Group #24907648 — subgroups

Subgroups — #24907648

Order 10624.

324 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 121 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	C₂²×#13050574	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₈₃⋊C₈)	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₈₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#19127456	C₂	1	2	yes	maximal, complemented
5	C₄×C₂²⋊C₄×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13050574	—	2	4	no	—
7	C₂×#13050574	—	2	2	no	—
8	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₈₃⋊C₈	C₂²	1	2	yes	—
11	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
12	C₂²⋊(C₈₃⋊C₈)	—	2	2	no	—
13	C₂³×C₄×C₈₃	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊C₄×C₈₃	C₄	1	2	yes	—
15	C₂×C₄²×C₈₃	C₂²	1	2	yes	—
16	#95×C₈₃	C₄	1	1	yes	characteristic
17	#95×C₈₃	C₄	1	1	yes	characteristic
18	#13050574	—	2	8	no	—
19	C₂×C₈₃⋊C₈	—	2	2	no	—
20	C₂×C₈₃⋊C₈	—	2	2	no	—
21	C₂×C₈₃⋊C₈	—	2	1	no	—
22	C₂×C₈₃⋊C₈	—	2	1	no	—
23	C₂×C₈₃⋊C₈	—	2	2	no	—
24	C₂×C₈₃⋊C₈	—	2	2	no	—
25	C₂×C₈₃⋊C₈	D₄	1	4	yes	—
26	C₂⁴×C₈₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₈₃	—	2	2	no	—
28	C₂²×C₄×C₈₃	—	2	2	no	—
29	C₂²×C₄×C₈₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₈₃	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₈₃	D₄	1	2	yes	—
33	C₂²×C₄×C₈₃	Q₈	1	2	yes	—
34	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
35	C₂²⋊C₄×C₈₃	—	2	4	no	—
36	C₄²×C₈₃	—	2	2	no	—
37	C₄²×C₈₃	—	2	2	no	—
38	C₂³×C₈₃	—	2	2	no	—
39	C₂³×C₈₃	—	2	2	no	—
40	C₂³×C₈₃	—	2	1	no	—
41	C₂³×C₈₃	—	2	1	no	—
42	C₂³×C₈₃	C₄⋊C₄	1	2	yes	—
43	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₈₃	—	2	2	no	—
45	C₂×C₄×C₈₃	—	2	2	no	—
46	C₂×C₄×C₈₃	—	2	2	no	—
47	C₂×C₄×C₈₃	—	2	2	no	—
48	C₂×C₄×C₈₃	—	2	4	no	—
49	C₂×C₄×C₈₃	—	2	2	no	—
50	C₂×C₄×C₈₃	—	2	2	no	—
51	C₂×C₄×C₈₃	—	2	2	no	—
52	C₂×C₄×C₈₃	—	2	2	no	—
53	C₂×C₄×C₈₃	C₄⋊C₄	1	2	yes	—
54	C₂×C₄×C₈₃	M₁₆	1	4	yes	complemented
55	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×C₈₃	#41	1	2	yes	—
59	C₈₃⋊C₈	—	2	4	no	—
60	C₈₃⋊C₈	—	4	1	no	—
61	C₈₃⋊C₈	—	4	1	no	—
62	C₂²×C₈₃	—	2	2	no	—
63	C₂²×C₈₃	—	2	2	no	—
64	C₂²×C₈₃	—	2	2	no	—
65	C₂²×C₈₃	—	4	2	no	—
66	C₂²×C₈₃	—	4	2	no	—
67	C₂²×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₈₃	C₄×D₄	1	2	yes	—
69	C₂²×C₈₃	C₂×M₁₆	1	2	yes	—
70	C₂²×C₈₃	#131	1	1	yes	characteristic
71	C₂²×C₈₃	#131	1	1	yes	characteristic
72	C₄×C₈₃	—	2	2	no	—
73	C₄×C₈₃	—	2	4	no	—
74	C₄×C₈₃	—	4	2	no	—
75	C₄×C₈₃	#120	1	1	yes	characteristic
76	C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
