Group #35297434 — subgroups

Subgroups — #35297434

Order 21248.

336 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 127 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	#24906222	C₂	1	2	yes	maximal
2	#24906201	C₂	1	2	yes	maximal
3	C₈₃⋊#1309	C₂	1	2	yes	maximal
4	C₈₃×#2960	C₂	1	1	yes	maximal, characteristic
5	#19127725	C₂²	1	4	yes	—
6	#19127454	—	2	2	no	—
7	#19127454	—	2	2	no	—
8	#19127447	—	2	4	no	—
9	#533×C₈₃	C₂²	1	1	yes	characteristic
10	#446×C₈₃	—	2	2	no	—
11	C₈×Q₈×C₈₃	—	2	2	no	—
12	#424×C₈₃	C₄	1	2	yes	—
13	#423×C₈₃	C₂²	1	1	yes	characteristic
14	#405×C₈₃	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
16	C₂×#13050574	—	2	4	no	—
17	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
18	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
19	C₂×M₁₆×C₈₃	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
22	C₄⋊Q₈×C₈₃	D₄	1	2	yes	—
23	#125×C₈₃	Q₈	1	2	yes	—
24	C₂²⋊Q₈×C₈₃	—	2	2	no	—
25	C₄×Q₈×C₈₃	—	2	2	no	—
26	#117×C₈₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₈₃	—	2	4	no	—
29	C₄⋊C₈×C₈₃	—	2	2	no	—
30	#97×C₈₃	—	2	2	no	—
31	C₄×C₈×C₈₃	—	2	2	no	—
32	C₄×C₈×C₈₃	D₄	1	2	yes	—
33	C₄×C₈×C₈₃	D₄	1	2	yes	—
34	#13050574	—	8	4	no	—
35	C₂×C₈₃⋊C₈	—	4	4	no	—
36	C₂×Q₈×C₈₃	—	2	2	no	—
37	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₈₃	#41	1	2	yes	—
39	M₁₆×C₈₃	—	4	2	no	—
40	C₂×C₈×C₈₃	—	2	2	no	—
41	C₂×C₈×C₈₃	—	2	2	no	—
42	C₂×C₈×C₈₃	—	4	1	no	—
43	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
45	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₄⋊C₄×C₈₃	—	2	2	no	—
47	C₄⋊C₄×C₈₃	—	2	4	no	—
48	C₄⋊C₄×C₈₃	C₄⋊C₄	1	4	yes	—
49	C₂²⋊C₄×C₈₃	—	2	2	no	—
50	C₄²×C₈₃	—	2	2	no	—
51	C₄²×C₈₃	C₂×D₄	1	2	yes	—
52	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
54	C₂³×C₈₃	#117	1	1	yes	characteristic
55	Q₈×C₈₃	—	4	2	no	—
56	Q₈×C₈₃	—	4	2	no	—
57	C₂×C₄×C₈₃	—	2	1	no	—
58	C₂×C₄×C₈₃	—	2	1	no	—
59	C₂×C₄×C₈₃	—	2	2	no	—
60	C₂×C₄×C₈₃	—	2	2	no	—
61	C₂×C₄×C₈₃	—	2	4	no	—
62	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₈₃	#121	1	2	yes	—
65	C₂×C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
66	C₈×C₈₃	—	2	1	no	—
67	C₈×C₈₃	—	2	2	no	—
68	C₈×C₈₃	—	4	2	no	—
69	C₈×C₈₃	#101	1	2	yes	—
70	C₈₃⋊C₈	—	8	4	no	—
71	C₂²×C₈₃	—	2	1	no	—
72	C₂²×C₈₃	—	2	2	no	—
73	C₂²×C₈₃	#380	1	1	yes	characteristic
74	C₄×C₈₃	—	2	1	no	—
75	C₄×C₈₃	—	2	2	no	—
76	C₄×C₈₃	—	4	4	no	—
77	C₄×C₈₃	C₂×#101	1	1	yes	characteristic
78	C₄×C₈₃	#413	1	1	yes	characteristic
79	C₄×C₈₃	#472	1	2	yes	—
80	C₄×C₈₃	#473	1	2	yes	—
81	#8639	—	83	1	no	maximal, Sylow
82	C₂×C₈₃	—	4	1	no	—
