Group #22590142 — subgroups

Subgroups — #22590142

Order 9088.

152 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 73 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	C₄×C₈×D₇₁	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂×D₇₁)⋊C₁₆	C₂	1	2	yes	maximal
4	#17985774	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₇₁	C₄	1	1	yes	characteristic
7	C₂×C₈×D₇₁	C₂²	1	1	yes	characteristic
8	C₄²×D₇₁	C₄	1	1	yes	characteristic
9	C₂×C₇₁⋊C₁₆	C₂²	1	2	yes	—
10	C₄×C₇₁⋊C₈	C₄	1	1	yes	characteristic
11	C₈×Dic₇₁	C₄	1	1	yes	characteristic
12	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₇₁	C₂²	1	2	yes	—
14	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₇₁	C₈	1	1	yes	characteristic
16	C₂×C₄×D₇₁	C₈	1	1	yes	characteristic
17	C₂×C₄×D₇₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₈×D₇₁	—	2	1	no	—
21	C₈×D₇₁	—	2	1	no	—
22	C₈×D₇₁	—	2	2	no	—
23	C₄×Dic₇₁	C₈	1	1	yes	characteristic
24	C₄×Dic₇₁	C₈	1	1	yes	characteristic
25	C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
26	C₇₁⋊C₁₆	—	2	2	no	—
27	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
29	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₁₆×C₇₁	—	2	2	no	—
31	C₂²×D₇₁	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×Dic₇₁	C₂×C₈	1	1	yes	characteristic
36	C₂×Dic₇₁	C₂×C₈	1	1	yes	characteristic
37	C₂×Dic₇₁	C₂×C₈	1	1	yes	characteristic
38	C₄×D₇₁	—	2	2	no	—
39	C₄×D₇₁	—	2	2	no	—
40	C₄×D₇₁	—	2	1	no	—
41	C₄×D₇₁	—	2	1	no	—
42	C₈×C₇₁	—	2	1	no	—
43	C₈×C₇₁	#41	1	1	yes	characteristic
44	C₈×C₇₁	#41	1	1	yes	characteristic
45	C₇₁⋊C₈	—	2	1	no	—
46	C₇₁⋊C₈	M₁₆	1	2	yes	—
47	C₂²×C₇₁	C₂²×C₈	1	1	yes	characteristic
48	C₂×D₇₁	—	2	1	no	—
49	C₂×D₇₁	—	2	1	no	—
50	C₂×D₇₁	M₃₂	1	2	yes	—
51	C₄×C₇₁	—	2	1	no	—
52	C₄×C₇₁	M₃₂	1	2	yes	—
53	C₄×C₇₁	#117	1	1	yes	characteristic
54	C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
55	Dic₇₁	—	2	1	no	—
56	Dic₇₁	—	2	1	no	—
57	Dic₇₁	—	2	1	no	—
58	C₂×C₇₁	#431	1	1	yes	characteristic
59	C₂×C₇₁	C₂×M₃₂	1	1	yes	characteristic
60	C₂×C₇₁	C₂×M₃₂	1	1	yes	characteristic
61	D₇₁	—	2	2	no	—
62	#2163	—	71	1	no	maximal, Sylow
63	C₇₁	#2163	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄×C₈	—	71	1	no	—
65	C₄⋊C₁₆	—	71	1	no	—
66	C₄⋊C₁₆	D₇₁	1	1	yes	characteristic, complemented
67	C₂²⋊C₁₆	—	71	2	no	—
68	#346	—	71	2	no	—
69	C₂²×C₈	—	71	1	no	—
70	C₂²×C₈	—	71	1	no	—
71	C₂×C₄²	—	71	1	no	—
72	C₂×C₁₆	—	71	2	no	—
73	C₂×C₁₆	C₂×D₇₁	1	2	yes	—
74	C₄×C₈	—	71	1	no	—
75	C₄×C₈	—	71	1	no	—
76	C₄×C₈	—	71	1	no	—
77	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
78	C₂²×C₄	—	71	1	no	—
79	C₂²×C₄	—	71	1	no	—
80	C₂²×C₄	—	71	1	no	—
81	C₂×C₈	—	71	1	no	—
82	C₂×C₈	—	71	1	no	—
83	C₂×C₈	—	142	2	no	—
84	C₂×C₈	—	142	1	no	—
85	C₂×C₈	—	142	1	no	—
86	C₂×C₈	C₄×D₇₁	1	1	yes	characteristic
87	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
88	C₄²	—	71	1	no	—
89	C₄²	—	71	1	no	—
90	C₄²	—	71	1	no	—
91	C₄²	C₄×D₇₁	1	1	yes	characteristic
92	C₁₆	—	2	2	no	—
93	C₁₆	—	142	2	no	—
94	C₂³	—	71	1	no	—
95	C₂×C₄	—	71	1	no	—
96	C₂×C₄	—	71	1	no	—
97	C₂×C₄	—	71	1	no	—
98	C₂×C₄	—	142	2	no	—
99	C₂×C₄	—	142	2	no	—
100	C₂×C₄	—	142	1	no	—
101	C₂×C₄	—	142	1	no	—
102	C₂×C₄	C₈×D₇₁	1	1	yes	characteristic
103	C₂×C₄	C₈×D₇₁	1	1	yes	characteristic
104	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
105	C₈	—	2	1	no	—
106	C₈	—	71	2	no	—
107	C₈	—	142	1	no	—
108	C₈	#11765864	1	1	yes	characteristic
109	C₈	#11765866	1	1	yes	characteristic
110	C₂²	—	71	2	no	—
111	C₂²	—	142	1	no	—
112	C₂²	—	142	1	no	—
113	C₂²	C₂×C₈×D₇₁	1	1	yes	characteristic
114	C₄	—	2	1	no	—
115	C₄	—	142	1	no	—
116	C₄	—	142	1	no	—
117	C₄	—	142	1	no	—
118	C₄	#14556067	1	2	yes	—
119	C₄	#14556086	1	1	yes	characteristic
120	C₄	M₁₆×D₇₁	1	1	yes	characteristic
121	C₂	—	142	2	no	—
122	C₂	#17986051	1	1	yes	characteristic
123	C₂	M₃₂×D₇₁	1	1	yes	characteristic
124	C₂	C₂×#14556067	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₈×D₇₁

