Group #31789869 — subgroups

Subgroups — #31789869

Order 17536.

396 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 217 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×C₄×Dic₂₇₄	C₂	1	1	yes	maximal, characteristic
2	C₈×Dic₂₇₄	C₂	1	4	yes	maximal, complemented
3	#22577527	C₂	1	4	yes	maximal, complemented
4	#22577456	C₂	1	1	yes	maximal, characteristic
5	#22577453	C₂	1	4	yes	maximal
6	#431×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₇₄	C₄	1	1	yes	characteristic
8	C₄×Dic₂₇₄	C₄	1	2	yes	—
9	C₄×Dic₂₇₄	C₄	1	2	yes	—
10	C₄×Dic₂₇₄	C₂²	1	2	yes	—
11	C₄×Dic₂₇₄	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₁₃₇	C₄	1	1	yes	characteristic
13	C₂×#14555489	C₄	1	2	yes	—
14	C₂×C₄×Dic₁₃₇	C₂²	1	2	yes	—
15	C₄⋊#11764040	C₂²	1	2	yes	—
16	#17984017	C₂²	1	8	yes	—
17	C₂²⋊#11764040	C₂²	1	2	yes	—
18	C₄×#11764040	C₂²	1	2	yes	—
19	C₈×Dic₁₃₇	C₂²	1	8	yes	—
20	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
22	C₂²⋊C₈×C₁₃₇	C₂²	1	2	yes	—
23	C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
24	C₂×Dic₂₇₄	C₈	1	8	yes	complemented
25	C₂×Dic₂₇₄	C₂×C₄	1	2	yes	—
26	C₂×Dic₂₇₄	C₂×C₄	1	2	yes	—
27	C₂²×Dic₁₃₇	C₂×C₄	1	2	yes	—
28	C₂²×Dic₁₃₇	C₂×C₄	1	2	yes	—
29	C₂×#11764040	C₂³	1	4	yes	—
30	C₄⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
31	C₄⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
32	#14555489	C₂×C₄	1	4	yes	—
33	#14555489	C₂×C₄	1	4	yes	—
34	C₄×Dic₁₃₇	C₂×C₄	1	4	yes	—
35	C₄×Dic₁₃₇	C₂³	1	4	yes	—
36	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
39	C₂×C₈×C₁₃₇	C₂³	1	4	yes	—
40	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
42	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
43	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
44	Dic₂₇₄	C₂×C₈	1	8	yes	—
45	Dic₂₇₄	C₂×C₈	1	8	yes	—
46	C₂×Dic₁₃₇	—	4	2	no	—
47	C₂×Dic₁₃₇	C₂×C₈	1	8	yes	—
48	C₂×Dic₁₃₇	C₂²×C₄	1	4	yes	—
49	C₂×Dic₁₃₇	C₂²×C₄	1	4	yes	—
50	#11764040	—	2	4	no	—
51	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃₇	—	2	2	no	—
53	C₂×C₄×C₁₃₇	—	4	1	no	—
54	C₂×C₄×C₁₃₇	C₂×C₈	1	4	yes	—
55	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
61	C₈×C₁₃₇	—	2	4	no	—
62	C₂²×C₁₃₇	—	2	1	no	—
63	C₂²×C₁₃₇	—	2	1	no	—
64	C₂²×C₁₃₇	C₂²×C₈	1	2	yes	—
65	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₁₃₇	—	2	1	no	—
67	C₄×C₁₃₇	—	2	1	no	—
68	C₄×C₁₃₇	—	2	1	no	—
69	C₄×C₁₃₇	C₂²×C₈	1	2	yes	—
70	C₄×C₁₃₇	C₂²×C₈	1	2	yes	—
71	C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
72	Dic₁₃₇	—	2	4	no	—
73	Dic₁₃₇	C₂²×C₈	1	8	yes	—
