Subgroups — #11928936

Order 1248.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₅₆	C₂	1	6	yes	maximal, complemented
2	C₃×C₄²×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#88128	—	3	1	no	maximal, Hall
4	C₂²×D₃₉	C₂²	1	4	yes	—
5	C₂×C₃×C₄×C₁₃	C₂²	1	3	yes	—
6	D₁₅₆	—	2	12	no	—
7	C₂×D₅₂	—	3	6	no	—
8	C₄²×C₁₃	D₃	1	1	yes	characteristic, complemented
9	C₂²×C₃×C₁₃	C₂³	1	1	yes	characteristic
10	C₂×D₃₉	—	2	12	no	—
11	C₃×C₄×C₁₃	D₄	1	6	yes	complemented
12	C₂²×D₁₃	—	3	4	no	—
13	C₂×C₄×C₁₃	C₂×D₃	1	3	yes	—
14	D₅₂	—	6	12	no	—
15	#874	—	13	1	no	maximal, Hall
16	C₂×C₃×C₁₃	C₂×D₄	1	3	yes	—
17	D₃₉	—	4	4	no	—
18	C₂²×C₁₃	C₂²×D₃	1	1	yes	characteristic
19	C₂×D₁₃	—	6	12	no	—
20	C₄×C₁₃	D₁₂	1	6	yes	complemented
21	C₂×D₁₂	—	13	6	no	—
22	C₃×C₄²	D₁₃	1	1	yes	characteristic, complemented
23	C₃×C₁₃	#127	1	1	yes	characteristic, complemented, Hall
24	#127	—	39	1	no	Sylow
25	C₂×C₁₃	C₂×D₁₂	1	3	yes	—
26	D₁₃	—	12	4	no	—
27	C₂²×D₃	—	13	4	no	—
28	C₂×C₃×C₄	C₂×D₁₃	1	3	yes	—
29	D₁₂	—	26	12	no	—
30	C₂×D₄	—	39	6	no	—
31	C₄²	D₃₉	1	1	yes	characteristic, complemented
32	C₁₃	#874	1	1	yes	characteristic, complemented, Sylow
33	C₂²×C₃	C₂²×D₁₃	1	1	yes	characteristic
34	C₂×D₃	—	26	12	no	—
35	C₃×C₄	D₅₂	1	6	yes	complemented
36	C₂³	—	39	4	no	—
37	D₄	—	78	12	no	—
38	C₂×C₄	C₂×D₃₉	1	3	yes	—
39	C₂×C₃	C₂×D₅₂	1	3	yes	—
40	D₃	—	52	4	no	—
41	C₂²	—	78	12	no	—
42	C₂²	C₂²×D₃₉	1	1	yes	characteristic
43	C₄	D₁₅₆	1	6	yes	complemented
44	C₃	#88128	1	1	yes	characteristic, complemented, Sylow
45	C₂	—	156	4	no	—
46	C₂	C₂×D₁₅₆	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃₉
commutator, socle, cosocle	C₂²×C₃×C₁₃	C₂³
Fitting	C₃×C₄²×C₁₃	C₂
hypercenter	C₄²	D₃₉
hypocenter	C₃×C₁₃	#127
LCS	G ▷ C₂²×C₃×C₁₃ ▷ C₃×C₁₃
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₅₆	C₂	semidirect
2. C₃×C₄²×C₁₃	C₂	semidirect
8. C₄²×C₁₃	D₃	semidirect
11. C₃×C₄×C₁₃	D₄	semidirect
20. C₄×C₁₃	D₁₂	semidirect
22. C₃×C₄²	D₁₃	semidirect
23. C₃×C₁₃	#127	semidirect
31. C₄²	D₃₉	semidirect
32. C₁₃	#874	semidirect
35. C₃×C₄	D₅₂	semidirect
43. C₄	D₁₅₆	semidirect
44. C₃	#88128	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₅₆	2
C₃×C₄²×C₁₃	2
#88128	3
#874	13