Group #16844446 — subgroups

Subgroups — #16844446

Order 3968.

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

normal subgroups only

This group has 254 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	#14521073	C₂	1	2	yes	maximal, complemented
2	#14520798	C₂	1	2	yes	maximal, complemented
3	#14520793	C₂	1	2	yes	maximal, complemented
4	C₃₁×#532	C₂	1	1	yes	maximal, characteristic
5	C₂×#41×C₃₁	C₄	1	2	yes	—
6	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₃₁×#127	C₂²	1	1	yes	characteristic
8	C₃₁×#124	C₄	1	2	yes	—
9	C₃₁×#121	—	2	2	no	—
10	C₄×D₄×C₃₁	—	2	2	no	—
11	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×#92474	—	2	2	no	—
13	C₄⋊(C₃₁⋊C₈)	—	2	2	no	—
14	C₄⋊(C₃₁⋊C₈)	C₂²	1	4	yes	—
15	#41×C₃₁	—	4	4	no	—
16	C₂×Q₈×C₃₁	C₂×C₄	1	2	yes	—
17	C₂×D₄×C₃₁	—	2	2	no	—
18	C₂×D₄×C₃₁	—	2	2	no	—
19	C₂×D₄×C₃₁	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₃₁	—	2	2	no	—
21	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₃₁	D₄	1	2	yes	—
23	C₄⋊C₄×C₃₁	—	2	2	no	—
24	C₂²⋊C₄×C₃₁	—	2	4	no	—
25	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₃₁	D₄	1	2	yes	—
27	C₄²×C₃₁	C₂³	1	1	yes	characteristic
28	#92474	—	4	4	no	—
29	C₂×C₃₁⋊C₈	—	2	4	no	—
30	C₂³×C₃₁	—	2	2	no	—
31	C₂³×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
32	Q₈×C₃₁	—	4	2	no	—
33	D₄×C₃₁	—	4	2	no	—
34	D₄×C₃₁	—	4	4	no	—
35	D₄×C₃₁	—	4	2	no	—
36	D₄×C₃₁	—	4	2	no	—
37	C₂×C₄×C₃₁	—	2	2	no	—
38	C₂×C₄×C₃₁	—	2	2	no	—
39	C₂×C₄×C₃₁	—	2	2	no	—
40	C₂×C₄×C₃₁	—	2	2	no	—
41	C₂×C₄×C₃₁	—	4	4	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	#1560	—	31	1	no	maximal, Sylow
48	C₂²×C₃₁	—	2	1	no	—
49	C₂²×C₃₁	—	2	2	no	—
50	C₂²×C₃₁	—	4	2	no	—
51	C₂²×C₃₁	—	4	4	no	—
52	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₃₁	—	2	1	no	—
54	C₄×C₃₁	—	2	2	no	—
55	C₄×C₃₁	—	4	2	no	—
56	C₄×C₃₁	#100	1	2	yes	—
57	C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
58	C₄×C₃₁	#137	1	2	yes	—
59	#532	D₃₁	1	1	yes	characteristic
60	#423	—	31	2	no	—
61	#333	—	31	2	no	—
62	#331	—	31	2	no	—
63	C₂×C₃₁	—	4	1	no	—
64	C₂×C₃₁	—	8	2	no	—
65	C₂×C₃₁	C₂×#100	1	1	yes	characteristic
66	C₂×C₃₁	#417	1	2	yes	—
67	C₂×#41	Dic₃₁	1	2	yes	—
68	C₂×M₁₆	—	62	2	no	—
69	C₄⋊Q₈	C₂×D₃₁	1	1	yes	characteristic
70	#127	C₂×D₃₁	1	1	yes	characteristic
71	#124	Dic₃₁	1	2	yes	—
72	#121	—	2	2	no	—
73	C₄×D₄	—	2	2	no	—
74	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
75	C₄⋊C₈	—	31	4	no	—
76	C₄⋊C₈	—	62	2	no	—
77	C₃₁	#1560	1	1	yes	characteristic, complemented, Sylow
78	#41	—	4	4	no	—
79	C₂×Q₈	C₂×Dic₃₁	1	2	yes	—
80	C₂×D₄	—	2	2	no	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	C₂×Dic₃₁	1	2	yes	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
85	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
86	M₁₆	—	124	4	no	—
87	C₂×C₈	—	62	4	no	—
88	C₄⋊C₄	—	2	2	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	C₂²⋊Dic₃₁	1	1	yes	characteristic
95	Q₈	—	4	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	4	no	—
98	D₄	—	4	2	no	—
99	D₄	—	4	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	4	4	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	1	no	—
111	C₂²	—	2	2	no	—
112	C₂²	—	4	2	no	—
113	C₂²	—	4	4	no	—
114	C₂²	C₂×C₂²⋊Dic₃₁	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	2	no	—
117	C₄	—	4	2	no	—
118	C₄	#11758421	1	2	yes	—
119	C₄	#11758526	1	2	yes	—
120	C₄	#11758528	1	2	yes	—
121	C₂	—	4	1	no	—
122	C₂	—	8	2	no	—
123	C₂	C₂×#11758421	1	1	yes	characteristic
124	C₂	#14521050	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521073

