Group #35313754 — subgroups

Subgroups — #35313754

Order 21248.

288 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 119 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	#24909266	C₂	1	2	yes	maximal
2	#24905405	C₂	1	4	yes	maximal
3	C₈₃×#2960	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₈₃⋊C₁₆)	—	2	2	no	—
5	C₂²⋊(C₈₃⋊C₁₆)	C₂²	1	4	yes	—
6	#19127477	—	2	2	no	—
7	#533×C₈₃	C₄	1	1	yes	characteristic
8	#446×C₈₃	—	2	2	no	—
9	C₈×Q₈×C₈₃	—	2	2	no	—
10	#424×C₈₃	C₂²	1	2	yes	—
11	#423×C₈₃	C₄	1	1	yes	characteristic
12	#405×C₈₃	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
14	C₂×C₈₃⋊C₁₆	—	2	4	no	—
15	C₂×M₁₆×C₈₃	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
18	C₄⋊Q₈×C₈₃	C₈	1	2	yes	—
19	#125×C₈₃	C₈	1	2	yes	—
20	C₂²⋊Q₈×C₈₃	—	2	2	no	—
21	C₄×Q₈×C₈₃	—	2	2	no	—
22	#117×C₈₃	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
24	C₄⋊C₈×C₈₃	—	2	4	no	—
25	C₄⋊C₈×C₈₃	—	2	2	no	—
26	#97×C₈₃	—	2	2	no	—
27	C₄×C₈×C₈₃	—	2	2	no	—
28	C₄×C₈×C₈₃	D₄	1	2	yes	—
29	C₄×C₈×C₈₃	D₄	1	2	yes	—
30	C₈₃⋊C₁₆	—	4	4	no	—
31	C₂×Q₈×C₈₃	—	2	2	no	—
32	C₂²×C₄×C₈₃	M₁₆	1	2	yes	—
33	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
34	M₁₆×C₈₃	—	4	2	no	—
35	C₂×C₈×C₈₃	—	2	2	no	—
36	C₂×C₈×C₈₃	—	2	2	no	—
37	C₂×C₈×C₈₃	—	4	1	no	—
38	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
41	C₄⋊C₄×C₈₃	—	2	4	no	—
42	C₄⋊C₄×C₈₃	—	2	2	no	—
43	C₄⋊C₄×C₈₃	C₂×C₈	1	4	yes	—
44	C₂²⋊C₄×C₈₃	—	2	2	no	—
45	C₄²×C₈₃	—	2	2	no	—
46	C₄²×C₈₃	C₂²⋊C₄	1	2	yes	—
47	C₄²×C₈₃	C₂×C₈	1	1	yes	characteristic
48	C₄²×C₈₃	C₂×C₈	1	1	yes	characteristic
49	C₂³×C₈₃	C₂×M₁₆	1	1	yes	characteristic
50	Q₈×C₈₃	—	4	4	no	—
51	C₂×C₄×C₈₃	—	2	1	no	—
52	C₂×C₄×C₈₃	—	2	1	no	—
53	C₂×C₄×C₈₃	—	2	4	no	—
54	C₂×C₄×C₈₃	—	4	2	no	—
55	C₂×C₄×C₈₃	C₂²⋊C₈	1	2	yes	—
56	C₂×C₄×C₈₃	C₂²⋊C₈	1	2	yes	—
57	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
59	C₈×C₈₃	—	2	2	no	—
60	C₈×C₈₃	—	2	1	no	—
61	C₈×C₈₃	—	4	2	no	—
62	C₈×C₈₃	#101	1	2	yes	—
63	C₂²×C₈₃	—	2	2	no	—
64	C₂²×C₈₃	—	2	1	no	—
65	C₂²×C₈₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
66	C₄×C₈₃	—	2	2	no	—
67	C₄×C₈₃	—	2	2	no	—
68	C₄×C₈₃	—	2	1	no	—
69	C₄×C₈₃	—	4	4	no	—
70	C₄×C₈₃	#410	1	1	yes	characteristic
71	C₄×C₈₃	C₂×#101	1	1	yes	characteristic
72	#11497	—	83	1	no	maximal, Sylow
73	C₂×C₈₃	—	4	1	no	—
74	C₂×C₈₃	#1461	1	1	yes	characteristic
75	C₂×C₈₃	#2117	1	2	yes	—
