Subgroups — #13099519

Order 1600.

256 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11717189	C₂	1	1	yes	maximal, characteristic
2	C₅²×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#6029	C₂	1	1	yes	maximal, characteristic
4	C₂×#6045	C₄	1	2	yes	—
5	C₂×C₅²×C₈	—	2	1	no	—
6	C₄²×C₅²	C₂²	1	1	yes	characteristic
7	C₄⋊#1039	—	2	1	no	—
8	C₄⋊#1039	—	2	1	no	—
9	C₂×#6029	—	2	1	no	—
10	#64701	—	5	6	no	maximal
11	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
13	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
14	C₂×#1039	—	2	2	no	—
15	#6045	—	4	2	no	—
16	C₅²×C₈	—	4	1	no	—
17	#6029	—	4	1	no	—
18	#4015	—	5	6	no	—
19	C₅×C₄⋊C₈	D₅	1	6	yes	—
20	C₄⋊#168	—	5	6	no	—
21	C₂²×C₅²	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₅²	—	2	1	no	—
23	C₄×C₅²	SD₁₆	1	1	yes	characteristic
24	C₄×C₅²	SD₁₆	1	1	yes	characteristic
25	C₄×C₅²	Q₁₆	1	1	yes	characteristic
26	C₄×C₅²	Q₁₆	1	1	yes	characteristic
27	#1039	—	4	2	no	—
28	C₂×Dic₁₀	—	5	12	no	—
29	C₂×C₅×C₈	—	2	6	no	—
30	C₄²×C₅	C₂×D₅	1	6	yes	—
31	C₄⋊Dic₅	—	10	6	no	—
32	C₄⋊Dic₅	—	10	6	no	—
33	C₂×#168	—	10	6	no	—
34	#333	—	25	1	no	Sylow
35	C₂×C₅²	#100	1	1	yes	characteristic
36	C₂×C₅²	#103	1	1	yes	characteristic
37	C₂×C₅²	#103	1	1	yes	characteristic
38	C₂×C₄×C₅	C₄×D₅	1	6	yes	—
39	C₂×C₄×C₅	D₂₀	1	6	yes	—
40	C₂×C₄×C₅	C₂²⋊D₅	1	6	yes	—
41	C₂×Dic₅	—	10	12	no	—
42	Dic₁₀	—	20	12	no	—
43	C₅×C₈	—	4	6	no	—
44	#168	—	20	6	no	—
45	C₄⋊Q₈	—	25	1	no	—
46	C₄⋊C₈	—	25	1	no	—
47	C₄⋊C₈	C₅⋊D₅	1	1	yes	characteristic
48	C₅²	#333	1	1	yes	characteristic, complemented, Sylow
49	C₂²×C₅	#685	1	6	yes	—
50	C₄×C₅	—	2	6	no	—
51	C₄×C₅	#677	1	6	yes	—
52	C₄×C₅	Dic₂₀	1	6	yes	—
53	C₄×C₅	#687	1	6	yes	—
54	C₄×C₅	#689	1	6	yes	—
55	Dic₅	—	20	12	no	—
56	C₂×Q₈	—	25	2	no	—
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	50	1	no	—
59	C₄⋊C₄	—	50	1	no	—
60	C₄⋊C₄	—	50	1	no	—
61	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
62	C₂×C₅	#3942	1	6	yes	—
63	C₂×C₅	#3948	1	6	yes	—
64	C₂×C₅	#3955	1	6	yes	—
65	Q₈	—	100	2	no	—
66	C₂×C₄	—	50	2	no	—
67	C₂×C₄	C₄×C₅⋊D₅	1	1	yes	characteristic
68	C₂×C₄	C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
69	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
70	C₈	—	4	1	no	—
71	C₈	—	100	1	no	—
72	C₅	#64701	1	6	yes	complemented
73	C₂²	#87808	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	100	2	no	—
76	C₄	#87800	1	1	yes	characteristic
77	C₄	#87802	1	1	yes	characteristic
78	C₄	#87810	1	1	yes	characteristic
79	C₄	#87812	1	1	yes	characteristic
80	C₂	#11716677	1	1	yes	characteristic
81	C₂	#11716683	1	1	yes	characteristic
82	C₂	#11716690	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#87808
commutator	C₂×C₄×C₅²	C₂×C₄
Frattini	C₄²	C₂×C₅⋊D₅
Fitting	C₅²×C₄⋊C₈	C₂
socle, prime radical	C₂²×C₅²	C₂²⋊C₄
cosocle	C₄²×C₅²	C₂²
hypercenter	C₄⋊C₈	C₅⋊D₅
hypocenter	C₅²	#333
LCS	G ▷ C₂×C₄×C₅² ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
48. C₅²	#333	semidirect
72. C₅	#64701	semidirect

Maximal subgroups

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

subgroup	index
#11717189	2
C₅²×C₄⋊C₈	2
C₄⋊#6029	2
#64701	5