Group #16846648 — subgroups

Subgroups — #16846648

Order 3968.

292 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 145 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×C₄×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#14520804	C₂	1	4	yes	maximal
4	C₂×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₃₁	C₄	1	1	yes	characteristic
6	C₄×Q₈×C₃₁	—	2	2	no	—
7	C₄×Q₈×C₃₁	—	2	2	no	—
8	C₂×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
9	C₂×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
10	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
11	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
13	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
14	C₄⋊(C₃₁⋊C₈)	—	2	2	no	—
15	C₄×C₃₁⋊C₈	—	2	2	no	—
16	C₂×Q₈×C₃₁	—	2	2	no	—
17	C₂×Q₈×C₃₁	—	2	2	no	—
18	C₂×Q₈×C₃₁	C₈	1	4	yes	complemented
19	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₃₁	—	2	4	no	—
25	C₄⋊C₄×C₃₁	—	2	2	no	—
26	C₄²×C₃₁	—	2	2	no	—
27	C₄²×C₃₁	—	2	2	no	—
28	C₄²×C₃₁	D₄	1	2	yes	—
29	C₄²×C₃₁	D₄	1	2	yes	—
30	C₂×C₃₁⋊C₈	—	2	2	no	—
31	C₂×C₃₁⋊C₈	—	2	2	no	—
32	C₂×C₃₁⋊C₈	—	2	2	no	—
33	C₂×C₃₁⋊C₈	—	2	4	no	—
34	C₂×C₃₁⋊C₈	Q₈	1	4	yes	complemented
35	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
36	Q₈×C₃₁	—	2	8	no	—
37	C₂×C₄×C₃₁	—	2	2	no	—
38	C₂×C₄×C₃₁	—	2	2	no	—
39	C₂×C₄×C₃₁	—	2	2	no	—
40	C₂×C₄×C₃₁	—	2	2	no	—
41	C₂×C₄×C₃₁	—	2	4	no	—
42	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₃₁	C₂×C₈	1	4	yes	—
45	C₂×C₄×C₃₁	C₂×C₈	1	2	yes	—
46	C₂×C₄×C₃₁	M₁₆	1	2	yes	—
47	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₃₁	#41	1	2	yes	—
51	C₃₁⋊C₈	—	2	4	no	—
52	C₃₁⋊C₈	—	4	2	no	—
53	#1984	—	31	1	no	maximal, Sylow
54	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₃₁	C₄×Q₈	1	2	yes	—
56	C₂²×C₃₁	C₂²×C₈	1	1	yes	characteristic
57	C₂²×C₃₁	C₂×M₁₆	1	1	yes	characteristic
58	C₂²×C₃₁	#131	1	2	yes	—
59	C₄×C₃₁	—	2	4	no	—
60	C₄×C₃₁	—	2	2	no	—
61	C₄×C₃₁	—	4	2	no	—
62	C₄×C₃₁	C₂²⋊C₈	1	4	yes	—
63	C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
64	C₄×C₃₁	#124	1	1	yes	characteristic
65	C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×Q₈	D₃₁	1	1	yes	characteristic
67	C₂×C₄⋊C₈	—	31	1	no	—
68	C₂×C₄×C₈	—	31	1	no	—
69	#336	—	31	4	no	—
