Subgroups — C₅₉×#17612 #28429721

Order 15104.

930 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#3495	C₂	1	1	yes	maximal, characteristic
2	C₅₉×#2924	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₉×#2347	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₉×#1521	C₂	1	2	yes	maximal, complemented
5	C₅₉×#1520	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₉×#1515	C₂	1	2	yes	maximal, complemented
7	C₅₉×#1513	C₂	1	1	yes	maximal, characteristic
8	C₅₉×#1509	C₂	1	2	yes	maximal, complemented
9	C₅₉×#1508	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₉×#1503	C₂	1	2	yes	maximal, complemented
11	C₅₉×#1501	C₂	1	1	yes	maximal, characteristic
12	C₅₉×#556	—	2	1	no	—
13	C₅₉×#554	—	2	1	no	—
14	C₅₉×#552	—	2	2	no	—
15	C₅₉×#552	—	2	1	no	—
16	C₅₉×#552	—	2	1	no	—
17	C₅₉×#551	—	2	2	no	—
18	C₅₉×#544	C₄	1	1	yes	characteristic, complemented
19	C₅₉×#542	C₄	1	2	yes	complemented
20	C₅₉×#541	C₄	1	2	yes	complemented
21	C₅₉×#536	C₄	1	1	yes	characteristic, complemented
22	C₅₉×#533	C₂²	1	2	yes	—
23	C₅₉×#529	C₂²	1	2	yes	—
24	C₅₉×#525	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
26	C₅₉×#520	C₄	1	1	yes	characteristic, complemented
27	C₅₉×#519	C₄	1	1	yes	characteristic, complemented
28	C₂×#117×C₅₉	C₂²	1	1	yes	characteristic
29	C₅₉×#433	—	2	2	no	—
30	C₅₉×#432	C₂²	1	2	yes	—
31	C₅₉×#431	C₂²	1	2	yes	—
32	C₅₉×#424	—	2	1	no	—
33	C₅₉×#407	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
35	C₅₉×#405	—	2	1	no	—
36	C₅₉×#389	—	2	1	no	—
37	C₅₉×#388	—	2	2	no	—
38	C₅₉×#383	C₂²	1	2	yes	—
39	C₅₉×#380	C₂²	1	1	yes	characteristic
40	C₄×C₄⋊C₄×C₅₉	—	2	1	no	—
41	C₄×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
42	C₄×C₂²⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
43	C₅₉×#328	C₂²	1	1	yes	characteristic
44	C₅₉×#328	C₂²	1	1	yes	characteristic
45	C₅₉×#328	C₂²	1	2	yes	—
46	C₅₉×#328	C₂²	1	2	yes	—
47	C₅₉×#328	C₂²	1	2	yes	—
48	C₅₉×#328	C₂²	1	2	yes	—
49	C₅₉×#328	C₂²	1	2	yes	—
50	C₅₉×#327	C₂²	1	1	yes	characteristic
51	C₅₉×#327	C₂²	1	1	yes	characteristic
52	C₅₉×#327	C₂²	1	2	yes	—
53	C₂×#41×C₅₉	C₂×C₄	1	1	yes	characteristic
54	C₂²×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
55	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
56	C₂×M₁₆×C₅₉	—	2	1	no	—
57	C₂²×C₈×C₅₉	—	2	1	no	—
58	C₄⋊Q₈×C₅₉	—	2	1	no	—
59	C₄⋊Q₈×C₅₉	C₂×C₄	1	2	yes	—
60	C₄⋊Q₈×C₅₉	C₂×C₄	1	2	yes	—
61	#126×C₅₉	—	2	2	no	—
62	#126×C₅₉	—	2	2	no	—
63	#126×C₅₉	—	2	2	no	—
64	#126×C₅₉	—	2	2	no	—
65	#125×C₅₉	—	2	2	no	—
66	#125×C₅₉	—	2	2	no	—
67	#125×C₅₉	—	2	1	no	—
68	#125×C₅₉	—	2	2	no	—
69	#125×C₅₉	C₂×C₄	1	2	yes	—
70	#124×C₅₉	—	2	2	no	—
71	#124×C₅₉	C₂×C₄	1	2	yes	—
72	#123×C₅₉	—	2	1	no	—
