Subgroups — C₅₉×#26480 #28438589

Order 15104.

850 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₉×#498	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×SD₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#2452	C₂	1	1	yes	maximal, characteristic
4	C₅₉×#2451	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#2066	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₉×#2065	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#2024	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₉×#2023	C₂	1	1	yes	maximal, characteristic
9	C₅₉×#1968	C₂	1	1	yes	maximal, characteristic
10	C₅₉×#1968	C₂	1	1	yes	maximal, characteristic
11	C₅₉×#1941	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₉×#1941	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅₉×#1865	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅₉×#1864	C₂	1	1	yes	maximal, characteristic
15	C₈×C₄⋊C₄×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×SD₁₆×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊Q₈×C₅₉	—	2	1	no	—
20	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
22	C₅₉×#498	—	2	4	no	—
23	C₄×SD₁₆×C₅₉	—	2	4	no	—
24	C₂×#106×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×#106×C₅₉	C₂²	1	1	yes	characteristic
26	C₂×#106×C₅₉	C₂²	1	1	yes	characteristic
27	C₂×#106×C₅₉	C₂²	1	1	yes	characteristic
28	C₂×#106×C₅₉	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
30	C₂×#103×C₅₉	C₂²	1	1	yes	characteristic
31	C₂×#103×C₅₉	C₂²	1	1	yes	characteristic
32	C₂×#103×C₅₉	C₂²	1	1	yes	characteristic
33	C₂×#102×C₅₉	C₂²	1	1	yes	characteristic
34	C₂×#102×C₅₉	C₂²	1	1	yes	characteristic
35	C₂×#102×C₅₉	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
39	C₅₉×#398	—	2	1	no	—
40	C₅₉×#396	—	2	1	no	—
41	C₅₉×#395	—	2	1	no	—
42	C₅₉×#393	—	2	1	no	—
43	C₅₉×#391	—	2	1	no	—
44	C₅₉×#391	C₂²	1	1	yes	characteristic
45	C₅₉×#391	C₂²	1	1	yes	characteristic
46	C₅₉×#390	C₂²	1	1	yes	characteristic
47	C₅₉×#390	C₂²	1	1	yes	characteristic
48	C₅₉×#389	C₂²	1	1	yes	characteristic
49	C₅₉×#389	C₂²	1	1	yes	characteristic
50	C₅₉×#389	C₂²	1	1	yes	characteristic
51	C₅₉×#389	C₂²	1	1	yes	characteristic
52	C₅₉×#387	—	2	1	no	—
53	C₅₉×#385	—	2	1	no	—
54	C₄×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
55	C₅₉×#340	C₂²	1	1	yes	characteristic
56	C₅₉×#340	C₂²	1	1	yes	characteristic
57	C₅₉×#340	C₂²	1	1	yes	characteristic
58	C₅₉×#340	C₂²	1	1	yes	characteristic
59	C₅₉×#336	C₂²	1	1	yes	characteristic
60	C₅₉×#336	C₂²	1	1	yes	characteristic
61	C₂²×Q₈×C₅₉	—	2	1	no	—
62	C₂²×Q₈×C₅₉	—	2	1	no	—
63	C₂²×Q₈×C₅₉	C₂³	1	1	yes	characteristic
