Subgroups — C₅₉×#24751 #28436860

Order 15104.

1076 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₉×#465	C₂	1	2	yes	maximal
2	C₂×C₅₉×#461	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄⋊C₈×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₉×#2857	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#121×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₅₉×#2066	C₂	1	2	yes	maximal, complemented
7	C₅₉×#2055	C₂	1	2	yes	maximal, complemented
8	C₅₉×#1981	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₉×#1948	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₉×#1892	C₂	1	2	yes	maximal, complemented
11	C₅₉×#1802	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₅₉	C₂²	1	2	yes	—
13	C₂³×C₈×C₅₉	C₂²	1	2	yes	—
14	C₂×C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₅₉	C₂²	1	2	yes	—
16	C₂×#121×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₅₉	—	2	1	no	—
18	C₂×C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
19	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
20	C₅₉×#465	—	2	8	no	—
21	C₅₉×#461	—	2	4	no	—
22	C₂×#106×C₅₉	C₂²	1	2	yes	—
23	C₂×C₄⋊C₈×C₅₉	—	2	4	no	—
24	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₅₉	C₂²	1	2	yes	—
26	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
27	C₂×#103×C₅₉	C₂²	1	2	yes	—
28	C₂×#103×C₅₉	C₂²	1	2	yes	—
29	C₂×#103×C₅₉	C₂²	1	2	yes	—
30	C₂×#102×C₅₉	C₂²	1	2	yes	—
31	C₅₉×#391	C₂²	1	2	yes	—
32	C₅₉×#390	C₂²	1	1	yes	characteristic
33	C₅₉×#389	—	2	1	no	—
34	C₅₉×#389	—	2	1	no	—
35	C₅₉×#389	C₂²	1	1	yes	characteristic
36	C₅₉×#388	—	2	1	no	—
37	C₅₉×#386	—	2	1	no	—
38	C₅₉×#384	C₂²	1	1	yes	characteristic
39	C₅₉×#380	—	2	1	no	—
40	C₅₉×#380	—	2	1	no	—
41	C₅₉×#380	C₂²	1	1	yes	characteristic
42	C₄×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
43	C₄×C₂²⋊C₄×C₅₉	—	2	1	no	—
44	C₅₉×#340	C₂²	1	2	yes	—
45	C₅₉×#340	C₂²	1	2	yes	—
46	C₂²×Q₈×C₅₉	C₂³	1	2	yes	—
47	C₂²×D₄×C₅₉	—	2	1	no	—
48	C₂²×D₄×C₅₉	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₅₉	—	2	1	no	—
50	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
51	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
52	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
53	C₂×SD₁₆×C₅₉	—	2	4	no	—
54	C₂×SD₁₆×C₅₉	—	2	4	no	—
55	C₂×SD₁₆×C₅₉	—	4	2	no	—
56	C₂²×C₈×C₅₉	—	2	4	no	—
57	C₂²×C₈×C₅₉	—	2	4	no	—
58	C₂²×C₈×C₅₉	—	4	2	no	—
59	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
