Subgroups — C₅₉×#17537 #28429646

Order 15104.

1116 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₉×#529	C₂	1	1	yes	maximal, characteristic
2	C₅₉×#2920	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#117×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₉×#1509	C₂	1	4	yes	maximal, complemented
5	C₅₉×#1508	C₂	1	2	yes	maximal, complemented
6	C₅₉×#1507	C₂	1	2	yes	maximal
7	C₅₉×#1500	C₂	1	4	yes	maximal, complemented
8	C₅₉×#529	—	2	1	no	—
9	C₅₉×#529	—	2	1	no	—
10	C₅₉×#529	—	2	1	no	—
11	C₅₉×#529	—	2	1	no	—
12	C₅₉×#529	C₄	1	2	yes	complemented
13	C₅₉×#529	C₄	1	2	yes	complemented
14	C₅₉×#529	C₂²	1	2	yes	—
15	C₅₉×#529	C₂²	1	2	yes	—
16	C₂×#126×C₅₉	—	2	1	no	—
17	C₂×#125×C₅₉	C₄	1	1	yes	characteristic, complemented
18	C₂×#124×C₅₉	C₄	1	1	yes	characteristic, complemented
19	C₂×#123×C₅₉	—	2	1	no	—
20	C₂×C₂²⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×#121×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₅₉	C₄	1	1	yes	characteristic, complemented
23	C₂×C₄×D₄×C₅₉	—	2	1	no	—
24	C₂×C₄×D₄×C₅₉	C₄	1	1	yes	characteristic, complemented
25	C₂×#117×C₅₉	—	2	1	no	—
26	C₂×#117×C₅₉	—	2	1	no	—
27	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
28	C₅₉×#432	—	2	1	no	—
29	C₅₉×#432	C₂²	1	2	yes	complemented
30	C₅₉×#431	—	2	1	no	—
31	C₅₉×#431	C₂²	1	2	yes	complemented
32	C₅₉×#423	—	2	1	no	—
33	C₅₉×#407	C₂²	1	2	yes	—
34	C₄×M₁₆×C₅₉	—	2	1	no	—
35	C₄×C₄⋊C₄×C₅₉	—	2	1	no	—
36	C₄×C₄⋊C₄×C₅₉	C₂²	1	2	yes	complemented
37	C₄×C₂²⋊C₄×C₅₉	—	2	1	no	—
38	C₄×C₂²⋊C₄×C₅₉	C₂²	1	2	yes	—
39	C₅₉×#376	C₂²	1	2	yes	complemented
40	C₄³×C₅₉	—	2	1	no	—
41	C₅₉×#328	C₂²	1	8	yes	complemented
42	C₅₉×#327	C₂²	1	8	yes	—
43	C₂²×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×D₄×C₅₉	—	2	1	no	—
45	C₂²×D₄×C₅₉	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₅₉	—	2	1	no	—
47	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
50	C₂×M₁₆×C₅₉	—	2	2	no	—
51	#126×C₅₉	—	2	2	no	—
52	#126×C₅₉	—	2	2	no	—
53	#126×C₅₉	—	2	2	no	—
54	#126×C₅₉	—	2	2	no	—
55	#125×C₅₉	—	2	1	no	—
56	#125×C₅₉	—	2	1	no	—
57	#125×C₅₉	C₂×C₄	1	2	yes	—
58	#125×C₅₉	C₂×C₄	1	2	yes	—
59	#124×C₅₉	—	2	1	no	—
60	#124×C₅₉	—	2	1	no	—
61	#124×C₅₉	C₂×C₄	1	2	yes	—
62	#124×C₅₉	C₂×C₄	1	2	yes	—
63	#123×C₅₉	—	2	2	no	—
64	#123×C₅₉	—	2	2	no	—
65	#123×C₅₉	—	2	2	no	—
66	#123×C₅₉	—	2	2	no	—
67	C₂²⋊Q₈×C₅₉	—	2	1	no	—
68	C₂²⋊Q₈×C₅₉	—	2	1	no	—
69	C₂²⋊Q₈×C₅₉	C₂×C₄	1	2	yes	—
70	C₂²⋊Q₈×C₅₉	C₂³	1	2	yes	—
71	#121×C₅₉	—	2	1	no	—
72	#121×C₅₉	—	2	1	no	—
73	#121×C₅₉	C₂×C₄	1	2	yes	—
74	#121×C₅₉	C₂³	1	2	yes	—
75	C₄×Q₈×C₅₉	—	2	1	no	—
76	C₄×Q₈×C₅₉	—	2	1	no	—
77	C₄×Q₈×C₅₉	C₂×C₄	1	2	yes	—
78	C₄×Q₈×C₅₉	C₂×C₄	1	2	yes	—
79	C₄×D₄×C₅₉	—	2	2	no	—
