Group #16856171 — subgroups

Subgroups — #16856171

Order 3968.

552 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 157 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×#11758528	C₂	1	4	yes	maximal, complemented
2	C₂×#11758446	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521444	C₂	1	4	yes	maximal, complemented
4	#14521428	C₂	1	2	yes	maximal, complemented
5	#14521421	C₂	1	2	yes	maximal, complemented
6	C₄×#92474	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#532	C₂	1	1	yes	maximal, characteristic
8	C₂×#41×C₃₁	C₂²	1	2	yes	—
9	C₂²×Dic₆₂	C₂²	1	2	yes	—
10	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₃₁×#127	C₂²	1	1	yes	characteristic
12	C₃₁×#124	C₂²	1	2	yes	—
13	C₃₁×#121	—	2	2	no	—
14	C₄×D₄×C₃₁	—	2	2	no	—
15	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
16	#11758528	—	2	16	no	—
17	C₂×#92482	C₂²	1	8	yes	—
18	C₂×#92480	C₂²	1	8	yes	—
19	C₂×C₄⋊Dic₃₁	—	2	2	no	—
20	C₂×#92474	C₂²	1	2	yes	—
21	#11758446	—	2	2	no	—
22	#11758446	C₂²	1	2	yes	—
23	#11758446	C₂²	1	2	yes	—
24	#11758391	C₂²	1	2	yes	—
25	C₄×C₃₁⋊C₈	C₂²	1	2	yes	—
26	#41×C₃₁	—	2	8	no	—
27	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
28	C₂×D₄×C₃₁	—	2	2	no	—
29	C₂×D₄×C₃₁	—	2	2	no	—
30	C₂×D₄×C₃₁	C₂³	1	2	yes	—
31	C₂²×C₄×C₃₁	—	2	2	no	—
32	C₂²×C₄×C₃₁	D₄	1	2	yes	—
33	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
34	C₂²×Dic₃₁	—	2	2	no	—
35	C₂×Dic₆₂	—	2	8	no	—
36	C₂×Dic₆₂	—	2	2	no	—
37	C₂×Dic₆₂	C₂³	1	4	yes	—
38	C₄⋊C₄×C₃₁	—	2	2	no	—
39	C₂²⋊C₄×C₃₁	—	2	4	no	—
40	C₄²×C₃₁	D₄	1	2	yes	—
41	C₄²×C₃₁	C₂³	1	1	yes	characteristic
42	C₄²×C₃₁	C₂³	1	1	yes	characteristic
43	#92482	—	2	16	no	—
44	#92480	—	2	16	no	—
45	C₄⋊Dic₃₁	—	2	4	no	—
46	C₄⋊Dic₃₁	—	2	4	no	—
47	#92474	D₄	1	8	yes	complemented
48	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
49	C₂³×C₃₁	—	2	2	no	—
50	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₃₁	—	2	4	no	—
52	D₄×C₃₁	—	2	4	no	—
53	D₄×C₃₁	—	4	4	no	—
54	D₄×C₃₁	—	4	4	no	—
55	C₂×C₄×C₃₁	—	2	4	no	—
56	C₂×C₄×C₃₁	—	2	2	no	—
57	C₂×C₄×C₃₁	—	4	4	no	—
58	C₂×C₄×C₃₁	C₂×D₄	1	4	yes	—
59	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
63	C₂×Dic₃₁	—	2	4	no	—
64	C₂×Dic₃₁	—	4	4	no	—
65	Dic₆₂	—	2	8	no	—
66	Dic₆₂	—	4	4	no	—
67	C₃₁⋊C₈	C₂×D₄	1	8	yes	—
68	#3152	—	31	1	no	maximal, Sylow
69	C₂²×C₃₁	—	2	2	no	—
70	C₂²×C₃₁	—	4	2	no	—
71	C₂²×C₃₁	—	4	4	no	—
72	C₂²×C₃₁	#127	1	2	yes	—
73	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
74	C₄×C₃₁	—	2	2	no	—
75	C₄×C₃₁	—	4	2	no	—
76	C₄×C₃₁	#127	1	2	yes	—
77	C₄×C₃₁	#137	1	4	yes	complemented