73	#123×C₅₉	—	2	2	no	—
74	#123×C₅₉	—	2	2	no	—
75	#123×C₅₉	—	2	2	no	—
76	C₂²⋊Q₈×C₅₉	—	2	1	no	—
77	C₂²⋊Q₈×C₅₉	—	2	1	no	—
78	C₂²⋊Q₈×C₅₉	—	2	2	no	—
79	C₂²⋊Q₈×C₅₉	—	2	1	no	—
80	C₂²⋊Q₈×C₅₉	—	2	1	no	—
81	C₂²⋊Q₈×C₅₉	—	2	2	no	—
82	C₂²⋊Q₈×C₅₉	—	2	2	no	—
83	C₂²⋊Q₈×C₅₉	C₂×C₄	1	2	yes	—
84	C₂²⋊Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
85	C₂²⋊Q₈×C₅₉	C₂³	1	2	yes	—
86	C₂²⋊Q₈×C₅₉	C₂³	1	1	yes	characteristic
87	#121×C₅₉	C₂×C₄	1	1	yes	characteristic
88	#121×C₅₉	C₂³	1	1	yes	characteristic
89	#120×C₅₉	—	2	1	no	—
90	C₄×Q₈×C₅₉	—	2	1	no	—
91	C₄×Q₈×C₅₉	—	2	1	no	—
92	C₄×Q₈×C₅₉	—	2	2	no	—
93	C₄×Q₈×C₅₉	C₂×C₄	1	2	yes	—
94	C₄×Q₈×C₅₉	C₂×C₄	1	2	yes	—
95	C₄×Q₈×C₅₉	C₂×C₄	1	2	yes	—
96	C₄×D₄×C₅₉	—	2	1	no	—
97	C₄×D₄×C₅₉	—	2	1	no	—
98	C₄×D₄×C₅₉	—	2	2	no	—
99	C₄×D₄×C₅₉	—	2	2	no	—
100	C₄×D₄×C₅₉	C₂×C₄	1	2	yes	—
101	#117×C₅₉	—	2	2	no	—
102	#117×C₅₉	—	2	1	no	—
103	#117×C₅₉	—	2	2	no	—
104	#117×C₅₉	—	2	2	no	—
105	#117×C₅₉	C₂×C₄	1	1	yes	characteristic
106	#117×C₅₉	C₂×C₄	1	1	yes	characteristic
107	#117×C₅₉	C₂×C₄	1	1	yes	characteristic
108	#117×C₅₉	C₂×C₄	1	2	yes	—
109	#117×C₅₉	D₄	1	1	yes	characteristic
110	#117×C₅₉	D₄	1	1	yes	characteristic
111	#117×C₅₉	D₄	1	2	yes	—
112	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
113	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
114	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
115	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
116	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
117	C₂×C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
118	C₂×C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
119	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
120	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
121	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
122	C₂×C₂²⋊C₄×C₅₉	D₄	1	2	yes	—
123	C₂×C₄²×C₅₉	—	2	1	no	—
124	C₂×C₄²×C₅₉	—	2	2	no	—
125	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
126	C₄⋊C₈×C₅₉	—	2	2	no	—
127	C₄⋊C₈×C₅₉	—	2	2	no	—
128	C₂²⋊C₈×C₅₉	—	2	1	no	—
129	C₂²⋊C₈×C₅₉	—	2	1	no	—
130	C₂²⋊C₈×C₅₉	C₂³	1	1	yes	characteristic
131	C₂²⋊C₈×C₅₉	C₂³	1	1	yes	characteristic
132	C₂²⋊C₈×C₅₉	C₂³	1	2	yes	—
133	#97×C₅₉	—	2	2	no	—
134	C₄×C₈×C₅₉	—	2	2	no	—
135	#95×C₅₉	—	2	2	no	—
136	#95×C₅₉	—	2	2	no	—
137	#95×C₅₉	C₂³	1	1	yes	characteristic
138	#95×C₅₉	C₂³	1	1	yes	characteristic
139	#95×C₅₉	C₂³	1	2	yes	—
140	C₂⁴×C₅₉	C₂×D₄	1	1	yes	characteristic
141	#41×C₅₉	—	4	2	no	—
142	C₂×Q₈×C₅₉	—	2	1	no	—
143	C₂×Q₈×C₅₉	—	2	2	no	—
144	C₂×Q₈×C₅₉	—	4	2	no	—
145	C₂×Q₈×C₅₉	C₂²×C₄	1	2	yes	—
146	C₂×Q₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
147	C₂×D₄×C₅₉	—	2	1	no	—
148	C₂×D₄×C₅₉	—	2	1	no	—
