Group #35324940 — subgroups

Subgroups — #35324940

Order 21248.

170 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 75 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	M₃₂×Dic₈₃	C₂	1	1	yes	maximal, characteristic
2	#24905660	C₂	1	2	yes	maximal, complemented
3	#24905603	C₂	1	1	yes	maximal, characteristic
4	#24905598	C₂	1	2	yes	maximal, complemented
5	C₈₃×#1400	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581732	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₈₃	C₄	1	1	yes	characteristic
8	#19127535	C₂²	1	2	yes	—
9	#19127476	C₄	1	1	yes	characteristic
10	#19127476	C₂²	1	1	yes	characteristic
11	C₁₆×Dic₈₃	C₄	1	1	yes	characteristic
12	C₁₆×Dic₈₃	C₂²	1	1	yes	characteristic
13	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
14	M₆₄×C₈₃	C₂²	1	2	yes	—
15	C₂²×C₈₃⋊C₈	C₈	1	1	yes	characteristic
16	C₂×C₄×Dic₈₃	C₈	1	1	yes	characteristic
17	#14581732	—	2	2	no	—
18	C₂×C₈₃⋊C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈₃⋊C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₈×Dic₈₃	C₈	1	1	yes	characteristic
21	C₈×Dic₈₃	C₈	1	1	yes	characteristic
22	C₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
23	C₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
24	C₈₃⋊C₃₂	—	2	2	no	—
25	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
26	M₃₂×C₈₃	—	2	2	no	—
27	C₂×C₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₈₃	C₂³	1	1	yes	characteristic
29	C₃₂×C₈₃	—	2	2	no	—
30	C₂²×Dic₈₃	—	2	1	no	—
31	C₂×C₈₃⋊C₈	—	2	2	no	—
32	C₂×C₈₃⋊C₈	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈₃⋊C₈	C₂×C₈	1	1	yes	characteristic
34	C₄×Dic₈₃	—	2	1	no	—
35	C₄×Dic₈₃	C₂×C₈	1	1	yes	characteristic
36	C₄×Dic₈₃	C₂×C₈	1	1	yes	characteristic
37	C₈₃⋊C₁₆	—	2	1	no	—
38	C₈₃⋊C₁₆	M₁₆	1	2	yes	—
39	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
40	C₂×C₈×C₈₃	—	2	1	no	—
41	C₂×C₈×C₈₃	—	2	1	no	—
42	C₂×C₈×C₈₃	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
44	C₁₆×C₈₃	—	2	1	no	—
45	C₁₆×C₈₃	#41	1	2	yes	—
46	C₂³×C₈₃	M₃₂	1	1	yes	characteristic
47	C₂×C₄×C₈₃	—	2	1	no	—
48	C₂×C₄×C₈₃	—	2	1	no	—
49	C₂×C₄×C₈₃	M₃₂	1	1	yes	characteristic
50	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
51	C₂×Dic₈₃	—	2	1	no	—
52	C₂×Dic₈₃	—	2	1	no	—
53	C₂×Dic₈₃	—	4	2	no	—
54	C₈×C₈₃	—	2	1	no	—
55	C₈×C₈₃	#117	1	1	yes	characteristic
56	C₈×C₈₃	C₂×M₁₆	1	1	yes	characteristic
