Group #31789508 — subgroups

Subgroups — #31789508

Order 17536.

484 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 189 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×#17984094	C₂	1	2	yes	maximal, complemented
2	#22577953	C₂	1	1	yes	maximal, characteristic
3	#22577476	C₂	1	8	yes	maximal, complemented
4	#22577427	C₂	1	1	yes	maximal, characteristic, complemented
5	#22577369	C₂	1	2	yes	maximal, complemented
6	C₂×#97×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14555510	C₂²	1	2	yes	—
8	#17984094	—	2	8	no	—
9	C₂×#14555494	C₂²	1	2	yes	—
10	C₂×#14555493	C₂²	1	4	yes	—
11	C₂×C₈×D₁₃₇	C₂²	1	4	yes	—
12	#17984088	C₄	1	1	yes	characteristic
13	#17984085	C₄	1	1	yes	characteristic
14	#17984076	C₄	1	2	yes	—
15	#17984062	C₄	1	4	yes	—
16	C₄×D₅₄₈	C₂²	1	4	yes	—
17	C₄⋊#11764040	C₂²	1	4	yes	—
18	#17984000	C₂²	1	8	yes	—
19	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
22	#97×C₁₃₇	C₂²	1	4	yes	—
23	#14555510	—	2	4	no	—
24	#14555510	—	2	4	no	—
25	C₂×Dic₂₇₄	C₂×C₄	1	2	yes	—
26	C₂×D₅₄₈	C₂×C₄	1	2	yes	—
27	C₂×D₅₄₈	C₂×C₄	1	4	yes	—
28	C₂×C₂²⋊D₁₃₇	C₂×C₄	1	4	yes	—
29	C₂×C₄×D₁₃₇	C₂³	1	4	yes	—
30	#14555494	—	2	4	no	—
31	#14555493	—	2	8	no	—
32	C₂×#11764040	C₂³	1	4	yes	—
33	C₈×D₁₃₇	—	2	8	no	—
34	C₄⋊Dic₁₃₇	C₂×C₄	1	4	yes	—
35	#14555487	C₂×C₄	1	8	yes	—
36	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
38	C₂×C₈×C₁₃₇	—	2	2	no	—
39	C₂×C₈×C₁₃₇	D₄	1	4	yes	complemented
40	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
41	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
42	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₁₃₇	C₂³	1	2	yes	—
45	C₂²×D₁₃₇	C₂²×C₄	1	4	yes	—
46	Dic₂₇₄	—	2	4	no	—
47	D₅₄₈	—	2	4	no	—
48	D₅₄₈	—	4	4	no	—
49	C₂²⋊D₁₃₇	—	2	8	no	—
50	C₂×Dic₁₃₇	C₂²×C₄	1	4	yes	—
51	C₄×D₁₃₇	—	2	4	no	—
52	C₄×D₁₃₇	—	2	4	no	—
53	#11764040	—	2	4	no	—
54	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₃₇	—	4	2	no	—
56	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₃₇	#41	1	2	yes	—
61	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
62	C₈×C₁₃₇	—	2	2	no	—
63	C₈×C₁₃₇	C₂×D₄	1	4	yes	—
64	C₂²×C₁₃₇	—	2	2	no	—
65	C₂²×C₁₃₇	C₄×D₄	1	2	yes	—
66	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
67	C₂×D₁₃₇	—	2	4	no	—
68	C₂×D₁₃₇	—	2	8	no	—
69	C₄×C₁₃₇	—	2	2	no	—
70	C₄×C₁₃₇	—	2	2	no	—
71	C₄×C₁₃₇	C₄×D₄	1	2	yes	—
72	C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
73	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
74	Dic₁₃₇	—	2	4	no	—
75	C₂×C₁₃₇	—	2	2	no	—
