Group #22589856 — subgroups

Subgroups — #22589856

Order 9088.

222 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 65 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	#17986048	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986047	C₂	1	2	yes	maximal, complemented
3	#17985963	C₂	1	2	yes	maximal
4	C₇₁⋊#338	C₂	1	1	yes	maximal, characteristic, complemented
5	#407×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#11765849	C₂²	1	2	yes	—
7	C₂×#11765849	C₂²	1	2	yes	—
8	#14556064	—	2	2	no	—
9	#14556063	—	2	4	no	—
10	C₂²⋊(C₇₁⋊C₈)	—	2	4	no	—
11	C₂³×C₄×C₇₁	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
14	C₂²⋊C₈×C₇₁	—	2	2	no	—
15	#11765849	—	2	2	no	—
16	#11765849	—	2	2	no	—
17	#11765849	—	2	2	no	—
18	#11765849	—	2	2	no	—
19	C₂×C₇₁⋊C₈	—	2	2	no	—
20	C₂×C₇₁⋊C₈	D₄	1	4	yes	—
21	C₂⁴×C₇₁	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₇₁	—	2	1	no	—
23	C₂²×C₄×C₇₁	—	2	2	no	—
24	C₂²×C₄×C₇₁	—	2	2	no	—
25	C₂²×C₄×C₇₁	—	2	1	no	—
26	C₂²×C₄×C₇₁	Q₈	1	1	yes	characteristic
27	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
28	M₁₆×C₇₁	—	2	2	no	—
29	M₁₆×C₇₁	—	2	2	no	—
30	C₂×C₈×C₇₁	—	2	1	no	—
31	C₂×C₈×C₇₁	D₄	1	2	yes	—
32	C₂³×C₇₁	—	2	2	no	—
33	C₂³×C₇₁	—	2	1	no	—
34	C₂³×C₇₁	—	4	2	no	—
35	C₂³×C₇₁	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₇₁	—	2	2	no	—
37	C₂×C₄×C₇₁	—	2	1	no	—
38	C₂×C₄×C₇₁	—	4	1	no	—
39	C₂×C₄×C₇₁	—	4	2	no	—
40	C₂×C₄×C₇₁	—	4	1	no	—
41	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
42	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₇₁	#41	1	2	yes	—
44	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
45	C₈×C₇₁	—	2	2	no	—
46	C₈×C₇₁	—	4	1	no	—
47	C₇₁⋊C₈	—	2	4	no	—
48	C₇₁⋊C₈	—	4	2	no	—
49	C₂²×C₇₁	—	2	4	no	—
50	C₂²×C₇₁	—	2	2	no	—
51	C₂²×C₇₁	—	4	1	no	—
52	C₂²×C₇₁	—	4	2	no	—
53	C₂²×C₇₁	—	4	1	no	—
54	C₂²×C₇₁	—	4	1	no	—
55	C₂²×C₇₁	C₂²⋊Q₈	1	2	yes	—
56	C₂²×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
57	C₄×C₇₁	—	4	1	no	—
58	C₄×C₇₁	#123	1	1	yes	characteristic
59	C₄×C₇₁	#123	1	2	yes	—
60	C₄×C₇₁	#127	1	1	yes	characteristic
61	C₂×C₇₁	—	2	2	no	—
62	C₂×C₇₁	—	2	1	no	—
63	C₂×C₇₁	—	4	2	no	—
64	C₂×C₇₁	#399	1	1	yes	characteristic
65	#2091	—	71	1	no	maximal, Sylow
66	C₇₁	#2091	1	1	yes	characteristic, complemented, Sylow
67	#430	—	71	1	no	—
68	#430	—	71	2	no	—
69	#407	—	71	2	no	—
70	#407	D₇₁	1	1	yes	characteristic
71	#338	—	71	1	no	—
72	C₂³×C₄	C₂×D₇₁	1	1	yes	characteristic
73	C₂×M₁₆	—	71	2	no	—
74	C₂×M₁₆	—	71	2	no	—
75	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
76	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
77	#108	—	142	4	no	—
78	#108	—	142	2	no	—
79	C₂²⋊C₈	—	2	2	no	—
80	C₂²⋊C₈	—	142	4	no	—
81	C₂⁴	Dic₁₄₂	1	1	yes	characteristic
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	2	1	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	2	1	no	—
86	C₂²×C₄	Dic₁₄₂	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
88	M₁₆	—	2	2	no	—
89	M₁₆	—	2	2	no	—
90	M₁₆	—	142	2	no	—
91	M₁₆	—	142	2	no	—
92	M₁₆	—	142	2	no	—
93	M₁₆	—	142	2	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	71	4	no	—
96	C₂×C₈	—	142	2	no	—
97	C₂×C₈	C₂²⋊D₇₁	1	2	yes	—
98	C₂³	—	2	2	no	—
99	C₂³	—	2	1	no	—
100	C₂³	—	4	2	no	—
101	C₂³	C₂×Dic₁₄₂	1	1	yes	characteristic
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	D₄×D₇₁	1	2	yes	—
108	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
109	C₂×C₄	#11765864	1	2	yes	—
110	C₂×C₄	#11765865	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	4	1	no	—
113	C₈	—	284	2	no	—
114	C₈	—	142	4	no	—
115	C₂²	—	2	4	no	—
116	C₂²	—	2	2	no	—
117	C₂²	—	4	1	no	—
118	C₂²	—	4	2	no	—
119	C₂²	—	4	1	no	—
120	C₂²	—	4	1	no	—
121	C₂²	#14556105	1	2	yes	—
122	C₂²	#14556107	1	1	yes	characteristic
123	C₄	—	4	1	no	—
124	C₄	#14556109	1	1	yes	characteristic
125	C₄	#14556110	1	2	yes	—
126	C₄	#14556124	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	—	2	1	no	—
129	C₂	—	4	2	no	—
130	C₂	#17985935	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986048

