Group #13103106 — subgroups

Subgroups — #13103106

Order 1600.

608 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 139 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×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	#11717271	C₂	1	2	yes	maximal, complemented
3	#11717256	C₂	1	1	yes	maximal, characteristic, complemented
4	#11716704	C₂	1	2	yes	maximal, complemented
5	#11716700	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅²×#41	—	2	2	no	—
7	C₅²×#41	C₂²	1	2	yes	—
8	C₂×C₅²×D₄	—	2	2	no	—
9	C₂×C₅²×D₄	—	2	2	no	—
10	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
11	C₂×#6045	C₂²	1	1	yes	characteristic, complemented
12	C₂²⋊#1039	—	2	2	no	—
13	#87812	—	2	2	no	—
14	#87810	—	2	2	no	—
15	C₄×#1039	C₂²	1	1	yes	characteristic, complemented
16	#87804	C₂²	1	2	yes	complemented
17	#65531	—	5	6	no	maximal
18	C₂³×C₅²	—	2	2	no	—
19	C₂³×C₅²	D₄	1	2	yes	—
20	C₅²×Q₈	D₄	1	2	yes	—
21	C₅²×D₄	—	2	4	no	—
22	C₅²×D₄	—	2	2	no	—
23	C₅²×D₄	—	2	2	no	—
24	C₅²×D₄	D₄	1	2	yes	—
25	C₂×C₄×C₅²	—	2	2	no	—
26	C₂×C₄×C₅²	—	2	2	no	—
27	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
28	C₂×#1039	—	2	1	no	—
29	C₂×#1039	—	2	1	no	—
30	#6045	—	2	2	no	—
31	#6029	—	2	2	no	—
32	C₅×H(5,𝔽₂)	D₅	1	6	yes	—
33	#4097	—	5	12	no	—
34	#4082	—	5	6	no	—
35	#3969	—	5	12	no	—
36	#3965	—	5	6	no	—
37	C₂²×C₅²	—	2	2	no	—
38	C₂²×C₅²	—	2	2	no	—
39	C₂²×C₅²	—	4	2	no	—
40	C₂²×C₅²	—	8	2	no	—
41	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
42	C₄×C₅²	—	2	2	no	—
43	C₄×C₅²	C₂×D₄	1	2	yes	—
44	#1039	—	4	1	no	—
45	#1039	—	4	1	no	—
46	C₅×#41	—	2	12	no	—
47	C₅×#41	C₂×D₅	1	12	yes	—
48	C₂×C₅×D₄	—	2	12	no	—
49	C₂×C₅×D₄	—	2	12	no	—
50	C₂×C₅×D₄	C₂×D₅	1	6	yes	—
51	C₂×Dic₁₀	—	5	6	no	—
52	C₂²⋊Dic₅	—	10	12	no	—
53	#689	—	10	12	no	—
54	#687	—	10	12	no	—
55	C₄×Dic₅	—	5	6	no	—
56	#681	—	5	12	no	—
57	#455	—	25	1	no	Sylow
58	C₂×C₅²	—	2	1	no	—
59	C₂×C₅²	—	4	2	no	—
60	C₂×C₅²	—	4	2	no	—
61	C₂×C₅²	#120	1	1	yes	characteristic
62	C₂³×C₅	—	2	12	no	—
63	C₂³×C₅	C₂²⋊D₅	1	12	yes	—
64	C₅×Q₈	C₂²⋊D₅	1	12	yes	—
65	C₅×D₄	—	2	24	no	—
66	C₅×D₄	—	2	12	no	—
67	C₅×D₄	—	2	12	no	—
68	C₅×D₄	C₂²⋊D₅	1	12	yes	—
69	C₂×C₄×C₅	—	2	12	no	—
70	C₂×C₄×C₅	—	2	12	no	—
71	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
72	C₂×Dic₅	—	10	6	no	—
73	C₂×Dic₅	—	10	6	no	—
74	Dic₁₀	—	10	12	no	—
75	#168	—	10	12	no	—
76	H(5,𝔽₂)	C₅⋊D₅	1	1	yes	characteristic
77	#137	—	25	2	no	—
78	#124	—	25	1	no	—
79	#104	—	25	2	no	—
80	#100	—	25	1	no	—
81	C₅²	#455	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₅	—	2	12	no	—
83	C₂²×C₅	—	2	12	no	—
84	C₂²×C₅	—	4	12	no	—
85	C₂²×C₅	—	8	12	no	—
86	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
87	C₄×C₅	—	2	12	no	—
88	C₄×C₅	C₂×C₂²⋊D₅	1	12	yes	—
89	Dic₅	—	20	6	no	—
90	Dic₅	—	20	6	no	—
91	#41	—	2	2	no	—
92	#41	C₂×C₅⋊D₅	1	2	yes	—
93	C₂×Q₈	—	25	1	no	—
94	C₂×D₄	—	2	2	no	—
95	C₂×D₄	—	2	2	no	—
96	C₂×D₄	C₂×C₅⋊D₅	1	1	yes	characteristic
97	Q₁₆	—	50	2	no	—
98	SD₁₆	—	50	2	no	—
99	M₁₆	—	25	2	no	—
100	C₂²⋊C₄	—	50	2	no	—
101	C₄²	—	25	1	no	—
102	C₂×C₅	—	2	6	no	—
103	C₂×C₅	—	4	12	no	—
104	C₂×C₅	—	4	12	no	—
105	C₂×C₅	#4099	1	6	yes	—
106	C₂³	—	2	2	no	—
107	C₂³	C₂²⋊(C₅⋊D₅)	1	2	yes	—
108	Q₈	—	50	2	no	—
109	Q₈	C₂²⋊(C₅⋊D₅)	1	2	yes	—
110	D₄	—	2	4	no	—
111	D₄	—	2	2	no	—
112	D₄	—	2	2	no	—
113	D₄	C₂²⋊(C₅⋊D₅)	1	2	yes	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	50	1	no	—
117	C₂×C₄	—	50	1	no	—
118	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
119	C₈	—	50	2	no	—
120	C₅	#65531	1	6	yes	complemented
121	C₂²	—	2	2	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	4	2	no	—
124	C₂²	—	8	2	no	—
125	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	100	1	no	—
128	C₄	—	100	1	no	—
129	C₄	C₂×C₂²⋊(C₅⋊D₅)	1	2	yes	—
130	C₂	—	2	1	no	—
131	C₂	—	4	2	no	—
132	C₂	—	4	2	no	—
133	C₂	#11717273	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×H(5,𝔽₂)

