Group #22586191 — subgroups

Subgroups — #22586191

Order 9088.

160 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 85 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	#17986052	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₄×M₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇₁⋊C₈	C₄	1	2	yes	—
5	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
6	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
7	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
8	C₂²⋊(C₇₁⋊C₈)	C₄	1	2	yes	—
9	C₄×C₇₁⋊C₈	—	2	1	no	—
10	C₂×M₁₆×C₇₁	C₄	1	2	yes	—
11	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
12	#97×C₇₁	—	2	1	no	—
13	C₄×C₈×C₇₁	—	2	1	no	—
14	C₂×C₇₁⋊C₈	—	2	2	no	—
15	C₂×C₇₁⋊C₈	—	2	2	no	—
16	C₂×C₇₁⋊C₈	—	4	2	no	—
17	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
20	M₁₆×C₇₁	—	2	2	no	—
21	M₁₆×C₇₁	C₈	1	4	yes	complemented
22	C₂×C₈×C₇₁	—	2	2	no	—
23	C₄²×C₇₁	D₄	1	1	yes	characteristic
24	C₄²×C₇₁	D₄	1	1	yes	characteristic
25	C₄²×C₇₁	D₄	1	1	yes	characteristic
26	C₄²×C₇₁	Q₈	1	1	yes	characteristic
27	C₂³×C₇₁	C₄²	1	1	yes	characteristic
28	C₂×C₄×C₇₁	—	2	1	no	—
29	C₂×C₄×C₇₁	—	2	1	no	—
30	C₂×C₄×C₇₁	—	4	1	no	—
31	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₇₁	C₂×C₈	1	2	yes	—
38	C₂×C₄×C₇₁	M₁₆	1	2	yes	—
39	C₈×C₇₁	—	2	4	no	—
40	C₇₁⋊C₈	—	4	2	no	—
41	C₇₁⋊C₈	—	4	2	no	—
42	C₂²×C₇₁	—	2	1	no	—
43	C₂²×C₇₁	—	2	1	no	—
44	C₂²×C₇₁	#95	1	1	yes	characteristic
45	C₂²×C₇₁	C₄×C₈	1	1	yes	characteristic
46	C₂²×C₇₁	#97	1	1	yes	characteristic
47	C₄×C₇₁	—	2	1	no	—
48	C₄×C₇₁	—	2	1	no	—
49	C₄×C₇₁	—	2	1	no	—
50	C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
51	C₄×C₇₁	#100	1	1	yes	characteristic
52	C₄×C₇₁	#101	1	1	yes	characteristic
53	C₄×C₇₁	C₄⋊C₈	1	2	yes	—
54	C₂×C₇₁	—	2	1	no	—
55	C₂×C₇₁	—	2	1	no	—
56	C₂×C₇₁	#336	1	1	yes	characteristic
57	C₂×C₇₁	#343	1	1	yes	characteristic
58	C₂×C₇₁	#344	1	1	yes	characteristic
59	#1283	—	71	1	no	maximal, Sylow
60	C₇₁	#1283	1	1	yes	characteristic, complemented, Sylow
61	#431	—	71	1	no	—
62	C₂×C₄⋊C₈	—	71	1	no	—
63	C₄×M₁₆	D₇₁	1	1	yes	characteristic
64	C₂×M₁₆	Dic₇₁	1	2	yes	—
65	C₂²×C₈	—	71	2	no	—
66	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
67	C₄⋊C₈	—	142	1	no	—
68	C₄⋊C₈	—	142	1	no	—
69	C₄⋊C₈	—	142	1	no	—
70	C₂²⋊C₈	—	71	2	no	—
71	#97	—	2	1	no	—
72	C₄×C₈	—	2	1	no	—
73	C₄×C₈	—	142	1	no	—
74	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
75	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
76	C₂²×C₄	C₂×Dic₇₁	1	1	yes	characteristic
77	M₁₆	—	2	2	no	—
78	M₁₆	C₇₁⋊C₈	1	4	yes	complemented
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	284	2	no	—
81	C₂×C₈	—	142	2	no	—
82	C₂×C₈	—	142	2	no	—
83	C₄²	Dic₁₄₂	1	1	yes	characteristic
84	C₄²	D₂₈₄	1	1	yes	characteristic
85	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
86	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
87	C₂³	C₄×Dic₇₁	1	1	yes	characteristic
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	4	1	no	—
91	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
92	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
93	C₂×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
94	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
95	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
96	C₂×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
97	C₂×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
98	C₂×C₄	#11765849	1	2	yes	—
99	C₈	—	2	4	no	—
100	C₈	—	284	2	no	—
101	C₈	—	284	2	no	—
102	C₂²	—	2	1	no	—
103	C₂²	—	2	1	no	—
104	C₂²	C₇₁⋊#95	1	1	yes	characteristic
105	C₂²	C₄×C₇₁⋊C₈	1	1	yes	characteristic
106	C₂²	#14556039	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
111	C₄	#14556044	1	1	yes	characteristic
112	C₄	#14556046	1	1	yes	characteristic
113	C₄	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
114	C₂	—	2	1	no	—
115	C₂	—	2	1	no	—
116	C₂	#17985754	1	1	yes	characteristic
117	C₂	#17985767	1	1	yes	characteristic
118	C₂	#17985771	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986052