64	C₂²×D₄×C₅₉	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₅₉	—	2	1	no	—
66	C₂×SD₁₆×C₅₉	—	2	2	no	—
67	C₂×SD₁₆×C₅₉	—	2	2	no	—
68	C₂×SD₁₆×C₅₉	Q₈	1	4	yes	complemented
69	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
73	C₄⋊Q₈×C₅₉	—	2	2	no	—
74	C₄⋊Q₈×C₅₉	—	2	2	no	—
75	C₄⋊Q₈×C₅₉	—	4	1	no	—
76	C₄⋊Q₈×C₅₉	—	4	1	no	—
77	C₂²⋊Q₈×C₅₉	—	4	4	no	—
78	C₄×Q₈×C₅₉	—	2	2	no	—
79	C₄×Q₈×C₅₉	—	4	1	no	—
80	C₄×D₄×C₅₉	—	2	2	no	—
81	C₄×D₄×C₅₉	—	4	1	no	—
82	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
83	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
84	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
85	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
86	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
87	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
88	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
89	C₂×C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
97	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
98	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
99	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
100	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
101	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
102	C₂×C₄²×C₅₉	—	2	1	no	—
103	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
106	#106×C₅₉	—	2	2	no	—
107	#106×C₅₉	—	2	2	no	—
108	#106×C₅₉	—	2	2	no	—
109	#106×C₅₉	—	2	2	no	—
110	#106×C₅₉	D₄	1	4	yes	—
111	C₄⋊C₈×C₅₉	—	2	2	no	—
112	#103×C₅₉	—	2	2	no	—
113	#103×C₅₉	—	2	2	no	—
114	#103×C₅₉	—	4	1	no	—
115	#102×C₅₉	—	2	2	no	—
116	#102×C₅₉	—	2	2	no	—
117	#102×C₅₉	—	4	1	no	—
118	C₄×C₈×C₅₉	—	2	2	no	—
119	C₄×C₈×C₅₉	—	2	2	no	—
120	C₄×C₈×C₅₉	—	2	2	no	—
121	#95×C₅₉	—	2	1	no	—
122	#95×C₅₉	—	2	1	no	—
123	#95×C₅₉	—	2	1	no	—
124	#95×C₅₉	—	2	1	no	—
125	#95×C₅₉	—	2	1	no	—
126	#95×C₅₉	—	2	1	no	—
127	#95×C₅₉	—	2	1	no	—
128	#95×C₅₉	—	2	1	no	—
129	#95×C₅₉	D₄	1	1	yes	characteristic
130	#95×C₅₉	D₄	1	1	yes	characteristic
131	C₂⁴×C₅₉	—	2	1	no	—
132	C₂×Q₈×C₅₉	—	2	1	no	—
133	C₂×Q₈×C₅₉	—	2	1	no	—
134	C₂×Q₈×C₅₉	—	2	1	no	—
135	C₂×Q₈×C₅₉	—	2	1	no	—
136	C₂×Q₈×C₅₉	—	2	1	no	—
137	C₂×Q₈×C₅₉	—	4	2	no	—
138	C₂×Q₈×C₅₉	—	4	2	no	—
139	C₂×Q₈×C₅₉	—	4	2	no	—
140	C₂×Q₈×C₅₉	—	4	2	no	—
141	C₂×Q₈×C₅₉	—	8	1	no	—
142	C₂×Q₈×C₅₉	—	8	1	no	—
143	C₂×Q₈×C₅₉	C₂×Q₈	1	2	yes	—
144	C₂×D₄×C₅₉	—	2	1	no	—
145	C₂×D₄×C₅₉	—	2	1	no	—
146	C₂×D₄×C₅₉	—	2	1	no	—
147	C₂×D₄×C₅₉	—	2	1	no	—
148	C₂×D₄×C₅₉	—	2	1	no	—
149	C₂×D₄×C₅₉	C₂×Q₈	1	2	yes	—
150	C₂²×C₄×C₅₉	—	2	1	no	—
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	1	no	—
