Group #22588355 — subgroups

Subgroups — #22588355

Order 9088.

292 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 125 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×C₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556052	C₂	1	2	yes	maximal
3	#17985865	C₂	1	1	yes	maximal, characteristic
4	#17985761	C₂	1	2	yes	maximal
5	C₂×C₄×Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₇₁	C₂²	1	2	yes	—
8	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
9	#14556052	—	2	4	no	—
10	C₇₁⋊#95	—	2	1	no	—
11	C₇₁⋊#95	—	2	1	no	—
12	C₂²×Q₈×C₇₁	C₂²	1	1	yes	characteristic
13	C₄×Q₈×C₇₁	—	2	2	no	—
14	C₄×Q₈×C₇₁	C₄	1	4	yes	complemented
15	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
16	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₇₁	—	2	1	no	—
18	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
19	C₂²×Dic₇₁	—	2	2	no	—
20	C₂×C₇₁⋊C₈	—	2	2	no	—
21	C₂×C₇₁⋊C₈	—	2	2	no	—
22	C₂×C₇₁⋊C₈	—	2	2	no	—
23	C₄⋊Dic₇₁	—	2	2	no	—
24	C₄⋊Dic₇₁	—	2	2	no	—
25	C₂×Q₈×C₇₁	—	2	1	no	—
26	C₂×Q₈×C₇₁	—	2	1	no	—
27	C₂×Q₈×C₇₁	—	2	1	no	—
28	C₂×Q₈×C₇₁	C₂×C₄	1	2	yes	—
29	C₂×Q₈×C₇₁	D₄	1	2	yes	—
30	C₂×Q₈×C₇₁	Q₈	1	2	yes	—
31	C₂²×C₄×C₇₁	—	2	1	no	—
32	C₂²×C₄×C₇₁	—	2	1	no	—
33	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₇₁	—	2	4	no	—
37	C₄⋊C₄×C₇₁	—	2	1	no	—
38	C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
39	C₄²×C₇₁	—	2	4	no	—
40	C₄²×C₇₁	C₂×C₄	1	2	yes	—
41	C₄²×C₇₁	D₄	1	2	yes	—
42	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₇₁	—	2	2	no	—
44	Q₈×C₇₁	—	2	2	no	—
45	Q₈×C₇₁	—	2	2	no	—
46	Q₈×C₇₁	C₄⋊C₄	1	4	yes	—
47	C₂×C₄×C₇₁	—	2	2	no	—
48	C₂×C₄×C₇₁	—	2	2	no	—
49	C₂×C₄×C₇₁	—	2	2	no	—
50	C₂×C₄×C₇₁	—	2	2	no	—
51	C₂×C₄×C₇₁	—	2	2	no	—
52	C₂×C₄×C₇₁	—	2	2	no	—
53	C₂×C₄×C₇₁	—	4	2	no	—
54	C₂×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
55	C₂×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
56	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
61	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
62	C₂×Dic₇₁	—	4	2	no	—
63	C₂×Dic₇₁	—	4	2	no	—
64	C₂×Dic₇₁	—	4	2	no	—
65	C₇₁⋊C₈	—	4	2	no	—
66	C₂²×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₇₁	#121	1	1	yes	characteristic
68	C₂²×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂²×C₇₁	#135	1	2	yes	—
70	C₂²×C₇₁	#137	1	2	yes	—
71	C₄×C₇₁	—	2	4	no	—
72	C₄×C₇₁	—	2	2	no	—
73	C₄×C₇₁	—	2	2	no	—
74	C₄×C₇₁	—	4	2	no	—
75	C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₄×C₇₁	#117	1	1	yes	characteristic
77	C₄×C₇₁	#121	1	1	yes	characteristic
78	C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
79	Dic₇₁	—	8	2	no	—
80	C₂×C₇₁	#380	1	1	yes	characteristic
81	C₂×C₇₁	#416	1	1	yes	characteristic
