Subgroups — C₅₉×#23590 #28435699

Order 15104.

1074 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#3461	C₂	1	1	yes	maximal, characteristic
2	C₄×#137×C₅₉	C₂	1	2	yes	maximal, complemented
3	C₄×C₄⋊Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₉×#2060	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#2039	C₂	1	1	yes	maximal, characteristic
6	C₅₉×#2038	C₂	1	2	yes	maximal, complemented
7	C₅₉×#2028	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₉×#2007	C₂	1	1	yes	maximal, characteristic
9	C₅₉×#2005	C₂	1	2	yes	maximal, complemented
10	C₄×#104×C₅₉	C₂	1	2	yes	maximal, complemented
11	C₄×#101×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137×C₅₉	C₂²	1	2	yes	—
13	C₅₉×#544	C₂²	1	1	yes	characteristic
14	C₅₉×#542	—	2	2	no	—
15	C₅₉×#541	—	2	2	no	—
16	C₅₉×#539	—	2	2	no	—
17	C₅₉×#537	C₂²	1	1	yes	characteristic
18	C₅₉×#536	C₂²	1	2	yes	—
19	C₅₉×#533	—	2	2	no	—
20	C₂×C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₅₉×#529	—	2	2	no	—
22	C₅₉×#529	—	2	2	no	—
23	C₄×#41×C₅₉	C₂²	1	2	yes	—
24	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
26	C₅₉×#441	—	2	2	no	—
27	C₅₉×#440	—	2	2	no	—
28	C₄×Q₁₆×C₅₉	—	2	2	no	—
29	C₄×SD₁₆×C₅₉	—	2	2	no	—
30	C₅₉×#428	C₂²	1	2	yes	—
31	C₂×#104×C₅₉	C₂²	1	2	yes	—
32	C₂×#104×C₅₉	C₂²	1	2	yes	—
33	C₅₉×#419	C₂²	1	2	yes	—
34	C₅₉×#417	C₂²	1	2	yes	—
35	C₂×#101×C₅₉	C₂²	1	2	yes	—
36	C₄×M₁₆×C₅₉	C₂²	1	2	yes	—
37	C₅₉×#391	C₂²	1	1	yes	characteristic
38	C₅₉×#391	C₂²	1	1	yes	characteristic
39	C₅₉×#389	—	2	2	no	—
40	C₅₉×#384	C₂²	1	1	yes	characteristic
41	C₄×C₄⋊C₄×C₅₉	—	2	2	no	—
42	C₄³×C₅₉	C₂²	1	1	yes	characteristic
43	C₅₉×#343	C₂²	1	1	yes	characteristic
44	C₅₉×#343	C₂²	1	1	yes	characteristic
45	C₅₉×#339	C₂²	1	2	yes	—
46	C₅₉×#339	C₂²	1	2	yes	—
47	C₂×#41×C₅₉	C₂³	1	2	yes	—
48	C₂²×Q₈×C₅₉	C₂³	1	1	yes	characteristic
49	C₂²×Q₈×C₅₉	C₂³	1	1	yes	characteristic
50	#137×C₅₉	—	2	4	no	—
51	#137×C₅₉	—	2	4	no	—
52	C₂×Q₁₆×C₅₉	—	2	2	no	—
53	C₂×SD₁₆×C₅₉	—	2	2	no	—
54	C₂×M₁₆×C₅₉	C₂³	1	2	yes	—
55	C₂×M₁₆×C₅₉	C₂³	1	2	yes	—
56	C₄⋊Q₈×C₅₉	—	2	2	no	—
57	C₄⋊Q₈×C₅₉	—	2	2	no	—
58	C₄⋊Q₈×C₅₉	—	2	2	no	—
59	C₄⋊Q₈×C₅₉	—	2	2	no	—
60	#126×C₅₉	—	2	4	no	—
61	#126×C₅₉	—	2	4	no	—
62	#125×C₅₉	—	2	2	no	—
63	#125×C₅₉	—	2	4	no	—
64	#125×C₅₉	—	2	2	no	—
65	#124×C₅₉	—	2	2	no	—
