Group #16846852 — subgroups

Subgroups — #16846852

Order 3968.

290 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 93 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×#11758446	C₂	1	1	yes	maximal, characteristic
2	C₂×#11758423	C₂	1	2	yes	maximal
3	#14520811	C₂	1	2	yes	maximal
4	#14520804	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#391	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂²×Dic₆₂	—	2	1	no	—
8	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₃₁	—	2	1	no	—
11	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
12	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
13	#11758446	—	2	4	no	—
14	C₃₁×#95	—	2	1	no	—
15	C₃₁×#95	—	2	1	no	—
16	#11758423	—	2	4	no	—
17	C₂×Q₈×C₃₁	—	2	1	no	—
18	C₂×Q₈×C₃₁	—	2	2	no	—
19	C₂×Q₈×C₃₁	—	2	2	no	—
20	C₂×Q₈×C₃₁	Q₈	1	2	yes	complemented
21	C₂²×C₄×C₃₁	—	2	1	no	—
22	C₂²×C₄×C₃₁	—	2	1	no	—
23	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
26	C₂²×Dic₃₁	—	2	2	no	—
27	C₂×Dic₆₂	—	2	4	no	—
28	C₂×Dic₆₂	—	4	2	no	—
29	C₂×Dic₆₂	—	4	2	no	—
30	C₄⋊C₄×C₃₁	—	2	2	no	—
31	C₄²×C₃₁	—	2	2	no	—
32	C₄⋊Dic₃₁	—	2	4	no	—
33	C₄⋊Dic₃₁	—	2	2	no	—
34	C₄⋊Dic₃₁	D₄	1	4	yes	—
35	C₂×C₃₁⋊C₈	—	2	2	no	—
36	C₂×C₃₁⋊C₈	—	2	2	no	—
37	C₂×C₃₁⋊C₈	—	2	2	no	—
38	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₃₁	—	2	4	no	—
40	Q₈×C₃₁	—	4	2	no	—
41	C₂×C₄×C₃₁	—	2	1	no	—
42	C₂×C₄×C₃₁	—	2	1	no	—
43	C₂×C₄×C₃₁	—	2	1	no	—
44	C₂×C₄×C₃₁	—	2	1	no	—
45	C₂×C₄×C₃₁	—	2	2	no	—
46	C₂×C₄×C₃₁	—	4	1	no	—
47	C₂×C₄×C₃₁	—	4	1	no	—
48	C₂×C₄×C₃₁	—	4	1	no	—
49	C₂×C₄×C₃₁	—	4	1	no	—
50	C₂×C₄×C₃₁	—	4	1	no	—
51	C₂×C₄×C₃₁	SD₁₆	1	2	yes	—
52	C₂×C₄×C₃₁	Q₁₆	1	2	yes	—
53	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
56	C₂×C₄×C₃₁	#41	1	2	yes	—
57	C₂×Dic₃₁	—	2	4	no	—
58	C₂×Dic₃₁	—	4	2	no	—
59	C₂×Dic₃₁	—	4	2	no	—
60	Dic₆₂	—	4	8	no	—
61	C₃₁⋊C₈	—	4	2	no	—
62	#2024	—	31	1	no	maximal, Sylow
63	C₂²×C₃₁	#120	1	1	yes	characteristic
64	C₂²×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
65	C₂²×C₃₁	#124	1	1	yes	characteristic
66	C₂²×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
67	C₂²×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
68	C₂²×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₂²×C₃₁	#137	1	1	yes	characteristic
70	C₄×C₃₁	—	2	2	no	—
71	C₄×C₃₁	—	2	2	no	—
72	C₄×C₃₁	—	4	2	no	—
73	C₄×C₃₁	—	8	1	no	—
74	C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
75	C₄×C₃₁	#124	1	2	yes	—
76	Dic₃₁	—	4	4	no	—
77	C₂×C₄⋊Q₈	—	31	1	no	—
78	C₂×#103	—	31	2	no	—
