Subgroups — #28509709

Order 15104.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21426371	C₂	1	2	yes	maximal
2	#21422296	C₂	1	4	yes	maximal
3	C₄×C₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic
4	#16837807	—	2	2	no	—
5	#16837228	—	2	2	no	—
6	#16837228	C₂²	1	4	yes	—
7	C₂×C₃₂×C₅₉	C₂²	1	2	yes	—
8	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
9	#14273727	—	4	2	no	—
10	#14273661	—	2	4	no	—
11	#14273661	—	4	2	no	—
12	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
13	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
14	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
15	C₃₂×C₅₉	Q₈	1	4	yes	complemented
16	C₂×Dic₁₁₈	—	8	2	no	—
17	C₂×C₈×C₅₉	D₈	1	1	yes	characteristic
18	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
19	C₄²×C₅₉	D₈	1	1	yes	characteristic
20	C₄⋊Dic₅₉	—	4	4	no	—
21	C₄⋊Dic₅₉	—	8	2	no	—
22	C₁₆×C₅₉	Q₁₆	1	2	yes	—
23	C₁₆×C₅₉	C₂×Q₈	1	2	yes	—
24	C₂×C₄×C₅₉	D₁₆	1	1	yes	characteristic
25	C₂×C₄×C₅₉	D₁₆	1	1	yes	characteristic
26	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
27	C₂×Dic₅₉	—	8	4	no	—
28	Dic₁₁₈	—	16	4	no	—
29	C₈×C₅₉	Q₃₂	1	2	yes	—
30	C₈×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
31	C₈×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
32	#13731	—	59	1	no	maximal, Sylow
33	C₂²×C₅₉	C₂×D₁₆	1	1	yes	characteristic
34	C₄×C₅₉	D₃₂	1	2	yes	—
35	C₄×C₅₉	Q₆₄	1	2	yes	—
36	C₄×C₅₉	#500	1	1	yes	characteristic
37	C₄×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
38	Dic₅₉	—	16	4	no	—
39	#2252	—	59	2	no	—
40	#1423	—	59	4	no	—
41	C₄×C₃₂	D₅₉	1	1	yes	characteristic
42	C₂×C₅₉	#2252	1	1	yes	characteristic
43	C₂×C₅₉	C₂×D₃₂	1	1	yes	characteristic
44	C₂×C₅₉	C₂×Q₆₄	1	1	yes	characteristic
45	#500	—	118	2	no	—
46	C₂×C₃₂	C₂×D₅₉	1	2	yes	—
47	#366	—	59	4	no	—
48	#366	—	118	2	no	—
49	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
50	C₅₉	#13731	1	1	yes	characteristic, complemented, Sylow
51	C₄⋊Q₈	—	236	2	no	—
52	C₂×C₁₆	D₂₃₆	1	1	yes	characteristic
53	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
54	#107	—	118	4	no	—
55	#107	—	236	2	no	—
56	C₄×C₈	D₂₃₆	1	1	yes	characteristic
57	C₃₂	Dic₁₁₈	1	4	yes	complemented
58	C₂×Q₈	—	472	2	no	—
59	C₂×C₈	D₄₇₂	1	1	yes	characteristic
60	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
61	C₄⋊C₄	—	472	2	no	—
62	C₄⋊C₄	—	236	4	no	—
63	C₄²	D₄₇₂	1	1	yes	characteristic
64	C₁₆	Dic₂₃₆	1	2	yes	—
65	C₁₆	C₂×Dic₁₁₈	1	2	yes	—
66	Q₈	—	944	4	no	—
67	C₂×C₄	—	472	4	no	—
68	C₂×C₄	D₉₄₄	1	1	yes	characteristic
69	C₂×C₄	D₉₄₄	1	1	yes	characteristic
70	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
71	C₈	Dic₄₇₂	1	2	yes	—
72	C₈	#14273727	1	1	yes	characteristic
73	C₈	C₂×Dic₂₃₆	1	1	yes	characteristic
74	C₂²	C₂×D₉₄₄	1	1	yes	characteristic
75	C₄	—	944	4	no	—
76	C₄	D₁₈₈₈	1	2	yes	—
77	C₄	Dic₉₄₄	1	2	yes	—
78	C₄	#16837807	1	1	yes	characteristic
79	C₄	C₂×Dic₄₇₂	1	1	yes	characteristic
80	C₂	#21426371	1	1	yes	characteristic
81	C₂	C₂×D₁₈₈₈	1	1	yes	characteristic
82	C₂	C₂×Dic₉₄₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₉₄₄
commutator, cosocle	C₂×C₁₆×C₅₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₅₉
Fitting	C₄×C₃₂×C₅₉	C₂
socle, prime radical	C₂²×C₅₉	C₂×D₁₆
hypercenter	C₄×C₃₂	D₅₉
hypocenter	C₅₉	#13731
LCS	G ▷ C₂×C₁₆×C₅₉ ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
15. C₃₂×C₅₉	Q₈	semidirect
50. C₅₉	#13731	semidirect
57. C₃₂	Dic₁₁₈	semidirect

Maximal subgroups

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

subgroup	index
#21426371	2
#21422296	2
C₄×C₃₂×C₅₉	2
#13731	59