Group #16846487 — subgroups

Subgroups — #16846487

Order 3968.

200 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 101 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×#11758407	C₂	1	2	yes	maximal
2	#14521073	C₂	1	1	yes	maximal, characteristic
3	C₄×#92474	C₂	1	1	yes	maximal, characteristic
4	#14520814	C₂	1	2	yes	maximal
5	C₂×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
7	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×#92474	C₂²	1	2	yes	—
9	C₂×#92474	C₂²	1	1	yes	characteristic
10	C₂×#92474	C₂²	1	1	yes	characteristic
11	C₃₁×C₄⋊C₈	C₄	1	2	yes	—
12	C₃₁×C₄⋊C₈	C₄	1	2	yes	—
13	#11758407	—	2	4	no	—
14	C₄⋊(C₃₁⋊C₈)	—	2	1	no	—
15	C₄⋊(C₃₁⋊C₈)	—	2	1	no	—
16	#11758391	—	2	1	no	—
17	C₄×C₃₁⋊C₈	—	2	1	no	—
18	C₂²×C₄×C₃₁	Q₈	1	1	yes	characteristic
19	C₂²×C₄×C₃₁	Q₈	1	1	yes	characteristic
20	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
21	C₂×C₈×C₃₁	—	2	2	no	—
22	C₂×C₈×C₃₁	—	2	2	no	—
23	C₂×C₈×C₃₁	C₂×C₄	1	4	yes	—
24	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₃₁	D₄	1	1	yes	characteristic
27	C₄²×C₃₁	D₄	1	1	yes	characteristic
28	#92474	—	2	1	no	—
29	#92474	—	2	1	no	—
30	#92474	—	2	2	no	—
31	#92474	—	2	2	no	—
32	#92474	D₄	1	2	yes	—
33	#92474	Q₈	1	2	yes	—
34	C₂×C₃₁⋊C₈	—	2	1	no	—
35	C₂×C₃₁⋊C₈	—	2	1	no	—
36	C₂×C₃₁⋊C₈	—	2	2	no	—
37	C₂³×C₃₁	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₃₁	—	2	2	no	—
39	C₂×C₄×C₃₁	—	4	1	no	—
40	C₂×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
49	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
50	C₈×C₃₁	—	2	4	no	—
51	C₃₁⋊C₈	—	2	2	no	—
52	C₃₁⋊C₈	—	2	2	no	—
53	C₃₁⋊C₈	—	4	2	no	—
54	#1950	—	31	1	no	maximal, Sylow
55	C₂²×C₃₁	—	2	1	no	—
56	C₂²×C₃₁	—	2	1	no	—
57	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₃₁	#125	1	1	yes	characteristic
59	C₂²×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
60	C₄×C₃₁	—	2	1	no	—
61	C₄×C₃₁	—	2	1	no	—
62	C₄×C₃₁	—	2	1	no	—
63	C₄×C₃₁	#108	1	2	yes	complemented
64	C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
65	C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
66	C₄×C₃₁	#121	1	1	yes	characteristic
67	C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×#108	—	31	2	no	—
69	#423	—	31	1	no	—
70	C₂×C₄⋊C₈	D₃₁	1	1	yes	characteristic
71	C₄×M₁₆	—	31	1	no	—
72	#341	—	31	2	no	—
