Group #13022984 — subgroups

Subgroups — #13022984

Order 1280.

322 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 152 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	C₄×#3990	C₂	1	2	yes	maximal
2	C₈²×D₅	C₂	1	1	yes	maximal, characteristic
3	#10598514	C₂	1	4	yes	maximal, complemented
4	C₂×#3990	C₄	1	4	yes	—
5	C₄×C₈×D₅	C₄	1	2	yes	—
6	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
7	#64886	C₄	1	4	yes	—
8	C₄×#674	C₂²	1	4	yes	—
9	C₅×C₈²	C₄	1	1	yes	characteristic
10	C₈×#168	C₄	1	1	yes	characteristic
11	C₈×#168	C₂²	1	2	yes	—
12	#11402	—	5	1	no	maximal, Sylow
13	C₂×C₈×D₅	C₈	1	4	yes	—
14	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
15	C₄²×D₅	C₂×C₄	1	1	yes	characteristic
16	C₂×#674	C₂×C₄	1	8	yes	—
17	#3990	—	2	8	no	—
18	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
19	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
20	C₈×Dic₅	C₈	1	4	yes	—
21	C₈×Dic₅	C₂×C₄	1	2	yes	—
22	C₄×#168	C₂×C₄	1	2	yes	—
23	C₄×#168	C₂³	1	1	yes	characteristic
24	C₄×M₃₂	—	5	2	no	—
25	C₂×C₈²	—	5	1	no	—
26	#1312	—	5	4	no	—
27	C₂×C₄×D₅	C₄²	1	1	yes	characteristic
28	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
29	C₂×C₅×C₈	C₄²	1	2	yes	—
30	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
31	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
32	C₄×Dic₅	C₄²	1	1	yes	characteristic
33	C₄×Dic₅	C₂×C₈	1	2	yes	—
34	C₂×#168	C₂×C₈	1	4	yes	—
35	C₂×#168	C₂²×C₄	1	2	yes	—
36	C₈×D₅	—	2	8	no	—
37	C₈×D₅	M₁₆	1	8	yes	—
38	#674	—	2	8	no	—
39	C₂×M₃₂	—	5	4	no	—
40	C₂×C₄×C₈	—	5	1	no	—
41	C₂×C₄×C₈	—	5	2	no	—
42	#346	—	5	4	no	—
43	C₄×C₁₆	—	5	4	no	—
44	C₈²	—	5	2	no	—
45	C₈²	—	5	1	no	—
46	C₈²	AGL(1,𝔽₅)	1	1	yes	characteristic
47	C₂²×D₅	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
49	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
50	C₂×Dic₅	C₄×C₈	1	2	yes	—
51	C₂×Dic₅	C₄×C₈	1	1	yes	characteristic
52	C₄×D₅	—	2	4	no	—
53	C₄×D₅	#97	1	4	yes	—
54	C₅×C₈	—	2	4	no	—
55	C₅×C₈	#97	1	4	yes	—
56	#168	M₃₂	1	8	yes	complemented
57	#168	C₂×M₁₆	1	4	yes	—
58	C₂²×C₈	—	5	2	no	—
59	C₂²×C₈	—	5	4	no	—
60	C₂×C₄²	—	5	1	no	—
61	M₃₂	—	10	8	no	—
62	C₂×C₁₆	—	5	8	no	—
63	C₄×C₈	—	5	4	no	—
64	C₄×C₈	—	5	2	no	—
65	C₄×C₈	—	5	1	no	—
66	C₄×C₈	—	5	2	no	—
67	C₄×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
68	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
69	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
70	C₂×D₅	—	2	2	no	—
71	C₂×D₅	#322	1	2	yes	—
72	C₄×C₅	C₂×#97	1	2	yes	—
73	C₄×C₅	C₂×M₃₂	1	4	yes	—
74	Dic₅	—	2	2	no	—
75	Dic₅	#322	1	2	yes	—
76	C₂²×C₄	—	5	1	no	—
77	C₂²×C₄	—	5	2	no	—
78	C₂×C₈	—	5	4	no	—
79	C₂×C₈	—	5	2	no	—
80	C₂×C₈	—	5	8	no	—
81	C₂×C₈	—	10	8	no	—
82	C₂×C₈	D₅⋊C₈	1	4	yes	—
83	C₂×C₈	C₄×AGL(1,𝔽₅)	1	2	yes	—
84	C₄²	—	5	2	no	—
85	C₄²	—	5	1	no	—
86	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
87	C₁₆	—	10	8	no	—
88	C₂×C₅	C₂×#322	1	1	yes	characteristic
89	C₂×C₅	C₄×M₃₂	1	2	yes	—
90	D₅	—	2	2	no	—
91	C₂³	—	5	1	no	—
92	C₂×C₄	—	5	1	no	—
93	C₂×C₄	—	5	4	no	—
94	C₂×C₄	—	5	2	no	—
95	C₂×C₄	—	10	4	no	—
96	C₂×C₄	C₂×D₅⋊C₈	1	2	yes	—
97	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
98	C₈	—	2	4	no	—
99	C₈	—	5	8	no	—
100	C₈	—	5	4	no	—
101	C₈	#3992	1	4	yes	—
102	C₅	#11402	1	1	yes	characteristic, complemented, Sylow
103	C₂²	—	5	2	no	—
104	C₂²	—	10	2	no	—
105	C₂²	C₄×D₅⋊C₈	1	1	yes	characteristic
106	C₄	—	5	2	no	—
107	C₄	—	10	2	no	—
108	C₄	D₅⋊M₃₂	1	4	yes	—
109	C₄	C₂×#3992	1	2	yes	—
110	C₂	—	10	2	no	—
111	C₂	#10603807	1	1	yes	characteristic
112	C₂	#10603825	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#3990

