Group #22589801 — subgroups

Subgroups — #22589801

Order 9088.

204 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 59 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	#17986004	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556045	C₂	1	1	yes	maximal, characteristic
4	#17985771	C₂	1	1	yes	maximal, characteristic, complemented
5	#17985769	C₂	1	1	yes	maximal, characteristic, complemented
6	#17985755	C₂	1	1	yes	maximal, characteristic, complemented
7	#418×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#11765849	C₂²	1	1	yes	characteristic
9	C₂×#11765849	C₂²	1	1	yes	characteristic
10	C₂×#11765849	C₂²	1	1	yes	characteristic
11	#14556087	C₂²	1	1	yes	characteristic
12	#14556063	—	2	2	no	—
13	#14556049	—	2	2	no	—
14	#14556045	—	2	2	no	—
15	C₂²×D₄×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
17	#117×C₇₁	C₂²	1	1	yes	characteristic
18	#102×C₇₁	—	2	2	no	—
19	#11765849	—	2	1	no	—
20	#11765849	—	2	1	no	—
21	#11765849	—	2	1	no	—
22	#11765849	—	2	1	no	—
23	#11765849	—	2	1	no	—
24	#11765849	—	2	1	no	—
25	C₂×C₇₁⋊C₈	—	2	1	no	—
26	C₂×C₇₁⋊C₈	—	2	1	no	—
27	C₂×C₇₁⋊C₈	D₄	1	2	yes	—
28	C₄⋊Dic₇₁	—	2	1	no	—
29	C₂²⋊Dic₇₁	—	2	1	no	—
30	C₄×Dic₇₁	—	2	1	no	—
31	C₂⁴×C₇₁	—	2	1	no	—
32	C₂×D₄×C₇₁	—	2	1	no	—
33	C₂×D₄×C₇₁	—	2	1	no	—
34	C₂×D₄×C₇₁	—	2	1	no	—
35	C₂×D₄×C₇₁	—	2	1	no	—
36	C₂×D₄×C₇₁	—	2	1	no	—
37	C₂×D₄×C₇₁	D₄	1	2	yes	—
38	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
39	M₁₆×C₇₁	—	2	1	no	—
40	M₁₆×C₇₁	—	2	1	no	—
41	C₂×C₈×C₇₁	—	2	1	no	—
42	C₄⋊C₄×C₇₁	—	2	1	no	—
43	C₂²⋊C₄×C₇₁	—	2	1	no	—
44	C₄²×C₇₁	—	2	1	no	—
45	C₂³×C₇₁	—	2	2	no	—
46	C₂³×C₇₁	—	4	1	no	—
47	C₂³×C₇₁	—	4	1	no	—
48	C₂³×C₇₁	—	8	2	no	—
49	C₂³×C₇₁	#41	1	1	yes	characteristic
50	D₄×C₇₁	—	2	2	no	—
51	D₄×C₇₁	—	2	2	no	—
52	D₄×C₇₁	—	4	1	no	—
53	D₄×C₇₁	—	4	1	no	—
54	C₂×C₄×C₇₁	—	4	1	no	—
55	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
58	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
59	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
60	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
61	C₂×Dic₇₁	—	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	—	4	2	no	—
68	C₂²×C₇₁	—	8	2	no	—
69	C₂²×C₇₁	—	8	2	no	—
70	C₂²×C₇₁	—	8	2	no	—
71	C₂²×C₇₁	#123	1	1	yes	characteristic
72	C₂²×C₇₁	#124	1	1	yes	characteristic
73	C₂²×C₇₁	#126	1	1	yes	characteristic
74	C₄×C₇₁	—	8	1	no	—
75	C₄×C₇₁	#121	1	1	yes	characteristic
76	C₄×C₇₁	#123	1	1	yes	characteristic
77	C₄×C₇₁	#123	1	1	yes	characteristic
78	C₄×C₇₁	#126	1	1	yes	characteristic
79	Dic₇₁	—	8	1	no	—
80	C₂×C₇₁	—	2	1	no	—
81	C₂×C₇₁	—	2	1	no	—
82	C₂×C₇₁	—	2	1	no	—
83	C₂×C₇₁	—	8	2	no	—
84	C₂×C₇₁	#397	1	1	yes	characteristic
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	D₇₁	1	1	yes	characteristic
90	C₂×#100	—	71	1	no	—
91	#344	—	71	1	no	—
92	#344	—	71	1	no	—
93	H(3,ℤ/4)	—	71	1	no	—
94	C₂²×D₄	C₂×D₇₁	1	1	yes	characteristic
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	—	71	1	no	—
100	#117	C₂×D₇₁	1	1	yes	characteristic
101	#108	—	142	2	no	—
102	#102	—	2	2	no	—
103	#102	—	142	2	no	—
104	#100	—	142	2	no	—
105	C₂⁴	—	2	1	no	—
106	C₂×D₄	—	2	1	no	—
107	C₂×D₄	—	2	1	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	2	1	no	—
111	C₂×D₄	C₂²⋊D₇₁	1	2	yes	—
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	—	142	1	no	—
127	C₂²⋊C₄	—	2	1	no	—
128	C₂²⋊C₄	—	142	1	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	142	1	no	—
131	C₂³	—	2	2	no	—
132	C₂³	—	4	1	no	—
133	C₂³	—	4	1	no	—
134	C₂³	—	8	2	no	—
135	C₂³	#11765864	1	1	yes	characteristic
136	D₄	—	2	2	no	—
137	D₄	—	2	2	no	—
138	D₄	—	4	1	no	—
139	D₄	—	4	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	284	1	no	—
142	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
143	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
144	C₂×C₄	#11765864	1	1	yes	characteristic
145	C₂×C₄	#11765864	1	1	yes	characteristic
146	C₂×C₄	#11765865	1	1	yes	characteristic
147	C₂×C₄	#11765865	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	—	4	2	no	—
154	C₂²	—	8	2	no	—
155	C₂²	—	8	2	no	—
156	C₂²	—	8	2	no	—
157	C₂²	#14556109	1	1	yes	characteristic
158	C₂²	#14556114	1	1	yes	characteristic
159	C₂²	#14556121	1	1	yes	characteristic
160	C₄	—	8	1	no	—
161	C₄	—	568	1	no	—
162	C₄	#14556100	1	1	yes	characteristic
163	C₄	#14556110	1	1	yes	characteristic
164	C₄	#14556111	1	1	yes	characteristic
165	C₄	#14556121	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	—	8	2	no	—
170	C₂	#17985927	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986047

