Group #31783033 — subgroups

Subgroups — #31783033

Order 17536.

132 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 47 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	#22577804	C₂	1	1	yes	maximal, characteristic
2	#22577781	C₂	1	1	yes	maximal, characteristic
3	#22577208	C₂	1	1	yes	maximal, characteristic
4	#22577207	C₂	1	1	yes	maximal, characteristic
5	#22577206	C₂	1	1	yes	maximal, characteristic
6	#22577178	C₂	1	1	yes	maximal, characteristic
7	#358×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₅₄₈	C₂²	1	1	yes	characteristic
9	C₂×#14555500	—	2	1	no	—
10	#17984086	—	2	1	no	—
11	#17984079	—	2	1	no	—
12	C₂×#14555485	C₂²	1	1	yes	characteristic
13	#17984022	C₂²	1	1	yes	characteristic
14	#17984019	—	2	1	no	—
15	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
18	#107×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₇₄	—	2	1	no	—
20	Dic₅₄₈	—	2	1	no	—
21	Dic₅₄₈	—	2	1	no	—
22	#14555500	—	4	2	no	—
23	C₂×#11764040	D₄	1	1	yes	characteristic
24	C₄⋊Dic₁₃₇	—	2	1	no	—
25	C₄⋊Dic₁₃₇	—	4	1	no	—
26	#14555489	—	4	1	no	—
27	#14555489	—	4	1	no	—
28	#14555489	—	4	1	no	—
29	C₄×Dic₁₃₇	D₄	1	1	yes	characteristic
30	#14555485	—	2	1	no	—
31	C₂×Q₈×C₁₃₇	—	2	1	no	—
32	Q₁₆×C₁₃₇	—	2	1	no	—
33	Q₁₆×C₁₃₇	—	2	1	no	—
34	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₁₃₇	—	2	1	no	—
36	C₁₆×C₁₃₇	—	2	1	no	—
37	Dic₂₇₄	—	4	1	no	—
38	Dic₂₇₄	—	4	1	no	—
39	C₂×Dic₁₃₇	—	4	1	no	—
40	C₂×Dic₁₃₇	—	4	1	no	—
41	C₂×Dic₁₃₇	D₈	1	1	yes	characteristic
42	C₂×Dic₁₃₇	D₈	1	1	yes	characteristic
43	#11764040	SD₁₆	1	2	yes	—
44	Q₈×C₁₃₇	—	4	1	no	—
45	Q₈×C₁₃₇	—	4	1	no	—
46	C₂×C₄×C₁₃₇	—	4	1	no	—
47	C₂×C₄×C₁₃₇	—	4	1	no	—
48	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
49	C₈×C₁₃₇	#41	1	1	yes	characteristic
50	C₈×C₁₃₇	#41	1	1	yes	characteristic
51	C₂²×C₁₃₇	C₂×D₈	1	1	yes	characteristic
52	C₄×C₁₃₇	—	8	1	no	—
53	C₄×C₁₃₇	—	8	1	no	—
54	C₄×C₁₃₇	#124	1	1	yes	characteristic
55	C₄×C₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