73	C₂×C₃₁	—	2	1	no	—
74	C₂×C₃₁	—	2	1	no	—
75	C₂×C₃₁	#389	1	1	yes	characteristic
76	C₂×C₃₁	C₂×#108	1	1	yes	characteristic
77	C₂×C₃₁	#430	1	1	yes	characteristic
78	C₂×M₁₆	—	31	1	no	—
79	C₂×M₁₆	—	31	1	no	—
80	C₂×M₁₆	—	31	2	no	—
81	C₂²×C₈	C₂×D₃₁	1	2	yes	—
82	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
83	#108	—	62	4	no	—
84	C₄⋊C₈	—	62	1	no	—
85	C₄⋊C₈	—	62	1	no	—
86	C₄⋊C₈	Dic₃₁	1	2	yes	—
87	C₄⋊C₈	Dic₃₁	1	2	yes	—
88	#97	—	62	1	no	—
89	C₄×C₈	—	62	1	no	—
90	C₃₁	#1950	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₄	Dic₆₂	1	1	yes	characteristic
92	C₂²×C₄	Dic₆₂	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
94	M₁₆	—	31	2	no	—
95	M₁₆	—	31	2	no	—
96	M₁₆	—	62	1	no	—
97	M₁₆	—	62	1	no	—
98	M₁₆	—	62	2	no	—
99	M₁₆	—	62	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	62	1	no	—
103	C₂×C₈	—	62	1	no	—
104	C₂×C₈	—	62	2	no	—
105	C₂×C₈	C₂×Dic₃₁	1	4	yes	—
106	C₄²	D₁₂₄	1	1	yes	characteristic
107	C₄²	D₁₂₄	1	1	yes	characteristic
108	C₄²	C₂×Dic₃₁	1	1	yes	characteristic
109	C₄²	C₂×Dic₃₁	1	1	yes	characteristic
110	C₂³	C₂×Dic₆₂	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	C₄⋊Dic₃₁	1	1	yes	characteristic
114	C₂×C₄	C₄⋊Dic₃₁	1	1	yes	characteristic
115	C₂×C₄	C₄⋊Dic₃₁	1	1	yes	characteristic
116	C₂×C₄	C₄⋊Dic₃₁	1	1	yes	characteristic
117	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
118	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
119	C₂×C₄	#92497	1	1	yes	characteristic
120	C₂×C₄	Q₈×D₃₁	1	1	yes	characteristic
121	C₂×C₄	#92499	1	1	yes	characteristic
122	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
123	C₈	—	2	4	no	—
124	C₈	—	62	2	no	—
125	C₈	—	62	2	no	—
126	C₈	—	124	2	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	#11758465	1	1	yes	characteristic
130	C₂²	#11758467	1	1	yes	characteristic
131	C₂²	C₂×C₄⋊Dic₃₁	1	1	yes	characteristic
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	#11758407	1	2	yes	complemented
136	C₄	#11758472	1	1	yes	characteristic
137	C₄	#11758474	1	1	yes	characteristic
138	C₄	D₄×Dic₃₁	1	1	yes	characteristic
139	C₄	Q₈×Dic₃₁	1	1	yes	characteristic
140	C₂	—	2	1	no	—
141	C₂	—	2	1	no	—
142	C₂	#14520948	1	1	yes	characteristic
143	C₂	C₂×#11758407	1	1	yes	characteristic
144	C₂	#14521098	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758407

