Subgroups — C₅₉×#18593 #28430702

Order 15104.

916 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#3511	C₂	1	1	yes	maximal, characteristic
2	C₅₉×#3254	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₉×#3165	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₉×#3107	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#1654	C₂	1	2	yes	maximal, complemented
6	C₅₉×#1652	C₂	1	2	yes	maximal, complemented
7	C₅₉×#1650	C₂	1	2	yes	maximal, complemented
8	C₅₉×#1647	C₂	1	2	yes	maximal, complemented
9	C₅₉×#1546	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₉×#1489	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₉×#1484	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₉×#556	C₂²	1	2	yes	—
13	C₅₉×#555	—	2	1	no	—
14	C₅₉×#554	—	2	1	no	—
15	C₅₉×#553	—	2	2	no	—
16	C₅₉×#548	—	2	2	no	—
17	C₅₉×#547	C₂²	1	2	yes	—
18	C₅₉×#542	—	2	2	no	—
19	C₅₉×#537	—	2	1	no	—
20	C₅₉×#533	—	2	1	no	—
21	C₅₉×#533	C₂²	1	1	yes	characteristic
22	C₅₉×#532	—	2	1	no	—
23	C₅₉×#532	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₅₉×#520	—	2	1	no	—
26	C₄×#41×C₅₉	C₂²	1	1	yes	characteristic
27	C₅₉×#501	—	2	2	no	—
28	C₅₉×#500	C₂²	1	2	yes	—
29	C₅₉×#498	C₂²	1	2	yes	—
30	C₅₉×#485	—	2	1	no	—
31	C₅₉×#477	—	2	1	no	—
32	C₅₉×#475	C₂²	1	1	yes	characteristic
33	C₅₉×#474	C₂²	1	2	yes	—
34	C₅₉×#474	C₂²	1	1	yes	characteristic
35	C₅₉×#468	—	2	2	no	—
36	C₅₉×#462	C₂²	1	1	yes	characteristic
37	C₅₉×#461	C₂²	1	2	yes	—
38	C₅₉×#461	C₂²	1	1	yes	characteristic
39	C₅₉×#460	—	2	1	no	—
40	C₅₉×#452	—	2	1	no	—
41	C₄×Q₁₆×C₅₉	—	2	1	no	—
42	C₄×SD₁₆×C₅₉	—	2	1	no	—
43	C₅₉×#431	C₂²	1	1	yes	characteristic
44	C₅₉×#428	—	2	2	no	—
45	C₅₉×#423	C₂²	1	1	yes	characteristic
46	C₄×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
47	C₅₉×#332	C₂²	1	2	yes	—
48	C₅₉×#332	C₂²	1	2	yes	—
49	C₅₉×Q₈⋊C₈	C₂²	1	2	yes	—
50	C₅₉×D₄⋊C₈	C₂²	1	2	yes	—
51	C₅₉×D₄⋊C₈	C₂²	1	2	yes	—
52	C₅₉×D₄⋊C₈	C₂²	1	2	yes	—
53	C₂×#41×C₅₉	—	2	2	no	—
54	C₂×#41×C₅₉	—	2	1	no	—
55	C₂×#41×C₅₉	D₄	1	1	yes	characteristic
56	C₂²×Q₈×C₅₉	—	2	1	no	—
57	C₂×Q₁₆×C₅₉	—	2	1	no	—
58	C₂×SD₁₆×C₅₉	—	2	1	no	—
59	C₂×SD₁₆×C₅₉	—	2	2	no	—
60	C₂×M₁₆×C₅₉	—	2	1	no	—
61	C₂×M₁₆×C₅₉	—	2	1	no	—
