Subgroups — C₅₉×#62249 #28474358

Order 15104.

1902 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#3506	C₂	1	2	yes	maximal, complemented
2	C₅₉×#3342	C₂	1	2	yes	maximal, complemented
3	C₅₉×#3341	C₂	1	2	yes	maximal, complemented
4	C₅₉×#3304	C₂	1	4	yes	maximal, complemented
5	C₅₉×#3303	C₂	1	4	yes	maximal, complemented
6	C₅₉×#3287	C₂	1	2	yes	maximal, complemented
7	C₅₉×#3283	C₂	1	2	yes	maximal, complemented
8	C₅₉×#3246	C₂	1	2	yes	maximal, complemented
9	C₅₉×#3191	C₂	1	4	yes	maximal, complemented
10	C₅₉×#3170	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₉×#3169	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₉×#3050	C₂	1	4	yes	maximal, complemented
13	C₂×C₈×D₄×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#135×C₅₉	C₂²	1	4	yes	complemented
15	C₂³×C₈×C₅₉	C₂²	1	2	yes	—
16	C₅₉×#559	—	2	4	no	—
17	C₅₉×#556	C₂²	1	4	yes	—
18	C₅₉×#555	C₂²	1	4	yes	—
19	C₅₉×#553	—	2	4	no	—
20	C₅₉×#550	C₂²	1	4	yes	—
21	C₅₉×#547	C₂²	1	4	yes	—
22	C₅₉×#538	—	2	4	no	—
23	C₅₉×#535	—	2	2	no	—
24	C₅₉×#533	C₂²	1	2	yes	—
25	C₅₉×#532	C₂²	1	2	yes	—
26	C₅₉×#525	C₂²	1	8	yes	—
27	C₅₉×#520	—	2	2	no	—
28	C₂×C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
29	C₅₉×#500	C₂²	1	1	yes	characteristic, complemented
30	C₅₉×#500	C₂²	1	1	yes	characteristic, complemented
31	C₅₉×#494	C₂²	1	1	yes	characteristic, complemented
32	C₅₉×#493	C₂²	1	1	yes	characteristic, complemented
33	C₅₉×#487	C₂²	1	2	yes	complemented
34	C₅₉×#486	C₂²	1	2	yes	complemented
35	C₅₉×#485	C₂²	1	8	yes	complemented
36	C₅₉×#482	C₂²	1	8	yes	complemented
37	C₅₉×#476	C₂²	1	4	yes	complemented
38	C₅₉×#475	C₂²	1	4	yes	complemented
39	C₅₉×#467	C₂²	1	2	yes	—
40	C₅₉×#467	C₂²	1	2	yes	—
41	C₅₉×#467	C₂²	1	4	yes	complemented
42	C₅₉×#466	C₂²	1	2	yes	—
43	C₅₉×#466	C₂²	1	2	yes	—
44	C₅₉×#466	C₂²	1	4	yes	complemented
45	C₅₉×#465	C₂²	1	8	yes	—
46	C₅₉×#465	C₂²	1	8	yes	complemented
47	C₅₉×#461	C₂²	1	4	yes	complemented
48	C₅₉×#460	C₂²	1	4	yes	complemented
49	C₅₉×#452	C₂²	1	8	yes	complemented
50	C₅₉×#449	C₂²	1	8	yes	complemented
51	C₄×Q₁₆×C₅₉	C₂²	1	2	yes	complemented
52	C₄×Q₁₆×C₅₉	C₂²	1	2	yes	complemented
53	C₄×D₈×C₅₉	C₂²	1	2	yes	complemented
54	C₄×D₈×C₅₉	C₂²	1	2	yes	complemented
55	C₈×D₄×C₅₉	C₂²	1	2	yes	—
56	C₈×D₄×C₅₉	C₂²	1	2	yes	—