57	C₈₃⋊C₈	—	2	1	no	—
58	C₈₃⋊C₈	M₃₂	1	2	yes	—
59	C₂²×C₈₃	—	2	1	no	—
60	C₂²×C₈₃	—	4	1	no	—
61	C₂²×C₈₃	C₂×M₃₂	1	1	yes	characteristic
62	C₄×C₈₃	—	2	1	no	—
63	C₄×C₈₃	#431	1	1	yes	characteristic
64	C₄×C₈₃	C₂×M₃₂	1	1	yes	characteristic
65	Dic₈₃	—	2	2	no	—
66	Dic₈₃	—	4	1	no	—
67	#13642	—	83	1	no	maximal, Sylow
68	C₂×C₈₃	—	2	1	no	—
69	C₂×C₈₃	—	4	1	no	—
70	C₂×C₈₃	#2163	1	1	yes	characteristic
71	C₄×M₃₂	—	83	1	no	—
72	#1422	—	83	2	no	—
73	#1400	—	83	1	no	—
74	#1400	D₈₃	1	1	yes	characteristic
75	#1398	—	83	2	no	—
76	C₈₃	#13642	1	1	yes	characteristic, complemented, Sylow
77	C₂×M₃₂	—	83	1	no	—
78	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
79	C₂×C₄×C₈	—	83	1	no	—
80	M₆₄	—	83	2	no	—
81	M₆₄	C₂×D₈₃	1	2	yes	—
82	#346	—	83	1	no	—
83	#346	—	83	1	no	—
84	C₄×C₁₆	—	83	1	no	—
85	C₄×C₁₆	—	83	1	no	—
86	C₂²×C₈	—	83	1	no	—
87	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
88	C₂×C₄²	—	83	1	no	—
89	M₃₂	—	2	2	no	—
90	M₃₂	—	166	2	no	—
91	C₂×C₁₆	—	83	1	no	—
92	C₂×C₁₆	—	83	1	no	—
93	C₂×C₁₆	C₄×D₈₃	1	1	yes	characteristic
94	C₂×C₁₆	C₂²×D₈₃	1	1	yes	characteristic
95	C₄×C₈	—	83	1	no	—
96	C₄×C₈	—	83	1	no	—
97	C₄×C₈	—	83	1	no	—
98	C₄×C₈	—	83	1	no	—
99	C₃₂	—	2	2	no	—
100	C₃₂	—	166	2	no	—
101	C₂²×C₄	—	166	1	no	—
102	C₂²×C₄	C₈×D₈₃	1	1	yes	characteristic
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	83	1	no	—
106	C₂×C₈	—	83	1	no	—
107	C₂×C₈	—	166	2	no	—
108	C₂×C₈	C₈×D₈₃	1	1	yes	characteristic
109	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
110	C₄²	—	83	1	no	—
111	C₄²	—	83	1	no	—
112	C₄²	—	166	1	no	—
113	C₁₆	—	2	1	no	—
114	C₁₆	—	83	2	no	—
115	C₁₆	—	166	1	no	—
116	C₁₆	#13050589	1	2	yes	—
117	C₂³	#14581731	1	1	yes	characteristic
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	332	2	no	—
121	C₂×C₄	—	166	1	no	—
122	C₂×C₄	—	166	1	no	—
123	C₂×C₄	#14581731	1	1	yes	characteristic
124	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	83	2	no	—
127	C₈	—	166	1	no	—
128	C₈	#14581748	1	1	yes	characteristic
129	C₈	M₁₆×D₈₃	1	1	yes	characteristic
130	C₂²	—	2	1	no	—
131	C₂²	—	4	1	no	—
132	C₂²	C₂×#14581731	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	332	1	no	—
135	C₄	—	166	2	no	—
136	C₄	#19127751	1	1	yes	characteristic
137	C₄	M₃₂×D₈₃	1	1	yes	characteristic
138	C₂	—	2	1	no	—
139	C₂	—	4	1	no	—
140	C₂	#24909263	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₃₂×Dic₈₃