60	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
61	C₄⋊Q₈×C₅₉	—	2	2	no	—
62	C₄⋊Q₈×C₅₉	—	2	2	no	—
63	C₂²⋊Q₈×C₅₉	—	2	4	no	—
64	C₂²⋊Q₈×C₅₉	—	2	4	no	—
65	#121×C₅₉	—	2	2	no	—
66	#121×C₅₉	D₄	1	4	yes	—
67	C₄×D₄×C₅₉	—	2	2	no	—
68	C₄×D₄×C₅₉	—	2	2	no	—
69	C₄×D₄×C₅₉	—	4	4	no	—
70	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
71	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
72	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
73	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
74	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
75	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
76	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
79	C₂×C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
80	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
81	C₂×C₂²⋊C₄×C₅₉	—	2	2	no	—
82	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
83	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
84	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
85	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
86	C₂×C₄²×C₅₉	—	2	2	no	—
87	C₂×C₄²×C₅₉	—	2	2	no	—
88	C₂×C₄²×C₅₉	—	2	1	no	—
89	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
90	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
91	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
92	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
93	#106×C₅₉	—	2	4	no	—
94	C₄⋊C₈×C₅₉	—	2	4	no	—
95	C₄⋊C₈×C₅₉	—	2	2	no	—
96	C₄⋊C₈×C₅₉	D₄	1	4	yes	complemented
97	#103×C₅₉	—	2	4	no	—
98	#103×C₅₉	—	2	4	no	—
99	#103×C₅₉	—	4	2	no	—
100	#102×C₅₉	—	2	4	no	—
101	#95×C₅₉	—	2	1	no	—
102	#95×C₅₉	—	2	1	no	—
103	#95×C₅₉	—	2	1	no	—
104	#95×C₅₉	—	2	2	no	—
105	#95×C₅₉	—	2	2	no	—
106	C₂⁴×C₅₉	—	2	1	no	—
107	C₂⁴×C₅₉	C₂×D₄	1	1	yes	characteristic
108	C₂×Q₈×C₅₉	—	2	2	no	—
109	C₂×Q₈×C₅₉	—	2	2	no	—
110	C₂×Q₈×C₅₉	—	2	2	no	—
111	C₂×Q₈×C₅₉	—	2	2	no	—
112	C₂×Q₈×C₅₉	—	2	2	no	—
113	C₂×Q₈×C₅₉	—	2	2	no	—
114	C₂×D₄×C₅₉	—	2	4	no	—
115	C₂×D₄×C₅₉	—	2	1	no	—
116	C₂×D₄×C₅₉	—	2	1	no	—
117	C₂×D₄×C₅₉	—	2	1	no	—
118	C₂×D₄×C₅₉	—	2	1	no	—
119	C₂×D₄×C₅₉	—	2	1	no	—
120	C₂×D₄×C₅₉	—	4	2	no	—
121	C₂×D₄×C₅₉	—	4	2	no	—
122	C₂×D₄×C₅₉	C₂×D₄	1	2	yes	—
123	C₂²×C₄×C₅₉	—	2	2	no	—
124	C₂²×C₄×C₅₉	—	2	2	no	—
125	C₂²×C₄×C₅₉	—	2	1	no	—
126	C₂²×C₄×C₅₉	—	2	1	no	—
127	C₂²×C₄×C₅₉	—	2	1	no	—
128	C₂²×C₄×C₅₉	—	2	1	no	—
129	C₂²×C₄×C₅₉	—	2	2	no	—
130	C₂²×C₄×C₅₉	—	2	1	no	—
131	C₂²×C₄×C₅₉	—	2	2	no	—
132	C₂²×C₄×C₅₉	—	2	1	no	—