57	C₈×D₄×C₅₉	C₂²	1	2	yes	—
58	C₈×D₄×C₅₉	C₂²	1	2	yes	—
59	C₅₉×#427	C₂²	1	4	yes	complemented
60	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
61	C₅₉×#416	C₂²	1	8	yes	complemented
62	C₂×C₂²⋊C₈×C₅₉	C₂²	1	2	yes	—
63	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
64	C₂×#41×C₅₉	—	2	2	no	—
65	C₂×#41×C₅₉	C₂³	1	8	yes	—
66	C₂²×D₄×C₅₉	D₄	1	1	yes	characteristic
67	C₂³×C₄×C₅₉	C₂³	1	2	yes	—
68	#135×C₅₉	—	4	8	no	—
69	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
70	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
71	C₂×SD₁₆×C₅₉	C₂³	1	8	yes	—
72	C₂×D₈×C₅₉	C₂³	1	2	yes	—
73	C₂×D₈×C₅₉	C₂³	1	2	yes	—
74	C₂²×C₈×C₅₉	—	4	4	no	—
75	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₅₉	C₂³	1	4	yes	—
77	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
79	C₂²×C₈×C₅₉	C₂³	1	4	yes	—
80	C₄⋊Q₈×C₅₉	C₂³	1	2	yes	—
81	C₄⋊Q₈×C₅₉	C₂³	1	2	yes	—
82	C₄⋊Q₈×C₅₉	C₂³	1	2	yes	—
83	#127×C₅₉	—	2	4	no	—
84	#127×C₅₉	C₂³	1	2	yes	—
85	#126×C₅₉	—	2	8	no	—
86	#125×C₅₉	—	2	2	no	—
87	#124×C₅₉	—	2	4	no	—
88	#124×C₅₉	—	2	8	no	—
89	#124×C₅₉	—	2	2	no	—
90	#123×C₅₉	—	2	8	no	—
91	#123×C₅₉	—	2	8	no	—
92	C₂²⋊Q₈×C₅₉	—	2	2	no	—
93	C₂²⋊Q₈×C₅₉	C₂³	1	8	yes	—
94	C₂²⋊Q₈×C₅₉	C₂³	1	4	yes	—
95	#121×C₅₉	—	2	4	no	—
96	#121×C₅₉	—	2	4	no	—
97	#121×C₅₉	—	2	2	no	—
98	#121×C₅₉	—	2	8	no	—
99	#121×C₅₉	C₂³	1	8	yes	—
100	#121×C₅₉	C₂³	1	4	yes	—
101	#120×C₅₉	—	2	8	no	—
102	C₄×Q₈×C₅₉	—	2	4	no	—
103	C₄×Q₈×C₅₉	—	2	2	no	—
104	C₄×Q₈×C₅₉	C₂³	1	4	yes	—
105	C₄×D₄×C₅₉	—	2	8	no	—
106	C₄×D₄×C₅₉	—	2	8	no	—
107	C₄×D₄×C₅₉	—	2	4	no	—
108	C₄×D₄×C₅₉	—	2	2	no	—
109	C₄×D₄×C₅₉	D₄	1	2	yes	—
110	C₄×D₄×C₅₉	D₄	1	2	yes	—
111	C₄×D₄×C₅₉	C₂³	1	4	yes	—
112	C₄×D₄×C₅₉	C₂³	1	2	yes	—
113	C₄×D₄×C₅₉	C₂³	1	2	yes	—
114	#117×C₅₉	—	2	2	no	—
115	#117×C₅₉	—	2	4	no	—
116	#117×C₅₉	C₂³	1	8	yes	—
117	C₂×C₄⋊C₄×C₅₉	—	2	4	no	—
118	C₂×C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
119	C₂×C₂²⋊C₄×C₅₉	D₄	1	2	yes	—
120	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
121	#107×C₅₉	C₂³	1	2	yes	—
122	#107×C₅₉	C₂³	1	2	yes	—
123	#107×C₅₉	C₂³	1	2	yes	—
124	#107×C₅₉	C₂³	1	2	yes	—
125	#106×C₅₉	C₂³	1	8	yes	—
126	C₄⋊C₈×C₅₉	C₂³	1	2	yes	—
127	C₄⋊C₈×C₅₉	C₂³	1	2	yes	—
128	#103×C₅₉	C₂³	1	4	yes	—
129	#103×C₅₉	C₂³	1	8	yes	—
130	#103×C₅₉	C₂³	1	4	yes	—