74	C₂×C₁₃₇	—	2	2	no	—
75	C₂×C₁₃₇	#519	1	1	yes	characteristic
76	C₂×C₁₃₇	C₂³×C₈	1	1	yes	characteristic
77	C₂×C₁₃₇	#568	1	1	yes	characteristic
78	C₁₃₇	#2982	1	1	yes	characteristic, complemented, Sylow
79	#2982	—	137	1	no	maximal, Sylow
80	C₂×C₄×Q₈	—	137	1	no	—
81	C₈×Q₈	—	137	4	no	—
82	C₈×Q₈	—	137	4	no	—
83	#431	—	137	1	no	—
84	#431	—	137	4	no	—
85	#431	D₁₃₇	1	1	yes	characteristic
86	C₂²×Q₈	—	137	1	no	—
87	C₄×Q₈	—	137	2	no	—
88	C₄×Q₈	—	137	2	no	—
89	C₄×Q₈	—	137	2	no	—
90	C₄×Q₈	—	137	2	no	—
91	C₂×C₄⋊C₄	—	137	2	no	—
92	C₂×C₄⋊C₄	—	137	1	no	—
93	C₂×C₄²	—	137	2	no	—
94	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
95	C₄⋊C₈	—	137	8	no	—
96	C₄⋊C₈	—	137	2	no	—
97	C₄⋊C₈	C₂×D₁₃₇	1	2	yes	—
98	C₂²⋊C₈	—	137	2	no	—
99	C₂²⋊C₈	C₂×D₁₃₇	1	2	yes	—
100	C₄×C₈	—	137	8	no	—
101	C₄×C₈	—	137	2	no	—
102	C₄×C₈	C₂×D₁₃₇	1	2	yes	—
103	C₂×Q₈	—	137	8	no	—
104	C₂×Q₈	—	137	2	no	—
105	C₂×Q₈	—	137	2	no	—
106	C₂²×C₄	—	137	2	no	—
107	C₂²×C₄	—	137	2	no	—
108	C₂²×C₄	C₄×D₁₃₇	1	1	yes	characteristic
109	C₂²×C₄	C₄×D₁₃₇	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
111	C₂×C₈	—	137	4	no	—
112	C₂×C₈	C₂²×D₁₃₇	1	4	yes	—
113	C₄⋊C₄	—	137	4	no	—
114	C₄⋊C₄	—	137	4	no	—
115	C₄⋊C₄	—	137	2	no	—
116	C₄⋊C₄	—	137	2	no	—
117	C₄²	—	137	4	no	—
118	C₄²	—	137	4	no	—
119	C₄²	C₄×D₁₃₇	1	1	yes	characteristic
120	C₄²	C₄×D₁₃₇	1	1	yes	characteristic
121	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
122	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
123	C₂³	C₂×C₄×D₁₃₇	1	1	yes	characteristic
124	Q₈	—	137	8	no	—
125	Q₈	—	137	8	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	548	2	no	—
129	C₂×C₄	—	137	4	no	—
130	C₂×C₄	—	137	8	no	—
131	C₂×C₄	—	137	4	no	—
132	C₂×C₄	C₈×D₁₃₇	1	4	yes	—
133	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
134	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
135	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
136	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
138	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
139	C₈	—	2	4	no	—
140	C₈	—	274	4	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	C₂×C₈×D₁₃₇	1	2	yes	—
144	C₂²	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	274	4	no	—
149	C₄	—	137	8	no	—
150	C₄	C₂×C₈×D₁₃₇	1	2	yes	—
151	C₄	C₂×C₈×D₁₃₇	1	2	yes	—
152	C₄	#17984135	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	#22577880	1	1	yes	characteristic
155	C₂	C₂²×C₈×D₁₃₇	1	1	yes	characteristic
156	C₂	#22578169	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₂₇₄

