Subgroups — #28503513

Order 15104.

478 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837178	C₂	1	3	yes	maximal
2	C₂×C₄×C₅₉⋊C₁₆	C₂	1	3	yes	maximal
3	C₅₉×#1726	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
5	C₂²×C₅₉⋊C₁₆	C₂²	1	3	yes	—
6	#16837178	C₄	1	12	yes	complemented
7	C₄×C₅₉⋊C₁₆	C₄	1	12	yes	complemented
8	C₂×#97×C₅₉	C₄	1	3	yes	—
9	C₂×C₄×C₈×C₅₉	C₂²	1	3	yes	—
10	C₅₉×#376	C₄	1	1	yes	characteristic
11	C₂×C₅₉⋊C₁₆	C₂×C₄	1	6	yes	—
12	C₂×C₅₉⋊C₁₆	C₂×C₄	1	6	yes	—
13	C₂×C₅₉⋊C₁₆	C₂×C₄	1	12	yes	—
14	C₂²×C₈×C₅₉	C₂×C₄	1	3	yes	—
15	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₅₉	C₂×C₄	1	3	yes	—
17	#97×C₅₉	C₈	1	12	yes	—
18	C₄×C₈×C₅₉	C₂×C₄	1	6	yes	—
19	C₄×C₈×C₅₉	C₂×C₄	1	6	yes	—
20	#95×C₅₉	C₈	1	4	yes	—
21	C₂²×C₄×C₅₉	C₂×C₈	1	6	yes	—
22	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
23	C₂×C₈×C₅₉	C₄²	1	6	yes	—
24	C₂×C₈×C₅₉	C₂×C₈	1	12	yes	—
25	C₂×C₈×C₅₉	C₂²×C₄	1	3	yes	—
26	C₂×C₈×C₅₉	C₂²×C₄	1	3	yes	—
27	C₄²×C₅₉	C₄²	1	6	yes	—
28	C₄²×C₅₉	C₂×C₈	1	6	yes	—
29	C₅₉⋊C₁₆	C₄²	1	4	yes	—
30	C₅₉⋊C₁₆	C₄²	1	12	yes	—
31	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
32	C₂×C₄×C₅₉	—	2	12	no	—
33	C₂×C₄×C₅₉	C₄×C₈	1	12	yes	—
34	C₂×C₄×C₅₉	C₂×C₄²	1	3	yes	—
35	C₂×C₄×C₅₉	C₂²×C₈	1	3	yes	—
36	C₈×C₅₉	C₄×C₈	1	12	yes	—
37	C₈×C₅₉	C₂×C₄²	1	1	yes	characteristic
38	C₈×C₅₉	C₂×C₄²	1	3	yes	—
39	#12539	—	59	1	no	maximal, Sylow
40	C₂²×C₅₉	C₂×C₄×C₈	1	3	yes	—
41	C₂²×C₅₉	#504	1	4	yes	—
42	C₄×C₅₉	—	2	12	no	—
43	C₄×C₅₉	C₄³	1	1	yes	characteristic
44	C₄×C₅₉	C₂×C₄×C₈	1	3	yes	—
45	C₂×#346	—	59	3	no	—
46	C₂×C₄×C₁₆	—	59	3	no	—
47	#1726	D₅₉	1	1	yes	characteristic
48	C₂×C₅₉	C₄²×C₈	1	1	yes	characteristic
49	C₂×C₅₉	#2108	1	6	yes	—
50	C₂²×C₁₆	—	59	1	no	—
51	C₂²×C₁₆	—	59	3	no	—
52	C₂×#97	Dic₅₉	1	3	yes	—
53	C₂×C₄×C₈	C₂×D₅₉	1	3	yes	—
54	#376	Dic₅₉	1	1	yes	characteristic
55	#346	—	59	12	no	—
56	C₄×C₁₆	—	59	12	no	—
57	C₅₉	#12539	1	1	yes	characteristic, complemented, Sylow
58	C₂²×C₈	C₂×Dic₅₉	1	3	yes	—
59	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
60	C₂×C₄²	C₂×Dic₅₉	1	3	yes	—
61	C₂×C₁₆	—	59	6	no	—
62	C₂×C₁₆	—	59	6	no	—
63	C₂×C₁₆	—	59	12	no	—
64	#97	C₅₉⋊C₈	1	12	yes	—
65	C₄×C₈	C₄×D₅₉	1	6	yes	—
66	C₄×C₈	C₄×D₅₉	1	6	yes	—
67	#95	C₅₉⋊C₈	1	4	yes	—
68	C₂²×C₄	C₂×C₅₉⋊C₈	1	6	yes	—
69	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
70	C₂×C₈	C₂×C₅₉⋊C₈	1	12	yes	—
71	C₂×C₈	C₄×Dic₅₉	1	6	yes	—
72	C₂×C₈	C₂×C₄×D₅₉	1	3	yes	—
73	C₂×C₈	C₂×C₄×D₅₉	1	3	yes	—
74	C₄²	C₂×C₅₉⋊C₈	1	6	yes	—
75	C₄²	C₄×Dic₅₉	1	6	yes	—
76	C₁₆	—	59	4	no	—
77	C₁₆	—	59	12	no	—
78	C₂³	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
79	C₂×C₄	—	2	12	no	—
80	C₂×C₄	C₄×C₅₉⋊C₈	1	12	yes	—
81	C₂×C₄	C₂×C₄×Dic₅₉	1	3	yes	—
82	C₂×C₄	C₂²×C₅₉⋊C₈	1	3	yes	—
83	C₈	C₄×C₅₉⋊C₈	1	12	yes	—
84	C₈	C₄²×D₅₉	1	1	yes	characteristic
85	C₈	C₄²×D₅₉	1	3	yes	—
86	C₂²	C₂×C₄×C₅₉⋊C₈	1	3	yes	—
87	C₂²	#16837819	1	4	yes	—
88	C₄	—	2	12	no	—
89	C₄	C₄²×Dic₅₉	1	1	yes	characteristic
90	C₄	C₂×C₄×C₅₉⋊C₈	1	3	yes	—
91	C₂	C₄²×C₅₉⋊C₈	1	1	yes	characteristic
92	C₂	#21425693	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₉
commutator	C₂×C₅₉	C₄²×C₈
Fitting	C₅₉×#1726	C₂
socle, prime radical	C₂³×C₅₉	C₂²×C₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1726	D₅₉
hypocenter	C₅₉	#12539
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
6. #16837178	C₄	semidirect
7. C₄×C₅₉⋊C₁₆	C₄	semidirect
57. C₅₉	#12539	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837178	2
C₂×C₄×C₅₉⋊C₁₆	2
C₅₉×#1726	2
#12539	59