Group #16845562 — subgroups

Subgroups — C₄⋊C₈×Dic₃₁ #16845562

Order 3968.

316 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 177 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×C₈×Dic₃₁	C₂	1	2	yes	maximal
3	C₄²×Dic₃₁	C₂	1	1	yes	maximal, characteristic
4	#14520803	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×C₄×Dic₃₁	C₄	1	2	yes	—
9	C₂×C₄×Dic₃₁	C₄	1	2	yes	—
10	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
12	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
13	C₃₁×C₄⋊C₈	C₄	1	4	yes	complemented
14	C₈×Dic₃₁	—	2	4	no	—
15	C₄⋊(C₃₁⋊C₈)	C₄	1	4	yes	complemented
16	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
19	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
20	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
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	2	yes	—
25	C₄²×C₃₁	C₂×C₄	1	2	yes	—
26	C₄×Dic₃₁	—	2	1	no	—
27	C₄×Dic₃₁	—	2	1	no	—
28	C₄×Dic₃₁	—	2	4	no	—
29	C₄×Dic₃₁	C₈	1	8	yes	complemented
30	C₄×Dic₃₁	D₄	1	2	yes	—
31	C₄×Dic₃₁	Q₈	1	2	yes	—
32	C₂×C₃₁⋊C₈	—	2	2	no	—
33	C₂×C₃₁⋊C₈	—	2	2	no	—
34	C₂×C₃₁⋊C₈	C₂×C₄	1	4	yes	—
35	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₃₁	—	2	2	no	—
37	C₂×C₄×C₃₁	C₄²	1	2	yes	—
38	C₂×C₄×C₃₁	C₄²	1	2	yes	—
39	C₂×C₄×C₃₁	C₂×C₈	1	4	yes	—
40	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₃₁	C₂²×C₄	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	#41	1	1	yes	characteristic
45	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
46	C₂×Dic₃₁	—	2	2	no	—
47	C₂×Dic₃₁	—	2	2	no	—
48	C₂×Dic₃₁	C₄⋊C₄	1	4	yes	—
49	C₂×Dic₃₁	C₂×C₈	1	4	yes	—
50	C₂×Dic₃₁	C₂×C₈	1	4	yes	—
51	C₂×Dic₃₁	M₁₆	1	4	yes	—
52	C₈×C₃₁	—	2	4	no	—
53	C₃₁⋊C₈	—	2	4	no	—
54	C₄×C₄⋊C₈	—	31	1	no	maximal, Sylow
55	C₂²×C₃₁	C₂×C₄²	1	1	yes	characteristic
56	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₃₁	#117	1	1	yes	characteristic
58	C₂²×C₃₁	C₂²×C₈	1	2	yes	—
59	C₂²×C₃₁	C₂×M₁₆	1	2	yes	—
60	C₄×C₃₁	—	2	2	no	—
61	C₄×C₃₁	C₄×C₈	1	4	yes	complemented
62	C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
63	C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
64	C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
65	C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
66	Dic₃₁	—	2	4	no	—
67	Dic₃₁	C₄⋊C₈	1	8	yes	direct product factor
68	C₂×C₄⋊C₈	—	31	1	no	—
69	C₂×C₄⋊C₈	D₃₁	1	1	yes	characteristic
