Group #16854817 — subgroups

Subgroups — #16854817

Order 3968.

314 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 197 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	#14521180	C₂	1	8	yes	maximal, complemented
2	#14521110	C₂	1	4	yes	maximal
3	#14521106	C₂	1	2	yes	maximal
4	C₃₁×#533	C₂	1	1	yes	maximal, characteristic
5	C₃₁×C₄⋊Q₈	C₄	1	2	yes	—
6	C₃₁×#125	C₄	1	2	yes	—
7	C₃₁×C₂²⋊Q₈	C₄	1	4	yes	—
8	C₄×Q₈×C₃₁	C₂²	1	4	yes	—
9	C₃₁×#117	C₂²	1	2	yes	—
10	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²⋊(C₃₁⋊C₈)	C₂²	1	4	yes	—
12	C₄⋊(C₃₁⋊C₈)	C₂²	1	4	yes	—
13	C₄⋊(C₃₁⋊C₈)	C₂²	1	8	yes	—
14	#11758391	C₂²	1	4	yes	—
15	#11758391	C₂²	1	4	yes	—
16	C₄×C₃₁⋊C₈	C₂²	1	4	yes	—
17	C₂×Q₈×C₃₁	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
21	C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
22	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
23	C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
24	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₃₁	C₂³	1	4	yes	—
27	C₄²×C₃₁	C₂³	1	2	yes	—
28	C₂×C₃₁⋊C₈	C₂³	1	8	yes	—
29	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
30	Q₈×C₃₁	—	4	4	no	—
31	C₂×C₄×C₃₁	—	2	2	no	—
32	C₂×C₄×C₃₁	—	4	2	no	—
33	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
35	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
36	C₂×C₄×C₃₁	C₂²×C₄	1	8	yes	—
37	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
38	C₃₁⋊C₈	—	2	8	no	—
39	#2987	—	31	1	no	maximal, Sylow
40	C₂²×C₃₁	—	2	1	no	—
41	C₂²×C₃₁	—	2	2	no	—
42	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
43	C₄×C₃₁	—	2	1	no	—
44	C₄×C₃₁	—	2	2	no	—
45	C₄×C₃₁	—	2	2	no	—
46	C₄×C₃₁	—	2	8	no	—
47	C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
48	#533	D₃₁	1	1	yes	characteristic
49	#446	—	31	8	no	—
50	#433	—	31	4	no	—
51	#432	—	31	2	no	—
52	C₂×C₃₁	—	4	1	no	—
53	C₂×C₃₁	#519	1	1	yes	characteristic
54	C₂×C₃₁	#568	1	2	yes	—
55	C₄⋊Q₈	Dic₃₁	1	2	yes	—
56	#125	Dic₃₁	1	2	yes	—
57	C₂²⋊Q₈	Dic₃₁	1	4	yes	—
58	C₄×Q₈	C₂×D₃₁	1	4	yes	—
59	#117	C₂×D₃₁	1	2	yes	—
60	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
61	C₄⋊C₈	—	31	4	no	—
62	C₄⋊C₈	—	31	8	no	—
63	C₂²⋊C₈	—	31	4	no	—
64	#97	—	31	4	no	—
65	#97	—	31	4	no	—
66	C₄×C₈	—	31	4	no	—
67	C₃₁	#2987	1	1	yes	characteristic, complemented, Sylow
68	C₂×Q₈	C₂×Dic₃₁	1	4	yes	—
69	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
70	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
71	C₂×C₈	—	31	8	no	—
72	C₄⋊C₄	C₂×Dic₃₁	1	4	yes	—
73	C₄⋊C₄	C₂×Dic₃₁	1	8	yes	—
74	C₄⋊C₄	C₂×Dic₃₁	1	4	yes	—
75	C₂²⋊C₄	C₂×Dic₃₁	1	4	yes	—
76	C₄²	C₂×Dic₃₁	1	1	yes	characteristic
77	C₄²	C₂×Dic₃₁	1	1	yes	characteristic
78	C₄²	C₂²×D₃₁	1	4	yes	—
79	C₄²	C₂²×D₃₁	1	2	yes	—
80	C₂³	C₂²×Dic₃₁	1	1	yes	characteristic
81	Q₈	—	4	4	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	4	2	no	—
84	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
85	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
86	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
87	C₂×C₄	C₂²×Dic₃₁	1	8	yes	—
88	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
89	C₈	—	62	8	no	—
90	C₂²	—	2	1	no	—
91	C₂²	—	2	2	no	—
92	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	2	no	—
95	C₄	—	2	2	no	—
96	C₄	—	2	8	no	—
97	C₄	#11758564	1	2	yes	—
98	C₂	—	4	1	no	—
99	C₂	#14521527	1	1	yes	characteristic
100	C₂	#14521816	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521180