76	#2960	D₈₃	1	1	yes	characteristic
77	#2163	—	83	2	no	—
78	#1323	—	83	4	no	—
79	C₈₃	#11497	1	1	yes	characteristic, complemented, Sylow
80	#533	Dic₈₃	1	1	yes	characteristic
81	#446	—	2	2	no	—
82	C₈×Q₈	—	2	2	no	—
83	#424	C₂×D₈₃	1	2	yes	—
84	#423	Dic₈₃	1	1	yes	characteristic
85	#405	Dic₈₃	1	2	yes	—
86	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
87	C₄⋊C₁₆	—	166	2	no	—
88	C₂²⋊C₁₆	—	83	4	no	—
89	#346	—	166	2	no	—
90	C₂×M₁₆	C₂×Dic₈₃	1	2	yes	—
91	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
92	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
93	C₄⋊Q₈	C₈₃⋊C₈	1	2	yes	—
94	#125	C₈₃⋊C₈	1	2	yes	—
95	C₂²⋊Q₈	—	2	2	no	—
96	C₄×Q₈	—	2	2	no	—
97	#117	C₂×Dic₈₃	1	2	yes	—
98	C₂×C₄²	C₂×Dic₈₃	1	1	yes	characteristic
99	C₂×C₁₆	—	166	4	no	—
100	C₄⋊C₈	—	2	4	no	—
101	C₄⋊C₈	—	2	2	no	—
102	#97	—	2	2	no	—
103	C₄×C₈	—	2	2	no	—
104	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
105	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
106	C₂×Q₈	—	2	2	no	—
107	C₂²×C₄	#13050574	1	2	yes	—
108	C₂²×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
109	M₁₆	—	4	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	4	1	no	—
113	C₂×C₈	C₂²⋊Dic₈₃	1	1	yes	characteristic
114	C₂×C₈	C₂²⋊Dic₈₃	1	1	yes	characteristic
115	C₂×C₈	C₂×C₂²⋊D₈₃	1	2	yes	—
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	C₂×C₈₃⋊C₈	1	4	yes	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₄²	—	2	2	no	—
121	C₄²	C₂²⋊Dic₈₃	1	2	yes	—
122	C₄²	C₂×C₈₃⋊C₈	1	1	yes	characteristic
123	C₄²	C₂×C₈₃⋊C₈	1	1	yes	characteristic
124	C₁₆	—	332	4	no	—
125	C₂³	C₂×#13050574	1	1	yes	characteristic
126	Q₈	—	4	4	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	C₂²⋊(C₈₃⋊C₈)	1	2	yes	—
132	C₂×C₄	C₂²⋊(C₈₃⋊C₈)	1	2	yes	—
133	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
134	C₂×C₄	C₂²×C₈₃⋊C₈	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	2	1	no	—
137	C₈	—	4	2	no	—
138	C₈	#14581711	1	2	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	C₂×C₂²⋊(C₈₃⋊C₈)	1	1	yes	characteristic
142	C₄	—	2	2	no	—
143	C₄	—	2	2	no	—
144	C₄	—	2	1	no	—
145	C₄	—	4	4	no	—
146	C₄	#19127675	1	1	yes	characteristic
147	C₄	C₂×#14581711	1	1	yes	characteristic
148	C₂	—	4	1	no	—
149	C₂	#24905787	1	1	yes	characteristic
150	C₂	#24909091	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24909266

