Group #31781736 — subgroups

Subgroups — #31781736

Order 17536.

292 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 121 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	C₄×C₄⋊Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#22577166	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₇₄	—	2	1	no	—
6	#17984088	—	2	2	no	—
7	#17984088	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₁₃₇	—	2	1	no	—
9	C₂×C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
11	C₂×C₄×Dic₁₃₇	—	2	1	no	—
12	#17983995	—	2	1	no	—
13	C₂²×C₈×C₁₃₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₁₃₇	—	2	2	no	—
16	C₂×Dic₂₇₄	—	2	4	no	—
17	C₂×Dic₂₇₄	—	4	4	no	—
18	C₂²×Dic₁₃₇	—	2	2	no	—
19	C₂²×Dic₁₃₇	—	2	2	no	—
20	C₄⋊Dic₁₃₇	—	2	4	no	—
21	C₄⋊Dic₁₃₇	—	2	2	no	—
22	C₄⋊Dic₁₃₇	—	2	2	no	—
23	C₄⋊Dic₁₃₇	C₂×C₄	1	4	yes	—
24	C₄⋊Dic₁₃₇	Q₈	1	4	yes	complemented
25	C₄×Dic₁₃₇	—	4	2	no	—
26	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
29	C₂×C₈×C₁₃₇	—	2	2	no	—
30	C₂×C₈×C₁₃₇	—	2	4	no	—
31	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
32	C₄²×C₁₃₇	D₄	1	2	yes	—
33	Dic₂₇₄	—	4	8	no	—
34	C₂×Dic₁₃₇	—	2	4	no	—
35	C₂×Dic₁₃₇	—	2	4	no	—
36	C₂×Dic₁₃₇	—	4	2	no	—
37	C₂×Dic₁₃₇	—	4	2	no	—
38	C₂×Dic₁₃₇	—	4	4	no	—
39	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₃₇	—	2	2	no	—
41	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₃₇	SD₁₆	1	2	yes	—
44	C₂×C₄×C₁₃₇	Q₁₆	1	2	yes	—
45	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₁₃₇	#41	1	2	yes	—
49	C₈×C₁₃₇	—	4	2	no	—
50	C₂²×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
52	C₂²×C₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
53	C₂²×C₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
54	C₂²×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₁₃₇	—	2	2	no	—
56	C₄×C₁₃₇	#103	1	4	yes	—
57	C₄×C₁₃₇	C₄×Q₈	1	2	yes	—
58	C₄×C₁₃₇	#124	1	1	yes	characteristic
59	C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
60	Dic₁₃₇	—	4	4	no	—
61	Dic₁₃₇	—	4	4	no	—
62	C₂×C₁₃₇	#391	1	1	yes	characteristic
63	C₂×C₁₃₇	C₂×#103	1	1	yes	characteristic
64	C₂×C₁₃₇	#418	1	1	yes	characteristic
65	C₂×C₁₃₇	#476	1	2	yes	—
66	C₂×C₁₃₇	#477	1	2	yes	—
67	C₁₃₇	#1987	1	1	yes	characteristic, complemented, Sylow
68	#1987	—	137	1	no	maximal, Sylow
69	C₂×C₄⋊Q₈	—	137	1	no	—
70	C₂×C₄⋊C₈	D₁₃₇	1	1	yes	characteristic
71	C₄×C₄⋊C₄	—	137	1	no	—
72	#340	—	137	4	no	—
73	C₂²×Q₈	—	274	1	no	—
74	C₂²×C₈	C₂×D₁₃₇	1	2	yes	—
75	C₄⋊Q₈	—	274	2	no	—
76	C₄⋊Q₈	—	137	4	no	—
77	C₂×C₄⋊C₄	—	274	1	no	—
78	C₂×C₄⋊C₄	—	137	2	no	—
79	C₂×C₄⋊C₄	—	137	2	no	—
80	C₂×C₄²	—	274	1	no	—
81	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
82	C₄⋊C₈	—	2	2	no	—
83	#95	—	274	1	no	—
84	C₂×Q₈	—	274	4	no	—
85	C₂×Q₈	—	548	4	no	—
86	C₂²×C₄	—	274	2	no	—
87	C₂²×C₄	—	274	2	no	—
88	C₂²×C₄	D₅₄₈	1	1	yes	characteristic
89	C₂²×C₄	D₅₄₈	1	1	yes	characteristic
90	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	4	no	—
93	C₄⋊C₄	—	274	4	no	—
94	C₄⋊C₄	—	274	2	no	—
95	C₄⋊C₄	—	274	2	no	—
96	C₄⋊C₄	—	137	4	no	—
97	C₄⋊C₄	—	137	4	no	—
98	C₄²	—	548	2	no	—
99	C₄²	C₄×D₁₃₇	1	2	yes	—
100	C₄²	C₂²⋊D₁₃₇	1	2	yes	—
101	C₂³	C₂×D₅₄₈	1	1	yes	characteristic
102	Q₈	—	548	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	274	4	no	—
105	C₂×C₄	—	274	4	no	—
106	C₂×C₄	—	548	2	no	—
107	C₂×C₄	—	548	4	no	—
108	C₂×C₄	—	548	2	no	—
109	C₂×C₄	#14555487	1	2	yes	—
110	C₂×C₄	#14555487	1	2	yes	—
111	C₂×C₄	#14555499	1	2	yes	—
112	C₂×C₄	Dic₅₄₈	1	2	yes	—
113	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
114	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
115	C₂×C₄	Q₈×D₁₃₇	1	2	yes	—
116	C₂×C₄	#14555509	1	2	yes	—
117	C₈	—	4	2	no	—
118	C₂²	C₂×#14555487	1	1	yes	characteristic
119	C₂²	#17984068	1	2	yes	—
120	C₂²	C₂×#14555499	1	1	yes	characteristic
121	C₂²	C₂×Dic₅₄₈	1	1	yes	characteristic
122	C₂²	#17984116	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	—	548	4	no	—
125	C₄	—	548	4	no	—
126	C₄	#17984013	1	4	yes	—
127	C₄	#17984053	1	2	yes	—
128	C₄	#17984075	1	1	yes	characteristic
129	C₄	#17984086	1	1	yes	characteristic
130	C₂	#22577308	1	1	yes	characteristic
131	C₂	C₂×#17984013	1	1	yes	characteristic
132	C₂	#22577409	1	1	yes	characteristic
133	C₂	#22577694	1	2	yes	—
134	C₂	#22577699	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984088

