Subgroups — C₅₉×#26413 #28438522

Order 15104.

746 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#2439	C₂	1	1	yes	maximal, characteristic
2	C₅₉×#2362	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#2101	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₉×#2087	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#2068	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₉×#2024	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1988	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₉×#1920	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₉×#1892	C₂	1	1	yes	maximal, characteristic
10	C₅₉×#1873	C₂	1	1	yes	maximal, characteristic
11	C₅₉×#1865	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₉×#1854	C₂	1	1	yes	maximal, characteristic
13	C₅₉×#1850	C₂	1	1	yes	maximal, characteristic
14	C₅₉×#1805	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#1735	C₂	1	1	yes	maximal, characteristic
16	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×#117×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×#117×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×#106×C₅₉	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
26	C₂×#103×C₅₉	C₂²	1	1	yes	characteristic
27	C₂×#103×C₅₉	C₂²	1	1	yes	characteristic
28	C₂×#103×C₅₉	C₂²	1	1	yes	characteristic
29	C₂×#103×C₅₉	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
31	C₅₉×#396	—	2	1	no	—
32	C₅₉×#393	—	2	1	no	—
33	C₅₉×#391	—	2	1	no	—
34	C₅₉×#391	C₂²	1	1	yes	characteristic
35	C₅₉×#389	C₂²	1	1	yes	characteristic
36	C₅₉×#389	C₂²	1	1	yes	characteristic
37	C₅₉×#389	C₂²	1	1	yes	characteristic
38	C₅₉×#388	—	2	1	no	—
39	C₅₉×#388	—	2	1	no	—
40	C₅₉×#387	—	2	1	no	—
41	C₅₉×#387	—	2	1	no	—
42	C₅₉×#382	C₂²	1	1	yes	characteristic
43	C₅₉×#380	C₂²	1	1	yes	characteristic
44	C₄×C₄⋊C₄×C₅₉	—	2	1	no	—
45	C₄×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
46	C₄×C₂²⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
47	C₅₉×#340	C₂²	1	1	yes	characteristic
48	C₅₉×#340	C₂²	1	1	yes	characteristic
49	C₅₉×#340	C₂²	1	1	yes	characteristic
50	C₅₉×#340	C₂²	1	1	yes	characteristic
51	C₅₉×#340	C₂²	1	1	yes	characteristic
52	C₅₉×#340	C₂²	1	1	yes	characteristic
53	C₅₉×#340	C₂²	1	1	yes	characteristic
54	C₅₉×#340	C₂²	1	1	yes	characteristic
55	C₅₉×#340	C₂²	1	1	yes	characteristic
56	C₅₉×#340	C₂²	1	1	yes	characteristic
57	C₅₉×#336	C₂²	1	1	yes	characteristic
58	C₅₉×#336	C₂²	1	1	yes	characteristic
59	C₂²×Q₈×C₅₉	—	2	1	no	—
60	C₂²×Q₈×C₅₉	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₅₉	—	2	2	no	—