149	C₂×D₄×C₅₉	—	2	1	no	—
150	C₂×D₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
151	C₂²×C₄×C₅₉	—	2	1	no	—
152	C₂²×C₄×C₅₉	—	2	1	no	—
153	C₂²×C₄×C₅₉	—	2	1	no	—
154	C₂²×C₄×C₅₉	—	2	1	no	—
155	C₂²×C₄×C₅₉	—	2	2	no	—
156	C₂²×C₄×C₅₉	—	2	1	no	—
157	C₂²×C₄×C₅₉	—	2	2	no	—
158	C₂²×C₄×C₅₉	—	4	2	no	—
159	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
160	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
162	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
165	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
167	M₁₆×C₅₉	—	4	2	no	—
168	C₂×C₈×C₅₉	—	2	1	no	—
169	C₂×C₈×C₅₉	—	2	2	no	—
170	C₂×C₈×C₅₉	—	2	1	no	—
171	C₂×C₈×C₅₉	—	4	2	no	—
172	C₄⋊C₄×C₅₉	—	2	1	no	—
173	C₄⋊C₄×C₅₉	—	2	2	no	—
174	C₄⋊C₄×C₅₉	—	2	2	no	—
175	C₄⋊C₄×C₅₉	—	2	2	no	—
176	C₄⋊C₄×C₅₉	—	2	2	no	—
177	C₄⋊C₄×C₅₉	—	2	2	no	—
178	C₄⋊C₄×C₅₉	—	2	2	no	—
179	C₄⋊C₄×C₅₉	—	2	2	no	—
180	C₄⋊C₄×C₅₉	—	2	2	no	—
181	C₄⋊C₄×C₅₉	—	2	1	no	—
182	C₄⋊C₄×C₅₉	—	2	2	no	—
183	C₄⋊C₄×C₅₉	—	2	2	no	—
184	C₄⋊C₄×C₅₉	—	2	1	no	—
185	C₄⋊C₄×C₅₉	—	4	2	no	—
186	C₄⋊C₄×C₅₉	—	4	4	no	—
187	C₄⋊C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
188	C₄⋊C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
189	C₄⋊C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
190	C₄⋊C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
191	C₄⋊C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
192	C₄⋊C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
193	C₄⋊C₄×C₅₉	C₂²×C₄	1	2	yes	—
194	C₄⋊C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
195	C₄⋊C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
196	C₂²⋊C₄×C₅₉	—	2	2	no	—
197	C₂²⋊C₄×C₅₉	—	2	1	no	—
198	C₂²⋊C₄×C₅₉	—	2	1	no	—
199	C₂²⋊C₄×C₅₉	—	2	2	no	—
200	C₂²⋊C₄×C₅₉	—	2	1	no	—
201	C₂²⋊C₄×C₅₉	—	2	2	no	—
202	C₂²⋊C₄×C₅₉	—	2	2	no	—
203	C₂²⋊C₄×C₅₉	—	2	1	no	—
204	C₂²⋊C₄×C₅₉	—	2	2	no	—
205	C₂²⋊C₄×C₅₉	—	2	1	no	—
206	C₂²⋊C₄×C₅₉	—	4	2	no	—
207	C₂²⋊C₄×C₅₉	—	8	1	no	—
208	C₂²⋊C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
209	C₂²⋊C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
210	C₂²⋊C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
211	C₂²⋊C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
212	C₄²×C₅₉	—	2	2	no	—
213	C₄²×C₅₉	—	2	2	no	—
214	C₄²×C₅₉	—	2	2	no	—
215	C₄²×C₅₉	—	2	1	no	—
216	C₄²×C₅₉	—	4	2	no	—
217	C₄²×C₅₉	—	8	2	no	—
218	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
219	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
220	C₄²×C₅₉	C₂×D₄	1	2	yes	—
221	C₄²×C₅₉	C₂×D₄	1	2	yes	—
222	C₂³×C₅₉	—	2	1	no	—
223	C₂³×C₅₉	—	2	1	no	—
224	C₂³×C₅₉	—	2	1	no	—
225	C₂³×C₅₉	—	4	1	no	—
226	C₂³×C₅₉	—	4	1	no	—