79	#391	D₃₁	1	1	yes	characteristic
80	#340	—	31	2	no	—
81	#336	—	31	1	no	—
82	C₂×C₃₁	#393	1	1	yes	characteristic
83	C₂×C₃₁	#450	1	1	yes	characteristic
84	C₂×C₃₁	#451	1	1	yes	characteristic
85	C₂×C₃₁	#475	1	1	yes	characteristic
86	C₂×C₃₁	#477	1	1	yes	characteristic
87	C₂×C₃₁	#487	1	1	yes	characteristic
88	C₂×C₃₁	#489	1	1	yes	characteristic
89	C₂²×Q₈	—	62	1	no	—
90	C₂²×Q₈	C₂×D₃₁	1	1	yes	characteristic
91	C₂²×C₈	—	31	2	no	—
92	C₄⋊Q₈	—	62	4	no	—
93	C₂×C₄⋊C₄	—	31	2	no	—
94	C₂×C₄⋊C₄	—	62	1	no	—
95	C₂×C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
96	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
97	#103	—	62	4	no	—
98	#95	—	2	1	no	—
99	#95	—	2	1	no	—
100	C₃₁	#2024	1	1	yes	characteristic, complemented, Sylow
101	C₂×Q₈	—	2	1	no	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	62	4	no	—
105	C₂×Q₈	—	124	2	no	—
106	C₂×Q₈	—	124	2	no	—
107	C₂×Q₈	Dic₆₂	1	2	yes	complemented
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	—	62	2	no	—
111	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
112	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
114	C₂×C₈	—	62	2	no	—
115	C₂×C₈	—	62	2	no	—
116	C₂×C₈	—	62	2	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	31	4	no	—
119	C₄⋊C₄	—	62	4	no	—
120	C₄⋊C₄	—	62	2	no	—
121	C₄²	—	2	2	no	—
122	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
123	Q₈	—	2	4	no	—
124	Q₈	—	4	2	no	—
125	Q₈	—	124	8	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	62	4	no	—
137	C₂×C₄	—	124	2	no	—
138	C₂×C₄	—	124	2	no	—
139	C₂×C₄	#92481	1	2	yes	—
140	C₂×C₄	#92482	1	2	yes	—
141	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
142	C₂×C₄	#92495	1	1	yes	characteristic
143	C₂×C₄	D₄×D₃₁	1	2	yes	—
144	C₂×C₄	#92497	1	2	yes	—
145	C₈	—	124	2	no	—
146	C₂²	#11758501	1	1	yes	characteristic
147	C₂²	#11758513	1	1	yes	characteristic
148	C₂²	#11758514	1	1	yes	characteristic
149	C₂²	C₂×#92481	1	1	yes	characteristic
150	C₂²	C₂×#92482	1	1	yes	characteristic
151	C₂²	#11758526	1	1	yes	characteristic
152	C₂²	#11758528	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	4	2	no	—
156	C₄	—	8	1	no	—
157	C₄	—	124	4	no	—
158	C₄	#11758455	1	2	yes	—
159	C₄	#11758462	1	2	yes	—
160	C₂	#14520961	1	1	yes	characteristic
161	C₂	#14521199	1	1	yes	characteristic
162	C₂	#14521204	1	1	yes	characteristic
163	C₂	#14521334	1	1	yes	characteristic
164	C₂	#14521344	1	1	yes	characteristic
165	C₂	#14521399	1	1	yes	characteristic
166	C₂	#14521408	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758446

