Group #31796705 — subgroups

Subgroups — #31796705

Order 17536.

260 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 106 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×#17984155	C₂	1	2	yes	maximal
2	#22578269	C₂	1	2	yes	maximal
3	#22578269	C₂	1	2	yes	maximal
4	C₄²×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11764047	C₂²	1	4	yes	—
6	#17984155	—	2	4	no	—
7	C₂×C₄×Dic₁₃₇	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₁₃₇	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₁₃₇	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₁₃₇	C₄	1	1	yes	characteristic
14	C₂×#11764047	—	2	4	no	—
15	C₂×#11764047	—	2	4	no	—
16	C₂×#11764047	—	2	4	no	—
17	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×Dic₁₃₇	C₂³	1	1	yes	characteristic
21	C₄×Dic₁₃₇	—	2	2	no	—
22	C₄×Dic₁₃₇	—	2	2	no	—
23	C₄×Dic₁₃₇	—	2	2	no	—
24	C₄×Dic₁₃₇	—	2	2	no	—
25	C₄×Dic₁₃₇	C₈	1	4	yes	complemented
26	C₄×Dic₁₃₇	D₄	1	2	yes	—
27	C₄×Dic₁₃₇	Q₈	1	2	yes	—
28	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₁₃₇	—	2	2	no	—
32	#11764047	—	4	4	no	—
33	C₂×Dic₁₃₇	—	2	1	no	—
34	C₂×Dic₁₃₇	—	2	1	no	—
35	C₂×Dic₁₃₇	—	2	1	no	—
36	C₂×Dic₁₃₇	—	2	1	no	—
37	C₂×Dic₁₃₇	C₄⋊C₄	1	2	yes	—
38	C₂×Dic₁₃₇	C₂×C₈	1	2	yes	—
39	C₂×Dic₁₃₇	C₂×C₈	1	2	yes	—
40	C₂×Dic₁₃₇	M₁₆	1	2	yes	—
41	C₂×Dic₁₃₇	M₁₆	1	2	yes	—
42	C₂×Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
43	C₂×Dic₁₃₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×Dic₁₃₇	#41	1	2	yes	—
45	C₂×Dic₁₃₇	#41	1	2	yes	—
46	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₃₇	—	2	1	no	—
48	C₂×C₄×C₁₃₇	—	2	1	no	—
49	C₂×C₄×C₁₃₇	—	2	1	no	—
50	C₂×C₄×C₁₃₇	—	2	1	no	—
51	C₂×C₄×C₁₃₇	—	2	1	no	—
52	C₂×C₄×C₁₃₇	—	2	1	no	—
53	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
54	C₂×C₄×C₁₃₇	C₂×C₈	1	2	yes	—
55	C₂×C₄×C₁₃₇	M₁₆	1	2	yes	—
56	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₃₇	#117	1	1	yes	characteristic
58	C₂²×C₁₃₇	#117	1	1	yes	characteristic
59	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
60	C₂²×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
61	C₂²×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
62	C₂²×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
63	C₄×C₁₃₇	—	2	2	no	—
64	C₄×C₁₃₇	—	2	2	no	—
65	C₄×C₁₃₇	—	2	2	no	—
66	Dic₁₃₇	—	2	2	no	—
67	Dic₁₃₇	—	2	2	no	—
68	Dic₁₃₇	C₄⋊C₈	1	4	yes	—
69	Dic₁₃₇	#124	1	2	yes	—
70	Dic₁₃₇	#125	1	2	yes	—
71	C₂×C₁₃₇	#382	1	1	yes	characteristic
