Group #22590493 — subgroups

Subgroups — #22590493

Order 9088.

130 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 55 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	#17986404	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₂₈₄	C₂	1	1	yes	maximal, characteristic
3	#17985825	C₂	1	2	yes	maximal
4	#17985804	C₂	1	2	yes	maximal
5	C₄⋊C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₈₄	C₂²	1	1	yes	characteristic
7	#14556128	—	2	1	no	—
8	C₄×Dic₁₄₂	—	2	1	no	—
9	#14556062	—	2	1	no	—
10	#14556062	C₂²	1	2	yes	—
11	#14556062	C₂²	1	1	yes	characteristic
12	#14556053	—	2	1	no	—
13	C₂×C₁₆×C₇₁	C₂²	1	2	yes	—
14	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₄₂	—	2	1	no	—
16	C₂×Dic₁₄₂	—	4	1	no	—
17	Dic₂₈₄	—	2	1	no	—
18	Dic₂₈₄	Q₈	1	2	yes	complemented
19	C₄⋊Dic₇₁	—	2	1	no	—
20	C₄⋊Dic₇₁	—	2	2	no	—
21	C₄⋊Dic₇₁	—	4	1	no	—
22	Dic₇₁⋊C₄	—	4	1	no	—
23	C₄×Dic₇₁	—	4	1	no	—
24	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
25	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
26	C₄²×C₇₁	D₄	1	1	yes	characteristic
27	C₁₆×C₇₁	—	2	2	no	—
28	C₂×C₄×C₇₁	D₈	1	1	yes	characteristic
29	C₂×C₄×C₇₁	D₈	1	1	yes	characteristic
30	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₇₁	—	4	1	no	—
32	C₂×Dic₇₁	—	4	1	no	—
33	C₂×Dic₇₁	—	4	2	no	—
34	Dic₁₄₂	—	2	2	no	—
35	Dic₁₄₂	—	4	1	no	—
36	Dic₁₄₂	—	8	2	no	—
37	C₈×C₇₁	—	2	1	no	—
38	C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
39	C₈×C₇₁	#41	1	1	yes	characteristic
40	C₂²×C₇₁	C₂×D₈	1	1	yes	characteristic
41	C₄×C₇₁	—	2	1	no	—
42	C₄×C₇₁	Q₃₂	1	2	yes	—
43	C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
44	C₄×C₇₁	#137	1	1	yes	characteristic
45	Dic₇₁	—	4	2	no	—
46	Dic₇₁	—	8	2	no	—
47	Dic₇₁	—	8	1	no	—
48	C₂×C₇₁	#474	1	1	yes	characteristic
49	C₂×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
50	C₂×C₇₁	#509	1	1	yes	characteristic
51	#2227	—	71	1	no	maximal, Sylow
52	C₇₁	#2227	1	1	yes	characteristic, complemented, Sylow
53	#500	—	71	1	no	—
54	C₄×Q₁₆	—	71	1	no	—
55	#366	—	71	2	no	—
56	C₄⋊C₁₆	D₇₁	1	1	yes	characteristic
57	#358	—	71	2	no	—
58	C₂×Q₁₆	—	71	1	no	—
59	C₄⋊Q₈	—	142	1	no	—
60	C₄×Q₈	—	142	1	no	—
61	C₂×C₁₆	C₂×D₇₁	1	2	yes	—
62	#107	—	71	2	no	—
63	#107	—	71	1	no	—
64	#107	—	142	1	no	—
65	#103	—	142	1	no	—
66	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
67	C₂×Q₈	—	284	1	no	—
68	C₂×Q₈	—	142	1	no	—
69	Q₁₆	—	71	2	no	—
70	Q₁₆	—	142	1	no	—
71	C₂×C₈	D₂₈₄	1	1	yes	characteristic
72	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
73	C₄⋊C₄	—	284	1	no	—
74	C₄⋊C₄	—	284	1	no	—
75	C₄⋊C₄	—	142	1	no	—
76	C₄⋊C₄	—	142	2	no	—
77	C₄²	—	284	1	no	—
78	C₄²	D₂₈₄	1	1	yes	characteristic
79	C₁₆	—	2	2	no	—
80	Q₈	—	284	1	no	—
81	Q₈	—	568	2	no	—
82	Q₈	—	142	2	no	—
83	C₂×C₄	—	284	1	no	—
84	C₂×C₄	—	284	1	no	—
85	C₂×C₄	—	284	2	no	—
86	C₂×C₄	D₅₆₈	1	1	yes	characteristic
87	C₂×C₄	D₅₆₈	1	1	yes	characteristic
88	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
89	C₈	—	2	1	no	—
90	C₈	Q₈×D₇₁	1	1	yes	characteristic
91	C₈	#11765864	1	1	yes	characteristic
92	C₂²	C₂×D₅₆₈	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	284	2	no	—
95	C₄	—	568	2	no	—
96	C₄	—	568	1	no	—
97	C₄	Dic₅₆₈	1	2	yes	—
98	C₄	#14556108	1	1	yes	characteristic
99	C₄	#14556161	1	1	yes	characteristic
100	C₂	#17986280	1	1	yes	characteristic
101	C₂	C₂×Dic₅₆₈	1	1	yes	characteristic
102	C₂	#17986437	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986404

