Group #22588239 — subgroups

Subgroups — #22588239

Order 9088.

272 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 133 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	C₂×#14556060	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556038	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×Dic₇₁	C₂	1	2	yes	maximal
4	#17985761	C₂	1	2	yes	maximal
5	C₂×#107×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₇₁	C₂²	1	2	yes	—
8	C₂×C₄×Dic₇₁	—	2	2	no	—
9	C₂×C₄×Dic₇₁	C₂²	1	1	yes	characteristic
10	#14556060	C₄	1	4	yes	complemented
11	#14556038	C₄	1	4	yes	complemented
12	C₇₁⋊#95	—	2	2	no	—
13	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
15	#107×C₇₁	C₄	1	4	yes	complemented
16	C₂²×Dic₇₁	—	2	2	no	—
17	C₂²×Dic₇₁	D₄	1	2	yes	—
18	C₂×C₇₁⋊C₈	—	2	1	no	—
19	C₂×C₇₁⋊C₈	—	2	1	no	—
20	C₂×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
21	C₄⋊Dic₇₁	—	2	2	no	—
22	C₄⋊Dic₇₁	C₂×C₄	1	4	yes	—
23	C₄×Dic₇₁	—	4	4	no	—
24	C₄×Dic₇₁	C₂×C₄	1	2	yes	—
25	C₄×Dic₇₁	Q₈	1	2	yes	—
26	C₂²×C₄×C₇₁	—	2	2	no	—
27	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
31	C₄⋊C₄×C₇₁	—	2	2	no	—
32	C₄⋊C₄×C₇₁	C₂×C₄	1	4	yes	—
33	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₇₁	—	2	4	no	—
35	C₂×C₄×C₇₁	—	4	2	no	—
36	C₂×C₄×C₇₁	—	4	2	no	—
37	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
42	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
43	C₂×Dic₇₁	—	2	2	no	—
44	C₂×Dic₇₁	—	2	2	no	—
45	C₂×Dic₇₁	—	2	4	no	—
46	C₂×Dic₇₁	—	4	2	no	—
47	C₂×Dic₇₁	—	4	2	no	—
48	C₂×Dic₇₁	C₄⋊C₄	1	4	yes	—
49	C₈×C₇₁	C₄²	1	4	yes	complemented
50	C₇₁⋊C₈	—	2	2	no	—
51	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₇₁	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₇₁	C₄×D₄	1	1	yes	characteristic
54	C₂²×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
55	C₂²×C₇₁	#137	1	2	yes	—
56	C₄×C₇₁	—	4	4	no	—
57	C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
58	C₄×C₇₁	#117	1	1	yes	characteristic
59	C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
60	C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
61	Dic₇₁	—	2	4	no	—
62	Dic₇₁	—	4	4	no	—
63	C₂×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₇₁	#428	1	2	yes	—
65	C₂×C₇₁	#440	1	2	yes	—
66	C₂×C₇₁	#441	1	1	yes	characteristic
67	C₂×C₇₁	#442	1	1	yes	characteristic
68	#1776	—	71	1	no	maximal, Sylow
69	C₇₁	#1776	1	1	yes	characteristic, complemented, Sylow
70	C₂×#107	D₇₁	1	1	yes	characteristic
71	C₂×#106	—	71	1	no	—
72	C₂×#97	—	71	1	no	—
73	C₄×C₄⋊C₄	—	71	2	no	—
74	#340	—	71	2	no	—
75	C₂²×C₈	—	71	1	no	—
76	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
77	C₂×C₄⋊C₄	—	71	2	no	—
78	C₂×C₄⋊C₄	C₂×D₇₁	1	2	yes	—
79	C₂×C₄²	—	71	1	no	—
80	C₂×C₄²	—	142	2	no	—
81	#107	Dic₇₁	1	4	yes	complemented
82	#106	—	71	4	no	—
83	#97	—	71	4	no	—
84	#95	—	142	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	—	71	2	no	—
87	C₂²×C₄	—	142	2	no	—
88	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
89	C₂×C₈	—	71	2	no	—
90	C₂×C₈	—	142	1	no	—
91	C₂×C₈	—	142	1	no	—
92	C₂×C₈	C₄×D₇₁	1	2	yes	—
93	C₂×C₈	C₄×D₇₁	1	2	yes	—
94	C₂×C₈	C₂×Dic₇₁	1	2	yes	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	71	4	no	—
97	C₄⋊C₄	—	142	2	no	—
98	C₄⋊C₄	C₂×Dic₇₁	1	4	yes	—
99	C₄²	—	284	4	no	—
100	C₄²	—	71	2	no	—
101	C₄²	—	71	2	no	—
102	C₂³	D₄×D₇₁	1	1	yes	characteristic
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	284	2	no	—
107	C₂×C₄	—	284	2	no	—
108	C₂×C₄	—	71	4	no	—
109	C₂×C₄	—	142	4	no	—
110	C₂×C₄	—	142	2	no	—
111	C₂×C₄	—	142	2	no	—
112	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
113	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
114	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
115	C₂×C₄	Q₈×D₇₁	1	1	yes	characteristic
116	C₂×C₄	#11765864	1	1	yes	characteristic
117	C₂×C₄	#11765866	1	1	yes	characteristic
118	C₈	—	142	2	no	—
119	C₈	C₄×Dic₇₁	1	4	yes	complemented
120	C₂²	C₄⋊C₄×D₇₁	1	1	yes	characteristic
121	C₂²	#14556090	1	1	yes	characteristic
122	C₂²	D₄×Dic₇₁	1	1	yes	characteristic
123	C₂²	#14556152	1	2	yes	—
124	C₂²	#14556160	1	2	yes	—
125	C₄	—	4	4	no	—
126	C₄	—	284	4	no	—
127	C₄	—	142	4	no	—
128	C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
129	C₄	#14556086	1	1	yes	characteristic
130	C₄	#14556093	1	1	yes	characteristic
131	C₄	Q₈×Dic₇₁	1	1	yes	characteristic
132	C₂	C₄⋊C₄×Dic₇₁	1	1	yes	characteristic
133	C₂	#17986040	1	2	yes	—
134	C₂	#17986098	1	2	yes	—
135	C₂	#17986106	1	1	yes	characteristic
136	C₂	#17986111	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556060

