Group #22586272 — subgroups

Subgroups — #22586272

Order 9088.

170 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 41 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	#17985725	C₂	1	2	yes	maximal, complemented
2	#390×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
4	C₂²×D₄×C₇₁	C₄	1	1	yes	characteristic
5	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
6	C₂×C₂²⋊C₄×C₇₁	—	2	2	no	—
7	C₂×C₄²×C₇₁	C₄	1	1	yes	characteristic
8	C₂×C₇₁⋊C₈	—	4	2	no	—
9	C₂⁴×C₇₁	—	2	1	no	—
10	C₂×D₄×C₇₁	—	2	1	no	—
11	C₂×D₄×C₇₁	—	4	1	no	—
12	C₂×D₄×C₇₁	—	4	1	no	—
13	C₂×D₄×C₇₁	C₈	1	2	yes	complemented
14	C₂²×C₄×C₇₁	—	2	1	no	—
15	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₇₁	D₄	1	2	yes	—
17	C₄⋊C₄×C₇₁	—	4	1	no	—
18	C₂²⋊C₄×C₇₁	—	4	4	no	—
19	C₄²×C₇₁	—	4	1	no	—
20	C₂³×C₇₁	—	2	2	no	—
21	C₂³×C₇₁	—	4	2	no	—
22	C₂³×C₇₁	—	4	2	no	—
23	C₂³×C₇₁	—	4	1	no	—
24	C₂³×C₇₁	—	4	1	no	—
25	C₂³×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
26	D₄×C₇₁	—	4	2	no	—
27	D₄×C₇₁	—	4	2	no	—
28	C₂×C₄×C₇₁	—	2	1	no	—
29	C₂×C₄×C₇₁	—	2	1	no	—
30	C₂×C₄×C₇₁	—	2	2	no	—
31	C₂×C₄×C₇₁	—	2	2	no	—
32	C₂×C₄×C₇₁	—	4	1	no	—
33	C₂×C₄×C₇₁	—	4	1	no	—
34	C₂×C₄×C₇₁	—	4	2	no	—
35	C₂×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×C₇₁	M₁₆	1	1	yes	characteristic
37	C₇₁⋊C₈	—	8	2	no	—
38	C₂²×C₇₁	—	2	1	no	—
39	C₂²×C₇₁	—	2	1	no	—
40	C₂²×C₇₁	—	4	4	no	—
41	C₂²×C₇₁	—	4	2	no	—
42	C₂²×C₇₁	—	8	2	no	—
43	C₂²×C₇₁	—	8	1	no	—
44	C₂²×C₇₁	—	8	1	no	—
45	C₂²×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
46	C₂²×C₇₁	#99	1	1	yes	characteristic
47	C₂²×C₇₁	#100	1	1	yes	characteristic
48	C₄×C₇₁	—	2	1	no	—
49	C₄×C₇₁	—	2	1	no	—
50	C₄×C₇₁	—	4	2	no	—
51	C₄×C₇₁	—	4	2	no	—
52	C₂×C₇₁	—	2	1	no	—
53	C₂×C₇₁	—	2	1	no	—
54	C₂×C₇₁	—	8	2	no	—
55	C₂×C₇₁	#323	1	1	yes	characteristic
56	C₂×C₇₁	#353	1	1	yes	characteristic
57	C₂×C₇₁	#355	1	1	yes	characteristic
58	#1318	—	71	1	no	maximal, Sylow
59	C₇₁	#1318	1	1	yes	characteristic, complemented, Sylow
60	#390	D₇₁	1	1	yes	characteristic
61	#324	—	71	2	no	—
62	C₂²×D₄	Dic₇₁	1	1	yes	characteristic
63	C₂×C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
64	C₂×C₂²⋊C₄	—	2	2	no	—
65	C₂×C₄²	Dic₇₁	1	1	yes	characteristic
66	C₂²⋊C₈	—	142	2	no	—
67	C₂⁴	—	2	1	no	—
68	C₂×D₄	—	2	1	no	—
69	C₂×D₄	—	4	1	no	—
70	C₂×D₄	—	4	1	no	—
71	C₂×D₄	C₇₁⋊C₈	1	2	yes	complemented
72	C₂²×C₄	—	2	1	no	—
73	C₂²×C₄	C₂×Dic₇₁	1	1	yes	characteristic
74	C₂²×C₄	C₂²⋊D₇₁	1	2	yes	—
75	C₂×C₈	—	284	2	no	—
76	C₄⋊C₄	—	4	1	no	—
77	C₂²⋊C₄	—	4	4	no	—
78	C₄²	—	4	1	no	—
79	C₂³	—	2	2	no	—
80	C₂³	—	4	2	no	—
81	C₂³	—	4	2	no	—
82	C₂³	—	4	1	no	—
83	C₂³	—	4	1	no	—
84	C₂³	C₂²⋊Dic₇₁	1	1	yes	characteristic
85	D₄	—	4	2	no	—
86	D₄	—	4	2	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	4	1	no	—
92	C₂×C₄	—	4	1	no	—
93	C₂×C₄	—	4	2	no	—
94	C₂×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
95	C₂×C₄	#11765849	1	1	yes	characteristic
96	C₈	—	568	2	no	—
97	C₂²	—	2	1	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	4	4	no	—
100	C₂²	—	4	2	no	—
101	C₂²	—	8	2	no	—
102	C₂²	—	8	1	no	—
103	C₂²	—	8	1	no	—
104	C₂²	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
105	C₂²	#14556043	1	1	yes	characteristic
106	C₂²	#14556045	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	—	4	2	no	—
110	C₄	—	4	2	no	—
111	C₂	—	2	1	no	—
112	C₂	—	2	1	no	—
113	C₂	—	8	2	no	—
114	C₂	#17985723	1	1	yes	characteristic
115	C₂	#17985792	1	1	yes	characteristic
116	C₂	#17985796	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17985725

