Subgroups — #21422395

Order 7552.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837453	C₂	1	3	yes	maximal
2	#16837124	C₂	1	3	yes	maximal, complemented
3	C₈²×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄²×D₅₉	C₂²	1	1	yes	characteristic
5	Dic₅₉⋊C₈	C₄	1	6	yes	—
6	(C₂×D₅₉)⋊C₈	C₄	1	6	yes	—
7	C₄×C₅₉⋊C₈	C₂²	1	3	yes	—
8	C₄×C₈×C₅₉	C₂²	1	3	yes	—
9	C₂×C₄×D₅₉	C₂×C₄	1	3	yes	—
10	C₂×C₈×C₅₉	C₂×C₄	1	6	yes	—
11	C₄²×C₅₉	C₂³	1	1	yes	characteristic
12	C₄×Dic₅₉	C₂×C₄	1	3	yes	—
13	C₂×C₅₉⋊C₈	C₂×C₄	1	6	yes	—
14	C₂²×D₅₉	C₄²	1	1	yes	characteristic
15	C₂×C₄×C₅₉	C₂²×C₄	1	3	yes	—
16	C₂×Dic₅₉	C₄²	1	3	yes	—
17	C₄×D₅₉	—	2	6	no	—
18	C₈×C₅₉	—	2	6	no	—
19	C₅₉⋊C₈	—	2	6	no	—
20	C₂²×C₅₉	C₂×C₄²	1	1	yes	characteristic
21	C₂×D₅₉	—	2	3	no	—
22	C₄×C₅₉	#131	1	6	yes	—
23	Dic₅₉	—	2	3	no	—
24	#1454	—	59	1	no	maximal, Sylow
25	C₂×C₅₉	#405	1	3	yes	—
26	D₅₉	—	4	1	no	—
27	#431	—	59	3	no	—
28	#322	—	59	3	no	—
29	C₈²	D₅₉	1	1	yes	characteristic, complemented
30	C₅₉	#1454	1	1	yes	characteristic, complemented, Sylow
31	C₂×C₄²	—	59	1	no	—
32	C₄⋊C₈	—	59	6	no	—
33	C₂²⋊C₈	—	59	6	no	—
34	C₄×C₈	—	59	3	no	—
35	C₄×C₈	C₂×D₅₉	1	3	yes	—
36	C₂²×C₄	—	59	3	no	—
37	C₂×C₈	—	59	6	no	—
38	C₂×C₈	C₄×D₅₉	1	6	yes	—
39	C₄²	—	59	3	no	—
40	C₄²	C₂²×D₅₉	1	1	yes	characteristic
41	C₂³	—	59	1	no	—
42	C₂×C₄	—	59	3	no	—
43	C₂×C₄	—	118	6	no	—
44	C₂×C₄	C₂×C₄×D₅₉	1	3	yes	—
45	C₈	—	2	6	no	—
46	C₈	—	118	6	no	—
47	C₂²	—	118	3	no	—
48	C₂²	C₄²×D₅₉	1	1	yes	characteristic
49	C₄	—	118	3	no	—
50	C₄	#14273733	1	6	yes	—
51	C₂	—	236	1	no	—
52	C₂	#16837356	1	3	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₅₉
commutator, socle	C₂²×C₅₉	C₂×C₄²
Fitting	C₈²×C₅₉	C₂
hypercenter	C₈²	D₅₉
hypocenter	C₅₉	#1454
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
2. #16837124	C₂	semidirect
3. C₈²×C₅₉	C₂	semidirect
29. C₈²	D₅₉	semidirect
30. C₅₉	#1454	semidirect