Group #22589802 — subgroups

Subgroups — #22589802

Order 9088.

204 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 57 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	#17986047	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986003	C₂	1	1	yes	maximal, characteristic
3	#17986003	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556044	C₂	1	1	yes	maximal, characteristic
5	#17985769	C₂	1	1	yes	maximal, characteristic, complemented
6	#17985769	C₂	1	1	yes	maximal, characteristic, complemented
7	H(3,ℤ/4)×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₂₈₄	C₂²	1	1	yes	characteristic
9	C₂×#11765849	C₂²	1	1	yes	characteristic
10	C₂×#11765849	C₂²	1	1	yes	characteristic
11	C₂×#11765849	C₂²	1	1	yes	characteristic
12	#14556063	—	2	2	no	—
13	#14556048	—	2	2	no	—
14	#14556048	—	2	2	no	—
15	#14556044	—	2	2	no	—
16	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
17	#117×C₇₁	C₂²	1	1	yes	characteristic
18	#117×C₇₁	C₂²	1	1	yes	characteristic
19	C₂³×D₇₁	—	2	1	no	—
20	C₂×D₂₈₄	—	2	1	no	—
21	C₂×D₂₈₄	—	2	1	no	—
22	C₂×D₂₈₄	—	2	1	no	—
23	C₂×D₂₈₄	—	2	1	no	—
24	C₂×D₂₈₄	—	2	1	no	—
25	C₂×D₂₈₄	D₄	1	2	yes	—
26	#11765849	—	2	1	no	—
27	#11765849	—	2	1	no	—
28	#11765849	—	2	1	no	—
29	#11765849	—	2	1	no	—
30	#11765849	—	2	1	no	—
31	#11765849	—	2	1	no	—
32	C₂×C₇₁⋊C₈	—	2	1	no	—
33	C₂×C₇₁⋊C₈	—	2	1	no	—
34	C₂×C₇₁⋊C₈	D₄	1	2	yes	—
35	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
36	M₁₆×C₇₁	—	2	1	no	—
37	M₁₆×C₇₁	—	2	1	no	—
38	C₂×C₈×C₇₁	—	2	1	no	—
39	C₄⋊C₄×C₇₁	—	2	1	no	—
40	C₄⋊C₄×C₇₁	—	2	1	no	—
41	C₂²⋊C₄×C₇₁	—	2	1	no	—
42	C₂²⋊C₄×C₇₁	—	2	1	no	—
43	C₄²×C₇₁	—	2	1	no	—
44	C₄²×C₇₁	—	2	1	no	—
45	C₂³×C₇₁	#41	1	1	yes	characteristic
46	C₂²×D₇₁	—	2	2	no	—
47	C₂²×D₇₁	—	4	1	no	—
48	C₂²×D₇₁	—	4	1	no	—
49	C₂²×D₇₁	—	8	2	no	—
50	C₂×C₄×C₇₁	—	4	1	no	—
51	C₂×C₄×C₇₁	—	4	1	no	—
52	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
55	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
56	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
57	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
58	D₂₈₄	—	2	2	no	—
59	D₂₈₄	—	2	2	no	—
60	D₂₈₄	—	4	1	no	—
61	D₂₈₄	—	4	1	no	—
62	C₈×C₇₁	—	4	1	no	—
63	C₇₁⋊C₈	—	2	2	no	—
64	C₇₁⋊C₈	—	4	1	no	—
65	C₇₁⋊C₈	—	4	1	no	—
66	C₂²×C₇₁	—	4	1	no	—
67	C₂²×C₇₁	#123	1	1	yes	characteristic
68	C₂²×C₇₁	#124	1	1	yes	characteristic
69	C₂²×C₇₁	#126	1	1	yes	characteristic
70	C₂×D₇₁	—	4	2	no	—
71	C₂×D₇₁	—	8	2	no	—
72	C₂×D₇₁	—	8	2	no	—
73	C₂×D₇₁	—	8	2	no	—
74	C₄×C₇₁	—	8	1	no	—
75	C₄×C₇₁	—	8	1	no	—
76	C₄×C₇₁	#121	1	1	yes	characteristic
77	C₄×C₇₁	#123	1	1	yes	characteristic