66	#124×C₅₉	—	2	2	no	—
67	#123×C₅₉	—	2	4	no	—
68	#123×C₅₉	—	2	4	no	—
69	C₂²⋊Q₈×C₅₉	—	2	4	no	—
70	C₂²⋊Q₈×C₅₉	—	2	2	no	—
71	C₂²⋊Q₈×C₅₉	—	2	2	no	—
72	C₂²⋊Q₈×C₅₉	—	2	2	no	—
73	C₂²⋊Q₈×C₅₉	—	2	2	no	—
74	#121×C₅₉	—	2	2	no	—
75	#121×C₅₉	—	2	2	no	—
76	C₄×Q₈×C₅₉	—	2	2	no	—
77	C₄×Q₈×C₅₉	—	2	2	no	—
78	C₄×Q₈×C₅₉	—	2	2	no	—
79	C₄×Q₈×C₅₉	—	2	2	no	—
80	C₄×Q₈×C₅₉	—	2	2	no	—
81	C₄×Q₈×C₅₉	D₄	1	2	yes	—
82	C₄×Q₈×C₅₉	D₄	1	2	yes	—
83	C₄×D₄×C₅₉	—	2	2	no	—
84	C₄×D₄×C₅₉	—	2	2	no	—
85	C₄×D₄×C₅₉	—	2	2	no	—
86	C₄×D₄×C₅₉	D₄	1	2	yes	—
87	C₄×D₄×C₅₉	D₄	1	2	yes	—
88	#117×C₅₉	—	2	4	no	—
89	#117×C₅₉	—	2	2	no	—
90	#117×C₅₉	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
92	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
93	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
94	C₂×C₄⋊C₄×C₅₉	D₄	1	2	yes	—
95	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
97	C₂×C₄²×C₅₉	—	2	1	no	—
98	C₂×C₄²×C₅₉	—	2	2	no	—
99	C₂×C₄²×C₅₉	—	2	1	no	—
100	C₂×C₄²×C₅₉	—	2	1	no	—
101	C₂×C₄²×C₅₉	D₄	1	2	yes	—
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	#106×C₅₉	—	2	2	no	—
107	#104×C₅₉	—	2	4	no	—
108	#104×C₅₉	—	2	4	no	—
109	#103×C₅₉	—	2	2	no	—
110	#103×C₅₉	—	2	2	no	—
111	#103×C₅₉	—	2	2	no	—
112	#102×C₅₉	—	2	2	no	—
113	#101×C₅₉	—	2	4	no	—
114	#97×C₅₉	—	2	2	no	—
115	C₄×C₈×C₅₉	—	2	2	no	—
116	#95×C₅₉	—	2	2	no	—
117	#95×C₅₉	—	2	2	no	—
118	#41×C₅₉	—	2	2	no	—
119	#41×C₅₉	—	2	2	no	—
120	#41×C₅₉	—	2	2	no	—
121	#41×C₅₉	—	2	2	no	—
122	C₂×Q₈×C₅₉	—	2	2	no	—
123	C₂×Q₈×C₅₉	—	2	2	no	—
124	C₂×Q₈×C₅₉	—	2	2	no	—
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	2	no	—
129	C₂×Q₈×C₅₉	—	2	2	no	—
130	C₂×Q₈×C₅₉	C₂×D₄	1	2	yes	—
131	C₂×Q₈×C₅₉	#41	1	2	yes	—
132	C₂×D₄×C₅₉	—	2	2	no	—
133	C₂×D₄×C₅₉	—	2	2	no	—
134	C₂×D₄×C₅₉	C₂×D₄	1	2	yes	—
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	2	no	—
139	C₂²×C₄×C₅₉	—	2	2	no	—
140	C₂²×C₄×C₅₉	—	2	2	no	—
141	C₂²×C₄×C₅₉	—	2	1	no	—
142	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
143	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
145	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
147	Q₁₆×C₅₉	—	2	8	no	—
148	SD₁₆×C₅₉	—	2	8	no	—
149	M₁₆×C₅₉	—	2	2	no	—
150	M₁₆×C₅₉	—	2	2	no	—
151	M₁₆×C₅₉	—	2	2	no	—
152	M₁₆×C₅₉	#41	1	4	yes	—
153	C₂×C₈×C₅₉	—	2	2	no	—