131	#102×C₅₉	C₂³	1	4	yes	—
132	#102×C₅₉	C₂³	1	8	yes	—
133	#102×C₅₉	C₂³	1	4	yes	—
134	C₂²⋊C₈×C₅₉	C₂³	1	4	yes	—
135	C₂²⋊C₈×C₅₉	C₂³	1	4	yes	—
136	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
137	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
138	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
139	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
140	C₂⁴×C₅₉	C₂×D₄	1	2	yes	—
141	#41×C₅₉	—	4	8	no	—
142	#41×C₅₉	—	4	8	no	—
143	#41×C₅₉	—	4	8	no	—
144	C₂×Q₈×C₅₉	—	2	2	no	—
145	C₂×Q₈×C₅₉	—	2	2	no	—
146	C₂×Q₈×C₅₉	C₂⁴	1	4	yes	—
147	C₂×D₄×C₅₉	—	2	4	no	—
148	C₂×D₄×C₅₉	—	2	8	no	—
149	C₂×D₄×C₅₉	—	2	8	no	—
150	C₂×D₄×C₅₉	—	2	4	no	—
151	C₂×D₄×C₅₉	—	2	2	no	—
152	C₂×D₄×C₅₉	—	2	2	no	—
153	C₂×D₄×C₅₉	C₂×D₄	1	2	yes	—
154	C₂×D₄×C₅₉	C₂×D₄	1	2	yes	—
155	C₂×D₄×C₅₉	C₂⁴	1	4	yes	—
156	C₂²×C₄×C₅₉	—	2	4	no	—
157	C₂²×C₄×C₅₉	—	2	8	no	—
158	C₂²×C₄×C₅₉	—	4	2	no	—
159	C₂²×C₄×C₅₉	—	4	2	no	—
160	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
162	C₂²×C₄×C₅₉	C₂×D₄	1	4	yes	—
163	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₅₉	C₂⁴	1	4	yes	—
165	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
166	Q₁₆×C₅₉	—	2	4	no	—
167	Q₁₆×C₅₉	—	4	2	no	—
168	SD₁₆×C₅₉	—	4	8	no	—
169	D₈×C₅₉	—	2	4	no	—
170	D₈×C₅₉	—	4	2	no	—
171	C₂×C₈×C₅₉	—	2	2	no	—
172	C₂×C₈×C₅₉	—	2	8	no	—
173	C₂×C₈×C₅₉	—	4	4	no	—
174	C₂×C₈×C₅₉	—	4	2	no	—
175	C₂×C₈×C₅₉	—	4	1	no	—
176	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
177	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
178	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
179	C₂×C₈×C₅₉	C₂⁴	1	4	yes	—
180	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
181	C₄⋊C₄×C₅₉	—	2	2	no	—
182	C₄⋊C₄×C₅₉	—	2	8	no	—
183	C₄⋊C₄×C₅₉	—	2	2	no	—
184	C₄⋊C₄×C₅₉	—	2	4	no	—
185	C₄⋊C₄×C₅₉	—	2	2	no	—
186	C₄⋊C₄×C₅₉	—	2	2	no	—
187	C₄⋊C₄×C₅₉	—	2	4	no	—
188	C₄⋊C₄×C₅₉	C₂×D₄	1	2	yes	—
189	C₄⋊C₄×C₅₉	C₂×D₄	1	2	yes	—
190	C₄⋊C₄×C₅₉	C₂⁴	1	4	yes	—
191	C₄⋊C₄×C₅₉	C₂⁴	1	4	yes	—
192	C₂²⋊C₄×C₅₉	—	2	4	no	—
193	C₂²⋊C₄×C₅₉	—	2	8	no	—
194	C₂²⋊C₄×C₅₉	—	2	8	no	—
195	C₂²⋊C₄×C₅₉	—	2	8	no	—
196	C₂²⋊C₄×C₅₉	—	2	4	no	—
197	C₂²⋊C₄×C₅₉	—	2	4	no	—
198	C₂²⋊C₄×C₅₉	C₂×D₄	1	4	yes	—
199	C₂²⋊C₄×C₅₉	C₂×D₄	1	4	yes	—
200	C₄²×C₅₉	—	2	8	no	—
201	C₄²×C₅₉	—	2	4	no	—
202	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
