Group #31796967 — subgroups

Subgroups — #31796967

Order 17536.

292 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 88 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	#22578404	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#10587739	C₂	1	1	yes	maximal, characteristic
3	#22578276	C₂	1	4	yes	maximal
4	C₄⋊Q₈×D₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14555518	C₂²	1	2	yes	—
6	C₂×C₄⋊#10587739	C₂²	1	2	yes	—
7	C₂×C₄×#10587739	—	2	1	no	—
8	#17984155	—	2	1	no	—
9	C₄⋊#11764047	—	2	1	no	—
10	(C₂×#10587739)⋊C₄	—	2	1	no	—
11	C₂×Q₈×D₁₃₇	—	2	1	no	—
12	#17984088	C₄	1	1	yes	characteristic, complemented
13	#17984087	—	2	1	no	—
14	#17984087	C₄	1	2	yes	complemented
15	#17984086	—	2	1	no	—
16	C₄⋊C₄×D₁₃₇	—	2	1	no	—
17	C₄⋊C₄×D₁₃₇	C₂²	1	2	yes	—
18	C₄²×D₁₃₇	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₁₃₇	C₄	1	1	yes	characteristic, complemented
20	C₂²×#10587739	—	2	2	no	—
21	#14555518	—	2	4	no	—
22	C₂×#11764047	—	2	2	no	—
23	C₄⋊#10587739	—	2	2	no	—
24	C₄⋊#10587739	Q₈	1	4	yes	complemented
25	C₄×#10587739	—	4	2	no	—
26	C₂×Dic₂₇₄	—	2	2	no	—
27	C₂×Dic₂₇₄	—	2	1	no	—
28	Q₈×D₁₃₇	—	4	4	no	—
29	C₂×C₄×D₁₃₇	—	2	2	no	—
30	C₂×C₄×D₁₃₇	D₄	1	1	yes	characteristic
31	C₂×C₄×D₁₃₇	D₄	1	1	yes	characteristic
32	C₂×C₄×D₁₃₇	C₂³	1	1	yes	characteristic
33	C₄⋊Dic₁₃₇	—	2	1	no	—
34	C₄⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
35	#14555489	—	2	2	no	—
36	#14555489	—	2	2	no	—
37	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
38	C₄×Dic₁₃₇	D₄	1	1	yes	characteristic
39	C₄×Dic₁₃₇	D₄	1	1	yes	characteristic
40	C₂×Q₈×C₁₃₇	—	2	1	no	—
41	C₄⋊C₄×C₁₃₇	—	2	1	no	—
42	C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
43	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
44	C₂×#10587739	—	2	4	no	—
45	C₂×#10587739	—	4	2	no	—
46	C₂×#10587739	—	4	2	no	—
47	#11764047	—	4	2	no	—
48	C₂²×D₁₃₇	C₂×D₄	1	1	yes	characteristic
49	Dic₂₇₄	—	4	4	no	—
50	Dic₂₇₄	—	4	2	no	—
51	C₂×Dic₁₃₇	—	2	2	no	—
52	C₂×Dic₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×Dic₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
55	C₄×D₁₃₇	—	2	2	no	—
56	C₄×D₁₃₇	—	4	4	no	—
57	C₄×D₁₃₇	SD₁₆	1	2	yes	—
58	C₄×D₁₃₇	Q₁₆	1	2	yes	—
59	C₄×D₁₃₇	C₂×Q₈	1	2	yes	—
60	C₄×D₁₃₇	#41	1	2	yes	—
61	Q₈×C₁₃₇	—	4	2	no	—
62	C₂×C₄×C₁₃₇	—	2	2	no	—
63	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×D₁₃₇	C₂²⋊Q₈	1	2	yes	—
68	C₂×D₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×D₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×D₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
71	#10587739	—	4	4	no	—
72	C₄×C₁₃₇	—	2	1	no	—
73	C₄×C₁₃₇	—	4	2	no	—
74	C₄×C₁₃₇	#103	1	2	yes	—
75	C₄×C₁₃₇	C₄×Q₈	1	2	yes	—
76	Dic₁₃₇	—	2	1	no	—
