Group #35320820 — subgroups

Subgroups — #35320820

Order 21248.

382 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 167 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×#19127521	C₂	1	4	yes	maximal
2	#24909219	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×Dic₈₃	C₂	1	1	yes	maximal, characteristic
4	C₈₃×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂×#14581732	C₂²	1	2	yes	—
6	C₂×C₄×C₈₃⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₈₃	C₄	1	2	yes	—
8	C₂×C₈×Dic₈₃	C₂²	1	2	yes	—
9	C₄²×Dic₈₃	C₄	1	1	yes	characteristic
10	#19127521	—	2	8	no	—
11	C₄⋊(C₈₃⋊C₁₆)	—	2	2	no	—
12	C₂×M₃₂×C₈₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₈₃	—	2	2	no	—
15	C₂²×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₈₃	—	2	2	no	—
17	C₂×C₄×Dic₈₃	C₂×C₄	1	2	yes	—
18	#14581732	—	2	4	no	—
19	C₂×C₈₃⋊C₁₆	—	2	2	no	—
20	C₄×C₈₃⋊C₈	C₈	1	2	yes	—
21	C₄×C₈₃⋊C₈	C₈	1	2	yes	—
22	C₈×Dic₈₃	—	2	4	no	—
23	C₈×Dic₈₃	D₄	1	4	yes	—
24	C₈×Dic₈₃	Q₈	1	4	yes	—
25	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
28	M₃₂×C₈₃	—	2	4	no	—
29	C₂×C₁₆×C₈₃	—	2	2	no	—
30	C₄×C₈×C₈₃	D₄	1	2	yes	—
31	C₄×C₈×C₈₃	Q₈	1	2	yes	—
32	C₂²×Dic₈₃	—	2	2	no	—
33	C₂×C₈₃⋊C₈	C₄⋊C₄	1	4	yes	—
34	C₂×C₈₃⋊C₈	C₂×C₈	1	4	yes	—
35	C₂×C₈₃⋊C₈	M₁₆	1	4	yes	—
36	C₄×Dic₈₃	—	2	8	no	—
37	C₄×Dic₈₃	—	2	2	no	—
38	C₄×Dic₈₃	C₄⋊C₄	1	4	yes	—
39	C₈₃⋊C₁₆	—	4	2	no	—
40	C₂²×C₄×C₈₃	M₁₆	1	2	yes	—
41	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₈₃	—	2	2	no	—
43	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
45	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₈₃	C₂×Q₈	1	2	yes	—
47	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₈₃	C₄⋊C₄	1	2	yes	—
49	C₄²×C₈₃	C₂×C₈	1	1	yes	characteristic
50	C₄²×C₈₃	C₂×C₈	1	1	yes	characteristic
51	C₁₆×C₈₃	—	4	2	no	—
52	C₂³×C₈₃	C₂×M₁₆	1	1	yes	characteristic
53	C₂×C₄×C₈₃	—	2	4	no	—
54	C₂×C₄×C₈₃	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₈₃	C₄⋊C₈	1	2	yes	—
56	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₈₃	C₂×M₁₆	1	2	yes	—
60	C₂×Dic₈₃	—	2	8	no	—
61	C₂×Dic₈₃	—	2	4	no	—
62	C₈×C₈₃	—	2	2	no	—
63	C₈×C₈₃	#127	1	1	yes	characteristic
64	C₈×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₈×C₈₃	C₄⋊Q₈	1	2	yes	—
66	C₈₃⋊C₈	C₄⋊C₈	1	8	yes	—
67	C₂²×C₈₃	—	2	2	no	—
68	C₂²×C₈₃	C₂×C₄⋊C₈	1	1	yes	characteristic
69	C₂²×C₈₃	#423	1	2	yes	—
70	C₄×C₈₃	—	2	2	no	—
71	C₄×C₈₃	#364	1	4	yes	complemented
72	C₄×C₈₃	#384	1	1	yes	characteristic
73	C₄×C₈₃	C₂×C₄⋊C₈	1	2	yes	—
74	C₄×C₈₃	#423	1	1	yes	characteristic
75	Dic₈₃	—	2	8	no	—
76	#12900	—	83	1	no	maximal, Sylow
77	C₂×C₈₃	—	2	2	no	—
78	C₂×C₈₃	#1842	1	1	yes	characteristic
