Subgroups — C₅²×#384 #13103121

Order 1600.

744 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×C₄⋊C₄	C₂	1	6	yes	maximal
2	C₂×C₄²×C₅²	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₅²	C₂²	1	4	yes	—
4	C₂²×C₄×C₅²	C₂²	1	3	yes	—
5	C₅²×C₄⋊C₄	—	2	12	no	—
6	C₄²×C₅²	C₄	1	4	yes	complemented
7	C₅×#384	C₅	1	6	yes	maximal, direct product factor
8	C₂³×C₅²	C₂³	1	1	yes	characteristic
9	C₂×C₄×C₅²	—	2	12	no	—
10	C₂×C₄×C₅²	C₂×C₄	1	6	yes	—
11	C₂×C₄×C₅²	D₄	1	6	yes	—
12	C₂×C₄×C₅²	Q₈	1	6	yes	—
13	C₂×C₅×C₄⋊C₄	C₂×C₅	1	36	yes	—
14	C₂×C₄²×C₅	C₂×C₅	1	6	yes	—
15	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
16	C₂²×C₅²	C₂×D₄	1	3	yes	—
17	C₂²×C₅²	C₂×Q₈	1	3	yes	—
18	C₄×C₅²	—	4	4	no	—
19	C₄×C₅²	C₄⋊C₄	1	12	yes	complemented
20	C₂²×C₄×C₅	C₂²×C₅	1	24	yes	—
21	C₂²×C₄×C₅	C₂²×C₅	1	18	yes	—
22	C₅×C₄⋊C₄	—	2	72	no	—
23	C₄²×C₅	C₄×C₅	1	24	yes	complemented
24	#384	C₅²	1	1	yes	characteristic, direct product factor, Sylow
25	C₂×C₅²	C₂×C₄⋊C₄	1	3	yes	—
26	C₂×C₅²	#127	1	1	yes	characteristic
27	C₂×C₅²	C₄⋊Q₈	1	3	yes	—
28	C₂³×C₅	C₂³×C₅	1	6	yes	—
29	C₂×C₄×C₅	—	2	72	no	—
30	C₂×C₄×C₅	C₂×C₄×C₅	1	36	yes	—
31	C₂×C₄×C₅	C₅×D₄	1	36	yes	—
32	C₂×C₄×C₅	C₅×Q₈	1	36	yes	—
33	C₂×C₄⋊C₄	C₂×C₅²	1	6	yes	—
34	C₂×C₄²	C₂×C₅²	1	1	yes	characteristic
35	C₅²	#384	1	1	yes	characteristic, direct product factor, Sylow
36	C₂²×C₅	C₂²×C₄×C₅	1	6	yes	—
37	C₂²×C₅	C₂×C₅×D₄	1	18	yes	—
38	C₂²×C₅	C₂×C₅×Q₈	1	18	yes	—
39	C₄×C₅	—	4	24	no	—
40	C₄×C₅	C₅×C₄⋊C₄	1	72	yes	complemented
41	C₂²×C₄	C₂²×C₅²	1	4	yes	—
42	C₂²×C₄	C₂²×C₅²	1	3	yes	—
43	C₄⋊C₄	—	2	12	no	—
44	C₄²	C₄×C₅²	1	4	yes	complemented
45	C₂×C₅	C₂×C₅×C₄⋊C₄	1	18	yes	—
46	C₂×C₅	C₅×#127	1	6	yes	—
47	C₂×C₅	C₅×C₄⋊Q₈	1	18	yes	—
48	C₂³	C₂³×C₅²	1	1	yes	characteristic
49	C₂×C₄	—	2	12	no	—
50	C₂×C₄	C₂×C₄×C₅²	1	6	yes	—
51	C₂×C₄	C₅²×D₄	1	6	yes	—
52	C₂×C₄	C₅²×Q₈	1	6	yes	—
53	C₅	C₅×#384	1	6	yes	direct product factor
54	C₂²	C₂²×C₄×C₅²	1	1	yes	characteristic
55	C₂²	C₂×C₅²×D₄	1	3	yes	—
56	C₂²	C₂×C₅²×Q₈	1	3	yes	—
57	C₄	—	4	4	no	—
58	C₄	C₅²×C₄⋊C₄	1	12	yes	complemented
59	C₂	C₂×C₅²×C₄⋊C₄	1	3	yes	—
60	C₂	C₅²×#127	1	1	yes	characteristic
61	C₂	C₅²×C₄⋊Q₈	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅²	C₂³
commutator	C₂²	C₂²×C₄×C₅²
Frattini, cosocle	C₂³	C₂³×C₅²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅² ◁ G

Complements

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

N	H ≅ G/N	type
6. C₄²×C₅²	C₄	semidirect
7. C₅×#384	C₅	direct (N × H)
19. C₄×C₅²	C₄⋊C₄	semidirect
23. C₄²×C₅	C₄×C₅	semidirect
24. #384	C₅²	direct (N × H)
35. C₅²	#384	direct (N × H)
40. C₄×C₅	C₅×C₄⋊C₄	semidirect
44. C₄²	C₄×C₅²	semidirect
53. C₅	C₅×#384	direct (N × H)
58. C₄	C₅²×C₄⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅²×C₄⋊C₄	2
C₂×C₄²×C₅²	2
C₅×#384	5

Subgroups — C52×#384 #13103121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×#384 #13103121