133	C₂²×C₄×C₅₉	—	2	1	no	—
134	C₂²×C₄×C₅₉	—	2	2	no	—
135	C₂²×C₄×C₅₉	—	2	2	no	—
136	C₂²×C₄×C₅₉	—	4	1	no	—
137	C₂²×C₄×C₅₉	—	4	2	no	—
138	C₂²×C₄×C₅₉	—	4	2	no	—
139	C₂²×C₄×C₅₉	—	4	2	no	—
140	C₂²×C₄×C₅₉	SD₁₆	1	4	yes	complemented
141	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₅₉	#41	1	2	yes	—
148	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
149	SD₁₆×C₅₉	—	4	8	no	—
150	C₂×C₈×C₅₉	—	2	2	no	—
151	C₂×C₈×C₅₉	—	2	2	no	—
152	C₂×C₈×C₅₉	—	2	2	no	—
153	C₂×C₈×C₅₉	—	2	2	no	—
154	C₂×C₈×C₅₉	—	2	2	no	—
155	C₂×C₈×C₅₉	—	4	8	no	—
156	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
157	C₄⋊C₄×C₅₉	—	2	1	no	—
158	C₄⋊C₄×C₅₉	—	2	2	no	—
159	C₄⋊C₄×C₅₉	—	2	1	no	—
160	C₄⋊C₄×C₅₉	—	2	1	no	—
161	C₄⋊C₄×C₅₉	—	2	2	no	—
162	C₄⋊C₄×C₅₉	—	2	1	no	—
163	C₄⋊C₄×C₅₉	—	2	1	no	—
164	C₄⋊C₄×C₅₉	—	4	2	no	—
165	C₄⋊C₄×C₅₉	—	4	4	no	—
166	C₄⋊C₄×C₅₉	—	4	2	no	—
167	C₄⋊C₄×C₅₉	—	4	2	no	—
168	C₄⋊C₄×C₅₉	—	4	2	no	—
169	C₄⋊C₄×C₅₉	—	4	2	no	—
170	C₄⋊C₄×C₅₉	C₂×D₄	1	2	yes	—
171	C₂²⋊C₄×C₅₉	—	2	4	no	—
172	C₂²⋊C₄×C₅₉	—	4	2	no	—
173	C₂²⋊C₄×C₅₉	—	4	4	no	—
174	C₂²⋊C₄×C₅₉	—	4	2	no	—
175	C₂²⋊C₄×C₅₉	—	4	2	no	—
176	C₂²⋊C₄×C₅₉	—	4	2	no	—
177	C₄²×C₅₉	—	2	2	no	—
178	C₄²×C₅₉	—	2	2	no	—
179	C₄²×C₅₉	—	2	1	no	—
180	C₄²×C₅₉	—	2	1	no	—
181	C₄²×C₅₉	—	4	2	no	—
182	C₄²×C₅₉	C₂×D₄	1	2	yes	—
183	C₄²×C₅₉	#41	1	2	yes	—
184	C₂³×C₅₉	—	2	1	no	—
185	C₂³×C₅₉	—	2	1	no	—
186	C₂³×C₅₉	—	2	1	no	—
187	C₂³×C₅₉	—	2	1	no	—
188	C₂³×C₅₉	—	2	2	no	—
189	C₂³×C₅₉	—	4	1	no	—
190	C₂³×C₅₉	—	4	1	no	—
191	C₂³×C₅₉	—	4	1	no	—
192	C₂³×C₅₉	—	4	1	no	—
193	C₂³×C₅₉	—	4	1	no	—
194	C₂³×C₅₉	—	4	1	no	—
195	C₂³×C₅₉	#121	1	2	yes	—
196	C₂³×C₅₉	C₂×SD₁₆	1	2	yes	—
197	C₂³×C₅₉	#137	1	2	yes	—
198	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
199	Q₈×C₅₉	—	4	2	no	—
200	Q₈×C₅₉	—	4	2	no	—
201	Q₈×C₅₉	—	4	2	no	—
202	Q₈×C₅₉	—	4	2	no	—
203	D₄×C₅₉	—	2	2	no	—
204	D₄×C₅₉	—	2	2	no	—
205	D₄×C₅₉	—	4	8	no	—
206	D₄×C₅₉	—	4	1	no	—
207	D₄×C₅₉	—	4	1	no	—
208	C₂×C₄×C₅₉	—	2	1	no	—
209	C₂×C₄×C₅₉	—	2	1	no	—
210	C₂×C₄×C₅₉	—	2	1	no	—
211	C₂×C₄×C₅₉	—	2	1	no	—
212	C₂×C₄×C₅₉	—	2	1	no	—
213	C₂×C₄×C₅₉	—	2	1	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	—	2	2	no	—
218	C₂×C₄×C₅₉	—	2	8	no	—
219	C₂×C₄×C₅₉	—	2	1	no	—
220	C₂×C₄×C₅₉	—	2	2	no	—
221	C₂×C₄×C₅₉	—	2	1	no	—
222	C₂×C₄×C₅₉	—	2	1	no	—
223	C₂×C₄×C₅₉	—	4	4	no	—
224	C₂×C₄×C₅₉	—	4	1	no	—
225	C₂×C₄×C₅₉	—	4	1	no	—
226	C₂×C₄×C₅₉	—	4	2	no	—
227	C₂×C₄×C₅₉	—	4	1	no	—
228	C₂×C₄×C₅₉	—	4	2	no	—