77	C₄×C₈₃	#123	1	1	yes	characteristic
78	C₂×C₈₃	—	4	2	no	—
79	C₂×C₈₃	#385	1	1	yes	characteristic
80	C₂×C₈₃	#423	1	1	yes	characteristic
81	C₂×C₈₃	#424	1	1	yes	characteristic
82	C₂×C₈₃	#435	1	2	yes	—
83	C₂×C₈₃	#436	1	2	yes	—
84	#1851	—	83	1	no	maximal, Sylow
85	C₈₃	#1851	1	1	yes	characteristic, complemented, Sylow
86	C₂²×M₁₆	—	83	1	no	—
87	C₂×C₄⋊C₈	—	83	2	no	—
88	C₂×C₂²⋊C₈	—	83	1	no	—
89	C₄×C₂²⋊C₄	D₈₃	1	1	yes	characteristic
90	#336	—	83	2	no	—
91	C₂³×C₄	C₂×D₈₃	1	1	yes	characteristic
92	C₂×M₁₆	—	166	4	no	—
93	C₂×M₁₆	—	166	2	no	—
94	C₂²×C₈	—	83	2	no	—
95	C₂²×C₈	—	83	1	no	—
96	C₂²×C₈	—	83	1	no	—
97	C₂×C₂²⋊C₄	Dic₈₃	1	2	yes	—
98	C₂×C₄²	C₂×D₈₃	1	2	yes	—
99	C₄⋊C₈	—	166	4	no	—
100	C₂²⋊C₈	—	166	2	no	—
101	#95	Dic₈₃	1	1	yes	characteristic
102	#95	Dic₈₃	1	1	yes	characteristic
103	C₂⁴	C₂×Dic₈₃	1	1	yes	characteristic
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	Dic₁₆₆	1	2	yes	—
107	C₂²×C₄	D₃₃₂	1	2	yes	—
108	C₂²×C₄	C₂×Dic₈₃	1	2	yes	—
109	C₂²×C₄	C₂×Dic₈₃	1	1	yes	characteristic
110	C₂²×C₄	C₂×Dic₈₃	1	2	yes	—
111	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
112	M₁₆	—	166	8	no	—
113	C₂×C₈	—	83	4	no	—
114	C₂×C₈	—	166	2	no	—
115	C₂×C₈	—	166	1	no	—
116	C₂×C₈	—	166	1	no	—
117	C₂×C₈	—	166	2	no	—
118	C₂×C₈	—	166	2	no	—
119	C₂×C₈	—	166	2	no	—
120	C₂²⋊C₄	—	2	4	no	—
121	C₄²	—	2	2	no	—
122	C₄²	—	2	2	no	—
123	C₂³	—	2	2	no	—
124	C₂³	—	2	1	no	—
125	C₂³	—	2	1	no	—
126	C₂³	—	2	2	no	—
127	C₂³	C₄⋊Dic₈₃	1	2	yes	—
128	C₂³	C₂²×Dic₈₃	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	C₄⋊Dic₈₃	1	2	yes	—
139	C₂×C₄	#13050574	1	4	yes	complemented
140	C₂×C₄	D₄×D₈₃	1	2	yes	—
141	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
142	C₂×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
143	C₂×C₄	#13050589	1	2	yes	—
144	C₈	—	332	1	no	—
145	C₈	—	332	1	no	—
146	C₈	—	166	4	no	—
147	C₂²	—	2	2	no	—
148	C₂²	—	2	2	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	4	2	no	—
151	C₂²	—	4	2	no	—
152	C₂²	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
153	C₂²	D₄×Dic₈₃	1	2	yes	—
154	C₂²	C₂×#13050574	1	2	yes	—
155	C₂²	#14581799	1	1	yes	characteristic
156	C₂²	#14581799	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	2	4	no	—
159	C₄	—	4	2	no	—
160	C₄	#14581762	1	1	yes	characteristic
161	C₄	#14581769	1	2	yes	—
162	C₄	#14581776	1	1	yes	characteristic
163	C₂	—	4	2	no	—
164	C₂	#19127576	1	1	yes	characteristic
165	C₂	#19127725	1	1	yes	characteristic
166	C₂	#19127728	1	1	yes	characteristic
167	C₂	#19127773	1	2	yes	—
168	C₂	#19127779	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#13050574

