Subgroups — C₅₉×#35208 #28447317

Order 15104.

1588 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#2863	C₂	1	2	yes	maximal
2	C₅₉×#2859	C₂	1	2	yes	maximal
3	C₅₉×#2666	C₂	1	4	yes	maximal
4	C₅₉×#2582	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#2567	C₂	1	1	yes	maximal, characteristic
6	C₅₉×#2567	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#2552	C₂	1	1	yes	maximal, characteristic
8	C₅₉×#2511	C₂	1	2	yes	maximal
9	C₅₉×#2507	C₂	1	2	yes	maximal
10	C₅₉×#2506	C₂	1	2	yes	maximal
11	C₅₉×#2462	C₂	1	4	yes	maximal
12	C₅₉×#2453	C₂	1	2	yes	maximal
13	C₅₉×#2330	C₂	1	2	yes	maximal
14	C₅₉×#2320	C₂	1	2	yes	maximal
15	C₄×C₄⋊Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
16	C₅₉×#2295	C₂	1	1	yes	maximal, characteristic
17	C₄²×Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
18	C₂×C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×#125×C₅₉	C₂²	1	2	yes	—
20	C₂×#125×C₅₉	C₂²	1	4	yes	—
21	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₅₉	C₂²	1	2	yes	—
25	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
26	C₅₉×#400	C₂²	1	4	yes	—
27	C₅₉×#400	C₂²	1	4	yes	—
28	C₅₉×#398	C₂²	1	4	yes	—
29	C₅₉×#398	C₂²	1	4	yes	—
30	C₅₉×#398	C₂²	1	4	yes	—
31	C₅₉×#398	C₂²	1	4	yes	—
32	C₅₉×#395	C₂²	1	4	yes	—
33	C₅₉×#395	C₂²	1	4	yes	—
34	C₅₉×#391	C₂²	1	1	yes	characteristic
35	C₅₉×#391	C₂²	1	2	yes	—
36	C₅₉×#391	C₂²	1	2	yes	—
37	C₅₉×#391	C₂²	1	1	yes	characteristic
38	C₅₉×#391	C₂²	1	2	yes	—
39	C₅₉×#391	C₂²	1	2	yes	—
40	C₅₉×#391	C₂²	1	2	yes	—
41	C₅₉×#389	C₂²	1	2	yes	—
42	C₅₉×#389	C₂²	1	2	yes	—
43	C₅₉×#389	C₂²	1	2	yes	—
44	C₅₉×#389	C₂²	1	2	yes	—
45	C₅₉×#389	C₂²	1	1	yes	characteristic
46	C₅₉×#389	C₂²	1	2	yes	—
47	C₅₉×#389	C₂²	1	2	yes	—
48	C₅₉×#389	C₂²	1	2	yes	—
49	C₅₉×#389	C₂²	1	4	yes	—
50	C₅₉×#389	C₂²	1	2	yes	—
51	C₅₉×#389	C₂²	1	1	yes	characteristic
52	C₅₉×#389	C₂²	1	2	yes	—
53	C₅₉×#389	C₂²	1	2	yes	—
54	C₅₉×#389	C₂²	1	4	yes	—
55	C₅₉×#389	C₂²	1	4	yes	—
56	C₅₉×#387	C₂²	1	4	yes	—
57	C₅₉×#387	C₂²	1	4	yes	—
58	C₅₉×#387	C₂²	1	4	yes	—
59	C₅₉×#387	C₂²	1	4	yes	—
60	C₅₉×#387	C₂²	1	2	yes	—
61	C₅₉×#387	C₂²	1	2	yes	—
62	C₅₉×#387	C₂²	1	4	yes	—
63	C₅₉×#387	C₂²	1	4	yes	—
64	C₅₉×#384	C₂²	1	2	yes	—
65	C₅₉×#384	C₂²	1	2	yes	—
66	C₅₉×#384	C₂²	1	2	yes	—
67	C₅₉×#384	C₂²	1	1	yes	characteristic
68	C₅₉×#383	C₂²	1	4	yes	—
69	C₅₉×#382	C₂²	1	4	yes	—
70	C₄×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
71	C₄×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
72	C₄×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
