Group #22597446 — subgroups

Subgroups — #22597446

Order 9088.

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	#17986130	C₂	1	8	yes	maximal, complemented
2	#17986060	C₂	1	4	yes	maximal
3	#17986056	C₂	1	2	yes	maximal
4	#533×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₄⋊(C₇₁⋊C₈)	C₂²	1	4	yes	—
6	C₄⋊(C₇₁⋊C₈)	C₂²	1	8	yes	—
7	C₂²⋊(C₇₁⋊C₈)	C₂²	1	4	yes	—
8	#14556039	C₂²	1	4	yes	—
9	#14556039	C₂²	1	4	yes	—
10	C₄×C₇₁⋊C₈	C₂²	1	4	yes	—
11	C₄⋊Q₈×C₇₁	C₄	1	2	yes	—
12	#125×C₇₁	C₄	1	2	yes	—
13	C₂²⋊Q₈×C₇₁	C₄	1	4	yes	—
14	C₄×Q₈×C₇₁	C₂²	1	4	yes	—
15	#117×C₇₁	C₂²	1	2	yes	—
16	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×C₇₁⋊C₈	C₂³	1	8	yes	—
18	C₂×Q₈×C₇₁	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₇₁	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₇₁	C₂×C₄	1	4	yes	—
22	C₄⋊C₄×C₇₁	C₂×C₄	1	8	yes	—
23	C₄⋊C₄×C₇₁	C₂×C₄	1	4	yes	—
24	C₂²⋊C₄×C₇₁	C₂×C₄	1	4	yes	—
25	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₇₁	C₂³	1	2	yes	—
28	C₄²×C₇₁	C₂³	1	4	yes	—
29	C₂³×C₇₁	C₂²×C₄	1	1	yes	characteristic
30	Q₈×C₇₁	—	4	4	no	—
31	C₂×C₄×C₇₁	—	2	2	no	—
32	C₂×C₄×C₇₁	—	4	2	no	—
33	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₇₁	C₂²×C₄	1	2	yes	—
35	C₂×C₄×C₇₁	C₂²×C₄	1	2	yes	—
36	C₂×C₄×C₇₁	C₂²×C₄	1	8	yes	—
37	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
38	C₇₁⋊C₈	—	2	8	no	—
39	C₂²×C₇₁	—	2	1	no	—
40	C₂²×C₇₁	—	2	2	no	—
41	C₂²×C₇₁	C₂³×C₄	1	1	yes	characteristic
42	C₄×C₇₁	—	2	1	no	—
43	C₄×C₇₁	—	2	2	no	—
44	C₄×C₇₁	—	2	2	no	—
45	C₄×C₇₁	—	2	8	no	—
46	C₄×C₇₁	2₋¹⁺⁴	1	2	yes	—
47	C₂×C₇₁	—	4	1	no	—
48	C₂×C₇₁	#519	1	1	yes	characteristic
49	C₂×C₇₁	#568	1	2	yes	—
50	#2987	—	71	1	no	maximal, Sylow
51	C₇₁	#2987	1	1	yes	characteristic, complemented, Sylow
52	#533	D₇₁	1	1	yes	characteristic
53	#446	—	71	8	no	—
54	#433	—	71	4	no	—
55	#432	—	71	2	no	—
56	C₄⋊Q₈	Dic₇₁	1	2	yes	—
57	#125	Dic₇₁	1	2	yes	—
58	C₂²⋊Q₈	Dic₇₁	1	4	yes	—
59	C₄×Q₈	C₂×D₇₁	1	4	yes	—
60	#117	C₂×D₇₁	1	2	yes	—
61	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
62	C₄⋊C₈	—	71	4	no	—
63	C₄⋊C₈	—	71	8	no	—
64	C₂²⋊C₈	—	71	4	no	—
65	#97	—	71	4	no	—
66	#97	—	71	4	no	—
67	C₄×C₈	—	71	4	no	—
68	C₂×Q₈	C₂×Dic₇₁	1	4	yes	—
69	C₂²×C₄	C₂×Dic₇₁	1	2	yes	—
70	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
71	C₂×C₈	—	71	8	no	—
72	C₄⋊C₄	C₂×Dic₇₁	1	4	yes	—
73	C₄⋊C₄	C₂×Dic₇₁	1	8	yes	—
74	C₄⋊C₄	C₂×Dic₇₁	1	4	yes	—
75	C₂²⋊C₄	C₂×Dic₇₁	1	4	yes	—
76	C₄²	C₂×Dic₇₁	1	1	yes	characteristic
77	C₄²	C₂×Dic₇₁	1	1	yes	characteristic
78	C₄²	C₂²×D₇₁	1	4	yes	—
79	C₄²	C₂²×D₇₁	1	2	yes	—
80	C₂³	C₂²×Dic₇₁	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×Dic₇₁	1	1	yes	characteristic
85	C₂×C₄	C₂²×Dic₇₁	1	2	yes	—
86	C₂×C₄	C₂²×Dic₇₁	1	2	yes	—
87	C₂×C₄	C₂²×Dic₇₁	1	8	yes	—
88	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
89	C₈	—	142	8	no	—
90	C₂²	—	2	1	no	—
91	C₂²	—	2	2	no	—
92	C₂²	C₂³×Dic₇₁	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	#14556176	1	2	yes	—
98	C₂	—	4	1	no	—
99	C₂	#17986477	1	1	yes	characteristic
100	C₂	#17986766	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17986130

