Subgroups — #12030504

Order 1280.

1278 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄³×C₅	C₂	1	1	yes	maximal, characteristic
2	#10602327	C₂	1	14	yes	maximal
3	C₂²×C₄²×C₅	C₂²	1	7	yes	—
4	C₄³×C₅	C₄	1	8	yes	complemented
5	C₂×#3963	C₂²	1	28	yes	—
6	#14691	—	5	1	no	maximal, Sylow
7	C₂³×C₄×C₅	C₂³	1	7	yes	—
8	C₂³×Dic₅	C₂³	1	8	yes	—
9	C₂×C₄²×C₅	—	2	28	no	—
10	C₂×C₄²×C₅	C₂×C₄	1	28	yes	—
11	#3963	—	2	56	no	—
12	C₂×C₄³	D₅	1	1	yes	characteristic
13	#1434	—	5	14	no	—
14	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
15	C₂²×C₄×C₅	—	2	28	no	—
16	C₂²×C₄×C₅	C₂²×C₄	1	14	yes	—
17	C₂²×C₄×C₅	#41	1	28	yes	—
18	C₂²×Dic₅	—	2	56	no	—
19	C₄²×C₅	—	2	56	no	—
20	C₂²×C₄²	C₂×D₅	1	7	yes	—
21	C₂×#95	—	5	28	no	—
22	C₄³	Dic₅	1	8	yes	complemented
23	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
24	C₂³×C₅	C₂×#41	1	14	yes	—
25	C₂×C₄×C₅	—	2	28	no	—
26	C₂×C₄×C₅	—	2	56	no	—
27	C₂×C₄×C₅	#117	1	28	yes	—
28	C₂×Dic₅	—	4	56	no	—
29	C₂³×C₄	—	5	8	no	—
30	C₂³×C₄	C₂²×D₅	1	7	yes	—
31	C₂×C₄²	—	2	28	no	—
32	C₂×C₄²	C₂×Dic₅	1	28	yes	—
33	#95	—	10	56	no	—
34	C₂²×C₅	C₂×#117	1	7	yes	—
35	C₂²×C₅	#529	1	28	yes	—
36	C₄×C₅	—	2	28	no	—
37	Dic₅	—	8	8	no	—
38	C₂⁴	C₂³×D₅	1	1	yes	characteristic
39	C₂²×C₄	—	2	28	no	—
40	C₂²×C₄	—	10	56	no	—
41	C₂²×C₄	#709	1	28	yes	—
42	C₂²×C₄	C₂²×Dic₅	1	14	yes	—
43	C₄²	—	2	56	no	—
44	C₂×C₅	#2283	1	7	yes	—
45	C₂×C₅	#2866	1	8	yes	—
46	C₂³	C₂×#709	1	14	yes	—
47	C₂³	C₂³×Dic₅	1	1	yes	characteristic
48	C₂×C₄	—	2	56	no	—
49	C₂×C₄	—	2	28	no	—
50	C₂×C₄	—	20	56	no	—
51	C₂×C₄	#4072	1	28	yes	—
52	C₅	#14691	1	1	yes	characteristic, complemented, Sylow
53	C₂²	#65813	1	28	yes	—
54	C₂²	C₂×#4072	1	7	yes	—
55	C₄	—	2	28	no	—
56	C₄	—	40	8	no	—
57	C₂	#10605048	1	8	yes	—
58	C₂	#10613791	1	7	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₅
commutator	C₂³×C₅	C₂³×C₄
Fitting	C₂×C₄³×C₅	C₂
socle, prime radical, cosocle	C₂⁴×C₅	C₂⁴
hypercenter	C₂×C₄³	D₅
hypocenter	C₅	#14691
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. C₄³×C₅	C₄	semidirect
22. C₄³	Dic₅	semidirect
52. C₅	#14691	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄³×C₅	2
#10602327	2
#14691	5