73	C₄×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
74	C₄×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
75	C₄×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
76	C₄×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
77	C₄×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
78	C₄×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
79	C₄×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
80	C₅₉×#376	C₂²	1	2	yes	—
81	C₄³×C₅₉	C₂²	1	1	yes	characteristic
82	C₄³×C₅₉	C₂²	1	2	yes	—
83	C₂²×Q₈×C₅₉	C₂³	1	1	yes	characteristic
84	C₂²×Q₈×C₅₉	C₂³	1	1	yes	characteristic
85	C₂²×Q₈×C₅₉	C₂³	1	2	yes	—
86	C₄⋊Q₈×C₅₉	—	2	4	no	—
87	#125×C₅₉	—	2	8	no	—
88	#125×C₅₉	—	4	8	no	—
89	C₄×Q₈×C₅₉	—	2	4	no	—
90	C₄×Q₈×C₅₉	—	2	4	no	—
91	C₄×Q₈×C₅₉	—	2	4	no	—
92	C₄×Q₈×C₅₉	—	4	4	no	—
93	C₄×Q₈×C₅₉	Q₈	1	8	yes	complemented
94	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
97	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
98	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
106	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
107	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
108	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
110	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
111	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
112	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
113	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
114	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
115	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
116	C₂×C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
117	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
118	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
119	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
122	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
123	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
124	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
125	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
126	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
127	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
128	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
129	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
130	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
131	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
132	C₂×C₄²×C₅₉	C₂³	1	4	yes	—
133	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
135	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
