Group #22587710 — subgroups

Subgroups — #22587710

Order 9088.

156 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 61 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	#17986379	C₂	1	1	yes	maximal, characteristic
2	#17986284	C₂	1	2	yes	maximal
3	#17985752	C₂	1	1	yes	maximal, characteristic
4	#17985746	C₂	1	2	yes	maximal
5	C₈×Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₈₄	—	2	1	no	—
7	C₂×Dic₂₈₄	—	2	1	no	—
8	#14556128	C₂²	1	2	yes	—
9	#14556059	—	2	2	no	—
10	C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
11	#14556052	—	2	2	no	—
12	C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₇₁	—	2	1	no	—
14	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₇₁	—	2	1	no	—
16	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₄₂	—	2	2	no	—
18	Dic₂₈₄	—	4	2	no	—
19	Dic₂₈₄	—	4	2	no	—
20	C₂×C₇₁⋊C₈	—	2	2	no	—
21	C₄⋊Dic₇₁	—	2	2	no	—
22	C₄⋊Dic₇₁	—	2	2	no	—
23	C₂×Q₈×C₇₁	D₄	1	1	yes	characteristic
24	C₂×C₈×C₇₁	—	2	1	no	—
25	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
26	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
27	C₄⋊C₄×C₇₁	—	2	1	no	—
28	C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
29	C₄²×C₇₁	—	2	1	no	—
30	C₄²×C₇₁	C₂³	1	1	yes	characteristic
31	Q₈×C₇₁	—	2	1	no	—
32	Q₈×C₇₁	Q₁₆	1	2	yes	complemented
33	C₂×C₄×C₇₁	—	2	1	no	—
34	C₂×C₄×C₇₁	—	2	1	no	—
35	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
38	C₂×Dic₇₁	—	4	2	no	—
39	Dic₁₄₂	—	4	2	no	—
40	Dic₁₄₂	—	4	2	no	—
41	C₈×C₇₁	—	2	1	no	—
42	C₈×C₇₁	—	4	1	no	—
43	C₈×C₇₁	SD₁₆	1	2	yes	—
44	C₇₁⋊C₈	—	4	2	no	—
45	C₂²×C₇₁	#121	1	1	yes	characteristic
46	C₄×C₇₁	—	2	2	no	—
47	C₄×C₇₁	—	2	1	no	—
48	C₄×C₇₁	—	4	1	no	—
49	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
50	C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
51	C₄×C₇₁	#135	1	1	yes	characteristic
52	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
53	Dic₇₁	—	8	2	no	—
54	C₂×C₇₁	#460	1	1	yes	characteristic
55	C₂×C₇₁	#467	1	1	yes	characteristic
56	C₂×C₇₁	#473	1	1	yes	characteristic
57	#1670	—	71	1	no	maximal, Sylow
58	C₇₁	#1670	1	1	yes	characteristic, complemented, Sylow
59	#494	—	71	1	no	—
60	#475	—	71	2	no	—
61	C₈×Q₈	D₇₁	1	1	yes	characteristic
62	#335	—	71	1	no	—
63	#332	—	71	2	no	—
64	C₂×Q₁₆	—	142	1	no	—
65	C₂×Q₁₆	—	142	1	no	—
66	C₄⋊Q₈	—	71	2	no	—
67	C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
68	#106	—	142	2	no	—
69	C₄⋊C₈	—	2	1	no	—
70	C₄⋊C₈	—	71	2	no	—
71	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
72	#103	—	142	2	no	—
73	C₄×C₈	—	2	1	no	—
74	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
75	C₂×Q₈	—	142	2	no	—
76	C₂×Q₈	D₂₈₄	1	1	yes	characteristic
77	Q₁₆	—	284	2	no	—
78	Q₁₆	—	284	2	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	142	2	no	—
81	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
82	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
83	C₄⋊C₄	—	2	1	no	—
84	C₄⋊C₄	—	142	2	no	—
85	C₄⋊C₄	—	142	2	no	—
86	C₄⋊C₄	D₂₈₄	1	1	yes	characteristic
87	C₄²	—	2	1	no	—
88	C₄²	C₂²×D₇₁	1	1	yes	characteristic
89	Q₈	—	2	1	no	—
90	Q₈	—	284	2	no	—
91	Q₈	—	284	2	no	—
92	Q₈	Dic₂₈₄	1	2	yes	complemented
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	284	2	no	—
96	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
97	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
98	C₂×C₄	#11765865	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	—	4	1	no	—
101	C₈	—	284	2	no	—
102	C₈	#11765852	1	2	yes	—
103	C₂²	#14556102	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	2	1	no	—
106	C₄	—	4	1	no	—
107	C₄	—	568	2	no	—
108	C₄	C₂×#11765852	1	1	yes	characteristic
109	C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
110	C₄	#14556150	1	1	yes	characteristic
111	C₄	#14556153	1	1	yes	characteristic
112	C₂	#17986199	1	1	yes	characteristic
113	C₂	#17986240	1	1	yes	characteristic
114	C₂	#17986274	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986379

