Subgroups — C₃×C₄×C₁₃×#3861 #37524821

Order 22932.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₁₃×#3861	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₃×#3861	C₃	1	3	yes	maximal, direct product factor
3	C₃×C₄×C₇²×C₁₃	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₁₃×#3861	C₄	1	1	yes	characteristic, direct product factor, Hall
5	C₂×C₁₃×#3861	C₂×C₃	1	3	yes	—
6	C₂×C₃×C₇²×C₁₃	C₂×C₃	1	1	yes	characteristic
7	C₃×C₄×C₁₃×ΓL(1,𝔽₈)	—	7	8	no	maximal
8	C₄×C₇²×C₁₃	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃×C₇²×C₁₃	C₃×C₄	1	1	yes	characteristic, complemented
10	C₁₃×#3861	C₃×C₄	1	3	yes	direct product factor
11	C₃×C₄×#3861	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂×C₃×C₁₃×ΓL(1,𝔽₈)	—	7	8	no	—
13	C₂×C₇²×C₁₃	C₂×C₃²	1	1	yes	characteristic
14	C₃×C₄×C₇×C₁₃	ΓL(1,𝔽₈)	1	8	yes	complemented
15	C₄×C₁₃×ΓL(1,𝔽₈)	—	7	24	no	—
16	C₂×C₃×#3861	C₂×C₁₃	1	1	yes	characteristic
17	C₃×C₁₃×ΓL(1,𝔽₈)	—	7	8	no	—
18	C₇²×C₁₃	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
19	C₃×C₄×C₇²	C₃×C₁₃	1	1	yes	characteristic, complemented
20	C₄×#3861	C₃×C₁₃	1	3	yes	direct product factor
21	C₂×C₃×C₇×C₁₃	C₂×ΓL(1,𝔽₈)	1	8	yes	—
22	C₂×C₁₃×ΓL(1,𝔽₈)	—	7	24	no	—
23	C₃²×C₄×C₁₃	—	49	1	no	Hall
24	C₃×#3861	C₄×C₁₃	1	1	yes	characteristic, direct product factor, Hall
25	C₄×C₇×C₁₃	C₃×ΓL(1,𝔽₈)	1	8	yes	complemented
26	C₂×C₃×C₇²	C₂×C₃×C₁₃	1	1	yes	characteristic
27	C₂×#3861	C₂×C₃×C₁₃	1	3	yes	—
28	C₃×C₇×C₁₃	C₄×ΓL(1,𝔽₈)	1	8	yes	complemented
29	C₁₃×ΓL(1,𝔽₈)	—	7	24	no	—
30	C₃×C₄×ΓL(1,𝔽₈)	—	7	8	no	—
31	C₂×C₃²×C₁₃	—	49	1	no	—
32	C₄×C₇²	C₃²×C₁₃	1	1	yes	characteristic, complemented, Hall
33	C₂×C₇×C₁₃	C₂×C₃×ΓL(1,𝔽₈)	1	8	yes	—
34	C₃×C₄×C₁₃	—	49	3	no	—
35	C₃×C₄×C₁₃	#3861	1	1	yes	characteristic, direct product factor
36	C₃×C₇²	C₃×C₄×C₁₃	1	1	yes	characteristic, complemented
37	#3861	C₃×C₄×C₁₃	1	3	yes	direct product factor
38	C₂×C₃×ΓL(1,𝔽₈)	—	7	8	no	—
39	C₃²×C₁₃	—	49	1	no	Hall
40	C₂×C₇²	C₂×C₃²×C₁₃	1	1	yes	characteristic
41	C₇×C₁₃	C₃×C₄×ΓL(1,𝔽₈)	1	8	yes	complemented
42	C₃×C₄×C₇	C₁₃×ΓL(1,𝔽₈)	1	8	yes	complemented
43	C₄×ΓL(1,𝔽₈)	—	7	24	no	—
44	C₂×C₃×C₁₃	—	49	3	no	—
45	C₂×C₃×C₁₃	C₂×#3861	1	1	yes	characteristic
46	C₃×ΓL(1,𝔽₈)	—	7	8	no	—
47	C₄×C₁₃	C₃×#3861	1	1	yes	characteristic, direct product factor, Hall
48	C₇²	C₃²×C₄×C₁₃	1	1	yes	characteristic, complemented, Sylow
49	C₂×C₃×C₇	C₂×C₁₃×ΓL(1,𝔽₈)	1	8	yes	—
50	C₂×ΓL(1,𝔽₈)	—	7	24	no	—
51	C₃×C₁₃	—	49	3	no	—
52	C₃×C₁₃	C₄×#3861	1	1	yes	characteristic, direct product factor
53	C₃²×C₄	—	49	1	no	Hall
54	C₄×C₇	C₃×C₁₃×ΓL(1,𝔽₈)	1	8	yes	complemented
55	C₂×C₁₃	C₂×C₃×#3861	1	1	yes	characteristic
56	C₃×C₇	C₄×C₁₃×ΓL(1,𝔽₈)	1	8	yes	complemented
57	ΓL(1,𝔽₈)	—	7	24	no	—
58	C₂×C₃²	—	49	1	no	—
59	C₂×C₇	C₂×C₃×C₁₃×ΓL(1,𝔽₈)	1	8	yes	—
60	C₁₃	C₃×C₄×#3861	1	1	yes	characteristic, direct product factor, Sylow
61	C₃×C₄	—	49	3	no	—
62	C₃×C₄	C₁₃×#3861	1	1	yes	characteristic, direct product factor
63	C₃²	—	49	1	no	Sylow
64	C₇	C₃×C₄×C₁₃×ΓL(1,𝔽₈)	1	8	yes	complemented
65	C₂×C₃	—	49	3	no	—
66	C₂×C₃	C₂×C₁₃×#3861	1	1	yes	characteristic
67	C₄	C₃×C₁₃×#3861	1	1	yes	characteristic, direct product factor, Sylow
68	C₃	—	49	3	no	—
69	C₃	C₄×C₁₃×#3861	1	1	yes	characteristic, direct product factor
70	C₂	C₂×C₃×C₁₃×#3861	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₃×C₄×C₁₃	#3861
commutator, hypocenter	C₇²	C₃²×C₄×C₁₃
Frattini	C₂	C₂×C₃×C₁₃×#3861
Fitting	C₃×C₄×C₇²×C₁₃	C₃
socle	C₂×C₃×C₇²×C₁₃	C₂×C₃
prime radical	C₂×C₃×C₁₃×#3861	C₂
cosocle	C₂×C₇²	C₂×C₃²×C₁₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₃×C₄×C₁₃