229	C₂×C₄×C₅₉	—	4	2	no	—
230	C₂×C₄×C₅₉	—	4	1	no	—
231	C₂×C₄×C₅₉	—	4	2	no	—
232	C₂×C₄×C₅₉	—	4	1	no	—
233	C₂×C₄×C₅₉	—	4	1	no	—
234	C₂×C₄×C₅₉	—	4	2	no	—
235	C₂×C₄×C₅₉	—	4	1	no	—
236	C₂×C₄×C₅₉	—	4	1	no	—
237	C₂×C₄×C₅₉	—	4	2	no	—
238	C₂×C₄×C₅₉	—	8	4	no	—
239	C₂×C₄×C₅₉	#121	1	2	yes	—
240	C₂×C₄×C₅₉	#121	1	2	yes	—
241	C₂×C₄×C₅₉	#121	1	2	yes	—
242	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
243	C₂×C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
244	C₂×C₄×C₅₉	#135	1	2	yes	—
245	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
246	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
247	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
248	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
249	C₂×C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
250	C₂×C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
251	C₈×C₅₉	—	2	4	no	—
252	C₈×C₅₉	—	4	2	no	—
253	#24751	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
254	C₂²×C₅₉	—	2	2	no	—
255	C₂²×C₅₉	—	2	2	no	—
256	C₂²×C₅₉	—	2	2	no	—
257	C₂²×C₅₉	—	2	2	no	—
258	C₂²×C₅₉	—	2	2	no	—
259	C₂²×C₅₉	—	2	2	no	—
260	C₂²×C₅₉	—	4	2	no	—
261	C₂²×C₅₉	—	4	2	no	—
262	C₂²×C₅₉	—	4	2	no	—
263	C₂²×C₅₉	—	8	1	no	—
264	C₂²×C₅₉	—	8	1	no	—
265	C₂²×C₅₉	—	8	1	no	—
266	C₂²×C₅₉	—	8	1	no	—
267	C₂²×C₅₉	#461	1	4	yes	complemented
268	C₂²×C₅₉	C₂×#121	1	1	yes	characteristic
269	C₂²×C₅₉	C₂×#121	1	1	yes	characteristic
270	C₂²×C₅₉	#537	1	1	yes	characteristic
271	C₂²×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
272	C₂²×C₅₉	C₂×#135	1	1	yes	characteristic
273	C₂²×C₅₉	C₂×#137	1	1	yes	characteristic
274	C₂²×C₅₉	#575	1	1	yes	characteristic
275	C₄×C₅₉	—	2	2	no	—
276	C₄×C₅₉	—	2	2	no	—
277	C₄×C₅₉	—	2	2	no	—
278	C₄×C₅₉	—	4	2	no	—
279	C₄×C₅₉	—	4	1	no	—
280	C₄×C₅₉	—	8	2	no	—
281	C₄×C₅₉	—	8	1	no	—
282	C₄×C₅₉	—	8	2	no	—
283	C₄×C₅₉	—	8	1	no	—
284	C₄×C₅₉	#465	1	4	yes	complemented
285	C₄×C₅₉	D₄²	1	1	yes	characteristic
286	C₄×C₅₉	#547	1	1	yes	characteristic
287	C₄×C₅₉	#548	1	1	yes	characteristic
288	C₄×C₅₉	#553	1	1	yes	characteristic
289	C₂×#465	C₂×C₅₉	1	2	yes	—
290	C₂×#461	C₂×C₅₉	1	1	yes	characteristic, complemented
291	C₂²×C₄⋊C₈	C₂×C₅₉	1	1	yes	characteristic, complemented
292	#2857	C₂×C₅₉	1	1	yes	characteristic, complemented
293	C₄×#121	C₂×C₅₉	1	1	yes	characteristic
294	#2066	C₂×C₅₉	1	2	yes	complemented
295	#2055	C₂×C₅₉	1	2	yes	complemented
296	#1981	C₂×C₅₉	1	1	yes	characteristic, complemented
