Group #22587510 — subgroups

Subgroups — #22587510

Order 9088.

194 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 53 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	#17986719	C₂	1	1	yes	maximal, characteristic
2	#17986349	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986291	C₂	1	1	yes	maximal, characteristic, complemented
4	#17985746	C₂	1	1	yes	maximal, characteristic, complemented
5	#17985729	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
7	#476×C₇₁	C₂	1	1	yes	maximal, characteristic
8	#14556128	C₂²	1	1	yes	characteristic
9	#14556121	—	2	2	no	—
10	#14556115	—	2	1	no	—
11	#14556107	—	2	1	no	—
12	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
13	#14556093	—	2	1	no	—
14	#14556087	—	2	1	no	—
15	#14556061	—	2	1	no	—
16	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
17	#14556052	—	2	2	no	—
18	#14556051	—	2	1	no	—
19	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
21	C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
22	#106×C₇₁	—	2	1	no	—
23	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
24	#102×C₇₁	—	2	1	no	—
25	C₂×Dic₁₄₂	—	2	1	no	—
26	C₂×Dic₁₄₂	D₄	1	1	yes	characteristic
27	C₂×C₇₁⋊C₈	—	2	1	no	—
28	C₂×C₇₁⋊C₈	—	2	1	no	—
29	C₄⋊Dic₇₁	—	2	1	no	—
30	C₄⋊Dic₇₁	D₄	1	2	yes	—
31	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
32	Dic₇₁⋊C₄	—	2	2	no	—
33	Dic₇₁⋊C₄	—	2	1	no	—
34	C₂²⋊Dic₇₁	—	2	2	no	—
35	C₂²⋊Dic₇₁	—	2	2	no	—
36	C₄×Dic₇₁	—	2	2	no	—
37	C₄×Dic₇₁	—	2	1	no	—
38	C₂×Q₈×C₇₁	—	2	1	no	—
39	C₂×D₄×C₇₁	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₁	—	2	1	no	—
41	C₂×C₈×C₇₁	—	2	1	no	—
42	C₄⋊C₄×C₇₁	—	2	1	no	—
43	C₄⋊C₄×C₇₁	—	2	1	no	—
44	C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
45	C₂²⋊C₄×C₇₁	—	2	1	no	—
46	C₄²×C₇₁	C₂³	1	1	yes	characteristic
47	C₂³×C₇₁	—	2	1	no	—
48	Q₈×C₇₁	—	4	2	no	—
49	D₄×C₇₁	—	2	1	no	—
50	D₄×C₇₁	—	2	1	no	—
51	C₂×C₄×C₇₁	—	2	1	no	—
52	C₂×C₄×C₇₁	—	4	1	no	—
53	C₂×C₄×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	C₂×D₄	1	1	yes	characteristic
58	C₂×Dic₇₁	—	2	1	no	—
59	C₂×Dic₇₁	—	2	1	no	—
60	C₂×Dic₇₁	—	2	2	no	—
61	Dic₁₄₂	—	2	1	no	—
62	Dic₁₄₂	—	2	1	no	—
63	Dic₁₄₂	—	4	1	no	—
64	Dic₁₄₂	—	4	1	no	—
65	C₈×C₇₁	—	4	1	no	—
66	C₇₁⋊C₈	—	4	1	no	—
67	C₇₁⋊C₈	—	4	1	no	—
68	C₂²×C₇₁	—	4	1	no	—
69	C₂²×C₇₁	—	4	1	no	—
70	C₂²×C₇₁	—	4	1	no	—
71	C₂²×C₇₁	#120	1	1	yes	characteristic
72	C₄×C₇₁	—	2	1	no	—
73	C₄×C₇₁	—	4	1	no	—
74	C₄×C₇₁	—	8	1	no	—
75	C₄×C₇₁	#135	1	1	yes	characteristic
76	C₄×C₇₁	#135	1	1	yes	characteristic
77	C₄×C₇₁	#137	1	1	yes	characteristic