227	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
228	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
229	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
230	Q₈×C₅₉	—	4	2	no	—
231	Q₈×C₅₉	—	4	1	no	—
232	Q₈×C₅₉	—	4	2	no	—
233	Q₈×C₅₉	—	8	2	no	—
234	D₄×C₅₉	—	4	1	no	—
235	D₄×C₅₉	—	4	2	no	—
236	D₄×C₅₉	—	4	1	no	—
237	D₄×C₅₉	—	4	1	no	—
238	D₄×C₅₉	—	8	1	no	—
239	C₂×C₄×C₅₉	—	2	2	no	—
240	C₂×C₄×C₅₉	—	2	2	no	—
241	C₂×C₄×C₅₉	—	2	2	no	—
242	C₂×C₄×C₅₉	—	2	2	no	—
243	C₂×C₄×C₅₉	—	2	2	no	—
244	C₂×C₄×C₅₉	—	2	2	no	—
245	C₂×C₄×C₅₉	—	2	1	no	—
246	C₂×C₄×C₅₉	—	2	1	no	—
247	C₂×C₄×C₅₉	—	2	2	no	—
248	C₂×C₄×C₅₉	—	2	1	no	—
249	C₂×C₄×C₅₉	—	2	2	no	—
250	C₂×C₄×C₅₉	—	4	2	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	2	no	—
256	C₂×C₄×C₅₉	—	4	1	no	—
257	C₂×C₄×C₅₉	—	4	2	no	—
258	C₂×C₄×C₅₉	—	4	1	no	—
259	C₂×C₄×C₅₉	—	4	1	no	—
260	C₂×C₄×C₅₉	—	4	2	no	—
261	C₂×C₄×C₅₉	—	4	2	no	—
262	C₂×C₄×C₅₉	—	4	2	no	—
263	C₂×C₄×C₅₉	—	4	1	no	—
264	C₂×C₄×C₅₉	—	8	2	no	—
265	C₂×C₄×C₅₉	—	8	1	no	—
266	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	2	yes	—
267	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	2	yes	—
268	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	2	yes	—
269	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	2	yes	—
270	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
271	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
272	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
273	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
274	C₈×C₅₉	—	4	1	no	—
275	C₈×C₅₉	—	4	1	no	—
276	C₈×C₅₉	—	4	2	no	—
277	#17612	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
278	C₂²×C₅₉	—	2	1	no	—
279	C₂²×C₅₉	—	2	1	no	—
280	C₂²×C₅₉	—	2	1	no	—
281	C₂²×C₅₉	—	2	1	no	—
282	C₂²×C₅₉	—	2	1	no	—
283	C₂²×C₅₉	—	2	1	no	—
284	C₂²×C₅₉	—	2	1	no	—
285	C₂²×C₅₉	—	2	1	no	—
286	C₂²×C₅₉	—	4	1	no	—
287	C₂²×C₅₉	—	4	1	no	—
288	C₂²×C₅₉	—	4	1	no	—
289	C₂²×C₅₉	—	8	1	no	—
290	C₂²×C₅₉	—	8	1	no	—
291	C₂²×C₅₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
292	C₂²×C₅₉	#577	1	1	yes	characteristic
293	C₂²×C₅₉	#578	1	1	yes	characteristic
294	C₄×C₅₉	—	2	1	no	—
295	C₄×C₅₉	—	2	1	no	—
296	C₄×C₅₉	—	2	1	no	—
297	C₄×C₅₉	—	2	2	no	—
298	C₄×C₅₉	—	2	2	no	—
299	C₄×C₅₉	—	2	2	no	—
300	C₄×C₅₉	—	2	2	no	—
301	C₄×C₅₉	—	4	2	no	—
302	C₄×C₅₉	—	4	1	no	—
303	C₄×C₅₉	—	4	2	no	—
304	C₄×C₅₉	—	4	1	no	—
305	C₄×C₅₉	—	4	2	no	—
306	C₄×C₅₉	—	4	1	no	—
307	C₄×C₅₉	—	8	2	no	—
308	C₄×C₅₉	—	8	1	no	—
309	C₄×C₅₉	#410	1	2	yes	—
310	#3495	C₂×C₅₉	1	1	yes	characteristic
311	#2924	C₂×C₅₉	1	1	yes	characteristic, complemented
312	#2347	C₂×C₅₉	1	1	yes	characteristic, complemented
313	#1521	C₂×C₅₉	1	2	yes	complemented
