Group #13102939 — subgroups

Subgroups — #13102939

Order 1600.

512 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 141 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	C₅²×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×#87810	C₂	1	2	yes	maximal, complemented
3	C₄×#6045	C₂	1	1	yes	maximal, characteristic, complemented
4	#11716699	C₂	1	2	yes	maximal, complemented
5	C₄⋊#6029	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅²×D₄	—	2	1	no	—
7	C₂×C₅²×D₄	—	2	1	no	—
8	C₂×C₅²×D₄	C₂²	1	2	yes	—
9	C₂×#6045	C₂²	1	1	yes	characteristic
10	C₄²×C₅²	C₂²	1	1	yes	characteristic
11	#87810	—	2	4	no	—
12	C₄⋊#1039	C₂²	1	1	yes	characteristic
13	#87806	—	2	1	no	—
14	C₄×#1039	—	2	1	no	—
15	C₂×#6029	C₂²	1	2	yes	—
16	#65364	—	5	6	no	maximal
17	C₂³×C₅²	—	2	2	no	—
18	C₅²×D₄	—	2	2	no	—
19	C₅²×D₄	—	2	2	no	—
20	C₅²×D₄	—	4	2	no	—
21	C₅²×D₄	—	4	2	no	—
22	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
23	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
24	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
25	C₂×#1039	—	2	1	no	—
26	C₂×#1039	—	2	1	no	—
27	#6045	—	2	1	no	—
28	#6045	D₄	1	2	yes	complemented
29	#6029	—	2	2	no	—
30	C₅×#127	D₅	1	6	yes	—
31	C₂×#687	—	5	12	no	—
32	C₄×Dic₁₀	—	5	6	no	—
33	#3964	—	5	12	no	—
34	C₄⋊#168	—	5	6	no	—
35	C₂²×C₅²	—	4	2	no	—
36	C₂²×C₅²	—	4	2	no	—
37	C₂²×C₅²	—	4	2	no	—
38	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
39	C₄×C₅²	—	2	1	no	—
40	C₄×C₅²	SD₁₆	1	2	yes	complemented
41	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
42	C₄×C₅²	#41	1	1	yes	characteristic
43	#1039	—	2	2	no	—
44	#1039	—	4	1	no	—
45	C₂×C₅×D₄	—	2	6	no	—
46	C₂×C₅×D₄	—	2	6	no	—
47	C₂×C₅×D₄	C₂×D₅	1	12	yes	—
48	C₂×Dic₁₀	—	5	6	no	—
49	C₄²×C₅	C₂×D₅	1	6	yes	—
50	#687	—	10	24	no	—
51	C₄⋊Dic₅	—	5	6	no	—
52	#683	—	10	6	no	—
53	C₄×Dic₅	—	10	6	no	—
54	C₂×#168	—	5	12	no	—
55	#460	—	25	1	no	Sylow
56	C₂×C₅²	—	8	2	no	—
57	C₂×C₅²	#121	1	1	yes	characteristic
58	C₂×C₅²	C₂×SD₁₆	1	1	yes	characteristic
59	C₂×C₅²	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂³×C₅	—	2	12	no	—
61	C₅×D₄	—	2	12	no	—
62	C₅×D₄	—	2	12	no	—
63	C₅×D₄	—	4	12	no	—
64	C₅×D₄	—	4	12	no	—
65	C₂×C₄×C₅	C₂²⋊D₅	1	6	yes	—
66	C₂×C₄×C₅	C₂²⋊D₅	1	6	yes	—
67	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
68	C₂×Dic₅	—	10	6	no	—
69	C₂×Dic₅	—	10	6	no	—
70	Dic₁₀	—	5	12	no	—
71	Dic₁₀	—	10	6	no	—
72	#168	—	10	12	no	—
73	C₂×SD₁₆	—	25	2	no	—
74	#127	C₅⋊D₅	1	1	yes	characteristic
75	C₄×Q₈	—	25	1	no	—
76	C₄⋊C₈	—	25	1	no	—
77	#102	—	25	2	no	—
78	C₅²	#460	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₅	—	4	12	no	—
80	C₂²×C₅	—	4	12	no	—
81	C₂²×C₅	—	4	12	no	—
82	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
83	C₄×C₅	—	2	6	no	—
84	C₄×C₅	#687	1	12	yes	complemented
85	C₄×C₅	D₄×D₅	1	6	yes	—
86	C₄×C₅	#711	1	6	yes	—
87	Dic₅	—	10	12	no	—
88	Dic₅	—	20	6	no	—
89	C₂×Q₈	—	25	1	no	—
90	C₂×D₄	—	2	1	no	—
91	C₂×D₄	—	2	1	no	—
92	C₂×D₄	C₂×C₅⋊D₅	1	2	yes	—
93	SD₁₆	—	50	4	no	—
94	C₂×C₈	—	25	2	no	—
95	C₄⋊C₄	—	25	1	no	—
96	C₄⋊C₄	—	50	1	no	—
97	C₄²	—	50	1	no	—
98	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
99	C₂×C₅	—	8	12	no	—
100	C₂×C₅	#4078	1	6	yes	—
101	C₂×C₅	C₂×#687	1	6	yes	—
102	C₂×C₅	#4084	1	6	yes	—
103	C₂³	—	2	2	no	—
104	Q₈	—	25	2	no	—
105	Q₈	—	50	1	no	—
106	D₄	—	2	2	no	—
107	D₄	—	2	2	no	—
108	D₄	—	4	2	no	—
109	D₄	—	4	2	no	—
110	C₂×C₄	—	50	1	no	—
111	C₂×C₄	—	50	1	no	—
112	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
113	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
114	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
115	C₈	—	50	2	no	—
116	C₅	#65364	1	6	yes	complemented
117	C₂²	—	4	2	no	—
118	C₂²	—	4	2	no	—
119	C₂²	—	4	2	no	—
120	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	50	2	no	—
123	C₄	—	100	1	no	—
124	C₄	#87810	1	2	yes	complemented
125	C₄	D₄×C₅⋊D₅	1	1	yes	characteristic
126	C₄	#87902	1	1	yes	characteristic
127	C₂	—	8	2	no	—
128	C₂	#11717252	1	1	yes	characteristic
129	C₂	C₂×#87810	1	1	yes	characteristic
130	C₂	#11717258	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×#127

