Group #16854808 — subgroups

Subgroups — #16854808

Order 3968.

322 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 161 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	#14521606	C₂	1	1	yes	maximal, characteristic
2	#14521177	C₂	1	4	yes	maximal, complemented
3	#14521176	C₂	1	2	yes	maximal, complemented
4	#14521173	C₂	1	2	yes	maximal, complemented
5	#14521107	C₂	1	4	yes	maximal
6	#14521102	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#432	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	#11758503	C₂²	1	2	yes	—
10	#11758501	C₄	1	2	yes	—
11	#11758501	C₄	1	2	yes	—
12	#11758447	C₄	1	2	yes	—
13	#11758446	C₄	1	1	yes	characteristic
14	#11758446	C₄	1	1	yes	characteristic
15	C₄×Dic₆₂	C₂²	1	2	yes	—
16	C₄×Dic₆₂	C₂²	1	2	yes	—
17	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
18	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
19	C₃₁×#97	C₂²	1	2	yes	—
20	C₂²⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
21	#11758403	C₂²	1	4	yes	—
22	Dic₃₁⋊C₈	C₂²	1	4	yes	—
23	Dic₃₁⋊C₈	C₂²	1	4	yes	—
24	C₈×Dic₃₁	C₂²	1	4	yes	—
25	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
26	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
27	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
28	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
31	C₂×Dic₆₂	C₂×C₄	1	2	yes	—
32	C₂×Dic₆₂	C₂×C₄	1	2	yes	—
33	C₂×C₈×C₃₁	C₂³	1	4	yes	—
34	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₃₁	C₂³	1	1	yes	characteristic
37	C₄²×C₃₁	C₂³	1	1	yes	characteristic
38	C₂²⋊Dic₃₁	C₂×C₄	1	4	yes	—
39	C₄⋊Dic₃₁	C₂×C₄	1	2	yes	—
40	C₄⋊Dic₃₁	C₂×C₄	1	2	yes	—
41	C₄⋊Dic₃₁	C₂×C₄	1	2	yes	—
42	C₄⋊Dic₃₁	C₂×C₄	1	2	yes	—
43	Dic₃₁⋊C₄	C₂×C₄	1	4	yes	—
44	Dic₃₁⋊C₄	C₂×C₄	1	4	yes	—
45	C₄×Dic₃₁	C₂³	1	4	yes	—
46	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
47	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
48	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₃₁	—	2	2	no	—
50	C₂×C₄×C₃₁	—	2	2	no	—
51	C₂×C₄×C₃₁	—	4	1	no	—
52	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
58	C₂×Dic₃₁	C₂²×C₄	1	4	yes	—
59	C₂×Dic₃₁	C₂²×C₄	1	4	yes	—
60	Dic₆₂	—	2	4	no	—
61	Dic₆₂	—	4	2	no	—
62	C₈×C₃₁	—	2	4	no	—
63	C₃₁⋊C₈	—	2	2	no	—
64	C₃₁⋊C₈	—	2	2	no	—
65	#2985	—	31	1	no	maximal, Sylow
66	C₂²×C₃₁	—	2	1	no	—
67	C₂²×C₃₁	—	2	1	no	—
68	C₂²×C₃₁	—	2	1	no	—
69	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
