Group #16845610 — subgroups

Subgroups — #16845610

Order 3968.

272 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 133 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×#11758405	C₂	1	1	yes	maximal, characteristic
2	C₂×#11758403	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×Dic₃₁	C₂	1	2	yes	maximal
4	#14520811	C₂	1	2	yes	maximal
5	C₂×C₃₁×#107	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
9	C₂×C₄×Dic₃₁	—	2	2	no	—
10	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
11	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₃₁×#107	C₄	1	4	yes	complemented
13	C₃₁⋊#95	—	2	2	no	—
14	#11758405	C₄	1	4	yes	complemented
15	#11758403	C₄	1	4	yes	complemented
16	C₂²×C₄×C₃₁	—	2	2	no	—
17	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
18	C₂²×Dic₃₁	—	2	2	no	—
19	C₂²×Dic₃₁	D₄	1	2	yes	—
20	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
23	C₄⋊C₄×C₃₁	—	2	2	no	—
24	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
25	C₄⋊Dic₃₁	—	2	2	no	—
26	C₄⋊Dic₃₁	C₂×C₄	1	4	yes	—
27	C₄×Dic₃₁	—	4	4	no	—
28	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
29	C₄×Dic₃₁	Q₈	1	2	yes	—
30	C₂×C₃₁⋊C₈	—	2	1	no	—
31	C₂×C₃₁⋊C₈	—	2	1	no	—
32	C₂×C₃₁⋊C₈	C₂×C₄	1	2	yes	—
33	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₃₁	—	2	4	no	—
35	C₂×C₄×C₃₁	—	4	2	no	—
36	C₂×C₄×C₃₁	—	4	2	no	—
37	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
42	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
43	C₂×Dic₃₁	—	2	2	no	—
44	C₂×Dic₃₁	—	2	2	no	—
45	C₂×Dic₃₁	—	2	4	no	—
46	C₂×Dic₃₁	—	4	2	no	—
47	C₂×Dic₃₁	—	4	2	no	—
48	C₂×Dic₃₁	C₄⋊C₄	1	4	yes	—
49	C₈×C₃₁	C₄²	1	4	yes	complemented
50	C₃₁⋊C₈	—	2	2	no	—
51	#1776	—	31	1	no	maximal, Sylow
52	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₃₁	C₄×D₄	1	1	yes	characteristic
54	C₂²×C₃₁	C₄×D₄	1	1	yes	characteristic
55	C₂²×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
56	C₂²×C₃₁	#137	1	2	yes	—
57	C₄×C₃₁	—	4	4	no	—
58	C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
59	C₄×C₃₁	#117	1	1	yes	characteristic
60	C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
61	C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
62	Dic₃₁	—	2	4	no	—
63	Dic₃₁	—	4	4	no	—
64	C₂×#107	D₃₁	1	1	yes	characteristic
65	C₂×#106	—	31	1	no	—
66	C₂×#97	—	31	1	no	—
67	C₄×C₄⋊C₄	—	31	2	no	—
68	#340	—	31	2	no	—
69	C₂×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₃₁	#428	1	2	yes	—
71	C₂×C₃₁	#440	1	2	yes	—
72	C₂×C₃₁	#441	1	1	yes	characteristic
73	C₂×C₃₁	#442	1	1	yes	characteristic
74	C₂²×C₈	—	31	1	no	—
75	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
76	C₂×C₄⋊C₄	—	31	2	no	—
77	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
78	C₂×C₄²	—	31	1	no	—
79	C₂×C₄²	—	62	2	no	—
80	#107	Dic₃₁	1	4	yes	complemented
81	#106	—	31	4	no	—
82	#97	—	31	4	no	—
83	#95	—	62	2	no	—
84	C₃₁	#1776	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	—	31	2	no	—
87	C₂²×C₄	—	62	2	no	—
88	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
89	C₂×C₈	—	31	2	no	—
90	C₂×C₈	—	62	1	no	—
91	C₂×C₈	—	62	1	no	—
92	C₂×C₈	C₄×D₃₁	1	2	yes	—
93	C₂×C₈	C₄×D₃₁	1	2	yes	—
94	C₂×C₈	C₂×Dic₃₁	1	2	yes	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	31	4	no	—
97	C₄⋊C₄	—	62	2	no	—
98	C₄⋊C₄	C₂×Dic₃₁	1	4	yes	—
99	C₄²	—	31	2	no	—
100	C₄²	—	31	2	no	—
101	C₄²	—	124	4	no	—
102	C₂³	D₄×D₃₁	1	1	yes	characteristic
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	31	4	no	—
107	C₂×C₄	—	62	2	no	—
108	C₂×C₄	—	62	2	no	—
109	C₂×C₄	—	62	4	no	—
110	C₂×C₄	—	124	2	no	—
111	C₂×C₄	—	124	2	no	—
112	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
113	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
114	C₂×C₄	#92497	1	1	yes	characteristic
115	C₂×C₄	Q₈×D₃₁	1	1	yes	characteristic
116	C₂×C₄	#92499	1	1	yes	characteristic
117	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
118	C₈	—	62	2	no	—
119	C₈	C₄×Dic₃₁	1	4	yes	complemented
120	C₂²	C₄⋊C₄×D₃₁	1	1	yes	characteristic
121	C₂²	#11758470	1	1	yes	characteristic
122	C₂²	#11758488	1	2	yes	—
123	C₂²	#11758495	1	2	yes	—
124	C₂²	D₄×Dic₃₁	1	1	yes	characteristic
125	C₄	—	4	4	no	—
126	C₄	—	62	4	no	—
127	C₄	—	124	4	no	—
128	C₄	#11758464	1	1	yes	characteristic
129	C₄	#11758469	1	1	yes	characteristic
130	C₄	C₂×C₄×Dic₃₁	1	1	yes	characteristic
131	C₄	Q₈×Dic₃₁	1	1	yes	characteristic
132	C₂	C₄⋊C₄×Dic₃₁	1	1	yes	characteristic
133	C₂	#14521090	1	2	yes	—
134	C₂	#14521148	1	2	yes	—
135	C₂	#14521156	1	1	yes	characteristic
136	C₂	#14521161	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758405

