Group #16845461 — subgroups

Subgroups — #16845461

Order 3968.

240 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 113 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×#11758403	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₃₁	C₂	1	1	yes	maximal, characteristic
3	#14520895	C₂	1	1	yes	maximal, characteristic
4	#14520804	C₂	1	1	yes	maximal, characteristic
5	#14520803	C₂	1	1	yes	maximal, characteristic
6	#14520803	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#336	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⋊#95	C₄	1	2	yes	—
16	C₃₁⋊#95	C₄	1	2	yes	—
17	#11758403	C₄	1	4	yes	complemented
18	C₈×Dic₃₁	C₄	1	4	yes	complemented
19	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
21	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
22	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₃₁	—	2	1	no	—
25	C₂×C₈×C₃₁	—	2	1	no	—
26	C₂×C₈×C₃₁	—	2	1	no	—
27	C₂×C₈×C₃₁	—	2	1	no	—
28	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
30	C₄²×C₃₁	—	2	1	no	—
31	C₄²×C₃₁	—	2	1	no	—
32	C₄×Dic₃₁	—	2	1	no	—
33	C₄×Dic₃₁	—	2	1	no	—
34	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
35	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
36	C₂×C₃₁⋊C₈	—	2	1	no	—
37	C₂×C₃₁⋊C₈	—	2	1	no	—
38	C₂×C₃₁⋊C₈	—	2	1	no	—
39	C₂×C₃₁⋊C₈	—	2	1	no	—
40	C₂×C₃₁⋊C₈	C₂×C₄	1	2	yes	—
41	C₂×C₃₁⋊C₈	C₂×C₄	1	2	yes	—
42	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₃₁	—	2	2	no	—
44	C₂×C₄×C₃₁	—	2	2	no	—
45	C₂×C₄×C₃₁	—	2	2	no	—
46	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
49	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
50	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
51	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
52	C₂×Dic₃₁	—	2	2	no	—
53	C₂×Dic₃₁	—	2	2	no	—
54	C₂×Dic₃₁	—	2	2	no	—
55	C₂×Dic₃₁	—	2	2	no	—
56	C₂×Dic₃₁	—	2	2	no	—
57	C₂×Dic₃₁	C₄²	1	2	yes	—
58	C₂×Dic₃₁	C₄²	1	2	yes	—
59	C₈×C₃₁	—	2	2	no	—
60	C₈×C₃₁	—	2	2	no	—
61	C₃₁⋊C₈	—	2	2	no	—
62	C₃₁⋊C₈	—	2	2	no	—
63	#1746	—	31	1	no	maximal, Sylow
64	C₂²×C₃₁	C₂×C₄²	1	1	yes	characteristic
65	C₂²×C₃₁	#117	1	1	yes	characteristic
66	C₂²×C₃₁	#117	1	1	yes	characteristic
67	C₂²×C₃₁	#131	1	2	yes	—
68	C₂²×C₃₁	#131	1	2	yes	—
69	C₄×C₃₁	—	4	2	no	—
70	C₄×C₃₁	#117	1	1	yes	characteristic
71	C₄×C₃₁	#117	1	1	yes	characteristic
72	C₄×C₃₁	#117	1	1	yes	characteristic
73	C₄×C₃₁	#117	1	1	yes	characteristic
74	Dic₃₁	—	2	2	no	—
75	Dic₃₁	—	2	2	no	—
76	Dic₃₁	—	4	2	no	—
77	C₂×#97	—	31	1	no	—
78	C₂×C₄×C₈	—	31	1	no	—
79	#376	—	31	1	no	—
80	#336	—	31	1	no	—
81	#336	—	31	1	no	—
82	#336	—	31	1	no	—
83	#336	D₃₁	1	1	yes	characteristic
84	C₂×C₃₁	#376	1	1	yes	characteristic
85	C₂×C₃₁	#405	1	1	yes	characteristic
86	C₂×C₃₁	#405	1	1	yes	characteristic
87	C₂×C₃₁	#433	1	2	yes	—
88	C₂×C₃₁	#433	1	2	yes	—
89	C₂²×C₈	—	31	1	no	—
90	C₂²×C₈	—	31	1	no	—
91	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
92	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
93	C₂×C₄²	—	31	1	no	—
94	C₂×C₄²	—	31	1	no	—
95	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
96	#97	—	31	4	no	—
97	C₄×C₈	—	31	4	no	—
98	#95	—	31	2	no	—
99	#95	—	31	2	no	—
100	C₃₁	#1746	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₄	—	31	1	no	—
102	C₂²×C₄	—	31	1	no	—
103	C₂²×C₄	—	31	2	no	—
104	C₂²×C₄	C₄×D₃₁	1	2	yes	—
105	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	31	2	no	—
111	C₂×C₈	—	31	2	no	—
112	C₂×C₈	—	62	1	no	—
113	C₂×C₈	—	62	1	no	—
114	C₂×C₈	—	62	1	no	—
115	C₂×C₈	—	62	1	no	—
116	C₂×C₈	C₄×D₃₁	1	2	yes	—
117	C₂×C₈	C₄×D₃₁	1	2	yes	—
118	C₄²	—	2	1	no	—
119	C₄²	—	2	1	no	—
120	C₄²	—	31	2	no	—
121	C₄²	—	31	2	no	—
122	C₄²	—	62	1	no	—
123	C₄²	—	62	1	no	—
124	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	31	2	no	—
129	C₂×C₄	—	31	2	no	—
130	C₂×C₄	—	62	2	no	—
131	C₂×C₄	—	62	2	no	—
132	C₂×C₄	—	62	2	no	—
133	C₂×C₄	—	62	2	no	—
134	C₂×C₄	—	62	2	no	—
135	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
136	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
137	C₂×C₄	#92497	1	1	yes	characteristic
138	C₂×C₄	#92497	1	1	yes	characteristic
139	C₂×C₄	#92497	1	1	yes	characteristic
140	C₂×C₄	#92499	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	—	2	2	no	—
143	C₈	—	62	2	no	—
144	C₈	—	62	2	no	—
145	C₂²	C₄²×D₃₁	1	1	yes	characteristic
146	C₂²	#11758454	1	1	yes	characteristic
147	C₂²	#11758469	1	1	yes	characteristic
148	C₂²	#11758478	1	2	yes	—
149	C₂²	#11758484	1	2	yes	—
150	C₄	—	4	2	no	—
151	C₄	—	62	2	no	—
152	C₄	—	62	2	no	—
153	C₄	—	124	2	no	—
154	C₄	#11758454	1	1	yes	characteristic
155	C₄	#11758454	1	1	yes	characteristic
156	C₄	#11758469	1	1	yes	characteristic
157	C₄	#11758469	1	1	yes	characteristic
158	C₂	#14520894	1	1	yes	characteristic
159	C₂	#14521003	1	1	yes	characteristic
160	C₂	#14521006	1	1	yes	characteristic
161	C₂	#14521107	1	2	yes	—
162	C₂	#14521109	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758403