156	C₂²×C₄×C₅₉	—	2	1	no	—
157	C₂²×C₄×C₅₉	—	4	2	no	—
158	C₂²×C₄×C₅₉	—	4	2	no	—
159	C₂²×C₄×C₅₉	—	8	1	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	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	C₂⁴	1	1	yes	characteristic
167	SD₁₆×C₅₉	—	2	8	no	—
168	C₂×C₈×C₅₉	—	2	1	no	—
169	C₂×C₈×C₅₉	—	2	1	no	—
170	C₂×C₈×C₅₉	—	2	1	no	—
171	C₂×C₈×C₅₉	—	2	1	no	—
172	C₂×C₈×C₅₉	—	2	1	no	—
173	C₂×C₈×C₅₉	—	2	1	no	—
174	C₂×C₈×C₅₉	—	2	1	no	—
175	C₂×C₈×C₅₉	—	2	1	no	—
176	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
177	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
178	C₂×C₈×C₅₉	#41	1	2	yes	—
179	C₂×C₈×C₅₉	#41	1	2	yes	—
180	C₄⋊C₄×C₅₉	—	2	2	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	—	2	1	no	—
185	C₄⋊C₄×C₅₉	—	2	1	no	—
186	C₄⋊C₄×C₅₉	—	2	1	no	—
187	C₄⋊C₄×C₅₉	—	2	1	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	—	4	2	no	—
192	C₄⋊C₄×C₅₉	—	4	2	no	—
193	C₄⋊C₄×C₅₉	—	4	2	no	—
194	C₄⋊C₄×C₅₉	—	4	2	no	—
195	C₄⋊C₄×C₅₉	—	4	2	no	—
196	C₄⋊C₄×C₅₉	—	8	1	no	—
197	C₄⋊C₄×C₅₉	—	8	1	no	—
198	C₄⋊C₄×C₅₉	SD₁₆	1	4	yes	complemented
199	C₄⋊C₄×C₅₉	C₂×D₄	1	2	yes	—
200	C₄⋊C₄×C₅₉	C₂×D₄	1	2	yes	—
201	C₂²⋊C₄×C₅₉	—	4	2	no	—
202	C₂²⋊C₄×C₅₉	—	4	2	no	—
203	C₂²⋊C₄×C₅₉	—	4	2	no	—
204	C₂²⋊C₄×C₅₉	—	4	2	no	—
205	C₂²⋊C₄×C₅₉	—	8	1	no	—
206	C₄²×C₅₉	—	2	1	no	—
207	C₄²×C₅₉	—	2	1	no	—
208	C₄²×C₅₉	—	2	1	no	—
209	C₄²×C₅₉	—	2	1	no	—
210	C₄²×C₅₉	—	2	1	no	—
211	C₄²×C₅₉	—	2	1	no	—
212	C₄²×C₅₉	—	4	2	no	—
213	C₂³×C₅₉	—	2	2	no	—
214	C₂³×C₅₉	—	4	1	no	—
215	C₂³×C₅₉	—	4	1	no	—
216	C₂³×C₅₉	—	4	1	no	—
217	C₂³×C₅₉	—	4	1	no	—
218	C₂³×C₅₉	—	4	1	no	—
219	C₂³×C₅₉	—	4	1	no	—
220	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
221	Q₈×C₅₉	—	2	2	no	—
222	Q₈×C₅₉	—	2	2	no	—
223	Q₈×C₅₉	—	4	1	no	—
224	Q₈×C₅₉	—	4	1	no	—
225	Q₈×C₅₉	—	8	4	no	—
226	Q₈×C₅₉	—	8	4	no	—
227	D₄×C₅₉	—	2	2	no	—
228	D₄×C₅₉	—	2	2	no	—
229	D₄×C₅₉	—	4	1	no	—
230	D₄×C₅₉	—	4	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	1	no	—
235	C₂×C₄×C₅₉	—	2	1	no	—
236	C₂×C₄×C₅₉	—	2	1	no	—
237	C₂×C₄×C₅₉	—	2	1	no	—
238	C₂×C₄×C₅₉	—	2	1	no	—
239	C₂×C₄×C₅₉	—	2	1	no	—
240	C₂×C₄×C₅₉	—	2	1	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	1	no	—
245	C₂×C₄×C₅₉	—	2	1	no	—
246	C₂×C₄×C₅₉	—	2	1	no	—
247	C₂×C₄×C₅₉	—	2	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	1	no	—
253	C₂×C₄×C₅₉	—	4	1	no	—
254	C₂×C₄×C₅₉	—	4	1	no	—
255	C₂×C₄×C₅₉	—	4	1	no	—
256	C₂×C₄×C₅₉	—	4	1	no	—
257	C₂×C₄×C₅₉	—	4	1	no	—
258	C₂×C₄×C₅₉	—	4	1	no	—
259	C₂×C₄×C₅₉	—	4	1	no	—
260	C₂×C₄×C₅₉	—	4	1	no	—
261	C₂×C₄×C₅₉	—	4	1	no	—
262	C₂×C₄×C₅₉	—	4	1	no	—
263	C₂×C₄×C₅₉	—	4	1	no	—
264	C₂×C₄×C₅₉	—	4	1	no	—