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₈₃⋊C₈)

C₂

yes

maximal, complemented

C₂×C₂²⋊(C₈₃⋊C₈)

C₂

yes

maximal, characteristic

#19127456

C₂

yes

maximal, complemented

C₄×C₂²⋊C₄×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

—

C₂×#13050574

—

C₂²×C₈₃⋊C₈

C₂²

yes

characteristic

C₂²×C₈₃⋊C₈

C₂²

yes

characteristic

C₂²×C₈₃⋊C₈

C₂²

yes

—

C₄⋊(C₈₃⋊C₈)

—

C₂²⋊(C₈₃⋊C₈)

—

C₂³×C₄×C₈₃

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₈₃

C₄

yes

—

C₂×C₄²×C₈₃

C₂²

yes

—

#95×C₈₃

C₄

yes

characteristic

#95×C₈₃

C₄

yes

characteristic

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

D₄

yes

—

C₂⁴×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

D₄

yes

—

C₂²×C₄×C₈₃

Q₈

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

C₄⋊C₄

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

C₄⋊C₄

yes

—

C₂×C₄×C₈₃

M₁₆

yes

complemented

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₃

#41

yes

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₈₃

C₄×D₄

yes

—

C₂²×C₈₃

C₂×M₁₆

yes

—

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#120

yes

characteristic

C₄×C₈₃

C₂²⋊Q₈

yes

—

C₄×C₈₃

#123

yes

characteristic

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₂×C₈₃

#436

yes

—

#1851

—

maximal, Sylow

C₈₃

#1851

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

C₄×C₂²⋊C₄

D₈₃

yes

characteristic

#336

—

C₂³×C₄

C₂×D₈₃

yes

characteristic

C₂×M₁₆

—

166

—

C₂×M₁₆

—

166

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₂²⋊C₄

Dic₈₃

yes

—

C₂×C₄²

C₂×D₈₃

yes

—

C₄⋊C₈

—

166

—

100

C₂²⋊C₈

—

166

—

101

#95

Dic₈₃

yes

characteristic

102

#95

Dic₈₃

yes

characteristic

103

C₂⁴

C₂×Dic₈₃

yes

characteristic

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

Dic₁₆₆

yes

—

107

C₂²×C₄

D₃₃₂

yes

—

108

C₂²×C₄

C₂×Dic₈₃

yes

—

109

C₂²×C₄

C₂×Dic₈₃

yes

characteristic

110

C₂²×C₄

C₂×Dic₈₃

yes

—

111

C₂²×C₄

C₂²×D₈₃

yes

characteristic

112

M₁₆

—

166

—

113

C₂×C₈

—

114

C₂×C₈

—

166

—

115

C₂×C₈

—

166

—

116

C₂×C₈

—

166

—

117

C₂×C₈

—

166

—

118

C₂×C₈

—

166

—

119

C₂×C₈

—

166

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

C₄⋊Dic₈₃

yes

—

128

C₂³

C₂²×Dic₈₃

yes

characteristic

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

C₄⋊Dic₈₃

yes

—

139

C₂×C₄

#13050574

yes

complemented

140

C₂×C₄

D₄×D₈₃

yes

—

141

C₂×C₄

C₂×D₃₃₂

yes

characteristic

142

C₂×C₄

C₂×Dic₁₆₆

yes

characteristic

143

C₂×C₄

#13050589

yes

—

144

C₈

—

332

—

145

C₈

—

332

—

146

C₈

—

166

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

C₂×C₄⋊Dic₈₃

yes

characteristic

153

C₂²

D₄×Dic₈₃

yes

—

154

C₂²

C₂×#13050574

yes

—

155

C₂²

#14581799

yes

characteristic

156

C₂²

#14581799

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

#14581762

yes

characteristic

161

C₄

#14581769

yes

—

162

C₄

#14581776

yes

characteristic

163

C₂

—

164

C₂

#19127576

yes

characteristic

165

C₂

#19127725

yes

characteristic

166

C₂

#19127728

yes

characteristic

167

C₂

#19127773

yes

—

168

C₂

#19127779

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₈₃
commutator, socle	C₂³×C₈₃	C₂²×C₄
Fitting	C₄×C₂²⋊C₄×C₈₃	C₂
prime radical	C₂⁴×C₈₃	C₂×C₄
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	C₄×C₂²⋊C₄	D₈₃
hypocenter	C₈₃	#1851
LCS	G ▷ C₂³×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₄

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₈₃

commutator, socle

C₂³×C₈₃

C₂²×C₄

Fitting

C₄×C₂²⋊C₄×C₈₃

C₂

prime radical

C₂⁴×C₈₃

C₂×C₄

cosocle

C₂²×C₄×C₈₃

C₂³

hypercenter

C₄×C₂²⋊C₄

D₈₃

C₈₃

G ▷ C₂³×C₈₃ ▷ C₈₃

C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₄

N	H ≅ G/N	type
2. C₂×C₄⋊(C₈₃⋊C₈)	C₂	semidirect
4. #19127456	C₂	semidirect
54. C₂×C₄×C₈₃	M₁₆	semidirect
85. C₈₃	#1851	semidirect
139. C₂×C₄	#13050574	semidirect

H ≅ G/N

type

2. C₂×C₄⋊(C₈₃⋊C₈)

C₂

semidirect

4. #19127456

C₂

semidirect

54. C₂×C₄×C₈₃

M₁₆

semidirect

85. C₈₃

#1851

semidirect

139. C₂×C₄

#13050574

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#13050574	2
C₂×C₄⋊(C₈₃⋊C₈)	2
C₂×C₂²⋊(C₈₃⋊C₈)	2
#19127456	2
C₄×C₂²⋊C₄×C₈₃	2
#1851	83

subgroup

index

C₂²×#13050574

C₂×C₄⋊(C₈₃⋊C₈)

C₂×C₂²⋊(C₈₃⋊C₈)

#19127456

C₄×C₂²⋊C₄×C₈₃

#1851

Subgroups — #24907648

Conjugacy classes of subgroups

Complements

Maximal subgroups