Subgroups — #28482440

Order 15104.

402 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21424247	C₂	1	2	yes	maximal
2	#21423841	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837153	C₂	1	1	yes	maximal, characteristic
4	C₂×#16837152	C₂	1	2	yes	maximal
5	C₂×C₅₉×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×Dic₅₉⋊C₈	—	2	4	no	—
8	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₄²×Dic₅₉	C₂²	1	1	yes	characteristic
10	#16837156	C₄	1	2	yes	—
11	#16837156	C₄	1	2	yes	—
12	#16837153	—	2	2	no	—
13	#16837152	—	2	4	no	—
14	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×#334	—	2	2	no	—
17	C₂²×C₅₉⋊C₈	—	2	2	no	—
18	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₅₉	D₄	1	2	yes	—
21	C₂×C₄×Dic₅₉	Q₈	1	2	yes	—
22	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
24	Dic₅₉⋊C₈	—	4	8	no	—
25	C₄×C₅₉⋊C₈	D₄	1	2	yes	—
26	C₄×C₅₉⋊C₈	Q₈	1	2	yes	—
27	C₂²×C₈×C₅₉	—	2	2	no	—
28	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₅₉	—	2	2	no	—
32	C₄⋊C₈×C₅₉	—	2	2	no	—
33	C₄×C₈×C₅₉	D₄	1	2	yes	—
34	C₄×C₈×C₅₉	Q₈	1	2	yes	—
35	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
38	C₂²×Dic₅₉	C₄⋊C₄	1	4	yes	—
39	C₂×C₈×C₅₉	—	2	2	no	—
40	C₂×C₈×C₅₉	—	2	2	no	—
41	C₂×C₈×C₅₉	—	4	2	no	—
42	C₂×C₈×C₅₉	—	4	4	no	—
43	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
45	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
49	C₄×Dic₅₉	—	2	2	no	—
50	C₄×Dic₅₉	—	2	2	no	—
51	C₄×Dic₅₉	—	4	2	no	—
52	C₄×Dic₅₉	SD₁₆	1	8	yes	—
53	C₂×C₅₉⋊C₈	—	2	2	no	—
54	C₂×C₅₉⋊C₈	—	2	2	no	—
55	C₂×C₅₉⋊C₈	—	4	2	no	—
56	C₂×C₅₉⋊C₈	—	4	4	no	—
57	C₂×C₅₉⋊C₈	C₄⋊C₄	1	4	yes	—
58	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₅₉	—	2	2	no	—
60	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
63	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₅₉	#131	1	2	yes	—
67	C₂×C₄×C₅₉	#131	1	2	yes	—
68	C₂×C₄×C₅₉	C₂×SD₁₆	1	4	yes	—
69	C₂×Dic₅₉	—	2	4	no	—
70	C₂×Dic₅₉	—	2	4	no	—
71	C₂×Dic₅₉	#106	1	8	yes	—
72	C₈×C₅₉	—	2	2	no	—
73	C₈×C₅₉	—	2	2	no	—
74	C₈×C₅₉	—	8	2	no	—
75	C₅₉⋊C₈	—	2	4	no	—
76	C₅₉⋊C₈	—	8	2	no	—
77	#8809	—	59	1	no	maximal, Sylow
78	C₂²×C₅₉	#384	1	1	yes	characteristic
79	C₂²×C₅₉	#424	1	2	yes	—
80	C₂²×C₅₉	C₂×#106	1	2	yes	—
81	C₂²×C₅₉	#430	1	2	yes	—
82	C₄×C₅₉	—	2	1	no	—
83	C₄×C₅₉	—	2	1	no	—
84	C₄×C₅₉	#424	1	2	yes	—
85	C₄×C₅₉	#424	1	1	yes	characteristic
86	C₄×C₅₉	#424	1	1	yes	characteristic
87	C₄×C₅₉	#492	1	2	yes	—
88	C₄×C₅₉	#498	1	2	yes	—
89	Dic₅₉	—	2	8	no	—
90	#1842	—	59	2	no	—
91	#1744	—	59	1	no	—
92	C₂×#334	—	59	2	no	—
93	C₂×#334	—	59	1	no	—
94	C₂×#334	D₅₉	1	1	yes	characteristic
95	C₂×C₅₉	#1570	1	4	yes	—
96	C₂×C₅₉	#1843	1	1	yes	characteristic
97	C₂×C₅₉	#1844	1	1	yes	characteristic
98	C₂×C₅₉	#1849	1	1	yes	characteristic
99	C₂×C₄⋊C₈	—	59	2	no	—
100	C₂×C₄⋊C₈	—	118	4	no	—
101	C₂×C₄⋊C₈	C₂×D₅₉	1	2	yes	—
102	C₂×C₄×C₈	—	59	1	no	—
103	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
104	C₄³	—	59	1	no	—
105	#336	—	59	2	no	—
106	#336	—	59	2	no	—
107	#334	—	2	2	no	—
