Group #22589812 — subgroups

Subgroups — #22589812

Order 9088.

230 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 91 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×#14556061	C₂	1	2	yes	maximal
2	#17985897	C₂	1	1	yes	maximal, characteristic
3	#17985761	C₂	1	2	yes	maximal
4	#17985754	C₂	1	1	yes	maximal, characteristic
5	#384×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
9	C₂×Dic₇₁⋊C₄	—	2	1	no	—
10	#14556061	—	2	4	no	—
11	C₇₁⋊#95	—	2	1	no	—
12	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
13	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
14	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
16	C₂²×Dic₇₁	—	2	1	no	—
17	C₂²×Dic₇₁	—	2	1	no	—
18	C₂×C₇₁⋊C₈	—	2	2	no	—
19	C₂×C₇₁⋊C₈	—	2	2	no	—
20	C₂×C₇₁⋊C₈	Q₈	1	4	yes	—
21	C₄⋊Dic₇₁	—	2	1	no	—
22	C₄⋊Dic₇₁	—	2	2	no	—
23	C₄⋊Dic₇₁	D₄	1	2	yes	—
24	Dic₇₁⋊C₄	—	4	2	no	—
25	C₂²×C₄×C₇₁	—	2	2	no	—
26	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₇₁	—	2	2	no	—
30	C₄⋊C₄×C₇₁	—	2	2	no	—
31	C₄⋊C₄×C₇₁	—	4	2	no	—
32	C₄⋊C₄×C₇₁	—	4	2	no	—
33	C₄²×C₇₁	—	2	2	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	1	no	—
38	C₂×C₄×C₇₁	—	2	1	no	—
39	C₂×C₄×C₇₁	—	4	2	no	—
40	C₂×C₄×C₇₁	—	4	2	no	—
41	C₂×C₄×C₇₁	—	4	2	no	—
42	C₂×C₄×C₇₁	D₈	1	2	yes	—
43	C₂×C₄×C₇₁	Q₁₆	1	2	yes	—
44	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₇₁	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
47	C₂×C₄×C₇₁	#41	1	2	yes	—
48	C₂×Dic₇₁	—	2	2	no	—
49	C₂×Dic₇₁	—	4	1	no	—
50	C₂×Dic₇₁	—	4	1	no	—
51	C₂×Dic₇₁	—	4	1	no	—
52	C₂×Dic₇₁	—	4	1	no	—
53	C₂×Dic₇₁	—	4	1	no	—
54	C₇₁⋊C₈	—	2	4	no	—
55	C₂²×C₇₁	#121	1	1	yes	characteristic
56	C₂²×C₇₁	#123	1	1	yes	characteristic
57	C₂²×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
58	C₂²×C₇₁	C₂×D₈	1	1	yes	characteristic
59	C₂²×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
60	C₂²×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂²×C₇₁	#137	1	1	yes	characteristic
62	C₄×C₇₁	—	2	2	no	—
63	C₄×C₇₁	—	2	2	no	—
64	C₄×C₇₁	—	8	2	no	—
65	C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
66	C₄×C₇₁	#125	1	2	yes	—
67	Dic₇₁	—	4	2	no	—
68	Dic₇₁	—	8	1	no	—
69	C₂×C₇₁	#398	1	1	yes	characteristic
70	C₂×C₇₁	#459	1	1	yes	characteristic
71	C₂×C₇₁	#462	1	1	yes	characteristic
72	C₂×C₇₁	#480	1	1	yes	characteristic
73	C₂×C₇₁	#484	1	1	yes	characteristic
74	C₂×C₇₁	#500	1	1	yes	characteristic
75	C₂×C₇₁	#501	1	1	yes	characteristic
76	#2083	—	71	1	no	maximal, Sylow
77	C₇₁	#2083	1	1	yes	characteristic, complemented, Sylow
78	C₂×#107	—	71	2	no	—
79	#389	—	71	1	no	—
80	#384	D₇₁	1	1	yes	characteristic
81	#340	—	71	2	no	—
82	#336	—	71	1	no	—
83	C₂²×C₈	—	71	2	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	2	1	no	—
86	C₂×C₄⋊C₄	—	71	1	no	—
87	C₂×C₄⋊C₄	—	71	1	no	—
88	C₂×C₄⋊C₄	—	142	1	no	—
89	C₂×C₄⋊C₄	C₂×D₇₁	1	2	yes	—
90	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
91	#107	—	142	4	no	—
92	#95	—	142	1	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	142	1	no	—
95	C₂²×C₄	—	142	1	no	—
96	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
99	C₂×C₈	—	71	4	no	—
100	C₂×C₈	—	142	2	no	—
101	C₂×C₈	—	142	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	284	2	no	—
107	C₄⋊C₄	—	71	2	no	—
108	C₄⋊C₄	—	142	2	no	—
109	C₄⋊C₄	—	142	1	no	—
110	C₄²	—	2	2	no	—
111	C₂³	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	284	1	no	—
120	C₂×C₄	—	284	1	no	—
121	C₂×C₄	—	284	1	no	—
122	C₂×C₄	—	284	1	no	—
123	C₂×C₄	—	284	1	no	—
124	C₂×C₄	—	142	2	no	—
125	C₂×C₄	#11765850	1	2	yes	—
126	C₂×C₄	#11765855	1	2	yes	—
127	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₇₁	1	2	yes	—
129	C₂×C₄	#11765864	1	1	yes	characteristic
130	C₂×C₄	#11765865	1	2	yes	—
131	C₈	—	142	4	no	—
132	C₂²	#14556099	1	1	yes	characteristic
133	C₂²	#14556111	1	1	yes	characteristic
134	C₂²	#14556126	1	1	yes	characteristic
135	C₂²	C₂×#11765850	1	1	yes	characteristic
136	C₂²	C₂×#11765855	1	1	yes	characteristic
137	C₂²	#14556154	1	1	yes	characteristic
138	C₂²	#14556159	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	2	2	no	—
141	C₄	—	8	2	no	—
142	C₄	—	284	2	no	—
143	C₄	—	568	1	no	—
144	C₄	#14556106	1	2	yes	—
145	C₄	#14556118	1	2	yes	—
146	C₂	#17985930	1	1	yes	characteristic
147	C₂	#17986191	1	1	yes	characteristic
148	C₂	#17986206	1	1	yes	characteristic
149	C₂	#17986312	1	1	yes	characteristic
150	C₂	#17986334	1	1	yes	characteristic
151	C₂	#17986402	1	1	yes	characteristic
152	C₂	#17986405	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556061