C₂

yes

maximal

C₈²×D₅

C₂

yes

maximal, characteristic

#10598514

C₂

yes

maximal, complemented

C₂×#3990

C₄

yes

—

C₄×C₈×D₅

C₄

yes

—

C₄×C₈×D₅

C₂²

yes

characteristic

#64886

C₄

yes

—

C₄×#674

C₂²

yes

—

C₅×C₈²

C₄

yes

characteristic

C₈×#168

C₄

yes

characteristic

C₈×#168

C₂²

yes

—

#11402

—

maximal, Sylow

C₂×C₈×D₅

C₈

yes

—

C₂×C₈×D₅

C₂×C₄

yes

—

C₄²×D₅

C₂×C₄

yes

characteristic

C₂×#674

C₂×C₄

yes

—

#3990

—

C₄×C₅×C₈

C₂×C₄

yes

characteristic

C₄×C₅×C₈

C₂×C₄

yes

—

C₈×Dic₅

C₈

yes

—

C₈×Dic₅

C₂×C₄

yes

—

C₄×#168

C₂×C₄

yes

—

C₄×#168

C₂³

yes

characteristic

C₄×M₃₂

—

C₂×C₈²

—

#1312

—

C₂×C₄×D₅

C₄²

yes

characteristic

C₂×C₄×D₅

C₂×C₈

yes

—

C₂×C₅×C₈

C₄²

yes

—

C₂×C₅×C₈

C₂×C₈

yes

—

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄×Dic₅

C₄²

yes

characteristic

C₄×Dic₅

C₂×C₈

yes

—

C₂×#168

C₂×C₈

yes

—

C₂×#168

C₂²×C₄

yes

—

C₈×D₅

—

C₈×D₅

M₁₆

yes

—

#674

—

C₂×M₃₂

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#346

—

C₄×C₁₆

—

C₈²

—

C₈²

—

C₈²

AGL(1,𝔽₅)

yes

characteristic

C₂²×D₅

C₄×C₈

yes

characteristic

C₂×C₄×C₅

C₂×C₄²

yes

characteristic

C₂×C₄×C₅

C₂²×C₈

yes

—

C₂×Dic₅

C₄×C₈

yes

—

C₂×Dic₅

C₄×C₈

yes

characteristic

C₄×D₅

—

C₄×D₅

#97

yes

—

C₅×C₈

—

C₅×C₈

#97

yes

—

#168

M₃₂

yes

complemented

#168

C₂×M₁₆

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

—

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂²×C₅

C₂×C₄×C₈

yes

characteristic

C₂×D₅

—

C₂×D₅

#322

yes

—

C₄×C₅

C₂×#97

yes

—

C₄×C₅

C₂×M₃₂

yes

—

Dic₅

—

Dic₅

#322

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₅⋊C₈

yes

—

C₂×C₈

C₄×AGL(1,𝔽₅)

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×AGL(1,𝔽₅)

yes

characteristic

C₁₆

—

C₂×C₅

C₂×#322

yes

characteristic

C₂×C₅

C₄×M₃₂

yes

—

D₅

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₅⋊C₈

yes

—

C₂×C₄

C₂×C₄×AGL(1,𝔽₅)

yes

characteristic

C₈

—

C₈

—

100

C₈

—

101

C₈

#3992

yes

—

102

C₅

#11402

yes

characteristic, complemented, Sylow

103

C₂²

—

104

C₂²

—

105

C₂²

C₄×D₅⋊C₈

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

D₅⋊M₃₂

yes

—

109

C₄

C₂×#3992

yes

—

110

C₂

—

111

C₂

#10603807

yes

characteristic

112

C₂

#10603825

yes

—

	subgroup	G/H
center, Frattini	C₄²	C₂²×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂×C₄×C₈
Fitting	C₅×C₈²	C₄
prime radical	C₂²×D₅	C₄×C₈
cosocle	C₄×#168	C₂³
hypercenter	C₈²	AGL(1,𝔽₅)
hypocenter	C₅	#11402
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₄² ◁ C₈²

subgroup

G/H

center, Frattini

C₄²

C₂²×AGL(1,𝔽₅)

commutator, socle

C₂²×C₅

C₂×C₄×C₈

Fitting

C₅×C₈²

C₄

prime radical

C₂²×D₅

C₄×C₈

C₂³

C₈²

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
3. #10598514	C₂	semidirect
56. #168	M₃₂	semidirect
102. C₅	#11402	semidirect

H ≅ G/N

type

3. #10598514

C₂

semidirect

56. #168

M₃₂

semidirect

102. C₅

#11402

semidirect

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

subgroup	index
C₄×#3990	2
C₈²×D₅	2
#10598514	2
#11402	5

subgroup

index

C₄×#3990

C₈²×D₅

#10598514

#11402

Subgroups — #13022984

Conjugacy classes of subgroups

Complements

Maximal subgroups