Group #22589083 — subgroups

Subgroups — #22589083

Order 9088.

240 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 109 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×#14556062	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556060	C₂	1	1	yes	maximal, characteristic
3	#17985897	C₂	1	2	yes	maximal
4	#17985762	C₂	1	2	yes	maximal
5	C₂×#97×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₇₁	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₇₁	C₂²	1	2	yes	—
8	C₂×Dic₇₁⋊C₄	—	2	2	no	—
9	#14556062	—	2	2	no	—
10	#14556060	C₄	1	4	yes	complemented
11	C₇₁⋊#95	—	2	2	no	—
12	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
15	#97×C₇₁	—	2	2	no	—
16	C₂²×Dic₇₁	—	2	2	no	—
17	C₂²×Dic₇₁	—	2	2	no	—
18	C₄⋊Dic₇₁	—	2	2	no	—
19	C₄⋊Dic₇₁	—	2	2	no	—
20	C₄⋊Dic₇₁	—	2	2	no	—
21	C₄⋊Dic₇₁	C₂×C₄	1	4	yes	—
22	Dic₇₁⋊C₄	—	4	4	no	—
23	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₇₁	—	2	1	no	—
27	C₂×C₈×C₇₁	—	2	1	no	—
28	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₇₁	D₄	1	2	yes	—
30	C₂×C₈×C₇₁	Q₈	1	2	yes	—
31	C₂×C₈×C₇₁	Q₈	1	2	yes	—
32	C₄²×C₇₁	—	2	1	no	—
33	C₄²×C₇₁	—	2	1	no	—
34	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₇₁	—	2	1	no	—
36	C₂×C₄×C₇₁	—	2	1	no	—
37	C₂×C₄×C₇₁	—	2	2	no	—
38	C₂×C₄×C₇₁	—	2	2	no	—
39	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
44	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
45	C₂×Dic₇₁	—	2	4	no	—
46	C₂×Dic₇₁	—	4	2	no	—
47	C₂×Dic₇₁	—	4	2	no	—
48	C₂×Dic₇₁	—	4	2	no	—
49	C₂×Dic₇₁	—	4	2	no	—
50	C₂×Dic₇₁	—	4	2	no	—
51	C₈×C₇₁	—	2	2	no	—
52	C₈×C₇₁	C₄⋊C₄	1	4	yes	complemented
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	—	4	1	no	—
59	C₄×C₇₁	—	4	1	no	—
60	C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
62	C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
63	C₄×C₇₁	#125	1	1	yes	characteristic
64	Dic₇₁	—	4	4	no	—
65	Dic₇₁	—	8	2	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	—	71	1	no	—
74	C₂×#106	—	71	1	no	—
75	C₂×#97	D₇₁	1	1	yes	characteristic
76	#389	—	71	2	no	—
77	#340	—	71	2	no	—
78	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
79	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
80	C₂×C₄⋊C₄	—	71	2	no	—
81	C₂×C₄⋊C₄	—	71	2	no	—
82	C₂×C₄⋊C₄	—	142	2	no	—
83	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
84	#107	—	142	2	no	—
85	#106	—	71	4	no	—
86	#97	—	2	2	no	—
87	#95	—	142	2	no	—
88	C₂²×C₄	—	142	2	no	—
89	C₂²×C₄	—	142	2	no	—
90	C₂²×C₄	D₂₈₄	1	1	yes	characteristic
91	C₂²×C₄	D₂₈₄	1	1	yes	characteristic
92	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	Dic₁₄₂	1	2	yes	—
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	—	284	4	no	—
100	C₄⋊C₄	—	71	4	no	—
101	C₄⋊C₄	—	142	2	no	—
102	C₄⋊C₄	—	142	2	no	—
103	C₄⋊C₄	—	142	2	no	—
104	C₄²	—	2	1	no	—
105	C₄²	—	2	1	no	—
106	C₂³	C₂×D₂₈₄	1	1	yes	characteristic
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	284	2	no	—
112	C₂×C₄	—	284	2	no	—
113	C₂×C₄	—	284	2	no	—
114	C₂×C₄	—	284	2	no	—
115	C₂×C₄	—	284	2	no	—
116	C₂×C₄	—	142	4	no	—
117	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
119	C₂×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
120	C₂×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
121	C₂×C₄	#11765865	1	1	yes	characteristic
122	C₂×C₄	#11765865	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	Dic₇₁⋊C₄	1	4	yes	complemented
125	C₂²	C₄×D₂₈₄	1	1	yes	characteristic
126	C₂²	#14556102	1	1	yes	characteristic
127	C₂²	#14556128	1	1	yes	characteristic
128	C₂²	#14556156	1	2	yes	—
129	C₂²	#14556161	1	2	yes	—
130	C₄	—	4	1	no	—
131	C₄	—	4	1	no	—
132	C₄	—	284	4	no	—
133	C₄	—	568	2	no	—
134	C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
135	C₄	C₄×Dic₁₄₂	1	1	yes	characteristic
136	C₄	#14556107	1	1	yes	characteristic
137	C₄	#14556120	1	1	yes	characteristic
138	C₂	#17985897	1	1	yes	characteristic
139	C₂	#17986101	1	2	yes	—
140	C₂	#17986252	1	1	yes	characteristic
141	C₂	#17986257	1	1	yes	characteristic
142	C₂	#17986408	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556062

