Group #22586436 — subgroups

Subgroups — #22586436

Order 9088.

132 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 73 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	C₂×#14556064	C₂	1	1	yes	maximal, characteristic
2	#17986037	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#11765849	—	2	1	no	—
5	#14556087	—	2	1	no	—
6	#14556064	—	2	1	no	—
7	#14556064	C₄	1	2	yes	complemented
8	#14556062	C₄	1	2	yes	—
9	#14556060	—	2	1	no	—
10	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×C₇₁	—	2	1	no	—
12	C₂×C₁₆×C₇₁	—	2	1	no	—
13	C₂×C₁₆×C₇₁	C₄	1	2	yes	complemented
14	#11765849	—	2	2	no	—
15	#11765849	—	4	1	no	—
16	C₂×C₇₁⋊C₈	—	4	1	no	—
17	C₄⋊Dic₇₁	—	2	2	no	—
18	C₂²⋊Dic₇₁	—	4	1	no	—
19	C₄×Dic₇₁	—	4	1	no	—
20	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
21	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
25	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
26	C₂×C₈×C₇₁	Q₈	1	1	yes	characteristic
27	C₁₆×C₇₁	—	2	2	no	—
28	C₂³×C₇₁	SD₁₆	1	1	yes	characteristic
29	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₇₁	D₈	1	1	yes	characteristic
33	C₂×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
34	C₂×C₄×C₇₁	Q₁₆	1	1	yes	characteristic
35	C₂×Dic₇₁	—	4	2	no	—
36	C₈×C₇₁	C₄²	1	1	yes	characteristic
37	C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
39	C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
40	C₇₁⋊C₈	—	4	2	no	—
41	C₂²×C₇₁	—	2	2	no	—
42	C₂²×C₇₁	#102	1	1	yes	characteristic
43	C₂²×C₇₁	#103	1	1	yes	characteristic
44	C₂²×C₇₁	#106	1	1	yes	characteristic
45	C₄×C₇₁	#95	1	1	yes	characteristic
46	C₄×C₇₁	#102	1	1	yes	characteristic
47	C₄×C₇₁	#103	1	1	yes	characteristic
48	C₄×C₇₁	#107	1	1	yes	characteristic
49	Dic₇₁	—	8	2	no	—
50	C₂×C₇₁	—	2	1	no	—
51	C₂×C₇₁	—	2	1	no	—
52	C₂×C₇₁	#340	1	1	yes	characteristic
53	C₂×C₇₁	#359	1	2	yes	—
54	#1382	—	71	1	no	maximal, Sylow
55	C₇₁	#1382	1	1	yes	characteristic, complemented, Sylow
56	C₂²×C₁₆	D₇₁	1	1	yes	characteristic
57	C₂×#108	—	71	1	no	—
58	#427	—	71	1	no	—
59	C₂×M₁₆	—	142	1	no	—
60	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
61	#117	—	142	1	no	—
62	C₂×C₁₆	—	2	1	no	—
63	C₂×C₁₆	—	2	1	no	—
64	C₂×C₁₆	Dic₇₁	1	2	yes	complemented
65	#108	—	71	2	no	—
66	#108	—	142	1	no	—
67	#107	—	71	2	no	—
68	#106	—	142	1	no	—
69	C₂²×C₄	D₂₈₄	1	1	yes	characteristic
70	M₁₆	—	284	1	no	—
71	M₁₆	—	142	2	no	—
72	C₂×C₈	—	284	1	no	—
73	C₂×C₈	Dic₁₄₂	1	1	yes	characteristic
74	C₂×C₈	C₄×D₇₁	1	1	yes	characteristic
75	C₂×C₈	C₄×D₇₁	1	1	yes	characteristic
76	C₂×C₈	C₂×Dic₇₁	1	1	yes	characteristic
77	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
78	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
79	C₄⋊C₄	—	142	2	no	—
80	C₂²⋊C₄	—	284	1	no	—
81	C₄²	—	284	1	no	—
82	C₁₆	—	2	2	no	—
83	C₂³	#11765854	1	1	yes	characteristic
84	C₂×C₄	—	284	2	no	—
85	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
86	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
87	C₂×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
88	C₂×C₄	D₅₆₈	1	1	yes	characteristic
89	C₂×C₄	#11765854	1	1	yes	characteristic
90	C₂×C₄	Dic₂₈₄	1	1	yes	characteristic
91	C₈	—	284	2	no	—
92	C₈	C₄×Dic₇₁	1	1	yes	characteristic
93	C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
94	C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
95	C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
96	C₂²	—	2	2	no	—
97	C₂²	#14556050	1	1	yes	characteristic
98	C₂²	#14556053	1	1	yes	characteristic
99	C₂²	#14556060	1	1	yes	characteristic
100	C₄	—	568	2	no	—
101	C₄	C₇₁⋊#95	1	1	yes	characteristic
102	C₄	#14556050	1	1	yes	characteristic
103	C₄	#14556053	1	1	yes	characteristic
104	C₄	#14556062	1	1	yes	characteristic
105	C₂	—	2	1	no	—
106	C₂	—	2	1	no	—
107	C₂	#17985762	1	1	yes	characteristic
108	C₂	#17985807	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556064

