Group #22587096 — subgroups

Subgroups — #22587096

Order 9088.

184 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 109 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	#17986023	C₂	1	2	yes	maximal
2	#17985750	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#11765849	—	2	2	no	—
5	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
6	C₄⋊(C₇₁⋊C₈)	C₂²	1	4	yes	—
7	C₂²×C₈×C₇₁	C₄	1	2	yes	—
8	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₇₁	C₄	1	1	yes	characteristic
10	C₄×C₈×C₇₁	C₄	1	1	yes	characteristic
11	C₄×C₈×C₇₁	C₂²	1	2	yes	—
12	#11765849	—	4	4	no	—
13	C₂×C₇₁⋊C₈	—	2	4	no	—
14	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₇₁	Q₈	1	1	yes	characteristic
17	C₂×C₈×C₇₁	—	2	4	no	—
18	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
20	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₇₁	D₄	1	1	yes	characteristic
22	C₄²×C₇₁	Q₈	1	1	yes	characteristic
23	C₄²×C₇₁	C₂³	1	1	yes	characteristic
24	C₂³×C₇₁	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₇₁	—	2	2	no	—
26	C₂×C₄×C₇₁	—	2	2	no	—
27	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₇₁	SD₁₆	1	4	yes	—
31	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
34	C₈×C₇₁	—	2	2	no	—
35	C₈×C₇₁	M₁₆	1	4	yes	complemented
36	C₇₁⋊C₈	—	4	4	no	—
37	C₂²×C₇₁	—	2	1	no	—
38	C₂²×C₇₁	—	2	1	no	—
39	C₂²×C₇₁	#106	1	2	yes	—
40	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₄×C₇₁	—	2	1	no	—
42	C₄×C₇₁	—	2	1	no	—
43	C₄×C₇₁	#106	1	2	yes	—
44	C₄×C₇₁	#108	1	2	yes	—
45	C₄×C₇₁	C₂×M₁₆	1	2	yes	—
46	C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
47	C₂×C₇₁	—	2	2	no	—
48	C₂×C₇₁	#423	1	1	yes	characteristic
49	C₂×C₇₁	C₂×#106	1	1	yes	characteristic
50	C₂×C₇₁	C₂×#108	1	1	yes	characteristic
51	#1566	—	71	1	no	maximal, Sylow
52	C₇₁	#1566	1	1	yes	characteristic, complemented, Sylow
53	#423	—	71	2	no	—
54	C₂×C₄×C₈	D₇₁	1	1	yes	characteristic
55	#334	—	71	4	no	—
56	C₂×M₁₆	—	142	2	no	—
57	C₂²×C₈	Dic₇₁	1	2	yes	—
58	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
59	C₄⋊C₈	—	71	4	no	—
60	C₄⋊C₈	—	142	2	no	—
61	C₄×C₈	Dic₇₁	1	1	yes	characteristic
62	C₄×C₈	Dic₇₁	1	1	yes	characteristic
63	C₄×C₈	C₂×D₇₁	1	2	yes	—
64	C₂²×C₄	Dic₁₄₂	1	1	yes	characteristic
65	C₂²×C₄	D₂₈₄	1	1	yes	characteristic
66	C₂²×C₄	C₂×Dic₇₁	1	1	yes	characteristic
67	M₁₆	—	284	4	no	—
68	C₂×C₈	—	2	4	no	—
69	C₂×C₈	—	142	4	no	—
70	C₂×C₈	C₂×Dic₇₁	1	2	yes	—
71	C₂×C₈	C₂×Dic₇₁	1	2	yes	—
72	C₄²	Dic₁₄₂	1	1	yes	characteristic
73	C₄²	D₂₈₄	1	1	yes	characteristic
74	C₄²	C₂×Dic₇₁	1	1	yes	characteristic
75	C₄²	C₂²×D₇₁	1	1	yes	characteristic
76	C₂³	C₄⋊Dic₇₁	1	1	yes	characteristic
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
80	C₂×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
81	C₂×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
82	C₂×C₄	#11765854	1	4	yes	—
83	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
84	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
85	C₂×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
86	C₈	—	2	2	no	—
87	C₈	—	284	4	no	—
88	C₈	#11765849	1	4	yes	complemented
89	C₂²	—	2	1	no	—
90	C₂²	—	2	1	no	—
91	C₂²	#14556060	1	2	yes	—
92	C₂²	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	1	no	—
95	C₄	#14556060	1	2	yes	—
96	C₄	#14556064	1	2	yes	—
97	C₄	C₂×#11765849	1	2	yes	—
98	C₄	C₂×#11765854	1	2	yes	—
99	C₂	—	2	2	no	—
100	C₂	#17986023	1	1	yes	characteristic
101	C₂	C₂×#14556060	1	1	yes	characteristic
102	C₂	C₂×#14556064	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986023