83	C₂×C₈₃	#1801	1	1	yes	characteristic
84	C₂×C₈₃	#1816	1	2	yes	—
85	#2960	D₈₃	1	1	yes	characteristic
86	#1567	—	83	2	no	—
87	#1561	—	83	2	no	—
88	#1309	—	83	2	no	—
89	C₈₃	#8639	1	1	yes	characteristic, complemented, Sylow
90	#533	C₂×D₈₃	1	1	yes	characteristic
91	#446	—	2	2	no	—
92	C₈×Q₈	—	2	2	no	—
93	#424	Dic₈₃	1	2	yes	—
94	#423	—	83	4	no	—
95	#423	C₂×D₈₃	1	1	yes	characteristic
96	#405	Dic₈₃	1	2	yes	—
97	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
98	#335	—	166	2	no	—
99	#335	—	166	2	no	—
100	#332	—	166	4	no	—
101	C₂×M₁₆	—	166	4	no	—
102	C₂×M₁₆	C₂×Dic₈₃	1	2	yes	—
103	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
104	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
105	C₄⋊Q₈	D₃₃₂	1	2	yes	—
106	#125	Dic₁₆₆	1	2	yes	—
107	C₂²⋊Q₈	—	2	2	no	—
108	C₄×Q₈	—	2	2	no	—
109	#117	C₂×Dic₈₃	1	2	yes	—
110	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
111	C₄⋊C₈	—	2	4	no	—
112	C₄⋊C₈	—	2	2	no	—
113	C₄⋊C₈	—	166	4	no	—
114	C₄⋊C₈	—	166	4	no	—
115	#97	—	2	2	no	—
116	C₄×C₈	—	2	2	no	—
117	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
118	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
119	C₂×Q₈	—	2	2	no	—
120	C₂²×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
121	C₂²×C₄	#13050590	1	2	yes	—
122	M₁₆	—	4	2	no	—
123	M₁₆	—	664	4	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	—	332	4	no	—
128	C₂×C₈	C₂²⋊Dic₈₃	1	1	yes	characteristic
129	C₂×C₈	C₂²⋊Dic₈₃	1	2	yes	—
130	C₂×C₈	C₂²⋊Dic₈₃	1	1	yes	characteristic
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	2	4	no	—
133	C₄⋊C₄	C₄⋊Dic₈₃	1	4	yes	—
134	C₂²⋊C₄	—	2	2	no	—
135	C₄²	—	2	2	no	—
136	C₄²	C₂×C₂²⋊D₈₃	1	2	yes	—
137	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
138	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
139	C₂³	#14581751	1	1	yes	characteristic
140	Q₈	—	4	2	no	—
141	Q₈	—	4	2	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
148	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
149	C₂×C₄	#14581766	1	2	yes	—
150	C₂×C₄	#14581771	1	2	yes	—
151	C₈	—	2	1	no	—
152	C₈	—	2	2	no	—
153	C₈	—	4	2	no	—
154	C₈	—	664	4	no	—
155	C₈	#14581711	1	2	yes	—
156	C₂²	—	2	2	no	—
157	C₂²	—	2	1	no	—
158	C₂²	#19127561	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	1	no	—
161	C₄	—	4	4	no	—
162	C₄	C₂×#14581711	1	1	yes	characteristic
163	C₄	#19127685	1	1	yes	characteristic
164	C₄	#19127971	1	2	yes	—
165	C₄	#19127976	1	2	yes	—
166	C₂	—	4	1	no	—
167	C₂	#24907459	1	1	yes	characteristic
168	C₂	#24907527	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24906222

