Group #31793431 — subgroups

Subgroups — #31793431

Order 17536.

370 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 141 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	#22578132	C₂	1	1	yes	maximal, characteristic, complemented
2	#22577755	C₂	1	3	yes	maximal, complemented
3	#22577685	C₂	1	3	yes	maximal, complemented
4	#22577608	C₂	1	3	yes	maximal
5	Q₈×#11764040	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14555498	C₂	1	3	yes	maximal
7	D₄×Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14555498	C₂²	1	3	yes	—
9	#17984088	C₂²	1	3	yes	—
10	#17984079	—	2	3	no	—
11	Q₈×Dic₁₃₇	—	2	1	no	—
12	C₄×Dic₂₇₄	C₂²	1	3	yes	—
13	#17984019	C₂²	1	3	yes	—
14	C₄⋊#11764040	C₂²	1	3	yes	—
15	#17984011	C₂²	1	6	yes	—
16	#17984011	C₂²	1	3	yes	—
17	#17984009	C₂²	1	1	yes	characteristic
18	C₄×#11764040	C₂²	1	3	yes	—
19	C₂²×Q₈×C₁₃₇	—	2	1	no	—
20	C₄⋊Q₈×C₁₃₇	C₂²	1	3	yes	—
21	C₂²⋊Q₈×C₁₃₇	—	2	3	no	—
22	C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₁₃₇	C₂²	1	3	yes	—
24	C₂×Dic₂₇₄	C₂³	1	3	yes	—
25	#14555498	—	2	6	no	—
26	C₂×#11764040	C₂³	1	1	yes	characteristic
27	C₂×#11764040	C₂³	1	3	yes	—
28	C₄⋊Dic₁₃₇	—	2	3	no	—
29	C₄⋊Dic₁₃₇	—	2	3	no	—
30	C₄⋊Dic₁₃₇	C₂³	1	1	yes	characteristic
31	#14555489	—	2	3	no	—
32	C₄×Dic₁₃₇	—	2	3	no	—
33	C₂×Q₈×C₁₃₇	—	2	3	no	—
34	C₂×Q₈×C₁₃₇	—	4	4	no	—
35	C₂×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
36	C₂×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₁₃₇	—	2	3	no	—
38	C₄⋊C₄×C₁₃₇	—	2	3	no	—
39	C₄⋊C₄×C₁₃₇	D₄	1	3	yes	—
40	C₄⋊C₄×C₁₃₇	C₂³	1	3	yes	—
41	C₂²⋊C₄×C₁₃₇	—	2	3	no	—
42	C₄²×C₁₃₇	C₂³	1	3	yes	—
43	Dic₂₇₄	—	2	3	no	—
44	Dic₂₇₄	—	2	3	no	—
45	C₂×Dic₁₃₇	—	2	1	no	—
46	C₂×Dic₁₃₇	—	2	3	no	—
47	#11764040	—	2	3	no	—
48	#11764040	—	2	1	no	—
49	C₂³×C₁₃₇	—	2	1	no	—
50	Q₈×C₁₃₇	—	4	6	no	—
51	Q₈×C₁₃₇	SD₁₆	1	4	yes	complemented
52	D₄×C₁₃₇	—	2	1	no	—
53	D₄×C₁₃₇	#41	1	2	yes	—
54	C₂×C₄×C₁₃₇	—	2	3	no	—
55	C₂×C₄×C₁₃₇	—	4	6	no	—
56	C₂×C₄×C₁₃₇	C₂×D₄	1	3	yes	—
57	C₂×C₄×C₁₃₇	C₂×D₄	1	3	yes	—
58	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
59	C₂²×C₁₃₇	—	2	2	no	—
60	C₂²×C₁₃₇	—	4	1	no	—
61	C₂²×C₁₃₇	—	4	1	no	—
62	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₁₃₇	—	2	3	no	—
64	C₄×C₁₃₇	—	4	3	no	—
65	C₄×C₁₃₇	C₂×SD₁₆	1	6	yes	—
66	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
67	C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
