Group #22597052 — subgroups

Subgroups — #22597052

Order 9088.

500 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 229 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×#14556135	C₂	1	2	yes	maximal, complemented
2	#17986077	C₂	1	4	yes	maximal, complemented
3	D₄×C₇₁⋊C₈	C₂	1	4	yes	maximal, complemented
4	#17986023	C₂	1	1	yes	maximal, characteristic, complemented
5	#17985966	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#532×C₇₁	C₂	1	1	yes	maximal, characteristic
8	#14556135	—	2	8	no	—
9	C₂×#11765849	C₂²	1	4	yes	—
10	C₂×#11765849	C₂²	1	2	yes	—
11	C₂²×C₇₁⋊C₈	C₂²	1	4	yes	—
12	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
14	C₄⋊(C₇₁⋊C₈)	C₂²	1	4	yes	—
15	C₂²⋊(C₇₁⋊C₈)	C₂²	1	8	yes	—
16	C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
17	C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
18	C₂×#41×C₇₁	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₇₁	C₄	1	1	yes	characteristic
20	#127×C₇₁	C₄	1	1	yes	characteristic
21	#124×C₇₁	C₄	1	2	yes	—
22	#121×C₇₁	C₄	1	4	yes	—
23	C₄×D₄×C₇₁	C₂²	1	4	yes	—
24	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
25	#11765849	—	2	4	no	—
26	#11765849	—	2	8	no	—
27	C₂×C₇₁⋊C₈	—	2	2	no	—
28	C₂×C₇₁⋊C₈	—	2	8	no	—
29	C₂×C₇₁⋊C₈	D₄	1	4	yes	complemented
30	C₂×C₇₁⋊C₈	C₂³	1	2	yes	—
31	C₂×C₇₁⋊C₈	C₂³	1	4	yes	—
32	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
33	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
34	#41×C₇₁	—	2	4	no	—
35	#41×C₇₁	—	2	4	no	—
36	C₂×Q₈×C₇₁	C₂×C₄	1	2	yes	—
37	C₂×D₄×C₇₁	C₂×C₄	1	4	yes	—
38	C₂×D₄×C₇₁	C₂×C₄	1	4	yes	—
39	C₂×D₄×C₇₁	C₂×C₄	1	2	yes	—
40	C₂²×C₄×C₇₁	C₂×C₄	1	2	yes	—
41	C₂²×C₄×C₇₁	C₂³	1	4	yes	—
42	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₇₁	C₂×C₄	1	4	yes	—
44	C₂²⋊C₄×C₇₁	C₂×C₄	1	8	yes	—
45	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
47	C₄²×C₇₁	C₂³	1	2	yes	—
48	C₂³×C₇₁	C₂²×C₄	1	4	yes	—
49	C₂³×C₇₁	C₂²×C₄	1	1	yes	characteristic
50	Q₈×C₇₁	—	2	4	no	—
51	D₄×C₇₁	—	2	8	no	—
52	D₄×C₇₁	—	2	4	no	—
53	D₄×C₇₁	—	2	8	no	—
54	C₂×C₄×C₇₁	—	2	4	no	—
55	C₂×C₄×C₇₁	—	2	4	no	—
56	C₂×C₄×C₇₁	—	2	4	no	—
57	C₂×C₄×C₇₁	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₇₁	C₂²×C₄	1	4	yes	—
59	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₇₁	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₇₁	#41	1	2	yes	—
63	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
64	C₇₁⋊C₈	—	2	4	no	—
65	C₇₁⋊C₈	—	2	2	no	—
66	C₇₁⋊C₈	C₂×D₄	1	2	yes	—
67	C₇₁⋊C₈	C₂×D₄	1	2	yes	—
68	C₂²×C₇₁	—	2	2	no	—
69	C₂²×C₇₁	—	2	4	no	—
70	C₂²×C₇₁	—	2	8	no	—
71	C₂²×C₇₁	C₄×D₄	1	2	yes	—
72	C₂²×C₇₁	C₂³×C₄	1	1	yes	characteristic
73	C₄×C₇₁	—	2	4	no	—
74	C₄×C₇₁	—	2	2	no	—
75	C₄×C₇₁	C₄×D₄	1	2	yes	—
76	C₄×C₇₁	#131	1	4	yes	complemented
