Group #16845570 — subgroups

Subgroups — #16845570

Order 3968.

260 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 133 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⋊(C₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×#11758403	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×Dic₃₁	C₂	1	1	yes	maximal, characteristic
4	#14520895	C₂	1	1	yes	maximal, characteristic
5	#14520803	C₂	1	1	yes	maximal, characteristic
6	#14520803	C₂	1	1	yes	maximal, characteristic
7	C₂×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
13	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
15	C₃₁×C₄⋊C₈	C₄	1	4	yes	complemented
16	C₃₁⋊#95	C₄	1	2	yes	—
17	C₃₁⋊#95	C₄	1	2	yes	—
18	#11758403	—	2	2	no	—
19	C₈×Dic₃₁	—	2	2	no	—
20	C₄⋊(C₃₁⋊C₈)	C₄	1	4	yes	complemented
21	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
24	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
25	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₃₁	—	2	1	no	—
27	C₂×C₈×C₃₁	—	2	1	no	—
28	C₂×C₈×C₃₁	—	2	1	no	—
29	C₂×C₈×C₃₁	—	2	1	no	—
30	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
32	C₄²×C₃₁	C₂×C₄	1	2	yes	—
33	C₄²×C₃₁	C₂×C₄	1	2	yes	—
34	C₄×Dic₃₁	—	2	1	no	—
35	C₄×Dic₃₁	—	2	1	no	—
36	C₄×Dic₃₁	D₄	1	2	yes	—
37	C₄×Dic₃₁	Q₈	1	2	yes	—
38	C₂×C₃₁⋊C₈	—	2	1	no	—
39	C₂×C₃₁⋊C₈	—	2	1	no	—
40	C₂×C₃₁⋊C₈	—	2	1	no	—
41	C₂×C₃₁⋊C₈	—	2	1	no	—
42	C₂×C₃₁⋊C₈	C₂×C₄	1	2	yes	—
43	C₂×C₃₁⋊C₈	C₂×C₄	1	2	yes	—
44	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₃₁	—	2	2	no	—
46	C₂×C₄×C₃₁	—	2	2	no	—
47	C₂×C₄×C₃₁	C₄²	1	2	yes	—
48	C₂×C₄×C₃₁	C₄²	1	2	yes	—
49	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
54	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
55	C₂×Dic₃₁	—	2	2	no	—
56	C₂×Dic₃₁	—	2	2	no	—
57	C₂×Dic₃₁	—	2	2	no	—
58	C₂×Dic₃₁	—	2	2	no	—
59	C₂×Dic₃₁	—	2	2	no	—
60	C₂×Dic₃₁	C₄⋊C₄	1	4	yes	—
61	C₈×C₃₁	—	2	2	no	—
62	C₈×C₃₁	—	2	2	no	—
63	C₃₁⋊C₈	—	2	2	no	—
64	C₃₁⋊C₈	—	2	2	no	—
65	#1768	—	31	1	no	maximal, Sylow
66	C₂²×C₃₁	C₂×C₄²	1	1	yes	characteristic
67	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₃₁	#117	1	1	yes	characteristic
69	C₂²×C₃₁	#131	1	2	yes	—
70	C₂²×C₃₁	#131	1	2	yes	—
71	C₄×C₃₁	—	2	2	no	—
72	C₄×C₃₁	—	2	2	no	—
73	C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
74	C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
75	C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
76	C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
77	Dic₃₁	—	2	4	no	—
78	Dic₃₁	—	4	2	no	—
79	C₂×C₄⋊C₈	—	31	1	no	—
80	C₂×C₄⋊C₈	D₃₁	1	1	yes	characteristic
81	C₂×#97	—	31	1	no	—
