Group #22587859 — subgroups

Subgroups — #22587859

Order 9088.

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

normal subgroups only

This group has 152 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	#17986210	C₂	1	2	yes	maximal
3	#17986129	C₂	1	1	yes	maximal, characteristic
4	#17985746	C₂	1	2	yes	maximal
5	#334×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₈₄	—	2	2	no	—
7	#14556128	C₂²	1	2	yes	—
8	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
9	#14556062	—	2	1	no	—
10	#14556062	—	2	1	no	—
11	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
12	Dic₇₁⋊C₈	—	2	1	no	—
13	#14556053	—	2	2	no	—
14	#14556038	—	2	1	no	—
15	C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
16	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₄₂	—	2	2	no	—
18	C₂×Dic₁₄₂	D₄	1	1	yes	characteristic
19	Dic₂₈₄	—	4	4	no	—
20	C₂×C₇₁⋊C₈	—	2	1	no	—
21	C₄⋊Dic₇₁	—	2	2	no	—
22	C₄⋊Dic₇₁	—	2	2	no	—
23	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
24	Dic₇₁⋊C₄	—	2	1	no	—
25	C₄×Dic₇₁	—	2	1	no	—
26	C₂×C₈×C₇₁	—	2	2	no	—
27	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
28	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
29	C₄²×C₇₁	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
33	C₂×Dic₇₁	—	2	1	no	—
34	C₂×Dic₇₁	—	2	1	no	—
35	C₂×Dic₇₁	—	4	2	no	—
36	Dic₁₄₂	—	2	2	no	—
37	Dic₁₄₂	—	4	2	no	—
38	Dic₁₄₂	—	4	2	no	—
39	C₈×C₇₁	—	2	1	no	—
40	C₈×C₇₁	—	4	2	no	—
41	C₈×C₇₁	Q₁₆	1	2	yes	complemented
42	C₇₁⋊C₈	—	4	1	no	—
43	C₂²×C₇₁	#121	1	1	yes	characteristic
44	C₄×C₇₁	—	2	1	no	—
45	C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
46	C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
47	C₄×C₇₁	#137	1	1	yes	characteristic
48	Dic₇₁	—	4	1	no	—
49	Dic₇₁	—	4	1	no	—
50	Dic₇₁	—	8	2	no	—
51	C₂×C₇₁	#462	1	1	yes	characteristic
52	C₂×C₇₁	#469	1	1	yes	characteristic
53	C₂×C₇₁	#473	1	1	yes	characteristic
54	#1694	—	71	1	no	maximal, Sylow
55	C₇₁	#1694	1	1	yes	characteristic, complemented, Sylow
56	#500	—	71	1	no	—
57	#462	—	71	2	no	—
58	#446	—	71	1	no	—
59	#334	D₇₁	1	1	yes	characteristic
60	#332	—	71	2	no	—
61	C₂×Q₁₆	—	142	2	no	—
62	C₄⋊Q₈	—	71	2	no	—
63	C₄×Q₈	—	71	1	no	—
64	#107	—	142	1	no	—
65	#107	—	142	1	no	—
66	C₄⋊C₈	—	71	1	no	—
67	C₄⋊C₈	—	142	1	no	—
68	C₄⋊C₈	C₂×D₇₁	1	2	yes	—
69	#103	—	142	2	no	—
70	#97	—	142	1	no	—
71	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
72	C₂×Q₈	—	71	1	no	—
73	C₂×Q₈	—	142	2	no	—
74	Q₁₆	—	284	4	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	142	1	no	—
77	C₂×C₈	D₂₈₄	1	1	yes	characteristic
78	C₂×C₈	D₂₈₄	1	1	yes	characteristic
79	C₄⋊C₄	—	71	1	no	—
80	C₄⋊C₄	—	142	2	no	—
81	C₄⋊C₄	—	142	2	no	—
82	C₄⋊C₄	—	142	1	no	—
83	C₄²	—	142	1	no	—
84	C₄²	C₂²×D₇₁	1	1	yes	characteristic
85	Q₈	—	284	2	no	—
86	Q₈	—	284	2	no	—
87	Q₈	—	142	2	no	—
88	C₂×C₄	—	284	2	no	—
89	C₂×C₄	—	142	1	no	—
90	C₂×C₄	—	142	1	no	—
91	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
92	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
93	C₂×C₄	#11765866	1	1	yes	characteristic
94	C₈	—	2	1	no	—
95	C₈	—	4	2	no	—
96	C₈	—	284	1	no	—
97	C₈	Dic₂₈₄	1	2	yes	complemented
98	C₂²	#14556103	1	1	yes	characteristic
99	C₄	—	2	1	no	—
100	C₄	—	284	1	no	—
101	C₄	—	284	1	no	—
102	C₄	—	568	2	no	—
103	C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
104	C₄	#14556155	1	2	yes	—
105	C₄	#14556161	1	1	yes	characteristic
106	C₂	#17986210	1	1	yes	characteristic
107	C₂	#17986253	1	1	yes	characteristic
108	C₂	#17986275	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986404