C₂

yes

maximal, complemented

#17986060

C₂

yes

maximal

#17986056

C₂

yes

maximal

#533×C₇₁

C₂

yes

maximal, characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

C₂²⋊(C₇₁⋊C₈)

C₂²

yes

—

#14556039

C₂²

yes

—

#14556039

C₂²

yes

—

C₄×C₇₁⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₇₁

C₄

yes

—

#125×C₇₁

C₄

yes

—

C₂²⋊Q₈×C₇₁

C₄

yes

—

C₄×Q₈×C₇₁

C₂²

yes

—

#117×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₂×C₇₁⋊C₈

C₂³

yes

—

C₂×Q₈×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

C₂×C₄

yes

—

C₄⋊C₄×C₇₁

C₂×C₄

yes

—

C₄⋊C₄×C₇₁

C₂×C₄

yes

—

C₂²⋊C₄×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

C₂³

yes

—

C₄²×C₇₁

C₂³

yes

—

C₂³×C₇₁

C₂²×C₄

yes

characteristic

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

—

C₂×C₄×C₇₁

C₂⁴

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂³×C₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

2₋¹⁺⁴

yes

—

C₂×C₇₁

—

C₂×C₇₁

#519

yes

characteristic

C₂×C₇₁

#568

yes

—

#2987

—

maximal, Sylow

C₇₁

#2987

yes

characteristic, complemented, Sylow

#533

D₇₁

yes

—

—

—

C₄⋊Q₈

Dic₇₁

yes

—

#125

Dic₇₁

yes

—

C₂²⋊Q₈

Dic₇₁

yes

—

C₄×Q₈

C₂×D₇₁

yes

—

#117

C₂×D₇₁

yes

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₂×Q₈

C₂×Dic₇₁

yes

—

C₂²×C₄

C₂×Dic₇₁

yes

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

C₂×C₈

—

C₄⋊C₄

C₂×Dic₇₁

yes

—

C₄⋊C₄

C₂×Dic₇₁

yes

—

C₄⋊C₄

C₂×Dic₇₁

yes

—

C₂²⋊C₄

C₂×Dic₇₁

yes

—

C₄²

C₂×Dic₇₁

yes

characteristic

C₄²

C₂×Dic₇₁

yes

characteristic

C₄²

C₂²×D₇₁

yes

—

C₄²

C₂²×D₇₁

yes

—

C₂³

C₂²×Dic₇₁

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

C₂×C₄

C₂²×Dic₇₁

yes

—

C₂×C₄

C₂²×Dic₇₁

yes

—

C₂×C₄

C₂²×Dic₇₁

yes

—

C₂×C₄

C₂³×D₇₁

yes

characteristic

C₈

—

142

—

C₂²

—

C₂²

—

C₂²

C₂³×Dic₇₁

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

#14556176

yes

—

C₂

—

C₂

#17986477

yes

characteristic

100

C₂

#17986766

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇₁
commutator, socle	C₂²×C₇₁	C₂³×C₄
Fitting	#533×C₇₁	C₂
hypercenter	#533	D₇₁
hypocenter	C₇₁	#2987
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ #533

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₇₁

commutator, socle

C₂²×C₇₁

C₂³×C₄

Fitting

#533×C₇₁

C₂

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂×C₄ ◁ #533

N	H ≅ G/N	type
1. #17986130	C₂	semidirect
51. C₇₁	#2987	semidirect

H ≅ G/N

type

1. #17986130

C₂

semidirect

51. C₇₁

#2987

semidirect

Subgroups — #22597446

Conjugacy classes of subgroups

Complements