Subgroups — #50303953

Order 38220.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇²×C₅⋊(D₁₃⋊C₃)	C₂	1	1	yes	maximal, characteristic
2	C₇⋊(C₇⋊#63121)	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅×C₇²×C₁₃⋊C₃	C₄	1	1	yes	characteristic, complemented, Hall
4	#21439664	—	5	1	no	maximal, Hall
5	C₇²×D₆₅	C₂×C₃	1	1	yes	characteristic
6	#19129749	—	7	8	no	maximal
7	C₇²×D₁₃⋊C₃	—	5	1	no	—
8	C₅×C₇²×C₁₃	C₃×C₄	1	1	yes	characteristic, complemented, Hall
9	C₃×C₇⋊(C₇⋊AGL(1,𝔽₅))	—	13	1	no	maximal, Hall
10	C₇×C₅⋊(D₁₃⋊C₃)	D₇	1	8	yes	—
11	C₇⋊(C₇⋊#261)	—	5	1	no	Hall
12	C₇²×C₁₃⋊C₃	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
13	C₇⋊#63121	—	7	8	no	—
14	C₃×C₇²×D₅	—	13	1	no	—
15	C₅×C₇×C₁₃⋊C₃	Dic₇	1	8	yes	complemented
16	C₇²×D₁₃	—	5	1	no	—
17	#11763979	—	35	8	no	—
18	C₇⋊(C₇⋊AGL(1,𝔽₅))	—	13	1	no	Hall
19	C₇×D₆₅	C₃×D₇	1	8	yes	—
20	#63121⋊C₃	—	49	1	no	Hall
21	C₃×C₅×C₇²	—	13	1	no	Hall
22	C₇²×C₁₃	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
23	C₃×C₇⋊Dic₇	—	65	1	no	Hall
24	C₇×D₁₃⋊C₃	—	5	8	no	—
25	C₇²×D₅	—	13	1	no	—
26	C₅×C₇×C₁₃	C₃×Dic₇	1	8	yes	complemented
27	C₃×C₇⋊AGL(1,𝔽₅)	—	91	8	no	—
28	C₅⋊(D₁₃⋊C₃)	C₇⋊D₇	1	1	yes	characteristic
29	C₇⋊#261	—	35	8	no	—
30	C₂×C₃×C₇²	—	65	1	no	—
31	C₇×C₁₃⋊C₃	C₇⋊AGL(1,𝔽₅)	1	8	yes	complemented
32	#63121	—	49	1	no	Hall
33	C₅×C₇²	AGL(1,𝔽₁₃)	1	1	yes	characteristic, complemented, Hall
34	C₃×C₇×D₅	—	13	8	no	—
35	C₇⋊Dic₇	—	65	1	no	Hall
36	C₅×C₁₃⋊C₃	C₇⋊Dic₇	1	1	yes	characteristic, complemented, Hall
37	C₇×D₁₃	—	5	8	no	—
38	AGL(1,𝔽₁₃)	—	245	1	no	Hall
39	C₃×C₇²	—	13	1	no	Hall
40	C₇⋊AGL(1,𝔽₅)	—	91	8	no	—
41	D₆₅	C₃×C₇⋊D₇	1	1	yes	characteristic
42	C₃×C₅×C₇	—	13	8	no	—
43	C₂×C₇²	—	65	1	no	—
44	C₇×C₁₃	C₃×C₇⋊AGL(1,𝔽₅)	1	8	yes	complemented
45	C₃×Dic₇	—	455	8	no	—
46	D₁₃⋊C₃	—	5	1	no	—
47	C₇×D₅	—	13	8	no	—
48	C₅×C₁₃	C₃×C₇⋊Dic₇	1	1	yes	characteristic, complemented, Hall
49	C₃×AGL(1,𝔽₅)	—	637	1	no	Hall
50	#261	—	245	1	no	Hall
51	C₇²	#63121⋊C₃	1	1	yes	characteristic, complemented, Sylow
52	C₂×C₃×C₇	—	65	8	no	—
53	C₁₃⋊C₃	C₇⋊(C₇⋊AGL(1,𝔽₅))	1	1	yes	characteristic, complemented, Hall
54	C₅×C₇	#11763979	1	8	yes	complemented
55	C₃×D₅	—	13	1	no	—
56	Dic₇	—	455	8	no	—
57	D₁₃	—	5	1	no	—
58	C₃×C₇	—	13	8	no	—
59	AGL(1,𝔽₅)	—	637	1	no	Hall
60	C₃×C₅	—	13	1	no	Hall
61	C₂×C₇	—	65	8	no	—
62	C₁₃	C₃×C₇⋊(C₇⋊AGL(1,𝔽₅))	1	1	yes	characteristic, complemented, Sylow
63	C₃×C₄	—	3185	1	no	Hall
64	D₅	—	13	1	no	—
65	C₇	#19129749	1	8	yes	complemented
66	C₂×C₃	—	65	1	no	—
67	C₅	#21439664	1	1	yes	characteristic, complemented, Sylow
68	C₄	—	3185	1	no	Sylow
69	C₃	—	13	1	no	Sylow
70	C₂	—	65	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅×C₇²×C₁₃	C₃×C₄
prime radical	C₇²×C₅⋊(D₁₃⋊C₃)	C₂
cosocle	C₇²×D₆₅	C₂×C₃
LCS	G ▷ C₅×C₇²×C₁₃
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. C₇⋊(C₇⋊#63121)	C₃	semidirect
3. C₅×C₇²×C₁₃⋊C₃	C₄	semidirect
8. C₅×C₇²×C₁₃	C₃×C₄	semidirect
12. C₇²×C₁₃⋊C₃	AGL(1,𝔽₅)	semidirect
15. C₅×C₇×C₁₃⋊C₃	Dic₇	semidirect
22. C₇²×C₁₃	C₃×AGL(1,𝔽₅)	semidirect
26. C₅×C₇×C₁₃	C₃×Dic₇	semidirect
31. C₇×C₁₃⋊C₃	C₇⋊AGL(1,𝔽₅)	semidirect
33. C₅×C₇²	AGL(1,𝔽₁₃)	semidirect
36. C₅×C₁₃⋊C₃	C₇⋊Dic₇	semidirect
44. C₇×C₁₃	C₃×C₇⋊AGL(1,𝔽₅)	semidirect
48. C₅×C₁₃	C₃×C₇⋊Dic₇	semidirect
51. C₇²	#63121⋊C₃	semidirect
53. C₁₃⋊C₃	C₇⋊(C₇⋊AGL(1,𝔽₅))	semidirect
54. C₅×C₇	#11763979	semidirect
62. C₁₃	C₃×C₇⋊(C₇⋊AGL(1,𝔽₅))	semidirect
65. C₇	#19129749	semidirect
67. C₅	#21439664	semidirect

Maximal subgroups

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

subgroup	index
C₇²×C₅⋊(D₁₃⋊C₃)	2
C₇⋊(C₇⋊#63121)	3
#21439664	5
#19129749	7
C₃×C₇⋊(C₇⋊AGL(1,𝔽₅))	13