56	Dic₁₃₇	—	2	1	no	—
57	Dic₁₃₇	—	2	1	no	—
58	Dic₁₃₇	—	8	1	no	—
59	Dic₁₃₇	—	8	1	no	—
60	C₂×C₁₃₇	#486	1	1	yes	characteristic
61	C₂×C₁₃₇	#510	1	1	yes	characteristic
62	C₂×C₁₃₇	#510	1	1	yes	characteristic
63	C₁₃₇	#2245	1	1	yes	characteristic, complemented, Sylow
64	#2245	—	137	1	no	maximal, Sylow
65	#499	—	137	1	no	—
66	#494	—	137	1	no	—
67	#358	—	137	1	no	—
68	#358	—	137	1	no	—
69	#358	—	137	1	no	—
70	#358	D₁₃₇	1	1	yes	characteristic
71	#346	—	137	1	no	—
72	C₂×Q₁₆	—	274	1	no	—
73	C₂×Q₁₆	—	137	1	no	—
74	C₂×Q₁₆	C₂×D₁₃₇	1	1	yes	characteristic
75	C₄⋊Q₈	—	274	1	no	—
76	#125	—	274	1	no	—
77	C₂×C₁₆	—	137	1	no	—
78	C₂×C₁₆	C₂×D₁₃₇	1	1	yes	characteristic
79	#107	—	137	1	no	—
80	#107	C₂×D₁₃₇	1	1	yes	characteristic
81	#106	—	274	1	no	—
82	C₄×C₈	—	137	1	no	—
83	C₂×Q₈	—	2	1	no	—
84	C₂×Q₈	—	274	1	no	—
85	Q₁₆	—	2	1	no	—
86	Q₁₆	—	2	1	no	—
87	Q₁₆	—	274	1	no	—
88	Q₁₆	—	274	1	no	—
89	Q₁₆	—	548	2	no	—
90	C₂×C₈	—	137	1	no	—
91	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
92	C₄⋊C₄	—	2	1	no	—
93	C₄⋊C₄	—	274	1	no	—
94	C₄⋊C₄	—	548	1	no	—
95	C₄⋊C₄	—	548	1	no	—
96	C₄⋊C₄	—	548	1	no	—
97	C₄⋊C₄	—	548	1	no	—
98	C₄²	—	137	1	no	—
99	C₁₆	—	2	1	no	—
100	C₁₆	—	274	1	no	—
101	Q₈	—	4	1	no	—
102	Q₈	—	4	1	no	—
103	Q₈	—	548	1	no	—
104	Q₈	—	548	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	548	1	no	—
108	C₂×C₄	—	548	1	no	—
109	C₂×C₄	—	137	1	no	—
110	C₂×C₄	—	137	1	no	—
111	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
112	C₈	—	137	2	no	—
113	C₈	#14555509	1	1	yes	characteristic
114	C₈	#14555510	1	1	yes	characteristic
115	C₂²	D₈×D₁₃₇	1	1	yes	characteristic
116	C₄	—	8	1	no	—
117	C₄	—	8	1	no	—
118	C₄	—	274	1	no	—
119	C₄	—	274	1	no	—
120	C₄	—	1096	1	no	—
121	C₄	—	1096	1	no	—
122	C₄	#17984074	1	1	yes	characteristic
123	C₄	SD₁₆×D₁₃₇	1	1	yes	characteristic
124	C₂	#22577748	1	1	yes	characteristic
125	C₂	#22577842	1	1	yes	characteristic
126	C₂	#22577845	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22577804

