984088	C₂²	1	2	yes	—
10	#17984081	—	2	4	no	—
11	#17984080	C₂²	1	2	yes	—
12	#17984078	—	2	4	no	—
13	#17984067	C₂²	1	4	yes	—
14	C₄×Dic₂₇₄	C₂²	1	4	yes	—
15	#17984022	C₂²	1	2	yes	—
16	#17984020	C₂²	1	2	yes	—
17	#17984013	C₂²	1	4	yes	—
18	#17984013	C₂²	1	4	yes	—
19	C₂×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
21	#117×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₃₇	C₂²	1	4	yes	—
23	C₂×Dic₂₇₄	C₂³	1	4	yes	—
24	Dic₅₄₈	—	4	2	no	—
25	Dic₅₄₈	—	4	2	no	—
26	C₄⋊Dic₁₃₇	—	2	4	no	—
27	C₄⋊Dic₁₃₇	C₂³	1	4	yes	—
28	#14555489	—	2	4	no	—
29	#14555489	—	2	4	no	—
30	#14555489	—	2	4	no	—
31	C₂²⋊Dic₁₃₇	—	2	4	no	—
32	C₄×Dic₁₃₇	—	2	4	no	—
33	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
34	M₁₆×C₁₃₇	—	4	1	no	—
35	C₂×C₈×C₁₃₇	—	4	1	no	—
36	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
37	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
38	C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
39	C₂²⋊C₄×C₁₃₇	D₄	1	2	yes	—
40	C₄²×C₁₃₇	C₂³	1	2	yes	—
41	Dic₂₇₄	—	2	4	no	—
42	Dic₂₇₄	—	2	4	no	—
43	C₂×Dic₁₃₇	—	2	4	no	—
44	C₂×Dic₁₃₇	—	2	4	no	—
45	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₃₇	—	2	1	no	—
47	C₂×C₄×C₁₃₇	—	2	1	no	—
48	C₂×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
49	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
51	C₈×C₁₃₇	—	2	2	no	—
52	C₈×C₁₃₇	—	2	2	no	—
53	C₂²×C₁₃₇	—	2	1	no	—
54	C₂²×C₁₃₇	—	2	2	no	—
55	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
56	C₄×C₁₃₇	—	2	1	no	—
57	C₄×C₁₃₇	—	2	4	no	—
58	C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
59	C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
60	Dic₁₃₇	—	4	4	no	—
61	Dic₁₃₇	—	4	4	no	—
62	C₂×C₁₃₇	—	4	1	no	—
63	C₂×C₁₃₇	#537	1	1	yes	characteristic
64	C₂×C₁₃₇	#578	1	2	yes	—
65	C₁₃₇	#3261	1	1	yes	characteristic, complemented, Sylow
66	#3261	—	137	1	no	maximal, Sylow
67	#541	—	137	2	no	—
68	#479	—	137	4	no	—
69	#470	—	137	2	no	—
70	#467	—	137	2	no	—
71	#462	—	137	4	no	—
72	#424	D₁₃₇	1	1	yes	characteristic
73	C₂×Q₁₆	—	137	2	no	—
74	C₂×Q₁₆	—	137	2	no	—
75	C₂×M₁₆	C₂×D₁₃₇	1	1	yes	characteristic
76	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
77	C₄⋊Q₈	—	137	2	no	—
78	#126	—	274	4	no	—
79	#125	—	274	4	no	—
80	#125	—	137	2	no	—
81	C₂²⋊Q₈	—	137	4	no	—
82	C₄×Q₈	—	137	4	no	—
83	#117	C₂×D₁₃₇	1	1	yes	characteristic
84	#107	—	137	2	no	—
85	#106	—	137	2	no	—
86	C₄⋊C₈	C₂×D₁₃₇	1	4	yes	—
87	#103	—	137	4	no	—
88	#103	—	137	4	no	—
89	C₂×Q₈	—	137	4	no	—
90	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
91	Q₁₆	—	548	2	no	—
92	Q₁₆	—	548	2	no	—
93	M₁₆	—	4	1	no	—
94	C₂×C₈	—	4	1	no	—
95	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
96	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
97	C₄⋊C₄	—	274	4	no	—
98	C₄⋊C₄	—	274	4	no	—
99	C₄⋊C₄	—	274	4	no	—
100	C₄⋊C₄	—	274	4	no	—
101	C₄⋊C₄	—	137	4	no	—
102	C₄⋊C₄	D₅₄₈	1	2	yes	—
103	C₂²⋊C₄	—	274	4	no	—
104	C₂²⋊C₄	D₅₄₈	1	2	yes	—
105	C₄²	—	274	4	no	—
106	C₄²	C₂²×D₁₃₇	1	2	yes	—
107	C₂³	C₂×D₅₄₈	1	1	yes	characteristic
108	Q₈	—	274	4	no	—
109	Q₈	—	274	4	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	274	4	no	—
113	C₂×C₄	—	274	4	no	—
114	C₂×C₄	C₂×D₅₄₈	1	4	yes	—
115	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
116	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
117	C₈	—	2	2	no	—
118	C₈	—	2	2	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	1	no	—
121	C₂²	C₂²×D₅₄₈	1	1	yes	characteristic
122	C₄	—	2	4	no	—
123	C₄	—	2	1	no	—
124	C₄	—	548	4	no	—
125	C₄	—	548	4	no	—
126	C₄	#17984133	1	1	yes	characteristic
127	C₄	#17984136	1	1	yes	characteristic
128	C₂	—	4	1	no	—
129	C₂	#22577984	1	1	yes	characteristic
130	C₂	#22578220	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22578015

