Group #22588482 — subgroups

Subgroups — #22588482

Order 9088.

240 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₄²×Dic₇₁	C₂	1	1	yes	maximal, characteristic
2	#17985754	C₂	1	1	yes	maximal, characteristic
3	#17985753	C₂	1	2	yes	maximal
4	#17985753	C₂	1	2	yes	maximal
5	#336×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×Dic₇₁	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₇₁	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₇₁	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₇₁	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₇₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₇₁	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₇₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
15	C₂²×Dic₇₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₇₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×Dic₇₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×Dic₇₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₇₁⋊C₈	—	2	2	no	—
20	C₂×C₇₁⋊C₈	—	2	2	no	—
21	C₂×C₇₁⋊C₈	—	2	2	no	—
22	C₄×Dic₇₁	—	2	2	no	—
23	C₄×Dic₇₁	—	2	1	no	—
24	C₄×Dic₇₁	—	2	2	no	—
25	C₄×Dic₇₁	—	2	1	no	—
26	C₄×Dic₇₁	—	2	2	no	—
27	C₄×Dic₇₁	—	2	2	no	—
28	C₄×Dic₇₁	C₈	1	4	yes	complemented
29	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
32	C₂×C₈×C₇₁	—	2	2	no	—
33	C₂×C₈×C₇₁	—	2	2	no	—
34	C₂×C₈×C₇₁	—	2	2	no	—
35	C₄²×C₇₁	—	2	2	no	—
36	C₂³×C₇₁	C₂²×C₄	1	1	yes	characteristic
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	—	2	1	no	—
41	C₂×C₄×C₇₁	C₂×C₈	1	2	yes	—
42	C₂×C₄×C₇₁	M₁₆	1	2	yes	—
43	C₂×C₄×C₇₁	#41	1	2	yes	—
44	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
45	C₂×C₄×C₇₁	#41	1	2	yes	—
46	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
47	C₂×Dic₇₁	—	2	1	no	—
48	C₂×Dic₇₁	—	2	1	no	—
49	C₂×Dic₇₁	—	2	1	no	—
50	C₂×Dic₇₁	—	2	1	no	—
51	C₂×Dic₇₁	—	2	1	no	—
52	C₂×Dic₇₁	—	2	1	no	—
53	C₂×Dic₇₁	—	2	1	no	—
54	C₂×Dic₇₁	—	2	1	no	—
55	C₂×Dic₇₁	C₂×C₈	1	2	yes	—
56	C₂×Dic₇₁	C₂×C₈	1	2	yes	—
57	C₂×Dic₇₁	M₁₆	1	2	yes	—
58	C₂×Dic₇₁	M₁₆	1	2	yes	—
59	C₈×C₇₁	—	4	2	no	—
60	C₇₁⋊C₈	—	4	2	no	—
61	C₂²×C₇₁	#117	1	1	yes	characteristic
62	C₂²×C₇₁	#117	1	1	yes	characteristic
63	C₂²×C₇₁	#117	1	1	yes	characteristic
64	C₂²×C₇₁	C₂²×C₈	1	1	yes	characteristic
65	C₂²×C₇₁	C₂×M₁₆	1	1	yes	characteristic
66	C₂²×C₇₁	C₂×M₁₆	1	1	yes	characteristic
67	C₂²×C₇₁	C₂×M₁₆	1	1	yes	characteristic
68	C₄×C₇₁	—	2	2	no	—
69	C₄×C₇₁	—	2	2	no	—
70	C₄×C₇₁	#126	1	2	yes	—
71	C₄×C₇₁	#126	1	2	yes	—
72	Dic₇₁	—	2	2	no	—
73	Dic₇₁	—	2	2	no	—
74	Dic₇₁	—	2	2	no	—
75	Dic₇₁	—	2	2	no	—
76	C₂×C₇₁	#383	1	1	yes	characteristic
