Group #24905777 — subgroups

Subgroups — #24905777

Order 10624.

264 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 109 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	#19127754	C₂	1	2	yes	maximal, complemented
2	#19127427	C₂	1	2	yes	maximal, complemented
3	#19127425	C₂	1	2	yes	maximal
4	C₂×#117×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
6	C₂²⋊(C₈₃⋊C₈)	C₂²	1	4	yes	—
7	C₄×C₈₃⋊C₈	—	2	2	no	—
8	C₂³×C₄×C₈₃	C₄	1	1	yes	characteristic
9	#117×C₈₃	—	2	2	no	—
10	#117×C₈₃	—	2	2	no	—
11	C₂×C₄⋊C₄×C₈₃	C₄	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊C₄×C₈₃	C₄	1	1	yes	characteristic
14	C₂×C₂²⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₈₃	C₄	1	1	yes	characteristic
16	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
17	C₂×C₈₃⋊C₈	—	2	4	no	—
18	C₂⁴×C₈₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₈₃	—	2	1	no	—
20	C₂²×C₄×C₈₃	—	2	1	no	—
21	C₂²×C₄×C₈₃	—	2	2	no	—
22	C₂²×C₄×C₈₃	C₈	1	4	yes	complemented
23	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₈₃	—	2	2	no	—
30	C₄⋊C₄×C₈₃	—	2	2	no	—
31	C₂²⋊C₄×C₈₃	—	2	2	no	—
32	C₂²⋊C₄×C₈₃	—	2	2	no	—
33	C₄²×C₈₃	—	2	1	no	—
34	C₄²×C₈₃	—	2	1	no	—
35	C₄²×C₈₃	D₄	1	2	yes	—
36	C₄²×C₈₃	D₄	1	2	yes	—
37	C₂³×C₈₃	—	2	1	no	—
38	C₂³×C₈₃	—	2	1	no	—
39	C₂³×C₈₃	—	4	1	no	—
40	C₂³×C₈₃	—	4	1	no	—
41	C₂³×C₈₃	C₂×C₈	1	2	yes	—
42	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₈₃	—	2	1	no	—
44	C₂×C₄×C₈₃	—	2	1	no	—
45	C₂×C₄×C₈₃	—	2	4	no	—
46	C₂×C₄×C₈₃	—	2	4	no	—
47	C₂×C₄×C₈₃	—	2	1	no	—
48	C₂×C₄×C₈₃	—	2	1	no	—
49	C₂×C₄×C₈₃	—	4	1	no	—
50	C₂×C₄×C₈₃	—	4	1	no	—
51	C₂×C₄×C₈₃	—	4	1	no	—
52	C₂×C₄×C₈₃	—	4	1	no	—
53	C₂×C₄×C₈₃	C₂²⋊C₄	1	2	yes	—
54	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₈₃	C₂×C₈	1	2	yes	—
57	C₂×C₄×C₈₃	C₂×C₈	1	2	yes	—
58	C₂×C₄×C₈₃	M₁₆	1	2	yes	—
59	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
60	C₈₃⋊C₈	—	4	4	no	—
61	C₂²×C₈₃	—	2	1	no	—
62	C₂²×C₈₃	—	2	1	no	—
63	C₂²×C₈₃	—	2	2	no	—
64	C₂²×C₈₃	—	4	2	no	—
65	C₂²×C₈₃	—	4	2	no	—
66	C₂²×C₈₃	—	4	1	no	—
67	C₂²×C₈₃	—	4	1	no	—
68	C₂²×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₈₃	C₂²×C₈	1	1	yes	characteristic
70	C₂²×C₈₃	C₂×M₁₆	1	1	yes	characteristic
71	C₄×C₈₃	—	2	2	no	—
72	C₄×C₈₃	—	2	2	no	—
73	C₄×C₈₃	—	2	2	no	—
74	C₄×C₈₃	—	4	1	no	—
75	C₄×C₈₃	—	4	1	no	—
76	C₄×C₈₃	C₂²⋊C₈	1	2	yes	—
77	C₄×C₈₃	C₂²⋊C₈	1	2	yes	—
78	C₂×C₈₃	—	2	1	no	—
79	C₂×C₈₃	—	2	1	no	—
80	C₂×C₈₃	—	4	2	no	—
81	C₂×C₈₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