80	C₄×D₄×C₅₉	—	2	2	no	—
81	C₄×D₄×C₅₉	—	2	2	no	—
82	C₄×D₄×C₅₉	—	2	2	no	—
83	C₄×D₄×C₅₉	—	2	1	no	—
84	C₄×D₄×C₅₉	—	2	1	no	—
85	C₄×D₄×C₅₉	C₂×C₄	1	2	yes	—
86	C₄×D₄×C₅₉	C₂×C₄	1	2	yes	—
87	#117×C₅₉	—	2	8	no	—
88	#117×C₅₉	—	2	2	no	—
89	#117×C₅₉	—	2	2	no	—
90	#117×C₅₉	—	2	2	no	—
91	#117×C₅₉	—	2	2	no	—
92	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
93	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
94	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
95	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
96	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
97	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
99	C₂×C₂²⋊C₄×C₅₉	—	2	2	no	—
100	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
101	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
102	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
103	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
104	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
105	C₂×C₄²×C₅₉	—	2	1	no	—
106	C₂×C₄²×C₅₉	—	2	2	no	—
107	C₂×C₄²×C₅₉	—	2	2	no	—
108	C₂×C₄²×C₅₉	—	2	2	no	—
109	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
110	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
111	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
112	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
113	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
114	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
115	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
116	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
117	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
118	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
119	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
121	C₄⋊C₈×C₅₉	—	2	2	no	—
122	C₄⋊C₈×C₅₉	—	2	2	no	—
123	C₂²⋊C₈×C₅₉	—	2	2	no	—
124	C₂²⋊C₈×C₅₉	C₂³	1	4	yes	—
125	#97×C₅₉	—	2	2	no	—
126	C₄×C₈×C₅₉	—	2	2	no	—
127	#95×C₅₉	—	2	2	no	—
128	#95×C₅₉	C₂³	1	4	yes	—
129	C₂⁴×C₅₉	—	2	1	no	—
130	C₂⁴×C₅₉	C₂×D₄	1	1	yes	characteristic
131	C₂×Q₈×C₅₉	—	2	1	no	—
132	C₂×Q₈×C₅₉	—	4	2	no	—
133	C₂×Q₈×C₅₉	C₂²×C₄	1	2	yes	—
134	C₂×D₄×C₅₉	—	2	2	no	—
135	C₂×D₄×C₅₉	—	2	2	no	—
136	C₂×D₄×C₅₉	—	2	1	no	—
137	C₂×D₄×C₅₉	—	4	2	no	—
138	C₂×D₄×C₅₉	—	4	4	no	—
139	C₂×D₄×C₅₉	C₂²×C₄	1	2	yes	—
140	C₂²×C₄×C₅₉	—	2	4	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	4	no	—
145	C₂²×C₄×C₅₉	—	2	2	no	—
146	C₂²×C₄×C₅₉	—	2	2	no	—
147	C₂²×C₄×C₅₉	—	4	2	no	—
148	C₂²×C₄×C₅₉	—	4	2	no	—
149	C₂²×C₄×C₅₉	—	4	4	no	—
150	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
151	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
153	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