62	C₄⋊Q₈×C₅₉	—	2	1	no	—
63	C₄⋊Q₈×C₅₉	—	2	1	no	—
64	C₄⋊Q₈×C₅₉	D₄	1	1	yes	characteristic
65	C₄⋊Q₈×C₅₉	D₄	1	2	yes	—
66	C₄⋊Q₈×C₅₉	C₂³	1	1	yes	characteristic
67	C₄⋊Q₈×C₅₉	C₂³	1	1	yes	characteristic
68	C₄⋊Q₈×C₅₉	C₂³	1	2	yes	—
69	#127×C₅₉	D₄	1	1	yes	characteristic
70	#126×C₅₉	—	2	4	no	—
71	#125×C₅₉	—	2	2	no	—
72	#125×C₅₉	—	2	1	no	—
73	#124×C₅₉	—	2	4	no	—
74	#124×C₅₉	—	2	2	no	—
75	#124×C₅₉	—	2	1	no	—
76	#123×C₅₉	—	2	4	no	—
77	C₂²⋊Q₈×C₅₉	—	2	2	no	—
78	C₂²⋊Q₈×C₅₉	—	2	4	no	—
79	C₂²⋊Q₈×C₅₉	—	2	2	no	—
80	C₂²⋊Q₈×C₅₉	—	2	2	no	—
81	C₂²⋊Q₈×C₅₉	—	2	1	no	—
82	C₂²⋊Q₈×C₅₉	—	2	1	no	—
83	#121×C₅₉	—	2	2	no	—
84	#121×C₅₉	—	2	2	no	—
85	#121×C₅₉	—	2	4	no	—
86	#121×C₅₉	—	2	2	no	—
87	#121×C₅₉	—	2	1	no	—
88	#121×C₅₉	D₄	1	2	yes	—
89	C₄×Q₈×C₅₉	—	2	2	no	—
90	C₄×Q₈×C₅₉	—	2	2	no	—
91	C₄×Q₈×C₅₉	—	2	1	no	—
92	C₄×Q₈×C₅₉	—	2	1	no	—
93	C₄×Q₈×C₅₉	D₄	1	1	yes	characteristic
94	C₄×Q₈×C₅₉	C₂³	1	1	yes	characteristic
95	C₄×D₄×C₅₉	—	2	2	no	—
96	C₄×D₄×C₅₉	—	2	4	no	—
97	C₄×D₄×C₅₉	—	2	2	no	—
98	C₄×D₄×C₅₉	—	2	1	no	—
99	C₄×D₄×C₅₉	—	2	1	no	—
100	C₄×D₄×C₅₉	—	2	1	no	—
101	C₄×D₄×C₅₉	D₄	1	1	yes	characteristic
102	C₄×D₄×C₅₉	C₂³	1	1	yes	characteristic
103	C₄×D₄×C₅₉	C₂³	1	2	yes	—
104	#117×C₅₉	—	2	2	no	—
105	#117×C₅₉	—	2	2	no	—
106	#117×C₅₉	—	2	1	no	—
107	#117×C₅₉	—	2	1	no	—
108	#117×C₅₉	D₄	1	1	yes	characteristic
109	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
110	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
111	C₂×C₄⋊C₄×C₅₉	D₄	1	2	yes	—
112	C₂×C₄²×C₅₉	—	2	1	no	—
113	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
114	#107×C₅₉	—	2	4	no	—
115	#107×C₅₉	—	2	1	no	—
116	#107×C₅₉	—	2	2	no	—
117	#106×C₅₉	—	2	4	no	—
118	#106×C₅₉	—	2	1	no	—
119	C₄⋊C₈×C₅₉	—	2	1	no	—
120	C₄⋊C₈×C₅₉	C₂³	1	2	yes	—
121	C₄⋊C₈×C₅₉	C₂³	1	1	yes	characteristic
122	C₄⋊C₈×C₅₉	C₂³	1	1	yes	characteristic
123	#103×C₅₉	—	2	1	no	—
124	#103×C₅₉	—	2	1	no	—
125	#103×C₅₉	—	2	1	no	—
126	#103×C₅₉	—	2	2	no	—
127	#102×C₅₉	—	2	1	no	—
128	#102×C₅₉	—	2	2	no	—
129	C₂²⋊C₈×C₅₉	—	2	1	no	—
130	#97×C₅₉	—	2	1	no	—
131	C₄×C₈×C₅₉	—	2	1	no	—
132	C₄×C₈×C₅₉	C₂³	1	2	yes	—
133	#41×C₅₉	—	4	1	no	—
134	#41×C₅₉	—	4	1	no	—
135	#41×C₅₉	—	4	2	no	—
136	#41×C₅₉	—	4	2	no	—
