Subgroups — C₅²×H(3,𝔽₅)⋊C₅ #30626921

Order 15625.

11632 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅⁵	C₅	1	1	yes	maximal, characteristic, complemented
2	C₅²×H(3,𝔽₅)	C₅	1	5	yes	maximal, complemented
3	C₅×H(3,𝔽₅)⋊C₅	C₅	1	150	yes	maximal, direct product factor
4	C₅⁴	—	5	125	no	—
5	C₅⁴	—	5	24	no	—
6	C₅⁴	—	5	1	no	—
7	C₅⁴	—	5	5	no	—
8	C₅⁴	C₅²	1	1	yes	characteristic
9	C₅⁴	C₅²	1	30	yes	complemented
10	C₅×H(3,𝔽₅)	—	5	120	no	—
11	C₅×H(3,𝔽₅)	C₅²	1	150	yes	complemented
12	H(3,𝔽₅)⋊C₅	C₅²	1	625	yes	direct product factor
13	C₅³	—	5	750	no	—
14	C₅³	—	5	3000	no	—
15	C₅³	—	5	125	no	—
16	C₅³	—	5	25	no	—
17	C₅³	—	5	120	no	—
18	C₅³	—	5	30	no	—
19	C₅³	—	5	150	no	—
20	C₅³	—	25	125	no	—
21	C₅³	H(3,𝔽₅)	1	24	yes	—
22	C₅³	H(3,𝔽₅)	1	1	yes	characteristic
23	C₅³	C₅³	1	6	yes	—
24	C₅³	C₅³	1	25	yes	complemented
25	H(3,𝔽₅)	—	5	600	no	—
26	H(3,𝔽₅)	C₅³	1	125	yes	complemented
27	C₅²	—	5	125	no	—
28	C₅²	—	5	3000	no	—
29	C₅²	—	5	750	no	—
30	C₅²	—	5	120	no	—
31	C₅²	—	5	30	no	—
32	C₅²	—	5	25	no	—
33	C₅²	—	5	125	no	—
34	C₅²	—	25	750	no	—
35	C₅²	H(3,𝔽₅)⋊C₅	1	25	yes	direct product factor
36	C₅²	C₅×H(3,𝔽₅)	1	24	yes	—
37	C₅²	C₅×H(3,𝔽₅)	1	6	yes	—
38	C₅²	C₅⁴	1	1	yes	characteristic
39	C₅	—	5	125	no	—
40	C₅	—	5	24	no	—
41	C₅	—	5	1	no	—
42	C₅	—	25	125	no	—
43	C₅	C₅×H(3,𝔽₅)⋊C₅	1	30	yes	direct product factor
44	C₅	C₅²×H(3,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₅³	H(3,𝔽₅)
commutator, Frattini, cosocle	C₅²	C₅⁴
LCS	G ▷ C₅² ▷ C₅ ▷ C₁
UCS	C₁ ◁ C₅³ ◁ C₅⁴ ◁ G
LECS	G ▷ C₅² ▷ C₅ ▷ C₁

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.