70	C₂×C₃₁	#391	1	1	yes	characteristic
71	C₂×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
72	C₂×C₃₁	#408	1	1	yes	characteristic
73	C₂×C₃₁	C₈×Q₈	1	2	yes	—
74	C₂×C₃₁	#446	1	2	yes	—
75	C₂²×Q₈	Dic₃₁	1	1	yes	characteristic
76	C₂²×C₈	—	31	2	no	—
77	C₂²×C₈	—	31	2	no	—
78	C₄×Q₈	—	2	2	no	—
79	C₄×Q₈	—	2	2	no	—
80	C₂×C₄⋊C₄	Dic₃₁	1	1	yes	characteristic
81	C₂×C₄⋊C₄	Dic₃₁	1	2	yes	—
82	C₂×C₄²	C₂×D₃₁	1	2	yes	—
83	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
84	C₄⋊C₈	—	62	2	no	—
85	C₄×C₈	—	62	2	no	—
86	C₃₁	#1984	1	1	yes	characteristic, complemented, Sylow
87	C₂×Q₈	—	2	2	no	—
88	C₂×Q₈	—	2	2	no	—
89	C₂×Q₈	C₃₁⋊C₈	1	4	yes	complemented
90	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
91	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
92	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
93	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
94	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
95	C₂×C₈	—	31	4	no	—
96	C₂×C₈	—	62	2	no	—
97	C₂×C₈	—	62	2	no	—
98	C₂×C₈	—	62	4	no	—
99	C₂×C₈	—	62	2	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄²	—	2	2	no	—
103	C₄²	—	2	2	no	—
104	C₄²	C₂²⋊D₃₁	1	2	yes	—
105	C₄²	C₂²⋊D₃₁	1	2	yes	—
106	C₂³	C₂²×Dic₃₁	1	1	yes	characteristic
107	Q₈	—	2	8	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	C₂×C₃₁⋊C₈	1	4	yes	—
114	C₂×C₄	C₂×C₃₁⋊C₈	1	2	yes	—
115	C₂×C₄	#92474	1	2	yes	—
116	C₂×C₄	C₂²⋊Dic₃₁	1	2	yes	—
117	C₂×C₄	C₂²⋊Dic₃₁	1	2	yes	—
118	C₂×C₄	Q₈×D₃₁	1	2	yes	—
119	C₂×C₄	#92499	1	2	yes	—
120	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
122	C₈	—	62	4	no	—
123	C₈	—	124	2	no	—
124	C₂²	C₂²×C₃₁⋊C₈	1	1	yes	characteristic
125	C₂²	C₂×#92474	1	1	yes	characteristic
126	C₂²	Q₈×Dic₃₁	1	2	yes	—
127	C₂²	#11758525	1	2	yes	—
128	C₂²	C₂×C₂²⋊Dic₃₁	1	1	yes	characteristic
129	C₄	—	2	4	no	—
130	C₄	—	2	2	no	—
131	C₄	—	4	2	no	—
132	C₄	C₂²⋊(C₃₁⋊C₈)	1	4	yes	—
133	C₄	#11758521	1	1	yes	characteristic
134	C₄	#11758523	1	2	yes	—
135	C₄	#11758524	1	1	yes	characteristic
136	C₂	#14520956	1	1	yes	characteristic
137	C₂	C₂×C₂²⋊(C₃₁⋊C₈)	1	1	yes	characteristic
138	C₂	#14521016	1	1	yes	characteristic
139	C₂	Q₈×C₃₁⋊C₈	1	2	yes	—
140	C₂	#14521180	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₃₁⋊C₈)