154	C₂×C₈×C₅₉	—	2	2	no	—
155	C₄⋊C₄×C₅₉	—	2	8	no	—
156	C₄⋊C₄×C₅₉	—	2	4	no	—
157	C₄⋊C₄×C₅₉	—	2	4	no	—
158	C₄⋊C₄×C₅₉	—	2	2	no	—
159	C₄⋊C₄×C₅₉	—	2	2	no	—
160	C₄⋊C₄×C₅₉	—	2	4	no	—
161	C₄⋊C₄×C₅₉	—	2	4	no	—
162	C₄⋊C₄×C₅₉	—	2	2	no	—
163	C₄⋊C₄×C₅₉	—	2	2	no	—
164	C₄⋊C₄×C₅₉	—	2	2	no	—
165	C₄⋊C₄×C₅₉	—	2	2	no	—
166	C₄⋊C₄×C₅₉	—	4	4	no	—
167	C₄⋊C₄×C₅₉	—	4	2	no	—
168	C₄⋊C₄×C₅₉	C₂×D₄	1	2	yes	—
169	C₄⋊C₄×C₅₉	C₂×D₄	1	2	yes	—
170	C₂²⋊C₄×C₅₉	—	2	4	no	—
171	C₂²⋊C₄×C₅₉	—	2	4	no	—
172	C₂²⋊C₄×C₅₉	—	2	4	no	—
173	C₂²⋊C₄×C₅₉	—	2	2	no	—
174	C₂²⋊C₄×C₅₉	—	2	2	no	—
175	C₂²⋊C₄×C₅₉	—	2	2	no	—
176	C₄²×C₅₉	—	2	4	no	—
177	C₄²×C₅₉	—	2	2	no	—
178	C₄²×C₅₉	—	2	2	no	—
179	C₄²×C₅₉	—	2	2	no	—
180	C₄²×C₅₉	—	2	2	no	—
181	C₄²×C₅₉	—	2	2	no	—
182	C₄²×C₅₉	—	2	1	no	—
183	C₄²×C₅₉	—	2	2	no	—
184	C₄²×C₅₉	—	4	2	no	—
185	C₄²×C₅₉	—	4	1	no	—
186	C₄²×C₅₉	—	4	1	no	—
187	C₄²×C₅₉	C₂×D₄	1	2	yes	—
188	C₄²×C₅₉	C₂×D₄	1	2	yes	—
189	C₄²×C₅₉	#41	1	2	yes	—
190	C₂³×C₅₉	—	2	2	no	—
191	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
192	Q₈×C₅₉	—	2	4	no	—
193	Q₈×C₅₉	—	2	2	no	—
194	Q₈×C₅₉	—	2	2	no	—
195	Q₈×C₅₉	—	4	2	no	—
196	Q₈×C₅₉	—	4	2	no	—
197	Q₈×C₅₉	—	4	2	no	—
198	Q₈×C₅₉	—	8	2	no	—
199	D₄×C₅₉	—	2	2	no	—
200	D₄×C₅₉	—	2	2	no	—
201	D₄×C₅₉	—	4	2	no	—
202	D₄×C₅₉	—	4	2	no	—
203	D₄×C₅₉	—	8	2	no	—
204	C₂×C₄×C₅₉	—	2	2	no	—
205	C₂×C₄×C₅₉	—	2	2	no	—
206	C₂×C₄×C₅₉	—	2	2	no	—
207	C₂×C₄×C₅₉	—	2	2	no	—
208	C₂×C₄×C₅₉	—	2	2	no	—
209	C₂×C₄×C₅₉	—	2	2	no	—
210	C₂×C₄×C₅₉	—	2	2	no	—
211	C₂×C₄×C₅₉	—	2	2	no	—
212	C₂×C₄×C₅₉	—	2	2	no	—
213	C₂×C₄×C₅₉	—	2	2	no	—
214	C₂×C₄×C₅₉	—	2	2	no	—
215	C₂×C₄×C₅₉	—	2	2	no	—
216	C₂×C₄×C₅₉	—	2	2	no	—
217	C₂×C₄×C₅₉	—	4	1	no	—
218	C₂×C₄×C₅₉	—	4	1	no	—
219	C₂×C₄×C₅₉	—	4	1	no	—
220	C₂×C₄×C₅₉	—	4	1	no	—
221	C₂×C₄×C₅₉	—	4	1	no	—
222	C₂×C₄×C₅₉	—	4	2	no	—
223	C₂×C₄×C₅₉	—	4	2	no	—
224	C₂×C₄×C₅₉	—	4	4	no	—
225	C₂×C₄×C₅₉	—	4	2	no	—
226	C₂×C₄×C₅₉	—	4	2	no	—
227	C₂×C₄×C₅₉	—	4	1	no	—
228	C₂×C₄×C₅₉	—	8	2	no	—
229	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
230	C₂×C₄×C₅₉	#120	1	2	yes	—
231	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
232	C₂×C₄×C₅₉	C₂²×D₄	1	2	yes	—
233	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
234	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
