Group #13102769 — subgroups

Subgroups — M₃₂×C₅⋊D₅ #13102769

Order 1600.

360 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 125 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	C₂×C₈×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
2	C₅²×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#11716679	C₂	1	1	yes	maximal, characteristic, complemented
4	#11716665	C₂	1	2	yes	maximal, complemented
5	C₁₆×C₅⋊D₅	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₅⋊D₅	C₄	1	1	yes	characteristic
7	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
8	C₂×#6029	C₄	1	1	yes	characteristic
9	C₈×C₅⋊D₅	C₄	1	1	yes	characteristic
10	C₈×C₅⋊D₅	C₄	1	1	yes	characteristic
11	C₈×C₅⋊D₅	C₂²	1	1	yes	characteristic
12	C₈×C₅⋊D₅	C₂²	1	1	yes	characteristic
13	C₅²×C₁₆	C₂²	1	2	yes	complemented
14	#87756	C₂²	1	2	yes	complemented
15	D₅×M₃₂	—	5	6	no	maximal
16	C₂²×C₅⋊D₅	C₈	1	1	yes	characteristic
17	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
18	C₂×#1039	C₈	1	1	yes	characteristic
19	C₄×C₅⋊D₅	C₈	1	1	yes	characteristic
20	C₄×C₅⋊D₅	C₈	1	1	yes	characteristic
21	C₄×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
22	C₄×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
23	C₅²×C₈	C₂×C₄	1	1	yes	characteristic
24	C₅²×C₈	C₂³	1	1	yes	characteristic
25	#6029	C₂×C₄	1	1	yes	characteristic
26	#6029	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×D₅	—	5	6	no	—
28	C₅×M₃₂	D₅	1	6	yes	complemented
29	#3944	—	5	6	no	—
30	#3930	—	5	12	no	—
31	D₅×C₁₆	—	5	12	no	—
32	C₂²×C₅²	C₂×C₈	1	1	yes	characteristic
33	C₂×C₅⋊D₅	—	2	2	no	—
34	C₂×C₅⋊D₅	C₂×C₈	1	1	yes	characteristic
35	C₂×C₅⋊D₅	C₂×C₈	1	1	yes	characteristic
36	C₄×C₅²	C₂×C₈	1	1	yes	characteristic
37	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
38	#1039	C₂×C₈	1	1	yes	characteristic
39	#1039	C₂×C₈	1	1	yes	characteristic
40	C₂×C₄×D₅	—	5	6	no	—
41	C₂×C₅×C₈	C₂×D₅	1	6	yes	—
42	C₂×#168	—	5	6	no	—
43	C₈×D₅	—	5	6	no	—
44	C₈×D₅	—	5	6	no	—
45	C₈×D₅	—	5	6	no	—
46	C₈×D₅	—	5	6	no	—
47	C₅×C₁₆	C₂×D₅	1	12	yes	complemented
48	#672	—	5	12	no	—
49	C₂×M₃₂	—	25	1	no	Sylow
50	C₂×C₅²	—	2	1	no	—
51	C₂×C₅²	C₂²×C₈	1	1	yes	characteristic
52	C₅⋊D₅	—	2	1	no	—
53	C₅⋊D₅	M₃₂	1	2	yes	direct product factor
54	C₂²×D₅	—	5	6	no	—
55	C₂×C₄×C₅	C₄×D₅	1	6	yes	—
56	C₂×Dic₅	—	5	6	no	—
57	C₄×D₅	—	5	6	no	—
58	C₄×D₅	—	5	6	no	—
59	C₄×D₅	—	5	6	no	—
60	C₄×D₅	—	5	6	no	—
61	C₅×C₈	C₄×D₅	1	6	yes	—
62	C₅×C₈	C₂²×D₅	1	6	yes	—
63	#168	—	5	6	no	—
64	#168	—	5	6	no	—
65	C₂²×C₈	—	25	1	no	—
66	M₃₂	—	25	1	no	—
67	M₃₂	—	25	2	no	—
68	M₃₂	C₅⋊D₅	1	1	yes	characteristic, direct product factor
69	C₂×C₁₆	—	25	2	no	—
70	C₅²	C₂×M₃₂	1	1	yes	characteristic, complemented, Sylow
71	C₂²×C₅	C₈×D₅	1	6	yes	—
72	C₂×D₅	—	5	6	no	—
73	C₂×D₅	—	5	6	no	—
74	C₂×D₅	—	10	12	no	—
75	C₄×C₅	C₈×D₅	1	6	yes	—
76	C₄×C₅	C₂×C₄×D₅	1	6	yes	—
77	Dic₅	—	5	6	no	—
78	Dic₅	—	5	6	no	—
79	C₂²×C₄	—	25	1	no	—
80	C₂×C₈	—	25	1	no	—
81	C₂×C₈	—	25	1	no	—
82	C₂×C₈	—	25	1	no	—
83	C₂×C₈	—	25	1	no	—
84	C₂×C₈	—	25	1	no	—
85	C₂×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
86	C₁₆	—	25	2	no	—
87	C₁₆	C₂×C₅⋊D₅	1	2	yes	complemented
88	C₂×C₅	—	2	6	no	—
89	C₂×C₅	C₂×C₈×D₅	1	6	yes	—
90	D₅	—	5	12	no	—
91	D₅	—	10	6	no	—
92	C₂³	—	25	1	no	—
93	C₂×C₄	—	25	1	no	—
94	C₂×C₄	—	25	1	no	—
95	C₂×C₄	—	25	1	no	—
96	C₂×C₄	—	25	1	no	—
97	C₂×C₄	—	25	1	no	—
98	C₂×C₄	C₄×C₅⋊D₅	1	1	yes	characteristic
99	C₈	—	25	1	no	—
100	C₈	—	25	1	no	—
101	C₈	C₄×C₅⋊D₅	1	1	yes	characteristic
102	C₈	C₂²×C₅⋊D₅	1	1	yes	characteristic
103	C₅	D₅×M₃₂	1	6	yes	complemented
104	C₂²	—	25	1	no	—
105	C₂²	—	25	1	no	—
106	C₂²	—	50	2	no	—
107	C₂²	C₈×C₅⋊D₅	1	1	yes	characteristic
108	C₄	—	25	1	no	—
109	C₄	—	25	1	no	—
110	C₄	C₈×C₅⋊D₅	1	1	yes	characteristic
111	C₄	C₂×C₄×C₅⋊D₅	1	1	yes	characteristic
112	C₂	—	2	1	no	—
113	C₂	—	25	2	no	—
114	C₂	—	50	1	no	—
115	C₂	C₂×C₈×C₅⋊D₅	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×C₅⋊D₅

