Group #16844482 — subgroups

Subgroups — #16844482

Order 3968.

168 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 97 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	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#14520800	C₂	1	4	yes	maximal, complemented
4	C₃₁×#431	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
6	C₂×#92474	—	2	1	no	—
7	C₂²×C₃₁⋊C₈	—	2	1	no	—
8	C₃₁×C₄⋊C₈	C₄	1	2	yes	—
9	C₃₁×C₂²⋊C₈	C₄	1	2	yes	—
10	C₄×C₈×C₃₁	C₂²	1	2	yes	—
11	C₄⋊(C₃₁⋊C₈)	—	2	1	no	—
12	C₄⋊(C₃₁⋊C₈)	—	2	1	no	—
13	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
14	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
15	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₃₁	Q₈	1	1	yes	characteristic
18	C₂×C₈×C₃₁	C₂×C₄	1	4	yes	—
19	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₃₁	D₄	1	1	yes	characteristic
21	C₄²×C₃₁	Q₈	1	1	yes	characteristic
22	C₄²×C₃₁	C₂³	1	1	yes	characteristic
23	#92474	—	4	2	no	—
24	C₂×C₃₁⋊C₈	—	2	2	no	—
25	C₂×C₃₁⋊C₈	—	2	2	no	—
26	C₂×C₃₁⋊C₈	—	4	2	no	—
27	C₂³×C₃₁	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₃₁	—	2	2	no	—
29	C₂×C₄×C₃₁	—	4	1	no	—
30	C₂×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₃₁	SD₁₆	1	2	yes	—
34	C₂×C₄×C₃₁	SD₁₆	1	2	yes	—
35	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
38	C₈×C₃₁	—	2	4	no	—
39	C₃₁⋊C₈	—	4	2	no	—
40	C₃₁⋊C₈	—	4	2	no	—
41	#1570	—	31	1	no	maximal, Sylow
42	C₂²×C₃₁	—	2	1	no	—
43	C₂²×C₃₁	—	2	1	no	—
44	C₂²×C₃₁	#106	1	2	yes	—
45	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₄×C₃₁	—	2	1	no	—
47	C₄×C₃₁	—	2	1	no	—
48	C₄×C₃₁	—	2	1	no	—
49	C₄×C₃₁	#106	1	2	yes	—
50	C₄×C₃₁	#131	1	2	yes	—
51	C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
52	C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
53	#431	D₃₁	1	1	yes	characteristic
54	#423	—	31	1	no	—
55	C₂×C₄⋊C₈	—	31	1	no	—
56	#334	—	31	4	no	—
57	C₂×C₃₁	—	2	2	no	—
58	C₂×C₃₁	#424	1	1	yes	characteristic
59	C₂×C₃₁	C₂×#106	1	1	yes	characteristic
60	C₂×C₃₁	#430	1	1	yes	characteristic
61	C₂×M₁₆	—	62	1	no	—
62	C₂²×C₈	—	62	1	no	—
63	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
64	C₄⋊C₈	—	31	2	no	—
65	C₄⋊C₈	—	31	2	no	—
66	C₄⋊C₈	—	62	1	no	—
67	C₄⋊C₈	—	62	1	no	—
68	C₄⋊C₈	Dic₃₁	1	2	yes	—
69	C₂²⋊C₈	Dic₃₁	1	2	yes	—
70	C₄×C₈	C₂×D₃₁	1	2	yes	—
71	C₃₁	#1570	1	1	yes	characteristic, complemented, Sylow
72	C₂²×C₄	Dic₆₂	1	1	yes	characteristic
73	C₂²×C₄	D₁₂₄	1	1	yes	characteristic
74	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
75	M₁₆	—	124	2	no	—
76	C₂×C₈	—	62	2	no	—
77	C₂×C₈	—	62	2	no	—
78	C₂×C₈	—	124	2	no	—
79	C₂×C₈	C₂×Dic₃₁	1	4	yes	—
80	C₄²	Dic₆₂	1	1	yes	characteristic
81	C₄²	D₁₂₄	1	1	yes	characteristic
82	C₄²	C₂×Dic₃₁	1	1	yes	characteristic
83	C₄²	C₂²×D₃₁	1	1	yes	characteristic
84	C₂³	C₄⋊Dic₃₁	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	#92470	1	2	yes	—
88	C₂×C₄	#92470	1	2	yes	—
89	C₂×C₄	C₄⋊Dic₃₁	1	1	yes	characteristic
90	C₂×C₄	C₄⋊Dic₃₁	1	1	yes	characteristic
91	C₂×C₄	C₄⋊Dic₃₁	1	1	yes	characteristic
92	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
93	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
94	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
95	C₈	—	2	4	no	—
96	C₈	—	124	2	no	—
97	C₈	—	124	2	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	#11758405	1	2	yes	—
101	C₂²	C₂×C₄⋊Dic₃₁	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	#11758405	1	2	yes	—
106	C₄	C₂×#92470	1	1	yes	characteristic
107	C₄	C₂×#92470	1	1	yes	characteristic
108	C₄	#11758525	1	2	yes	—
109	C₂	—	2	2	no	—
110	C₂	#14521076	1	1	yes	characteristic
111	C₂	C₂×#11758405	1	1	yes	characteristic
112	C₂	#14521098	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521073