C₂

yes

maximal

#17985750

C₂

yes

maximal, complemented

C₂×C₄×C₈×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

C₂²×C₈×C₇₁

C₄

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₄×C₈×C₇₁

C₄

yes

characteristic

C₄×C₈×C₇₁

C₄

yes

characteristic

C₄×C₈×C₇₁

C₂²

yes

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

Q₈

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

Q₈

yes

characteristic

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

SD₁₆

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

M₁₆

yes

complemented

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#106

yes

—

C₂²×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#106

yes

—

C₄×C₇₁

#108

yes

—

C₄×C₇₁

C₂×M₁₆

yes

—

C₄×C₇₁

C₂×SD₁₆

yes

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#423

—

C₂×C₄×C₈

D₇₁

yes

characteristic

#334

—

C₂×M₁₆

—

142

—

C₂²×C₈

Dic₇₁

yes

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

142

—

C₄×C₈

Dic₇₁

yes

characteristic

C₄×C₈

Dic₇₁

yes

characteristic

C₄×C₈

C₂×D₇₁

yes

—

C₂²×C₄

Dic₁₄₂

yes

characteristic

C₂²×C₄

D₂₈₄

yes

characteristic

C₂²×C₄

C₂×Dic₇₁

yes

characteristic

M₁₆

—

284

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

C₂×Dic₇₁

yes

—

C₂×C₈

C₂×Dic₇₁

yes

—

C₄²

Dic₁₄₂

yes

characteristic

C₄²

D₂₈₄

yes

characteristic

C₄²

C₂×Dic₇₁

yes

characteristic

C₄²

C₂²×D₇₁

yes

characteristic

C₂³

C₄⋊Dic₇₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₇₁

yes

characteristic

C₂×C₄

C₄⋊Dic₇₁

yes

characteristic

C₂×C₄

C₄⋊Dic₇₁

yes

characteristic

C₂×C₄

#11765854

yes

—

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

C₂×C₄

C₂×D₂₈₄

yes

characteristic

C₂×C₄

C₂×Dic₁₄₂

yes

characteristic

C₈

—

C₈

—

284

—

C₈

#11765849

yes

complemented

C₂²

—

C₂²

—

C₂²

#14556060

yes

—

C₂²

C₂×C₄⋊Dic₇₁

yes

characteristic

C₄

—

C₄

—

C₄

#14556060

yes

—

C₄

#14556064

yes

—

C₄

C₂×#11765849

yes

—

C₄

C₂×#11765854

yes

—

C₂

—

100

C₂

#17986023

yes

characteristic

101

C₂

C₂×#14556060

yes

characteristic

102

C₂

C₂×#14556064

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₄²	C₂²×D₇₁
Fitting	C₂×C₄×C₈×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₄⋊C₄
hypercenter	C₂×C₄×C₈	D₇₁
hypocenter	C₇₁	#1566
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₂₈₄

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₄²

C₂²×D₇₁

Fitting

C₂×C₄×C₈×C₇₁

C₂

socle

C₂²×C₇₁

C₂×C₄⋊C₄

hypercenter

C₂×C₄×C₈

D₇₁

hypocenter

C₇₁

#1566

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #17985750	C₂	semidirect
35. C₈×C₇₁	M₁₆	semidirect
52. C₇₁	#1566	semidirect
88. C₈	#11765849	semidirect

H ≅ G/N

type

2. #17985750

C₂

semidirect

35. C₈×C₇₁

M₁₆

semidirect

52. C₇₁

#1566

semidirect

88. C₈

#11765849

semidirect

Subgroups — #22587096

Conjugacy classes of subgroups

Complements