Subgroups — C₅₉×#26481 #28438590

Order 15104.

798 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₉×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×SD₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#2454	C₂	1	1	yes	maximal, characteristic
4	C₅₉×#2453	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#2068	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₉×#2067	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#2026	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₉×#2025	C₂	1	1	yes	maximal, characteristic
9	C₅₉×#1968	C₂	1	1	yes	maximal, characteristic
10	C₅₉×#1968	C₂	1	1	yes	maximal, characteristic
11	C₅₉×#1941	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₉×#1941	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅₉×#1863	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅₉×#1862	C₂	1	1	yes	maximal, characteristic
15	C₈×C₄⋊C₄×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×SD₁₆×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×#125×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×#125×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
21	C₅₉×#499	—	2	4	no	—
22	C₄×SD₁₆×C₅₉	—	2	4	no	—
23	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
24	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×#106×C₅₉	C₂²	1	1	yes	characteristic
26	C₂×#106×C₅₉	C₂²	1	1	yes	characteristic
27	C₂×#106×C₅₉	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
29	C₂×#103×C₅₉	C₂²	1	1	yes	characteristic
30	C₂×#103×C₅₉	C₂²	1	1	yes	characteristic
31	C₂×#103×C₅₉	C₂²	1	1	yes	characteristic
32	C₂×#102×C₅₉	C₂²	1	1	yes	characteristic
33	C₂×#102×C₅₉	C₂²	1	1	yes	characteristic
34	C₂×#102×C₅₉	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
38	C₅₉×#400	—	2	1	no	—
39	C₅₉×#399	—	2	1	no	—
40	C₅₉×#398	—	2	1	no	—
41	C₅₉×#396	—	2	1	no	—
42	C₅₉×#391	C₂²	1	1	yes	characteristic
43	C₅₉×#391	C₂²	1	1	yes	characteristic
44	C₅₉×#390	C₂²	1	1	yes	characteristic
45	C₅₉×#390	C₂²	1	1	yes	characteristic
46	C₅₉×#389	—	2	1	no	—
47	C₅₉×#389	C₂²	1	1	yes	characteristic
48	C₅₉×#389	C₂²	1	1	yes	characteristic
49	C₅₉×#389	C₂²	1	1	yes	characteristic
50	C₅₉×#389	C₂²	1	1	yes	characteristic
51	C₅₉×#387	—	2	1	no	—
52	C₅₉×#385	—	2	1	no	—
53	C₅₉×#380	—	2	1	no	—
54	C₄×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
55	C₅₉×#340	C₂²	1	1	yes	characteristic
56	C₅₉×#340	C₂²	1	1	yes	characteristic
57	C₅₉×#340	C₂²	1	1	yes	characteristic
58	C₅₉×#340	C₂²	1	1	yes	characteristic
