Group #22586237 — subgroups

Subgroups — #22586237

Order 9088.

148 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 77 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	#17986023	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#11765849	—	2	1	no	—
5	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
6	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
7	C₄⋊(C₇₁⋊C₈)	C₄	1	2	yes	—
8	C₂²⋊(C₇₁⋊C₈)	C₄	1	2	yes	—
9	C₄×C₇₁⋊C₈	—	2	1	no	—
10	C₂²×C₈×C₇₁	—	2	1	no	—
11	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₇₁	—	2	1	no	—
13	C₄⋊C₈×C₇₁	C₄	1	2	yes	—
14	#11765849	—	4	1	no	—
15	#11765849	—	4	1	no	—
16	C₂×C₇₁⋊C₈	—	2	2	no	—
17	C₂×C₇₁⋊C₈	—	2	2	no	—
18	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₇₁	Q₈	1	1	yes	characteristic
21	C₂×C₈×C₇₁	—	2	2	no	—
22	C₂×C₈×C₇₁	—	4	1	no	—
23	C₂×C₈×C₇₁	—	4	1	no	—
24	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₇₁	D₄	1	1	yes	characteristic
27	C₄²×C₇₁	D₄	1	1	yes	characteristic
28	C₂³×C₇₁	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₇₁	—	2	2	no	—
30	C₂×C₄×C₇₁	—	4	1	no	—
31	C₂×C₄×C₇₁	C₄²	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	C₂²⋊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	D₈	1	1	yes	characteristic
38	C₂×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
39	C₂×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
40	C₂×C₄×C₇₁	Q₁₆	1	1	yes	characteristic
41	C₈×C₇₁	—	4	2	no	—
42	C₇₁⋊C₈	—	4	2	no	—
43	C₇₁⋊C₈	—	4	2	no	—
44	C₂²×C₇₁	—	2	1	no	—
45	C₂²×C₇₁	—	2	1	no	—
46	C₂²×C₇₁	#95	1	1	yes	characteristic
47	C₂²×C₇₁	#106	1	1	yes	characteristic
48	C₂²×C₇₁	#107	1	1	yes	characteristic
49	C₄×C₇₁	—	2	1	no	—
50	C₄×C₇₁	—	2	1	no	—
51	C₄×C₇₁	—	2	1	no	—
52	C₄×C₇₁	#102	1	1	yes	characteristic
53	C₄×C₇₁	#102	1	1	yes	characteristic
54	C₄×C₇₁	#103	1	1	yes	characteristic
55	C₄×C₇₁	#103	1	1	yes	characteristic
56	C₄×C₇₁	#108	1	2	yes	—
57	C₂×C₇₁	—	2	1	no	—
58	C₂×C₇₁	—	2	1	no	—
59	C₂×C₇₁	#340	1	1	yes	characteristic
60	C₂×C₇₁	#341	1	1	yes	characteristic
61	C₂×C₇₁	#344	1	1	yes	characteristic
62	#1301	—	71	1	no	maximal, Sylow
63	C₇₁	#1301	1	1	yes	characteristic, complemented, Sylow
64	#431	—	71	1	no	—
65	#423	—	71	1	no	—
66	C₂×C₄⋊C₈	D₇₁	1	1	yes	characteristic
67	C₂×M₁₆	—	142	1	no	—
68	C₂²×C₈	—	2	1	no	—
69	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
70	C₄⋊C₈	—	2	1	no	—
71	C₄⋊C₈	—	71	2	no	—
72	C₄⋊C₈	—	142	1	no	—
73	C₄⋊C₈	—	142	1	no	—
74	C₄⋊C₈	Dic₇₁	1	2	yes	—
75	C₂²⋊C₈	—	71	2	no	—
76	C₄×C₈	—	142	1	no	—
77	C₂²×C₄	Dic₁₄₂	1	1	yes	characteristic
78	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
80	M₁₆	—	284	1	no	—
81	M₁₆	—	284	1	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	4	1	no	—
84	C₂×C₈	—	4	1	no	—
85	C₂×C₈	—	142	2	no	—
86	C₂×C₈	—	142	2	no	—
87	C₄²	C₄×D₇₁	1	1	yes	characteristic
88	C₄²	D₂₈₄	1	1	yes	characteristic
89	C₄²	C₂×Dic₇₁	1	1	yes	characteristic
90	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
91	C₂³	Dic₇₁⋊C₄	1	1	yes	characteristic
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	C₄×Dic₇₁	1	1	yes	characteristic
95	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
96	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
97	C₂×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
98	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
99	C₂×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
100	C₂×C₄	#11765850	1	1	yes	characteristic
101	C₂×C₄	#11765852	1	1	yes	characteristic
102	C₂×C₄	#11765853	1	1	yes	characteristic
103	C₂×C₄	#11765855	1	1	yes	characteristic
104	C₈	—	4	2	no	—
105	C₈	—	284	2	no	—
106	C₈	—	284	2	no	—
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	C₇₁⋊#95	1	1	yes	characteristic
110	C₂²	#14556059	1	1	yes	characteristic
111	C₂²	#14556061	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	#14556048	1	1	yes	characteristic
116	C₄	#14556049	1	1	yes	characteristic
117	C₄	#14556051	1	1	yes	characteristic
118	C₄	#14556052	1	1	yes	characteristic
119	C₄	#14556064	1	2	yes	—
120	C₂	—	2	1	no	—
121	C₂	—	2	1	no	—
122	C₂	#17985761	1	1	yes	characteristic
123	C₂	#17985764	1	1	yes	characteristic
124	C₂	#17985771	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986052