77	Dic₁₃₇	—	4	2	no	—
78	Dic₁₃₇	#103	1	2	yes	—
79	Dic₁₃₇	#124	1	1	yes	characteristic
80	Dic₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₁₃₇	#391	1	1	yes	characteristic
82	C₂×C₁₃₇	C₂×#103	1	1	yes	characteristic
83	C₂×C₁₃₇	#418	1	1	yes	characteristic
84	D₁₃₇	#476	1	2	yes	—
85	D₁₃₇	#477	1	2	yes	—
86	C₁₃₇	#1987	1	1	yes	characteristic, complemented, Sylow
87	#1987	—	137	1	no	maximal, Sylow
88	C₂×C₄⋊Q₈	—	137	1	no	—
89	C₂×C₄⋊C₈	—	137	1	no	—
90	C₄×C₄⋊C₄	—	137	1	no	—
91	#340	—	137	4	no	—
92	C₂²×Q₈	—	274	1	no	—
93	C₂²×C₈	—	137	2	no	—
94	C₄⋊Q₈	—	274	1	no	—
95	C₄⋊Q₈	—	274	1	no	—
96	C₄⋊Q₈	—	137	2	no	—
97	C₄⋊Q₈	—	137	1	no	—
98	C₄⋊Q₈	#10587739	1	1	yes	characteristic, complemented
99	C₂×C₄⋊C₄	—	274	1	no	—
100	C₂×C₄⋊C₄	—	137	2	no	—
101	C₂×C₄⋊C₄	—	137	2	no	—
102	C₂×C₄²	—	274	1	no	—
103	C₂×C₄²	—	137	1	no	—
104	C₄⋊C₈	—	274	1	no	—
105	C₄⋊C₈	—	274	1	no	—
106	#95	—	274	1	no	—
107	C₂×Q₈	—	2	1	no	—
108	C₂×Q₈	—	274	2	no	—
109	C₂×Q₈	—	274	1	no	—
110	C₂×Q₈	—	548	4	no	—
111	C₂²×C₄	—	274	2	no	—
112	C₂²×C₄	—	274	2	no	—
113	C₂²×C₄	—	137	1	no	—
114	C₂²×C₄	—	137	1	no	—
115	C₂²×C₄	—	137	1	no	—
116	C₂×C₈	—	274	2	no	—
117	C₂×C₈	—	274	4	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	274	2	no	—
120	C₄⋊C₄	—	274	2	no	—
121	C₄⋊C₄	—	274	2	no	—
122	C₄⋊C₄	—	274	1	no	—
123	C₄⋊C₄	—	137	4	no	—
124	C₄⋊C₄	—	137	2	no	—
125	C₄⋊C₄	C₂×#10587739	1	2	yes	—
126	C₄²	—	548	2	no	—
127	C₄²	—	137	1	no	—
128	C₄²	—	137	1	no	—
129	C₄²	—	137	1	no	—
130	C₄²	C₂×#10587739	1	1	yes	characteristic
131	C₂³	—	137	1	no	—
132	Q₈	—	4	2	no	—
133	Q₈	—	548	2	no	—
134	Q₈	—	548	4	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	274	2	no	—
137	C₂×C₄	—	274	4	no	—
138	C₂×C₄	—	274	2	no	—
139	C₂×C₄	—	548	4	no	—
140	C₂×C₄	—	548	2	no	—
141	C₂×C₄	—	548	2	no	—
142	C₂×C₄	—	137	1	no	—
143	C₂×C₄	—	137	1	no	—
144	C₂×C₄	—	137	1	no	—
145	C₂×C₄	—	137	2	no	—
146	C₂×C₄	—	137	2	no	—
147	C₂×C₄	—	137	2	no	—
148	C₂×C₄	—	137	2	no	—
149	C₂×C₄	C₂²⋊#10587739	1	1	yes	characteristic
150	C₂×C₄	C₂²⋊#10587739	1	1	yes	characteristic
151	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
152	C₈	—	548	2	no	—
153	C₂²	—	137	2	no	—
154	C₂²	—	137	1	no	—
155	C₂²	—	137	1	no	—
156	C₂²	—	137	2	no	—
157	C₂²	C₂×C₂²⋊#10587739	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	4	2	no	—
160	C₄	—	274	1	no	—
161	C₄	—	548	4	no	—
162	C₄	—	548	2	no	—
163	C₄	—	137	1	no	—
164	C₄	—	137	1	no	—
165	C₄	—	137	2	no	—
166	C₄	#17984151	1	2	yes	—
167	C₄	Q₈×#10587739	1	2	yes	—
168	C₂	—	137	2	no	—
169	C₂	—	137	2	no	—
170	C₂	(C₂×#10587739)⋊Q₈	1	1	yes	characteristic
171	C₂	C₂×#17984151	1	1	yes	characteristic
172	C₂	#22578395	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22578404

