Group #16845827 — subgroups

Subgroups — #16845827

Order 3968.

260 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 129 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	C₄²×Dic₃₁	C₂	1	1	yes	maximal, characteristic
3	#14520803	C₂	1	4	yes	maximal
4	C₂×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
6	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₃₁	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₃₁	C₄	1	2	yes	—
9	C₂×C₄×Dic₃₁	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
11	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
12	C₃₁×C₄⋊C₈	—	2	2	no	—
13	C₄⋊(C₃₁⋊C₈)	—	2	2	no	—
14	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
17	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
18	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₃₁	—	2	2	no	—
20	C₂×C₈×C₃₁	—	2	4	no	—
21	C₄²×C₃₁	D₄	1	2	yes	—
22	C₄²×C₃₁	Q₈	1	2	yes	—
23	C₄×Dic₃₁	—	2	2	no	—
24	C₄×Dic₃₁	—	2	2	no	—
25	C₄×Dic₃₁	—	2	2	no	—
26	C₄×Dic₃₁	—	2	4	no	—
27	C₄×Dic₃₁	C₈	1	4	yes	complemented
28	C₂×C₃₁⋊C₈	—	2	2	no	—
29	C₂×C₃₁⋊C₈	—	2	4	no	—
30	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₃₁	—	2	2	no	—
32	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
33	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
34	C₂×C₄×C₃₁	C₂×C₈	1	2	yes	—
35	C₂×C₄×C₃₁	M₁₆	1	2	yes	—
36	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₃₁	#41	1	2	yes	—
39	C₂×C₄×C₃₁	#41	1	2	yes	—
40	C₂×Dic₃₁	—	2	2	no	—
41	C₂×Dic₃₁	—	2	2	no	—
42	C₂×Dic₃₁	—	2	2	no	—
43	C₂×Dic₃₁	—	2	2	no	—
44	C₂×Dic₃₁	C₂×C₈	1	4	yes	—
45	C₂×Dic₃₁	M₁₆	1	4	yes	—
46	C₈×C₃₁	—	4	2	no	—
47	C₃₁⋊C₈	—	4	2	no	—
48	#1831	—	31	1	no	maximal, Sylow
49	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₃₁	#117	1	2	yes	—
51	C₂²×C₃₁	C₂²×C₈	1	1	yes	characteristic
52	C₂²×C₃₁	C₂×M₁₆	1	1	yes	characteristic
53	C₂²×C₃₁	C₂×M₁₆	1	2	yes	—
54	C₄×C₃₁	—	2	2	no	—
55	C₄×C₃₁	C₄⋊C₈	1	4	yes	—
56	C₄×C₃₁	#124	1	1	yes	characteristic
57	C₄×C₃₁	#124	1	1	yes	characteristic
58	C₄×C₃₁	#125	1	2	yes	—
59	Dic₃₁	—	2	4	no	—
60	Dic₃₁	—	2	4	no	—
61	C₂×C₄⋊C₈	—	31	1	no	—
62	C₂×C₄⋊C₈	D₃₁	1	1	yes	characteristic
63	C₄³	—	31	1	no	—
64	#336	—	31	4	no	—
65	C₂×C₃₁	#382	1	1	yes	characteristic
66	C₂×C₃₁	C₂×C₄⋊C₈	1	1	yes	characteristic
67	C₂×C₃₁	#423	1	1	yes	characteristic
68	C₂×C₃₁	#431	1	2	yes	—
69	C₂×C₃₁	#432	1	2	yes	—
70	C₂²×C₈	—	31	2	no	—
71	C₂²×C₈	C₂×D₃₁	1	2	yes	—
72	C₂×C₄²	—	31	1	no	—
73	C₂×C₄²	—	31	2	no	—
74	C₂×C₄²	—	31	1	no	—
75	C₂×C₄²	—	31	2	no	—
76	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
77	C₄⋊C₈	—	2	2	no	—
78	C₄⋊C₈	—	62	2	no	—
79	C₃₁	#1831	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₄	—	31	2	no	—
81	C₂²×C₄	—	31	2	no	—
82	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
83	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
84	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
85	C₂×C₈	—	2	4	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	62	4	no	—
88	C₂×C₈	—	62	2	no	—
89	C₄²	—	31	4	no	—
90	C₄²	—	62	4	no	—
91	C₄²	—	62	2	no	—
92	C₄²	—	62	2	no	—
93	C₄²	—	62	2	no	—
94	C₄²	Dic₆₂	1	2	yes	—
95	C₄²	C₂²⋊D₃₁	1	2	yes	—
96	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	31	4	no	—
99	C₂×C₄	—	31	4	no	—
100	C₂×C₄	—	62	2	no	—
101	C₂×C₄	—	62	2	no	—
102	C₂×C₄	—	62	2	no	—
103	C₂×C₄	—	62	2	no	—
104	C₂×C₄	C₈×D₃₁	1	2	yes	—
105	C₂×C₄	#92469	1	2	yes	—
106	C₂×C₄	Dic₃₁⋊C₄	1	2	yes	—
107	C₂×C₄	Dic₃₁⋊C₄	1	2	yes	—
108	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
109	C₂×C₄	#92497	1	2	yes	—
110	C₂×C₄	#92499	1	2	yes	—
111	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
112	C₈	—	4	2	no	—
113	C₈	—	124	2	no	—
114	C₂²	#11758469	1	2	yes	—
115	C₂²	C₂×C₈×D₃₁	1	1	yes	characteristic
116	C₂²	C₂×#92469	1	1	yes	characteristic
117	C₂²	M₁₆×D₃₁	1	2	yes	—
118	C₂²	C₂×Dic₃₁⋊C₄	1	1	yes	characteristic
119	C₄	—	2	2	no	—
120	C₄	—	62	4	no	—
121	C₄	—	62	4	no	—
122	C₄	Dic₃₁⋊C₈	1	4	yes	—
123	C₄	#11758467	1	2	yes	—
124	C₄	#11758513	1	1	yes	characteristic
125	C₄	#11758524	1	1	yes	characteristic
126	C₂	#14520914	1	1	yes	characteristic
127	C₂	C₂×Dic₃₁⋊C₈	1	1	yes	characteristic
128	C₂	#14521072	1	1	yes	characteristic
129	C₂	#14521101	1	2	yes	—
130	C₂	#14521105	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₃₁⋊C₈)