C₂

yes

maximal, characteristic

#17986037

C₂

yes

maximal, characteristic

C₂²×C₁₆×C₇₁

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

complemented

#14556062

C₄

yes

—

#14556060

—

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

C₄

yes

complemented

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

Q₈

yes

characteristic

C₁₆×C₇₁

—

C₂³×C₇₁

SD₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

D₈

yes

characteristic

C₂×C₄×C₇₁

SD₁₆

yes

characteristic

C₂×C₄×C₇₁

Q₁₆

yes

characteristic

C₂×Dic₇₁

—

C₈×C₇₁

C₄²

yes

characteristic

C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#340

yes

characteristic

C₂×C₇₁

#359

yes

—

#1382

—

maximal, Sylow

C₇₁

#1382

yes

characteristic, complemented, Sylow

C₂²×C₁₆

D₇₁

yes

characteristic

C₂×#108

—

#427

—

C₂×M₁₆

—

142

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

#117

—

142

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

Dic₇₁

yes

complemented

#108

—

#108

—

142

—

#107

—

#106

—

142

—

C₂²×C₄

D₂₈₄

yes

characteristic

M₁₆

—

284

—

M₁₆

—

142

—

C₂×C₈

—

284

—

C₂×C₈

Dic₁₄₂

yes

characteristic

C₂×C₈

C₄×D₇₁

yes

characteristic

C₂×C₈

C₄×D₇₁

yes

characteristic

C₂×C₈

C₂×Dic₇₁

yes

characteristic

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₄⋊C₄

—

142

—

C₂²⋊C₄

—

284

—

C₄²

—

284

—

C₁₆

—

C₂³

#11765854

yes

characteristic

C₂×C₄

—

284

—

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₇₁

yes

characteristic

C₂×C₄

D₅₆₈

yes

characteristic

C₂×C₄

#11765854

yes

characteristic

C₂×C₄

Dic₂₈₄

yes

characteristic

C₈

—

284

—

C₈

C₄×Dic₇₁

yes

characteristic

C₈

C₂²⋊Dic₇₁

yes

characteristic

C₈

Dic₇₁⋊C₄

yes

characteristic

C₈

Dic₇₁⋊C₄

yes

characteristic

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

100

C₄

—

568

—

101

C₄

C₇₁⋊#95

yes

characteristic

102

C₄

#14556050

yes

characteristic

103

C₄

#14556053

yes

characteristic

104

C₄

#14556062

yes

characteristic

105

C₂

—

106

C₂

—

107

C₂

#17985762

yes

characteristic

108

C₂

#17985807

yes

—

	subgroup	G/H
center	C₂×C₄	D₅₆₈
commutator	C₈×C₇₁	C₄²
Frattini	C₂²×C₈	C₂×D₇₁
Fitting	C₂²×C₁₆×C₇₁	C₂
socle	C₂²×C₇₁	#102
hypercenter	C₂²×C₁₆	D₇₁
hypocenter	C₇₁	#1382
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂×C₄

D₅₆₈

commutator

C₈×C₇₁

C₄²

Frattini

C₂²×C₈

C₂×D₇₁

Fitting

C₂²×C₁₆×C₇₁

C₂

socle

C₂²×C₇₁

#102

hypercenter

C₂²×C₁₆

D₇₁

hypocenter

C₇₁

#1382

LCS

G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
7. #14556064	C₄	semidirect
13. C₂×C₁₆×C₇₁	C₄	semidirect
55. C₇₁	#1382	semidirect
64. C₂×C₁₆	Dic₇₁	semidirect

H ≅ G/N

type

7. #14556064

C₄

semidirect

13. C₂×C₁₆×C₇₁

C₄

semidirect

55. C₇₁

#1382

semidirect

64. C₂×C₁₆

Dic₇₁

semidirect

Subgroups — #22586436

Conjugacy classes of subgroups

Complements