154	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
157	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
158	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
159	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
160	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
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	1	yes	characteristic
165	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
168	M₁₆×C₅₉	—	4	4	no	—
169	C₂×C₈×C₅₉	—	2	4	no	—
170	C₄⋊C₄×C₅₉	—	2	8	no	—
171	C₄⋊C₄×C₅₉	—	2	2	no	—
172	C₄⋊C₄×C₅₉	—	2	2	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	1	no	—
179	C₄⋊C₄×C₅₉	—	2	1	no	—
180	C₄⋊C₄×C₅₉	—	2	2	no	—
181	C₄⋊C₄×C₅₉	C₂²×C₄	1	2	yes	—
182	C₄⋊C₄×C₅₉	C₂²×C₄	1	2	yes	—
183	C₂²⋊C₄×C₅₉	—	2	8	no	—
184	C₂²⋊C₄×C₅₉	—	2	2	no	—
185	C₂²⋊C₄×C₅₉	—	2	2	no	—
186	C₂²⋊C₄×C₅₉	—	2	2	no	—
187	C₂²⋊C₄×C₅₉	—	2	2	no	—
188	C₂²⋊C₄×C₅₉	—	2	2	no	—
189	C₂²⋊C₄×C₅₉	—	2	2	no	—
190	C₂²⋊C₄×C₅₉	—	2	2	no	—
191	C₂²⋊C₄×C₅₉	—	2	2	no	—
192	C₂²⋊C₄×C₅₉	—	2	2	no	—
193	C₂²⋊C₄×C₅₉	—	2	2	no	—
194	C₄²×C₅₉	—	2	8	no	—
195	C₄²×C₅₉	—	2	2	no	—
196	C₄²×C₅₉	—	2	2	no	—
197	C₄²×C₅₉	—	4	4	no	—
198	C₄²×C₅₉	—	4	4	no	—
199	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
200	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
201	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
202	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
203	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
204	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
205	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
206	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
207	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
208	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
209	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
210	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
211	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
212	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
213	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
214	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
215	C₂³×C₅₉	—	2	2	no	—
216	C₂³×C₅₉	—	2	2	no	—
217	C₂³×C₅₉	—	2	2	no	—
218	C₂³×C₅₉	—	4	1	no	—
219	C₂³×C₅₉	—	4	2	no	—
220	C₂³×C₅₉	—	4	2	no	—
221	C₂³×C₅₉	—	4	2	no	—
222	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
223	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
224	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
225	Q₈×C₅₉	—	4	2	no	—
226	Q₈×C₅₉	—	4	2	no	—
227	D₄×C₅₉	—	4	2	no	—
228	D₄×C₅₉	—	4	4	no	—
229	D₄×C₅₉	—	4	4	no	—
230	D₄×C₅₉	—	4	2	no	—
231	C₂×C₄×C₅₉	—	2	2	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	2	no	—
236	C₂×C₄×C₅₉	—	2	2	no	—
