Group #24903644 — subgroups

Subgroups — C₈₃×#1959 #24903644

Order 10624.

344 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 128 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×#124×C₈₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#103×C₈₃	C₂	1	2	yes	maximal, complemented
3	C₂×#102×C₈₃	C₂	1	2	yes	maximal
4	C₂×#97×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#384×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₈×C₈₃	C₂²	1	1	yes	characteristic
7	C₂²×D₄×C₈₃	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₈₃	C₂²	1	2	yes	—
9	#124×C₈₃	—	2	2	no	—
10	#124×C₈₃	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
12	C₂×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
13	C₂×C₂²⋊C₄×C₈₃	—	2	2	no	—
14	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
15	#103×C₈₃	—	2	4	no	—
16	#102×C₈₃	—	2	4	no	—
17	#97×C₈₃	—	2	2	no	—
18	C₂⁴×C₈₃	—	2	1	no	—
19	C₂×Q₈×C₈₃	—	2	1	no	—
20	C₂×Q₈×C₈₃	—	2	2	no	—
21	C₂×Q₈×C₈₃	—	2	2	no	—
22	C₂×Q₈×C₈₃	C₂×C₄	1	2	yes	—
23	C₂×D₄×C₈₃	—	2	2	no	—
24	C₂×D₄×C₈₃	—	2	1	no	—
25	C₂×D₄×C₈₃	—	2	2	no	—
26	C₂×D₄×C₈₃	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₈₃	—	2	1	no	—
28	C₂²×C₄×C₈₃	—	2	2	no	—
29	C₂²×C₄×C₈₃	D₄	1	2	yes	—
30	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₈₃	—	2	2	no	—
32	C₂×C₈×C₈₃	—	2	2	no	—
33	C₂×C₈×C₈₃	D₄	1	4	yes	—
34	C₄⋊C₄×C₈₃	—	2	2	no	—
35	C₄⋊C₄×C₈₃	—	2	2	no	—
36	C₄⋊C₄×C₈₃	—	4	4	no	—
37	C₂²⋊C₄×C₈₃	—	2	8	no	—
38	C₄²×C₈₃	C₂×C₄	1	2	yes	—
39	C₄²×C₈₃	D₄	1	2	yes	—
40	C₂³×C₈₃	—	2	2	no	—
41	C₂³×C₈₃	—	4	1	no	—
42	C₂³×C₈₃	—	4	1	no	—
43	C₂³×C₈₃	—	4	2	no	—
44	C₂³×C₈₃	—	4	2	no	—
45	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₈₃	—	2	4	no	—
47	Q₈×C₈₃	—	4	2	no	—
48	D₄×C₈₃	—	2	4	no	—
49	D₄×C₈₃	—	4	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	—	4	1	no	—
54	C₂×C₄×C₈₃	—	4	1	no	—
55	C₂×C₄×C₈₃	—	4	2	no	—
56	C₂×C₄×C₈₃	—	4	4	no	—
57	C₂×C₄×C₈₃	C₂²⋊C₄	1	4	yes	—
58	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₈₃	#41	1	2	yes	—
62	C₈×C₈₃	—	2	4	no	—
63	C₂²×C₈₃	—	4	2	no	—
64	C₂²×C₈₃	—	4	4	no	—
65	C₂²×C₈₃	—	8	2	no	—
66	C₂²×C₈₃	—	8	1	no	—
67	C₂²×C₈₃	—	8	1	no	—
68	C₂²×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₈₃	#124	1	1	yes	characteristic
70	C₂²×C₈₃	#127	1	1	yes	characteristic
71	C₂²×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
72	C₂²×C₈₃	#137	1	2	yes	—
73	C₄×C₈₃	—	2	4	no	—
74	C₄×C₈₃	—	4	2	no	—
75	C₄×C₈₃	—	8	2	no	—
76	C₄×C₈₃	C₄×D₄	1	2	yes	—
77	C₄×C₈₃	#121	1	2	yes	—
78	C₂×C₈₃	—	8	2	no	—
79	C₂×C₈₃	#390	1	1	yes	characteristic