C₂

yes

maximal, characteristic

C₂×C₈×Dic₃₁

C₂

yes

maximal, characteristic

#14520895

C₂

yes

maximal, characteristic

#14520804

C₂

yes

maximal, characteristic

#14520803

C₂

yes

maximal, characteristic

#14520803

C₂

yes

maximal, characteristic

C₃₁×#336

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⋊#95

C₄

yes

—

C₃₁⋊#95

C₄

yes

—

#11758403

C₄

yes

complemented

C₈×Dic₃₁

C₄

yes

complemented

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

C₂×C₄

yes

—

C₂²×Dic₃₁

C₂×C₄

yes

characteristic

C₂²×Dic₃₁

C₂×C₄

yes

characteristic

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

—

C₄²×C₃₁

—

C₄×Dic₃₁

—

C₄×Dic₃₁

—

C₄×Dic₃₁

C₂×C₄

yes

—

C₄×Dic₃₁

C₂×C₄

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

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

C₂×C₄×C₃₁

yes

C₂×C₄×C₃₁

yes

C₂×C₄×C₃₁

yes

C₂×C₄×C₃₁

yes

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

C₄²

yes

—

C₂×Dic₃₁

C₄²

yes

—

C₈×C₃₁

—

C₈×C₃₁

—

C₃₁⋊C₈

—

C₃₁⋊C₈

—

#1746

—

maximal, Sylow

C₂²×C₃₁

C₂×C₄²

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

—

C₂²×C₃₁

#131

yes

—

C₄×C₃₁

—

C₄×C₃₁

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

C₂×#97

—

C₂×C₄×C₈

—

—

—

—

—

D₃₁

yes

C₂×C₃₁

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

#97

—

C₄×C₈

—

#95

—

#95

—

100

C₃₁

#1746

yes

characteristic, complemented, Sylow

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₄×D₃₁

yes

—

105

C₂²×C₄

C₂²×D₃₁

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

C₄×D₃₁

yes

—

117

C₂×C₈

C₄×D₃₁

yes

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

124

C₂³

C₂×C₄×D₃₁

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

136

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

137

C₂×C₄

#92497

yes

characteristic

138

C₂×C₄

#92497

yes

characteristic

139

C₂×C₄

#92497

yes

characteristic

140

C₂×C₄

#92499

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₂²

C₄²×D₃₁

yes

characteristic

146

C₂²

#11758454

yes

characteristic

147

C₂²

#11758469

yes

characteristic

148

C₂²

#11758478

yes

—

149

C₂²

#11758484

yes

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

124

—

154

C₄

#11758454

yes

characteristic

155

C₄

#11758454

yes

characteristic

156

C₄

#11758469

yes

characteristic

157

C₄

#11758469

yes

characteristic

158

C₂

#14520894

yes

characteristic

159

C₂

#14521003

yes

characteristic

160

C₂

#14521006

yes

characteristic

161

C₂

#14521107

yes

—

162

C₂

#14521109

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃₁
commutator	C₂²×C₃₁	C₂×C₄²
Fitting	C₃₁×#336	C₂
socle, prime radical	C₂³×C₃₁	C₂²×C₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	#336	D₃₁
hypocenter	C₃₁	#1746
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₃₁

commutator

C₂²×C₃₁

C₂×C₄²

Fitting

C₃₁×#336

C₂

socle, prime radical

C₂³×C₃₁

C₂²×C₄

cosocle

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
17. #11758403	C₄	semidirect
18. C₈×Dic₃₁	C₄	semidirect
100. C₃₁	#1746	semidirect

H ≅ G/N

type

17. #11758403

C₄

semidirect

18. C₈×Dic₃₁

C₄

semidirect

100. C₃₁

#1746

semidirect

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

subgroup	index
C₂×#11758403	2
C₂×C₈×Dic₃₁	2
#14520895	2
#14520804	2
#14520803	2
C₃₁×#336	2
#1746	31

subgroup

index

C₂×C₈×Dic₃₁

C₃₁×#336

Subgroups — #16845461

Conjugacy classes of subgroups

Complements

Maximal subgroups