C₂

yes

maximal

#22577711

C₂

yes

maximal, complemented

#22577661

C₂

yes

maximal

#22577644

C₂

yes

maximal

#22577614

C₂

yes

maximal, complemented

#424×C₁₃₇

C₂

yes

maximal, characteristic

C₂×Dic₅₄₈

C₂²

yes

—

C₂×Dic₅₄₈

C₂²

yes

—

#17984088

C₂²

yes

—

#17984081

—

#17984080

C₂²

yes

—

#17984078

—

#17984067

C₂²

yes

—

C₄×Dic₂₇₄

C₂²

yes

—

#17984022

C₂²

yes

—

#17984020

C₂²

yes

—

#17984013

C₂²

yes

—

#17984013

C₂²

yes

—

C₂×M₁₆×C₁₃₇

C₂²

yes

characteristic

C₂²×C₈×C₁₃₇

C₂²

yes

characteristic

#117×C₁₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₇

C₂²

yes

—

C₂×Dic₂₇₄

C₂³

yes

—

Dic₅₄₈

—

Dic₅₄₈

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

C₂³

yes

—

#14555489

—

#14555489

—

#14555489

—

C₂²⋊Dic₁₃₇

—

C₄×Dic₁₃₇

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

M₁₆×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

C₂³

yes

—

C₂×C₈×C₁₃₇

C₂³

yes

—

C₄⋊C₄×C₁₃₇

D₄

yes

—

C₂²⋊C₄×C₁₃₇

D₄

yes

—

C₄²×C₁₃₇

C₂³

yes

—

Dic₂₇₄

—

Dic₂₇₄

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂³×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂⁴

yes

characteristic

C₈×C₁₃₇

—

C₈×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂²×D₄

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

H(5,𝔽₂)

yes

characteristic

C₄×C₁₃₇

2₋¹⁺⁴

yes

characteristic

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

#537

yes

characteristic

C₂×C₁₃₇

#578

yes

—

C₁₃₇

#3261

yes

characteristic, complemented, Sylow

#3261

—

137

maximal, Sylow

#541

—

137

—

#479

—

137

—

#470

—

137

—

#467

—

137

—

#462

—

137

—

#424

D₁₃₇

yes

characteristic

C₂×Q₁₆

—

137

—

C₂×Q₁₆

—

137

—

C₂×M₁₆

C₂×D₁₃₇

yes

characteristic

C₂²×C₈

C₂×D₁₃₇

yes

characteristic

C₄⋊Q₈

—

137

—

#126

—

274

—

#125

—

274

—

#125

—

137

—

C₂²⋊Q₈

—

137

—

C₄×Q₈

—

137

—

#117

C₂×D₁₃₇

yes

characteristic

#107

—

137

—

#106

—

137

—

C₄⋊C₈

C₂×D₁₃₇

yes

—

#103

—

137

—

#103

—

137

—

C₂×Q₈

—

137

—

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

Q₁₆

—

548

—

Q₁₆

—

548

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²×D₁₃₇

yes

—

C₂×C₈

C₂²×D₁₃₇

yes

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

100

C₄⋊C₄

—

274

—

101

C₄⋊C₄

—

137

—

102

C₄⋊C₄

D₅₄₈

yes

—

103

C₂²⋊C₄

—

274

—

104

C₂²⋊C₄

D₅₄₈

yes

—

105

C₄²

—

274

—

106

C₄²

C₂²×D₁₃₇

yes

—

107

C₂³

C₂×D₅₄₈

yes

characteristic

108

Q₈

—

274

—

109

Q₈

—

274

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

274

—

113

C₂×C₄

—

274

—

114

C₂×C₄

C₂×D₅₄₈

yes

—

115

C₂×C₄

C₂×D₅₄₈

yes

characteristic

116

C₂×C₄

C₂³×D₁₃₇

yes

characteristic

117

C₈

—

118

C₈

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂²×D₅₄₈

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

548

—

125

C₄

—

548

—

126

C₄

#17984133

yes

characteristic

127

C₄

#17984136

yes

characteristic

128

C₂

—

129

C₂

#22577984

yes

characteristic

130

C₂

#22578220

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₅₄₈
commutator, cosocle	C₂×C₄×C₁₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	#424×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂²×D₄
prime radical	C₂³×C₁₃₇	C₂×D₄
hypercenter	#424	D₁₃₇
hypocenter	C₁₃₇	#3261
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #424

subgroup

G/H

center

C₂²

C₂²×D₅₄₈

commutator, cosocle

C₂×C₄×C₁₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₇

Fitting

#424×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 ◁ #117 ◁ #424

N	H ≅ G/N	type
2. #22577711	C₂	semidirect
5. #22577614	C₂	semidirect
65. C₁₃₇	#3261	semidirect

H ≅ G/N

type

2. #22577711

C₂

semidirect

5. #22577614

C₂

semidirect

65. C₁₃₇

#3261

semidirect

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

subgroup	index
#22578015	2
#22577711	2
#22577661	2
#22577644	2
#22577614	2
#424×C₁₃₇	2
#3261	137

subgroup

index

#424×C₁₃₇

137

Subgroups — #31792243

Conjugacy classes of subgroups

Complements

Maximal subgroups