Group #24908720 — subgroups

Subgroups — #24908720

Order 10624.

296 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₂²×D₆₆₄	C₂	1	1	yes	maximal, characteristic
2	#19127727	C₂	1	1	yes	maximal, characteristic, complemented
3	#19127707	C₂	1	2	yes	maximal, complemented
4	C₂×#14581708	C₂	1	2	yes	maximal, complemented
5	#341×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₃₂	C₂²	1	2	yes	—
7	C₂×D₆₆₄	—	2	4	no	—
8	C₂×D₆₆₄	—	2	2	no	—
9	C₂×#13050574	C₂²	1	1	yes	characteristic
10	#14581751	C₂²	1	1	yes	characteristic
11	Dic₈₃⋊C₈	—	2	2	no	—
12	#14581714	—	2	4	no	—
13	#14581708	—	2	4	no	—
14	C₂×M₁₆×C₈₃	C₂²	1	2	yes	—
15	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
16	C₂³×D₈₃	—	2	2	no	—
17	C₂×D₃₃₂	—	2	2	no	—
18	C₂×D₃₃₂	—	2	2	no	—
19	C₂×D₃₃₂	—	2	2	no	—
20	C₂×D₃₃₂	—	2	2	no	—
21	C₂×D₃₃₂	—	2	2	no	—
22	C₂×D₃₃₂	D₄	1	4	yes	—
23	D₆₆₄	—	2	8	no	—
24	#13050574	—	2	2	no	—
25	C₂×C₈₃⋊C₈	—	2	1	no	—
26	C₄⋊Dic₈₃	D₄	1	2	yes	—
27	C₂²⋊Dic₈₃	—	2	1	no	—
28	C₄×Dic₈₃	—	2	1	no	—
29	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
30	M₁₆×C₈₃	—	2	2	no	—
31	M₁₆×C₈₃	—	2	2	no	—
32	C₂×C₈×C₈₃	—	2	2	no	—
33	C₂×C₈×C₈₃	—	2	2	no	—
34	C₂×C₈×C₈₃	D₄	1	2	yes	—
35	C₂³×C₈₃	#41	1	1	yes	characteristic
36	C₂²×D₈₃	—	2	4	no	—
37	C₂²×D₈₃	—	4	2	no	—
38	C₂²×D₈₃	—	4	2	no	—
39	C₂²×D₈₃	—	8	4	no	—
40	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₈₃	#41	1	2	yes	—
44	C₂×Dic₈₃	—	2	2	no	—
45	D₃₃₂	—	2	4	no	—
46	D₃₃₂	—	2	4	no	—
47	D₃₃₂	—	4	2	no	—
48	D₃₃₂	—	4	2	no	—
49	C₈×C₈₃	—	2	2	no	—
50	C₈×C₈₃	—	4	2	no	—
51	C₈₃⋊C₈	—	4	1	no	—
52	C₂²×C₈₃	—	2	2	no	—
53	C₂²×C₈₃	#121	1	1	yes	characteristic
54	C₂²×C₈₃	#123	1	2	yes	—
55	C₂×D₈₃	—	4	4	no	—
56	C₂×D₈₃	—	8	4	no	—
57	C₂×D₈₃	—	8	4	no	—
58	C₂×D₈₃	—	8	4	no	—
59	C₄×C₈₃	#120	1	1	yes	characteristic
60	C₄×C₈₃	#121	1	2	yes	—
61	C₄×C₈₃	#124	1	1	yes	characteristic
62	Dic₈₃	—	4	2	no	—
63	C₂×C₈₃	—	2	2	no	—
64	C₂×C₈₃	#394	1	1	yes	characteristic
65	C₂×C₈₃	#472	1	2	yes	—
66	D₈₃	—	8	4	no	—
67	#2045	—	83	1	no	maximal, Sylow
68	C₈₃	#2045	1	1	yes	characteristic, complemented, Sylow
69	C₂²×D₈	—	83	1	no	—
70	#424	—	83	1	no	—
71	#418	—	83	2	no	—
72	C₂×#100	—	83	2	no	—
73	#341	D₈₃	1	1	yes	characteristic, complemented
74	C₂²×D₄	—	83	2	no	—
75	C₂×D₈	—	166	4	no	—
76	C₂×D₈	—	166	2	no	—
77	C₂×M₁₆	—	83	1	no	—
78	C₂×M₁₆	C₂×D₈₃	1	2	yes	—
79	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
80	#117	—	83	1	no	—
81	C₄⋊C₈	—	166	2	no	—
82	#102	—	166	4	no	—
83	#100	—	166	4	no	—
84	C₂⁴	—	166	2	no	—
85	C₂×D₄	—	83	4	no	—
86	C₂×D₄	—	166	2	no	—
87	C₂×D₄	—	166	2	no	—
88	C₂×D₄	—	166	2	no	—
89	C₂×D₄	—	166	2	no	—
90	C₂×D₄	—	166	2	no	—
91	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
92	D₈	—	166	8	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	166	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	166	1	no	—
99	C₂×C₈	D₃₃₂	1	2	yes	—
100	C₄⋊C₄	—	83	2	no	—
101	C₂²⋊C₄	—	166	1	no	—
102	C₄²	—	166	1	no	—
103	C₂³	—	332	2	no	—
104	C₂³	—	332	2	no	—
105	C₂³	—	664	4	no	—
106	C₂³	—	166	4	no	—
107	C₂³	#13050591	1	1	yes	characteristic
108	D₄	—	332	2	no	—
109	D₄	—	332	2	no	—
110	D₄	—	166	4	no	—
111	D₄	—	166	4	no	—
112	C₂×C₄	—	166	2	no	—
113	C₂×C₄	D₄×D₈₃	1	2	yes	—
114	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
115	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
116	C₂×C₄	#13050590	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	332	1	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	332	4	no	—
122	C₂²	—	664	4	no	—
123	C₂²	—	664	4	no	—
124	C₂²	—	664	4	no	—
125	C₂²	#14581767	1	1	yes	characteristic
126	C₂²	#14581777	1	2	yes	—
127	C₄	—	332	2	no	—
128	C₄	#14581762	1	1	yes	characteristic
129	C₄	#14581765	1	2	yes	—
130	C₄	#14581780	1	1	yes	characteristic
131	C₂	—	2	2	no	—
132	C₂	—	664	4	no	—
133	C₂	#19127619	1	1	yes	characteristic
134	C₂	#19127972	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₆₆₄

