Group #31794158 — subgroups

Subgroups — C₂×#22578016 #31794158

Order 17536.

1004 conjugacy classes of subgroups; only the 485 normal subgroups are shown.

normal subgroups only

This group has 1004 conjugacy classes of subgroups — too many to draw. Only its 485 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	#22578016	C₂	1	16	yes	maximal, direct product factor
2	C₂×#17984088	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17984083	C₂	1	4	yes	maximal
4	C₂×#17984079	C₂	1	4	yes	maximal, complemented
5	C₂×#17984068	C₂	1	2	yes	maximal
6	C₂×#17984053	C₂	1	2	yes	maximal, complemented
7	C₂×#17984045	C₂	1	1	yes	maximal, characteristic
8	C₂×#125×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Dic₂₇₄	C₂²	1	2	yes	—
10	C₂²×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
11	#17984088	C₂²	1	8	yes	complemented
12	#17984083	C₂²	1	32	yes	—
13	#17984079	C₂²	1	32	yes	complemented
14	#17984068	C₂²	1	16	yes	—
15	#17984053	C₂²	1	16	yes	complemented
16	#17984045	C₂²	1	8	yes	—
17	C₂×C₄⋊Dic₁₃₇	C₂²	1	4	yes	—
18	C₂×C₄⋊Dic₁₃₇	C₂²	1	4	yes	—
19	C₂×#14555489	C₂²	1	2	yes	—
20	C₂×#14555489	C₂²	1	4	yes	—
21	C₂×#14555487	C₂²	1	2	yes	—
22	C₂×#14555487	C₂²	1	4	yes	—
23	C₂×C₄×Dic₁₃₇	C₂²	1	4	yes	—
24	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
25	#125×C₁₃₇	C₂²	1	8	yes	complemented
26	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	4	yes	—
27	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
28	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
29	C₂³×D₁₃₇	C₂³	1	1	yes	characteristic
30	C₂×Dic₂₇₄	—	2	8	no	—
31	C₂×Dic₂₇₄	C₂³	1	8	yes	—
32	C₂²×Dic₁₃₇	C₂³	1	2	yes	—
33	C₂²×Dic₁₃₇	C₂³	1	4	yes	—
34	C₂²×Dic₁₃₇	C₂³	1	1	yes	characteristic
35	C₂×C₄×D₁₃₇	—	2	4	no	—
36	C₂×C₄×D₁₃₇	C₂³	1	4	yes	—
37	C₄⋊Dic₁₃₇	C₂³	1	16	yes	—
38	C₄⋊Dic₁₃₇	C₂³	1	16	yes	—
39	#14555489	C₂³	1	8	yes	—
40	#14555489	C₂³	1	16	yes	—
41	#14555487	C₂³	1	8	yes	—
42	#14555487	C₂³	1	16	yes	—
43	C₄×Dic₁₃₇	C₂³	1	4	yes	—
44	C₄×Dic₁₃₇	C₂³	1	16	yes	—
45	C₂²×C₄×C₁₃₇	C₂³	1	4	yes	—
46	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
47	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
48	C₄⋊C₄×C₁₃₇	C₂³	1	8	yes	—
49	C₄⋊C₄×C₁₃₇	C₂³	1	16	yes	—
50	C₄²×C₁₃₇	C₂³	1	4	yes	—
51	C₂²×D₁₃₇	—	2	2	no	—
52	C₂²×D₁₃₇	—	2	4	no	—
53	C₂²×D₁₃₇	C₂⁴	1	2	yes	—
54	Dic₂₇₄	—	2	16	no	—
55	C₂×Dic₁₃₇	—	2	2	no	—
56	C₂×Dic₁₃₇	—	2	8	no	—
57	C₂×Dic₁₃₇	—	2	4	no	—
58	C₂×Dic₁₃₇	C₂⁴	1	4	yes	—
59	C₂×Dic₁₃₇	C₂⁴	1	8	yes	—
60	C₂×Dic₁₃₇	C₂⁴	1	2	yes	—
61	C₄×D₁₃₇	—	2	8	no	—
62	C₂³×C₁₃₇	C₂⁴	1	1	yes	characteristic
63	C₂×C₄×C₁₃₇	—	2	8	no	—
64	C₂×C₄×C₁₃₇	—	2	4	no	—
65	C₂×C₄×C₁₃₇	#41	1	4	yes	—
66	C₂×C₄×C₁₃₇	C₂⁴	1	4	yes	—
67	C₂×C₄×C₁₃₇	C₂⁴	1	8	yes	—
68	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
69	C₂²×C₁₃₇	C₂×#41	1	2	yes	—
70	C₂²×C₁₃₇	2₋¹⁺⁴	1	4	yes	—
71	C₂²×C₁₃₇	C₂⁵	1	1	yes	characteristic