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₁₃₇

C₂

yes

maximal, characteristic

#22577166

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₃₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₇₄

—

#17984088

—

#17984088

C₄

yes

complemented

C₂×C₄⋊Dic₁₃₇

—

C₂×C₄⋊Dic₁₃₇

C₂²

yes

—

C₂×C₄⋊Dic₁₃₇

C₂²

yes

—

C₂×C₄×Dic₁₃₇

—

#17983995

—

C₂²×C₈×C₁₃₇

C₂²

yes

—

C₂×C₄²×C₁₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₇

—

C₂×Dic₂₇₄

—

C₂×Dic₂₇₄

—

C₂²×Dic₁₃₇

—

C₂²×Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

C₂×C₄

yes

—

C₄⋊Dic₁₃₇

Q₈

yes

complemented

C₄×Dic₁₃₇

—

C₂²×C₄×C₁₃₇

D₄

yes

characteristic

C₂²×C₄×C₁₃₇

D₄

yes

characteristic

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₄²×C₁₃₇

C₂×C₄

yes

—

C₄²×C₁₃₇

D₄

yes

—

Dic₂₇₄

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂³×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂²⋊C₄

yes

—

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×Q₈

yes

—

C₂×C₄×C₁₃₇

#41

yes

—

C₈×C₁₃₇

—

C₂²×C₁₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃₇

C₂²⋊Q₈

yes

—

C₂²×C₁₃₇

C₂×SD₁₆

yes

characteristic

C₂²×C₁₃₇

C₂×Q₁₆

yes

characteristic

C₂²×C₁₃₇

AGL(1,ℤ/8)