80	C₂×C₈₃	#417	1	2	yes	—
81	C₂×C₈₃	#489	1	2	yes	—
82	C₂×C₈₃	#496	1	1	yes	characteristic
83	C₂×C₈₃	#497	1	1	yes	characteristic
84	#1959	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₈₃	#1959	1	1	yes	characteristic, direct product factor, Sylow
86	C₂×#124	C₂×C₈₃	1	1	yes	characteristic
87	C₂×#103	C₂×C₈₃	1	2	yes	complemented
88	C₂×#102	C₂×C₈₃	1	2	yes	—
89	C₂×#97	C₂×C₈₃	1	1	yes	characteristic, complemented
90	#384	C₂×C₈₃	1	1	yes	characteristic, complemented
91	C₂²×Q₈	C₂²×C₈₃	1	1	yes	characteristic
92	C₂²×D₄	C₂²×C₈₃	1	1	yes	characteristic
93	C₂²×C₈	C₂²×C₈₃	1	2	yes	—
94	#124	—	2	2	no	—
95	#124	C₄×C₈₃	1	4	yes	complemented
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	C₂²×C₈₃	1	2	yes	—
98	C₂×C₂²⋊C₄	—	2	2	no	—
99	C₂×C₄²	C₂²×C₈₃	1	1	yes	characteristic
100	#103	—	2	4	no	—
101	#102	—	2	4	no	—
102	#97	—	2	2	no	—
103	C₂⁴	—	2	1	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	2	2	no	—
107	C₂×Q₈	C₂×C₄×C₈₃	1	2	yes	—
108	C₂×D₄	—	2	2	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	2	2	no	—
111	C₂×D₄	C₂×C₄×C₈₃	1	2	yes	—
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	D₄×C₈₃	1	2	yes	—
115	C₂²×C₄	C₂³×C₈₃	1	1	yes	characteristic
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	D₄×C₈₃	1	4	yes	—
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	4	4	no	—
122	C₂²⋊C₄	—	2	8	no	—
123	C₄²	C₂×C₄×C₈₃	1	2	yes	—
124	C₄²	D₄×C₈₃	1	2	yes	—
125	C₂³	—	2	2	no	—
126	C₂³	—	4	1	no	—
127	C₂³	—	4	1	no	—
128	C₂³	—	4	2	no	—
129	C₂³	—	4	2	no	—
130	C₂³	C₂×D₄×C₈₃	1	1	yes	characteristic
131	Q₈	—	2	4	no	—
132	Q₈	—	4	2	no	—
133	D₄	—	2	4	no	—
134	D₄	—	4	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	4	4	no	—
142	C₂×C₄	C₂²⋊C₄×C₈₃	1	4	yes	—
143	C₂×C₄	C₂²×C₄×C₈₃	1	1	yes	characteristic
144	C₂×C₄	C₂×D₄×C₈₃	1	2	yes	—
145	C₂×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
146	C₂×C₄	#41×C₈₃	1	2	yes	—
147	C₈	—	2	4	no	—
148	C₂²	—	4	2	no	—
149	C₂²	—	4	4	no	—
150	C₂²	—	8	2	no	—
151	C₂²	—	8	1	no	—
152	C₂²	—	8	1	no	—
153	C₂²	C₂×C₂²⋊C₄×C₈₃	1	1	yes	characteristic
154	C₂²	#124×C₈₃	1	1	yes	characteristic
155	C₂²	#127×C₈₃	1	1	yes	characteristic
156	C₂²	AGL(1,ℤ/8)×C₈₃	1	2	yes	—
157	C₂²	#137×C₈₃	1	2	yes	—
158	C₄	—	2	4	no	—
159	C₄	—	4	2	no	—
160	C₄	—	8	2	no	—
161	C₄	C₄×D₄×C₈₃	1	2	yes	—
162	C₄	#121×C₈₃	1	2	yes	—
163	C₂	—	8	2	no	—
164	C₂	#390×C₈₃	1	1	yes	characteristic
165	C₂	#417×C₈₃	1	2	yes	—
166	C₂	#489×C₈₃	1	2	yes	—
167	C₂	#496×C₈₃	1	1	yes	characteristic
168	C₂	#497×C₈₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#124×C₈₃