137	#41×C₅₉	—	8	2	no	—
138	#41×C₅₉	—	8	2	no	—
139	C₂×Q₈×C₅₉	—	2	1	no	—
140	C₂×Q₈×C₅₉	—	2	2	no	—
141	C₂×Q₈×C₅₉	—	2	2	no	—
142	C₂×Q₈×C₅₉	—	2	1	no	—
143	C₂×Q₈×C₅₉	—	2	1	no	—
144	C₂×Q₈×C₅₉	—	8	1	no	—
145	C₂×Q₈×C₅₉	—	8	1	no	—
146	C₂×Q₈×C₅₉	C₂×D₄	1	1	yes	characteristic
147	C₂×D₄×C₅₉	—	2	1	no	—
148	C₂×D₄×C₅₉	—	2	2	no	—
149	C₂×D₄×C₅₉	—	2	4	no	—
150	C₂×D₄×C₅₉	—	2	1	no	—
151	C₂×D₄×C₅₉	—	2	1	no	—
152	C₂×D₄×C₅₉	C₂×D₄	1	1	yes	characteristic
153	C₂×D₄×C₅₉	C₂×D₄	1	2	yes	—
154	C₂²×C₄×C₅₉	—	2	4	no	—
155	C₂²×C₄×C₅₉	—	2	2	no	—
156	C₂²×C₄×C₅₉	—	2	1	no	—
157	C₂²×C₄×C₅₉	—	2	1	no	—
158	C₂²×C₄×C₅₉	—	2	1	no	—
159	C₂²×C₄×C₅₉	—	2	2	no	—
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	Q₁₆×C₅₉	—	4	2	no	—
164	SD₁₆×C₅₉	—	4	2	no	—
165	SD₁₆×C₅₉	—	8	2	no	—
166	M₁₆×C₅₉	—	2	4	no	—
167	M₁₆×C₅₉	—	8	1	no	—
168	C₂×C₈×C₅₉	—	2	2	no	—
169	C₂×C₈×C₅₉	—	2	1	no	—
170	C₂×C₈×C₅₉	—	2	1	no	—
171	C₂×C₈×C₅₉	—	2	2	no	—
172	C₄⋊C₄×C₅₉	—	2	4	no	—
173	C₄⋊C₄×C₅₉	—	2	4	no	—
174	C₄⋊C₄×C₅₉	—	2	2	no	—
175	C₄⋊C₄×C₅₉	—	2	2	no	—
176	C₄⋊C₄×C₅₉	—	2	1	no	—
177	C₄⋊C₄×C₅₉	—	2	1	no	—
178	C₄⋊C₄×C₅₉	—	2	2	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	—	4	1	no	—
185	C₄⋊C₄×C₅₉	—	4	1	no	—
186	C₄⋊C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
187	C₄⋊C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
188	C₄⋊C₄×C₅₉	C₂×D₄	1	2	yes	—
189	C₄⋊C₄×C₅₉	C₂×D₄	1	4	yes	—
190	C₂²⋊C₄×C₅₉	—	2	4	no	—
191	C₂²⋊C₄×C₅₉	—	2	4	no	—
192	C₂²⋊C₄×C₅₉	—	2	4	no	—
193	C₂²⋊C₄×C₅₉	—	2	1	no	—
194	C₂²⋊C₄×C₅₉	—	2	1	no	—
195	C₂²⋊C₄×C₅₉	—	2	2	no	—
196	C₂²⋊C₄×C₅₉	—	2	1	no	—
197	C₂²⋊C₄×C₅₉	—	2	2	no	—
198	C₂²⋊C₄×C₅₉	—	4	1	no	—
199	C₄²×C₅₉	—	2	4	no	—
200	C₄²×C₅₉	—	2	1	no	—
201	C₄²×C₅₉	—	2	1	no	—
202	C₄²×C₅₉	—	2	1	no	—
203	C₄²×C₅₉	—	4	1	no	—
204	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
205	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
206	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
207	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
208	C₂³×C₅₉	—	2	1	no	—
209	C₂³×C₅₉	—	2	2	no	—
210	C₂³×C₅₉	#120	1	1	yes	characteristic
211	Q₈×C₅₉	—	2	1	no	—
212	Q₈×C₅₉	—	2	1	no	—
213	Q₈×C₅₉	—	4	2	no	—
214	Q₈×C₅₉	—	4	2	no	—
215	Q₈×C₅₉	—	4	2	no	—
