Subgroups — C₅₉×#18590 #28430699

Order 15104.

1102 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#3499	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×#1651	C₂	1	2	yes	maximal, complemented
7	C₅₉×#1649	C₂	1	2	yes	maximal, complemented
8	C₅₉×#1647	C₂	1	2	yes	maximal, complemented
9	C₅₉×#1549	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	C₂×2₋¹⁺⁴×C₅₉	—	2	1	no	—
13	D₄×Q₈×C₅₉	—	2	1	no	—
14	D₄×Q₈×C₅₉	C₂²	1	2	yes	—
15	C₅₉×#547	—	2	2	no	—
16	C₅₉×#547	—	2	1	no	—
17	C₅₉×#547	—	2	2	no	—
18	C₅₉×#547	C₂²	1	2	yes	—
19	C₅₉×#536	—	2	2	no	—
20	C₅₉×#536	—	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	—	2	1	no	—
25	C₂×C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
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	2₋¹⁺⁴×C₅₉	—	4	4	no	—
54	2₋¹⁺⁴×C₅₉	—	4	4	no	—
55	C₂×#41×C₅₉	—	2	2	no	—
56	C₂×#41×C₅₉	—	2	4	no	—
57	C₂×#41×C₅₉	—	2	2	no	—
58	C₂×#41×C₅₉	—	2	1	no	—
59	C₂×#41×C₅₉	D₄	1	1	yes	characteristic
60	C₂²×Q₈×C₅₉	—	2	2	no	—
61	C₂²×Q₈×C₅₉	—	2	2	no	—
62	C₂²×Q₈×C₅₉	—	2	1	no	—
63	C₂×Q₁₆×C₅₉	—	2	1	no	—
64	C₂×SD₁₆×C₅₉	—	2	1	no	—
65	C₂×SD₁₆×C₅₉	—	2	2	no	—
66	C₂×M₁₆×C₅₉	—	2	1	no	—
67	C₂×M₁₆×C₅₉	—	2	1	no	—
68	C₄⋊Q₈×C₅₉	—	2	2	no	—
69	C₄⋊Q₈×C₅₉	—	2	2	no	—
70	C₄⋊Q₈×C₅₉	—	2	2	no	—
71	C₄⋊Q₈×C₅₉	—	2	1	no	—
72	C₄⋊Q₈×C₅₉	—	2	1	no	—
73	C₄⋊Q₈×C₅₉	D₄	1	1	yes	characteristic
74	C₄⋊Q₈×C₅₉	D₄	1	2	yes	—
75	C₄⋊Q₈×C₅₉	C₂³	1	1	yes	characteristic
76	C₄⋊Q₈×C₅₉	C₂³	1	1	yes	characteristic
77	C₄⋊Q₈×C₅₉	C₂³	1	2	yes	—
78	#127×C₅₉	D₄	1	1	yes	characteristic
79	#125×C₅₉	—	2	1	no	—
80	#124×C₅₉	—	2	2	no	—
81	#124×C₅₉	—	2	1	no	—
82	#123×C₅₉	—	2	4	no	—
83	#123×C₅₉	—	2	4	no	—
84	#123×C₅₉	—	2	4	no	—
85	C₂²⋊Q₈×C₅₉	—	2	4	no	—
86	C₂²⋊Q₈×C₅₉	—	2	2	no	—
87	C₂²⋊Q₈×C₅₉	—	2	4	no	—
88	C₂²⋊Q₈×C₅₉	—	2	2	no	—
89	C₂²⋊Q₈×C₅₉	—	2	1	no	—
90	C₂²⋊Q₈×C₅₉	—	2	1	no	—
91	#121×C₅₉	—	2	2	no	—
92	#121×C₅₉	—	2	2	no	—
93	#121×C₅₉	—	2	1	no	—
94	#121×C₅₉	D₄	1	2	yes	—
95	C₄×Q₈×C₅₉	—	2	1	no	—
96	C₄×Q₈×C₅₉	—	2	1	no	—
97	C₄×Q₈×C₅₉	D₄	1	1	yes	characteristic
98	C₄×Q₈×C₅₉	C₂³	1	1	yes	characteristic
99	C₄×D₄×C₅₉	—	2	2	no	—
100	C₄×D₄×C₅₉	—	2	2	no	—
