Subgroups — #3573×C₂₂₃ #43252588

Order 28544.

1424 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#585×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	#568×C₂₂₃	C₂	1	15	yes	maximal, complemented
3	C₂×#131×C₂₂₃	C₂	1	15	yes	maximal, complemented
4	2₋¹⁺⁴×C₂₂₃	C₄	1	6	yes	—
5	H(5,𝔽₂)×C₂₂₃	C₄	1	10	yes	—
6	C₂×#41×C₂₂₃	C₂²	1	15	yes	—
7	#131×C₂₂₃	C₂²	1	20	yes	complemented
8	#131×C₂₂₃	C₂²	1	60	yes	complemented
9	C₂×M₁₆×C₂₂₃	C₂²	1	45	yes	complemented
10	C₂²×C₈×C₂₂₃	C₂²	1	15	yes	complemented
11	#41×C₂₂₃	C₂×C₄	1	60	yes	—
12	#41×C₂₂₃	C₂³	1	20	yes	—
13	C₂×Q₈×C₂₂₃	C₂×C₄	1	15	yes	—
14	C₂×D₄×C₂₂₃	C₂×C₄	1	45	yes	—
15	C₂²×C₄×C₂₂₃	C₂³	1	15	yes	—
16	M₁₆×C₂₂₃	C₂³	1	60	yes	—
17	C₂×C₈×C₂₂₃	C₂³	1	45	yes	—
18	C₂×C₈×C₂₂₃	C₂³	1	15	yes	—
19	C₂³×C₂₂₃	C₂²×C₄	1	15	yes	—
20	Q₈×C₂₂₃	C₂²×C₄	1	20	yes	—
21	D₄×C₂₂₃	C₂²×C₄	1	60	yes	—
22	C₂×C₄×C₂₂₃	C₂²×C₄	1	45	yes	—
23	C₂×C₄×C₂₂₃	C₂⁴	1	15	yes	—
24	C₈×C₂₂₃	C₂⁴	1	15	yes	—
25	C₈×C₂₂₃	C₂⁴	1	1	yes	characteristic
26	C₂²×C₂₂₃	—	4	15	no	—
27	C₂²×C₂₂₃	C₂³×C₄	1	15	yes	—
28	C₄×C₂₂₃	C₂³×C₄	1	15	yes	—
29	C₄×C₂₂₃	C₂⁵	1	1	yes	characteristic
30	C₂×C₂₂₃	—	2	15	no	—
31	C₂×C₂₂₃	C₂⁴×C₄	1	1	yes	characteristic
32	C₂₂₃	#3573	1	1	yes	characteristic, direct product factor, Sylow
33	#3573	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	#585	C₂×C₂₂₃	1	1	yes	characteristic
35	#568	C₂×C₂₂₃	1	15	yes	complemented
36	C₂×#131	C₂×C₂₂₃	1	15	yes	complemented
37	2₋¹⁺⁴	C₄×C₂₂₃	1	6	yes	—
38	H(5,𝔽₂)	C₄×C₂₂₃	1	10	yes	—
39	C₂×#41	C₂²×C₂₂₃	1	15	yes	—
40	#131	C₂²×C₂₂₃	1	20	yes	complemented
41	#131	C₂²×C₂₂₃	1	60	yes	complemented
42	C₂×M₁₆	C₂²×C₂₂₃	1	45	yes	complemented
43	C₂²×C₈	C₂²×C₂₂₃	1	15	yes	complemented
44	#41	C₂×C₄×C₂₂₃	1	60	yes	—
45	#41	C₂³×C₂₂₃	1	20	yes	—
46	C₂×Q₈	C₂×C₄×C₂₂₃	1	15	yes	—
47	C₂×D₄	C₂×C₄×C₂₂₃	1	45	yes	—
48	C₂²×C₄	C₂³×C₂₂₃	1	15	yes	—
49	M₁₆	C₂³×C₂₂₃	1	60	yes	—
50	C₂×C₈	C₂³×C₂₂₃	1	15	yes	—
51	C₂×C₈	C₂³×C₂₂₃	1	45	yes	—
52	C₂³	C₂²×C₄×C₂₂₃	1	15	yes	—
53	Q₈	C₂²×C₄×C₂₂₃	1	20	yes	—
54	D₄	C₂²×C₄×C₂₂₃	1	60	yes	—
55	C₂×C₄	C₂²×C₄×C₂₂₃	1	45	yes	—
56	C₂×C₄	C₂⁴×C₂₂₃	1	15	yes	—
57	C₈	C₂⁴×C₂₂₃	1	1	yes	characteristic
58	C₈	C₂⁴×C₂₂₃	1	15	yes	—
59	C₂²	—	4	15	no	—
60	C₂²	C₂³×C₄×C₂₂₃	1	15	yes	—
61	C₄	C₂³×C₄×C₂₂₃	1	15	yes	—
62	C₄	C₂⁵×C₂₂₃	1	1	yes	characteristic
63	C₂	—	2	15	no	—
64	C₂	C₂⁴×C₄×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₂₂₃	C₂⁴
commutator	C₂	C₂⁴×C₄×C₂₂₃
Frattini, cosocle	C₄	C₂⁵×C₂₂₃
socle	C₂×C₂₂₃	C₂⁴×C₄
prime radical	#585×C₂₂₃	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. #568×C₂₂₃	C₂	semidirect
3. C₂×#131×C₂₂₃	C₂	semidirect
7. #131×C₂₂₃	C₂²	semidirect
8. #131×C₂₂₃	C₂²	semidirect
9. C₂×M₁₆×C₂₂₃	C₂²	semidirect
10. C₂²×C₈×C₂₂₃	C₂²	semidirect
32. C₂₂₃	#3573	direct (N × H)
33. #3573	C₂₂₃	direct (N × H)
35. #568	C₂×C₂₂₃	semidirect
36. C₂×#131	C₂×C₂₂₃	semidirect
40. #131	C₂²×C₂₂₃	semidirect
41. #131	C₂²×C₂₂₃	semidirect
42. C₂×M₁₆	C₂²×C₂₂₃	semidirect
43. C₂²×C₈	C₂²×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#585×C₂₂₃	2
#568×C₂₂₃	2
C₂×#131×C₂₂₃	2
#3573	223

Subgroups — #3573×C223 #43252588

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3573×C₂₂₃ #43252588