Group #22587900 — subgroups

Subgroups — #22587900

Order 9088.

176 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 49 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	#17986345	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986205	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986191	C₂	1	1	yes	maximal, characteristic
4	#17986076	C₂	1	1	yes	maximal, characteristic, complemented
5	#17985749	C₂	1	1	yes	maximal, characteristic, complemented
6	#17985744	C₂	1	1	yes	maximal, characteristic, complemented
7	#331×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11765854	—	2	1	no	—
9	C₂×#11765850	—	2	1	no	—
10	C₂×#11765848	—	2	1	no	—
11	#14556128	C₂²	1	1	yes	characteristic
12	C₄×D₂₈₄	C₂²	1	1	yes	characteristic
13	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
14	#14556053	—	2	1	no	—
15	#14556049	—	2	1	no	—
16	#14556049	—	2	1	no	—
17	#14556049	—	2	1	no	—
18	(C₂×D₇₁)⋊C₈	—	2	1	no	—
19	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
20	#127×C₇₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₄₂	—	2	1	no	—
24	C₂×D₂₈₄	D₄	1	1	yes	characteristic
25	C₂×C₄×D₇₁	—	2	1	no	—
26	#11765854	—	4	2	no	—
27	#11765850	—	4	2	no	—
28	#11765848	—	4	2	no	—
29	C₂×C₇₁⋊C₈	—	2	1	no	—
30	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
31	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
32	C₄⋊Dic₇₁	—	2	1	no	—
33	C₄⋊Dic₇₁	—	2	1	no	—
34	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
35	(C₂×D₇₁)⋊C₄	—	2	1	no	—
36	C₂×D₄×C₇₁	—	2	1	no	—
37	C₂×D₄×C₇₁	—	2	1	no	—
38	C₂×D₄×C₇₁	—	2	1	no	—
39	C₂×C₈×C₇₁	—	2	1	no	—
40	C₂×C₈×C₇₁	—	2	1	no	—
41	C₄²×C₇₁	C₂³	1	1	yes	characteristic
42	C₂³×C₇₁	—	4	1	no	—
43	C₂²×D₇₁	—	2	1	no	—
44	D₄×C₇₁	—	4	1	no	—
45	D₄×C₇₁	—	4	1	no	—
46	D₄×C₇₁	—	4	1	no	—
47	D₄×C₇₁	—	4	1	no	—
48	D₄×C₇₁	—	8	1	no	—
49	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
52	C₂×Dic₇₁	—	2	1	no	—
53	C₂×Dic₇₁	—	4	1	no	—
54	D₂₈₄	—	2	1	no	—
55	D₂₈₄	—	2	1	no	—
56	C₄×D₇₁	—	4	1	no	—
57	C₄×D₇₁	—	4	1	no	—
58	Dic₁₄₂	—	4	1	no	—
59	Dic₁₄₂	—	4	1	no	—
60	C₈×C₇₁	—	4	1	no	—
61	C₈×C₇₁	—	4	1	no	—
62	C₇₁⋊C₈	—	2	1	no	—
63	C₇₁⋊C₈	—	2	1	no	—
64	C₇₁⋊C₈	—	4	1	no	—
65	C₂²×C₇₁	—	8	1	no	—
66	C₂²×C₇₁	—	8	1	no	—
67	C₂²×C₇₁	—	8	1	no	—
68	C₂²×C₇₁	#121	1	1	yes	characteristic
69	C₂×D₇₁	—	4	1	no	—
70	C₂×D₇₁	—	4	1	no	—
71	C₂×D₇₁	—	4	1	no	—
72	C₄×C₇₁	—	2	1	no	—
73	C₄×C₇₁	#135	1	1	yes	characteristic
74	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
