Group #24916180 — subgroups

Subgroups — #24916180

Order 10624.

484 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 181 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×#14581800	C₂	1	2	yes	maximal, complemented
2	#19128251	C₂	1	1	yes	maximal, characteristic
3	#19127771	C₂	1	8	yes	maximal, complemented
4	#19127669	C₂	1	2	yes	maximal, complemented
5	C₂×#14581703	C₂	1	1	yes	maximal, characteristic, complemented
6	#423×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13050590	C₂²	1	2	yes	—
8	#14581800	—	2	8	no	—
9	C₂×#13050573	C₂²	1	4	yes	—
10	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₈×D₈₃	C₂²	1	4	yes	—
13	#14581792	C₄	1	1	yes	characteristic
14	#14581789	C₄	1	1	yes	characteristic
15	#14581780	C₄	1	2	yes	—
16	#14581766	C₄	1	4	yes	—
17	C₄×D₃₃₂	C₂²	1	4	yes	—
18	(C₂×D₈₃)⋊C₈	C₂²	1	8	yes	—
19	#14581703	C₂²	1	4	yes	—
20	C₂×M₁₆×C₈₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₈₃	C₂²	1	4	yes	—
23	#13050590	—	2	4	no	—
24	#13050590	—	2	4	no	—
25	C₂×Dic₁₆₆	C₂×C₄	1	2	yes	—
26	C₂×D₃₃₂	C₂×C₄	1	2	yes	—
27	C₂×D₃₃₂	C₂×C₄	1	4	yes	—
28	C₂×C₂²⋊D₈₃	C₂×C₄	1	4	yes	—
29	C₂×C₄×D₈₃	C₂³	1	4	yes	—
30	#13050573	—	2	8	no	—
31	C₂×C₈₃⋊C₈	—	2	2	no	—
32	C₂×C₈₃⋊C₈	D₄	1	4	yes	complemented
33	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
34	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
35	C₈×D₈₃	—	2	8	no	—
36	C₄⋊Dic₈₃	C₂×C₄	1	4	yes	—
37	(C₂×D₈₃)⋊C₄	C₂×C₄	1	8	yes	—
38	C₂²×C₄×C₈₃	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
40	M₁₆×C₈₃	—	2	4	no	—
41	C₂×C₈×C₈₃	C₂³	1	4	yes	—
42	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₈₃	C₂³	1	2	yes	—
45	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×D₈₃	C₂²×C₄	1	4	yes	—
47	C₂×C₄×C₈₃	—	4	2	no	—
48	C₂×C₄×C₈₃	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₈₃	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₈₃	#41	1	2	yes	—
53	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
54	C₂²⋊D₈₃	—	2	8	no	—
55	C₂×Dic₈₃	C₂²×C₄	1	4	yes	—
56	D₃₃₂	—	2	4	no	—
57	D₃₃₂	—	4	4	no	—
58	C₄×D₈₃	—	2	4	no	—
59	C₄×D₈₃	—	2	4	no	—
60	Dic₁₆₆	—	2	4	no	—
61	C₈×C₈₃	—	2	4	no	—
62	C₈₃⋊C₈	—	2	2	no	—
63	C₈₃⋊C₈	C₂×D₄	1	4	yes	—
64	C₂²×C₈₃	—	2	2	no	—
65	C₂²×C₈₃	C₄×D₄	1	2	yes	—
66	C₂²×C₈₃	C₂³×C₄	1	1	yes	characteristic
67	C₂×D₈₃	—	2	4	no	—
68	C₂×D₈₃	—	2	8	no	—
69	C₄×C₈₃	—	2	2	no	—
70	C₄×C₈₃	—	2	2	no	—
71	C₄×C₈₃	C₄×D₄	1	2	yes	—
72	C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
