Group #16846434 — subgroups

Subgroups — #16846434

Order 3968.

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

normal subgroups only

This group has 248 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×C₄⋊(C₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₃₁⋊C₈	C₂	1	2	yes	maximal
3	C₄×C₄⋊Dic₃₁	C₂	1	1	yes	maximal, characteristic
4	#14520803	C₂	1	2	yes	maximal
5	C₂×C₃₁×#97	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₃₁	C₄	1	2	yes	—
10	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
11	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
12	C₃₁×#97	—	2	2	no	—
13	C₃₁⋊#95	C₄	1	1	yes	characteristic
14	C₃₁⋊#95	C₄	1	1	yes	characteristic
15	Dic₃₁⋊C₈	—	2	4	no	—
16	C₄⋊(C₃₁⋊C₈)	—	2	2	no	—
17	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
20	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
21	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₃₁	—	2	1	no	—
23	C₂×C₈×C₃₁	—	2	1	no	—
24	C₂×C₈×C₃₁	—	2	1	no	—
25	C₂×C₈×C₃₁	—	2	1	no	—
26	C₂×C₈×C₃₁	D₄	1	2	yes	—
27	C₂×C₈×C₃₁	Q₈	1	2	yes	—
28	C₄²×C₃₁	D₄	1	2	yes	—
29	C₄²×C₃₁	Q₈	1	2	yes	—
30	C₄⋊Dic₃₁	—	2	4	no	—
31	C₄×Dic₃₁	—	2	2	no	—
32	C₄×Dic₃₁	—	2	2	no	—
33	C₂×C₃₁⋊C₈	—	2	2	no	—
34	C₂×C₃₁⋊C₈	—	2	2	no	—
35	C₂×C₃₁⋊C₈	—	2	2	no	—
36	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₃₁	—	2	2	no	—
38	C₂×C₄×C₃₁	—	2	2	no	—
39	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
41	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
46	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
47	C₂×Dic₃₁	—	2	2	no	—
48	C₂×Dic₃₁	—	2	2	no	—
49	C₂×Dic₃₁	—	2	2	no	—
50	C₂×Dic₃₁	—	2	2	no	—
51	C₂×Dic₃₁	—	2	2	no	—
52	C₂×Dic₃₁	M₁₆	1	4	yes	—
53	C₈×C₃₁	—	2	2	no	—
54	C₈×C₃₁	—	2	2	no	—
55	C₃₁⋊C₈	—	4	2	no	—
56	#1940	—	31	1	no	maximal, Sylow
57	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₃₁	C₄×D₄	1	1	yes	characteristic
59	C₂²×C₃₁	C₄×Q₈	1	1	yes	characteristic
60	C₂²×C₃₁	C₂×M₁₆	1	2	yes	—
61	C₂²×C₃₁	#131	1	2	yes	—
62	C₄×C₃₁	—	2	2	no	—
63	C₄×C₃₁	—	2	2	no	—
64	C₄×C₃₁	#121	1	1	yes	characteristic
65	C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
66	C₄×C₃₁	#125	1	1	yes	characteristic
67	C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
68	Dic₃₁	—	2	4	no	—
69	Dic₃₁	—	4	2	no	—
70	C₂×C₄⋊C₈	—	31	2	no	—
71	C₂×C₄⋊C₈	—	31	1	no	—
72	C₂×#97	D₃₁	1	1	yes	characteristic
73	C₄×C₄⋊C₄	—	31	1	no	—
74	#336	—	31	2	no	—
75	C₂×C₃₁	#389	1	1	yes	characteristic
76	C₂×C₃₁	#423	1	1	yes	characteristic
77	C₂×C₃₁	#424	1	1	yes	characteristic
78	C₂×C₃₁	#435	1	2	yes	—
79	C₂×C₃₁	#446	1	2	yes	—
80	C₂²×C₈	—	31	2	no	—
81	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
82	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
83	C₂×C₄⋊C₄	—	31	2	no	—
84	C₂×C₄²	—	31	2	no	—
85	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
86	C₄⋊C₈	—	62	4	no	—
87	C₄⋊C₈	—	62	2	no	—
88	#97	—	2	2	no	—
89	#95	—	31	1	no	—
90	#95	—	31	1	no	—
91	C₃₁	#1940	1	1	yes	characteristic, complemented, Sylow
92	C₂²×C₄	—	31	2	no	—
93	C₂²×C₄	—	31	2	no	—
94	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
95	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	62	2	no	—
102	C₂×C₈	—	62	2	no	—
103	C₂×C₈	—	62	2	no	—
104	C₂×C₈	Dic₆₂	1	2	yes	—
105	C₂×C₈	D₁₂₄	1	2	yes	—
106	C₄⋊C₄	—	62	4	no	—
107	C₄²	—	62	2	no	—
108	C₄²	—	62	2	no	—
109	C₄²	Dic₆₂	1	2	yes	—
110	C₄²	C₂²⋊D₃₁	1	2	yes	—
111	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	31	4	no	—
115	C₂×C₄	—	62	2	no	—
116	C₂×C₄	—	62	2	no	—
117	C₂×C₄	—	62	2	no	—
118	C₂×C₄	—	62	2	no	—
119	C₂×C₄	—	62	2	no	—
120	C₂×C₄	Dic₃₁⋊C₄	1	2	yes	—
121	C₂×C₄	Dic₃₁⋊C₄	1	2	yes	—
122	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
123	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
124	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
125	C₂×C₄	#92495	1	1	yes	characteristic
126	C₂×C₄	#92495	1	1	yes	characteristic
127	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	—	2	2	no	—
130	C₈	—	124	2	no	—
131	C₂²	C₄×Dic₆₂	1	1	yes	characteristic
132	C₂²	C₄×D₁₂₄	1	1	yes	characteristic
133	C₂²	M₁₆×D₃₁	1	2	yes	—
134	C₂²	#11758484	1	2	yes	—
135	C₂²	C₂×Dic₃₁⋊C₄	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	62	4	no	—
139	C₄	—	124	2	no	—
140	C₄	#11758446	1	1	yes	characteristic
141	C₄	#11758447	1	1	yes	characteristic
142	C₄	#11758501	1	1	yes	characteristic
143	C₄	#11758507	1	1	yes	characteristic
144	C₂	#14520947	1	1	yes	characteristic
145	C₂	#14521072	1	1	yes	characteristic
146	C₂	#14521077	1	1	yes	characteristic
147	C₂	#14521122	1	2	yes	—
148	C₂	#14521177	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₃₁⋊C₈)