C₂

yes

maximal, characteristic

#22577781

C₂

yes

maximal, characteristic

#22577208

C₂

yes

maximal, characteristic

#22577207

C₂

yes

maximal, characteristic

#22577206

C₂

yes

maximal, characteristic

#22577178

C₂

yes

maximal, characteristic

#358×C₁₃₇

C₂

yes

maximal, characteristic

C₂×Dic₅₄₈

C₂²

yes

—

—

—

C₂²

yes

characteristic

#17984022

C₂²

yes

characteristic

#17984019

—

C₈×Dic₁₃₇

C₂²

yes

characteristic

C₂×Q₁₆×C₁₃₇

C₂²

yes

characteristic

C₂×C₁₆×C₁₃₇

C₂²

yes

characteristic

#107×C₁₃₇

C₂²

yes

characteristic

C₂×Dic₂₇₄

—

Dic₅₄₈

—

Dic₅₄₈

—

#14555500

—

C₂×#11764040

D₄

yes

characteristic

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

#14555489

—

#14555489

—

#14555489

—

C₄×Dic₁₃₇

D₄

yes

characteristic

#14555485

—

C₂×Q₈×C₁₃₇

—

Q₁₆×C₁₃₇

—

Q₁₆×C₁₃₇

—

C₂×C₈×C₁₃₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₇

—

C₁₆×C₁₃₇

—

Dic₂₇₄

—

Dic₂₇₄

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

D₈

yes

characteristic

C₂×Dic₁₃₇

D₈

yes

characteristic

#11764040

SD₁₆

yes

—

Q₈×C₁₃₇

—

Q₈×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

C₈×C₁₃₇

yes

C₈×C₁₃₇

yes

C₂²×C₁₃₇

C₂×D₈

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#124

yes

characteristic

C₄×C₁₃₇

C₂×SD₁₆

yes

characteristic

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₁₃₇

yes

characteristic, complemented, Sylow

#2245

—

137

maximal, Sylow

#499

—

137

—

#494

—

137

—

#358

—

137

—

#358

—

137

—

#358

—

137

—

#358

D₁₃₇

yes

characteristic

#346

—

137

—

C₂×Q₁₆

—

274

—

C₂×Q₁₆

—

137

—

C₂×Q₁₆

C₂×D₁₃₇

yes

characteristic

C₄⋊Q₈

—

274

—

#125

—

274

—

C₂×C₁₆

—

137

—

C₂×C₁₆

C₂×D₁₃₇

yes

characteristic

#107

—

137

—

#107

C₂×D₁₃₇

yes

characteristic

#106

—

274

—

C₄×C₈

—

137

—

C₂×Q₈

—

C₂×Q₈

—

274

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

274

—

Q₁₆

—

274

—

Q₁₆

—

548

—

C₂×C₈

—

137

—

C₂×C₈

C₂²×D₁₃₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

548

—

C₄⋊C₄

—

548

—

C₄⋊C₄

—

548

—

C₄⋊C₄

—

548

—

C₄²

—

137

—

C₁₆

—

100

C₁₆

—

274

—

101

Q₈

—

102

Q₈

—

103

Q₈

—

548

—

104

Q₈

—

548

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

548

—

108

C₂×C₄

—

548

—

109

C₂×C₄

—

137

—

110

C₂×C₄

—

137

—

111

C₂×C₄

D₄×D₁₃₇

yes

characteristic

112

C₈

—

137

—

113

C₈

#14555509

yes

characteristic

114

C₈

#14555510

yes

characteristic

115

C₂²

D₈×D₁₃₇

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

274

—

119

C₄

—

274

—

120

C₄

—

1096

—

121

C₄

—

1096

—

122

C₄

#17984074

yes

characteristic

123

C₄

SD₁₆×D₁₃₇

yes

characteristic

124

C₂

#22577748

yes

characteristic

125

C₂

#22577842

yes

characteristic

126

C₂

#22577845

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₃₇
commutator, cosocle	C₂×C₈×C₁₃₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₃₇
Fitting	#358×C₁₃₇	C₂
socle, prime radical	C₂²×C₁₃₇	C₂×D₈
hypercenter	#358	D₁₃₇
hypocenter	C₁₃₇	#2245
LCS	G ▷ C₂×C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₁₃₇

commutator, cosocle

C₂×C₈×C₁₃₇

C₂³

Frattini

C₂×C₈

C₂²×D₁₃₇

Fitting

#358×C₁₃₇

C₂

socle, prime radical

C₂²×C₁₃₇

C₂×D₈

D₁₃₇

C₁₃₇

G ▷ C₂×C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
63. C₁₃₇	#2245	semidirect

H ≅ G/N

type

63. C₁₃₇

#2245

semidirect

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

subgroup	index
#22577804	2
#22577781	2
#22577208	2
#22577207	2
#22577206	2
#22577178	2
#358×C₁₃₇	2
#2245	137

subgroup

index

#358×C₁₃₇

137

Subgroups — #31783033

Conjugacy classes of subgroups

Complements

Maximal subgroups