Group #35313435 — subgroups

Subgroups — #35313435

Order 21248.

306 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 163 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	#24909260	C₂	1	2	yes	maximal
2	C₈²×D₈₃	C₂	1	1	yes	maximal, characteristic
3	#24905381	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₆×C₈₃⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₈₃×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₈×D₈₃	C₄	1	1	yes	characteristic
7	C₄×C₈×D₈₃	C₄	1	1	yes	characteristic
8	C₄×C₈×D₈₃	C₂²	1	1	yes	characteristic
9	Dic₈₃⋊C₁₆	C₄	1	4	yes	—
10	(C₂×D₈₃)⋊C₁₆	C₄	1	4	yes	—
11	C₄×C₈₃⋊C₁₆	C₂²	1	2	yes	—
12	C₈×C₈₃⋊C₈	C₄	1	1	yes	characteristic
13	C₈×C₈₃⋊C₈	C₄	1	1	yes	characteristic
14	C₈×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₈₃	C₂²	1	2	yes	—
16	C₈²×C₈₃	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₈₃	C₈	1	2	yes	—
18	C₂×C₈×D₈₃	C₈	1	2	yes	—
19	C₂×C₈×D₈₃	C₂×C₄	1	2	yes	—
20	C₄²×D₈₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈₃⋊C₁₆	C₂×C₄	1	4	yes	—
22	C₄×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₈×Dic₈₃	C₈	1	2	yes	—
26	C₈×Dic₈₃	C₈	1	2	yes	—
27	C₈×Dic₈₃	C₂×C₄	1	2	yes	—
28	C₂×C₁₆×C₈₃	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
32	C₂×C₄×D₈₃	C₄²	1	1	yes	characteristic
33	C₂×C₄×D₈₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×D₈₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈₃⋊C₈	C₄²	1	2	yes	—
36	C₂×C₈₃⋊C₈	C₂×C₈	1	2	yes	—
37	C₂×C₈₃⋊C₈	C₂×C₈	1	2	yes	—
38	C₈×D₈₃	—	2	4	no	—
39	C₈×D₈₃	—	2	4	no	—
40	C₈×D₈₃	C₁₆	1	8	yes	complemented
41	C₄×Dic₈₃	C₄²	1	1	yes	characteristic
42	C₄×Dic₈₃	C₂×C₈	1	1	yes	characteristic
43	C₄×Dic₈₃	C₂×C₈	1	1	yes	characteristic
44	C₈₃⋊C₁₆	—	2	4	no	—
45	C₂×C₈×C₈₃	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₈₃	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
48	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
49	C₁₆×C₈₃	—	2	4	no	—
50	C₂²×D₈₃	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
53	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
54	C₂×Dic₈₃	C₄×C₈	1	1	yes	characteristic
55	C₂×Dic₈₃	C₄×C₈	1	1	yes	characteristic
56	C₂×Dic₈₃	C₄×C₈	1	1	yes	characteristic
57	C₄×D₈₃	—	2	2	no	—
58	C₄×D₈₃	—	2	2	no	—
59	C₄×D₈₃	C₂×C₁₆	1	4	yes	—
60	C₈×C₈₃	—	2	2	no	—
61	C₈×C₈₃	C₂×C₁₆	1	4	yes	—
62	C₈×C₈₃	#131	1	4	yes	—
63	C₈₃⋊C₈	—	2	2	no	—
64	C₈₃⋊C₈	C₂×C₁₆	1	4	yes	—
65	C₈₃⋊C₈	M₃₂	1	4	yes	—
66	C₂²×C₈₃	C₂×C₄×C₈	1	1	yes	characteristic
67	C₂×D₈₃	—	2	1	no	—
68	C₂×D₈₃	—	2	1	no	—
69	C₂×D₈₃	C₄×C₁₆	1	2	yes	—
70	C₄×C₈₃	#405	1	2	yes	—
71	C₄×C₈₃	C₂²×C₁₆	1	2	yes	—
