Group #35325311 — subgroups

Subgroups — #35325311

Order 21248.

158 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 57 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	#24909740	C₂	1	1	yes	maximal, characteristic
2	Q₃₂×Dic₈₃	C₂	1	1	yes	maximal, characteristic
3	#24905665	C₂	1	1	yes	maximal, characteristic
4	#24905662	C₂	1	1	yes	maximal, characteristic
5	Dic₈₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic
6	#24905644	C₂	1	1	yes	maximal, characteristic
7	C₈₃×#1416	C₂	1	1	yes	maximal, characteristic
8	#19128102	—	2	1	no	—
9	Q₁₆×Dic₈₃	—	2	1	no	—
10	C₂×C₈₃⋊C₃₂	C₂²	1	1	yes	characteristic
11	#19127528	—	2	1	no	—
12	#19127527	C₂²	1	1	yes	characteristic
13	#19127527	C₂²	1	1	yes	characteristic
14	#19127505	—	2	1	no	—
15	C₁₆×Dic₈₃	C₂²	1	1	yes	characteristic
16	C₂×Q₃₂×C₈₃	C₂²	1	1	yes	characteristic
17	C₂×C₃₂×C₈₃	C₂²	1	1	yes	characteristic
18	#366×C₈₃	C₂²	1	1	yes	characteristic
19	#14581783	—	4	1	no	—
20	Q₈×Dic₈₃	—	4	1	no	—
21	C₂×C₈₃⋊C₁₆	D₄	1	1	yes	characteristic
22	#14581726	—	2	1	no	—
23	#14581726	—	2	1	no	—
24	#14581723	—	4	1	no	—
25	#14581716	—	4	1	no	—
26	C₈×Dic₈₃	D₄	1	1	yes	characteristic
27	C₈₃⋊C₃₂	—	2	1	no	—
28	C₂×Q₁₆×C₈₃	—	2	1	no	—
29	Q₃₂×C₈₃	—	2	1	no	—
30	Q₃₂×C₈₃	Q₈	1	2	yes	—
31	C₂×C₁₆×C₈₃	C₂³	1	1	yes	characteristic
32	#107×C₈₃	—	2	1	no	—
33	C₃₂×C₈₃	—	2	1	no	—
34	C₂×C₈₃⋊C₈	D₈	1	1	yes	characteristic
35	C₄⋊Dic₈₃	—	4	1	no	—
36	C₄⋊Dic₈₃	—	4	1	no	—
37	C₄⋊Dic₈₃	—	8	1	no	—
38	C₄⋊Dic₈₃	—	8	1	no	—
39	Dic₈₃⋊C₄	—	8	1	no	—
40	C₄×Dic₈₃	—	8	1	no	—
41	C₄×Dic₈₃	D₈	1	1	yes	characteristic
42	C₈₃⋊C₁₆	—	2	1	no	—
43	C₂×Q₈×C₈₃	—	4	1	no	—
44	Q₁₆×C₈₃	—	2	2	no	—
45	Q₁₆×C₈₃	—	4	1	no	—
46	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
47	C₄⋊C₄×C₈₃	—	4	1	no	—
48	C₁₆×C₈₃	C₂×Q₈	1	1	yes	characteristic
49	C₁₆×C₈₃	#41	1	1	yes	characteristic
50	Q₈×C₈₃	—	4	2	no	—
51	Q₈×C₈₃	—	8	1	no	—
52	C₂×C₄×C₈₃	—	8	1	no	—
53	C₂×C₄×C₈₃	—	8	1	no	—
54	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
55	C₂×Dic₈₃	—	8	1	no	—
56	C₂×Dic₈₃	—	8	1	no	—
57	C₂×Dic₈₃	D₁₆	1	1	yes	characteristic
58	C₂×Dic₈₃	D₁₆	1	1	yes	characteristic
59	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
60	C₈×C₈₃	#137	1	1	yes	characteristic
61	C₈₃⋊C₈	—	2	1	no	—
62	C₂²×C₈₃	C₂×D₁₆	1	1	yes	characteristic
63	C₄×C₈₃	—	8	2	no	—
64	C₄×C₈₃	—	16	1	no	—
65	C₄×C₈₃	#474	1	1	yes	characteristic
