Group #24908830 — subgroups

Subgroups — #24908830

Order 10624.

208 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 69 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×#14581727	C₂	1	1	yes	maximal, characteristic
2	#19127744	C₂	1	1	yes	maximal, characteristic
3	#19127728	C₂	1	1	yes	maximal, characteristic
4	C₂×#14581716	C₂	1	1	yes	maximal, characteristic
5	C₂×#14581711	C₂	1	1	yes	maximal, characteristic
6	#19127469	C₂	1	1	yes	maximal, characteristic
7	#419×C₈₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13050574	C₂²	1	1	yes	characteristic
9	C₂×#13050574	C₂²	1	1	yes	characteristic
10	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
12	#14581727	—	2	2	no	—
13	#14581725	—	2	1	no	—
14	#14581723	—	2	1	no	—
15	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
16	#14581716	—	2	2	no	—
17	#14581711	—	2	2	no	—
18	C₂²×Q₈×C₈₃	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
20	#117×C₈₃	C₂²	1	1	yes	characteristic
21	#103×C₈₃	—	2	1	no	—
22	#103×C₈₃	—	2	1	no	—
23	C₂²×Dic₈₃	—	2	1	no	—
24	#13050574	—	2	1	no	—
25	#13050574	—	2	1	no	—
26	#13050574	—	2	1	no	—
27	#13050574	D₄	1	2	yes	—
28	C₂×C₈₃⋊C₈	—	2	1	no	—
29	C₂×C₈₃⋊C₈	—	2	1	no	—
30	C₂×C₈₃⋊C₈	—	2	1	no	—
31	C₂×C₈₃⋊C₈	—	2	1	no	—
32	C₂×C₈₃⋊C₈	D₄	1	2	yes	—
33	C₄⋊Dic₈₃	—	2	1	no	—
34	C₄⋊Dic₈₃	—	2	1	no	—
35	C₂×Q₈×C₈₃	—	2	1	no	—
36	C₂×Q₈×C₈₃	—	2	1	no	—
37	C₂×Q₈×C₈₃	—	2	1	no	—
38	C₂×Q₈×C₈₃	—	2	1	no	—
39	C₂×Q₈×C₈₃	—	2	1	no	—
40	C₂×Q₈×C₈₃	Q₈	1	2	yes	—
41	C₂²×C₄×C₈₃	—	2	1	no	—
42	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
43	M₁₆×C₈₃	—	2	1	no	—
44	M₁₆×C₈₃	—	2	1	no	—
45	C₂×C₈×C₈₃	—	2	1	no	—
46	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
47	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
48	C₂²⋊C₄×C₈₃	—	2	1	no	—
49	C₄²×C₈₃	—	2	1	no	—
50	C₂³×C₈₃	#41	1	1	yes	characteristic
51	Q₈×C₈₃	—	2	2	no	—
52	Q₈×C₈₃	—	2	2	no	—
53	Q₈×C₈₃	—	4	1	no	—
54	Q₈×C₈₃	—	4	1	no	—
55	C₂×C₄×C₈₃	—	2	1	no	—
56	C₂×C₄×C₈₃	—	2	1	no	—
57	C₂×C₄×C₈₃	—	2	2	no	—
58	C₂×C₄×C₈₃	—	4	1	no	—
59	C₂×C₄×C₈₃	—	4	1	no	—
60	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
65	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
66	C₂×Dic₈₃	—	4	1	no	—
67	C₂×Dic₈₃	—	8	1	no	—
68	C₈×C₈₃	—	4	1	no	—
69	C₈₃⋊C₈	—	2	2	no	—
70	C₈₃⋊C₈	—	2	2	no	—
71	C₈₃⋊C₈	—	4	1	no	—
72	C₂²×C₈₃	—	2	1	no	—
73	C₂²×C₈₃	—	2	1	no	—
74	C₂²×C₈₃	#121	1	1	yes	characteristic
75	C₂²×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂²×C₈₃	#126	1	1	yes	characteristic
77	C₄×C₈₃	—	4	2	no	—
78	C₄×C₈₃	—	4	1	no	—
79	C₄×C₈₃	—	4	1	no	—
80	C₄×C₈₃	#121	1	1	yes	characteristic
81	C₄×C₈₃	#121	1	1	yes	characteristic
82	C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
83	C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
84	Dic₈₃	—	8	1	no	—
