Subgroups — #11717426

Order 800.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#711	C₂	1	1	yes	maximal, characteristic
2	#87853	C₂	1	2	yes	maximal, complemented
3	C₂×#6032	C₂	1	2	yes	maximal, complemented
4	#87849	C₂	1	1	yes	maximal, characteristic, complemented
5	D₅⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅²×D₄	C₄	1	1	yes	characteristic
7	C₅×C₂²⋊D₅	C₄	1	2	yes	—
8	C₂×C₅×Dic₅	C₂²	1	2	yes	—
9	C₄×C₅×D₅	C₂²	1	1	yes	characteristic
10	C₅×Dic₁₀	C₄	1	1	yes	characteristic
11	#6032	C₂²	1	1	yes	characteristic
12	#6032	C₂²	1	1	yes	characteristic
13	#6032	C₂²	1	2	yes	—
14	#4131	—	5	1	no	maximal
15	#4094	—	5	1	no	maximal
16	C₂²×C₅²	C₂×C₄	1	2	yes	—
17	C₂×C₅×D₅	C₂×C₄	1	1	yes	characteristic
18	C₄×C₅²	C₂×C₄	1	1	yes	characteristic
19	C₅×Dic₅	C₂×C₄	1	2	yes	—
20	C₅×Dic₅	C₂³	1	1	yes	characteristic
21	C₅×#41	—	5	1	no	—
22	#711	D₅	1	1	yes	characteristic
23	#704	—	5	2	no	—
24	C₂×#170	—	5	2	no	—
25	#700	—	5	1	no	—
26	D₅⋊C₈	—	5	1	no	—
27	#681	—	5	1	no	—
28	#681	—	5	2	no	—
29	C₂×#168	—	5	1	no	—
30	C₂×#168	—	5	2	no	—
31	C₂×C₅²	—	2	2	no	—
32	C₂×C₅²	C₂²×C₄	1	1	yes	characteristic
33	C₅×D₅	—	2	1	no	—
34	C₅×Q₈	—	5	1	no	—
35	C₅×D₄	—	4	1	no	—
36	C₅×D₄	—	5	2	no	—
37	C₅×D₄	Dic₅	1	1	yes	characteristic
38	C₅×D₄	AGL(1,𝔽₅)	1	1	yes	characteristic
39	C₂×C₄×C₅	—	5	2	no	—
40	C₂×C₄×C₅	—	5	1	no	—
41	C₂²⋊D₅	Dic₅	1	2	yes	—
42	C₂×Dic₅	C₂×D₅	1	2	yes	—
43	C₄×D₅	C₂×D₅	1	1	yes	characteristic
44	Dic₁₀	Dic₅	1	1	yes	characteristic
45	#170	—	5	1	no	—
46	#170	—	5	1	no	—
47	#170	—	5	2	no	—
48	#168	—	5	1	no	—
49	#168	—	5	1	no	—
50	#168	—	5	2	no	—
51	#131	—	25	1	no	Sylow
52	C₅²	#131	1	1	yes	characteristic, complemented, Sylow
53	C₂²×C₅	—	4	2	no	—
54	C₂²×C₅	—	5	1	no	—
55	C₂²×C₅	C₂×Dic₅	1	2	yes	—
56	C₂²×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
57	C₂×D₅	C₂×Dic₅	1	1	yes	characteristic
58	C₄×C₅	—	4	1	no	—
59	C₄×C₅	—	5	2	no	—
60	C₄×C₅	—	5	1	no	—
61	C₄×C₅	C₂×Dic₅	1	1	yes	characteristic
62	C₄×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
63	Dic₅	C₂×Dic₅	1	2	yes	—
64	Dic₅	C₂²×D₅	1	1	yes	characteristic
65	#41	—	5	1	no	—
66	M₁₆	—	25	1	no	—
67	M₁₆	—	25	2	no	—
68	C₂×C₈	—	25	1	no	—
69	C₂×C₈	—	25	2	no	—
70	C₂×C₅	—	2	2	no	—
71	C₂×C₅	—	2	2	no	—
72	C₂×C₅	—	4	1	no	—
73	C₂×C₅	—	8	2	no	—
74	C₂×C₅	—	10	1	no	—
75	C₂×C₅	C₂²×Dic₅	1	1	yes	characteristic
76	C₂×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
77	D₅	—	2	1	no	—
78	Q₈	—	5	1	no	—
79	D₄	—	5	2	no	—
80	D₄	#1042	1	1	yes	characteristic
81	C₂×C₄	—	5	2	no	—
82	C₂×C₄	—	5	1	no	—
83	C₈	—	25	1	no	—
84	C₈	—	25	1	no	—
85	C₈	—	25	2	no	—
86	C₅	—	4	1	no	—
87	C₅	#4094	1	1	yes	characteristic, complemented
88	C₅	#4131	1	1	yes	characteristic, complemented
89	C₂²	—	5	1	no	—
90	C₂²	C₂×#1042	1	2	yes	—
91	C₄	—	5	1	no	—
92	C₄	—	5	2	no	—
93	C₄	C₂×#1042	1	1	yes	characteristic
94	C₂	—	2	2	no	—
95	C₂	—	10	1	no	—
96	C₂	C₂²×#1042	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×#1042
commutator, socle	C₂×C₅²	C₂²×C₄
Fitting	C₅²×D₄	C₄
prime radical	C₅×#711	C₂
cosocle	C₅×Dic₅	C₂³
hypercenter	D₄	#1042
hypocenter	C₅²	#131
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ D₄

Complements

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

N	H ≅ G/N	type
2. #87853	C₂	semidirect
3. C₂×#6032	C₂	semidirect
4. #87849	C₂	semidirect
5. D₅⋊#168	C₂	semidirect
52. C₅²	#131	semidirect
87. C₅	#4094	semidirect
88. C₅	#4131	semidirect

Maximal subgroups

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

subgroup	index
C₅×#711	2
#87853	2
C₂×#6032	2
#87849	2
D₅⋊#168	2
#4131	5
#4094	5