Group #24906703 — subgroups

Subgroups — #24906703

Order 10624.

194 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 55 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	#19128429	C₂	1	1	yes	maximal, characteristic
2	#19128104	C₂	1	1	yes	maximal, characteristic
3	#19127983	C₂	1	1	yes	maximal, characteristic
4	#19127448	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊(C₈₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#19127431	C₂	1	1	yes	maximal, characteristic
7	#462×C₈₃	C₂	1	1	yes	maximal, characteristic
8	#14581792	C₂²	1	1	yes	characteristic
9	#14581790	—	2	1	no	—
10	#14581784	—	2	1	no	—
11	#14581782	—	2	2	no	—
12	C₄×Dic₁₆₆	—	2	1	no	—
13	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
14	#14581757	—	2	1	no	—
15	#14581725	—	2	2	no	—
16	#14581723	—	2	1	no	—
17	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
18	#14581715	—	2	1	no	—
19	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×Q₁₆×C₈₃	—	2	1	no	—
21	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
24	#103×C₈₃	—	2	1	no	—
25	C₂×Dic₁₆₆	—	2	1	no	—
26	C₂×Dic₁₆₆	D₄	1	1	yes	characteristic
27	C₂×C₈₃⋊C₈	—	2	1	no	—
28	C₂×C₈₃⋊C₈	—	2	1	no	—
29	C₄⋊Dic₈₃	—	2	1	no	—
30	C₄⋊Dic₈₃	D₄	1	2	yes	—
31	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
32	Dic₈₃⋊C₄	—	2	2	no	—
33	Dic₈₃⋊C₄	—	2	2	no	—
34	Dic₈₃⋊C₄	—	2	2	no	—
35	Dic₈₃⋊C₄	—	2	1	no	—
36	C₄×Dic₈₃	—	2	2	no	—
37	C₄×Dic₈₃	—	2	1	no	—
38	C₂×Q₈×C₈₃	—	2	1	no	—
39	C₂×Q₈×C₈₃	D₄	1	1	yes	characteristic
40	Q₁₆×C₈₃	—	4	2	no	—
41	C₂×C₈×C₈₃	—	2	1	no	—
42	C₄⋊C₄×C₈₃	—	2	1	no	—
43	C₄⋊C₄×C₈₃	—	2	1	no	—
44	C₄⋊C₄×C₈₃	—	2	1	no	—
45	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
46	C₄²×C₈₃	—	2	1	no	—
47	C₄²×C₈₃	C₂³	1	1	yes	characteristic
48	Q₈×C₈₃	—	2	1	no	—
49	Q₈×C₈₃	—	2	1	no	—
50	Q₈×C₈₃	—	4	1	no	—
51	Q₈×C₈₃	—	4	1	no	—
52	C₂×C₄×C₈₃	—	2	1	no	—
53	C₂×C₄×C₈₃	—	2	1	no	—
54	C₂×C₄×C₈₃	—	4	1	no	—
55	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
58	C₂×Dic₈₃	—	2	1	no	—
59	C₂×Dic₈₃	—	2	1	no	—
60	C₂×Dic₈₃	—	2	2	no	—
61	Dic₁₆₆	—	2	1	no	—
62	Dic₁₆₆	—	4	1	no	—
63	Dic₁₆₆	—	4	1	no	—
64	Dic₁₆₆	Q₁₆	1	2	yes	complemented
65	C₈×C₈₃	—	4	1	no	—
66	C₈₃⋊C₈	—	2	2	no	—
67	C₈₃⋊C₈	—	4	1	no	—
68	C₂²×C₈₃	#120	1	1	yes	characteristic
69	C₄×C₈₃	—	2	1	no	—
70	C₄×C₈₃	—	4	1	no	—
71	C₄×C₈₃	—	4	1	no	—
72	C₄×C₈₃	—	8	1	no	—
73	C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
74	C₄×C₈₃	#135	1	1	yes	characteristic
75	C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