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₃₁⋊C₈)

C₂

yes

maximal, characteristic

#14520800

C₂

yes

maximal, complemented

C₃₁×#431

C₂

yes

maximal, characteristic

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×#92474

—

C₂²×C₃₁⋊C₈

—

C₃₁×C₄⋊C₈

C₄

yes

—

C₃₁×C₂²⋊C₈

C₄

yes

—

C₄×C₈×C₃₁

C₂²

yes

—

C₄⋊(C₃₁⋊C₈)

—

C₄⋊(C₃₁⋊C₈)

—

C₄⋊(C₃₁⋊C₈)

C₂²

yes

—

C₄⋊(C₃₁⋊C₈)

C₂²

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

Q₈

yes

characteristic

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

D₄

yes

characteristic

C₄²×C₃₁

Q₈

yes

characteristic

C₄²×C₃₁

C₂³

yes

characteristic

#92474

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₁

SD₁₆

yes

—

C₂×C₄×C₃₁

SD₁₆

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₈×C₃₁

—

C₃₁⋊C₈

—

C₃₁⋊C₈

—

#1570

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#106

yes

—

C₂²×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#106

yes

—

C₄×C₃₁

#131

yes

—

C₄×C₃₁

C₂×SD₁₆

yes

characteristic

C₄×C₃₁

C₂×SD₁₆

yes

characteristic

#431

D₃₁

yes

characteristic

#423

—

C₂×C₄⋊C₈

—

#334

—

C₂×C₃₁

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

Dic₃₁

yes

—

C₂²⋊C₈

Dic₃₁

yes

—

C₄×C₈

C₂×D₃₁

yes

—

C₃₁

#1570

yes

characteristic, complemented, Sylow

C₂²×C₄

Dic₆₂

yes

characteristic

C₂²×C₄

D₁₂₄

yes

characteristic

C₂²×C₄

C₂×Dic₃₁

yes

characteristic

M₁₆

—

124

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

124

—

C₂×C₈

C₂×Dic₃₁

yes

—

C₄²

Dic₆₂

yes

characteristic

C₄²

D₁₂₄

yes

characteristic

C₄²

C₂×Dic₃₁

yes

characteristic

C₄²

C₂²×D₃₁

yes

characteristic

C₂³

C₄⋊Dic₃₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#92470

yes

—

C₂×C₄

#92470

yes

—

C₂×C₄

C₄⋊Dic₃₁

yes

characteristic

C₂×C₄

C₄⋊Dic₃₁

yes

characteristic

C₂×C₄

C₄⋊Dic₃₁

yes

characteristic

C₂×C₄

C₂×Dic₆₂

yes

characteristic

C₂×C₄

C₂×D₁₂₄

yes

characteristic

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

C₈

—

C₈

—

124

—

C₈

—

124

—

C₂²

—

C₂²

—

100

C₂²

#11758405

yes

—

101

C₂²

C₂×C₄⋊Dic₃₁

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#11758405

yes

—

106

C₄

C₂×#92470

yes

characteristic

107

C₄

C₂×#92470

yes

characteristic

108

C₄

#11758525

yes

—

109

C₂

—

110

C₂

#14521076

yes

characteristic

111

C₂

C₂×#11758405

yes

characteristic

112

C₂

#14521098

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₂₄
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₄²	C₂²×D₃₁
Fitting	C₃₁×#431	C₂
socle	C₂²×C₃₁	C₂×C₄⋊C₄
prime radical	C₂³×C₃₁	C₄⋊C₄
cosocle	C₄²×C₃₁	C₂³
hypercenter	#431	D₃₁
hypocenter	C₃₁	#1570
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₁₂₄

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₄²

C₂²×D₃₁

Fitting

C₃₁×#431

C₂

socle

C₂²×C₃₁

C₂×C₄⋊C₄

prime radical

C₂³×C₃₁

C₄⋊C₄

cosocle

C₄²×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #14520800	C₂	semidirect
71. C₃₁	#1570	semidirect

H ≅ G/N

type

3. #14520800

C₂

semidirect

71. C₃₁

#1570

semidirect

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

subgroup	index
#14521073	2
C₂×C₄⋊(C₃₁⋊C₈)	2
#14520800	2
C₃₁×#431	2
#1570	31

subgroup

index

#14521073

C₂×C₄⋊(C₃₁⋊C₈)

#14520800

C₃₁×#431

#1570

Subgroups — #16844482

Conjugacy classes of subgroups

Complements

Maximal subgroups