Subgroups — C₂×#21422230 #28508862

Order 15104.

254 conjugacy classes of subgroups.

The subgroup lattice has 114 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅₉⋊C₃₂	C₂	1	2	yes	maximal
2	#21422230	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₉⋊C₃₂	C₄	1	8	yes	complemented
5	C₂×C₅₉⋊C₃₂	C₂²	1	4	yes	—
6	C₂²×C₁₆×C₅₉	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₅₉	C₄	1	1	yes	characteristic
9	C₄×C₁₆×C₅₉	C₄	1	2	yes	—
10	C₄×C₁₆×C₅₉	C₂²	1	2	yes	—
11	C₅₉⋊C₃₂	C₂×C₄	1	8	yes	complemented
12	C₂²×C₈×C₅₉	C₈	1	1	yes	characteristic
13	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄²×C₅₉	C₈	1	1	yes	characteristic
15	C₂×C₁₆×C₅₉	C₈	1	4	yes	—
16	C₂×C₁₆×C₅₉	C₂×C₄	1	4	yes	—
17	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
19	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
20	C₄×C₈×C₅₉	C₈	1	2	yes	—
21	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₅₉	C₁₆	1	2	yes	—
23	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
24	C₂×C₈×C₅₉	C₁₆	1	4	yes	—
25	C₂×C₈×C₅₉	C₄²	1	2	yes	—
26	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
27	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
28	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
29	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
30	C₄²×C₅₉	C₁₆	1	2	yes	—
31	C₄²×C₅₉	C₂×C₈	1	2	yes	—
32	C₁₆×C₅₉	C₄²	1	2	yes	—
33	C₁₆×C₅₉	C₂×C₈	1	4	yes	—
34	C₁₆×C₅₉	C₂²×C₄	1	2	yes	—
35	C₂³×C₅₉	C₂×C₁₆	1	1	yes	characteristic
36	C₂×C₄×C₅₉	—	2	4	no	—
37	C₂×C₄×C₅₉	C₄×C₈	1	2	yes	—
38	C₂×C₄×C₅₉	C₂×C₁₆	1	4	yes	—
39	C₂×C₄×C₅₉	C₂×C₁₆	1	1	yes	characteristic
40	C₂×C₄×C₅₉	C₂×C₁₆	1	2	yes	—
41	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
42	C₈×C₅₉	C₄×C₈	1	2	yes	—
43	C₈×C₅₉	C₂×C₁₆	1	4	yes	—
44	C₈×C₅₉	C₂×C₄²	1	1	yes	characteristic
45	C₈×C₅₉	C₂²×C₈	1	1	yes	characteristic
46	C₂×#1397	—	59	1	no	maximal, Sylow
47	C₂²×C₅₉	C₄×C₁₆	1	2	yes	—
48	C₂²×C₅₉	M₆₄	1	4	yes	—
49	C₂²×C₅₉	C₂²×C₁₆	1	1	yes	characteristic
50	C₄×C₅₉	—	2	4	no	—
51	C₄×C₅₉	C₄×C₁₆	1	2	yes	—
52	C₄×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
53	C₄×C₅₉	C₂²×C₁₆	1	1	yes	characteristic
54	C₂²×C₃₂	—	59	2	no	—
55	C₂×C₄×C₁₆	D₅₉	1	1	yes	characteristic
56	#1397	—	59	4	no	—
57	C₂×C₅₉	#1397	1	4	yes	complemented
58	C₂×C₅₉	C₂×C₄×C₁₆	1	1	yes	characteristic
59	C₂×C₅₉	C₂×M₆₄	1	2	yes	—
60	C₂²×C₁₆	Dic₅₉	1	1	yes	characteristic
61	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
62	C₂×C₄×C₈	Dic₅₉	1	1	yes	characteristic
63	C₂×C₃₂	—	59	8	no	—
64	C₂×C₃₂	—	59	4	no	—
65	C₄×C₁₆	Dic₅₉	1	2	yes	—