63	C₂²×C₈×C₅₉	—	4	1	no	—
64	C₂²×C₈×C₅₉	—	4	1	no	—
65	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
69	C₄⋊Q₈×C₅₉	—	2	2	no	—
70	C₄⋊Q₈×C₅₉	—	4	1	no	—
71	C₂²⋊Q₈×C₅₉	—	2	2	no	—
72	C₂²⋊Q₈×C₅₉	Q₈	1	4	yes	complemented
73	C₄×Q₈×C₅₉	—	2	2	no	—
74	C₄×Q₈×C₅₉	—	4	1	no	—
75	#117×C₅₉	—	2	2	no	—
76	#117×C₅₉	—	2	2	no	—
77	#117×C₅₉	—	4	1	no	—
78	#117×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	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₅₉	C₂³	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×C₂²⋊C₄×C₅₉	—	2	1	no	—
93	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
94	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
95	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
96	C₂×C₂²⋊C₄×C₅₉	D₄	1	1	yes	characteristic
97	C₂×C₂²⋊C₄×C₅₉	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₅₉	—	2	1	no	—
99	C₂×C₄²×C₅₉	—	2	1	no	—
100	C₂×C₄²×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	#107×C₅₉	—	4	1	no	—
105	#106×C₅₉	—	2	2	no	—
106	C₄⋊C₈×C₅₉	—	4	1	no	—
107	C₄⋊C₈×C₅₉	—	4	1	no	—
108	#103×C₅₉	—	2	2	no	—
109	#103×C₅₉	—	4	1	no	—
110	#103×C₅₉	—	4	1	no	—
111	#103×C₅₉	—	4	1	no	—
112	C₂²⋊C₈×C₅₉	—	2	2	no	—
113	#95×C₅₉	—	2	1	no	—
114	#95×C₅₉	—	2	1	no	—
115	#95×C₅₉	—	2	1	no	—
116	#95×C₅₉	—	2	1	no	—
117	#95×C₅₉	—	2	1	no	—
118	#95×C₅₉	—	2	1	no	—
119	#95×C₅₉	—	2	1	no	—
120	#95×C₅₉	—	2	1	no	—
121	#95×C₅₉	—	2	1	no	—
122	#95×C₅₉	D₄	1	1	yes	characteristic
123	#95×C₅₉	D₄	1	1	yes	characteristic
124	C₂⁴×C₅₉	C₂×D₄	1	1	yes	characteristic
125	C₂×Q₈×C₅₉	—	2	1	no	—
126	C₂×Q₈×C₅₉	—	2	1	no	—
127	C₂×Q₈×C₅₉	—	2	1	no	—
128	C₂×Q₈×C₅₉	—	2	1	no	—
129	C₂×Q₈×C₅₉	—	2	1	no	—
130	C₂×Q₈×C₅₉	—	4	2	no	—
131	C₂×Q₈×C₅₉	—	4	2	no	—
132	C₂×Q₈×C₅₉	—	8	1	no	—
133	C₂×Q₈×C₅₉	C₂×Q₈	1	2	yes	—
134	C₂²×C₄×C₅₉	—	2	1	no	—
135	C₂²×C₄×C₅₉	—	2	1	no	—
136	C₂²×C₄×C₅₉	—	2	1	no	—
137	C₂²×C₄×C₅₉	—	2	1	no	—
138	C₂²×C₄×C₅₉	—	2	1	no	—
139	C₂²×C₄×C₅₉	—	2	1	no	—
140	C₂²×C₄×C₅₉	—	2	1	no	—
141	C₂²×C₄×C₅₉	—	2	1	no	—
142	C₂²×C₄×C₅₉	—	2	1	no	—
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	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₅₉	C₂×Q₈	1	2	yes	—
152	C₂²×C₄×C₅₉	#41	1	2	yes	—
153	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
154	C₂×C₈×C₅₉	—	2	1	no	—
155	C₂×C₈×C₅₉	—	2	1	no	—
156	C₂×C₈×C₅₉	—	2	1	no	—
157	C₂×C₈×C₅₉	—	2	1	no	—
158	C₂×C₈×C₅₉	—	2	1	no	—
159	C₂×C₈×C₅₉	—	2	1	no	—
160	C₂×C₈×C₅₉	—	2	1	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	—	4	4	no	—
167	C₄⋊C₄×C₅₉	—	2	4	no	—
168	C₄⋊C₄×C₅₉	—	2	1	no	—
169	C₄⋊C₄×C₅₉	—	2	1	no	—
170	C₄⋊C₄×C₅₉	—	2	1	no	—
171	C₄⋊C₄×C₅₉	—	2	1	no	—