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#103

yes

—

C₄×C₁₃₇

C₄×Q₈

yes

—

C₄×C₁₃₇

#124

yes

characteristic

C₄×C₁₃₇

C₄⋊Q₈

yes

characteristic

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

—

C₂×C₁₃₇

#477

yes

—

C₁₃₇

#1987

yes

characteristic, complemented, Sylow

#1987

—

137

maximal, Sylow

C₂×C₄⋊Q₈

—

137

—

C₂×C₄⋊C₈

D₁₃₇

yes

characteristic

C₄×C₄⋊C₄

—

137

—

#340

—

137

—

C₂²×Q₈

—

274

—

C₂²×C₈

C₂×D₁₃₇

yes

—

C₄⋊Q₈

—

274

—

C₄⋊Q₈

—

137

—

C₂×C₄⋊C₄

—

274

—

C₂×C₄⋊C₄

—

137

—

C₂×C₄⋊C₄

—

137

—

C₂×C₄²

—

274

—

C₂×C₄²

C₂×D₁₃₇

yes

characteristic

C₄⋊C₈

—

#95

—

274

—

C₂×Q₈

—

274

—

C₂×Q₈

—

548

—

C₂²×C₄

—

274

—

C₂²×C₄

—

274

—

C₂²×C₄

D₅₄₈

yes

characteristic

C₂²×C₄

D₅₄₈

yes

characteristic

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

274

—

C₄⋊C₄

—

137

—

C₄⋊C₄

—

137

—

C₄²

—

548

—

C₄²

C₄×D₁₃₇

yes

—

100

C₄²

C₂²⋊D₁₃₇

yes

—

101

C₂³

C₂×D₅₄₈

yes

characteristic

102

Q₈

—

548

—

103

C₂×C₄

—

104

C₂×C₄

—

274

—

105

C₂×C₄

—

274

—

106

C₂×C₄

—

548

—

107

C₂×C₄

—

548

—

108

C₂×C₄

—

548

—

109

C₂×C₄

#14555487

yes

—

110

C₂×C₄

#14555487

yes

—

111

C₂×C₄

#14555499

yes

—

112

C₂×C₄

Dic₅₄₈

yes

—

113

C₂×C₄

C₂×C₄×D₁₃₇

yes

characteristic

114

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

characteristic

115

C₂×C₄

Q₈×D₁₃₇

yes

—

116

C₂×C₄

#14555509

yes

—

117

C₈

—

118

C₂²

C₂×#14555487

yes

characteristic

119

C₂²

#17984068

yes

—

120

C₂²

C₂×#14555499

yes

characteristic

121

C₂²

C₂×Dic₅₄₈

yes

characteristic

122

C₂²

#17984116

yes

—

123

C₄

—

124

C₄

—

548

—

125

C₄

—

548

—

126

C₄

#17984013

yes

—

127

C₄

#17984053

yes

—

128

C₄

#17984075

yes

characteristic

129

C₄

#17984086

yes

characteristic

130

C₂

#22577308

yes

characteristic

131

C₂

C₂×#17984013

yes

characteristic

132

C₂

#22577409

yes

characteristic

133

C₂

#22577694

yes

—

134

C₂

#22577699

yes

—

	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	#1987
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

#1987

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
24. C₄⋊Dic₁₃₇	Q₈	semidirect
67. C₁₃₇	#1987	semidirect

H ≅ G/N

type

7. #17984088

C₄

semidirect

24. C₄⋊Dic₁₃₇

Q₈

semidirect

67. C₁₃₇

#1987

semidirect

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

subgroup	index
C₂×#17984088	2
C₄×C₄⋊Dic₁₃₇	2
#22577166	2
C₂×C₄⋊C₈×C₁₃₇	2
#1987	137

subgroup

index

C₂×#17984088

C₄×C₄⋊Dic₁₃₇

#22577166

C₂×C₄⋊C₈×C₁₃₇

#1987

137

Subgroups — #31781736

Conjugacy classes of subgroups

Complements

Maximal subgroups