72	C₂×D₁₃₇	—	2	2	no	—
73	C₂×D₁₃₇	—	2	4	no	—
74	C₂×D₁₃₇	—	4	4	no	—
75	C₄×C₁₃₇	—	2	4	no	—
76	C₄×C₁₃₇	—	2	8	no	—
77	C₄×C₁₃₇	C₂×#41	1	4	yes	—
78	Dic₁₃₇	—	2	2	no	—
79	Dic₁₃₇	—	2	8	no	—
80	Dic₁₃₇	—	2	4	no	—
81	C₂×C₁₃₇	#541	1	4	yes	complemented
82	C₂×C₁₃₇	C₂²×#41	1	1	yes	characteristic
83	C₂×C₁₃₇	C₂×2₋¹⁺⁴	1	2	yes	—
84	D₁₃₇	—	4	2	no	—
85	C₁₃₇	C₂×#541	1	1	yes	characteristic, complemented, Sylow
86	C₂×#541	—	137	1	no	maximal, Sylow
87	#541	—	137	16	no	—
88	C₂×C₄⋊Q₈	—	137	1	no	—
89	C₂×#126	—	137	4	no	—
90	C₂×#125	—	137	4	no	—
91	C₂×#125	D₁₃₇	1	1	yes	characteristic, complemented
92	C₂×C₂²⋊Q₈	—	137	2	no	—
93	C₂×C₄×Q₈	—	137	2	no	—
94	C₂×#117	—	137	1	no	—
95	C₂²×Q₈	—	137	2	no	—
96	C₂³×C₄	—	137	1	no	—
97	C₄⋊Q₈	—	137	8	no	—
98	#126	—	137	32	no	—
99	#125	—	137	32	no	—
100	#125	C₂×D₁₃₇	1	8	yes	complemented
101	C₂²⋊Q₈	—	137	16	no	—
102	C₄×Q₈	—	137	16	no	—
103	#117	—	137	8	no	—
104	C₂×C₄⋊C₄	—	137	2	no	—
105	C₂×C₄⋊C₄	—	137	4	no	—
106	C₂×C₄⋊C₄	—	137	4	no	—
107	C₂×C₄⋊C₄	—	137	4	no	—
108	C₂×C₄⋊C₄	C₂×D₁₃₇	1	2	yes	—
109	C₂×C₄⋊C₄	C₂×D₁₃₇	1	4	yes	—
110	C₂×C₂²⋊C₄	—	137	2	no	—
111	C₂×C₂²⋊C₄	—	137	4	no	—
112	C₂×C₄²	—	137	1	no	—
113	C₂×C₄²	—	137	4	no	—
114	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
115	C₂⁴	—	137	1	no	—
116	C₂×Q₈	—	274	8	no	—
117	C₂×Q₈	—	137	8	no	—
118	C₂²×C₄	—	274	4	no	—
119	C₂²×C₄	—	137	2	no	—
120	C₂²×C₄	—	137	4	no	—
121	C₂²×C₄	—	137	1	no	—
122	C₂²×C₄	—	137	4	no	—
123	C₂²×C₄	C₂²×D₁₃₇	1	4	yes	—
124	C₂²×C₄	C₂²×D₁₃₇	1	2	yes	—
125	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
126	C₄⋊C₄	—	137	8	no	—
127	C₄⋊C₄	—	137	16	no	—
128	C₄⋊C₄	—	137	16	no	—
129	C₄⋊C₄	—	137	16	no	—
130	C₄⋊C₄	C₂²×D₁₃₇	1	8	yes	—
131	C₄⋊C₄	C₂²×D₁₃₇	1	16	yes	—
132	C₂²⋊C₄	—	137	8	no	—
133	C₂²⋊C₄	—	137	16	no	—
134	C₄²	—	137	4	no	—
135	C₄²	—	137	16	no	—
136	C₄²	C₂²×D₁₃₇	1	4	yes	—
137	C₂³	—	274	2	no	—
138	C₂³	—	274	4	no	—
139	C₂³	—	137	2	no	—
140	C₂³	C₂³×D₁₃₇	1	1	yes	characteristic
141	Q₈	—	274	16	no	—
142	C₂×C₄	—	2	8	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	274	2	no	—
145	C₂×C₄	—	274	8	no	—
146	C₂×C₄	—	274	4	no	—
147	C₂×C₄	—	274	8	no	—
148	C₂×C₄	—	137	4	no	—
149	C₂×C₄	—	137	8	no	—
150	C₂×C₄	—	137	2	no	—
151	C₂×C₄	#14555511	1	4	yes	—
152	C₂×C₄	C₂³×D₁₃₇	1	8	yes	—
153	C₂×C₄	C₂³×D₁₃₇	1	4	yes	—
154	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
155	C₂²	—	274	2	no	—
156	C₂²	—	274	4	no	—
157	C₂²	—	548	4	no	—
158	C₂²	C₂×#14555511	1	2	yes	—
159	C₂²	#17984135	1	4	yes	—
160	C₂²	C₂⁴×D₁₃₇	1	1	yes	characteristic
161	C₄	—	2	8	no	—
162	C₄	—	2	4	no	—
163	C₄	—	274	2	no	—
164	C₄	—	274	8	no	—
165	C₄	—	274	4	no	—
166	C₄	C₂×#14555511	1	4	yes	—
167	C₂	—	548	2	no	—
168	C₂	#22578016	1	4	yes	direct product factor
169	C₂	C₂²×#14555511	1	1	yes	characteristic
170	C₂	C₂×#17984135	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22578016