70	C₄×C₃₁	—	2	1	no	—
71	C₄×C₃₁	—	2	1	no	—
72	C₄×C₃₁	—	2	1	no	—
73	C₄×C₃₁	—	2	1	no	—
74	C₄×C₃₁	#131	1	2	yes	—
75	C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
76	Dic₃₁	—	2	4	no	—
77	Dic₃₁	—	2	4	no	—
78	#533	—	31	1	no	—
79	#446	—	31	2	no	—
80	#446	—	31	4	no	—
81	C₈×Q₈	—	31	2	no	—
82	#433	—	31	4	no	—
83	#432	D₃₁	1	1	yes	characteristic
84	#431	—	31	1	no	—
85	C₂×C₃₁	—	4	1	no	—
86	C₂×C₃₁	#519	1	1	yes	characteristic
87	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
88	C₂×C₃₁	#568	1	1	yes	characteristic
89	C₄⋊Q₈	—	31	1	no	—
90	C₄⋊Q₈	—	31	1	no	—
91	#125	—	31	2	no	—
92	C₂²⋊Q₈	—	31	2	no	—
93	C₂²⋊Q₈	—	31	2	no	—
94	C₄×Q₈	—	31	2	no	—
95	C₄×Q₈	—	31	2	no	—
96	#117	—	31	2	no	—
97	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
98	C₄⋊C₈	—	31	4	no	—
99	C₄⋊C₈	—	31	4	no	—
100	C₄⋊C₈	—	31	2	no	—
101	C₄⋊C₈	C₂×D₃₁	1	2	yes	—
102	C₂²⋊C₈	—	31	2	no	—
103	C₂²⋊C₈	C₂×D₃₁	1	2	yes	—
104	#97	—	31	4	no	—
105	#97	C₂×D₃₁	1	2	yes	—
106	C₄×C₈	—	31	4	no	—
107	C₄×C₈	—	31	1	no	—
108	C₄×C₈	—	31	1	no	—
109	C₃₁	#2985	1	1	yes	characteristic, complemented, Sylow
110	C₂×Q₈	—	31	2	no	—
111	C₂×Q₈	—	31	2	no	—
112	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
113	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
115	C₂×C₈	—	31	2	no	—
116	C₂×C₈	—	31	2	no	—
117	C₂×C₈	C₂²×D₃₁	1	4	yes	—
118	C₄⋊C₄	—	31	4	no	—
119	C₄⋊C₄	—	31	4	no	—
120	C₄⋊C₄	—	31	2	no	—
121	C₄⋊C₄	—	31	2	no	—
122	C₄⋊C₄	—	31	2	no	—
123	C₄⋊C₄	—	31	2	no	—
124	C₂²⋊C₄	—	31	4	no	—
125	C₄²	—	31	4	no	—
126	C₄²	C₄×D₃₁	1	1	yes	characteristic
127	C₄²	C₄×D₃₁	1	1	yes	characteristic
128	C₄²	C₂²×D₃₁	1	1	yes	characteristic
129	C₄²	C₂²×D₃₁	1	1	yes	characteristic
130	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
131	Q₈	—	62	4	no	—
132	Q₈	—	124	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	31	4	no	—
137	C₂×C₄	—	31	4	no	—
138	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
143	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
144	C₈	—	2	4	no	—
145	C₈	—	62	2	no	—
146	C₈	—	62	2	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	62	4	no	—
156	C₄	—	62	4	no	—
157	C₄	#11758484	1	2	yes	—
158	C₄	#11758567	1	2	yes	—
159	C₂	—	4	1	no	—
160	C₂	#14521526	1	1	yes	characteristic
161	C₂	C₂×#11758484	1	1	yes	characteristic
162	C₂	#14521815	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521606

