Group #24904660 — subgroups

Subgroups — C₈₃×#2975 #24904660

Order 10624.

452 conjugacy classes of subgroups; only the 268 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 268 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×C₄×D₄×C₈₃	C₂	1	1	yes	maximal, characteristic
2	#436×C₈₃	C₂	1	8	yes	maximal, complemented
3	#431×C₈₃	C₂	1	2	yes	maximal, complemented
4	#407×C₈₃	C₂	1	4	yes	maximal, complemented
5	C₂²×D₄×C₈₃	C₄	1	1	yes	characteristic
6	C₂³×C₄×C₈₃	C₂²	1	2	yes	—
7	C₂×M₁₆×C₈₃	C₂²	1	8	yes	complemented
8	C₄×D₄×C₈₃	C₄	1	4	yes	—
9	C₄×D₄×C₈₃	C₂²	1	4	yes	—
10	C₂×C₄⋊C₄×C₈₃	C₄	1	1	yes	characteristic
11	C₂×C₂²⋊C₄×C₈₃	C₄	1	2	yes	—
12	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₈₃	C₂²	1	4	yes	complemented
14	C₂²⋊C₈×C₈₃	C₂²	1	4	yes	—
15	C₂²⋊C₈×C₈₃	C₂²	1	8	yes	complemented
16	C₄×C₈×C₈₃	C₂²	1	4	yes	complemented
17	C₂⁴×C₈₃	C₂×C₄	1	2	yes	—
18	C₂×D₄×C₈₃	—	2	4	no	—
19	C₂×D₄×C₈₃	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₈₃	—	4	4	no	—
21	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₈₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₈₃	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₈₃	C₂³	1	4	yes	—
27	M₁₆×C₈₃	—	4	8	no	—
28	C₂×C₈×C₈₃	C₂³	1	8	yes	—
29	C₄⋊C₄×C₈₃	C₂×C₄	1	4	yes	—
30	C₂²⋊C₄×C₈₃	C₂×C₄	1	8	yes	—
31	C₄²×C₈₃	C₂×C₄	1	2	yes	—
32	C₄²×C₈₃	C₂³	1	2	yes	—
33	C₂³×C₈₃	—	2	4	no	—
34	C₂³×C₈₃	—	4	2	no	—
35	C₂³×C₈₃	—	4	2	no	—
36	C₂³×C₈₃	C₂²×C₄	1	4	yes	—
37	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
38	D₄×C₈₃	—	2	8	no	—
39	C₂×C₄×C₈₃	—	2	8	no	—
40	C₂×C₄×C₈₃	—	2	2	no	—
41	C₂×C₄×C₈₃	—	4	4	no	—
42	C₂×C₄×C₈₃	—	4	2	no	—
43	C₂×C₄×C₈₃	—	4	1	no	—
44	C₂×C₄×C₈₃	M₁₆	1	4	yes	complemented
45	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₈₃	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₈₃	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₈₃	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
50	C₈×C₈₃	—	2	8	no	—
51	C₂²×C₈₃	—	2	1	no	—
52	C₂²×C₈₃	—	2	1	no	—
53	C₂²×C₈₃	—	2	4	no	—
54	C₂²×C₈₃	—	2	4	no	—
55	C₂²×C₈₃	—	2	4	no	—
56	C₂²×C₈₃	—	4	8	no	—
57	C₂²×C₈₃	C₂×M₁₆	1	2	yes	—
58	C₂²×C₈₃	C₂³×C₄	1	1	yes	characteristic
59	C₄×C₈₃	—	2	1	no	—
60	C₄×C₈₃	—	2	4	no	—
61	C₄×C₈₃	—	2	2	no	—
62	C₄×C₈₃	C₂×M₁₆	1	4	yes	—
63	C₄×C₈₃	H(5,𝔽₂)	1	2	yes	—
64	C₂×C₈₃	—	2	2	no	—
65	C₂×C₈₃	—	4	4	no	—
