Subgroups — C₅×#6729 #11925265

Order 1200.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×SL(2,𝔽₅)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×GL(2,𝔽₃)	—	5	1	no	maximal
3	#6729	C₅	1	1	yes	maximal, characteristic, direct product factor
4	C₅×#170	—	6	1	no	maximal
5	C₅×#67	—	10	1	no	maximal
6	C₅×SL(2,𝔽₃)	—	5	1	no	—
7	SL(2,𝔽₅)	C₂×C₅	1	1	yes	characteristic, complemented
8	C₅×Dic₅	—	6	1	no	—
9	C₅×SD₁₆	—	15	1	no	—
10	C₂²×C₃×C₅	—	10	1	no	—
11	C₂×C₅×D₃	—	10	1	no	—
12	C₅×Dic₃	—	10	1	no	—
13	C₂×C₅²	—	6	1	no	—
14	GL(2,𝔽₃)	—	5	1	no	Hall
15	C₅×Q₈	—	5	1	no	—
16	C₅×D₄	—	15	1	no	—
17	#170	—	6	1	no	—
18	C₅×C₈	—	15	1	no	—
19	C₂×C₃×C₅	—	10	1	no	—
20	C₂×C₃×C₅	—	20	1	no	—
21	C₅×D₃	—	20	1	no	—
22	C₅²	—	6	1	no	Sylow
23	#67	—	10	1	no	—
24	SL(2,𝔽₃)	—	5	1	no	—
25	C₂²×C₅	—	10	1	no	—
26	C₄×C₅	—	15	1	no	—
27	Dic₅	—	6	1	no	—
28	SD₁₆	—	15	1	no	Sylow
29	C₃×C₅	—	10	1	no	—
30	C₂²×C₃	—	10	1	no	—
31	C₂×D₃	—	10	1	no	—
32	Dic₃	—	10	1	no	—
33	C₂×C₅	—	6	1	no	—
34	C₂×C₅	—	20	1	no	—
35	C₂×C₅	—	24	1	no	—
36	C₂×C₅	S₅	1	1	yes	characteristic
37	Q₈	—	5	1	no	—
38	D₄	—	15	1	no	—
39	C₈	—	15	1	no	—
40	C₂×C₃	—	10	1	no	—
41	C₂×C₃	—	20	1	no	—
42	D₃	—	20	1	no	—
43	C₅	—	6	1	no	—
44	C₅	—	24	1	no	—
45	C₅	#6729	1	1	yes	characteristic, direct product factor
46	C₂²	—	10	1	no	—
47	C₄	—	15	1	no	—
48	C₃	—	10	1	no	Sylow
49	C₂	—	20	1	no	—
50	C₂	C₅×S₅	1	1	yes	characteristic

	subgroup	G/H
center, Fitting, socle, hypercenter	C₂×C₅	S₅
commutator, cosocle, hypocenter	SL(2,𝔽₅)	C₂×C₅
Frattini	C₂	C₅×S₅
LCS	G ▷ SL(2,𝔽₅)
UCS	C₁ ◁ C₂×C₅

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.