76	C₂×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
77	C₂×C₁₃₇	#568	1	2	yes	—
78	D₁₃₇	—	4	4	no	—
79	C₁₃₇	#2932	1	1	yes	characteristic, complemented, Sylow
80	#2932	—	137	1	no	maximal, Sylow
81	C₂×#131	—	137	2	no	—
82	#532	—	137	1	no	—
83	#435	—	137	8	no	—
84	#423	—	137	1	no	—
85	#408	—	137	2	no	—
86	C₂×#97	D₁₃₇	1	1	yes	characteristic, complemented
87	C₂×#41	—	137	2	no	—
88	#131	—	274	8	no	—
89	C₂×M₁₆	—	137	2	no	—
90	C₂×M₁₆	—	137	4	no	—
91	C₂²×C₈	—	137	4	no	—
92	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
93	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
94	C₄⋊Q₈	—	137	1	no	—
95	#127	—	137	1	no	—
96	#124	—	137	2	no	—
97	#121	—	137	4	no	—
98	C₄×D₄	—	137	4	no	—
99	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
100	C₄⋊C₈	—	137	4	no	—
101	C₂²⋊C₈	—	137	8	no	—
102	#97	C₂×D₁₃₇	1	4	yes	—
103	#41	—	274	4	no	—
104	#41	—	274	4	no	—
105	C₂×Q₈	—	137	2	no	—
106	C₂×D₄	—	137	4	no	—
107	C₂×D₄	—	137	2	no	—
108	C₂×D₄	—	137	4	no	—
109	C₂²×C₄	—	137	4	no	—
110	C₂²×C₄	C₄×D₁₃₇	1	2	yes	—
111	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
112	M₁₆	—	274	4	no	—
113	M₁₆	—	274	8	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	274	8	no	—
116	C₂×C₈	—	137	4	no	—
117	C₂×C₈	D₅₄₈	1	4	yes	complemented
118	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
119	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
120	C₄⋊C₄	—	137	4	no	—
121	C₂²⋊C₄	—	137	8	no	—
122	C₄²	C₄×D₁₃₇	1	1	yes	characteristic
123	C₄²	C₄×D₁₃₇	1	1	yes	characteristic
124	C₄²	C₂²×D₁₃₇	1	2	yes	—
125	C₂³	—	137	4	no	—
126	C₂³	C₂×C₄×D₁₃₇	1	1	yes	characteristic
127	Q₈	—	274	4	no	—
128	D₄	—	274	8	no	—
129	D₄	—	274	4	no	—
130	D₄	—	548	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	274	4	no	—
133	C₂×C₄	—	274	4	no	—
134	C₂×C₄	—	137	4	no	—
135	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
136	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
138	C₂×C₄	C₂×D₅₄₈	1	2	yes	—
139	C₂×C₄	#14555510	1	2	yes	—
140	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	—	274	4	no	—
143	C₈	C₂×D₅₄₈	1	4	yes	—
144	C₂²	—	2	2	no	—
145	C₂²	—	274	4	no	—
146	C₂²	—	274	8	no	—
147	C₂²	C₄×D₅₄₈	1	2	yes	—
148	C₂²	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	274	4	no	—
152	C₄	C₄×D₅₄₈	1	2	yes	—
153	C₄	C₂²×D₅₄₈	1	1	yes	characteristic
154	C₄	C₂×#14555510	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	548	4	no	—
157	C₂	C₂×C₄×D₅₄₈	1	1	yes	characteristic
158	C₂	#22578169	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984094