101	C₄×D₄×C₅₉	—	2	1	no	—
102	C₄×D₄×C₅₉	—	2	1	no	—
103	C₄×D₄×C₅₉	—	2	1	no	—
104	C₄×D₄×C₅₉	D₄	1	1	yes	characteristic
105	C₄×D₄×C₅₉	C₂³	1	1	yes	characteristic
106	C₄×D₄×C₅₉	C₂³	1	2	yes	—
107	#117×C₅₉	—	2	1	no	—
108	#117×C₅₉	—	2	1	no	—
109	#117×C₅₉	D₄	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
111	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
112	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
113	C₂×C₄⋊C₄×C₅₉	D₄	1	2	yes	—
114	C₂×C₄²×C₅₉	—	2	1	no	—
115	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
116	#107×C₅₉	—	2	4	no	—
117	#107×C₅₉	—	2	1	no	—
118	#107×C₅₉	—	2	2	no	—
119	#106×C₅₉	—	2	4	no	—
120	#106×C₅₉	—	2	1	no	—
121	C₄⋊C₈×C₅₉	—	2	1	no	—
122	C₄⋊C₈×C₅₉	C₂³	1	2	yes	—
123	C₄⋊C₈×C₅₉	C₂³	1	1	yes	characteristic
124	C₄⋊C₈×C₅₉	C₂³	1	1	yes	characteristic
125	#103×C₅₉	—	2	1	no	—
126	#103×C₅₉	—	2	1	no	—
127	#103×C₅₉	—	2	1	no	—
128	#103×C₅₉	—	2	2	no	—
129	#102×C₅₉	—	2	1	no	—
130	#102×C₅₉	—	2	2	no	—
131	C₂²⋊C₈×C₅₉	—	2	1	no	—
132	#97×C₅₉	—	2	1	no	—
133	C₄×C₈×C₅₉	—	2	1	no	—
134	C₄×C₈×C₅₉	C₂³	1	2	yes	—
135	#41×C₅₉	—	2	2	no	—
136	#41×C₅₉	—	2	2	no	—
137	#41×C₅₉	—	4	8	no	—
138	#41×C₅₉	—	4	2	no	—
139	#41×C₅₉	—	4	2	no	—
140	#41×C₅₉	—	4	16	no	—
141	#41×C₅₉	—	4	4	no	—
142	#41×C₅₉	—	4	4	no	—
143	C₂×Q₈×C₅₉	—	2	4	no	—
144	C₂×Q₈×C₅₉	—	2	2	no	—
145	C₂×Q₈×C₅₉	—	2	1	no	—
146	C₂×Q₈×C₅₉	—	2	2	no	—
147	C₂×Q₈×C₅₉	—	2	1	no	—
148	C₂×Q₈×C₅₉	—	2	1	no	—
149	C₂×Q₈×C₅₉	—	4	8	no	—
150	C₂×Q₈×C₅₉	—	4	2	no	—
151	C₂×Q₈×C₅₉	—	4	2	no	—
152	C₂×Q₈×C₅₉	—	4	4	no	—
153	C₂×Q₈×C₅₉	—	4	4	no	—
154	C₂×Q₈×C₅₉	C₂×D₄	1	1	yes	characteristic
155	C₂×D₄×C₅₉	—	2	1	no	—
156	C₂×D₄×C₅₉	—	2	4	no	—
157	C₂×D₄×C₅₉	—	2	2	no	—
158	C₂×D₄×C₅₉	—	2	1	no	—
159	C₂×D₄×C₅₉	—	2	1	no	—
160	C₂×D₄×C₅₉	C₂×D₄	1	2	yes	—
161	C₂×D₄×C₅₉	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₅₉	—	2	1	no	—
163	C₂²×C₄×C₅₉	—	2	4	no	—
164	C₂²×C₄×C₅₉	—	2	4	no	—
165	C₂²×C₄×C₅₉	—	2	2	no	—
166	C₂²×C₄×C₅₉	—	2	1	no	—
167	C₂²×C₄×C₅₉	—	2	1	no	—
168	C₂²×C₄×C₅₉	—	2	2	no	—
169	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
172	Q₁₆×C₅₉	—	4	2	no	—
173	SD₁₆×C₅₉	—	4	2	no	—
