Subgroups — #13100331

Order 1600.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11716891	C₂	1	2	yes	maximal
2	C₄×#6029	C₂	1	1	yes	maximal, characteristic
3	C₄²×C₅²	C₄	1	1	yes	characteristic
4	C₂×#6029	C₄	1	1	yes	characteristic
5	C₂×#6029	C₂²	1	1	yes	characteristic
6	#87765	—	2	2	no	—
7	#64861	—	5	2	no	maximal
8	C₂×C₄×C₅²	—	2	1	no	—
9	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
10	#6029	C₈	1	2	yes	—
11	#6029	D₄	1	1	yes	characteristic
12	#6029	Q₈	1	1	yes	characteristic
13	#3998	—	5	4	no	—
14	C₄×#168	—	5	2	no	—
15	C₄×#168	—	10	2	no	—
16	C₂²×C₅²	M₁₆	1	1	yes	characteristic
17	C₄×C₅²	—	2	2	no	—
18	C₄×C₅²	C₄⋊C₄	1	1	yes	characteristic
19	C₄×C₅²	C₂×C₈	1	1	yes	characteristic
20	C₄²×C₅	—	2	2	no	—
21	C₄²×C₅	AGL(1,𝔽₅)	1	2	yes	—
22	C₂×#168	—	5	2	no	—
23	C₂×#168	—	5	2	no	—
24	C₂×#168	—	10	2	no	—
25	C₂×#168	—	10	2	no	—
26	#674	—	10	4	no	—
27	#364	—	25	1	no	Sylow
28	C₂×C₅²	—	2	1	no	—
29	C₂×C₅²	C₄⋊C₈	1	1	yes	characteristic
30	C₂×C₄×C₅	—	2	4	no	—
31	C₂×C₄×C₅	—	2	2	no	—
32	C₂×C₄×C₅	—	2	2	no	—
33	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
34	#168	—	5	4	no	—
35	#168	—	5	2	no	—
36	#168	—	5	2	no	—
37	#168	—	10	4	no	—
38	#168	—	10	2	no	—
39	#168	—	10	2	no	—
40	M₃₂	—	25	2	no	—
41	C₄×C₈	—	25	1	no	—
42	C₅²	#364	1	1	yes	characteristic, complemented, Sylow
43	C₂²×C₅	—	2	2	no	—
44	C₂²×C₅	#704	1	2	yes	—
45	C₄×C₅	—	2	8	no	—
46	C₄×C₅	—	2	4	no	—
47	C₄×C₅	—	2	2	no	—
48	C₄×C₅	—	2	2	no	—
49	C₄×C₅	D₅⋊C₈	1	2	yes	—
50	C₄×C₅	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
51	C₂×C₈	—	25	1	no	—
52	C₂×C₈	—	25	1	no	—
53	C₄²	#1044	1	1	yes	characteristic
54	C₁₆	—	50	2	no	—
55	C₂×C₅	—	2	4	no	—
56	C₂×C₅	—	2	2	no	—
57	C₂×C₅	—	2	2	no	—
58	C₂×C₅	#4004	1	2	yes	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	C₂×#1044	1	1	yes	characteristic
61	C₈	—	25	2	no	—
62	C₈	—	25	1	no	—
63	C₈	—	25	1	no	—
64	C₅	—	2	2	no	—
65	C₅	#64861	1	2	yes	complemented
66	C₂²	#87867	1	1	yes	characteristic
67	C₄	—	2	2	no	—
68	C₄	#87862	1	1	yes	characteristic
69	C₄	C₄⋊#1044	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	#11716897	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄⋊#1044
commutator	C₄×C₅²	C₂×C₈
Frattini	C₂×C₄	C₂×#1044
Fitting	C₄²×C₅²	C₄
socle	C₂×C₅²	C₄⋊C₈
prime radical	C₂²×C₅²	M₁₆
cosocle	C₂×#6029	C₂²
hypercenter	C₄²	#1044
hypocenter	C₅²	#364
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
42. C₅²	#364	semidirect
65. C₅	#64861	semidirect

Maximal subgroups

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

subgroup	index
#11716891	2
C₄×#6029	2
#64861	5