136	#95×C₅₉	C₂³	1	2	yes	—
137	#95×C₅₉	C₂³	1	2	yes	—
138	#95×C₅₉	C₂³	1	2	yes	—
139	#95×C₅₉	C₂³	1	2	yes	—
140	#95×C₅₉	C₂³	1	2	yes	—
141	#95×C₅₉	C₂³	1	2	yes	—
142	#95×C₅₉	C₂³	1	2	yes	—
143	#95×C₅₉	C₂³	1	4	yes	—
144	#95×C₅₉	C₂³	1	2	yes	—
145	#95×C₅₉	C₂³	1	2	yes	—
146	#95×C₅₉	C₂³	1	2	yes	—
147	#95×C₅₉	C₂³	1	4	yes	—
148	#95×C₅₉	C₂³	1	4	yes	—
149	#95×C₅₉	C₂³	1	4	yes	—
150	#95×C₅₉	C₂³	1	4	yes	—
151	#95×C₅₉	C₂³	1	4	yes	—
152	#95×C₅₉	C₂³	1	4	yes	—
153	#95×C₅₉	C₂³	1	4	yes	—
154	#95×C₅₉	C₂³	1	4	yes	—
155	#95×C₅₉	C₂³	1	4	yes	—
156	C₂×Q₈×C₅₉	—	2	2	no	—
157	C₂×Q₈×C₅₉	—	2	2	no	—
158	C₂×Q₈×C₅₉	—	2	2	no	—
159	C₂×Q₈×C₅₉	—	2	4	no	—
160	C₂×Q₈×C₅₉	—	2	2	no	—
161	C₂×Q₈×C₅₉	—	2	2	no	—
162	C₂×Q₈×C₅₉	—	4	2	no	—
163	C₂×Q₈×C₅₉	—	4	2	no	—
164	C₂×Q₈×C₅₉	C₂×Q₈	1	4	yes	—
165	C₂²×C₄×C₅₉	C₂⁴	1	2	yes	—
166	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
167	C₂²×C₄×C₅₉	C₂⁴	1	2	yes	—
168	C₂²×C₄×C₅₉	C₂⁴	1	2	yes	—
169	C₂²×C₄×C₅₉	C₂⁴	1	4	yes	—
170	C₂²×C₄×C₅₉	C₂⁴	1	4	yes	—
171	C₂²×C₄×C₅₉	C₂⁴	1	2	yes	—
172	C₂²×C₄×C₅₉	C₂⁴	1	2	yes	—
173	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
174	C₂²×C₄×C₅₉	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
176	C₂²×C₄×C₅₉	C₂⁴	1	2	yes	—
177	C₂²×C₄×C₅₉	C₂⁴	1	2	yes	—
178	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
179	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
180	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
181	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
182	C₄⋊C₄×C₅₉	—	2	4	no	—
183	C₄⋊C₄×C₅₉	—	2	4	no	—
184	C₄⋊C₄×C₅₉	—	2	2	no	—
185	C₄⋊C₄×C₅₉	—	2	2	no	—
186	C₄⋊C₄×C₅₉	—	2	4	no	—
187	C₄⋊C₄×C₅₉	—	2	8	no	—
188	C₄⋊C₄×C₅₉	—	2	8	no	—
189	C₄⋊C₄×C₅₉	—	2	4	no	—
190	C₄⋊C₄×C₅₉	—	2	4	no	—
191	C₄⋊C₄×C₅₉	—	2	4	no	—
192	C₄⋊C₄×C₅₉	—	2	8	no	—
193	C₄⋊C₄×C₅₉	—	2	4	no	—
194	C₄⋊C₄×C₅₉	—	2	2	no	—
195	C₄⋊C₄×C₅₉	—	2	4	no	—
196	C₄⋊C₄×C₅₉	—	2	2	no	—
197	C₄⋊C₄×C₅₉	—	4	4	no	—
198	C₄⋊C₄×C₅₉	—	4	2	no	—
199	C₄⋊C₄×C₅₉	—	4	4	no	—
200	C₄⋊C₄×C₅₉	—	4	2	no	—
201	C₄⋊C₄×C₅₉	—	4	4	no	—
202	C₄⋊C₄×C₅₉	—	4	4	no	—
203	C₄⋊C₄×C₅₉	—	4	4	no	—
204	C₄⋊C₄×C₅₉	—	4	4	no	—
205	C₄⋊C₄×C₅₉	C₂×Q₈	1	8	yes	—
206	C₄⋊C₄×C₅₉	C₂×Q₈	1	4	yes	—
207	C₄⋊C₄×C₅₉	#41	1	4	yes	—
208	C₄²×C₅₉	—	2	4	no	—
209	C₄²×C₅₉	—	2	2	no	—
210	C₄²×C₅₉	—	2	4	no	—
211	C₄²×C₅₉	—	2	4	no	—
212	C₄²×C₅₉	—	2	4	no	—
213	C₄²×C₅₉	—	2	2	no	—
214	C₄²×C₅₉	—	2	4	no	—
215	C₄²×C₅₉	—	2	4	no	—
216	C₄²×C₅₉	—	2	2	no	—
217	C₄²×C₅₉	—	2	2	no	—
218	C₄²×C₅₉	—	4	2	no	—
219	C₄²×C₅₉	—	4	1	no	—
220	C₄²×C₅₉	—	4	4	no	—
221	C₄²×C₅₉	C₂×Q₈	1	8	yes	—
222	C₄²×C₅₉	C₂×Q₈	1	4	yes	—
223	C₄²×C₅₉	#41	1	4	yes	—
