Group #13100402 — subgroups

Subgroups — #13100402

Order 1600.

174 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 44 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	#11717268	C₂	1	1	yes	maximal, characteristic
2	#11716891	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#87765	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅²×#41	C₄	1	1	yes	characteristic
5	#87804	—	2	1	no	—
6	#87804	C₄	1	1	yes	characteristic
7	C₂×#6029	—	2	1	no	—
8	C₂×#6029	C₂²	1	1	yes	characteristic
9	#87765	—	2	1	no	—
10	#87765	—	2	1	no	—
11	#64932	—	5	2	no	maximal
12	C₅²×Q₈	C₈	1	1	yes	characteristic
13	C₅²×D₄	—	2	1	no	—
14	C₅²×D₄	C₈	1	1	yes	characteristic
15	C₂×C₄×C₅²	—	2	1	no	—
16	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
17	#6029	—	2	1	no	—
18	#6029	D₄	1	1	yes	characteristic
19	#6029	D₄	1	1	yes	characteristic
20	#4094	—	5	2	no	—
21	#4094	—	10	2	no	—
22	#3998	—	5	2	no	—
23	C₂×#674	—	5	2	no	—
24	C₂²×C₅²	—	2	1	no	—
25	C₂²×C₅²	M₁₆	1	1	yes	characteristic
26	C₄×C₅²	—	2	1	no	—
27	C₄×C₅²	C₂²⋊C₄	1	1	yes	characteristic
28	C₄×C₅²	C₂×C₈	1	1	yes	characteristic
29	C₅×#41	—	2	2	no	—
30	C₅×#41	AGL(1,𝔽₅)	1	2	yes	—
31	#681	—	5	2	no	—
32	#681	—	10	2	no	—
33	#681	—	10	2	no	—
34	#681	—	10	4	no	—
35	C₂×#168	—	5	2	no	—
36	C₂×#168	—	10	2	no	—
37	C₂×#168	—	10	2	no	—
38	C₂×#168	—	10	4	no	—
39	#674	—	10	2	no	—
40	#674	—	10	2	no	—
41	#350	—	25	1	no	Sylow
42	C₂×C₅²	—	2	1	no	—
43	C₂×C₅²	—	4	1	no	—
44	C₂×C₅²	C₂²⋊C₈	1	1	yes	characteristic
45	C₅×Q₈	—	2	2	no	—
46	C₅×Q₈	#170	1	2	yes	—
47	C₅×D₄	—	2	4	no	—
48	C₅×D₄	—	2	2	no	—
49	C₅×D₄	—	2	2	no	—
50	C₅×D₄	#170	1	2	yes	—
51	C₂×C₄×C₅	—	2	4	no	—
52	C₂×C₄×C₅	—	2	2	no	—
53	C₂×C₄×C₅	—	2	2	no	—
54	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
55	#168	—	5	2	no	—
56	#168	—	5	2	no	—
57	#168	—	10	2	no	—
58	#168	—	10	2	no	—
59	#168	—	10	4	no	—
60	#168	—	10	2	no	—
61	#131	—	25	1	no	—
62	M₃₂	—	25	1	no	—
63	C₂×C₁₆	—	25	1	no	—
64	C₅²	#350	1	1	yes	characteristic, complemented, Sylow
65	C₂²×C₅	—	2	4	no	—
66	C₂²×C₅	—	2	2	no	—
67	C₂²×C₅	—	2	2	no	—
68	C₂²×C₅	#704	1	2	yes	—
69	C₄×C₅	—	2	4	no	—
70	C₄×C₅	—	2	2	no	—
71	C₄×C₅	—	2	2	no	—
72	C₄×C₅	—	2	2	no	—
73	C₄×C₅	C₂×#170	1	2	yes	—
74	C₄×C₅	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
75	#41	#1044	1	1	yes	characteristic
76	M₁₆	—	25	1	no	—
77	M₁₆	—	50	1	no	—
78	C₂×C₈	—	25	1	no	—
79	C₂×C₈	—	50	1	no	—
80	C₁₆	—	50	1	no	—
81	C₁₆	—	50	1	no	—
82	C₂×C₅	—	2	2	no	—
83	C₂×C₅	—	2	2	no	—
84	C₂×C₅	—	4	2	no	—
85	C₂×C₅	—	4	4	no	—
86	C₂×C₅	—	4	2	no	—
87	C₂×C₅	C₂²⋊#170	1	2	yes	—
88	Q₈	#6034	1	1	yes	characteristic
89	D₄	—	2	1	no	—
90	D₄	#6034	1	1	yes	characteristic
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	C₂×#1044	1	1	yes	characteristic
93	C₈	—	25	1	no	—
94	C₈	—	25	1	no	—
95	C₈	—	50	1	no	—
96	C₅	—	2	2	no	—
97	C₅	#64932	1	2	yes	complemented
98	C₂²	—	2	1	no	—
99	C₂²	#87867	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	C₂×#6034	1	1	yes	characteristic
102	C₄	C₂²⋊#1044	1	1	yes	characteristic
103	C₂	—	2	1	no	—
104	C₂	—	4	1	no	—
105	C₂	C₂²⋊#6034	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11717268