77	C₂×C₇₁	#431	1	1	yes	characteristic
78	C₂×C₇₁	#431	1	1	yes	characteristic
79	C₂×C₇₁	#431	1	1	yes	characteristic
80	C₂×C₇₁	#432	1	1	yes	characteristic
81	C₂×C₇₁	#432	1	1	yes	characteristic
82	C₂×C₇₁	#432	1	1	yes	characteristic
83	#1839	—	71	1	no	maximal, Sylow
84	C₇₁	#1839	1	1	yes	characteristic, complemented, Sylow
85	C₄³	—	71	1	no	—
86	#336	—	71	2	no	—
87	#336	—	71	2	no	—
88	#336	—	71	1	no	—
89	#336	D₇₁	1	1	yes	characteristic
90	C₂²×C₈	—	71	2	no	—
91	C₂²×C₈	C₂×D₇₁	1	2	yes	—
92	C₂×C₄²	—	71	1	no	—
93	C₂×C₄²	—	71	1	no	—
94	C₂×C₄²	—	71	1	no	—
95	C₂×C₄²	—	71	1	no	—
96	C₂×C₄²	—	71	1	no	—
97	C₂×C₄²	—	71	1	no	—
98	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
99	C₂²×C₄	—	71	1	no	—
100	C₂²×C₄	—	71	1	no	—
101	C₂²×C₄	—	71	1	no	—
102	C₂²×C₄	—	71	1	no	—
103	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
104	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
105	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	142	2	no	—
110	C₂×C₈	—	142	2	no	—
111	C₂×C₈	—	142	2	no	—
112	C₄²	—	2	2	no	—
113	C₄²	—	71	4	no	—
114	C₄²	—	142	2	no	—
115	C₄²	—	142	2	no	—
116	C₄²	—	142	2	no	—
117	C₄²	—	142	1	no	—
118	C₄²	—	142	2	no	—
119	C₄²	—	142	1	no	—
120	C₂³	C₂×C₄×D₇₁	1	1	yes	characteristic
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	71	2	no	—
126	C₂×C₄	—	71	2	no	—
127	C₂×C₄	—	71	2	no	—
128	C₂×C₄	—	71	2	no	—
129	C₂×C₄	—	142	1	no	—
130	C₂×C₄	—	142	1	no	—
131	C₂×C₄	—	142	1	no	—
132	C₂×C₄	—	142	1	no	—
133	C₂×C₄	—	142	1	no	—
134	C₂×C₄	—	142	1	no	—
135	C₂×C₄	—	142	1	no	—
136	C₂×C₄	—	142	1	no	—
137	C₂×C₄	C₈×D₇₁	1	2	yes	—
138	C₂×C₄	#11765848	1	2	yes	—
139	C₂×C₄	#11765864	1	1	yes	characteristic
140	C₂×C₄	#11765864	1	2	yes	—
141	C₂×C₄	#11765865	1	2	yes	—
142	C₂×C₄	#11765866	1	1	yes	characteristic
143	C₈	—	4	2	no	—
144	C₈	—	284	2	no	—
145	C₂²	#14556084	1	1	yes	characteristic
146	C₂²	#14556085	1	1	yes	characteristic
147	C₂²	#14556086	1	1	yes	characteristic
148	C₂²	C₂×C₈×D₇₁	1	1	yes	characteristic
149	C₂²	M₁₆×D₇₁	1	1	yes	characteristic
150	C₂²	M₁₆×D₇₁	1	1	yes	characteristic
151	C₂²	C₂×#11765848	1	1	yes	characteristic
152	C₄	—	2	2	no	—
153	C₄	—	2	2	no	—
154	C₄	—	142	2	no	—
155	C₄	—	142	2	no	—
156	C₄	—	142	2	no	—
157	C₄	—	142	2	no	—
158	C₄	#14556121	1	2	yes	—
159	C₄	#14556123	1	2	yes	—
160	C₂	#17985866	1	1	yes	characteristic
161	C₂	#17986049	1	1	yes	characteristic
162	C₂	#17986050	1	1	yes	characteristic
163	C₂	#17986051	1	1	yes	characteristic
164	C₂	#17986053	1	1	yes	characteristic
165	C₂	#17986054	1	1	yes	characteristic
166	C₂	#17986055	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄²×Dic₇₁

