Subgroups — #13104826

Order 1600.

480 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#1044	C₂	1	6	yes	maximal
2	C₄²×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
3	C₂²×#1044	C₂²	1	4	yes	—
4	C₂×C₄×C₅⋊D₅	C₂²	1	3	yes	—
5	C₄⋊#1044	—	2	12	no	—
6	C₄²×C₅²	C₄	1	1	yes	characteristic, complemented
7	C₄×#1039	C₄	1	3	yes	complemented
8	#65688	—	5	2	no	maximal
9	C₂²×C₅⋊D₅	C₂³	1	1	yes	characteristic
10	C₂×#1044	—	2	12	no	—
11	C₂×C₄×C₅²	C₂×C₄	1	3	yes	—
12	C₂×#1039	C₂×C₄	1	3	yes	—
13	C₄×C₅⋊D₅	D₄	1	6	yes	—
14	C₄×C₅⋊D₅	Q₈	1	6	yes	—
15	C₂×C₄⋊AGL(1,𝔽₅)	—	5	12	no	—
16	C₄²×D₅	—	5	2	no	—
17	C₄²×D₅	—	10	2	no	—
18	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
19	C₂×C₅⋊D₅	C₂×D₄	1	3	yes	—
20	C₂×C₅⋊D₅	C₂×Q₈	1	3	yes	—
21	#1044	—	4	4	no	—
22	C₄×C₅²	C₄⋊C₄	1	6	yes	complemented
23	#1039	C₄⋊C₄	1	6	yes	complemented
24	C₂²×AGL(1,𝔽₅)	—	5	8	no	—
25	C₂×C₄×D₅	—	5	6	no	—
26	C₂×C₄×D₅	—	10	6	no	—
27	C₄⋊AGL(1,𝔽₅)	—	10	24	no	—
28	C₄²×C₅	—	2	2	no	—
29	C₄²×C₅	AGL(1,𝔽₅)	1	2	yes	complemented
30	C₄×Dic₅	—	5	6	no	—
31	C₄×Dic₅	—	10	6	no	—
32	#384	—	25	1	no	Sylow
33	C₂×C₅²	C₂×C₄⋊C₄	1	3	yes	—
34	C₅⋊D₅	#127	1	1	yes	characteristic
35	C₅⋊D₅	C₄⋊Q₈	1	3	yes	—
36	C₂²×D₅	—	5	2	no	—
37	C₂²×D₅	—	10	2	no	—
38	C₂×AGL(1,𝔽₅)	—	10	24	no	—
39	C₂×C₄×C₅	—	2	6	no	—
40	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	6	yes	—
41	C₂×Dic₅	—	5	6	no	—
42	C₂×Dic₅	—	10	6	no	—
43	C₄×D₅	—	5	12	no	—
44	C₄×D₅	—	5	12	no	—
45	C₄×D₅	—	10	12	no	—
46	C₄×D₅	—	10	12	no	—
47	C₂×C₄⋊C₄	—	25	6	no	—
48	C₂×C₄²	—	25	1	no	—
49	C₅²	#384	1	1	yes	characteristic, complemented, Sylow
50	C₂²×C₅	—	2	2	no	—
51	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
52	C₂×D₅	—	5	6	no	—
53	C₂×D₅	—	5	6	no	—
54	C₂×D₅	—	10	6	no	—
55	C₂×D₅	—	10	6	no	—
56	AGL(1,𝔽₅)	—	20	8	no	—
57	C₄×C₅	—	2	12	no	—
58	C₄×C₅	C₄⋊AGL(1,𝔽₅)	1	12	yes	complemented
59	Dic₅	—	5	12	no	—
60	Dic₅	—	10	12	no	—
61	C₂²×C₄	—	25	3	no	—
62	C₂²×C₄	—	25	4	no	—
63	C₄⋊C₄	—	50	12	no	—
64	C₄²	—	25	3	no	—
65	C₄²	#1044	1	1	yes	characteristic, complemented
66	C₂×C₅	—	2	6	no	—
67	C₂×C₅	C₂×C₄⋊AGL(1,𝔽₅)	1	6	yes	—
68	D₅	—	5	2	no	—
69	D₅	—	5	6	no	—
70	D₅	—	10	2	no	—
71	D₅	—	10	6	no	—
72	C₂³	—	25	1	no	—
73	C₂×C₄	—	25	3	no	—
74	C₂×C₄	—	25	6	no	—
75	C₂×C₄	—	25	6	no	—
76	C₂×C₄	—	50	12	no	—
77	C₂×C₄	C₂×#1044	1	3	yes	—
78	C₅	—	2	2	no	—
79	C₅	#65688	1	2	yes	complemented
80	C₂²	—	25	3	no	—
81	C₂²	—	25	3	no	—
82	C₂²	C₂²×#1044	1	1	yes	characteristic
83	C₄	—	25	6	no	—
84	C₄	—	100	4	no	—
85	C₄	C₄⋊#1044	1	6	yes	complemented
86	C₂	—	25	1	no	—
87	C₂	—	25	3	no	—
88	C₂	C₂×C₄⋊#1044	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#1044
commutator, socle	C₂²×C₅²	C₂²×C₄
Fitting	C₄²×C₅²	C₄
prime radical, cosocle	C₂²×C₅⋊D₅	C₂³
hypercenter	C₄²	#1044
hypocenter	C₅²	#384
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
6. C₄²×C₅²	C₄	semidirect
7. C₄×#1039	C₄	semidirect
22. C₄×C₅²	C₄⋊C₄	semidirect
23. #1039	C₄⋊C₄	semidirect
29. C₄²×C₅	AGL(1,𝔽₅)	semidirect
49. C₅²	#384	semidirect
58. C₄×C₅	C₄⋊AGL(1,𝔽₅)	semidirect
65. C₄²	#1044	semidirect
79. C₅	#65688	semidirect
85. C₄	C₄⋊#1044	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#1044	2
C₄²×C₅⋊D₅	2
#65688	5