78	C₄×C₃₁	C₂²×D₄	1	2	yes	—
79	Dic₃₁	—	4	4	no	—
80	C₂×#137	—	31	4	no	—
81	#532	D₃₁	1	1	yes	characteristic
82	C₂×C₄⋊Q₈	—	31	1	no	—
83	#497	—	31	4	no	—
84	#494	—	31	2	no	—
85	#492	—	31	2	no	—
86	C₄×M₁₆	—	31	1	no	—
87	C₂×C₃₁	—	2	2	no	—
88	C₂×C₃₁	—	8	2	no	—
89	C₂×C₃₁	C₂×#127	1	1	yes	characteristic
90	C₂×C₃₁	C₂×#137	1	2	yes	—
91	C₂×#41	C₂×D₃₁	1	2	yes	—
92	C₂²×Q₈	—	31	2	no	—
93	#137	—	62	16	no	—
94	C₂×Q₁₆	—	31	8	no	—
95	C₂×SD₁₆	—	31	8	no	—
96	C₂×M₁₆	—	31	2	no	—
97	C₄⋊Q₈	—	31	2	no	—
98	C₄⋊Q₈	—	31	2	no	—
99	C₄⋊Q₈	—	62	2	no	—
100	C₄⋊Q₈	C₂×D₃₁	1	1	yes	characteristic
101	#127	C₂×D₃₁	1	1	yes	characteristic
102	#124	C₂×D₃₁	1	2	yes	—
103	#121	—	2	2	no	—
104	C₄×D₄	—	2	2	no	—
105	C₂×C₄⋊C₄	—	62	2	no	—
106	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
107	#97	—	31	2	no	—
108	C₄×C₈	—	31	2	no	—
109	C₃₁	#3152	1	1	yes	characteristic, complemented, Sylow
110	#41	—	2	8	no	—
111	C₂×Q₈	—	31	4	no	—
112	C₂×Q₈	—	62	2	no	—
113	C₂×Q₈	—	62	8	no	—
114	C₂×Q₈	C₂²×D₃₁	1	2	yes	—
115	C₂×D₄	—	2	2	no	—
116	C₂×D₄	—	2	2	no	—
117	C₂×D₄	C₂²×D₃₁	1	2	yes	—
118	C₂²×C₄	—	2	2	no	—
119	C₂²×C₄	—	62	2	no	—
120	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
121	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
122	Q₁₆	—	62	16	no	—
123	SD₁₆	—	62	16	no	—
124	M₁₆	—	31	8	no	—
125	C₂×C₈	—	31	4	no	—
126	C₄⋊C₄	—	2	2	no	—
127	C₄⋊C₄	—	62	4	no	—
128	C₄⋊C₄	—	62	4	no	—
129	C₂²⋊C₄	—	2	4	no	—
130	C₄²	C₂²⋊D₃₁	1	2	yes	—
131	C₄²	C₂²×D₃₁	1	1	yes	characteristic
132	C₄²	C₂²×D₃₁	1	1	yes	characteristic
133	C₂³	—	2	2	no	—
134	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
135	Q₈	—	2	4	no	—
136	Q₈	—	62	8	no	—
137	Q₈	—	124	4	no	—
138	D₄	—	2	4	no	—
139	D₄	—	4	4	no	—
140	D₄	—	4	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	4	4	no	—
144	C₂×C₄	—	62	4	no	—
145	C₂×C₄	—	124	4	no	—
146	C₂×C₄	D₄×D₃₁	1	4	yes	—
147	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
148	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
149	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
150	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
151	C₈	—	31	8	no	—
152	C₂²	—	2	2	no	—
153	C₂²	—	4	4	no	—
154	C₂²	—	4	2	no	—
155	C₂²	#11758517	1	2	yes	—
156	C₂²	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	4	2	no	—
159	C₄	—	124	4	no	—
160	C₄	#11758517	1	2	yes	—
161	C₄	#11758528	1	4	yes	complemented
162	C₄	C₂×D₄×D₃₁	1	2	yes	—
163	C₂	—	2	2	no	—
164	C₂	—	8	2	no	—
165	C₂	C₂×#11758517	1	1	yes	characteristic
166	C₂	C₂×#11758528	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758528

