Subgroups — #50304201

Order 38220.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×D₁₃×#3862	C₂	1	1	yes	maximal, characteristic
2	C₇⋊(C₇⋊#63118)	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅×C₁₃×#3862	C₄	1	1	yes	characteristic, complemented, Hall
4	#21439605	—	5	1	no	maximal, Hall
5	C₅×C₇²×D₁₃	C₂×C₃	1	1	yes	characteristic
6	ΓL(1,𝔽₈)⋊#63118	—	7	2	no	maximal
7	D₁₃×#3862	D₅	1	1	yes	characteristic
8	C₅×C₇²×C₁₃	C₃×C₄	1	1	yes	characteristic, complemented, Hall
9	#15668735	—	13	1	no	maximal, Hall
10	C₅×ΓL(1,𝔽₈)×D₁₃	—	7	2	no	—
11	C₇⋊(C₇⋊#261)	—	5	1	no	Hall
12	C₁₃×#3862	Dic₅	1	1	yes	characteristic, complemented, Hall
13	C₇⋊#63118	—	7	2	no	—
14	C₇⋊#63118	—	21	2	no	—
15	C₂×C₅×#3862	—	13	1	no	—
16	C₅×C₁₃×ΓL(1,𝔽₈)	—	7	2	no	—
17	C₇²×D₁₃	C₃×D₅	1	1	yes	characteristic
18	ΓL(1,𝔽₈)⋊#261	—	35	2	no	—
19	C₇⋊Dic₃₅	—	13	1	no	Hall
20	C₅×C₇×D₁₃	—	3	2	no	—
21	C₅×C₇×D₁₃	AGL(1,𝔽₇)	1	2	yes	—
22	C₃×#63118	—	49	1	no	Hall
23	C₅×#3862	#261	1	1	yes	characteristic, complemented, Hall
24	C₇²×C₁₃	C₃×Dic₅	1	1	yes	characteristic, complemented, Hall
25	#10596875	—	65	1	no	Hall
26	ΓL(1,𝔽₈)×D₁₃	—	7	2	no	—
27	C₂×C₅×C₇²	—	13	1	no	—
28	C₅×C₇×C₁₃	—	3	2	no	—
29	C₅×C₇×C₁₃	ΓL(1,𝔽₈)⋊C₄	1	2	yes	complemented
30	ΓL(1,𝔽₈)⋊Dic₅	—	91	2	no	—
31	C₃×C₅×D₁₃	—	49	1	no	—
32	C₇⋊#261	—	35	2	no	—
33	C₇⋊#261	—	105	2	no	—
34	C₂×#3862	—	13	1	no	—
35	C₁₃×ΓL(1,𝔽₈)	—	7	2	no	—
36	#63118	—	49	1	no	Hall
37	C₅×C₇²	C₃×#261	1	1	yes	characteristic, complemented, Hall
38	C₂×C₅×ΓL(1,𝔽₈)	—	91	2	no	—
39	C₇⋊Dic₇	—	65	1	no	Hall
40	C₃×C₅×C₁₃	—	49	1	no	Hall
41	C₇×D₁₃	—	3	2	no	—
42	C₇×D₁₃	C₅⋊AGL(1,𝔽₇)	1	2	yes	—
43	C₃×#261	—	245	1	no	Hall
44	#3862	#63118	1	1	yes	characteristic, complemented, Hall
45	Dic₃₅	—	273	2	no	—
46	Dic₃₅	—	91	2	no	—
47	C₅×D₁₃	#64425	1	1	yes	characteristic
48	C₅×ΓL(1,𝔽₈)	—	7	2	no	—
49	C₂×C₇²	—	13	1	no	—
50	C₇×C₁₃	—	3	2	no	—
51	C₇×C₁₃	ΓL(1,𝔽₈)⋊Dic₅	1	2	yes	complemented
52	ΓL(1,𝔽₈)⋊C₄	—	455	2	no	—
53	C₃×D₁₃	—	49	1	no	—
54	C₂×C₅×C₇	—	13	2	no	—
55	C₂×C₅×C₇	—	39	2	no	—
56	C₅×C₁₃	#10596875	1	1	yes	characteristic, complemented, Hall
57	C₃×Dic₅	—	637	1	no	Hall
58	#261	—	245	1	no	Hall
59	C₇²	C₃×#63118	1	1	yes	characteristic, complemented, Sylow
60	C₂×ΓL(1,𝔽₈)	—	91	2	no	—
61	C₃×C₁₃	—	49	1	no	Hall
62	C₅×C₇	—	3	2	no	—
63	C₅×C₇	ΓL(1,𝔽₈)⋊#261	1	2	yes	complemented
64	C₂×C₃×C₅	—	637	1	no	—
65	Dic₇	—	1365	2	no	—
66	Dic₇	—	455	2	no	—
67	D₁₃	C₅⋊#64425	1	1	yes	characteristic
68	ΓL(1,𝔽₈)	—	7	2	no	—
69	Dic₅	—	637	1	no	Hall
70	C₃×C₅	—	49	1	no	Hall
71	C₂×C₇	—	13	2	no	—
72	C₂×C₇	—	39	2	no	—
73	C₁₃	#15668735	1	1	yes	characteristic, complemented, Sylow
74	C₃×C₄	—	3185	1	no	Hall
75	C₂×C₅	—	13	1	no	—
76	C₇	—	3	2	no	—
77	C₇	ΓL(1,𝔽₈)⋊#63118	1	2	yes	complemented
78	C₂×C₃	—	637	1	no	—
79	C₅	#21439605	1	1	yes	characteristic, complemented, Sylow
80	C₄	—	3185	1	no	Sylow
81	C₃	—	49	1	no	Sylow
82	C₂	—	13	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅×C₇²×C₁₃	C₃×C₄
prime radical	C₅×D₁₃×#3862	C₂
cosocle	C₅×C₇²×D₁₃	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⋊#63118)	C₃	semidirect
3. C₅×C₁₃×#3862	C₄	semidirect
8. C₅×C₇²×C₁₃	C₃×C₄	semidirect
12. C₁₃×#3862	Dic₅	semidirect
23. C₅×#3862	#261	semidirect
24. C₇²×C₁₃	C₃×Dic₅	semidirect
29. C₅×C₇×C₁₃	ΓL(1,𝔽₈)⋊C₄	semidirect
37. C₅×C₇²	C₃×#261	semidirect
44. #3862	#63118	semidirect
51. C₇×C₁₃	ΓL(1,𝔽₈)⋊Dic₅	semidirect
56. C₅×C₁₃	#10596875	semidirect
59. C₇²	C₃×#63118	semidirect
63. C₅×C₇	ΓL(1,𝔽₈)⋊#261	semidirect
73. C₁₃	#15668735	semidirect
77. C₇	ΓL(1,𝔽₈)⋊#63118	semidirect
79. C₅	#21439605	semidirect

Maximal subgroups

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

subgroup	index
C₅×D₁₃×#3862	2
C₇⋊(C₇⋊#63118)	3
#21439605	5
ΓL(1,𝔽₈)⋊#63118	7
#15668735	13