C₂

yes

maximal, characteristic

#11717271

C₂

yes

maximal, complemented

#11717256

C₂

yes

maximal, characteristic, complemented

#11716704

C₂

yes

maximal, complemented

#11716700

C₂

yes

maximal, characteristic, complemented

C₅²×#41

—

C₅²×#41

C₂²

yes

—

C₂×C₅²×D₄

—

C₂×C₅²×D₄

—

C₂×C₅²×D₄

C₂²

yes

characteristic

C₂×#6045

C₂²

yes

characteristic, complemented

C₂²⋊#1039

—

#87812

—

#87810

—

C₄×#1039

C₂²

yes

characteristic, complemented

#87804

C₂²

yes

complemented

#65531

—

maximal

C₂³×C₅²

—

C₂³×C₅²

D₄

yes

—

C₅²×Q₈

D₄

yes

—

C₅²×D₄

—

C₅²×D₄

—

C₅²×D₄

—

C₅²×D₄

D₄

yes

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

C₂³

yes

—

—

—

—

C₅×H(5,𝔽₂)

D₅

yes

—

—

—

—

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

—

C₄×C₅²

C₂×D₄

yes

—

#1039

—

#1039

—

C₅×#41

—

C₅×#41

C₂×D₅

yes

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

C₂×D₅

yes

—

C₂×Dic₁₀

—

C₂²⋊Dic₅

—

#689

—

#687

—

C₄×Dic₅

—

#681

—

#455

—

Sylow

C₂×C₅²

—

C₂×C₅²

—

C₂×C₅²

—

C₂×C₅²

#120

yes

characteristic

C₂³×C₅

—

C₂³×C₅

C₂²⋊D₅

yes

—

C₅×Q₈

C₂²⋊D₅

yes

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

C₂²⋊D₅

yes

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

#168

—

H(5,𝔽₂)

C₅⋊D₅

yes

—

—

—

—

C₅²

#455

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂×C₂²⋊D₅

yes

—

C₄×C₅

—

C₄×C₅

C₂×C₂²⋊D₅

yes

—

Dic₅

—

Dic₅

—

#41

—

#41

C₂×C₅⋊D₅

yes

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×C₅⋊D₅

yes

characteristic

Q₁₆

—

SD₁₆

—

M₁₆

—

100

C₂²⋊C₄

—

101

C₄²

—

102

C₂×C₅

—

103

C₂×C₅

—

104

C₂×C₅

—

105

C₂×C₅

#4099

yes

—

106

C₂³

—

107

C₂³

C₂²⋊(C₅⋊D₅)

yes

—

108

Q₈

—

109

Q₈

C₂²⋊(C₅⋊D₅)

yes

—

110

D₄

—

111

D₄

—

112

D₄

—

113

D₄

C₂²⋊(C₅⋊D₅)

yes

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

119

C₈

—

120

C₅

#65531

yes

complemented

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

126

C₄

—

127

C₄

—

100

—

128

C₄

—

100

—

129

C₄

C₂×C₂²⋊(C₅⋊D₅)

yes

—

130

C₂

—

131

C₂

—

132

C₂

—

133

C₂

#11717273

yes

characteristic

	subgroup	G/H
center	C₂	#11717273
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting, prime radical	C₅²×H(5,𝔽₂)	C₂
socle	C₂×C₅²	#120
hypercenter	H(5,𝔽₂)	C₅⋊D₅
hypocenter	C₅²	#455
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

#11717273

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂²×C₅⋊D₅

Fitting, prime radical

C₅²×H(5,𝔽₂)

C₂

C₂×C₅²

C₅⋊D₅

C₅²

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ H(5,𝔽₂)

N	H ≅ G/N	type
2. #11717271	C₂	semidirect
3. #11717256	C₂	semidirect
4. #11716704	C₂	semidirect
5. #11716700	C₂	semidirect
11. C₂×#6045	C₂²	semidirect
15. C₄×#1039	C₂²	semidirect
16. #87804	C₂²	semidirect
81. C₅²	#455	semidirect
120. C₅	#65531	semidirect

H ≅ G/N

type

2. #11717271

C₂

semidirect

3. #11717256

C₂

semidirect

4. #11716704

C₂

semidirect

5. #11716700

C₂

semidirect

11. C₂×#6045

C₂²

semidirect

15. C₄×#1039

C₂²

semidirect

16. #87804

C₂²

semidirect

81. C₅²

#455

semidirect

120. C₅

#65531

semidirect

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

subgroup	index
C₅²×H(5,𝔽₂)	2
#11717271	2
#11717256	2
#11716704	2
#11716700	2
#65531	5

subgroup

index

C₅²×H(5,𝔽₂)

Subgroups — #13103106

Conjugacy classes of subgroups

Complements

Maximal subgroups