174	SD₁₆×C₅₉	—	8	2	no	—
175	M₁₆×C₅₉	—	2	4	no	—
176	M₁₆×C₅₉	—	8	1	no	—
177	C₂×C₈×C₅₉	—	2	2	no	—
178	C₂×C₈×C₅₉	—	2	1	no	—
179	C₂×C₈×C₅₉	—	2	1	no	—
180	C₂×C₈×C₅₉	—	2	2	no	—
181	C₄⋊C₄×C₅₉	—	2	4	no	—
182	C₄⋊C₄×C₅₉	—	2	4	no	—
183	C₄⋊C₄×C₅₉	—	2	1	no	—
184	C₄⋊C₄×C₅₉	—	2	2	no	—
185	C₄⋊C₄×C₅₉	—	2	2	no	—
186	C₄⋊C₄×C₅₉	—	2	1	no	—
187	C₄⋊C₄×C₅₉	—	2	2	no	—
188	C₄⋊C₄×C₅₉	—	2	1	no	—
189	C₄⋊C₄×C₅₉	—	2	1	no	—
190	C₄⋊C₄×C₅₉	—	2	1	no	—
191	C₄⋊C₄×C₅₉	—	2	1	no	—
192	C₄⋊C₄×C₅₉	—	2	1	no	—
193	C₄⋊C₄×C₅₉	—	4	1	no	—
194	C₄⋊C₄×C₅₉	—	4	1	no	—
195	C₄⋊C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
196	C₄⋊C₄×C₅₉	C₂×D₄	1	2	yes	—
197	C₄⋊C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
198	C₄⋊C₄×C₅₉	C₂×D₄	1	4	yes	—
199	C₂²⋊C₄×C₅₉	—	2	4	no	—
200	C₂²⋊C₄×C₅₉	—	2	4	no	—
201	C₂²⋊C₄×C₅₉	—	2	1	no	—
202	C₂²⋊C₄×C₅₉	—	2	1	no	—
203	C₂²⋊C₄×C₅₉	—	2	2	no	—
204	C₂²⋊C₄×C₅₉	—	2	1	no	—
205	C₂²⋊C₄×C₅₉	—	2	2	no	—
206	C₂²⋊C₄×C₅₉	—	4	1	no	—
207	C₄²×C₅₉	—	2	1	no	—
208	C₄²×C₅₉	—	2	1	no	—
209	C₄²×C₅₉	—	2	1	no	—
210	C₄²×C₅₉	—	4	1	no	—
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	C₂⁴	1	1	yes	characteristic
215	C₂³×C₅₉	—	2	1	no	—
216	C₂³×C₅₉	—	2	2	no	—
217	C₂³×C₅₉	#120	1	1	yes	characteristic
218	Q₈×C₅₉	—	2	1	no	—
219	Q₈×C₅₉	—	4	4	no	—
220	Q₈×C₅₉	—	4	2	no	—
221	Q₈×C₅₉	—	4	8	no	—
222	Q₈×C₅₉	—	4	2	no	—
223	Q₈×C₅₉	—	4	2	no	—
224	Q₈×C₅₉	—	4	1	no	—
225	Q₈×C₅₉	—	4	1	no	—
226	Q₈×C₅₉	—	4	1	no	—
227	Q₈×C₅₉	—	4	1	no	—
228	Q₈×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
229	D₄×C₅₉	—	2	4	no	—
230	D₄×C₅₉	—	2	1	no	—
231	D₄×C₅₉	—	2	2	no	—
232	D₄×C₅₉	—	2	2	no	—
233	D₄×C₅₉	—	4	2	no	—
234	D₄×C₅₉	—	4	4	no	—
235	D₄×C₅₉	—	4	8	no	—
236	D₄×C₅₉	—	4	2	no	—
237	D₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
238	C₂×C₄×C₅₉	—	2	4	no	—
239	C₂×C₄×C₅₉	—	2	1	no	—
240	C₂×C₄×C₅₉	—	2	1	no	—
241	C₂×C₄×C₅₉	—	2	1	no	—
242	C₂×C₄×C₅₉	—	2	1	no	—
243	C₂×C₄×C₅₉	—	2	1	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	4	no	—
248	C₂×C₄×C₅₉	—	2	2	no	—
249	C₂×C₄×C₅₉	—	4	8	no	—
250	C₂×C₄×C₅₉	—	4	4	no	—
251	C₂×C₄×C₅₉	—	4	8	no	—
252	C₂×C₄×C₅₉	—	4	2	no	—