314	#1520	C₂×C₅₉	1	1	yes	characteristic, complemented
315	#1515	C₂×C₅₉	1	2	yes	complemented
316	#1513	C₂×C₅₉	1	1	yes	characteristic
317	#1509	C₂×C₅₉	1	2	yes	complemented
318	#1508	C₂×C₅₉	1	1	yes	characteristic, complemented
319	#1503	C₂×C₅₉	1	2	yes	complemented
320	#1501	C₂×C₅₉	1	1	yes	characteristic
321	C₂×C₅₉	—	2	1	no	—
322	C₂×C₅₉	—	2	1	no	—
323	C₂×C₅₉	—	4	1	no	—
324	C₂×C₅₉	—	4	1	no	—
325	C₂×C₅₉	—	8	1	no	—
326	C₂×C₅₉	C₂×#410	1	1	yes	characteristic
327	C₂×C₅₉	#2898	1	1	yes	characteristic
328	C₂×C₅₉	#2901	1	1	yes	characteristic
329	#556	—	2	1	no	—
330	#554	—	2	1	no	—
331	#552	—	2	2	no	—
332	#552	—	2	1	no	—
333	#552	—	2	1	no	—
334	#551	—	2	2	no	—
335	#544	C₄×C₅₉	1	1	yes	characteristic, complemented
336	#542	C₄×C₅₉	1	2	yes	complemented
337	#541	C₄×C₅₉	1	2	yes	complemented
338	#536	C₄×C₅₉	1	1	yes	characteristic, complemented
339	#533	C₂²×C₅₉	1	2	yes	—
340	#529	C₂²×C₅₉	1	2	yes	—
341	#525	C₂²×C₅₉	1	1	yes	characteristic
342	C₂×C₂²⋊Q₈	C₂²×C₅₉	1	1	yes	characteristic
343	#520	C₄×C₅₉	1	1	yes	characteristic, complemented
344	#519	C₄×C₅₉	1	1	yes	characteristic, complemented
345	C₂×#117	C₂²×C₅₉	1	1	yes	characteristic
346	#433	—	2	2	no	—
347	#432	C₂²×C₅₉	1	2	yes	—
348	#431	C₂²×C₅₉	1	2	yes	—
349	#424	—	2	1	no	—
350	#407	C₂²×C₅₉	1	1	yes	characteristic
351	C₂×C₂²⋊C₈	C₂²×C₅₉	1	1	yes	characteristic
352	#405	—	2	1	no	—
353	#389	—	2	1	no	—
354	#388	—	2	2	no	—
355	#383	C₂²×C₅₉	1	2	yes	—
356	#380	C₂²×C₅₉	1	1	yes	characteristic
357	C₄×C₄⋊C₄	—	2	1	no	—
358	C₄×C₄⋊C₄	C₂²×C₅₉	1	2	yes	—
359	C₄×C₂²⋊C₄	C₂²×C₅₉	1	1	yes	characteristic
360	#328	C₂²×C₅₉	1	1	yes	characteristic
361	#328	C₂²×C₅₉	1	1	yes	characteristic
362	#328	C₂²×C₅₉	1	2	yes	—
363	#328	C₂²×C₅₉	1	2	yes	—
364	#328	C₂²×C₅₉	1	2	yes	—
365	#328	C₂²×C₅₉	1	2	yes	—
366	#328	C₂²×C₅₉	1	2	yes	—
367	#327	C₂²×C₅₉	1	1	yes	characteristic
368	#327	C₂²×C₅₉	1	1	yes	characteristic
369	#327	C₂²×C₅₉	1	2	yes	—
370	C₅₉	#17612	1	1	yes	characteristic, direct product factor, Sylow
371	C₂×#41	C₂×C₄×C₅₉	1	1	yes	characteristic
372	C₂²×Q₈	C₂×C₄×C₅₉	1	1	yes	characteristic
373	C₂³×C₄	C₂³×C₅₉	1	1	yes	characteristic
374	C₂×M₁₆	—	2	1	no	—
375	C₂²×C₈	—	2	1	no	—
376	C₄⋊Q₈	—	2	1	no	—
377	C₄⋊Q₈	C₂×C₄×C₅₉	1	2	yes	—
378	C₄⋊Q₈	C₂×C₄×C₅₉	1	2	yes	—
379	#126	—	2	2	no	—
380	#126	—	2	2	no	—
381	#126	—	2	2	no	—
382	#126	—	2	2	no	—
383	#125	—	2	2	no	—
384	#125	—	2	2	no	—
385	#125	—	2	1	no	—
386	#125	—	2	2	no	—
387	#125	C₂×C₄×C₅₉	1	2	yes	—
388	#124	—	2	2	no	—
389	#124	C₂×C₄×C₅₉	1	2	yes	—
390	#123	—	2	1	no	—
391	#123	—	2	2	no	—
392	#123	—	2	2	no	—
393	#123	—	2	2	no	—
394	C₂²⋊Q₈	—	2	1	no	—
395	C₂²⋊Q₈	—	2	1	no	—
396	C₂²⋊Q₈	—	2	2	no	—
397	C₂²⋊Q₈	—	2	1	no	—