85	C₂×C₈₃	—	2	1	no	—
86	C₂×C₈₃	—	2	1	no	—
87	C₂×C₈₃	#396	1	1	yes	characteristic
88	C₂×C₈₃	#471	1	1	yes	characteristic
89	C₂×C₈₃	#473	1	1	yes	characteristic
90	#2064	—	83	1	no	maximal, Sylow
91	C₈₃	#2064	1	1	yes	characteristic, complemented, Sylow
92	C₂×#108	—	83	1	no	—
93	#428	—	83	1	no	—
94	#424	—	83	1	no	—
95	#419	D₈₃	1	1	yes	characteristic
96	C₂×#103	—	83	1	no	—
97	C₂×#101	—	83	1	no	—
98	#343	—	83	1	no	—
99	C₂²×Q₈	C₂×D₈₃	1	1	yes	characteristic
100	C₂×M₁₆	—	83	1	no	—
101	C₂×M₁₆	—	83	1	no	—
102	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic
103	C₂²×C₈	—	83	1	no	—
104	#117	C₂×D₈₃	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	83	1	no	—
106	#108	—	166	2	no	—
107	#107	—	166	1	no	—
108	#106	—	166	1	no	—
109	C₄⋊C₈	—	166	2	no	—
110	#103	—	2	1	no	—
111	#103	—	2	1	no	—
112	#103	—	166	2	no	—
113	#101	—	166	2	no	—
114	C₂×Q₈	—	2	1	no	—
115	C₂×Q₈	—	2	1	no	—
116	C₂×Q₈	—	2	1	no	—
117	C₂×Q₈	—	2	1	no	—
118	C₂×Q₈	—	2	1	no	—
119	C₂×Q₈	Dic₁₆₆	1	2	yes	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	166	1	no	—
122	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
123	M₁₆	—	2	1	no	—
124	M₁₆	—	2	1	no	—
125	M₁₆	—	83	2	no	—
126	M₁₆	—	166	1	no	—
127	M₁₆	—	166	1	no	—
128	M₁₆	—	166	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	83	2	no	—
131	C₂×C₈	—	166	1	no	—
132	C₂×C₈	—	166	1	no	—
133	C₂×C₈	—	166	1	no	—
134	C₂×C₈	—	166	1	no	—
135	C₄⋊C₄	—	166	1	no	—
136	C₄⋊C₄	—	166	1	no	—
137	C₄⋊C₄	C₂²⋊D₈₃	1	1	yes	characteristic
138	C₄⋊C₄	C₂²⋊D₈₃	1	1	yes	characteristic
139	C₂²⋊C₄	—	2	1	no	—
140	C₄²	—	2	1	no	—
141	C₂³	#13050589	1	1	yes	characteristic
142	Q₈	—	2	2	no	—
143	Q₈	—	2	2	no	—
144	Q₈	—	4	1	no	—
145	Q₈	—	4	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	332	1	no	—
152	C₂×C₄	—	664	1	no	—
153	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
154	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
155	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
156	C₂×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
157	C₂×C₄	#13050589	1	1	yes	characteristic
158	C₂×C₄	#13050590	1	1	yes	characteristic
159	C₈	—	4	1	no	—
160	C₈	—	332	1	no	—
161	C₈	—	166	2	no	—
162	C₈	—	166	2	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	#14581764	1	1	yes	characteristic
166	C₂²	#14581769	1	1	yes	characteristic
167	C₂²	#14581785	1	1	yes	characteristic
168	C₄	—	4	2	no	—
169	C₄	—	4	1	no	—
170	C₄	—	4	1	no	—
171	C₄	—	664	1	no	—
172	C₄	#14581763	1	1	yes	characteristic
173	C₄	#14581765	1	1	yes	characteristic
174	C₄	#14581769	1	1	yes	characteristic
175	C₄	#14581771	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	#19127624	1	1	yes	characteristic
179	C₂	#19127966	1	1	yes	characteristic
180	C₂	#19127974	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581727

