Subgroups — #28478054

Order 15104.

526 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21423842	C₂	1	2	yes	maximal
2	C₂×#16837124	C₂	1	4	yes	maximal
3	C₄²×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₅₉⋊C₈	C₂²	1	4	yes	—
5	#16837157	C₄	1	8	yes	—
6	#16837124	C₄	1	16	yes	complemented
7	C₂×C₄×C₈×C₅₉	C₄	1	4	yes	—
8	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
9	C₄³×C₅₉	C₂²	1	1	yes	characteristic
10	C₂²×C₅₉⋊C₈	C₂×C₄	1	8	yes	—
11	C₄×C₅₉⋊C₈	C₂×C₄	1	16	yes	—
12	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
13	C₂×C₄²×C₅₉	C₂×C₄	1	4	yes	—
14	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₅₉	—	2	8	no	—
17	C₄×C₈×C₅₉	C₂×C₄	1	8	yes	—
18	C₂²×C₄×C₅₉	C₄²	1	4	yes	—
19	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
21	C₂×C₈×C₅₉	—	2	8	no	—
22	C₂×C₈×C₅₉	C₄²	1	8	yes	—
23	C₄²×C₅₉	—	2	8	no	—
24	C₄²×C₅₉	M₁₆	1	8	yes	—
25	C₄²×C₅₉	C₂²×C₄	1	4	yes	—
26	C₂×C₅₉⋊C₈	—	2	16	no	—
27	C₂×C₅₉⋊C₈	C₄²	1	16	yes	—
28	C₂³×C₅₉	C₂×C₄²	1	1	yes	characteristic
29	C₂×C₄×C₅₉	—	2	8	no	—
30	C₂×C₄×C₅₉	#97	1	8	yes	—
31	C₂×C₄×C₅₉	C₂×C₄²	1	4	yes	—
32	C₂×C₄×C₅₉	C₂×C₄²	1	2	yes	—
33	C₂×C₄×C₅₉	C₂×M₁₆	1	4	yes	—
34	C₂×C₄×C₅₉	#131	1	8	yes	—
35	C₈×C₅₉	—	2	8	no	—
36	C₅₉⋊C₈	—	2	16	no	—
37	#7843	—	59	1	no	maximal, Sylow
38	C₂²×C₅₉	C₄³	1	1	yes	characteristic
39	C₂²×C₅₉	C₂×#97	1	2	yes	—
40	C₂²×C₅₉	#405	1	4	yes	—
41	C₄×C₅₉	—	2	8	no	—
42	C₄×C₅₉	C₄×M₁₆	1	4	yes	—
43	C₄×C₅₉	#405	1	8	yes	—
44	#1744	—	59	2	no	—
45	C₄²×C₈	D₅₉	1	1	yes	characteristic
46	C₂×#322	—	59	4	no	—
47	C₂×C₅₉	#1452	1	4	yes	—
48	C₂×C₅₉	C₄×#97	1	1	yes	characteristic
49	C₂×C₅₉	#1726	1	2	yes	—
50	C₂×C₄×C₈	—	59	4	no	—
51	C₂×C₄×C₈	Dic₅₉	1	4	yes	—
52	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
53	C₄³	C₂×D₅₉	1	1	yes	characteristic
54	#336	—	59	8	no	—
55	#322	—	59	16	no	—
56	C₅₉	#7843	1	1	yes	characteristic, complemented, Sylow
57	C₂²×C₈	—	59	8	no	—
58	C₂²×C₈	C₂×Dic₅₉	1	4	yes	—
59	C₂×C₄²	C₄×D₅₉	1	4	yes	—
60	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
61	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
62	C₄×C₈	—	2	8	no	—
63	C₄×C₈	—	59	16	no	—
64	C₄×C₈	C₄×D₅₉	1	8	yes	—
65	C₂²×C₄	C₄×Dic₅₉	1	4	yes	—
66	C₂²×C₄	C₂×C₄×D₅₉	1	2	yes	—
67	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
68	C₂×C₈	—	2	8	no	—
69	C₂×C₈	—	59	16	no	—
70	C₂×C₈	—	118	16	no	—
71	C₂×C₈	C₄×Dic₅₉	1	8	yes	—
72	C₄²	—	2	8	no	—
73	C₄²	#11745664	1	8	yes	—
74	C₄²	C₂×C₄×D₅₉	1	4	yes	—
75	C₂³	C₂×C₄×Dic₅₉	1	1	yes	characteristic
76	C₂×C₄	—	2	8	no	—
77	C₂×C₄	#14273638	1	8	yes	—
78	C₂×C₄	C₄²×D₅₉	1	4	yes	—
79	C₂×C₄	C₂×C₄×Dic₅₉	1	2	yes	—
80	C₂×C₄	C₂×#11745664	1	4	yes	—
81	C₂×C₄	#14273733	1	8	yes	—
82	C₈	—	2	8	no	—
83	C₈	—	118	16	no	—
84	C₂²	C₄²×Dic₅₉	1	1	yes	characteristic
85	C₂²	C₂×#14273638	1	2	yes	—
86	C₂²	#16837357	1	4	yes	—
87	C₄	—	2	8	no	—
88	C₄	C₄×#11745664	1	4	yes	—
89	C₄	#16837356	1	8	yes	—
90	C₂	#21422388	1	4	yes	—
91	C₂	C₄×#14273638	1	1	yes	characteristic
92	C₂	#21423782	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₉
commutator	C₂²×C₅₉	C₄³
Fitting	C₄²×C₈×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₄²
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₄²×C₈	D₅₉
hypocenter	C₅₉	#7843
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
6. #16837124	C₄	semidirect
56. C₅₉	#7843	semidirect

Maximal subgroups

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

subgroup	index
#21423842	2
C₂×#16837124	2
C₄²×C₈×C₅₉	2
#7843	59