Group #22588455 — subgroups

Subgroups — #22588455

Order 9088.

260 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 117 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×Dic₇₁⋊C₈	C₂	1	2	yes	maximal
2	C₄²×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	#17985754	C₂	1	1	yes	maximal, characteristic
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	Dic₇₁⋊C₈	—	2	4	no	—
14	C₂²×C₈×C₇₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₇₁	C₂²	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	C₂²×Dic₇₁	C₂×C₄	1	1	yes	characteristic
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	C₄×Dic₇₁	—	2	2	no	—
26	C₄×Dic₇₁	—	2	2	no	—
27	C₄×Dic₇₁	C₈	1	4	yes	complemented
28	C₄×Dic₇₁	D₄	1	2	yes	—
29	C₄×Dic₇₁	Q₈	1	2	yes	—
30	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₇₁	—	2	2	no	—
34	C₂×C₈×C₇₁	—	2	2	no	—
35	C₂×C₈×C₇₁	—	2	2	no	—
36	C₄²×C₇₁	—	2	2	no	—
37	C₂³×C₇₁	C₂²×C₄	1	1	yes	characteristic
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	—	2	1	no	—
42	C₂×C₄×C₇₁	C₂×C₈	1	2	yes	—
43	C₂×C₄×C₇₁	M₁₆	1	2	yes	—
44	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₇₁	#41	1	2	yes	—
47	C₂×C₄×C₇₁	#41	1	2	yes	—
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	C₄⋊C₄	1	2	yes	—
55	C₂×Dic₇₁	C₄⋊C₄	1	2	yes	—
56	C₂×Dic₇₁	C₂×C₈	1	2	yes	—
57	C₂×Dic₇₁	C₂×C₈	1	2	yes	—
58	C₂×Dic₇₁	M₁₆	1	2	yes	—
59	C₂×Dic₇₁	M₁₆	1	2	yes	—
60	C₈×C₇₁	—	4	2	no	—
61	C₇₁⋊C₈	—	4	2	no	—
62	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₇₁	#117	1	1	yes	characteristic
64	C₂²×C₇₁	#117	1	1	yes	characteristic
65	C₂²×C₇₁	C₂²×C₈	1	1	yes	characteristic
66	C₂²×C₇₁	C₂×M₁₆	1	1	yes	characteristic
67	C₂²×C₇₁	C₂×M₁₆	1	1	yes	characteristic
68	C₂²×C₇₁	C₂×M₁₆	1	1	yes	characteristic
69	C₄×C₇₁	—	2	2	no	—
70	C₄×C₇₁	—	2	2	no	—
71	C₄×C₇₁	#124	1	2	yes	—
72	C₄×C₇₁	#125	1	2	yes	—
73	Dic₇₁	—	2	2	no	—
74	Dic₇₁	—	2	2	no	—
75	Dic₇₁	—	2	2	no	—
76	Dic₇₁	C₄⋊C₈	1	4	yes	—
77	C₂×C₇₁	#382	1	1	yes	characteristic
78	C₂×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
79	C₂×C₇₁	#423	1	1	yes	characteristic
80	C₂×C₇₁	#431	1	1	yes	characteristic
81	C₂×C₇₁	#431	1	1	yes	characteristic
82	C₂×C₇₁	#432	1	1	yes	characteristic
83	C₂×C₇₁	#432	1	1	yes	characteristic
84	#1831	—	71	1	no	maximal, Sylow
85	C₇₁	#1831	1	1	yes	characteristic, complemented, Sylow
86	C₂×C₄⋊C₈	—	71	2	no	—
87	C₄³	—	71	1	no	—
88	#336	—	71	2	no	—
89	#336	—	71	1	no	—
90	#336	D₇₁	1	1	yes	characteristic
91	C₂²×C₈	—	71	2	no	—
92	C₂²×C₈	C₂×D₇₁	1	2	yes	—
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	—	71	1	no	—
99	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
100	C₄⋊C₈	—	142	4	no	—
101	C₂²×C₄	—	71	1	no	—
102	C₂²×C₄	—	71	1	no	—
103	C₂²×C₄	—	71	1	no	—
104	C₂²×C₄	—	71	1	no	—
105	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
106	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
107	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	142	2	no	—
112	C₂×C₈	—	142	2	no	—
113	C₂×C₈	—	142	2	no	—
114	C₄²	—	2	2	no	—
115	C₄²	—	71	4	no	—
116	C₄²	—	71	2	no	—
117	C₄²	—	71	2	no	—
118	C₄²	—	142	2	no	—
119	C₄²	—	142	2	no	—
120	C₄²	—	142	2	no	—
121	C₄²	—	142	2	no	—
122	C₂³	C₂×C₄×D₇₁	1	1	yes	characteristic
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	71	2	no	—
128	C₂×C₄	—	71	2	no	—
129	C₂×C₄	—	71	2	no	—
130	C₂×C₄	—	71	2	no	—
131	C₂×C₄	—	71	2	no	—
132	C₂×C₄	—	71	2	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	—	142	1	no	—
138	C₂×C₄	—	142	1	no	—
139	C₂×C₄	C₈×D₇₁	1	2	yes	—
140	C₂×C₄	#11765848	1	2	yes	—
141	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
142	C₂×C₄	Q₈×D₇₁	1	1	yes	characteristic
143	C₂×C₄	#11765864	1	2	yes	—
144	C₂×C₄	#11765865	1	2	yes	—
145	C₈	—	4	2	no	—
146	C₈	—	284	2	no	—
147	C₂²	C₄⋊C₄×D₇₁	1	1	yes	characteristic
148	C₂²	#14556084	1	1	yes	characteristic
149	C₂²	#14556085	1	1	yes	characteristic
150	C₂²	C₂×C₈×D₇₁	1	1	yes	characteristic
151	C₂²	M₁₆×D₇₁	1	1	yes	characteristic
152	C₂²	M₁₆×D₇₁	1	1	yes	characteristic
153	C₂²	C₂×#11765848	1	1	yes	characteristic
154	C₄	—	2	2	no	—
155	C₄	—	2	2	no	—
156	C₄	—	71	4	no	—
157	C₄	—	142	2	no	—
158	C₄	—	142	2	no	—
159	C₄	—	142	2	no	—
160	C₄	#14556114	1	2	yes	—
161	C₄	#14556118	1	2	yes	—
162	C₂	#17985863	1	1	yes	characteristic
163	C₂	C₄⋊C₈×D₇₁	1	1	yes	characteristic
164	C₂	#17986021	1	1	yes	characteristic
165	C₂	#17986049	1	1	yes	characteristic
166	C₂	#17986050	1	1	yes	characteristic
167	C₂	#17986053	1	1	yes	characteristic
168	C₂	#17986054	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₇₁⋊C₈