C₂

yes

maximal, direct product factor

C₂×#17984088

C₂

yes

maximal, characteristic, complemented

C₂×#17984083

C₂

yes

maximal

C₂×#17984079

C₂

yes

maximal, complemented

C₂×#17984068

C₂

yes

maximal

C₂×#17984053

C₂

yes

maximal, complemented

C₂×#17984045

C₂

yes

maximal, characteristic

C₂×#125×C₁₃₇

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₂₇₄

C₂²

yes

—

C₂²×C₄×D₁₃₇

C₂²

yes

characteristic

#17984088

C₂²

yes

complemented

#17984083

C₂²

yes

—

#17984079

C₂²

yes

complemented

#17984068

C₂²

yes

—

#17984053

C₂²

yes

complemented

#17984045

C₂²

yes

—

C₂×C₄⋊Dic₁₃₇

C₂²

yes

—

C₂×C₄⋊Dic₁₃₇

C₂²

yes

—

C₂×#14555489

C₂²

yes

—

C₂×#14555489

C₂²

yes

—

C₂×#14555487

C₂²

yes

—

C₂×#14555487

C₂²

yes

—

C₂×C₄×Dic₁₃₇

C₂²

yes

—

C₂×C₄×Dic₁₃₇

C₂²

yes

characteristic

#125×C₁₃₇

C₂²

yes

complemented

C₂×C₄⋊C₄×C₁₃₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₃₇

C₂²

yes

—

C₂×C₄²×C₁₃₇

C₂²

yes

characteristic

C₂³×D₁₃₇

C₂³

yes

characteristic

C₂×Dic₂₇₄

—

C₂×Dic₂₇₄

C₂³

yes

—

C₂²×Dic₁₃₇

C₂³

yes

—

C₂²×Dic₁₃₇

C₂³

yes

—

C₂²×Dic₁₃₇

C₂³

yes

characteristic

C₂×C₄×D₁₃₇

—

C₂×C₄×D₁₃₇

C₂³

yes

—

C₄⋊Dic₁₃₇

C₂³

yes

—

C₄⋊Dic₁₃₇

C₂³

yes

—

#14555489

C₂³

yes

—

#14555489

C₂³

yes

—

#14555487

C₂³

yes

—

#14555487

C₂³

yes

—

C₄×Dic₁₃₇

C₂³

yes

—

C₄×Dic₁₃₇

C₂³

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₇

C₂³

yes

—

C₄⋊C₄×C₁₃₇

C₂³

yes

—

C₄²×C₁₃₇

C₂³

yes

—

C₂²×D₁₃₇

—

C₂²×D₁₃₇

—

C₂²×D₁₃₇

C₂⁴

yes

—

Dic₂₇₄

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

C₂⁴

yes

—

C₂×Dic₁₃₇

C₂⁴

yes

—

C₂×Dic₁₃₇

C₂⁴

yes

—

C₄×D₁₃₇

—

C₂³×C₁₃₇

C₂⁴

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

#41

yes

—

C₂×C₄×C₁₃₇

C₂⁴

yes

—

C₂×C₄×C₁₃₇

C₂⁴

yes

—

C₂×C₄×C₁₃₇

C₂⁴

yes

—

C₂²×C₁₃₇

C₂×#41

yes

—

C₂²×C₁₃₇

2₋¹⁺⁴

yes

—

C₂²×C₁₃₇

C₂⁵

yes

characteristic

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₂×D₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₂×#41

