Subgroups — D₅²×#3861 #28413329

Order 14700.

480 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅⋊D₅×#3861	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×D₅×#3861	C₂	1	2	yes	maximal, complemented
3	C₇²×D₅²	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₅²×#3861	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×D₅×#3861	—	5	2	no	maximal
6	C₇²×C₅⋊D₅	C₂×C₃	1	1	yes	characteristic, complemented
7	C₅×C₇²×D₅	C₂×C₃	1	2	yes	complemented
8	D₅²×ΓL(1,𝔽₈)	—	7	8	no	maximal
9	C₂×C₅×#3861	—	5	2	no	—
10	D₅×#3861	—	5	2	no	—
11	D₅×#3861	—	10	2	no	—
12	D₅×#3861	D₅	1	2	yes	direct product factor
13	C₅²×C₇²	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
14	ΓL(1,𝔽₈)×C₅⋊D₅	—	7	8	no	—
15	C₅×D₅×ΓL(1,𝔽₈)	—	7	16	no	—
16	C₂×C₇²×D₅	—	5	2	no	—
17	C₅×#3861	—	2	2	no	—
18	C₅×#3861	C₂×D₅	1	2	yes	complemented
19	C₇×D₅²	ΓL(1,𝔽₈)	1	8	yes	complemented
20	C₂²×#3861	—	25	1	no	Hall
21	C₅²×ΓL(1,𝔽₈)	—	7	8	no	—
22	C₂×C₅×C₇²	—	5	2	no	—
23	C₇²×D₅	—	5	2	no	—
24	C₇²×D₅	—	10	2	no	—
25	C₇²×D₅	C₃×D₅	1	2	yes	complemented
26	C₂×D₅×ΓL(1,𝔽₈)	—	35	16	no	—
27	C₇×C₅⋊D₅	C₂×ΓL(1,𝔽₈)	1	8	yes	complemented
28	C₅×C₇×D₅	C₂×ΓL(1,𝔽₈)	1	16	yes	complemented
29	C₃×D₅²	—	49	1	no	Hall
30	C₂×#3861	—	5	2	no	—
31	C₂×#3861	—	25	1	no	—
32	C₅×C₇²	—	2	2	no	—
33	C₅×C₇²	C₂×C₃×D₅	1	2	yes	complemented
34	C₂×C₅×ΓL(1,𝔽₈)	—	35	16	no	—
35	D₅×ΓL(1,𝔽₈)	—	7	16	no	—
36	D₅×ΓL(1,𝔽₈)	—	35	16	no	—
37	D₅×ΓL(1,𝔽₈)	—	70	16	no	—
38	C₂²×C₇²	—	25	1	no	Hall
39	C₅²×C₇	C₂²×ΓL(1,𝔽₈)	1	8	yes	complemented
40	C₃×C₅⋊D₅	—	49	1	no	—
41	C₃×C₅×D₅	—	49	2	no	—
42	#3861	D₅²	1	1	yes	characteristic, direct product factor, Hall
43	C₂×C₇×D₅	—	5	16	no	—
44	C₅×ΓL(1,𝔽₈)	—	7	16	no	—
45	C₅×ΓL(1,𝔽₈)	—	14	16	no	—
46	D₅²	#3861	1	1	yes	characteristic, direct product factor, Hall
47	C₂×C₇²	—	5	2	no	—
48	C₂×C₇²	—	25	1	no	—
49	C₂²×ΓL(1,𝔽₈)	—	175	8	no	—
50	C₃×C₅²	—	49	1	no	Hall
51	C₂×C₅×C₇	—	5	16	no	—
52	C₇×D₅	—	5	16	no	—
53	C₇×D₅	—	10	16	no	—
54	C₇×D₅	D₅×ΓL(1,𝔽₈)	1	16	yes	complemented
55	C₂×C₃×D₅	—	245	2	no	—
56	C₅⋊D₅	C₂×#3861	1	1	yes	characteristic, complemented
57	C₅×D₅	C₂×#3861	1	2	yes	complemented
