Group #16844335 — subgroups

Subgroups — #16844335

Order 3968.

248 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 77 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	#14521106	C₂	1	2	yes	maximal, complemented
2	#14520789	C₂	1	4	yes	maximal, complemented
3	C₂×C₃₁×#124	C₂	1	1	yes	maximal, characteristic
4	C₂²×Q₈×C₃₁	C₄	1	1	yes	characteristic
5	C₂²×D₄×C₃₁	C₄	1	1	yes	characteristic
6	C₃₁×#124	—	2	2	no	—
7	C₃₁×#124	C₄	1	2	yes	—
8	C₃₁×#124	C₂²	1	2	yes	—
9	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
10	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²⋊(C₃₁⋊C₈)	—	2	2	no	—
12	C₄⋊(C₃₁⋊C₈)	—	2	2	no	—
13	#11758391	C₂²	1	4	yes	complemented
14	C₂⁴×C₃₁	—	2	1	no	—
15	C₂×Q₈×C₃₁	—	2	1	no	—
16	C₂×Q₈×C₃₁	—	4	2	no	—
17	C₂×Q₈×C₃₁	C₂×C₄	1	2	yes	—
18	C₂×D₄×C₃₁	—	2	1	no	—
19	C₂×D₄×C₃₁	—	4	2	no	—
20	C₂×D₄×C₃₁	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₃₁	—	2	1	no	—
22	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₃₁	D₄	1	2	yes	—
24	C₂²⋊C₄×C₃₁	—	2	4	no	—
25	C₂²⋊C₄×C₃₁	—	2	4	no	—
26	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₃₁	D₄	1	2	yes	—
28	C₄²×C₃₁	C₂³	1	1	yes	characteristic
29	C₂×C₃₁⋊C₈	—	2	4	no	—
30	C₂³×C₃₁	—	2	2	no	—
31	C₂³×C₃₁	—	4	4	no	—
32	C₂³×C₃₁	—	4	2	no	—
33	C₂³×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
34	Q₈×C₃₁	—	4	2	no	—
35	Q₈×C₃₁	—	4	2	no	—
36	D₄×C₃₁	—	4	2	no	—
37	D₄×C₃₁	—	4	2	no	—
38	C₂×C₄×C₃₁	—	2	2	no	—
39	C₂×C₄×C₃₁	—	2	2	no	—
40	C₂×C₄×C₃₁	—	4	2	no	—
41	C₂×C₄×C₃₁	—	4	2	no	—
42	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
46	C₃₁⋊C₈	—	4	4	no	—
47	#1533	—	31	1	no	maximal, Sylow
48	C₂²×C₃₁	—	2	2	no	—
49	C₂²×C₃₁	—	4	2	no	—
50	C₂²×C₃₁	—	4	4	no	—
51	C₂²×C₃₁	—	8	4	no	—
52	C₂²×C₃₁	#100	1	2	yes	—
53	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₄×C₃₁	—	2	1	no	—
55	C₄×C₃₁	—	2	1	no	—
56	C₄×C₃₁	—	2	2	no	—
57	C₄×C₃₁	—	2	2	no	—
58	C₄×C₃₁	—	4	2	no	—
59	C₂×#124	D₃₁	1	1	yes	characteristic
60	#432	—	31	2	no	—
61	#329	—	31	4	no	—
62	C₂×C₃₁	—	2	2	no	—
63	C₂×C₃₁	—	8	2	no	—
64	C₂×C₃₁	C₂×#100	1	1	yes	characteristic
65	C₂×C₃₁	#421	1	2	yes	—
66	C₂²×Q₈	Dic₃₁	1	1	yes	characteristic
67	C₂²×D₄	Dic₃₁	1	1	yes	characteristic
68	#124	—	2	2	no	—
69	#124	Dic₃₁	1	2	yes	—
70	#124	C₂×D₃₁	1	2	yes	—
71	C₂×C₂²⋊C₄	—	2	2	no	—
72	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
73	C₄⋊C₈	—	62	2	no	—
74	C₂²⋊C₈	—	62	2	no	—
75	#97	—	31	4	no	—
76	C₃₁	#1533	1	1	yes	characteristic, complemented, Sylow
77	C₂⁴	—	2	1	no	—
78	C₂×Q₈	—	2	1	no	—
79	C₂×Q₈	—	4	2	no	—
80	C₂×Q₈	C₂×Dic₃₁	1	2	yes	—
81	C₂×D₄	—	2	1	no	—
82	C₂×D₄	—	4	2	no	—
83	C₂×D₄	C₂×Dic₃₁	1	2	yes	—
84	C₂²×C₄	—	2	1	no	—
85	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
87	C₂×C₈	—	62	4	no	—
88	C₂²⋊C₄	—	2	4	no	—
89	C₂²⋊C₄	—	2	4	no	—
90	C₄²	C₂×Dic₃₁	1	1	yes	characteristic
91	C₄²	C₂²⋊D₃₁	1	2	yes	—
92	C₄²	C₂²×D₃₁	1	1	yes	characteristic
93	C₂³	—	2	2	no	—
94	C₂³	—	4	4	no	—
95	C₂³	—	4	2	no	—
96	C₂³	C₂²⋊Dic₃₁	1	1	yes	characteristic
97	Q₈	—	4	2	no	—
98	Q₈	—	4	2	no	—
99	D₄	—	4	2	no	—
100	D₄	—	4	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	C₂²⋊Dic₃₁	1	2	yes	—
106	C₂×C₄	C₂²⋊Dic₃₁	1	1	yes	characteristic
107	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
108	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
109	C₈	—	124	4	no	—
110	C₂²	—	2	2	no	—
111	C₂²	—	4	2	no	—
112	C₂²	—	4	4	no	—
113	C₂²	—	8	4	no	—
114	C₂²	#11758421	1	2	yes	—
115	C₂²	C₂×C₂²⋊Dic₃₁	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	4	2	no	—
121	C₂	—	2	2	no	—
122	C₂	—	8	2	no	—
123	C₂	C₂×#11758421	1	1	yes	characteristic
124	C₂	#14521066	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521106