82	C₂×C₇₁	#419	1	1	yes	characteristic
83	C₂×C₇₁	#464	1	2	yes	—
84	C₂×C₇₁	#479	1	2	yes	—
85	#1805	—	71	1	no	maximal, Sylow
86	C₇₁	#1805	1	1	yes	characteristic, complemented, Sylow
87	C₂×C₄×Q₈	D₇₁	1	1	yes	characteristic
88	C₂×C₄⋊C₈	—	71	1	no	—
89	C₂×#103	—	71	2	no	—
90	#382	—	71	1	no	—
91	#340	—	71	2	no	—
92	C₂²×Q₈	C₂×D₇₁	1	1	yes	characteristic
93	C₂²×C₈	—	71	2	no	—
94	C₄×Q₈	—	2	2	no	—
95	C₄×Q₈	Dic₇₁	1	4	yes	complemented
96	C₂×C₄⋊C₄	—	2	1	no	—
97	C₂×C₄⋊C₄	—	71	2	no	—
98	C₂×C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
99	C₂×C₄²	—	2	1	no	—
100	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
101	C₄⋊C₈	—	142	2	no	—
102	#103	—	142	4	no	—
103	#95	—	142	1	no	—
104	#95	—	142	1	no	—
105	C₂×Q₈	—	2	1	no	—
106	C₂×Q₈	—	2	1	no	—
107	C₂×Q₈	—	2	1	no	—
108	C₂×Q₈	Dic₁₄₂	1	2	yes	—
109	C₂×Q₈	D₂₈₄	1	2	yes	—
110	C₂×Q₈	C₂×Dic₇₁	1	2	yes	—
111	C₂²×C₄	—	2	1	no	—
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	—	142	2	no	—
114	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
115	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
117	C₂×C₈	—	142	2	no	—
118	C₂×C₈	—	142	2	no	—
119	C₂×C₈	—	142	2	no	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	142	2	no	—
123	C₄⋊C₄	—	142	2	no	—
124	C₄⋊C₄	C₂×Dic₇₁	1	2	yes	—
125	C₄²	—	2	4	no	—
126	C₄²	C₂×Dic₇₁	1	2	yes	—
127	C₄²	C₂²⋊D₇₁	1	2	yes	—
128	C₂³	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
129	Q₈	—	2	2	no	—
130	Q₈	—	2	2	no	—
131	Q₈	—	2	2	no	—
132	Q₈	C₄⋊Dic₇₁	1	4	yes	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	284	2	no	—
141	C₂×C₄	—	284	2	no	—
142	C₂×C₄	—	284	2	no	—
143	C₂×C₄	C₂²⋊Dic₇₁	1	2	yes	—
144	C₂×C₄	C₂²⋊Dic₇₁	1	2	yes	—
145	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
147	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
148	C₂×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
149	C₂×C₄	#11765865	1	1	yes	characteristic
150	C₂×C₄	#11765865	1	1	yes	characteristic
151	C₈	—	284	2	no	—
152	C₂²	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
153	C₂²	#14556102	1	1	yes	characteristic
154	C₂²	#14556107	1	1	yes	characteristic
155	C₂²	#14556148	1	2	yes	—
156	C₂²	#14556159	1	2	yes	—
157	C₄	—	2	4	no	—
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	4	2	no	—
161	C₄	—	568	2	no	—
162	C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
163	C₄	#14556087	1	1	yes	characteristic
164	C₄	#14556102	1	1	yes	characteristic
165	C₄	#14556107	1	1	yes	characteristic
166	C₂	#17985859	1	1	yes	characteristic
167	C₂	#17985995	1	1	yes	characteristic
168	C₂	#17986008	1	1	yes	characteristic
169	C₂	#17986221	1	2	yes	—
170	C₂	#17986306	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₇₁⋊C₈)