398	C₂²⋊Q₈	—	2	1	no	—
399	C₂²⋊Q₈	—	2	2	no	—
400	C₂²⋊Q₈	—	2	2	no	—
401	C₂²⋊Q₈	C₂×C₄×C₅₉	1	2	yes	—
402	C₂²⋊Q₈	C₂×C₄×C₅₉	1	1	yes	characteristic
403	C₂²⋊Q₈	C₂³×C₅₉	1	2	yes	—
404	C₂²⋊Q₈	C₂³×C₅₉	1	1	yes	characteristic
405	#121	C₂×C₄×C₅₉	1	1	yes	characteristic
406	#121	C₂³×C₅₉	1	1	yes	characteristic
407	#120	—	2	1	no	—
408	C₄×Q₈	—	2	1	no	—
409	C₄×Q₈	—	2	1	no	—
410	C₄×Q₈	—	2	2	no	—
411	C₄×Q₈	C₂×C₄×C₅₉	1	2	yes	—
412	C₄×Q₈	C₂×C₄×C₅₉	1	2	yes	—
413	C₄×Q₈	C₂×C₄×C₅₉	1	2	yes	—
414	C₄×D₄	—	2	1	no	—
415	C₄×D₄	—	2	1	no	—
416	C₄×D₄	—	2	2	no	—
417	C₄×D₄	—	2	2	no	—
418	C₄×D₄	C₂×C₄×C₅₉	1	2	yes	—
419	#117	—	2	2	no	—
420	#117	—	2	1	no	—
421	#117	—	2	2	no	—
422	#117	—	2	2	no	—
423	#117	C₂×C₄×C₅₉	1	1	yes	characteristic
424	#117	C₂×C₄×C₅₉	1	1	yes	characteristic
425	#117	C₂×C₄×C₅₉	1	1	yes	characteristic
426	#117	C₂×C₄×C₅₉	1	2	yes	—
427	#117	D₄×C₅₉	1	1	yes	characteristic
428	#117	D₄×C₅₉	1	2	yes	—
429	#117	D₄×C₅₉	1	1	yes	characteristic
430	C₂×C₄⋊C₄	—	2	1	no	—
431	C₂×C₄⋊C₄	—	2	1	no	—
432	C₂×C₄⋊C₄	—	2	1	no	—
433	C₂×C₄⋊C₄	—	2	2	no	—
434	C₂×C₄⋊C₄	C₂×C₄×C₅₉	1	1	yes	characteristic
435	C₂×C₄⋊C₄	D₄×C₅₉	1	1	yes	characteristic
436	C₂×C₄⋊C₄	D₄×C₅₉	1	1	yes	characteristic
437	C₂×C₂²⋊C₄	—	2	1	no	—
438	C₂×C₂²⋊C₄	—	2	1	no	—
439	C₂×C₂²⋊C₄	—	2	1	no	—
440	C₂×C₂²⋊C₄	D₄×C₅₉	1	2	yes	—
441	C₂×C₄²	—	2	1	no	—
442	C₂×C₄²	—	2	2	no	—
443	C₂×C₄²	C₂³×C₅₉	1	2	yes	—
444	C₄⋊C₈	—	2	2	no	—
445	C₄⋊C₈	—	2	2	no	—
446	C₂²⋊C₈	—	2	1	no	—
447	C₂²⋊C₈	—	2	1	no	—
448	C₂²⋊C₈	C₂³×C₅₉	1	1	yes	characteristic
449	C₂²⋊C₈	C₂³×C₅₉	1	1	yes	characteristic
450	C₂²⋊C₈	C₂³×C₅₉	1	2	yes	—
451	#97	—	2	2	no	—
452	C₄×C₈	—	2	2	no	—
453	#95	—	2	2	no	—
454	#95	—	2	2	no	—
455	#95	C₂³×C₅₉	1	1	yes	characteristic
456	#95	C₂³×C₅₉	1	1	yes	characteristic
457	#95	C₂³×C₅₉	1	2	yes	—
458	C₂⁴	C₂×D₄×C₅₉	1	1	yes	characteristic
459	#41	—	4	2	no	—
460	C₂×Q₈	—	2	1	no	—
461	C₂×Q₈	—	2	2	no	—
462	C₂×Q₈	—	4	2	no	—
463	C₂×Q₈	C₂²×C₄×C₅₉	1	2	yes	—
464	C₂×Q₈	C₂²×C₄×C₅₉	1	1	yes	characteristic
465	C₂×D₄	—	2	1	no	—
466	C₂×D₄	—	2	1	no	—
467	C₂×D₄	—	2	1	no	—
468	C₂×D₄	C₂²×C₄×C₅₉	1	1	yes	characteristic
469	C₂²×C₄	—	2	1	no	—
470	C₂²×C₄	—	2	1	no	—
471	C₂²×C₄	—	2	1	no	—
472	C₂²×C₄	—	2	1	no	—
473	C₂²×C₄	—	2	2	no	—
474	C₂²×C₄	—	2	1	no	—
475	C₂²×C₄	—	2	2	no	—
476	C₂²×C₄	—	4	2	no	—
477	C₂²×C₄	C₂²×C₄×C₅₉	1	1	yes	characteristic
478	C₂²×C₄	C₂²×C₄×C₅₉	1	1	yes	characteristic
479	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
480	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
481	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
482	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
483	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
