Group #22589081 — subgroups

Subgroups — #22589081

Order 9088.

240 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 105 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	#17985895	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×Dic₇₁⋊C₄	—	2	2	no	—
9	C₂×C₄×Dic₇₁	C₂²	1	1	yes	characteristic
10	#14556060	C₄	1	4	yes	complemented
11	#14556038	—	2	2	no	—
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	—	2	2	no	—
16	C₂²×Dic₇₁	—	2	2	no	—
17	C₂²×Dic₇₁	D₄	1	1	yes	characteristic
18	C₂²×Dic₇₁	D₄	1	1	yes	characteristic
19	C₂×C₇₁⋊C₈	—	2	1	no	—
20	C₂×C₇₁⋊C₈	—	2	1	no	—
21	C₂×C₇₁⋊C₈	Q₈	1	2	yes	—
22	C₄⋊Dic₇₁	—	2	2	no	—
23	C₄⋊Dic₇₁	C₂×C₄	1	4	yes	—
24	Dic₇₁⋊C₄	—	4	4	no	—
25	C₄×Dic₇₁	—	2	1	no	—
26	C₄×Dic₇₁	—	2	1	no	—
27	C₂²×C₄×C₇₁	—	2	2	no	—
28	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
29	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₇₁	D₄	1	2	yes	—
31	C₂×C₈×C₇₁	Q₈	1	2	yes	—
32	C₄⋊C₄×C₇₁	—	2	2	no	—
33	C₄⋊C₄×C₇₁	—	2	2	no	—
34	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₇₁	—	4	2	no	—
36	C₂×C₄×C₇₁	—	4	2	no	—
37	C₂×C₄×C₇₁	—	4	2	no	—
38	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
43	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
44	C₂×Dic₇₁	—	2	4	no	—
45	C₂×Dic₇₁	—	2	1	no	—
46	C₂×Dic₇₁	—	2	1	no	—
47	C₂×Dic₇₁	—	2	2	no	—
48	C₂×Dic₇₁	—	2	2	no	—
49	C₂×Dic₇₁	—	4	2	no	—
50	C₂×Dic₇₁	—	4	2	no	—
51	C₈×C₇₁	C₄⋊C₄	1	4	yes	complemented
52	C₇₁⋊C₈	—	2	2	no	—
53	C₂²×C₇₁	C₄×D₄	1	1	yes	characteristic
54	C₂²×C₇₁	#121	1	1	yes	characteristic
55	C₂²×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
56	C₂²×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
57	C₂²×C₇₁	#137	1	2	yes	—
58	C₄×C₇₁	—	8	2	no	—
59	C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
61	C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
62	C₄×C₇₁	#125	1	1	yes	characteristic
63	Dic₇₁	—	4	4	no	—
64	Dic₇₁	—	4	1	no	—
65	Dic₇₁	—	4	1	no	—
66	C₂×C₇₁	#389	1	1	yes	characteristic
67	C₂×C₇₁	#440	1	2	yes	—
68	C₂×C₇₁	#469	1	1	yes	characteristic
69	C₂×C₇₁	#470	1	1	yes	characteristic
70	C₂×C₇₁	#501	1	2	yes	—
71	#1944	—	71	1	no	maximal, Sylow
72	C₇₁	#1944	1	1	yes	characteristic, complemented, Sylow
73	C₂×#107	D₇₁	1	1	yes	characteristic
74	C₂×#106	—	71	1	no	—
75	C₂×#97	—	71	1	no	—
76	#389	—	71	2	no	—
77	#340	—	71	2	no	—
78	C₂²×C₈	—	71	1	no	—
79	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
80	C₂×C₄⋊C₄	—	71	2	no	—
81	C₂×C₄⋊C₄	—	142	2	no	—
82	C₂×C₄⋊C₄	C₂×D₇₁	1	2	yes	—
83	C₂×C₄²	—	71	1	no	—
84	#107	—	2	2	no	—
85	#106	—	71	4	no	—
86	#97	—	142	2	no	—
87	#95	—	142	2	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	—	71	1	no	—
90	C₂²×C₄	—	71	1	no	—
91	C₂²×C₄	—	142	2	no	—
92	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
93	C₂×C₈	—	71	2	no	—
94	C₂×C₈	—	142	1	no	—
95	C₂×C₈	—	142	1	no	—
96	C₂×C₈	Dic₁₄₂	1	2	yes	—
97	C₂×C₈	C₄×D₇₁	1	2	yes	—
98	C₂×C₈	C₂²⋊D₇₁	1	2	yes	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	284	4	no	—
102	C₄⋊C₄	—	71	4	no	—
103	C₄⋊C₄	—	142	2	no	—
104	C₄²	—	142	1	no	—
105	C₄²	—	142	1	no	—
106	C₂³	D₄×D₇₁	1	1	yes	characteristic
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	284	2	no	—
111	C₂×C₄	—	284	2	no	—
112	C₂×C₄	—	142	2	no	—
113	C₂×C₄	—	142	4	no	—
114	C₂×C₄	—	142	1	no	—
115	C₂×C₄	—	142	1	no	—
116	C₂×C₄	—	142	2	no	—
117	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
119	C₂×C₄	Q₈×D₇₁	1	1	yes	characteristic
120	C₂×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
121	C₂×C₄	#11765864	1	1	yes	characteristic
122	C₂×C₄	#11765866	1	1	yes	characteristic
123	C₈	—	142	2	no	—
124	C₈	Dic₇₁⋊C₄	1	4	yes	complemented
125	C₂²	#14556090	1	1	yes	characteristic
126	C₂²	#14556099	1	1	yes	characteristic
127	C₂²	#14556127	1	1	yes	characteristic
128	C₂²	#14556152	1	2	yes	—
129	C₂²	#14556160	1	2	yes	—
130	C₄	—	8	2	no	—
131	C₄	—	284	4	no	—
132	C₄	—	284	1	no	—
133	C₄	—	284	1	no	—
134	C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
135	C₄	#14556093	1	1	yes	characteristic
136	C₄	#14556104	1	1	yes	characteristic
137	C₄	#14556119	1	1	yes	characteristic
138	C₂	#17985895	1	1	yes	characteristic
139	C₂	#17986098	1	2	yes	—
140	C₂	#17986249	1	1	yes	characteristic
141	C₂	#17986254	1	1	yes	characteristic
142	C₂	#17986407	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556060

