Group #16844425 — subgroups

Subgroups — #16844425

Order 3968.

332 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 85 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	#14520793	C₂	1	4	yes	maximal, complemented
3	C₂×C₃₁×#127	C₂	1	1	yes	maximal, characteristic
4	C₂²×D₄×C₃₁	—	2	2	no	—
5	C₂²×D₄×C₃₁	C₄	1	2	yes	—
6	C₃₁×#127	—	2	2	no	—
7	C₃₁×#127	C₄	1	2	yes	—
8	C₃₁×#127	C₂²	1	2	yes	—
9	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²⋊(C₃₁⋊C₈)	—	2	2	no	—
11	C₄⋊(C₃₁⋊C₈)	C₂²	1	4	yes	complemented
12	#11758391	—	2	2	no	—
13	C₂⁴×C₃₁	—	2	2	no	—
14	C₂×D₄×C₃₁	—	2	2	no	—
15	C₂×D₄×C₃₁	—	2	4	no	—
16	C₂×D₄×C₃₁	—	2	4	no	—
17	C₂×D₄×C₃₁	—	4	8	no	—
18	C₂×D₄×C₃₁	—	4	4	no	—
19	C₂×D₄×C₃₁	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₃₁	D₄	1	2	yes	—
22	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₃₁	D₄	1	2	yes	—
24	C₄²×C₃₁	C₂³	1	1	yes	characteristic
25	C₂×C₃₁⋊C₈	—	2	4	no	—
26	C₂³×C₃₁	—	2	4	no	—
27	C₂³×C₃₁	—	4	8	no	—
28	C₂³×C₃₁	—	4	4	no	—
29	C₂³×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
30	D₄×C₃₁	—	4	4	no	—
31	D₄×C₃₁	—	4	4	no	—
32	D₄×C₃₁	—	4	8	no	—
33	D₄×C₃₁	—	4	8	no	—
34	C₂×C₄×C₃₁	—	2	2	no	—
35	C₂×C₄×C₃₁	—	2	4	no	—
36	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
40	C₃₁⋊C₈	—	4	4	no	—
41	#1555	—	31	1	no	maximal, Sylow
42	C₂²×C₃₁	—	2	2	no	—
43	C₂²×C₃₁	—	4	4	no	—
44	C₂²×C₃₁	—	4	8	no	—
45	C₂²×C₃₁	—	8	8	no	—
46	C₂²×C₃₁	#100	1	2	yes	—
47	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₄×C₃₁	—	2	1	no	—
49	C₄×C₃₁	—	2	1	no	—
50	C₄×C₃₁	—	2	2	no	—
51	C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
52	C₂×#127	D₃₁	1	1	yes	characteristic
53	#432	—	31	2	no	—
54	#331	—	31	4	no	—
55	C₂×C₃₁	—	2	2	no	—
56	C₂×C₃₁	—	8	4	no	—
57	C₂×C₃₁	C₂×#100	1	1	yes	characteristic
58	C₂×C₃₁	#418	1	2	yes	—
59	C₂²×D₄	—	2	2	no	—
60	C₂²×D₄	Dic₃₁	1	2	yes	—
61	#127	—	2	2	no	—
62	#127	Dic₃₁	1	2	yes	—
63	#127	C₂×D₃₁	1	2	yes	—
64	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
65	C₄⋊C₈	—	31	4	no	—
66	C₂²⋊C₈	—	62	2	no	—
67	#97	—	62	2	no	—
68	C₃₁	#1555	1	1	yes	characteristic, complemented, Sylow
69	C₂⁴	—	2	2	no	—
70	C₂×D₄	—	2	2	no	—
71	C₂×D₄	—	2	4	no	—
72	C₂×D₄	—	2	4	no	—
73	C₂×D₄	—	4	4	no	—
74	C₂×D₄	—	4	8	no	—
75	C₂×D₄	C₂×Dic₃₁	1	4	yes	—
76	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
77	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
78	C₂×C₈	—	62	4	no	—
79	C₄²	C₂×Dic₃₁	1	1	yes	characteristic
80	C₄²	C₂²⋊D₃₁	1	2	yes	—
81	C₄²	C₂²×D₃₁	1	1	yes	characteristic
82	C₂³	—	2	4	no	—
83	C₂³	—	4	8	no	—
84	C₂³	—	4	4	no	—
85	C₂³	C₂²⋊Dic₃₁	1	1	yes	characteristic
86	D₄	—	4	4	no	—
87	D₄	—	4	4	no	—
88	D₄	—	4	8	no	—
89	D₄	—	4	8	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	C₂²⋊Dic₃₁	1	2	yes	—
93	C₂×C₄	C₂²⋊Dic₃₁	1	1	yes	characteristic
94	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
95	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
96	C₈	—	124	4	no	—
97	C₂²	—	2	2	no	—
98	C₂²	—	4	4	no	—
99	C₂²	—	4	8	no	—
100	C₂²	—	8	8	no	—
101	C₂²	#11758421	1	2	yes	—
102	C₂²	C₂×C₂²⋊Dic₃₁	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	—	2	2	no	—
106	C₄	#11758509	1	4	yes	—
107	C₂	—	2	2	no	—
108	C₂	—	8	4	no	—
109	C₂	C₂×#11758421	1	1	yes	characteristic
110	C₂	#14521054	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521106