108	#334	—	118	4	no	—
109	#334	—	118	2	no	—
110	C₅₉	#8809	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	—	59	2	no	—
113	C₂²×C₈	—	118	2	no	—
114	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
115	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
116	C₂×C₄²	—	59	2	no	—
117	C₂×C₄²	—	59	2	no	—
118	C₂×C₄²	—	59	2	no	—
119	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	2	2	no	—
122	C₄⋊C₈	—	118	2	no	—
123	C₄⋊C₈	—	118	2	no	—
124	C₄⋊C₈	—	236	8	no	—
125	C₄×C₈	—	59	2	no	—
126	C₄×C₈	—	59	2	no	—
127	C₄×C₈	Dic₁₁₈	1	2	yes	—
128	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
129	C₂²×C₄	—	59	4	no	—
130	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
131	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
132	C₂²×C₄	Q₈×D₅₉	1	1	yes	characteristic
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	4	4	no	—
137	C₂×C₈	—	59	4	no	—
138	C₂×C₈	—	118	2	no	—
139	C₂×C₈	—	118	2	no	—
140	C₂×C₈	—	236	2	no	—
141	C₂×C₈	—	236	4	no	—
142	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
143	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
144	C₄²	—	59	8	no	—
145	C₄²	—	118	2	no	—
146	C₄²	—	118	2	no	—
147	C₄²	—	236	2	no	—
148	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
149	C₄²	D₄×D₅₉	1	1	yes	characteristic
150	C₄²	Q₈×D₅₉	1	1	yes	characteristic
151	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
152	C₂³	C₄⋊C₄×D₅₉	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	59	8	no	—
155	C₂×C₄	—	118	4	no	—
156	C₂×C₄	—	118	4	no	—
157	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
158	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
159	C₂×C₄	#14273723	1	1	yes	characteristic
160	C₂×C₄	#14273725	1	1	yes	characteristic
161	C₂×C₄	#14273726	1	1	yes	characteristic
162	C₂×C₄	#14273726	1	1	yes	characteristic
163	C₂×C₄	#14273733	1	2	yes	—
164	C₂×C₄	#14273735	1	2	yes	—
165	C₂×C₄	SD₁₆×D₅₉	1	4	yes	—
166	C₈	—	2	2	no	—
167	C₈	—	2	2	no	—
168	C₈	—	8	2	no	—
169	C₈	—	118	4	no	—
170	C₈	—	472	2	no	—
171	C₂²	#16837271	1	1	yes	characteristic
172	C₂²	#16837427	1	2	yes	—
173	C₂²	#106×D₅₉	1	2	yes	—
174	C₂²	#16837449	1	2	yes	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	118	8	no	—
178	C₄	#16837427	1	2	yes	—
179	C₄	#16837428	1	1	yes	characteristic
180	C₄	#16837430	1	1	yes	characteristic
181	C₄	#16837773	1	2	yes	—
182	C₄	#16837799	1	2	yes	—
183	C₂	#21422865	1	4	yes	—
184	C₂	#21424249	1	1	yes	characteristic
185	C₂	#21424251	1	1	yes	characteristic
186	C₂	#21424269	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₉
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₅₉×#334	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×#334	D₅₉
hypocenter	C₅₉	#8809
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
110. C₅₉	#8809	semidirect

Maximal subgroups

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

subgroup	index
#21424247	2
#21423841	2
C₂×#16837153	2
C₂×#16837152	2
C₂×C₅₉×#334	2
#8809	59