70	C₂×C₄×C₈	—	31	2	no	—
71	C₄³	—	31	1	no	—
72	#336	—	31	2	no	—
73	C₂×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₃₁	C₂×C₄×C₈	1	1	yes	characteristic
75	C₂×C₃₁	C₄×M₁₆	1	1	yes	characteristic
76	C₂×C₃₁	C₂×C₄⋊C₈	1	2	yes	—
77	C₂×C₃₁	#431	1	2	yes	—
78	C₂²×C₈	—	31	2	no	—
79	C₂²×C₈	C₂×D₃₁	1	2	yes	—
80	C₂×C₄²	—	31	2	no	—
81	C₂×C₄²	—	31	1	no	—
82	C₂×C₄²	—	31	2	no	—
83	C₂×C₄²	—	31	1	no	—
84	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
85	C₄⋊C₈	—	31	4	no	—
86	C₄⋊C₈	Dic₃₁	1	4	yes	direct product factor
87	C₄×C₈	—	62	4	no	—
88	C₃₁	C₄×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₄	—	31	2	no	—
90	C₂²×C₄	—	31	2	no	—
91	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
92	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	31	4	no	—
97	C₂×C₈	—	62	2	no	—
98	C₂×C₈	—	62	2	no	—
99	C₂×C₈	C₂×Dic₃₁	1	4	yes	—
100	C₄²	—	31	8	no	—
101	C₄²	—	31	2	no	—
102	C₄²	—	31	2	no	—
103	C₄²	—	62	4	no	—
104	C₄²	—	62	1	no	—
105	C₄²	—	62	1	no	—
106	C₄²	C₄×D₃₁	1	2	yes	—
107	C₄²	C₂×Dic₃₁	1	2	yes	—
108	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	31	4	no	—
111	C₂×C₄	—	31	4	no	—
112	C₂×C₄	—	31	4	no	—
113	C₂×C₄	—	31	4	no	—
114	C₂×C₄	—	62	2	no	—
115	C₂×C₄	—	62	2	no	—
116	C₂×C₄	C₈×D₃₁	1	4	yes	—
117	C₂×C₄	C₄×Dic₃₁	1	2	yes	—
118	C₂×C₄	C₄×Dic₃₁	1	2	yes	—
119	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
120	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
121	C₂×C₄	#92497	1	1	yes	characteristic
122	C₂×C₄	Q₈×D₃₁	1	1	yes	characteristic
123	C₂×C₄	#92499	1	1	yes	characteristic
124	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
125	C₈	—	2	4	no	—
126	C₈	—	62	4	no	—
127	C₂²	C₄⋊C₄×D₃₁	1	1	yes	characteristic
128	C₂²	#11758469	1	1	yes	characteristic
129	C₂²	C₂×C₈×D₃₁	1	2	yes	—
130	C₂²	M₁₆×D₃₁	1	2	yes	—
131	C₂²	C₂×C₄×Dic₃₁	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	31	8	no	—
134	C₄	—	62	4	no	—
135	C₄	C₈×Dic₃₁	1	4	yes	complemented
136	C₄	#11758464	1	1	yes	characteristic
137	C₄	#11758470	1	1	yes	characteristic
138	C₄	D₄×Dic₃₁	1	1	yes	characteristic
139	C₄	Q₈×Dic₃₁	1	1	yes	characteristic
140	C₂	C₄⋊C₄×Dic₃₁	1	1	yes	characteristic
141	C₂	C₂×C₈×Dic₃₁	1	1	yes	characteristic
142	C₂	M₁₆×Dic₃₁	1	1	yes	characteristic
143	C₂	C₄⋊C₈×D₃₁	1	2	yes	—
144	C₂	#14521101	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₃₁⋊C₈)