66	C₂×C₈₃	#518	1	1	yes	characteristic
67	C₂×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
68	C₂×C₈₃	#568	1	1	yes	characteristic
69	#2975	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₈₃	#2975	1	1	yes	characteristic, direct product factor, Sylow
71	C₂×C₄×D₄	C₂×C₈₃	1	1	yes	characteristic
72	#436	C₂×C₈₃	1	8	yes	complemented
73	#431	C₂×C₈₃	1	2	yes	complemented
74	#407	C₂×C₈₃	1	4	yes	complemented
75	C₂²×D₄	C₄×C₈₃	1	1	yes	characteristic
76	C₂³×C₄	C₂²×C₈₃	1	2	yes	—
77	C₂×M₁₆	C₂²×C₈₃	1	8	yes	complemented
78	C₄×D₄	C₄×C₈₃	1	4	yes	—
79	C₄×D₄	C₂²×C₈₃	1	4	yes	—
80	C₂×C₄⋊C₄	C₄×C₈₃	1	1	yes	characteristic
81	C₂×C₂²⋊C₄	C₄×C₈₃	1	2	yes	—
82	C₂×C₄²	C₂²×C₈₃	1	1	yes	characteristic
83	C₄⋊C₈	C₂²×C₈₃	1	4	yes	complemented
84	C₂²⋊C₈	C₂²×C₈₃	1	4	yes	—
85	C₂²⋊C₈	C₂²×C₈₃	1	8	yes	complemented
86	C₄×C₈	C₂²×C₈₃	1	4	yes	complemented
87	C₂⁴	C₂×C₄×C₈₃	1	2	yes	—
88	C₂×D₄	—	2	4	no	—
89	C₂×D₄	C₂×C₄×C₈₃	1	4	yes	—
90	C₂²×C₄	—	4	4	no	—
91	C₂²×C₄	C₂×C₄×C₈₃	1	1	yes	characteristic
92	C₂²×C₄	C₂×C₄×C₈₃	1	2	yes	—
93	C₂²×C₄	C₂×C₄×C₈₃	1	4	yes	—
94	C₂²×C₄	C₂×C₄×C₈₃	1	1	yes	characteristic
95	C₂²×C₄	C₂³×C₈₃	1	1	yes	characteristic
96	C₂²×C₄	C₂³×C₈₃	1	4	yes	—
97	M₁₆	—	4	8	no	—
98	C₂×C₈	C₂³×C₈₃	1	8	yes	—
99	C₄⋊C₄	C₂×C₄×C₈₃	1	4	yes	—
100	C₂²⋊C₄	C₂×C₄×C₈₃	1	8	yes	—
101	C₄²	C₂×C₄×C₈₃	1	2	yes	—
102	C₄²	C₂³×C₈₃	1	2	yes	—
103	C₂³	—	2	4	no	—
104	C₂³	—	4	2	no	—
105	C₂³	—	4	2	no	—
106	C₂³	C₂²×C₄×C₈₃	1	4	yes	—
107	C₂³	C₂²×C₄×C₈₃	1	1	yes	characteristic
108	D₄	—	2	8	no	—
109	C₂×C₄	—	2	8	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	M₁₆×C₈₃	1	4	yes	complemented
115	C₂×C₄	C₂²×C₄×C₈₃	1	1	yes	characteristic
116	C₂×C₄	C₂²×C₄×C₈₃	1	2	yes	—
117	C₂×C₄	C₂²×C₄×C₈₃	1	4	yes	—
118	C₂×C₄	C₂²×C₄×C₈₃	1	2	yes	—
119	C₂×C₄	C₂⁴×C₈₃	1	1	yes	characteristic
120	C₈	—	2	8	no	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	4	no	—
124	C₂²	—	2	4	no	—
125	C₂²	—	2	4	no	—
126	C₂²	—	4	8	no	—
127	C₂²	C₂×M₁₆×C₈₃	1	2	yes	—
128	C₂²	C₂³×C₄×C₈₃	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2	4	no	—
131	C₄	—	2	2	no	—
132	C₄	C₂×M₁₆×C₈₃	1	4	yes	—
133	C₄	H(5,𝔽₂)×C₈₃	1	2	yes	—
134	C₂	—	2	2	no	—
135	C₂	—	4	4	no	—
136	C₂	#518×C₈₃	1	1	yes	characteristic
137	C₂	C₂²×M₁₆×C₈₃	1	1	yes	characteristic
138	C₂	#568×C₈₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×D₄×C₈₃