216	Q₈×C₅₉	—	4	1	no	—
217	Q₈×C₅₉	—	4	1	no	—
218	Q₈×C₅₉	—	4	1	no	—
219	Q₈×C₅₉	—	4	1	no	—
220	Q₈×C₅₉	—	8	2	no	—
221	D₄×C₅₉	—	2	1	no	—
222	D₄×C₅₉	—	2	1	no	—
223	D₄×C₅₉	—	2	2	no	—
224	D₄×C₅₉	—	2	2	no	—
225	D₄×C₅₉	—	4	2	no	—
226	D₄×C₅₉	—	4	2	no	—
227	D₄×C₅₉	—	4	4	no	—
228	D₄×C₅₉	—	4	2	no	—
229	D₄×C₅₉	—	8	4	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	2	no	—
235	C₂×C₄×C₅₉	—	2	1	no	—
236	C₂×C₄×C₅₉	—	2	1	no	—
237	C₂×C₄×C₅₉	—	2	2	no	—
238	C₂×C₄×C₅₉	—	2	1	no	—
239	C₂×C₄×C₅₉	—	2	4	no	—
240	C₂×C₄×C₅₉	—	4	2	no	—
241	C₂×C₄×C₅₉	—	4	2	no	—
242	C₂×C₄×C₅₉	—	4	2	no	—
243	C₂×C₄×C₅₉	—	4	1	no	—
244	C₂×C₄×C₅₉	—	4	1	no	—
245	C₂×C₄×C₅₉	—	4	2	no	—
246	C₂×C₄×C₅₉	—	4	1	no	—
247	C₂×C₄×C₅₉	—	4	1	no	—
248	C₂×C₄×C₅₉	—	8	2	no	—
249	C₂×C₄×C₅₉	—	8	4	no	—
250	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
251	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
252	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
253	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
254	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
255	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
256	C₈×C₅₉	—	2	4	no	—
257	C₈×C₅₉	—	4	1	no	—
258	C₈×C₅₉	—	4	1	no	—
259	C₈×C₅₉	—	4	2	no	—
260	#18593	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
261	C₂²×C₅₉	—	2	1	no	—
262	C₂²×C₅₉	—	2	1	no	—
263	C₂²×C₅₉	—	2	1	no	—
264	C₂²×C₅₉	—	4	1	no	—
265	C₂²×C₅₉	—	4	2	no	—
266	C₂²×C₅₉	—	4	1	no	—
267	C₂²×C₅₉	—	4	2	no	—
268	C₂²×C₅₉	—	4	1	no	—
269	C₂²×C₅₉	—	4	2	no	—
270	C₂²×C₅₉	C₂×#120	1	1	yes	characteristic
271	C₄×C₅₉	—	2	1	no	—
272	C₄×C₅₉	—	2	1	no	—
273	C₄×C₅₉	—	2	1	no	—
274	C₄×C₅₉	—	4	1	no	—
275	C₄×C₅₉	—	4	2	no	—
276	C₄×C₅₉	—	4	1	no	—
277	C₄×C₅₉	—	4	4	no	—
278	C₄×C₅₉	—	4	1	no	—
279	C₄×C₅₉	—	8	1	no	—
280	C₄×C₅₉	—	8	1	no	—
281	C₄×C₅₉	#456	1	2	yes	—
282	C₄×C₅₉	#575	1	1	yes	characteristic
283	C₄×C₅₉	#575	1	1	yes	characteristic
284	C₄×C₅₉	#576	1	1	yes	characteristic
285	C₄×C₅₉	#577	1	1	yes	characteristic
286	#3511	C₂×C₅₉	1	1	yes	characteristic
287	#3254	C₂×C₅₉	1	1	yes	characteristic, complemented
288	#3165	C₂×C₅₉	1	1	yes	characteristic, complemented
289	#3107	C₂×C₅₉	1	1	yes	characteristic, complemented
