Group #10616438 — subgroups

Subgroups — C₅×#2843 #10616438

Order 640.

348 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 184 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	C₅×#400	C₂	1	3	yes	maximal
2	C₅×#398	C₂	1	6	yes	maximal
3	C₅×#395	C₂	1	3	yes	maximal
4	C₅×#384	C₂	1	1	yes	maximal, characteristic
5	C₅×#383	C₂	1	2	yes	maximal
6	C₂²×C₅×Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₅×C₄⋊C₄	C₂²	1	3	yes	—
8	C₂×C₅×C₄⋊C₄	C₂²	1	3	yes	—
9	C₂×C₅×C₄⋊C₄	C₂²	1	6	yes	—
10	C₂×C₅×C₄⋊C₄	C₂²	1	3	yes	—
11	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₅×#95	C₂²	1	6	yes	—
13	C₅×#95	C₂²	1	6	yes	—
14	C₅×#95	C₂²	1	3	yes	—
15	C₅×#95	C₂²	1	3	yes	—
16	#2843	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂×C₅×Q₈	—	4	3	no	—
18	C₂²×C₄×C₅	C₂³	1	3	yes	—
19	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₅	C₂³	1	2	yes	—
21	C₂²×C₄×C₅	C₂³	1	6	yes	—
22	C₂²×C₄×C₅	C₂³	1	3	yes	—
23	C₅×C₄⋊C₄	—	4	3	no	—
24	C₅×C₄⋊C₄	—	4	3	no	—
25	C₅×C₄⋊C₄	—	4	3	no	—
26	C₅×C₄⋊C₄	—	4	6	no	—
27	C₄²×C₅	Q₈	1	4	yes	complemented
28	#400	C₂×C₅	1	3	yes	—
29	#398	C₂×C₅	1	6	yes	—
30	#395	C₂×C₅	1	3	yes	—
31	#384	C₂×C₅	1	1	yes	characteristic
32	#383	C₂×C₅	1	2	yes	—
33	C₂³×C₅	C₂⁴	1	1	yes	characteristic
34	C₅×Q₈	—	4	4	no	—
35	C₂×C₄×C₅	—	2	3	no	—
36	C₂×C₄×C₅	—	2	3	no	—
37	C₂×C₄×C₅	—	2	6	no	—
38	C₂×C₄×C₅	—	2	3	no	—
39	C₂×C₄×C₅	—	2	6	no	—
40	C₂×C₄×C₅	—	2	3	no	—
41	C₂×C₄×C₅	—	2	6	no	—
42	C₂×C₄×C₅	—	2	12	no	—
43	C₂×C₄×C₅	C₂×Q₈	1	6	yes	—
44	C₂²×Q₈	C₂²×C₅	1	1	yes	characteristic
45	C₂×C₄⋊C₄	C₂²×C₅	1	3	yes	—
46	C₂×C₄⋊C₄	C₂²×C₅	1	3	yes	—
47	C₂×C₄⋊C₄	C₂²×C₅	1	6	yes	—
48	C₂×C₄⋊C₄	C₂²×C₅	1	3	yes	—
49	C₂×C₄²	C₂²×C₅	1	1	yes	characteristic
50	#95	C₂²×C₅	1	6	yes	—
51	#95	C₂²×C₅	1	6	yes	—
52	#95	C₂²×C₅	1	3	yes	—
53	#95	C₂²×C₅	1	3	yes	—
54	C₂²×C₅	C₂²×Q₈	1	1	yes	characteristic
55	C₂²×C₅	H(5,𝔽₂)	1	3	yes	—
56	C₂²×C₅	2₋¹⁺⁴	1	3	yes	—
57	C₄×C₅	—	2	6	no	—
58	C₄×C₅	—	4	3	no	—
59	C₄×C₅	—	4	1	no	—
60	C₄×C₅	—	4	2	no	—
61	C₄×C₅	—	4	6	no	—
62	C₂×Q₈	—	4	3	no	—
63	C₂²×C₄	C₂³×C₅	1	6	yes	—
64	C₂²×C₄	C₂³×C₅	1	2	yes	—
65	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
66	C₂²×C₄	C₂³×C₅	1	3	yes	—
67	C₂²×C₄	C₂³×C₅	1	3	yes	—
68	C₄⋊C₄	—	4	3	no	—
69	C₄⋊C₄	—	4	3	no	—
70	C₄⋊C₄	—	4	3	no	—
71	C₄⋊C₄	—	4	6	no	—
72	C₄²	C₅×Q₈	1	4	yes	complemented
73	C₂×C₅	#544	1	3	yes	—
74	C₂×C₅	#560	1	1	yes	characteristic
75	C₂×C₅	#563	1	3	yes	—
76	C₂³	C₂⁴×C₅	1	1	yes	characteristic
77	Q₈	—	4	4	no	—
78	C₂×C₄	—	2	3	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	6	no	—
81	C₂×C₄	—	2	3	no	—
82	C₂×C₄	—	2	6	no	—
83	C₂×C₄	—	2	3	no	—
84	C₂×C₄	—	2	6	no	—
85	C₂×C₄	—	2	12	no	—
86	C₂×C₄	C₂×C₅×Q₈	1	6	yes	—
87	C₅	#2843	1	1	yes	characteristic, direct product factor, Sylow
88	C₂²	C₂²×C₅×Q₈	1	1	yes	characteristic
89	C₂²	C₅×H(5,𝔽₂)	1	3	yes	—
90	C₂²	C₅×2₋¹⁺⁴	1	3	yes	—
91	C₄	—	2	6	no	—
92	C₄	—	4	1	no	—
93	C₄	—	4	2	no	—
94	C₄	—	4	3	no	—
95	C₄	—	4	6	no	—
96	C₂	C₅×#544	1	3	yes	—
97	C₂	C₅×#560	1	1	yes	characteristic
98	C₂	C₅×#563	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#400