C₂

yes

maximal, characteristic

C₂×C₈×Dic₃₁

C₂

yes

maximal

C₄²×Dic₃₁

C₂

yes

maximal, characteristic

#14520803

C₂

yes

maximal

C₂×C₃₁×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂²×C₈×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄×Dic₃₁

C₄

yes

—

C₂×C₄×Dic₃₁

C₄

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

characteristic

C₂×C₄×Dic₃₁

C₂²

yes

characteristic

C₂²×C₃₁⋊C₈

C₂²

yes

—

C₃₁×C₄⋊C₈

C₄

yes

complemented

C₈×Dic₃₁

—

C₄⋊(C₃₁⋊C₈)

C₄

yes

complemented

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×C₄

yes

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄×Dic₃₁

—

C₄×Dic₃₁

—

C₄×Dic₃₁

—

C₄×Dic₃₁

C₈

yes

complemented

C₄×Dic₃₁

D₄

yes

—

C₄×Dic₃₁

Q₈

yes

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

C₂×C₄

yes

—

C₂³×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₄²

yes

—

C₂×C₄×C₃₁

C₄²

yes

—

C₂×C₄×C₃₁

C₂×C₈

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

C₂×C₄×C₃₁

yes

C₂×C₄×C₃₁

yes

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

C₄⋊C₄

yes

—

C₂×Dic₃₁

C₂×C₈

yes

—

C₂×Dic₃₁

C₂×C₈

yes

—

C₂×Dic₃₁

M₁₆

yes

—

C₈×C₃₁

—

C₃₁⋊C₈

—

C₄×C₄⋊C₈

—

maximal, Sylow

C₂²×C₃₁

C₂×C₄²

yes

characteristic

C₂²×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₁

#117

yes

characteristic

C₂²×C₃₁

C₂²×C₈

yes

—

C₂²×C₃₁

C₂×M₁₆

yes

—

C₄×C₃₁

—

C₄×C₃₁

C₄×C₈

yes

complemented

C₄×C₃₁

C₄×D₄

yes

characteristic

C₄×C₃₁

C₄×D₄

yes

characteristic

C₄×C₃₁

C₄×Q₈

yes

characteristic

C₄×C₃₁

C₄×Q₈

yes

characteristic

Dic₃₁

—

Dic₃₁

C₄⋊C₈

yes

direct product factor

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₃₁

yes

characteristic

C₂×C₄×C₈

—

C₄³

—

#336

—

C₂×C₃₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₃₁

C₂×C₄×C₈

yes

characteristic

C₂×C₃₁

C₄×M₁₆

yes

characteristic

C₂×C₃₁

C₂×C₄⋊C₈

yes

—

C₂×C₃₁

#431

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

Dic₃₁

yes

direct product factor

C₄×C₈

—

C₃₁

C₄×C₄⋊C₈

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

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₃₁

yes

—

100

C₄²

—

101

C₄²

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

C₄×D₃₁

yes

—

107

C₄²

C₂×Dic₃₁

yes

—

108

C₂³

C₂×C₄×D₃₁

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₈×D₃₁

yes

—

117

C₂×C₄

C₄×Dic₃₁

yes

—

118

C₂×C₄

C₄×Dic₃₁

yes

—

119

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

120

C₂×C₄

D₄×D₃₁

yes

characteristic

121

C₂×C₄

#92497

yes

characteristic

122

C₂×C₄

Q₈×D₃₁

yes

characteristic

123

C₂×C₄

#92499

yes

characteristic

124

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

125

C₈

—

126

C₈

—

127

C₂²

C₄⋊C₄×D₃₁

yes

characteristic

128

C₂²

#11758469

yes

characteristic

129

C₂²

C₂×C₈×D₃₁

yes

—

130

C₂²

M₁₆×D₃₁

yes

—

131

C₂²

C₂×C₄×Dic₃₁

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

C₈×Dic₃₁

yes

complemented

136

C₄

#11758464

yes

characteristic

137

C₄

#11758470

yes

characteristic

138

C₄

D₄×Dic₃₁

yes

characteristic

139

C₄

Q₈×Dic₃₁

yes

characteristic

140

C₂

C₄⋊C₄×Dic₃₁

yes

characteristic

141

C₂

C₂×C₈×Dic₃₁

yes

characteristic

142

C₂

M₁₆×Dic₃₁

yes

characteristic

143

C₂

C₄⋊C₈×D₃₁

yes

—

144

C₂

#14521101

yes

—

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

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₃₁

commutator

C₂×C₃₁

C₂×C₄×C₈

Fitting

C₂×C₃₁×C₄⋊C₈

C₂

socle, prime radical

C₂³×C₃₁

C₂²×C₄

cosocle

C₂²×C₄×C₃₁

C₂³

hypercenter

C₂×C₄⋊C₈

D₃₁

hypocenter

C₃₁

C₄×C₄⋊C₈

LCS

G ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
13. C₃₁×C₄⋊C₈	C₄	semidirect
15. C₄⋊(C₃₁⋊C₈)	C₄	semidirect
29. C₄×Dic₃₁	C₈	semidirect
61. C₄×C₃₁	C₄×C₈	semidirect
67. Dic₃₁	C₄⋊C₈	direct (N × H)
86. C₄⋊C₈	Dic₃₁	direct (N × H)
88. C₃₁	C₄×C₄⋊C₈	semidirect
135. C₄	C₈×Dic₃₁	semidirect

H ≅ G/N

type

13. C₃₁×C₄⋊C₈

C₄

semidirect

15. C₄⋊(C₃₁⋊C₈)

C₄

semidirect

29. C₄×Dic₃₁

C₈

semidirect

61. C₄×C₃₁

C₄×C₈

semidirect

67. Dic₃₁

C₄⋊C₈

direct (N × H)

86. C₄⋊C₈

Dic₃₁

direct (N × H)

88. C₃₁

C₄×C₄⋊C₈

semidirect

135. C₄

C₈×Dic₃₁

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×C₈×Dic₃₁	2
C₄²×Dic₃₁	2
#14520803	2
C₂×C₃₁×C₄⋊C₈	2
C₄×C₄⋊C₈	31

subgroup

index

C₂×C₄⋊(C₃₁⋊C₈)

C₂×C₈×Dic₃₁

C₄²×Dic₃₁

#14520803

C₂×C₃₁×C₄⋊C₈

C₄×C₄⋊C₈

Subgroups — C₄⋊C₈×Dic₃₁ #16845562

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4⋊C8×Dic31 #16845562

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊C₈×Dic₃₁ #16845562