82	C₂×C₈₃	#410	1	1	yes	characteristic
83	C₂×C₈₃	#413	1	1	yes	characteristic
84	#1458	—	83	1	no	maximal, Sylow
85	C₈₃	#1458	1	1	yes	characteristic, complemented, Sylow
86	C₂×#117	D₈₃	1	1	yes	characteristic
87	#431	—	83	2	no	—
88	#324	—	83	2	no	—
89	#323	—	83	2	no	—
90	C₂³×C₄	Dic₈₃	1	1	yes	characteristic
91	#117	—	2	2	no	—
92	#117	—	2	2	no	—
93	C₂×C₄⋊C₄	Dic₈₃	1	1	yes	characteristic
94	C₂×C₄⋊C₄	C₂×D₈₃	1	1	yes	characteristic
95	C₂×C₂²⋊C₄	Dic₈₃	1	1	yes	characteristic
96	C₂×C₂²⋊C₄	C₂×D₈₃	1	1	yes	characteristic
97	C₂×C₄²	Dic₈₃	1	1	yes	characteristic
98	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
99	C₄⋊C₈	—	166	2	no	—
100	C₂²⋊C₈	—	83	4	no	—
101	C₄×C₈	—	166	2	no	—
102	C₂⁴	C₂×Dic₈₃	1	1	yes	characteristic
103	C₂²×C₄	—	2	1	no	—
104	C₂²×C₄	—	2	1	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	C₈₃⋊C₈	1	4	yes	complemented
107	C₂²×C₄	C₂×Dic₈₃	1	1	yes	characteristic
108	C₂²×C₄	C₂×Dic₈₃	1	1	yes	characteristic
109	C₂²×C₄	C₂×Dic₈₃	1	1	yes	characteristic
110	C₂²×C₄	C₂×Dic₈₃	1	1	yes	characteristic
111	C₂²×C₄	C₂×Dic₈₃	1	1	yes	characteristic
112	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
113	C₂×C₈	—	166	4	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₂²⋊C₄	—	2	2	no	—
117	C₂²⋊C₄	—	2	2	no	—
118	C₄²	—	2	1	no	—
119	C₄²	—	2	1	no	—
120	C₄²	C₂²⋊D₈₃	1	2	yes	—
121	C₄²	C₂²⋊D₈₃	1	2	yes	—
122	C₂³	—	2	1	no	—
123	C₂³	—	2	1	no	—
124	C₂³	—	4	1	no	—
125	C₂³	—	4	1	no	—
126	C₂³	C₂×C₈₃⋊C₈	1	2	yes	—
127	C₂³	C₂²×Dic₈₃	1	1	yes	characteristic
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	C₂²⋊Dic₈₃	1	2	yes	—
139	C₂×C₄	C₂²⋊Dic₈₃	1	1	yes	characteristic
140	C₂×C₄	C₂²⋊Dic₈₃	1	1	yes	characteristic
141	C₂×C₄	C₂×C₈₃⋊C₈	1	2	yes	—
142	C₂×C₄	C₂×C₈₃⋊C₈	1	2	yes	—
143	C₂×C₄	#13050574	1	2	yes	—
144	C₂×C₄	C₂×C₂²⋊D₈₃	1	2	yes	—
145	C₈	—	332	4	no	—
146	C₂²	—	2	1	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	2	no	—
149	C₂²	—	4	2	no	—
150	C₂²	—	4	2	no	—
151	C₂²	—	4	1	no	—
152	C₂²	—	4	1	no	—
153	C₂²	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
154	C₂²	C₂²×C₈₃⋊C₈	1	1	yes	characteristic
155	C₂²	C₂×#13050574	1	1	yes	characteristic
156	C₄	—	2	2	no	—
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	4	1	no	—
160	C₄	—	4	1	no	—
161	C₄	C₂²⋊(C₈₃⋊C₈)	1	2	yes	—
162	C₄	C₂²⋊(C₈₃⋊C₈)	1	2	yes	—
163	C₂	—	2	1	no	—
164	C₂	—	2	1	no	—
165	C₂	—	4	2	no	—
166	C₂	C₂×C₂²⋊(C₈₃⋊C₈)	1	1	yes	characteristic
167	C₂	#19127675	1	1	yes	characteristic
168	C₂	#19127685	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127754