290	#1654	C₂×C₅₉	1	2	yes	complemented
291	#1652	C₂×C₅₉	1	2	yes	complemented
292	#1650	C₂×C₅₉	1	2	yes	complemented
293	#1647	C₂×C₅₉	1	2	yes	complemented
294	#1546	C₂×C₅₉	1	1	yes	characteristic, complemented
295	#1489	C₂×C₅₉	1	1	yes	characteristic, complemented
296	#1484	C₂×C₅₉	1	1	yes	characteristic, complemented
297	C₂×C₅₉	—	4	1	no	—
298	C₂×C₅₉	—	8	1	no	—
299	C₂×C₅₉	—	8	2	no	—
300	C₂×C₅₉	#3005	1	1	yes	characteristic
301	C₂×C₅₉	#3012	1	1	yes	characteristic
302	C₂×C₅₉	C₂×#456	1	1	yes	characteristic
303	#556	C₂²×C₅₉	1	2	yes	—
304	#555	—	2	1	no	—
305	#554	—	2	1	no	—
306	#553	—	2	2	no	—
307	#548	—	2	2	no	—
308	#547	C₂²×C₅₉	1	2	yes	—
309	#542	—	2	2	no	—
310	#537	—	2	1	no	—
311	#533	—	2	1	no	—
312	#533	C₂²×C₅₉	1	1	yes	characteristic
313	#532	—	2	1	no	—
314	#532	C₂²×C₅₉	1	1	yes	characteristic
315	C₂×C₄⋊Q₈	C₂²×C₅₉	1	1	yes	characteristic
316	#520	—	2	1	no	—
317	C₄×#41	C₂²×C₅₉	1	1	yes	characteristic
318	#501	—	2	2	no	—
319	#500	C₂²×C₅₉	1	2	yes	—
320	#498	C₂²×C₅₉	1	2	yes	—
321	#485	—	2	1	no	—
322	#477	—	2	1	no	—
323	#475	C₂²×C₅₉	1	1	yes	characteristic
324	#474	C₂²×C₅₉	1	1	yes	characteristic
325	#474	C₂²×C₅₉	1	2	yes	—
326	#468	—	2	2	no	—
327	#462	C₂²×C₅₉	1	1	yes	characteristic
328	#461	C₂²×C₅₉	1	1	yes	characteristic
329	#461	C₂²×C₅₉	1	2	yes	—
330	#460	—	2	1	no	—
331	#452	—	2	1	no	—
332	C₄×Q₁₆	—	2	1	no	—
333	C₄×SD₁₆	—	2	1	no	—
334	#431	C₂²×C₅₉	1	1	yes	characteristic
335	#428	—	2	2	no	—
336	#423	C₂²×C₅₉	1	1	yes	characteristic
337	C₄×M₁₆	C₂²×C₅₉	1	1	yes	characteristic
338	#332	C₂²×C₅₉	1	2	yes	—
339	#332	C₂²×C₅₉	1	2	yes	—
340	Q₈⋊C₈	C₂²×C₅₉	1	2	yes	—
341	D₄⋊C₈	C₂²×C₅₉	1	2	yes	—
342	D₄⋊C₈	C₂²×C₅₉	1	2	yes	—
343	D₄⋊C₈	C₂²×C₅₉	1	2	yes	—
344	C₅₉	#18593	1	1	yes	characteristic, direct product factor, Sylow
345	C₂×#41	—	2	2	no	—
346	C₂×#41	—	2	1	no	—
347	C₂×#41	D₄×C₅₉	1	1	yes	characteristic
348	C₂²×Q₈	—	2	1	no	—
349	C₂×Q₁₆	—	2	1	no	—
350	C₂×SD₁₆	—	2	1	no	—
351	C₂×SD₁₆	—	2	2	no	—
352	C₂×M₁₆	—	2	1	no	—
353	C₂×M₁₆	—	2	1	no	—
354	C₄⋊Q₈	—	2	1	no	—
355	C₄⋊Q₈	—	2	1	no	—
356	C₄⋊Q₈	D₄×C₅₉	1	2	yes	—
357	C₄⋊Q₈	D₄×C₅₉	1	1	yes	characteristic
358	C₄⋊Q₈	C₂³×C₅₉	1	1	yes	characteristic
359	C₄⋊Q₈	C₂³×C₅₉	1	1	yes	characteristic