C₂

yes

maximal

#24906201

C₂

yes

maximal

C₈₃⋊#1309

C₂

yes

maximal

C₈₃×#2960

C₂

yes

maximal, characteristic

#19127725

C₂²

yes

—

#19127454

—

#19127454

—

#19127447

—

#533×C₈₃

C₂²

yes

characteristic

#446×C₈₃

—

C₈×Q₈×C₈₃

—

#424×C₈₃

C₄

yes

—

#423×C₈₃

C₂²

yes

characteristic

#405×C₈₃

C₄

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×#13050574

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

—

C₂×M₁₆×C₈₃

C₂×C₄

yes

—

C₂²×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄⋊Q₈×C₈₃

D₄

yes

—

#125×C₈₃

Q₈

yes

—

C₂²⋊Q₈×C₈₃

—

C₄×Q₈×C₈₃

—

#117×C₈₃

C₂×C₄

yes

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄⋊C₈×C₈₃

—

C₄⋊C₈×C₈₃

—

#97×C₈₃

—

C₄×C₈×C₈₃

—

C₄×C₈×C₈₃

D₄

yes

—

C₄×C₈×C₈₃

D₄

yes

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×Q₈×C₈₃

—

C₂²×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₃

#41

yes

—

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₈₃

C₂²⋊C₄

yes

—

C₂×C₈×C₈₃

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

C₄⋊C₄

yes

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

C₂×D₄

yes

—

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₄²×C₈₃

C₂×Q₈

yes

characteristic

C₂³×C₈₃

#117

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

#121

yes

—

C₂×C₄×C₈₃

C₂²⋊Q₈

yes

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

#101

yes

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#380

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

—

C₄×C₈₃

#473

yes

—

#8639

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

#1801

yes

characteristic

C₂×C₈₃

#1816

yes

—

#2960

D₈₃

yes

—

—

—

C₈₃

#8639

yes

characteristic, complemented, Sylow

#533

C₂×D₈₃

yes

characteristic

#446

—

C₈×Q₈

—

#424

Dic₈₃

yes

—

#423

—

#423

C₂×D₈₃

yes

characteristic

#405

Dic₈₃

yes

—

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

#335

—

166

—

#335

—

166

—

100

#332

—

166

—

101

C₂×M₁₆

—

166

—

102

C₂×M₁₆

C₂×Dic₈₃

yes

—

103

C₂²×C₈

C₂×Dic₈₃

yes

characteristic

104

C₂²×C₈

C₂×Dic₈₃

yes

characteristic

105

C₄⋊Q₈

D₃₃₂

yes

—

106

#125

Dic₁₆₆

yes

—

107

C₂²⋊Q₈

—

108

C₄×Q₈

—

109

#117

C₂×Dic₈₃

yes

—

110

C₂×C₄²

C₂²×D₈₃

yes

characteristic

111

C₄⋊C₈

—

112

C₄⋊C₈

—

113

C₄⋊C₈

—

166

—

114

C₄⋊C₈

—

166

—

115

#97

—

116

C₄×C₈

—

117

C₄×C₈

C₂²⋊D₈₃

yes

—

118

C₄×C₈

C₂²⋊D₈₃

yes

—

119

C₂×Q₈

—

120

C₂²×C₄

C₂²×Dic₈₃

yes

characteristic

121

C₂²×C₄

#13050590

yes

—

122

M₁₆

—

123

M₁₆

—

664

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

332

—

128

C₂×C₈

C₂²⋊Dic₈₃

yes

characteristic

129

C₂×C₈

C₂²⋊Dic₈₃

yes

—

130

C₂×C₈

C₂²⋊Dic₈₃

yes

characteristic

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

C₄⋊Dic₈₃

yes

—

134

C₂²⋊C₄

—

135

C₄²

—

136

C₄²

C₂×C₂²⋊D₈₃

yes

—

137

C₄²

C₂×D₃₃₂

yes

characteristic

138

C₄²

C₂×Dic₁₆₆

yes

characteristic

139

C₂³

#14581751

yes

characteristic

140

Q₈

—

141

Q₈

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

C₂×C₂²⋊Dic₈₃

yes

characteristic

148

C₂×C₄

C₂×C₄⋊Dic₈₃

yes

characteristic

149

C₂×C₄

#14581766

yes

—

150

C₂×C₄

#14581771

yes

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

664

—

155

C₈

#14581711

yes

—

156

C₂²

—

157

C₂²

—

158

C₂²

#19127561

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

C₂×#14581711

yes

characteristic

163

C₄

#19127685

yes

characteristic

164

C₄

#19127971

yes

—

165

C₄

#19127976

yes

—

166

C₂

—

167

C₂

#24907459

yes

characteristic

168

C₂

#24907527

yes

—

	subgroup	G/H
center	C₂²	#19127561
commutator	C₂²×C₄×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#2960	C₂
socle	C₂²×C₈₃	#380
prime radical	C₂³×C₈₃	#117
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#2960	D₈₃
hypocenter	C₈₃	#8639
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

subgroup

G/H

center

C₂²

#19127561

commutator

C₂²×C₄×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#2960

C₂

C₂²×C₈₃

C₂³×C₈₃

C₂×C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂²×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

N	H ≅ G/N	type
89. C₈₃	#8639	semidirect

H ≅ G/N

type

89. C₈₃

#8639

semidirect

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

subgroup	index
#24906222	2
#24906201	2
C₈₃⋊#1309	2
C₈₃×#2960	2
#8639	83

subgroup

index

#24906222

#24906201

C₈₃⋊#1309

C₈₃×#2960

#8639

Subgroups — #35297434

Conjugacy classes of subgroups

Complements

Maximal subgroups