172	C₄⋊C₄×C₅₉	—	2	1	no	—
173	C₄⋊C₄×C₅₉	—	2	1	no	—
174	C₄⋊C₄×C₅₉	—	2	1	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	—	4	2	no	—
180	C₄⋊C₄×C₅₉	—	4	2	no	—
181	C₄⋊C₄×C₅₉	—	4	2	no	—
182	C₄⋊C₄×C₅₉	—	4	2	no	—
183	C₄⋊C₄×C₅₉	—	8	1	no	—
184	C₄⋊C₄×C₅₉	C₂×Q₈	1	2	yes	—
185	C₄⋊C₄×C₅₉	#41	1	2	yes	—
186	C₄⋊C₄×C₅₉	#41	1	2	yes	—
187	C₂²⋊C₄×C₅₉	—	2	4	no	—
188	C₂²⋊C₄×C₅₉	—	2	2	no	—
189	C₂²⋊C₄×C₅₉	—	4	1	no	—
190	C₂²⋊C₄×C₅₉	—	4	2	no	—
191	C₂²⋊C₄×C₅₉	—	4	2	no	—
192	C₂²⋊C₄×C₅₉	—	8	1	no	—
193	C₄²×C₅₉	—	2	1	no	—
194	C₄²×C₅₉	—	2	1	no	—
195	C₄²×C₅₉	—	2	1	no	—
196	C₄²×C₅₉	—	2	1	no	—
197	C₄²×C₅₉	—	4	2	no	—
198	C₄²×C₅₉	—	4	2	no	—
199	C₄²×C₅₉	—	4	2	no	—
200	C₄²×C₅₉	—	4	2	no	—
201	C₂³×C₅₉	—	2	1	no	—
202	C₂³×C₅₉	—	2	1	no	—
203	C₂³×C₅₉	—	2	1	no	—
204	C₂³×C₅₉	—	2	1	no	—
205	C₂³×C₅₉	—	2	1	no	—
206	C₂³×C₅₉	—	2	1	no	—
207	C₂³×C₅₉	C₂²⋊Q₈	1	2	yes	—
208	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
209	Q₈×C₅₉	—	2	2	no	—
210	Q₈×C₅₉	—	2	2	no	—
211	Q₈×C₅₉	—	4	1	no	—
212	Q₈×C₅₉	—	4	1	no	—
213	Q₈×C₅₉	—	8	4	no	—
214	C₂×C₄×C₅₉	—	2	1	no	—
215	C₂×C₄×C₅₉	—	2	1	no	—
216	C₂×C₄×C₅₉	—	2	1	no	—
217	C₂×C₄×C₅₉	—	2	1	no	—
218	C₂×C₄×C₅₉	—	2	2	no	—
219	C₂×C₄×C₅₉	—	2	2	no	—
220	C₂×C₄×C₅₉	—	2	2	no	—
221	C₂×C₄×C₅₉	—	2	2	no	—
222	C₂×C₄×C₅₉	—	2	2	no	—
223	C₂×C₄×C₅₉	—	2	2	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	2	no	—
228	C₂×C₄×C₅₉	—	2	2	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	1	no	—
233	C₂×C₄×C₅₉	—	2	1	no	—
234	C₂×C₄×C₅₉	—	2	1	no	—
235	C₂×C₄×C₅₉	—	4	1	no	—
236	C₂×C₄×C₅₉	—	4	1	no	—
237	C₂×C₄×C₅₉	—	4	1	no	—
238	C₂×C₄×C₅₉	—	4	1	no	—
239	C₂×C₄×C₅₉	—	4	1	no	—
240	C₂×C₄×C₅₉	—	4	1	no	—
241	C₂×C₄×C₅₉	—	4	1	no	—
242	C₂×C₄×C₅₉	—	4	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	C₂²⋊Q₈	1	2	yes	—
256	C₂×C₄×C₅₉	#135	1	2	yes	—
257	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
258	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
259	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
260	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
261	C₂×C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
262	C₂×C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
263	C₈×C₅₉	—	4	1	no	—
264	C₈×C₅₉	—	4	1	no	—
265	C₈×C₅₉	—	4	1	no	—
266	C₈×C₅₉	—	4	1	no	—
267	#26413	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
268	C₂²×C₅₉	—	2	2	no	—
269	C₂²×C₅₉	—	2	2	no	—
270	C₂²×C₅₉	—	2	2	no	—
271	C₂²×C₅₉	—	4	1	no	—
272	C₂²×C₅₉	—	4	1	no	—
273	C₂²×C₅₉	—	4	1	no	—
274	C₂²×C₅₉	—	4	1	no	—
275	C₂²×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
276	C₂²×C₅₉	#546	1	1	yes	characteristic