C₂

yes

maximal, complemented

#14520789

C₂

yes

maximal, complemented

C₂×C₃₁×#124

C₂

yes

maximal, characteristic

C₂²×Q₈×C₃₁

C₄

yes

characteristic

C₂²×D₄×C₃₁

C₄

yes

characteristic

C₃₁×#124

—

C₃₁×#124

C₄

yes

—

C₃₁×#124

C₂²

yes

—

C₂×C₂²⋊C₄×C₃₁

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂²⋊(C₃₁⋊C₈)

—

C₄⋊(C₃₁⋊C₈)

—

#11758391

C₂²

yes

complemented

C₂⁴×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

C₂×C₄

yes

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

D₄

yes

—

C₂²⋊C₄×C₃₁

—

C₂²⋊C₄×C₃₁

—

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

D₄

yes

—

C₄²×C₃₁

C₂³

yes

characteristic

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

C₂²⋊C₄

yes

characteristic

Q₈×C₃₁

—

Q₈×C₃₁

—

D₄×C₃₁

—

D₄×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

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₃₁⋊C₈

—

#1533

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#100

yes

—

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₂×#124

D₃₁

yes

characteristic

#432

—

#329

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

C₂×#100

yes

characteristic

C₂×C₃₁

#421

yes

—

C₂²×Q₈

Dic₃₁

yes

characteristic

C₂²×D₄

Dic₃₁

yes

characteristic

#124

—

#124

Dic₃₁

yes

—

#124

C₂×D₃₁

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₃₁

#1533

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₃₁

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₃₁

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₃₁

yes

characteristic

C₂²×C₄

C₂²⋊D₃₁

yes

—

C₂×C₈

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×Dic₃₁

yes

characteristic

C₄²

C₂²⋊D₃₁

yes

—

C₄²

C₂²×D₃₁

yes

characteristic

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊Dic₃₁

yes

characteristic

Q₈

—

Q₈

—

D₄

—

100

D₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²⋊Dic₃₁

yes

—

106

C₂×C₄

C₂²⋊Dic₃₁

yes

characteristic

107

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

108

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

109

C₈

—

124

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

#11758421

yes

—

115

C₂²

C₂×C₂²⋊Dic₃₁

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₂

—

122

C₂

—

123

C₂

C₂×#11758421

yes

characteristic

124

C₂

#14521066

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₃₁
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₄²	C₂²×D₃₁
Fitting	C₂×C₃₁×#124	C₂
socle	C₂²×C₃₁	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₃₁	C₄
cosocle	C₄²×C₃₁	C₂³
hypercenter	C₂×#124	D₃₁
hypocenter	C₃₁	#1533
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#124

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₃₁

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₄²

C₂²×D₃₁

Fitting

C₂×C₃₁×#124

C₂

socle

C₂²×C₃₁

C₂×C₂²⋊C₄

prime radical

C₂²×D₄×C₃₁

C₄

cosocle

C₄²×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#124

N	H ≅ G/N	type
1. #14521106	C₂	semidirect
2. #14520789	C₂	semidirect
13. #11758391	C₂²	semidirect
76. C₃₁	#1533	semidirect

H ≅ G/N

type

1. #14521106

C₂

semidirect

2. #14520789

C₂

semidirect

13. #11758391

C₂²

semidirect

76. C₃₁

#1533

semidirect

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

subgroup	index
#14521106	2
#14520789	2
C₂×C₃₁×#124	2
#1533	31

subgroup

index

#14521106

#14520789

C₂×C₃₁×#124

#1533

Subgroups — #16844335

Conjugacy classes of subgroups

Complements

Maximal subgroups