C₂

yes

maximal, characteristic

C₂×Dic₃₁⋊C₈

C₂

yes

maximal

C₄×C₄⋊Dic₃₁

C₂

yes

maximal, characteristic

#14520803

C₂

yes

maximal

C₂×C₃₁×#97

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

C₄

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

—

C₂²×C₃₁⋊C₈

C₂²

yes

—

C₃₁×#97

—

C₃₁⋊#95

C₄

yes

characteristic

C₃₁⋊#95

C₄

yes

characteristic

Dic₃₁⋊C₈

—

C₄⋊(C₃₁⋊C₈)

—

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

D₄

yes

—

C₂×C₈×C₃₁

Q₈

yes

—

C₄²×C₃₁

D₄

yes

—

C₄²×C₃₁

Q₈

yes

—

C₄⋊Dic₃₁

—

C₄×Dic₃₁

—

C₄×Dic₃₁

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₄⋊C₄

yes

—

C₂×C₄×C₃₁

C₄⋊C₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

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

M₁₆

yes

—

C₈×C₃₁

—

C₈×C₃₁

—

C₃₁⋊C₈

—

#1940

—

maximal, Sylow

C₂²×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₁

C₄×D₄

yes

characteristic

C₂²×C₃₁

C₄×Q₈

yes

characteristic

C₂²×C₃₁

C₂×M₁₆

yes

—

C₂²×C₃₁

#131

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#121

yes

characteristic

C₄×C₃₁

C₂²⋊Q₈

yes

characteristic

C₄×C₃₁

#125

yes

characteristic

C₄×C₃₁

C₄⋊Q₈

yes

characteristic

Dic₃₁

—

Dic₃₁

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

D₃₁

yes

characteristic

C₄×C₄⋊C₄

—

#336

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂×C₃₁

#446

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃₁

yes

characteristic

C₂²×C₈

C₂×D₃₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#95

—

#95

—

C₃₁

#1940

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₃₁

yes

characteristic

C₂²×C₄

C₄×D₃₁

yes

characteristic

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

Dic₆₂

yes

—

105

C₂×C₈

D₁₂₄

yes

—

106

C₄⋊C₄

—

107

C₄²

—

108

C₄²

—

109

C₄²

Dic₆₂

yes

—

110

C₄²

C₂²⋊D₃₁

yes

—

111

C₂³

C₂×C₄×D₃₁

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

Dic₃₁⋊C₄

yes

—

121

C₂×C₄

Dic₃₁⋊C₄

yes

—

122

C₂×C₄

C₂×Dic₆₂

yes

characteristic

123

C₂×C₄

C₂×Dic₆₂

yes

characteristic

124

C₂×C₄

C₂×D₁₂₄

yes

characteristic

125

C₂×C₄

#92495

yes

characteristic

126

C₂×C₄

#92495

yes

characteristic

127

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

124

—

131

C₂²

C₄×Dic₆₂

yes

characteristic

132

C₂²

C₄×D₁₂₄

yes

characteristic

133

C₂²

M₁₆×D₃₁

yes

—

134

C₂²

#11758484

yes

—

135

C₂²

C₂×Dic₃₁⋊C₄

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

124

—

140

C₄

#11758446

yes

characteristic

141

C₄

#11758447

yes

characteristic

142

C₄

#11758501

yes

characteristic

143

C₄

#11758507

yes

characteristic

144

C₂

#14520947

yes

characteristic

145

C₂

#14521072

yes

characteristic

146

C₂

#14521077

yes

characteristic

147

C₂

#14521122

yes

—

148

C₂

#14521177

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃₁
commutator, socle, prime radical	C₂³×C₃₁	C₂²×C₄
Fitting	C₂×C₃₁×#97	C₂
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂×#97	D₃₁
hypocenter	C₃₁	#1940
LCS	G ▷ C₂³×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₃₁

commutator, socle, prime radical

C₂³×C₃₁

C₂²×C₄

Fitting

C₂×C₃₁×#97

C₂

cosocle

C₂²×C₄×C₃₁

C₂³

hypercenter

C₂×#97

D₃₁

C₃₁

G ▷ C₂³×C₃₁ ▷ C₃₁

C₁ ◁ C₂²×C₄ ◁ C₂×#97

N	H ≅ G/N	type
91. C₃₁	#1940	semidirect

H ≅ G/N

type

91. C₃₁

#1940

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×Dic₃₁⋊C₈	2
C₄×C₄⋊Dic₃₁	2
#14520803	2
C₂×C₃₁×#97	2
#1940	31

subgroup

index

C₂×C₄⋊(C₃₁⋊C₈)

C₂×Dic₃₁⋊C₈

C₄×C₄⋊Dic₃₁

#14520803

C₂×C₃₁×#97

#1940

Subgroups — #16846434

Conjugacy classes of subgroups

Complements

Maximal subgroups