277	C₂²×C₅₉	#546	1	1	yes	characteristic
278	C₂²×C₅₉	C₂×#135	1	1	yes	characteristic
279	C₂²×C₅₉	#575	1	1	yes	characteristic
280	C₂²×C₅₉	#577	1	1	yes	characteristic
281	C₂²×C₅₉	#578	1	1	yes	characteristic
282	C₄×C₅₉	—	2	2	no	—
283	C₄×C₅₉	—	2	2	no	—
284	C₄×C₅₉	—	4	2	no	—
285	C₄×C₅₉	—	4	2	no	—
286	C₄×C₅₉	—	4	2	no	—
287	C₄×C₅₉	—	4	1	no	—
288	C₄×C₅₉	—	4	2	no	—
289	C₄×C₅₉	—	4	1	no	—
290	C₄×C₅₉	—	4	1	no	—
291	C₄×C₅₉	—	8	1	no	—
292	C₄×C₅₉	—	8	1	no	—
293	C₄×C₅₉	—	8	1	no	—
294	C₄×C₅₉	—	8	1	no	—
295	C₄×C₅₉	#529	1	1	yes	characteristic
296	C₄×C₅₉	#538	1	1	yes	characteristic
297	C₄×C₅₉	#554	1	1	yes	characteristic
298	C₄×C₅₉	#554	1	1	yes	characteristic
299	#2439	C₂×C₅₉	1	1	yes	characteristic
300	#2362	C₂×C₅₉	1	1	yes	characteristic
301	#2101	C₂×C₅₉	1	1	yes	characteristic, complemented
302	#2087	C₂×C₅₉	1	1	yes	characteristic, complemented
303	#2068	C₂×C₅₉	1	1	yes	characteristic, complemented
304	#2024	C₂×C₅₉	1	1	yes	characteristic, complemented
305	#1988	C₂×C₅₉	1	1	yes	characteristic, complemented
306	#1920	C₂×C₅₉	1	1	yes	characteristic, complemented
307	#1892	C₂×C₅₉	1	1	yes	characteristic
308	#1873	C₂×C₅₉	1	1	yes	characteristic
309	#1865	C₂×C₅₉	1	1	yes	characteristic, complemented
310	#1854	C₂×C₅₉	1	1	yes	characteristic
311	#1850	C₂×C₅₉	1	1	yes	characteristic
312	#1805	C₂×C₅₉	1	1	yes	characteristic, complemented
313	#1735	C₂×C₅₉	1	1	yes	characteristic
314	C₂×C₅₉	—	4	2	no	—
315	C₂×C₅₉	#2450	1	1	yes	characteristic
316	C₂×C₅₉	#3076	1	1	yes	characteristic
317	C₂×C₅₉	#3082	1	1	yes	characteristic
318	C₂×C₅₉	#3296	1	1	yes	characteristic
319	C₂×C₅₉	#3302	1	1	yes	characteristic
320	C₂×C₅₉	#3313	1	1	yes	characteristic
321	C₂×C₅₉	#3321	1	1	yes	characteristic
322	C₂³×C₈	C₂²×C₅₉	1	1	yes	characteristic
323	C₂×C₄⋊Q₈	C₂²×C₅₉	1	1	yes	characteristic
324	C₂×C₂²⋊Q₈	C₂²×C₅₉	1	1	yes	characteristic
325	C₂×C₄×Q₈	C₂²×C₅₉	1	1	yes	characteristic
326	C₂×#117	C₂²×C₅₉	1	1	yes	characteristic
327	C₂×#117	C₂²×C₅₉	1	1	yes	characteristic
328	C₂×#107	C₂²×C₅₉	1	1	yes	characteristic
329	C₂×#106	C₂²×C₅₉	1	1	yes	characteristic
330	C₂×C₄⋊C₈	C₂²×C₅₉	1	1	yes	characteristic
331	C₂×C₄⋊C₈	C₂²×C₅₉	1	1	yes	characteristic
332	C₂×#103	C₂²×C₅₉	1	1	yes	characteristic
333	C₂×#103	C₂²×C₅₉	1	1	yes	characteristic
334	C₂×#103	C₂²×C₅₉	1	1	yes	characteristic
335	C₂×#103	C₂²×C₅₉	1	1	yes	characteristic
336	C₂×C₂²⋊C₈	C₂²×C₅₉	1	1	yes	characteristic
337	#396	—	2	1	no	—
338	#393	—	2	1	no	—
339	#391	—	2	1	no	—
340	#391	C₂²×C₅₉	1	1	yes	characteristic
341	#389	C₂²×C₅₉	1	1	yes	characteristic