C₂

yes

maximal, characteristic

#17986023

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

C₄

yes

—

C₂²⋊(C₇₁⋊C₈)

C₄

yes

—

C₄×C₇₁⋊C₈

—

C₂²×C₈×C₇₁

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

C₄

yes

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

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×C₈×C₇₁

—

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₂³×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄²

yes

characteristic

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

D₈

yes

characteristic

C₂×C₄×C₇₁

SD₁₆

yes

characteristic

C₂×C₄×C₇₁

SD₁₆

yes

characteristic

C₂×C₄×C₇₁

Q₁₆

yes

characteristic

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#431

—

#423

—

C₂×C₄⋊C₈

D₇₁

yes

characteristic

C₂×M₁₆

—

142

—

C₂²×C₈

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

142

—

C₄⋊C₈

—

142

—

C₄⋊C₈

Dic₇₁

yes

—

C₂²⋊C₈

—

C₄×C₈

—

142

—

C₂²×C₄

Dic₁₄₂

yes

characteristic

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

M₁₆

—

284

—

M₁₆

—

284

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₄²

C₄×D₇₁

yes

characteristic

C₄²

D₂₈₄

yes

characteristic

C₄²

C₂×Dic₇₁

yes

characteristic

C₄²

C₂²⋊D₇₁

yes

characteristic

C₂³

Dic₇₁⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₇₁

yes

characteristic

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

C₄⋊Dic₇₁

yes

characteristic

100

C₂×C₄

#11765850

yes

characteristic

101

C₂×C₄

#11765852

yes

characteristic

102

C₂×C₄

#11765853

yes

characteristic

103

C₂×C₄

#11765855

yes

characteristic

104

C₈

—

105

C₈

—

284

—

106

C₈

—

284

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₇₁⋊#95

yes

characteristic

110

C₂²

#14556059

yes

characteristic

111

C₂²

#14556061

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

#14556048

yes

characteristic

116

C₄

#14556049

yes

characteristic

117

C₄

#14556051

yes

characteristic

118

C₄

#14556052

yes

characteristic

119

C₄

#14556064

yes

—

120

C₂

—

121

C₂

—

122

C₂

#17985761

yes

characteristic

123

C₂

#17985764

yes

characteristic

124

C₂

#17985771

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₇₁)⋊C₄
commutator	C₂×C₄×C₇₁	C₄²
Frattini	C₂×C₄²	C₂×D₇₁
Fitting	C₂×C₄⋊C₈×C₇₁	C₂
socle	C₂²×C₇₁	#95
hypercenter	C₂×C₄⋊C₈	D₇₁
hypocenter	C₇₁	#1301
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

(C₂×D₇₁)⋊C₄

commutator

C₂×C₄×C₇₁

C₄²

Frattini

C₂×C₄²

C₂×D₇₁

Fitting

C₂×C₄⋊C₈×C₇₁

C₂

socle

C₂²×C₇₁

#95

hypercenter

C₂×C₄⋊C₈

D₇₁

hypocenter

C₇₁

#1301

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
63. C₇₁	#1301	semidirect

H ≅ G/N

type

63. C₇₁

#1301

semidirect

Subgroups — #22586237

Conjugacy classes of subgroups

Complements