59	C₅₉×#336	C₂²	1	1	yes	characteristic
60	C₅₉×#336	C₂²	1	1	yes	characteristic
61	C₂²×Q₈×C₅₉	C₂³	1	1	yes	characteristic
62	C₂²×D₄×C₅₉	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₅₉	—	2	1	no	—
64	C₂×SD₁₆×C₅₉	—	2	2	no	—
65	C₂×SD₁₆×C₅₉	—	2	2	no	—
66	C₂×SD₁₆×C₅₉	Q₈	1	4	yes	—
67	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
71	#125×C₅₉	—	2	2	no	—
72	#125×C₅₉	—	2	2	no	—
73	#125×C₅₉	—	4	1	no	—
74	#125×C₅₉	—	4	1	no	—
75	C₄×Q₈×C₅₉	—	2	2	no	—
76	C₄×Q₈×C₅₉	—	4	1	no	—
77	C₄×D₄×C₅₉	—	2	2	no	—
78	C₄×D₄×C₅₉	—	4	1	no	—
79	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
80	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
81	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
82	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
83	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
84	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
85	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
86	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
87	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
88	C₂×C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
96	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
97	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
98	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
99	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
100	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
101	C₂×C₄²×C₅₉	—	2	1	no	—
102	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
105	#107×C₅₉	—	2	2	no	—
106	#107×C₅₉	—	2	2	no	—
107	#106×C₅₉	—	2	2	no	—
108	#106×C₅₉	—	2	2	no	—
109	#106×C₅₉	D₄	1	4	yes	—
110	C₄⋊C₈×C₅₉	—	2	2	no	—
111	#103×C₅₉	—	2	2	no	—
112	#103×C₅₉	—	2	2	no	—
113	#103×C₅₉	—	4	1	no	—
114	#102×C₅₉	—	2	2	no	—
115	#102×C₅₉	—	2	2	no	—
116	#102×C₅₉	—	4	1	no	—
117	C₄×C₈×C₅₉	—	2	2	no	—
118	C₄×C₈×C₅₉	—	2	2	no	—
119	C₄×C₈×C₅₉	—	2	2	no	—
120	#95×C₅₉	—	2	1	no	—
121	#95×C₅₉	—	2	1	no	—
122	#95×C₅₉	—	2	1	no	—
123	#95×C₅₉	—	2	1	no	—
124	#95×C₅₉	—	2	1	no	—
125	#95×C₅₉	—	2	1	no	—
126	#95×C₅₉	—	2	1	no	—
127	#95×C₅₉	—	2	1	no	—
128	#95×C₅₉	D₄	1	1	yes	characteristic
129	#95×C₅₉	D₄	1	1	yes	characteristic
130	C₂⁴×C₅₉	—	2	1	no	—
131	C₂×Q₈×C₅₉	—	2	1	no	—
132	C₂×Q₈×C₅₉	—	2	1	no	—
133	C₂×Q₈×C₅₉	—	2	1	no	—
134	C₂×Q₈×C₅₉	—	2	1	no	—
135	C₂×Q₈×C₅₉	—	2	1	no	—
136	C₂×Q₈×C₅₉	C₂×Q₈	1	2	yes	—
137	C₂×D₄×C₅₉	—	2	1	no	—
138	C₂×D₄×C₅₉	—	2	1	no	—
139	C₂×D₄×C₅₉	—	2	1	no	—
140	C₂×D₄×C₅₉	—	2	1	no	—
141	C₂×D₄×C₅₉	—	2	1	no	—
142	C₂×D₄×C₅₉	C₂×Q₈	1	2	yes	—
