Subgroups — C₂×C₃×H(5,𝔽₂) #5975

Order 192.

898 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×H(5,𝔽₂)	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₃×#41	C₂	1	6	yes	maximal, complemented
3	C₂²×C₃×D₄	C₂	1	9	yes	maximal, complemented
4	C₂×H(5,𝔽₂)	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂⁴×C₃	C₂²	1	6	yes	complemented
6	C₃×#41	C₂²	1	48	yes	complemented
7	C₂×C₃×Q₈	C₂²	1	2	yes	complemented
8	C₂×C₃×D₄	C₂²	1	18	yes	complemented
9	C₂×C₃×D₄	C₂²	1	72	yes	complemented
10	C₂²×C₃×C₄	C₂²	1	9	yes	complemented
11	H(5,𝔽₂)	C₂×C₃	1	16	yes	direct product factor
12	C₂×#41	C₂×C₃	1	6	yes	complemented
13	C₂²×D₄	C₂×C₃	1	9	yes	complemented
14	C₂³×C₃	—	4	12	no	—
15	C₂³×C₃	C₂³	1	9	yes	—
16	C₂³×C₃	C₂³	1	24	yes	complemented
17	C₃×Q₈	C₂³	1	8	yes	complemented
18	C₃×D₄	C₂³	1	72	yes	complemented
19	C₂×C₃×C₄	C₂³	1	6	yes	—
20	C₂×C₃×C₄	C₂³	1	36	yes	complemented
21	C₂⁴	C₂²×C₃	1	6	yes	complemented
22	#41	C₂²×C₃	1	48	yes	complemented
23	C₂×Q₈	C₂²×C₃	1	2	yes	complemented
24	C₂×D₄	C₂²×C₃	1	18	yes	complemented
25	C₂×D₄	C₂²×C₃	1	72	yes	complemented
26	C₂²×C₄	C₂²×C₃	1	9	yes	complemented
27	C₂²×C₃	—	2	18	no	—
28	C₂²×C₃	—	4	24	no	—
29	C₂²×C₃	C₂⁴	1	18	yes	—
30	C₂²×C₃	C₂⁴	1	1	yes	characteristic
31	C₃×C₄	C₂⁴	1	12	yes	—
32	C₂³	—	4	12	no	—
33	C₂³	C₂³×C₃	1	9	yes	—
34	C₂³	C₂³×C₃	1	24	yes	complemented
35	Q₈	C₂³×C₃	1	8	yes	complemented
36	D₄	C₂³×C₃	1	72	yes	complemented
37	C₂×C₄	C₂³×C₃	1	6	yes	—
38	C₂×C₄	C₂³×C₃	1	36	yes	complemented
39	C₂×C₃	—	2	18	no	—
40	C₂×C₃	H(5,𝔽₂)	1	2	yes	direct product factor
41	C₂×C₃	C₂⁵	1	1	yes	characteristic
42	C₂²	—	2	18	no	—
43	C₂²	—	4	24	no	—
44	C₂²	C₂⁴×C₃	1	18	yes	—
45	C₂²	C₂⁴×C₃	1	1	yes	characteristic
46	C₄	C₂⁴×C₃	1	12	yes	—
47	C₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic, direct product factor, Sylow
48	C₂	—	2	18	no	—
49	C₂	C₃×H(5,𝔽₂)	1	2	yes	direct product factor
50	C₂	C₂⁵×C₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁵×C₃
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×H(5,𝔽₂)	C₂	direct (N × H)
2. C₂×C₃×#41	C₂	semidirect
3. C₂²×C₃×D₄	C₂	semidirect
4. C₂×H(5,𝔽₂)	C₃	direct (N × H)
5. C₂⁴×C₃	C₂²	semidirect
6. C₃×#41	C₂²	semidirect
7. C₂×C₃×Q₈	C₂²	semidirect
8. C₂×C₃×D₄	C₂²	semidirect
9. C₂×C₃×D₄	C₂²	semidirect
10. C₂²×C₃×C₄	C₂²	semidirect
11. H(5,𝔽₂)	C₂×C₃	direct (N × H)
12. C₂×#41	C₂×C₃	semidirect
13. C₂²×D₄	C₂×C₃	semidirect
16. C₂³×C₃	C₂³	semidirect
17. C₃×Q₈	C₂³	semidirect
18. C₃×D₄	C₂³	semidirect
20. C₂×C₃×C₄	C₂³	semidirect
21. C₂⁴	C₂²×C₃	semidirect
22. #41	C₂²×C₃	semidirect
23. C₂×Q₈	C₂²×C₃	semidirect
24. C₂×D₄	C₂²×C₃	semidirect
25. C₂×D₄	C₂²×C₃	semidirect
26. C₂²×C₄	C₂²×C₃	semidirect
34. C₂³	C₂³×C₃	semidirect
35. Q₈	C₂³×C₃	semidirect
36. D₄	C₂³×C₃	semidirect
38. C₂×C₄	C₂³×C₃	semidirect
40. C₂×C₃	H(5,𝔽₂)	direct (N × H)
47. C₃	C₂×H(5,𝔽₂)	direct (N × H)
49. C₂	C₃×H(5,𝔽₂)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×H(5,𝔽₂)	2
C₂×C₃×#41	2
C₂²×C₃×D₄	2
C₂×H(5,𝔽₂)	3

Subgroups — C2×C3×H(5,𝔽2) #5975

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×H(5,𝔽₂) #5975