Group #22587047 — subgroups

Subgroups — #22587047

Order 9088.

198 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 65 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	#17986554	C₂	1	1	yes	maximal, characteristic
2	#17986055	C₂	1	1	yes	maximal, characteristic
3	#17986051	C₂	1	1	yes	maximal, characteristic
4	#17985747	C₂	1	1	yes	maximal, characteristic, complemented
5	#17985747	C₂	1	1	yes	maximal, characteristic, complemented
6	#17985745	C₂	1	1	yes	maximal, characteristic, complemented
7	#332×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#14556128	C₂²	1	1	yes	characteristic
9	#14556119	—	2	1	no	—
10	#14556108	C₄	1	2	yes	—
11	#14556104	—	2	1	no	—
12	Q₈×Dic₇₁	—	2	1	no	—
13	#14556093	C₄	1	2	yes	—
14	#14556086	—	2	1	no	—
15	C₄²×D₇₁	C₂²	1	1	yes	characteristic
16	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
17	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
18	(C₂×D₇₁)⋊C₈	—	2	1	no	—
19	(C₂×D₇₁)⋊C₈	—	2	1	no	—
20	#14556038	—	2	1	no	—
21	C₈×Dic₇₁	—	2	1	no	—
22	C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
25	C₂×Dic₁₄₂	C₂×C₄	1	2	yes	—
26	C₂×C₄×D₇₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×D₇₁	D₄	1	1	yes	characteristic
28	C₂×C₄×D₇₁	D₄	1	1	yes	characteristic
29	C₂×C₇₁⋊C₈	—	2	1	no	—
30	C₂×C₇₁⋊C₈	—	2	1	no	—
31	C₄⋊Dic₇₁	—	2	1	no	—
32	C₄⋊Dic₇₁	—	2	1	no	—
33	C₄⋊Dic₇₁	—	2	2	no	—
34	Dic₇₁⋊C₄	—	2	2	no	—
35	(C₂×D₇₁)⋊C₄	—	2	2	no	—
36	C₄×Dic₇₁	—	2	2	no	—
37	C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
38	C₄×Dic₇₁	D₄	1	1	yes	characteristic
39	C₄×Dic₇₁	D₄	1	1	yes	characteristic
40	C₂×Q₈×C₇₁	—	2	1	no	—
41	C₂×C₈×C₇₁	—	2	1	no	—
42	C₂×C₈×C₇₁	—	2	1	no	—
43	C₄⋊C₄×C₇₁	—	2	1	no	—
44	C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
45	C₄²×C₇₁	C₂³	1	1	yes	characteristic
46	C₂²×D₇₁	C₂²⋊C₄	1	1	yes	characteristic
47	Q₈×C₇₁	—	4	1	no	—
48	Q₈×C₇₁	—	4	1	no	—
49	C₂×C₄×C₇₁	—	2	2	no	—
50	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₇₁	—	2	2	no	—
54	C₂×Dic₇₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Dic₇₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₇₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₄×D₇₁	—	2	1	no	—
58	C₄×D₇₁	—	2	1	no	—
59	C₄×D₇₁	—	2	1	no	—
60	C₄×D₇₁	—	2	1	no	—
61	C₄×D₇₁	—	4	1	no	—
62	Dic₁₄₂	—	4	2	no	—
63	C₈×C₇₁	—	4	1	no	—
64	C₈×C₇₁	—	4	1	no	—
65	C₇₁⋊C₈	—	4	1	no	—
66	C₇₁⋊C₈	—	4	1	no	—
67	C₂²×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×D₇₁	—	2	1	no	—
69	C₂×D₇₁	—	2	1	no	—
70	C₂×D₇₁	—	2	1	no	—
71	C₄×C₇₁	—	2	1	no	—
72	C₄×C₇₁	—	4	2	no	—
73	C₄×C₇₁	#135	1	1	yes	characteristic
74	C₄×C₇₁	#135	1	1	yes	characteristic
75	C₄×C₇₁	#137	1	1	yes	characteristic