224	C₂³×C₅₉	C₂⁵	1	1	yes	characteristic
225	Q₈×C₅₉	—	2	8	no	—
226	Q₈×C₅₉	—	4	8	no	—
227	Q₈×C₅₉	—	4	4	no	—
228	C₂×C₄×C₅₉	—	2	1	no	—
229	C₂×C₄×C₅₉	—	2	2	no	—
230	C₂×C₄×C₅₉	—	2	2	no	—
231	C₂×C₄×C₅₉	—	2	1	no	—
232	C₂×C₄×C₅₉	—	2	2	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	2	no	—
237	C₂×C₄×C₅₉	—	2	2	no	—
238	C₂×C₄×C₅₉	—	2	1	no	—
239	C₂×C₄×C₅₉	—	2	1	no	—
240	C₂×C₄×C₅₉	—	2	2	no	—
241	C₂×C₄×C₅₉	—	2	1	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	2	no	—
246	C₂×C₄×C₅₉	—	2	2	no	—
247	C₂×C₄×C₅₉	—	2	2	no	—
248	C₂×C₄×C₅₉	—	2	4	no	—
249	C₂×C₄×C₅₉	—	2	4	no	—
250	C₂×C₄×C₅₉	—	2	2	no	—
251	C₂×C₄×C₅₉	—	2	4	no	—
252	C₂×C₄×C₅₉	—	2	4	no	—
253	C₂×C₄×C₅₉	—	2	2	no	—
254	C₂×C₄×C₅₉	—	2	2	no	—
255	C₂×C₄×C₅₉	—	2	1	no	—
256	C₂×C₄×C₅₉	—	2	2	no	—
257	C₂×C₄×C₅₉	—	2	2	no	—
258	C₂×C₄×C₅₉	—	2	4	no	—
259	C₂×C₄×C₅₉	—	2	2	no	—
260	C₂×C₄×C₅₉	—	2	2	no	—
261	C₂×C₄×C₅₉	—	2	2	no	—
262	C₂×C₄×C₅₉	—	2	1	no	—
263	C₂×C₄×C₅₉	—	2	2	no	—
264	C₂×C₄×C₅₉	—	2	2	no	—
265	C₂×C₄×C₅₉	—	2	4	no	—
266	C₂×C₄×C₅₉	C₂²×Q₈	1	2	yes	—
267	C₂×C₄×C₅₉	C₂²×Q₈	1	2	yes	—
268	C₂×C₄×C₅₉	C₂²×Q₈	1	4	yes	—
269	C₂×C₄×C₅₉	C₂²×Q₈	1	4	yes	—
270	C₂×C₄×C₅₉	C₂²×Q₈	1	2	yes	—
271	C₂×C₄×C₅₉	C₂×#41	1	4	yes	—
272	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
273	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
274	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
275	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
276	C₂×C₄×C₅₉	H(5,𝔽₂)	1	2	yes	—
277	C₂×C₄×C₅₉	2₋¹⁺⁴	1	2	yes	—
278	C₂×C₄×C₅₉	2₋¹⁺⁴	1	2	yes	—
279	#35208	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
280	C₂²×C₅₉	C₂³×Q₈	1	1	yes	characteristic
281	C₂²×C₅₉	C₂²×#41	1	1	yes	characteristic
282	C₂²×C₅₉	C₂²×#41	1	1	yes	characteristic
283	C₂²×C₅₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
284	C₂²×C₅₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
285	C₂²×C₅₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
286	C₂²×C₅₉	#585	1	1	yes	characteristic
287	C₄×C₅₉	—	2	4	no	—
288	C₄×C₅₉	—	2	2	no	—
289	C₄×C₅₉	—	2	2	no	—
290	C₄×C₅₉	—	2	2	no	—
291	C₄×C₅₉	—	2	4	no	—
292	C₄×C₅₉	—	2	4	no	—
293	C₄×C₅₉	—	2	2	no	—
294	C₄×C₅₉	—	4	2	no	—
295	C₄×C₅₉	—	4	2	no	—
296	C₄×C₅₉	—	4	2	no	—
297	C₄×C₅₉	—	4	4	no	—
298	C₄×C₅₉	—	4	1	no	—
299	C₄×C₅₉	—	4	2	no	—
300	C₄×C₅₉	—	4	2	no	—
301	C₄×C₅₉	—	4	4	no	—
302	C₄×C₅₉	#533	1	4	yes	—
303	C₄×C₅₉	#541	1	4	yes	—
304	#2863	C₂×C₅₉	1	2	yes	—
305	#2859	C₂×C₅₉	1	2	yes	—
306	#2666	C₂×C₅₉	1	4	yes	—
307	#2582	C₂×C₅₉	1	1	yes	characteristic
308	#2567	C₂×C₅₉	1	1	yes	characteristic