58	C₇²	C₃×D₅²	1	1	yes	characteristic, complemented, Sylow
59	C₂×ΓL(1,𝔽₈)	—	35	16	no	—
60	C₂×ΓL(1,𝔽₈)	—	175	8	no	—
61	C₅×C₇	—	2	16	no	—
62	C₅×C₇	C₂×D₅×ΓL(1,𝔽₈)	1	16	yes	complemented
63	C₂×C₃×C₅	—	245	2	no	—
64	C₃×D₅	—	49	2	no	—
65	C₃×D₅	—	490	2	no	—
66	C₃×D₅	—	245	2	no	—
67	C₂²×C₇	—	25	8	no	—
68	C₅²	C₂²×#3861	1	1	yes	characteristic, complemented, Sylow
69	ΓL(1,𝔽₈)	—	7	8	no	—
70	C₂×D₅	—	5	2	no	—
71	C₃×C₅	—	49	2	no	—
72	C₃×C₅	—	98	2	no	—
73	C₂×C₇	—	5	16	no	—
74	C₂×C₇	—	25	8	no	—
75	C₂²×C₃	—	1225	1	no	Hall
76	C₂×C₅	—	5	2	no	—
77	D₅	—	5	2	no	—
78	D₅	—	10	2	no	—
79	D₅	D₅×#3861	1	2	yes	direct product factor
80	C₇	D₅²×ΓL(1,𝔽₈)	1	8	yes	complemented
81	C₂×C₃	—	1225	1	no	—
82	C₂×C₃	—	245	2	no	—
83	C₅	—	2	2	no	—
84	C₅	C₂×D₅×#3861	1	2	yes	complemented
85	C₂²	—	25	1	no	Sylow
86	C₃	—	49	1	no	Sylow
87	C₂	—	5	2	no	—
88	C₂	—	25	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅²×C₇²	C₂²×C₃
LCS	G ▷ C₅²×C₇²
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₅⋊D₅×#3861	C₂	semidirect
2. C₅×D₅×#3861	C₂	semidirect
3. C₇²×D₅²	C₃	semidirect
4. C₅²×#3861	C₂²	semidirect
6. C₇²×C₅⋊D₅	C₂×C₃	semidirect
7. C₅×C₇²×D₅	C₂×C₃	semidirect
12. D₅×#3861	D₅	direct (N × H)
13. C₅²×C₇²	C₂²×C₃	semidirect
18. C₅×#3861	C₂×D₅	semidirect
19. C₇×D₅²	ΓL(1,𝔽₈)	semidirect
25. C₇²×D₅	C₃×D₅	semidirect
27. C₇×C₅⋊D₅	C₂×ΓL(1,𝔽₈)	semidirect
28. C₅×C₇×D₅	C₂×ΓL(1,𝔽₈)	semidirect
33. C₅×C₇²	C₂×C₃×D₅	semidirect
39. C₅²×C₇	C₂²×ΓL(1,𝔽₈)	semidirect
42. #3861	D₅²	direct (N × H)
46. D₅²	#3861	direct (N × H)
54. C₇×D₅	D₅×ΓL(1,𝔽₈)	semidirect
56. C₅⋊D₅	C₂×#3861	semidirect
57. C₅×D₅	C₂×#3861	semidirect
58. C₇²	C₃×D₅²	semidirect
62. C₅×C₇	C₂×D₅×ΓL(1,𝔽₈)	semidirect
68. C₅²	C₂²×#3861	semidirect
79. D₅	D₅×#3861	direct (N × H)
80. C₇	D₅²×ΓL(1,𝔽₈)	semidirect
84. C₅	C₂×D₅×#3861	semidirect

Maximal subgroups

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

subgroup	index
C₅⋊D₅×#3861	2
C₅×D₅×#3861	2
C₇²×D₅²	3
C₂×D₅×#3861	5
D₅²×ΓL(1,𝔽₈)	7

Subgroups — D52×#3861 #28413329

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅²×#3861 #28413329