C₂

yes

maximal, characteristic

C₄×C₄⋊#10587739

C₂

yes

maximal, characteristic

#22578276

C₂

yes

maximal

C₄⋊Q₈×D₁₃₇

C₂

yes

maximal, characteristic

C₂×#14555518

C₂²

yes

—

C₂×C₄⋊#10587739

C₂²

yes

—

—

—

—

—

C₂×Q₈×D₁₃₇

—

#17984088

C₄

yes

characteristic, complemented

#17984087

—

#17984087

C₄

yes

complemented

#17984086

—

C₄⋊C₄×D₁₃₇

—

C₄⋊C₄×D₁₃₇

C₂²

yes

—

C₄²×D₁₃₇

C₂²

yes

characteristic

C₄⋊Q₈×C₁₃₇

C₄

yes

characteristic, complemented

—

—

—

—

Q₈

yes

complemented

C₄×#10587739

—

C₂×Dic₂₇₄

—

C₂×Dic₂₇₄

—

Q₈×D₁₃₇

—

C₂×C₄×D₁₃₇

—

C₂×C₄×D₁₃₇

D₄

yes

characteristic

C₂×C₄×D₁₃₇

D₄

yes

characteristic

C₂×C₄×D₁₃₇

C₂³

yes

characteristic

C₄⋊Dic₁₃₇

—

C₄⋊Dic₁₃₇

C₂×C₄

yes

—

#14555489

—

#14555489

—

C₄×Dic₁₃₇

C₂×C₄

yes

characteristic

C₄×Dic₁₃₇

D₄

yes

characteristic

C₄×Dic₁₃₇

D₄

yes

characteristic

C₂×Q₈×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

C₂×C₄

yes

—

C₄²×C₁₃₇

C₂×C₄

yes

—

—

—

—

C₂²×D₁₃₇

C₂×D₄

yes

characteristic

Dic₂₇₄

—

Dic₂₇₄

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₃₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₃₇

C₂×D₄

yes

characteristic

C₄×D₁₃₇

—

C₄×D₁₃₇

—

C₄×D₁₃₇

SD₁₆

yes

—

C₄×D₁₃₇

Q₁₆

yes

—

C₄×D₁₃₇

C₂×Q₈

yes

—

C₄×D₁₃₇

#41

yes

—

Q₈×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂²×C₁₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₃₇

C₂²⋊Q₈

yes

—

C₂×D₁₃₇

C₂×SD₁₆

yes

characteristic

C₂×D₁₃₇

C₂×Q₁₆

yes

characteristic

C₂×D₁₃₇

AGL(1,ℤ/8)