237	C₂×C₄×C₅₉	—	2	2	no	—
238	C₂×C₄×C₅₉	—	2	2	no	—
239	C₂×C₄×C₅₉	—	2	2	no	—
240	C₂×C₄×C₅₉	—	2	2	no	—
241	C₂×C₄×C₅₉	—	2	8	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	—	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	4	no	—
253	C₂×C₄×C₅₉	—	4	2	no	—
254	C₂×C₄×C₅₉	—	4	2	no	—
255	C₂×C₄×C₅₉	—	4	4	no	—
256	C₂×C₄×C₅₉	—	4	2	no	—
257	C₂×C₄×C₅₉	—	4	4	no	—
258	C₂×C₄×C₅₉	—	4	4	no	—
259	C₂×C₄×C₅₉	#104	1	4	yes	—
260	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
261	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
262	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
263	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
264	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
265	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
266	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
267	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
268	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
269	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
270	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
271	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
272	C₈×C₅₉	—	4	4	no	—
273	#17537	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
274	C₂²×C₅₉	—	2	1	no	—
275	C₂²×C₅₉	—	2	1	no	—
276	C₂²×C₅₉	—	2	1	no	—
277	C₂²×C₅₉	—	2	1	no	—
278	C₂²×C₅₉	—	2	1	no	—
279	C₂²×C₅₉	—	2	1	no	—
280	C₂²×C₅₉	—	2	1	no	—
281	C₂²×C₅₉	—	4	4	no	—
282	C₂²×C₅₉	—	4	2	no	—
283	C₂²×C₅₉	—	4	2	no	—
284	C₂²×C₅₉	—	4	2	no	—
285	C₂²×C₅₉	—	8	4	no	—
286	C₂²×C₅₉	C₂×#104	1	2	yes	—
287	C₂²×C₅₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
288	C₂²×C₅₉	#575	1	2	yes	—
289	C₄×C₅₉	—	2	1	no	—
290	C₄×C₅₉	—	2	1	no	—
291	C₄×C₅₉	—	2	1	no	—
292	C₄×C₅₉	—	2	1	no	—
293	C₄×C₅₉	—	2	1	no	—
294	C₄×C₅₉	—	2	1	no	—
295	C₄×C₅₉	—	2	1	no	—
296	C₄×C₅₉	—	2	1	no	—
297	C₄×C₅₉	—	2	1	no	—
298	C₄×C₅₉	—	4	8	no	—
299	C₄×C₅₉	—	4	2	no	—
300	C₄×C₅₉	—	4	2	no	—
301	C₄×C₅₉	—	4	2	no	—
302	C₄×C₅₉	#410	1	2	yes	—
303	C₄×C₅₉	C₂×#104	1	2	yes	—
304	C₄×C₅₉	C₂×#104	1	2	yes	—
305	C₂×#529	C₂×C₅₉	1	1	yes	characteristic
306	#2920	C₂×C₅₉	1	1	yes	characteristic, complemented
307	C₄×#117	C₂×C₅₉	1	1	yes	characteristic, complemented
308	#1509	C₂×C₅₉	1	4	yes	complemented
309	#1508	C₂×C₅₉	1	2	yes	complemented
310	#1507	C₂×C₅₉	1	2	yes	—
311	#1500	C₂×C₅₉	1	4	yes	complemented
312	C₂×C₅₉	—	2	2	no	—
313	C₂×C₅₉	—	2	2	no	—
314	C₂×C₅₉	—	4	1	no	—
315	C₂×C₅₉	—	8	2	no	—
316	C₂×C₅₉	C₂×#410	1	1	yes	characteristic
317	C₂×C₅₉	#2896	1	1	yes	characteristic
318	C₂×C₅₉	C₂²×#104	1	1	yes	characteristic
319	#529	—	2	1	no	—
320	#529	—	2	1	no	—
321	#529	—	2	1	no	—
322	#529	—	2	1	no	—
323	#529	C₄×C₅₉	1	2	yes	complemented