265	C₂×C₄×C₅₉	—	4	1	no	—
266	C₂×C₄×C₅₉	—	8	4	no	—
267	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
268	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
269	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
270	C₂×C₄×C₅₉	#135	1	2	yes	—
271	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
272	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
273	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
274	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
275	C₂×C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
276	C₂×C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
277	C₈×C₅₉	—	2	2	no	—
278	C₈×C₅₉	—	4	1	no	—
279	C₈×C₅₉	—	4	1	no	—
280	C₈×C₅₉	C₂²⋊Q₈	1	4	yes	complemented
281	#26480	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
282	C₂²×C₅₉	—	4	2	no	—
283	C₂²×C₅₉	—	4	2	no	—
284	C₂²×C₅₉	—	4	2	no	—
285	C₂²×C₅₉	—	8	1	no	—
286	C₂²×C₅₉	—	8	1	no	—
287	C₂²×C₅₉	—	8	1	no	—
288	C₂²×C₅₉	—	8	1	no	—
289	C₂²×C₅₉	#532	1	1	yes	characteristic
290	C₂²×C₅₉	D₄²	1	1	yes	characteristic
291	C₂²×C₅₉	D₄×Q₈	1	1	yes	characteristic
292	C₂²×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
293	C₂²×C₅₉	C₂×#135	1	1	yes	characteristic
294	C₂²×C₅₉	#577	1	1	yes	characteristic
295	C₂²×C₅₉	#577	1	1	yes	characteristic
296	C₄×C₅₉	—	2	2	no	—
297	C₄×C₅₉	—	2	2	no	—
298	C₄×C₅₉	—	2	2	no	—
299	C₄×C₅₉	—	2	2	no	—
300	C₄×C₅₉	—	4	1	no	—
301	C₄×C₅₉	—	4	2	no	—
302	C₄×C₅₉	—	4	2	no	—
303	C₄×C₅₉	—	4	1	no	—
304	C₄×C₅₉	—	4	2	no	—
305	C₄×C₅₉	—	8	1	no	—
306	C₄×C₅₉	—	8	1	no	—
307	C₄×C₅₉	—	8	1	no	—
308	C₄×C₅₉	—	8	1	no	—
309	C₄×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
310	C₄×C₅₉	#542	1	1	yes	characteristic
311	C₄×C₅₉	#548	1	1	yes	characteristic
312	C₄×C₅₉	#554	1	1	yes	characteristic
313	C₂×#498	C₂×C₅₉	1	1	yes	characteristic, complemented
314	C₂×C₄×SD₁₆	C₂×C₅₉	1	1	yes	characteristic
315	#2452	C₂×C₅₉	1	1	yes	characteristic
316	#2451	C₂×C₅₉	1	1	yes	characteristic, complemented
317	#2066	C₂×C₅₉	1	1	yes	characteristic, complemented
318	#2065	C₂×C₅₉	1	1	yes	characteristic
319	#2024	C₂×C₅₉	1	1	yes	characteristic, complemented
320	#2023	C₂×C₅₉	1	1	yes	characteristic
321	#1968	C₂×C₅₉	1	1	yes	characteristic
322	#1968	C₂×C₅₉	1	1	yes	characteristic
323	#1941	C₂×C₅₉	1	1	yes	characteristic, complemented
324	#1941	C₂×C₅₉	1	1	yes	characteristic, complemented
325	#1865	C₂×C₅₉	1	1	yes	characteristic, complemented
326	#1864	C₂×C₅₉	1	1	yes	characteristic
327	C₈×C₄⋊C₄	C₂×C₅₉	1	1	yes	characteristic, complemented
328	C₂×C₅₉	—	8	2	no	—
329	C₂×C₅₉	#2452	1	1	yes	characteristic