C₂

yes

maximal, characteristic

C₅²×M₃₂

C₂

yes

maximal, characteristic, complemented

#11716679

C₂

yes

maximal, characteristic, complemented

#11716665

C₂

yes

maximal, complemented

C₁₆×C₅⋊D₅

C₂

yes

maximal, complemented

C₂×C₄×C₅⋊D₅

C₄

yes

characteristic

C₂×C₅²×C₈

C₂²

yes

characteristic

C₂×#6029

C₄

yes

characteristic

C₈×C₅⋊D₅

C₄

yes

characteristic

C₈×C₅⋊D₅

C₄

yes

characteristic

C₈×C₅⋊D₅

C₂²

yes

characteristic

C₈×C₅⋊D₅

C₂²

yes

characteristic

C₅²×C₁₆

C₂²

yes

complemented

#87756

C₂²

yes

complemented

D₅×M₃₂

—

maximal

C₂²×C₅⋊D₅

C₈

yes

characteristic

C₂×C₄×C₅²

C₂×C₄

yes

characteristic

C₂×#1039

C₈

yes

characteristic

C₄×C₅⋊D₅

C₈

yes

characteristic

C₄×C₅⋊D₅

C₈

yes

characteristic

C₄×C₅⋊D₅

C₂×C₄

yes

characteristic

C₄×C₅⋊D₅

C₂×C₄

yes

characteristic

C₅²×C₈

C₂×C₄

yes

characteristic

C₅²×C₈

C₂³

yes

characteristic

#6029

C₂×C₄

yes

characteristic

#6029

C₂×C₄

yes

characteristic

C₂×C₈×D₅

—

C₅×M₃₂

D₅

yes

complemented

#3944

—

#3930

—

D₅×C₁₆

—

C₂²×C₅²

C₂×C₈

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

C₂×C₈

yes

characteristic

C₂×C₅⋊D₅

C₂×C₈

yes

characteristic

C₄×C₅²

C₂×C₈

yes

characteristic

C₄×C₅²

C₂²×C₄

yes

characteristic

#1039

C₂×C₈

yes

characteristic

#1039

C₂×C₈

yes

characteristic

C₂×C₄×D₅

—

C₂×C₅×C₈

C₂×D₅

yes

—

C₂×#168

—

C₈×D₅

—

C₈×D₅

—

C₈×D₅

—

C₈×D₅

—

C₅×C₁₆

C₂×D₅

yes

complemented

#672

—

C₂×M₃₂

—

Sylow

C₂×C₅²

—

C₂×C₅²

C₂²×C₈

yes

characteristic

C₅⋊D₅

—

C₅⋊D₅

M₃₂

yes

direct product factor

C₂²×D₅

—

C₂×C₄×C₅

C₄×D₅

yes

—

C₂×Dic₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₅×C₈

C₄×D₅

yes

—

C₅×C₈

C₂²×D₅

yes

—

#168

—

#168

—

C₂²×C₈

—

M₃₂

—

M₃₂

—

M₃₂

C₅⋊D₅

yes

characteristic, direct product factor

C₂×C₁₆

—

C₅²

C₂×M₃₂

yes

characteristic, complemented, Sylow

C₂²×C₅

C₈×D₅

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

C₈×D₅

yes

—

C₄×C₅

C₂×C₄×D₅

yes

—

Dic₅

—

Dic₅

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₅⋊D₅

yes

characteristic

C₁₆

—

C₁₆

C₂×C₅⋊D₅

yes

complemented

C₂×C₅

—

C₂×C₅

C₂×C₈×D₅

yes

—

D₅

—

D₅