C₂

yes

maximal, characteristic

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

characteristic

C₂×C₄×Dic₃₁

C₄

yes

—

C₂×C₄×Dic₃₁

C₄

yes

characteristic

C₂×C₄×Dic₃₁

C₂²

yes

—

C₂²×C₃₁⋊C₈

C₂²

yes

—

C₃₁×C₄⋊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

—

C₄²×C₃₁

D₄

yes

—

C₄²×C₃₁

Q₈

yes

—

C₄×Dic₃₁

—

C₄×Dic₃₁

—

C₄×Dic₃₁

—

C₄×Dic₃₁

—

C₄×Dic₃₁

C₈

yes

complemented

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₄⋊C₄

yes

—

C₂×C₄×C₃₁

C₄⋊C₄

yes

—

C₂×C₄×C₃₁

C₂×C₈

yes

—

C₂×C₄×C₃₁

M₁₆

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

C₂×C₈

yes

—

C₂×Dic₃₁

M₁₆

yes

—

C₈×C₃₁

—

C₃₁⋊C₈

—

#1831

—

maximal, Sylow

C₂²×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₁

#117

yes

—

C₂²×C₃₁

C₂²×C₈

yes

characteristic

C₂²×C₃₁

C₂×M₁₆

yes

characteristic

C₂²×C₃₁

C₂×M₁₆

yes

—

C₄×C₃₁

—

C₄×C₃₁

C₄⋊C₈

yes

—

C₄×C₃₁

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

—

Dic₃₁

—

Dic₃₁

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₃₁

yes

characteristic

C₄³

—

#336

—

C₂×C₃₁

#382

yes

characteristic

C₂×C₃₁

C₂×C₄⋊C₈

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂×C₃₁

#432

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

—

C₃₁

#1831

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

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

Dic₆₂

yes

—

C₄²

C₂²⋊D₃₁

yes

—

C₂³

C₂×C₄×D₃₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₈×D₃₁

yes

—

105

C₂×C₄

#92469

yes

—

106

C₂×C₄

Dic₃₁⋊C₄

yes

—

107

C₂×C₄

Dic₃₁⋊C₄

yes

—

108

C₂×C₄

C₂×Dic₆₂

yes

characteristic

109

C₂×C₄

#92497

yes

—

110

C₂×C₄

#92499

yes

—

111

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

112

C₈

—

113

C₈

—

124

—

114

C₂²

#11758469

yes

—

115

C₂²

C₂×C₈×D₃₁

yes

characteristic

116

C₂²

C₂×#92469

yes

characteristic

117

C₂²

M₁₆×D₃₁

yes

—

118

C₂²

C₂×Dic₃₁⋊C₄

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

Dic₃₁⋊C₈

yes

—

123

C₄

#11758467

yes

—

124

C₄

#11758513

yes

characteristic

125

C₄

#11758524

yes

characteristic

126

C₂

#14520914

yes

characteristic

127

C₂

C₂×Dic₃₁⋊C₈

yes

characteristic

128

C₂

#14521072

yes

characteristic

129

C₂

#14521101

yes

—

130

C₂

#14521105

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃₁
commutator	C₂²×C₃₁	C₂²×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	#1831
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₃₁

commutator

C₂²×C₃₁

C₂²×C₈

Fitting

C₂×C₃₁×C₄⋊C₈

C₂

socle, prime radical

C₂³×C₃₁

C₂²×C₄

cosocle

C₂²×C₄×C₃₁

C₂³

hypercenter

C₂×C₄⋊C₈

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
27. C₄×Dic₃₁	C₈	semidirect
79. C₃₁	#1831	semidirect

H ≅ G/N

type

27. C₄×Dic₃₁

C₈

semidirect

79. C₃₁

#1831

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
C₄²×Dic₃₁	2
#14520803	2
C₂×C₃₁×C₄⋊C₈	2
#1831	31

subgroup

index

C₂×C₄⋊(C₃₁⋊C₈)

C₄²×Dic₃₁

#14520803

C₂×C₃₁×C₄⋊C₈

#1831

Subgroups — #16845827

Conjugacy classes of subgroups

Complements

Maximal subgroups