Subgroups — #21432494

Order 7552.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14273727	C₂	1	6	yes	maximal
2	#16837272	C₂	1	8	yes	maximal
3	C₄³×C₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₁₁₈	C₂²	1	4	yes	—
5	#14273727	—	2	24	no	—
6	C₂×C₄⋊Dic₅₉	C₂²	1	24	yes	—
7	C₂×C₄²×C₅₉	C₂²	1	3	yes	—
8	C₂×C₄²×C₅₉	C₂²	1	4	yes	—
9	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
10	C₂²×C₄×C₅₉	C₂³	1	6	yes	—
11	C₂²×Dic₅₉	C₂³	1	8	yes	—
12	C₂×Dic₁₁₈	—	2	24	no	—
13	C₄²×C₅₉	D₄	1	12	yes	—
14	C₄²×C₅₉	Q₈	1	16	yes	complemented
15	C₄⋊Dic₅₉	—	2	48	no	—
16	C₂³×C₅₉	C₂⁴	1	1	yes	characteristic
17	C₂×C₄×C₅₉	C₂×D₄	1	6	yes	—
18	C₂×C₄×C₅₉	C₂×D₄	1	12	yes	—
19	C₂×C₄×C₅₉	C₂×Q₈	1	24	yes	—
20	C₂×Dic₅₉	—	2	24	no	—
21	Dic₁₁₈	—	4	16	no	—
22	C₂²×C₅₉	C₂²×D₄	1	3	yes	—
23	C₂²×C₅₉	C₂²×Q₈	1	4	yes	—
24	C₄×C₅₉	#127	1	4	yes	—
25	C₄×C₅₉	C₄⋊Q₈	1	24	yes	complemented
26	Dic₅₉	—	4	8	no	—
27	#2868	—	59	1	no	maximal, Sylow
28	C₂×C₅₉	C₂×#127	1	1	yes	characteristic
29	C₂×C₅₉	C₂×C₄⋊Q₈	1	6	yes	—
30	C₂×C₄⋊Q₈	—	59	6	no	—
31	#384	—	59	8	no	—
32	C₄³	D₅₉	1	1	yes	characteristic
33	C₅₉	#2868	1	1	yes	characteristic, complemented, Sylow
34	C₂²×Q₈	—	59	4	no	—
35	C₄⋊Q₈	—	118	24	no	—
36	C₂×C₄⋊C₄	—	59	24	no	—
37	C₂×C₄²	C₂×D₅₉	1	4	yes	—
38	C₂×C₄²	C₂×D₅₉	1	3	yes	—
39	C₂×Q₈	—	118	24	no	—
40	C₂²×C₄	—	59	8	no	—
41	C₂²×C₄	C₂²×D₅₉	1	6	yes	—
42	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
43	C₄⋊C₄	—	118	48	no	—
44	C₄²	Dic₁₁₈	1	16	yes	complemented
45	C₄²	D₂₃₆	1	12	yes	—
46	C₂³	C₂³×D₅₉	1	1	yes	characteristic
47	Q₈	—	236	16	no	—
48	C₂×C₄	—	118	24	no	—
49	C₂×C₄	C₂×Dic₁₁₈	1	24	yes	—
50	C₂×C₄	C₂×D₂₃₆	1	6	yes	—
51	C₂×C₄	C₂×D₂₃₆	1	12	yes	—
52	C₂²	C₂²×D₂₃₆	1	3	yes	—
53	C₂²	C₂²×Dic₁₁₈	1	4	yes	—
54	C₄	—	236	8	no	—
55	C₄	#14273724	1	4	yes	—
56	C₄	#14273727	1	24	yes	complemented
57	C₂	C₂×#14273724	1	1	yes	characteristic
58	C₂	C₂×#14273727	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₉
commutator, socle	C₂³×C₅₉	C₂⁴
Fitting	C₄³×C₅₉	C₂
hypercenter	C₄³	D₅₉
hypocenter	C₅₉	#2868
LCS	G ▷ C₂³×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₄³

Complements

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

N	H ≅ G/N	type
14. C₄²×C₅₉	Q₈	semidirect
25. C₄×C₅₉	C₄⋊Q₈	semidirect
33. C₅₉	#2868	semidirect
44. C₄²	Dic₁₁₈	semidirect
56. C₄	#14273727	semidirect