Subgroups — C₂²×#66774 #13062871

Order 1360.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₅×C₁₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#66774	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₅×C₁₇	C₄	1	1	yes	characteristic, complemented
4	C₂×D₅×C₁₇	C₄	1	3	yes	complemented
5	C₂×D₅×C₁₇	C₂²	1	3	yes	—
6	#66774	C₂²	1	4	yes	direct product factor
7	C₂²×Dic₁₇	—	5	1	no	maximal, Hall
8	C₂×C₅×C₁₇	C₂×C₄	1	3	yes	complemented
9	D₅×C₁₇	C₂×C₄	1	3	yes	complemented
10	D₅×C₁₇	C₂³	1	1	yes	characteristic
11	C₂³×C₁₇	—	5	1	no	—
12	C₂×Dic₁₇	—	5	6	no	—
13	C₅×C₁₇	C₂²×C₄	1	1	yes	characteristic, complemented, Hall
14	C₂²×AGL(1,𝔽₅)	—	17	1	no	maximal, Hall
15	C₂²×C₁₇	—	5	3	no	—
16	C₂²×C₁₇	—	5	3	no	—
17	C₂²×C₁₇	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
18	Dic₁₇	—	5	4	no	—
19	C₂²×D₅	D₁₇	1	1	yes	characteristic
20	C₂×AGL(1,𝔽₅)	—	17	6	no	—
21	C₂×C₁₇	—	5	1	no	—
22	C₂×C₁₇	—	5	3	no	—
23	C₂×C₁₇	C₂×AGL(1,𝔽₅)	1	3	yes	complemented
24	C₂²×C₅	Dic₁₇	1	1	yes	characteristic, complemented
25	C₂×D₅	Dic₁₇	1	3	yes	complemented
26	C₂×D₅	C₂×D₁₇	1	3	yes	—
27	AGL(1,𝔽₅)	—	17	4	no	—
28	C₁₇	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
29	C₂²×C₄	—	85	1	no	Sylow
30	C₂×C₅	C₂×Dic₁₇	1	3	yes	complemented
31	D₅	C₂×Dic₁₇	1	3	yes	complemented
32	D₅	C₂²×D₁₇	1	1	yes	characteristic
33	C₂³	—	5	1	no	—
34	C₂×C₄	—	85	6	no	—
35	C₅	C₂²×Dic₁₇	1	1	yes	characteristic, complemented, Sylow
36	C₂²	—	5	3	no	—
37	C₂²	—	5	3	no	—
38	C₂²	#66774	1	1	yes	characteristic, direct product factor
39	C₄	—	85	4	no	—
40	C₂	—	5	1	no	—
41	C₂	—	5	3	no	—
42	C₂	C₂×#66774	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²	#66774
commutator, hypocenter	C₅×C₁₇	C₂²×C₄
Frattini	C₁	G
Fitting, socle	C₂²×C₅×C₁₇	C₄
prime radical	C₂²×D₅×C₁₇	C₂
cosocle	D₅×C₁₇	C₂³
LCS	G ▷ C₅×C₁₇
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
2. C₂×#66774	C₂	direct (N × H)
3. C₂²×C₅×C₁₇	C₄	semidirect
4. C₂×D₅×C₁₇	C₄	semidirect
6. #66774	C₂²	direct (N × H)
8. C₂×C₅×C₁₇	C₂×C₄	semidirect
9. D₅×C₁₇	C₂×C₄	semidirect
13. C₅×C₁₇	C₂²×C₄	semidirect
17. C₂²×C₁₇	AGL(1,𝔽₅)	semidirect
23. C₂×C₁₇	C₂×AGL(1,𝔽₅)	semidirect
24. C₂²×C₅	Dic₁₇	semidirect
25. C₂×D₅	Dic₁₇	semidirect
28. C₁₇	C₂²×AGL(1,𝔽₅)	semidirect
30. C₂×C₅	C₂×Dic₁₇	semidirect
31. D₅	C₂×Dic₁₇	semidirect
35. C₅	C₂²×Dic₁₇	semidirect
38. C₂²	#66774	direct (N × H)
42. C₂	C₂×#66774	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×D₅×C₁₇	2
C₂×#66774	2
C₂²×Dic₁₇	5
C₂²×AGL(1,𝔽₅)	17

Subgroups — C22×#66774 #13062871

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#66774 #13062871