Subgroups — #11928128

Order 1248.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₄⋊Dic₃₉	C₂	1	2	yes	maximal
3	#88057	—	3	1	no	maximal, Hall
4	C₂×C₃×C₄×C₁₃	C₂²	1	1	yes	characteristic
5	C₂×Dic₃₉	—	2	2	no	—
6	C₃×C₈×C₁₃	C₄	1	2	yes	complemented
7	C₂×C₈×C₁₃	D₃	1	1	yes	characteristic
8	C₄⋊Dic₁₃	—	3	2	no	—
9	C₂²×C₃×C₁₃	D₄	1	1	yes	characteristic
10	C₃×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
11	C₃×C₄×C₁₃	Q₈	1	1	yes	characteristic
12	Dic₃₉	—	4	2	no	—
13	C₂×C₄×C₁₃	C₂×D₃	1	1	yes	characteristic
14	C₂×Dic₁₃	—	6	2	no	—
15	C₈×C₁₃	Dic₃	1	2	yes	complemented
16	#817	—	13	1	no	maximal, Hall
17	C₂×C₃×C₁₃	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₃×C₁₃	SD₁₆	1	2	yes	—
19	C₂²×C₁₃	D₁₂	1	1	yes	characteristic
20	C₄×C₁₃	Dic₆	1	1	yes	characteristic
21	C₄×C₁₃	C₂×Dic₃	1	1	yes	characteristic
22	Dic₁₃	—	12	2	no	—
23	C₂×C₃×C₈	D₁₃	1	1	yes	characteristic
24	C₄⋊Dic₃	—	13	2	no	—
25	C₃×C₁₃	#106	1	1	yes	characteristic, complemented, Hall
26	#106	—	39	1	no	Sylow
27	C₂×C₁₃	#204	1	2	yes	—
28	C₂×C₁₃	C₄⋊Dic₃	1	1	yes	characteristic
29	C₂×C₃×C₄	C₂×D₁₃	1	1	yes	characteristic
30	C₂×Dic₃	—	26	2	no	—
31	C₃×C₈	Dic₁₃	1	2	yes	complemented
32	C₂×C₈	D₃₉	1	1	yes	characteristic
33	C₄⋊C₄	—	39	2	no	—
34	C₁₃	#817	1	1	yes	characteristic, complemented, Sylow
35	C₂²×C₃	D₅₂	1	1	yes	characteristic
36	C₃×C₄	Dic₂₆	1	1	yes	characteristic
37	C₃×C₄	C₂×Dic₁₃	1	1	yes	characteristic
38	Dic₃	—	52	2	no	—
39	C₂×C₄	—	78	2	no	—
40	C₂×C₄	C₂×D₃₉	1	1	yes	characteristic
41	C₈	Dic₃₉	1	2	yes	complemented
42	C₂×C₃	#6095	1	2	yes	—
43	C₂×C₃	C₄⋊Dic₁₃	1	1	yes	characteristic
44	C₂²	D₁₅₆	1	1	yes	characteristic
45	C₄	—	156	2	no	—
46	C₄	Dic₇₈	1	1	yes	characteristic
47	C₄	C₂×Dic₃₉	1	1	yes	characteristic
48	C₃	#88057	1	1	yes	characteristic, complemented, Sylow
49	C₂	#10597658	1	2	yes	—
50	C₂	C₄⋊Dic₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₅₆
commutator	C₃×C₄×C₁₃	C₂×C₄
Frattini	C₂×C₄	C₂×D₃₉
Fitting	C₂×C₃×C₈×C₁₃	C₂
socle, prime radical	C₂²×C₃×C₁₃	D₄
cosocle	C₂×C₃×C₄×C₁₃	C₂²
hypercenter	C₂×C₈	D₃₉
hypocenter	C₃×C₁₃	#106
LCS	G ▷ C₃×C₄×C₁₃ ▷ C₂×C₃×C₁₃ ▷ C₃×C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
6. C₃×C₈×C₁₃	C₄	semidirect
15. C₈×C₁₃	Dic₃	semidirect
25. C₃×C₁₃	#106	semidirect
31. C₃×C₈	Dic₁₃	semidirect
34. C₁₃	#817	semidirect
41. C₈	Dic₃₉	semidirect
48. C₃	#88057	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₈×C₁₃	2
C₄⋊Dic₃₉	2
#88057	3
#817	13