82	C₂×C₄×C₈	—	31	1	no	—
83	#376	—	31	1	no	—
84	#336	—	31	1	no	—
85	#336	—	31	1	no	—
86	C₂×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₃₁	#405	1	1	yes	characteristic
88	C₂×C₃₁	#405	1	1	yes	characteristic
89	C₂×C₃₁	#424	1	2	yes	—
90	C₂×C₃₁	#433	1	2	yes	—
91	C₂²×C₈	—	31	1	no	—
92	C₂²×C₈	—	31	1	no	—
93	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
94	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
95	C₂×C₄²	—	31	1	no	—
96	C₂×C₄²	—	31	1	no	—
97	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
98	C₄⋊C₈	—	31	4	no	—
99	C₄⋊C₈	Dic₃₁	1	4	yes	complemented
100	#97	—	62	2	no	—
101	C₄×C₈	—	62	2	no	—
102	#95	—	31	2	no	—
103	#95	—	31	2	no	—
104	C₃₁	#1768	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₄	—	31	2	no	—
106	C₂²×C₄	—	31	2	no	—
107	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
108	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
109	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	31	2	no	—
115	C₂×C₈	—	31	2	no	—
116	C₂×C₈	—	62	1	no	—
117	C₂×C₈	—	62	1	no	—
118	C₂×C₈	—	62	1	no	—
119	C₂×C₈	—	62	1	no	—
120	C₂×C₈	C₂×Dic₃₁	1	2	yes	—
121	C₂×C₈	C₂×Dic₃₁	1	2	yes	—
122	C₄²	—	31	2	no	—
123	C₄²	—	31	2	no	—
124	C₄²	—	62	1	no	—
125	C₄²	—	62	1	no	—
126	C₄²	C₄×D₃₁	1	2	yes	—
127	C₄²	C₂×Dic₃₁	1	2	yes	—
128	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	31	4	no	—
132	C₂×C₄	—	62	2	no	—
133	C₂×C₄	—	62	2	no	—
134	C₂×C₄	—	62	2	no	—
135	C₂×C₄	—	62	2	no	—
136	C₂×C₄	—	62	2	no	—
137	C₂×C₄	C₄×Dic₃₁	1	2	yes	—
138	C₂×C₄	C₄×Dic₃₁	1	2	yes	—
139	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
140	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
141	C₂×C₄	#92497	1	1	yes	characteristic
142	C₂×C₄	Q₈×D₃₁	1	1	yes	characteristic
143	C₂×C₄	#92499	1	1	yes	characteristic
144	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
145	C₈	—	2	2	no	—
146	C₈	—	2	2	no	—
147	C₈	—	62	2	no	—
148	C₈	—	62	2	no	—
149	C₂²	C₄⋊C₄×D₃₁	1	1	yes	characteristic
150	C₂²	#11758469	1	1	yes	characteristic
151	C₂²	#11758478	1	2	yes	—
152	C₂²	#11758484	1	2	yes	—
153	C₂²	C₂×C₄×Dic₃₁	1	1	yes	characteristic
154	C₄	—	2	2	no	—
155	C₄	—	2	2	no	—
156	C₄	—	62	4	no	—
157	C₄	—	124	2	no	—
158	C₄	#11758464	1	1	yes	characteristic
159	C₄	#11758470	1	1	yes	characteristic
160	C₄	D₄×Dic₃₁	1	1	yes	characteristic
161	C₄	Q₈×Dic₃₁	1	1	yes	characteristic
162	C₂	C₄⋊C₄×Dic₃₁	1	1	yes	characteristic
163	C₂	#14521004	1	1	yes	characteristic
164	C₂	#14521007	1	1	yes	characteristic
165	C₂	#14521074	1	2	yes	—
166	C₂	#14521109	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₃₁⋊C₈)