143	C₂²×C₄×C₅₉	—	2	1	no	—
144	C₂²×C₄×C₅₉	—	2	1	no	—
145	C₂²×C₄×C₅₉	—	2	1	no	—
146	C₂²×C₄×C₅₉	—	2	1	no	—
147	C₂²×C₄×C₅₉	—	2	1	no	—
148	C₂²×C₄×C₅₉	—	2	1	no	—
149	C₂²×C₄×C₅₉	—	2	1	no	—
150	C₂²×C₄×C₅₉	—	4	2	no	—
151	C₂²×C₄×C₅₉	—	4	2	no	—
152	C₂²×C₄×C₅₉	—	8	1	no	—
153	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
160	SD₁₆×C₅₉	—	2	8	no	—
161	C₂×C₈×C₅₉	—	2	1	no	—
162	C₂×C₈×C₅₉	—	2	1	no	—
163	C₂×C₈×C₅₉	—	2	1	no	—
164	C₂×C₈×C₅₉	—	2	1	no	—
165	C₂×C₈×C₅₉	—	2	1	no	—
166	C₂×C₈×C₅₉	—	2	1	no	—
167	C₂×C₈×C₅₉	—	2	1	no	—
168	C₂×C₈×C₅₉	—	2	1	no	—
169	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
170	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
171	C₂×C₈×C₅₉	#41	1	2	yes	—
172	C₂×C₈×C₅₉	#41	1	2	yes	—
173	C₄⋊C₄×C₅₉	—	2	2	no	—
174	C₄⋊C₄×C₅₉	—	2	2	no	—
175	C₄⋊C₄×C₅₉	—	2	1	no	—
176	C₄⋊C₄×C₅₉	—	2	1	no	—
177	C₄⋊C₄×C₅₉	—	2	1	no	—
178	C₄⋊C₄×C₅₉	—	2	1	no	—
179	C₄⋊C₄×C₅₉	—	2	1	no	—
180	C₄⋊C₄×C₅₉	—	2	1	no	—
181	C₄⋊C₄×C₅₉	—	2	1	no	—
182	C₄⋊C₄×C₅₉	—	2	1	no	—
183	C₄⋊C₄×C₅₉	—	2	1	no	—
184	C₄⋊C₄×C₅₉	—	2	1	no	—
185	C₄⋊C₄×C₅₉	—	4	2	no	—
186	C₄⋊C₄×C₅₉	—	4	2	no	—
187	C₄⋊C₄×C₅₉	—	4	2	no	—
188	C₄⋊C₄×C₅₉	—	4	2	no	—
189	C₄⋊C₄×C₅₉	—	4	2	no	—
190	C₄⋊C₄×C₅₉	—	4	2	no	—
191	C₄⋊C₄×C₅₉	—	4	2	no	—
192	C₄⋊C₄×C₅₉	—	8	1	no	—
193	C₄⋊C₄×C₅₉	—	8	1	no	—
194	C₄⋊C₄×C₅₉	C₂×D₄	1	2	yes	—
195	C₄⋊C₄×C₅₉	C₂×D₄	1	2	yes	—
196	C₂²⋊C₄×C₅₉	—	4	2	no	—
197	C₂²⋊C₄×C₅₉	—	4	2	no	—
198	C₂²⋊C₄×C₅₉	—	8	1	no	—
199	C₂²⋊C₄×C₅₉	—	8	1	no	—
200	C₂²⋊C₄×C₅₉	—	8	1	no	—
201	C₄²×C₅₉	—	2	1	no	—
202	C₄²×C₅₉	—	2	1	no	—
203	C₄²×C₅₉	—	2	1	no	—
204	C₄²×C₅₉	—	2	1	no	—
205	C₄²×C₅₉	—	2	1	no	—
206	C₄²×C₅₉	—	2	1	no	—
207	C₄²×C₅₉	—	4	2	no	—
208	C₂³×C₅₉	—	2	2	no	—
209	C₂³×C₅₉	—	4	1	no	—
210	C₂³×C₅₉	—	4	1	no	—
211	C₂³×C₅₉	—	4	1	no	—
212	C₂³×C₅₉	—	4	1	no	—
213	C₂³×C₅₉	—	4	1	no	—
214	C₂³×C₅₉	—	4	1	no	—
215	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
216	Q₈×C₅₉	—	2	2	no	—
217	Q₈×C₅₉	—	2	2	no	—
218	Q₈×C₅₉	—	4	1	no	—
219	Q₈×C₅₉	—	4	1	no	—
220	D₄×C₅₉	—	2	2	no	—
221	D₄×C₅₉	—	2	2	no	—
222	D₄×C₅₉	—	4	1	no	—
223	D₄×C₅₉	—	4	1	no	—
224	C₂×C₄×C₅₉	—	2	1	no	—
225	C₂×C₄×C₅₉	—	2	1	no	—
226	C₂×C₄×C₅₉	—	2	1	no	—
227	C₂×C₄×C₅₉	—	2	1	no	—
228	C₂×C₄×C₅₉	—	2	1	no	—
229	C₂×C₄×C₅₉	—	2	1	no	—
230	C₂×C₄×C₅₉	—	2	1	no	—
231	C₂×C₄×C₅₉	—	2	1	no	—
232	C₂×C₄×C₅₉	—	2	2	no	—
233	C₂×C₄×C₅₉	—	2	2	no	—
234	C₂×C₄×C₅₉	—	2	2	no	—
235	C₂×C₄×C₅₉	—	2	1	no	—
236	C₂×C₄×C₅₉	—	2	1	no	—
237	C₂×C₄×C₅₉	—	2	1	no	—
238	C₂×C₄×C₅₉	—	2	1	no	—
239	C₂×C₄×C₅₉	—	2	1	no	—
240	C₂×C₄×C₅₉	—	2	1	no	—
241	C₂×C₄×C₅₉	—	2	1	no	—