342	#389	C₂²×C₅₉	1	1	yes	characteristic
343	#389	C₂²×C₅₉	1	1	yes	characteristic
344	#388	—	2	1	no	—
345	#388	—	2	1	no	—
346	#387	—	2	1	no	—
347	#387	—	2	1	no	—
348	#382	C₂²×C₅₉	1	1	yes	characteristic
349	#380	C₂²×C₅₉	1	1	yes	characteristic
350	C₄×C₄⋊C₄	—	2	1	no	—
351	C₄×C₄⋊C₄	C₂²×C₅₉	1	1	yes	characteristic
352	C₄×C₂²⋊C₄	C₂²×C₅₉	1	1	yes	characteristic
353	#340	C₂²×C₅₉	1	1	yes	characteristic
354	#340	C₂²×C₅₉	1	1	yes	characteristic
355	#340	C₂²×C₅₉	1	1	yes	characteristic
356	#340	C₂²×C₅₉	1	1	yes	characteristic
357	#340	C₂²×C₅₉	1	1	yes	characteristic
358	#340	C₂²×C₅₉	1	1	yes	characteristic
359	#340	C₂²×C₅₉	1	1	yes	characteristic
360	#340	C₂²×C₅₉	1	1	yes	characteristic
361	#340	C₂²×C₅₉	1	1	yes	characteristic
362	#340	C₂²×C₅₉	1	1	yes	characteristic
363	#336	C₂²×C₅₉	1	1	yes	characteristic
364	#336	C₂²×C₅₉	1	1	yes	characteristic
365	C₅₉	#26413	1	1	yes	characteristic, direct product factor, Sylow
366	C₂²×Q₈	—	2	1	no	—
367	C₂²×Q₈	C₂³×C₅₉	1	1	yes	characteristic
368	C₂³×C₄	C₂³×C₅₉	1	1	yes	characteristic
369	C₂²×C₈	—	2	2	no	—
370	C₂²×C₈	—	4	1	no	—
371	C₂²×C₈	—	4	1	no	—
372	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
373	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
374	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
375	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
376	C₄⋊Q₈	—	2	2	no	—
377	C₄⋊Q₈	—	4	1	no	—
378	C₂²⋊Q₈	—	2	2	no	—
379	C₂²⋊Q₈	Q₈×C₅₉	1	4	yes	complemented
380	C₄×Q₈	—	2	2	no	—
381	C₄×Q₈	—	4	1	no	—
382	#117	—	2	2	no	—
383	#117	—	2	2	no	—
384	#117	—	4	1	no	—
385	#117	—	4	1	no	—
386	C₂×C₄⋊C₄	—	2	1	no	—
387	C₂×C₄⋊C₄	—	2	1	no	—
388	C₂×C₄⋊C₄	—	2	1	no	—
389	C₂×C₄⋊C₄	—	2	1	no	—
390	C₂×C₄⋊C₄	—	2	1	no	—
391	C₂×C₄⋊C₄	—	2	1	no	—
392	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
393	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
394	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
395	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
396	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
397	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
398	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
399	C₂×C₂²⋊C₄	—	2	1	no	—
400	C₂×C₂²⋊C₄	—	2	1	no	—
401	C₂×C₂²⋊C₄	—	2	1	no	—
402	C₂×C₂²⋊C₄	—	2	1	no	—
403	C₂×C₂²⋊C₄	D₄×C₅₉	1	1	yes	characteristic
404	C₂×C₂²⋊C₄	D₄×C₅₉	1	1	yes	characteristic
405	C₂×C₄²	—	2	1	no	—
406	C₂×C₄²	—	2	1	no	—
407	C₂×C₄²	—	2	1	no	—
408	C₂×C₄²	—	2	1	no	—
409	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
410	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
411	#107	—	4	1	no	—
412	#106	—	2	2	no	—
