Group #16846536 — subgroups

Subgroups — #16846536

Order 3968.

412 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 113 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×#11758420	C₂	1	4	yes	maximal, complemented
2	C₂×#11758391	C₂	1	1	yes	maximal, characteristic, complemented
3	#14520915	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#127	C₂	1	1	yes	maximal, characteristic
5	C₂²×D₄×C₃₁	—	2	2	no	—
6	C₂²×D₄×C₃₁	C₂²	1	2	yes	—
7	C₃₁×#127	—	2	2	no	—
8	C₃₁×#127	C₄	1	4	yes	complemented
9	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
11	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
12	#11758420	—	2	8	no	—
13	C₃₁⋊#95	—	2	2	no	—
14	#11758391	—	2	2	no	—
15	C₂⁴×C₃₁	—	2	2	no	—
16	C₂×D₄×C₃₁	—	2	8	no	—
17	C₂×D₄×C₃₁	—	2	4	no	—
18	C₂×D₄×C₃₁	—	2	2	no	—
19	C₂×D₄×C₃₁	—	2	4	no	—
20	C₂×D₄×C₃₁	—	4	8	no	—
21	C₂×D₄×C₃₁	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₃₁	D₄	1	2	yes	—
23	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
24	C₂²×Dic₃₁	—	2	2	no	—
25	C₄²×C₃₁	C₂×C₄	1	2	yes	—
26	C₄²×C₃₁	D₄	1	2	yes	—
27	C₄⋊Dic₃₁	—	2	4	no	—
28	C₂×C₃₁⋊C₈	—	2	2	no	—
29	C₂×C₃₁⋊C₈	—	2	2	no	—
30	C₂×C₃₁⋊C₈	D₄	1	4	yes	complemented
31	C₂³×C₃₁	—	2	4	no	—
32	C₂³×C₃₁	—	4	2	no	—
33	C₂³×C₃₁	—	4	2	no	—
34	C₂³×C₃₁	—	4	8	no	—
35	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
36	D₄×C₃₁	—	2	8	no	—
37	D₄×C₃₁	—	4	16	no	—
38	D₄×C₃₁	—	4	4	no	—
39	C₂×C₄×C₃₁	—	2	4	no	—
40	C₂×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
41	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₃₁	#41	1	2	yes	—
45	C₂×Dic₃₁	—	4	2	no	—
46	C₂×Dic₃₁	—	4	4	no	—
47	C₃₁⋊C₈	—	2	4	no	—
48	#1961	—	31	1	no	maximal, Sylow
49	C₂²×C₃₁	—	4	4	no	—
50	C₂²×C₃₁	—	4	8	no	—
51	C₂²×C₃₁	—	8	4	no	—
52	C₂²×C₃₁	—	8	4	no	—
53	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₃₁	#124	1	1	yes	characteristic
55	C₂²×C₃₁	#127	1	1	yes	characteristic
56	C₂²×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
57	C₄×C₃₁	—	2	4	no	—
58	C₄×C₃₁	C₄×D₄	1	2	yes	—
59	C₄×C₃₁	#121	1	2	yes	—
60	Dic₃₁	—	8	2	no	—
61	C₂×#127	D₃₁	1	1	yes	characteristic
62	C₂×#102	—	31	4	no	—
63	C₂×#97	—	31	1	no	—
64	#382	—	31	1	no	—
65	C₂×C₃₁	—	8	4	no	—
66	C₂×C₃₁	#390	1	1	yes	characteristic
67	C₂×C₃₁	#418	1	2	yes	—
68	C₂×C₃₁	#490	1	2	yes	—
69	C₂×C₃₁	#496	1	2	yes	—
70	C₂²×D₄	—	2	2	no	—
71	C₂²×D₄	C₂×D₃₁	1	2	yes	—
72	C₂²×C₈	—	31	2	no	—
73	#127	—	2	2	no	—
74	#127	Dic₃₁	1	4	yes	complemented
75	C₂×C₄⋊C₄	—	31	2	no	—
76	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
77	#102	—	62	8	no	—
78	#97	—	62	2	no	—
79	#95	—	62	2	no	—
80	C₃₁	#1961	1	1	yes	characteristic, complemented, Sylow
81	C₂⁴	—	2	2	no	—
82	C₂×D₄	—	2	8	no	—
83	C₂×D₄	—	2	4	no	—
84	C₂×D₄	—	2	4	no	—
85	C₂×D₄	—	2	2	no	—
86	C₂×D₄	—	4	8	no	—
87	C₂×D₄	C₂×Dic₃₁	1	4	yes	—
88	C₂²×C₄	—	62	2	no	—
89	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
90	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
91	C₂×C₈	—	31	4	no	—
92	C₂×C₈	—	62	2	no	—
93	C₂×C₈	—	62	2	no	—
94	C₄⋊C₄	—	62	4	no	—
95	C₄²	C₂×Dic₃₁	1	2	yes	—
96	C₄²	C₂²⋊D₃₁	1	2	yes	—
97	C₂³	—	2	4	no	—
98	C₂³	—	4	8	no	—
99	C₂³	—	4	2	no	—
100	C₂³	—	4	2	no	—
101	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
102	D₄	—	2	8	no	—
103	D₄	—	4	16	no	—
104	D₄	—	4	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	124	4	no	—
107	C₂×C₄	—	124	2	no	—
108	C₂×C₄	C₂²⋊Dic₃₁	1	4	yes	—
109	C₂×C₄	D₄×D₃₁	1	2	yes	—
110	C₂×C₄	#92497	1	2	yes	—
111	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
113	C₈	—	62	4	no	—
114	C₂²	—	4	4	no	—
115	C₂²	—	4	8	no	—
116	C₂²	—	8	4	no	—
117	C₂²	—	8	4	no	—
118	C₂²	#11758509	1	4	yes	—
119	C₂²	#11758513	1	1	yes	characteristic
120	C₂²	#11758517	1	1	yes	characteristic
121	C₂²	C₂×C₂²⋊Dic₃₁	1	1	yes	characteristic
122	C₄	—	2	4	no	—
123	C₄	—	248	2	no	—
124	C₄	D₄×Dic₃₁	1	2	yes	—
125	C₄	#11758515	1	2	yes	—
126	C₂	—	8	4	no	—
127	C₂	#14520952	1	1	yes	characteristic
128	C₂	#14521054	1	2	yes	—
129	C₂	#14521413	1	2	yes	—
130	C₂	#14521436	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758420