C₂

yes

maximal, complemented

#22577953

C₂

yes

maximal, characteristic

#22577476

C₂

yes

maximal, complemented

#22577427

C₂

yes

maximal, characteristic, complemented

#22577369

C₂

yes

maximal, complemented

C₂×#97×C₁₃₇

C₂

yes

maximal, characteristic, complemented

C₂×#14555510

C₂²

yes

—

#17984094

—

C₂×#14555494

C₂²

yes

—

C₂×#14555493

C₂²

yes

—

C₂×C₈×D₁₃₇

C₂²

yes

—

#17984088

C₄

yes

characteristic

#17984085

C₄

yes

characteristic

#17984076

C₄

yes

—

#17984062

C₄

yes

—

C₄×D₅₄₈

C₂²

yes

—

C₄⋊#11764040

C₂²

yes

—

#17984000

C₂²

yes

—

C₂²×C₈×C₁₃₇

C₂²

yes

characteristic

C₂²×C₈×C₁₃₇

C₂²

yes

characteristic

C₂×C₄²×C₁₃₇

C₂²

yes

characteristic

#97×C₁₃₇

C₂²

yes

—

#14555510

—

#14555510

—

C₂×Dic₂₇₄

C₂×C₄

yes

—

C₂×D₅₄₈

C₂×C₄

yes

—

C₂×D₅₄₈

C₂×C₄

yes

—

C₂×C₂²⋊D₁₃₇

C₂×C₄

yes

—

C₂×C₄×D₁₃₇

C₂³

yes

—

#14555494

—

#14555493

—

C₂×#11764040

C₂³

yes

—

C₈×D₁₃₇

—

C₄⋊Dic₁₃₇

C₂×C₄

yes

—

#14555487

C₂×C₄

yes

—

C₂²×C₄×C₁₃₇

C₂×C₄

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

D₄

yes

complemented

C₂×C₈×C₁₃₇

C₂³

yes

—

C₂×C₈×C₁₃₇

C₂³

yes

—

C₄²×C₁₃₇

C₂×C₄

yes

characteristic

C₄²×C₁₃₇

C₂×C₄

yes

characteristic

C₄²×C₁₃₇

C₂³

yes

—

C₂²×D₁₃₇

C₂²×C₄

yes

—

Dic₂₇₄

—

D₅₄₈

—

D₅₄₈

—

C₂²⋊D₁₃₇

—

C₂×Dic₁₃₇

C₂²×C₄

yes

—

C₄×D₁₃₇

—

C₄×D₁₃₇

—

#11764040

—

C₂³×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂×C₄×C₁₃₇

#41

yes

—

C₂×C₄×C₁₃₇

C₂⁴

yes

characteristic

C₈×C₁₃₇

—

C₈×C₁₃₇

C₂×D₄

yes

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₄×D₄

yes

—

C₂²×C₁₃₇

C₂³×C₄

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₄×D₄

yes

—

C₄×C₁₃₇

C₂²×D₄

yes

characteristic

C₄×C₁₃₇

C₂×#41

yes

characteristic

Dic₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

C₂×C₄×D₄

yes

characteristic

C₂×C₁₃₇

#568

yes

—

D₁₃₇

—

C₁₃₇

#2932

yes

characteristic, complemented, Sylow

#2932

—

137

maximal, Sylow

C₂×#131

—

137

—

#532

—

137

—

#435

—

137

—

#423

—

137

—

#408

—

137

—

C₂×#97

D₁₃₇

yes

characteristic, complemented

C₂×#41

—

137

—

#131

—

274

—

C₂×M₁₆

—

137

—

C₂×M₁₆

—

137

—

C₂²×C₈

—

137

—

C₂²×C₈

C₂×D₁₃₇

yes

characteristic

C₂²×C₈

C₂×D₁₃₇

yes

characteristic

C₄⋊Q₈

—

137

—

#127

—

137

—

#124

—

137

—

#121

—

137

—

C₄×D₄

—

137

—

C₂×C₄²

C₂×D₁₃₇

yes

characteristic

100

C₄⋊C₈

—

137

—

101

C₂²⋊C₈

—

137

—

102

#97

C₂×D₁₃₇

yes

—

103

#41

—

274

—

104

#41

—

274

—

105

C₂×Q₈

—

137

—

106

C₂×D₄

—

137

—

107

C₂×D₄

—

137

—

108

C₂×D₄

—

137

—

109

C₂²×C₄

—