297	#1948	C₂×C₅₉	1	1	yes	characteristic, complemented
298	#1892	C₂×C₅₉	1	2	yes	complemented
299	#1802	C₂×C₅₉	1	1	yes	characteristic, complemented
300	C₂×C₅₉	—	2	4	no	—
301	C₂×C₅₉	—	8	2	no	—
302	C₂×C₅₉	#2416	1	1	yes	characteristic
303	C₂×C₅₉	C₂×#461	1	1	yes	characteristic
304	C₂×C₅₉	#3035	1	1	yes	characteristic
305	C₂×C₅₉	C₂×#465	1	1	yes	characteristic
306	C₂×C₅₉	#3056	1	1	yes	characteristic
307	C₂×C₅₉	#3246	1	1	yes	characteristic
308	C₂×C₅₉	#3248	1	1	yes	characteristic
309	C₂²×SD₁₆	C₂²×C₅₉	1	2	yes	—
310	C₂³×C₈	C₂²×C₅₉	1	2	yes	—
311	C₂×C₄⋊Q₈	C₂²×C₅₉	1	1	yes	characteristic
312	C₂×C₂²⋊Q₈	C₂²×C₅₉	1	2	yes	—
313	C₂×#121	C₂²×C₅₉	1	1	yes	characteristic
314	C₂×C₄×D₄	—	2	1	no	—
315	C₂×C₄×D₄	C₂²×C₅₉	1	1	yes	characteristic
316	C₂²×C₄²	C₂²×C₅₉	1	1	yes	characteristic
317	#465	—	2	8	no	—
318	#461	—	2	4	no	—
319	C₂×#106	C₂²×C₅₉	1	2	yes	—
320	C₂×C₄⋊C₈	—	2	4	no	—
321	C₂×C₄⋊C₈	C₂²×C₅₉	1	2	yes	—
322	C₂×C₄⋊C₈	C₂²×C₅₉	1	1	yes	characteristic
323	C₂×C₄⋊C₈	C₂²×C₅₉	1	1	yes	characteristic
324	C₂×#103	C₂²×C₅₉	1	2	yes	—
325	C₂×#103	C₂²×C₅₉	1	2	yes	—
326	C₂×#103	C₂²×C₅₉	1	2	yes	—
327	C₂×#102	C₂²×C₅₉	1	2	yes	—
328	#391	C₂²×C₅₉	1	2	yes	—
329	#390	C₂²×C₅₉	1	1	yes	characteristic
330	#389	—	2	1	no	—
331	#389	—	2	1	no	—
332	#389	C₂²×C₅₉	1	1	yes	characteristic
333	#388	—	2	1	no	—
334	#386	—	2	1	no	—
335	#384	C₂²×C₅₉	1	1	yes	characteristic
336	#380	—	2	1	no	—
337	#380	—	2	1	no	—
338	#380	C₂²×C₅₉	1	1	yes	characteristic
339	C₄×C₄⋊C₄	C₂²×C₅₉	1	1	yes	characteristic
340	C₄×C₂²⋊C₄	—	2	1	no	—
341	#340	C₂²×C₅₉	1	2	yes	—
342	#340	C₂²×C₅₉	1	2	yes	—
343	C₅₉	#24751	1	1	yes	characteristic, direct product factor, Sylow
344	C₂²×Q₈	C₂³×C₅₉	1	2	yes	—
345	C₂²×D₄	—	2	1	no	—
346	C₂²×D₄	C₂³×C₅₉	1	1	yes	characteristic
347	C₂³×C₄	—	2	1	no	—
348	C₂³×C₄	D₄×C₅₉	1	1	yes	characteristic
349	C₂³×C₄	D₄×C₅₉	1	1	yes	characteristic
350	C₂³×C₄	C₂³×C₅₉	1	1	yes	characteristic
351	C₂×SD₁₆	—	2	4	no	—
352	C₂×SD₁₆	—	2	4	no	—
353	C₂×SD₁₆	—	4	2	no	—
354	C₂²×C₈	—	2	4	no	—
355	C₂²×C₈	—	2	4	no	—
356	C₂²×C₈	—	4	2	no	—
357	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
358	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
359	C₄⋊Q₈	—	2	2	no	—
360	C₄⋊Q₈	—	2	2	no	—
361	C₂²⋊Q₈	—	2	4	no	—
362	C₂²⋊Q₈	—	2	4	no	—
363	#121	—	2	2	no	—
364	#121	D₄×C₅₉	1	4	yes	—
365	C₄×D₄	—	2	2	no	—
366	C₄×D₄	—	2	2	no	—
367	C₄×D₄	—	4	4	no	—
368	C₂×C₄⋊C₄	—	2	1	no	—
369	C₂×C₄⋊C₄	—	2	2	no	—
