Subgroups — C₂²×C₃×C₁₃⋊C₃×C₄₉ #37524311

Order 22932.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₁₃⋊C₃×C₄₉	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₁₃⋊C₃×C₄₉	C₃	1	3	yes	maximal, direct product factor
3	C₂²×C₃×C₁₃×C₄₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₁₃⋊C₃×C₄₉	C₂²	1	1	yes	characteristic, direct product factor, Hall
5	C₂×C₁₃⋊C₃×C₄₉	C₂×C₃	1	9	yes	direct product factor
6	C₂×C₃×C₁₃×C₄₉	C₂×C₃	1	3	yes	complemented
7	C₂²×C₃×C₇×C₁₃⋊C₃	C₇	1	1	yes	maximal, characteristic
8	C₂²×C₁₃×C₄₉	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃×C₁₃×C₄₉	C₂²×C₃	1	1	yes	characteristic, complemented
10	C₁₃⋊C₃×C₄₉	C₂²×C₃	1	3	yes	direct product factor
11	C₂²×C₃²×C₄₉	—	13	1	no	maximal, Hall
12	C₂×C₃×C₇×C₁₃⋊C₃	C₂×C₇	1	3	yes	—
13	C₂×C₁₃×C₄₉	C₂×C₃²	1	3	yes	complemented
14	C₂²×C₃×C₇×C₁₃	C₃×C₇	1	1	yes	characteristic
15	C₂²×C₇×C₁₃⋊C₃	C₃×C₇	1	3	yes	—
16	C₂×C₃²×C₄₉	—	13	3	no	—
17	C₃×C₇×C₁₃⋊C₃	C₂²×C₇	1	1	yes	characteristic
18	C₁₃×C₄₉	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
19	C₂²×C₃×C₄₉	—	13	3	no	—
20	C₂²×C₃×C₄₉	C₁₃⋊C₃	1	1	yes	characteristic, direct product factor
21	C₂×C₃×C₇×C₁₃	C₂×C₃×C₇	1	3	yes	—
22	C₂×C₇×C₁₃⋊C₃	C₂×C₃×C₇	1	9	yes	—
23	C₂²×C₃×C₁₃⋊C₃	C₄₉	1	1	yes	characteristic, direct product factor, Hall
24	C₃²×C₄₉	—	13	1	no	Hall
25	C₂²×C₇×C₁₃	C₃²×C₇	1	1	yes	characteristic
26	C₂×C₃×C₄₉	—	13	9	no	—
27	C₂×C₃×C₄₉	C₂×C₁₃⋊C₃	1	3	yes	direct product factor
28	C₃×C₇×C₁₃	C₂²×C₃×C₇	1	1	yes	characteristic
29	C₇×C₁₃⋊C₃	C₂²×C₃×C₇	1	3	yes	—
30	C₂²×C₃²×C₇	—	13	1	no	—
31	C₂×C₃×C₁₃⋊C₃	C₂×C₄₉	1	3	yes	direct product factor
32	C₂²×C₄₉	C₃×C₁₃⋊C₃	1	1	yes	characteristic, direct product factor, Hall
33	C₂×C₇×C₁₃	C₂×C₃²×C₇	1	3	yes	—
34	C₂²×C₃×C₁₃	C₃×C₄₉	1	1	yes	characteristic, complemented
35	C₂²×C₁₃⋊C₃	C₃×C₄₉	1	3	yes	direct product factor
36	C₃×C₄₉	—	13	3	no	—
37	C₃×C₄₉	C₂²×C₁₃⋊C₃	1	1	yes	characteristic, direct product factor
38	C₂×C₃²×C₇	—	13	3	no	—
39	C₃×C₁₃⋊C₃	C₂²×C₄₉	1	1	yes	characteristic, direct product factor, Hall
40	C₂×C₄₉	C₂×C₃×C₁₃⋊C₃	1	3	yes	direct product factor
41	C₇×C₁₃	C₂²×C₃²×C₇	1	1	yes	characteristic
42	C₂²×C₃×C₇	—	13	3	no	—
43	C₂²×C₃×C₇	C₇×C₁₃⋊C₃	1	1	yes	characteristic
44	C₂×C₃×C₁₃	C₂×C₃×C₄₉	1	3	yes	complemented
45	C₂×C₁₃⋊C₃	C₂×C₃×C₄₉	1	9	yes	direct product factor
46	C₃²×C₇	—	13	1	no	—
47	C₂²×C₁₃	C₃²×C₄₉	1	1	yes	characteristic, complemented, Hall
48	C₄₉	C₂²×C₃×C₁₃⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
49	C₂×C₃×C₇	—	13	9	no	—
50	C₂×C₃×C₇	C₂×C₇×C₁₃⋊C₃	1	3	yes	—
51	C₃×C₁₃	C₂²×C₃×C₄₉	1	1	yes	characteristic, complemented
52	C₁₃⋊C₃	C₂²×C₃×C₄₉	1	3	yes	direct product factor
53	C₂²×C₃²	—	13	1	no	Hall
54	C₂²×C₇	C₃×C₇×C₁₃⋊C₃	1	1	yes	characteristic
55	C₂×C₁₃	C₂×C₃²×C₄₉	1	3	yes	complemented
56	C₃×C₇	—	13	3	no	—
57	C₃×C₇	C₂²×C₇×C₁₃⋊C₃	1	1	yes	characteristic
58	C₂×C₃²	—	13	3	no	—
59	C₂×C₇	C₂×C₃×C₇×C₁₃⋊C₃	1	3	yes	—
60	C₁₃	C₂²×C₃²×C₄₉	1	1	yes	characteristic, complemented, Sylow
61	C₂²×C₃	—	13	3	no	—
62	C₂²×C₃	C₁₃⋊C₃×C₄₉	1	1	yes	characteristic, direct product factor
63	C₃²	—	13	1	no	Sylow
64	C₇	C₂²×C₃×C₇×C₁₃⋊C₃	1	1	yes	characteristic
65	C₂×C₃	—	13	9	no	—
66	C₂×C₃	C₂×C₁₃⋊C₃×C₄₉	1	3	yes	direct product factor
67	C₂²	C₃×C₁₃⋊C₃×C₄₉	1	1	yes	characteristic, direct product factor, Sylow
68	C₃	—	13	3	no	—
69	C₃	C₂²×C₁₃⋊C₃×C₄₉	1	1	yes	characteristic, direct product factor
70	C₂	C₂×C₃×C₁₃⋊C₃×C₄₉	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃×C₄₉	C₁₃⋊C₃
commutator, hypocenter	C₁₃	C₂²×C₃²×C₄₉
Frattini	C₇	C₂²×C₃×C₇×C₁₃⋊C₃
Fitting	C₂²×C₃×C₁₃×C₄₉	C₃
socle	C₂²×C₃×C₇×C₁₃	C₃×C₇
prime radical	C₂²×C₃×C₇×C₁₃⋊C₃	C₇
cosocle	C₇×C₁₃	C₂²×C₃²×C₇
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₂²×C₃×C₄₉