68	Dic₁₃₇	—	4	3	no	—
69	Dic₁₃₇	—	4	1	no	—
70	C₂×C₁₃₇	—	4	2	no	—
71	C₂×C₁₃₇	#548	1	1	yes	characteristic
72	C₂×C₁₃₇	C₂²×SD₁₆	1	1	yes	characteristic
73	C₂×C₁₃₇	#578	1	1	yes	characteristic
74	C₁₃₇	#3359	1	1	yes	characteristic, complemented, Sylow
75	#3359	—	137	1	no	maximal, Sylow
76	#557	—	137	1	no	—
77	D₄×Q₈	D₁₃₇	1	1	yes	characteristic
78	#487	—	137	3	no	—
79	#475	—	137	3	no	—
80	#461	—	137	3	no	—
81	C₈×Q₈	—	137	1	no	—
82	C₄×SD₁₆	—	137	3	no	—
83	C₂²×Q₈	—	2	1	no	—
84	C₂×SD₁₆	—	137	3	no	—
85	C₄⋊Q₈	—	137	3	no	—
86	C₄⋊Q₈	C₂×D₁₃₇	1	3	yes	—
87	#125	—	274	3	no	—
88	C₂²⋊Q₈	—	2	3	no	—
89	C₄×Q₈	—	274	1	no	—
90	C₄×Q₈	—	137	3	no	—
91	C₄×Q₈	C₂×D₁₃₇	1	1	yes	characteristic
92	C₄×D₄	C₂×D₁₃₇	1	3	yes	—
93	#106	—	137	3	no	—
94	C₄⋊C₈	—	137	3	no	—
95	#103	—	137	3	no	—
96	#103	—	137	6	no	—
97	#102	—	137	1	no	—
98	C₄×C₈	—	137	3	no	—
99	C₂×Q₈	—	2	3	no	—
100	C₂×Q₈	—	4	4	no	—
101	C₂×Q₈	—	137	3	no	—
102	C₂×Q₈	C₂²⋊D₁₃₇	1	1	yes	characteristic
103	C₂×D₄	C₂²×D₁₃₇	1	1	yes	characteristic
104	C₂²×C₄	—	2	3	no	—
105	SD₁₆	—	274	6	no	—
106	C₂×C₈	—	137	1	no	—
107	C₂×C₈	—	137	3	no	—
108	C₄⋊C₄	—	2	3	no	—
109	C₄⋊C₄	—	274	3	no	—
110	C₄⋊C₄	—	274	3	no	—
111	C₄⋊C₄	—	274	3	no	—
112	C₄⋊C₄	—	137	1	no	—
113	C₄⋊C₄	C₂²⋊D₁₃₇	1	3	yes	—
114	C₄⋊C₄	C₂²×D₁₃₇	1	3	yes	—
115	C₂²⋊C₄	—	2	3	no	—
116	C₄²	—	274	3	no	—
117	C₄²	C₂²×D₁₃₇	1	3	yes	—
118	C₂³	—	2	1	no	—
119	Q₈	—	4	6	no	—
120	Q₈	—	274	3	no	—
121	Q₈	—	274	3	no	—
122	Q₈	#14555498	1	4	yes	complemented
123	D₄	—	2	1	no	—
124	D₄	#14555511	1	2	yes	—
125	C₂×C₄	—	2	3	no	—
126	C₂×C₄	—	4	6	no	—
127	C₂×C₄	—	274	1	no	—
128	C₂×C₄	—	274	3	no	—
129	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	3	yes	—
130	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	3	yes	—
131	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
132	C₈	—	274	3	no	—
133	C₈	—	274	1	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	4	1	no	—
137	C₂²	C₂²×C₂²⋊D₁₃₇	1	1	yes	characteristic
138	C₄	—	2	3	no	—
139	C₄	—	4	3	no	—
140	C₄	—	548	1	no	—
141	C₄	—	548	3	no	—
142	C₄	C₂×#14555498	1	6	yes	—
143	C₄	C₂×#14555511	1	1	yes	characteristic
144	C₄	#17984136	1	1	yes	characteristic
145	C₂	—	4	2	no	—
146	C₂	#22578064	1	1	yes	characteristic
147	C₂	C₂²×#14555498	1	1	yes	characteristic
148	C₂	#22578218	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22578132

