Group #22598991 — subgroups

Subgroups — #22598991

Order 9088.

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

normal subgroups only

This group has 386 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	#17986605	C₂	1	2	yes	maximal, complemented
2	#17986376	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986365	C₂	1	2	yes	maximal, complemented
4	#17986253	C₂	1	2	yes	maximal, complemented
5	#17986236	C₂	1	2	yes	maximal, complemented
6	#17986110	C₂	1	2	yes	maximal, complemented
7	#17986081	C₂	1	2	yes	maximal, complemented
8	#17985953	C₂	1	1	yes	maximal, characteristic, complemented
9	#499×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₅₆₈	C₂²	1	2	yes	—
11	C₂×D₅₆₈	C₂²	1	2	yes	—
12	C₂×#11765848	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₇₁	C₂²	1	1	yes	characteristic
14	#14556125	C₂²	1	2	yes	—
15	#14556113	—	2	4	no	—
16	#14556103	—	2	4	no	—
17	#14556103	C₂²	1	4	yes	—
18	#14556090	C₂²	1	4	yes	—
19	#14556085	C₂²	1	1	yes	characteristic
20	#14556048	C₂²	1	4	yes	—
21	#14556048	C₂²	1	4	yes	—
22	#14556039	C₂²	1	1	yes	characteristic
23	#14556038	C₂²	1	1	yes	characteristic
24	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
25	#125×C₇₁	C₂²	1	2	yes	—
26	#107×C₇₁	C₂²	1	2	yes	—
27	#106×C₇₁	C₂²	1	2	yes	—
28	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
29	C₂×D₂₈₄	—	2	4	no	—
30	C₂×D₂₈₄	—	2	4	no	—
31	C₂×D₂₈₄	C₂³	1	4	yes	—
32	C₂×C₄×D₇₁	—	2	4	no	—
33	C₂×C₄×D₇₁	C₂³	1	1	yes	characteristic
34	D₅₆₈	—	2	4	no	—
35	D₅₆₈	—	2	4	no	—
36	#11765848	—	2	2	no	—
37	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
38	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
39	C₈×D₇₁	—	2	2	no	—
40	Dic₇₁⋊C₄	D₄	1	2	yes	—
41	(C₂×D₇₁)⋊C₄	—	2	4	no	—
42	(C₂×D₇₁)⋊C₄	—	2	4	no	—
43	(C₂×D₇₁)⋊C₄	D₄	1	2	yes	—
44	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
46	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₇₁	—	2	4	no	—
48	C₄⋊C₄×C₇₁	C₂³	1	4	yes	—
49	C₄²×C₇₁	C₂³	1	1	yes	characteristic
50	C₂²×D₇₁	—	2	4	no	—
51	C₂²×D₇₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₁	—	2	4	no	—
53	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₇₁	C₂×D₄	1	2	yes	—
56	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
57	D₂₈₄	—	2	4	no	—
58	D₂₈₄	—	2	4	no	—
59	D₂₈₄	—	8	4	no	—
60	D₂₈₄	—	8	4	no	—
61	C₄×D₇₁	—	2	1	no	—
62	C₄×D₇₁	—	2	1	no	—
63	C₄×D₇₁	—	8	4	no	—
64	C₈×C₇₁	#41	1	2	yes	—
65	C₈×C₇₁	#41	1	2	yes	—
66	C₇₁⋊C₈	—	2	1	no	—
67	C₇₁⋊C₈	—	2	1	no	—
68	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
69	C₂×D₇₁	—	2	1	no	—
70	C₂×D₇₁	—	2	2	no	—
71	C₂×D₇₁	—	4	4	no	—
72	C₂×D₇₁	—	4	4	no	—