203	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
204	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
205	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
206	C₂³×C₅₉	—	2	4	no	—
207	C₂³×C₅₉	—	2	4	no	—
208	C₂³×C₅₉	—	4	2	no	—
209	C₂³×C₅₉	—	4	2	no	—
210	C₂³×C₅₉	C₂²×D₄	1	4	yes	—
211	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
212	Q₈×C₅₉	—	2	4	no	—
213	Q₈×C₅₉	—	2	4	no	—
214	Q₈×C₅₉	—	4	4	no	—
215	Q₈×C₅₉	—	8	2	no	—
216	D₄×C₅₉	—	2	4	no	—
217	D₄×C₅₉	—	2	4	no	—
218	D₄×C₅₉	—	2	4	no	—
219	D₄×C₅₉	—	2	4	no	—
220	D₄×C₅₉	—	4	8	no	—
221	D₄×C₅₉	—	4	4	no	—
222	D₄×C₅₉	—	8	2	no	—
223	D₄×C₅₉	—	8	8	no	—
224	D₄×C₅₉	—	8	8	no	—
225	C₂×C₄×C₅₉	—	2	1	no	—
226	C₂×C₄×C₅₉	—	2	4	no	—
227	C₂×C₄×C₅₉	—	2	1	no	—
228	C₂×C₄×C₅₉	—	2	4	no	—
229	C₂×C₄×C₅₉	—	2	4	no	—
230	C₂×C₄×C₅₉	—	2	4	no	—
231	C₂×C₄×C₅₉	—	2	4	no	—
232	C₂×C₄×C₅₉	—	4	4	no	—
233	C₂×C₄×C₅₉	—	4	2	no	—
234	C₂×C₄×C₅₉	—	4	1	no	—
235	C₂×C₄×C₅₉	—	4	8	no	—
236	C₂×C₄×C₅₉	—	8	8	no	—
237	C₂×C₄×C₅₉	#135	1	4	yes	complemented
238	C₂×C₄×C₅₉	C₂²×D₄	1	4	yes	—
239	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
240	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
242	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
243	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
244	C₂×C₄×C₅₉	C₂⁵	1	1	yes	characteristic
245	C₈×C₅₉	—	2	1	no	—
246	C₈×C₅₉	—	2	4	no	—
247	C₈×C₅₉	—	2	1	no	—
248	C₈×C₅₉	—	2	1	no	—
249	C₈×C₅₉	H(5,𝔽₂)	1	2	yes	—
250	#62249	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
251	C₂²×C₅₉	—	2	1	no	—
252	C₂²×C₅₉	—	2	1	no	—
253	C₂²×C₅₉	—	2	4	no	—
254	C₂²×C₅₉	—	2	4	no	—
255	C₂²×C₅₉	—	2	4	no	—
256	C₂²×C₅₉	—	4	8	no	—
257	C₂²×C₅₉	—	4	4	no	—
258	C₂²×C₅₉	—	4	4	no	—
259	C₂²×C₅₉	—	4	4	no	—
260	C₂²×C₅₉	C₂×#135	1	2	yes	—
261	C₂²×C₅₉	C₂³×D₄	1	1	yes	characteristic
262	C₄×C₅₉	—	2	1	no	—
263	C₄×C₅₉	—	2	1	no	—
264	C₄×C₅₉	—	2	1	no	—
265	C₄×C₅₉	—	2	4	no	—
266	C₄×C₅₉	—	4	4	no	—
267	C₄×C₅₉	—	4	4	no	—
268	C₄×C₅₉	—	4	4	no	—
269	C₄×C₅₉	C₂×#135	1	2	yes	—
270	C₄×C₅₉	C₂×#135	1	2	yes	—
271	C₄×C₅₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
272	C₄×C₅₉	#585	1	1	yes	characteristic
273	#3506	C₂×C₅₉	1	2	yes	complemented
274	#3342	C₂×C₅₉	1	2	yes	complemented
275	#3341	C₂×C₅₉	1	2	yes	complemented