77	C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
78	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
79	C₂×C₇₁	—	2	2	no	—
80	C₂×C₇₁	—	4	4	no	—
81	C₂×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
82	C₂×C₇₁	C₂×#131	1	2	yes	—
83	#2931	—	71	1	no	maximal, Sylow
84	C₇₁	#2931	1	1	yes	characteristic, complemented, Sylow
85	C₂×#131	—	71	2	no	—
86	#532	D₇₁	1	1	yes	characteristic
87	#436	—	71	4	no	—
88	C₈×D₄	—	71	4	no	—
89	#423	—	71	1	no	—
90	#408	—	71	2	no	—
91	C₂×C₄×C₈	—	71	1	no	—
92	C₂×#41	C₂×D₇₁	1	2	yes	—
93	#131	—	142	8	no	—
94	C₂×M₁₆	—	71	2	no	—
95	C₂×M₁₆	—	71	4	no	—
96	C₂²×C₈	—	71	1	no	—
97	C₂²×C₈	—	71	4	no	—
98	C₂²×C₈	—	71	1	no	—
99	C₄⋊Q₈	Dic₇₁	1	1	yes	characteristic
100	#127	Dic₇₁	1	1	yes	characteristic
101	#124	Dic₇₁	1	2	yes	—
102	#121	Dic₇₁	1	4	yes	—
103	C₄×D₄	C₂×D₇₁	1	4	yes	—
104	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
105	C₄⋊C₈	—	71	4	no	—
106	C₂²⋊C₈	—	71	8	no	—
107	C₄×C₈	—	71	2	no	—
108	C₄×C₈	—	71	2	no	—
109	#41	—	2	4	no	—
110	#41	—	2	4	no	—
111	C₂×Q₈	C₂×Dic₇₁	1	2	yes	—
112	C₂×D₄	C₂×Dic₇₁	1	4	yes	—
113	C₂×D₄	C₂×Dic₇₁	1	4	yes	—
114	C₂×D₄	C₂×Dic₇₁	1	2	yes	—
115	C₂²×C₄	C₂×Dic₇₁	1	2	yes	—
116	C₂²×C₄	C₂²×D₇₁	1	4	yes	—
117	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
118	M₁₆	—	142	8	no	—
119	M₁₆	—	142	4	no	—
120	C₂×C₈	—	71	4	no	—
121	C₂×C₈	—	71	1	no	—
122	C₂×C₈	—	71	1	no	—
123	C₂×C₈	—	71	2	no	—
124	C₂×C₈	—	71	4	no	—
125	C₂×C₈	—	142	8	no	—
126	C₂×C₈	—	142	2	no	—
127	C₄⋊C₄	C₂×Dic₇₁	1	4	yes	—
128	C₂²⋊C₄	C₂×Dic₇₁	1	8	yes	—
129	C₄²	C₂×Dic₇₁	1	1	yes	characteristic
130	C₄²	C₂×Dic₇₁	1	1	yes	characteristic
131	C₄²	C₂²×D₇₁	1	2	yes	—
132	C₂³	C₂²×Dic₇₁	1	4	yes	—
133	C₂³	C₂²×Dic₇₁	1	1	yes	characteristic
134	Q₈	—	2	4	no	—
135	D₄	—	2	8	no	—
136	D₄	—	2	4	no	—
137	D₄	—	2	8	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	C₂²×Dic₇₁	1	2	yes	—
142	C₂×C₄	C₂²×Dic₇₁	1	4	yes	—
143	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
144	C₂×C₄	C₂²×Dic₇₁	1	2	yes	—
145	C₂×C₄	D₄×D₇₁	1	2	yes	—
146	C₂×C₄	#11765864	1	2	yes	—
147	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
148	C₈	—	71	2	no	—
149	C₈	—	71	2	no	—
150	C₈	—	142	4	no	—
151	C₈	—	142	2	no	—
152	C₂²	—	2	2	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	2	8	no	—
155	C₂²	D₄×Dic₇₁	1	2	yes	—
156	C₂²	C₂³×Dic₇₁	1	1	yes	characteristic
157	C₄	—	2	4	no	—
158	C₄	—	2	2	no	—
159	C₄	D₄×Dic₇₁	1	2	yes	—
160	C₄	#14556135	1	4	yes	complemented
161	C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
162	C₄	C₂×#11765864	1	1	yes	characteristic
163	C₂	—	2	2	no	—
164	C₂	—	4	4	no	—
165	C₂	C₂×D₄×Dic₇₁	1	1	yes	characteristic
166	C₂	C₂×#14556135	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556135