C₂

yes

maximal, characteristic

#19127727

C₂

yes

maximal, characteristic, complemented

#19127707

C₂

yes

maximal, complemented

C₂×#14581708

C₂

yes

maximal, complemented

#341×C₈₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₃₂

C₂²

yes

—

C₂×D₆₆₄

—

C₂×D₆₆₄

—

C₂×#13050574

C₂²

yes

characteristic

#14581751

C₂²

yes

characteristic

Dic₈₃⋊C₈

—

#14581714

—

#14581708

—

C₂×M₁₆×C₈₃

C₂²

yes

—

C₂²×C₈×C₈₃

C₂²

yes

characteristic

C₂³×D₈₃

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

D₄

yes

—

D₆₆₄

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

D₄

yes

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

M₁₆×C₈₃

—

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

D₄

yes

—

C₂³×C₈₃

#41

yes

characteristic

C₂²×D₈₃

—

C₂²×D₈₃

—

C₂²×D₈₃

—

C₂²×D₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

—

C₂×Dic₈₃

—

D₃₃₂

—

D₃₃₂

—

D₃₃₂

—

D₃₃₂

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#121

yes

characteristic

C₂²×C₈₃

#123

yes

—

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

#120

yes

characteristic

C₄×C₈₃

#121

yes

—

C₄×C₈₃

#124

yes

characteristic

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#394

yes

characteristic

C₂×C₈₃

#472

yes

—

D₈₃

—

#2045

—

maximal, Sylow

C₈₃

#2045

yes

characteristic, complemented, Sylow

C₂²×D₈

—

—

—

—

D₈₃

yes

characteristic, complemented

C₂²×D₄

—

C₂×D₈

—

166

—

C₂×D₈

—

166

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₃

yes

—

C₂²×C₈

C₂×D₈₃

yes

characteristic

#117

—

C₄⋊C₈

—

166

—

#102

—

166

—

#100

—

166

—

C₂⁴

—

166

—

C₂×D₄

—

C₂×D₄

—

166

—

C₂×D₄

—

166

—

C₂×D₄

—

166

—

C₂×D₄

—

166

—

C₂×D₄

—

166

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

D₈

—

166

—

M₁₆

—

M₁₆

—

M₁₆

—

166

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

166

—

C₂×C₈

D₃₃₂

yes

—

100

C₄⋊C₄

—

101

C₂²⋊C₄

—

166

—

102

C₄²

—

166

—

103

C₂³

—

332

—

104

C₂³

—

332

—

105

C₂³

—

664

—

106

C₂³

—

166

—

107

C₂³

#13050591

yes

characteristic

108

D₄

—

332

—

109

D₄

—

332

—

110

D₄

—

166

—

111

D₄

—

166

—

112

C₂×C₄

—

166

—

113

C₂×C₄

D₄×D₈₃

yes

—

114

C₂×C₄

D₄×D₈₃

yes

characteristic

115

C₂×C₄

C₂×D₃₃₂

yes

characteristic

116

C₂×C₄

#13050590

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

332

—

120

C₂²

—

121

C₂²

—

332

—

122

C₂²

—

664

—

123

C₂²

—

664

—

124

C₂²

—

664

—

125

C₂²

#14581767

yes

characteristic

126

C₂²

#14581777

yes

—

127

C₄

—

332

—

128

C₄

#14581762

yes

characteristic

129

C₄

#14581765

yes

—

130

C₄

#14581780

yes

characteristic

131

C₂

—

132

C₂

—

664

—

133

C₂

#19127619

yes

characteristic

134

C₂

#19127972

yes

—

	subgroup	G/H
center	C₂²	#14581767
commutator, cosocle	C₂²×C₄×C₈₃	C₂³
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#341×C₈₃	C₂
socle	C₂²×C₈₃	#121
prime radical	C₂²×D₆₆₄	C₂
hypercenter	#341	D₈₃
hypocenter	C₈₃	#2045
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#14581767

commutator, cosocle

C₂²×C₄×C₈₃

C₂³

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#341×C₈₃

C₂

socle

C₂²×C₈₃

#121

prime radical

C₂²×D₆₆₄

C₂

D₈₃

C₈₃

G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
2. #19127727	C₂	semidirect
3. #19127707	C₂	semidirect
4. C₂×#14581708	C₂	semidirect
5. #341×C₈₃	C₂	semidirect
68. C₈₃	#2045	semidirect
73. #341	D₈₃	semidirect

H ≅ G/N

type

2. #19127727

C₂

semidirect

3. #19127707

C₂

semidirect

4. C₂×#14581708

C₂

semidirect

5. #341×C₈₃

C₂

semidirect

68. C₈₃

#2045

semidirect

73. #341

D₈₃

semidirect

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

subgroup	index
C₂²×D₆₆₄	2
#19127727	2
#19127707	2
C₂×#14581708	2
#341×C₈₃	2
#2045	83

subgroup

index

C₂²×D₆₆₄

#341×C₈₃

Subgroups — #24908720

Conjugacy classes of subgroups

Complements

Maximal subgroups