Subgroups — #50303626

Order 38220.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#14274263	C₂	1	1	yes	maximal, characteristic
2	C₁₃×C₇⋊Dic₃₅	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅×#14274263	C₄	1	1	yes	characteristic, complemented, Hall
4	#21439524	—	5	1	no	maximal, Hall
5	C₂×C₅×C₇²×C₁₃	C₂×C₃	1	1	yes	characteristic
6	#19129701	—	7	8	no	maximal
7	C₂×#14274263	D₅	1	1	yes	characteristic
8	C₅×C₇²×C₁₃	C₃×C₄	1	1	yes	characteristic, complemented, Hall
9	#15668734	—	13	1	no	maximal, Hall
10	C₂×C₅×#63270	—	7	8	no	—
11	C₁₃×C₇⋊Dic₇	—	5	1	no	Hall
12	#14274263	Dic₅	1	1	yes	characteristic, complemented, Hall
13	C₁₃×Dic₃₅	—	7	8	no	—
14	C₂×C₅×#3861	—	13	1	no	—
15	C₅×#63270	—	7	8	no	—
16	C₂×C₇²×C₁₃	C₃×D₅	1	1	yes	characteristic
17	#11763961	—	35	8	no	—
18	C₇⋊Dic₃₅	C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
19	C₂×C₅×C₇×C₁₃	AGL(1,𝔽₇)	1	8	yes	—
20	Dic₅×C₁₃⋊C₃	—	49	1	no	Hall
21	C₅×#3861	—	13	1	no	Hall
22	C₇²×C₁₃	C₃×Dic₅	1	1	yes	characteristic, complemented, Hall
23	#10596874	—	65	1	no	Hall
24	C₂×#63270	—	7	8	no	—
25	C₂×C₅×C₇²	C₂×C₁₃⋊C₃	1	1	yes	characteristic
26	C₅×C₇×C₁₃	ΓL(1,𝔽₈)⋊C₄	1	8	yes	complemented
27	ΓL(1,𝔽₈)⋊Dic₅	—	91	8	no	—
28	C₂×C₅×C₁₃⋊C₃	—	49	1	no	—
29	C₁₃×Dic₇	—	35	8	no	—
30	C₂×#3861	—	13	1	no	—
31	#63270	—	7	8	no	—
32	C₁₃×Dic₅	—	49	1	no	Hall
33	C₅×C₇²	C₄×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
34	C₂×C₅×ΓL(1,𝔽₈)	—	91	8	no	—
35	C₇⋊Dic₇	—	5	1	no	Hall
36	C₅×C₁₃⋊C₃	—	49	1	no	Hall
37	C₂×C₇×C₁₃	C₅⋊AGL(1,𝔽₇)	1	8	yes	—
38	C₄×C₁₃⋊C₃	—	245	1	no	Hall
39	#3861	—	13	1	no	Hall
40	Dic₃₅	—	7	8	no	—
41	C₂×C₅×C₁₃	#64424	1	1	yes	characteristic
42	C₅×ΓL(1,𝔽₈)	—	91	8	no	—
43	C₂×C₇²	D₅×C₁₃⋊C₃	1	1	yes	characteristic
44	C₇×C₁₃	ΓL(1,𝔽₈)⋊Dic₅	1	8	yes	complemented
45	ΓL(1,𝔽₈)⋊C₄	—	455	8	no	—
46	C₂×C₁₃⋊C₃	—	49	1	no	—
47	C₂×C₅×C₇	#10587720	1	8	yes	—
48	C₅×C₁₃	#10596874	1	1	yes	characteristic, complemented, Hall
49	C₃×Dic₅	—	637	1	no	Hall
50	C₄×C₁₃	—	245	1	no	Hall
51	C₇²	Dic₅×C₁₃⋊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	#11763961	1	8	yes	complemented
55	C₂×C₃×C₅	—	637	1	no	—
56	Dic₇	—	35	8	no	—
57	C₂×C₁₃	C₅⋊#64424	1	1	yes	characteristic
58	ΓL(1,𝔽₈)	—	91	8	no	—
59	Dic₅	—	49	1	no	Hall
60	C₃×C₅	—	637	1	no	Hall
61	C₂×C₇	C₅⋊#10587720	1	8	yes	—
62	C₁₃	#15668734	1	1	yes	characteristic, complemented, Sylow
63	C₃×C₄	—	3185	1	no	Hall
64	C₂×C₅	#16838448	1	1	yes	characteristic
65	C₇	#19129701	1	8	yes	complemented
66	C₂×C₃	—	637	1	no	—
67	C₅	#21439524	1	1	yes	characteristic, complemented, Sylow
68	C₄	—	245	1	no	Sylow
69	C₃	—	637	1	no	Sylow
70	C₂	C₅⋊#16838448	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₅⋊#16838448
commutator, hypocenter	C₅×C₇²×C₁₃	C₃×C₄
Fitting, socle, cosocle	C₂×C₅×C₇²×C₁₃	C₂×C₃
prime radical	C₂×C₅×#14274263	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. C₁₃×C₇⋊Dic₃₅	C₃	semidirect
3. C₅×#14274263	C₄	semidirect
8. C₅×C₇²×C₁₃	C₃×C₄	semidirect
12. #14274263	Dic₅	semidirect
18. C₇⋊Dic₃₅	C₁₃⋊C₃	semidirect
22. C₇²×C₁₃	C₃×Dic₅	semidirect
26. C₅×C₇×C₁₃	ΓL(1,𝔽₈)⋊C₄	semidirect
33. C₅×C₇²	C₄×C₁₃⋊C₃	semidirect
44. C₇×C₁₃	ΓL(1,𝔽₈)⋊Dic₅	semidirect
48. C₅×C₁₃	#10596874	semidirect
51. C₇²	Dic₅×C₁₃⋊C₃	semidirect
54. C₅×C₇	#11763961	semidirect
62. C₁₃	#15668734	semidirect
65. C₇	#19129701	semidirect
67. C₅	#21439524	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#14274263	2
C₁₃×C₇⋊Dic₃₅	3
#21439524	5
#19129701	7
#15668734	13