324	#529	C₄×C₅₉	1	2	yes	complemented
325	#529	C₂²×C₅₉	1	2	yes	—
326	#529	C₂²×C₅₉	1	2	yes	—
327	C₂×#126	—	2	1	no	—
328	C₂×#125	C₄×C₅₉	1	1	yes	characteristic, complemented
329	C₂×#124	C₄×C₅₉	1	1	yes	characteristic, complemented
330	C₂×#123	—	2	1	no	—
331	C₂×C₂²⋊Q₈	C₂²×C₅₉	1	1	yes	characteristic
332	C₂×#121	C₂²×C₅₉	1	1	yes	characteristic
333	C₂×C₄×Q₈	C₄×C₅₉	1	1	yes	characteristic, complemented
334	C₂×C₄×D₄	—	2	1	no	—
335	C₂×C₄×D₄	C₄×C₅₉	1	1	yes	characteristic, complemented
336	C₂×#117	—	2	1	no	—
337	C₂×#117	—	2	1	no	—
338	C₂²×C₄²	C₂²×C₅₉	1	1	yes	characteristic
339	#432	—	2	1	no	—
340	#432	C₂²×C₅₉	1	2	yes	complemented
341	#431	—	2	1	no	—
342	#431	C₂²×C₅₉	1	2	yes	complemented
343	#423	—	2	1	no	—
344	#407	C₂²×C₅₉	1	2	yes	—
345	C₄×M₁₆	—	2	1	no	—
346	C₄×C₄⋊C₄	—	2	1	no	—
347	C₄×C₄⋊C₄	C₂²×C₅₉	1	2	yes	complemented
348	C₄×C₂²⋊C₄	—	2	1	no	—
349	C₄×C₂²⋊C₄	C₂²×C₅₉	1	2	yes	—
350	#376	C₂²×C₅₉	1	2	yes	complemented
351	C₄³	—	2	1	no	—
352	#328	C₂²×C₅₉	1	8	yes	complemented
353	#327	C₂²×C₅₉	1	8	yes	—
354	C₅₉	#17537	1	1	yes	characteristic, direct product factor, Sylow
355	C₂²×Q₈	C₂×C₄×C₅₉	1	1	yes	characteristic
356	C₂²×D₄	—	2	1	no	—
357	C₂²×D₄	C₂×C₄×C₅₉	1	1	yes	characteristic
358	C₂³×C₄	—	2	1	no	—
359	C₂³×C₄	D₄×C₅₉	1	1	yes	characteristic
360	C₂³×C₄	D₄×C₅₉	1	1	yes	characteristic
361	C₂³×C₄	C₂³×C₅₉	1	1	yes	characteristic
362	C₂×M₁₆	—	2	2	no	—
363	#126	—	2	2	no	—
364	#126	—	2	2	no	—
365	#126	—	2	2	no	—
366	#126	—	2	2	no	—
367	#125	—	2	1	no	—
368	#125	—	2	1	no	—
369	#125	C₂×C₄×C₅₉	1	2	yes	—
370	#125	C₂×C₄×C₅₉	1	2	yes	—
371	#124	—	2	1	no	—
372	#124	—	2	1	no	—
373	#124	C₂×C₄×C₅₉	1	2	yes	—
374	#124	C₂×C₄×C₅₉	1	2	yes	—
375	#123	—	2	2	no	—
376	#123	—	2	2	no	—
377	#123	—	2	2	no	—
378	#123	—	2	2	no	—
379	C₂²⋊Q₈	—	2	1	no	—
380	C₂²⋊Q₈	—	2	1	no	—
381	C₂²⋊Q₈	C₂×C₄×C₅₉	1	2	yes	—
382	C₂²⋊Q₈	C₂³×C₅₉	1	2	yes	—
383	#121	—	2	1	no	—
384	#121	—	2	1	no	—
385	#121	C₂×C₄×C₅₉	1	2	yes	—
386	#121	C₂³×C₅₉	1	2	yes	—
387	C₄×Q₈	—	2	1	no	—
388	C₄×Q₈	—	2	1	no	—
389	C₄×Q₈	C₂×C₄×C₅₉	1	2	yes	—
390	C₄×Q₈	C₂×C₄×C₅₉	1	2	yes	—
391	C₄×D₄	—	2	2	no	—
392	C₄×D₄	—	2	2	no	—
393	C₄×D₄	—	2	2	no	—
394	C₄×D₄	—	2	2	no	—
395	C₄×D₄	—	2	1	no	—
396	C₄×D₄	—	2	1	no	—
397	C₄×D₄	C₂×C₄×C₅₉	1	2	yes	—
398	C₄×D₄	C₂×C₄×C₅₉	1	2	yes	—
399	#117	—	2	8	no	—
400	#117	—	2	2	no	—
401	#117	—	2	2	no	—
402	#117	—	2	2	no	—
403	#117	—	2	2	no	—
404	C₂×C₄⋊C₄	—	2	1	no	—
405	C₂×C₄⋊C₄	—	2	1	no	—
406	C₂×C₄⋊C₄	—	2	1	no	—
407	C₂×C₄⋊C₄	—	2	1	no	—
408	C₂×C₄⋊C₄	—	2	2	no	—
409	C₂×C₄⋊C₄	C₂×C₄×C₅₉	1	1	yes	characteristic
410	C₂×C₄⋊C₄	C₂×C₄×C₅₉	1	1	yes	characteristic
411	C₂×C₂²⋊C₄	—	2	2	no	—
412	C₂×C₂²⋊C₄	—	2	1	no	—
413	C₂×C₂²⋊C₄	—	2	1	no	—