C₂

yes

maximal, complemented

#17986077

C₂

yes

maximal, complemented

D₄×C₇₁⋊C₈

C₂

yes

maximal, complemented

#17986023

C₂

yes

maximal, characteristic, complemented

#17985966

C₂

yes

maximal, complemented

C₂×C₄×C₇₁⋊C₈

C₂

yes

maximal, characteristic, complemented

#532×C₇₁

C₂

yes

maximal, characteristic

#14556135

—

C₂×#11765849

C₂²

yes

—

C₂×#11765849

C₂²

yes

—

C₂²×C₇₁⋊C₈

C₂²

yes

—

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

C₂²⋊(C₇₁⋊C₈)

C₂²

yes

—

C₄×C₇₁⋊C₈

C₂²

yes

—

C₄×C₇₁⋊C₈

C₂²

yes

—

C₂×#41×C₇₁

C₂²

yes

—

C₄⋊Q₈×C₇₁

C₄

yes

characteristic

#127×C₇₁

C₄

yes

characteristic

#124×C₇₁

C₄

yes

—

#121×C₇₁

C₄

yes

—

C₄×D₄×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₄

yes

complemented

C₂×C₇₁⋊C₈

C₂³

yes

—

C₂×C₇₁⋊C₈

C₂³

yes

—

C₂×C₇₁⋊C₈

C₂³

yes

characteristic

C₂×C₇₁⋊C₈

C₂³

yes

characteristic

#41×C₇₁

—

#41×C₇₁

—

C₂×Q₈×C₇₁

C₂×C₄

yes

—

C₂×D₄×C₇₁

C₂×C₄

yes

—

C₂×D₄×C₇₁

C₂×C₄

yes

—

C₂×D₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂³

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

C₂×C₄

yes

—

C₂²⋊C₄×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

C₂³

yes

—

C₂³×C₇₁

C₂²×C₄

yes

—

C₂³×C₇₁

C₂²×C₄

yes

characteristic

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²×C₄

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

#41

yes

—

C₂×C₄×C₇₁

C₂⁴

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

C₂×D₄

yes

—

C₇₁⋊C₈

C₂×D₄

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₄×D₄

yes

—

C₂²×C₇₁

C₂³×C₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₄×D₄

yes

—

C₄×C₇₁

#131

yes

complemented

C₄×C₇₁

C₂²×D₄

yes

characteristic

C₄×C₇₁

C₂×#41

yes

characteristic

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

C₂×C₄×D₄

yes

characteristic

C₂×C₇₁

C₂×#131

yes

—

#2931

—

maximal, Sylow

C₇₁

#2931

yes

characteristic, complemented, Sylow

C₂×#131

—

#532

D₇₁

yes

characteristic

#436

—

C₈×D₄

—

#423

—

#408

—

C₂×C₄×C₈

—

C₂×#41

C₂×D₇₁

yes

—

#131

—

142

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

Dic₇₁

yes

characteristic

100

#127

Dic₇₁

yes

characteristic

101

#124

Dic₇₁

yes

—

102

#121

Dic₇₁

yes

—

103

C₄×D₄

C₂×D₇₁

yes

—

104

C₂×C₄²

C₂×D₇₁

yes

characteristic

105

C₄⋊C₈

—

106

C₂²⋊C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

#41

—

110

#41

—

111

C₂×Q₈