C₂

yes

maximal, characteristic

#17985754

C₂

yes

maximal, characteristic

#17985753

C₂

yes

maximal

#17985753

C₂

yes

maximal

#336×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

—

C₂×C₄×Dic₇₁

C₄

yes

characteristic

C₂×C₄×Dic₇₁

C₄

yes

characteristic

C₂×C₄×Dic₇₁

C₄

yes

characteristic

C₂×C₄×Dic₇₁

C₄

yes

characteristic

C₂×C₄×Dic₇₁

C₂²

yes

characteristic

C₂×C₄×Dic₇₁

C₂²

yes

characteristic

C₂²×C₈×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₂²×Dic₇₁

C₂×C₄

yes

characteristic

C₂²×Dic₇₁

C₂×C₄

yes

characteristic

C₂²×Dic₇₁

C₂×C₄

yes

characteristic

C₂²×Dic₇₁

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

C₈

yes

complemented

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₈

yes

—

C₂×C₄×C₇₁

M₁₆

yes

—

C₂×C₄×C₇₁

#41

yes

—

C₂×C₄×C₇₁

#41

yes

characteristic

C₂×C₄×C₇₁

#41

yes

—

C₂×C₄×C₇₁

#41

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

C₂×C₈

yes

—

C₂×Dic₇₁

C₂×C₈

yes

—

C₂×Dic₇₁

M₁₆

yes

—

C₂×Dic₇₁

M₁₆

yes

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

C₂²×C₈

yes

characteristic

C₂²×C₇₁

C₂×M₁₆

yes

characteristic

C₂²×C₇₁

C₂×M₁₆

yes

characteristic

C₂²×C₇₁

C₂×M₁₆

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#126

yes

—

C₄×C₇₁

#126

yes

—

Dic₇₁

—

Dic₇₁

—

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

C₄³

—

—

—

—

D₇₁

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×D₇₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₄×D₇₁

yes

characteristic

104

C₂²×C₄

C₄×D₇₁

yes

characteristic

105

C₂²×C₄

C₂²×D₇₁

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

142

—

110

C₂×C₈

—

142

—

111

C₂×C₈

—

142

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

142

—

115

C₄²

—

142

—

116

C₄²

—

142

—

117

C₄²

—

142

—

118

C₄²

—

142

—

119

C₄²

—

142

—

120

C₂³

C₂×C₄×D₇₁

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

142

—

130

C₂×C₄

—

142

—

131

C₂×C₄

—

142

—

132

C₂×C₄

—

142

—

133

C₂×C₄

—

142

—

134

C₂×C₄

—

142

—

135

C₂×C₄

—

142

—

136

C₂×C₄

—

142

—

137

C₂×C₄

C₈×D₇₁

yes

—

138

C₂×C₄

#11765848

yes

—

139

C₂×C₄

#11765864

yes

characteristic

140

C₂×C₄

#11765864

yes

—

141

C₂×C₄

#11765865

yes

—

142

C₂×C₄

#11765866

yes

characteristic

143

C₈

—

144

C₈

—

284

—

145

C₂²

#14556084

yes

characteristic

146

C₂²

#14556085

yes

characteristic

147

C₂²

#14556086

yes

characteristic

148

C₂²

C₂×C₈×D₇₁

yes

characteristic

149

C₂²

M₁₆×D₇₁

yes

characteristic

150

C₂²

M₁₆×D₇₁

yes

characteristic

151

C₂²

C₂×#11765848

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

142

—

155

C₄

—

142

—

156

C₄

—

142

—

157

C₄

—

142

—

158

C₄

#14556121

yes

—

159

C₄

#14556123

yes

—

160

C₂

#17985866

yes

characteristic

161

C₂

#17986049

yes

characteristic

162

C₂

#17986050

yes

characteristic

163

C₂

#17986051

yes

characteristic

164

C₂

#17986053

yes

characteristic

165

C₂

#17986054

yes

characteristic

166

C₂

#17986055

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₇₁
commutator	C₂²×C₇₁	C₂²×C₈
Fitting	#336×C₇₁	C₂
socle	C₂³×C₇₁	C₂²×C₄
hypercenter	#336	D₇₁
hypocenter	C₇₁	#1839
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₇₁

commutator

C₂²×C₇₁

C₂²×C₈

Fitting

#336×C₇₁

C₂

socle

C₂³×C₇₁

C₂²×C₄

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
28. C₄×Dic₇₁	C₈	semidirect
84. C₇₁	#1839	semidirect

H ≅ G/N

type

28. C₄×Dic₇₁

C₈

semidirect

84. C₇₁

#1839

semidirect

Subgroups — #22588482

Conjugacy classes of subgroups

Complements