235	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
236	C₈×C₅₉	—	2	4	no	—
237	C₈×C₅₉	—	4	2	no	—
238	#23590	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
239	C₂²×C₅₉	—	2	1	no	—
240	C₂²×C₅₉	—	2	1	no	—
241	C₂²×C₅₉	—	4	2	no	—
242	C₂²×C₅₉	—	4	2	no	—
243	C₂²×C₅₉	—	4	2	no	—
244	C₂²×C₅₉	C₂×#120	1	1	yes	characteristic
245	C₂²×C₅₉	#525	1	1	yes	characteristic
246	C₂²×C₅₉	#532	1	1	yes	characteristic
247	C₄×C₅₉	—	2	4	no	—
248	C₄×C₅₉	—	2	1	no	—
249	C₄×C₅₉	—	2	2	no	—
250	C₄×C₅₉	—	4	2	no	—
251	C₄×C₅₉	—	4	2	no	—
252	C₄×C₅₉	—	4	2	no	—
253	C₄×C₅₉	—	4	2	no	—
254	C₄×C₅₉	—	4	2	no	—
255	C₄×C₅₉	—	4	2	no	—
256	C₄×C₅₉	—	4	2	no	—
257	C₄×C₅₉	—	8	1	no	—
258	C₄×C₅₉	#456	1	2	yes	—
259	C₄×C₅₉	#525	1	2	yes	—
260	C₄×C₅₉	#532	1	2	yes	—
261	#3461	C₂×C₅₉	1	1	yes	characteristic
262	C₄×#137	C₂×C₅₉	1	2	yes	complemented
263	C₄×C₄⋊Q₈	C₂×C₅₉	1	1	yes	characteristic, complemented
264	#2060	C₂×C₅₉	1	1	yes	characteristic, complemented
265	#2039	C₂×C₅₉	1	1	yes	characteristic
266	#2038	C₂×C₅₉	1	2	yes	complemented
267	#2028	C₂×C₅₉	1	1	yes	characteristic, complemented
268	#2007	C₂×C₅₉	1	1	yes	characteristic
269	#2005	C₂×C₅₉	1	2	yes	complemented
270	C₄×#104	C₂×C₅₉	1	2	yes	complemented
271	C₄×#101	C₂×C₅₉	1	1	yes	characteristic, complemented
272	C₂×C₅₉	—	2	1	no	—
273	C₂×C₅₉	—	2	1	no	—
274	C₂×C₅₉	—	8	2	no	—
275	C₂×C₅₉	#2388	1	1	yes	characteristic
276	C₂×C₅₉	C₂×#456	1	1	yes	characteristic
277	C₂×C₅₉	#3018	1	1	yes	characteristic
278	C₂×#137	C₂²×C₅₉	1	2	yes	—
279	#544	C₂²×C₅₉	1	1	yes	characteristic
280	#542	—	2	2	no	—
281	#541	—	2	2	no	—
282	#539	—	2	2	no	—
283	#537	C₂²×C₅₉	1	1	yes	characteristic
284	#536	C₂²×C₅₉	1	2	yes	—
285	#533	—	2	2	no	—
286	C₂×C₄⋊Q₈	C₂²×C₅₉	1	1	yes	characteristic
287	#529	—	2	2	no	—
288	#529	—	2	2	no	—
289	C₄×#41	C₂²×C₅₉	1	2	yes	—
290	C₂×C₄×Q₈	C₂²×C₅₉	1	1	yes	characteristic
291	C₂×C₄×Q₈	C₂²×C₅₉	1	1	yes	characteristic
292	#441	—	2	2	no	—
293	#440	—	2	2	no	—
294	C₄×Q₁₆	—	2	2	no	—
295	C₄×SD₁₆	—	2	2	no	—
296	#428	C₂²×C₅₉	1	2	yes	—
297	C₂×#104	C₂²×C₅₉	1	2	yes	—
298	C₂×#104	C₂²×C₅₉	1	2	yes	—
299	#419	C₂²×C₅₉	1	2	yes	—
300	#417	C₂²×C₅₉	1	2	yes	—
301	C₂×#101	C₂²×C₅₉	1	2	yes	—
302	C₄×M₁₆	C₂²×C₅₉	1	2	yes	—
303	#391	C₂²×C₅₉	1	1	yes	characteristic
304	#391	C₂²×C₅₉	1	1	yes	characteristic
