Subgroups — #50303815

Order 38220.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅×C₁₃×#3862	C₂	1	1	yes	maximal, characteristic
2	C₇×C₇⋊#63120	—	3	1	no	maximal, Hall
3	C₅×C₁₃×#3862	C₄	1	1	yes	characteristic, complemented, Hall
4	#21439563	—	5	1	no	maximal, Hall
5	C₇²×D₅×C₁₃	D₃	1	1	yes	characteristic
6	C₂×C₁₃×#3862	—	5	1	no	—
7	C₅×C₇²×C₁₃	Dic₃	1	1	yes	characteristic, complemented, Hall
8	#15668773	—	13	1	no	maximal, Hall
9	D₅×C₁₃×ΓL(1,𝔽₈)	—	14	1	no	—
10	C₇×Dic₉₁	—	15	1	no	Hall
11	C₁₃×#3862	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
12	C₇⋊#63120	—	3	1	no	—
13	C₇×#63120	—	21	1	no	—
14	D₅×#3862	D₁₃	1	1	yes	characteristic
15	C₅×C₁₃×ΓL(1,𝔽₈)	—	14	1	no	—
16	C₂×C₇²×C₁₃	—	5	1	no	—
17	C₇×C₇⋊AGL(1,𝔽₅)	—	39	1	no	Hall
18	C₇×D₅×C₁₃	—	2	1	no	—
19	C₇×D₅×C₁₃	—	3	1	no	—
20	C₇×D₅×C₁₃	—	3	1	no	—
21	C₃⋊#63120	—	49	1	no	maximal, Hall
22	C₅×#3862	Dic₁₃	1	1	yes	characteristic, complemented, Hall
23	C₇²×C₁₃	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented, Hall
24	#10596868	—	65	1	no	Hall
25	C₂×C₁₃×ΓL(1,𝔽₈)	—	70	1	no	—
26	C₇²×D₅	D₃₉	1	1	yes	characteristic
27	C₅×C₇×C₁₃	—	2	1	no	—
28	C₅×C₇×C₁₃	—	3	1	no	—
29	C₅×C₇×C₁₃	—	3	1	no	—
30	C₃×D₅×C₁₃	—	49	1	no	—
31	Dic₉₁	—	15	1	no	—
32	C₇×Dic₁₃	—	105	1	no	—
33	C₂×#3862	—	5	1	no	—
34	C₁₃×ΓL(1,𝔽₈)	—	14	1	no	—
35	#63120	—	21	1	no	Hall
36	C₅×C₇²	Dic₃₉	1	1	yes	characteristic, complemented, Hall
37	D₅×ΓL(1,𝔽₈)	—	14	1	no	—
38	C₇×Dic₇	—	195	1	no	Hall
39	C₃×C₅×C₁₃	—	49	1	no	Hall
40	C₂×C₇×C₁₃	—	10	1	no	—
41	C₂×C₇×C₁₃	—	15	1	no	—
42	C₂×C₇×C₁₃	—	15	1	no	—
43	Dic₃₉	—	245	1	no	Hall
44	#3862	#63120	1	1	yes	characteristic, complemented, Hall
45	C₇⋊AGL(1,𝔽₅)	—	39	1	no	—
46	C₇×AGL(1,𝔽₅)	—	273	1	no	—
47	D₅×C₁₃	#64418	1	1	yes	characteristic
48	C₅×ΓL(1,𝔽₈)	—	14	1	no	—
49	C₂×C₇²	—	5	1	no	—
50	C₇×C₁₃	—	2	1	no	—
51	C₇×C₁₃	—	3	1	no	—
52	C₇×C₁₃	—	3	1	no	—
53	C₂×C₃×C₁₃	—	245	1	no	—
54	C₇×D₅	—	2	1	no	—
55	C₇×D₅	—	3	1	no	—
56	C₇×D₅	—	3	1	no	—
57	C₅×C₁₃	#10596868	1	1	yes	characteristic, complemented, Hall
58	ΓL(1,𝔽₁₆)	—	637	1	no	Hall
59	Dic₁₃	—	105	1	no	Hall
60	C₇²	C₃⋊#63120	1	1	yes	characteristic, complemented, Sylow
61	C₂×ΓL(1,𝔽₈)	—	70	1	no	—
62	C₃×C₁₃	—	49	1	no	Hall
63	C₅×C₇	—	2	1	no	—
64	C₅×C₇	—	3	1	no	—
65	C₅×C₇	—	3	1	no	—
66	C₃×D₅	—	49	1	no	—
67	C₄×C₇	—	1365	1	no	—
68	Dic₇	—	195	1	no	—
69	C₂×C₁₃	—	5	1	no	—
70	ΓL(1,𝔽₈)	—	14	1	no	—
71	AGL(1,𝔽₅)	—	273	1	no	Hall
72	C₃×C₅	—	49	1	no	Hall
73	C₂×C₇	—	10	1	no	—
74	C₂×C₇	—	15	1	no	—
75	C₂×C₇	—	15	1	no	—
76	C₁₃	#15668773	1	1	yes	characteristic, complemented, Sylow
77	Dic₃	—	3185	1	no	Hall
78	D₅	C₁₃⋊#64418	1	1	yes	characteristic
79	C₇	—	2	1	no	—
80	C₇	—	3	1	no	—
81	C₇	—	3	1	no	—
82	C₂×C₃	—	245	1	no	—
83	C₅	#21439563	1	1	yes	characteristic, complemented, Sylow
84	C₄	—	1365	1	no	Sylow
85	C₃	—	49	1	no	Sylow
86	C₂	—	5	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, hypocenter	C₅×C₁₃×#3862	C₄
Fitting, socle	C₅×C₇²×C₁₃	Dic₃
prime radical, cosocle	D₅×C₁₃×#3862	C₂
LCS	G ▷ C₅×C₁₃×#3862
UCS	C₁

Complements

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

N	H ≅ G/N	type
3. C₅×C₁₃×#3862	C₄	semidirect
7. C₅×C₇²×C₁₃	Dic₃	semidirect
11. C₁₃×#3862	AGL(1,𝔽₅)	semidirect
22. C₅×#3862	Dic₁₃	semidirect
23. C₇²×C₁₃	ΓL(1,𝔽₁₆)	semidirect
36. C₅×C₇²	Dic₃₉	semidirect
44. #3862	#63120	semidirect
57. C₅×C₁₃	#10596868	semidirect
60. C₇²	C₃⋊#63120	semidirect
76. C₁₃	#15668773	semidirect
83. C₅	#21439563	semidirect

Maximal subgroups

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

subgroup	index
D₅×C₁₃×#3862	2
C₇×C₇⋊#63120	3
#21439563	5
#15668773	13
C₃⋊#63120	49