413	C₄⋊C₈	—	4	1	no	—
414	C₄⋊C₈	—	4	1	no	—
415	#103	—	2	2	no	—
416	#103	—	4	1	no	—
417	#103	—	4	1	no	—
418	#103	—	4	1	no	—
419	C₂²⋊C₈	—	2	2	no	—
420	#95	—	2	1	no	—
421	#95	—	2	1	no	—
422	#95	—	2	1	no	—
423	#95	—	2	1	no	—
424	#95	—	2	1	no	—
425	#95	—	2	1	no	—
426	#95	—	2	1	no	—
427	#95	—	2	1	no	—
428	#95	—	2	1	no	—
429	#95	D₄×C₅₉	1	1	yes	characteristic
430	#95	D₄×C₅₉	1	1	yes	characteristic
431	C₂⁴	C₂×D₄×C₅₉	1	1	yes	characteristic
432	C₂×Q₈	—	2	1	no	—
433	C₂×Q₈	—	2	1	no	—
434	C₂×Q₈	—	2	1	no	—
435	C₂×Q₈	—	2	1	no	—
436	C₂×Q₈	—	2	1	no	—
437	C₂×Q₈	—	4	2	no	—
438	C₂×Q₈	—	4	2	no	—
439	C₂×Q₈	—	8	1	no	—
440	C₂×Q₈	C₂×Q₈×C₅₉	1	2	yes	—
441	C₂²×C₄	—	2	1	no	—
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	2	1	no	—
445	C₂²×C₄	—	2	1	no	—
446	C₂²×C₄	—	2	1	no	—
447	C₂²×C₄	—	2	1	no	—
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	2	1	no	—
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	—	2	1	no	—
452	C₂²×C₄	—	2	1	no	—
453	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
454	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
455	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
456	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
457	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
458	C₂²×C₄	C₂×Q₈×C₅₉	1	2	yes	—
459	C₂²×C₄	#41×C₅₉	1	2	yes	—
460	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	—	2	1	no	—
468	C₂×C₈	—	2	1	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	2	1	no	—
473	C₂×C₈	—	4	4	no	—
474	C₄⋊C₄	—	2	4	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	1	no	—
486	C₄⋊C₄	—	4	2	no	—
487	C₄⋊C₄	—	4	2	no	—
488	C₄⋊C₄	—	4	2	no	—
489	C₄⋊C₄	—	4	2	no	—
490	C₄⋊C₄	—	8	1	no	—
491	C₄⋊C₄	C₂×Q₈×C₅₉	1	2	yes	—
492	C₄⋊C₄	#41×C₅₉	1	2	yes	—
493	C₄⋊C₄	#41×C₅₉	1	2	yes	—
494	C₂²⋊C₄	—	2	4	no	—
495	C₂²⋊C₄	—	2	2	no	—
496	C₂²⋊C₄	—	4	1	no	—
497	C₂²⋊C₄	—	4	2	no	—
498	C₂²⋊C₄	—	4	2	no	—
499	C₂²⋊C₄	—	8	1	no	—
500	C₄²	—	2	1	no	—
501	C₄²	—	2	1	no	—
502	C₄²	—	2	1	no	—
503	C₄²	—	2	1	no	—
504	C₄²	—	4	2	no	—
505	C₄²	—	4	2	no	—
506	C₄²	—	4	2	no	—
507	C₄²	—	4	2	no	—
508	C₂³	—	2	1	no	—
509	C₂³	—	2	1	no	—
510	C₂³	—	2	1	no	—
511	C₂³	—	2	1	no	—
512	C₂³	—	2	1	no	—
513	C₂³	—	2	1	no	—
514	C₂³	C₂²⋊Q₈×C₅₉	1	2	yes	—
515	C₂³	C₂²×D₄×C₅₉	1	1	yes	characteristic
516	Q₈	—	2	2	no	—
517	Q₈	—	2	2	no	—
518	Q₈	—	4	1	no	—
519	Q₈	—	4	1	no	—
520	Q₈	—	8	4	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	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	—	4	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	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	C₂²⋊Q₈×C₅₉	1	2	yes	—