78	C₄×C₇₁	#123	1	1	yes	characteristic
79	C₄×C₇₁	#126	1	1	yes	characteristic
80	C₂×C₇₁	—	2	1	no	—
81	C₂×C₇₁	—	2	1	no	—
82	C₂×C₇₁	—	2	1	no	—
83	C₂×C₇₁	#397	1	1	yes	characteristic
84	D₇₁	—	8	2	no	—
85	#2081	—	71	1	no	maximal, Sylow
86	C₇₁	#2081	1	1	yes	characteristic, complemented, Sylow
87	#430	—	71	1	no	—
88	#418	—	71	1	no	—
89	#418	—	71	1	no	—
90	C₂×#100	—	71	1	no	—
91	#344	—	71	1	no	—
92	#344	—	71	1	no	—
93	H(3,ℤ/4)	D₇₁	1	1	yes	characteristic, complemented
94	C₂²×D₄	—	71	1	no	—
95	C₂×M₁₆	—	71	1	no	—
96	C₂×M₁₆	—	71	1	no	—
97	C₂×M₁₆	—	71	1	no	—
98	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
99	#117	C₂×D₇₁	1	1	yes	characteristic
100	#117	C₂×D₇₁	1	1	yes	characteristic
101	#108	—	142	2	no	—
102	#102	—	142	2	no	—
103	#102	—	142	2	no	—
104	#100	—	142	2	no	—
105	C₂⁴	—	142	1	no	—
106	C₂×D₄	—	71	2	no	—
107	C₂×D₄	—	142	1	no	—
108	C₂×D₄	—	142	1	no	—
109	C₂×D₄	—	142	1	no	—
110	C₂×D₄	—	142	1	no	—
111	C₂×D₄	—	142	1	no	—
112	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
113	M₁₆	—	2	1	no	—
114	M₁₆	—	2	1	no	—
115	M₁₆	—	142	1	no	—
116	M₁₆	—	142	1	no	—
117	M₁₆	—	142	1	no	—
118	M₁₆	—	142	1	no	—
119	M₁₆	—	142	1	no	—
120	M₁₆	—	142	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	71	2	no	—
123	C₂×C₈	—	142	1	no	—
124	C₂×C₈	—	142	1	no	—
125	C₄⋊C₄	—	2	1	no	—
126	C₄⋊C₄	—	2	1	no	—
127	C₂²⋊C₄	—	2	1	no	—
128	C₂²⋊C₄	—	2	1	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	2	1	no	—
131	C₂³	—	284	1	no	—
132	C₂³	—	284	1	no	—
133	C₂³	—	568	2	no	—
134	C₂³	—	142	2	no	—
135	C₂³	#11765864	1	1	yes	characteristic
136	D₄	—	284	1	no	—
137	D₄	—	284	1	no	—
138	D₄	—	142	2	no	—
139	D₄	—	142	2	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
143	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
144	C₂×C₄	#11765865	1	1	yes	characteristic
145	C₂×C₄	#11765865	1	1	yes	characteristic
146	C₂×C₄	#11765865	1	1	yes	characteristic
147	C₂×C₄	#11765866	1	1	yes	characteristic
148	C₈	—	4	1	no	—
149	C₈	—	284	1	no	—
150	C₈	—	284	1	no	—
151	C₈	—	142	2	no	—
152	C₂²	—	4	1	no	—
153	C₂²	—	284	2	no	—
154	C₂²	—	568	2	no	—
155	C₂²	—	568	2	no	—
156	C₂²	—	568	2	no	—
157	C₂²	#14556110	1	1	yes	characteristic
158	C₂²	#14556114	1	1	yes	characteristic
159	C₂²	#14556123	1	1	yes	characteristic
160	C₄	—	8	1	no	—
161	C₄	—	8	1	no	—
162	C₄	#14556101	1	1	yes	characteristic
163	C₄	#14556113	1	1	yes	characteristic
164	C₄	#14556113	1	1	yes	characteristic
165	C₄	#14556122	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	—	568	2	no	—
170	C₂	#17985928	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986047