253	C₂×C₄×C₅₉	—	4	2	no	—
254	C₂×C₄×C₅₉	—	4	1	no	—
255	C₂×C₄×C₅₉	—	4	1	no	—
256	C₂×C₄×C₅₉	—	4	2	no	—
257	C₂×C₄×C₅₉	—	4	1	no	—
258	C₂×C₄×C₅₉	—	4	1	no	—
259	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
260	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
261	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
262	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
263	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
264	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
265	C₈×C₅₉	—	2	4	no	—
266	C₈×C₅₉	—	4	1	no	—
267	C₈×C₅₉	—	4	1	no	—
268	C₈×C₅₉	—	4	2	no	—
269	#18590	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
270	C₂²×C₅₉	—	2	1	no	—
271	C₂²×C₅₉	—	2	2	no	—
272	C₂²×C₅₉	—	2	1	no	—
273	C₂²×C₅₉	—	2	1	no	—
274	C₂²×C₅₉	—	4	1	no	—
275	C₂²×C₅₉	—	4	1	no	—
276	C₂²×C₅₉	—	4	2	no	—
277	C₂²×C₅₉	—	4	2	no	—
278	C₂²×C₅₉	—	4	2	no	—
279	C₂²×C₅₉	C₂×#120	1	1	yes	characteristic
280	C₄×C₅₉	—	2	1	no	—
281	C₄×C₅₉	—	2	2	no	—
282	C₄×C₅₉	—	2	1	no	—
283	C₄×C₅₉	—	2	1	no	—
284	C₄×C₅₉	—	4	4	no	—
285	C₄×C₅₉	—	4	2	no	—
286	C₄×C₅₉	—	4	1	no	—
287	C₄×C₅₉	—	4	1	no	—
288	C₄×C₅₉	—	8	1	no	—
289	C₄×C₅₉	—	8	1	no	—
290	C₄×C₅₉	#456	1	2	yes	—
291	C₄×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
292	C₄×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
293	C₄×C₅₉	#577	1	1	yes	characteristic
294	C₄×C₅₉	#578	1	1	yes	characteristic
295	#3499	C₂×C₅₉	1	1	yes	characteristic
296	#3254	C₂×C₅₉	1	1	yes	characteristic, complemented
297	#3165	C₂×C₅₉	1	1	yes	characteristic, complemented
298	#3107	C₂×C₅₉	1	1	yes	characteristic, complemented
299	#1654	C₂×C₅₉	1	2	yes	complemented
300	#1651	C₂×C₅₉	1	2	yes	complemented
301	#1649	C₂×C₅₉	1	2	yes	complemented
302	#1647	C₂×C₅₉	1	2	yes	complemented
303	#1549	C₂×C₅₉	1	1	yes	characteristic, complemented
304	#1489	C₂×C₅₉	1	1	yes	characteristic, complemented
305	#1484	C₂×C₅₉	1	1	yes	characteristic, complemented
306	C₂×C₅₉	—	4	1	no	—
307	C₂×C₅₉	—	4	2	no	—
308	C₂×C₅₉	—	8	2	no	—
309	C₂×C₅₉	#3003	1	1	yes	characteristic
310	C₂×C₅₉	#3013	1	1	yes	characteristic
