Subgroups — C₅×C₁₃⋊C₃×C₇⋊Dic₇ #50303604

Order 38220.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₇²×C₁₃⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₅×C₁₃×C₇⋊Dic₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅×C₇²×C₁₃⋊C₃	C₄	1	1	yes	characteristic, complemented, Hall
4	C₁₃⋊C₃×C₇⋊Dic₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₅×C₇²×C₁₃	C₂×C₃	1	1	yes	characteristic
6	C₅×Dic₇×C₁₃⋊C₃	—	7	8	no	maximal
7	C₂×C₇²×C₁₃⋊C₃	C₂×C₅	1	1	yes	characteristic
8	C₅×C₇²×C₁₃	C₃×C₄	1	1	yes	characteristic, complemented, Hall
9	C₃×C₅×C₇⋊Dic₇	—	13	1	no	maximal, Hall
10	C₂×C₅×C₇×C₁₃⋊C₃	D₇	1	8	yes	—
11	C₁₃×C₇⋊Dic₇	C₃×C₅	1	1	yes	characteristic, complemented, Hall
12	C₇²×C₁₃⋊C₃	C₄×C₅	1	1	yes	characteristic, complemented, Hall
13	C₅×C₁₃×Dic₇	—	7	8	no	—
14	C₂×C₃×C₅×C₇²	—	13	1	no	—
15	C₅×C₇×C₁₃⋊C₃	Dic₇	1	8	yes	complemented
16	C₂×C₇²×C₁₃	C₂×C₃×C₅	1	1	yes	characteristic
17	Dic₇×C₁₃⋊C₃	—	7	8	no	—
18	C₅×C₇⋊Dic₇	C₁₃⋊C₃	1	1	yes	characteristic, direct product factor, Hall
19	C₂×C₅×C₇×C₁₃	C₃×D₇	1	8	yes	—
20	C₄×C₅×C₁₃⋊C₃	—	49	1	no	Hall
21	C₃×C₅×C₇²	—	13	1	no	Hall
22	C₇²×C₁₃	C₃×C₄×C₅	1	1	yes	characteristic, complemented, Hall
23	C₃×C₇⋊Dic₇	—	13	1	no	Hall
24	C₂×C₇×C₁₃⋊C₃	C₅×D₇	1	8	yes	—
25	C₂×C₅×C₇²	C₂×C₁₃⋊C₃	1	1	yes	characteristic
26	C₅×C₇×C₁₃	C₃×Dic₇	1	8	yes	complemented
27	C₃×C₅×Dic₇	—	91	8	no	—
28	C₂×C₅×C₁₃⋊C₃	C₇⋊D₇	1	1	yes	characteristic
29	C₁₃×Dic₇	—	7	8	no	—
30	C₂×C₃×C₇²	—	13	1	no	—
31	C₇×C₁₃⋊C₃	C₅×Dic₇	1	8	yes	complemented
32	C₄×C₅×C₁₃	—	49	1	no	Hall
33	C₅×C₇²	C₄×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
34	C₂×C₃×C₅×C₇	—	13	8	no	—
35	C₇⋊Dic₇	C₅×C₁₃⋊C₃	1	1	yes	characteristic, direct product factor, Hall
36	C₅×C₁₃⋊C₃	C₇⋊Dic₇	1	1	yes	characteristic, direct product factor, Hall
37	C₂×C₇×C₁₃	C₃×C₅×D₇	1	8	yes	—
38	C₄×C₁₃⋊C₃	—	49	1	no	Hall
39	C₃×C₇²	—	13	1	no	Hall
40	C₅×Dic₇	—	7	8	no	—
41	C₂×C₅×C₁₃	C₃×C₇⋊D₇	1	1	yes	characteristic
42	C₃×C₅×C₇	—	13	8	no	—
43	C₂×C₇²	C₂×C₅×C₁₃⋊C₃	1	1	yes	characteristic
44	C₇×C₁₃	C₃×C₅×Dic₇	1	8	yes	complemented
45	C₃×Dic₇	—	91	8	no	—
46	C₂×C₁₃⋊C₃	C₅×C₇⋊D₇	1	1	yes	characteristic
47	C₂×C₅×C₇	D₇×C₁₃⋊C₃	1	8	yes	—
48	C₅×C₁₃	C₃×C₇⋊Dic₇	1	1	yes	characteristic, complemented, Hall
49	C₃×C₄×C₅	—	637	1	no	Hall
50	C₄×C₁₃	—	49	1	no	Hall
51	C₇²	C₄×C₅×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Sylow
52	C₂×C₃×C₇	—	13	8	no	—
53	C₁₃⋊C₃	C₅×C₇⋊Dic₇	1	1	yes	characteristic, direct product factor, Hall
54	C₅×C₇	Dic₇×C₁₃⋊C₃	1	8	yes	complemented
55	C₂×C₃×C₅	—	13	1	no	—
56	Dic₇	—	7	8	no	—
57	C₂×C₁₃	C₃×C₅×C₇⋊D₇	1	1	yes	characteristic
58	C₃×C₇	—	13	8	no	—
59	C₄×C₅	—	49	1	no	Hall
60	C₃×C₅	—	13	1	no	Hall
61	C₂×C₇	C₅×D₇×C₁₃⋊C₃	1	8	yes	—
62	C₁₃	C₃×C₅×C₇⋊Dic₇	1	1	yes	characteristic, complemented, Sylow
63	C₃×C₄	—	637	1	no	Hall
64	C₂×C₅	C₁₃⋊C₃×C₇⋊D₇	1	1	yes	characteristic
65	C₇	C₅×Dic₇×C₁₃⋊C₃	1	8	yes	complemented
66	C₂×C₃	—	13	1	no	—
67	C₅	C₁₃⋊C₃×C₇⋊Dic₇	1	1	yes	characteristic, direct product factor, Sylow
68	C₄	—	49	1	no	Sylow
69	C₃	—	13	1	no	Sylow
70	C₂	C₅×C₁₃⋊C₃×C₇⋊D₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₅	C₁₃⋊C₃×C₇⋊D₇
commutator, hypocenter	C₇²×C₁₃	C₃×C₄×C₅
Frattini	C₂	C₅×C₁₃⋊C₃×C₇⋊D₇
Fitting, socle	C₂×C₅×C₇²×C₁₃	C₂×C₃
prime radical	C₂×C₅×C₇²×C₁₃⋊C₃	C₂
cosocle	C₂×C₇²×C₁₃	C₂×C₃×C₅
LCS	G ▷ C₇²×C₁₃
UCS	C₁ ◁ C₂×C₅

