Group #31778686 — subgroups

Subgroups — #31778686

Order 17536.

148 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 77 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	#22577456	C₂	1	1	yes	maximal, characteristic
2	#22577427	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555494	—	2	1	no	—
5	C₄⋊#11764040	—	2	1	no	—
6	C₄⋊#11764040	—	2	1	no	—
7	C₄⋊#11764040	C₄	1	2	yes	—
8	C₂²⋊#11764040	C₄	1	2	yes	—
9	C₄×#11764040	—	2	1	no	—
10	C₂²×C₈×C₁₃₇	—	2	1	no	—
11	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₁₃₇	—	2	1	no	—
13	C₄⋊C₈×C₁₃₇	C₄	1	2	yes	—
14	#14555494	—	4	1	no	—
15	#14555494	—	4	1	no	—
16	C₂×#11764040	—	2	2	no	—
17	C₂×#11764040	—	2	2	no	—
18	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₃₇	Q₈	1	1	yes	characteristic
21	C₂×C₈×C₁₃₇	—	2	2	no	—
22	C₂×C₈×C₁₃₇	—	4	1	no	—
23	C₂×C₈×C₁₃₇	—	4	1	no	—
24	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
27	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
28	#11764040	—	4	2	no	—
29	#11764040	—	4	2	no	—
30	C₂³×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₃₇	—	2	2	no	—
32	C₂×C₄×C₁₃₇	—	4	1	no	—
33	C₂×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
34	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₃₇	D₈	1	1	yes	characteristic
40	C₂×C₄×C₁₃₇	SD₁₆	1	1	yes	characteristic
41	C₂×C₄×C₁₃₇	SD₁₆	1	1	yes	characteristic
42	C₂×C₄×C₁₃₇	Q₁₆	1	1	yes	characteristic
43	C₈×C₁₃₇	—	4	2	no	—
44	C₂²×C₁₃₇	—	2	1	no	—
45	C₂²×C₁₃₇	—	2	1	no	—
46	C₂²×C₁₃₇	#95	1	1	yes	characteristic
47	C₂²×C₁₃₇	#106	1	1	yes	characteristic
48	C₂²×C₁₃₇	#107	1	1	yes	characteristic
49	C₄×C₁₃₇	—	2	1	no	—
50	C₄×C₁₃₇	—	2	1	no	—
51	C₄×C₁₃₇	—	2	1	no	—
52	C₄×C₁₃₇	#102	1	1	yes	characteristic
53	C₄×C₁₃₇	#102	1	1	yes	characteristic
54	C₄×C₁₃₇	#103	1	1	yes	characteristic
55	C₄×C₁₃₇	#103	1	1	yes	characteristic
56	C₄×C₁₃₇	#108	1	2	yes	—
57	C₂×C₁₃₇	—	2	1	no	—
58	C₂×C₁₃₇	—	2	1	no	—
59	C₂×C₁₃₇	#340	1	1	yes	characteristic
60	C₂×C₁₃₇	#341	1	1	yes	characteristic
61	C₂×C₁₃₇	#344	1	1	yes	characteristic
62	C₁₃₇	#1301	1	1	yes	characteristic, complemented, Sylow
63	#1301	—	137	1	no	maximal, Sylow
64	#431	—	137	1	no	—
65	#423	—	137	1	no	—
66	C₂×C₄⋊C₈	D₁₃₇	1	1	yes	characteristic
67	C₂×M₁₆	—	274	1	no	—
68	C₂²×C₈	—	2	1	no	—
69	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
70	C₄⋊C₈	—	2	1	no	—
71	C₄⋊C₈	—	274	1	no	—
72	C₄⋊C₈	—	274	1	no	—
73	C₄⋊C₈	—	137	2	no	—
74	C₄⋊C₈	Dic₁₃₇	1	2	yes	—
75	C₂²⋊C₈	—	137	2	no	—
76	C₄×C₈	—	274	1	no	—
77	C₂²×C₄	C₄×D₁₃₇	1	1	yes	characteristic
78	C₂²×C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
79	C₂²×C₄	Dic₂₇₄	1	1	yes	characteristic
80	M₁₆	—	548	1	no	—
81	M₁₆	—	548	1	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	4	1	no	—
84	C₂×C₈	—	4	1	no	—
85	C₂×C₈	—	274	2	no	—
86	C₂×C₈	—	274	2	no	—
87	C₄²	C₄×D₁₃₇	1	1	yes	characteristic
88	C₄²	C₂×Dic₁₃₇	1	1	yes	characteristic
89	C₄²	C₂²⋊D₁₃₇	1	1	yes	characteristic
90	C₄²	D₅₄₈	1	1	yes	characteristic
91	C₂³	#14555489	1	1	yes	characteristic
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	C₄×Dic₁₃₇	1	1	yes	characteristic
95	C₂×C₄	#14555487	1	1	yes	characteristic
96	C₂×C₄	#14555487	1	1	yes	characteristic
97	C₂×C₄	C₂²⋊Dic₁₃₇	1	1	yes	characteristic
98	C₂×C₄	#14555489	1	1	yes	characteristic
99	C₂×C₄	C₄⋊Dic₁₃₇	1	1	yes	characteristic
100	C₂×C₄	#14555495	1	1	yes	characteristic
101	C₂×C₄	#14555497	1	1	yes	characteristic
102	C₂×C₄	#14555498	1	1	yes	characteristic
103	C₂×C₄	#14555500	1	1	yes	characteristic
104	C₈	—	4	2	no	—
105	C₈	—	548	2	no	—
106	C₈	—	548	2	no	—
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	#17983995	1	1	yes	characteristic
110	C₂²	#17984019	1	1	yes	characteristic
111	C₂²	#17984021	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	#17984008	1	1	yes	characteristic
116	C₄	#17984009	1	1	yes	characteristic
117	C₄	#17984011	1	1	yes	characteristic
118	C₄	#17984012	1	1	yes	characteristic
119	C₄	#17984024	1	2	yes	—
120	C₂	—	2	1	no	—
121	C₂	—	2	1	no	—
122	C₂	#22577165	1	1	yes	characteristic
123	C₂	#22577168	1	1	yes	characteristic
124	C₂	#22577175	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22577456