C₂

yes

maximal, characteristic

#19127744

C₂

yes

maximal, characteristic

#19127728

C₂

yes

maximal, characteristic

C₂×#14581716

C₂

yes

maximal, characteristic

C₂×#14581711

C₂

yes

maximal, characteristic

#19127469

C₂

yes

maximal, characteristic

#419×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

C₂²

yes

characteristic

C₂×#13050574

C₂²

yes

characteristic

C₂²×C₈₃⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₃

C₂²

yes

—

—

—

C₄⋊(C₈₃⋊C₈)

—

#14581716

—

#14581711

—

C₂²×Q₈×C₈₃

C₂²

yes

characteristic

C₂×M₁₆×C₈₃

C₂²

yes

characteristic

#117×C₈₃

C₂²

yes

characteristic

#103×C₈₃

—

#103×C₈₃

—

C₂²×Dic₈₃

—

—

—

—

D₄

yes

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

D₄

yes

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

Q₈

yes

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

M₁₆×C₈₃

—

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₄⋊C₄×C₈₃

D₄

yes

characteristic

C₄⋊C₄×C₈₃

D₄

yes

characteristic

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₂³×C₈₃

#41

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×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×C₄×C₈₃

C₂×Q₈

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#121

yes

characteristic

C₂²×C₈₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₈₃

#126

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

C₂²⋊Q₈

yes

characteristic

C₄×C₈₃

C₂²⋊Q₈

yes

characteristic

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

—

—

—

D₈₃

yes

—

—

—

C₂²×Q₈

C₂×D₈₃

yes

characteristic

100

C₂×M₁₆

—

101

C₂×M₁₆

—

102

C₂×M₁₆

C₂×D₈₃

yes

characteristic

103

C₂²×C₈

—

104

#117

C₂×D₈₃

yes

characteristic

105

C₂×C₄⋊C₄

—

106

#108

—

166

—

107

#107

—

166

—

108

#106

—

166

—

109

C₄⋊C₈

—

166

—

110

#103

—

111

#103

—

112

#103

—

166

—

113

#101

—

166

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

Dic₁₆₆

yes

—

120

C₂²×C₄

—

121

C₂²×C₄

—

166

—

122

C₂²×C₄

C₂²×D₈₃

yes

characteristic

123

M₁₆

—

124

M₁₆

—

125

M₁₆

—

126

M₁₆

—

166

—

127

M₁₆

—

166

—

128

M₁₆

—

166

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

166

—

132

C₂×C₈

—

166

—

133

C₂×C₈

—

166

—

134

C₂×C₈

—

166

—

135

C₄⋊C₄

—

166

—

136

C₄⋊C₄

—

166

—

137

C₄⋊C₄

C₂²⋊D₈₃

yes

characteristic

138

C₄⋊C₄

C₂²⋊D₈₃

yes

characteristic

139

C₂²⋊C₄

—

140

C₄²

—

141

C₂³

#13050589

yes

characteristic

142

Q₈

—

143

Q₈

—

144

Q₈

—

145

Q₈

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

332

—

152

C₂×C₄

—

664

—

153

C₂×C₄

D₄×D₈₃

yes

characteristic

154

C₂×C₄

D₄×D₈₃

yes

characteristic

155

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

156

C₂×C₄

C₂×Dic₁₆₆

yes

characteristic

157

C₂×C₄

#13050589

yes

characteristic

158

C₂×C₄

#13050590

yes

characteristic

159

C₈

—

160

C₈

—

332

—

161

C₈

—

166

—

162

C₈

—

166

—

163

C₂²

—

164

C₂²

—

165

C₂²

#14581764

yes

characteristic

166

C₂²

#14581769

yes

characteristic

167

C₂²

#14581785

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

664

—

172

C₄

#14581763

yes

characteristic

173

C₄

#14581765

yes

characteristic

174

C₄

#14581769

yes

characteristic

175

C₄

#14581771

yes

characteristic

176

C₂

—

177

C₂

—

178

C₂

#19127624

yes

characteristic

179

C₂

#19127966

yes

characteristic

180

C₂

#19127974

yes

characteristic

	subgroup	G/H
center	C₂²	#14581764
commutator, cosocle	C₂²×C₄×C₈₃	C₂³
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#419×C₈₃	C₂
socle	C₂²×C₈₃	#121
prime radical	C₂³×C₈₃	#41
hypercenter	#419	D₈₃
hypocenter	C₈₃	#2064
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

subgroup

G/H

center

C₂²

#14581764

commutator, cosocle

C₂²×C₄×C₈₃

C₂³

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#419×C₈₃

C₂

C₂²×C₈₃

C₂³×C₈₃

D₈₃

C₈₃

G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

N	H ≅ G/N	type
91. C₈₃	#2064	semidirect

H ≅ G/N

type

91. C₈₃

#2064

semidirect

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

subgroup	index
C₂×#14581727	2
#19127744	2
#19127728	2
C₂×#14581716	2
C₂×#14581711	2
#19127469	2
#419×C₈₃	2
#2064	83

subgroup

index

#419×C₈₃

Subgroups — #24908830

Conjugacy classes of subgroups

Complements

Maximal subgroups