72	C₂×C₁₃₇	C₂×C₄⋊C₈	1	1	yes	characteristic
73	C₂×C₁₃₇	#423	1	1	yes	characteristic
74	C₂×C₁₃₇	#431	1	1	yes	characteristic
75	C₂×C₁₃₇	#431	1	1	yes	characteristic
76	C₂×C₁₃₇	#432	1	1	yes	characteristic
77	C₂×C₁₃₇	#432	1	1	yes	characteristic
78	C₁₃₇	#1831	1	1	yes	characteristic, complemented, Sylow
79	#1831	—	137	1	no	maximal, Sylow
80	C₂×C₄⋊C₈	—	137	2	no	—
81	C₄³	—	137	1	no	—
82	#336	—	137	2	no	—
83	#336	—	137	2	no	—
84	C₂²×C₈	—	137	4	no	—
85	C₂×C₄²	—	137	1	no	—
86	C₂×C₄²	—	137	1	no	—
87	C₂×C₄²	—	137	1	no	—
88	C₂×C₄²	—	137	1	no	—
89	C₂×C₄²	—	137	1	no	—
90	C₂×C₄²	—	137	1	no	—
91	C₂×C₄²	#10587739	1	1	yes	characteristic
92	C₄⋊C₈	—	274	4	no	—
93	C₂²×C₄	—	137	1	no	—
94	C₂²×C₄	—	137	1	no	—
95	C₂²×C₄	—	137	1	no	—
96	C₂²×C₄	—	137	1	no	—
97	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
98	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
99	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
100	C₂×C₈	—	274	4	no	—
101	C₂×C₈	—	274	4	no	—
102	C₂×C₈	—	274	4	no	—
103	C₄²	—	2	2	no	—
104	C₄²	—	274	2	no	—
105	C₄²	—	274	2	no	—
106	C₄²	—	274	2	no	—
107	C₄²	—	274	2	no	—
108	C₄²	—	137	2	no	—
109	C₄²	—	137	2	no	—
110	C₄²	—	137	4	no	—
111	C₂³	C₂²×#10587739	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	274	1	no	—
119	C₂×C₄	—	274	1	no	—
120	C₂×C₄	—	274	1	no	—
121	C₂×C₄	—	274	1	no	—
122	C₂×C₄	—	137	2	no	—
123	C₂×C₄	—	137	2	no	—
124	C₂×C₄	—	137	1	no	—
125	C₂×C₄	—	137	2	no	—
126	C₂×C₄	—	137	1	no	—
127	C₂×C₄	—	137	2	no	—
128	C₂×C₄	—	137	2	no	—
129	C₂×C₄	—	137	2	no	—
130	C₂×C₄	—	137	2	no	—
131	C₂×C₄	C₄⋊#10587739	1	2	yes	—
132	C₂×C₄	#14555518	1	2	yes	—
133	C₂×C₄	#14555519	1	2	yes	—
134	C₈	—	548	4	no	—
135	C₂²	C₂×C₄⋊#10587739	1	1	yes	characteristic
136	C₂²	#17984167	1	1	yes	characteristic
137	C₂²	#17984167	1	1	yes	characteristic
138	C₂²	C₂×#14555518	1	1	yes	characteristic
139	C₂²	C₂×#14555519	1	1	yes	characteristic
140	C₂²	D₁₃₇⋊M₁₆	1	1	yes	characteristic
141	C₂²	D₁₃₇⋊M₁₆	1	1	yes	characteristic
142	C₄	—	2	2	no	—
143	C₄	—	2	2	no	—
144	C₄	—	2	2	no	—
145	C₄	—	274	2	no	—
146	C₄	—	274	2	no	—
147	C₄	—	137	2	no	—
148	C₄	—	137	2	no	—
149	C₄	—	137	4	no	—
150	C₂	#22578347	1	1	yes	characteristic
151	C₂	#22578404	1	1	yes	characteristic
152	C₂	#22578406	1	1	yes	characteristic
153	C₂	#22578420	1	1	yes	characteristic
154	C₂	#22578421	1	1	yes	characteristic
155	C₂	#22578423	1	1	yes	characteristic
156	C₂	#22578426	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984155