yes

—

#10587739

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#103

yes

—

C₄×C₁₃₇

C₄×Q₈

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

#103

yes

—

Dic₁₃₇

#124

yes

characteristic

Dic₁₃₇

C₄⋊Q₈

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

D₁₃₇

yes

—

D₁₃₇

#477

yes

—

C₁₃₇

#1987

yes

characteristic, complemented, Sylow

#1987

—

137

maximal, Sylow

C₂×C₄⋊Q₈

—

137

—

C₂×C₄⋊C₈

—

137

—

C₄×C₄⋊C₄

—

137

—

#340

—

137

—

C₂²×Q₈

—

274

—

C₂²×C₈

—

137

—

C₄⋊Q₈

—

274

—

C₄⋊Q₈

—

274

—

C₄⋊Q₈

—

137

—

C₄⋊Q₈

—

137

—

C₄⋊Q₈

#10587739

yes

characteristic, complemented

C₂×C₄⋊C₄

—

274

—

100

C₂×C₄⋊C₄

—

137

—

101

C₂×C₄⋊C₄

—

137

—

102

C₂×C₄²

—

274

—

103

C₂×C₄²

—

137

—

104

C₄⋊C₈

—

274

—

105

C₄⋊C₈

—

274

—

106

#95

—

274

—

107

C₂×Q₈

—

108

C₂×Q₈

—

274

—

109

C₂×Q₈

—

274

—

110

C₂×Q₈

—

548

—

111

C₂²×C₄

—

274

—

112

C₂²×C₄

—

274

—

113

C₂²×C₄

—

137

—

114

C₂²×C₄

—

137

—

115

C₂²×C₄

—

137

—

116

C₂×C₈

—

274

—

117

C₂×C₈

—

274

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

274

—

120

C₄⋊C₄

—

274

—

121

C₄⋊C₄

—

274

—

122

C₄⋊C₄

—

274

—

123

C₄⋊C₄

—

137

—

124

C₄⋊C₄

—

137

—

125

C₄⋊C₄

C₂×#10587739

yes

—

126

C₄²

—

548

—

127

C₄²

—

137

—

128

C₄²

—

137

—

129

C₄²

—

137

—

130

C₄²

C₂×#10587739

yes

characteristic

131

C₂³

—

137

—

132

Q₈

—

133

Q₈

—

548

—

134

Q₈

—

548

—

135

C₂×C₄

—

136

C₂×C₄

—

274

—

137

C₂×C₄

—

274

—

138

C₂×C₄

—

274

—

139

C₂×C₄

—

548

—

140

C₂×C₄

—

548

—

141

C₂×C₄

—

548

—

142

C₂×C₄

—

137

—

143

C₂×C₄

—

137

—

144

C₂×C₄

—

137

—

145

C₂×C₄

—

137

—

146

C₂×C₄

—

137

—

147

C₂×C₄

—

137

—

148

C₂×C₄

—

137

—

149

C₂×C₄

C₂²⋊#10587739

yes

characteristic

150

C₂×C₄

C₂²⋊#10587739

yes

characteristic

151

C₂×C₄

C₂²×#10587739

yes

characteristic

152

C₈

—

548

—

153

C₂²

—

137

—

154

C₂²

—

137

—

155

C₂²

—

137

—

156

C₂²

—

137

—

157

C₂²

C₂×C₂²⋊#10587739

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

274

—

161

C₄

—

548

—

162

C₄

—

548

—

163

C₄

—

137

—

164

C₄

—

137

—

165

C₄

—

137

—

166

C₄

#17984151

yes

—

167

C₄

Q₈×#10587739

yes

—

168

C₂

—

137

—

169

C₂

—

137

—

170

C₂

(C₂×#10587739)⋊Q₈

yes

characteristic

171

C₂

C₂×#17984151

yes

characteristic

172

C₂

#22578395

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#10587739
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#10587739
Fitting	C₄⋊Q₈×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₃₇	C₂×D₄
cosocle	C₂×C₄×D₁₃₇	C₂³
hypercenter	C₄⋊Q₈	#10587739
hypocenter	C₁₃₇	#1987
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#10587739

commutator

C₂×C₄×C₁₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#10587739

Fitting

C₄⋊Q₈×C₁₃₇

C₄

socle

C₂²×C₁₃₇

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₃₇

C₂×D₄

cosocle

C₂×C₄×D₁₃₇

C₂³

C₄⋊Q₈

C₁₃₇

G ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
12. #17984088	C₄	semidirect
14. #17984087	C₄	semidirect
19. C₄⋊Q₈×C₁₃₇	C₄	semidirect
24. C₄⋊#10587739	Q₈	semidirect
86. C₁₃₇	#1987	semidirect
98. C₄⋊Q₈	#10587739	semidirect

H ≅ G/N

type

12. #17984088

C₄

semidirect

14. #17984087

C₄

semidirect

19. C₄⋊Q₈×C₁₃₇

C₄

semidirect

24. C₄⋊#10587739

Q₈

semidirect

86. C₁₃₇

#1987

semidirect

98. C₄⋊Q₈

#10587739

semidirect

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

subgroup	index
#22578404	2
C₄×C₄⋊#10587739	2
#22578276	2
C₄⋊Q₈×D₁₃₇	2
#1987	137

subgroup

index

C₄⋊Q₈×D₁₃₇

137

Subgroups — #31796967

Conjugacy classes of subgroups

Complements

Maximal subgroups