C₂

yes

maximal, characteristic

C₂×#14556038

C₂

yes

maximal, characteristic

C₄⋊C₄×Dic₇₁

C₂

yes

maximal

#17985761

C₂

yes

maximal

C₂×#107×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₁

C₂²

yes

—

C₂×C₄×Dic₇₁

—

C₂×C₄×Dic₇₁

C₂²

yes

characteristic

#14556060

C₄

yes

complemented

#14556038

C₄

yes

complemented

C₇₁⋊#95

—

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₁

C₂²

yes

—

#107×C₇₁

C₄

yes

complemented

C₂²×Dic₇₁

—

C₂²×Dic₇₁

D₄

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

C₂×C₄

yes

—

C₄×Dic₇₁

—

C₄×Dic₇₁

C₂×C₄

yes

—

C₄×Dic₇₁

Q₈

yes

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

C₂×C₄

yes

—

C₂³×C₇₁

C₂×D₄

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

C₂×Q₈

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

C₄⋊C₄

yes

—

C₈×C₇₁

C₄²

yes

complemented

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₁

C₄×D₄

yes

characteristic

C₂²×C₇₁

C₄×D₄

yes

characteristic

C₂²×C₇₁

AGL(1,ℤ/8)

yes

—

C₂²×C₇₁

#137

yes

—

C₄×C₇₁

—

C₄×C₇₁

C₂×C₄²

yes

characteristic

C₄×C₇₁

#117

yes

characteristic

C₄×C₇₁

C₄×Q₈

yes

characteristic

C₄×C₇₁

C₄×Q₈

yes

characteristic

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₇₁

#428

yes

—

C₂×C₇₁

#440

yes

—

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₂×#107

D₇₁

yes

characteristic

C₂×#106

—

C₂×#97

—

C₄×C₄⋊C₄

—

#340

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

142

—

#107

Dic₇₁

yes

—

—

—

142

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

142

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

C₄×D₇₁

yes

—

C₂×C₈

C₄×D₇₁

yes

—

C₂×C₈

C₂×Dic₇₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

142

—

C₄⋊C₄

C₂×Dic₇₁

yes

—

C₄²

—

284

—

100

C₄²

—

101

C₄²

—

102

C₂³

D₄×D₇₁

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

284

—

107

C₂×C₄

—

284

—

108

C₂×C₄

—

109

C₂×C₄

—

142

—

110

C₂×C₄

—

142

—

111

C₂×C₄

—

142

—

112

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

113

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

114

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

115

C₂×C₄

Q₈×D₇₁

yes

characteristic

116

C₂×C₄

#11765864

yes

characteristic

117

C₂×C₄

#11765866

yes

characteristic

118

C₈

—

142

—

119

C₈

C₄×Dic₇₁

yes

complemented

120

C₂²

C₄⋊C₄×D₇₁

yes

characteristic

121

C₂²

#14556090

yes

characteristic

122

C₂²

D₄×Dic₇₁

yes

characteristic

123

C₂²

#14556152

yes

—

124

C₂²

#14556160

yes

—

125

C₄

—

126

C₄

—

284

—

127

C₄

—

142

—

128

C₄

C₂×C₄×Dic₇₁

yes

characteristic

129

C₄

#14556086

yes

characteristic

130

C₄

#14556093

yes

characteristic

131

C₄

Q₈×Dic₇₁

yes

characteristic

132

C₂

C₄⋊C₄×Dic₇₁

yes

characteristic

133

C₂

#17986040

yes

—

134

C₂

#17986098

yes

—

135

C₂

#17986106

yes

characteristic

136

C₂

#17986111

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₇₁
commutator	C₄×C₇₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	C₂×#107×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₄
hypercenter	C₂×#107	D₇₁
hypocenter	C₇₁	#1776
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₇₁

commutator

C₄×C₇₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

C₂×#107×C₇₁

C₂

socle

C₂³×C₇₁

C₂×D₄

D₇₁

C₇₁

G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
10. #14556060	C₄	semidirect
11. #14556038	C₄	semidirect
15. #107×C₇₁	C₄	semidirect
49. C₈×C₇₁	C₄²	semidirect
69. C₇₁	#1776	semidirect
81. #107	Dic₇₁	semidirect
119. C₈	C₄×Dic₇₁	semidirect

H ≅ G/N

type

10. #14556060

C₄

semidirect

11. #14556038

C₄

semidirect

15. #107×C₇₁

C₄

semidirect

49. C₈×C₇₁

C₄²

semidirect

69. C₇₁

#1776

semidirect

81. #107

Dic₇₁

semidirect

119. C₈

C₄×Dic₇₁

semidirect

Subgroups — #22588239

Conjugacy classes of subgroups

Complements