Subgroups — #585×C₂₂₃ #28383297

Order 14272.

810 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	2₋¹⁺⁴×C₂₂₃	C₂	1	6	yes	maximal, complemented
2	H(5,𝔽₂)×C₂₂₃	C₂	1	10	yes	maximal, complemented
3	C₂×#41×C₂₂₃	C₂	1	15	yes	maximal, complemented
4	#41×C₂₂₃	C₂²	1	20	yes	complemented
5	#41×C₂₂₃	C₂²	1	60	yes	complemented
6	C₂×Q₈×C₂₂₃	C₂²	1	15	yes	complemented
7	C₂×D₄×C₂₂₃	C₂²	1	45	yes	complemented
8	C₂²×C₄×C₂₂₃	C₂²	1	15	yes	complemented
9	C₂³×C₂₂₃	C₂³	1	15	yes	—
10	Q₈×C₂₂₃	C₂³	1	20	yes	—
11	D₄×C₂₂₃	C₂³	1	60	yes	—
12	C₂×C₄×C₂₂₃	C₂³	1	45	yes	—
13	C₂×C₄×C₂₂₃	C₂³	1	15	yes	—
14	C₂²×C₂₂₃	—	4	15	no	—
15	C₂²×C₂₂₃	C₂⁴	1	15	yes	—
16	C₄×C₂₂₃	C₂⁴	1	15	yes	—
17	C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
18	C₂×C₂₂₃	—	2	15	no	—
19	C₂×C₂₂₃	C₂⁵	1	1	yes	characteristic
20	C₂₂₃	#585	1	1	yes	characteristic, direct product factor, Sylow
21	#585	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	2₋¹⁺⁴	C₂×C₂₂₃	1	6	yes	complemented
23	H(5,𝔽₂)	C₂×C₂₂₃	1	10	yes	complemented
24	C₂×#41	C₂×C₂₂₃	1	15	yes	complemented
25	#41	C₂²×C₂₂₃	1	20	yes	complemented
26	#41	C₂²×C₂₂₃	1	60	yes	complemented
27	C₂×Q₈	C₂²×C₂₂₃	1	15	yes	complemented
28	C₂×D₄	C₂²×C₂₂₃	1	45	yes	complemented
29	C₂²×C₄	C₂²×C₂₂₃	1	15	yes	complemented
30	C₂³	C₂³×C₂₂₃	1	15	yes	—
31	Q₈	C₂³×C₂₂₃	1	20	yes	—
32	D₄	C₂³×C₂₂₃	1	60	yes	—
33	C₂×C₄	C₂³×C₂₂₃	1	15	yes	—
34	C₂×C₄	C₂³×C₂₂₃	1	45	yes	—
35	C₂²	—	4	15	no	—
36	C₂²	C₂⁴×C₂₂₃	1	15	yes	—
37	C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
38	C₄	C₂⁴×C₂₂₃	1	15	yes	—
39	C₂	—	2	15	no	—
40	C₂	C₂⁵×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₂₃	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁵×C₂₂₃
socle	C₂×C₂₂₃	C₂⁵
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. 2₋¹⁺⁴×C₂₂₃	C₂	semidirect
2. H(5,𝔽₂)×C₂₂₃	C₂	semidirect
3. C₂×#41×C₂₂₃	C₂	semidirect
4. #41×C₂₂₃	C₂²	semidirect
5. #41×C₂₂₃	C₂²	semidirect
6. C₂×Q₈×C₂₂₃	C₂²	semidirect
7. C₂×D₄×C₂₂₃	C₂²	semidirect
8. C₂²×C₄×C₂₂₃	C₂²	semidirect
20. C₂₂₃	#585	direct (N × H)
21. #585	C₂₂₃	direct (N × H)
22. 2₋¹⁺⁴	C₂×C₂₂₃	semidirect
23. H(5,𝔽₂)	C₂×C₂₂₃	semidirect
24. C₂×#41	C₂×C₂₂₃	semidirect
25. #41	C₂²×C₂₂₃	semidirect
26. #41	C₂²×C₂₂₃	semidirect
27. C₂×Q₈	C₂²×C₂₂₃	semidirect
28. C₂×D₄	C₂²×C₂₂₃	semidirect
29. C₂²×C₄	C₂²×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
2₋¹⁺⁴×C₂₂₃	2
H(5,𝔽₂)×C₂₂₃	2
C₂×#41×C₂₂₃	2
#585	223

Subgroups — #585×C223 #28383297

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #585×C₂₂₃ #28383297