242	C₂×C₄×C₅₉	—	2	1	no	—
243	C₂×C₄×C₅₉	—	4	1	no	—
244	C₂×C₄×C₅₉	—	4	1	no	—
245	C₂×C₄×C₅₉	—	4	1	no	—
246	C₂×C₄×C₅₉	—	4	1	no	—
247	C₂×C₄×C₅₉	—	4	1	no	—
248	C₂×C₄×C₅₉	—	4	1	no	—
249	C₂×C₄×C₅₉	—	4	1	no	—
250	C₂×C₄×C₅₉	—	4	1	no	—
251	C₂×C₄×C₅₉	—	4	1	no	—
252	C₂×C₄×C₅₉	—	4	1	no	—
253	C₂×C₄×C₅₉	—	4	1	no	—
254	C₂×C₄×C₅₉	—	4	1	no	—
255	C₂×C₄×C₅₉	—	4	1	no	—
256	C₂×C₄×C₅₉	—	4	1	no	—
257	C₂×C₄×C₅₉	—	4	1	no	—
258	C₂×C₄×C₅₉	—	4	1	no	—
259	C₂×C₄×C₅₉	—	4	1	no	—
260	C₂×C₄×C₅₉	—	4	1	no	—
261	C₂×C₄×C₅₉	—	8	4	no	—
262	C₂×C₄×C₅₉	#135	1	2	yes	—
263	C₂×C₄×C₅₉	#135	1	2	yes	—
264	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
265	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
266	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
267	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
268	C₂×C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
269	C₂×C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
270	C₈×C₅₉	—	2	2	no	—
271	C₈×C₅₉	—	4	1	no	—
272	C₈×C₅₉	—	4	1	no	—
273	C₈×C₅₉	C₂²⋊Q₈	1	4	yes	—
274	#26481	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
275	C₂²×C₅₉	—	4	2	no	—
276	C₂²×C₅₉	—	4	2	no	—
277	C₂²×C₅₉	—	4	2	no	—
278	C₂²×C₅₉	—	8	1	no	—
279	C₂²×C₅₉	—	8	1	no	—
280	C₂²×C₅₉	—	8	1	no	—
281	C₂²×C₅₉	—	8	1	no	—
282	C₂²×C₅₉	#532	1	1	yes	characteristic
283	C₂²×C₅₉	D₄²	1	1	yes	characteristic
284	C₂²×C₅₉	D₄×Q₈	1	1	yes	characteristic
285	C₂²×C₅₉	C₂×#135	1	1	yes	characteristic
286	C₂²×C₅₉	C₂×#135	1	1	yes	characteristic
287	C₂²×C₅₉	#577	1	1	yes	characteristic
288	C₂²×C₅₉	#577	1	1	yes	characteristic
289	C₄×C₅₉	—	2	2	no	—
290	C₄×C₅₉	—	2	2	no	—
291	C₄×C₅₉	—	4	1	no	—
292	C₄×C₅₉	—	4	2	no	—
293	C₄×C₅₉	—	4	1	no	—
294	C₄×C₅₉	—	4	2	no	—
295	C₄×C₅₉	—	4	1	no	—
296	C₄×C₅₉	—	4	1	no	—
297	C₄×C₅₉	—	4	2	no	—
298	C₄×C₅₉	—	8	1	no	—
299	C₄×C₅₉	—	8	1	no	—
300	C₄×C₅₉	—	8	1	no	—
301	C₄×C₅₉	—	8	1	no	—
302	C₄×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
303	C₄×C₅₉	#542	1	1	yes	characteristic
304	C₄×C₅₉	#548	1	1	yes	characteristic
305	C₄×C₅₉	#554	1	1	yes	characteristic
306	C₂×#499	C₂×C₅₉	1	1	yes	characteristic, complemented
307	C₂×C₄×SD₁₆	C₂×C₅₉	1	1	yes	characteristic
308	#2454	C₂×C₅₉	1	1	yes	characteristic
309	#2453	C₂×C₅₉	1	1	yes	characteristic, complemented
310	#2068	C₂×C₅₉	1	1	yes	characteristic, complemented
311	#2067	C₂×C₅₉	1	1	yes	characteristic
312	#2026	C₂×C₅₉	1	1	yes	characteristic, complemented
313	#2025	C₂×C₅₉	1	1	yes	characteristic
314	#1968	C₂×C₅₉	1	1	yes	characteristic