C₂

yes

maximal

#24905405

C₂

yes

maximal

C₈₃×#2960

C₂

yes

maximal, characteristic

C₄⋊(C₈₃⋊C₁₆)

—

C₂²⋊(C₈₃⋊C₁₆)

C₂²

yes

—

#19127477

—

#533×C₈₃

C₄

yes

characteristic

#446×C₈₃

—

C₈×Q₈×C₈₃

—

#424×C₈₃

C₂²

yes

—

#423×C₈₃

C₄

yes

characteristic

#405×C₈₃

C₄

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×C₈₃⋊C₁₆

—

C₂×M₁₆×C₈₃

C₂×C₄

yes

—

C₂²×C₈×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₃

C₂³

yes

characteristic

C₄⋊Q₈×C₈₃

C₈

yes

—

#125×C₈₃

C₈

yes

—

C₂²⋊Q₈×C₈₃

—

C₄×Q₈×C₈₃

—

#117×C₈₃

C₂×C₄

yes

—

C₂×C₄²×C₈₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₈₃

—

C₄⋊C₈×C₈₃

—

#97×C₈₃

—

C₄×C₈×C₈₃

—

C₄×C₈×C₈₃

D₄

yes

—

C₄×C₈×C₈₃

D₄

yes

—

C₈₃⋊C₁₆

—

C₂×Q₈×C₈₃

—

C₂²×C₄×C₈₃

M₁₆

yes

—

C₂²×C₄×C₈₃

C₂²×C₄

yes

characteristic

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₈₃

C₂×D₄

yes

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

C₂×C₈

yes

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

C₂²⋊C₄

yes

—

C₄²×C₈₃

C₂×C₈

yes

characteristic

C₄²×C₈₃

C₂×C₈

yes

characteristic

C₂³×C₈₃

C₂×M₁₆

yes

characteristic

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²⋊C₈

yes

—

C₂×C₄×C₈₃

C₂²⋊C₈

yes

—

C₂×C₄×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₈

yes

characteristic

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

#101

yes

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

—

C₂×C₈₃

—

C₂×C₈₃

#1461

yes

characteristic

C₂×C₈₃

#2117

yes

—

#2960

D₈₃

yes

characteristic

#2163

—

#1323

—

C₈₃

#11497

yes

characteristic, complemented, Sylow

#533

Dic₈₃

yes

characteristic

#446

—

C₈×Q₈

—

#424

C₂×D₈₃

yes

—

#423

Dic₈₃

yes

characteristic

#405

Dic₈₃

yes

—

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

C₄⋊C₁₆

—

166

—

C₂²⋊C₁₆

—

#346

—

166

—

C₂×M₁₆

C₂×Dic₈₃

yes

—

C₂²×C₈

C₂×Dic₈₃

yes

characteristic

C₂²×C₈

C₂²×D₈₃

yes

characteristic

C₄⋊Q₈

C₈₃⋊C₈

yes

—

#125

C₈₃⋊C₈

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×Dic₈₃

yes

—

C₂×C₄²

C₂×Dic₈₃

yes

characteristic

C₂×C₁₆

—

166

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

#97

—

103

C₄×C₈

—

104

C₄×C₈

C₂²⋊D₈₃

yes

—

105

C₄×C₈

C₂²⋊D₈₃

yes

—

106

C₂×Q₈

—

107

C₂²×C₄

#13050574

yes

—

108

C₂²×C₄

C₂²×Dic₈₃

yes

characteristic

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₂²⋊Dic₈₃

yes

characteristic

114

C₂×C₈

C₂²⋊Dic₈₃

yes

characteristic

115

C₂×C₈

C₂×C₂²⋊D₈₃

yes

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

C₂×C₈₃⋊C₈

yes

—

119

C₂²⋊C₄

—

120

C₄²

—

121

C₄²

C₂²⋊Dic₈₃

yes

—

122

C₄²

C₂×C₈₃⋊C₈

yes

characteristic

123

C₄²

C₂×C₈₃⋊C₈

yes

characteristic

124

C₁₆

—

332

—

125

C₂³

C₂×#13050574

yes

characteristic

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₂²⋊(C₈₃⋊C₈)

yes

—

132

C₂×C₄

C₂²⋊(C₈₃⋊C₈)

yes

—

133

C₂×C₄

C₂×C₂²⋊Dic₈₃

yes

characteristic

134

C₂×C₄

C₂²×C₈₃⋊C₈

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

#14581711

yes

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂×C₂²⋊(C₈₃⋊C₈)

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

#19127675

yes

characteristic

147

C₄

C₂×#14581711

yes

characteristic

148

C₂

—

149

C₂

#24905787

yes

characteristic

150

C₂

#24909091

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#2960	C₂
socle	C₂²×C₈₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₈₃	C₂×M₁₆
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#2960	D₈₃
hypocenter	C₈₃	#11497
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2960

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊Dic₈₃

commutator

C₂×C₄×C₈₃

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₈₃

Fitting

C₈₃×#2960

C₂

socle

C₂²×C₈₃

C₂×C₂²⋊C₈

prime radical

C₂³×C₈₃

C₂×M₁₆

cosocle

C₂²×C₈×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2960

N	H ≅ G/N	type
79. C₈₃	#11497	semidirect

H ≅ G/N

type

79. C₈₃

#11497

semidirect

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

subgroup	index
#24909266	2
#24905405	2
C₈₃×#2960	2
#11497	83

subgroup

index

#24909266

#24905405

C₈₃×#2960

#11497

Subgroups — #35313754

Conjugacy classes of subgroups

Complements

Maximal subgroups