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₁₃⋊C₃×C₄₉	C₂	direct (N × H)
2. C₂²×C₁₃⋊C₃×C₄₉	C₃	direct (N × H)
3. C₂²×C₃×C₁₃×C₄₉	C₃	semidirect
4. C₃×C₁₃⋊C₃×C₄₉	C₂²	direct (N × H)
5. C₂×C₁₃⋊C₃×C₄₉	C₂×C₃	direct (N × H)
6. C₂×C₃×C₁₃×C₄₉	C₂×C₃	semidirect
8. C₂²×C₁₃×C₄₉	C₃²	semidirect
9. C₃×C₁₃×C₄₉	C₂²×C₃	semidirect
10. C₁₃⋊C₃×C₄₉	C₂²×C₃	direct (N × H)
13. C₂×C₁₃×C₄₉	C₂×C₃²	semidirect
18. C₁₃×C₄₉	C₂²×C₃²	semidirect
20. C₂²×C₃×C₄₉	C₁₃⋊C₃	direct (N × H)
23. C₂²×C₃×C₁₃⋊C₃	C₄₉	direct (N × H)
27. C₂×C₃×C₄₉	C₂×C₁₃⋊C₃	direct (N × H)
31. C₂×C₃×C₁₃⋊C₃	C₂×C₄₉	direct (N × H)
32. C₂²×C₄₉	C₃×C₁₃⋊C₃	direct (N × H)
34. C₂²×C₃×C₁₃	C₃×C₄₉	semidirect
35. C₂²×C₁₃⋊C₃	C₃×C₄₉	direct (N × H)
37. C₃×C₄₉	C₂²×C₁₃⋊C₃	direct (N × H)
39. C₃×C₁₃⋊C₃	C₂²×C₄₉	direct (N × H)
40. C₂×C₄₉	C₂×C₃×C₁₃⋊C₃	direct (N × H)
44. C₂×C₃×C₁₃	C₂×C₃×C₄₉	semidirect
45. C₂×C₁₃⋊C₃	C₂×C₃×C₄₉	direct (N × H)
47. C₂²×C₁₃	C₃²×C₄₉	semidirect
48. C₄₉	C₂²×C₃×C₁₃⋊C₃	direct (N × H)
51. C₃×C₁₃	C₂²×C₃×C₄₉	semidirect
52. C₁₃⋊C₃	C₂²×C₃×C₄₉	direct (N × H)
55. C₂×C₁₃	C₂×C₃²×C₄₉	semidirect
60. C₁₃	C₂²×C₃²×C₄₉	semidirect
62. C₂²×C₃	C₁₃⋊C₃×C₄₉	direct (N × H)
66. C₂×C₃	C₂×C₁₃⋊C₃×C₄₉	direct (N × H)
67. C₂²	C₃×C₁₃⋊C₃×C₄₉	direct (N × H)
69. C₃	C₂²×C₁₃⋊C₃×C₄₉	direct (N × H)
70. C₂	C₂×C₃×C₁₃⋊C₃×C₄₉	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×C₁₃⋊C₃×C₄₉	2
C₂²×C₁₃⋊C₃×C₄₉	3
C₂²×C₃×C₁₃×C₄₉	3
C₂²×C₃×C₇×C₁₃⋊C₃	7
C₂²×C₃²×C₄₉	13

Subgroups — C22×C3×C13⋊C3×C49 #37524311

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₁₃⋊C₃×C₄₉ #37524311