C₂

yes

maximal

C₄²×Dic₇₁

C₂

yes

maximal, characteristic

#17985754

C₂

yes

maximal, characteristic

#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

Dic₇₁⋊C₈

—

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

C₈

yes

complemented

C₄×Dic₇₁

D₄

yes

—

C₄×Dic₇₁

Q₈

yes

—

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

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₁

#41

yes

—

C₂×C₄×C₇₁

#41

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

C₄⋊C₄

yes

—

C₂×Dic₇₁

C₄⋊C₄

yes

—

C₂×Dic₇₁

C₂×C₈

yes

—

C₂×Dic₇₁

C₂×C₈

yes

—

C₂×Dic₇₁

M₁₆

yes

—

C₂×Dic₇₁

M₁₆

yes

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×C₄⋊C₄

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

#124

yes

—

C₄×C₇₁

#125

yes

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

C₄⋊C₈

yes

—

C₂×C₇₁

#382

yes

characteristic

C₂×C₇₁

C₂×C₄⋊C₈

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₄³

—

#336

—

#336

—

#336

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

100

C₄⋊C₈

—

142

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

C₄×D₇₁

yes

characteristic

106

C₂²×C₄

C₄×D₇₁

yes

characteristic

107

C₂²×C₄

C₂²×D₇₁

yes

characteristic

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

142

—

112

C₂×C₈

—

142

—

113

C₂×C₈

—

142

—

114

C₄²

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

142

—

119

C₄²

—

142

—

120

C₄²

—

142

—

121

C₄²

—

142

—

122

C₂³

C₂×C₄×D₇₁

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

142

—

134

C₂×C₄

—

142

—

135

C₂×C₄

—

142

—

136

C₂×C₄

—

142

—

137

C₂×C₄

—

142

—

138

C₂×C₄

—

142

—

139

C₂×C₄

C₈×D₇₁

yes

—

140

C₂×C₄

#11765848

yes

—

141

C₂×C₄

D₄×D₇₁

yes

characteristic

142

C₂×C₄

Q₈×D₇₁

yes

characteristic

143

C₂×C₄

#11765864

yes

—

144

C₂×C₄

#11765865

yes

—

145

C₈

—

146

C₈

—

284

—

147

C₂²

C₄⋊C₄×D₇₁

yes

characteristic

148

C₂²

#14556084

yes

characteristic

149

C₂²

#14556085

yes

characteristic

150

C₂²

C₂×C₈×D₇₁

yes

characteristic

151

C₂²

M₁₆×D₇₁

yes

characteristic

152

C₂²

M₁₆×D₇₁

yes

characteristic

153

C₂²

C₂×#11765848

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

142

—

158

C₄

—

142

—

159

C₄

—

142

—

160

C₄

#14556114

yes

—

161

C₄

#14556118

yes

—

162

C₂

#17985863

yes

characteristic

163

C₂

C₄⋊C₈×D₇₁

yes

characteristic

164

C₂

#17986021

yes

characteristic

165

C₂

#17986049

yes

characteristic

166

C₂

#17986050

yes

characteristic

167

C₂

#17986053

yes

characteristic

168

C₂

#17986054

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	#1831
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
27. C₄×Dic₇₁	C₈	semidirect
85. C₇₁	#1831	semidirect

H ≅ G/N

type

27. C₄×Dic₇₁

C₈

semidirect

85. C₇₁

#1831

semidirect

Subgroups — #22588455

Conjugacy classes of subgroups

Complements