C₂

yes

maximal, characteristic

#17986284

C₂

yes

maximal

#17985752

C₂

yes

maximal, characteristic

#17985746

C₂

yes

maximal

C₈×Q₈×C₇₁

C₂

yes

maximal, characteristic

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

—

#14556128

C₂²

yes

—

#14556059

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

#14556052

—

C₄×Q₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₄×C₈×C₇₁

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

Dic₂₈₄

—

Dic₂₈₄

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂×Q₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

Q₁₆

yes

complemented

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

#41

yes

characteristic

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

SD₁₆

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

#121

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₄×C₇₁

C₂×Q₁₆

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#494

—

#475

—

C₈×Q₈

D₇₁

yes

characteristic

#335

—

#332

—

C₂×Q₁₆

—

142

—

C₂×Q₁₆

—

142

—

C₄⋊Q₈

—

C₄×Q₈

C₂×D₇₁

yes

characteristic

#106

—

142

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₇₁

yes

characteristic

#103

—

142

—

C₄×C₈

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×Q₈

—

142

—

C₂×Q₈

D₂₈₄

yes

characteristic

Q₁₆

—

284

—

Q₁₆

—

284

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄⋊C₄

D₂₈₄

yes

characteristic

C₄²

—

C₄²

C₂²×D₇₁

yes

characteristic

Q₈

—

Q₈

—

284

—

Q₈

—

284

—

Q₈

Dic₂₈₄

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

284

—

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

C₂×C₄

C₂×D₂₈₄

yes

characteristic

C₂×C₄

#11765865

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

284

—

102

C₈

#11765852

yes

—

103

C₂²

#14556102

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

568

—

108

C₄

C₂×#11765852

yes

characteristic

109

C₄

C₂×Dic₂₈₄

yes

characteristic

110

C₄

#14556150

yes

characteristic

111

C₄

#14556153

yes

characteristic

112

C₂

#17986199

yes

characteristic

113

C₂

#17986240

yes

characteristic

114

C₂

#17986274

yes

characteristic

	subgroup	G/H
center	C₂²	#14556102
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	C₈×Q₈×C₇₁	C₂
socle	C₂²×C₇₁	#121
hypercenter	C₈×Q₈	D₇₁
hypocenter	C₇₁	#1670
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

subgroup

G/H

center

C₂²

#14556102

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

C₈×Q₈×C₇₁

C₂

socle

C₂²×C₇₁

#121

hypercenter

C₈×Q₈

D₇₁

hypocenter

C₇₁

#1670

LCS

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

N	H ≅ G/N	type
32. Q₈×C₇₁	Q₁₆	semidirect
58. C₇₁	#1670	semidirect
92. Q₈	Dic₂₈₄	semidirect

H ≅ G/N

type

32. Q₈×C₇₁

Q₁₆

semidirect

58. C₇₁

#1670

semidirect

92. Q₈

Dic₂₈₄

semidirect

Subgroups — #22587710

Conjugacy classes of subgroups

Complements