311	C₂×C₅₉	C₂×#456	1	1	yes	characteristic
312	C₂×2₋¹⁺⁴	—	2	1	no	—
313	D₄×Q₈	—	2	1	no	—
314	D₄×Q₈	C₂²×C₅₉	1	2	yes	—
315	#547	—	2	2	no	—
316	#547	—	2	1	no	—
317	#547	—	2	2	no	—
318	#547	C₂²×C₅₉	1	2	yes	—
319	#536	—	2	2	no	—
320	#536	—	2	1	no	—
321	#533	C₂²×C₅₉	1	1	yes	characteristic
322	#532	—	2	1	no	—
323	#532	C₂²×C₅₉	1	1	yes	characteristic
324	C₂×C₄⋊Q₈	—	2	1	no	—
325	C₂×C₄⋊Q₈	C₂²×C₅₉	1	1	yes	characteristic
326	C₄×#41	C₂²×C₅₉	1	1	yes	characteristic
327	#501	—	2	2	no	—
328	#500	C₂²×C₅₉	1	2	yes	—
329	#498	C₂²×C₅₉	1	2	yes	—
330	#485	—	2	1	no	—
331	#477	—	2	1	no	—
332	#475	C₂²×C₅₉	1	1	yes	characteristic
333	#474	C₂²×C₅₉	1	1	yes	characteristic
334	#474	C₂²×C₅₉	1	2	yes	—
335	#468	—	2	2	no	—
336	#462	C₂²×C₅₉	1	1	yes	characteristic
337	#461	C₂²×C₅₉	1	1	yes	characteristic
338	#461	C₂²×C₅₉	1	2	yes	—
339	#460	—	2	1	no	—
340	#452	—	2	1	no	—
341	C₄×Q₁₆	—	2	1	no	—
342	C₄×SD₁₆	—	2	1	no	—
343	#431	C₂²×C₅₉	1	1	yes	characteristic
344	#428	—	2	2	no	—
345	#423	C₂²×C₅₉	1	1	yes	characteristic
346	C₄×M₁₆	C₂²×C₅₉	1	1	yes	characteristic
347	#332	C₂²×C₅₉	1	2	yes	—
348	#332	C₂²×C₅₉	1	2	yes	—
349	Q₈⋊C₈	C₂²×C₅₉	1	2	yes	—
350	D₄⋊C₈	C₂²×C₅₉	1	2	yes	—
351	D₄⋊C₈	C₂²×C₅₉	1	2	yes	—
352	D₄⋊C₈	C₂²×C₅₉	1	2	yes	—
353	C₅₉	#18590	1	1	yes	characteristic, direct product factor, Sylow
354	2₋¹⁺⁴	—	4	4	no	—
355	2₋¹⁺⁴	—	4	4	no	—
356	C₂×#41	—	2	2	no	—
357	C₂×#41	—	2	4	no	—
358	C₂×#41	—	2	2	no	—
359	C₂×#41	—	2	1	no	—
360	C₂×#41	D₄×C₅₉	1	1	yes	characteristic
361	C₂²×Q₈	—	2	2	no	—
362	C₂²×Q₈	—	2	2	no	—
363	C₂²×Q₈	—	2	1	no	—
364	C₂×Q₁₆	—	2	1	no	—
365	C₂×SD₁₆	—	2	1	no	—
366	C₂×SD₁₆	—	2	2	no	—
367	C₂×M₁₆	—	2	1	no	—
368	C₂×M₁₆	—	2	1	no	—
369	C₄⋊Q₈	—	2	2	no	—
370	C₄⋊Q₈	—	2	2	no	—
371	C₄⋊Q₈	—	2	2	no	—
372	C₄⋊Q₈	—	2	1	no	—
373	C₄⋊Q₈	—	2	1	no	—
374	C₄⋊Q₈	D₄×C₅₉	1	1	yes	characteristic
375	C₄⋊Q₈	D₄×C₅₉	1	2	yes	—
376	C₄⋊Q₈	C₂³×C₅₉	1	1	yes	characteristic
377	C₄⋊Q₈	C₂³×C₅₉	1	1	yes	characteristic
378	C₄⋊Q₈	C₂³×C₅₉	1	2	yes	—
379	#127	D₄×C₅₉	1	1	yes	characteristic
380	#125	—	2	1	no	—
381	#124	—	2	2	no	—
382	#124	—	2	1	no	—
383	#123	—	2	4	no	—
384	#123	—	2	4	no	—