Complements

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

N	H ≅ G/N	type
2. C₄×C₁₃×#3861	C₃	direct (N × H)
3. C₃×C₄×C₇²×C₁₃	C₃	semidirect
4. C₃×C₁₃×#3861	C₄	direct (N × H)
8. C₄×C₇²×C₁₃	C₃²	semidirect
9. C₃×C₇²×C₁₃	C₃×C₄	semidirect
10. C₁₃×#3861	C₃×C₄	direct (N × H)
11. C₃×C₄×#3861	C₁₃	direct (N × H)
14. C₃×C₄×C₇×C₁₃	ΓL(1,𝔽₈)	semidirect
18. C₇²×C₁₃	C₃²×C₄	semidirect
19. C₃×C₄×C₇²	C₃×C₁₃	semidirect
20. C₄×#3861	C₃×C₁₃	direct (N × H)
24. C₃×#3861	C₄×C₁₃	direct (N × H)
25. C₄×C₇×C₁₃	C₃×ΓL(1,𝔽₈)	semidirect
28. C₃×C₇×C₁₃	C₄×ΓL(1,𝔽₈)	semidirect
32. C₄×C₇²	C₃²×C₁₃	semidirect
35. C₃×C₄×C₁₃	#3861	direct (N × H)
36. C₃×C₇²	C₃×C₄×C₁₃	semidirect
37. #3861	C₃×C₄×C₁₃	direct (N × H)
41. C₇×C₁₃	C₃×C₄×ΓL(1,𝔽₈)	semidirect
42. C₃×C₄×C₇	C₁₃×ΓL(1,𝔽₈)	semidirect
47. C₄×C₁₃	C₃×#3861	direct (N × H)
48. C₇²	C₃²×C₄×C₁₃	semidirect
52. C₃×C₁₃	C₄×#3861	direct (N × H)
54. C₄×C₇	C₃×C₁₃×ΓL(1,𝔽₈)	semidirect
56. C₃×C₇	C₄×C₁₃×ΓL(1,𝔽₈)	semidirect
60. C₁₃	C₃×C₄×#3861	direct (N × H)
62. C₃×C₄	C₁₃×#3861	direct (N × H)
64. C₇	C₃×C₄×C₁₃×ΓL(1,𝔽₈)	semidirect
67. C₄	C₃×C₁₃×#3861	direct (N × H)
69. C₃	C₄×C₁₃×#3861	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×#3861	2
C₄×C₁₃×#3861	3
C₃×C₄×C₇²×C₁₃	3
C₃×C₄×C₁₃×ΓL(1,𝔽₈)	7
C₃×C₄×#3861	13

Subgroups — C3×C4×C13×#3861 #37524821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×C₁₃×#3861 #37524821