73	C₂×D₇₁	—	4	4	no	—
74	C₄×C₇₁	—	2	2	no	—
75	C₄×C₇₁	—	4	4	no	—
76	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
77	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
78	Dic₇₁	—	2	1	no	—
79	Dic₇₁	—	2	2	no	—
80	C₂×C₇₁	#532	1	1	yes	characteristic
81	C₂×C₇₁	#576	1	2	yes	—
82	D₇₁	—	4	1	no	—
83	D₇₁	—	8	4	no	—
84	#3176	—	71	1	no	maximal, Sylow
85	C₇₁	#3176	1	1	yes	characteristic, complemented, Sylow
86	#540	—	71	2	no	—
87	#499	D₇₁	1	1	yes	characteristic, complemented
88	#493	—	71	1	no	—
89	#490	—	71	2	no	—
90	#469	—	71	2	no	—
91	#466	—	71	2	no	—
92	#442	—	71	2	no	—
93	C₄×D₈	—	71	2	no	—
94	#405	—	71	1	no	—
95	C₂×D₈	—	71	2	no	—
96	C₂×D₈	—	71	2	no	—
97	C₂×M₁₆	—	71	1	no	—
98	C₂²×C₈	—	71	1	no	—
99	#127	—	71	2	no	—
100	#125	C₂×D₇₁	1	2	yes	—
101	#123	—	142	4	no	—
102	#121	—	71	4	no	—
103	#121	—	142	4	no	—
104	C₄×D₄	—	71	4	no	—
105	#117	—	71	1	no	—
106	#107	C₂×D₇₁	1	2	yes	—
107	#106	C₂×D₇₁	1	2	yes	—
108	#102	—	71	4	no	—
109	#102	—	71	4	no	—
110	#97	—	71	1	no	—
111	#97	—	71	1	no	—
112	C₄×C₈	—	71	1	no	—
113	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
114	C₂×D₄	—	71	4	no	—
115	C₂×D₄	—	142	4	no	—
116	C₂×D₄	—	142	4	no	—
117	C₂²×C₄	—	71	1	no	—
118	C₂²×C₄	—	142	4	no	—
119	D₈	—	142	4	no	—
120	D₈	—	142	4	no	—
121	M₁₆	—	142	2	no	—
122	C₂×C₈	—	71	1	no	—
123	C₂×C₈	—	71	1	no	—
124	C₂×C₈	—	142	2	no	—
125	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
126	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
127	C₄⋊C₄	—	2	4	no	—
128	C₄⋊C₄	—	71	2	no	—
129	C₄⋊C₄	C₂²×D₇₁	1	4	yes	—
130	C₂²⋊C₄	—	71	2	no	—
131	C₂²⋊C₄	—	142	4	no	—
132	C₂²⋊C₄	—	142	4	no	—
133	C₄²	—	71	1	no	—
134	C₄²	C₂²×D₇₁	1	1	yes	characteristic
135	C₂³	—	71	1	no	—
136	C₂³	—	142	4	no	—
137	D₄	—	568	4	no	—
138	D₄	—	568	4	no	—
139	D₄	—	142	4	no	—
140	D₄	—	142	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	568	4	no	—
143	C₂×C₄	—	71	2	no	—
144	C₂×C₄	—	71	1	no	—
145	C₂×C₄	—	142	1	no	—
146	C₂×C₄	—	142	1	no	—
147	C₂×C₄	D₄×D₇₁	1	2	yes	—
148	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
149	C₈	—	142	1	no	—
150	C₈	—	142	1	no	—
151	C₈	#11765866	1	2	yes	—
152	C₈	#11765866	1	2	yes	—
153	C₂²	—	284	4	no	—
154	C₂²	—	284	4	no	—
155	C₂²	—	284	4	no	—
156	C₂²	—	142	2	no	—
157	C₂²	—	142	1	no	—
158	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	4	4	no	—
161	C₄	—	142	2	no	—
162	C₄	—	142	1	no	—
163	C₄	C₂×#11765866	1	1	yes	characteristic
164	C₄	C₂×#11765866	1	1	yes	characteristic
165	C₂	—	284	1	no	—
166	C₂	—	568	4	no	—
167	C₂	#17986550	1	1	yes	characteristic
168	C₂	#17986801	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17986605