360	C₄⋊Q₈	C₂³×C₅₉	1	2	yes	—
361	#127	D₄×C₅₉	1	1	yes	characteristic
362	#126	—	2	4	no	—
363	#125	—	2	2	no	—
364	#125	—	2	1	no	—
365	#124	—	2	4	no	—
366	#124	—	2	2	no	—
367	#124	—	2	1	no	—
368	#123	—	2	4	no	—
369	C₂²⋊Q₈	—	2	2	no	—
370	C₂²⋊Q₈	—	2	4	no	—
371	C₂²⋊Q₈	—	2	2	no	—
372	C₂²⋊Q₈	—	2	2	no	—
373	C₂²⋊Q₈	—	2	1	no	—
374	C₂²⋊Q₈	—	2	1	no	—
375	#121	—	2	2	no	—
376	#121	—	2	2	no	—
377	#121	—	2	4	no	—
378	#121	—	2	2	no	—
379	#121	—	2	1	no	—
380	#121	D₄×C₅₉	1	2	yes	—
381	C₄×Q₈	—	2	2	no	—
382	C₄×Q₈	—	2	2	no	—
383	C₄×Q₈	—	2	1	no	—
384	C₄×Q₈	—	2	1	no	—
385	C₄×Q₈	D₄×C₅₉	1	1	yes	characteristic
386	C₄×Q₈	C₂³×C₅₉	1	1	yes	characteristic
387	C₄×D₄	—	2	2	no	—
388	C₄×D₄	—	2	4	no	—
389	C₄×D₄	—	2	2	no	—
390	C₄×D₄	—	2	1	no	—
391	C₄×D₄	—	2	1	no	—
392	C₄×D₄	—	2	1	no	—
393	C₄×D₄	D₄×C₅₉	1	1	yes	characteristic
394	C₄×D₄	C₂³×C₅₉	1	1	yes	characteristic
395	C₄×D₄	C₂³×C₅₉	1	2	yes	—
396	#117	—	2	2	no	—
397	#117	—	2	2	no	—
398	#117	—	2	1	no	—
399	#117	—	2	1	no	—
400	#117	D₄×C₅₉	1	1	yes	characteristic
401	C₂×C₄⋊C₄	—	2	1	no	—
402	C₂×C₄⋊C₄	—	2	2	no	—
403	C₂×C₄⋊C₄	D₄×C₅₉	1	2	yes	—
404	C₂×C₄²	—	2	1	no	—
405	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
406	#107	—	2	4	no	—
407	#107	—	2	1	no	—
408	#107	—	2	2	no	—
409	#106	—	2	4	no	—
410	#106	—	2	1	no	—
411	C₄⋊C₈	—	2	1	no	—
412	C₄⋊C₈	C₂³×C₅₉	1	1	yes	characteristic
413	C₄⋊C₈	C₂³×C₅₉	1	1	yes	characteristic
414	C₄⋊C₈	C₂³×C₅₉	1	2	yes	—
415	#103	—	2	1	no	—
416	#103	—	2	1	no	—
417	#103	—	2	1	no	—
418	#103	—	2	2	no	—
419	#102	—	2	1	no	—
420	#102	—	2	2	no	—
421	C₂²⋊C₈	—	2	1	no	—
422	#97	—	2	1	no	—
423	C₄×C₈	—	2	1	no	—
424	C₄×C₈	C₂³×C₅₉	1	2	yes	—
425	#41	—	4	1	no	—
426	#41	—	4	1	no	—
427	#41	—	4	2	no	—
428	#41	—	4	2	no	—
429	#41	—	8	2	no	—
430	#41	—	8	2	no	—
431	C₂×Q₈	—	2	1	no	—
432	C₂×Q₈	—	2	2	no	—
433	C₂×Q₈	—	2	2	no	—
434	C₂×Q₈	—	2	1	no	—
435	C₂×Q₈	—	2	1	no	—
436	C₂×Q₈	—	8	1	no	—
437	C₂×Q₈	—	8	1	no	—
438	C₂×Q₈	C₂×D₄×C₅₉	1	1	yes	characteristic
439	C₂×D₄	—	2	1	no	—
440	C₂×D₄	—	2	2	no	—
441	C₂×D₄	—	2	4	no	—
442	C₂×D₄	—	2	1	no	—
443	C₂×D₄	—	2	1	no	—
444	C₂×D₄	C₂×D₄×C₅₉	1	1	yes	characteristic
445	C₂×D₄	C₂×D₄×C₅₉	1	2	yes	—
446	C₂²×C₄	—	2	4	no	—
447	C₂²×C₄	—	2	2	no	—
