Group #31780901 — subgroups

Subgroups — #31780901

Order 17536.

236 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 81 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	#22577881	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984007	C₂	1	1	yes	maximal, characteristic
3	#22577175	C₂	1	2	yes	maximal
4	#22577174	C₂	1	2	yes	maximal
5	C₂×#101×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555494	C₂²	1	2	yes	—
7	Q₈×Dic₁₃₇	—	2	2	no	—
8	Q₈×Dic₁₃₇	—	2	2	no	—
9	#17984047	C₄	1	4	yes	—
10	#17984047	C₂²	1	1	yes	characteristic
11	#17984047	C₂²	1	1	yes	characteristic
12	#17984007	—	2	2	no	—
13	C₂²×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₁₃₇	C₂²	1	2	yes	—
15	#101×C₁₃₇	—	2	2	no	—
16	#14555494	—	2	2	no	—
17	#14555494	—	2	2	no	—
18	C₂×#11764040	—	2	2	no	—
19	C₄⋊Dic₁₃₇	—	2	4	no	—
20	C₄⋊Dic₁₃₇	—	2	1	no	—
21	C₄⋊Dic₁₃₇	—	2	1	no	—
22	C₂²⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
23	C₂²⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
24	C₄×Dic₁₃₇	—	2	4	no	—
25	C₄×Dic₁₃₇	—	2	1	no	—
26	C₄×Dic₁₃₇	—	2	1	no	—
27	C₂×Q₈×C₁₃₇	—	2	2	no	—
28	C₂×Q₈×C₁₃₇	—	2	2	no	—
29	C₂×Q₈×C₁₃₇	D₄	1	2	yes	—
30	C₂×Q₈×C₁₃₇	D₄	1	2	yes	—
31	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
33	M₁₆×C₁₃₇	—	2	2	no	—
34	M₁₆×C₁₃₇	—	2	2	no	—
35	C₂×C₈×C₁₃₇	—	2	2	no	—
36	C₂×Dic₁₃₇	—	2	2	no	—
37	C₂×Dic₁₃₇	—	2	2	no	—
38	#11764040	—	4	2	no	—
39	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
40	Q₈×C₁₃₇	—	2	4	no	—
41	Q₈×C₁₃₇	—	2	4	no	—
42	C₂×C₄×C₁₃₇	—	2	2	no	—
43	C₂×C₄×C₁₃₇	—	2	2	no	—
44	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	4	yes	—
45	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₃₇	#41	1	2	yes	—
48	C₂×C₄×C₁₃₇	#41	1	2	yes	—
49	C₈×C₁₃₇	—	4	2	no	—
50	C₂²×C₁₃₇	—	4	1	no	—
51	C₂²×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₁₃₇	#117	1	1	yes	characteristic
53	C₂²×C₁₃₇	#117	1	1	yes	characteristic
54	C₄×C₁₃₇	—	2	4	no	—
55	C₄×C₁₃₇	#123	1	2	yes	—
56	C₄×C₁₃₇	#123	1	2	yes	—
57	Dic₁₃₇	—	4	2	no	—
58	Dic₁₃₇	—	4	2	no	—
59	C₂×C₁₃₇	—	2	1	no	—
60	C₂×C₁₃₇	—	2	1	no	—
61	C₂×C₁₃₇	—	2	1	no	—
62	C₂×C₁₃₇	#381	1	1	yes	characteristic
63	C₁₃₇	#1830	1	1	yes	characteristic, complemented, Sylow
64	#1830	—	137	1	no	maximal, Sylow
65	#519	—	137	1	no	—
66	C₂×#101	—	137	1	no	—
67	C₂×#101	D₁₃₇	1	1	yes	characteristic
68	#344	—	137	2	no	—
69	#344	—	137	2	no	—
70	C₂²×Q₈	C₂×D₁₃₇	1	1	yes	characteristic
71	C₂×M₁₆	—	137	2	no	—
72	C₂×M₁₆	C₂×D₁₃₇	1	2	yes	—
73	C₄×Q₈	—	274	2	no	—
74	C₄×Q₈	—	274	2	no	—
75	#117	—	137	4	no	—
76	#117	—	137	1	no	—
77	#117	—	137	1	no	—
78	#101	—	2	2	no	—
79	#101	—	274	2	no	—
80	C₂×Q₈	—	2	2	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	C₂²⋊D₁₃₇	1	2	yes	—
83	C₂×Q₈	D₅₄₈	1	2	yes	—
84	C₂²×C₄	C₄×D₁₃₇	1	2	yes	—
85	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
86	M₁₆	—	2	2	no	—
87	M₁₆	—	2	2	no	—
88	M₁₆	—	274	2	no	—
89	M₁₆	—	274	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	274	2	no	—
92	C₄⋊C₄	—	274	4	no	—
93	C₄⋊C₄	—	274	1	no	—
94	C₄⋊C₄	—	274	1	no	—
95	C₂²⋊C₄	—	137	2	no	—
96	C₂²⋊C₄	—	137	2	no	—
97	C₄²	—	274	4	no	—
98	C₄²	—	274	1	no	—
99	C₄²	—	274	1	no	—
100	C₂³	C₂×C₄×D₁₃₇	1	1	yes	characteristic
101	Q₈	—	2	4	no	—
102	Q₈	—	2	4	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	274	2	no	—
106	C₂×C₄	—	274	2	no	—
107	C₂×C₄	#14555487	1	4	yes	—
108	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
109	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
110	C₂×C₄	#14555509	1	2	yes	—
111	C₂×C₄	#14555509	1	2	yes	—
112	C₈	—	4	2	no	—
113	C₈	—	548	2	no	—
114	C₂²	—	4	1	no	—
115	C₂²	C₂×#14555487	1	1	yes	characteristic
116	C₂²	#17984044	1	1	yes	characteristic
117	C₂²	#17984044	1	1	yes	characteristic
118	C₄	—	2	4	no	—
119	C₄	—	548	2	no	—
120	C₄	—	548	2	no	—
121	C₄	#17984069	1	2	yes	—
122	C₄	#17984072	1	2	yes	—
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	—	2	1	no	—
126	C₂	#22577264	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22577881