C₂

yes

maximal, characteristic

#436×C₈₃

C₂

yes

maximal, complemented

#431×C₈₃

C₂

yes

maximal, complemented

#407×C₈₃

C₂

yes

maximal, complemented

C₂²×D₄×C₈₃

C₄

yes

characteristic

C₂³×C₄×C₈₃

C₂²

yes

—

C₂×M₁₆×C₈₃

C₂²

yes

complemented

C₄×D₄×C₈₃

C₄

yes

—

C₄×D₄×C₈₃

C₂²

yes

—

C₂×C₄⋊C₄×C₈₃

C₄

yes

characteristic

C₂×C₂²⋊C₄×C₈₃

C₄

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₂²

yes

complemented

C₂²⋊C₈×C₈₃

C₂²

yes

—

C₂²⋊C₈×C₈₃

C₂²

yes

complemented

C₄×C₈×C₈₃

C₂²

yes

complemented

C₂⁴×C₈₃

C₂×C₄

yes

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₂²×C₄×C₈₃

C₂³

yes

—

M₁₆×C₈₃

—

C₂×C₈×C₈₃

C₂³

yes

—

C₄⋊C₄×C₈₃

C₂×C₄

yes

—

C₂²⋊C₄×C₈₃

C₂×C₄

yes

—

C₄²×C₈₃

C₂×C₄

yes

—

C₄²×C₈₃

C₂³

yes

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

—

C₂³×C₈₃

C₂²×C₄

yes

—

C₂³×C₈₃

C₂²×C₄

yes

characteristic

D₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

M₁₆

yes

complemented

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₄

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

—

C₂×C₄×C₈₃

C₂⁴

yes

characteristic

C₈×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×M₁₆

yes

—

C₂²×C₈₃

C₂³×C₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×M₁₆

yes

—

C₄×C₈₃

H(5,𝔽₂)

