Subgroups — #38760325

Order 25284.

72 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
2	C₇×Dic₃₀₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₄₃×#3861	C₄	1	1	yes	characteristic, complemented, Hall
4	C₂×C₇²×C₄₃	C₂×C₃	1	1	yes	characteristic
5	ΓL(1,𝔽₈)⋊Dic₄₃	—	7	1	no	maximal
6	ΓL(1,𝔽₈)×Dic₄₃	—	7	1	no	maximal
7	C₇²×C₄₃	C₃×C₄	1	1	yes	characteristic, complemented, Hall
8	C₂×ΓL(1,𝔽₈)×C₄₃	—	7	1	no	—
9	C₂×ΓL(1,𝔽₈)×C₄₃	—	7	1	no	—
10	C₂×ΓL(1,𝔽₈)×C₄₃	—	14	3	no	—
11	Dic₃₀₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
12	C₇×Dic₄₃	—	7	1	no	—
13	ΓL(1,𝔽₈)×C₄₃	—	7	1	no	—
14	ΓL(1,𝔽₈)×C₄₃	—	7	1	no	—
15	ΓL(1,𝔽₈)×C₄₃	—	14	3	no	—
16	C₂×C₇×C₄₃	—	2	3	no	—
17	C₂×C₇×C₄₃	AGL(1,𝔽₇)	1	1	yes	characteristic
18	C₂×C₇×C₄₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
19	#10596873	—	43	1	no	maximal, Hall
20	C₃×Dic₄₃	—	49	1	no	Hall
21	C₇×C₄₃	—	2	3	no	—
22	C₇×C₄₃	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, complemented
23	C₇×C₄₃	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
24	C₂×#3861	D₄₃	1	1	yes	characteristic
25	C₂×C₃×C₄₃	—	98	1	no	—
26	C₇×Dic₇	—	43	1	no	Hall
27	Dic₄₃	—	7	1	no	Hall
28	#3861	Dic₄₃	1	1	yes	characteristic, complemented, Hall
29	C₃×C₄₃	—	98	1	no	Hall
30	C₂×C₇²	C₃×D₄₃	1	1	yes	characteristic
31	C₂×C₄₃	#64423	1	1	yes	characteristic
32	C₄×ΓL(1,𝔽₈)	—	301	1	no	—
33	ΓL(1,𝔽₈)⋊C₄	—	301	1	no	—
34	C₇²	C₃×Dic₄₃	1	1	yes	characteristic, complemented, Sylow
35	C₄₃	#10596873	1	1	yes	characteristic, complemented, Sylow
36	C₂×ΓL(1,𝔽₈)	—	7	1	no	—
37	C₂×ΓL(1,𝔽₈)	—	14	3	no	—
38	C₂×ΓL(1,𝔽₈)	—	14	1	no	—
39	C₄×C₇	—	301	1	no	—
40	Dic₇	—	43	1	no	—
41	ΓL(1,𝔽₈)	—	7	1	no	—
42	ΓL(1,𝔽₈)	—	14	3	no	—
43	ΓL(1,𝔽₈)	—	14	1	no	—
44	C₂×C₇	—	2	3	no	—
45	C₂×C₇	ΓL(1,𝔽₈)×D₄₃	1	1	yes	characteristic
46	C₂×C₇	ΓL(1,𝔽₈)⋊D₄₃	1	1	yes	characteristic
47	C₃×C₄	—	2107	1	no	Hall
48	C₇	—	2	3	no	—
49	C₇	ΓL(1,𝔽₈)×Dic₄₃	1	1	yes	characteristic, complemented
50	C₇	ΓL(1,𝔽₈)⋊Dic₄₃	1	1	yes	characteristic, complemented
51	C₂×C₃	—	98	1	no	—
52	C₄	—	301	1	no	Sylow
53	C₃	—	98	1	no	Sylow
54	C₂	#27162418	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	#27162418
commutator, hypocenter	C₇²×C₄₃	C₃×C₄
Fitting, socle, cosocle	C₂×C₇²×C₄₃	C₂×C₃
prime radical	C₂×C₄₃×#3861	C₂
LCS	G ▷ C₇²×C₄₃
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. C₇×Dic₃₀₁	C₃	semidirect
3. C₄₃×#3861	C₄	semidirect
7. C₇²×C₄₃	C₃×C₄	semidirect
11. Dic₃₀₁	ΓL(1,𝔽₈)	semidirect
22. C₇×C₄₃	ΓL(1,𝔽₈)⋊C₄	semidirect
23. C₇×C₄₃	C₄×ΓL(1,𝔽₈)	semidirect
28. #3861	Dic₄₃	semidirect
34. C₇²	C₃×Dic₄₃	semidirect
35. C₄₃	#10596873	semidirect
49. C₇	ΓL(1,𝔽₈)×Dic₄₃	semidirect
50. C₇	ΓL(1,𝔽₈)⋊Dic₄₃	semidirect

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
C₇×Dic₃₀₁	3
ΓL(1,𝔽₈)⋊Dic₄₃	7
ΓL(1,𝔽₈)×Dic₄₃	7
#10596873	43