315	#1968	C₂×C₅₉	1	1	yes	characteristic
316	#1941	C₂×C₅₉	1	1	yes	characteristic, complemented
317	#1941	C₂×C₅₉	1	1	yes	characteristic, complemented
318	#1863	C₂×C₅₉	1	1	yes	characteristic, complemented
319	#1862	C₂×C₅₉	1	1	yes	characteristic
320	C₈×C₄⋊C₄	C₂×C₅₉	1	1	yes	characteristic, complemented
321	C₂×C₅₉	—	8	2	no	—
322	C₂×C₅₉	#2452	1	1	yes	characteristic
323	C₂×C₅₉	#3168	1	1	yes	characteristic
324	C₂×C₅₉	#3174	1	1	yes	characteristic
325	C₂×C₅₉	#3282	1	1	yes	characteristic
326	C₂×C₅₉	#3284	1	1	yes	characteristic
327	C₂×C₅₉	#3381	1	1	yes	characteristic
328	C₂×C₅₉	#3381	1	1	yes	characteristic
329	C₂²×SD₁₆	C₂²×C₅₉	1	1	yes	characteristic
330	C₂×#125	C₂²×C₅₉	1	1	yes	characteristic
331	C₂×#125	C₂²×C₅₉	1	1	yes	characteristic
332	C₂×C₄×Q₈	C₂²×C₅₉	1	1	yes	characteristic
333	C₂×C₄×D₄	C₂²×C₅₉	1	1	yes	characteristic
334	#499	—	2	4	no	—
335	C₄×SD₁₆	—	2	4	no	—
336	C₂×#107	C₂²×C₅₉	1	1	yes	characteristic
337	C₂×#107	C₂²×C₅₉	1	1	yes	characteristic
338	C₂×#106	C₂²×C₅₉	1	1	yes	characteristic
339	C₂×#106	C₂²×C₅₉	1	1	yes	characteristic
340	C₂×#106	C₂²×C₅₉	1	1	yes	characteristic
341	C₂×C₄⋊C₈	C₂²×C₅₉	1	1	yes	characteristic
342	C₂×#103	C₂²×C₅₉	1	1	yes	characteristic
343	C₂×#103	C₂²×C₅₉	1	1	yes	characteristic
344	C₂×#103	C₂²×C₅₉	1	1	yes	characteristic
345	C₂×#102	C₂²×C₅₉	1	1	yes	characteristic
346	C₂×#102	C₂²×C₅₉	1	1	yes	characteristic
347	C₂×#102	C₂²×C₅₉	1	1	yes	characteristic
348	C₂×C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic
349	C₂×C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic
350	C₂×C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic
351	#400	—	2	1	no	—
352	#399	—	2	1	no	—
353	#398	—	2	1	no	—
354	#396	—	2	1	no	—
355	#391	C₂²×C₅₉	1	1	yes	characteristic
356	#391	C₂²×C₅₉	1	1	yes	characteristic
357	#390	C₂²×C₅₉	1	1	yes	characteristic
358	#390	C₂²×C₅₉	1	1	yes	characteristic
359	#389	—	2	1	no	—
360	#389	C₂²×C₅₉	1	1	yes	characteristic
361	#389	C₂²×C₅₉	1	1	yes	characteristic
362	#389	C₂²×C₅₉	1	1	yes	characteristic
363	#389	C₂²×C₅₉	1	1	yes	characteristic
364	#387	—	2	1	no	—
365	#385	—	2	1	no	—
366	#380	—	2	1	no	—
367	C₄×C₄⋊C₄	C₂²×C₅₉	1	1	yes	characteristic
368	#340	C₂²×C₅₉	1	1	yes	characteristic
369	#340	C₂²×C₅₉	1	1	yes	characteristic
370	#340	C₂²×C₅₉	1	1	yes	characteristic
371	#340	C₂²×C₅₉	1	1	yes	characteristic
372	#336	C₂²×C₅₉	1	1	yes	characteristic
373	#336	C₂²×C₅₉	1	1	yes	characteristic
374	C₅₉	#26481	1	1	yes	characteristic, direct product factor, Sylow
375	C₂²×Q₈	C₂³×C₅₉	1	1	yes	characteristic