C₂

yes

maximal, characteristic

C₂×#103×C₈₃

C₂

yes

maximal, complemented

C₂×#102×C₈₃

C₂

yes

maximal

C₂×#97×C₈₃

C₂

yes

maximal, characteristic, complemented

#384×C₈₃

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₈₃

C₂²

yes

characteristic

C₂²×D₄×C₈₃

C₂²

yes

characteristic

C₂²×C₈×C₈₃

C₂²

yes

—

#124×C₈₃

—

#124×C₈₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄⋊C₄×C₈₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₈₃

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

#103×C₈₃

—

#102×C₈₃

—

#97×C₈₃

—

C₂⁴×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

C₂×C₄

yes

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

D₄

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

D₄

yes

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

C₂×C₄

yes

—

C₄²×C₈₃

D₄

yes

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

C₂×D₄

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

—

D₄×C₈₃

—

D₄×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

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

—

C₈×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×C₂²⋊C₄

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

—

C₂²×C₈₃

#137

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₄×D₄

yes

—

C₄×C₈₃

#121

yes

—

C₂×C₈₃

—

C₂×C₈₃

#390

yes

characteristic

C₂×C₈₃

#417

yes

—

C₂×C₈₃

#489

yes

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₈₃

yes

maximal, characteristic, direct product factor, Sylow

C₈₃

#1959

yes

characteristic, direct product factor, Sylow

C₂×#124

C₂×C₈₃

yes

characteristic

C₂×#103

C₂×C₈₃

yes

complemented

C₂×#102

C₂×C₈₃

yes

—

C₂×#97

C₂×C₈₃

yes

characteristic, complemented

#384

C₂×C₈₃

yes

characteristic, complemented

C₂²×Q₈

C₂²×C₈₃

yes

characteristic

C₂²×D₄

C₂²×C₈₃

yes

characteristic

C₂²×C₈

C₂²×C₈₃

yes

—

#124

—

#124

C₄×C₈₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₈₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂²×C₈₃

yes

characteristic

100

#103

—

101

#102

—

102

#97

—

103

C₂⁴

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

C₂×C₄×C₈₃

yes

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

C₂×C₄×C₈₃

yes

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

D₄×C₈₃

yes

—

115

C₂²×C₄

C₂³×C₈₃

yes

characteristic

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

D₄×C₈₃

yes

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₂²⋊C₄

—

123

C₄²

C₂×C₄×C₈₃

yes

—

124

C₄²

D₄×C₈₃

yes

—

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

—

130

C₂³

C₂×D₄×C₈₃

yes

characteristic

131

Q₈

—

132

Q₈

—

133

D₄

—

134

D₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

C₂²⋊C₄×C₈₃

yes

—

143

C₂×C₄

C₂²×C₄×C₈₃

yes

characteristic

144

C₂×C₄

C₂×D₄×C₈₃

yes

—

145

C₂×C₄

C₂×D₄×C₈₃

yes

characteristic

146

C₂×C₄

#41×C₈₃

yes

—

147

C₈

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

C₂×C₂²⋊C₄×C₈₃

yes

characteristic

154

C₂²

#124×C₈₃

yes

characteristic

155

C₂²

#127×C₈₃

yes

characteristic

156

C₂²

AGL(1,ℤ/8)×C₈₃

yes

—

157

C₂²

#137×C₈₃

yes

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

C₄×D₄×C₈₃

yes

—

162

C₄

#121×C₈₃

yes

—

163

C₂

—

164

C₂

#390×C₈₃

yes

characteristic

165

C₂

#417×C₈₃

yes

—

166

C₂

#489×C₈₃

yes

—

167

C₂

#496×C₈₃

yes

characteristic

168

C₂

#497×C₈₃

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₈₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₈₃
Frattini, cosocle	C₂²×C₄	C₂³×C₈₃
prime radical	C₂²×D₄×C₈₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₃ ◁ C₂×C₄²×C₈₃ ◁ G

subgroup

G/H

center, socle

C₂³×C₈₃

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₈₃

Frattini, cosocle

C₂²×C₄

C₂³×C₈₃

prime radical

C₂²×D₄×C₈₃

C₂²

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₈₃ ◁ C₂×C₄²×C₈₃ ◁ G

N	H ≅ G/N	type
2. C₂×#103×C₈₃	C₂	semidirect
4. C₂×#97×C₈₃	C₂	semidirect
5. #384×C₈₃	C₂	semidirect
10. #124×C₈₃	C₄	semidirect
84. #1959	C₈₃	direct (N × H)
85. C₈₃	#1959	direct (N × H)
87. C₂×#103	C₂×C₈₃	semidirect
89. C₂×#97	C₂×C₈₃	semidirect
90. #384	C₂×C₈₃	semidirect
95. #124	C₄×C₈₃	semidirect

H ≅ G/N

type

2. C₂×#103×C₈₃

C₂

semidirect

4. C₂×#97×C₈₃

C₂

semidirect

5. #384×C₈₃

C₂

semidirect

10. #124×C₈₃

C₄

semidirect

84. #1959

C₈₃

direct (N × H)

85. C₈₃

#1959

direct (N × H)

87. C₂×#103

C₂×C₈₃

semidirect

89. C₂×#97

C₂×C₈₃

semidirect

90. #384

C₂×C₈₃

semidirect

95. #124

C₄×C₈₃

semidirect

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

subgroup	index
C₂×#124×C₈₃	2
C₂×#103×C₈₃	2
C₂×#102×C₈₃	2
C₂×#97×C₈₃	2
#384×C₈₃	2
#1959	83

subgroup

index

C₂×#124×C₈₃

C₂×#103×C₈₃

C₂×#102×C₈₃

C₂×#97×C₈₃

#384×C₈₃

#1959

Subgroups — C₈₃×#1959 #24903644

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C83×#1959 #24903644

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₃×#1959 #24903644