yes

—

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

—

C₂×C₁₃₇

#541

yes

complemented

C₂×C₁₃₇

C₂²×#41

yes

characteristic

C₂×C₁₃₇

C₂×2₋¹⁺⁴

yes

—

D₁₃₇

—

C₁₃₇

C₂×#541

yes

characteristic, complemented, Sylow

C₂×#541

—

137

maximal, Sylow

#541

—

137

—

C₂×C₄⋊Q₈

—

137

—

C₂×#126

—

137

—

C₂×#125

—

137

—

C₂×#125

D₁₃₇

yes

characteristic, complemented

C₂×C₂²⋊Q₈

—

137

—

C₂×C₄×Q₈

—

137

—

C₂×#117

—

137

—

C₂²×Q₈

—

137

—

C₂³×C₄

—

137

—

C₄⋊Q₈

—

137

—

#126

—

137

—

#125

—

137

—

100

#125

C₂×D₁₃₇

yes

complemented

101

C₂²⋊Q₈

—

137

—

102

C₄×Q₈

—

137

—

103

#117

—

137

—

104

C₂×C₄⋊C₄

—

137

—

105

C₂×C₄⋊C₄

—

137

—

106

C₂×C₄⋊C₄

—

137

—

107

C₂×C₄⋊C₄

—

137

—

108

C₂×C₄⋊C₄

C₂×D₁₃₇

yes

—

109

C₂×C₄⋊C₄

C₂×D₁₃₇

yes

—

110

C₂×C₂²⋊C₄

—

137

—

111

C₂×C₂²⋊C₄

—

137

—

112

C₂×C₄²

—

137

—

113

C₂×C₄²

—

137

—

114

C₂×C₄²

C₂×D₁₃₇

yes

characteristic

115

C₂⁴

—

137

—

116

C₂×Q₈

—

274

—

117

C₂×Q₈

—

137

—

118

C₂²×C₄

—

274

—

119

C₂²×C₄

—

137

—

120

C₂²×C₄

—

137

—

121

C₂²×C₄

—

137

—

122

C₂²×C₄

—

137

—

123

C₂²×C₄

C₂²×D₁₃₇

yes

—

124

C₂²×C₄

C₂²×D₁₃₇

yes

—

125

C₂²×C₄

C₂²×D₁₃₇

yes

characteristic

126

C₄⋊C₄

—

137

—

127

C₄⋊C₄

—

137

—

128

C₄⋊C₄

—

137

—

129

C₄⋊C₄

—

137

—

130

C₄⋊C₄

C₂²×D₁₃₇

yes

—

131

C₄⋊C₄

C₂²×D₁₃₇

yes

—

132

C₂²⋊C₄

—

137

—

133

C₂²⋊C₄

—

137

—

134

C₄²

—

137

—

135

C₄²

—

137

—

136

C₄²

C₂²×D₁₃₇

yes

—

137

C₂³

—

274

—

138

C₂³

—

274

—

139

C₂³

—

137

—

140

C₂³

C₂³×D₁₃₇

yes

characteristic

141

Q₈

—

274

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

274

—

145

C₂×C₄

—

274

—

146

C₂×C₄

—

274

—

147

C₂×C₄

—

274

—

148

C₂×C₄

—

137

—

149

C₂×C₄

—

137

—

150

C₂×C₄

—

137

—

151

C₂×C₄

#14555511

yes

—

152

C₂×C₄

C₂³×D₁₃₇

yes

—

153

C₂×C₄

C₂³×D₁₃₇

yes

—

154

C₂×C₄

C₂³×D₁₃₇

yes

—

155

C₂²

—

274

—

156

C₂²

—

274

—

157

C₂²

—

548

—

158

C₂²

C₂×#14555511

yes

—

159

C₂²

#17984135

yes

—