370	C₂×C₄⋊C₄	—	2	1	no	—
371	C₂×C₄⋊C₄	—	2	1	no	—
372	C₂×C₄⋊C₄	—	2	1	no	—
373	C₂×C₄⋊C₄	—	2	1	no	—
374	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
375	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
376	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
377	C₂×C₄⋊C₄	C₂³×C₅₉	1	1	yes	characteristic
378	C₂×C₂²⋊C₄	—	2	1	no	—
379	C₂×C₂²⋊C₄	—	2	1	no	—
380	C₂×C₂²⋊C₄	—	2	2	no	—
381	C₂×C₂²⋊C₄	—	2	1	no	—
382	C₂×C₂²⋊C₄	—	2	1	no	—
383	C₂×C₂²⋊C₄	—	2	1	no	—
384	C₂×C₄²	—	2	2	no	—
385	C₂×C₄²	—	2	2	no	—
386	C₂×C₄²	—	2	1	no	—
387	C₂×C₄²	D₄×C₅₉	1	1	yes	characteristic
388	C₂×C₄²	D₄×C₅₉	1	1	yes	characteristic
389	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
390	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
391	#106	—	2	4	no	—
392	C₄⋊C₈	—	2	4	no	—
393	C₄⋊C₈	—	2	2	no	—
394	C₄⋊C₈	D₄×C₅₉	1	4	yes	complemented
395	#103	—	2	4	no	—
396	#103	—	2	4	no	—
397	#103	—	4	2	no	—
398	#102	—	2	4	no	—
399	#95	—	2	1	no	—
400	#95	—	2	1	no	—
401	#95	—	2	1	no	—
402	#95	—	2	2	no	—
403	#95	—	2	2	no	—
404	C₂⁴	—	2	1	no	—
405	C₂⁴	C₂×D₄×C₅₉	1	1	yes	characteristic
406	C₂×Q₈	—	2	2	no	—
407	C₂×Q₈	—	2	2	no	—
408	C₂×Q₈	—	2	2	no	—
409	C₂×Q₈	—	2	2	no	—
410	C₂×Q₈	—	2	2	no	—
411	C₂×Q₈	—	2	2	no	—
412	C₂×D₄	—	2	4	no	—
413	C₂×D₄	—	2	1	no	—
414	C₂×D₄	—	2	1	no	—
415	C₂×D₄	—	2	1	no	—
416	C₂×D₄	—	2	1	no	—
417	C₂×D₄	—	2	1	no	—
418	C₂×D₄	—	4	2	no	—
419	C₂×D₄	—	4	2	no	—
420	C₂×D₄	C₂×D₄×C₅₉	1	2	yes	—
421	C₂²×C₄	—	2	2	no	—
422	C₂²×C₄	—	2	2	no	—
423	C₂²×C₄	—	2	1	no	—
424	C₂²×C₄	—	2	1	no	—
425	C₂²×C₄	—	2	1	no	—
426	C₂²×C₄	—	2	1	no	—
427	C₂²×C₄	—	2	2	no	—
428	C₂²×C₄	—	2	1	no	—
429	C₂²×C₄	—	2	2	no	—
430	C₂²×C₄	—	2	1	no	—
431	C₂²×C₄	—	2	1	no	—
432	C₂²×C₄	—	2	2	no	—
433	C₂²×C₄	—	2	2	no	—
434	C₂²×C₄	—	4	1	no	—
435	C₂²×C₄	—	4	2	no	—
436	C₂²×C₄	—	4	2	no	—
437	C₂²×C₄	—	4	2	no	—
438	C₂²×C₄	SD₁₆×C₅₉	1	4	yes	complemented
439	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
440	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
441	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
442	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
443	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
444	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
445	C₂²×C₄	#41×C₅₉	1	2	yes	—
446	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
447	SD₁₆	—	4	8	no	—
448	C₂×C₈	—	2	2	no	—
449	C₂×C₈	—	2	2	no	—
450	C₂×C₈	—	2	2	no	—
451	C₂×C₈	—	2	2	no	—
452	C₂×C₈	—	2	2	no	—