76	C₄×C₇₁	#137	1	1	yes	characteristic
77	Dic₇₁	—	2	1	no	—
78	Dic₇₁	—	2	1	no	—
79	Dic₇₁	—	2	1	no	—
80	Dic₇₁	—	4	2	no	—
81	C₂×C₇₁	#413	1	1	yes	characteristic
82	C₂×C₇₁	#416	1	1	yes	characteristic
83	C₂×C₇₁	#419	1	1	yes	characteristic
84	D₇₁	—	4	1	no	—
85	#1554	—	71	1	no	maximal, Sylow
86	C₇₁	#1554	1	1	yes	characteristic, complemented, Sylow
87	#533	—	71	1	no	—
88	#432	—	71	1	no	—
89	#431	—	71	1	no	—
90	#333	—	71	1	no	—
91	#333	—	71	1	no	—
92	#332	—	71	1	no	—
93	#332	D₇₁	1	1	yes	characteristic, complemented
94	C₄⋊Q₈	—	71	1	no	—
95	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
96	#125	—	142	1	no	—
97	C₂²⋊Q₈	—	71	2	no	—
98	C₂²⋊Q₈	—	142	1	no	—
99	C₄×Q₈	—	71	2	no	—
100	C₄×Q₈	—	142	1	no	—
101	#117	—	142	1	no	—
102	C₂×C₄²	—	71	1	no	—
103	C₄⋊C₈	—	71	1	no	—
104	C₄⋊C₈	—	71	1	no	—
105	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
106	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
107	C₂²⋊C₈	—	142	1	no	—
108	C₂²⋊C₈	—	142	1	no	—
109	#97	—	142	1	no	—
110	C₄×C₈	—	142	1	no	—
111	C₂×Q₈	—	2	1	no	—
112	C₂×Q₈	—	71	2	no	—
113	C₂²×C₄	—	71	1	no	—
114	C₂²×C₄	—	71	1	no	—
115	C₂²×C₄	—	71	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	142	1	no	—
119	C₂×C₈	—	142	1	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	142	2	no	—
122	C₄⋊C₄	—	142	2	no	—
123	C₄⋊C₄	—	142	1	no	—
124	C₄⋊C₄	—	142	1	no	—
125	C₄⋊C₄	C₄×D₇₁	1	2	yes	—
126	C₂²⋊C₄	—	142	2	no	—
127	C₄²	—	71	1	no	—
128	C₄²	—	71	1	no	—
129	C₄²	—	71	1	no	—
130	C₄²	—	142	2	no	—
131	C₄²	C₂²×D₇₁	1	1	yes	characteristic
132	C₂³	—	71	1	no	—
133	Q₈	—	4	1	no	—
134	Q₈	—	4	1	no	—
135	Q₈	—	284	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	284	1	no	—
138	C₂×C₄	—	71	1	no	—
139	C₂×C₄	—	71	1	no	—
140	C₂×C₄	—	71	1	no	—
141	C₂×C₄	—	142	2	no	—
142	C₂×C₄	—	142	1	no	—
143	C₂×C₄	—	142	1	no	—
144	C₂×C₄	—	142	1	no	—
145	C₂×C₄	—	142	1	no	—
146	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
147	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
148	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	4	1	no	—
151	C₈	—	284	1	no	—
152	C₈	—	284	1	no	—
153	C₂²	—	142	1	no	—
154	C₂²	—	142	1	no	—
155	C₂²	—	142	1	no	—
156	C₂²	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	4	2	no	—
159	C₄	—	284	2	no	—
160	C₄	—	142	1	no	—
161	C₄	—	142	1	no	—
162	C₄	—	142	1	no	—
163	C₄	#14556147	1	1	yes	characteristic
164	C₄	#14556149	1	1	yes	characteristic
165	C₄	#14556157	1	1	yes	characteristic
166	C₄	#14556160	1	1	yes	characteristic
167	C₂	—	284	1	no	—
168	C₂	#17985982	1	1	yes	characteristic
169	C₂	#17985992	1	1	yes	characteristic
170	C₂	#17986006	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986554