C₂

yes

maximal

C₅×#398

C₂

yes

maximal

C₅×#395

C₂

yes

maximal

C₅×#384

C₂

yes

maximal, characteristic

C₅×#383

C₂

yes

maximal

C₂²×C₅×Q₈

C₂²

yes

characteristic

C₂×C₅×C₄⋊C₄

C₂²

yes

—

C₂×C₅×C₄⋊C₄

C₂²

yes

—

C₂×C₅×C₄⋊C₄

C₂²

yes

—

C₂×C₅×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₅

C₂²

yes

characteristic

C₅×#95

C₂²

yes

—

C₅×#95

C₂²

yes

—

C₅×#95

C₂²

yes

—

C₅×#95

C₂²

yes

—

#2843

C₅

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₅×Q₈

—

C₂²×C₄×C₅

C₂³

yes

—

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂²×C₄×C₅

C₂³

yes

—

C₂²×C₄×C₅

C₂³

yes

—

C₂²×C₄×C₅

C₂³

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

Q₈

yes

complemented

#400

C₂×C₅

yes

—

#398

C₂×C₅

yes

—

#395

C₂×C₅

yes

—

#384

C₂×C₅

yes

characteristic

#383

C₂×C₅

yes

—

C₂³×C₅

C₂⁴

yes

characteristic

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×Q₈

yes

—

C₂²×Q₈

C₂²×C₅

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₅

yes

—

C₂×C₄⋊C₄

C₂²×C₅

yes

—

C₂×C₄⋊C₄

C₂²×C₅

yes

—

C₂×C₄⋊C₄

C₂²×C₅

yes

—

C₂×C₄²

C₂²×C₅

yes

characteristic

#95

C₂²×C₅

yes

—

#95

C₂²×C₅

yes

—

#95

C₂²×C₅

yes

—

#95

C₂²×C₅

yes

—

C₂²×C₅

C₂²×Q₈

yes

characteristic

C₂²×C₅

H(5,𝔽₂)

yes

—

C₂²×C₅

2₋¹⁺⁴

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₂×Q₈

—

C₂²×C₄

C₂³×C₅

yes

—

C₂²×C₄

C₂³×C₅

yes

—

C₂²×C₄

C₂³×C₅

yes

characteristic

C₂²×C₄

C₂³×C₅

yes

—

C₂²×C₄

C₂³×C₅

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₅×Q₈

yes

complemented

C₂×C₅

#544

yes

—

C₂×C₅

#560

yes

characteristic

C₂×C₅

#563

yes

—

C₂³

C₂⁴×C₅

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₅×Q₈

yes

—

C₅

#2843

yes

characteristic, direct product factor, Sylow

C₂²

C₂²×C₅×Q₈

yes

characteristic

C₂²

C₅×H(5,𝔽₂)

yes

—

C₂²

C₅×2₋¹⁺⁴

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₂

C₅×#544

yes

—

C₂

C₅×#560

yes

characteristic

C₂

C₅×#563

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₅

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₅

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₅ ◁ G

N	H ≅ G/N	type
16. #2843	C₅	direct (N × H)
27. C₄²×C₅	Q₈	semidirect
72. C₄²	C₅×Q₈	semidirect
87. C₅	#2843	direct (N × H)

H ≅ G/N

type

16. #2843

C₅

direct (N × H)

27. C₄²×C₅

Q₈

semidirect

72. C₄²

C₅×Q₈

semidirect

87. C₅

#2843

direct (N × H)

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

subgroup	index
C₅×#400	2
C₅×#398	2
C₅×#395	2
C₅×#384	2
C₅×#383	2
#2843	5

subgroup

index

Subgroups — C₅×#2843 #10616438

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#2843 #10616438

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#2843 #10616438