Subgroups — #50303653

Order 38220.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#14274262	C₂	1	1	yes	maximal, characteristic
2	C₇²×Dic₆₅	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅×#14274262	C₄	1	1	yes	characteristic, complemented, Hall
4	#21439550	—	5	1	no	maximal, Hall
5	C₂×C₅×C₇²×C₁₃	C₂×C₃	1	1	yes	characteristic
6	#19129698	—	7	8	no	maximal
7	C₂×#14274262	D₅	1	1	yes	characteristic
8	C₅×C₇²×C₁₃	C₃×C₄	1	1	yes	characteristic, complemented, Hall
9	Dic₅×#3861	—	13	1	no	maximal, Hall
10	C₂×C₅×#63271	—	7	8	no	—
11	C₇²×Dic₁₃	—	5	1	no	Hall
12	#14274262	Dic₅	1	1	yes	characteristic, complemented, Hall
13	C₇×Dic₆₅	ΓL(1,𝔽₈)	1	8	yes	complemented
14	C₂×C₅×#3861	—	13	1	no	—
15	C₅×#63271	—	7	8	no	—
16	C₂×C₇²×C₁₃	C₃×D₅	1	1	yes	characteristic
17	#11763958	—	35	8	no	—
18	C₇²×Dic₅	—	13	1	no	Hall
19	C₂×C₅×C₇×C₁₃	C₂×ΓL(1,𝔽₈)	1	8	yes	—
20	Dic₆₅⋊C₃	—	49	1	no	Hall
21	C₅×#3861	—	13	1	no	Hall
22	C₇²×C₁₃	C₃×Dic₅	1	1	yes	characteristic, complemented, Hall
23	C₄×#3861	—	65	1	no	Hall
24	C₂×#63271	—	7	8	no	—
25	C₂×C₅×C₇²	D₁₃⋊C₃	1	1	yes	characteristic
26	C₅×C₇×C₁₃	C₄×ΓL(1,𝔽₈)	1	8	yes	complemented
27	Dic₅×ΓL(1,𝔽₈)	—	91	8	no	—
28	C₂×C₅×C₁₃⋊C₃	—	49	1	no	—
29	C₇×Dic₁₃	—	5	8	no	—
30	C₂×#3861	—	13	1	no	—
31	#63271	—	7	8	no	—
32	Dic₆₅	#3861	1	1	yes	characteristic, complemented, Hall
33	C₅×C₇²	Dic₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
34	C₂×C₅×ΓL(1,𝔽₈)	—	91	8	no	—
35	C₄×C₇²	—	65	1	no	Hall
36	C₅×C₁₃⋊C₃	—	49	1	no	Hall
37	C₂×C₇×C₁₃	D₅×ΓL(1,𝔽₈)	1	8	yes	—
38	Dic₁₃⋊C₃	—	245	1	no	Hall
39	#3861	—	13	1	no	Hall
40	C₇×Dic₅	—	13	8	no	—
41	C₂×C₅×C₁₃	C₂×#3861	1	1	yes	characteristic
42	C₅×ΓL(1,𝔽₈)	—	91	8	no	—
43	C₂×C₇²	C₅⋊(D₁₃⋊C₃)	1	1	yes	characteristic
44	C₇×C₁₃	Dic₅×ΓL(1,𝔽₈)	1	8	yes	complemented
45	C₄×ΓL(1,𝔽₈)	—	455	8	no	—
46	C₂×C₁₃⋊C₃	—	49	1	no	—
47	C₂×C₅×C₇	#10587717	1	8	yes	—
48	C₅×C₁₃	C₄×#3861	1	1	yes	characteristic, complemented, Hall
49	C₃×Dic₅	—	637	1	no	Hall
50	Dic₁₃	—	5	1	no	Hall
51	C₇²	Dic₆₅⋊C₃	1	1	yes	characteristic, complemented, Sylow
52	C₂×ΓL(1,𝔽₈)	—	91	8	no	—
53	C₁₃⋊C₃	—	49	1	no	Hall
54	C₅×C₇	#11763958	1	8	yes	complemented
55	C₂×C₃×C₅	—	637	1	no	—
56	C₄×C₇	—	65	8	no	—
57	C₂×C₁₃	D₅×#3861	1	1	yes	characteristic
58	ΓL(1,𝔽₈)	—	91	8	no	—
59	Dic₅	—	13	1	no	Hall
60	C₃×C₅	—	637	1	no	Hall
61	C₂×C₇	C₅⋊#10587717	1	8	yes	—
62	C₁₃	Dic₅×#3861	1	1	yes	characteristic, complemented, Sylow
63	C₃×C₄	—	3185	1	no	Hall
64	C₂×C₅	#16838474	1	1	yes	characteristic
65	C₇	#19129698	1	8	yes	complemented
66	C₂×C₃	—	637	1	no	—
67	C₅	#21439550	1	1	yes	characteristic, complemented, Sylow
68	C₄	—	65	1	no	Sylow
69	C₃	—	637	1	no	Sylow
70	C₂	C₅⋊#16838474	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₅⋊#16838474
commutator, hypocenter	C₅×C₇²×C₁₃	C₃×C₄
Fitting, socle, cosocle	C₂×C₅×C₇²×C₁₃	C₂×C₃
prime radical	C₂×C₅×#14274262	C₂
LCS	G ▷ C₅×C₇²×C₁₃
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. C₇²×Dic₆₅	C₃	semidirect
3. C₅×#14274262	C₄	semidirect
8. C₅×C₇²×C₁₃	C₃×C₄	semidirect
12. #14274262	Dic₅	semidirect
13. C₇×Dic₆₅	ΓL(1,𝔽₈)	semidirect
22. C₇²×C₁₃	C₃×Dic₅	semidirect
26. C₅×C₇×C₁₃	C₄×ΓL(1,𝔽₈)	semidirect
32. Dic₆₅	#3861	semidirect
33. C₅×C₇²	Dic₁₃⋊C₃	semidirect
44. C₇×C₁₃	Dic₅×ΓL(1,𝔽₈)	semidirect
48. C₅×C₁₃	C₄×#3861	semidirect
51. C₇²	Dic₆₅⋊C₃	semidirect
54. C₅×C₇	#11763958	semidirect
62. C₁₃	Dic₅×#3861	semidirect
65. C₇	#19129698	semidirect
67. C₅	#21439550	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#14274262	2
C₇²×Dic₆₅	3
#21439550	5
#19129698	7
Dic₅×#3861	13