C₂

yes

maximal, characteristic

C₂×#14556060

C₂

yes

maximal, characteristic

#17985897

C₂

yes

maximal

#17985762

C₂

yes

maximal

C₂×#97×C₇₁

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₇₁

C₂²

yes

—

C₂×C₄⋊Dic₇₁

C₂²

yes

—

C₂×Dic₇₁⋊C₄

—

#14556062

—

#14556060

C₄

yes

complemented

C₇₁⋊#95

—

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₄²×C₇₁

C₂²

yes

characteristic

#97×C₇₁

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

C₂×C₄

yes

—

Dic₇₁⋊C₄

—

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₈×C₇₁

D₄

yes

—

C₂×C₈×C₇₁

Q₈

yes

—

C₂×C₈×C₇₁

Q₈

yes

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

—

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

—

C₈×C₇₁

—

C₈×C₇₁

C₄⋊C₄

yes

complemented

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

—

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

—

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

—

C₂×#106

—

C₂×#97

D₇₁

yes

characteristic

#389

—

#340

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

C₂²×C₈

C₂×D₇₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

#107

—

142

—

#106

—

#97

—

#95

—

142

—

C₂²×C₄

—

142

—

C₂²×C₄

—

142

—

C₂²×C₄

D₂₈₄

yes

characteristic

C₂²×C₄

D₂₈₄

yes

characteristic

C₂²×C₄

C₂²×D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₄₂

yes

—

C₂×C₈

Dic₁₄₂

yes

—

C₂×C₈

C₄×D₇₁

yes

—

C₂×C₈

C₂²⋊D₇₁

yes

—

C₄⋊C₄

—

284

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

142

—

102

C₄⋊C₄

—

142

—

103

C₄⋊C₄

—

142

—

104

C₄²

—

105

C₄²

—

106

C₂³

C₂×D₂₈₄

yes

characteristic

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

284

—

112

C₂×C₄

—

284

—

113

C₂×C₄

—

284

—

114

C₂×C₄

—

284

—

115

C₂×C₄

—

284

—

116

C₂×C₄

—

142

—

117

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

119

C₂×C₄

C₂×Dic₁₄₂

yes

characteristic

120

C₂×C₄

C₂×Dic₁₄₂

yes

characteristic

121

C₂×C₄

#11765865

yes

characteristic

122

C₂×C₄

#11765865

yes

characteristic

123

C₈

—

124

C₈

Dic₇₁⋊C₄

yes

complemented

125

C₂²

C₄×D₂₈₄

yes

characteristic

126

C₂²

#14556102

yes

characteristic

127

C₂²

#14556128

yes

characteristic

128

C₂²

#14556156

yes

—

129

C₂²

#14556161

yes

—

130

C₄

—

131

C₄

—

132

C₄

—

284

—

133

C₄

—

568

—

134

C₄

C₂×Dic₇₁⋊C₄

yes

characteristic

135

C₄

C₄×Dic₁₄₂

yes

characteristic

136

C₄

#14556107

yes

characteristic

137

C₄

#14556120

yes

characteristic

138

C₂

#17985897

yes

characteristic

139

C₂

#17986101

yes

—

140

C₂

#17986252

yes

characteristic

141

C₂

#17986257

yes

characteristic

142

C₂

#17986408

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	C₂×#97×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₄
hypercenter	C₂×#97	D₇₁
hypocenter	C₇₁	#1944
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₂₈₄

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

C₂×#97×C₇₁

C₂

socle

C₂³×C₇₁

C₂×D₄

hypercenter

C₂×#97

D₇₁

hypocenter

C₇₁

#1944

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

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

H ≅ G/N

type

10. #14556060

C₄

semidirect

52. C₈×C₇₁

C₄⋊C₄

semidirect

72. C₇₁

#1944

semidirect

124. C₈

Dic₇₁⋊C₄

semidirect

Subgroups — #22589083

Conjugacy classes of subgroups

Complements