385	#123	—	2	4	no	—
386	C₂²⋊Q₈	—	2	4	no	—
387	C₂²⋊Q₈	—	2	2	no	—
388	C₂²⋊Q₈	—	2	4	no	—
389	C₂²⋊Q₈	—	2	2	no	—
390	C₂²⋊Q₈	—	2	1	no	—
391	C₂²⋊Q₈	—	2	1	no	—
392	#121	—	2	2	no	—
393	#121	—	2	2	no	—
394	#121	—	2	1	no	—
395	#121	D₄×C₅₉	1	2	yes	—
396	C₄×Q₈	—	2	1	no	—
397	C₄×Q₈	—	2	1	no	—
398	C₄×Q₈	D₄×C₅₉	1	1	yes	characteristic
399	C₄×Q₈	C₂³×C₅₉	1	1	yes	characteristic
400	C₄×D₄	—	2	2	no	—
401	C₄×D₄	—	2	2	no	—
402	C₄×D₄	—	2	1	no	—
403	C₄×D₄	—	2	1	no	—
404	C₄×D₄	—	2	1	no	—
405	C₄×D₄	D₄×C₅₉	1	1	yes	characteristic
406	C₄×D₄	C₂³×C₅₉	1	1	yes	characteristic
407	C₄×D₄	C₂³×C₅₉	1	2	yes	—
408	#117	—	2	1	no	—
409	#117	—	2	1	no	—
410	#117	D₄×C₅₉	1	1	yes	characteristic
411	C₂×C₄⋊C₄	—	2	2	no	—
412	C₂×C₄⋊C₄	—	2	1	no	—
413	C₂×C₄⋊C₄	—	2	2	no	—
414	C₂×C₄⋊C₄	D₄×C₅₉	1	2	yes	—
415	C₂×C₄²	—	2	1	no	—
416	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
417	#107	—	2	4	no	—
418	#107	—	2	1	no	—
419	#107	—	2	2	no	—
420	#106	—	2	4	no	—
421	#106	—	2	1	no	—
422	C₄⋊C₈	—	2	1	no	—
423	C₄⋊C₈	C₂³×C₅₉	1	1	yes	characteristic
424	C₄⋊C₈	C₂³×C₅₉	1	1	yes	characteristic
425	C₄⋊C₈	C₂³×C₅₉	1	2	yes	—
426	#103	—	2	1	no	—
427	#103	—	2	1	no	—
428	#103	—	2	1	no	—
429	#103	—	2	2	no	—
430	#102	—	2	1	no	—
431	#102	—	2	2	no	—
432	C₂²⋊C₈	—	2	1	no	—
433	#97	—	2	1	no	—
434	C₄×C₈	—	2	1	no	—
435	C₄×C₈	C₂³×C₅₉	1	2	yes	—
436	#41	—	2	2	no	—
437	#41	—	2	2	no	—
438	#41	—	4	8	no	—
439	#41	—	4	2	no	—
440	#41	—	4	2	no	—
441	#41	—	4	16	no	—
442	#41	—	4	4	no	—
443	#41	—	4	4	no	—
444	C₂×Q₈	—	2	4	no	—
445	C₂×Q₈	—	2	2	no	—
446	C₂×Q₈	—	2	1	no	—
447	C₂×Q₈	—	2	2	no	—
448	C₂×Q₈	—	2	1	no	—
449	C₂×Q₈	—	2	1	no	—
450	C₂×Q₈	—	4	8	no	—
451	C₂×Q₈	—	4	2	no	—
452	C₂×Q₈	—	4	2	no	—
453	C₂×Q₈	—	4	4	no	—
454	C₂×Q₈	—	4	4	no	—
455	C₂×Q₈	C₂×D₄×C₅₉	1	1	yes	characteristic
456	C₂×D₄	—	2	1	no	—
457	C₂×D₄	—	2	4	no	—
458	C₂×D₄	—	2	2	no	—
459	C₂×D₄	—	2	1	no	—
460	C₂×D₄	—	2	1	no	—
461	C₂×D₄	C₂×D₄×C₅₉	1	2	yes	—
462	C₂×D₄	C₂×D₄×C₅₉	1	1	yes	characteristic
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	2	4	no	—
465	C₂²×C₄	—	2	4	no	—
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	2	1	no	—