C₂

yes

maximal, characteristic, complemented

#17986047

C₂

yes

maximal, complemented

#17985963

C₂

yes

maximal

C₇₁⋊#338

C₂

yes

maximal, characteristic, complemented

#407×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

C₂²

yes

—

C₂×#11765849

C₂²

yes

—

#14556064

—

#14556063

—

C₂²⋊(C₇₁⋊C₈)

—

C₂³×C₄×C₇₁

C₂²

yes

characteristic

C₂×M₁₆×C₇₁

C₂²

yes

characteristic

C₂×M₁₆×C₇₁

C₂²

yes

characteristic

C₂²⋊C₈×C₇₁

—

—

—

—

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₄

yes

—

C₂⁴×C₇₁

Q₈

yes

characteristic

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

Q₈

yes

characteristic

C₂²×C₄×C₇₁

C₂³

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

D₄

yes

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

#41

yes

—

C₂×C₄×C₇₁

#41

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂²⋊Q₈

yes

—

C₂²×C₇₁

C₂²⋊Q₈

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

#123

yes

characteristic

C₄×C₇₁

#123

yes

—

C₄×C₇₁

#127

yes

characteristic

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

—

—

—

D₇₁

yes

characteristic

#338

—

C₂³×C₄

C₂×D₇₁

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₁

yes

characteristic

C₂×M₁₆

C₂×D₇₁

yes

characteristic

#108

—

142

—

#108

—

142

—

C₂²⋊C₈

—

C₂²⋊C₈

—

142

—

C₂⁴

Dic₁₄₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₁₄₂

yes

characteristic

C₂²×C₄

C₂²×D₇₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

142

—

M₁₆

—

142

—

M₁₆

—

142

—

M₁₆

—

142

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

C₂²⋊D₇₁

yes

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

C₂×Dic₁₄₂

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

D₄×D₇₁

yes

—

108

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

109

C₂×C₄

#11765864

yes

—

110

C₂×C₄

#11765865

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

284

—

114

C₈

—

142

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

#14556105

yes

—

122

C₂²

#14556107

yes

characteristic

123

C₄

—

124

C₄

#14556109

yes

characteristic

125

C₄

#14556110

yes

—

126

C₄

#14556124

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#17985935

yes

characteristic

	subgroup	G/H
center	C₄	#14556124
commutator	C₂²×C₄×C₇₁	C₂³
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	#407×C₇₁	C₂
socle	C₂×C₇₁	#399
hypercenter	#407	D₇₁
hypocenter	C₇₁	#2091
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #407

subgroup

G/H

center

C₄

#14556124

commutator

C₂²×C₄×C₇₁

C₂³

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

#407×C₇₁

C₂

C₂×C₇₁

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #407

N	H ≅ G/N	type
1. #17986048	C₂	semidirect
2. #17986047	C₂	semidirect
4. C₇₁⋊#338	C₂	semidirect
66. C₇₁	#2091	semidirect

H ≅ G/N

type

1. #17986048

C₂

semidirect

2. #17986047

C₂

semidirect

4. C₇₁⋊#338

C₂

semidirect

66. C₇₁

#2091

semidirect

Subgroups — #22589856

Conjugacy classes of subgroups

Complements