Subgroups — #11716989

Order 800.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#1039	C₂	1	1	yes	maximal, characteristic
2	C₅×#683	C₂	1	2	yes	maximal
3	#87781	C₂	1	1	yes	maximal, characteristic
4	#87781	C₂	1	1	yes	maximal, characteristic
5	#87780	C₂	1	2	yes	maximal
6	C₂×C₄×C₅²	C₂²	1	1	yes	characteristic
7	C₂×#1039	C₂²	1	1	yes	characteristic
8	C₂×#1039	C₂²	1	1	yes	characteristic
9	C₂×C₅×Dic₅	C₂²	1	2	yes	—
10	C₂×C₅×Dic₅	C₂²	1	2	yes	—
11	#4035	—	5	2	no	maximal
12	C₂²×C₅²	C₂³	1	1	yes	characteristic
13	C₄×C₅²	—	2	1	no	—
14	#1039	—	2	1	no	—
15	#1039	Q₈	1	2	yes	—
16	C₅×Dic₅	—	2	2	no	—
17	C₅×Dic₅	—	2	2	no	—
18	C₅×C₄⋊C₄	—	5	2	no	—
19	C₄⋊Dic₅	—	5	2	no	—
20	#683	—	5	2	no	—
21	#683	—	5	2	no	—
22	#683	—	5	2	no	—
23	#683	D₅	1	2	yes	—
24	C₄×Dic₅	—	5	2	no	—
25	C₄×Dic₅	—	10	2	no	—
26	C₂×C₅²	C₂×Q₈	1	1	yes	characteristic
27	C₂×C₅²	#41	1	1	yes	characteristic
28	C₂×C₅²	#41	1	1	yes	characteristic
29	C₂×C₄×C₅	—	2	2	no	—
30	C₂×C₄×C₅	—	5	2	no	—
31	C₂×C₄×C₅	—	5	2	no	—
32	C₂×C₄×C₅	C₂×D₅	1	2	yes	—
33	C₂×Dic₅	—	5	2	no	—
34	C₂×Dic₅	—	5	2	no	—
35	C₂×Dic₅	—	10	2	no	—
36	C₂×Dic₅	—	10	2	no	—
37	C₂×Dic₅	C₂×D₅	1	2	yes	—
38	C₂×Dic₅	C₂×D₅	1	2	yes	—
39	#125	—	25	1	no	Sylow
40	C₅²	#125	1	1	yes	characteristic, complemented, Sylow
41	C₂²×C₅	—	2	2	no	—
42	C₂²×C₅	C₂²×D₅	1	2	yes	—
43	C₄×C₅	—	2	4	no	—
44	C₄×C₅	—	2	2	no	—
45	C₄×C₅	—	10	2	no	—
46	C₄×C₅	—	10	2	no	—
47	Dic₅	—	2	2	no	—
48	Dic₅	—	2	2	no	—
49	Dic₅	—	5	4	no	—
50	Dic₅	—	10	4	no	—
51	Dic₅	—	10	4	no	—
52	Dic₅	—	10	2	no	—
53	C₄⋊C₄	—	5	2	no	—
54	C₄⋊C₄	—	25	1	no	—
55	C₄⋊C₄	—	25	1	no	—
56	C₄⋊C₄	—	25	2	no	—
57	C₄²	—	25	1	no	—
58	C₂×C₅	—	2	2	no	—
59	C₂×C₅	—	2	2	no	—
60	C₂×C₅	—	2	2	no	—
61	C₂×C₅	#709	1	2	yes	—
62	C₂×C₅	#711	1	2	yes	—
63	C₂×C₅	Q₈×D₅	1	2	yes	—
64	C₂×C₄	—	5	2	no	—
65	C₂×C₄	—	5	2	no	—
66	C₂×C₄	—	25	1	no	—
67	C₂×C₄	—	25	1	no	—
68	C₂×C₄	D₅²	1	1	yes	characteristic
69	C₅	—	2	2	no	—
70	C₅	#4035	1	2	yes	complemented
71	C₂²	C₂×D₅²	1	1	yes	characteristic
72	C₄	—	2	1	no	—
73	C₄	—	10	2	no	—
74	C₄	—	10	2	no	—
75	C₄	—	25	2	no	—
76	C₄	—	50	1	no	—
77	C₂	#87872	1	1	yes	characteristic
78	C₂	#87873	1	1	yes	characteristic
79	C₂	#87880	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×D₅²
commutator, socle, prime radical, cosocle	C₂²×C₅²	C₂³
Fitting	C₂×C₄×C₅²	C₂²
hypercenter	C₂×C₄	D₅²
hypocenter	C₅²	#125
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
40. C₅²	#125	semidirect
70. C₅	#4035	semidirect

Maximal subgroups

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

subgroup	index
C₄×#1039	2
C₅×#683	2
#87781	2
#87780	2
#4035	5