C₂

yes

maximal, characteristic

C₂×#14556038

C₂

yes

maximal, characteristic

#17985895

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×Dic₇₁⋊C₄

—

C₂×C₄×Dic₇₁

C₂²

yes

characteristic

#14556060

C₄

yes

complemented

#14556038

—

C₇₁⋊#95

—

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₁

C₂²

yes

—

#107×C₇₁

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

D₄

yes

characteristic

C₂²×Dic₇₁

D₄

yes

characteristic

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

Q₈

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

C₂×C₄

yes

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₈×C₇₁

D₄

yes

—

C₂×C₈×C₇₁

Q₈

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

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

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

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

—

C₂×Dic₇₁

—

C₈×C₇₁

C₄⋊C₄

yes

complemented

C₇₁⋊C₈

—

C₂²×C₇₁

C₄×D₄

yes

characteristic

C₂²×C₇₁

#121

yes

characteristic

C₂²×C₇₁

C₄⋊Q₈

yes

characteristic

C₂²×C₇₁

AGL(1,ℤ/8)

yes

—

C₂²×C₇₁

#137

yes

—

C₄×C₇₁

—

C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₁

C₄×Q₈

yes

characteristic

C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

C₄×C₇₁

#125

yes

characteristic

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

#389

yes

characteristic

C₂×C₇₁

#440

yes

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

#1944

—

maximal, Sylow

C₇₁

#1944

yes

characteristic, complemented, Sylow

C₂×#107

D₇₁

yes

characteristic

C₂×#106

—

C₂×#97

—

#389

—

#340

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄⋊C₄

C₂×D₇₁

yes

—

C₂×C₄²

—

#107

—

#106

—

#97

—

142

—

#95

—

142

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

142

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

Dic₁₄₂

yes

—

C₂×C₈

C₄×D₇₁

yes

—

C₂×C₈

C₂²⋊D₇₁

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

284

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

142

—

104

C₄²

—

142

—

105

C₄²

—

142

—

106

C₂³

D₄×D₇₁

yes

characteristic

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

284

—

111

C₂×C₄

—

284

—

112

C₂×C₄

—

142

—

113

C₂×C₄

—

142

—

114

C₂×C₄

—

142

—

115

C₂×C₄

—

142

—

116

C₂×C₄

—

142

—

117

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

119

C₂×C₄

Q₈×D₇₁

yes

characteristic

120

C₂×C₄

C₂×Dic₁₄₂

yes

characteristic

121

C₂×C₄

#11765864

yes

characteristic

122

C₂×C₄

#11765866

yes

characteristic

123

C₈

—

142

—

124

C₈

Dic₇₁⋊C₄

yes

complemented

125

C₂²

#14556090

yes

characteristic

126

C₂²

#14556099

yes

characteristic

127

C₂²

#14556127

yes

characteristic

128

C₂²

#14556152

yes

—

129

C₂²

#14556160

yes

—

130

C₄

—

131

C₄

—

284

—

132

C₄

—

284

—

133

C₄

—

284

—

134

C₄

C₂×Dic₇₁⋊C₄

yes

characteristic

135

C₄

#14556093

yes

characteristic

136

C₄

#14556104

yes

characteristic

137

C₄

#14556119

yes

characteristic

138

C₂

#17985895

yes

characteristic

139

C₂

#17986098

yes

—

140

C₂

#17986249

yes

characteristic

141

C₂

#17986254

yes

characteristic

142

C₂

#17986407

yes

—

	subgroup	G/H
center	C₂³	D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	C₂×#107×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₄
hypercenter	C₂×#107	D₇₁
hypocenter	C₇₁	#1944
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

C₂×#107×C₇₁

C₂

socle

C₂³×C₇₁

C₂×D₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
10. #14556060	C₄	semidirect
51. C₈×C₇₁	C₄⋊C₄	semidirect
72. C₇₁	#1944	semidirect
124. C₈	Dic₇₁⋊C₄	semidirect

H ≅ G/N

type

10. #14556060

C₄

semidirect

51. C₈×C₇₁

C₄⋊C₄

semidirect

72. C₇₁

#1944

semidirect

124. C₈

Dic₇₁⋊C₄

semidirect

Subgroups — #22589081

Conjugacy classes of subgroups

Complements