Subgroups — #125×C₅₂₃ #30660259

Order 16736.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊C₄×C₅₂₃	C₂	1	4	yes	maximal
2	C₄⋊C₄×C₅₂₃	C₂	1	2	yes	maximal
3	C₄²×C₅₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₅₂₃	C₂²	1	2	yes	—
5	C₂×C₄×C₅₂₃	C₂²	1	4	yes	—
6	C₂×C₄×C₅₂₃	C₂²	1	1	yes	characteristic
7	C₂²×C₅₂₃	C₂³	1	1	yes	characteristic
8	C₄×C₅₂₃	—	2	2	no	—
9	C₄×C₅₂₃	—	2	4	no	—
10	C₄×C₅₂₃	Q₈	1	2	yes	—
11	C₂×C₅₂₃	C₂×Q₈	1	1	yes	characteristic
12	C₂×C₅₂₃	#41	1	2	yes	—
13	C₅₂₃	#125	1	1	yes	characteristic, direct product factor, Sylow
14	#125	C₅₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₄⋊C₄	C₂×C₅₂₃	1	4	yes	—
16	C₄⋊C₄	C₂×C₅₂₃	1	2	yes	—
17	C₄²	C₂×C₅₂₃	1	1	yes	characteristic
18	C₂×C₄	C₂²×C₅₂₃	1	2	yes	—
19	C₂×C₄	C₂²×C₅₂₃	1	4	yes	—
20	C₂×C₄	C₂²×C₅₂₃	1	1	yes	characteristic
21	C₂²	C₂³×C₅₂₃	1	1	yes	characteristic
22	C₄	—	2	2	no	—
23	C₄	—	2	4	no	—
24	C₄	Q₈×C₅₂₃	1	2	yes	—
25	C₂	C₂×Q₈×C₅₂₃	1	1	yes	characteristic
26	C₂	#41×C₅₂₃	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₅₂₃	C₂³
commutator, Frattini, cosocle	C₂²	C₂³×C₅₂₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₂₃ ◁ G

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

N	H ≅ G/N	type
13. C₅₂₃	#125	direct (N × H)
14. #125	C₅₂₃	direct (N × H)

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