563	C₂×C₄	#135×C₅₉	1	2	yes	—
564	C₂×C₄	C₂²×Q₈×C₅₉	1	1	yes	characteristic
565	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
566	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
567	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
568	C₂×C₄	H(5,𝔽₂)×C₅₉	1	1	yes	characteristic
569	C₂×C₄	H(5,𝔽₂)×C₅₉	1	1	yes	characteristic
570	C₈	—	4	1	no	—
571	C₈	—	4	1	no	—
572	C₈	—	4	1	no	—
573	C₈	—	4	1	no	—
574	C₂²	—	2	2	no	—
575	C₂²	—	2	2	no	—
576	C₂²	—	2	2	no	—
577	C₂²	—	4	1	no	—
578	C₂²	—	4	1	no	—
579	C₂²	—	4	1	no	—
580	C₂²	—	4	1	no	—
581	C₂²	C₂×C₂²⋊Q₈×C₅₉	1	1	yes	characteristic
582	C₂²	C₅₉×#546	1	1	yes	characteristic
583	C₂²	C₅₉×#546	1	1	yes	characteristic
584	C₂²	C₂×#135×C₅₉	1	1	yes	characteristic
585	C₂²	C₅₉×#575	1	1	yes	characteristic
586	C₂²	C₅₉×#577	1	1	yes	characteristic
587	C₂²	C₅₉×#578	1	1	yes	characteristic
588	C₄	—	2	2	no	—
589	C₄	—	2	2	no	—
590	C₄	—	4	2	no	—
591	C₄	—	4	1	no	—
592	C₄	—	4	2	no	—
593	C₄	—	4	1	no	—
594	C₄	—	4	1	no	—
595	C₄	—	4	2	no	—
596	C₄	—	4	2	no	—
597	C₄	—	8	1	no	—
598	C₄	—	8	1	no	—
599	C₄	—	8	1	no	—
600	C₄	—	8	1	no	—
601	C₄	C₅₉×#529	1	1	yes	characteristic
602	C₄	C₅₉×#538	1	1	yes	characteristic
603	C₄	C₅₉×#554	1	1	yes	characteristic
604	C₄	C₅₉×#554	1	1	yes	characteristic
605	C₂	—	4	2	no	—
606	C₂	C₅₉×#2450	1	1	yes	characteristic
607	C₂	C₅₉×#3076	1	1	yes	characteristic
608	C₂	C₅₉×#3082	1	1	yes	characteristic
609	C₂	C₅₉×#3296	1	1	yes	characteristic
610	C₂	C₅₉×#3302	1	1	yes	characteristic
611	C₂	C₅₉×#3313	1	1	yes	characteristic
612	C₂	C₅₉×#3321	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₅₉×#2101	C₂	semidirect
4. C₅₉×#2087	C₂	semidirect
5. C₅₉×#2068	C₂	semidirect
6. C₅₉×#2024	C₂	semidirect
7. C₅₉×#1988	C₂	semidirect
8. C₅₉×#1920	C₂	semidirect
11. C₅₉×#1865	C₂	semidirect
14. C₅₉×#1805	C₂	semidirect
72. C₂²⋊Q₈×C₅₉	Q₈	semidirect
267. #26413	C₅₉	direct (N × H)
301. #2101	C₂×C₅₉	semidirect
302. #2087	C₂×C₅₉	semidirect
303. #2068	C₂×C₅₉	semidirect
304. #2024	C₂×C₅₉	semidirect
305. #1988	C₂×C₅₉	semidirect
306. #1920	C₂×C₅₉	semidirect
309. #1865	C₂×C₅₉	semidirect
312. #1805	C₂×C₅₉	semidirect
365. C₅₉	#26413	direct (N × H)
379. C₂²⋊Q₈	Q₈×C₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₅₉×#2439	2
C₅₉×#2362	2
C₅₉×#2101	2
C₅₉×#2087	2
C₅₉×#2068	2
C₅₉×#2024	2
C₅₉×#1988	2
C₅₉×#1920	2
C₅₉×#1892	2
C₅₉×#1873	2
C₅₉×#1865	2
C₅₉×#1854	2
C₅₉×#1850	2
C₅₉×#1805	2
C₅₉×#1735	2
#26413	59

Subgroups — C59×#26413 #28438522

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#26413 #28438522