Group #22589749 — subgroups

Subgroups — #22589749

Order 9088.

238 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 87 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×#14556051	C₂	1	2	yes	maximal
2	#17985905	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×Dic₁₄₂	C₂²	1	1	yes	characteristic
7	C₂²×C₇₁⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
9	#14556051	—	2	4	no	—
10	C₇₁⋊#95	—	2	1	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	C₂×Dic₁₄₂	—	2	2	no	—
17	C₂×Dic₁₄₂	—	2	2	no	—
18	C₂×Dic₁₄₂	—	2	1	no	—
19	C₂×Dic₁₄₂	D₄	1	2	yes	—
20	C₂²×Dic₇₁	—	2	1	no	—
21	C₂²×Dic₇₁	—	2	1	no	—
22	C₂×C₇₁⋊C₈	—	2	2	no	—
23	C₂×C₇₁⋊C₈	—	2	2	no	—
24	C₂×C₇₁⋊C₈	—	2	2	no	—
25	C₄⋊Dic₇₁	—	2	2	no	—
26	C₂²×C₄×C₇₁	—	2	2	no	—
27	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₇₁	—	2	2	no	—
31	C₄⋊C₄×C₇₁	—	2	2	no	—
32	C₄⋊C₄×C₇₁	—	4	2	no	—
33	C₄⋊C₄×C₇₁	—	4	2	no	—
34	C₄²×C₇₁	—	2	2	no	—
35	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
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	—	2	1	no	—
40	C₂×C₄×C₇₁	—	4	2	no	—
41	C₂×C₄×C₇₁	—	4	2	no	—
42	C₂×C₄×C₇₁	—	4	2	no	—
43	C₂×C₄×C₇₁	SD₁₆	1	2	yes	—
44	C₂×C₄×C₇₁	Q₁₆	1	2	yes	—
45	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
47	C₂×C₄×C₇₁	#41	1	2	yes	—
48	C₂×C₄×C₇₁	#41	1	2	yes	—
49	C₂×Dic₇₁	—	2	2	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	C₂×Dic₇₁	—	4	1	no	—
55	Dic₁₄₂	—	2	4	no	—
56	Dic₁₄₂	—	4	2	no	—
57	C₇₁⋊C₈	—	4	2	no	—
58	C₂²×C₇₁	#121	1	1	yes	characteristic
59	C₂²×C₇₁	#123	1	1	yes	characteristic
60	C₂²×C₇₁	#124	1	1	yes	characteristic
61	C₂²×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
62	C₂²×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
63	C₂²×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂²×C₇₁	#137	1	1	yes	characteristic
65	C₄×C₇₁	—	2	2	no	—
66	C₄×C₇₁	—	2	2	no	—
67	C₄×C₇₁	—	8	2	no	—
68	C₄×C₇₁	#123	1	2	yes	—
69	C₄×C₇₁	#126	1	2	yes	—
70	Dic₇₁	—	4	2	no	—
71	Dic₇₁	—	8	1	no	—
72	C₂×C₇₁	#397	1	1	yes	characteristic
73	C₂×C₇₁	#460	1	1	yes	characteristic
74	C₂×C₇₁	#462	1	1	yes	characteristic
75	C₂×C₇₁	#483	1	1	yes	characteristic
76	C₂×C₇₁	#484	1	1	yes	characteristic
77	C₂×C₇₁	#487	1	1	yes	characteristic
78	C₂×C₇₁	#489	1	1	yes	characteristic
79	#2072	—	71	1	no	maximal, Sylow
80	C₇₁	#2072	1	1	yes	characteristic, complemented, Sylow
81	C₂×#103	—	71	2	no	—
82	#391	—	71	1	no	—
83	#384	D₇₁	1	1	yes	characteristic
84	#340	—	71	2	no	—
85	#336	—	71	1	no	—
86	C₂²×Q₈	—	71	1	no	—
87	C₂²×C₈	—	71	2	no	—
88	C₂×C₄⋊C₄	—	2	1	no	—
89	C₂×C₄⋊C₄	—	2	1	no	—
90	C₂×C₄⋊C₄	—	71	1	no	—
91	C₂×C₄⋊C₄	C₂×D₇₁	1	2	yes	—
92	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
93	#103	—	142	4	no	—
94	#95	—	142	1	no	—
95	#95	—	142	1	no	—
96	C₂×Q₈	—	71	2	no	—
97	C₂×Q₈	—	142	2	no	—
98	C₂×Q₈	—	142	2	no	—
99	C₂×Q₈	—	142	1	no	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	—	142	1	no	—
102	C₂²×C₄	—	142	1	no	—
103	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
104	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
105	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
106	C₂×C₈	—	142	2	no	—
107	C₂×C₈	—	142	2	no	—
108	C₂×C₈	—	142	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	142	2	no	—
114	C₄²	—	2	2	no	—
115	C₂³	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
116	Q₈	—	284	2	no	—
117	Q₈	—	142	4	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	284	1	no	—
126	C₂×C₄	—	284	1	no	—
127	C₂×C₄	—	284	1	no	—
128	C₂×C₄	—	284	1	no	—
129	C₂×C₄	—	284	1	no	—
130	C₂×C₄	—	142	2	no	—
131	C₂×C₄	#11765853	1	2	yes	—
132	C₂×C₄	#11765855	1	2	yes	—
133	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
134	C₂×C₄	#11765864	1	1	yes	characteristic
135	C₂×C₄	#11765865	1	2	yes	—
136	C₂×C₄	#11765866	1	2	yes	—
137	C₈	—	284	2	no	—
138	C₂²	#14556099	1	1	yes	characteristic
139	C₂²	#14556111	1	1	yes	characteristic
140	C₂²	#14556117	1	1	yes	characteristic
141	C₂²	C₂×#11765853	1	1	yes	characteristic
142	C₂²	C₂×#11765855	1	1	yes	characteristic
143	C₂²	#14556154	1	1	yes	characteristic
144	C₂²	#14556159	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	8	2	no	—
148	C₄	—	284	2	no	—
149	C₄	—	568	1	no	—
150	C₄	#14556113	1	2	yes	—
151	C₄	#14556123	1	2	yes	—
152	C₂	#17985925	1	1	yes	characteristic
153	C₂	#17986196	1	1	yes	characteristic
154	C₂	#17986206	1	1	yes	characteristic
155	C₂	#17986329	1	1	yes	characteristic
156	C₂	#17986334	1	1	yes	characteristic
157	C₂	#17986351	1	1	yes	characteristic
158	C₂	#17986361	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556051