C₂

yes

maximal, complemented

#19127427

C₂

yes

maximal, complemented

#19127425

C₂

yes

maximal

C₂×#117×C₈₃

C₂

yes

maximal, characteristic

C₄⋊(C₈₃⋊C₈)

—

C₂²⋊(C₈₃⋊C₈)

C₂²

yes

—

C₄×C₈₃⋊C₈

—

C₂³×C₄×C₈₃

C₄

yes

characteristic

#117×C₈₃

—

#117×C₈₃

—

C₂×C₄⋊C₄×C₈₃

C₄

yes

characteristic

C₂×C₄⋊C₄×C₈₃

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₈₃

C₄

yes

characteristic

C₂×C₂²⋊C₄×C₈₃

C₂²

yes

characteristic

C₂×C₄²×C₈₃

C₄

yes

characteristic

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₂×C₈₃⋊C₈

—

C₂⁴×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₈

yes

complemented

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

D₄

yes

—

C₄²×C₈₃

D₄

yes

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

C₂×C₈

yes

—

C₂³×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²⋊C₄

yes

—

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×C₈

yes

—

C₂×C₄×C₈₃

C₂×C₈

yes

—

C₂×C₄×C₈₃

M₁₆

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₈₃

C₂²×C₈

yes

characteristic

C₂²×C₈₃

C₂×M₁₆

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂²⋊C₈

yes

—

C₄×C₈₃

C₂²⋊C₈

yes

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

C₂×C₂²⋊C₈

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₂×#117

D₈₃

yes

—

—

—

C₂³×C₄

Dic₈₃

yes

characteristic

#117

—

#117

—

C₂×C₄⋊C₄

Dic₈₃

yes

characteristic

C₂×C₄⋊C₄

C₂×D₈₃

yes

characteristic

C₂×C₂²⋊C₄

Dic₈₃

yes

characteristic

C₂×C₂²⋊C₄

C₂×D₈₃

yes

characteristic

C₂×C₄²

Dic₈₃

yes

characteristic

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

166

—

100

C₂²⋊C₈

—

101

C₄×C₈

—

166

—

102

C₂⁴

C₂×Dic₈₃

yes

characteristic

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

C₈₃⋊C₈

yes

complemented

107

C₂²×C₄

C₂×Dic₈₃

yes

characteristic

108

C₂²×C₄

C₂×Dic₈₃

yes

characteristic

109

C₂²×C₄

C₂×Dic₈₃

yes

characteristic

110

C₂²×C₄

C₂×Dic₈₃

yes

characteristic

111

C₂²×C₄

C₂×Dic₈₃

yes

characteristic

112

C₂²×C₄

C₂²×D₈₃

yes

characteristic

113

C₂×C₈

—

166

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₂²⋊D₈₃

yes

—

121

C₄²

C₂²⋊D₈₃

yes

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

C₂×C₈₃⋊C₈

yes

—

127

C₂³

C₂²×Dic₈₃

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

C₂²⋊Dic₈₃

yes

—

139

C₂×C₄

C₂²⋊Dic₈₃

yes

characteristic

140

C₂×C₄

C₂²⋊Dic₈₃

yes

characteristic

141

C₂×C₄

C₂×C₈₃⋊C₈

yes

—

142

C₂×C₄

C₂×C₈₃⋊C₈

yes

—

143

C₂×C₄

#13050574

yes

—

144

C₂×C₄

C₂×C₂²⋊D₈₃

yes

—

145

C₈

—

332

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

C₂×C₂²⋊Dic₈₃

yes

characteristic

154

C₂²

C₂²×C₈₃⋊C₈

yes

characteristic

155

C₂²

C₂×#13050574

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

C₂²⋊(C₈₃⋊C₈)

yes

—

162

C₄

C₂²⋊(C₈₃⋊C₈)

yes

—

163

C₂

—

164

C₂

—

165

C₂

—

166

C₂

C₂×C₂²⋊(C₈₃⋊C₈)

yes

characteristic

167

C₂

#19127675

yes

characteristic

168

C₂

#19127685

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₈₃
commutator	C₂²×C₈₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	C₂×#117×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₈₃	C₂×C₄
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	C₂×#117	D₈₃
hypocenter	C₈₃	#1458
LCS	G ▷ C₂²×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#117

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₈₃

commutator

C₂²×C₈₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

C₂×#117×C₈₃

C₂

socle

C₂²×C₈₃

C₂×C₂²⋊C₄

prime radical

C₂⁴×C₈₃

C₂×C₄

cosocle

C₂²×C₄×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂²×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#117

N	H ≅ G/N	type
1. #19127754	C₂	semidirect
2. #19127427	C₂	semidirect
22. C₂²×C₄×C₈₃	C₈	semidirect
85. C₈₃	#1458	semidirect
106. C₂²×C₄	C₈₃⋊C₈	semidirect

H ≅ G/N

type

1. #19127754

C₂

semidirect

2. #19127427

C₂

semidirect

22. C₂²×C₄×C₈₃

C₈

semidirect

85. C₈₃

#1458

semidirect

106. C₂²×C₄

C₈₃⋊C₈

semidirect

Subgroups — #24905777

Conjugacy classes of subgroups

Complements