330	C₂×C₅₉	#3167	1	1	yes	characteristic
331	C₂×C₅₉	#3173	1	1	yes	characteristic
332	C₂×C₅₉	D₄×SD₁₆	1	1	yes	characteristic
333	C₂×C₅₉	#3282	1	1	yes	characteristic
334	C₂×C₅₉	Q₈×SD₁₆	1	1	yes	characteristic
335	C₂×C₅₉	#3381	1	1	yes	characteristic
336	C₂²×SD₁₆	C₂²×C₅₉	1	1	yes	characteristic
337	C₂×C₄⋊Q₈	C₂²×C₅₉	1	1	yes	characteristic
338	C₂×C₄⋊Q₈	C₂²×C₅₉	1	1	yes	characteristic
339	C₂×C₂²⋊Q₈	—	2	1	no	—
340	C₂×C₄×Q₈	C₂²×C₅₉	1	1	yes	characteristic
341	C₂×C₄×D₄	C₂²×C₅₉	1	1	yes	characteristic
342	#498	—	2	4	no	—
343	C₄×SD₁₆	—	2	4	no	—
344	C₂×#106	C₂²×C₅₉	1	1	yes	characteristic
345	C₂×#106	C₂²×C₅₉	1	1	yes	characteristic
346	C₂×#106	C₂²×C₅₉	1	1	yes	characteristic
347	C₂×#106	C₂²×C₅₉	1	1	yes	characteristic
348	C₂×#106	C₂²×C₅₉	1	1	yes	characteristic
349	C₂×C₄⋊C₈	C₂²×C₅₉	1	1	yes	characteristic
350	C₂×#103	C₂²×C₅₉	1	1	yes	characteristic
351	C₂×#103	C₂²×C₅₉	1	1	yes	characteristic
352	C₂×#103	C₂²×C₅₉	1	1	yes	characteristic
353	C₂×#102	C₂²×C₅₉	1	1	yes	characteristic
354	C₂×#102	C₂²×C₅₉	1	1	yes	characteristic
355	C₂×#102	C₂²×C₅₉	1	1	yes	characteristic
356	C₂×C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic
357	C₂×C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic
358	C₂×C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic
359	#398	—	2	1	no	—
360	#396	—	2	1	no	—
361	#395	—	2	1	no	—
362	#393	—	2	1	no	—
363	#391	—	2	1	no	—
364	#391	C₂²×C₅₉	1	1	yes	characteristic
365	#391	C₂²×C₅₉	1	1	yes	characteristic
366	#390	C₂²×C₅₉	1	1	yes	characteristic
367	#390	C₂²×C₅₉	1	1	yes	characteristic
368	#389	C₂²×C₅₉	1	1	yes	characteristic
369	#389	C₂²×C₅₉	1	1	yes	characteristic
370	#389	C₂²×C₅₉	1	1	yes	characteristic
371	#389	C₂²×C₅₉	1	1	yes	characteristic
372	#387	—	2	1	no	—
373	#385	—	2	1	no	—
374	C₄×C₄⋊C₄	C₂²×C₅₉	1	1	yes	characteristic
375	#340	C₂²×C₅₉	1	1	yes	characteristic
376	#340	C₂²×C₅₉	1	1	yes	characteristic
377	#340	C₂²×C₅₉	1	1	yes	characteristic
378	#340	C₂²×C₅₉	1	1	yes	characteristic
379	#336	C₂²×C₅₉	1	1	yes	characteristic
380	#336	C₂²×C₅₉	1	1	yes	characteristic
381	C₅₉	#26480	1	1	yes	characteristic, direct product factor, Sylow
382	C₂²×Q₈	—	2	1	no	—
383	C₂²×Q₈	—	2	1	no	—
384	C₂²×Q₈	C₂³×C₅₉	1	1	yes	characteristic
385	C₂²×D₄	C₂³×C₅₉	1	1	yes	characteristic
386	C₂³×C₄	—	2	1	no	—
387	C₂×SD₁₆	—	2	2	no	—
388	C₂×SD₁₆	—	2	2	no	—
389	C₂×SD₁₆	Q₈×C₅₉	1	4	yes	complemented
390	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
391	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
392	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
393	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