453	C₂×C₈	—	4	8	no	—
454	C₂×C₈	C₂×D₄×C₅₉	1	4	yes	—
455	C₄⋊C₄	—	2	1	no	—
456	C₄⋊C₄	—	2	2	no	—
457	C₄⋊C₄	—	2	1	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	2	2	no	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	—	4	2	no	—
463	C₄⋊C₄	—	4	4	no	—
464	C₄⋊C₄	—	4	2	no	—
465	C₄⋊C₄	—	4	2	no	—
466	C₄⋊C₄	—	4	2	no	—
467	C₄⋊C₄	—	4	2	no	—
468	C₄⋊C₄	C₂×D₄×C₅₉	1	2	yes	—
469	C₂²⋊C₄	—	2	4	no	—
470	C₂²⋊C₄	—	4	2	no	—
471	C₂²⋊C₄	—	4	4	no	—
472	C₂²⋊C₄	—	4	2	no	—
473	C₂²⋊C₄	—	4	2	no	—
474	C₂²⋊C₄	—	4	2	no	—
475	C₄²	—	2	2	no	—
476	C₄²	—	2	2	no	—
477	C₄²	—	2	1	no	—
478	C₄²	—	2	1	no	—
479	C₄²	—	4	2	no	—
480	C₄²	C₂×D₄×C₅₉	1	2	yes	—
481	C₄²	#41×C₅₉	1	2	yes	—
482	C₂³	—	2	1	no	—
483	C₂³	—	2	1	no	—
484	C₂³	—	2	1	no	—
485	C₂³	—	2	1	no	—
486	C₂³	—	2	2	no	—
487	C₂³	—	4	1	no	—
488	C₂³	—	4	1	no	—
489	C₂³	—	4	1	no	—
490	C₂³	—	4	1	no	—
491	C₂³	—	4	1	no	—
492	C₂³	—	4	1	no	—
493	C₂³	#121×C₅₉	1	2	yes	—
494	C₂³	C₂×SD₁₆×C₅₉	1	2	yes	—
495	C₂³	#137×C₅₉	1	2	yes	—
496	C₂³	C₂²×D₄×C₅₉	1	1	yes	characteristic
497	Q₈	—	4	2	no	—
498	Q₈	—	4	2	no	—
499	Q₈	—	4	2	no	—
500	Q₈	—	4	2	no	—
501	D₄	—	2	2	no	—
502	D₄	—	2	2	no	—
503	D₄	—	4	8	no	—
504	D₄	—	4	1	no	—
505	D₄	—	4	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	8	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	1	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	4	4	no	—
522	C₂×C₄	—	4	1	no	—
523	C₂×C₄	—	4	1	no	—
524	C₂×C₄	—	4	2	no	—
525	C₂×C₄	—	4	1	no	—
526	C₂×C₄	—	4	2	no	—
527	C₂×C₄	—	4	2	no	—
528	C₂×C₄	—	4	1	no	—
529	C₂×C₄	—	4	2	no	—
530	C₂×C₄	—	4	1	no	—
531	C₂×C₄	—	4	1	no	—
532	C₂×C₄	—	4	2	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	2	no	—
536	C₂×C₄	—	8	4	no	—
537	C₂×C₄	#121×C₅₉	1	2	yes	—
538	C₂×C₄	#121×C₅₉	1	2	yes	—
539	C₂×C₄	#121×C₅₉	1	2	yes	—
540	C₂×C₄	C₂×SD₁₆×C₅₉	1	2	yes	—
541	C₂×C₄	C₂×SD₁₆×C₅₉	1	2	yes	—
542	C₂×C₄	#135×C₅₉	1	2	yes	—
543	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
544	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
545	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
546	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
547	C₂×C₄	H(5,𝔽₂)×C₅₉	1	1	yes	characteristic
548	C₂×C₄	2₋¹⁺⁴×C₅₉	1	1	yes	characteristic
549	C₈	—	2	4	no	—
550	C₈	—	4	2	no	—
551	C₂²	—	2	2	no	—
552	C₂²	—	2	2	no	—
553	C₂²	—	2	2	no	—
554	C₂²	—	2	2	no	—
555	C₂²	—	2	2	no	—
556	C₂²	—	2	2	no	—
557	C₂²	—	4	2	no	—
558	C₂²	—	4	2	no	—
559	C₂²	—	4	2	no	—
560	C₂²	—	8	1	no	—