C₂

yes

maximal

#17985897

C₂

yes

maximal, characteristic

#17985761

C₂

yes

maximal

#17985754

C₂

yes

maximal, characteristic

#384×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₇₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₁

C₂²

yes

characteristic

C₂×Dic₇₁⋊C₄

—

#14556061

—

C₇₁⋊#95

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₂²×Dic₇₁

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

Q₈

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

D₄

yes

—

Dic₇₁⋊C₄

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

D₈

yes

—

C₂×C₄×C₇₁

Q₁₆

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

—

C₂×C₄×C₇₁

#41

yes

characteristic

C₂×C₄×C₇₁

#41

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

C₄⋊Q₈

yes

characteristic

C₂²×C₇₁

C₂×D₈

yes

characteristic

C₂²×C₇₁

C₂×Q₁₆

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂²⋊Q₈

yes

—

C₄×C₇₁

#125

yes

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₂×#107

—

#389

—

#384

D₇₁

yes

characteristic

#340

—

#336

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄⋊C₄

C₂×D₇₁

yes

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

#107

—

142

—

#95

—

142

—

C₂²×C₄

—

C₂²×C₄

—

142

—

C₂²×C₄

—

142

—

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

C₂²×C₄

C₂²×D₇₁

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

142

—

101

C₂×C₈

—

142

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

284

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

142

—

109

C₄⋊C₄

—

142

—

110

C₄²

—

111

C₂³

C₂×C₂²⋊D₇₁

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

284

—

120

C₂×C₄

—

284

—

121

C₂×C₄

—

284

—

122

C₂×C₄

—

284

—

123

C₂×C₄

—

284

—

124

C₂×C₄

—

142

—

125

C₂×C₄

#11765850

yes

—

126

C₂×C₄

#11765855

yes

—

127

C₂×C₄

D₄×D₇₁

yes

characteristic

128

C₂×C₄

Q₈×D₇₁

yes

—

129

C₂×C₄

#11765864

yes

characteristic

130

C₂×C₄

#11765865

yes

—

131

C₈

—

142

—

132

C₂²

#14556099

yes

characteristic

133

C₂²

#14556111

yes

characteristic

134

C₂²

#14556126

yes

characteristic

135

C₂²

C₂×#11765850

yes

characteristic

136

C₂²

C₂×#11765855

yes

characteristic

137

C₂²

#14556154

yes

characteristic

138

C₂²

#14556159

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

284

—

143

C₄

—

568

—

144

C₄

#14556106

yes

—

145

C₄

#14556118

yes

—

146

C₂

#17985930

yes

characteristic

147

C₂

#17986191

yes

characteristic

148

C₂

#17986206

yes

characteristic

149

C₂

#17986312

yes

characteristic

150

C₂

#17986334

yes

characteristic

151

C₂

#17986402

yes

characteristic

152

C₂

#17986405

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₁
commutator	C₂²×C₄×C₇₁	C₂³
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	#384×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₄
hypercenter	#384	D₇₁
hypocenter	C₇₁	#2083
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₁

commutator

C₂²×C₄×C₇₁

C₂³

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

#384×C₇₁

C₂

socle

C₂³×C₇₁

C₂×D₄

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
77. C₇₁	#2083	semidirect

H ≅ G/N

type

77. C₇₁

#2083

semidirect

Subgroups — #22589812

Conjugacy classes of subgroups

Complements