C₂

yes

maximal, characteristic

#17986055

C₂

yes

maximal, characteristic

#17986051

C₂

yes

maximal, characteristic

#17985747

C₂

yes

maximal, characteristic, complemented

#17985747

C₂

yes

maximal, characteristic, complemented

#17985745

C₂

yes

maximal, characteristic, complemented

#332×C₇₁

C₂

yes

maximal, characteristic, complemented

#14556128

C₂²

yes

characteristic

#14556119

—

#14556108

C₄

yes

—

#14556104

—

Q₈×Dic₇₁

—

#14556093

C₄

yes

—

#14556086

—

C₄²×D₇₁

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

(C₂×D₇₁)⋊C₈

—

(C₂×D₇₁)⋊C₈

—

#14556038

—

C₈×Dic₇₁

—

C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

C₂×C₄

yes

—

C₂×C₄×D₇₁

C₂×C₄

yes

characteristic

C₂×C₄×D₇₁

D₄

yes

characteristic

C₂×C₄×D₇₁

D₄

yes

characteristic

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

(C₂×D₇₁)⋊C₄

—

C₄×Dic₇₁

—

C₄×Dic₇₁

C₂×C₄

yes

characteristic

C₄×Dic₇₁

D₄

yes

characteristic

C₄×Dic₇₁

D₄

yes

characteristic

C₂×Q₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂²×D₇₁

C₂²⋊C₄

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₁

C₂²⋊C₄

yes

characteristic

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

—

#1554

—

maximal, Sylow

C₇₁

#1554

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₇₁

yes

characteristic, complemented

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₁

yes

characteristic

#125

—

142

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

142

—

C₄×Q₈

—

100

C₄×Q₈

—

142

—

101

#117

—

142

—

102

C₂×C₄²

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄⋊C₈

C₂×D₇₁

yes

characteristic

106

C₄⋊C₈

C₂×D₇₁

yes

characteristic

107

C₂²⋊C₈

—

142

—

108

C₂²⋊C₈

—

142

—

109

#97

—

142

—

110

C₄×C₈

—

142

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

142

—

119

C₂×C₈

—

142

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

142

—

122

C₄⋊C₄

—

142

—

123

C₄⋊C₄

—

142

—

124

C₄⋊C₄

—

142

—

125

C₄⋊C₄

C₄×D₇₁

yes

—

126

C₂²⋊C₄

—

142

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

142

—

131

C₄²

C₂²×D₇₁

yes

characteristic

132

C₂³

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

284

—

136

C₂×C₄

—

137

C₂×C₄

—

284

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

—

142

C₂×C₄

—

142

—

143

C₂×C₄

—

142

—

144

C₂×C₄

—

142

—

145

C₂×C₄

—

142

—

146

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

147

C₂×C₄

D₄×D₇₁

yes

characteristic

148

C₂×C₄

D₄×D₇₁

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

284

—

152

C₈

—

284

—

153

C₂²

—

142

—

154

C₂²

—

142

—

155

C₂²

—

142

—

156

C₂²

C₂²⋊C₄×D₇₁

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

284

—

160

C₄

—

142

—

161

C₄

—

142

—

162

C₄

—

142

—

163

C₄

#14556147

yes

characteristic

164

C₄

#14556149

yes

characteristic

165

C₄

#14556157

yes

characteristic

166

C₄

#14556160

yes

characteristic

167

C₂

—

284

—

168

C₂

#17985982

yes

characteristic

169

C₂

#17985992

yes

characteristic

170

C₂

#17986006

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₄²	C₂²×D₇₁
Fitting	#332×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₂²⋊C₄
hypercenter	#332	D₇₁
hypocenter	C₇₁	#1554
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₄²

C₂²×D₇₁

Fitting

#332×C₇₁

C₂

socle

C₂²×C₇₁

C₂×C₂²⋊C₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
4. #17985747	C₂	semidirect
5. #17985747	C₂	semidirect
6. #17985745	C₂	semidirect
7. #332×C₇₁	C₂	semidirect
86. C₇₁	#1554	semidirect
93. #332	D₇₁	semidirect

H ≅ G/N

type

4. #17985747

C₂

semidirect

5. #17985747

C₂

semidirect

6. #17985745

C₂

semidirect

7. #332×C₇₁

C₂

semidirect

86. C₇₁

#1554

semidirect

93. #332

D₇₁

semidirect

Subgroups — #22587047

Conjugacy classes of subgroups

Complements