Subgroups — C₂²×#700 #66250

Order 320.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×#170	C₂	1	2	yes	maximal, complemented
3	C₂×#700	C₂	1	12	yes	maximal, direct product factor
4	C₂³×D₅	C₄	1	1	yes	characteristic
5	C₂²×C₄×C₅	C₄	1	1	yes	characteristic
6	C₂²×Dic₅	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₅	C₄	1	6	yes	—
8	C₂×C₄×D₅	C₂²	1	6	yes	—
9	C₂×#170	C₂²	1	12	yes	complemented
10	#700	C₂²	1	16	yes	direct product factor
11	C₂²×M₁₆	—	5	1	no	maximal, Sylow
12	C₂³×C₅	C₂×C₄	1	1	yes	characteristic
13	C₂²×D₅	—	2	4	no	—
14	C₂²×D₅	C₂×C₄	1	6	yes	—
15	C₂×C₄×C₅	C₂×C₄	1	6	yes	—
16	C₂×Dic₅	C₂×C₄	1	3	yes	—
17	C₂×Dic₅	C₂³	1	3	yes	—
18	C₄×D₅	C₂×C₄	1	12	yes	—
19	C₄×D₅	C₂³	1	4	yes	—
20	#170	C₂³	1	8	yes	complemented
21	C₂³×C₄	—	5	1	no	—
22	C₂×M₁₆	—	5	12	no	—
23	C₂²×C₈	—	5	2	no	—
24	C₂²×C₅	M₁₆	1	4	yes	complemented
25	C₂²×C₅	C₂²×C₄	1	3	yes	—
26	C₂×D₅	—	2	12	no	—
27	C₂×D₅	C₂²×C₄	1	4	yes	—
28	C₄×C₅	C₂²×C₄	1	4	yes	—
29	Dic₅	C₂²×C₄	1	3	yes	—
30	Dic₅	C₂⁴	1	1	yes	characteristic
31	C₂⁴	—	5	1	no	—
32	C₂²×C₄	—	5	1	no	—
33	C₂²×C₄	—	5	6	no	—
34	C₂²×C₄	—	5	6	no	—
35	C₂²×C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
36	M₁₆	—	5	16	no	—
37	C₂×C₈	—	5	12	no	—
38	C₂×C₅	C₂×M₁₆	1	6	yes	complemented
39	C₂×C₅	C₂³×C₄	1	1	yes	characteristic
40	D₅	—	2	4	no	—
41	C₂³	—	5	6	no	—
42	C₂³	—	10	4	no	—
43	C₂³	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
44	C₂×C₄	—	5	3	no	—
45	C₂×C₄	—	5	3	no	—
46	C₂×C₄	—	5	4	no	—
47	C₂×C₄	—	5	12	no	—
48	C₂×C₄	C₂×AGL(1,𝔽₅)	1	6	yes	—
49	C₈	—	5	8	no	—
50	C₅	C₂²×M₁₆	1	1	yes	characteristic, complemented, Sylow
51	C₂²	—	5	4	no	—
52	C₂²	—	10	12	no	—
53	C₂²	#700	1	4	yes	direct product factor
54	C₂²	C₂²×AGL(1,𝔽₅)	1	3	yes	—
55	C₄	—	5	1	no	—
56	C₄	—	5	3	no	—
57	C₄	C₂²×AGL(1,𝔽₅)	1	4	yes	—
58	C₂	—	10	4	no	—
59	C₂	C₂×#700	1	6	yes	direct product factor
60	C₂	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂³×C₄
Frattini	C₂	C₂³×AGL(1,𝔽₅)
Fitting	C₂²×C₄×C₅	C₄
socle	C₂³×C₅	C₂×C₄
prime radical	C₂³×D₅	C₄
cosocle	Dic₅	C₂⁴
hypercenter	C₂²×C₄	AGL(1,𝔽₅)
hypocenter	C₅	C₂²×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×#170	C₂	semidirect
3. C₂×#700	C₂	direct (N × H)
9. C₂×#170	C₂²	semidirect
10. #700	C₂²	direct (N × H)
20. #170	C₂³	semidirect
24. C₂²×C₅	M₁₆	semidirect
38. C₂×C₅	C₂×M₁₆	semidirect
50. C₅	C₂²×M₁₆	semidirect
53. C₂²	#700	direct (N × H)
59. C₂	C₂×#700	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₅	2
C₂²×#170	2
C₂×#700	2
C₂²×M₁₆	5

Subgroups — C22×#700 #66250

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#700 #66250