Complements

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

N	H ≅ G/N	type
2. C₅×C₁₃×C₇⋊Dic₇	C₃	semidirect
3. C₅×C₇²×C₁₃⋊C₃	C₄	semidirect
4. C₁₃⋊C₃×C₇⋊Dic₇	C₅	direct (N × H)
8. C₅×C₇²×C₁₃	C₃×C₄	semidirect
11. C₁₃×C₇⋊Dic₇	C₃×C₅	semidirect
12. C₇²×C₁₃⋊C₃	C₄×C₅	semidirect
15. C₅×C₇×C₁₃⋊C₃	Dic₇	semidirect
18. C₅×C₇⋊Dic₇	C₁₃⋊C₃	direct (N × H)
22. C₇²×C₁₃	C₃×C₄×C₅	semidirect
26. C₅×C₇×C₁₃	C₃×Dic₇	semidirect
31. C₇×C₁₃⋊C₃	C₅×Dic₇	semidirect
33. C₅×C₇²	C₄×C₁₃⋊C₃	semidirect
35. C₇⋊Dic₇	C₅×C₁₃⋊C₃	direct (N × H)
36. C₅×C₁₃⋊C₃	C₇⋊Dic₇	direct (N × H)
44. C₇×C₁₃	C₃×C₅×Dic₇	semidirect
48. C₅×C₁₃	C₃×C₇⋊Dic₇	semidirect
51. C₇²	C₄×C₅×C₁₃⋊C₃	semidirect
53. C₁₃⋊C₃	C₅×C₇⋊Dic₇	direct (N × H)
54. C₅×C₇	Dic₇×C₁₃⋊C₃	semidirect
62. C₁₃	C₃×C₅×C₇⋊Dic₇	semidirect
65. C₇	C₅×Dic₇×C₁₃⋊C₃	semidirect
67. C₅	C₁₃⋊C₃×C₇⋊Dic₇	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×C₅×C₇²×C₁₃⋊C₃	2
C₅×C₁₃×C₇⋊Dic₇	3
C₁₃⋊C₃×C₇⋊Dic₇	5
C₅×Dic₇×C₁₃⋊C₃	7
C₃×C₅×C₇⋊Dic₇	13

Subgroups — C5×C13⋊C3×C7⋊Dic7 #50303604

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₁₃⋊C₃×C₇⋊Dic₇ #50303604