yes

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#518

yes

characteristic

C₂×C₈₃

C₂²×M₁₆

yes

C₂×C₈₃

yes

C₈₃

yes

maximal, characteristic, direct product factor, Sylow

C₈₃

#2975

yes

characteristic, direct product factor, Sylow

C₂×C₄×D₄

C₂×C₈₃

yes

characteristic

#436

C₂×C₈₃

yes

complemented

#431

C₂×C₈₃

yes

complemented

#407

C₂×C₈₃

yes

complemented

C₂²×D₄

C₄×C₈₃

yes

characteristic

C₂³×C₄

C₂²×C₈₃

yes

—

C₂×M₁₆

C₂²×C₈₃

yes

complemented

C₄×D₄

C₄×C₈₃

yes

—

C₄×D₄

C₂²×C₈₃

yes

—

C₂×C₄⋊C₄

C₄×C₈₃

yes

characteristic

C₂×C₂²⋊C₄

C₄×C₈₃

yes

—

C₂×C₄²

C₂²×C₈₃

yes

characteristic

C₄⋊C₈

C₂²×C₈₃

yes

complemented

C₂²⋊C₈

C₂²×C₈₃

yes

—

C₂²⋊C₈

C₂²×C₈₃

yes

complemented

C₄×C₈

C₂²×C₈₃

yes

complemented

C₂⁴

C₂×C₄×C₈₃

yes

—

C₂×D₄

—

C₂×D₄

C₂×C₄×C₈₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₈₃

yes

characteristic

C₂²×C₄

C₂×C₄×C₈₃

yes

—

C₂²×C₄

C₂×C₄×C₈₃

yes

—

C₂²×C₄

C₂×C₄×C₈₃

yes

characteristic

C₂²×C₄

C₂³×C₈₃

yes

characteristic

C₂²×C₄

C₂³×C₈₃

yes

—

M₁₆

—

C₂×C₈

C₂³×C₈₃

yes

—

C₄⋊C₄

C₂×C₄×C₈₃

yes

—

100

C₂²⋊C₄

C₂×C₄×C₈₃

yes

—

101

C₄²

C₂×C₄×C₈₃

yes

—

102

C₄²

C₂³×C₈₃

yes

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

106

C₂³

C₂²×C₄×C₈₃

yes

—

107

C₂³

C₂²×C₄×C₈₃

yes

characteristic

108

D₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

M₁₆×C₈₃

yes

complemented

115

C₂×C₄

C₂²×C₄×C₈₃

yes

characteristic

116

C₂×C₄

C₂²×C₄×C₈₃

yes

—

117

C₂×C₄

C₂²×C₄×C₈₃

yes

—

118

C₂×C₄

C₂²×C₄×C₈₃

yes

—

119

C₂×C₄

C₂⁴×C₈₃

yes

characteristic

120

C₈

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

C₂×M₁₆×C₈₃

yes

—

128

C₂²

C₂³×C₄×C₈₃

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

C₂×M₁₆×C₈₃

yes

—

133

C₄

H(5,𝔽₂)×C₈₃

yes

—

134

C₂

—

135

C₂

—

136

C₂

#518×C₈₃

yes

characteristic

137

C₂

C₂²×M₁₆×C₈₃

yes

characteristic

138

C₂

#568×C₈₃

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₈₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₈₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₈₃
socle	C₂²×C₈₃	C₂³×C₄
prime radical	C₂²×D₄×C₈₃	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈₃ ◁ G

subgroup

G/H

center

C₂×C₄×C₈₃

C₂⁴

commutator

C₂²

C₂³×C₄×C₈₃

Frattini, cosocle

C₂×C₄

C₂⁴×C₈₃

socle

C₂²×C₈₃

C₂³×C₄

prime radical

C₂²×D₄×C₈₃

C₄

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₈₃ ◁ G

N	H ≅ G/N	type
2. #436×C₈₃	C₂	semidirect
3. #431×C₈₃	C₂	semidirect
4. #407×C₈₃	C₂	semidirect
7. C₂×M₁₆×C₈₃	C₂²	semidirect
13. C₄⋊C₈×C₈₃	C₂²	semidirect
15. C₂²⋊C₈×C₈₃	C₂²	semidirect
16. C₄×C₈×C₈₃	C₂²	semidirect
44. C₂×C₄×C₈₃	M₁₆	semidirect
69. #2975	C₈₃	direct (N × H)
70. C₈₃	#2975	direct (N × H)
72. #436	C₂×C₈₃	semidirect
73. #431	C₂×C₈₃	semidirect
74. #407	C₂×C₈₃	semidirect
77. C₂×M₁₆	C₂²×C₈₃	semidirect
83. C₄⋊C₈	C₂²×C₈₃	semidirect
85. C₂²⋊C₈	C₂²×C₈₃	semidirect
86. C₄×C₈	C₂²×C₈₃	semidirect
114. C₂×C₄	M₁₆×C₈₃	semidirect

H ≅ G/N

type

2. #436×C₈₃

C₂

semidirect

3. #431×C₈₃

C₂

semidirect

4. #407×C₈₃

C₂

semidirect

7. C₂×M₁₆×C₈₃

C₂²

semidirect

13. C₄⋊C₈×C₈₃

C₂²

semidirect

15. C₂²⋊C₈×C₈₃

C₂²

semidirect

16. C₄×C₈×C₈₃

C₂²

semidirect

44. C₂×C₄×C₈₃

M₁₆

semidirect

69. #2975

C₈₃

direct (N × H)

70. C₈₃

#2975

direct (N × H)

72. #436

C₂×C₈₃

semidirect

73. #431

C₂×C₈₃

semidirect

74. #407

C₂×C₈₃

semidirect

77. C₂×M₁₆

C₂²×C₈₃

semidirect

83. C₄⋊C₈

C₂²×C₈₃

semidirect

85. C₂²⋊C₈

C₂²×C₈₃

semidirect

86. C₄×C₈

C₂²×C₈₃

semidirect

114. C₂×C₄

M₁₆×C₈₃

semidirect

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

subgroup	index
C₂×C₄×D₄×C₈₃	2
#436×C₈₃	2
#431×C₈₃	2
#407×C₈₃	2
#2975	83

subgroup

index

C₂×C₄×D₄×C₈₃

#436×C₈₃

#431×C₈₃

#407×C₈₃

#2975

Subgroups — C₈₃×#2975 #24904660

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C83×#2975 #24904660

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₃×#2975 #24904660