C₂

yes

maximal, complemented

C₂×#11758391

C₂

yes

maximal, characteristic, complemented

#14520915

C₂

yes

maximal, characteristic, complemented

C₂×C₃₁×#127

C₂

yes

maximal, characteristic

C₂²×D₄×C₃₁

—

C₂²×D₄×C₃₁

C₂²

yes

—

C₃₁×#127

—

C₃₁×#127

C₄

yes

complemented

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂²×C₃₁⋊C₈

C₂²

yes

—

#11758420

—

C₃₁⋊#95

—

#11758391

—

C₂⁴×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

D₄

yes

—

C₄⋊Dic₃₁

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

D₄

yes

complemented

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

D₄×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₃₁⋊C₈

—

#1961

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

—

C₄×C₃₁

—

C₄×C₃₁

C₄×D₄

yes

—

C₄×C₃₁

#121

yes

—

Dic₃₁

—

C₂×#127

D₃₁

yes

characteristic

C₂×#102

—

C₂×#97

—

#382

—

C₂×C₃₁

—

C₂×C₃₁

#390

yes

characteristic

C₂×C₃₁

#418

yes

—

C₂×C₃₁

#490

yes

—

C₂×C₃₁

#496

yes

—

C₂²×D₄

—

C₂²×D₄

C₂×D₃₁

yes

—

C₂²×C₈

—

#127

—

#127

Dic₃₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₁

yes

—

—

—

C₃₁

#1961

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₃₁

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄²

C₂×Dic₃₁

yes

—

C₄²

C₂²⋊D₃₁

yes

—

C₂³

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

102

D₄

—

103

D₄

—

104

D₄

—

105

C₂×C₄

—

106

C₂×C₄

—

124

—

107

C₂×C₄

—

124

—

108

C₂×C₄

C₂²⋊Dic₃₁

yes

—

109

C₂×C₄

D₄×D₃₁

yes

—

110

C₂×C₄

#92497

yes

—

111

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

113

C₈

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

#11758509

yes

—

119

C₂²

#11758513

yes

characteristic

120

C₂²

#11758517

yes

characteristic

121

C₂²

C₂×C₂²⋊Dic₃₁

yes

characteristic

122

C₄

—

123

C₄

—

248

—

124

C₄

D₄×Dic₃₁

yes

—

125

C₄

#11758515

yes

—

126

C₂

—

127

C₂

#14520952

yes

characteristic

128

C₂

#14521054

yes

—

129

C₂

#14521413

yes

—

130

C₂

#14521436

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₁
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting, prime radical	C₂×C₃₁×#127	C₂
socle	C₂³×C₃₁	C₂×D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂×#127	D₃₁
hypocenter	C₃₁	#1961
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₁

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting, prime radical

C₂×C₃₁×#127

C₂

socle

C₂³×C₃₁

C₂×D₄

cosocle

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×#11758420	C₂	semidirect
2. C₂×#11758391	C₂	semidirect
3. #14520915	C₂	semidirect
8. C₃₁×#127	C₄	semidirect
30. C₂×C₃₁⋊C₈	D₄	semidirect
74. #127	Dic₃₁	semidirect
80. C₃₁	#1961	semidirect

H ≅ G/N

type

1. C₂×#11758420

C₂

semidirect

2. C₂×#11758391

C₂

semidirect

3. #14520915

C₂

semidirect

8. C₃₁×#127

C₄

semidirect

30. C₂×C₃₁⋊C₈

D₄

semidirect

74. #127

Dic₃₁

semidirect

80. C₃₁

#1961

semidirect

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

subgroup	index
C₂×#11758420	2
C₂×#11758391	2
#14520915	2
C₂×C₃₁×#127	2
#1961	31

subgroup

index

C₂×C₃₁×#127

Subgroups — #16846536

Conjugacy classes of subgroups

Complements

Maximal subgroups