468	C₂²×C₄	—	2	1	no	—
469	C₂²×C₄	—	2	2	no	—
470	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
471	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
472	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
473	Q₁₆	—	4	2	no	—
474	SD₁₆	—	4	2	no	—
475	SD₁₆	—	8	2	no	—
476	M₁₆	—	2	4	no	—
477	M₁₆	—	8	1	no	—
478	C₂×C₈	—	2	2	no	—
479	C₂×C₈	—	2	1	no	—
480	C₂×C₈	—	2	1	no	—
481	C₂×C₈	—	2	2	no	—
482	C₄⋊C₄	—	2	4	no	—
483	C₄⋊C₄	—	2	4	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	2	no	—
486	C₄⋊C₄	—	2	2	no	—
487	C₄⋊C₄	—	2	1	no	—
488	C₄⋊C₄	—	2	2	no	—
489	C₄⋊C₄	—	2	1	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	4	1	no	—
495	C₄⋊C₄	—	4	1	no	—
496	C₄⋊C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
497	C₄⋊C₄	C₂×D₄×C₅₉	1	2	yes	—
498	C₄⋊C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
499	C₄⋊C₄	C₂×D₄×C₅₉	1	4	yes	—
500	C₂²⋊C₄	—	2	4	no	—
501	C₂²⋊C₄	—	2	4	no	—
502	C₂²⋊C₄	—	2	1	no	—
503	C₂²⋊C₄	—	2	1	no	—
504	C₂²⋊C₄	—	2	2	no	—
505	C₂²⋊C₄	—	2	1	no	—
506	C₂²⋊C₄	—	2	2	no	—
507	C₂²⋊C₄	—	4	1	no	—
508	C₄²	—	2	1	no	—
509	C₄²	—	2	1	no	—
510	C₄²	—	2	1	no	—
511	C₄²	—	4	1	no	—
512	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
513	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
514	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
515	C₄²	C₂⁴×C₅₉	1	1	yes	characteristic
516	C₂³	—	2	1	no	—
517	C₂³	—	2	2	no	—
518	C₂³	#120×C₅₉	1	1	yes	characteristic
519	Q₈	—	2	1	no	—
520	Q₈	—	4	4	no	—
521	Q₈	—	4	2	no	—
522	Q₈	—	4	8	no	—
523	Q₈	—	4	2	no	—
524	Q₈	—	4	2	no	—
525	Q₈	—	4	1	no	—
526	Q₈	—	4	1	no	—
527	Q₈	—	4	1	no	—
528	Q₈	—	4	1	no	—
529	Q₈	AGL(1,ℤ/8)×C₅₉	1	2	yes	—
530	D₄	—	2	4	no	—
531	D₄	—	2	1	no	—
532	D₄	—	2	2	no	—
533	D₄	—	2	2	no	—
534	D₄	—	4	2	no	—
535	D₄	—	4	4	no	—
536	D₄	—	4	8	no	—
537	D₄	—	4	2	no	—
538	D₄	AGL(1,ℤ/8)×C₅₉	1	2	yes	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	4	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	4	8	no	—
551	C₂×C₄	—	4	4	no	—
552	C₂×C₄	—	4	8	no	—
553	C₂×C₄	—	4	2	no	—
554	C₂×C₄	—	4	2	no	—