C₂

yes

maximal, characteristic

C₄×Dic₂₈₄

C₂

yes

maximal, characteristic

#17985825

C₂

yes

maximal

#17985804

C₂

yes

maximal

C₄⋊C₁₆×C₇₁

C₂

yes

maximal, characteristic

C₂×Dic₂₈₄

C₂²

yes

characteristic

#14556128

—

C₄×Dic₁₄₂

—

#14556062

—

#14556062

C₂²

yes

—

#14556062

C₂²

yes

characteristic

#14556053

—

C₂×C₁₆×C₇₁

C₂²

yes

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

Dic₂₈₄

—

Dic₂₈₄

Q₈

yes

complemented

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₁₆×C₇₁

—

C₂×C₄×C₇₁

D₈

yes

characteristic

C₂×C₄×C₇₁

D₈

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

C₂×Q₈

yes

characteristic

C₈×C₇₁

#41

yes

characteristic

C₂²×C₇₁

C₂×D₈

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

Q₃₂

yes

—

C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

C₄×C₇₁

#137

yes

characteristic

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

#474

yes

characteristic

C₂×C₇₁

C₂×Q₃₂

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#500

—

C₄×Q₁₆

—

#366

—

C₄⋊C₁₆

D₇₁

yes

characteristic

#358

—

C₂×Q₁₆

—

C₄⋊Q₈

—

142

—

C₄×Q₈

—

142

—

C₂×C₁₆

C₂×D₇₁

yes

—

#107

—

#107

—

#107

—

142

—

#103

—

142

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×Q₈

—

284

—

C₂×Q₈

—

142

—

Q₁₆

—

Q₁₆

—

142

—

C₂×C₈

D₂₈₄

yes

characteristic

C₂×C₈

C₂²×D₇₁

yes

characteristic

C₄⋊C₄

—

284

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄²

—

284

—

C₄²

D₂₈₄

yes

characteristic

C₁₆

—

Q₈

—

284

—

Q₈

—

568

—

Q₈

—

142

—

C₂×C₄

—

284

—

C₂×C₄

—

284

—

C₂×C₄

—

284

—

C₂×C₄

D₅₆₈

yes

characteristic

C₂×C₄

D₅₆₈

yes

characteristic

C₂×C₄

C₂×D₂₈₄

yes

characteristic

C₈

—

C₈

Q₈×D₇₁

yes

characteristic

C₈

#11765864

yes

characteristic

C₂²

C₂×D₅₆₈

yes

characteristic

C₄

—

C₄

—

284

—

C₄

—

568

—

C₄

—

568

—

C₄

Dic₅₆₈

yes

—

C₄

yes

C₄

yes

100

C₂

#17986280

yes

characteristic

101

C₂

C₂×Dic₅₆₈

yes

characteristic

102

C₂

#17986437

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₅₆₈
commutator	C₂×C₈×C₇₁	C₂³
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	C₄⋊C₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₂×D₈
hypercenter	C₄⋊C₁₆	D₇₁
hypocenter	C₇₁	#2227
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

C₂×D₅₆₈

commutator

C₂×C₈×C₇₁

C₂³

Frattini

C₂×C₈

C₂²×D₇₁

Fitting

C₄⋊C₁₆×C₇₁

C₂

socle

C₂²×C₇₁

C₂×D₈

hypercenter

C₄⋊C₁₆

D₇₁

hypocenter

C₇₁

#2227

LCS

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
18. Dic₂₈₄	Q₈	semidirect
52. C₇₁	#2227	semidirect

H ≅ G/N

type

18. Dic₂₈₄

Q₈

semidirect

52. C₇₁

#2227

semidirect

Subgroups — #22590493

Conjugacy classes of subgroups

Complements