484	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
485	M₁₆	—	4	2	no	—
486	C₂×C₈	—	2	1	no	—
487	C₂×C₈	—	2	2	no	—
488	C₂×C₈	—	2	1	no	—
489	C₂×C₈	—	4	2	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	2	no	—
492	C₄⋊C₄	—	2	2	no	—
493	C₄⋊C₄	—	2	2	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	—	2	2	no	—
497	C₄⋊C₄	—	2	2	no	—
498	C₄⋊C₄	—	2	2	no	—
499	C₄⋊C₄	—	2	1	no	—
500	C₄⋊C₄	—	2	2	no	—
501	C₄⋊C₄	—	2	2	no	—
502	C₄⋊C₄	—	2	1	no	—
503	C₄⋊C₄	—	4	2	no	—
504	C₄⋊C₄	—	4	4	no	—
505	C₄⋊C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
506	C₄⋊C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
507	C₄⋊C₄	C₂²⋊C₄×C₅₉	1	2	yes	—
508	C₄⋊C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
509	C₄⋊C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
510	C₄⋊C₄	C₂²⋊C₄×C₅₉	1	2	yes	—
511	C₄⋊C₄	C₂²×C₄×C₅₉	1	2	yes	—
512	C₄⋊C₄	C₂²×C₄×C₅₉	1	1	yes	characteristic
513	C₄⋊C₄	C₂²×C₄×C₅₉	1	1	yes	characteristic
514	C₂²⋊C₄	—	2	2	no	—
515	C₂²⋊C₄	—	2	1	no	—
516	C₂²⋊C₄	—	2	1	no	—
517	C₂²⋊C₄	—	2	2	no	—
518	C₂²⋊C₄	—	2	1	no	—
519	C₂²⋊C₄	—	2	2	no	—
520	C₂²⋊C₄	—	2	2	no	—
521	C₂²⋊C₄	—	2	1	no	—
522	C₂²⋊C₄	—	2	2	no	—
523	C₂²⋊C₄	—	2	1	no	—
524	C₂²⋊C₄	—	4	2	no	—
525	C₂²⋊C₄	—	8	1	no	—
526	C₂²⋊C₄	C₂²⋊C₄×C₅₉	1	2	yes	—
527	C₂²⋊C₄	C₂²⋊C₄×C₅₉	1	2	yes	—
528	C₂²⋊C₄	C₂²⋊C₄×C₅₉	1	2	yes	—
529	C₂²⋊C₄	C₂²⋊C₄×C₅₉	1	2	yes	—
530	C₄²	—	2	2	no	—
531	C₄²	—	2	2	no	—
532	C₄²	—	2	2	no	—
533	C₄²	—	2	1	no	—
534	C₄²	—	4	2	no	—
535	C₄²	—	8	2	no	—
536	C₄²	C₂²×C₄×C₅₉	1	2	yes	—
537	C₄²	C₂²×C₄×C₅₉	1	2	yes	—
538	C₄²	C₂×D₄×C₅₉	1	2	yes	—
539	C₄²	C₂×D₄×C₅₉	1	2	yes	—
540	C₂³	—	2	1	no	—
541	C₂³	—	2	1	no	—
542	C₂³	—	2	1	no	—
543	C₂³	—	4	1	no	—
544	C₂³	—	4	1	no	—
545	C₂³	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
546	C₂³	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
547	C₂³	C₂²×D₄×C₅₉	1	1	yes	characteristic
548	Q₈	—	4	2	no	—
549	Q₈	—	4	1	no	—
550	Q₈	—	4	2	no	—
551	Q₈	—	8	2	no	—
552	D₄	—	4	1	no	—
553	D₄	—	4	2	no	—
554	D₄	—	4	1	no	—
555	D₄	—	4	1	no	—
556	D₄	—	8	1	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	1	no	—
564	C₂×C₄	—	2	1	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	4	2	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	2	no	—
574	C₂×C₄	—	4	1	no	—
575	C₂×C₄	—	4	2	no	—
576	C₂×C₄	—	4	1	no	—
577	C₂×C₄	—	4	1	no	—
578	C₂×C₄	—	4	2	no	—
579	C₂×C₄	—	4	2	no	—
580	C₂×C₄	—	4	2	no	—
581	C₂×C₄	—	4	1	no	—
582	C₂×C₄	—	8	2	no	—
583	C₂×C₄	—	8	1	no	—
584	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	2	yes	—
585	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	2	yes	—
586	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	2	yes	—
