Subgroups — C₄×D₅²⋊C₂ #11717387

Order 800.

163 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅²⋊C₂	C₂	1	1	yes	maximal, characteristic
2	C₄×D₅²	C₂	1	2	yes	maximal, complemented
3	C₄×#1044	C₂	1	1	yes	maximal, characteristic, complemented
4	#87836	C₂	1	1	yes	maximal, characteristic, complemented
5	#87835	C₂	1	2	yes	maximal, complemented
6	C₂×D₅²	C₂²	1	2	yes	—
7	C₂×#1044	C₂²	1	1	yes	characteristic
8	C₂×#1044	C₂²	1	1	yes	characteristic
9	D₅²⋊C₂	C₄	1	4	yes	direct product factor
10	C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
11	C₄×C₅×D₅	—	2	2	no	—
12	#6036	C₂²	1	2	yes	—
13	D₅×Dic₅	—	2	2	no	—
14	C₂×C₅⋊D₅	C₂³	1	1	yes	characteristic
15	C₂×C₅×D₅	—	2	2	no	—
16	D₅²	—	2	2	no	—
17	D₅²	C₂×C₄	1	4	yes	complemented
18	#1044	—	2	1	no	—
19	#1044	C₂×C₄	1	2	yes	complemented
20	C₄×C₅²	D₄	1	1	yes	characteristic, complemented
21	#1039	D₄	1	1	yes	characteristic, complemented
22	C₅×Dic₅	—	2	2	no	—
23	C₂×C₄×D₅	—	10	2	no	—
24	C₄×AGL(1,𝔽₅)	—	10	1	no	—
25	C₂×C₅²	C₂×D₄	1	1	yes	characteristic
26	C₅⋊D₅	C₂²×C₄	1	1	yes	characteristic
27	C₅⋊D₅	#41	1	1	yes	characteristic
28	C₅×D₅	—	2	4	no	—
29	C₂²×D₅	—	10	2	no	—
30	C₂×AGL(1,𝔽₅)	—	10	1	no	—
31	C₂×AGL(1,𝔽₅)	—	10	1	no	—
32	C₂×C₄×C₅	—	10	2	no	—
33	C₂×Dic₅	—	10	2	no	—
34	C₄×D₅	—	2	2	no	—
35	C₄×D₅	—	10	2	no	—
36	C₄×D₅	—	10	1	no	—
37	C₄×D₅	—	10	2	no	—
38	C₄×D₅	—	10	2	no	—
39	C₄×D₄	—	25	1	no	maximal, Sylow
40	C₅²	C₄×D₄	1	1	yes	characteristic, complemented, Sylow
41	C₂²×C₅	—	10	2	no	—
42	C₂×D₅	—	2	2	no	—
43	C₂×D₅	—	10	4	no	—
44	C₂×D₅	—	10	2	no	—
45	C₂×D₅	—	10	1	no	—
46	C₂×D₅	—	10	4	no	—
47	AGL(1,𝔽₅)	—	10	2	no	—
48	AGL(1,𝔽₅)	—	10	2	no	—
49	C₄×C₅	—	2	2	no	—
50	C₄×C₅	—	2	1	no	—
51	C₄×C₅	—	10	2	no	—
52	Dic₅	—	2	2	no	—
53	Dic₅	—	10	1	no	—
54	Dic₅	—	10	2	no	—
55	C₂×D₄	—	25	1	no	—
56	C₂²×C₄	—	25	2	no	—
57	C₄⋊C₄	—	25	1	no	—
58	C₂²⋊C₄	—	25	2	no	—
59	C₄²	—	25	1	no	—
60	C₂×C₅	—	2	2	no	—
61	C₂×C₅	—	2	1	no	—
62	C₂×C₅	—	10	4	no	—
63	D₅	—	2	4	no	—
64	D₅	—	10	1	no	—
65	D₅	—	10	2	no	—
66	D₅	—	10	1	no	—
67	D₅	—	10	2	no	—
68	C₂³	—	25	2	no	—
69	D₄	—	25	4	no	—
70	C₂×C₄	—	10	2	no	—
71	C₂×C₄	—	25	2	no	—
72	C₂×C₄	—	25	1	no	—
73	C₂×C₄	—	25	1	no	—
74	C₂×C₄	—	25	1	no	—
75	C₂×C₄	—	50	2	no	—
76	C₅	—	2	2	no	—
77	C₅	—	2	1	no	—
78	C₂²	—	10	2	no	—
79	C₂²	—	25	4	no	—
80	C₂²	—	25	1	no	—
81	C₂²	—	50	2	no	—
82	C₄	—	10	2	no	—
83	C₄	—	25	2	no	—
84	C₄	—	25	1	no	—
85	C₄	—	50	1	no	—
86	C₄	D₅²⋊C₂	1	1	yes	characteristic, direct product factor
87	C₂	—	10	4	no	—
88	C₂	—	25	1	no	—
89	C₂	—	25	1	no	—
90	C₂	C₂×D₅²⋊C₂	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄	D₅²⋊C₂
commutator	C₅⋊D₅	C₂²×C₄
Frattini	C₂	C₂×D₅²⋊C₂
Fitting	C₄×C₅²	D₄
socle	C₂×C₅²	C₂×D₄
prime radical	C₂×D₅²⋊C₂	C₂
cosocle	C₂×C₅⋊D₅	C₂³
hypocenter	C₅²	C₄×D₄
LCS	G ▷ C₅⋊D₅ ▷ C₅²
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₄×D₅²	C₂	semidirect
3. C₄×#1044	C₂	semidirect
4. #87836	C₂	semidirect
5. #87835	C₂	semidirect
9. D₅²⋊C₂	C₄	direct (N × H)
17. D₅²	C₂×C₄	semidirect
19. #1044	C₂×C₄	semidirect
20. C₄×C₅²	D₄	semidirect
21. #1039	D₄	semidirect
40. C₅²	C₄×D₄	semidirect
86. C₄	D₅²⋊C₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×D₅²⋊C₂	2
C₄×D₅²	2
C₄×#1044	2
#87836	2
#87835	2
C₄×D₄	25

Subgroups — C4×D52⋊C2 #11717387

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×D₅²⋊C₂ #11717387