C₂

yes

maximal, characteristic

#24905660

C₂

yes

maximal, complemented

#24905603

C₂

yes

maximal, characteristic

#24905598

C₂

yes

maximal, complemented

C₈₃×#1400

C₂

yes

maximal, characteristic

C₂×#14581732

C₄

yes

characteristic

C₂×C₈×Dic₈₃

C₄

yes

characteristic

#19127535

C₂²

yes

—

#19127476

C₄

yes

characteristic

#19127476

C₂²

yes

characteristic

C₁₆×Dic₈₃

C₄

yes

characteristic

C₁₆×Dic₈₃

C₂²

yes

characteristic

C₂×M₃₂×C₈₃

C₂²

yes

characteristic

M₆₄×C₈₃

C₂²

yes

—

C₂²×C₈₃⋊C₈

C₈

yes

characteristic

C₂×C₄×Dic₈₃

C₈

yes

characteristic

#14581732

—

C₂×C₈₃⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₈₃⋊C₁₆

C₂×C₄

yes

characteristic

C₈×Dic₈₃

C₈

yes

characteristic

C₈×Dic₈₃

C₈

yes

characteristic

C₈×Dic₈₃

C₂×C₄

yes

characteristic

C₈×Dic₈₃

C₂×C₄

yes

characteristic

C₈₃⋊C₃₂

—

C₂²×C₈×C₈₃

C₂×C₄

yes

characteristic

M₃₂×C₈₃

—

C₂×C₁₆×C₈₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₃

C₂³

yes

characteristic

C₃₂×C₈₃

—

C₂²×Dic₈₃

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

C₂×C₈

yes

characteristic

C₂×C₈₃⋊C₈

C₂×C₈

yes

characteristic

C₄×Dic₈₃

—

C₄×Dic₈₃

C₂×C₈

yes

characteristic

C₄×Dic₈₃

C₂×C₈

yes

characteristic

C₈₃⋊C₁₆

—

C₈₃⋊C₁₆

M₁₆

yes

—

C₂²×C₄×C₈₃

C₂×C₈

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×C₈

yes

characteristic

C₂×C₈×C₈₃

C₂²×C₄

yes

characteristic

C₁₆×C₈₃

—

C₁₆×C₈₃

#41

yes

—

C₂³×C₈₃

M₃₂

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

M₃₂

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₈

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈×C₈₃

#117

yes

characteristic

C₈×C₈₃

C₂×M₁₆

yes

characteristic

C₈₃⋊C₈

—

C₈₃⋊C₈

M₃₂

yes

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×M₃₂

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

#431

yes

characteristic

C₄×C₈₃

C₂×M₃₂

yes

characteristic

Dic₈₃

—

Dic₈₃

—

#13642

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#2163

yes

characteristic

C₄×M₃₂

—

#1422

—

#1400

—

#1400

D₈₃

yes

characteristic

#1398

—

C₈₃

#13642

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

C₂×D₈₃

yes

characteristic

C₂×C₄×C₈

—

M₆₄

—

M₆₄

C₂×D₈₃

yes

—

#346

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

C₂²×C₈

—

C₂²×C₈

C₄×D₈₃

yes

characteristic

C₂×C₄²

—

M₃₂

—

M₃₂

—

166

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₈₃

yes

characteristic

C₂×C₁₆

C₂²×D₈₃

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₃₂

—

100

C₃₂

—

166

—

101

C₂²×C₄

—

166

—

102

C₂²×C₄

C₈×D₈₃

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

166

—

108

C₂×C₈

C₈×D₈₃

yes

characteristic

109

C₂×C₈

C₂×C₄×D₈₃

yes

characteristic

110

C₄²

—

111

C₄²

—

112

C₄²

—

166

—

113

C₁₆

—

114

C₁₆

—

115

C₁₆

—

166

—

116

C₁₆

#13050589

yes

—

117

C₂³

#14581731

yes

characteristic

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

332

—

121

C₂×C₄

—

166

—

122

C₂×C₄

—

166

—

123

C₂×C₄

#14581731

yes

characteristic

124

C₂×C₄

C₂×C₈×D₈₃

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

166

—

128

C₈

#14581748

yes

characteristic

129

C₈

M₁₆×D₈₃

yes

characteristic

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×#14581731

yes

characteristic

133

C₄

—

134

C₄

—

332

—

135

C₄

—

166

—

136

C₄

#19127751

yes

characteristic

137

C₄

M₃₂×D₈₃

yes

characteristic

138

C₂

—

139

C₂

—

140

C₂

#24909263

yes

characteristic

	subgroup	G/H
center	C₈	#14581748
commutator	C₂×C₄×C₈₃	C₂²×C₈
Frattini	C₂×C₁₆	C₂²×D₈₃
Fitting	C₈₃×#1400	C₂
socle	C₂×C₈₃	#2163
prime radical	C₂³×C₈₃	M₃₂
cosocle	C₂×C₁₆×C₈₃	C₂³
hypercenter	#1400	D₈₃
hypocenter	C₈₃	#13642
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #1400

subgroup

G/H

center

C₈

#14581748

commutator

C₂×C₄×C₈₃

C₂²×C₈

Frattini

C₂×C₁₆

C₂²×D₈₃

Fitting

C₈₃×#1400

C₂

socle

C₂×C₈₃

#2163

prime radical

C₂³×C₈₃

M₃₂

cosocle

C₂×C₁₆×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #1400

N	H ≅ G/N	type
2. #24905660	C₂	semidirect
4. #24905598	C₂	semidirect
76. C₈₃	#13642	semidirect

H ≅ G/N

type

2. #24905660

C₂

semidirect

4. #24905598

C₂

semidirect

76. C₈₃

#13642

semidirect

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

subgroup	index
M₃₂×Dic₈₃	2
#24905660	2
#24905603	2
#24905598	2
C₈₃×#1400	2
#13642	83

subgroup

index

M₃₂×Dic₈₃

C₈₃×#1400

Subgroups — #35324940

Conjugacy classes of subgroups

Complements

Maximal subgroups