C₂

yes

maximal, characteristic

C₂×#11758403

C₂

yes

maximal, characteristic

C₄⋊C₄×Dic₃₁

C₂

yes

maximal

#14520811

C₂

yes

maximal

C₂×C₃₁×#107

C₂

yes

maximal, characteristic

C₂²×C₈×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄×Dic₃₁

—

C₂×C₄×Dic₃₁

C₂²

yes

characteristic

C₂²×C₃₁⋊C₈

C₂²

yes

characteristic

C₃₁×#107

C₄

yes

complemented

C₃₁⋊#95

—

#11758405

C₄

yes

complemented

#11758403

C₄

yes

complemented

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂²×Dic₃₁

D₄

yes

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

C₂×C₄

yes

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂×C₄

yes

—

C₄×Dic₃₁

—

C₄×Dic₃₁

C₂×C₄

yes

—

C₄×Dic₃₁

Q₈

yes

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

C₂×C₄

yes

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

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

C₄⋊C₄

yes

—

C₈×C₃₁

C₄²

yes

complemented

C₃₁⋊C₈

—

#1776

—

maximal, Sylow

C₂²×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₁

C₄×D₄

yes

characteristic

C₂²×C₃₁

C₄×D₄

yes

characteristic

C₂²×C₃₁

AGL(1,ℤ/8)

yes

—

C₂²×C₃₁

#137

yes

—

C₄×C₃₁

—

C₄×C₃₁

C₂×C₄²

yes

characteristic

C₄×C₃₁

#117

yes

characteristic

C₄×C₃₁

C₄×Q₈

yes

characteristic

C₄×C₃₁

C₄×Q₈

yes

characteristic

Dic₃₁

—

Dic₃₁

—

C₂×#107

D₃₁

yes

characteristic

C₂×#106

—

C₂×#97

—

C₄×C₄⋊C₄

—

#340

—

C₂×C₃₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₃₁

#428

yes

—

C₂×C₃₁

#440

yes

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂²×C₈

—

C₂²×C₈

C₂×D₃₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

#107

Dic₃₁

yes

—

—

—

C₃₁

#1776

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₃₁

yes

—

C₂×C₈

C₄×D₃₁

yes

—

C₂×C₈

C₂×Dic₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₃₁

yes

—

C₄²

—

100

C₄²

—

101

C₄²

—

124

—

102

C₂³

D₄×D₃₁

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

124

—

111

C₂×C₄

—

124

—

112

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

113

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

114

C₂×C₄

#92497

yes

characteristic

115

C₂×C₄

Q₈×D₃₁

yes

characteristic

116

C₂×C₄

#92499

yes

characteristic

117

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

118

C₈

—

119

C₈

C₄×Dic₃₁

yes

complemented

120

C₂²

C₄⋊C₄×D₃₁

yes

characteristic

121

C₂²

#11758470

yes

characteristic

122

C₂²

#11758488

yes

—

123

C₂²

#11758495

yes

—

124

C₂²

D₄×Dic₃₁

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

124

—

128

C₄

#11758464

yes

characteristic

129

C₄

#11758469

yes

characteristic

130

C₄

C₂×C₄×Dic₃₁

yes

characteristic

131

C₄

Q₈×Dic₃₁

yes

characteristic

132

C₂

C₄⋊C₄×Dic₃₁

yes

characteristic

133

C₂

#14521090

yes

—

134

C₂

#14521148

yes

—

135

C₂

#14521156

yes

characteristic

136

C₂

#14521161

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₃₁
commutator	C₄×C₃₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂×C₃₁×#107	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂×#107	D₃₁
hypocenter	C₃₁	#1776
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₃₁

commutator

C₄×C₃₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₂×C₃₁×#107

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

cosocle

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
12. C₃₁×#107	C₄	semidirect
14. #11758405	C₄	semidirect
15. #11758403	C₄	semidirect
49. C₈×C₃₁	C₄²	semidirect
80. #107	Dic₃₁	semidirect
84. C₃₁	#1776	semidirect
119. C₈	C₄×Dic₃₁	semidirect

H ≅ G/N

type

12. C₃₁×#107

C₄

semidirect

14. #11758405

C₄

semidirect

15. #11758403

C₄

semidirect

49. C₈×C₃₁

C₄²

semidirect

80. #107

Dic₃₁

semidirect

84. C₃₁

#1776

semidirect

119. C₈

C₄×Dic₃₁

semidirect

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

subgroup	index
C₂×#11758405	2
C₂×#11758403	2
C₄⋊C₄×Dic₃₁	2
#14520811	2
C₂×C₃₁×#107	2
#1776	31

subgroup

index

C₄⋊C₄×Dic₃₁

C₂×C₃₁×#107

Subgroups — #16845610

Conjugacy classes of subgroups

Complements

Maximal subgroups