79	C₂×C₈₃	C₂×#364	1	2	yes	—
80	C₂×#364	—	83	4	no	—
81	#2150	—	83	1	no	—
82	#2150	D₈₃	1	1	yes	characteristic
83	C₄²×C₈	—	83	1	no	—
84	C₈₃	#12900	1	1	yes	characteristic, complemented, Sylow
85	C₂×M₃₂	—	83	2	no	—
86	C₂×M₃₂	C₂×D₈₃	1	2	yes	—
87	C₂×C₄×C₈	—	83	2	no	—
88	C₂×C₄×C₈	—	83	1	no	—
89	C₂×C₄×C₈	—	83	2	no	—
90	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
91	C₄³	—	83	1	no	—
92	#364	—	166	8	no	—
93	C₄⋊C₁₆	—	2	2	no	—
94	C₄⋊C₁₆	—	166	2	no	—
95	C₂²×C₈	—	83	2	no	—
96	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
97	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
98	C₂×C₄²	—	83	2	no	—
99	C₂×C₄²	—	166	2	no	—
100	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
101	M₃₂	—	2	4	no	—
102	M₃₂	—	166	4	no	—
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	—	166	2	no	—
105	C₄×C₈	—	83	4	no	—
106	C₄×C₈	—	83	4	no	—
107	C₄×C₈	—	83	2	no	—
108	C₄×C₈	—	83	2	no	—
109	C₄×C₈	—	166	4	no	—
110	C₄×C₈	Dic₁₆₆	1	2	yes	—
111	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
112	C₂²×C₄	—	166	2	no	—
113	C₂²×C₄	#13050573	1	2	yes	—
114	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	83	4	no	—
117	C₂×C₈	—	83	4	no	—
118	C₂×C₈	—	83	4	no	—
119	C₂×C₈	Dic₈₃⋊C₄	1	2	yes	—
120	C₂×C₈	D₄×D₈₃	1	2	yes	—
121	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
122	C₂×C₈	Q₈×D₈₃	1	2	yes	—
123	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
124	C₄²	—	83	4	no	—
125	C₄²	—	166	8	no	—
126	C₄²	—	166	2	no	—
127	C₄²	Dic₈₃⋊C₄	1	2	yes	—
128	C₄²	C₈×D₈₃	1	1	yes	characteristic
129	C₄²	C₈×D₈₃	1	1	yes	characteristic
130	C₁₆	—	4	2	no	—
131	C₁₆	—	332	2	no	—
132	C₂³	C₂×#13050573	1	1	yes	characteristic
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	166	8	no	—
135	C₂×C₄	—	166	4	no	—
136	C₂×C₄	Dic₈₃⋊C₈	1	2	yes	—
137	C₂×C₄	Dic₈₃⋊C₈	1	2	yes	—
138	C₂×C₄	C₄⋊C₄×D₈₃	1	2	yes	—
139	C₂×C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
140	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
141	C₂×C₄	M₁₆×D₈₃	1	2	yes	—
142	C₈	—	2	2	no	—
143	C₈	—	83	8	no	—
144	C₈	#14581788	1	1	yes	characteristic
145	C₈	#14581790	1	1	yes	characteristic
146	C₈	#14581791	1	2	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	C₂×Dic₈₃⋊C₈	1	1	yes	characteristic
149	C₂²	#19127723	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	166	8	no	—
152	C₄	#19127521	1	4	yes	complemented
153	C₄	#19127570	1	1	yes	characteristic
154	C₄	C₄⋊C₈×D₈₃	1	2	yes	—
155	C₄	#19127724	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#24907614	1	1	yes	characteristic
158	C₂	C₂×#19127521	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19127521

