Group #24907749 — subgroups

Subgroups — #24907749

Order 10624.

260 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 89 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	#19127729	C₂	1	2	yes	maximal, complemented
2	#19127675	C₂	1	1	yes	maximal, characteristic
3	#19127668	C₂	1	1	yes	maximal, characteristic
4	#19127472	C₂	1	2	yes	maximal, complemented
5	#568×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13050574	C₂²	1	1	yes	characteristic
7	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
8	#14581751	C₂²	1	2	yes	—
9	Dic₈₃⋊C₈	—	2	4	no	—
10	#14581707	C₄	1	4	yes	—
11	C₂²⋊(C₈₃⋊C₈)	—	2	2	no	—
12	C₂×#41×C₈₃	C₂²	1	1	yes	characteristic
13	#131×C₈₃	—	2	4	no	—
14	C₂×M₁₆×C₈₃	—	2	2	no	—
15	C₂×M₁₆×C₈₃	C₂²	1	2	yes	—
16	#13050574	—	2	2	no	—
17	C₂×C₈₃⋊C₈	—	2	1	no	—
18	C₂×C₈₃⋊C₈	—	2	2	no	—
19	C₂×C₈₃⋊C₈	—	2	1	no	—
20	C₄⋊Dic₈₃	—	2	2	no	—
21	C₂²⋊Dic₈₃	C₂×C₄	1	4	yes	—
22	C₄×Dic₈₃	—	2	2	no	—
23	#41×C₈₃	—	2	2	no	—
24	#41×C₈₃	—	2	2	no	—
25	C₂×Q₈×C₈₃	Q₈	1	1	yes	characteristic
26	C₂×D₄×C₈₃	C₂×C₄	1	2	yes	—
27	C₂×D₄×C₈₃	Q₈	1	1	yes	characteristic
28	C₂²×C₄×C₈₃	D₄	1	2	yes	—
29	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
30	M₁₆×C₈₃	—	2	4	no	—
31	M₁₆×C₈₃	—	2	2	no	—
32	M₁₆×C₈₃	—	2	2	no	—
33	C₂×C₈×C₈₃	—	2	4	no	—
34	C₂×C₈×C₈₃	D₄	1	4	yes	—
35	C₂³×C₈₃	C₄⋊C₄	1	2	yes	—
36	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
37	Q₈×C₈₃	—	2	2	no	—
38	D₄×C₈₃	—	2	4	no	—
39	D₄×C₈₃	—	2	2	no	—
40	C₂×C₄×C₈₃	—	2	2	no	—
41	C₂×C₄×C₈₃	—	2	2	no	—
42	C₂×C₄×C₈₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₈₃	#41	1	2	yes	—
47	C₂×Dic₈₃	—	2	4	no	—
48	C₈×C₈₃	—	2	4	no	—
49	C₈₃⋊C₈	—	4	1	no	—
50	C₈₃⋊C₈	—	4	1	no	—
51	C₂²×C₈₃	—	2	2	no	—
52	C₂²×C₈₃	—	4	1	no	—
53	C₂²×C₈₃	—	4	2	no	—
54	C₂²×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₈₃	C₄×D₄	1	2	yes	—
56	C₄×C₈₃	—	2	2	no	—
57	C₄×C₈₃	#120	1	1	yes	characteristic
58	C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₈₃	#123	1	1	yes	characteristic
60	Dic₈₃	—	4	4	no	—
61	C₂×C₈₃	—	2	2	no	—
62	C₂×C₈₃	—	2	1	no	—
63	C₂×C₈₃	—	4	2	no	—
64	C₂×C₈₃	#385	1	1	yes	characteristic
65	#1868	—	83	1	no	maximal, Sylow
66	C₈₃	#1868	1	1	yes	characteristic, complemented, Sylow
67	#568	D₈₃	1	1	yes	characteristic
68	#424	—	83	2	no	—
69	#410	—	83	1	no	—
70	#408	—	83	1	no	—
71	#344	—	83	2	no	—
72	C₂×#41	C₂×D₈₃	1	1	yes	characteristic
73	#131	—	2	4	no	—
74	C₂×M₁₆	—	2	2	no	—
75	C₂×M₁₆	—	83	1	no	—
76	C₂×M₁₆	C₂×D₈₃	1	2	yes	—
77	C₂²×C₈	—	83	1	no	—
78	#117	—	83	2	no	—
79	C₄⋊C₈	—	166	4	no	—
80	#99	—	83	4	no	—
81	C₂²⋊C₈	—	166	2	no	—
82	#41	—	2	2	no	—
83	#41	—	2	2	no	—
84	C₂×Q₈	Dic₁₆₆	1	1	yes	characteristic
85	C₂×D₄	Dic₁₆₆	1	1	yes	characteristic
86	C₂×D₄	C₄×D₈₃	1	2	yes	—
87	C₂²×C₄	C₂²⋊D₈₃	1	2	yes	—
88	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
89	M₁₆	—	2	4	no	—
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	166	2	no	—
93	C₂×C₈	—	2	4	no	—
94	C₂×C₈	—	166	1	no	—
95	C₂×C₈	—	166	1	no	—
96	C₂×C₈	—	166	2	no	—
97	C₂×C₈	C₂²⋊D₈₃	1	4	yes	—
98	C₄⋊C₄	—	166	2	no	—
99	C₂²⋊C₄	—	83	4	no	—
100	C₄²	—	166	2	no	—
101	C₂³	Dic₈₃⋊C₄	1	2	yes	—
102	C₂³	C₂×C₄×D₈₃	1	1	yes	characteristic
103	Q₈	—	2	2	no	—
104	D₄	—	2	4	no	—
105	D₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	166	4	no	—
109	C₂×C₄	Dic₈₃⋊C₄	1	2	yes	—
110	C₂×C₄	C₂×C₂²⋊D₈₃	1	2	yes	—
111	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
112	C₂×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
113	C₂×C₄	#13050590	1	2	yes	—
114	C₈	—	2	4	no	—
115	C₈	—	332	1	no	—
116	C₈	—	332	1	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	4	1	no	—
119	C₂²	—	4	2	no	—
120	C₂²	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
121	C₂²	C₄×C₂²⋊D₈₃	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	332	4	no	—
124	C₄	#14581761	1	1	yes	characteristic
125	C₄	#14581771	1	2	yes	—
126	C₄	#14581775	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	—	2	1	no	—
129	C₂	—	4	2	no	—
130	C₂	#19127574	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127729