76	C₄×C₈₃	#137	1	1	yes	characteristic
77	Dic₈₃	—	2	2	no	—
78	Dic₈₃	—	4	2	no	—
79	Dic₈₃	—	4	1	no	—
80	C₂×C₈₃	#449	1	1	yes	characteristic
81	C₂×C₈₃	#451	1	1	yes	characteristic
82	C₂×C₈₃	#456	1	1	yes	characteristic
83	#1648	—	83	1	no	maximal, Sylow
84	C₈₃	#1648	1	1	yes	characteristic, complemented, Sylow
85	#557	—	83	1	no	—
86	#500	—	83	1	no	—
87	#475	—	83	1	no	—
88	#462	D₈₃	1	1	yes	characteristic
89	#332	—	83	1	no	—
90	Q₈⋊C₈	—	83	1	no	—
91	Q₈⋊C₈	—	83	1	no	—
92	C₂×Q₁₆	—	2	1	no	—
93	C₄⋊Q₈	—	83	1	no	—
94	C₄⋊Q₈	—	166	1	no	—
95	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
96	#125	—	166	1	no	—
97	#125	—	166	2	no	—
98	C₄×Q₈	—	83	1	no	—
99	C₄×Q₈	—	166	1	no	—
100	C₄×Q₈	—	166	1	no	—
101	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
102	#107	—	166	2	no	—
103	#106	—	166	1	no	—
104	C₄⋊C₈	—	83	1	no	—
105	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
106	#103	—	2	1	no	—
107	#103	—	166	1	no	—
108	C₄×C₈	—	83	1	no	—
109	C₂×Q₈	—	2	1	no	—
110	C₂×Q₈	—	83	1	no	—
111	C₂×Q₈	—	166	1	no	—
112	C₂×Q₈	C₂²⋊D₈₃	1	1	yes	characteristic
113	Q₁₆	—	4	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	166	1	no	—
116	C₂×C₈	—	166	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	83	1	no	—
121	C₄⋊C₄	—	83	2	no	—
122	C₄⋊C₄	—	166	2	no	—
123	C₄⋊C₄	—	166	2	no	—
124	C₄⋊C₄	—	166	2	no	—
125	C₄⋊C₄	—	166	1	no	—
126	C₄⋊C₄	—	166	1	no	—
127	C₄⋊C₄	C₂²⋊D₈₃	1	1	yes	characteristic
128	C₄²	—	2	1	no	—
129	C₄²	—	166	2	no	—
130	C₄²	—	166	1	no	—
131	C₄²	C₂²×D₈₃	1	1	yes	characteristic
132	Q₈	—	2	1	no	—
133	Q₈	—	2	1	no	—
134	Q₈	—	4	1	no	—
135	Q₈	—	4	1	no	—
136	Q₈	—	332	1	no	—
137	Q₈	—	332	1	no	—
138	Q₈	—	83	2	no	—
139	Q₈	—	166	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	166	1	no	—
144	C₂×C₄	—	166	2	no	—
145	C₂×C₄	—	166	1	no	—
146	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
147	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
148	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	332	1	no	—
151	C₈	—	166	2	no	—
152	C₂²	#14581760	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	4	1	no	—
156	C₄	—	8	1	no	—
157	C₄	—	332	2	no	—
158	C₄	—	332	1	no	—
159	C₄	—	166	2	no	—
160	C₄	Q₁₆×D₈₃	1	1	yes	characteristic
161	C₄	#14581812	1	1	yes	characteristic
162	C₄	#14581818	1	1	yes	characteristic
163	C₄	#14581824	1	1	yes	characteristic
164	C₂	#19127844	1	1	yes	characteristic
165	C₂	#19127855	1	1	yes	characteristic
166	C₂	#19127882	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128429