78	C₄×C₇₁	#137	1	1	yes	characteristic
79	Dic₇₁	—	4	1	no	—
80	Dic₇₁	—	4	2	no	—
81	Dic₇₁	—	4	1	no	—
82	C₂×C₇₁	—	8	1	no	—
83	C₂×C₇₁	#452	1	1	yes	characteristic
84	C₂×C₇₁	#452	1	1	yes	characteristic
85	C₂×C₇₁	#454	1	1	yes	characteristic
86	#1638	—	71	1	no	maximal, Sylow
87	C₇₁	#1638	1	1	yes	characteristic, complemented, Sylow
88	#556	—	71	1	no	—
89	#487	—	71	1	no	—
90	#477	—	71	1	no	—
91	#476	D₇₁	1	1	yes	characteristic
92	#332	—	71	1	no	—
93	Q₈⋊C₈	—	71	1	no	—
94	D₄⋊C₈	—	71	1	no	—
95	C₄⋊Q₈	—	71	1	no	—
96	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
97	#126	—	142	2	no	—
98	#124	—	142	1	no	—
99	C₂²⋊Q₈	—	142	1	no	—
100	C₄×Q₈	—	71	1	no	—
101	C₄×Q₈	—	142	1	no	—
102	C₄×D₄	C₂×D₇₁	1	1	yes	characteristic
103	#117	—	142	1	no	—
104	#107	—	142	1	no	—
105	#106	—	2	1	no	—
106	C₄⋊C₈	—	71	1	no	—
107	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
108	#103	—	142	1	no	—
109	#103	—	142	2	no	—
110	#102	—	2	1	no	—
111	C₄×C₈	—	71	1	no	—
112	C₂×Q₈	—	2	1	no	—
113	C₂×Q₈	—	71	1	no	—
114	C₂×Q₈	—	142	1	no	—
115	C₂×D₄	C₂²⋊D₇₁	1	1	yes	characteristic
116	C₂²×C₄	—	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	—	2	1	no	—
122	C₄⋊C₄	—	71	2	no	—
123	C₄⋊C₄	—	71	1	no	—
124	C₄⋊C₄	—	142	2	no	—
125	C₄⋊C₄	—	142	1	no	—
126	C₄⋊C₄	—	142	1	no	—
127	C₄⋊C₄	C₂²⋊D₇₁	1	1	yes	characteristic
128	C₂²⋊C₄	—	2	1	no	—
129	C₂²⋊C₄	—	142	2	no	—
130	C₂²⋊C₄	—	142	2	no	—
131	C₄²	—	142	2	no	—
132	C₄²	—	142	1	no	—
133	C₄²	C₂²×D₇₁	1	1	yes	characteristic
134	C₂³	—	2	1	no	—
135	Q₈	—	4	2	no	—
136	Q₈	—	284	1	no	—
137	Q₈	—	284	1	no	—
138	Q₈	—	142	1	no	—
139	Q₈	—	142	1	no	—
140	D₄	—	2	1	no	—
141	D₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	142	1	no	—
147	C₂×C₄	—	142	2	no	—
148	C₂×C₄	—	142	1	no	—
149	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
150	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
151	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
152	C₈	—	4	1	no	—
153	C₈	—	284	1	no	—
154	C₈	—	284	1	no	—
155	C₂²	—	4	1	no	—
156	C₂²	—	4	1	no	—
157	C₂²	—	4	1	no	—
158	C₂²	#14556096	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	4	1	no	—
161	C₄	—	8	1	no	—
162	C₄	—	284	1	no	—
163	C₄	—	284	2	no	—
164	C₄	—	284	1	no	—
165	C₄	#14556148	1	1	yes	characteristic
166	C₄	#14556149	1	1	yes	characteristic
167	C₄	#14556157	1	1	yes	characteristic
168	C₄	#14556159	1	1	yes	characteristic
169	C₂	—	8	1	no	—
170	C₂	#17986158	1	1	yes	characteristic
171	C₂	#17986159	1	1	yes	characteristic
172	C₂	#17986170	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986719

