Subgroups — #13100310

Order 1600.

224 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	#11716866	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#87764	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅²×#41	C₄	1	1	yes	characteristic
5	#87804	—	2	1	no	—
6	#87804	C₄	1	1	yes	characteristic
7	C₂×#6029	—	2	1	no	—
8	C₂×#6029	C₂²	1	1	yes	characteristic
9	#87764	—	2	1	no	—
10	#87764	—	2	1	no	—
11	#64932	—	5	6	no	maximal
12	C₅²×Q₈	C₈	1	1	yes	characteristic
13	C₅²×D₄	—	2	1	no	—
14	C₅²×D₄	C₈	1	1	yes	characteristic
15	C₂×C₄×C₅²	—	2	1	no	—
16	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
17	#6029	—	2	1	no	—
18	#6029	D₄	1	1	yes	characteristic
19	#6029	D₄	1	1	yes	characteristic
20	#4094	—	5	6	no	—
21	#3998	—	5	6	no	—
22	C₂×#674	—	5	6	no	—
23	C₂²×C₅²	—	2	1	no	—
24	C₂²×C₅²	M₁₆	1	1	yes	characteristic
25	C₄×C₅²	—	2	1	no	—
26	C₄×C₅²	C₂²⋊C₄	1	1	yes	characteristic
27	C₄×C₅²	C₂×C₈	1	1	yes	characteristic
28	C₅×#41	AGL(1,𝔽₅)	1	6	yes	—
29	#681	—	5	6	no	—
30	#681	—	10	6	no	—
31	C₂×#168	—	5	6	no	—
32	C₂×#168	—	10	6	no	—
33	#674	—	10	6	no	—
34	#674	—	10	6	no	—
35	#350	—	25	1	no	Sylow
36	C₂×C₅²	—	2	1	no	—
37	C₂×C₅²	—	4	1	no	—
38	C₂×C₅²	C₂²⋊C₈	1	1	yes	characteristic
39	C₅×Q₈	#170	1	6	yes	—
40	C₅×D₄	—	2	6	no	—
41	C₅×D₄	#170	1	6	yes	—
42	C₂×C₄×C₅	—	2	6	no	—
43	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	6	yes	—
44	#168	—	5	6	no	—
45	#168	—	5	6	no	—
46	#168	—	10	6	no	—
47	#131	—	25	1	no	—
48	M₃₂	—	25	1	no	—
49	C₂×C₁₆	—	25	1	no	—
50	C₅²	#350	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₅	—	2	6	no	—
52	C₂²×C₅	#704	1	6	yes	—
53	C₄×C₅	—	2	6	no	—
54	C₄×C₅	C₂×#170	1	6	yes	—
55	C₄×C₅	C₂²⋊AGL(1,𝔽₅)	1	6	yes	—
56	#41	#1043	1	1	yes	characteristic
57	M₁₆	—	25	1	no	—
58	M₁₆	—	50	1	no	—
59	C₂×C₈	—	25	1	no	—
60	C₂×C₈	—	50	1	no	—
61	C₁₆	—	50	1	no	—
62	C₁₆	—	50	1	no	—
63	C₂×C₅	—	2	6	no	—
64	C₂×C₅	—	4	6	no	—
65	C₂×C₅	C₂²⋊#170	1	6	yes	—
66	Q₈	#6033	1	1	yes	characteristic
67	D₄	—	2	1	no	—
68	D₄	#6033	1	1	yes	characteristic
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	C₂×#1043	1	1	yes	characteristic
71	C₈	—	25	1	no	—
72	C₈	—	25	1	no	—
73	C₈	—	50	1	no	—
74	C₅	#64932	1	6	yes	complemented
75	C₂²	—	2	1	no	—
76	C₂²	#87860	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	C₂×#6033	1	1	yes	characteristic
79	C₄	C₂²⋊#1043	1	1	yes	characteristic
80	C₂	—	2	1	no	—
81	C₂	—	4	1	no	—
82	C₂	C₂²⋊#6033	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊#1043
commutator	C₄×C₅²	C₂×C₈
Frattini	C₂×C₄	C₂×#1043
Fitting, prime radical	C₅²×#41	C₄
socle	C₂×C₅²	C₂²⋊C₈
cosocle	C₂×#6029	C₂²
hypercenter	#41	#1043
hypocenter	C₅²	#350
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
2. #11716866	C₂	semidirect
3. C₂×#87764	C₂	semidirect
50. C₅²	#350	semidirect
74. C₅	#64932	semidirect

Maximal subgroups

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

subgroup	index
#11717268	2
#11716866	2
C₂×#87764	2
#64932	5