C₂

yes

maximal

#22578269

C₂

yes

maximal

#22578269

C₂

yes

maximal

C₄²×Dic₁₃₇

C₂

yes

maximal, characteristic

C₂²×#11764047

C₂²

yes

—

#17984155

—

C₂×C₄×Dic₁₃₇

C₄

yes

characteristic

C₂×C₄×Dic₁₃₇

C₄

yes

characteristic

C₂×C₄×Dic₁₃₇

C₄

yes

characteristic

C₂×C₄×Dic₁₃₇

C₂²

yes

characteristic

C₂×C₄×Dic₁₃₇

C₂²

yes

characteristic

C₂×C₄×Dic₁₃₇

C₂²

yes

characteristic

C₂×C₄²×C₁₃₇

C₄

yes

—

—

—

C₂²×Dic₁₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₁₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₁₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₁₃₇

C₂³

yes

characteristic

C₄×Dic₁₃₇

—

C₄×Dic₁₃₇

—

C₄×Dic₁₃₇

—

C₄×Dic₁₃₇

—

C₄×Dic₁₃₇

C₈

yes

complemented

C₄×Dic₁₃₇

D₄

yes

—

C₄×Dic₁₃₇

Q₈

yes

—

C₂²×C₄×C₁₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₇

C₂×C₄

yes

characteristic

C₄²×C₁₃₇

—

#11764047

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

—

C₂×Dic₁₃₇

C₄⋊C₄

yes

—

C₂×Dic₁₃₇

C₂×C₈

yes

—

C₂×Dic₁₃₇

C₂×C₈

yes

—

C₂×Dic₁₃₇

M₁₆

yes

—

C₂×Dic₁₃₇

M₁₆

yes

—

C₂×Dic₁₃₇

C₂×D₄

yes

characteristic

C₂×Dic₁₃₇

C₂×Q₈

yes

characteristic

C₂×Dic₁₃₇

#41

yes

—

C₂×Dic₁₃₇

#41

yes

—

C₂³×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₃₇

C₂×C₈

yes

—

C₂×C₄×C₁₃₇

M₁₆

yes

—

C₂²×C₁₃₇

C₂×C₄⋊C₄

yes

C₂²×C₁₃₇

yes

C₂²×C₁₃₇

yes

C₂²×C₁₃₇

C₂²×C₈

yes

characteristic

C₂²×C₁₃₇

C₂×M₁₆

yes

characteristic

C₂²×C₁₃₇

C₂×M₁₆

yes

characteristic

C₂²×C₁₃₇

C₂×M₁₆

yes

characteristic

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

—

Dic₁₃₇

C₄⋊C₈

yes

—

Dic₁₃₇

#124

yes

—

Dic₁₃₇

#125

yes

—

C₂×C₁₃₇

#382

yes

characteristic

C₂×C₁₃₇

C₂×C₄⋊C₈

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₁₃₇

yes

characteristic, complemented, Sylow

#1831

—

137

maximal, Sylow

C₂×C₄⋊C₈

—

137

—

C₄³

—

137

—

#336

—

137

—

#336

—

137

—

C₂²×C₈

—

137

—

C₂×C₄²

—

137

—

C₂×C₄²

—

137

—

C₂×C₄²

—

137

—

C₂×C₄²

—

137

—

C₂×C₄²

—

137

—

C₂×C₄²

—

137

—

C₂×C₄²

#10587739

yes

characteristic

C₄⋊C₈

—

274

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

—

137

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

100

C₂×C₈

—

274

—

101

C₂×C₈

—

274

—

102

C₂×C₈

—

274

—

103

C₄²

—

104

C₄²

—

274

—

105

C₄²

—

274

—

106

C₄²

—

274

—

107

C₄²

—

274

—

108

C₄²

—

137

—

109

C₄²

—

137

—

110

C₄²

—

137

—

111

C₂³

C₂²×#10587739

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

274

—

119

C₂×C₄

—

274

—

120

C₂×C₄

—

274

—

121

C₂×C₄

—

274

—

122

C₂×C₄

—

137

—

123

C₂×C₄

—

137

—

124

C₂×C₄

—

137

—

125

C₂×C₄

—

137

—

126

C₂×C₄

—

137

—

127

C₂×C₄

—

137

—

128

C₂×C₄

—

137

—

129

C₂×C₄

—

137

—

130

C₂×C₄

—

137

—

131

C₂×C₄

C₄⋊#10587739

yes

—

132

C₂×C₄

#14555518

yes

—

133

C₂×C₄

#14555519

yes

—

134

C₈

—

548

—

135

C₂²

C₂×C₄⋊#10587739

yes

characteristic

136

C₂²

#17984167

yes

characteristic

137

C₂²

#17984167

yes

characteristic

138

C₂²

C₂×#14555518

yes

characteristic

139

C₂²

C₂×#14555519

yes

characteristic

140

C₂²

D₁₃₇⋊M₁₆

yes

characteristic

141

C₂²

D₁₃₇⋊M₁₆

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

274

—

146

C₄

—

274

—

147

C₄

—

137

—

148

C₄

—

137

—

149

C₄

—

137

—

150

C₂

#22578347

yes

characteristic

151

C₂

#22578404

yes

characteristic

152

C₂

#22578406

yes

characteristic

153

C₂

#22578420

yes

characteristic

154

C₂

#22578421

yes

characteristic

155

C₂

#22578423

yes

characteristic

156

C₂

#22578426

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×#10587739
commutator	C₂²×C₁₃₇	C₂²×C₈
Fitting	C₂×C₄²×C₁₃₇	C₄
socle, prime radical	C₂³×C₁₃₇	C₂²×C₄
cosocle	C₂²×Dic₁₃₇	C₂³
hypercenter	C₂×C₄²	#10587739
hypocenter	C₁₃₇	#1831
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

subgroup

G/H

center, Frattini

C₂³

C₂²×#10587739

commutator

C₂²×C₁₃₇

C₂²×C₈

Fitting

C₂×C₄²×C₁₃₇

C₄

socle, prime radical

C₂³×C₁₃₇

C₂²×C₄

cosocle

C₂²×Dic₁₃₇

C₂³

C₂×C₄²

C₁₃₇

G ▷ C₂²×C₁₃₇ ▷ C₁₃₇

C₁ ◁ C₂³ ◁ C₂×C₄²

N	H ≅ G/N	type
25. C₄×Dic₁₃₇	C₈	semidirect
78. C₁₃₇	#1831	semidirect

H ≅ G/N

type

25. C₄×Dic₁₃₇

C₈

semidirect

78. C₁₃₇

#1831

semidirect

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

subgroup	index
C₂×#17984155	2
#22578269	2
C₄²×Dic₁₃₇	2
#1831	137

subgroup

index

C₂×#17984155

#22578269

C₄²×Dic₁₃₇

#1831

137

Subgroups — #31796705

Conjugacy classes of subgroups

Complements

Maximal subgroups