66	C₄×C₁₆	C₂×D₅₉	1	2	yes	—
67	C₅₉	C₂×#1397	1	1	yes	characteristic, complemented, Sylow
68	C₂²×C₈	C₅₉⋊C₈	1	1	yes	characteristic
69	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
70	C₂×C₄²	C₅₉⋊C₈	1	1	yes	characteristic
71	C₂×C₁₆	C₅₉⋊C₈	1	4	yes	—
72	C₂×C₁₆	C₄×D₅₉	1	4	yes	—
73	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
74	C₂×C₁₆	C₂×Dic₅₉	1	1	yes	characteristic
75	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
76	C₄×C₈	C₅₉⋊C₈	1	2	yes	—
77	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
78	C₃₂	—	59	8	no	—
79	C₂²×C₄	C₅₉⋊C₁₆	1	2	yes	—
80	C₂²×C₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
81	C₂×C₈	C₅₉⋊C₁₆	1	4	yes	—
82	C₂×C₈	C₂×C₅₉⋊C₈	1	1	yes	characteristic
83	C₂×C₈	C₂×C₅₉⋊C₈	1	2	yes	—
84	C₂×C₈	C₂×C₅₉⋊C₈	1	2	yes	—
85	C₂×C₈	C₄×Dic₅₉	1	2	yes	—
86	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
87	C₄²	C₅₉⋊C₁₆	1	2	yes	—
88	C₄²	C₂×C₅₉⋊C₈	1	2	yes	—
89	C₁₆	C₂×C₅₉⋊C₈	1	4	yes	—
90	C₁₆	C₄×Dic₅₉	1	2	yes	—
91	C₁₆	C₂×C₄×D₅₉	1	2	yes	—
92	C₂³	C₂×C₅₉⋊C₁₆	1	1	yes	characteristic
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	C₄×C₅₉⋊C₈	1	2	yes	—
95	C₂×C₄	C₂×C₅₉⋊C₁₆	1	4	yes	—
96	C₂×C₄	C₂×C₅₉⋊C₁₆	1	1	yes	characteristic
97	C₂×C₄	C₂×C₅₉⋊C₁₆	1	2	yes	—
98	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
99	C₈	C₄×C₅₉⋊C₈	1	2	yes	—
100	C₈	C₂×C₅₉⋊C₁₆	1	4	yes	—
101	C₈	C₂×C₄×Dic₅₉	1	1	yes	characteristic
102	C₈	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
103	C₂²	C₄×C₅₉⋊C₁₆	1	2	yes	—
104	C₂²	#16837236	1	4	yes	—
105	C₂²	C₂²×C₅₉⋊C₁₆	1	1	yes	characteristic
106	C₄	—	2	4	no	—
107	C₄	C₄×C₅₉⋊C₁₆	1	2	yes	—
108	C₄	C₂×C₄×C₅₉⋊C₈	1	1	yes	characteristic
109	C₄	C₂²×C₅₉⋊C₁₆	1	1	yes	characteristic
110	C₂	#21422230	1	4	yes	direct product factor
111	C₂	C₂×C₄×C₅₉⋊C₁₆	1	1	yes	characteristic
112	C₂	C₂×#16837236	1	2	yes	—

	subgroup	G/H
center	C₂²×C₁₆	C₂×D₅₉
commutator	C₂×C₅₉	C₂×C₄×C₁₆
Frattini	C₂×C₁₆	C₂²×D₅₉
Fitting	C₂×C₄×C₁₆×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₁₆
cosocle	C₂×C₁₆×C₅₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₉
hypocenter	C₅₉	C₂×#1397
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #21422230	C₂	direct (N × H)
4. C₂×C₅₉⋊C₃₂	C₄	semidirect
11. C₅₉⋊C₃₂	C₂×C₄	semidirect
57. C₂×C₅₉	#1397	semidirect
67. C₅₉	C₂×#1397	semidirect
110. C₂	#21422230	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅₉⋊C₃₂	2
#21422230	2
C₂×C₄×C₁₆×C₅₉	2
C₂×#1397	59

Subgroups — C2×#21422230 #28508862

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21422230 #28508862