Subgroups — C₂²×C₃×M₁₆ #5896

Order 192.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×M₁₆	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₃×C₈	C₂	1	2	yes	maximal, complemented
4	C₂²×M₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂⁴×C₃	C₄	1	1	yes	characteristic
6	C₂²×C₃×C₄	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₄	C₄	1	6	yes	—
8	C₂²×C₃×C₄	C₂²	1	6	yes	—
9	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
10	C₃×M₁₆	C₂²	1	16	yes	direct product factor
11	C₂×C₃×C₈	C₂²	1	12	yes	complemented
12	C₂³×C₄	C₂×C₃	1	1	yes	characteristic
13	C₂×M₁₆	C₂×C₃	1	12	yes	direct product factor
14	C₂²×C₈	C₂×C₃	1	2	yes	complemented
15	C₂³×C₃	—	2	4	no	—
16	C₂³×C₃	C₂×C₄	1	6	yes	—
17	C₂³×C₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃×C₄	C₂×C₄	1	12	yes	—
19	C₂×C₃×C₄	C₂×C₄	1	6	yes	—
20	C₂×C₃×C₄	C₂×C₄	1	3	yes	—
21	C₂×C₃×C₄	C₂³	1	3	yes	—
22	C₂×C₃×C₄	C₂³	1	4	yes	—
23	C₃×C₈	C₂³	1	8	yes	complemented
24	C₂⁴	C₃×C₄	1	1	yes	characteristic
25	C₂²×C₄	C₃×C₄	1	6	yes	—
26	C₂²×C₄	C₃×C₄	1	1	yes	characteristic
27	C₂²×C₄	C₂²×C₃	1	1	yes	characteristic
28	C₂²×C₄	C₂²×C₃	1	6	yes	—
29	M₁₆	C₂²×C₃	1	16	yes	direct product factor
30	C₂×C₈	C₂²×C₃	1	12	yes	complemented
31	C₂²×C₃	—	2	12	no	—
32	C₂²×C₃	M₁₆	1	4	yes	direct product factor
33	C₂²×C₃	C₂²×C₄	1	4	yes	—
34	C₂²×C₃	C₂²×C₄	1	3	yes	—
35	C₃×C₄	C₂²×C₄	1	4	yes	—
36	C₃×C₄	C₂²×C₄	1	3	yes	—
37	C₃×C₄	C₂⁴	1	1	yes	characteristic
38	C₂³	—	2	4	no	—
39	C₂³	C₂×C₃×C₄	1	6	yes	—
40	C₂³	C₂×C₃×C₄	1	1	yes	characteristic
41	C₂×C₄	C₂×C₃×C₄	1	12	yes	—
42	C₂×C₄	C₂×C₃×C₄	1	3	yes	—
43	C₂×C₄	C₂×C₃×C₄	1	6	yes	—
44	C₂×C₄	C₂³×C₃	1	3	yes	—
45	C₂×C₄	C₂³×C₃	1	4	yes	—
46	C₈	C₂³×C₃	1	8	yes	complemented
47	C₂×C₃	—	2	4	no	—
48	C₂×C₃	C₂×M₁₆	1	6	yes	direct product factor
49	C₂×C₃	C₂³×C₄	1	1	yes	characteristic
50	C₂²	—	2	12	no	—
51	C₂²	C₃×M₁₆	1	4	yes	direct product factor
52	C₂²	C₂²×C₃×C₄	1	4	yes	—
53	C₂²	C₂²×C₃×C₄	1	3	yes	—
54	C₄	C₂²×C₃×C₄	1	3	yes	—
55	C₄	C₂²×C₃×C₄	1	4	yes	—
56	C₄	C₂⁴×C₃	1	1	yes	characteristic
57	C₃	C₂²×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
58	C₂	—	2	4	no	—
59	C₂	C₂×C₃×M₁₆	1	6	yes	direct product factor
60	C₂	C₂³×C₃×C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₂²
commutator	C₂	C₂³×C₃×C₄
Frattini, cosocle	C₄	C₂⁴×C₃
socle	C₂³×C₃	C₂×C₄
prime radical	C₂⁴×C₃	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×M₁₆	C₂	direct (N × H)
3. C₂²×C₃×C₈	C₂	semidirect
4. C₂²×M₁₆	C₃	direct (N × H)
10. C₃×M₁₆	C₂²	direct (N × H)
11. C₂×C₃×C₈	C₂²	semidirect
13. C₂×M₁₆	C₂×C₃	direct (N × H)
14. C₂²×C₈	C₂×C₃	semidirect
23. C₃×C₈	C₂³	semidirect
29. M₁₆	C₂²×C₃	direct (N × H)
30. C₂×C₈	C₂²×C₃	semidirect
32. C₂²×C₃	M₁₆	direct (N × H)
46. C₈	C₂³×C₃	semidirect
48. C₂×C₃	C₂×M₁₆	direct (N × H)
51. C₂²	C₃×M₁₆	direct (N × H)
57. C₃	C₂²×M₁₆	direct (N × H)
59. C₂	C₂×C₃×M₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₄	2
C₂×C₃×M₁₆	2
C₂²×C₃×C₈	2
C₂²×M₁₆	3

Subgroups — C22×C3×M16 #5896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×M₁₆ #5896