Subgroups — C₂×#27162418 #38760521

Order 25284.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₃×#3861	C₂	1	1	yes	maximal, characteristic, complemented
2	#27162418	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₇×D₃₀₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₄₃×#3861	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₇×D₃₀₁	C₂×C₃	1	2	yes	complemented
6	C₂×C₇²×C₄₃	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂×ΓL(1,𝔽₈)×D₄₃	—	7	1	no	maximal
8	C₂×ΓL(1,𝔽₈)⋊D₄₃	—	7	1	no	maximal
9	C₇²×C₄₃	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
10	C₂×ΓL(1,𝔽₈)×C₄₃	—	7	1	no	—
11	C₂×ΓL(1,𝔽₈)×C₄₃	—	14	3	no	—
12	C₂×ΓL(1,𝔽₈)×C₄₃	—	14	1	no	—
13	ΓL(1,𝔽₈)⋊D₄₃	—	7	2	no	—
14	ΓL(1,𝔽₈)×D₄₃	—	7	2	no	—
15	C₂×D₃₀₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
16	C₂×C₇×D₄₃	—	7	1	no	—
17	ΓL(1,𝔽₈)×C₄₃	—	7	1	no	—
18	ΓL(1,𝔽₈)×C₄₃	—	14	1	no	—
19	ΓL(1,𝔽₈)×C₄₃	—	14	3	no	—
20	C₂×C₇×C₄₃	—	2	3	no	—
21	C₂×C₇×C₄₃	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
22	C₂×C₇×C₄₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
23	D₃₀₁	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
24	C₇×D₄₃	—	7	2	no	—
25	C₂×#64423	—	43	1	no	maximal, Hall
26	C₂×C₃×D₄₃	—	49	1	no	Hall
27	C₇×C₄₃	—	2	3	no	—
28	C₇×C₄₃	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
29	C₇×C₄₃	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
30	C₂×#3861	D₄₃	1	1	yes	characteristic, complemented
31	#64423	—	43	2	no	—
32	C₂×C₃×C₄₃	—	98	1	no	—
33	C₃×D₄₃	—	49	2	no	—
34	C₂×C₇×D₇	—	43	1	no	Hall
35	C₂×D₄₃	—	7	1	no	Hall
36	#3861	C₂×D₄₃	1	1	yes	characteristic, complemented, Hall
37	C₃×C₄₃	—	98	1	no	Hall
38	C₂×C₇²	C₃×D₄₃	1	1	yes	characteristic, complemented
39	C₇×D₇	—	43	2	no	—
40	C₂×C₄₃	#64423	1	1	yes	characteristic, complemented
41	D₄₃	—	7	2	no	—
42	C₂²×ΓL(1,𝔽₈)	—	301	1	no	—
43	C₂×AGL(1,𝔽₇)	—	301	1	no	—
44	C₇²	C₂×C₃×D₄₃	1	1	yes	characteristic, complemented, Sylow
45	C₄₃	C₂×#64423	1	1	yes	characteristic, complemented, Sylow
46	C₂×ΓL(1,𝔽₈)	—	7	1	no	—
47	C₂×ΓL(1,𝔽₈)	—	7	1	no	—
48	C₂×ΓL(1,𝔽₈)	—	14	3	no	—
49	C₂×ΓL(1,𝔽₈)	—	301	2	no	—
50	AGL(1,𝔽₇)	—	301	2	no	—
51	C₂²×C₇	—	301	1	no	—
52	C₂×D₇	—	43	1	no	—
53	ΓL(1,𝔽₈)	—	7	1	no	—
54	ΓL(1,𝔽₈)	—	7	1	no	—
55	ΓL(1,𝔽₈)	—	14	3	no	—
56	C₂×C₇	—	2	3	no	—
57	C₂×C₇	—	301	2	no	—
58	C₂×C₇	ΓL(1,𝔽₈)×D₄₃	1	1	yes	characteristic, complemented
59	C₂×C₇	ΓL(1,𝔽₈)⋊D₄₃	1	1	yes	characteristic, complemented
60	D₇	—	43	2	no	—
61	C₂²×C₃	—	2107	1	no	Hall
62	C₇	—	2	3	no	—
63	C₇	C₂×ΓL(1,𝔽₈)⋊D₄₃	1	1	yes	characteristic, complemented
64	C₇	C₂×ΓL(1,𝔽₈)×D₄₃	1	1	yes	characteristic, complemented
65	C₂×C₃	—	2107	2	no	—
66	C₂×C₃	—	98	1	no	—
67	C₂²	—	301	1	no	Sylow
68	C₃	—	98	1	no	Sylow
69	C₂	—	301	2	no	—
70	C₂	#27162418	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	#27162418
commutator, cosocle, hypocenter	C₇²×C₄₃	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₇²×C₄₃	C₂×C₃
LCS	G ▷ C₇²×C₄₃
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₃×#3861	C₂	semidirect
2. #27162418	C₂	direct (N × H)
3. C₂×C₇×D₃₀₁	C₃	semidirect
4. C₄₃×#3861	C₂²	semidirect
5. C₇×D₃₀₁	C₂×C₃	semidirect
6. C₂×C₇²×C₄₃	C₂×C₃	semidirect
9. C₇²×C₄₃	C₂²×C₃	semidirect
15. C₂×D₃₀₁	ΓL(1,𝔽₈)	semidirect
21. C₂×C₇×C₄₃	AGL(1,𝔽₇)	semidirect
22. C₂×C₇×C₄₃	C₂×ΓL(1,𝔽₈)	semidirect
23. D₃₀₁	C₂×ΓL(1,𝔽₈)	semidirect
28. C₇×C₄₃	C₂×AGL(1,𝔽₇)	semidirect
29. C₇×C₄₃	C₂²×ΓL(1,𝔽₈)	semidirect
30. C₂×#3861	D₄₃	semidirect
36. #3861	C₂×D₄₃	semidirect
38. C₂×C₇²	C₃×D₄₃	semidirect
40. C₂×C₄₃	#64423	semidirect
44. C₇²	C₂×C₃×D₄₃	semidirect
45. C₄₃	C₂×#64423	semidirect
58. C₂×C₇	ΓL(1,𝔽₈)×D₄₃	semidirect
59. C₂×C₇	ΓL(1,𝔽₈)⋊D₄₃	semidirect
63. C₇	C₂×ΓL(1,𝔽₈)⋊D₄₃	semidirect
64. C₇	C₂×ΓL(1,𝔽₈)×D₄₃	semidirect
70. C₂	#27162418	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×C₄₃×#3861	2
#27162418	2
C₂×C₇×D₃₀₁	3
C₂×ΓL(1,𝔽₈)×D₄₃	7
C₂×ΓL(1,𝔽₈)⋊D₄₃	7
C₂×#64423	43

Subgroups — C2×#27162418 #38760521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27162418 #38760521