C₂

yes

maximal, characteristic, complemented

#17986003

C₂

yes

maximal, characteristic

#17986003

C₂

yes

maximal, characteristic

C₂×#14556044

C₂

yes

maximal, characteristic

#17985769

C₂

yes

maximal, characteristic, complemented

#17985769

C₂

yes

maximal, characteristic, complemented

H(3,ℤ/4)×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₈₄

C₂²

yes

characteristic

C₂×#11765849

C₂²

yes

characteristic

C₂×#11765849

C₂²

yes

characteristic

C₂×#11765849

C₂²

yes

—

—

—

—

C₂×M₁₆×C₇₁

C₂²

yes

characteristic

#117×C₇₁

C₂²

yes

characteristic

#117×C₇₁

C₂²

yes

characteristic

C₂³×D₇₁

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

C₂×D₂₈₄

D₄

yes

—

—

—

—

—

—

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₄

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

#41

yes

characteristic

C₂²×D₇₁

—

C₂²×D₇₁

—

C₂²×D₇₁

—

C₂²×D₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#397

yes

characteristic

D₇₁

—

#2081

—

maximal, Sylow

C₇₁

#2081

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₇₁

yes

characteristic, complemented

C₂²×D₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₁

yes

characteristic

#117

C₂×D₇₁

yes

characteristic

100

#117

C₂×D₇₁

yes

characteristic

101

#108

—

142

—

102

#102

—

142

—

103

#102

—

142

—

104

#100

—

142

—

105

C₂⁴

—

142

—

106

C₂×D₄

—

107

C₂×D₄

—

142

—

108

C₂×D₄

—

142

—

109

C₂×D₄

—

142

—

110

C₂×D₄

—

142

—

111

C₂×D₄

—

142

—

112

C₂²×C₄

C₂²×D₇₁

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

142

—

116

M₁₆

—

142

—

117

M₁₆

—

142

—

118

M₁₆

—

142

—

119

M₁₆

—

142

—

120

M₁₆

—

142

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

142

—

124

C₂×C₈

—

142

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₂²⋊C₄

—

128

C₂²⋊C₄

—

129

C₄²

—

130

C₄²

—

131

C₂³

—

284

—

132

C₂³

—

284

—

133

C₂³

—

568

—

134

C₂³

—

142

—

135

C₂³

#11765864

yes

characteristic

136

D₄

—

284

—

137

D₄

—

284

—

138

D₄

—

142

—

139

D₄

—

142

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

D₄×D₇₁

yes

characteristic

143

C₂×C₄

D₄×D₇₁

yes

characteristic

144

C₂×C₄

#11765865

yes

characteristic

145

C₂×C₄

#11765865

yes

characteristic

146

C₂×C₄

#11765865

yes

characteristic

147

C₂×C₄

#11765866

yes

characteristic

148

C₈

—

149

C₈

—

284

—

150

C₈

—

284

—

151

C₈

—

142

—

152

C₂²

—

153

C₂²

—

284

—

154

C₂²

—

568

—

155

C₂²

—

568

—

156

C₂²

—

568

—

157

C₂²

#14556110

yes

characteristic

158

C₂²

#14556114

yes

characteristic

159

C₂²

#14556123

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

#14556101

yes

characteristic

163

C₄

#14556113

yes

characteristic

164

C₄

#14556113

yes

characteristic

165

C₄

#14556122

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

—

169

C₂

—

568

—

170

C₂

#17985928

yes

characteristic

	subgroup	G/H
center	C₂	#17985928
commutator	C₂²×C₄×C₇₁	C₂³
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	H(3,ℤ/4)×C₇₁	C₂
socle	C₂×C₇₁	#397
hypercenter	H(3,ℤ/4)	D₇₁
hypocenter	C₇₁	#2081
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₂

#17985928

commutator

C₂²×C₄×C₇₁

C₂³

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

H(3,ℤ/4)×C₇₁

C₂

C₂×C₇₁

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
1. #17986047	C₂	semidirect
5. #17985769	C₂	semidirect
6. #17985769	C₂	semidirect
7. H(3,ℤ/4)×C₇₁	C₂	semidirect
86. C₇₁	#2081	semidirect
93. H(3,ℤ/4)	D₇₁	semidirect

H ≅ G/N

type

1. #17986047

C₂

semidirect

5. #17985769

C₂

semidirect

6. #17985769

C₂

semidirect

7. H(3,ℤ/4)×C₇₁

C₂

semidirect

86. C₇₁

#2081

semidirect

93. H(3,ℤ/4)

D₇₁

semidirect

Subgroups — #22589802

Conjugacy classes of subgroups

Complements