72	C₄×C₈₃	C₂×M₃₂	1	2	yes	—
73	Dic₈₃	—	2	1	no	—
74	Dic₈₃	—	2	1	no	—
75	Dic₈₃	C₄×C₁₆	1	2	yes	—
76	#11410	—	83	1	no	maximal, Sylow
77	C₂×C₈₃	#1450	1	1	yes	characteristic
78	C₂×C₈₃	C₂×C₄×C₁₆	1	1	yes	characteristic
79	C₂×C₈₃	C₄×M₃₂	1	1	yes	characteristic
80	D₈₃	—	2	2	no	—
81	#2162	—	83	2	no	—
82	C₂×C₈²	—	83	1	no	—
83	#1312	—	83	1	no	—
84	#1312	D₈₃	1	1	yes	characteristic, complemented
85	C₈×C₁₆	—	83	2	no	—
86	C₈₃	#11410	1	1	yes	characteristic, complemented, Sylow
87	C₂×C₄×C₈	—	83	1	no	—
88	C₂×C₄×C₈	—	83	1	no	—
89	C₂×C₄×C₈	—	83	1	no	—
90	C₄⋊C₁₆	—	83	4	no	—
91	C₂²⋊C₁₆	—	83	4	no	—
92	C₄×C₁₆	—	83	2	no	—
93	C₄×C₁₆	C₂×D₈₃	1	2	yes	—
94	C₈²	—	83	1	no	—
95	C₈²	—	83	1	no	—
96	C₈²	—	83	1	no	—
97	C₈²	C₂×D₈₃	1	1	yes	characteristic
98	C₂²×C₈	—	83	2	no	—
99	C₂²×C₈	—	83	2	no	—
100	C₂²×C₈	—	83	2	no	—
101	C₂×C₄²	—	83	1	no	—
102	C₂×C₁₆	—	83	4	no	—
103	C₂×C₁₆	C₄×D₈₃	1	4	yes	—
104	C₄×C₈	—	83	2	no	—
105	C₄×C₈	—	83	2	no	—
106	C₄×C₈	—	83	2	no	—
107	C₄×C₈	—	83	1	no	—
108	C₄×C₈	—	83	1	no	—
109	C₄×C₈	—	83	1	no	—
110	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
111	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
112	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
113	C₂²×C₄	—	83	1	no	—
114	C₂²×C₄	—	83	1	no	—
115	C₂²×C₄	—	83	1	no	—
116	C₂×C₈	—	83	2	no	—
117	C₂×C₈	—	83	2	no	—
118	C₂×C₈	—	83	8	no	—
119	C₂×C₈	—	83	2	no	—
120	C₂×C₈	—	166	4	no	—
121	C₂×C₈	—	166	4	no	—
122	C₂×C₈	C₈×D₈₃	1	2	yes	—
123	C₂×C₈	C₈×D₈₃	1	2	yes	—
124	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
125	C₄²	—	83	1	no	—
126	C₄²	—	83	1	no	—
127	C₄²	—	83	1	no	—
128	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
129	C₁₆	—	2	4	no	—
130	C₁₆	—	166	4	no	—
131	C₂³	—	83	1	no	—
132	C₂×C₄	—	83	1	no	—
133	C₂×C₄	—	83	4	no	—
134	C₂×C₄	—	83	1	no	—
135	C₂×C₄	—	83	1	no	—
136	C₂×C₄	—	166	2	no	—
137	C₂×C₄	—	166	2	no	—
138	C₂×C₄	C₄²×D₈₃	1	1	yes	characteristic
139	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
140	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	—	83	4	no	—
143	C₈	—	83	4	no	—
144	C₈	—	166	2	no	—
145	C₈	C₁₆×D₈₃	1	4	yes	—
146	C₈	#14581800	1	4	yes	—
147	C₂²	—	83	2	no	—
148	C₂²	—	166	1	no	—
149	C₂²	—	166	1	no	—
150	C₂²	C₄×C₈×D₈₃	1	1	yes	characteristic
151	C₄	—	83	2	no	—
152	C₄	—	166	1	no	—
153	C₄	—	166	1	no	—
154	C₄	#19127658	1	2	yes	—
155	C₄	C₂×C₁₆×D₈₃	1	2	yes	—
156	C₄	M₃₂×D₈₃	1	2	yes	—
157	C₂	—	166	2	no	—
158	C₂	#24905748	1	1	yes	characteristic
159	C₂	C₄×C₁₆×D₈₃	1	1	yes	characteristic
160	C₂	#24909056	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24909260

