Subgroups — C₂²×C₃×#14584108 #45691448

Order 33516.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#14584108	C₂	1	3	yes	maximal, direct product factor
2	C₂²×#14584108	C₃	1	3	yes	maximal, direct product factor
3	C₂²×C₃×C₁₉×C₄₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×#14584108	C₂²	1	1	yes	characteristic, direct product factor, Hall
5	C₂×#14584108	C₂×C₃	1	9	yes	direct product factor
6	C₂×C₃×C₁₉×C₄₉	C₂×C₃	1	3	yes	complemented
7	C₂²×C₃×#87705	—	7	1	no	maximal
8	C₂²×C₁₉×C₄₉	C₃²	1	1	yes	characteristic, complemented, Hall
9	#14584108	C₂²×C₃	1	3	yes	direct product factor
10	C₃×C₁₉×C₄₉	C₂²×C₃	1	1	yes	characteristic, complemented
11	C₂×C₃×#87705	—	7	3	no	—
12	C₂×C₁₉×C₄₉	C₂×C₃²	1	3	yes	complemented
13	C₂²×C₃×C₄₉⋊C₃	—	19	1	no	maximal, Hall
14	C₂²×C₃×C₇×C₁₉	ΓL(1,𝔽₈)	1	1	yes	characteristic
15	C₂²×#87705	—	7	3	no	—
16	C₃×#87705	—	7	1	no	—
17	C₁₉×C₄₉	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
18	C₂×C₃×C₄₉⋊C₃	—	19	3	no	—
19	C₂×C₃×C₇×C₁₉	C₂×ΓL(1,𝔽₈)	1	3	yes	—
20	C₂×#87705	—	7	9	no	—
21	C₂²×C₃×C₁₉⋊C₃	—	49	1	no	Hall
22	C₂²×C₃×C₄₉	C₁₉⋊C₃	1	1	yes	characteristic, complemented
23	C₂²×C₄₉⋊C₃	—	19	3	no	—
24	C₂²×C₇×C₁₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
25	C₃×C₄₉⋊C₃	—	19	1	no	Hall
26	C₃×C₇×C₁₉	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
27	#87705	—	7	3	no	—
28	C₂×C₃×C₁₉⋊C₃	—	49	3	no	—
29	C₂×C₃×C₄₉	C₂×C₁₉⋊C₃	1	3	yes	complemented
30	C₂×C₄₉⋊C₃	—	19	9	no	—
31	C₂×C₇×C₁₉	C₂×C₃×ΓL(1,𝔽₈)	1	3	yes	—
32	C₂²×C₃×ΓL(1,𝔽₈)	—	133	1	no	—
33	C₂²×C₃×C₁₉	C₄₉⋊C₃	1	1	yes	characteristic, complemented
34	C₂²×C₁₉⋊C₃	—	49	3	no	—
35	C₂²×C₄₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented, Hall
36	C₃×C₁₉⋊C₃	—	49	1	no	Hall
37	C₃×C₄₉	C₂²×C₁₉⋊C₃	1	1	yes	characteristic, complemented
38	C₄₉⋊C₃	—	19	3	no	—
39	C₇×C₁₉	C₂²×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
40	C₂×C₃×ΓL(1,𝔽₈)	—	133	3	no	—
41	C₂×C₃×C₁₉	C₂×C₄₉⋊C₃	1	3	yes	complemented
42	C₂×C₁₉⋊C₃	—	49	9	no	—
43	C₂×C₄₉	C₂×C₃×C₁₉⋊C₃	1	3	yes	complemented
44	C₂²×C₃×C₇	#87705	1	1	yes	characteristic
45	C₂²×ΓL(1,𝔽₈)	—	133	3	no	—
46	C₂²×C₁₉	C₃×C₄₉⋊C₃	1	1	yes	characteristic, complemented, Hall
47	C₃×ΓL(1,𝔽₈)	—	133	1	no	—
48	C₃×C₁₉	C₂²×C₄₉⋊C₃	1	1	yes	characteristic, complemented
49	C₁₉⋊C₃	—	49	3	no	—
50	C₄₉	C₂²×C₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented, Sylow
51	C₂×C₃×C₇	C₂×#87705	1	3	yes	—
52	C₂×ΓL(1,𝔽₈)	—	133	9	no	—
53	C₂×C₁₉	C₂×C₃×C₄₉⋊C₃	1	3	yes	complemented
54	C₂²×C₃²	—	931	1	no	Hall
55	C₂²×C₇	C₃×#87705	1	1	yes	characteristic
56	C₃×C₇	C₂²×#87705	1	1	yes	characteristic
57	ΓL(1,𝔽₈)	—	133	3	no	—
58	C₁₉	C₂²×C₃×C₄₉⋊C₃	1	1	yes	characteristic, complemented, Sylow
59	C₂×C₃²	—	931	3	no	—
60	C₂×C₇	C₂×C₃×#87705	1	3	yes	—
61	C₂²×C₃	—	931	3	no	—
62	C₂²×C₃	#14584108	1	1	yes	characteristic, direct product factor
63	C₃²	—	931	1	no	Sylow
64	C₇	C₂²×C₃×#87705	1	1	yes	characteristic
65	C₂×C₃	—	931	9	no	—
66	C₂×C₃	C₂×#14584108	1	3	yes	direct product factor
67	C₂²	C₃×#14584108	1	1	yes	characteristic, direct product factor, Sylow
68	C₃	—	931	3	no	—
69	C₃	C₂²×#14584108	1	1	yes	characteristic, direct product factor
70	C₂	C₂×C₃×#14584108	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃	#14584108
commutator, cosocle, hypocenter	C₁₉×C₄₉	C₂²×C₃²
Frattini	C₇	C₂²×C₃×#87705
Fitting	C₂²×C₃×C₁₉×C₄₉	C₃
socle	C₂²×C₃×C₇×C₁₉	ΓL(1,𝔽₈)
LCS	G ▷ C₁₉×C₄₉
UCS	C₁ ◁ C₂²×C₃