C₂

yes

maximal, characteristic

#22577427

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₁₃₇

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

—

C₂²⋊#11764040

C₄

yes

—

C₄×#11764040

—

C₂²×C₈×C₁₃₇

—

C₂×C₄²×C₁₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₇

—

C₄⋊C₈×C₁₃₇

C₄

yes

—

—

—

—

—

C₂²×C₄×C₁₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₇

D₄

yes

characteristic

C₂²×C₄×C₁₃₇

Q₈

yes

characteristic

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₄²×C₁₃₇

C₂×C₄

yes

characteristic

C₄²×C₁₃₇

C₂×C₄

yes

characteristic

C₄²×C₁₃₇

D₄

yes

characteristic

C₄²×C₁₃₇

D₄

yes

characteristic

#11764040

—

#11764040

—

C₂³×C₁₃₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₄²

yes

characteristic

C₂×C₄×C₁₃₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₇

D₈

yes

characteristic

C₂×C₄×C₁₃₇

SD₁₆

yes

characteristic

C₂×C₄×C₁₃₇

SD₁₆

yes

characteristic

C₂×C₄×C₁₃₇

Q₁₆

yes

characteristic

C₈×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

yes

C₂²×C₁₃₇

yes

C₂²×C₁₃₇

yes

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

—

C₂×C₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₁₃₇

yes

characteristic, complemented, Sylow

#1301

—

137

maximal, Sylow

#431

—

137

—

#423

—

137

—

C₂×C₄⋊C₈

D₁₃₇

yes

characteristic

C₂×M₁₆

—

274

—

C₂²×C₈

—

C₂×C₄²

C₂×D₁₃₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

274

—

C₄⋊C₈

—

274

—

C₄⋊C₈

—

137

—

C₄⋊C₈

Dic₁₃₇

yes

—

C₂²⋊C₈

—

137

—

C₄×C₈

—

274

—

C₂²×C₄

C₄×D₁₃₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₃₇

yes

characteristic

C₂²×C₄

Dic₂₇₄

yes

characteristic

M₁₆

—

548

—

M₁₆

—

548

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

274

—

C₂×C₈

—

274

—

C₄²

C₄×D₁₃₇

yes

characteristic

C₄²

C₂×Dic₁₃₇

yes

characteristic

C₄²

C₂²⋊D₁₃₇

yes

characteristic

C₄²

D₅₄₈

yes

characteristic

C₂³

#14555489

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₃₇

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Dic₁₃₇

yes

characteristic

C₂×C₄

#14555489

yes

characteristic

C₂×C₄

C₄⋊Dic₁₃₇

yes

characteristic

100

C₂×C₄

#14555495

yes

characteristic

101

C₂×C₄

#14555497

yes

characteristic

102

C₂×C₄

#14555498

yes

characteristic

103

C₂×C₄

#14555500

yes

characteristic

104

C₈

—

105

C₈

—

548

—

106

C₈

—

548

—

107

C₂²

—

108

C₂²

—

109

C₂²

#17983995

yes

characteristic

110

C₂²

#17984019

yes

characteristic

111

C₂²

#17984021

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

#17984008

yes

characteristic

116

C₄

#17984009

yes

characteristic

117

C₄

#17984011

yes

characteristic

118

C₄

#17984012

yes

characteristic

119

C₄

#17984024

yes

—

120

C₂

—

121

C₂

—

122

C₂

#22577165

yes

characteristic

123

C₂

#22577168

yes

characteristic

124

C₂

#22577175

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14555487
commutator	C₂×C₄×C₁₃₇	C₄²
Frattini	C₂×C₄²	C₂×D₁₃₇
Fitting	C₂×C₄⋊C₈×C₁₃₇	C₂
socle	C₂²×C₁₃₇	#95
prime radical	C₂³×C₁₃₇	C₄⋊C₄
cosocle	C₂×C₄²×C₁₃₇	C₂²
hypercenter	C₂×C₄⋊C₈	D₁₃₇
hypocenter	C₁₃₇	#1301
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

#14555487

commutator

C₂×C₄×C₁₃₇

C₄²

Frattini

C₂×C₄²

C₂×D₁₃₇

Fitting

C₂×C₄⋊C₈×C₁₃₇

C₂

socle

C₂²×C₁₃₇

#95

prime radical

C₂³×C₁₃₇

C₄⋊C₄

cosocle

C₂×C₄²×C₁₃₇

C₂²

hypercenter

C₂×C₄⋊C₈

D₁₃₇

hypocenter

C₁₃₇

#1301

LCS

G ▷ C₂×C₄×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
62. C₁₃₇	#1301	semidirect

H ≅ G/N

type

62. C₁₃₇

#1301

semidirect

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

subgroup	index
#22577456	2
#22577427	2
C₂×C₄⋊C₈×C₁₃₇	2
#1301	137

subgroup

index

#22577456

#22577427

C₂×C₄⋊C₈×C₁₃₇

#1301

137

Subgroups — #31778686

Conjugacy classes of subgroups

Complements

Maximal subgroups