Subgroups — C₂×#87828 #11717348

Order 800.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅×Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₂×#6032	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#170	C₂	1	1	yes	maximal, characteristic, complemented
4	#87828	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₅×D₅	C₄	1	1	yes	characteristic
6	C₂×#1039	C₄	1	1	yes	characteristic
7	C₂×C₅×Dic₅	C₂²	1	1	yes	characteristic
8	D₅×Dic₅	C₄	1	2	yes	—
9	D₅×Dic₅	C₂²	1	2	yes	—
10	#6032	C₂²	1	2	yes	complemented
11	C₅×#170	C₂²	1	2	yes	complemented
12	C₂×#704	—	5	1	no	maximal
13	C₂×#676	—	5	1	no	maximal
14	C₂²×C₅²	C₂×C₄	1	1	yes	characteristic
15	C₂×C₅×D₅	—	2	2	no	—
16	C₂×C₅×D₅	C₂×C₄	1	2	yes	—
17	#1039	C₂×C₄	1	2	yes	—
18	C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
19	C₅×Dic₅	C₂³	1	1	yes	characteristic
20	C₂²×Dic₅	—	5	1	no	—
21	C₂×C₄×D₅	—	5	1	no	—
22	#704	—	5	4	no	—
23	C₂×#170	—	5	1	no	—
24	C₂×#170	D₅	1	1	yes	characteristic, complemented
25	C₂×C₅×C₈	—	5	1	no	—
26	C₂×#168	—	5	1	no	—
27	#676	—	5	4	no	—
28	C₂×C₅²	M₁₆	1	2	yes	complemented
29	C₂×C₅²	C₂²×C₄	1	1	yes	characteristic
30	C₅×D₅	—	2	2	no	—
31	C₂³×C₅	—	5	1	no	—
32	C₂²×D₅	AGL(1,𝔽₅)	1	1	yes	characteristic
33	C₂×C₄×C₅	—	5	1	no	—
34	C₂×Dic₅	—	5	1	no	—
35	C₂×Dic₅	—	5	1	no	—
36	C₂×Dic₅	—	5	2	no	—
37	C₂×Dic₅	—	5	2	no	—
38	C₂×Dic₅	—	20	1	no	—
39	C₂×Dic₅	C₂×D₅	1	1	yes	characteristic
40	C₄×D₅	—	5	2	no	—
41	C₄×D₅	—	5	2	no	—
42	#170	—	5	2	no	—
43	#170	C₂×D₅	1	2	yes	complemented
44	C₅×C₈	—	5	2	no	—
45	#168	—	5	2	no	—
46	C₂×M₁₆	—	25	1	no	Sylow
47	C₅²	C₂×M₁₆	1	1	yes	characteristic, complemented, Sylow
48	C₂²×C₅	—	4	1	no	—
49	C₂²×C₅	—	5	2	no	—
50	C₂²×C₅	—	10	2	no	—
51	C₂²×C₅	C₄×D₅	1	1	yes	characteristic
52	C₂²×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
53	C₂×D₅	—	2	2	no	—
54	C₂×D₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
55	C₄×C₅	—	5	1	no	—
56	C₄×C₅	—	5	1	no	—
57	Dic₅	—	5	2	no	—
58	Dic₅	—	5	2	no	—
59	Dic₅	—	20	2	no	—
60	Dic₅	C₄×D₅	1	1	yes	characteristic
61	Dic₅	C₂²×D₅	1	1	yes	characteristic
62	C₂²×C₄	—	25	1	no	—
63	M₁₆	—	25	4	no	—
64	C₂×C₈	—	5	1	no	—
65	C₂×C₈	—	25	1	no	—
66	C₂×C₅	—	4	2	no	—
67	C₂×C₅	—	4	1	no	—
68	C₂×C₅	—	10	2	no	—
69	C₂×C₅	#676	1	2	yes	complemented
70	C₂×C₅	#704	1	2	yes	complemented
71	C₂×C₅	C₂×C₄×D₅	1	1	yes	characteristic
72	C₂×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
73	D₅	—	2	2	no	—
74	C₂³	—	5	1	no	—
75	C₂×C₄	—	5	1	no	—
76	C₂×C₄	—	25	2	no	—
77	C₂×C₄	—	25	2	no	—
78	C₂×C₄	—	25	1	no	—
79	C₈	—	5	2	no	—
80	C₈	—	25	2	no	—
81	C₅	—	4	1	no	—
82	C₅	C₂×#676	1	1	yes	characteristic, complemented
83	C₅	C₂×#704	1	1	yes	characteristic, complemented
84	C₂²	—	5	2	no	—
85	C₂²	—	10	2	no	—
86	C₂²	D₅×AGL(1,𝔽₅)	1	1	yes	characteristic
87	C₄	—	5	1	no	—
88	C₄	—	5	1	no	—
89	C₄	—	25	2	no	—
90	C₂	—	10	2	no	—
91	C₂	#87828	1	2	yes	direct product factor
92	C₂	C₂×D₅×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²	D₅×AGL(1,𝔽₅)
commutator	C₂×C₅²	C₂²×C₄
Frattini	C₂	C₂×D₅×AGL(1,𝔽₅)
Fitting, socle	C₂²×C₅²	C₂×C₄
prime radical	C₂²×C₅×D₅	C₄
cosocle	C₅×Dic₅	C₂³
hypocenter	C₅²	C₂×M₁₆
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
2. C₂×#6032	C₂	semidirect
3. C₂×C₅×#170	C₂	semidirect
4. #87828	C₂	direct (N × H)
10. #6032	C₂²	semidirect
11. C₅×#170	C₂²	semidirect
24. C₂×#170	D₅	semidirect
28. C₂×C₅²	M₁₆	semidirect
43. #170	C₂×D₅	semidirect
47. C₅²	C₂×M₁₆	semidirect
69. C₂×C₅	#676	semidirect
70. C₂×C₅	#704	semidirect
82. C₅	C₂×#676	semidirect
83. C₅	C₂×#704	semidirect
91. C₂	#87828	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×D₅×Dic₅	2
C₂×#6032	2
C₂×C₅×#170	2
#87828	2
C₂×#704	5
C₂×#676	5

Subgroups — C2×#87828 #11717348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#87828 #11717348