66	C₄×C₈₃	#510	1	1	yes	characteristic
67	Dic₈₃	—	2	1	no	—
68	Dic₈₃	—	2	1	no	—
69	Dic₈₃	—	16	1	no	—
70	Dic₈₃	—	16	1	no	—
71	#13708	—	83	1	no	maximal, Sylow
72	C₂×C₈₃	#2224	1	1	yes	characteristic
73	C₂×C₈₃	#2262	1	1	yes	characteristic
74	C₂×C₈₃	#2264	1	1	yes	characteristic
75	#2253	—	83	1	no	—
76	C₄×Q₃₂	—	83	1	no	—
77	#1424	—	83	1	no	—
78	#1423	—	83	1	no	—
79	C₄⋊C₃₂	—	83	1	no	—
80	#1416	—	83	1	no	—
81	#1416	D₈₃	1	1	yes	characteristic
82	C₈₃	#13708	1	1	yes	characteristic, complemented, Sylow
83	C₂×Q₃₂	C₂×D₈₃	1	1	yes	characteristic
84	#499	—	166	1	no	—
85	C₄×Q₁₆	—	166	1	no	—
86	C₂×C₃₂	—	83	1	no	—
87	C₂×C₃₂	C₂×D₈₃	1	1	yes	characteristic
88	#367	—	166	1	no	—
89	#366	—	83	1	no	—
90	#366	—	83	1	no	—
91	#366	C₂×D₈₃	1	1	yes	characteristic
92	#358	—	166	1	no	—
93	C₄×C₁₆	—	83	1	no	—
94	C₂×Q₁₆	—	2	1	no	—
95	#125	—	332	1	no	—
96	C₄×Q₈	—	332	1	no	—
97	Q₃₂	—	2	1	no	—
98	Q₃₂	Dic₁₆₆	1	2	yes	—
99	C₂×C₁₆	—	83	1	no	—
100	C₂×C₁₆	C₂²×D₈₃	1	1	yes	characteristic
101	#107	—	2	1	no	—
102	#107	—	166	1	no	—
103	#107	—	166	1	no	—
104	#106	—	332	1	no	—
105	#103	—	332	1	no	—
106	C₄×C₈	—	83	1	no	—
107	C₃₂	—	2	1	no	—
108	C₃₂	—	166	1	no	—
109	C₂×Q₈	—	4	1	no	—
110	Q₁₆	—	2	2	no	—
111	Q₁₆	—	4	1	no	—
112	C₂×C₈	—	83	1	no	—
113	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
114	C₄⋊C₄	—	4	1	no	—
115	C₄⋊C₄	—	332	1	no	—
116	C₄⋊C₄	—	332	1	no	—
117	C₄⋊C₄	—	664	1	no	—
118	C₄⋊C₄	—	664	1	no	—
119	C₄⋊C₄	—	664	1	no	—
120	C₄²	—	83	1	no	—
121	C₄²	—	664	1	no	—
122	C₁₆	—	166	1	no	—
123	C₁₆	C₂×Dic₁₆₆	1	1	yes	characteristic
124	C₁₆	#13050589	1	1	yes	characteristic
125	Q₈	—	4	2	no	—
126	Q₈	—	8	1	no	—
127	C₂×C₄	—	8	1	no	—
128	C₂×C₄	—	8	1	no	—
129	C₂×C₄	—	83	1	no	—
130	C₂×C₄	—	83	1	no	—
131	C₂×C₄	—	664	1	no	—
132	C₂×C₄	—	664	1	no	—
133	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
134	C₈	—	166	1	no	—
135	C₈	#14581769	1	1	yes	characteristic
136	C₈	#14581821	1	1	yes	characteristic
137	C₂²	D₁₆×D₈₃	1	1	yes	characteristic
138	C₄	—	8	2	no	—
139	C₄	—	16	1	no	—
140	C₄	—	1328	1	no	—
141	C₄	—	1328	1	no	—
142	C₄	—	166	1	no	—
143	C₄	—	166	1	no	—
144	C₄	#19127979	1	1	yes	characteristic
145	C₄	#19128140	1	1	yes	characteristic
146	C₂	#24909598	1	1	yes	characteristic
147	C₂	#24909771	1	1	yes	characteristic
148	C₂	#24909782	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24909740