C₂

yes

maximal, complemented

#390×C₇₁

C₂

yes

maximal, characteristic

C₂²⋊(C₇₁⋊C₈)

—

C₂²×D₄×C₇₁

C₄

yes

characteristic

C₂×C₄⋊C₄×C₇₁

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₇₁

—

C₂×C₄²×C₇₁

C₄

yes

characteristic

C₂×C₇₁⋊C₈

—

C₂⁴×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

C₈

yes

complemented

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

—

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

C₂²⋊C₄

yes

characteristic

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₇₁

M₁₆

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂²⋊C₈

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#390

D₇₁

yes

characteristic

#324

—

C₂²×D₄

Dic₇₁

yes

characteristic

C₂×C₄⋊C₄

C₂×D₇₁

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

Dic₇₁

yes

characteristic

C₂²⋊C₈

—

142

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₇₁⋊C₈

yes

complemented

C₂²×C₄

—

C₂²×C₄

C₂×Dic₇₁

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

—

C₂×C₈

—

284

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊Dic₇₁

yes

characteristic

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₇₁⋊C₈

yes

characteristic

C₂×C₄

#11765849

yes

characteristic

C₈

—

568

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₂²⋊(C₇₁⋊C₈)

yes

characteristic

105

C₂²

#14556043

yes

characteristic

106

C₂²

#14556045

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₂

—

112

C₂

—

113

C₂

—

114

C₂

#17985723

yes

characteristic

115

C₂

#17985792

yes

characteristic

116

C₂

#17985796

yes

characteristic

	subgroup	G/H
center	C₂²	#14556043
commutator	C₂×C₄×C₇₁	C₂×C₈
Frattini	C₂×C₄⋊C₄	C₂×D₇₁
Fitting	#390×C₇₁	C₂
socle	C₂²×C₇₁	#99
hypercenter	#390	D₇₁
hypocenter	C₇₁	#1318
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #390

subgroup

G/H

center

C₂²

#14556043

commutator

C₂×C₄×C₇₁

C₂×C₈

Frattini

C₂×C₄⋊C₄

C₂×D₇₁

Fitting

#390×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #390

N	H ≅ G/N	type
1. #17985725	C₂	semidirect
13. C₂×D₄×C₇₁	C₈	semidirect
59. C₇₁	#1318	semidirect
71. C₂×D₄	C₇₁⋊C₈	semidirect

H ≅ G/N

type

1. #17985725

C₂

semidirect

13. C₂×D₄×C₇₁

C₈

semidirect

59. C₇₁

#1318

semidirect

71. C₂×D₄

C₇₁⋊C₈

semidirect

Subgroups — #22586272

Conjugacy classes of subgroups

Complements