C₂

yes

maximal

C₈²×D₈₃

C₂

yes

maximal, characteristic

#24905381

C₂

yes

maximal, characteristic, complemented

C₁₆×C₈₃⋊C₈

C₂

yes

maximal, complemented

C₈₃×#1312

C₂

yes

maximal, characteristic, complemented

C₄×C₈×D₈₃

C₄

yes

characteristic

C₄×C₈×D₈₃

C₄

yes

characteristic

C₄×C₈×D₈₃

C₂²

yes

characteristic

Dic₈₃⋊C₁₆

C₄

yes

—

(C₂×D₈₃)⋊C₁₆

C₄

yes

—

C₄×C₈₃⋊C₁₆

C₂²

yes

—

C₈×C₈₃⋊C₈

C₄

yes

characteristic

C₈×C₈₃⋊C₈

C₄

yes

characteristic

C₈×C₈₃⋊C₈

C₂²

yes

characteristic

C₄×C₁₆×C₈₃

C₂²

yes

—

C₈²×C₈₃

C₂²

yes

characteristic

C₂×C₈×D₈₃

C₈

yes

—

C₂×C₈×D₈₃

C₈

yes

—

C₂×C₈×D₈₃

C₂×C₄

yes

—

C₄²×D₈₃

C₂×C₄

yes

characteristic

C₂×C₈₃⋊C₁₆

C₂×C₄

yes

—

C₄×C₈₃⋊C₈

C₂×C₄

yes

characteristic

C₄×C₈₃⋊C₈

C₂×C₄

yes

characteristic

C₄×C₈₃⋊C₈

C₂×C₄

yes

characteristic

C₈×Dic₈₃

C₈

yes

—

C₈×Dic₈₃

C₈

yes

—

C₈×Dic₈₃

C₂×C₄

yes

—

C₂×C₁₆×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄×C₈×C₈₃

C₂³

yes

characteristic

C₂×C₄×D₈₃

C₄²

yes

characteristic

C₂×C₄×D₈₃

C₂×C₈

yes

characteristic

C₂×C₄×D₈₃

C₂×C₈

yes

characteristic

C₂×C₈₃⋊C₈

C₄²

yes

—

C₂×C₈₃⋊C₈

C₂×C₈

yes

—

C₂×C₈₃⋊C₈

C₂×C₈

yes

—

C₈×D₈₃

—

C₈×D₈₃

—

C₈×D₈₃

C₁₆

yes

complemented

C₄×Dic₈₃

C₄²

yes

characteristic

C₄×Dic₈₃

C₂×C₈

yes

characteristic

C₄×Dic₈₃

C₂×C₈

yes

characteristic

C₈₃⋊C₁₆

—

C₂×C₈×C₈₃

C₂×C₈

yes

—

C₂×C₈×C₈₃

C₂×C₈

yes

—

C₂×C₈×C₈₃

C₂²×C₄

yes

—

C₄²×C₈₃

C₂²×C₄

yes

characteristic

C₁₆×C₈₃

—

C₂²×D₈₃

C₄×C₈

yes

characteristic

C₂×C₄×C₈₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₈

yes

characteristic

C₂×Dic₈₃

C₄×C₈

yes

characteristic

C₂×Dic₈₃

C₄×C₈

yes

characteristic

C₂×Dic₈₃

C₄×C₈

yes

characteristic

C₄×D₈₃

—

C₄×D₈₃

—

C₄×D₈₃

C₂×C₁₆

yes

—

C₈×C₈₃

—

C₈×C₈₃

C₂×C₁₆

yes

—

C₈×C₈₃

#131

yes

—

C₈₃⋊C₈

—

C₈₃⋊C₈

C₂×C₁₆

yes

—

C₈₃⋊C₈

M₃₂

yes

—

C₂²×C₈₃

C₂×C₄×C₈

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

C₄×C₁₆

yes

—

C₄×C₈₃

#405

yes

—

C₄×C₈₃

C₂²×C₁₆

yes

—

C₄×C₈₃

C₂×M₃₂

yes

—

Dic₈₃

—

Dic₈₃

—

Dic₈₃

C₄×C₁₆

yes

—

—

C₂×C₈₃

yes

C₂×C₈₃

C₂×C₄×C₁₆

yes

characteristic

C₂×C₈₃

C₄×M₃₂

yes

characteristic

D₈₃

—

#2162

—

C₂×C₈²

—

#1312

—

#1312

D₈₃

yes

characteristic, complemented

C₈×C₁₆

—

C₈₃

#11410

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₈₃

yes

—

C₈²

—

C₈²

—

C₈²

—

C₈²

C₂×D₈₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

100