C₂

yes

maximal, characteristic

C₂×#17984007

C₂

yes

maximal, characteristic

#22577175

C₂

yes

maximal

#22577174

C₂

yes

maximal

C₂×#101×C₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555494

C₂²

yes

—

Q₈×Dic₁₃₇

—

Q₈×Dic₁₃₇

—

#17984047

C₄

yes

—

#17984047

C₂²

yes

characteristic

#17984047

C₂²

yes

characteristic

#17984007

—

C₂²×Q₈×C₁₃₇

C₂²

yes

characteristic

C₂×M₁₆×C₁₃₇

C₂²

yes

—

#101×C₁₃₇

—

#14555494

—

#14555494

—

C₂×#11764040

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₂²⋊Dic₁₃₇

C₂×C₄

yes

—

C₂²⋊Dic₁₃₇

C₂×C₄

yes

—

C₄×Dic₁₃₇

—

C₄×Dic₁₃₇

—

C₄×Dic₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

D₄

yes

—

C₂×Q₈×C₁₃₇

D₄

yes

—

C₂²×C₄×C₁₃₇

C₂×C₄

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

M₁₆×C₁₃₇

—

M₁₆×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

#11764040

—

C₂³×C₁₃₇

C₂²×C₄

yes

characteristic

Q₈×C₁₃₇

—

Q₈×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

#41

yes

—

C₂×C₄×C₁₃₇

#41

yes

—

C₈×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂×C₂²⋊C₄

yes

C₂²×C₁₃₇

yes

C₂²×C₁₃₇

yes

C₄×C₁₃₇

—

C₄×C₁₃₇

#123

yes

—

C₄×C₁₃₇

#123

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

#381

yes

characteristic

C₁₃₇

#1830

yes

characteristic, complemented, Sylow

#1830

—

137

maximal, Sylow

#519

—

137

—

C₂×#101

—

137

—

C₂×#101

D₁₃₇

yes

characteristic

#344

—

137

—

#344

—

137

—

C₂²×Q₈

C₂×D₁₃₇

yes

characteristic

C₂×M₁₆

—

137

—

C₂×M₁₆

C₂×D₁₃₇

yes

—

C₄×Q₈

—

274

—

C₄×Q₈

—

274

—

#117

—

137

—

#117

—

137

—

#117

—

137

—

#101

—

#101

—

274

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₁₃₇

yes

—

C₂×Q₈

D₅₄₈

yes

—

C₂²×C₄

C₄×D₁₃₇

yes

—

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

274

—

M₁₆

—

274

—

C₂×C₈

—

C₂×C₈

—

274

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₂²⋊C₄

—

137

—

C₂²⋊C₄

—

137

—

C₄²

—

274

—

C₄²

—

274

—

C₄²

—

274

—

100

C₂³

C₂×C₄×D₁₃₇

yes

characteristic

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

274

—

106

C₂×C₄

—

274

—

107

C₂×C₄

#14555487

yes

—

108

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

characteristic

109

C₂×C₄

C₂×D₅₄₈

yes

characteristic

110

C₂×C₄

#14555509

yes

—

111

C₂×C₄

#14555509

yes

—

112

C₈

—

113

C₈

—

548

—

114

C₂²

—

115

C₂²

C₂×#14555487

yes

characteristic

116

C₂²

#17984044

yes

characteristic

117

C₂²

#17984044

yes

characteristic

118

C₄

—

119

C₄

—

548

—

120

C₄

—

548

—

121

C₄

#17984069

yes

—

122

C₄

#17984072

yes

—

123

C₂

—

124

C₂

—

125

C₂

—

126

C₂

#22577264

yes

characteristic

	subgroup	G/H
center	C₂	#22577264
commutator, prime radical	C₂³×C₁₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₇
Fitting	C₂×#101×C₁₃₇	C₂
socle	C₂×C₁₃₇	#381
cosocle	C₂²×C₄×C₁₃₇	C₂³
hypercenter	C₂×#101	D₁₃₇
hypocenter	C₁₃₇	#1830
LCS	G ▷ C₂³×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂

#22577264

commutator, prime radical

C₂³×C₁₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₇

Fitting

C₂×#101×C₁₃₇

C₂

socle

C₂×C₁₃₇

#381

cosocle

C₂²×C₄×C₁₃₇

C₂³

D₁₃₇

C₁₃₇

G ▷ C₂³×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
63. C₁₃₇	#1830	semidirect

H ≅ G/N

type

63. C₁₃₇

#1830

semidirect

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

subgroup	index
#22577881	2
C₂×#17984007	2
#22577175	2
#22577174	2
C₂×#101×C₁₃₇	2
#1830	137

subgroup

index

C₂×#101×C₁₃₇

137

Subgroups — #31780901

Conjugacy classes of subgroups

Complements

Maximal subgroups