305	#389	—	2	2	no	—
306	#384	C₂²×C₅₉	1	1	yes	characteristic
307	C₄×C₄⋊C₄	—	2	2	no	—
308	C₄³	C₂²×C₅₉	1	1	yes	characteristic
309	#343	C₂²×C₅₉	1	1	yes	characteristic
310	#343	C₂²×C₅₉	1	1	yes	characteristic
311	#339	C₂²×C₅₉	1	2	yes	—
312	#339	C₂²×C₅₉	1	2	yes	—
313	C₅₉	#23590	1	1	yes	characteristic, direct product factor, Sylow
314	C₂×#41	C₂³×C₅₉	1	2	yes	—
315	C₂²×Q₈	C₂³×C₅₉	1	1	yes	characteristic
316	C₂²×Q₈	C₂³×C₅₉	1	1	yes	characteristic
317	#137	—	2	4	no	—
318	#137	—	2	4	no	—
319	C₂×Q₁₆	—	2	2	no	—
320	C₂×SD₁₆	—	2	2	no	—
321	C₂×M₁₆	C₂³×C₅₉	1	2	yes	—
322	C₂×M₁₆	C₂³×C₅₉	1	2	yes	—
323	C₄⋊Q₈	—	2	2	no	—
324	C₄⋊Q₈	—	2	2	no	—
325	C₄⋊Q₈	—	2	2	no	—
326	C₄⋊Q₈	—	2	2	no	—
327	#126	—	2	4	no	—
328	#126	—	2	4	no	—
329	#125	—	2	2	no	—
330	#125	—	2	4	no	—
331	#125	—	2	2	no	—
332	#124	—	2	2	no	—
333	#124	—	2	2	no	—
334	#123	—	2	4	no	—
335	#123	—	2	4	no	—
336	C₂²⋊Q₈	—	2	4	no	—
337	C₂²⋊Q₈	—	2	2	no	—
338	C₂²⋊Q₈	—	2	2	no	—
339	C₂²⋊Q₈	—	2	2	no	—
340	C₂²⋊Q₈	—	2	2	no	—
341	#121	—	2	2	no	—
342	#121	—	2	2	no	—
343	C₄×Q₈	—	2	2	no	—
344	C₄×Q₈	—	2	2	no	—
345	C₄×Q₈	—	2	2	no	—
346	C₄×Q₈	—	2	2	no	—
347	C₄×Q₈	—	2	2	no	—
348	C₄×Q₈	D₄×C₅₉	1	2	yes	—
349	C₄×Q₈	D₄×C₅₉	1	2	yes	—
350	C₄×D₄	—	2	2	no	—
351	C₄×D₄	—	2	2	no	—
352	C₄×D₄	—	2	2	no	—
353	C₄×D₄	D₄×C₅₉	1	2	yes	—
354	C₄×D₄	D₄×C₅₉	1	2	yes	—
355	#117	—	2	4	no	—
356	#117	—	2	2	no	—
357	#117	C₂³×C₅₉	1	2	yes	—
358	C₂×C₄⋊C₄	—	2	2	no	—
359	C₂×C₄⋊C₄	—	2	1	no	—
360	C₂×C₄⋊C₄	—	2	2	no	—
361	C₂×C₄⋊C₄	D₄×C₅₉	1	2	yes	—
362	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
363	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
364	C₂×C₄²	—	2	1	no	—
365	C₂×C₄²	—	2	2	no	—
366	C₂×C₄²	—	2	1	no	—
367	C₂×C₄²	—	2	1	no	—
368	C₂×C₄²	D₄×C₅₉	1	2	yes	—
369	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
370	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
371	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
372	#107	—	2	2	no	—
373	#106	—	2	2	no	—
374	#104	—	2	4	no	—
375	#104	—	2	4	no	—
376	#103	—	2	2	no	—
377	#103	—	2	2	no	—
378	#103	—	2	2	no	—
379	#102	—	2	2	no	—
380	#101	—	2	4	no	—
381	#97	—	2	2	no	—
382	C₄×C₈	—	2	2	no	—
383	#95	—	2	2	no	—
384	#95	—	2	2	no	—
385	#41	—	2	2	no	—
386	#41	—	2	2	no	—
387	#41	—	2	2	no	—
388	#41	—	2	2	no	—
389	C₂×Q₈	—	2	2	no	—