555	C₂×C₄	—	4	2	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	#120×C₅₉	1	1	yes	characteristic
561	C₂×C₄	#120×C₅₉	1	1	yes	characteristic
562	C₂×C₄	#120×C₅₉	1	1	yes	characteristic
563	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
564	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
565	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
566	C₈	—	2	4	no	—
567	C₈	—	4	1	no	—
568	C₈	—	4	1	no	—
569	C₈	—	4	2	no	—
570	C₂²	—	2	1	no	—
571	C₂²	—	2	2	no	—
572	C₂²	—	2	1	no	—
573	C₂²	—	2	1	no	—
574	C₂²	—	4	1	no	—
575	C₂²	—	4	1	no	—
576	C₂²	—	4	2	no	—
577	C₂²	—	4	2	no	—
578	C₂²	—	4	2	no	—
579	C₂²	C₂×#120×C₅₉	1	1	yes	characteristic
580	C₄	—	2	1	no	—
581	C₄	—	2	2	no	—
582	C₄	—	2	1	no	—
583	C₄	—	2	1	no	—
584	C₄	—	4	4	no	—
585	C₄	—	4	2	no	—
586	C₄	—	4	1	no	—
587	C₄	—	4	1	no	—
588	C₄	—	8	1	no	—
589	C₄	—	8	1	no	—
590	C₄	C₅₉×#456	1	2	yes	—
591	C₄	C₂×AGL(1,ℤ/8)×C₅₉	1	1	yes	characteristic
592	C₄	C₂×AGL(1,ℤ/8)×C₅₉	1	1	yes	characteristic
593	C₄	C₅₉×#577	1	1	yes	characteristic
594	C₄	C₅₉×#578	1	1	yes	characteristic
595	C₂	—	4	1	no	—
596	C₂	—	4	2	no	—
597	C₂	—	8	2	no	—
598	C₂	C₅₉×#3003	1	1	yes	characteristic
599	C₂	C₅₉×#3013	1	1	yes	characteristic
600	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×#3499	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×#1651	C₂	semidirect
7. C₅₉×#1649	C₂	semidirect
8. C₅₉×#1647	C₂	semidirect
9. C₅₉×#1549	C₂	semidirect
10. C₅₉×#1489	C₂	semidirect
11. C₅₉×#1484	C₂	semidirect
269. #18590	C₅₉	direct (N × H)
296. #3254	C₂×C₅₉	semidirect
297. #3165	C₂×C₅₉	semidirect
298. #3107	C₂×C₅₉	semidirect
299. #1654	C₂×C₅₉	semidirect
300. #1651	C₂×C₅₉	semidirect
301. #1649	C₂×C₅₉	semidirect
302. #1647	C₂×C₅₉	semidirect
303. #1549	C₂×C₅₉	semidirect
304. #1489	C₂×C₅₉	semidirect
305. #1484	C₂×C₅₉	semidirect
353. C₅₉	#18590	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×#3499	2
C₅₉×#3254	2
C₅₉×#3165	2
C₅₉×#3107	2
C₅₉×#1654	2
C₅₉×#1651	2
C₅₉×#1649	2
C₅₉×#1647	2
C₅₉×#1549	2
C₅₉×#1489	2
C₅₉×#1484	2
#18590	59

Subgroups — C59×#18590 #28430699

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#18590 #28430699