394	C₄⋊Q₈	—	2	2	no	—
395	C₄⋊Q₈	—	2	2	no	—
396	C₄⋊Q₈	—	4	1	no	—
397	C₄⋊Q₈	—	4	1	no	—
398	C₂²⋊Q₈	—	4	4	no	—
399	C₄×Q₈	—	2	2	no	—
400	C₄×Q₈	—	4	1	no	—
401	C₄×D₄	—	2	2	no	—
402	C₄×D₄	—	4	1	no	—
403	C₂×C₄⋊C₄	—	2	1	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	1	no	—
409	C₂×C₄⋊C₄	—	2	1	no	—
410	C₂×C₄⋊C₄	D₄×C₅₉	1	1	yes	characteristic
411	C₂×C₄⋊C₄	D₄×C₅₉	1	1	yes	characteristic
412	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
413	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
414	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
415	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
416	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
417	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
418	C₂×C₂²⋊C₄	—	2	1	no	—
419	C₂×C₂²⋊C₄	—	2	1	no	—
420	C₂×C₂²⋊C₄	—	2	1	no	—
421	C₂×C₂²⋊C₄	—	2	1	no	—
422	C₂×C₂²⋊C₄	—	2	1	no	—
423	C₂×C₄²	—	2	1	no	—
424	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
425	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
426	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
427	#106	—	2	2	no	—
428	#106	—	2	2	no	—
429	#106	—	2	2	no	—
430	#106	—	2	2	no	—
431	#106	D₄×C₅₉	1	4	yes	—
432	C₄⋊C₈	—	2	2	no	—
433	#103	—	2	2	no	—
434	#103	—	2	2	no	—
435	#103	—	4	1	no	—
436	#102	—	2	2	no	—
437	#102	—	2	2	no	—
438	#102	—	4	1	no	—
439	C₄×C₈	—	2	2	no	—
440	C₄×C₈	—	2	2	no	—
441	C₄×C₈	—	2	2	no	—
442	#95	—	2	1	no	—
443	#95	—	2	1	no	—
444	#95	—	2	1	no	—
445	#95	—	2	1	no	—
446	#95	—	2	1	no	—
447	#95	—	2	1	no	—
448	#95	—	2	1	no	—
449	#95	—	2	1	no	—
450	#95	D₄×C₅₉	1	1	yes	characteristic
451	#95	D₄×C₅₉	1	1	yes	characteristic
452	C₂⁴	—	2	1	no	—
453	C₂×Q₈	—	2	1	no	—
454	C₂×Q₈	—	2	1	no	—
455	C₂×Q₈	—	2	1	no	—
456	C₂×Q₈	—	2	1	no	—
457	C₂×Q₈	—	2	1	no	—
458	C₂×Q₈	—	4	2	no	—
459	C₂×Q₈	—	4	2	no	—
460	C₂×Q₈	—	4	2	no	—
461	C₂×Q₈	—	4	2	no	—
462	C₂×Q₈	—	8	1	no	—
463	C₂×Q₈	—	8	1	no	—
464	C₂×Q₈	C₂×Q₈×C₅₉	1	2	yes	—
465	C₂×D₄	—	2	1	no	—
466	C₂×D₄	—	2	1	no	—
467	C₂×D₄	—	2	1	no	—
468	C₂×D₄	—	2	1	no	—
469	C₂×D₄	—	2	1	no	—
470	C₂×D₄	C₂×Q₈×C₅₉	1	2	yes	—
471	C₂²×C₄	—	2	1	no	—
472	C₂²×C₄	—	2	1	no	—
473	C₂²×C₄	—	2	1	no	—
474	C₂²×C₄	—	2	1	no	—
475	C₂²×C₄	—	2	1	no	—
476	C₂²×C₄	—	2	1	no	—
477	C₂²×C₄	—	2	1	no	—
478	C₂²×C₄	—	4	2	no	—
479	C₂²×C₄	—	4	2	no	—
480	C₂²×C₄	—	8	1	no	—
481	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
482	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
483	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
484	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