C₂

yes

maximal, complemented

#14520793

C₂

yes

maximal, complemented

C₂×C₃₁×#127

C₂

yes

maximal, characteristic

C₂²×D₄×C₃₁

—

C₂²×D₄×C₃₁

C₄

yes

—

C₃₁×#127

—

C₃₁×#127

C₄

yes

—

C₃₁×#127

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂²⋊(C₃₁⋊C₈)

—

C₄⋊(C₃₁⋊C₈)

C₂²

yes

complemented

#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

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

D₄

yes

—

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

D₄×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

D₄×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

—

#1555

—

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

AGL(1,ℤ/8)

yes

—

C₂×#127

D₃₁

yes

characteristic

#432

—

#331

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

C₂×#100

yes

characteristic

C₂×C₃₁

#418

yes

—

C₂²×D₄

—

C₂²×D₄

Dic₃₁

yes

—

#127

—

#127

Dic₃₁

yes

—

#127

C₂×D₃₁

yes

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₃₁

#1555

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂×Dic₃₁

yes

characteristic

C₂²×C₄

C₂²⋊D₃₁

yes

—

C₂×C₈

—

C₄²

C₂×Dic₃₁

yes

characteristic

C₄²

C₂²⋊D₃₁

yes

—

C₄²

C₂²×D₃₁

yes

characteristic

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊Dic₃₁

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₃₁

yes

—

C₂×C₄

C₂²⋊Dic₃₁

yes

characteristic

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

C₈

—

124

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

#11758421

yes

—

102

C₂²

C₂×C₂²⋊Dic₃₁

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

#11758509

yes

—

107

C₂

—

108

C₂

—

109

C₂

C₂×#11758421

yes

characteristic

110

C₂

#14521054

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₃₁
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₄²	C₂²×D₃₁
Fitting, prime radical	C₂×C₃₁×#127	C₂
socle	C₂²×C₃₁	C₂×C₂²⋊C₄
cosocle	C₄²×C₃₁	C₂³
hypercenter	C₂×#127	D₃₁
hypocenter	C₃₁	#1555
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₃₁

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₄²

C₂²×D₃₁

Fitting, prime radical

C₂×C₃₁×#127

C₂

socle

C₂²×C₃₁

C₂×C₂²⋊C₄

cosocle

C₄²×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. #14521106	C₂	semidirect
2. #14520793	C₂	semidirect
11. C₄⋊(C₃₁⋊C₈)	C₂²	semidirect
68. C₃₁	#1555	semidirect

H ≅ G/N

type

1. #14521106

C₂

semidirect

2. #14520793

C₂

semidirect

11. C₄⋊(C₃₁⋊C₈)

C₂²

semidirect

68. C₃₁

#1555

semidirect

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

subgroup	index
#14521106	2
#14520793	2
C₂×C₃₁×#127	2
#1555	31

subgroup

index

#14521106

#14520793

C₂×C₃₁×#127

#1555

Subgroups — #16844425

Conjugacy classes of subgroups

Complements

Maximal subgroups