C₂×Dic₇₁

yes

—

112

C₂×D₄

C₂×Dic₇₁

yes

—

113

C₂×D₄

C₂×Dic₇₁

yes

—

114

C₂×D₄

C₂×Dic₇₁

yes

—

115

C₂²×C₄

C₂×Dic₇₁

yes

—

116

C₂²×C₄

C₂²×D₇₁

yes

—

117

C₂²×C₄

C₂²×D₇₁

yes

characteristic

118

M₁₆

—

142

—

119

M₁₆

—

142

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

142

—

126

C₂×C₈

—

142

—

127

C₄⋊C₄

C₂×Dic₇₁

yes

—

128

C₂²⋊C₄

C₂×Dic₇₁

yes

—

129

C₄²

C₂×Dic₇₁

yes

characteristic

130

C₄²

C₂×Dic₇₁

yes

characteristic

131

C₄²

C₂²×D₇₁

yes

—

132

C₂³

C₂²×Dic₇₁

yes

—

133

C₂³

C₂²×Dic₇₁

yes

characteristic

134

Q₈

—

135

D₄

—

136

D₄

—

137

D₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

C₂²×Dic₇₁

yes

—

142

C₂×C₄

C₂²×Dic₇₁

yes

—

143

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

144

C₂×C₄

C₂²×Dic₇₁

yes

—

145

C₂×C₄

D₄×D₇₁

yes

—

146

C₂×C₄

#11765864

yes

—

147

C₂×C₄

C₂³×D₇₁

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

142

—

151

C₈

—

142

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

D₄×Dic₇₁

yes

—

156

C₂²

C₂³×Dic₇₁

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

D₄×Dic₇₁

yes

—

160

C₄

#14556135

yes

complemented

161

C₄

C₂×D₄×D₇₁

yes

characteristic

162

C₄

C₂×#11765864

yes

characteristic

163

C₂

—

164

C₂

—

165

C₂

C₂×D₄×Dic₇₁

yes

characteristic

166

C₂

C₂×#14556135

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇₁
commutator, socle	C₂²×C₇₁	C₂³×C₄
Fitting	#532×C₇₁	C₂
hypercenter	#532	D₇₁
hypocenter	C₇₁	#2931
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ #532

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₇₁

commutator, socle

C₂²×C₇₁

C₂³×C₄

Fitting

#532×C₇₁

C₂

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂×C₄ ◁ #532

N	H ≅ G/N	type
1. C₂×#14556135	C₂	semidirect
2. #17986077	C₂	semidirect
3. D₄×C₇₁⋊C₈	C₂	semidirect
4. #17986023	C₂	semidirect
5. #17985966	C₂	semidirect
6. C₂×C₄×C₇₁⋊C₈	C₂	semidirect
29. C₂×C₇₁⋊C₈	D₄	semidirect
76. C₄×C₇₁	#131	semidirect
84. C₇₁	#2931	semidirect
160. C₄	#14556135	semidirect

H ≅ G/N

type

1. C₂×#14556135

C₂

semidirect

2. #17986077

C₂

semidirect

3. D₄×C₇₁⋊C₈

C₂

semidirect

4. #17986023

C₂

semidirect

5. #17985966

C₂

semidirect

6. C₂×C₄×C₇₁⋊C₈

C₂

semidirect

29. C₂×C₇₁⋊C₈

D₄

semidirect

76. C₄×C₇₁

#131

semidirect

84. C₇₁

#2931

semidirect

160. C₄

#14556135

semidirect

Subgroups — #22597052

Conjugacy classes of subgroups

Complements