73	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
74	Dic₈₃	—	2	4	no	—
75	C₂×C₈₃	—	2	2	no	—
76	C₂×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
77	C₂×C₈₃	#568	1	2	yes	—
78	D₈₃	—	4	4	no	—
79	#2932	—	83	1	no	maximal, Sylow
80	C₈₃	#2932	1	1	yes	characteristic, complemented, Sylow
81	C₂×#131	—	83	2	no	—
82	#532	—	83	1	no	—
83	#435	—	83	8	no	—
84	#423	D₈₃	1	1	yes	characteristic, complemented
85	#408	—	83	2	no	—
86	C₂×#97	—	83	1	no	—
87	C₂×#41	—	83	2	no	—
88	#131	—	166	8	no	—
89	C₂×M₁₆	—	83	4	no	—
90	C₂×M₁₆	C₂×D₈₃	1	2	yes	—
91	C₂²×C₈	—	83	1	no	—
92	C₂²×C₈	—	83	1	no	—
93	C₂²×C₈	—	83	4	no	—
94	C₄⋊Q₈	—	83	1	no	—
95	#127	—	83	1	no	—
96	#124	—	83	2	no	—
97	#121	—	83	4	no	—
98	C₄×D₄	—	83	4	no	—
99	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
100	C₄⋊C₈	C₂×D₈₃	1	4	yes	—
101	C₂²⋊C₈	—	83	8	no	—
102	#97	—	83	4	no	—
103	#41	—	166	4	no	—
104	#41	—	166	4	no	—
105	C₂×Q₈	—	83	2	no	—
106	C₂×D₄	—	83	4	no	—
107	C₂×D₄	—	83	4	no	—
108	C₂×D₄	—	83	2	no	—
109	C₂²×C₄	—	83	4	no	—
110	C₂²×C₄	C₄×D₈₃	1	2	yes	—
111	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
112	M₁₆	—	2	4	no	—
113	M₁₆	—	166	8	no	—
114	C₂×C₈	—	83	4	no	—
115	C₂×C₈	—	83	2	no	—
116	C₂×C₈	—	83	2	no	—
117	C₂×C₈	—	166	8	no	—
118	C₂×C₈	—	166	2	no	—
119	C₂×C₈	C₂²×D₈₃	1	4	yes	—
120	C₄⋊C₄	—	83	4	no	—
121	C₂²⋊C₄	—	83	8	no	—
122	C₄²	C₄×D₈₃	1	1	yes	characteristic
123	C₄²	C₄×D₈₃	1	1	yes	characteristic
124	C₄²	C₂²×D₈₃	1	2	yes	—
125	C₂³	—	83	4	no	—
126	C₂³	C₂×C₄×D₈₃	1	1	yes	characteristic
127	Q₈	—	166	4	no	—
128	D₄	—	332	4	no	—
129	D₄	—	166	8	no	—
130	D₄	—	166	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	83	4	no	—
133	C₂×C₄	—	166	4	no	—
134	C₂×C₄	—	166	4	no	—
135	C₂×C₄	C₂×C₄×D₈₃	1	2	yes	—
136	C₂×C₄	C₂×C₄×D₈₃	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
138	C₂×C₄	D₄×D₈₃	1	2	yes	—
139	C₂×C₄	#13050591	1	2	yes	—
140	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
141	C₈	—	2	4	no	—
142	C₈	—	83	4	no	—
143	C₈	—	166	2	no	—
144	C₂²	—	2	2	no	—
145	C₂²	—	166	4	no	—
146	C₂²	—	166	8	no	—
147	C₂²	#14581754	1	2	yes	—
148	C₂²	C₂²×C₄×D₈₃	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	166	4	no	—
152	C₄	#14581754	1	2	yes	—
153	C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
154	C₄	C₂×#13050591	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	332	4	no	—
157	C₂	C₂×#14581754	1	1	yes	characteristic
158	C₂	#19128467	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581800