587	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	2	yes	—
588	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
589	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
590	C₂×C₄	C₂³×C₄×C₅₉	1	1	yes	characteristic
591	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
592	C₈	—	4	1	no	—
593	C₈	—	4	1	no	—
594	C₈	—	4	2	no	—
595	C₂²	—	2	1	no	—
596	C₂²	—	2	1	no	—
597	C₂²	—	2	1	no	—
598	C₂²	—	2	1	no	—
599	C₂²	—	2	1	no	—
600	C₂²	—	2	1	no	—
601	C₂²	—	2	1	no	—
602	C₂²	—	2	1	no	—
603	C₂²	—	4	1	no	—
604	C₂²	—	4	1	no	—
605	C₂²	—	4	1	no	—
606	C₂²	—	8	1	no	—
607	C₂²	—	8	1	no	—
608	C₂²	C₂²×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
609	C₂²	C₅₉×#577	1	1	yes	characteristic
610	C₂²	C₅₉×#578	1	1	yes	characteristic
611	C₄	—	2	1	no	—
612	C₄	—	2	1	no	—
613	C₄	—	2	1	no	—
614	C₄	—	2	2	no	—
615	C₄	—	2	2	no	—
616	C₄	—	2	2	no	—
617	C₄	—	2	2	no	—
618	C₄	—	4	2	no	—
619	C₄	—	4	1	no	—
620	C₄	—	4	2	no	—
621	C₄	—	4	1	no	—
622	C₄	—	4	2	no	—
623	C₄	—	4	1	no	—
624	C₄	—	8	2	no	—
625	C₄	—	8	1	no	—
626	C₄	C₅₉×#410	1	2	yes	—
627	C₂	—	2	1	no	—
628	C₂	—	2	1	no	—
629	C₂	—	4	1	no	—
630	C₂	—	4	1	no	—
631	C₂	—	8	1	no	—
632	C₂	C₂×C₅₉×#410	1	1	yes	characteristic
633	C₂	C₅₉×#2898	1	1	yes	characteristic
634	C₂	C₅₉×#2901	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₅₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₉
prime radical	C₅₉×#525	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₂×#117×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₉×#2924	C₂	semidirect
3. C₅₉×#2347	C₂	semidirect
4. C₅₉×#1521	C₂	semidirect
5. C₅₉×#1520	C₂	semidirect
6. C₅₉×#1515	C₂	semidirect
8. C₅₉×#1509	C₂	semidirect
9. C₅₉×#1508	C₂	semidirect
10. C₅₉×#1503	C₂	semidirect
18. C₅₉×#544	C₄	semidirect
19. C₅₉×#542	C₄	semidirect
20. C₅₉×#541	C₄	semidirect
21. C₅₉×#536	C₄	semidirect
26. C₅₉×#520	C₄	semidirect
27. C₅₉×#519	C₄	semidirect
277. #17612	C₅₉	direct (N × H)
311. #2924	C₂×C₅₉	semidirect
312. #2347	C₂×C₅₉	semidirect
313. #1521	C₂×C₅₉	semidirect
314. #1520	C₂×C₅₉	semidirect
315. #1515	C₂×C₅₉	semidirect
317. #1509	C₂×C₅₉	semidirect
318. #1508	C₂×C₅₉	semidirect
319. #1503	C₂×C₅₉	semidirect
335. #544	C₄×C₅₉	semidirect
336. #542	C₄×C₅₉	semidirect
337. #541	C₄×C₅₉	semidirect
338. #536	C₄×C₅₉	semidirect
343. #520	C₄×C₅₉	semidirect
344. #519	C₄×C₅₉	semidirect
370. C₅₉	#17612	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₉×#3495	2
C₅₉×#2924	2
C₅₉×#2347	2
C₅₉×#1521	2
C₅₉×#1520	2
C₅₉×#1515	2
C₅₉×#1513	2
C₅₉×#1509	2
C₅₉×#1508	2
C₅₉×#1503	2
C₅₉×#1501	2
#17612	59

Subgroups — C59×#17612 #28429721

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#17612 #28429721