C₂

yes

maximal, characteristic

C₄⋊(C₇₁⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

(C₂×D₇₁)⋊C₁₆

C₂

yes

maximal

#17985774

C₂

yes

maximal, complemented

C₄⋊C₁₆×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₇₁

C₄

yes

characteristic

C₂×C₈×D₇₁

C₂²

yes

characteristic

C₄²×D₇₁

C₄

yes

characteristic

C₂×C₇₁⋊C₁₆

C₂²

yes

—

C₄×C₇₁⋊C₈

C₄

yes

characteristic

C₈×Dic₇₁

C₄

yes

characteristic

C₈×Dic₇₁

C₂²

yes

characteristic

C₂×C₁₆×C₇₁

C₂²

yes

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₄×D₇₁

C₈

yes

characteristic

C₂×C₄×D₇₁

C₈

yes

characteristic

C₂×C₄×D₇₁

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₈×D₇₁

—

C₈×D₇₁

—

C₈×D₇₁

—

C₄×Dic₇₁

C₈

yes

characteristic

C₄×Dic₇₁

C₈

yes

characteristic

C₄×Dic₇₁

C₂×C₄

yes

characteristic

C₇₁⋊C₁₆

—

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₁₆×C₇₁

—

C₂²×D₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×Dic₇₁

C₂×C₈

yes

characteristic

C₂×Dic₇₁

C₂×C₈

yes

characteristic

C₂×Dic₇₁

C₂×C₈

yes

characteristic

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

—

C₈×C₇₁

—

C₈×C₇₁

yes

C₈×C₇₁

yes

C₇₁⋊C₈

—

C₇₁⋊C₈

M₁₆

yes

—

C₂²×C₇₁

C₂²×C₈

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

M₃₂

yes

—

C₄×C₇₁

—

C₄×C₇₁

M₃₂

yes

—

C₄×C₇₁

#117

yes

characteristic

C₄×C₇₁

C₂×M₁₆

yes

characteristic

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

#431

yes

characteristic

C₂×C₇₁

C₂×M₃₂

yes

characteristic

C₂×C₇₁

C₂×M₃₂

yes

characteristic

D₇₁

—

#2163

—

maximal, Sylow

C₇₁

#2163

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

D₇₁

yes

characteristic, complemented

C₂²⋊C₁₆

—

#346

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₁

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

C₄×D₇₁

yes

characteristic

C₂×C₈

C₂²×D₇₁

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄×D₇₁

yes

characteristic

C₁₆

—

C₁₆

—

142

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

142

—

C₂×C₄

—

142

—

100

C₂×C₄

—

142

—

101

C₂×C₄

—

142

—

102

C₂×C₄

C₈×D₇₁

yes

characteristic

103

C₂×C₄

C₈×D₇₁

yes

characteristic

104

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

—

142

—

108

C₈

#11765864

yes

characteristic

109

C₈

#11765866

yes

characteristic

110

C₂²

—

111

C₂²

—

142

—

112

C₂²

—

142

—

113

C₂²

C₂×C₈×D₇₁

yes

characteristic

114

C₄

—

115

C₄

—

142

—

116

C₄

—

142

—

117

C₄

—

142

—

118

C₄

#14556067

yes

—

119

C₄

#14556086

yes

characteristic

120

C₄

M₁₆×D₇₁

yes

characteristic

121

C₂

—

142

—

122

C₂

#17986051

yes

characteristic

123

C₂

M₃₂×D₇₁

yes

characteristic

124

C₂

C₂×#14556067

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₇₁
commutator, socle	C₂²×C₇₁	C₂²×C₈
Fitting	C₄⋊C₁₆×C₇₁	C₂
hypercenter	C₄⋊C₁₆	D₇₁
hypocenter	C₇₁	#2163
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₇₁

commutator, socle

C₂²×C₇₁

C₂²×C₈

Fitting

C₄⋊C₁₆×C₇₁

C₂

hypercenter

C₄⋊C₁₆

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
2. C₄⋊(C₇₁⋊C₁₆)	C₂	semidirect
4. #17985774	C₂	semidirect
5. C₄⋊C₁₆×C₇₁	C₂	semidirect
63. C₇₁	#2163	semidirect
66. C₄⋊C₁₆	D₇₁	semidirect

H ≅ G/N

type

2. C₄⋊(C₇₁⋊C₁₆)

C₂

semidirect

4. #17985774

C₂

semidirect

5. C₄⋊C₁₆×C₇₁

C₂

semidirect

63. C₇₁

#2163

semidirect

66. C₄⋊C₁₆

D₇₁

semidirect

Subgroups — #22590142

Conjugacy classes of subgroups

Complements