137

—

110

C₂²×C₄

C₄×D₁₃₇

yes

—

111

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

112

M₁₆

—

274

—

113

M₁₆

—

274

—

114

C₂×C₈

—

115

C₂×C₈

—

274

—

116

C₂×C₈

—

137

—

117

C₂×C₈

D₅₄₈

yes

complemented

118

C₂×C₈

C₂²×D₁₃₇

yes

—

119

C₂×C₈

C₂²×D₁₃₇

yes

—

120

C₄⋊C₄

—

137

—

121

C₂²⋊C₄

—

137

—

122

C₄²

C₄×D₁₃₇

yes

characteristic

123

C₄²

C₄×D₁₃₇

yes

characteristic

124

C₄²

C₂²×D₁₃₇

yes

—

125

C₂³

—

137

—

126

C₂³

C₂×C₄×D₁₃₇

yes

characteristic

127

Q₈

—

274

—

128

D₄

—

274

—

129

D₄

—

274

—

130

D₄

—

548

—

131

C₂×C₄

—

132

C₂×C₄

—

274

—

133

C₂×C₄

—

274

—

134

C₂×C₄

—

137

—

135

C₂×C₄

C₂×C₄×D₁₃₇

yes

—

136

C₂×C₄

C₂×C₄×D₁₃₇

yes

—

137

C₂×C₄

C₂×C₄×D₁₃₇

yes

characteristic

138

C₂×C₄

C₂×D₅₄₈

yes

—

139

C₂×C₄

#14555510

yes

—

140

C₂×C₄

C₂³×D₁₃₇

yes

characteristic

141

C₈

—

142

C₈

—

274

—

143

C₈

C₂×D₅₄₈

yes

—

144

C₂²

—

145

C₂²

—

274

—

146

C₂²

—

274

—

147

C₂²

C₄×D₅₄₈

yes

—

148

C₂²

C₂²×C₄×D₁₃₇

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

274

—

152

C₄

C₄×D₅₄₈

yes

—

153

C₄

C₂²×D₅₄₈

yes

characteristic

154

C₄

C₂×#14555510

yes

characteristic

155

C₂

—

156

C₂

—

548

—

157

C₂

C₂×C₄×D₅₄₈

yes

characteristic

158

C₂

#22578169

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂³×C₄
Fitting	C₂×#97×C₁₃₇	C₂
prime radical	#22577953	C₂
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	C₂×#97	D₁₃₇
hypocenter	C₁₃₇	#2932
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₃₇

commutator, socle

C₂²×C₁₃₇

C₂³×C₄

Fitting

C₂×#97×C₁₃₇

C₂

prime radical

#22577953

C₂

cosocle

C₂×C₄×C₁₃₇

C₂⁴

hypercenter

C₂×#97

D₁₃₇

C₁₃₇

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂×C₄ ◁ C₂×#97

N	H ≅ G/N	type
1. C₂×#17984094	C₂	semidirect
3. #22577476	C₂	semidirect
4. #22577427	C₂	semidirect
5. #22577369	C₂	semidirect
6. C₂×#97×C₁₃₇	C₂	semidirect
39. C₂×C₈×C₁₃₇	D₄	semidirect
79. C₁₃₇	#2932	semidirect
86. C₂×#97	D₁₃₇	semidirect
117. C₂×C₈	D₅₄₈	semidirect

H ≅ G/N

type

1. C₂×#17984094

C₂

semidirect

3. #22577476

C₂

semidirect

4. #22577427

C₂

semidirect

5. #22577369

C₂

semidirect

6. C₂×#97×C₁₃₇

C₂

semidirect

39. C₂×C₈×C₁₃₇

D₄

semidirect

79. C₁₃₇

#2932

semidirect

86. C₂×#97

D₁₃₇

semidirect

117. C₂×C₈

D₅₄₈

semidirect

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

subgroup	index
C₂×#17984094	2
#22577953	2
#22577476	2
#22577427	2
#22577369	2
C₂×#97×C₁₃₇	2
#2932	137

subgroup

index

C₂×#97×C₁₃₇

137

Subgroups — #31789508

Conjugacy classes of subgroups

Complements

Maximal subgroups