C₂

yes

maximal

#14521073

C₂

yes

maximal, characteristic

C₄×#92474

C₂

yes

maximal, characteristic

#14520814

C₂

yes

maximal

C₂×C₃₁×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂²×C₈×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×#92474

C₂²

yes

—

C₂×#92474

C₂²

yes

characteristic

C₂×#92474

C₂²

yes

characteristic

C₃₁×C₄⋊C₈

C₄

yes

—

C₃₁×C₄⋊C₈

C₄

yes

—

#11758407

—

C₄⋊(C₃₁⋊C₈)

—

C₄⋊(C₃₁⋊C₈)

—

#11758391

—

C₄×C₃₁⋊C₈

—

C₂²×C₄×C₃₁

Q₈

yes

characteristic

C₂²×C₄×C₃₁

Q₈

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

D₄

yes

characteristic

C₄²×C₃₁

D₄

yes

—

—

—

—

D₄

yes

—

#92474

Q₈

yes

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

C₂×C₄×C₃₁

yes

C₂×C₄×C₃₁

yes

C₈×C₃₁

—

C₃₁⋊C₈

—

C₃₁⋊C₈

—

C₃₁⋊C₈

—

#1950

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₁

#125

yes

characteristic

C₂²×C₃₁

C₄⋊Q₈

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#108

yes

complemented

C₄×C₃₁

C₄×D₄

yes

characteristic

C₄×C₃₁

C₄×Q₈

yes

characteristic

C₄×C₃₁

#121

yes

characteristic

C₄×C₃₁

C₂²⋊Q₈

yes

characteristic

C₂×#108

—

#423

—

C₂×C₄⋊C₈

D₃₁

yes

characteristic

C₄×M₁₆

—

#341

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₃₁

yes

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

Dic₃₁

yes

—

C₄⋊C₈

Dic₃₁

yes

—

#97

—

C₄×C₈

—

C₃₁

#1950

yes

characteristic, complemented, Sylow

C₂²×C₄

Dic₆₂

yes

characteristic

C₂²×C₄

Dic₆₂

yes

characteristic

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

C₂×Dic₃₁

yes

—

106

C₄²

D₁₂₄

yes

characteristic

107

C₄²

D₁₂₄

yes

characteristic

108

C₄²

C₂×Dic₃₁

yes

characteristic

109

C₄²

C₂×Dic₃₁

yes

characteristic

110

C₂³

C₂×Dic₆₂

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₄⋊Dic₃₁

yes

characteristic

114

C₂×C₄

C₄⋊Dic₃₁

yes

characteristic

115

C₂×C₄

C₄⋊Dic₃₁

yes

characteristic

116

C₂×C₄

C₄⋊Dic₃₁

yes

characteristic

117

C₂×C₄

C₂×D₁₂₄

yes

characteristic

118

C₂×C₄

D₄×D₃₁

yes

characteristic

119

C₂×C₄

#92497

yes

characteristic

120

C₂×C₄

Q₈×D₃₁

yes

characteristic

121

C₂×C₄

#92499

yes

characteristic

122

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

124

—

127

C₂²

—

128

C₂²

—

129

C₂²

#11758465

yes

characteristic

130

C₂²

#11758467

yes

characteristic

131

C₂²

C₂×C₄⋊Dic₃₁

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

#11758407

yes

complemented

136

C₄

#11758472

yes

characteristic

137

C₄

#11758474

yes

characteristic

138

C₄

D₄×Dic₃₁

yes

characteristic

139

C₄

Q₈×Dic₃₁

yes

characteristic

140

C₂

—

141

C₂

—

142

C₂

#14520948

yes

characteristic

143

C₂

C₂×#11758407

yes

characteristic

144

C₂

#14521098

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₂₄
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂×C₃₁×C₄⋊C₈	C₂
socle	C₂²×C₃₁	C₂×C₄⋊C₄
prime radical	C₂³×C₃₁	C₂×Q₈
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂×C₄⋊C₈	D₃₁
hypocenter	C₃₁	#1950
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

C₂×D₁₂₄

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₂×C₃₁×C₄⋊C₈

C₂

socle

C₂²×C₃₁

C₂×C₄⋊C₄

prime radical

C₂³×C₃₁

C₂×Q₈

cosocle

C₂²×C₄×C₃₁

C₂³

hypercenter

C₂×C₄⋊C₈

D₃₁

hypocenter

C₃₁

#1950

LCS

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
63. C₄×C₃₁	#108	semidirect
90. C₃₁	#1950	semidirect
135. C₄	#11758407	semidirect

H ≅ G/N

type

63. C₄×C₃₁

#108

semidirect

90. C₃₁

#1950

semidirect

135. C₄

#11758407

semidirect

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

subgroup	index
C₂×#11758407	2
#14521073	2
C₄×#92474	2
#14520814	2
C₂×C₃₁×C₄⋊C₈	2
#1950	31

subgroup

index

C₂×C₃₁×C₄⋊C₈

Subgroups — #16846487

Conjugacy classes of subgroups

Complements

Maximal subgroups