C₂

yes

maximal, characteristic

C₂×C₄×C₃₁⋊C₈

C₂

yes

maximal, characteristic

#14520804

C₂

yes

maximal

C₂×C₄×Q₈×C₃₁

C₂

yes

maximal, characteristic

C₂²×Q₈×C₃₁

C₄

yes

characteristic

C₄×Q₈×C₃₁

—

C₄×Q₈×C₃₁

—

C₂×C₄⋊C₄×C₃₁

C₄

yes

characteristic

C₂×C₄⋊C₄×C₃₁

C₄

yes

—

C₂×C₄²×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂²×C₃₁⋊C₈

C₂²

yes

—

C₂²×C₃₁⋊C₈

C₂²

yes

—

C₄⋊(C₃₁⋊C₈)

—

C₄×C₃₁⋊C₈

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

C₈

yes

complemented

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

D₄

yes

—

C₄²×C₃₁

D₄

yes

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

Q₈

yes

complemented

C₂³×C₃₁

C₂²×C₄

yes

characteristic

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

C₂×C₈

yes

—

C₂×C₄×C₃₁

C₂×C₈

yes

—

C₂×C₄×C₃₁

M₁₆

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₃₁⋊C₈

—

C₃₁⋊C₈

—

#1984

—

maximal, Sylow

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₁

C₄×Q₈

yes

—

C₂²×C₃₁

C₂²×C₈

yes

characteristic

C₂²×C₃₁

C₂×M₁₆

yes

characteristic

C₂²×C₃₁

#131

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂²⋊C₈

yes

—

C₄×C₃₁

C₂²⋊Q₈

yes

—

C₄×C₃₁

#124

yes

characteristic

C₄×C₃₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×Q₈

D₃₁

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

#336

—

C₂×C₃₁

#391

yes

characteristic

C₂×C₃₁

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₃₁

#408

yes

characteristic

C₂×C₃₁

C₈×Q₈

yes

—

C₂×C₃₁

#446

yes

—

C₂²×Q₈

Dic₃₁

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₃₁

yes

characteristic

C₂×C₄⋊C₄

Dic₃₁

yes

—

C₂×C₄²

C₂×D₃₁

yes

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

C₄×C₈

—

C₃₁

#1984

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₃₁⋊C₈

yes

complemented

C₂²×C₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂×Dic₃₁

yes

characteristic

C₂²×C₄

C₂×Dic₃₁

yes

characteristic

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄²

—

103

C₄²

—

104

C₄²

C₂²⋊D₃₁

yes

—

105

C₄²

C₂²⋊D₃₁

yes

—

106

C₂³

C₂²×Dic₃₁

yes

characteristic

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×C₃₁⋊C₈

yes

—

114

C₂×C₄

C₂×C₃₁⋊C₈

yes

—

115

C₂×C₄

#92474

yes

—

116

C₂×C₄

C₂²⋊Dic₃₁

yes

—

117

C₂×C₄

C₂²⋊Dic₃₁

yes

—

118

C₂×C₄

Q₈×D₃₁

yes

—

119

C₂×C₄

#92499

yes

—

120

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

122

C₈

—

123

C₈

—

124

—

124

C₂²

C₂²×C₃₁⋊C₈

yes

characteristic

125

C₂²

C₂×#92474

yes

characteristic

126

C₂²

Q₈×Dic₃₁

yes

—

127

C₂²

#11758525

yes

—

128

C₂²

C₂×C₂²⋊Dic₃₁

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

C₂²⋊(C₃₁⋊C₈)

yes

—

133

C₄

#11758521

yes

characteristic

134

C₄

#11758523

yes

—

135

C₄

#11758524

yes

characteristic

136

C₂

#14520956

yes

characteristic

137

C₂

C₂×C₂²⋊(C₃₁⋊C₈)

yes

characteristic

138

C₂

#14521016

yes

characteristic

139

C₂

Q₈×C₃₁⋊C₈

yes

—

140

C₂

#14521180

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃₁
commutator	C₂²×C₃₁	C₂²×C₈
Fitting	C₂×C₄×Q₈×C₃₁	C₂
socle, prime radical	C₂³×C₃₁	C₂²×C₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂×C₄×Q₈	D₃₁
hypocenter	C₃₁	#1984
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₃₁

commutator

C₂²×C₃₁

C₂²×C₈

Fitting

C₂×C₄×Q₈×C₃₁

C₂

socle, prime radical

C₂³×C₃₁

C₂²×C₄

cosocle

C₂²×C₄×C₃₁

C₂³

hypercenter

C₂×C₄×Q₈

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
18. C₂×Q₈×C₃₁	C₈	semidirect
34. C₂×C₃₁⋊C₈	Q₈	semidirect
86. C₃₁	#1984	semidirect
89. C₂×Q₈	C₃₁⋊C₈	semidirect

H ≅ G/N

type

18. C₂×Q₈×C₃₁

C₈

semidirect

34. C₂×C₃₁⋊C₈

Q₈

semidirect

86. C₃₁

#1984

semidirect

89. C₂×Q₈

C₃₁⋊C₈

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×C₄×C₃₁⋊C₈	2
#14520804	2
C₂×C₄×Q₈×C₃₁	2
#1984	31

subgroup

index

C₂×C₄⋊(C₃₁⋊C₈)

C₂×C₄×C₃₁⋊C₈

#14520804

C₂×C₄×Q₈×C₃₁

#1984

Subgroups — #16846648

Conjugacy classes of subgroups

Complements

Maximal subgroups