Subgroups — #11929127

Order 1248.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#41⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃⋊GL(2,𝔽₃)	C₂	1	2	yes	maximal, complemented
3	#88203	—	3	1	no	maximal, Hall
4	D₁₅₆	—	4	1	no	maximal
5	C₁₃×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
6	C₁₃×#41	D₃	1	1	yes	characteristic, complemented
7	C₂×D₅₂	—	3	1	no	—
8	#6106	—	3	2	no	—
9	#6104	—	3	2	no	—
10	#6099	—	3	1	no	—
11	C₂×D₃₉	—	4	2	no	—
12	C₃×C₄×C₁₃	—	4	1	no	—
13	C₂²×D₁₃	—	6	1	no	—
14	Q₈×C₁₃	C₂×D₃	1	1	yes	characteristic
15	D₄×C₁₃	—	3	1	no	—
16	C₂×C₄×C₁₃	—	3	1	no	—
17	D₅₂	—	3	2	no	—
18	D₅₂	—	6	1	no	—
19	#1055	—	3	2	no	—
20	#986	—	13	1	no	maximal, Hall
21	C₂×C₃×C₁₃	—	4	1	no	—
22	D₃₉	—	8	2	no	—
23	C₂²×C₁₃	—	3	1	no	—
24	C₂×D₁₃	—	6	2	no	—
25	C₂×D₁₃	—	12	2	no	—
26	C₄×C₁₃	—	3	1	no	—
27	C₄×C₁₃	S₄	1	1	yes	characteristic
28	#41⋊C₃	D₁₃	1	1	yes	characteristic, complemented
29	GL(2,𝔽₃)	—	13	2	no	—
30	C₃×C₁₃	—	4	1	no	Hall
31	AGL(1,ℤ/8)	—	39	1	no	Sylow
32	C₂×C₁₃	—	6	1	no	—
33	C₂×C₁₃	C₂×S₄	1	1	yes	characteristic
34	D₁₃	—	12	2	no	—
35	D₁₂	—	52	1	no	—
36	SL(2,𝔽₃)	C₂×D₁₃	1	1	yes	characteristic, complemented
37	#41	D₃₉	1	1	yes	characteristic, complemented
38	C₂×D₄	—	39	1	no	—
39	SD₁₆	—	39	2	no	—
40	D₈	—	39	2	no	—
41	M₁₆	—	39	1	no	—
42	C₁₃	#986	1	1	yes	characteristic, complemented, Sylow
43	C₂×D₃	—	52	2	no	—
44	C₃×C₄	—	4	1	no	—
45	C₂³	—	78	1	no	—
46	Q₈	C₂×D₃₉	1	1	yes	characteristic
47	D₄	—	3	1	no	—
48	D₄	—	39	2	no	—
49	D₄	—	78	1	no	—
50	C₂×C₄	—	3	1	no	—
51	C₈	—	39	2	no	—
52	C₂×C₃	—	4	1	no	—
53	D₃	—	104	2	no	—
54	C₂²	—	3	1	no	—
55	C₂²	—	78	2	no	—
56	C₂²	—	156	2	no	—
57	C₄	—	3	1	no	—
58	C₄	A₄⋊D₁₃	1	1	yes	characteristic
59	C₃	—	4	1	no	Sylow
60	C₂	—	6	1	no	—
61	C₂	—	156	2	no	—
62	C₂	C₂×A₄⋊D₁₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×A₄⋊D₁₃
commutator, cosocle, hypocenter	C₁₃×SL(2,𝔽₃)	C₂²
Fitting	C₁₃×#41	D₃
socle	C₂×C₁₃	C₂×S₄
hypercenter	C₄	A₄⋊D₁₃
LCS	G ▷ C₁₃×SL(2,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. C₁₃×#41⋊C₃	C₂	semidirect
2. C₁₃⋊GL(2,𝔽₃)	C₂	semidirect
5. C₁₃×SL(2,𝔽₃)	C₂²	semidirect
6. C₁₃×#41	D₃	semidirect
28. #41⋊C₃	D₁₃	semidirect
36. SL(2,𝔽₃)	C₂×D₁₃	semidirect
37. #41	D₃₉	semidirect
42. C₁₃	#986	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#41⋊C₃	2
C₁₃⋊GL(2,𝔽₃)	2
#88203	3
D₁₅₆	4
#986	13