160

C₂²

C₂⁴×D₁₃₇

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

274

—

164

C₄

—

274

—

165

C₄

—

274

—

166

C₄

C₂×#14555511

yes

—

167

C₂

—

548

—

168

C₂

#22578016

yes

direct product factor

169

C₂

C₂²×#14555511

yes

characteristic

170

C₂

C₂×#17984135

yes

—

	subgroup	G/H
center	C₂³	C₂³×D₁₃₇
commutator, cosocle	C₂²×C₁₃₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃₇
Fitting	C₂×#125×C₁₃₇	C₂
socle	C₂³×C₁₃₇	C₂⁴
prime radical	C₂³×D₁₃₇	C₂³
hypercenter	C₂×#125	D₁₃₇
hypocenter	C₁₃₇	C₂×#541
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂³×D₁₃₇

commutator, cosocle

C₂²×C₁₃₇

C₂⁵

Frattini

C₂²

C₂⁴×D₁₃₇

Fitting

C₂×#125×C₁₃₇

C₂

socle

C₂³×C₁₃₇

C₂⁴

prime radical

C₂³×D₁₃₇

C₂³

D₁₃₇

C₁₃₇

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
1. #22578016	C₂	direct (N × H)
2. C₂×#17984088	C₂	semidirect
4. C₂×#17984079	C₂	semidirect
6. C₂×#17984053	C₂	semidirect
8. C₂×#125×C₁₃₇	C₂	semidirect
11. #17984088	C₂²	semidirect
13. #17984079	C₂²	semidirect
15. #17984053	C₂²	semidirect
25. #125×C₁₃₇	C₂²	semidirect
81. C₂×C₁₃₇	#541	semidirect
85. C₁₃₇	C₂×#541	semidirect
91. C₂×#125	D₁₃₇	semidirect
100. #125	C₂×D₁₃₇	semidirect
168. C₂	#22578016	direct (N × H)

H ≅ G/N

type

1. #22578016

C₂

direct (N × H)

2. C₂×#17984088

C₂

semidirect

4. C₂×#17984079

C₂

semidirect

6. C₂×#17984053

C₂

semidirect

8. C₂×#125×C₁₃₇

C₂

semidirect

11. #17984088

C₂²

semidirect

13. #17984079

C₂²

semidirect

15. #17984053

C₂²

semidirect

25. #125×C₁₃₇

C₂²

semidirect

81. C₂×C₁₃₇

#541

semidirect

85. C₁₃₇

C₂×#541

semidirect

91. C₂×#125

D₁₃₇

semidirect

100. #125

C₂×D₁₃₇

semidirect

168. C₂

#22578016

direct (N × H)

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

subgroup	index
#22578016	2
C₂×#17984088	2
C₂×#17984083	2
C₂×#17984079	2
C₂×#17984068	2
C₂×#17984053	2
C₂×#17984045	2
C₂×#125×C₁₃₇	2
C₂×#541	137

subgroup

index

C₂×#125×C₁₃₇

137

Subgroups — C₂×#22578016 #31794158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22578016 #31794158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22578016 #31794158