376	C₂²×D₄	C₂³×C₅₉	1	1	yes	characteristic
377	C₂³×C₄	—	2	1	no	—
378	C₂×SD₁₆	—	2	2	no	—
379	C₂×SD₁₆	—	2	2	no	—
380	C₂×SD₁₆	Q₈×C₅₉	1	4	yes	—
381	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
382	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
383	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
384	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
385	#125	—	2	2	no	—
386	#125	—	2	2	no	—
387	#125	—	4	1	no	—
388	#125	—	4	1	no	—
389	C₄×Q₈	—	2	2	no	—
390	C₄×Q₈	—	4	1	no	—
391	C₄×D₄	—	2	2	no	—
392	C₄×D₄	—	4	1	no	—
393	C₂×C₄⋊C₄	—	2	1	no	—
394	C₂×C₄⋊C₄	—	2	1	no	—
395	C₂×C₄⋊C₄	—	2	1	no	—
396	C₂×C₄⋊C₄	—	2	1	no	—
397	C₂×C₄⋊C₄	—	2	1	no	—
398	C₂×C₄⋊C₄	—	2	1	no	—
399	C₂×C₄⋊C₄	—	2	1	no	—
400	C₂×C₄⋊C₄	—	2	1	no	—
401	C₂×C₄⋊C₄	—	2	1	no	—
402	C₂×C₄⋊C₄	D₄×C₅₉	1	1	yes	characteristic
403	C₂×C₄⋊C₄	D₄×C₅₉	1	1	yes	characteristic
404	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
405	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
406	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
407	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
408	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
409	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
410	C₂×C₂²⋊C₄	—	2	1	no	—
411	C₂×C₂²⋊C₄	—	2	1	no	—
412	C₂×C₂²⋊C₄	—	2	1	no	—
413	C₂×C₂²⋊C₄	—	2	1	no	—
414	C₂×C₂²⋊C₄	—	2	1	no	—
415	C₂×C₄²	—	2	1	no	—
416	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
417	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
418	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
419	#107	—	2	2	no	—
420	#107	—	2	2	no	—
421	#106	—	2	2	no	—
422	#106	—	2	2	no	—
423	#106	D₄×C₅₉	1	4	yes	—
424	C₄⋊C₈	—	2	2	no	—
425	#103	—	2	2	no	—
426	#103	—	2	2	no	—
427	#103	—	4	1	no	—
428	#102	—	2	2	no	—
429	#102	—	2	2	no	—
430	#102	—	4	1	no	—
431	C₄×C₈	—	2	2	no	—
432	C₄×C₈	—	2	2	no	—
433	C₄×C₈	—	2	2	no	—
434	#95	—	2	1	no	—
435	#95	—	2	1	no	—
436	#95	—	2	1	no	—
437	#95	—	2	1	no	—
438	#95	—	2	1	no	—
439	#95	—	2	1	no	—
440	#95	—	2	1	no	—
441	#95	—	2	1	no	—
442	#95	D₄×C₅₉	1	1	yes	characteristic
443	#95	D₄×C₅₉	1	1	yes	characteristic
444	C₂⁴	—	2	1	no	—
445	C₂×Q₈	—	2	1	no	—
446	C₂×Q₈	—	2	1	no	—
447	C₂×Q₈	—	2	1	no	—
448	C₂×Q₈	—	2	1	no	—
449	C₂×Q₈	—	2	1	no	—
450	C₂×Q₈	C₂×Q₈×C₅₉	1	2	yes	—
451	C₂×D₄	—	2	1	no	—
452	C₂×D₄	—	2	1	no	—
453	C₂×D₄	—	2	1	no	—
454	C₂×D₄	—	2	1	no	—
455	C₂×D₄	—	2	1	no	—
456	C₂×D₄	C₂×Q₈×C₅₉	1	2	yes	—
457	C₂²×C₄	—	2	1	no	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	1	no	—
