Subgroups — #28382181

Order 14196.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#63271	C₂	1	1	yes	maximal, characteristic
2	C₇⋊#10621725	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₁₃×#63271	C₄	1	1	yes	characteristic, complemented, Hall
4	C₂×C₇×C₁₃²	C₂×C₃	1	1	yes	characteristic
5	#14523132	—	7	1	no	maximal, Hall
6	C₇×C₁₃²	C₃×C₄	1	1	yes	characteristic, complemented, Hall
7	#11763964	—	13	1	no	maximal
8	ΓL(1,𝔽₈)⋊Dic₁₃	—	13	1	no	maximal
9	C₂×C₁₃×C₁₃⋊C₃	—	7	1	no	—
10	#10621725	—	7	1	no	Hall
11	C₂×#63271	D₁₃	1	1	yes	characteristic
12	C₂×C₁₃×ΓL(1,𝔽₈)	—	13	1	no	—
13	C₁₃×C₁₃⋊C₃	—	7	1	no	Hall
14	Dic₉₁	—	13	1	no	—
15	Dic₉₁	—	13	1	no	—
16	Dic₉₁	—	39	4	no	—
17	C₂×C₁₃²	AGL(1,𝔽₇)	1	1	yes	characteristic
18	#63271	Dic₁₃	1	1	yes	characteristic, complemented
19	C₁₃×ΓL(1,𝔽₈)	—	13	1	no	—
20	C₂×C₇×C₁₃	—	3	4	no	—
21	C₂×C₇×C₁₃	D₁₃⋊C₃	1	1	yes	characteristic
22	C₂×C₇×C₁₃	C₃×D₁₃	1	1	yes	characteristic
23	C₁₃²	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, complemented, Sylow
24	C₃×Dic₁₃	—	91	1	no	—
25	Dic₁₃⋊C₃	—	91	1	no	—
26	C₇×C₁₃	—	3	4	no	—
27	C₇×C₁₃	Dic₁₃⋊C₃	1	1	yes	characteristic, complemented
28	C₇×C₁₃	C₃×Dic₁₃	1	1	yes	characteristic, complemented
29	ΓL(1,𝔽₈)⋊C₄	—	169	1	no	Hall
30	C₂×C₃×C₁₃	—	91	1	no	—
31	C₂×C₁₃⋊C₃	—	7	1	no	—
32	Dic₁₃	—	273	4	no	—
33	Dic₁₃	—	91	1	no	—
34	Dic₁₃	—	91	1	no	—
35	C₂×ΓL(1,𝔽₈)	—	13	1	no	—
36	C₃×C₁₃	—	91	1	no	—
37	C₁₃⋊C₃	—	7	1	no	—
38	Dic₇	—	169	1	no	Hall
39	C₂×C₁₃	—	3	4	no	—
40	C₂×C₁₃	ΓL(1,𝔽₈)⋊D₁₃	1	1	yes	characteristic
41	C₂×C₁₃	#10587723	1	1	yes	characteristic
42	ΓL(1,𝔽₈)	—	13	1	no	Hall
43	C₂×C₇	(C₁₃⋊C₃)⋊D₁₃	1	1	yes	characteristic
44	C₁₃	—	3	4	no	—
45	C₁₃	ΓL(1,𝔽₈)⋊Dic₁₃	1	1	yes	characteristic, complemented
46	C₁₃	#11763964	1	1	yes	characteristic, complemented
47	C₃×C₄	—	1183	1	no	Hall
48	C₇	#14523132	1	1	yes	characteristic, complemented, Sylow
49	C₂×C₃	—	91	1	no	—
50	C₄	—	1183	1	no	Sylow
51	C₃	—	91	1	no	Sylow
52	C₂	#20296909	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	#20296909
commutator, hypocenter	C₇×C₁₃²	C₃×C₄
Fitting, socle, cosocle	C₂×C₇×C₁₃²	C₂×C₃
prime radical	C₂×C₁₃×#63271	C₂
LCS	G ▷ C₇×C₁₃²
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. C₇⋊#10621725	C₃	semidirect
3. C₁₃×#63271	C₄	semidirect
6. C₇×C₁₃²	C₃×C₄	semidirect
18. #63271	Dic₁₃	semidirect
23. C₁₃²	ΓL(1,𝔽₈)⋊C₄	semidirect
27. C₇×C₁₃	Dic₁₃⋊C₃	semidirect
28. C₇×C₁₃	C₃×Dic₁₃	semidirect
45. C₁₃	ΓL(1,𝔽₈)⋊Dic₁₃	semidirect
46. C₁₃	#11763964	semidirect
48. C₇	#14523132	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#63271	2
C₇⋊#10621725	3
#14523132	7
#11763964	13
ΓL(1,𝔽₈)⋊Dic₁₃	13