561	C₂²	—	8	1	no	—
562	C₂²	—	8	1	no	—
563	C₂²	—	8	1	no	—
564	C₂²	C₅₉×#461	1	4	yes	complemented
565	C₂²	C₂×#121×C₅₉	1	1	yes	characteristic
566	C₂²	C₂×#121×C₅₉	1	1	yes	characteristic
567	C₂²	C₅₉×#537	1	1	yes	characteristic
568	C₂²	C₂²×SD₁₆×C₅₉	1	1	yes	characteristic
569	C₂²	C₂×#135×C₅₉	1	1	yes	characteristic
570	C₂²	C₂×#137×C₅₉	1	1	yes	characteristic
571	C₂²	C₅₉×#575	1	1	yes	characteristic
572	C₄	—	2	2	no	—
573	C₄	—	2	2	no	—
574	C₄	—	2	2	no	—
575	C₄	—	4	2	no	—
576	C₄	—	4	1	no	—
577	C₄	—	8	2	no	—
578	C₄	—	8	1	no	—
579	C₄	—	8	2	no	—
580	C₄	—	8	1	no	—
581	C₄	C₅₉×#465	1	4	yes	complemented
582	C₄	D₄²×C₅₉	1	1	yes	characteristic
583	C₄	C₅₉×#547	1	1	yes	characteristic
584	C₄	C₅₉×#548	1	1	yes	characteristic
585	C₄	C₅₉×#553	1	1	yes	characteristic
586	C₂	—	2	4	no	—
587	C₂	—	8	2	no	—
588	C₂	C₅₉×#2416	1	1	yes	characteristic
589	C₂	C₂×C₅₉×#461	1	1	yes	characteristic
590	C₂	C₅₉×#3035	1	1	yes	characteristic
591	C₂	C₂×C₅₉×#465	1	1	yes	characteristic
592	C₂	C₅₉×#3056	1	1	yes	characteristic
593	C₂	C₅₉×#3246	1	1	yes	characteristic
594	C₂	C₅₉×#3248	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₉
prime radical	C₂×#121×C₅₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₉ ◁ C₂²×C₄²×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₉×#461	C₂	semidirect
3. C₂²×C₄⋊C₈×C₅₉	C₂	semidirect
4. C₅₉×#2857	C₂	semidirect
6. C₅₉×#2066	C₂	semidirect
7. C₅₉×#2055	C₂	semidirect
8. C₅₉×#1981	C₂	semidirect
9. C₅₉×#1948	C₂	semidirect
10. C₅₉×#1892	C₂	semidirect
11. C₅₉×#1802	C₂	semidirect
96. C₄⋊C₈×C₅₉	D₄	semidirect
140. C₂²×C₄×C₅₉	SD₁₆	semidirect
253. #24751	C₅₉	direct (N × H)
267. C₂²×C₅₉	#461	semidirect
284. C₄×C₅₉	#465	semidirect
290. C₂×#461	C₂×C₅₉	semidirect
291. C₂²×C₄⋊C₈	C₂×C₅₉	semidirect
292. #2857	C₂×C₅₉	semidirect
294. #2066	C₂×C₅₉	semidirect
295. #2055	C₂×C₅₉	semidirect
296. #1981	C₂×C₅₉	semidirect
297. #1948	C₂×C₅₉	semidirect
298. #1892	C₂×C₅₉	semidirect
299. #1802	C₂×C₅₉	semidirect
343. C₅₉	#24751	direct (N × H)
394. C₄⋊C₈	D₄×C₅₉	semidirect
438. C₂²×C₄	SD₁₆×C₅₉	semidirect
564. C₂²	C₅₉×#461	semidirect
581. C₄	C₅₉×#465	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₉×#465	2
C₂×C₅₉×#461	2
C₂²×C₄⋊C₈×C₅₉	2
C₅₉×#2857	2
C₄×#121×C₅₉	2
C₅₉×#2066	2
C₅₉×#2055	2
C₅₉×#1981	2
C₅₉×#1948	2
C₅₉×#1892	2
C₅₉×#1802	2
#24751	59

Subgroups — C59×#24751 #28436860

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#24751 #28436860