C₂

yes

maximal, complemented

#19127675

C₂

yes

maximal, characteristic

#19127668

C₂

yes

maximal, characteristic

#19127472

C₂

yes

maximal, complemented

#568×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

C₂²

yes

characteristic

C₂²×C₈₃⋊C₈

C₂²

yes

characteristic

#14581751

C₂²

yes

—

Dic₈₃⋊C₈

—

#14581707

C₄

yes

—

C₂²⋊(C₈₃⋊C₈)

—

C₂×#41×C₈₃

C₂²

yes

characteristic

#131×C₈₃

—

C₂×M₁₆×C₈₃

—

C₂×M₁₆×C₈₃

C₂²

yes

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₂²⋊Dic₈₃

C₂×C₄

yes

—

C₄×Dic₈₃

—

#41×C₈₃

—

#41×C₈₃

—

C₂×Q₈×C₈₃

Q₈

yes

characteristic

C₂×D₄×C₈₃

C₂×C₄

yes

—

C₂×D₄×C₈₃

Q₈

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

M₁₆×C₈₃

—

M₁₆×C₈₃

—

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

D₄

yes

—

C₂³×C₈₃

C₄⋊C₄

yes

—

C₂³×C₈₃

C₂²×C₄

yes

characteristic

Q₈×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₄⋊C₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₃

#41

yes

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₈₃

C₄×D₄

yes

—

C₄×C₈₃

—

C₄×C₈₃

#120

yes

characteristic

C₄×C₈₃

C₂²⋊Q₈

yes

—

C₄×C₈₃

#123

yes

characteristic

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#568

D₈₃

yes

—

—

—

—

C₂×#41

C₂×D₈₃

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₃

yes

—

C₂²×C₈

—

#117

—

C₄⋊C₈

—

166

—

#99

—

C₂²⋊C₈

—

166

—

#41

—

#41

—

C₂×Q₈

Dic₁₆₆

yes

characteristic

C₂×D₄

Dic₁₆₆

yes

characteristic

C₂×D₄

C₄×D₈₃

yes

—

C₂²×C₄

C₂²⋊D₈₃

yes

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

166

—

C₂×C₈

—

C₂×C₈

—

166

—

C₂×C₈

—

166

—

C₂×C₈

—

166

—

C₂×C₈

C₂²⋊D₈₃

yes

—

C₄⋊C₄

—

166

—

C₂²⋊C₄

—

100

C₄²

—

166

—

101

C₂³

Dic₈₃⋊C₄

yes

—

102

C₂³

C₂×C₄×D₈₃

yes

characteristic

103

Q₈

—

104

D₄

—

105

D₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

166

—

109

C₂×C₄

Dic₈₃⋊C₄

yes

—

110

C₂×C₄

C₂×C₂²⋊D₈₃

yes

—

111

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

112

C₂×C₄

C₂×Dic₁₆₆

yes

characteristic

113

C₂×C₄

#13050590

yes

—

114

C₈

—

115

C₈

—

332

—

116

C₈

—

332

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×Dic₈₃⋊C₄

yes

characteristic

121

C₂²

C₄×C₂²⋊D₈₃

yes

—

122

C₄

—

123

C₄

—

332

—

124

C₄

#14581761

yes

characteristic

125

C₄

#14581771

yes

—

126

C₄

#14581775

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#19127574

yes

characteristic

	subgroup	G/H
center	C₄	#14581761
commutator	C₂³×C₈₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#568×C₈₃	C₂
socle	C₂×C₈₃	#385
prime radical	C₂×#41×C₈₃	C₂²
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	#568	D₈₃
hypocenter	C₈₃	#1868
LCS	G ▷ C₂³×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #568

subgroup

G/H

center

C₄

#14581761

commutator

C₂³×C₈₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#568×C₈₃

C₂

socle

C₂×C₈₃

#385

prime radical

C₂×#41×C₈₃

C₂²

cosocle

C₂²×C₄×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂³×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #568

N	H ≅ G/N	type
1. #19127729	C₂	semidirect
4. #19127472	C₂	semidirect
66. C₈₃	#1868	semidirect

H ≅ G/N

type

1. #19127729

C₂

semidirect

4. #19127472

C₂

semidirect

66. C₈₃

#1868

semidirect

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

subgroup	index
#19127729	2
#19127675	2
#19127668	2
#19127472	2
#568×C₈₃	2
#1868	83

subgroup

index

#568×C₈₃

Subgroups — #24907749

Conjugacy classes of subgroups

Complements

Maximal subgroups