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×C₅⋊D₅

yes

characteristic

C₈

—

100

C₈

—

101

C₈

C₄×C₅⋊D₅

yes

characteristic

102

C₈

C₂²×C₅⋊D₅

yes

characteristic

103

C₅

D₅×M₃₂

yes

complemented

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₈×C₅⋊D₅

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

C₈×C₅⋊D₅

yes

characteristic

111

C₄

C₂×C₄×C₅⋊D₅

yes

characteristic

112

C₂

—

113

C₂

—

114

C₂

—

115

C₂

C₂×C₈×C₅⋊D₅

yes

characteristic

	subgroup	G/H
center, Frattini	C₈	C₂²×C₅⋊D₅
commutator, socle	C₂×C₅²	C₂²×C₈
Fitting	C₅²×M₃₂	C₂
prime radical	C₂²×C₅⋊D₅	C₈
cosocle	C₅²×C₈	C₂³
hypercenter	M₃₂	C₅⋊D₅
hypocenter	C₅²	C₂×M₃₂
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₈ ◁ M₃₂

subgroup

G/H

center, Frattini

C₈

C₂²×C₅⋊D₅

commutator, socle

C₂×C₅²

C₂²×C₈

Fitting

C₅²×M₃₂

C₂

prime radical

C₂²×C₅⋊D₅

C₈

cosocle

C₅²×C₈

C₂³

hypercenter

M₃₂

C₅⋊D₅

hypocenter

C₅²

C₂×M₃₂

LCS

G ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₈ ◁ M₃₂

N	H ≅ G/N	type
2. C₅²×M₃₂	C₂	semidirect
3. #11716679	C₂	semidirect
4. #11716665	C₂	semidirect
5. C₁₆×C₅⋊D₅	C₂	semidirect
13. C₅²×C₁₆	C₂²	semidirect
14. #87756	C₂²	semidirect
28. C₅×M₃₂	D₅	semidirect
47. C₅×C₁₆	C₂×D₅	semidirect
53. C₅⋊D₅	M₃₂	direct (N × H)
68. M₃₂	C₅⋊D₅	direct (N × H)
70. C₅²	C₂×M₃₂	semidirect
87. C₁₆	C₂×C₅⋊D₅	semidirect
103. C₅	D₅×M₃₂	semidirect

H ≅ G/N

type

2. C₅²×M₃₂

C₂

semidirect

3. #11716679

C₂

semidirect

4. #11716665

C₂

semidirect

5. C₁₆×C₅⋊D₅

C₂

semidirect

13. C₅²×C₁₆

C₂²

semidirect

14. #87756

C₂²

semidirect

28. C₅×M₃₂

D₅

semidirect

47. C₅×C₁₆

C₂×D₅

semidirect

53. C₅⋊D₅

M₃₂

direct (N × H)

68. M₃₂

C₅⋊D₅

direct (N × H)

70. C₅²

C₂×M₃₂

semidirect

87. C₁₆

C₂×C₅⋊D₅

semidirect

103. C₅

D₅×M₃₂

semidirect

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

subgroup	index
C₂×C₈×C₅⋊D₅	2
C₅²×M₃₂	2
#11716679	2
#11716665	2
C₁₆×C₅⋊D₅	2
D₅×M₃₂	5

subgroup

index

C₂×C₈×C₅⋊D₅

C₅²×M₃₂

#11716679

#11716665

C₁₆×C₅⋊D₅

D₅×M₃₂

Subgroups — M₃₂×C₅⋊D₅ #13102769

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M32×C5⋊D5 #13102769

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₃₂×C₅⋊D₅ #13102769