Subgroups — #11929164

Order 1248.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41⋊(C₁₃⋊C₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	D₁₃⋊SL(2,𝔽₃)	C₂	1	2	yes	maximal, complemented
3	#88257	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	Q₈⋊(C₁₃⋊C₃)	C₂²	1	1	yes	characteristic
5	C₄⋊(D₁₃⋊C₃)	—	4	1	no	maximal
6	C₁₃×#41	C₂×C₃	1	1	yes	characteristic, complemented
7	Q₈×D₁₃	C₂×C₃	1	2	yes	complemented
8	#6129	—	3	2	no	—
9	#6127	—	3	1	no	—
10	C₂×Dic₂₆	—	3	1	no	—
11	C₂×D₁₃⋊C₃	—	4	2	no	—
12	C₄×C₁₃⋊C₃	—	4	1	no	—
13	Q₈×C₁₃	C₂²×C₃	1	1	yes	characteristic
14	D₄×C₁₃	—	3	1	no	—
15	C₂×C₄×C₁₃	—	3	1	no	—
16	C₂²⋊D₁₃	—	3	2	no	—
17	C₂×Dic₁₃	—	3	2	no	—
18	D₅₂	A₄	1	1	yes	characteristic
19	C₄×D₁₃	—	3	2	no	—
20	Dic₂₆	—	3	1	no	—
21	Dic₂₆	—	3	2	no	—
22	2₋¹⁺⁴⋊C₃	—	13	1	no	maximal, Hall
23	C₂×C₁₃⋊C₃	—	4	1	no	—
24	D₁₃⋊C₃	—	8	2	no	—
25	C₂²×C₁₃	—	3	1	no	—
26	C₂×D₁₃	C₂×A₄	1	2	yes	—
27	C₄×C₁₃	—	3	1	no	—
28	C₄×C₁₃	C₂×A₄	1	1	yes	characteristic
29	Dic₁₃	—	3	2	no	—
30	#41⋊C₃	—	13	1	no	—
31	C₂×SL(2,𝔽₃)	—	13	2	no	—
32	C₁₃⋊C₃	—	4	1	no	Hall
33	2₋¹⁺⁴	—	13	1	no	Sylow
34	C₂×C₁₃	—	6	1	no	—
35	C₂×C₁₃	C₂²×A₄	1	1	yes	characteristic
36	D₁₃	—	2	2	no	—
37	C₃×D₄	—	52	1	no	—
38	SL(2,𝔽₃)	—	13	1	no	—
39	#41	—	39	1	no	—
40	#41	—	39	2	no	—
41	#41	D₁₃⋊C₃	1	1	yes	characteristic, complemented
42	C₂×Q₈	—	13	2	no	—
43	C₂×Q₈	—	39	1	no	—
44	C₁₃	2₋¹⁺⁴⋊C₃	1	1	yes	characteristic, complemented, Sylow
45	C₂²×C₃	—	52	2	no	—
46	C₃×C₄	—	52	1	no	—
47	Q₈	—	39	2	no	—
48	Q₈	—	39	1	no	—
49	Q₈	C₂×D₁₃⋊C₃	1	1	yes	characteristic
50	D₄	—	3	1	no	—
51	D₄	—	13	1	no	—
52	D₄	—	39	2	no	—
53	C₂×C₄	—	3	1	no	—
54	C₂×C₄	—	39	2	no	—
55	C₂×C₄	—	39	2	no	—
56	C₂×C₃	—	52	1	no	—
57	C₂×C₃	—	104	2	no	—
58	C₂²	—	3	1	no	—
59	C₂²	—	13	2	no	—
60	C₄	—	3	1	no	—
61	C₄	—	39	2	no	—
62	C₄	D₁₃⋊A₄	1	1	yes	characteristic
63	C₃	—	52	1	no	Sylow
64	C₂	—	6	1	no	—
65	C₂	—	26	2	no	—
66	C₂	C₂×D₁₃⋊A₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₃⋊A₄
commutator, cosocle, hypocenter	Q₈×C₁₃	C₂²×C₃
Fitting	C₁₃×#41	C₂×C₃
socle	C₂×C₁₃	C₂²×A₄
hypercenter	C₄	D₁₃⋊A₄
LCS	G ▷ Q₈×C₁₃
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. #41⋊(C₁₃⋊C₃)	C₂	semidirect
2. D₁₃⋊SL(2,𝔽₃)	C₂	semidirect
3. #88257	C₃	semidirect
6. C₁₃×#41	C₂×C₃	semidirect
7. Q₈×D₁₃	C₂×C₃	semidirect
41. #41	D₁₃⋊C₃	semidirect
44. C₁₃	2₋¹⁺⁴⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
#41⋊(C₁₃⋊C₃)	2
D₁₃⋊SL(2,𝔽₃)	2
#88257	3
C₄⋊(D₁₃⋊C₃)	4
2₋¹⁺⁴⋊C₃	13