C₂

yes

maximal, characteristic

C₂×#14556052

C₂

yes

maximal

#17985865

C₂

yes

maximal, characteristic

#17985761

C₂

yes

maximal

C₂×C₄×Q₈×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₇₁

C₂²

yes

—

C₄⋊(C₇₁⋊C₈)

—

#14556052

—

C₇₁⋊#95

—

C₇₁⋊#95

—

C₂²×Q₈×C₇₁

C₂²

yes

characteristic

C₄×Q₈×C₇₁

—

C₄×Q₈×C₇₁

C₄

yes

complemented

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

C₂²

yes

characteristic

C₂×C₄²×C₇₁

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

C₂×C₄

yes

—

C₂×Q₈×C₇₁

D₄

yes

—

C₂×Q₈×C₇₁

Q₈

yes

—

C₂²×C₄×C₇₁

—

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

C₂×C₄

yes

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

D₄

yes

—

C₂³×C₇₁

C₂×D₄

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₄

yes

—

C₂×C₄×C₇₁

C₂²⋊C₄

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₁

#121

yes

characteristic

C₂²×C₇₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₁

#135

yes

—

C₂²×C₇₁

#137

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×C₄⋊C₄

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₂×C₇₁

#479

yes

—

#1805

—

maximal, Sylow

C₇₁

#1805

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

D₇₁

yes

characteristic

C₂×C₄⋊C₈

—

C₂×#103

—

#382

—

#340

—

C₂²×Q₈

C₂×D₇₁

yes

characteristic

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

Dic₇₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₁

yes

characteristic

C₂×C₄²

—

100

C₂×C₄²

C₂×D₇₁

yes

characteristic

101

C₄⋊C₈

—

142

—

102

#103

—

142

—

103

#95

—

142

—

104

#95

—

142

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

Dic₁₄₂

yes

—

109

C₂×Q₈

D₂₈₄

yes

—

110

C₂×Q₈

C₂×Dic₇₁

yes

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

142

—

114

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

115

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

116

C₂²×C₄

C₂²×D₇₁

yes

characteristic

117

C₂×C₈

—

142

—

118

C₂×C₈

—

142

—

119

C₂×C₈

—

142

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

142

—

123

C₄⋊C₄

—

142

—

124

C₄⋊C₄

C₂×Dic₇₁

yes

—

125

C₄²

—

126

C₄²

C₂×Dic₇₁

yes

—

127

C₄²

C₂²⋊D₇₁

yes

—

128

C₂³

C₂×C₂²⋊D₇₁

yes

characteristic

129

Q₈

—

130

Q₈

—

131

Q₈

—

132

Q₈

C₄⋊Dic₇₁

yes

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

284

—

141

C₂×C₄

—

284

—

142

C₂×C₄

—

284

—

143

C₂×C₄

C₂²⋊Dic₇₁

yes

—

144

C₂×C₄

C₂²⋊Dic₇₁

yes

—

145

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

146

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

147

C₂×C₄

C₂×D₂₈₄

yes

characteristic

148

C₂×C₄

C₂×Dic₁₄₂

yes

characteristic

149

C₂×C₄

#11765865

yes

characteristic

150

C₂×C₄

#11765865

yes

characteristic

151

C₈

—

284

—

152

C₂²

C₂×C₂²⋊Dic₇₁

yes

characteristic

153

C₂²

#14556102

yes

characteristic

154

C₂²

#14556107

yes

characteristic

155

C₂²

#14556148

yes

—

156

C₂²

#14556159

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

568

—

162

C₄

C₂×C₄⋊Dic₇₁

yes

characteristic

163

C₄

#14556087

yes

characteristic

164

C₄

#14556102

yes

characteristic

165

C₄

#14556107

yes

characteristic

166

C₂

#17985859

yes

characteristic

167

C₂

#17985995

yes

characteristic

168

C₂

#17986008

yes

characteristic

169

C₂

#17986221

yes

—

170

C₂

#17986306

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	C₂×C₄×Q₈×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₄
hypercenter	C₂×C₄×Q₈	D₇₁
hypocenter	C₇₁	#1805
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

C₂×C₄×Q₈×C₇₁

C₂

socle

C₂³×C₇₁

C₂×D₄

hypercenter

C₂×C₄×Q₈

D₇₁

hypocenter

C₇₁

#1805

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
14. C₄×Q₈×C₇₁	C₄	semidirect
86. C₇₁	#1805	semidirect
95. C₄×Q₈	Dic₇₁	semidirect

H ≅ G/N

type

14. C₄×Q₈×C₇₁

C₄

semidirect

86. C₇₁

#1805

semidirect

95. C₄×Q₈

Dic₇₁

semidirect

Subgroups — #22588355

Conjugacy classes of subgroups

Complements