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	2	1	no	—
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	—	2	2	no	—
452	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
453	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
454	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
455	Q₁₆	—	4	2	no	—
456	SD₁₆	—	4	2	no	—
457	SD₁₆	—	8	2	no	—
458	M₁₆	—	2	4	no	—
459	M₁₆	—	8	1	no	—
460	C₂×C₈	—	2	2	no	—
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	2	2	no	—
464	C₄⋊C₄	—	2	4	no	—
465	C₄⋊C₄	—	2	4	no	—
466	C₄⋊C₄	—	2	2	no	—
467	C₄⋊C₄	—	2	2	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	2	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	4	1	no	—
477	C₄⋊C₄	—	4	1	no	—
478	C₄⋊C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
479	C₄⋊C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
480	C₄⋊C₄	C₂×D₄×C₅₉	1	2	yes	—
481	C₄⋊C₄	C₂×D₄×C₅₉	1	4	yes	—
482	C₂²⋊C₄	—	2	4	no	—
483	C₂²⋊C₄	—	2	4	no	—
484	C₂²⋊C₄	—	2	4	no	—
485	C₂²⋊C₄	—	2	1	no	—
486	C₂²⋊C₄	—	2	1	no	—
487	C₂²⋊C₄	—	2	2	no	—
488	C₂²⋊C₄	—	2	1	no	—
489	C₂²⋊C₄	—	2	2	no	—
490	C₂²⋊C₄	—	4	1	no	—
491	C₄²	—	2	4	no	—
492	C₄²	—	2	1	no	—
493	C₄²	—	2	1	no	—
494	C₄²	—	2	1	no	—
495	C₄²	—	4	1	no	—
496	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
497	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
498	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
499	C₄²	C₂⁴×C₅₉	1	1	yes	characteristic
500	C₂³	—	2	1	no	—
501	C₂³	—	2	2	no	—
502	C₂³	#120×C₅₉	1	1	yes	characteristic
503	Q₈	—	2	1	no	—
504	Q₈	—	2	1	no	—
505	Q₈	—	4	2	no	—
506	Q₈	—	4	2	no	—
507	Q₈	—	4	2	no	—
508	Q₈	—	4	1	no	—
509	Q₈	—	4	1	no	—
510	Q₈	—	4	1	no	—
511	Q₈	—	4	1	no	—
512	Q₈	—	8	2	no	—
513	D₄	—	2	1	no	—
514	D₄	—	2	1	no	—
515	D₄	—	2	2	no	—
516	D₄	—	2	2	no	—
517	D₄	—	4	2	no	—
518	D₄	—	4	2	no	—
519	D₄	—	4	4	no	—
520	D₄	—	4	2	no	—
521	D₄	—	8	4	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	4	2	no	—
533	C₂×C₄	—	4	2	no	—
534	C₂×C₄	—	4	2	no	—
535	C₂×C₄	—	4	2	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	8	2	no	—
541	C₂×C₄	—	8	4	no	—
542	C₂×C₄	#120×C₅₉	1	1	yes	characteristic
543	C₂×C₄	#120×C₅₉	1	1	yes	characteristic