309	#2567	C₂×C₅₉	1	1	yes	characteristic
310	#2552	C₂×C₅₉	1	1	yes	characteristic
311	#2511	C₂×C₅₉	1	2	yes	—
312	#2507	C₂×C₅₉	1	2	yes	—
313	#2506	C₂×C₅₉	1	2	yes	—
314	#2462	C₂×C₅₉	1	4	yes	—
315	#2453	C₂×C₅₉	1	2	yes	—
316	#2330	C₂×C₅₉	1	2	yes	—
317	#2320	C₂×C₅₉	1	2	yes	—
318	C₄×C₄⋊Q₈	C₂×C₅₉	1	1	yes	characteristic
319	#2295	C₂×C₅₉	1	1	yes	characteristic
320	C₄²×Q₈	C₂×C₅₉	1	1	yes	characteristic
321	C₂×C₅₉	C₂×#533	1	1	yes	characteristic
322	C₂×C₅₉	C₂×#541	1	1	yes	characteristic
323	C₂×C₅₉	#3460	1	1	yes	characteristic
324	C₂×C₅₉	#3502	1	1	yes	characteristic
325	C₂×C₅₉	#3503	1	1	yes	characteristic
326	C₂×C₅₉	#3515	1	1	yes	characteristic
327	C₂×C₅₉	#3522	1	1	yes	characteristic
328	C₂×C₄⋊Q₈	C₂²×C₅₉	1	1	yes	characteristic
329	C₂×#125	C₂²×C₅₉	1	2	yes	—
330	C₂×#125	C₂²×C₅₉	1	4	yes	—
331	C₂×C₄×Q₈	C₂²×C₅₉	1	1	yes	characteristic
332	C₂×C₄×Q₈	C₂²×C₅₉	1	1	yes	characteristic
333	C₂×C₄×Q₈	C₂²×C₅₉	1	1	yes	characteristic
334	C₂×C₄×Q₈	C₂²×C₅₉	1	2	yes	—
335	C₂×C₄×Q₈	C₂²×C₅₉	1	1	yes	characteristic
336	#400	C₂²×C₅₉	1	4	yes	—
337	#400	C₂²×C₅₉	1	4	yes	—
338	#398	C₂²×C₅₉	1	4	yes	—
339	#398	C₂²×C₅₉	1	4	yes	—
340	#398	C₂²×C₅₉	1	4	yes	—
341	#398	C₂²×C₅₉	1	4	yes	—
342	#395	C₂²×C₅₉	1	4	yes	—
343	#395	C₂²×C₅₉	1	4	yes	—
344	#391	C₂²×C₅₉	1	2	yes	—
345	#391	C₂²×C₅₉	1	2	yes	—
346	#391	C₂²×C₅₉	1	2	yes	—
347	#391	C₂²×C₅₉	1	2	yes	—
348	#391	C₂²×C₅₉	1	2	yes	—
349	#391	C₂²×C₅₉	1	1	yes	characteristic
350	#391	C₂²×C₅₉	1	1	yes	characteristic
351	#389	C₂²×C₅₉	1	2	yes	—
352	#389	C₂²×C₅₉	1	2	yes	—
353	#389	C₂²×C₅₉	1	2	yes	—
354	#389	C₂²×C₅₉	1	2	yes	—
355	#389	C₂²×C₅₉	1	2	yes	—
356	#389	C₂²×C₅₉	1	2	yes	—
357	#389	C₂²×C₅₉	1	2	yes	—
358	#389	C₂²×C₅₉	1	2	yes	—
359	#389	C₂²×C₅₉	1	2	yes	—
360	#389	C₂²×C₅₉	1	4	yes	—
361	#389	C₂²×C₅₉	1	4	yes	—
362	#389	C₂²×C₅₉	1	1	yes	characteristic
363	#389	C₂²×C₅₉	1	4	yes	—
364	#389	C₂²×C₅₉	1	2	yes	—
365	#389	C₂²×C₅₉	1	1	yes	characteristic
366	#387	C₂²×C₅₉	1	4	yes	—
367	#387	C₂²×C₅₉	1	4	yes	—
368	#387	C₂²×C₅₉	1	4	yes	—
369	#387	C₂²×C₅₉	1	4	yes	—
370	#387	C₂²×C₅₉	1	4	yes	—
371	#387	C₂²×C₅₉	1	4	yes	—
372	#387	C₂²×C₅₉	1	2	yes	—
373	#387	C₂²×C₅₉	1	2	yes	—
374	#384	C₂²×C₅₉	1	1	yes	characteristic
375	#384	C₂²×C₅₉	1	2	yes	—
376	#384	C₂²×C₅₉	1	2	yes	—
377	#384	C₂²×C₅₉	1	2	yes	—
378	#383	C₂²×C₅₉	1	4	yes	—
379	#382	C₂²×C₅₉	1	4	yes	—
380	C₄×C₄⋊C₄	C₂²×C₅₉	1	2	yes	—
381	C₄×C₄⋊C₄	C₂²×C₅₉	1	2	yes	—
382	C₄×C₄⋊C₄	C₂²×C₅₉	1	1	yes	characteristic
383	C₄×C₄⋊C₄	C₂²×C₅₉	1	1	yes	characteristic
384	C₄×C₄⋊C₄	C₂²×C₅₉	1	1	yes	characteristic
385	C₄×C₄⋊C₄	C₂²×C₅₉	1	2	yes	—
