Subgroups — C₂×C₃×2₋¹⁺⁴ #5976

Order 192.

794 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×2₋¹⁺⁴	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₃×#41	C₂	1	10	yes	maximal, complemented
3	C₂²×C₃×Q₈	C₂	1	5	yes	maximal, complemented
4	C₂×2₋¹⁺⁴	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₃×#41	C₂²	1	80	yes	complemented
6	C₂×C₃×Q₈	C₂²	1	10	yes	—
7	C₂×C₃×Q₈	C₂²	1	40	yes	complemented
8	C₂×C₃×D₄	C₂²	1	10	yes	—
9	C₂²×C₃×C₄	C₂²	1	15	yes	—
10	2₋¹⁺⁴	C₂×C₃	1	16	yes	direct product factor
11	C₂×#41	C₂×C₃	1	10	yes	complemented
12	C₂²×Q₈	C₂×C₃	1	5	yes	complemented
13	C₂³×C₃	C₂³	1	5	yes	—
14	C₃×Q₈	C₂³	1	40	yes	—
15	C₃×D₄	C₂³	1	40	yes	—
16	C₂×C₃×C₄	C₂³	1	60	yes	—
17	C₂×C₃×C₄	C₂³	1	10	yes	—
18	#41	C₂²×C₃	1	80	yes	complemented
19	C₂×Q₈	C₂²×C₃	1	10	yes	—
20	C₂×Q₈	C₂²×C₃	1	40	yes	complemented
21	C₂×D₄	C₂²×C₃	1	10	yes	—
22	C₂²×C₄	C₂²×C₃	1	15	yes	—
23	C₂²×C₃	—	2	10	no	—
24	C₂²×C₃	C₂⁴	1	10	yes	—
25	C₂²×C₃	C₂⁴	1	1	yes	characteristic
26	C₃×C₄	C₂⁴	1	20	yes	—
27	C₂³	C₂³×C₃	1	5	yes	—
28	Q₈	C₂³×C₃	1	40	yes	—
29	D₄	C₂³×C₃	1	40	yes	—
30	C₂×C₄	C₂³×C₃	1	10	yes	—
31	C₂×C₄	C₂³×C₃	1	60	yes	—
32	C₂×C₃	—	2	10	no	—
33	C₂×C₃	2₋¹⁺⁴	1	2	yes	direct product factor
34	C₂×C₃	C₂⁵	1	1	yes	characteristic
35	C₂²	—	2	10	no	—
36	C₂²	C₂⁴×C₃	1	10	yes	—
37	C₂²	C₂⁴×C₃	1	1	yes	characteristic
38	C₄	C₂⁴×C₃	1	20	yes	—
39	C₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic, direct product factor, Sylow
40	C₂	—	2	10	no	—
41	C₂	C₃×2₋¹⁺⁴	1	2	yes	direct product factor
42	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. C₃×2₋¹⁺⁴	C₂	direct (N × H)
2. C₂×C₃×#41	C₂	semidirect
3. C₂²×C₃×Q₈	C₂	semidirect
4. C₂×2₋¹⁺⁴	C₃	direct (N × H)
5. C₃×#41	C₂²	semidirect
7. C₂×C₃×Q₈	C₂²	semidirect
10. 2₋¹⁺⁴	C₂×C₃	direct (N × H)
11. C₂×#41	C₂×C₃	semidirect
12. C₂²×Q₈	C₂×C₃	semidirect
18. #41	C₂²×C₃	semidirect
20. C₂×Q₈	C₂²×C₃	semidirect
33. C₂×C₃	2₋¹⁺⁴	direct (N × H)
39. C₃	C₂×2₋¹⁺⁴	direct (N × H)
41. C₂	C₃×2₋¹⁺⁴	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×2₋¹⁺⁴	2
C₂×C₃×#41	2
C₂²×C₃×Q₈	2
C₂×2₋¹⁺⁴	3

Subgroups — C2×C3×2−1+4 #5976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×2₋¹⁺⁴ #5976