C₂

yes

maximal, complemented

C₂×#11758446

C₂

yes

maximal, characteristic, complemented

#14521444

C₂

yes

maximal, complemented

#14521428

C₂

yes

maximal, complemented

#14521421

C₂

yes

maximal, complemented

C₄×#92474

C₂

yes

maximal, characteristic, complemented

C₃₁×#532

C₂

yes

maximal, characteristic

C₂×#41×C₃₁

C₂²

yes

—

C₂²×Dic₆₂

C₂²

yes

—

C₃₁×C₄⋊Q₈

C₂²

yes

characteristic

C₃₁×#127

C₂²

yes

characteristic

C₃₁×#124

C₂²

yes

—

C₃₁×#121

—

C₄×D₄×C₃₁

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

#11758528

—

C₂×#92482

C₂²

yes

—

C₂×#92480

C₂²

yes

—

C₂×C₄⋊Dic₃₁

—

C₂×#92474

C₂²

yes

—

#11758446

—

#11758446

C₂²

yes

—

#11758446

C₂²

yes

—

#11758391

C₂²

yes

—

C₄×C₃₁⋊C₈

C₂²

yes

—

#41×C₃₁

—

C₂×Q₈×C₃₁

C₂³

yes

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

C₂³

yes

—

C₄⋊C₄×C₃₁

—

C₂²⋊C₄×C₃₁

—

C₄²×C₃₁

D₄

yes

—

C₄²×C₃₁

C₂³

yes

characteristic

C₄²×C₃₁

C₂³

yes

characteristic

#92482

—

#92480

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

#92474

D₄

yes

complemented

C₂×C₃₁⋊C₈

C₂³

yes

—

C₂³×C₃₁

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

Q₈×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

Dic₆₂

—

C₃₁⋊C₈

C₂×D₄

yes

—

#3152

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#127

yes

—

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#127

yes

—

C₄×C₃₁

#137

yes

complemented

C₄×C₃₁

C₂²×D₄

yes

—

Dic₃₁

—

C₂×#137

—

#532

D₃₁

yes

characteristic

C₂×C₄⋊Q₈

—

#497

—

#494

—

#492

—

C₄×M₁₆

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

C₂×#127

yes

characteristic

C₂×C₃₁

C₂×#137

yes

—

C₂×#41

C₂×D₃₁

yes

—

C₂²×Q₈

—

#137

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

100

C₄⋊Q₈

C₂×D₃₁

yes

characteristic

101

#127

C₂×D₃₁

yes

characteristic

102

#124

C₂×D₃₁

yes

—

103

#121

—

104

C₄×D₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄²

C₂×D₃₁

yes

characteristic

107

#97

—

108

C₄×C₈

—

109

C₃₁

#3152

yes

characteristic, complemented, Sylow

110

#41

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

C₂²×D₃₁

yes

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

C₂²×D₃₁

yes

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

C₂²⋊D₃₁

yes

—

121

C₂²×C₄

C₂²×D₃₁

yes

characteristic

122

Q₁₆

—

123

SD₁₆

—

124

M₁₆

—

125

C₂×C₈

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₂²⋊C₄

—

130

C₄²

C₂²⋊D₃₁

yes

—

131

C₄²

C₂²×D₃₁

yes

characteristic

132

C₄²

C₂²×D₃₁

yes

characteristic

133

C₂³

—

134

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

135

Q₈

—

136

Q₈

—

137

Q₈

—

124

—

138

D₄

—

139

D₄

—

140

D₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

124

—

146

C₂×C₄

D₄×D₃₁

yes

—

147

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

148

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

149

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

150

C₂×C₄

C₂³×D₃₁

yes

characteristic

151

C₈

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

#11758517

yes

—

156

C₂²

C₂²×C₂²⋊D₃₁

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

124

—

160

C₄

#11758517

yes

—

161

C₄

#11758528

yes

complemented

162

C₄

C₂×D₄×D₃₁

yes

—

163

C₂

—

164

C₂

—

165

C₂

C₂×#11758517

yes

characteristic

166

C₂

C₂×#11758528

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₃₁
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting, prime radical	C₃₁×#532	C₂
socle	C₂²×C₃₁	C₂²×D₄
hypercenter	#532	D₃₁
hypocenter	C₃₁	#3152
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₃₁

commutator, cosocle

C₂×C₄×C₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₁

Fitting, prime radical

C₃₁×#532

C₂

socle

C₂²×C₃₁

C₂²×D₄

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. C₂×#11758528	C₂	semidirect
2. C₂×#11758446	C₂	semidirect
3. #14521444	C₂	semidirect
4. #14521428	C₂	semidirect
5. #14521421	C₂	semidirect
6. C₄×#92474	C₂	semidirect
47. #92474	D₄	semidirect
77. C₄×C₃₁	#137	semidirect
109. C₃₁	#3152	semidirect
161. C₄	#11758528	semidirect

H ≅ G/N

type

1. C₂×#11758528

C₂

semidirect

2. C₂×#11758446

C₂

semidirect

3. #14521444

C₂

semidirect

4. #14521428

C₂

semidirect

5. #14521421

C₂

semidirect

6. C₄×#92474

C₂

semidirect

47. #92474

D₄

semidirect

77. C₄×C₃₁

#137

semidirect

109. C₃₁

#3152

semidirect

161. C₄

#11758528

semidirect

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

subgroup	index
C₂×#11758528	2
C₂×#11758446	2
#14521444	2
#14521428	2
#14521421	2
C₄×#92474	2
C₃₁×#532	2
#3152	31

subgroup

index

C₃₁×#532

Subgroups — #16856171

Conjugacy classes of subgroups

Complements

Maximal subgroups