C₂

yes

maximal, complemented

#14520798

C₂

yes

maximal, complemented

#14520793

C₂

yes

maximal, complemented

C₃₁×#532

C₂

yes

maximal, characteristic

C₂×#41×C₃₁

C₄

yes

—

C₃₁×C₄⋊Q₈

C₂²

yes

characteristic

C₃₁×#127

C₂²

yes

characteristic

C₃₁×#124

C₄

yes

—

C₃₁×#121

—

C₄×D₄×C₃₁

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×#92474

—

C₄⋊(C₃₁⋊C₈)

—

C₄⋊(C₃₁⋊C₈)

C₂²

yes

—

#41×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

—

C₄⋊C₄×C₃₁

—

C₂²⋊C₄×C₃₁

—

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

D₄

yes

—

C₄²×C₃₁

C₂³

yes

characteristic

#92474

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

—

C₂³×C₃₁

C₂²⋊C₄

yes

characteristic

Q₈×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

C₂×C₄×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

—

#1560

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#100

yes

—

C₄×C₃₁

AGL(1,ℤ/8)

yes

—

C₄×C₃₁

#137

yes

—

#532

D₃₁

yes

—

—

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

C₂×#100

yes

characteristic

C₂×C₃₁

#417

yes

—

C₂×#41

Dic₃₁

yes

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₃₁

yes

characteristic

#127

C₂×D₃₁

yes

characteristic

#124

Dic₃₁

yes

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₃₁

#1560

yes

characteristic, complemented, Sylow

#41

—

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

—

M₁₆

—

124

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×Dic₃₁

yes

characteristic

C₄²

C₂²⋊D₃₁

yes

—

C₄²

C₂²×D₃₁

yes

characteristic

C₂³

—

C₂³

C₂²⋊Dic₃₁

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

100

C₂×C₄

—

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

C₂×C₂²⋊Dic₃₁

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

#11758421

yes

—

119

C₄

#11758526

yes

—

120

C₄

#11758528

yes

—

121

C₂

—

122

C₂

—

123

C₂

C₂×#11758421

yes

characteristic

124

C₂

#14521050

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₃₁
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₄²	C₂²×D₃₁
Fitting, prime radical	C₃₁×#532	C₂
socle	C₂²×C₃₁	C₂×C₂²⋊C₄
cosocle	C₄²×C₃₁	C₂³
hypercenter	#532	D₃₁
hypocenter	C₃₁	#1560
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₃₁

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₄²

C₂²×D₃₁

Fitting, prime radical

C₃₁×#532

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 ◁ #532

N	H ≅ G/N	type
1. #14521073	C₂	semidirect
2. #14520798	C₂	semidirect
3. #14520793	C₂	semidirect
77. C₃₁	#1560	semidirect

H ≅ G/N

type

1. #14521073

C₂

semidirect

2. #14520798

C₂

semidirect

3. #14520793

C₂

semidirect

77. C₃₁

#1560

semidirect

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

subgroup	index
#14521073	2
#14520798	2
#14520793	2
C₃₁×#532	2
#1560	31

subgroup

index

C₃₁×#532

Subgroups — #16844446

Conjugacy classes of subgroups

Complements

Maximal subgroups