76	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
77	Dic₇₁	—	4	1	no	—
78	Dic₇₁	—	8	1	no	—
79	C₂×C₇₁	—	16	1	no	—
80	C₂×C₇₁	#463	1	1	yes	characteristic
81	C₂×C₇₁	#469	1	1	yes	characteristic
82	C₂×C₇₁	#471	1	1	yes	characteristic
83	D₇₁	—	8	1	no	—
84	#1701	—	71	1	no	maximal, Sylow
85	C₇₁	#1701	1	1	yes	characteristic, complemented, Sylow
86	#486	—	71	1	no	—
87	#461	—	71	1	no	—
88	#459	—	71	1	no	—
89	#436	—	71	1	no	—
90	#334	—	71	1	no	—
91	#332	—	71	1	no	—
92	#331	D₇₁	1	1	yes	characteristic, complemented
93	C₂×SD₁₆	—	142	1	no	—
94	C₂×D₈	—	142	1	no	—
95	C₂×M₁₆	—	142	1	no	—
96	C₄⋊Q₈	—	71	1	no	—
97	#127	C₂×D₇₁	1	1	yes	characteristic
98	C₄×D₄	—	71	1	no	—
99	C₄⋊C₈	—	71	1	no	—
100	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
101	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
102	#103	—	142	1	no	—
103	#102	—	142	1	no	—
104	#102	—	142	1	no	—
105	#102	—	142	1	no	—
106	C₂²⋊C₈	—	142	1	no	—
107	C₄×C₈	—	71	1	no	—
108	C₂×Q₈	—	142	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	2	1	no	—
111	C₂×D₄	—	2	1	no	—
112	C₂×D₄	—	71	1	no	—
113	C₂²×C₄	—	142	1	no	—
114	SD₁₆	—	284	2	no	—
115	D₈	—	284	2	no	—
116	M₁₆	—	284	2	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	71	1	no	—
120	C₂×C₈	—	71	1	no	—
121	C₂×C₈	—	142	1	no	—
122	C₄⋊C₄	—	71	1	no	—
123	C₄⋊C₄	—	142	1	no	—
124	C₄⋊C₄	—	142	1	no	—
125	C₂²⋊C₄	—	142	1	no	—
126	C₄²	C₂²×D₇₁	1	1	yes	characteristic
127	C₂³	—	4	1	no	—
128	C₂³	—	142	1	no	—
129	Q₈	—	284	1	no	—
130	Q₈	—	284	1	no	—
131	D₄	—	4	1	no	—
132	D₄	—	4	1	no	—
133	D₄	—	4	1	no	—
134	D₄	—	4	1	no	—
135	D₄	—	8	1	no	—
136	D₄	—	142	1	no	—
137	D₄	—	142	1	no	—
138	C₂×C₄	—	284	1	no	—
139	C₂×C₄	—	284	1	no	—
140	C₂×C₄	—	284	1	no	—
141	C₂×C₄	—	142	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	C₈	—	4	1	no	—
146	C₈	—	4	1	no	—
147	C₈	—	284	1	no	—
148	C₈	—	142	1	no	—
149	C₈	—	142	1	no	—
150	C₂²	—	8	1	no	—
151	C₂²	—	8	1	no	—
152	C₂²	—	8	1	no	—
153	C₂²	—	284	1	no	—
154	C₂²	—	284	1	no	—
155	C₂²	—	284	1	no	—
156	C₂²	#14556101	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	284	1	no	—
159	C₄	—	568	1	no	—
160	C₄	#14556149	1	1	yes	characteristic
161	C₄	#14556152	1	1	yes	characteristic
162	C₄	#14556152	1	1	yes	characteristic
163	C₄	#14556155	1	1	yes	characteristic
164	C₂	—	16	1	no	—
165	C₂	—	568	1	no	—
166	C₂	#17986213	1	1	yes	characteristic
167	C₂	#17986251	1	1	yes	characteristic
168	C₂	#17986261	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986345