390	C₂×Q₈	—	2	2	no	—
391	C₂×Q₈	—	2	2	no	—
392	C₂×Q₈	—	2	1	no	—
393	C₂×Q₈	—	2	1	no	—
394	C₂×Q₈	—	2	1	no	—
395	C₂×Q₈	—	2	2	no	—
396	C₂×Q₈	—	2	2	no	—
397	C₂×Q₈	C₂×D₄×C₅₉	1	2	yes	—
398	C₂×Q₈	#41×C₅₉	1	2	yes	—
399	C₂×D₄	—	2	2	no	—
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	C₂×D₄×C₅₉	1	2	yes	—
402	C₂²×C₄	—	2	1	no	—
403	C₂²×C₄	—	2	1	no	—
404	C₂²×C₄	—	2	1	no	—
405	C₂²×C₄	—	2	2	no	—
406	C₂²×C₄	—	2	2	no	—
407	C₂²×C₄	—	2	2	no	—
408	C₂²×C₄	—	2	1	no	—
409	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
410	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
411	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
412	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
413	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
414	Q₁₆	—	2	8	no	—
415	SD₁₆	—	2	8	no	—
416	M₁₆	—	2	2	no	—
417	M₁₆	—	2	2	no	—
418	M₁₆	—	2	2	no	—
419	M₁₆	#41×C₅₉	1	4	yes	—
420	C₂×C₈	—	2	2	no	—
421	C₂×C₈	—	2	2	no	—
422	C₄⋊C₄	—	2	8	no	—
423	C₄⋊C₄	—	2	4	no	—
424	C₄⋊C₄	—	2	4	no	—
425	C₄⋊C₄	—	2	2	no	—
426	C₄⋊C₄	—	2	2	no	—
427	C₄⋊C₄	—	2	4	no	—
428	C₄⋊C₄	—	2	4	no	—
429	C₄⋊C₄	—	2	2	no	—
430	C₄⋊C₄	—	2	2	no	—
431	C₄⋊C₄	—	2	2	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	—	4	4	no	—
434	C₄⋊C₄	—	4	2	no	—
435	C₄⋊C₄	C₂×D₄×C₅₉	1	2	yes	—
436	C₄⋊C₄	C₂×D₄×C₅₉	1	2	yes	—
437	C₂²⋊C₄	—	2	4	no	—
438	C₂²⋊C₄	—	2	4	no	—
439	C₂²⋊C₄	—	2	4	no	—
440	C₂²⋊C₄	—	2	2	no	—
441	C₂²⋊C₄	—	2	2	no	—
442	C₂²⋊C₄	—	2	2	no	—
443	C₄²	—	2	4	no	—
444	C₄²	—	2	2	no	—
445	C₄²	—	2	2	no	—
446	C₄²	—	2	2	no	—
447	C₄²	—	2	2	no	—
448	C₄²	—	2	2	no	—
449	C₄²	—	2	1	no	—
450	C₄²	—	2	2	no	—
451	C₄²	—	4	2	no	—
452	C₄²	—	4	1	no	—
453	C₄²	—	4	1	no	—
454	C₄²	C₂×D₄×C₅₉	1	2	yes	—
455	C₄²	C₂×D₄×C₅₉	1	2	yes	—
456	C₄²	#41×C₅₉	1	2	yes	—
457	C₂³	—	2	2	no	—
458	C₂³	C₂²×D₄×C₅₉	1	1	yes	characteristic
459	Q₈	—	2	4	no	—
460	Q₈	—	2	2	no	—
461	Q₈	—	2	2	no	—
462	Q₈	—	4	2	no	—
463	Q₈	—	4	2	no	—
464	Q₈	—	4	2	no	—
465	Q₈	—	8	2	no	—
466	D₄	—	2	2	no	—
467	D₄	—	2	2	no	—
468	D₄	—	4	2	no	—
469	D₄	—	4	2	no	—
470	D₄	—	8	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	2	2	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	4	1	no	—
485	C₂×C₄	—	4	1	no	—
486	C₂×C₄	—	4	1	no	—
487	C₂×C₄	—	4	1	no	—
488	C₂×C₄	—	4	1	no	—
489	C₂×C₄	—	4	4	no	—