414	C₂×C₂²⋊C₄	—	2	1	no	—
415	C₂×C₂²⋊C₄	—	2	1	no	—
416	C₂×C₂²⋊C₄	—	2	1	no	—
417	C₂×C₄²	—	2	1	no	—
418	C₂×C₄²	—	2	2	no	—
419	C₂×C₄²	—	2	2	no	—
420	C₂×C₄²	—	2	2	no	—
421	C₂×C₄²	C₂×C₄×C₅₉	1	1	yes	characteristic
422	C₂×C₄²	C₂×C₄×C₅₉	1	1	yes	characteristic
423	C₂×C₄²	C₂×C₄×C₅₉	1	1	yes	characteristic
424	C₂×C₄²	C₂×C₄×C₅₉	1	1	yes	characteristic
425	C₂×C₄²	D₄×C₅₉	1	1	yes	characteristic
426	C₂×C₄²	D₄×C₅₉	1	1	yes	characteristic
427	C₂×C₄²	D₄×C₅₉	1	1	yes	characteristic
428	C₂×C₄²	D₄×C₅₉	1	1	yes	characteristic
429	C₂×C₄²	D₄×C₅₉	1	1	yes	characteristic
430	C₂×C₄²	D₄×C₅₉	1	1	yes	characteristic
431	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
432	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
433	C₄⋊C₈	—	2	2	no	—
434	C₄⋊C₈	—	2	2	no	—
435	C₂²⋊C₈	—	2	2	no	—
436	C₂²⋊C₈	C₂³×C₅₉	1	4	yes	—
437	#97	—	2	2	no	—
438	C₄×C₈	—	2	2	no	—
439	#95	—	2	2	no	—
440	#95	C₂³×C₅₉	1	4	yes	—
441	C₂⁴	—	2	1	no	—
442	C₂⁴	C₂×D₄×C₅₉	1	1	yes	characteristic
443	C₂×Q₈	—	2	1	no	—
444	C₂×Q₈	—	4	2	no	—
445	C₂×Q₈	C₂²×C₄×C₅₉	1	2	yes	—
446	C₂×D₄	—	2	2	no	—
447	C₂×D₄	—	2	2	no	—
448	C₂×D₄	—	2	1	no	—
449	C₂×D₄	—	4	2	no	—
450	C₂×D₄	—	4	4	no	—
451	C₂×D₄	C₂²×C₄×C₅₉	1	2	yes	—
452	C₂²×C₄	—	2	4	no	—
453	C₂²×C₄	—	2	1	no	—
454	C₂²×C₄	—	2	1	no	—
455	C₂²×C₄	—	2	1	no	—
456	C₂²×C₄	—	2	4	no	—
457	C₂²×C₄	—	2	2	no	—
458	C₂²×C₄	—	2	2	no	—
459	C₂²×C₄	—	4	2	no	—
460	C₂²×C₄	—	4	2	no	—
461	C₂²×C₄	—	4	4	no	—
462	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
463	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
464	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
465	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
466	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
467	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
468	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
469	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
470	C₂²×C₄	C₂²×C₄×C₅₉	1	1	yes	characteristic
471	C₂²×C₄	C₂²×C₄×C₅₉	1	1	yes	characteristic
472	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
473	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
474	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
475	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
476	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
477	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
478	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
479	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
480	M₁₆	—	4	4	no	—
481	C₂×C₈	—	2	4	no	—
482	C₄⋊C₄	—	2	8	no	—
483	C₄⋊C₄	—	2	2	no	—
484	C₄⋊C₄	—	2	2	no	—
485	C₄⋊C₄	—	2	2	no	—
486	C₄⋊C₄	—	2	2	no	—
487	C₄⋊C₄	—	2	2	no	—
488	C₄⋊C₄	—	2	2	no	—