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₁⋊C₈)

C₂

yes

maximal, characteristic

C₄×M₁₆×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

C₄

yes

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₂²⋊(C₇₁⋊C₈)

C₄

yes

—

C₄×C₇₁⋊C₈

—

C₂×M₁₆×C₇₁

C₄

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

#97×C₇₁

—

C₄×C₈×C₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

C₈

yes

complemented

C₂×C₈×C₇₁

—

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

Q₈

yes

characteristic

C₂³×C₇₁

C₄²

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₄

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

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×C₈

yes

—

C₂×C₄×C₇₁

M₁₆

yes

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#95

yes

characteristic

C₂²×C₇₁

C₄×C₈

yes

characteristic

C₂²×C₇₁

#97

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂²⋊C₈

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

C₄⋊C₈

yes

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#431

—

C₂×C₄⋊C₈

—

C₄×M₁₆

D₇₁

yes

characteristic

C₂×M₁₆

Dic₇₁

yes

—

C₂²×C₈

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

142

—

C₄⋊C₈

—

142

—

C₄⋊C₈

—

142

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

142

—

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

C₂×Dic₇₁

yes

characteristic

M₁₆

—

M₁₆

C₇₁⋊C₈

yes

complemented

C₂×C₈

—

C₂×C₈

—

284

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₄²

Dic₁₄₂

yes

characteristic

C₄²

D₂₈₄

yes

characteristic

C₄²

C₂²⋊D₇₁

yes

characteristic

C₄²

C₂²⋊D₇₁

yes

characteristic

C₂³

C₄×Dic₇₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

C₂×C₄

C₂²⋊Dic₇₁

yes

characteristic

C₂×C₄

Dic₇₁⋊C₄

yes

characteristic

C₂×C₄

Dic₇₁⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₇₁

yes

characteristic

C₂×C₄

C₂×C₇₁⋊C₈

yes

—

C₂×C₄

#11765849

yes

—

C₈

—

100

C₈

—

284

—

101

C₈

—

284

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₇₁⋊#95

yes

characteristic

105

C₂²

C₄×C₇₁⋊C₈

yes

characteristic

106

C₂²

#14556039

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

C₂²⋊(C₇₁⋊C₈)

yes

—

111

C₄

#14556044

yes

characteristic

112

C₄

#14556046

yes

characteristic

113

C₄

C₄⋊(C₇₁⋊C₈)

yes

—

114

C₂

—

115

C₂

—

116

C₂

#17985754

yes

characteristic

117

C₂

#17985767

yes

characteristic

118

C₂

#17985771

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₇₁)⋊C₄
commutator	C₂²×C₇₁	C₄×C₈
Frattini	C₂×C₄²	C₂×D₇₁
Fitting	C₄×M₁₆×C₇₁	C₂
socle	C₂²×C₇₁	#95
hypercenter	C₄×M₁₆	D₇₁
hypocenter	C₇₁	#1283
LCS	G ▷ C₂²×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

(C₂×D₇₁)⋊C₄

commutator

C₂²×C₇₁

C₄×C₈

Frattini

C₂×C₄²

C₂×D₇₁

Fitting

C₄×M₁₆×C₇₁

C₂

socle

C₂²×C₇₁

#95

hypercenter

C₄×M₁₆

D₇₁

hypocenter

C₇₁

#1283

LCS

G ▷ C₂²×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
21. M₁₆×C₇₁	C₈	semidirect
60. C₇₁	#1283	semidirect
78. M₁₆	C₇₁⋊C₈	semidirect

H ≅ G/N

type

21. M₁₆×C₇₁

C₈

semidirect

60. C₇₁

#1283

semidirect

78. M₁₆

C₇₁⋊C₈

semidirect

Subgroups — #22586191

Conjugacy classes of subgroups

Complements