462	C₂²×C₄	—	2	1	no	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	4	2	no	—
465	C₂²×C₄	—	4	2	no	—
466	C₂²×C₄	—	8	1	no	—
467	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
468	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
469	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
470	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
471	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
472	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
473	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
474	SD₁₆	—	2	8	no	—
475	C₂×C₈	—	2	1	no	—
476	C₂×C₈	—	2	1	no	—
477	C₂×C₈	—	2	1	no	—
478	C₂×C₈	—	2	1	no	—
479	C₂×C₈	—	2	1	no	—
480	C₂×C₈	—	2	1	no	—
481	C₂×C₈	—	2	1	no	—
482	C₂×C₈	—	2	1	no	—
483	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
484	C₂×C₈	C₂×Q₈×C₅₉	1	2	yes	—
485	C₂×C₈	#41×C₅₉	1	2	yes	—
486	C₂×C₈	#41×C₅₉	1	2	yes	—
487	C₄⋊C₄	—	2	2	no	—
488	C₄⋊C₄	—	2	2	no	—
489	C₄⋊C₄	—	2	1	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	—	2	1	no	—
497	C₄⋊C₄	—	2	1	no	—
498	C₄⋊C₄	—	2	1	no	—
499	C₄⋊C₄	—	4	2	no	—
500	C₄⋊C₄	—	4	2	no	—
501	C₄⋊C₄	—	4	2	no	—
502	C₄⋊C₄	—	4	2	no	—
503	C₄⋊C₄	—	4	2	no	—
504	C₄⋊C₄	—	4	2	no	—
505	C₄⋊C₄	—	4	2	no	—
506	C₄⋊C₄	—	8	1	no	—
507	C₄⋊C₄	—	8	1	no	—
508	C₄⋊C₄	C₂×D₄×C₅₉	1	2	yes	—
509	C₄⋊C₄	C₂×D₄×C₅₉	1	2	yes	—
510	C₂²⋊C₄	—	4	2	no	—
511	C₂²⋊C₄	—	4	2	no	—
512	C₂²⋊C₄	—	8	1	no	—
513	C₂²⋊C₄	—	8	1	no	—
514	C₂²⋊C₄	—	8	1	no	—
515	C₄²	—	2	1	no	—
516	C₄²	—	2	1	no	—
517	C₄²	—	2	1	no	—
518	C₄²	—	2	1	no	—
519	C₄²	—	2	1	no	—
520	C₄²	—	2	1	no	—
521	C₄²	—	4	2	no	—
522	C₂³	—	2	2	no	—
523	C₂³	—	4	1	no	—
524	C₂³	—	4	1	no	—
525	C₂³	—	4	1	no	—
526	C₂³	—	4	1	no	—
527	C₂³	—	4	1	no	—
528	C₂³	—	4	1	no	—
529	C₂³	C₂²×D₄×C₅₉	1	1	yes	characteristic
530	Q₈	—	2	2	no	—
531	Q₈	—	2	2	no	—
532	Q₈	—	4	1	no	—
533	Q₈	—	4	1	no	—
534	D₄	—	2	2	no	—
535	D₄	—	2	2	no	—
536	D₄	—	4	1	no	—
537	D₄	—	4	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	1	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	—	4	1	no	—
561	C₂×C₄	—	4	1	no	—
562	C₂×C₄	—	4	1	no	—
563	C₂×C₄	—	4	1	no	—
564	C₂×C₄	—	4	1	no	—
565	C₂×C₄	—	4	1	no	—
566	C₂×C₄	—	4	1	no	—
567	C₂×C₄	—	4	1	no	—
568	C₂×C₄	—	4	1	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	—	4	1	no	—
571	C₂×C₄	—	4	1	no	—
572	C₂×C₄	—	4	1	no	—
573	C₂×C₄	—	4	1	no	—
574	C₂×C₄	—	4	1	no	—
575	C₂×C₄	—	8	4	no	—
576	C₂×C₄	#135×C₅₉	1	2	yes	—
577	C₂×C₄	#135×C₅₉	1	2	yes	—
578	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
579	C₂×C₄	C₂²×Q₈×C₅₉	1	1	yes	characteristic
