Subgroups — H(5,𝔽₂)×C₁₃⋊C₃ #11929250

Order 1248.

332 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×C₁₃⋊C₃	C₂	1	6	yes	maximal, complemented
2	C₂×D₄×C₁₃⋊C₃	C₂	1	9	yes	maximal, complemented
3	C₁₃×H(5,𝔽₂)	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂³×C₁₃⋊C₃	C₂²	1	6	yes	complemented
5	Q₈×C₁₃⋊C₃	C₂²	1	2	yes	complemented
6	D₄×C₁₃⋊C₃	C₂²	1	18	yes	complemented
7	C₂×C₄×C₁₃⋊C₃	C₂²	1	9	yes	complemented
8	C₁₃×#41	C₂×C₃	1	6	yes	complemented
9	C₂×D₄×C₁₃	C₂×C₃	1	9	yes	complemented
10	C₂²×C₁₃⋊C₃	—	4	6	no	—
11	C₂²×C₁₃⋊C₃	C₂³	1	9	yes	—
12	C₄×C₁₃⋊C₃	C₂³	1	6	yes	—
13	C₂³×C₁₃	C₂²×C₃	1	6	yes	complemented
14	Q₈×C₁₃	C₂²×C₃	1	2	yes	complemented
15	D₄×C₁₃	C₂²×C₃	1	18	yes	complemented
16	C₂×C₄×C₁₃	C₂²×C₃	1	9	yes	complemented
17	C₃×H(5,𝔽₂)	—	13	1	no	maximal, Hall
18	C₂×C₁₃⋊C₃	—	2	9	no	—
19	C₂×C₁₃⋊C₃	C₂⁴	1	1	yes	characteristic
20	C₂²×C₁₃	—	4	6	no	—
21	C₂²×C₁₃	C₂³×C₃	1	9	yes	—
22	C₄×C₁₃	C₂³×C₃	1	6	yes	—
23	C₃×#41	—	13	6	no	—
24	C₂×C₃×D₄	—	13	9	no	—
25	C₁₃⋊C₃	H(5,𝔽₂)	1	1	yes	characteristic, direct product factor, Hall
26	H(5,𝔽₂)	C₁₃⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
27	C₂×C₁₃	—	2	9	no	—
28	C₂×C₁₃	C₂⁴×C₃	1	1	yes	characteristic
29	C₂³×C₃	—	13	6	no	—
30	C₃×Q₈	—	13	2	no	—
31	C₃×D₄	—	13	18	no	—
32	C₂×C₃×C₄	—	13	9	no	—
33	#41	C₂×C₁₃⋊C₃	1	6	yes	complemented
34	C₂×D₄	C₂×C₁₃⋊C₃	1	9	yes	complemented
35	C₁₃	C₃×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
36	C₂²×C₃	—	13	9	no	—
37	C₂²×C₃	—	52	6	no	—
38	C₃×C₄	—	13	6	no	—
39	C₂³	C₂²×C₁₃⋊C₃	1	6	yes	complemented
40	Q₈	C₂²×C₁₃⋊C₃	1	2	yes	complemented
41	D₄	C₂²×C₁₃⋊C₃	1	18	yes	complemented
42	C₂×C₄	C₂²×C₁₃⋊C₃	1	9	yes	complemented
43	C₂×C₃	—	13	1	no	—
44	C₂×C₃	—	26	9	no	—
45	C₂²	—	4	6	no	—
46	C₂²	C₂³×C₁₃⋊C₃	1	9	yes	—
47	C₄	C₂³×C₁₃⋊C₃	1	6	yes	—
48	C₃	—	13	1	no	Sylow
49	C₂	—	2	9	no	—
50	C₂	C₂⁴×C₁₃⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×C₁₃⋊C₃
commutator, socle, cosocle	C₂×C₁₃	C₂⁴×C₃
Fitting	C₁₃×H(5,𝔽₂)	C₃
hypercenter	H(5,𝔽₂)	C₁₃⋊C₃
hypocenter	C₁₃	C₃×H(5,𝔽₂)
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. #41×C₁₃⋊C₃	C₂	semidirect
2. C₂×D₄×C₁₃⋊C₃	C₂	semidirect
3. C₁₃×H(5,𝔽₂)	C₃	semidirect
4. C₂³×C₁₃⋊C₃	C₂²	semidirect
5. Q₈×C₁₃⋊C₃	C₂²	semidirect
6. D₄×C₁₃⋊C₃	C₂²	semidirect
7. C₂×C₄×C₁₃⋊C₃	C₂²	semidirect
8. C₁₃×#41	C₂×C₃	semidirect
9. C₂×D₄×C₁₃	C₂×C₃	semidirect
13. C₂³×C₁₃	C₂²×C₃	semidirect
14. Q₈×C₁₃	C₂²×C₃	semidirect
15. D₄×C₁₃	C₂²×C₃	semidirect
16. C₂×C₄×C₁₃	C₂²×C₃	semidirect
25. C₁₃⋊C₃	H(5,𝔽₂)	direct (N × H)
26. H(5,𝔽₂)	C₁₃⋊C₃	direct (N × H)
33. #41	C₂×C₁₃⋊C₃	semidirect
34. C₂×D₄	C₂×C₁₃⋊C₃	semidirect
35. C₁₃	C₃×H(5,𝔽₂)	semidirect
39. C₂³	C₂²×C₁₃⋊C₃	semidirect
40. Q₈	C₂²×C₁₃⋊C₃	semidirect
41. D₄	C₂²×C₁₃⋊C₃	semidirect
42. C₂×C₄	C₂²×C₁₃⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
#41×C₁₃⋊C₃	2
C₂×D₄×C₁₃⋊C₃	2
C₁₃×H(5,𝔽₂)	3
C₃×H(5,𝔽₂)	13

Subgroups — H(5,𝔽2)×C13⋊C3 #11929250

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(5,𝔽₂)×C₁₃⋊C₃ #11929250