C₂²×C₈

—

101

C₂×C₄²

—

102

C₂×C₁₆

—

103

C₂×C₁₆

C₄×D₈₃

yes

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

C₄×D₈₃

yes

characteristic

111

C₄×C₈

C₄×D₈₃

yes

characteristic

112

C₄×C₈

C₂²×D₈₃

yes

characteristic

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

166

—

121

C₂×C₈

—

166

—

122

C₂×C₈

C₈×D₈₃

yes

—

123

C₂×C₈

C₈×D₈₃

yes

—

124

C₂×C₈

C₂×C₄×D₈₃

yes

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

C₂×C₄×D₈₃

yes

characteristic

129

C₁₆

—

130

C₁₆

—

166

—

131

C₂³

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

166

—

137

C₂×C₄

—

166

—

138

C₂×C₄

C₄²×D₈₃

yes

characteristic

139

C₂×C₄

C₂×C₈×D₈₃

yes

characteristic

140

C₂×C₄

C₂×C₈×D₈₃

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

166

—

145

C₈

C₁₆×D₈₃

yes

—

146

C₈

#14581800

yes

—

147

C₂²

—

148

C₂²

—

166

—

149

C₂²

—

166

—

150

C₂²

C₄×C₈×D₈₃

yes

characteristic

151

C₄

—

152

C₄

—

166

—

153

C₄

—

166

—

154

C₄

#19127658

yes

—

155

C₄

C₂×C₁₆×D₈₃

yes

—

156

C₄

M₃₂×D₈₃

yes

—

157

C₂

—

166

—

158

C₂

#24905748

yes

characteristic

159

C₂

C₄×C₁₆×D₈₃

yes

characteristic

160

C₂

#24909056

yes

characteristic

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₈₃
commutator	C₂×C₈₃	C₂×C₄×C₁₆
Fitting	C₈₃×#1312	C₂
socle	C₂²×C₈₃	C₂×C₄×C₈
prime radical	C₂²×D₈₃	C₄×C₈
cosocle	C₄×C₈×C₈₃	C₂³
hypercenter	#1312	D₈₃
hypocenter	C₈₃	#11410
LCS	G ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄×C₈ ◁ #1312

subgroup

G/H

center, Frattini

C₄×C₈

C₂²×D₈₃

commutator

C₂×C₈₃

C₂×C₄×C₁₆

Fitting

C₈₃×#1312

C₂

socle

C₂²×C₈₃

C₂×C₄×C₈

prime radical

C₂²×D₈₃

C₄×C₈

cosocle

C₄×C₈×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₈₃ ▷ C₈₃

C₁ ◁ C₄×C₈ ◁ #1312

N	H ≅ G/N	type
3. #24905381	C₂	semidirect
4. C₁₆×C₈₃⋊C₈	C₂	semidirect
5. C₈₃×#1312	C₂	semidirect
40. C₈×D₈₃	C₁₆	semidirect
84. #1312	D₈₃	semidirect
86. C₈₃	#11410	semidirect

H ≅ G/N

type

3. #24905381

C₂

semidirect

4. C₁₆×C₈₃⋊C₈

C₂

semidirect

5. C₈₃×#1312

C₂

semidirect

40. C₈×D₈₃

C₁₆

semidirect

84. #1312

D₈₃

semidirect

86. C₈₃

#11410

semidirect

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

subgroup	index
#24909260	2
C₈²×D₈₃	2
#24905381	2
C₁₆×C₈₃⋊C₈	2
C₈₃×#1312	2
#11410	83

subgroup

index

#24909260

C₈²×D₈₃

#24905381

C₁₆×C₈₃⋊C₈

C₈₃×#1312

#11410

Subgroups — #35313435

Conjugacy classes of subgroups

Complements

Maximal subgroups