Subgroups — #28476553

Order 15104.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21422265	C₂	1	2	yes	maximal
2	C₅₉×#1713	C₂	1	1	yes	maximal, characteristic
3	#16837201	—	2	2	no	—
4	C₅₉×#500	—	2	2	no	—
5	C₅₉×#494	C₄	1	1	yes	characteristic
6	C₅₉×#494	C₂²	1	1	yes	characteristic
7	C₈²×C₅₉	C₄	1	1	yes	characteristic
8	#14273646	—	4	2	no	—
9	C₂×Q₁₆×C₅₉	—	4	1	no	—
10	C₂×Q₁₆×C₅₉	—	4	1	no	—
11	C₄⋊Q₈×C₅₉	—	2	1	no	—
12	C₄⋊Q₈×C₅₉	D₄	1	2	yes	—
13	#107×C₅₉	—	4	4	no	—
14	C₄×C₈×C₅₉	—	2	1	no	—
15	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
16	C₂×Q₈×C₅₉	—	2	2	no	—
17	C₂×Q₈×C₅₉	—	4	1	no	—
18	Q₁₆×C₅₉	—	8	2	no	—
19	Q₁₆×C₅₉	—	8	2	no	—
20	C₂×C₈×C₅₉	—	2	2	no	—
21	C₂×C₈×C₅₉	—	2	1	no	—
22	C₄⋊C₄×C₅₉	—	4	2	no	—
23	C₄⋊C₄×C₅₉	—	4	2	no	—
24	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
25	#11745664	—	8	2	no	—
26	Q₈×C₅₉	—	8	2	no	—
27	Q₈×C₅₉	—	8	2	no	—
28	C₂×C₄×C₅₉	—	2	1	no	—
29	C₂×C₄×C₅₉	—	4	2	no	—
30	C₂×C₄×C₅₉	—	8	1	no	—
31	C₂×C₄×C₅₉	#99	1	1	yes	characteristic
32	C₈×C₅₉	—	2	4	no	—
33	C₈×C₅₉	—	2	2	no	—
34	C₅₉⋊C₈	—	16	2	no	—
35	#7532	—	59	1	no	maximal, Sylow
36	C₂²×C₅₉	#353	1	1	yes	characteristic
37	C₄×C₅₉	—	2	2	no	—
38	C₄×C₅₉	—	2	1	no	—
39	C₄×C₅₉	—	8	2	no	—
40	C₄×C₅₉	—	16	1	no	—
41	#1713	D₅₉	1	1	yes	characteristic
42	#1411	—	59	2	no	—
43	C₂×C₅₉	—	2	1	no	—
44	C₂×C₅₉	#1408	1	1	yes	characteristic
45	#500	—	2	2	no	—
46	#494	Dic₅₉	1	1	yes	characteristic
47	#494	C₂×D₅₉	1	1	yes	characteristic
48	#356	—	118	2	no	—
49	C₈²	Dic₅₉	1	1	yes	characteristic
50	C₅₉	#7532	1	1	yes	characteristic, complemented, Sylow
51	C₂×Q₁₆	—	4	1	no	—
52	C₂×Q₁₆	—	4	1	no	—
53	C₄⋊Q₈	—	2	1	no	—
54	C₄⋊Q₈	C₂²⋊D₅₉	1	2	yes	—
55	#107	—	4	4	no	—
56	#101	—	236	2	no	—
57	C₄×C₈	—	2	1	no	—
58	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
59	C₂×Q₈	—	2	2	no	—
60	C₂×Q₈	—	4	1	no	—
61	Q₁₆	—	8	2	no	—
62	Q₁₆	—	8	2	no	—
63	M₁₆	—	472	2	no	—
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	2	1	no	—
66	C₄⋊C₄	—	4	2	no	—
67	C₄⋊C₄	—	4	2	no	—
68	C₄²	C₂²⋊Dic₅₉	1	1	yes	characteristic
69	Q₈	—	8	2	no	—
70	Q₈	—	8	2	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	8	1	no	—
74	C₂×C₄	#14273642	1	1	yes	characteristic
75	C₈	—	2	4	no	—
76	C₈	—	2	2	no	—
77	C₈	—	944	2	no	—
78	C₂²	#16837195	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	2	1	no	—
81	C₄	—	8	2	no	—
82	C₄	—	16	1	no	—
83	C₂	—	2	1	no	—
84	C₂	#21422259	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#21422259
commutator	C₄×C₈×C₅₉	C₂×C₄
Frattini	#494	C₂×D₅₉
Fitting	C₅₉×#1713	C₂
socle	C₂×C₅₉	#1408
prime radical	C₂²×C₅₉	#353
cosocle	C₅₉×#494	C₂²
hypercenter	#1713	D₅₉
hypocenter	C₅₉	#7532
LCS	G ▷ C₄×C₈×C₅₉ ▷ C₄²×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #494 ◁ #1713

Complements

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

N	H ≅ G/N	type
50. C₅₉	#7532	semidirect

Maximal subgroups

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

subgroup	index
#21422265	2
C₅₉×#1713	2
#7532	59