C₂

yes

maximal, characteristic

C₂×#87810

C₂

yes

maximal, complemented

C₄×#6045

C₂

yes

maximal, characteristic, complemented

#11716699

C₂

yes

maximal, complemented

C₄⋊#6029

C₂

yes

maximal, characteristic, complemented

C₂×C₅²×D₄

—

C₂×C₅²×D₄

—

C₂×C₅²×D₄

C₂²

yes

—

C₂×#6045

C₂²

yes

characteristic

C₄²×C₅²

C₂²

yes

characteristic

#87810

—

C₄⋊#1039

C₂²

yes

—

—

C₂²

yes

—

#65364

—

maximal

C₂³×C₅²

—

C₅²×D₄

—

C₅²×D₄

—

C₅²×D₄

—

C₅²×D₄

—

C₂×C₄×C₅²

D₄

yes

characteristic

C₂×C₄×C₅²

D₄

yes

characteristic

C₂×C₄×C₅²

C₂³

yes

—

—

—

D₄

yes

complemented

#6029

—

C₅×#127

D₅

yes

—

C₂×#687

—

C₄×Dic₁₀

—

#3964

—

C₄⋊#168

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

—

C₄×C₅²

SD₁₆

yes

complemented

C₄×C₅²

C₂×D₄

yes

C₄×C₅²

yes

—

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

C₂×D₅

yes

—

C₂×Dic₁₀

—

C₄²×C₅

C₂×D₅

yes

—

#687

—

C₄⋊Dic₅

—

#683

—

C₄×Dic₅

—

C₂×#168

—

#460

—

Sylow

C₂×C₅²

—

C₂×C₅²

#121

yes

characteristic

C₂×C₅²

C₂×SD₁₆

yes

characteristic

C₂×C₅²

AGL(1,ℤ/8)

yes

characteristic

C₂³×C₅

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

#168

—

C₂×SD₁₆

—

#127

C₅⋊D₅

yes

characteristic

C₄×Q₈

—

C₄⋊C₈

—

#102

—

C₅²

#460

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂×C₂²⋊D₅

yes

—

C₄×C₅

—

C₄×C₅

#687

yes

complemented

C₄×C₅

D₄×D₅

yes

—

C₄×C₅

#711

yes

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×C₅⋊D₅

yes

—

SD₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂×C₅⋊D₅

yes

characteristic

C₂×C₅

—

100

C₂×C₅

#4078

yes

—

101

C₂×C₅

C₂×#687

yes

—

102

C₂×C₅

#4084

yes

—

103

C₂³

—

104

Q₈

—

105

Q₈

—

106

D₄

—

107

D₄

—

108

D₄

—

109

D₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

characteristic

113

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

characteristic

114

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

115

C₈

—

116

C₅

#65364

yes

complemented

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

100

—

124

C₄

#87810

yes

complemented

125

C₄

D₄×C₅⋊D₅

yes

characteristic

126

C₄

#87902

yes

characteristic

127

C₂

—

128

C₂

#11717252

yes

characteristic

129

C₂

C₂×#87810

yes

characteristic

130

C₂

#11717258

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊(C₅⋊D₅)
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting, prime radical	C₅²×#127	C₂
socle	C₂²×C₅²	C₂×D₄
hypercenter	#127	C₅⋊D₅
hypocenter	C₅²	#460
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

subgroup

G/H

center

C₂²

C₂×C₂²⋊(C₅⋊D₅)

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂²×C₅⋊D₅

Fitting, prime radical

C₅²×#127

C₂

socle

C₂²×C₅²

C₂×D₄

C₅⋊D₅

C₅²

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #127

N	H ≅ G/N	type
2. C₂×#87810	C₂	semidirect
3. C₄×#6045	C₂	semidirect
4. #11716699	C₂	semidirect
5. C₄⋊#6029	C₂	semidirect
28. #6045	D₄	semidirect
40. C₄×C₅²	SD₁₆	semidirect
78. C₅²	#460	semidirect
84. C₄×C₅	#687	semidirect
116. C₅	#65364	semidirect
124. C₄	#87810	semidirect

H ≅ G/N

type

2. C₂×#87810

C₂

semidirect

3. C₄×#6045

C₂

semidirect

4. #11716699

C₂

semidirect

5. C₄⋊#6029

C₂

semidirect

28. #6045

D₄

semidirect

40. C₄×C₅²

SD₁₆

semidirect

78. C₅²

#460

semidirect

84. C₄×C₅

#687

semidirect

116. C₅

#65364

semidirect

124. C₄

#87810

semidirect

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

subgroup	index
C₅²×#127	2
C₂×#87810	2
C₄×#6045	2
#11716699	2
C₄⋊#6029	2
#65364	5

subgroup

index

C₅²×#127

Subgroups — #13102939

Conjugacy classes of subgroups

Complements

Maximal subgroups