C₂

yes

maximal, characteristic

C₈×Dic₂₇₄

C₂

yes

maximal, complemented

#22577527

C₂

yes

maximal, complemented

#22577456

C₂

yes

maximal, characteristic

#22577453

C₂

yes

maximal

#431×C₁₃₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₇₄

C₄

yes

characteristic

C₄×Dic₂₇₄

C₄

yes

—

C₄×Dic₂₇₄

C₄

yes

—

C₄×Dic₂₇₄

C₂²

yes

—

C₄×Dic₂₇₄

C₂²

yes

—

C₂×C₄⋊Dic₁₃₇

C₄

yes

characteristic

C₂×#14555489

C₄

yes

—

C₂×C₄×Dic₁₃₇

C₂²

yes

—

C₄⋊#11764040

C₂²

yes

—

#17984017

C₂²

yes

—

C₂²⋊#11764040

C₂²

yes

—

C₄×#11764040

C₂²

yes

—

C₈×Dic₁₃₇

C₂²

yes

—

C₂×C₄²×C₁₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₇

C₂²

yes

—

C₂²⋊C₈×C₁₃₇

C₂²

yes

—

C₄×C₈×C₁₃₇

C₂²

yes

—

C₂×Dic₂₇₄

C₈

yes

complemented

C₂×Dic₂₇₄

C₂×C₄

yes

—

C₂×Dic₂₇₄

C₂×C₄

yes

—

C₂²×Dic₁₃₇

C₂×C₄

yes

—

C₂²×Dic₁₃₇

C₂×C₄

yes

—

C₂×#11764040

C₂³

yes

—

C₄⋊Dic₁₃₇

C₂×C₄

yes

—

C₄⋊Dic₁₃₇

C₂×C₄

yes

—

#14555489

C₂×C₄

yes

—

#14555489

C₂×C₄

yes

—

C₄×Dic₁₃₇

C₂×C₄

yes

—

C₄×Dic₁₃₇

C₂³

yes

—

C₂²×C₄×C₁₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

C₂×C₈×C₁₃₇

C₂³

yes

—

C₄²×C₁₃₇

C₂×C₄

yes

characteristic

C₄²×C₁₃₇

C₂×C₄

yes

characteristic

C₄²×C₁₃₇

C₂³

yes

characteristic

C₄²×C₁₃₇

C₂³

yes

characteristic

Dic₂₇₄

C₂×C₈

yes

—

Dic₂₇₄

C₂×C₈

yes

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

C₂×C₈

yes

—

C₂×Dic₁₃₇

C₂²×C₄

yes

—

C₂×Dic₁₃₇

C₂²×C₄

yes

—

#11764040

—

C₂³×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂×C₈

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂⁴

yes

characteristic

C₈×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂²×C₈

yes

—

C₂²×C₁₃₇

C₂³×C₄

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂²×C₈

yes

—

C₄×C₁₃₇

C₂²×C₈

yes

—

C₄×C₁₃₇

2₋¹⁺⁴

yes

—

Dic₁₃₇

—

Dic₁₃₇

C₂²×C₈

yes

—

C₂×C₁₃₇

—

C₂×C₁₃₇

#519

yes

characteristic

C₂×C₁₃₇

C₂³×C₈

yes

C₂×C₁₃₇

yes

C₁₃₇

yes

characteristic, complemented, Sylow

#2982

—

137

maximal, Sylow

C₂×C₄×Q₈

—

137

—

C₈×Q₈

—

137

—

C₈×Q₈

—

137

—

#431

—

137

—

#431

—

137

—

#431

D₁₃₇

yes

characteristic

C₂²×Q₈

—

137

—

C₄×Q₈

—

137

—

C₄×Q₈

—

137

—

C₄×Q₈

—

137

—

C₄×Q₈

—

137

—

C₂×C₄⋊C₄

—

137

—

C₂×C₄⋊C₄

—

137

—

C₂×C₄²

—

137

—

C₂×C₄²

C₂×D₁₃₇

yes

characteristic

C₄⋊C₈

—

137

—

C₄⋊C₈

—

137

—

C₄⋊C₈

C₂×D₁₃₇

yes

—

C₂²⋊C₈

—

137

—

C₂²⋊C₈

C₂×D₁₃₇

yes

—

100

C₄×C₈

—

137

—

101

C₄×C₈

—

137

—

102

C₄×C₈

C₂×D₁₃₇

yes

—

103

C₂×Q₈

—

137

—

104

C₂×Q₈

—

137

—

105

C₂×Q₈

—

137

—

106

C₂²×C₄

—

137

—

