Subgroups — C₂×C₃×#127 #5860

Order 192.

498 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×D₄	C₂	1	6	yes	maximal, complemented
2	C₃×#127	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₃×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#127	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂⁴×C₃	C₂²	1	4	yes	—
6	C₂×C₃×D₄	—	2	24	no	—
7	C₂×C₃×D₄	C₂²	1	24	yes	complemented
8	C₂²×C₃×C₄	C₂²	1	3	yes	—
9	C₃×C₄²	C₂²	1	4	yes	complemented
10	C₂²×D₄	C₂×C₃	1	6	yes	complemented
11	#127	C₂×C₃	1	8	yes	direct product factor
12	C₂×C₄²	C₂×C₃	1	1	yes	characteristic, complemented
13	C₂³×C₃	—	2	24	no	—
14	C₂³×C₃	C₂³	1	8	yes	—
15	C₂³×C₃	C₂³	1	1	yes	characteristic
16	C₃×D₄	—	2	48	no	—
17	C₂×C₃×C₄	D₄	1	12	yes	complemented
18	C₂×C₃×C₄	C₂³	1	6	yes	—
19	C₂⁴	C₂²×C₃	1	4	yes	—
20	C₂×D₄	—	2	24	no	—
21	C₂×D₄	C₂²×C₃	1	24	yes	complemented
22	C₂²×C₄	C₂²×C₃	1	3	yes	—
23	C₄²	C₂²×C₃	1	4	yes	complemented
24	C₂²×C₃	—	2	24	no	—
25	C₂²×C₃	—	4	16	no	—
26	C₂²×C₃	C₂×D₄	1	6	yes	—
27	C₂²×C₃	C₂⁴	1	1	yes	characteristic
28	C₃×C₄	C₂×D₄	1	12	yes	complemented
29	C₂³	—	2	24	no	—
30	C₂³	C₂³×C₃	1	8	yes	—
31	C₂³	C₂³×C₃	1	1	yes	characteristic
32	D₄	—	2	48	no	—
33	C₂×C₄	C₃×D₄	1	12	yes	complemented
34	C₂×C₄	C₂³×C₃	1	6	yes	—
35	C₂×C₃	—	4	8	no	—
36	C₂×C₃	#127	1	4	yes	direct product factor
37	C₂×C₃	C₂²×D₄	1	3	yes	—
38	C₂²	—	2	24	no	—
39	C₂²	—	4	16	no	—
40	C₂²	C₂×C₃×D₄	1	6	yes	—
41	C₂²	C₂⁴×C₃	1	1	yes	characteristic
42	C₄	C₂×C₃×D₄	1	12	yes	complemented
43	C₃	C₂×#127	1	1	yes	characteristic, direct product factor, Sylow
44	C₂	—	4	8	no	—
45	C₂	C₃×#127	1	4	yes	direct product factor
46	C₂	C₂²×C₃×D₄	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×D₄	C₂	semidirect
2. C₃×#127	C₂	direct (N × H)
3. C₂×C₃×C₄²	C₂	semidirect
4. C₂×#127	C₃	direct (N × H)
7. C₂×C₃×D₄	C₂²	semidirect
9. C₃×C₄²	C₂²	semidirect
10. C₂²×D₄	C₂×C₃	semidirect
11. #127	C₂×C₃	direct (N × H)
12. C₂×C₄²	C₂×C₃	semidirect
17. C₂×C₃×C₄	D₄	semidirect
21. C₂×D₄	C₂²×C₃	semidirect
23. C₄²	C₂²×C₃	semidirect
28. C₃×C₄	C₂×D₄	semidirect
33. C₂×C₄	C₃×D₄	semidirect
36. C₂×C₃	#127	direct (N × H)
42. C₄	C₂×C₃×D₄	semidirect
43. C₃	C₂×#127	direct (N × H)
45. C₂	C₃×#127	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×C₃×D₄	2
C₃×#127	2
C₂×C₃×C₄²	2
C₂×#127	3

Subgroups — C2×C3×#127 #5860

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#127 #5860