C₂

yes

maximal, characteristic

#14521177

C₂

yes

maximal, complemented

#14521176

C₂

yes

maximal, complemented

#14521173

C₂

yes

maximal, complemented

#14521107

C₂

yes

maximal

#14521102

C₂

yes

maximal, characteristic

C₃₁×#432

C₂

yes

maximal, characteristic

C₂×C₄²×C₃₁

C₂²

yes

characteristic

#11758503

C₂²

yes

—

#11758501

C₄

yes

—

#11758501

C₄

yes

—

#11758447

C₄

yes

—

#11758446

C₄

yes

characteristic

#11758446

C₄

yes

characteristic

C₄×Dic₆₂

C₂²

yes

—

C₄×Dic₆₂

C₂²

yes

—

C₃₁×C₄⋊C₈

C₂²

yes

—

C₃₁×C₂²⋊C₈

C₂²

yes

—

C₃₁×#97

C₂²

yes

—

C₂²⋊(C₃₁⋊C₈)

C₂²

yes

—

#11758403

C₂²

yes

—

Dic₃₁⋊C₈

C₂²

yes

—

Dic₃₁⋊C₈

C₂²

yes

—

C₈×Dic₃₁

C₂²

yes

—

C₄⋊(C₃₁⋊C₈)

C₂²

yes

—

C₄×C₃₁⋊C₈

C₂²

yes

characteristic

C₄×C₃₁⋊C₈

C₂²

yes

characteristic

C₂²×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂×Dic₆₂

C₂×C₄

yes

—

C₂×Dic₆₂

C₂×C₄

yes

—

C₂×C₈×C₃₁

C₂³

yes

—

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

C₂³

yes

characteristic

C₄²×C₃₁

C₂³

yes

characteristic

C₂²⋊Dic₃₁

C₂×C₄

yes

—

C₄⋊Dic₃₁

C₂×C₄

yes

—

C₄⋊Dic₃₁

C₂×C₄

yes

—

C₄⋊Dic₃₁

C₂×C₄

yes

—

C₄⋊Dic₃₁

C₂×C₄

yes

—

Dic₃₁⋊C₄

C₂×C₄

yes

—

Dic₃₁⋊C₄

C₂×C₄

yes

—

C₄×Dic₃₁

C₂³

yes

—

C₂×C₃₁⋊C₈

C₂³

yes

—

C₂×C₃₁⋊C₈

C₂³

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

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

C₂²×C₄

yes

—

C₂×Dic₃₁

C₂²×C₄

yes

—

Dic₆₂

—

Dic₆₂

—

C₈×C₃₁

—

C₃₁⋊C₈

—

C₃₁⋊C₈

—

#2985

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂³×C₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#131

yes

—

C₄×C₃₁

2₋¹⁺⁴

yes

—

Dic₃₁

—

Dic₃₁

—

#533

—

#446

—

#446

—

C₈×Q₈

—

#433

—

#432

D₃₁

yes

characteristic

#431

—

C₂×C₃₁

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₄⋊C₈

C₂×D₃₁

yes

—

102

C₂²⋊C₈

—

103

C₂²⋊C₈

C₂×D₃₁

yes

—

104

#97

—

105

#97

C₂×D₃₁

yes

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₃₁

#2985

yes

characteristic, complemented, Sylow

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂²×C₄

C₄×D₃₁

yes

characteristic

113

C₂²×C₄

C₄×D₃₁

yes

characteristic

114

C₂²×C₄

C₂²×D₃₁

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

C₂²×D₃₁

yes

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

C₄×D₃₁

yes

characteristic

127

C₄²

C₄×D₃₁

yes

characteristic

128

C₄²

C₂²×D₃₁

yes

characteristic

129

C₄²

C₂²×D₃₁

yes

characteristic

130

C₂³

C₂×C₄×D₃₁

yes

characteristic

131

Q₈

—

132

Q₈

—

124

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

139

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

140

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

141

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

142

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

143

C₂×C₄

C₂³×D₃₁

yes

characteristic

144

C₈

—

145

C₈

—

146

C₈

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

C₂²×C₄×D₃₁

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

#11758484

yes

—

158

C₄

#11758567

yes

—

159

C₂

—

160

C₂

#14521526

yes

characteristic

161

C₂

C₂×#11758484

yes

characteristic

162

C₂

#14521815

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₃₁×#432	C₂
prime radical	C₂³×C₃₁	C₂²×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#432	D₃₁
hypocenter	C₃₁	#2985
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #432

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₃₁

commutator, socle

C₂²×C₃₁

C₂³×C₄

Fitting

C₃₁×#432

C₂

prime radical

C₂³×C₃₁

C₂²×C₄

cosocle

C₂×C₄×C₃₁

C₂⁴

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂×C₄ ◁ #432

N	H ≅ G/N	type
2. #14521177	C₂	semidirect
3. #14521176	C₂	semidirect
4. #14521173	C₂	semidirect
109. C₃₁	#2985	semidirect

H ≅ G/N

type

2. #14521177

C₂

semidirect

3. #14521176

C₂

semidirect

4. #14521173

C₂

semidirect

109. C₃₁

#2985

semidirect

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

subgroup	index
#14521606	2
#14521177	2
#14521176	2
#14521173	2
#14521107	2
#14521102	2
C₃₁×#432	2
#2985	31

subgroup

index

C₃₁×#432

Subgroups — #16854808

Conjugacy classes of subgroups

Complements

Maximal subgroups