C₂

yes

maximal, characteristic

#17986210

C₂

yes

maximal

#17986129

C₂

yes

maximal, characteristic

#17985746

C₂

yes

maximal

#334×C₇₁

C₂

yes

maximal, characteristic

C₂×Dic₂₈₄

—

#14556128

C₂²

yes

—

C₄×Dic₁₄₂

C₂²

yes

characteristic

#14556062

—

#14556062

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

Dic₇₁⋊C₈

—

#14556053

—

#14556038

—

C₄⋊C₈×C₇₁

C₂²

yes

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

D₄

yes

characteristic

Dic₂₈₄

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

D₄

yes

characteristic

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

C₂³

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

#41

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

Q₁₆

yes

complemented

C₇₁⋊C₈

—

C₂²×C₇₁

#121

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

C₂×Q₁₆

yes

characteristic

C₄×C₇₁

AGL(1,ℤ/8)

yes

—

C₄×C₇₁

#137

yes

characteristic

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

—

—

—

D₇₁

yes

characteristic

#332

—

C₂×Q₁₆

—

142

—

C₄⋊Q₈

—

C₄×Q₈

—

#107

—

142

—

#107

—

142

—

C₄⋊C₈

—

C₄⋊C₈

—

142

—

C₄⋊C₈

C₂×D₇₁

yes

—

#103

—

142

—

#97

—

142

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

142

—

Q₁₆

—

284

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

D₂₈₄

yes

characteristic

C₂×C₈

D₂₈₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄²

—

142

—

C₄²

C₂²×D₇₁

yes

characteristic

Q₈

—

284

—

Q₈

—

284

—

Q₈

—

142

—

C₂×C₄

—

284

—

C₂×C₄

—

142

—

C₂×C₄

—

142

—

C₂×C₄

D₄×D₇₁

yes

characteristic

C₂×C₄

C₂×D₂₈₄

yes

characteristic

C₂×C₄

#11765866

yes

characteristic

C₈

—

C₈

—

C₈

—

284

—

C₈

Dic₂₈₄

yes

complemented

C₂²

#14556103

yes

characteristic

C₄

—

100

C₄

—

284

—

101

C₄

—

284

—

102

C₄

—

568

—

103

C₄

C₂×Dic₂₈₄

yes

characteristic

104

C₄

#14556155

yes

—

105

C₄

#14556161

yes

characteristic

106

C₂

#17986210

yes

characteristic

107

C₂

#17986253

yes

characteristic

108

C₂

#17986275

yes

characteristic

	subgroup	G/H
center	C₂²	#14556103
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#334×C₇₁	C₂
socle	C₂²×C₇₁	#121
hypercenter	#334	D₇₁
hypocenter	C₇₁	#1694
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#14556103

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#334×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
41. C₈×C₇₁	Q₁₆	semidirect
55. C₇₁	#1694	semidirect
97. C₈	Dic₂₈₄	semidirect

H ≅ G/N

type

41. C₈×C₇₁

Q₁₆

semidirect

55. C₇₁

#1694

semidirect

97. C₈

Dic₂₈₄

semidirect

Subgroups — #22587859

Conjugacy classes of subgroups

Complements