C₂

yes

maximal, characteristic

C₂×#11758423

C₂

yes

maximal

#14520811

C₂

yes

maximal

#14520804

C₂

yes

maximal, characteristic

C₃₁×#391

C₂

yes

maximal, characteristic

C₂²×Q₈×C₃₁

C₂²

yes

characteristic

C₂²×Dic₆₂

—

C₂×C₄⋊C₄×C₃₁

C₂²

yes

characteristic

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂²×C₃₁⋊C₈

C₂²

yes

—

#11758446

—

C₃₁×#95

—

C₃₁×#95

—

#11758423

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

Q₈

yes

complemented

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

D₄

yes

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

Q₈×C₃₁

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

SD₁₆

yes

—

C₂×C₄×C₃₁

Q₁₆

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

C₂×C₄×C₃₁

yes

C₂×C₄×C₃₁

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

C₃₁⋊C₈

—

—

C₂²×C₃₁

yes

C₂²×C₃₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₃₁

#124

yes

characteristic

C₂²×C₃₁

C₂×SD₁₆

yes

characteristic

C₂²×C₃₁

C₂×Q₁₆

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂²⋊Q₈

yes

—

C₄×C₃₁

#124

yes

—

Dic₃₁

—

C₂×C₄⋊Q₈

—

C₂×#103

—

#391

D₃₁

yes

characteristic

#340

—

#336

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂²×Q₈

—

C₂²×Q₈

C₂×D₃₁

yes

characteristic

C₂²×C₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

characteristic

C₂×C₄²

C₂×D₃₁

yes

—

—

—

100

C₃₁

#2024

yes

characteristic, complemented, Sylow

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

124

—

106

C₂×Q₈

—

124

—

107

C₂×Q₈

Dic₆₂

yes

complemented

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²⋊D₃₁

yes

characteristic

112

C₂²×C₄

C₂²⋊D₃₁

yes

characteristic

113

C₂²×C₄

C₂²×D₃₁

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄²

—

122

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

123

Q₈

—

124

Q₈

—

125

Q₈

—

124

—

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

—

136

C₂×C₄

—

137

C₂×C₄

—

124

—

138

C₂×C₄

—

124

—

139

C₂×C₄

#92481

yes

—

140

C₂×C₄

#92482

yes

—

141

C₂×C₄

C₂×Dic₆₂

yes

characteristic

142

C₂×C₄

#92495

yes

characteristic

143

C₂×C₄

D₄×D₃₁

yes

—

144

C₂×C₄

#92497

yes

—

145

C₈

—

124

—

146

C₂²

#11758501

yes

characteristic

147

C₂²

#11758513

yes

characteristic

148

C₂²

#11758514

yes

characteristic

149

C₂²

C₂×#92481

yes

characteristic

150

C₂²

C₂×#92482

yes

characteristic

151

C₂²

#11758526

yes

characteristic

152

C₂²

#11758528

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

124

—

158

C₄

#11758455

yes

—

159

C₄

#11758462

yes

—

160

C₂

#14520961

yes

characteristic

161

C₂

#14521199

yes

characteristic

162

C₂

#14521204

yes

characteristic

163

C₂

#14521334

yes

characteristic

164

C₂

#14521344

yes

characteristic

165

C₂

#14521399

yes

characteristic

166

C₂

#14521408

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₁
commutator, cosocle	C₂²×C₄×C₃₁	C₂³
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₃₁×#391	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
hypercenter	#391	D₃₁
hypocenter	C₃₁	#2024
LCS	G ▷ C₂²×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₁

commutator, cosocle

C₂²×C₄×C₃₁

C₂³

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₃₁×#391

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

D₃₁

C₃₁

G ▷ C₂²×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
20. C₂×Q₈×C₃₁	Q₈	semidirect
100. C₃₁	#2024	semidirect
107. C₂×Q₈	Dic₆₂	semidirect

H ≅ G/N

type

20. C₂×Q₈×C₃₁

Q₈

semidirect

100. C₃₁

#2024

semidirect

107. C₂×Q₈

Dic₆₂

semidirect

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

subgroup	index
C₂×#11758446	2
C₂×#11758423	2
#14520811	2
#14520804	2
C₃₁×#391	2
#2024	31

subgroup

index

C₃₁×#391

Subgroups — #16846852

Conjugacy classes of subgroups

Complements

Maximal subgroups