C₂

yes

maximal, complemented

#14521110

C₂

yes

maximal

#14521106

C₂

yes

maximal

C₃₁×#533

C₂

yes

maximal, characteristic

C₃₁×C₄⋊Q₈

C₄

yes

—

C₃₁×#125

C₄

yes

—

C₃₁×C₂²⋊Q₈

C₄

yes

—

C₄×Q₈×C₃₁

C₂²

yes

—

C₃₁×#117

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂²⋊(C₃₁⋊C₈)

C₂²

yes

—

C₄⋊(C₃₁⋊C₈)

C₂²

yes

—

C₄⋊(C₃₁⋊C₈)

C₂²

yes

—

#11758391

C₂²

yes

—

#11758391

C₂²

yes

—

C₄×C₃₁⋊C₈

C₂²

yes

—

C₂×Q₈×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₄⋊C₄×C₃₁

C₂×C₄

yes

—

C₄⋊C₄×C₃₁

C₂×C₄

yes

—

C₄⋊C₄×C₃₁

C₂×C₄

yes

—

C₂²⋊C₄×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

C₂³

yes

—

C₄²×C₃₁

C₂³

yes

—

C₂×C₃₁⋊C₈

C₂³

yes

—

C₂³×C₃₁

C₂²×C₄

yes

characteristic

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₃₁⋊C₈

—

#2987

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂³×C₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

2₋¹⁺⁴

yes

—

#533

D₃₁

yes

—

—

—

C₂×C₃₁

—

C₂×C₃₁

#519

yes

characteristic

C₂×C₃₁

#568

yes

—

C₄⋊Q₈

Dic₃₁

yes

—

#125

Dic₃₁

yes

—

C₂²⋊Q₈

Dic₃₁

yes

—

C₄×Q₈

C₂×D₃₁

yes

—

#117

C₂×D₃₁

yes

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₃₁

#2987

yes

characteristic, complemented, Sylow

C₂×Q₈

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₄⋊C₄

C₂×Dic₃₁

yes

—

C₄⋊C₄

C₂×Dic₃₁

yes

—

C₄⋊C₄

C₂×Dic₃₁

yes

—

C₂²⋊C₄

C₂×Dic₃₁

yes

—

C₄²

C₂×Dic₃₁

yes

characteristic

C₄²

C₂×Dic₃₁

yes

characteristic

C₄²

C₂²×D₃₁

yes

—

C₄²

C₂²×D₃₁

yes

—

C₂³

C₂²×Dic₃₁

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

C₂×C₄

C₂²×Dic₃₁

yes

—

C₂×C₄

C₂²×Dic₃₁

yes

—

C₂×C₄

C₂²×Dic₃₁

yes

—

C₂×C₄

C₂³×D₃₁

yes

characteristic

C₈

—

C₂²

—

C₂²

—

C₂²

C₂³×Dic₃₁

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

#11758564

yes

—

C₂

—

C₂

#14521527

yes

characteristic

100

C₂

#14521816

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₃₁×#533	C₂
prime radical	C₂³×C₃₁	C₂²×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#533	D₃₁
hypocenter	C₃₁	#2987
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #533

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₃₁

commutator, socle

C₂²×C₃₁

C₂³×C₄

Fitting

C₃₁×#533

C₂

prime radical

C₂³×C₃₁

C₂²×C₄

cosocle

C₂×C₄×C₃₁

C₂⁴

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂×C₄ ◁ #533

N	H ≅ G/N	type
1. #14521180	C₂	semidirect
67. C₃₁	#2987	semidirect

H ≅ G/N

type

1. #14521180

C₂

semidirect

67. C₃₁

#2987

semidirect

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

subgroup	index
#14521180	2
#14521110	2
#14521106	2
C₃₁×#533	2
#2987	31

subgroup

index

C₃₁×#533

Subgroups — #16854817

Conjugacy classes of subgroups

Complements

Maximal subgroups