Group #22590572 — subgroups

Subgroups — #22590572

Order 9088.

138 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 53 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	#17986401	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986383	C₂	1	1	yes	maximal, characteristic
3	#17985804	C₂	1	2	yes	maximal, complemented
4	#17985801	C₂	1	2	yes	maximal
5	C₄×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₂₈₄	C₂²	1	1	yes	characteristic
7	C₂×#11765854	—	2	1	no	—
8	C₂×D₅₆₈	C₂²	1	1	yes	characteristic
9	#14556120	—	2	1	no	—
10	#14556116	—	2	1	no	—
11	#14556062	C₂²	1	2	yes	—
12	#14556060	—	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×D₂₈₄	—	2	1	no	—
17	Dic₂₈₄	—	2	2	no	—
18	#11765854	—	4	2	no	—
19	D₅₆₈	—	2	2	no	—
20	C₄⋊Dic₇₁	—	2	2	no	—
21	Dic₇₁⋊C₄	—	4	2	no	—
22	(C₂×D₇₁)⋊C₄	—	4	2	no	—
23	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
24	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
25	C₄²×C₇₁	D₄	1	1	yes	characteristic
26	C₁₆×C₇₁	—	2	2	no	—
27	C₂²×D₇₁	—	4	1	no	—
28	C₂×C₄×C₇₁	D₈	1	2	yes	—
29	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
30	C₂×Dic₇₁	—	4	1	no	—
31	C₂×Dic₇₁	—	4	2	no	—
32	D₂₈₄	—	4	1	no	—
33	D₂₈₄	—	4	1	no	—
34	Dic₁₄₂	—	4	1	no	—
35	Dic₁₄₂	—	4	1	no	—
36	C₈×C₇₁	SD₁₆	1	2	yes	—
37	C₈×C₇₁	#41	1	2	yes	—
38	C₂²×C₇₁	C₂×D₈	1	1	yes	characteristic
39	C₂×D₇₁	—	8	1	no	—
40	C₂×D₇₁	—	8	2	no	—
41	C₄×C₇₁	—	2	2	no	—
42	C₄×C₇₁	#124	1	1	yes	characteristic
43	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
44	Dic₇₁	—	8	1	no	—
45	Dic₇₁	—	8	2	no	—
46	C₂×C₇₁	#486	1	1	yes	characteristic
47	C₂×C₇₁	#508	1	2	yes	—
48	D₇₁	—	16	1	no	—
49	#2242	—	71	1	no	maximal, Sylow
50	C₇₁	#2242	1	1	yes	characteristic, complemented, Sylow
51	#499	—	71	1	no	—
52	#495	—	71	1	no	—
53	#358	—	71	2	no	—
54	#357	—	71	2	no	—
55	C₄×C₁₆	D₇₁	1	1	yes	characteristic, complemented
56	C₂×Q₁₆	—	71	1	no	—
57	C₂×SD₁₆	—	142	1	no	—
58	C₂×D₈	—	71	1	no	—
59	#125	—	142	1	no	—
60	#124	—	142	1	no	—
61	C₂×C₁₆	C₂×D₇₁	1	2	yes	—
62	#107	—	71	2	no	—
63	#106	—	142	1	no	—
64	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
65	C₂×Q₈	—	142	1	no	—
66	C₂×D₄	—	142	1	no	—
67	Q₁₆	—	142	2	no	—
68	SD₁₆	—	284	2	no	—
69	D₈	—	142	2	no	—
70	C₂×C₈	D₂₈₄	1	1	yes	characteristic
71	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
72	C₄⋊C₄	—	284	2	no	—
73	C₄⋊C₄	—	142	2	no	—
74	C₂²⋊C₄	—	284	2	no	—
75	C₄²	D₂₈₄	1	1	yes	characteristic
76	C₁₆	—	2	2	no	—
77	C₂³	—	284	1	no	—
78	Q₈	—	284	1	no	—
79	Q₈	—	284	1	no	—
80	D₄	—	284	1	no	—
81	D₄	—	284	1	no	—
82	C₂×C₄	—	284	1	no	—
83	C₂×C₄	—	284	2	no	—
84	C₂×C₄	D₅₆₈	1	2	yes	—
85	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
86	C₈	#11765854	1	2	yes	—
87	C₈	#11765865	1	2	yes	—
88	C₂²	—	568	1	no	—
89	C₂²	—	568	2	no	—
90	C₂²	C₂×D₅₆₈	1	1	yes	characteristic
91	C₄	—	2	2	no	—
92	C₄	—	568	1	no	—
93	C₄	—	568	2	no	—
94	C₄	#14556116	1	1	yes	characteristic
95	C₄	C₂×#11765854	1	1	yes	characteristic
96	C₂	—	1136	1	no	—
97	C₂	#17986346	1	1	yes	characteristic
98	C₂	#17986431	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17986401