C₂

yes

maximal, characteristic, complemented

#17986205

C₂

yes

maximal, characteristic, complemented

#17986191

C₂

yes

maximal, characteristic

#17986076

C₂

yes

maximal, characteristic, complemented

#17985749

C₂

yes

maximal, characteristic, complemented

#17985744

C₂

yes

maximal, characteristic, complemented

#331×C₇₁

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

characteristic

C₄×D₂₈₄

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

—

—

—

(C₂×D₇₁)⋊C₈

—

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

#127×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×D₂₈₄

D₄

yes

characteristic

C₂×C₄×D₇₁

—

#11765854

—

#11765850

—

#11765848

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

D₄

yes

characteristic

(C₂×D₇₁)⋊C₄

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

—

C₂²×D₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

#41

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

D₂₈₄

—

D₂₈₄

—

C₄×D₇₁

—

C₄×D₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#121

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

—

#1701

—

maximal, Sylow

C₇₁

#1701

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₇₁

yes

characteristic, complemented

C₂×SD₁₆

—

142

—

C₂×D₈

—

142

—

C₂×M₁₆

—

142

—

C₄⋊Q₈

—

#127

C₂×D₇₁

yes

characteristic

C₄×D₄

—

C₄⋊C₈

—

100

C₄⋊C₈

C₂×D₇₁

yes

characteristic

101

C₄⋊C₈

C₂×D₇₁

yes

characteristic

102

#103

—

142

—

103

#102

—

142

—

104

#102

—

142

—

105

#102

—

142

—

106

C₂²⋊C₈

—

142

—

107

C₄×C₈

—

108

C₂×Q₈

—

142

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂²×C₄

—

142

—

114

SD₁₆

—

284

—

115

D₈

—

284

—

116

M₁₆

—

284

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

142

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

142

—

124

C₄⋊C₄

—

142

—

125

C₂²⋊C₄

—

142

—

126

C₄²

C₂²×D₇₁

yes

characteristic

127

C₂³

—

128

C₂³

—

142

—

129

Q₈

—

284

—

130

Q₈

—

284

—

131

D₄

—

132

D₄

—

133

D₄

—

134

D₄

—

135

D₄

—

136

D₄

—

142

—

137

D₄

—

142

—

138

C₂×C₄

—

284

—

139

C₂×C₄

—

284

—

140

C₂×C₄

—

284

—

141

C₂×C₄

—

142

—

142

C₂×C₄

D₄×D₇₁

yes

characteristic

143

C₂×C₄

D₄×D₇₁

yes

characteristic

144

C₂×C₄

#11765865

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

284

—

148

C₈

—

142

—

149

C₈

—

142

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

284

—

154

C₂²

—

284

—

155

C₂²

—

284

—

156

C₂²

#14556101

yes

characteristic

157

C₄

—

158

C₄

—

284

—

159

C₄

—

568

—

160

C₄

#14556149

yes

characteristic

161

C₄

#14556152

yes

characteristic

162

C₄

#14556152

yes

characteristic

163

C₄

#14556155

yes

characteristic

164

C₂

—

165

C₂

—

568

—

166

C₂

#17986213

yes

characteristic

167

C₂

#17986251

yes

characteristic

168

C₂

#17986261

yes

characteristic

	subgroup	G/H
center	C₂²	#14556101
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#331×C₇₁	C₂
socle	C₂²×C₇₁	#121
hypercenter	#331	D₇₁
hypocenter	C₇₁	#1701
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

#14556101

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#331×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
1. #17986345	C₂	semidirect
2. #17986205	C₂	semidirect
4. #17986076	C₂	semidirect
5. #17985749	C₂	semidirect
6. #17985744	C₂	semidirect
7. #331×C₇₁	C₂	semidirect
85. C₇₁	#1701	semidirect
92. #331	D₇₁	semidirect

H ≅ G/N

type

1. #17986345

C₂

semidirect

2. #17986205

C₂

semidirect

4. #17986076

C₂

semidirect

5. #17985749

C₂

semidirect

6. #17985744

C₂

semidirect

7. #331×C₇₁

C₂

semidirect

85. C₇₁

#1701

semidirect

92. #331

D₇₁

semidirect

Subgroups — #22587900

Conjugacy classes of subgroups

Complements