Subgroups — #13104929

Order 1600.

230 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11717268	C₂	1	1	yes	maximal, characteristic
2	#11716891	C₂	1	3	yes	maximal, complemented
3	C₂×#87765	C₂	1	3	yes	maximal, complemented
4	C₅²×#41	C₄	1	1	yes	characteristic
5	#87804	C₄	1	3	yes	—
6	C₂×#6029	C₂²	1	3	yes	—
7	#87765	C₂²	1	1	yes	characteristic
8	#87765	C₂²	1	3	yes	—
9	#65791	—	5	2	no	maximal
10	C₅²×Q₈	C₈	1	1	yes	characteristic
11	C₅²×D₄	C₈	1	3	yes	—
12	C₂×C₄×C₅²	C₂×C₄	1	3	yes	—
13	#6029	C₂×C₄	1	3	yes	—
14	#6029	C₂³	1	1	yes	characteristic
15	#4094	—	5	2	no	—
16	#4094	—	10	2	no	—
17	#3998	—	5	6	no	—
18	C₂×#674	—	5	6	no	—
19	C₂²×C₅²	C₂×C₈	1	3	yes	—
20	C₄×C₅²	C₂×C₈	1	3	yes	—
21	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
22	C₅×#41	—	2	2	no	—
23	C₅×#41	AGL(1,𝔽₅)	1	2	yes	—
24	#681	—	5	6	no	—
25	#681	—	10	6	no	—
26	C₂×#168	—	5	6	no	—
27	C₂×#168	—	10	6	no	—
28	#674	—	5	2	no	—
29	#674	—	5	6	no	—
30	#504	—	25	1	no	Sylow
31	C₂×C₅²	—	2	3	no	—
32	C₂×C₅²	C₂²×C₈	1	1	yes	characteristic
33	C₅×Q₈	—	2	2	no	—
34	C₅×Q₈	#170	1	2	yes	—
35	C₅×D₄	—	2	6	no	—
36	C₅×D₄	#170	1	6	yes	—
37	C₂×C₄×C₅	—	2	6	no	—
38	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	6	yes	—
39	#168	—	5	6	no	—
40	#168	—	5	2	no	—
41	#168	—	10	6	no	—
42	#168	—	10	2	no	—
43	#131	—	25	1	no	—
44	M₃₂	—	25	3	no	—
45	C₂×C₁₆	—	25	3	no	—
46	C₅²	#504	1	1	yes	characteristic, complemented, Sylow
47	C₂²×C₅	—	2	6	no	—
48	C₂²×C₅	C₂×#170	1	6	yes	—
49	C₄×C₅	—	2	6	no	—
50	C₄×C₅	—	2	2	no	—
51	C₄×C₅	C₂×#170	1	6	yes	—
52	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
53	#41	#1044	1	1	yes	characteristic
54	M₁₆	—	25	3	no	—
55	C₂×C₈	—	25	3	no	—
56	C₁₆	—	25	1	no	—
57	C₁₆	—	25	3	no	—
58	C₂×C₅	—	2	6	no	—
59	C₂×C₅	—	2	2	no	—
60	C₂×C₅	—	4	6	no	—
61	C₂×C₅	C₂²×#170	1	2	yes	—
62	Q₈	#6034	1	1	yes	characteristic
63	D₄	#6034	1	3	yes	—
64	C₂×C₄	C₂×#1044	1	3	yes	—
65	C₈	—	25	1	no	—
66	C₈	—	25	3	no	—
67	C₅	—	2	2	no	—
68	C₅	#65791	1	2	yes	complemented
69	C₂²	C₂×#6034	1	3	yes	—
70	C₄	C₂×#6034	1	3	yes	—
71	C₄	C₂²×#1044	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	C₂²×#6034	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×#1044
commutator, socle	C₂×C₅²	C₂²×C₈
Fitting, prime radical	C₅²×#41	C₄
cosocle	#6029	C₂³
hypercenter	#41	#1044
hypocenter	C₅²	#504
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
2. #11716891	C₂	semidirect
3. C₂×#87765	C₂	semidirect
46. C₅²	#504	semidirect
68. C₅	#65791	semidirect

Maximal subgroups

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

subgroup	index
#11717268	2
#11716891	2
C₂×#87765	2
#65791	5