580	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
581	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
582	C₂×C₄	H(5,𝔽₂)×C₅₉	1	1	yes	characteristic
583	C₂×C₄	2₋¹⁺⁴×C₅₉	1	1	yes	characteristic
584	C₈	—	2	2	no	—
585	C₈	—	4	1	no	—
586	C₈	—	4	1	no	—
587	C₈	C₂²⋊Q₈×C₅₉	1	4	yes	—
588	C₂²	—	4	2	no	—
589	C₂²	—	4	2	no	—
590	C₂²	—	4	2	no	—
591	C₂²	—	8	1	no	—
592	C₂²	—	8	1	no	—
593	C₂²	—	8	1	no	—
594	C₂²	—	8	1	no	—
595	C₂²	C₅₉×#532	1	1	yes	characteristic
596	C₂²	D₄²×C₅₉	1	1	yes	characteristic
597	C₂²	D₄×Q₈×C₅₉	1	1	yes	characteristic
598	C₂²	C₂×#135×C₅₉	1	1	yes	characteristic
599	C₂²	C₂×#135×C₅₉	1	1	yes	characteristic
600	C₂²	C₅₉×#577	1	1	yes	characteristic
601	C₂²	C₅₉×#577	1	1	yes	characteristic
602	C₄	—	2	2	no	—
603	C₄	—	2	2	no	—
604	C₄	—	4	1	no	—
605	C₄	—	4	2	no	—
606	C₄	—	4	1	no	—
607	C₄	—	4	1	no	—
608	C₄	—	4	2	no	—
609	C₄	—	4	1	no	—
610	C₄	—	4	2	no	—
611	C₄	—	8	1	no	—
612	C₄	—	8	1	no	—
613	C₄	—	8	1	no	—
614	C₄	—	8	1	no	—
615	C₄	C₂×C₂²⋊Q₈×C₅₉	1	1	yes	characteristic
616	C₄	C₅₉×#542	1	1	yes	characteristic
617	C₄	C₅₉×#548	1	1	yes	characteristic
618	C₄	C₅₉×#554	1	1	yes	characteristic
619	C₂	—	8	2	no	—
620	C₂	C₅₉×#2452	1	1	yes	characteristic
621	C₂	C₅₉×#3168	1	1	yes	characteristic
622	C₂	C₅₉×#3174	1	1	yes	characteristic
623	C₂	C₅₉×#3282	1	1	yes	characteristic
624	C₂	C₅₉×#3284	1	1	yes	characteristic
625	C₂	C₅₉×#3381	1	1	yes	characteristic
626	C₂	C₅₉×#3381	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₉
prime radical	C₂²×D₄×C₅₉	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₉ ◁ C₄×C₄⋊C₄×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₉×#499	C₂	semidirect
4. C₅₉×#2453	C₂	semidirect
5. C₅₉×#2068	C₂	semidirect
7. C₅₉×#2026	C₂	semidirect
11. C₅₉×#1941	C₂	semidirect
12. C₅₉×#1941	C₂	semidirect
13. C₅₉×#1863	C₂	semidirect
15. C₈×C₄⋊C₄×C₅₉	C₂	semidirect
274. #26481	C₅₉	direct (N × H)
306. C₂×#499	C₂×C₅₉	semidirect
309. #2453	C₂×C₅₉	semidirect
310. #2068	C₂×C₅₉	semidirect
312. #2026	C₂×C₅₉	semidirect
316. #1941	C₂×C₅₉	semidirect
317. #1941	C₂×C₅₉	semidirect
318. #1863	C₂×C₅₉	semidirect
320. C₈×C₄⋊C₄	C₂×C₅₉	semidirect
374. C₅₉	#26481	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₉×#499	2
C₂×C₄×SD₁₆×C₅₉	2
C₅₉×#2454	2
C₅₉×#2453	2
C₅₉×#2068	2
C₅₉×#2067	2
C₅₉×#2026	2
C₅₉×#2025	2
C₅₉×#1968	2
C₅₉×#1941	2
C₅₉×#1863	2
C₅₉×#1862	2
C₈×C₄⋊C₄×C₅₉	2
#26481	59

Subgroups — C59×#26481 #28438590

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#26481 #28438590