C₂

yes

maximal

#24909219

C₂

yes

maximal, characteristic

C₄×C₈×Dic₈₃

C₂

yes

maximal, characteristic

C₈₃×#2150

C₂

yes

maximal, characteristic

C₂×#14581732

C₂²

yes

—

C₂×C₄×C₈₃⋊C₈

C₄

yes

characteristic

C₂×C₈×Dic₈₃

C₄

yes

—

C₂×C₈×Dic₈₃

C₂²

yes

—

C₄²×Dic₈₃

C₄

yes

characteristic

#19127521

—

C₄⋊(C₈₃⋊C₁₆)

—

C₂×M₃₂×C₈₃

C₂²

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

C₄⋊C₁₆×C₈₃

—

C₂²×C₈₃⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₈₃

—

C₂×C₄×Dic₈₃

C₂×C₄

yes

—

#14581732

—

C₂×C₈₃⋊C₁₆

—

C₄×C₈₃⋊C₈

C₈

yes

—

C₄×C₈₃⋊C₈

C₈

yes

—

C₈×Dic₈₃

—

C₈×Dic₈₃

D₄

yes

—

C₈×Dic₈₃

Q₈

yes

—

C₂²×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₃

C₂³

yes

characteristic

C₂×C₄²×C₈₃

C₂×C₄

yes

characteristic

M₃₂×C₈₃

—

C₂×C₁₆×C₈₃

—

C₄×C₈×C₈₃

D₄

yes

—

C₄×C₈×C₈₃

Q₈

yes

—

C₂²×Dic₈₃

—

C₂×C₈₃⋊C₈

C₄⋊C₄

yes

—

C₂×C₈₃⋊C₈

C₂×C₈

yes

—

C₂×C₈₃⋊C₈

M₁₆

yes

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₄×Dic₈₃

C₄⋊C₄

yes

—

C₈₃⋊C₁₆

—

C₂²×C₄×C₈₃

M₁₆

yes

—

C₂²×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₄⋊C₄

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

C₂×Q₈

yes

—

C₂×C₈×C₈₃

C₂×Q₈

yes

characteristic

C₄²×C₈₃

C₄⋊C₄

yes

—

C₄²×C₈₃

C₂×C₈

yes

characteristic

C₄²×C₈₃

C₂×C₈

yes

characteristic

C₁₆×C₈₃

—

C₂³×C₈₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₄⋊C₈

yes

—

C₂×C₄×C₈₃

C₄⋊C₈

yes

—

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₈₃

C₂×M₁₆

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈×C₈₃

#127

yes

characteristic

C₈×C₈₃

C₄⋊Q₈

yes

characteristic

C₈×C₈₃

C₄⋊Q₈

yes

—

C₈₃⋊C₈

C₄⋊C₈

yes

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₈₃

#423

yes

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

C₂×C₄⋊C₈

yes

—

C₄×C₈₃

#423

yes

characteristic

Dic₈₃

—

#12900

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

#1842

yes

characteristic

C₂×C₈₃

C₂×#364

yes

—

C₂×#364

—

#2150

—

#2150

D₈₃

yes

characteristic

C₄²×C₈

—

C₈₃

#12900

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

C₂×D₈₃

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

C₄³

—

#364

—

166

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

166

—

C₂²×C₈

—

C₂²×C₈

C₄×D₈₃

yes

characteristic

C₂²×C₈

C₂²×D₈₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

166

—

100

C₂×C₄²

C₄×D₈₃

yes

characteristic

101

M₃₂

—

102

M₃₂

—

166

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

166

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

166

—

110

C₄×C₈

Dic₁₆₆

yes

—

111

C₄×C₈

C₂²⋊D₈₃

yes

—

112

C₂²×C₄

—

166

—

113

C₂²×C₄

#13050573

yes

—

114

C₂²×C₄

C₂×C₄×D₈₃

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

Dic₈₃⋊C₄

yes

—

120

C₂×C₈

D₄×D₈₃

yes

—

121

C₂×C₈

C₂×C₂²⋊D₈₃

yes

characteristic

122

C₂×C₈

Q₈×D₈₃

yes

—

123

C₂×C₈

C₂×Dic₁₆₆

yes

characteristic

124

C₄²

—

125

C₄²

—

166

—

126

C₄²

—

166

—

127

C₄²

Dic₈₃⋊C₄

yes

—

128

C₄²

C₈×D₈₃

yes

characteristic

129

C₄²

C₈×D₈₃

yes

characteristic

130

C₁₆

—

131

C₁₆

—

332

—

132

C₂³

C₂×#13050573

yes

characteristic

133

C₂×C₄

—

134

C₂×C₄

—

166

—

135

C₂×C₄

—

166

—

136

C₂×C₄

Dic₈₃⋊C₈

yes

—

137

C₂×C₄

Dic₈₃⋊C₈

yes

—

138

C₂×C₄

C₄⋊C₄×D₈₃

yes

—

139

C₂×C₄

C₂×Dic₈₃⋊C₄

yes

characteristic

140

C₂×C₄

C₂×C₈×D₈₃

yes

characteristic

141

C₂×C₄

M₁₆×D₈₃

yes

—

142

C₈

—

143

C₈

—

144

C₈

#14581788

yes

characteristic

145

C₈

#14581790

yes

characteristic

146

C₈

#14581791

yes

—

147

C₂²

—

148

C₂²

C₂×Dic₈₃⋊C₈

yes

characteristic

149

C₂²

#19127723

yes

—

150

C₄

—

151

C₄

—

166

—

152

C₄

#19127521

yes

complemented

153

C₄

#19127570

yes

characteristic

154

C₄

C₄⋊C₈×D₈₃

yes

—

155

C₄

#19127724

yes

characteristic

156

C₂

—

157

C₂

#24907614

yes

characteristic

158

C₂

C₂×#19127521

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#2150	C₂
socle	C₂²×C₈₃	C₂×C₄⋊C₈
prime radical	C₂³×C₈₃	C₂×M₁₆
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#2150	D₈₃
hypocenter	C₈₃	#12900
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂×C₈

C₂×C₂²⋊D₈₃

commutator

C₂×C₄×C₈₃

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₈₃

Fitting

C₈₃×#2150

C₂

socle

C₂²×C₈₃

C₂×C₄⋊C₈

prime radical

C₂³×C₈₃

C₂×M₁₆

cosocle

C₂²×C₈×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
71. C₄×C₈₃	#364	semidirect
84. C₈₃	#12900	semidirect
152. C₄	#19127521	semidirect

H ≅ G/N

type

71. C₄×C₈₃

#364

semidirect

84. C₈₃

#12900

semidirect

152. C₄

#19127521

semidirect

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

subgroup	index
C₂×#19127521	2
#24909219	2
C₄×C₈×Dic₈₃	2
C₈₃×#2150	2
#12900	83

subgroup

index

C₂×#19127521

#24909219

C₄×C₈×Dic₈₃

C₈₃×#2150

#12900

Subgroups — #35320820

Conjugacy classes of subgroups

Complements

Maximal subgroups