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#14584108	C₂	direct (N × H)
2. C₂²×#14584108	C₃	direct (N × H)
3. C₂²×C₃×C₁₉×C₄₉	C₃	semidirect
4. C₃×#14584108	C₂²	direct (N × H)
5. C₂×#14584108	C₂×C₃	direct (N × H)
6. C₂×C₃×C₁₉×C₄₉	C₂×C₃	semidirect
8. C₂²×C₁₉×C₄₉	C₃²	semidirect
9. #14584108	C₂²×C₃	direct (N × H)
10. C₃×C₁₉×C₄₉	C₂²×C₃	semidirect
12. C₂×C₁₉×C₄₉	C₂×C₃²	semidirect
17. C₁₉×C₄₉	C₂²×C₃²	semidirect
22. C₂²×C₃×C₄₉	C₁₉⋊C₃	semidirect
29. C₂×C₃×C₄₉	C₂×C₁₉⋊C₃	semidirect
33. C₂²×C₃×C₁₉	C₄₉⋊C₃	semidirect
35. C₂²×C₄₉	C₃×C₁₉⋊C₃	semidirect
37. C₃×C₄₉	C₂²×C₁₉⋊C₃	semidirect
41. C₂×C₃×C₁₉	C₂×C₄₉⋊C₃	semidirect
43. C₂×C₄₉	C₂×C₃×C₁₉⋊C₃	semidirect
46. C₂²×C₁₉	C₃×C₄₉⋊C₃	semidirect
48. C₃×C₁₉	C₂²×C₄₉⋊C₃	semidirect
50. C₄₉	C₂²×C₃×C₁₉⋊C₃	semidirect
53. C₂×C₁₉	C₂×C₃×C₄₉⋊C₃	semidirect
58. C₁₉	C₂²×C₃×C₄₉⋊C₃	semidirect
62. C₂²×C₃	#14584108	direct (N × H)
66. C₂×C₃	C₂×#14584108	direct (N × H)
67. C₂²	C₃×#14584108	direct (N × H)
69. C₃	C₂²×#14584108	direct (N × H)
70. C₂	C₂×C₃×#14584108	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#14584108	2
C₂²×#14584108	3
C₂²×C₃×C₁₉×C₄₉	3
C₂²×C₃×#87705	7
C₂²×C₃×C₄₉⋊C₃	19

Subgroups — C22×C3×#14584108 #45691448

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#14584108 #45691448