107

C₂²×C₄

—

137

—

108

C₂²×C₄

C₄×D₁₃₇

yes

characteristic

109

C₂²×C₄

C₄×D₁₃₇

yes

characteristic

110

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

111

C₂×C₈

—

137

—

112

C₂×C₈

C₂²×D₁₃₇

yes

—

113

C₄⋊C₄

—

137

—

114

C₄⋊C₄

—

137

—

115

C₄⋊C₄

—

137

—

116

C₄⋊C₄

—

137

—

117

C₄²

—

137

—

118

C₄²

—

137

—

119

C₄²

C₄×D₁₃₇

yes

characteristic

120

C₄²

C₄×D₁₃₇

yes

characteristic

121

C₄²

C₂²×D₁₃₇

yes

characteristic

122

C₄²

C₂²×D₁₃₇

yes

characteristic

123

C₂³

C₂×C₄×D₁₃₇

yes

characteristic

124

Q₈

—

137

—

125

Q₈

—

137

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

548

—

129

C₂×C₄

—

137

—

130

C₂×C₄

—

137

—

131

C₂×C₄

—

137

—

132

C₂×C₄

C₈×D₁₃₇

yes

—

133

C₂×C₄

C₂×C₄×D₁₃₇

yes

characteristic

134

C₂×C₄

C₂×C₄×D₁₃₇

yes

characteristic

135

C₂×C₄

C₂×C₄×D₁₃₇

yes

characteristic

136

C₂×C₄

C₂×C₄×D₁₃₇

yes

characteristic

137

C₂×C₄

C₂×C₄×D₁₃₇

yes

characteristic

138

C₂×C₄

C₂³×D₁₃₇

yes

characteristic

139

C₈

—

140

C₈

—

274

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×C₈×D₁₃₇

yes

—

144

C₂²

C₂²×C₄×D₁₃₇

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

274

—

149

C₄

—

137

—

150

C₄

C₂×C₈×D₁₃₇

yes

—

151

C₄

C₂×C₈×D₁₃₇

yes

—

152

C₄

#17984135

yes

—

153

C₂

—

154

C₂

#22577880

yes

characteristic

155

C₂

C₂²×C₈×D₁₃₇

yes

characteristic

156

C₂

#22578169

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₇
commutator	C₂×C₁₃₇	C₂³×C₈
Fitting	#431×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂³×C₄
prime radical	C₂³×C₁₃₇	C₂²×C₄
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	#431	D₁₃₇
hypocenter	C₁₃₇	#2982
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #431

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₃₇

commutator

C₂×C₁₃₇

C₂³×C₈

Fitting

#431×C₁₃₇

C₂

socle

C₂²×C₁₃₇

C₂³×C₄

prime radical

C₂³×C₁₃₇

C₂²×C₄

cosocle

C₂×C₄×C₁₃₇

C₂⁴

D₁₃₇

C₁₃₇

G ▷ C₂×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂×C₄ ◁ #431

N	H ≅ G/N	type
2. C₈×Dic₂₇₄	C₂	semidirect
3. #22577527	C₂	semidirect
24. C₂×Dic₂₇₄	C₈	semidirect
78. C₁₃₇	#2982	semidirect

H ≅ G/N

type

2. C₈×Dic₂₇₄

C₂

semidirect

3. #22577527

C₂

semidirect

24. C₂×Dic₂₇₄

C₈

semidirect

78. C₁₃₇

#2982

semidirect

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

subgroup	index
C₂×C₄×Dic₂₇₄	2
C₈×Dic₂₇₄	2
#22577527	2
#22577456	2
#22577453	2
#431×C₁₃₇	2
#2982	137

subgroup

index

C₂×C₄×Dic₂₇₄

C₈×Dic₂₇₄

#431×C₁₃₇

137

Subgroups — #31789869

Conjugacy classes of subgroups

Complements

Maximal subgroups