C₂

yes

maximal, characteristic

Q₃₂×Dic₈₃

C₂

yes

maximal, characteristic

#24905665

C₂

yes

maximal, characteristic

#24905662

C₂

yes

maximal, characteristic

Dic₈₃⋊C₃₂

C₂

yes

maximal, characteristic

#24905644

C₂

yes

maximal, characteristic

C₈₃×#1416

C₂

yes

maximal, characteristic

#19128102

—

Q₁₆×Dic₈₃

—

C₂×C₈₃⋊C₃₂

C₂²

yes

characteristic

#19127528

—

#19127527

C₂²

yes

characteristic

#19127527

C₂²

yes

characteristic

#19127505

—

C₁₆×Dic₈₃

C₂²

yes

characteristic

C₂×Q₃₂×C₈₃

C₂²

yes

characteristic

C₂×C₃₂×C₈₃

C₂²

yes

characteristic

#366×C₈₃

C₂²

yes

characteristic

#14581783

—

Q₈×Dic₈₃

—

C₂×C₈₃⋊C₁₆

D₄

yes

—

—

—

—

C₈×Dic₈₃

D₄

yes

characteristic

C₈₃⋊C₃₂

—

C₂×Q₁₆×C₈₃

—

Q₃₂×C₈₃

—

Q₃₂×C₈₃

Q₈

yes

—

C₂×C₁₆×C₈₃

C₂³

yes

characteristic

#107×C₈₃

—

C₃₂×C₈₃

—

C₂×C₈₃⋊C₈

D₈

yes

characteristic

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

Dic₈₃⋊C₄

—

C₄×Dic₈₃

—

C₄×Dic₈₃

D₈

yes

characteristic

C₈₃⋊C₁₆

—

C₂×Q₈×C₈₃

—

Q₁₆×C₈₃

—

Q₁₆×C₈₃

—

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₈₃

—

C₁₆×C₈₃

C₂×Q₈

yes

characteristic

C₁₆×C₈₃

#41

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₈

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

D₁₆

yes

characteristic

C₂×Dic₈₃

D₁₆

yes

characteristic

C₈×C₈₃

C₂²⋊Q₈

yes

characteristic

C₈×C₈₃

#137

yes

characteristic

C₈₃⋊C₈

—

C₂²×C₈₃

C₂×D₁₆

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₄×Q₃₂

—

#1424

—

#1423

—

C₄⋊C₃₂

—

#1416

—

#1416

D₈₃

yes

characteristic

C₈₃

#13708

yes

characteristic, complemented, Sylow

C₂×Q₃₂

C₂×D₈₃

yes

characteristic

#499

—

166

—

C₄×Q₁₆

—

166

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₈₃

yes

characteristic

#367

—

166

—

#366

—

#366

—

#366

C₂×D₈₃

yes

characteristic

#358

—

166

—

C₄×C₁₆

—

C₂×Q₁₆

—

#125

—

332

—

C₄×Q₈

—

332

—

Q₃₂

—

Q₃₂

Dic₁₆₆

yes

—

C₂×C₁₆

—

100

C₂×C₁₆

C₂²×D₈₃

yes

characteristic

101

#107

—

102

#107

—

166

—

103

#107

—

166

—

104

#106

—

332

—

105

#103

—

332

—

106

C₄×C₈

—

107

C₃₂

—

108

C₃₂

—

166

—

109

C₂×Q₈

—

110

Q₁₆

—

111

Q₁₆

—

112

C₂×C₈

—

113

C₂×C₈

D₄×D₈₃

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

332

—

116

C₄⋊C₄

—

332

—

117

C₄⋊C₄

—

664

—

118

C₄⋊C₄

—

664

—

119

C₄⋊C₄

—

664

—

120

C₄²

—

121

C₄²

—

664

—

122

C₁₆

—

166

—

123

C₁₆

C₂×Dic₁₆₆

yes

characteristic

124

C₁₆

#13050589

yes

characteristic

125

Q₈

—

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

664

—

132

C₂×C₄

—

664

—

133

C₂×C₄

D₈×D₈₃

yes

characteristic

134

C₈

—

166

—

135

C₈

#14581769

yes

characteristic

136

C₈

#14581821

yes

characteristic

137

C₂²

D₁₆×D₈₃

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

1328

—

141

C₄

—

1328

—

142

C₄

—

166

—

143

C₄

—

166

—

144

C₄

#19127979

yes

characteristic

145

C₄

#19128140

yes

characteristic

146

C₂

#24909598

yes

characteristic

147

C₂

#24909771

yes

characteristic

148

C₂

#24909782

yes

characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₈₃
commutator, cosocle	C₂×C₁₆×C₈₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₈₃
Fitting	C₈₃×#1416	C₂
socle, prime radical	C₂²×C₈₃	C₂×D₁₆
hypercenter	#1416	D₈₃
hypocenter	C₈₃	#13708
LCS	G ▷ C₂×C₁₆×C₈₃ ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

subgroup

G/H

center

C₂²

D₁₆×D₈₃

commutator, cosocle

C₂×C₁₆×C₈₃

C₂³

Frattini

C₂×C₁₆

C₂²×D₈₃

Fitting

C₈₃×#1416

C₂

socle, prime radical

C₂²×C₈₃

C₂×D₁₆

D₈₃

C₈₃

G ▷ C₂×C₁₆×C₈₃ ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

N	H ≅ G/N	type
82. C₈₃	#13708	semidirect

H ≅ G/N

type

82. C₈₃

#13708

semidirect

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

subgroup	index
#24909740	2
Q₃₂×Dic₈₃	2
#24905665	2
#24905662	2
Dic₈₃⋊C₃₂	2
#24905644	2
C₈₃×#1416	2
#13708	83

subgroup

index

Q₃₂×Dic₈₃

Dic₈₃⋊C₃₂

C₈₃×#1416

Subgroups — #35325311

Conjugacy classes of subgroups

Complements

Maximal subgroups