490	C₂×C₄	—	4	2	no	—
491	C₂×C₄	—	4	2	no	—
492	C₂×C₄	—	4	2	no	—
493	C₂×C₄	—	4	2	no	—
494	C₂×C₄	—	4	1	no	—
495	C₂×C₄	—	8	2	no	—
496	C₂×C₄	#120×C₅₉	1	1	yes	characteristic
497	C₂×C₄	#120×C₅₉	1	2	yes	—
498	C₂×C₄	#120×C₅₉	1	1	yes	characteristic
499	C₂×C₄	C₂²×D₄×C₅₉	1	2	yes	—
500	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
501	C₂×C₄	C₂×#41×C₅₉	1	2	yes	—
502	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
503	C₈	—	2	4	no	—
504	C₈	—	4	2	no	—
505	C₂²	—	2	1	no	—
506	C₂²	—	2	1	no	—
507	C₂²	—	4	2	no	—
508	C₂²	—	4	2	no	—
509	C₂²	—	4	2	no	—
510	C₂²	C₂×#120×C₅₉	1	1	yes	characteristic
511	C₂²	C₅₉×#525	1	1	yes	characteristic
512	C₂²	C₅₉×#532	1	1	yes	characteristic
513	C₄	—	2	4	no	—
514	C₄	—	2	1	no	—
515	C₄	—	2	2	no	—
516	C₄	—	4	2	no	—
517	C₄	—	4	2	no	—
518	C₄	—	4	2	no	—
519	C₄	—	4	2	no	—
520	C₄	—	4	2	no	—
521	C₄	—	4	2	no	—
522	C₄	—	4	2	no	—
523	C₄	—	8	1	no	—
524	C₄	C₅₉×#456	1	2	yes	—
525	C₄	C₅₉×#525	1	2	yes	—
526	C₄	C₅₉×#532	1	2	yes	—
527	C₂	—	2	1	no	—
528	C₂	—	2	1	no	—
529	C₂	—	8	2	no	—
530	C₂	C₅₉×#2388	1	1	yes	characteristic
531	C₂	C₂×C₅₉×#456	1	1	yes	characteristic
532	C₂	C₅₉×#3018	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₉	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₉
socle	C₂²×C₅₉	C₂×#120
prime radical	C₅₉×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₉ ◁ C₂×C₄²×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×#137×C₅₉	C₂	semidirect
3. C₄×C₄⋊Q₈×C₅₉	C₂	semidirect
4. C₅₉×#2060	C₂	semidirect
6. C₅₉×#2038	C₂	semidirect
7. C₅₉×#2028	C₂	semidirect
9. C₅₉×#2005	C₂	semidirect
10. C₄×#104×C₅₉	C₂	semidirect
11. C₄×#101×C₅₉	C₂	semidirect
238. #23590	C₅₉	direct (N × H)
262. C₄×#137	C₂×C₅₉	semidirect
263. C₄×C₄⋊Q₈	C₂×C₅₉	semidirect
264. #2060	C₂×C₅₉	semidirect
266. #2038	C₂×C₅₉	semidirect
267. #2028	C₂×C₅₉	semidirect
269. #2005	C₂×C₅₉	semidirect
270. C₄×#104	C₂×C₅₉	semidirect
271. C₄×#101	C₂×C₅₉	semidirect
313. C₅₉	#23590	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×#3461	2
C₄×#137×C₅₉	2
C₄×C₄⋊Q₈×C₅₉	2
C₅₉×#2060	2
C₅₉×#2039	2
C₅₉×#2038	2
C₅₉×#2028	2
C₅₉×#2007	2
C₅₉×#2005	2
C₄×#104×C₅₉	2
C₄×#101×C₅₉	2
#23590	59

Subgroups — C59×#23590 #28435699

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#23590 #28435699