Subgroups — C₂×C₃×#120 #5851

Order 192.

662 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×D₄	C₂	1	3	yes	maximal, complemented
3	C₃×#120	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×C₂²⋊C₄	C₂	1	3	yes	maximal, complemented
5	C₂×#120	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂⁴×C₃	—	2	12	no	—
7	C₂⁴×C₃	C₂²	1	3	yes	—
8	C₂⁴×C₃	C₂²	1	4	yes	complemented
9	C₂⁴×C₃	C₂²	1	1	yes	characteristic, complemented
10	C₂×C₃×D₄	—	2	12	no	—
11	C₂×C₃×D₄	C₂²	1	12	yes	complemented
12	C₂²×C₃×C₄	C₂²	1	3	yes	complemented
13	C₃×C₂²⋊C₄	C₂²	1	12	yes	complemented
14	C₂⁵	C₂×C₃	1	1	yes	characteristic, complemented
15	C₂²×D₄	C₂×C₃	1	3	yes	complemented
16	#120	C₂×C₃	1	8	yes	direct product factor
17	C₂×C₂²⋊C₄	C₂×C₃	1	3	yes	complemented
18	C₂³×C₃	—	2	32	no	—
19	C₂³×C₃	—	2	12	no	—
20	C₂³×C₃	—	2	24	no	—
21	C₂³×C₃	—	2	6	no	—
22	C₂³×C₃	D₄	1	12	yes	complemented
23	C₂³×C₃	C₂³	1	6	yes	—
24	C₂³×C₃	C₂³	1	1	yes	characteristic
25	C₂³×C₃	C₂³	1	2	yes	complemented
26	C₃×D₄	—	2	24	no	—
27	C₂×C₃×C₄	—	2	6	no	—
28	C₂×C₃×C₄	C₂³	1	6	yes	complemented
29	C₂⁴	—	2	12	no	—
30	C₂⁴	C₂²×C₃	1	3	yes	—
31	C₂⁴	C₂²×C₃	1	4	yes	complemented
32	C₂⁴	C₂²×C₃	1	1	yes	characteristic, complemented
33	C₂×D₄	—	2	12	no	—
34	C₂×D₄	C₂²×C₃	1	12	yes	complemented
35	C₂²×C₄	C₂²×C₃	1	3	yes	complemented
36	C₂²⋊C₄	C₂²×C₃	1	12	yes	complemented
37	C₂²×C₃	—	2	32	no	—
38	C₂²×C₃	—	2	24	no	—
39	C₂²×C₃	—	2	12	no	—
40	C₂²×C₃	—	2	6	no	—
41	C₂²×C₃	—	4	4	no	—
42	C₂²×C₃	C₂×D₄	1	6	yes	—
43	C₂²×C₃	C₂×D₄	1	12	yes	complemented
44	C₂²×C₃	C₂⁴	1	1	yes	characteristic
45	C₃×C₄	—	2	6	no	—
46	C₂³	—	2	32	no	—
47	C₂³	—	2	12	no	—
48	C₂³	—	2	24	no	—
49	C₂³	—	2	6	no	—
50	C₂³	C₃×D₄	1	12	yes	complemented
51	C₂³	C₂³×C₃	1	6	yes	—
52	C₂³	C₂³×C₃	1	1	yes	characteristic
53	C₂³	C₂³×C₃	1	2	yes	complemented
54	D₄	—	2	24	no	—
55	C₂×C₄	—	2	6	no	—
56	C₂×C₄	C₂³×C₃	1	6	yes	complemented
57	C₂×C₃	—	2	12	no	—
58	C₂×C₃	—	4	2	no	—
59	C₂×C₃	#120	1	4	yes	direct product factor
60	C₂×C₃	C₂²×D₄	1	3	yes	—
61	C₂²	—	2	32	no	—
62	C₂²	—	2	24	no	—
63	C₂²	—	2	12	no	—
64	C₂²	—	2	6	no	—
65	C₂²	—	4	4	no	—
66	C₂²	C₂×C₃×D₄	1	6	yes	—
67	C₂²	C₂×C₃×D₄	1	12	yes	complemented
68	C₂²	C₂⁴×C₃	1	1	yes	characteristic
69	C₄	—	2	6	no	—
70	C₃	C₂×#120	1	1	yes	characteristic, direct product factor, Sylow
71	C₂	—	2	12	no	—
72	C₂	—	4	2	no	—
73	C₂	C₃×#120	1	4	yes	direct product factor
74	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. C₂⁵×C₃	C₂	semidirect
2. C₂²×C₃×D₄	C₂	semidirect
3. C₃×#120	C₂	direct (N × H)
4. C₂×C₃×C₂²⋊C₄	C₂	semidirect
5. C₂×#120	C₃	direct (N × H)
8. C₂⁴×C₃	C₂²	semidirect
9. C₂⁴×C₃	C₂²	semidirect
11. C₂×C₃×D₄	C₂²	semidirect
12. C₂²×C₃×C₄	C₂²	semidirect
13. C₃×C₂²⋊C₄	C₂²	semidirect
14. C₂⁵	C₂×C₃	semidirect
15. C₂²×D₄	C₂×C₃	semidirect
16. #120	C₂×C₃	direct (N × H)
17. C₂×C₂²⋊C₄	C₂×C₃	semidirect
22. C₂³×C₃	D₄	semidirect
25. C₂³×C₃	C₂³	semidirect
28. C₂×C₃×C₄	C₂³	semidirect
31. C₂⁴	C₂²×C₃	semidirect
32. C₂⁴	C₂²×C₃	semidirect
34. C₂×D₄	C₂²×C₃	semidirect
35. C₂²×C₄	C₂²×C₃	semidirect
36. C₂²⋊C₄	C₂²×C₃	semidirect
43. C₂²×C₃	C₂×D₄	semidirect
50. C₂³	C₃×D₄	semidirect
53. C₂³	C₂³×C₃	semidirect
56. C₂×C₄	C₂³×C₃	semidirect
59. C₂×C₃	#120	direct (N × H)
67. C₂²	C₂×C₃×D₄	semidirect
70. C₃	C₂×#120	direct (N × H)
73. C₂	C₃×#120	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₃	2
C₂²×C₃×D₄	2
C₃×#120	2
C₂×C₃×C₂²⋊C₄	2
C₂×#120	3

Subgroups — C2×C3×#120 #5851

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#120 #5851