C₂

yes

maximal, characteristic, complemented

#22577755

C₂

yes

maximal, complemented

#22577685

C₂

yes

maximal, complemented

#22577608

C₂

yes

maximal

Q₈×#11764040

C₂

yes

maximal, characteristic, complemented

C₄×#14555498

C₂

yes

maximal

D₄×Q₈×C₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555498

C₂²

yes

—

#17984088

C₂²

yes

—

#17984079

—

Q₈×Dic₁₃₇

—

C₄×Dic₂₇₄

C₂²

yes

—

#17984019

C₂²

yes

—

C₄⋊#11764040

C₂²

yes

—

#17984011

C₂²

yes

—

#17984011

C₂²

yes

—

#17984009

C₂²

yes

characteristic

C₄×#11764040

C₂²

yes

—

C₂²×Q₈×C₁₃₇

—

C₄⋊Q₈×C₁₃₇

C₂²

yes

—

C₂²⋊Q₈×C₁₃₇

—

C₄×Q₈×C₁₃₇

C₂²

yes

characteristic

C₄×D₄×C₁₃₇

C₂²

yes

—

C₂×Dic₂₇₄

C₂³

yes

—

#14555498

—

C₂×#11764040

C₂³

yes

characteristic

C₂×#11764040

C₂³

yes

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

C₂³

yes

characteristic

#14555489

—

C₄×Dic₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

D₄

yes

characteristic

C₂×D₄×C₁₃₇

C₂³

yes

characteristic

C₂²×C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

D₄

yes

—

C₄⋊C₄×C₁₃₇

C₂³

yes

—

C₂²⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

C₂³

yes

—

Dic₂₇₄

—

Dic₂₇₄

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

#11764040

—

#11764040

—

C₂³×C₁₃₇

—

Q₈×C₁₃₇

—

Q₈×C₁₃₇

SD₁₆

yes

complemented

D₄×C₁₃₇

—

D₄×C₁₃₇

#41

yes

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂×C₄×C₁₃₇

C₂⁴

yes

characteristic

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂²×D₄

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×SD₁₆

yes

—

C₄×C₁₃₇

C₂×#41

yes

characteristic

C₄×C₁₃₇

2₋¹⁺⁴

yes

characteristic

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

#548

yes

characteristic

C₂×C₁₃₇

C₂²×SD₁₆

yes

C₂×C₁₃₇

yes

C₁₃₇

yes

characteristic, complemented, Sylow

#3359

—

137

maximal, Sylow

#557

—

137

—

D₄×Q₈

D₁₃₇

yes

characteristic

#487

—

137

—

#475

—

137

—

#461

—

137

—

C₈×Q₈

—

137

—

C₄×SD₁₆

—

137

—

C₂²×Q₈

—

C₂×SD₁₆

—

137

—

C₄⋊Q₈

—

137

—

C₄⋊Q₈

C₂×D₁₃₇

yes

—

#125

—

274

—

C₂²⋊Q₈

—

C₄×Q₈

—

274

—

C₄×Q₈

—

137

—

C₄×Q₈

C₂×D₁₃₇

yes

characteristic

C₄×D₄

C₂×D₁₃₇

yes

—

#106

—

137

—

C₄⋊C₈

—

137

—

#103

—

137

—

#103

—

137

—

#102

—

137

—

C₄×C₈

—

137

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

137

—

102

C₂×Q₈

C₂²⋊D₁₃₇

yes

characteristic

103

C₂×D₄

C₂²×D₁₃₇

yes

characteristic

104

C₂²×C₄

—

105

SD₁₆

—

274

—

106

C₂×C₈

—

137

—

107

C₂×C₈

—

137

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

274

—

110

C₄⋊C₄

—

274

—

111

C₄⋊C₄

—

274

—

112

C₄⋊C₄

—

137

—

113

C₄⋊C₄

C₂²⋊D₁₃₇

yes

—

114

C₄⋊C₄

C₂²×D₁₃₇

yes

—

115

C₂²⋊C₄

—

116

C₄²

—

274

—

117

C₄²

C₂²×D₁₃₇

yes

—

118

C₂³

—

119

Q₈

—

120

Q₈

—

274

—

121

Q₈

—

274

—

122

Q₈

#14555498

yes

complemented

123

D₄

—

124

D₄

#14555511

yes

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

274

—

128

C₂×C₄

—

274

—

129

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

—

130

C₂×C₄

C₂×C₂²⋊D₁₃₇

yes

—

131

C₂×C₄

C₂³×D₁₃₇

yes

characteristic

132

C₈

—

274

—

133

C₈

—

274

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂²×C₂²⋊D₁₃₇

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

548

—

141

C₄

—

548

—

142

C₄

C₂×#14555498

yes

—

143

C₄

C₂×#14555511

yes

characteristic

144

C₄

#17984136

yes

characteristic

145

C₂

—

146

C₂

#22578064

yes

characteristic

147

C₂

C₂²×#14555498

yes

characteristic

148

C₂

#22578218

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₃₇
commutator, cosocle	C₂×C₄×C₁₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	D₄×Q₈×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂²×D₄
prime radical	C₂×D₄×C₁₃₇	C₂³
hypercenter	D₄×Q₈	D₁₃₇
hypocenter	C₁₃₇	#3359
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₃₇

commutator, cosocle

C₂×C₄×C₁₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₇

Fitting

D₄×Q₈×C₁₃₇

C₂

socle

C₂²×C₁₃₇

C₂²×D₄

prime radical

C₂×D₄×C₁₃₇

C₂³

hypercenter

D₄×Q₈

D₁₃₇

hypocenter

C₁₃₇

#3359

LCS

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

N	H ≅ G/N	type
1. #22578132	C₂	semidirect
2. #22577755	C₂	semidirect
3. #22577685	C₂	semidirect
5. Q₈×#11764040	C₂	semidirect
51. Q₈×C₁₃₇	SD₁₆	semidirect
74. C₁₃₇	#3359	semidirect
122. Q₈	#14555498	semidirect

H ≅ G/N

type

1. #22578132

C₂

semidirect

2. #22577755

C₂

semidirect

3. #22577685

C₂

semidirect

5. Q₈×#11764040

C₂

semidirect

51. Q₈×C₁₃₇

SD₁₆

semidirect

74. C₁₃₇

#3359

semidirect

122. Q₈

#14555498

semidirect

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

subgroup	index
#22578132	2
#22577755	2
#22577685	2
#22577608	2
Q₈×#11764040	2
C₄×#14555498	2
D₄×Q₈×C₁₃₇	2
#3359	137

subgroup

index

D₄×Q₈×C₁₃₇

137

Subgroups — #31793431

Conjugacy classes of subgroups

Complements

Maximal subgroups