Subgroups — Dic₅² #87777

Order 400.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1039	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×Dic₅	C₂	1	2	yes	maximal
3	C₂²×C₅²	C₂²	1	1	yes	characteristic
4	#1039	C₄	1	2	yes	complemented
5	C₅×Dic₅	C₄	1	4	yes	complemented
6	C₄×Dic₅	—	5	2	no	maximal
7	C₂×C₅²	C₂×C₄	1	1	yes	characteristic
8	C₂×C₅²	C₂×C₄	1	2	yes	—
9	C₂×C₄×C₅	—	5	2	no	—
10	C₂×Dic₅	—	5	2	no	—
11	C₂×Dic₅	—	10	2	no	—
12	C₂×Dic₅	D₅	1	2	yes	—
13	C₅²	C₄²	1	1	yes	characteristic, complemented, Sylow
14	C₂²×C₅	—	2	2	no	—
15	C₂²×C₅	C₂×D₅	1	2	yes	—
16	C₄×C₅	—	5	4	no	—
17	Dic₅	—	5	4	no	—
18	Dic₅	—	10	4	no	—
19	Dic₅	Dic₅	1	4	yes	direct product factor
20	C₄²	—	25	1	no	Sylow
21	C₂×C₅	—	2	4	no	—
22	C₂×C₅	—	2	2	no	—
23	C₂×C₅	C₄×D₅	1	2	yes	—
24	C₂×C₅	C₄×D₅	1	2	yes	—
25	C₂×C₅	C₂×Dic₅	1	2	yes	—
26	C₂×C₄	—	5	2	no	—
27	C₂×C₄	—	25	1	no	—
28	C₅	—	2	2	no	—
29	C₅	C₄×Dic₅	1	2	yes	complemented
30	C₂²	D₅²	1	1	yes	characteristic
31	C₄	—	5	4	no	—
32	C₄	—	25	2	no	—
33	C₂	D₅×Dic₅	1	2	yes	—
34	C₂	#6036	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂²	D₅²
commutator, hypocenter	C₅²	C₄²
Fitting, socle, prime radical, cosocle	C₂²×C₅²	C₂²
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂²

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

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