C₂

yes

maximal, characteristic, complemented

#17986004

C₂

yes

maximal, characteristic

C₂×#14556045

C₂

yes

maximal, characteristic

#17985771

C₂

yes

maximal, characteristic, complemented

#17985769

C₂

yes

maximal, characteristic, complemented

#17985755

C₂

yes

maximal, characteristic, complemented

#418×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

C₂²

yes

characteristic

C₂×#11765849

C₂²

yes

characteristic

C₂×#11765849

C₂²

yes

characteristic

#14556087

C₂²

yes

—

—

—

C₂²×D₄×C₇₁

C₂²

yes

characteristic

C₂×M₁₆×C₇₁

C₂²

yes

characteristic

#117×C₇₁

C₂²

yes

characteristic

#102×C₇₁

—

—

—

—

—

—

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₄

yes

—

C₄⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₂⁴×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

D₄

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

#41

yes

characteristic

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×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

C₂×Dic₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#430

—

#418

—

#418

D₇₁

yes

—

—

—

—

C₂²×D₄

C₂×D₇₁

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₁

yes

characteristic

#117

—

100

#117

C₂×D₇₁

yes

characteristic

101

#108

—

142

—

102

#102

—

103

#102

—

142

—

104

#100

—

142

—

105

C₂⁴

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

C₂²⋊D₇₁

yes

—

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

—

142

—

127

C₂²⋊C₄

—

128

C₂²⋊C₄

—

142

—

129

C₄²

—

130

C₄²

—

142

—

131

C₂³

—

132

C₂³

—

133

C₂³

—

134

C₂³

—

135

C₂³

#11765864

yes

characteristic

136

D₄

—

137

D₄

—

138

D₄

—

139

D₄

—

140

C₂×C₄

—

141

C₂×C₄

—

284

—

142

C₂×C₄

D₄×D₇₁

yes

characteristic

143

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

144

C₂×C₄

#11765864

yes

characteristic

145

C₂×C₄

#11765864

yes

characteristic

146

C₂×C₄

#11765865

yes

characteristic

147

C₂×C₄

#11765865

yes

characteristic

148

C₈

—

149

C₈

—

284

—

150

C₈

—

284

—

151

C₈

—

142

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

#14556109

yes

characteristic

158

C₂²

#14556114

yes

characteristic

159

C₂²

#14556121

yes

characteristic

160

C₄

—

161

C₄

—

568

—

162

C₄

#14556100

yes

characteristic

163

C₄

#14556110

yes

characteristic

164

C₄

#14556111

yes

characteristic

165

C₄

#14556121

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

—

169

C₂

—

170

C₂

#17985927

yes

characteristic

	subgroup	G/H
center	C₂	#17985927
commutator	C₂²×C₄×C₇₁	C₂³
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	#418×C₇₁	C₂
socle	C₂×C₇₁	#397
hypercenter	#418	D₇₁
hypocenter	C₇₁	#2081
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #418

subgroup

G/H

center

C₂

#17985927

commutator

C₂²×C₄×C₇₁

C₂³

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

#418×C₇₁

C₂

C₂×C₇₁

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #418

N	H ≅ G/N	type
1. #17986047	C₂	semidirect
4. #17985771	C₂	semidirect
5. #17985769	C₂	semidirect
6. #17985755	C₂	semidirect
86. C₇₁	#2081	semidirect

H ≅ G/N

type

1. #17986047

C₂

semidirect

4. #17985771

C₂

semidirect

5. #17985769

C₂

semidirect

6. #17985755

C₂

semidirect

86. C₇₁

#2081

semidirect

Subgroups — #22589801

Conjugacy classes of subgroups

Complements