489	C₄⋊C₄	—	2	2	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	2	no	—
493	C₄⋊C₄	C₂²×C₄×C₅₉	1	2	yes	—
494	C₄⋊C₄	C₂²×C₄×C₅₉	1	2	yes	—
495	C₂²⋊C₄	—	2	8	no	—
496	C₂²⋊C₄	—	2	2	no	—
497	C₂²⋊C₄	—	2	2	no	—
498	C₂²⋊C₄	—	2	2	no	—
499	C₂²⋊C₄	—	2	2	no	—
500	C₂²⋊C₄	—	2	2	no	—
501	C₂²⋊C₄	—	2	2	no	—
502	C₂²⋊C₄	—	2	2	no	—
503	C₂²⋊C₄	—	2	2	no	—
504	C₂²⋊C₄	—	2	2	no	—
505	C₂²⋊C₄	—	2	2	no	—
506	C₄²	—	2	8	no	—
507	C₄²	—	2	2	no	—
508	C₄²	—	2	2	no	—
509	C₄²	—	4	4	no	—
510	C₄²	—	4	4	no	—
511	C₄²	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
512	C₄²	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
513	C₄²	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
514	C₄²	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
515	C₄²	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
516	C₄²	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
517	C₄²	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
518	C₄²	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
519	C₄²	C₂²×C₄×C₅₉	1	1	yes	characteristic
520	C₄²	C₂²×C₄×C₅₉	1	1	yes	characteristic
521	C₄²	C₂²×C₄×C₅₉	1	1	yes	characteristic
522	C₄²	C₂²×C₄×C₅₉	1	1	yes	characteristic
523	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
524	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
525	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
526	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
527	C₂³	—	2	2	no	—
528	C₂³	—	2	2	no	—
529	C₂³	—	2	2	no	—
530	C₂³	—	4	1	no	—
531	C₂³	—	4	2	no	—
532	C₂³	—	4	2	no	—
533	C₂³	—	4	2	no	—
534	C₂³	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
535	C₂³	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
536	C₂³	C₂²×D₄×C₅₉	1	1	yes	characteristic
537	Q₈	—	4	2	no	—
538	Q₈	—	4	2	no	—
539	D₄	—	4	2	no	—
540	D₄	—	4	4	no	—
541	D₄	—	4	4	no	—
542	D₄	—	4	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	8	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	2	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	4	no	—
565	C₂×C₄	—	4	2	no	—
566	C₂×C₄	—	4	2	no	—
567	C₂×C₄	—	4	2	no	—
568	C₂×C₄	—	4	4	no	—
569	C₂×C₄	—	4	4	no	—
570	C₂×C₄	—	4	4	no	—
571	C₂×C₄	#104×C₅₉	1	4	yes	—
572	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
573	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
574	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
575	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
576	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
577	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
578	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
579	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
580	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