C₂

yes

maximal, complemented

#19128251

C₂

yes

maximal, characteristic

#19127771

C₂

yes

maximal, complemented

#19127669

C₂

yes

maximal, complemented

C₂×#14581703

C₂

yes

maximal, characteristic, complemented

#423×C₈₃

C₂

yes

maximal, characteristic, complemented

C₂×#13050590

C₂²

yes

—

#14581800

—

C₂×#13050573

C₂²

yes

—

C₂²×C₈₃⋊C₈

C₂²

yes

characteristic

C₂²×C₈₃⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₈₃

C₂²

yes

—

#14581792

C₄

yes

characteristic

#14581789

C₄

yes

characteristic

#14581780

C₄

yes

—

#14581766

C₄

yes

—

C₄×D₃₃₂

C₂²

yes

—

(C₂×D₈₃)⋊C₈

C₂²

yes

—

#14581703

C₂²

yes

—

C₂×M₁₆×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₂²

yes

—

#13050590

—

#13050590

—

C₂×Dic₁₆₆

C₂×C₄

yes

—

C₂×D₃₃₂

C₂×C₄

yes

—

C₂×D₃₃₂

C₂×C₄

yes

—

C₂×C₂²⋊D₈₃

C₂×C₄

yes

—

C₂×C₄×D₈₃

C₂³

yes

—

#13050573

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

D₄

yes

complemented

C₂×C₈₃⋊C₈

C₂³

yes

—

C₂×C₈₃⋊C₈

C₂³

yes

—

C₈×D₈₃

—

C₄⋊Dic₈₃

C₂×C₄

yes

—

(C₂×D₈₃)⋊C₄

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

M₁₆×C₈₃

—

C₂×C₈×C₈₃

C₂³

yes

—

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄²×C₈₃

C₂³

yes

—

C₂³×C₈₃

C₂²×C₄

yes

characteristic

C₂²×D₈₃

C₂²×C₄

yes

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²×C₄

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

#41

yes

—

C₂×C₄×C₈₃

C₂⁴

yes

characteristic

C₂²⋊D₈₃

—

C₂×Dic₈₃

C₂²×C₄

yes

—

D₃₃₂

—

D₃₃₂

—

C₄×D₈₃

—

C₄×D₈₃

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

C₂×D₄

yes

—

C₂²×C₈₃

—

C₂²×C₈₃

C₄×D₄

yes

—

C₂²×C₈₃

C₂³×C₄

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₄×D₄

yes

—

C₄×C₈₃

C₂²×D₄

yes

characteristic

C₄×C₈₃

C₂×#41

yes

characteristic

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

C₂×C₄×D₄

yes

characteristic

C₂×C₈₃

#568

yes

—

D₈₃

—

#2932

—

maximal, Sylow

C₈₃

#2932

yes

characteristic, complemented, Sylow

—

—

—

D₈₃

yes

characteristic, complemented

#408

—

C₂×#97

—

C₂×#41

—

#131

—

166

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₃

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

#127

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

100

C₄⋊C₈

C₂×D₈₃

yes

—

101

C₂²⋊C₈

—

102

#97

—

103

#41

—

166

—

104

#41

—

166

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₄×D₈₃

yes

—

111

C₂²×C₄

C₂²×D₈₃

yes

characteristic

112

M₁₆

—

113

M₁₆

—

166

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

166

—

118

C₂×C₈

—

166

—

119

C₂×C₈

C₂²×D₈₃

yes

—

120

C₄⋊C₄

—

121

C₂²⋊C₄

—

122

C₄²

C₄×D₈₃

yes

characteristic

123

C₄²

C₄×D₈₃

yes

characteristic

124

C₄²

C₂²×D₈₃

yes

—

125

C₂³

—

126

C₂³

C₂×C₄×D₈₃

yes

characteristic

127

Q₈

—

166

—

128

D₄

—

332

—

129

D₄

—

166

—

130

D₄

—

166

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

166

—

134

C₂×C₄

—

166

—

135

C₂×C₄

C₂×C₄×D₈₃

yes

—

136

C₂×C₄

C₂×C₄×D₈₃

yes

—

137

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

138

C₂×C₄

D₄×D₈₃

yes

—

139

C₂×C₄

#13050591

yes

—

140

C₂×C₄

C₂³×D₈₃

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

166

—

144

C₂²

—

145

C₂²

—

166

—

146

C₂²

—

166

—

147

C₂²

#14581754

yes

—

148

C₂²

C₂²×C₄×D₈₃

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

166

—

152

C₄

#14581754

yes

—

153

C₄

C₂×D₄×D₈₃

yes

characteristic

154

C₄

C₂×#13050591

yes

characteristic

155

C₂

—

156

C₂

—

332

—

157

C₂

C₂×#14581754

yes

characteristic

158

C₂

#19128467

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₃
commutator, socle	C₂²×C₈₃	C₂³×C₄
Fitting	#423×C₈₃	C₂
prime radical	#19128251	C₂
cosocle	C₂×C₄×C₈₃	C₂⁴
hypercenter	#423	D₈₃
hypocenter	C₈₃	#2932
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₈₃

commutator, socle

C₂²×C₈₃

C₂³×C₄

Fitting

#423×C₈₃

C₂

prime radical

#19128251

C₂

cosocle

C₂×C₄×C₈₃

C₂⁴

D₈₃

C₈₃

G ▷ C₂²×C₈₃ ▷ C₈₃

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
1. C₂×#14581800	C₂	semidirect
3. #19127771	C₂	semidirect
4. #19127669	C₂	semidirect
5. C₂×#14581703	C₂	semidirect
6. #423×C₈₃	C₂	semidirect
32. C₂×C₈₃⋊C₈	D₄	semidirect
80. C₈₃	#2932	semidirect
84. #423	D₈₃	semidirect

H ≅ G/N

type

1. C₂×#14581800

C₂

semidirect

3. #19127771

C₂

semidirect

4. #19127669

C₂

semidirect

5. C₂×#14581703

C₂

semidirect

6. #423×C₈₃

C₂

semidirect

32. C₂×C₈₃⋊C₈

D₄

semidirect

80. C₈₃

#2932

semidirect

84. #423

D₈₃

semidirect

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

subgroup	index
C₂×#14581800	2
#19128251	2
#19127771	2
#19127669	2
C₂×#14581703	2
#423×C₈₃	2
#2932	83

subgroup

index

#423×C₈₃

Subgroups — #24916180

Conjugacy classes of subgroups

Complements

Maximal subgroups