C₂

yes

maximal, characteristic

#17986349

C₂

yes

maximal, characteristic, complemented

#17986291

C₂

yes

maximal, characteristic, complemented

#17985746

C₂

yes

maximal, characteristic, complemented

#17985729

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₇₁⋊C₈)

C₂

yes

maximal, characteristic

#476×C₇₁

C₂

yes

maximal, characteristic

#14556128

C₂²

yes

—

—

—

C₄×Dic₁₄₂

C₂²

yes

—

—

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

#14556052

—

#14556051

—

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

C₄×D₄×C₇₁

C₂²

yes

characteristic

#106×C₇₁

—

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

#102×C₇₁

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

D₄

yes

characteristic

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

D₄

yes

—

C₄⋊Dic₇₁

D₄

yes

characteristic

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

C₂²⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₂×Q₈×C₇₁

—

C₂×D₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

D₄

yes

characteristic

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

—

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#120

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

—

—

—

D₇₁

yes

characteristic

#332

—

Q₈⋊C₈

—

D₄⋊C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₁

yes

characteristic

#126

—

142

—

#124

—

142

—

C₂²⋊Q₈

—

142

—

100

C₄×Q₈

—

101

C₄×Q₈

—

142

—

102

C₄×D₄

C₂×D₇₁

yes

characteristic

103

#117

—

142

—

104

#107

—

142

—

105

#106

—

106

C₄⋊C₈

—

107

C₄⋊C₈

C₂×D₇₁

yes

characteristic

108

#103

—

142

—

109

#103

—

142

—

110

#102

—

111

C₄×C₈

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

—

142

—

115

C₂×D₄

C₂²⋊D₇₁

yes

characteristic

116

C₂²×C₄

—

117

C₂×C₈

—

118

C₂×C₈

—

142

—

119

C₂×C₈

—

142

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

142

—

125

C₄⋊C₄

—

142

—

126

C₄⋊C₄

—

142

—

127

C₄⋊C₄

C₂²⋊D₇₁

yes

characteristic

128

C₂²⋊C₄

—

129

C₂²⋊C₄

—

142

—

130

C₂²⋊C₄

—

142

—

131

C₄²

—

142

—

132

C₄²

—

142

—

133

C₄²

C₂²×D₇₁

yes

characteristic

134

C₂³

—

135

Q₈

—

136

Q₈

—

284

—

137

Q₈

—

284

—

138

Q₈

—

142

—

139

Q₈

—

142

—

140

D₄

—

141

D₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

142

—

147

C₂×C₄

—

142

—

148

C₂×C₄

—

142

—

149

C₂×C₄

D₄×D₇₁

yes

characteristic

150

C₂×C₄

D₄×D₇₁

yes

characteristic

151

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

152

C₈

—

153

C₈

—

284

—

154

C₈

—

284

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

#14556096

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

284

—

163

C₄

—

284

—

164

C₄

—

284

—

165

C₄

#14556148

yes

characteristic

166

C₄

#14556149

yes

characteristic

167

C₄

#14556157

yes

characteristic

168

C₄

#14556159

yes

characteristic

169

C₂

—

170

C₂

#17986158

yes

characteristic

171

C₂

#17986159

yes

characteristic

172

C₂

#17986170

yes

characteristic

	subgroup	G/H
center	C₂²	#14556096
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#476×C₇₁	C₂
socle	C₂²×C₇₁	#120
hypercenter	#476	D₇₁
hypocenter	C₇₁	#1638
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #476

subgroup

G/H

center

C₂²

#14556096

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#476×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #476

N	H ≅ G/N	type
2. #17986349	C₂	semidirect
3. #17986291	C₂	semidirect
4. #17985746	C₂	semidirect
5. #17985729	C₂	semidirect
87. C₇₁	#1638	semidirect

H ≅ G/N

type

2. #17986349

C₂

semidirect

3. #17986291

C₂

semidirect

4. #17985746

C₂

semidirect

5. #17985729

C₂

semidirect

87. C₇₁

#1638

semidirect

Subgroups — #22587510

Conjugacy classes of subgroups

Complements