581	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
582	C₂×C₄	C₂³×C₄×C₅₉	1	1	yes	characteristic
583	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
584	C₈	—	4	4	no	—
585	C₂²	—	2	1	no	—
586	C₂²	—	2	1	no	—
587	C₂²	—	2	1	no	—
588	C₂²	—	2	1	no	—
589	C₂²	—	2	1	no	—
590	C₂²	—	2	1	no	—
591	C₂²	—	2	1	no	—
592	C₂²	—	4	4	no	—
593	C₂²	—	4	2	no	—
594	C₂²	—	4	2	no	—
595	C₂²	—	4	2	no	—
596	C₂²	—	8	4	no	—
597	C₂²	C₂×#104×C₅₉	1	2	yes	—
598	C₂²	C₂²×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
599	C₂²	C₅₉×#575	1	2	yes	—
600	C₄	—	2	1	no	—
601	C₄	—	2	1	no	—
602	C₄	—	2	1	no	—
603	C₄	—	2	1	no	—
604	C₄	—	2	1	no	—
605	C₄	—	2	1	no	—
606	C₄	—	2	1	no	—
607	C₄	—	2	1	no	—
608	C₄	—	2	1	no	—
609	C₄	—	4	8	no	—
610	C₄	—	4	2	no	—
611	C₄	—	4	2	no	—
612	C₄	—	4	2	no	—
613	C₄	C₅₉×#410	1	2	yes	—
614	C₄	C₂×#104×C₅₉	1	2	yes	—
615	C₄	C₂×#104×C₅₉	1	2	yes	—
616	C₂	—	2	2	no	—
617	C₂	—	2	2	no	—
618	C₂	—	4	1	no	—
619	C₂	—	8	2	no	—
620	C₂	C₂×C₅₉×#410	1	1	yes	characteristic
621	C₂	C₅₉×#2896	1	1	yes	characteristic
622	C₂	C₂²×#104×C₅₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₉	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₅₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₉
socle	C₂²×C₅₉	C₂²×C₂²⋊C₄
prime radical	C₂×#121×C₅₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₉ ◁ C₂²×C₄²×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₉×#2920	C₂	semidirect
3. C₄×#117×C₅₉	C₂	semidirect
4. C₅₉×#1509	C₂	semidirect
5. C₅₉×#1508	C₂	semidirect
7. C₅₉×#1500	C₂	semidirect
12. C₅₉×#529	C₄	semidirect
13. C₅₉×#529	C₄	semidirect
17. C₂×#125×C₅₉	C₄	semidirect
18. C₂×#124×C₅₉	C₄	semidirect
22. C₂×C₄×Q₈×C₅₉	C₄	semidirect
24. C₂×C₄×D₄×C₅₉	C₄	semidirect
29. C₅₉×#432	C₂²	semidirect
31. C₅₉×#431	C₂²	semidirect
36. C₄×C₄⋊C₄×C₅₉	C₂²	semidirect
39. C₅₉×#376	C₂²	semidirect
41. C₅₉×#328	C₂²	semidirect
273. #17537	C₅₉	direct (N × H)
306. #2920	C₂×C₅₉	semidirect
307. C₄×#117	C₂×C₅₉	semidirect
308. #1509	C₂×C₅₉	semidirect
309. #1508	C₂×C₅₉	semidirect
311. #1500	C₂×C₅₉	semidirect
323. #529	C₄×C₅₉	semidirect
324. #529	C₄×C₅₉	semidirect
328. C₂×#125	C₄×C₅₉	semidirect
329. C₂×#124	C₄×C₅₉	semidirect
333. C₂×C₄×Q₈	C₄×C₅₉	semidirect
335. C₂×C₄×D₄	C₄×C₅₉	semidirect
340. #432	C₂²×C₅₉	semidirect
342. #431	C₂²×C₅₉	semidirect
347. C₄×C₄⋊C₄	C₂²×C₅₉	semidirect
350. #376	C₂²×C₅₉	semidirect
352. #328	C₂²×C₅₉	semidirect
354. C₅₉	#17537	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×#529	2
C₅₉×#2920	2
C₄×#117×C₅₉	2
C₅₉×#1509	2
C₅₉×#1508	2
C₅₉×#1507	2
C₅₉×#1500	2
#17537	59

Subgroups — C59×#17537 #28429646

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#17537 #28429646