C₂

yes

maximal, complemented

#17986376

C₂

yes

maximal, characteristic, complemented

#17986365

C₂

yes

maximal, complemented

#17986253

C₂

yes

maximal, complemented

#17986236

C₂

yes

maximal, complemented

#17986110

C₂

yes

maximal, complemented

#17986081

C₂

yes

maximal, complemented

#17985953

C₂

yes

maximal, characteristic, complemented

#499×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×D₅₆₈

C₂²

yes

—

C₂×D₅₆₈

C₂²

yes

—

C₂×#11765848

C₂²

yes

characteristic

C₂×C₈×D₇₁

C₂²

yes

characteristic

#14556125

C₂²

yes

—

#14556113

—

#14556103

—

#14556103

C₂²

yes

—

#14556090

C₂²

yes

—

#14556085

C₂²

yes

characteristic

#14556048

C₂²

yes

—

#14556048

C₂²

yes

—

#14556039

C₂²

yes

characteristic

#14556038

C₂²

yes

characteristic

C₈×Dic₇₁

C₂²

yes

characteristic

#125×C₇₁

C₂²

yes

—

#107×C₇₁

C₂²

yes

—

#106×C₇₁

C₂²

yes

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×D₂₈₄

—

C₂×D₂₈₄

—

C₂×D₂₈₄

C₂³

yes

—

C₂×C₄×D₇₁

—

C₂×C₄×D₇₁

C₂³

yes

characteristic

D₅₆₈

—

D₅₆₈

—

#11765848

—

C₂×C₇₁⋊C₈

C₂³

yes

characteristic

C₂×C₇₁⋊C₈

C₂³

yes

characteristic

C₈×D₇₁

—

Dic₇₁⋊C₄

D₄

yes

—

(C₂×D₇₁)⋊C₄

—

(C₂×D₇₁)⋊C₄

—

(C₂×D₇₁)⋊C₄

D₄

yes

—

C₄×Dic₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

C₂³

yes

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂²×D₇₁

—

C₂²×D₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂⁴

yes

characteristic

C₂×Dic₇₁

C₂×D₄

yes

—

C₂×Dic₇₁

C₂×D₄

yes

characteristic

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

—

C₈×C₇₁

#41

yes

—

C₈×C₇₁

#41

yes

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂²×D₄

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×#41

yes

characteristic

C₄×C₇₁

C₂×#41

yes

characteristic

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

#532

yes

characteristic

C₂×C₇₁

#576

yes

—

D₇₁

—

D₇₁

—

#3176

—

maximal, Sylow

C₇₁

#3176

yes

characteristic, complemented, Sylow

#540

—

#499

D₇₁

yes

characteristic, complemented

—

—

—

—

—

C₄×D₈

—

#405

—

C₂×D₈

—

C₂×D₈

—

C₂×M₁₆

—

C₂²×C₈

—

#127

—

100

#125

C₂×D₇₁

yes

—

101

#123

—

142

—

102

#121

—

103

#121

—

142

—

104

C₄×D₄

—

105

#117

—

106

#107

C₂×D₇₁

yes

—

107

#106

C₂×D₇₁

yes

—

108

#102

—

109

#102

—

110

#97

—

111

#97

—

112

C₄×C₈

—

113

C₄×C₈

C₂×D₇₁

yes

characteristic

114

C₂×D₄

—

115

C₂×D₄

—

142

—

116

C₂×D₄

—

142

—

117

C₂²×C₄

—

118

C₂²×C₄

—

142

—

119

D₈

—

142

—

120

D₈

—

142

—

121

M₁₆

—

142

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

142

—

125

C₂×C₈

C₂²×D₇₁

yes

characteristic

126

C₂×C₈

C₂²×D₇₁

yes

characteristic

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

C₂²×D₇₁

yes

—

130

C₂²⋊C₄

—

131

C₂²⋊C₄

—

142

—

132

C₂²⋊C₄

—

142

—

133

C₄²

—

134

C₄²

C₂²×D₇₁

yes

characteristic

135

C₂³

—

136

C₂³

—

142

—

137

D₄

—

568

—

138

D₄

—

568

—

139

D₄

—

142

—

140

D₄

—

142

—

141

C₂×C₄

—

142

C₂×C₄

—

568

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

142

—

146

C₂×C₄

—

142

—

147

C₂×C₄

D₄×D₇₁

yes

—

148

C₂×C₄

C₂³×D₇₁

yes

characteristic

149

C₈

—

142

—

150

C₈

—

142

—

151

C₈

#11765866

yes

—

152

C₈

#11765866

yes

—

153

C₂²

—

284

—

154

C₂²

—

284

—

155

C₂²

—

284

—

156

C₂²

—

142

—

157

C₂²

—

142

—

158

C₂²

C₂×D₄×D₇₁

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

142

—

162

C₄

—

142

—

163

C₄

C₂×#11765866

yes

characteristic

164

C₄

C₂×#11765866

yes

characteristic

165

C₂

—

284

—

166

C₂

—

568

—

167

C₂

#17986550

yes

characteristic

168

C₂

#17986801

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#499×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#499	D₇₁
hypocenter	C₇₁	#3176
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

subgroup

G/H

center

C₂²

C₂×D₄×D₇₁

commutator

C₂×C₄×C₇₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₁

Fitting

#499×C₇₁

C₂

socle

C₂²×C₇₁

C₂²×D₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #499

N	H ≅ G/N	type
1. #17986605	C₂	semidirect
2. #17986376	C₂	semidirect
3. #17986365	C₂	semidirect
4. #17986253	C₂	semidirect
5. #17986236	C₂	semidirect
6. #17986110	C₂	semidirect
7. #17986081	C₂	semidirect
8. #17985953	C₂	semidirect
9. #499×C₇₁	C₂	semidirect
85. C₇₁	#3176	semidirect
87. #499	D₇₁	semidirect

H ≅ G/N

type

1. #17986605

C₂

semidirect

2. #17986376

C₂

semidirect

3. #17986365

C₂

semidirect

4. #17986253

C₂

semidirect

5. #17986236

C₂

semidirect

6. #17986110

C₂

semidirect

7. #17986081

C₂

semidirect

8. #17985953

C₂

semidirect

9. #499×C₇₁

C₂

semidirect

85. C₇₁

#3176

semidirect

87. #499

D₇₁

semidirect

Subgroups — #22598991

Conjugacy classes of subgroups

Complements