386	C₄×C₄⋊C₄	C₂²×C₅₉	1	2	yes	—
387	C₄×C₄⋊C₄	C₂²×C₅₉	1	2	yes	—
388	C₄×C₄⋊C₄	C₂²×C₅₉	1	1	yes	characteristic
389	C₄×C₄⋊C₄	C₂²×C₅₉	1	2	yes	—
390	#376	C₂²×C₅₉	1	2	yes	—
391	C₄³	C₂²×C₅₉	1	1	yes	characteristic
392	C₄³	C₂²×C₅₉	1	2	yes	—
393	C₅₉	#35208	1	1	yes	characteristic, direct product factor, Sylow
394	C₂²×Q₈	C₂³×C₅₉	1	1	yes	characteristic
395	C₂²×Q₈	C₂³×C₅₉	1	1	yes	characteristic
396	C₂²×Q₈	C₂³×C₅₉	1	2	yes	—
397	C₄⋊Q₈	—	2	4	no	—
398	#125	—	2	8	no	—
399	#125	—	4	8	no	—
400	C₄×Q₈	—	2	4	no	—
401	C₄×Q₈	—	2	4	no	—
402	C₄×Q₈	—	2	4	no	—
403	C₄×Q₈	—	4	4	no	—
404	C₄×Q₈	Q₈×C₅₉	1	8	yes	complemented
405	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
406	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
407	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
408	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
409	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
410	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
411	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
412	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
413	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
414	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
415	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
416	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
417	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
418	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
419	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
420	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
421	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
422	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
423	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
424	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
425	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
426	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
427	C₂×C₄⋊C₄	C₂³×C₅₉	1	4	yes	—
428	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
429	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
430	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
431	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
432	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
433	C₂×C₄²	C₂³×C₅₉	1	2	yes	—
434	C₂×C₄²	C₂³×C₅₉	1	2	yes	—
435	C₂×C₄²	C₂³×C₅₉	1	2	yes	—
436	C₂×C₄²	C₂³×C₅₉	1	2	yes	—
437	C₂×C₄²	C₂³×C₅₉	1	2	yes	—
438	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
439	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