C₂

yes

maximal, characteristic

C₂×#11758403

C₂

yes

maximal, characteristic

C₂×C₈×Dic₃₁

C₂

yes

maximal, characteristic

#14520895

C₂

yes

maximal, characteristic

#14520803

C₂

yes

maximal, characteristic

#14520803

C₂

yes

maximal, characteristic

C₂×C₃₁×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂²×C₈×C₃₁

C₂²

yes

characteristic

C₂²×C₈×C₃₁

C₂²

yes

characteristic

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄×Dic₃₁

C₂²

yes

characteristic

C₂×C₄×Dic₃₁

C₂²

yes

characteristic

C₂²×C₃₁⋊C₈

C₂²

yes

characteristic

C₂²×C₃₁⋊C₈

C₂²

yes

characteristic

C₃₁×C₄⋊C₈

C₄

yes

complemented

C₃₁⋊#95

C₄

yes

—

C₃₁⋊#95

C₄

yes

—

#11758403

—

C₈×Dic₃₁

—

C₄⋊(C₃₁⋊C₈)

C₄

yes

complemented

C₂²×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

C₂×C₄

yes

—

C₂²×Dic₃₁

C₂×C₄

yes

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄×Dic₃₁

—

C₄×Dic₃₁

—

C₄×Dic₃₁

D₄

yes

—

C₄×Dic₃₁

Q₈

yes

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

C₂×C₄

yes

—

C₂×C₃₁⋊C₈

C₂×C₄

yes

—

C₂³×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₄²

yes

—

C₂×C₄×C₃₁

C₄²

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

C₂×C₄×C₃₁

yes

C₂×C₄×C₃₁

yes

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

C₄⋊C₄

yes

—

C₈×C₃₁

—

C₈×C₃₁

—

C₃₁⋊C₈

—

C₃₁⋊C₈

—

#1768

—

maximal, Sylow

C₂²×C₃₁

C₂×C₄²

yes

characteristic

C₂²×C₃₁

C₂×C₄⋊C₄

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

—

C₂²×C₃₁

#131

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₄×D₄

yes

characteristic

C₄×C₃₁

C₄×D₄

yes

characteristic

C₄×C₃₁

C₄×Q₈

yes

characteristic

C₄×C₃₁

C₄×Q₈

yes

characteristic

Dic₃₁

—

Dic₃₁

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₃₁

yes

characteristic

C₂×#97

—

C₂×C₄×C₈

—

#376

—

#336

—

#336

—

C₂×C₃₁

C₄×C₄⋊C₄

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂×C₃₁

#433

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃₁

yes

characteristic

C₂²×C₈

C₂×D₃₁

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

Dic₃₁

yes

complemented

100

#97

—

101

C₄×C₈

—

102

#95

—

103

#95

—

104

C₃₁

#1768

yes

characteristic, complemented, Sylow

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

C₄×D₃₁

yes

characteristic

108

C₂²×C₄

C₄×D₃₁

yes

characteristic

109

C₂²×C₄

C₂²×D₃₁

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

C₂×Dic₃₁

yes

—

121

C₂×C₈

C₂×Dic₃₁

yes

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

C₄×D₃₁

yes

—

127

C₄²

C₂×Dic₃₁

yes

—

128

C₂³

C₂×C₄×D₃₁

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₄×Dic₃₁

yes

—

138

C₂×C₄

C₄×Dic₃₁

yes

—

139

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

140

C₂×C₄

D₄×D₃₁

yes

characteristic

141

C₂×C₄

#92497

yes

characteristic

142

C₂×C₄

Q₈×D₃₁

yes

characteristic

143

C₂×C₄

#92499

yes

characteristic

144

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

149

C₂²

C₄⋊C₄×D₃₁

yes

characteristic

150

C₂²

#11758469

yes

characteristic

151

C₂²

#11758478

yes

—

152

C₂²

#11758484

yes

—

153

C₂²

C₂×C₄×Dic₃₁

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

124

—

158

C₄

#11758464

yes

characteristic

159

C₄

#11758470

yes

characteristic

160

C₄

D₄×Dic₃₁

yes

characteristic

161

C₄

Q₈×Dic₃₁

yes

characteristic

162

C₂

C₄⋊C₄×Dic₃₁

yes

characteristic

163

C₂

#14521004

yes

characteristic

164

C₂

#14521007

yes

characteristic

165

C₂

#14521074

yes

—

166

C₂

#14521109

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃₁
commutator	C₂²×C₃₁	C₂×C₄²
Fitting	C₂×C₃₁×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₃₁	C₂²×C₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂×C₄⋊C₈	D₃₁
hypocenter	C₃₁	#1768
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₃₁

commutator

C₂²×C₃₁

C₂×C₄²

Fitting

C₂×C₃₁×C₄⋊C₈

C₂

socle, prime radical

C₂³×C₃₁

C₂²×C₄

cosocle

C₂²×C₄×C₃₁

C₂³

hypercenter

C₂×C₄⋊C₈

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
15. C₃₁×C₄⋊C₈	C₄	semidirect
20. C₄⋊(C₃₁⋊C₈)	C₄	semidirect
99. C₄⋊C₈	Dic₃₁	semidirect
104. C₃₁	#1768	semidirect

H ≅ G/N

type

15. C₃₁×C₄⋊C₈

C₄

semidirect

20. C₄⋊(C₃₁⋊C₈)

C₄

semidirect

99. C₄⋊C₈

Dic₃₁

semidirect

104. C₃₁

#1768

semidirect

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

subgroup	index
C₂×C₄⋊(C₃₁⋊C₈)	2
C₂×#11758403	2
C₂×C₈×Dic₃₁	2
#14520895	2
#14520803	2
C₂×C₃₁×C₄⋊C₈	2
#1768	31

subgroup

index

C₂×C₄⋊(C₃₁⋊C₈)

C₂×C₈×Dic₃₁

C₂×C₃₁×C₄⋊C₈

Subgroups — #16845570

Conjugacy classes of subgroups

Complements

Maximal subgroups