C₂

yes

maximal, characteristic

#11716891

C₂

yes

maximal, characteristic, complemented

C₂×#87765

C₂

yes

maximal, characteristic, complemented

C₅²×#41

C₄

yes

characteristic

#87804

—

#87804

C₄

yes

characteristic

C₂×#6029

—

C₂×#6029

C₂²

yes

—

—

—

C₅²×Q₈

C₈

yes

characteristic

C₅²×D₄

—

C₅²×D₄

C₈

yes

characteristic

C₂×C₄×C₅²

—

C₂×C₄×C₅²

C₂×C₄

yes

characteristic

#6029

—

#6029

D₄

yes

characteristic

#6029

D₄

yes

—

—

—

—

C₂²×C₅²

—

C₂²×C₅²

M₁₆

yes

characteristic

C₄×C₅²

—

C₄×C₅²

C₂²⋊C₄

yes

characteristic

C₄×C₅²

C₂×C₈

yes

characteristic

C₅×#41

—

C₅×#41

AGL(1,𝔽₅)

yes

—

—

—

—

—

—

—

—

—

—

—

—

C₂×C₅²

—

C₂×C₅²

—

C₂×C₅²

C₂²⋊C₈

yes

characteristic

C₅×Q₈

—

C₅×Q₈

#170

yes

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

#170

yes

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×AGL(1,𝔽₅)

yes

—

—

—

—

—

—

—

—

M₃₂

—

C₂×C₁₆

—

C₅²

#350

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

#704

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₂×#170

yes

—

C₄×C₅

C₂²⋊AGL(1,𝔽₅)

yes

—

#41

#1044

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₁₆

—

C₁₆

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

C₂²⋊#170

yes

—

Q₈

#6034

yes

characteristic

D₄

—

D₄

#6034

yes

characteristic

C₂×C₄

—

C₂×C₄

C₂×#1044

yes

characteristic

C₈

—

C₈

—

C₈

—

C₅

—

C₅

#64932

yes

complemented

C₂²

—

C₂²

#87867

yes

characteristic

100

C₄

—

101

C₄

C₂×#6034

yes

characteristic

102

C₄

C₂²⋊#1044

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

C₂²⋊#6034

yes

characteristic

	subgroup	G/H
center	C₄	C₂²⋊#1044
commutator	C₄×C₅²	C₂×C₈
Frattini	C₂×C₄	C₂×#1044
Fitting, prime radical	C₅²×#41	C₄
socle	C₂×C₅²	C₂²⋊C₈
cosocle	C₂×#6029	C₂²
hypercenter	#41	#1044
hypocenter	C₅²	#350
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #41

subgroup

G/H

center

C₄

C₂²⋊#1044

commutator

C₄×C₅²

C₂×C₈

Frattini

C₂×C₄

C₂×#1044

Fitting, prime radical

C₅²×#41

C₄

socle

C₂×C₅²

C₂²⋊C₈

C₂²

C₅²

G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ #41

N	H ≅ G/N	type
2. #11716891	C₂	semidirect
3. C₂×#87765	C₂	semidirect
64. C₅²	#350	semidirect
97. C₅	#64932	semidirect

H ≅ G/N

type

2. #11716891

C₂

semidirect

3. C₂×#87765

C₂

semidirect

64. C₅²

#350

semidirect

97. C₅

#64932

semidirect

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

subgroup	index
#11717268	2
#11716891	2
C₂×#87765	2
#64932	5

subgroup

index

Subgroups — #13100402

Conjugacy classes of subgroups

Complements

Maximal subgroups