440	C₂×C₄²	C₂³×C₅₉	1	2	yes	—
441	C₂×C₄²	C₂³×C₅₉	1	4	yes	—
442	C₂×C₄²	C₂³×C₅₉	1	2	yes	—
443	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
444	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
445	C₂×C₄²	C₂³×C₅₉	1	2	yes	—
446	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
447	#95	C₂³×C₅₉	1	2	yes	—
448	#95	C₂³×C₅₉	1	2	yes	—
449	#95	C₂³×C₅₉	1	2	yes	—
450	#95	C₂³×C₅₉	1	2	yes	—
451	#95	C₂³×C₅₉	1	2	yes	—
452	#95	C₂³×C₅₉	1	2	yes	—
453	#95	C₂³×C₅₉	1	2	yes	—
454	#95	C₂³×C₅₉	1	4	yes	—
455	#95	C₂³×C₅₉	1	2	yes	—
456	#95	C₂³×C₅₉	1	2	yes	—
457	#95	C₂³×C₅₉	1	2	yes	—
458	#95	C₂³×C₅₉	1	4	yes	—
459	#95	C₂³×C₅₉	1	4	yes	—
460	#95	C₂³×C₅₉	1	4	yes	—
461	#95	C₂³×C₅₉	1	4	yes	—
462	#95	C₂³×C₅₉	1	4	yes	—
463	#95	C₂³×C₅₉	1	4	yes	—
464	#95	C₂³×C₅₉	1	4	yes	—
465	#95	C₂³×C₅₉	1	4	yes	—
466	#95	C₂³×C₅₉	1	4	yes	—
467	C₂×Q₈	—	2	2	no	—
468	C₂×Q₈	—	2	2	no	—
469	C₂×Q₈	—	2	2	no	—
470	C₂×Q₈	—	2	4	no	—
471	C₂×Q₈	—	2	2	no	—
472	C₂×Q₈	—	2	2	no	—
473	C₂×Q₈	—	4	2	no	—
474	C₂×Q₈	—	4	2	no	—
475	C₂×Q₈	C₂×Q₈×C₅₉	1	4	yes	—
476	C₂²×C₄	C₂⁴×C₅₉	1	2	yes	—
477	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
478	C₂²×C₄	C₂⁴×C₅₉	1	2	yes	—
479	C₂²×C₄	C₂⁴×C₅₉	1	2	yes	—
480	C₂²×C₄	C₂⁴×C₅₉	1	4	yes	—
481	C₂²×C₄	C₂⁴×C₅₉	1	4	yes	—
482	C₂²×C₄	C₂⁴×C₅₉	1	2	yes	—
483	C₂²×C₄	C₂⁴×C₅₉	1	2	yes	—
484	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
485	C₂²×C₄	C₂⁴×C₅₉	1	2	yes	—
486	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
487	C₂²×C₄	C₂⁴×C₅₉	1	2	yes	—
488	C₂²×C₄	C₂⁴×C₅₉	1	2	yes	—
489	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
490	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
491	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
492	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
493	C₄⋊C₄	—	2	4	no	—
494	C₄⋊C₄	—	2	4	no	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	—	2	2	no	—
497	C₄⋊C₄	—	2	4	no	—
498	C₄⋊C₄	—	2	8	no	—
499	C₄⋊C₄	—	2	8	no	—
500	C₄⋊C₄	—	2	4	no	—
501	C₄⋊C₄	—	2	4	no	—
502	C₄⋊C₄	—	2	4	no	—
503	C₄⋊C₄	—	2	8	no	—
504	C₄⋊C₄	—	2	4	no	—
505	C₄⋊C₄	—	2	2	no	—
506	C₄⋊C₄	—	2	4	no	—
507	C₄⋊C₄	—	2	2	no	—
508	C₄⋊C₄	—	4	2	no	—
509	C₄⋊C₄	—	4	4	no	—
510	C₄⋊C₄	—	4	4	no	—
511	C₄⋊C₄	—	4	4	no	—
512	C₄⋊C₄	—	4	4	no	—
513	C₄⋊C₄	—	4	4	no	—
514	C₄⋊C₄	—	4	2	no	—
515	C₄⋊C₄	—	4	4	no	—
516	C₄⋊C₄	C₂×Q₈×C₅₉	1	8	yes	—
517	C₄⋊C₄	C₂×Q₈×C₅₉	1	4	yes	—
518	C₄⋊C₄	#41×C₅₉	1	4	yes	—
519	C₄²	—	2	4	no	—
520	C₄²	—	2	2	no	—
521	C₄²	—	2	4	no	—
522	C₄²	—	2	4	no	—
523	C₄²	—	2	4	no	—
524	C₄²	—	2	2	no	—
525	C₄²	—	2	4	no	—
526	C₄²	—	2	4	no	—
