Subgroups — #11927993

Order 1248.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Dic₃₉	C₂	1	1	yes	maximal, characteristic
2	C₁₃×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#1055	C₂	1	1	yes	maximal, characteristic
4	#88048	—	3	1	no	maximal, Hall
5	C₂×C₃×C₄×C₁₃	C₂²	1	1	yes	characteristic
6	C₂×Dic₃₉	—	2	1	no	—
7	C₂×Dic₃×C₁₃	—	2	1	no	—
8	C₃×#1055	C₄	1	2	yes	complemented
9	C₁₃×C₄⋊C₄	—	3	1	no	—
10	C₄⋊Dic₁₃	—	3	1	no	—
11	C₂×#1055	D₃	1	1	yes	characteristic
12	C₂²×C₃×C₁₃	D₄	1	1	yes	characteristic
13	C₃×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
14	C₃×C₄×C₁₃	Q₈	1	1	yes	characteristic
15	Dic₃₉	—	4	1	no	—
16	Dic₃×C₁₃	—	4	1	no	—
17	C₂×C₄×C₁₃	—	6	1	no	—
18	C₂×C₄×C₁₃	C₂×D₃	1	1	yes	characteristic
19	C₂×Dic₁₃	—	6	1	no	—
20	#1055	Dic₃	1	2	yes	complemented
21	#817	—	13	1	no	maximal, Hall
22	C₂×C₃×C₁₃	C₄⋊C₄	1	1	yes	characteristic
23	C₂×C₃×C₁₃	SD₁₆	1	1	yes	characteristic
24	C₂×C₃×C₁₃	SD₁₆	1	1	yes	characteristic
25	C₂²×C₁₃	D₁₂	1	1	yes	characteristic
26	C₄×C₁₃	—	12	1	no	—
27	C₄×C₁₃	Dic₆	1	1	yes	characteristic
28	C₄×C₁₃	C₂×Dic₃	1	1	yes	characteristic
29	Dic₁₃	—	12	1	no	—
30	C₂×C₃×C₈	—	13	1	no	—
31	C₄⋊Dic₃	—	13	1	no	—
32	C₄⋊Dic₃	D₁₃	1	1	yes	characteristic
33	C₃×C₁₃	#106	1	1	yes	characteristic, complemented, Hall
34	#106	—	39	1	no	Sylow
35	C₂×C₁₃	#204	1	1	yes	characteristic
36	C₂×C₁₃	#204	1	1	yes	characteristic
37	C₂×C₁₃	C₄⋊Dic₃	1	1	yes	characteristic
38	C₂×C₃×C₄	C₂×D₁₃	1	1	yes	characteristic
39	C₂×Dic₃	—	2	1	no	—
40	C₂×Dic₃	—	26	1	no	—
41	C₃×C₈	—	13	2	no	—
42	C₂×C₈	—	13	1	no	—
43	C₄⋊C₄	—	3	1	no	—
44	C₄⋊C₄	—	39	1	no	—
45	C₁₃	#817	1	1	yes	characteristic, complemented, Sylow
46	C₂²×C₃	C₂²⋊D₁₃	1	1	yes	characteristic
47	C₃×C₄	Dic₂₆	1	1	yes	characteristic
48	C₃×C₄	C₄×D₁₃	1	1	yes	characteristic
49	Dic₃	—	4	1	no	—
50	Dic₃	—	52	1	no	—
51	C₂×C₄	—	6	1	no	—
52	C₂×C₄	—	78	1	no	—
53	C₂×C₄	D₃×D₁₃	1	1	yes	characteristic
54	C₈	—	13	2	no	—
55	C₂×C₃	#6101	1	1	yes	characteristic
56	C₂×C₃	#6105	1	1	yes	characteristic
57	C₂×C₃	#6106	1	1	yes	characteristic
58	C₂²	#64604	1	1	yes	characteristic
59	C₄	—	12	1	no	—
60	C₄	—	156	1	no	—
61	C₄	Dic₃×D₁₃	1	1	yes	characteristic
62	C₄	#64605	1	1	yes	characteristic
63	C₃	#88048	1	1	yes	characteristic, complemented, Sylow
64	C₂	#10597602	1	1	yes	characteristic
65	C₂	#10597603	1	1	yes	characteristic
66	C₂	#10597611	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#64604
commutator	C₃×C₄×C₁₃	C₂×C₄
Frattini, hypercenter	C₂×C₄	D₃×D₁₃
Fitting, cosocle	C₂×C₃×C₄×C₁₃	C₂²
socle, prime radical	C₂²×C₃×C₁₃	D₄
hypocenter	C₃×C₁₃	#106
LCS	G ▷ C₃×C₄×C₁₃ ▷ C₂×C₃×C₁₃ ▷ C₃×C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
8. C₃×#1055	C₄	semidirect
20. #1055	Dic₃	semidirect
33. C₃×C₁₃	#106	semidirect
45. C₁₃	#817	semidirect
63. C₃	#88048	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Dic₃₉	2
C₁₃×C₄⋊Dic₃	2
C₂×C₃×#1055	2
#88048	3
#817	13