C₂

yes

maximal, characteristic, complemented

#17986383

C₂

yes

maximal, characteristic

#17985804

C₂

yes

maximal, complemented

#17985801

C₂

yes

maximal

C₄×C₁₆×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₈₄

C₂²

yes

characteristic

C₂×#11765854

—

C₂×D₅₆₈

C₂²

yes

—

—

C₂²

yes

—

#14556060

—

C₂×C₁₆×C₇₁

C₂²

yes

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×D₂₈₄

—

Dic₂₈₄

—

#11765854

—

D₅₆₈

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

(C₂×D₇₁)⋊C₄

—

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₁₆×C₇₁

—

C₂²×D₇₁

—

C₂×C₄×C₇₁

D₈

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

D₂₈₄

—

D₂₈₄

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

SD₁₆

yes

—

C₈×C₇₁

#41

yes

—

C₂²×C₇₁

C₂×D₈

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#124

yes

characteristic

C₄×C₇₁

C₂×SD₁₆

yes

characteristic

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

#486

yes

characteristic

C₂×C₇₁

#508

yes

—

D₇₁

—

#2242

—

maximal, Sylow

C₇₁

#2242

yes

characteristic, complemented, Sylow

—

—

—

—

C₄×C₁₆

D₇₁

yes

characteristic, complemented

C₂×Q₁₆

—

C₂×SD₁₆

—

142

—

C₂×D₈

—

#125

—

142

—

#124

—

142

—

C₂×C₁₆

C₂×D₇₁

yes

—

#107

—

#106

—

142

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×Q₈

—

142

—

C₂×D₄

—

142

—

Q₁₆

—

142

—

SD₁₆

—

284

—

D₈

—

142

—

C₂×C₈

D₂₈₄

yes

characteristic

C₂×C₈

C₂²×D₇₁

yes

characteristic

C₄⋊C₄

—

284

—

C₄⋊C₄

—

142

—

C₂²⋊C₄

—

284

—

C₄²

D₂₈₄

yes

characteristic

C₁₆

—

C₂³

—

284

—

Q₈

—

284

—

Q₈

—

284

—

D₄

—

284

—

D₄

—

284

—

C₂×C₄

—

284

—

C₂×C₄

—

284

—

C₂×C₄

D₅₆₈

yes

—

C₂×C₄

C₂×D₂₈₄

yes

characteristic

C₈

#11765854

yes

—

C₈

#11765865

yes

—

C₂²

—

568

—

C₂²

—

568

—

C₂²

C₂×D₅₆₈

yes

characteristic

C₄

—

C₄

—

568

—

C₄

—

568

—

C₄

yes

C₄

yes

C₂

—

1136

—

C₂

#17986346

yes

characteristic

C₂

#17986431

yes

—

	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	#2242
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

#2242

LCS

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
1. #17986401	C₂	semidirect
3. #17985804	C₂	semidirect
5. C₄×C₁₆×C₇₁	C₂	semidirect
50. C₇₁	#2242	semidirect
55. C₄×C₁₆	D₇₁	semidirect

H ≅ G/N

type

1. #17986401

C₂

semidirect

3. #17985804

C₂

semidirect

5. C₄×C₁₆×C₇₁

C₂

semidirect

50. C₇₁

#2242

semidirect

55. C₄×C₁₆

D₇₁

semidirect

Subgroups — #22590572

Conjugacy classes of subgroups

Complements