C₂

yes

maximal

#17985905

C₂

yes

maximal, characteristic

#17985761

C₂

yes

maximal

#17985754

C₂

yes

maximal, characteristic

#384×C₇₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₂

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₇₁

C₂²

yes

characteristic

#14556051

—

C₇₁⋊#95

—

C₇₁⋊#95

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

D₄

yes

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

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

SD₁₆

yes

—

C₂×C₄×C₇₁

Q₁₆

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

—

C₂×C₄×C₇₁

#41

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₇₁⋊C₈

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

C₂×SD₁₆

yes

characteristic

C₂²×C₇₁

C₂×Q₁₆

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#123

yes

—

C₄×C₇₁

#126

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×#103

—

#391

—

#384

D₇₁

yes

characteristic

#340

—

#336

—

C₂²×Q₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₁

yes

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

#103

—

142

—

#95

—

142

—

#95

—

142

—

C₂×Q₈

—

C₂×Q₈

—

142

—

C₂×Q₈

—

142

—

C₂×Q₈

—

142

—

100

C₂²×C₄

—

101

C₂²×C₄

—

142

—

102

C₂²×C₄

—

142

—

103

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

104

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

105

C₂²×C₄

C₂²×D₇₁

yes

characteristic

106

C₂×C₈

—

142

—

107

C₂×C₈

—

142

—

108

C₂×C₈

—

142

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

142

—

114

C₄²

—

115

C₂³

C₂×C₂²⋊D₇₁

yes

characteristic

116

Q₈

—

284

—

117

Q₈

—

142

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

284

—

126

C₂×C₄

—

284

—

127

C₂×C₄

—

284

—

128

C₂×C₄

—

284

—

129

C₂×C₄

—

284

—

130

C₂×C₄

—

142

—

131

C₂×C₄

#11765853

yes

—

132

C₂×C₄

#11765855

yes

—

133

C₂×C₄

D₄×D₇₁

yes

characteristic

134

C₂×C₄

#11765864

yes

characteristic

135

C₂×C₄

#11765865

yes

—

136

C₂×C₄

#11765866

yes

—

137

C₈

—

284

—

138

C₂²

#14556099

yes

characteristic

139

C₂²

#14556111

yes

characteristic

140

C₂²

#14556117

yes

characteristic

141

C₂²

C₂×#11765853

yes

characteristic

142

C₂²

C₂×#11765855

yes

characteristic

143

C₂²

#14556154

yes

characteristic

144

C₂²

#14556159

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

284

—

149

C₄

—

568

—

150

C₄

#14556113

yes

—

151

C₄

#14556123

yes

—

152

C₂

#17985925

yes

characteristic

153

C₂

#17986196

yes

characteristic

154

C₂

#17986206

yes

characteristic

155

C₂

#17986329

yes

characteristic

156

C₂

#17986334

yes

characteristic

157

C₂

#17986351

yes

characteristic

158

C₂

#17986361

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	#2072
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
80. C₇₁	#2072	semidirect

H ≅ G/N

type

80. C₇₁

#2072

semidirect

Subgroups — #22589749

Conjugacy classes of subgroups

Complements