527	C₄²	—	2	2	no	—
528	C₄²	—	2	2	no	—
529	C₄²	—	4	4	no	—
530	C₄²	—	4	2	no	—
531	C₄²	—	4	1	no	—
532	C₄²	C₂×Q₈×C₅₉	1	8	yes	—
533	C₄²	C₂×Q₈×C₅₉	1	4	yes	—
534	C₄²	#41×C₅₉	1	4	yes	—
535	C₂³	C₂⁵×C₅₉	1	1	yes	characteristic
536	Q₈	—	2	8	no	—
537	Q₈	—	4	8	no	—
538	Q₈	—	4	4	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	4	no	—
560	C₂×C₄	—	2	4	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	4	no	—
563	C₂×C₄	—	2	4	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	4	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	1	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	2	4	no	—
577	C₂×C₄	C₂²×Q₈×C₅₉	1	2	yes	—
578	C₂×C₄	C₂²×Q₈×C₅₉	1	2	yes	—
579	C₂×C₄	C₂²×Q₈×C₅₉	1	4	yes	—
580	C₂×C₄	C₂²×Q₈×C₅₉	1	4	yes	—
581	C₂×C₄	C₂²×Q₈×C₅₉	1	2	yes	—
582	C₂×C₄	C₂×#41×C₅₉	1	4	yes	—
583	C₂×C₄	C₂×#41×C₅₉	1	2	yes	—
584	C₂×C₄	C₂×#41×C₅₉	1	2	yes	—
585	C₂×C₄	C₂×#41×C₅₉	1	2	yes	—
586	C₂×C₄	C₂×#41×C₅₉	1	2	yes	—
587	C₂×C₄	H(5,𝔽₂)×C₅₉	1	2	yes	—
588	C₂×C₄	2₋¹⁺⁴×C₅₉	1	2	yes	—
589	C₂×C₄	2₋¹⁺⁴×C₅₉	1	2	yes	—
590	C₂²	C₂³×Q₈×C₅₉	1	1	yes	characteristic
591	C₂²	C₂²×#41×C₅₉	1	1	yes	characteristic
592	C₂²	C₂²×#41×C₅₉	1	1	yes	characteristic
593	C₂²	C₂×H(5,𝔽₂)×C₅₉	1	1	yes	characteristic
594	C₂²	C₂×2₋¹⁺⁴×C₅₉	1	1	yes	characteristic
595	C₂²	C₂×2₋¹⁺⁴×C₅₉	1	1	yes	characteristic
596	C₂²	C₅₉×#585	1	1	yes	characteristic
597	C₄	—	2	4	no	—
598	C₄	—	2	2	no	—
599	C₄	—	2	4	no	—
600	C₄	—	2	2	no	—
601	C₄	—	2	2	no	—
602	C₄	—	2	2	no	—
603	C₄	—	2	4	no	—
604	C₄	—	4	1	no	—
605	C₄	—	4	2	no	—
606	C₄	—	4	2	no	—
607	C₄	—	4	4	no	—
608	C₄	—	4	2	no	—
609	C₄	—	4	4	no	—
610	C₄	—	4	2	no	—
611	C₄	—	4	2	no	—
612	C₄	C₅₉×#533	1	4	yes	—
613	C₄	C₅₉×#541	1	4	yes	—
614	C₂	C₂×C₅₉×#533	1	1	yes	characteristic
615	C₂	C₂×C₅₉×#541	1	1	yes	characteristic
616	C₂	C₅₉×#3460	1	1	yes	characteristic
617	C₂	C₅₉×#3502	1	1	yes	characteristic
618	C₂	C₅₉×#3503	1	1	yes	characteristic
619	C₂	C₅₉×#3515	1	1	yes	characteristic
620	C₂	C₅₉×#3522	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₉
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
93. C₄×Q₈×C₅₉	Q₈	semidirect
279. #35208	C₅₉	direct (N × H)
393. C₅₉	#35208	direct (N × H)
404. C₄×Q₈	Q₈×C₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₅₉×#2863	2
C₅₉×#2859	2
C₅₉×#2666	2
C₅₉×#2582	2
C₅₉×#2567	2
C₅₉×#2552	2
C₅₉×#2511	2
C₅₉×#2507	2
C₅₉×#2506	2
C₅₉×#2462	2
C₅₉×#2453	2
C₅₉×#2330	2
C₅₉×#2320	2
C₄×C₄⋊Q₈×C₅₉	2
C₅₉×#2295	2
C₄²×Q₈×C₅₉	2
#35208	59

Subgroups — C59×#35208 #28447317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#35208 #28447317