485	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
486	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
487	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
488	SD₁₆	—	2	8	no	—
489	C₂×C₈	—	2	1	no	—
490	C₂×C₈	—	2	1	no	—
491	C₂×C₈	—	2	1	no	—
492	C₂×C₈	—	2	1	no	—
493	C₂×C₈	—	2	1	no	—
494	C₂×C₈	—	2	1	no	—
495	C₂×C₈	—	2	1	no	—
496	C₂×C₈	—	2	1	no	—
497	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
498	C₂×C₈	C₂×Q₈×C₅₉	1	2	yes	—
499	C₂×C₈	#41×C₅₉	1	2	yes	—
500	C₂×C₈	#41×C₅₉	1	2	yes	—
501	C₄⋊C₄	—	2	2	no	—
502	C₄⋊C₄	—	2	1	no	—
503	C₄⋊C₄	—	2	1	no	—
504	C₄⋊C₄	—	2	1	no	—
505	C₄⋊C₄	—	2	1	no	—
506	C₄⋊C₄	—	2	1	no	—
507	C₄⋊C₄	—	2	1	no	—
508	C₄⋊C₄	—	2	1	no	—
509	C₄⋊C₄	—	2	1	no	—
510	C₄⋊C₄	—	2	1	no	—
511	C₄⋊C₄	—	2	1	no	—
512	C₄⋊C₄	—	4	2	no	—
513	C₄⋊C₄	—	4	2	no	—
514	C₄⋊C₄	—	4	2	no	—
515	C₄⋊C₄	—	4	2	no	—
516	C₄⋊C₄	—	4	2	no	—
517	C₄⋊C₄	—	8	1	no	—
518	C₄⋊C₄	—	8	1	no	—
519	C₄⋊C₄	SD₁₆×C₅₉	1	4	yes	complemented
520	C₄⋊C₄	C₂×D₄×C₅₉	1	2	yes	—
521	C₄⋊C₄	C₂×D₄×C₅₉	1	2	yes	—
522	C₂²⋊C₄	—	4	2	no	—
523	C₂²⋊C₄	—	4	2	no	—
524	C₂²⋊C₄	—	4	2	no	—
525	C₂²⋊C₄	—	4	2	no	—
526	C₂²⋊C₄	—	8	1	no	—
527	C₄²	—	2	1	no	—
528	C₄²	—	2	1	no	—
529	C₄²	—	2	1	no	—
530	C₄²	—	2	1	no	—
531	C₄²	—	2	1	no	—
532	C₄²	—	2	1	no	—
533	C₄²	—	4	2	no	—
534	C₂³	—	2	2	no	—
535	C₂³	—	4	1	no	—
536	C₂³	—	4	1	no	—
537	C₂³	—	4	1	no	—
538	C₂³	—	4	1	no	—
539	C₂³	—	4	1	no	—
540	C₂³	—	4	1	no	—
541	C₂³	C₂²×D₄×C₅₉	1	1	yes	characteristic
542	Q₈	—	2	2	no	—
543	Q₈	—	2	2	no	—
544	Q₈	—	4	1	no	—
545	Q₈	—	4	1	no	—
546	Q₈	—	8	4	no	—
547	Q₈	—	8	4	no	—
548	D₄	—	2	2	no	—
549	D₄	—	2	2	no	—
550	D₄	—	4	1	no	—
551	D₄	—	4	1	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	1	no	—
558	C₂×C₄	—	2	1	no	—
559	C₂×C₄	—	2	1	no	—
560	C₂×C₄	—	2	1	no	—
561	C₂×C₄	—	2	1	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	1	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	1	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	1	no	—
574	C₂×C₄	—	4	1	no	—
575	C₂×C₄	—	4	1	no	—
576	C₂×C₄	—	4	1	no	—
577	C₂×C₄	—	4	1	no	—
578	C₂×C₄	—	4	1	no	—
579	C₂×C₄	—	4	1	no	—
580	C₂×C₄	—	4	1	no	—
581	C₂×C₄	—	4	1	no	—
582	C₂×C₄	—	4	1	no	—
583	C₂×C₄	—	4	1	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	4	1	no	—
586	C₂×C₄	—	4	1	no	—
587	C₂×C₄	—	8	4	no	—
588	C₂×C₄	C₂×SD₁₆×C₅₉	1	2	yes	—
589	C₂×C₄	C₂×SD₁₆×C₅₉	1	2	yes	—
590	C₂×C₄	C₂×SD₁₆×C₅₉	1	2	yes	—
591	C₂×C₄	#135×C₅₉	1	2	yes	—
592	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
593	C₂×C₄	C₂²×Q₈×C₅₉	1	1	yes	characteristic
594	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
595	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