544	C₂×C₄	#120×C₅₉	1	1	yes	characteristic
545	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
546	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
547	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
548	C₈	—	2	4	no	—
549	C₈	—	4	1	no	—
550	C₈	—	4	1	no	—
551	C₈	—	4	2	no	—
552	C₂²	—	2	1	no	—
553	C₂²	—	2	1	no	—
554	C₂²	—	2	1	no	—
555	C₂²	—	4	1	no	—
556	C₂²	—	4	2	no	—
557	C₂²	—	4	1	no	—
558	C₂²	—	4	2	no	—
559	C₂²	—	4	1	no	—
560	C₂²	—	4	2	no	—
561	C₂²	C₂×#120×C₅₉	1	1	yes	characteristic
562	C₄	—	2	1	no	—
563	C₄	—	2	1	no	—
564	C₄	—	2	1	no	—
565	C₄	—	4	1	no	—
566	C₄	—	4	2	no	—
567	C₄	—	4	1	no	—
568	C₄	—	4	4	no	—
569	C₄	—	4	1	no	—
570	C₄	—	8	1	no	—
571	C₄	—	8	1	no	—
572	C₄	C₅₉×#456	1	2	yes	—
573	C₄	C₅₉×#575	1	1	yes	characteristic
574	C₄	C₅₉×#575	1	1	yes	characteristic
575	C₄	C₅₉×#576	1	1	yes	characteristic
576	C₄	C₅₉×#577	1	1	yes	characteristic
577	C₂	—	4	1	no	—
578	C₂	—	8	1	no	—
579	C₂	—	8	2	no	—
580	C₂	C₅₉×#3005	1	1	yes	characteristic
581	C₂	C₅₉×#3012	1	1	yes	characteristic
582	C₂	C₂×C₅₉×#456	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₉
prime radical	C₅₉×#3511	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₂×C₄²×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₉×#3254	C₂	semidirect
3. C₅₉×#3165	C₂	semidirect
4. C₅₉×#3107	C₂	semidirect
5. C₅₉×#1654	C₂	semidirect
6. C₅₉×#1652	C₂	semidirect
7. C₅₉×#1650	C₂	semidirect
8. C₅₉×#1647	C₂	semidirect
9. C₅₉×#1546	C₂	semidirect
10. C₅₉×#1489	C₂	semidirect
11. C₅₉×#1484	C₂	semidirect
260. #18593	C₅₉	direct (N × H)
287. #3254	C₂×C₅₉	semidirect
288. #3165	C₂×C₅₉	semidirect
289. #3107	C₂×C₅₉	semidirect
290. #1654	C₂×C₅₉	semidirect
291. #1652	C₂×C₅₉	semidirect
292. #1650	C₂×C₅₉	semidirect
293. #1647	C₂×C₅₉	semidirect
294. #1546	C₂×C₅₉	semidirect
295. #1489	C₂×C₅₉	semidirect
296. #1484	C₂×C₅₉	semidirect
344. C₅₉	#18593	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×#3511	2
C₅₉×#3254	2
C₅₉×#3165	2
C₅₉×#3107	2
C₅₉×#1654	2
C₅₉×#1652	2
C₅₉×#1650	2
C₅₉×#1647	2
C₅₉×#1546	2
C₅₉×#1489	2
C₅₉×#1484	2
#18593	59

Subgroups — C59×#18593 #28430702

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#18593 #28430702