Subgroups — C₂×(C₅⋊D₅)⋊#3862 #28413444

Order 14700.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²⋊#3862	C₂	1	1	yes	maximal, characteristic, complemented
2	(C₅⋊D₅)⋊#3862	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₇²×C₅⋊D₅	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₅²⋊#3862	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₇²×C₅⋊D₅	C₂×C₃	1	2	yes	complemented
6	C₂×C₅²×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂×(C₅⋊D₅)⋊ΓL(1,𝔽₈)	—	7	2	no	maximal
8	C₅²×C₇²	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
9	C₂×C₇⋊(C₅²⋊C₃)	—	7	2	no	—
10	(C₅⋊D₅)⋊ΓL(1,𝔽₈)	—	7	4	no	—
11	C₂×C₇²×D₅	—	15	2	no	—
12	C₂×C₇×C₅⋊D₅	—	3	2	no	—
13	C₂×C₇×C₅⋊D₅	ΓL(1,𝔽₈)	1	2	yes	complemented
14	C₂²×#3862	—	25	1	no	maximal, Hall
15	C₇⋊(C₅²⋊C₃)	—	7	2	no	—
16	C₂×C₅×C₇²	—	3	2	no	—
17	C₇²×D₅	—	15	4	no	—
18	C₂×C₅²×C₇	—	3	2	no	—
19	C₂×C₅²×C₇	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
20	C₇×C₅⋊D₅	—	3	4	no	—
21	C₇×C₅⋊D₅	C₂×ΓL(1,𝔽₈)	1	4	yes	complemented
22	C₂×(C₅⋊D₅)⋊C₃	—	49	1	no	Hall
23	C₂×#3862	—	25	2	no	—
24	C₂×#3862	—	25	1	no	—
25	C₅×C₇²	—	3	2	no	—
26	C₂²×C₇²	—	25	1	no	Hall
27	C₅²×C₇	—	3	2	no	—
28	C₅²×C₇	C₂²×ΓL(1,𝔽₈)	1	2	yes	complemented
29	C₂×C₅²⋊C₃	—	49	1	no	—
30	(C₅⋊D₅)⋊C₃	—	49	2	no	—
31	#3862	—	25	1	no	Hall
32	C₂×C₇×D₅	—	15	4	no	—
33	C₂×C₇×D₅	—	15	12	no	—
34	C₂×C₅⋊D₅	#3862	1	1	yes	characteristic, complemented, Hall
35	C₂×C₇²	—	25	2	no	—
36	C₂×C₇²	(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented
37	C₂²×ΓL(1,𝔽₈)	—	175	2	no	—
38	C₅²⋊C₃	—	49	1	no	Hall
39	C₂×C₅×C₇	—	3	12	no	—
40	C₂×C₅×C₇	—	3	4	no	—
41	C₇×D₅	—	15	8	no	—
42	C₇×D₅	—	15	24	no	—
43	C₂×C₅²	C₂×#3862	1	1	yes	characteristic, complemented
44	C₅⋊D₅	C₂×#3862	1	2	yes	complemented
45	C₇²	C₂×(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented, Sylow
46	C₂×ΓL(1,𝔽₈)	—	175	4	no	—
47	C₂×ΓL(1,𝔽₈)	—	175	2	no	—
48	C₅×C₇	—	3	12	no	—
49	C₅×C₇	—	3	4	no	—
50	C₂²×C₇	—	25	2	no	—
51	C₂²×C₇	—	75	2	no	—
52	C₅²	C₂²×#3862	1	1	yes	characteristic, complemented, Sylow
53	ΓL(1,𝔽₈)	—	175	2	no	—
54	C₂×D₅	—	15	2	no	—
55	C₂×C₇	—	3	2	no	—
56	C₂×C₇	—	25	4	no	—
57	C₂×C₇	—	75	4	no	—
58	C₂×C₇	(C₅⋊D₅)⋊ΓL(1,𝔽₈)	1	2	yes	complemented
59	C₂²×C₃	—	1225	1	no	Hall
60	C₂×C₅	—	3	2	no	—
61	D₅	—	15	4	no	—
62	C₇	—	3	2	no	—
63	C₇	C₂×(C₅⋊D₅)⋊ΓL(1,𝔽₈)	1	2	yes	complemented
64	C₂×C₃	—	1225	2	no	—
65	C₂×C₃	—	1225	1	no	—
66	C₅	—	3	2	no	—
67	C₂²	—	25	1	no	Sylow
68	C₃	—	1225	1	no	Sylow
69	C₂	—	25	2	no	—
70	C₂	(C₅⋊D₅)⋊#3862	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	(C₅⋊D₅)⋊#3862
commutator, cosocle, hypocenter	C₅²×C₇²	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₅²×C₇²	C₂×C₃
LCS	G ▷ C₅²×C₇²
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₅²⋊#3862	C₂	semidirect
2. (C₅⋊D₅)⋊#3862	C₂	direct (N × H)
3. C₂×C₇²×C₅⋊D₅	C₃	semidirect
4. C₅²⋊#3862	C₂²	semidirect
5. C₇²×C₅⋊D₅	C₂×C₃	semidirect
6. C₂×C₅²×C₇²	C₂×C₃	semidirect
8. C₅²×C₇²	C₂²×C₃	semidirect
13. C₂×C₇×C₅⋊D₅	ΓL(1,𝔽₈)	semidirect
19. C₂×C₅²×C₇	C₂×ΓL(1,𝔽₈)	semidirect
21. C₇×C₅⋊D₅	C₂×ΓL(1,𝔽₈)	semidirect
28. C₅²×C₇	C₂²×ΓL(1,𝔽₈)	semidirect
34. C₂×C₅⋊D₅	#3862	semidirect
36. C₂×C₇²	(C₅⋊D₅)⋊C₃	semidirect
43. C₂×C₅²	C₂×#3862	semidirect
44. C₅⋊D₅	C₂×#3862	semidirect
45. C₇²	C₂×(C₅⋊D₅)⋊C₃	semidirect
52. C₅²	C₂²×#3862	semidirect
58. C₂×C₇	(C₅⋊D₅)⋊ΓL(1,𝔽₈)	semidirect
63. C₇	C₂×(C₅⋊D₅)⋊ΓL(1,𝔽₈)	semidirect
70. C₂	(C₅⋊D₅)⋊#3862	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅²⋊#3862	2
(C₅⋊D₅)⋊#3862	2
C₂×C₇²×C₅⋊D₅	3
C₂×(C₅⋊D₅)⋊ΓL(1,𝔽₈)	7
C₂²×#3862	25

Subgroups — C2×(C5⋊D5)⋊#3862 #28413444

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×(C₅⋊D₅)⋊#3862 #28413444