C₂

yes

maximal, characteristic

#19128104

C₂

yes

maximal, characteristic

#19127983

C₂

yes

maximal, characteristic

#19127448

C₂

yes

maximal, characteristic

Q₈⋊(C₈₃⋊C₈)

C₂

yes

maximal, characteristic

#19127431

C₂

yes

maximal, characteristic

#462×C₈₃

C₂

yes

maximal, characteristic

#14581792

C₂²

yes

—

—

—

C₄×Dic₁₆₆

—

C₄×Dic₁₆₆

C₂²

yes

—

—

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

characteristic

#14581715

—

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₂×Q₁₆×C₈₃

—

C₄⋊Q₈×C₈₃

C₂²

yes

characteristic

C₄×Q₈×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₂²

yes

characteristic

#103×C₈₃

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

D₄

yes

characteristic

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

D₄

yes

—

C₄⋊Dic₈₃

D₄

yes

characteristic

Dic₈₃⋊C₄

—

Dic₈₃⋊C₄

—

Dic₈₃⋊C₄

—

Dic₈₃⋊C₄

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

D₄

yes

characteristic

Q₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

—

C₄²×C₈₃

C₂³

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

Q₁₆

yes

complemented

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

#120

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×Q₁₆

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

—

—

—

D₈₃

yes

characteristic

#332

—

Q₈⋊C₈

—

Q₈⋊C₈

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

166

—

C₄⋊Q₈

C₂×D₈₃

yes

characteristic

#125

—

166

—

#125

—

166

—

C₄×Q₈

—

C₄×Q₈

—

166

—

100

C₄×Q₈

—

166

—

101

C₄×Q₈

C₂×D₈₃

yes

characteristic

102

#107

—

166

—

103

#106

—

166

—

104

C₄⋊C₈

—

105

C₄⋊C₈

C₂×D₈₃

yes

characteristic

106

#103

—

107

#103

—

166

—

108

C₄×C₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

166

—

112

C₂×Q₈

C₂²⋊D₈₃

yes

characteristic

113

Q₁₆

—

114

C₂×C₈

—

115

C₂×C₈

—

166

—

116

C₂×C₈

—

166

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

166

—

123

C₄⋊C₄

—

166

—

124

C₄⋊C₄

—

166

—

125

C₄⋊C₄

—

166

—

126

C₄⋊C₄

—

166

—

127

C₄⋊C₄

C₂²⋊D₈₃

yes

characteristic

128

C₄²

—

129

C₄²

—

166

—

130

C₄²

—

166

—

131

C₄²

C₂²×D₈₃

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

Q₈

—

332

—

137

Q₈

—

332

—

138

Q₈

—

139

Q₈

—

166

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

166

—

144

C₂×C₄

—

166

—

145

C₂×C₄

—

166

—

146

C₂×C₄

D₄×D₈₃

yes

characteristic

147

C₂×C₄

D₄×D₈₃

yes

characteristic

148

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

149

C₈

—

150

C₈

—

332

—

151

C₈

—

166

—

152

C₂²

#14581760

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

332

—

158

C₄

—

332

—

159

C₄

—

166

—

160

C₄

Q₁₆×D₈₃

yes

characteristic

161

C₄

#14581812

yes

characteristic

162

C₄

#14581818

yes

characteristic

163

C₄

#14581824

yes

characteristic

164

C₂

#19127844

yes

characteristic

165

C₂

#19127855

yes

characteristic

166

C₂

#19127882

yes

characteristic

	subgroup	G/H
center	C₂²	#14581760
commutator, cosocle	C₄²×C₈₃	C₂³
Frattini	C₄²	C₂²×D₈₃
Fitting	#462×C₈₃	C₂
socle, prime radical	C₂²×C₈₃	#120
hypercenter	#462	D₈₃
hypocenter	C₈₃	#1648
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

subgroup

G/H

center

C₂²

#14581760

commutator, cosocle

C₄²×C₈₃

C₂³

Frattini

C₄²

C₂²×D₈₃

Fitting

#462×C₈₃

C₂

C₂²×C₈₃

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #462

N	H ≅ G/N	type
64. Dic₁₆₆	Q₁₆	semidirect
84. C₈₃	#1648	semidirect

H ≅ G/N

type

64. Dic₁₆₆

Q₁₆

semidirect

84. C₈₃

#1648

semidirect

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

subgroup	index
#19128429	2
#19128104	2
#19127983	2
#19127448	2
Q₈⋊(C₈₃⋊C₈)	2
#19127431	2
#462×C₈₃	2
#1648	83

subgroup

index

Q₈⋊(C₈₃⋊C₈)

#462×C₈₃

Subgroups — #24906703

Conjugacy classes of subgroups

Complements

Maximal subgroups