596	C₂×C₄	H(5,𝔽₂)×C₅₉	1	1	yes	characteristic
597	C₂×C₄	2₋¹⁺⁴×C₅₉	1	1	yes	characteristic
598	C₈	—	2	2	no	—
599	C₈	—	4	1	no	—
600	C₈	—	4	1	no	—
601	C₈	C₂²⋊Q₈×C₅₉	1	4	yes	complemented
602	C₂²	—	4	2	no	—
603	C₂²	—	4	2	no	—
604	C₂²	—	4	2	no	—
605	C₂²	—	8	1	no	—
606	C₂²	—	8	1	no	—
607	C₂²	—	8	1	no	—
608	C₂²	—	8	1	no	—
609	C₂²	C₅₉×#532	1	1	yes	characteristic
610	C₂²	D₄²×C₅₉	1	1	yes	characteristic
611	C₂²	D₄×Q₈×C₅₉	1	1	yes	characteristic
612	C₂²	C₂²×SD₁₆×C₅₉	1	1	yes	characteristic
613	C₂²	C₂×#135×C₅₉	1	1	yes	characteristic
614	C₂²	C₅₉×#577	1	1	yes	characteristic
615	C₂²	C₅₉×#577	1	1	yes	characteristic
616	C₄	—	2	2	no	—
617	C₄	—	2	2	no	—
618	C₄	—	2	2	no	—
619	C₄	—	2	2	no	—
620	C₄	—	4	2	no	—
621	C₄	—	4	1	no	—
622	C₄	—	4	2	no	—
623	C₄	—	4	1	no	—
624	C₄	—	4	2	no	—
625	C₄	—	8	1	no	—
626	C₄	—	8	1	no	—
627	C₄	—	8	1	no	—
628	C₄	—	8	1	no	—
629	C₄	C₂×C₂²⋊Q₈×C₅₉	1	1	yes	characteristic
630	C₄	C₅₉×#542	1	1	yes	characteristic
631	C₄	C₅₉×#548	1	1	yes	characteristic
632	C₄	C₅₉×#554	1	1	yes	characteristic
633	C₂	—	8	2	no	—
634	C₂	C₅₉×#2452	1	1	yes	characteristic
635	C₂	C₅₉×#3167	1	1	yes	characteristic
636	C₂	C₅₉×#3173	1	1	yes	characteristic
637	C₂	D₄×SD₁₆×C₅₉	1	1	yes	characteristic
638	C₂	C₅₉×#3282	1	1	yes	characteristic
639	C₂	Q₈×SD₁₆×C₅₉	1	1	yes	characteristic
640	C₂	C₅₉×#3381	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₉
prime radical	C₂²×D₄×C₅₉	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₉ ◁ C₄×C₄⋊C₄×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₉×#498	C₂	semidirect
4. C₅₉×#2451	C₂	semidirect
5. C₅₉×#2066	C₂	semidirect
7. C₅₉×#2024	C₂	semidirect
11. C₅₉×#1941	C₂	semidirect
12. C₅₉×#1941	C₂	semidirect
13. C₅₉×#1865	C₂	semidirect
15. C₈×C₄⋊C₄×C₅₉	C₂	semidirect
68. C₂×SD₁₆×C₅₉	Q₈	semidirect
198. C₄⋊C₄×C₅₉	SD₁₆	semidirect
280. C₈×C₅₉	C₂²⋊Q₈	semidirect
281. #26480	C₅₉	direct (N × H)
313. C₂×#498	C₂×C₅₉	semidirect
316. #2451	C₂×C₅₉	semidirect
317. #2066	C₂×C₅₉	semidirect
319. #2024	C₂×C₅₉	semidirect
323. #1941	C₂×C₅₉	semidirect
324. #1941	C₂×C₅₉	semidirect
325. #1865	C₂×C₅₉	semidirect
327. C₈×C₄⋊C₄	C₂×C₅₉	semidirect
381. C₅₉	#26480	direct (N × H)
389. C₂×SD₁₆	Q₈×C₅₉	semidirect
519. C₄⋊C₄	SD₁₆×C₅₉	semidirect
601. C₈	C₂²⋊Q₈×C₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₉×#498	2
C₂×C₄×SD₁₆×C₅₉	2
C₅₉×#2452	2
C₅₉×#2451	2
C₅₉×#2066	2
C₅₉×#2065	2
C₅₉×#2024	2
C₅₉×#2023	2
C₅₉×#1968	2
C₅₉×#1941	2
C₅₉×#1865	2
C₅₉×#1864	2
C₈×C₄⋊C₄×C₅₉	2
#26480	59

Subgroups — C59×#26480 #28438589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#26480 #28438589