276	#3304	C₂×C₅₉	1	4	yes	complemented
277	#3303	C₂×C₅₉	1	4	yes	complemented
278	#3287	C₂×C₅₉	1	2	yes	complemented
279	#3283	C₂×C₅₉	1	2	yes	complemented
280	#3246	C₂×C₅₉	1	2	yes	complemented
281	#3191	C₂×C₅₉	1	4	yes	complemented
282	#3170	C₂×C₅₉	1	1	yes	characteristic, complemented
283	#3169	C₂×C₅₉	1	1	yes	characteristic, complemented
284	#3050	C₂×C₅₉	1	4	yes	complemented
285	C₂×C₈×D₄	C₂×C₅₉	1	1	yes	characteristic, complemented
286	C₂×C₅₉	—	2	2	no	—
287	C₂×C₅₉	—	4	4	no	—
288	C₂×C₅₉	—	8	4	no	—
289	C₂×C₅₉	#3496	1	1	yes	characteristic
290	C₂×C₅₉	C₂²×#135	1	1	yes	characteristic
291	C₂×C₅₉	#3584	1	1	yes	characteristic
292	C₂×#135	C₂²×C₅₉	1	4	yes	complemented
293	C₂³×C₈	C₂²×C₅₉	1	2	yes	—
294	#559	—	2	4	no	—
295	#556	C₂²×C₅₉	1	4	yes	—
296	#555	C₂²×C₅₉	1	4	yes	—
297	#553	—	2	4	no	—
298	#550	C₂²×C₅₉	1	4	yes	—
299	#547	C₂²×C₅₉	1	4	yes	—
300	#538	—	2	4	no	—
301	#535	—	2	2	no	—
302	#533	C₂²×C₅₉	1	2	yes	—
303	#532	C₂²×C₅₉	1	2	yes	—
304	#525	C₂²×C₅₉	1	8	yes	—
305	#520	—	2	2	no	—
306	C₂×C₄×D₄	C₂²×C₅₉	1	1	yes	characteristic
307	#500	C₂²×C₅₉	1	1	yes	characteristic, complemented
308	#500	C₂²×C₅₉	1	1	yes	characteristic, complemented
309	#494	C₂²×C₅₉	1	1	yes	characteristic, complemented
310	#493	C₂²×C₅₉	1	1	yes	characteristic, complemented
311	#487	C₂²×C₅₉	1	2	yes	complemented
312	#486	C₂²×C₅₉	1	2	yes	complemented
313	#485	C₂²×C₅₉	1	8	yes	complemented
314	#482	C₂²×C₅₉	1	8	yes	complemented
315	#476	C₂²×C₅₉	1	4	yes	complemented
316	#475	C₂²×C₅₉	1	4	yes	complemented
317	#467	C₂²×C₅₉	1	2	yes	—
318	#467	C₂²×C₅₉	1	2	yes	—
319	#467	C₂²×C₅₉	1	4	yes	complemented
320	#466	C₂²×C₅₉	1	2	yes	—
321	#466	C₂²×C₅₉	1	2	yes	—
322	#466	C₂²×C₅₉	1	4	yes	complemented
323	#465	C₂²×C₅₉	1	8	yes	—
324	#465	C₂²×C₅₉	1	8	yes	complemented
325	#461	C₂²×C₅₉	1	4	yes	complemented
326	#460	C₂²×C₅₉	1	4	yes	complemented
327	#452	C₂²×C₅₉	1	8	yes	complemented
328	#449	C₂²×C₅₉	1	8	yes	complemented
329	C₄×Q₁₆	C₂²×C₅₉	1	2	yes	complemented
330	C₄×Q₁₆	C₂²×C₅₉	1	2	yes	complemented
331	C₄×D₈	C₂²×C₅₉	1	2	yes	complemented
332	C₄×D₈	C₂²×C₅₉	1	2	yes	complemented
333	C₈×D₄	C₂²×C₅₉	1	2	yes	—
334	C₈×D₄	C₂²×C₅₉	1	2	yes	—
335	C₈×D₄	C₂²×C₅₉	1	2	yes	—
336	C₈×D₄	C₂²×C₅₉	1	2	yes	—
337	#427	C₂²×C₅₉	1	4	yes	complemented
338	C₂×C₄⋊C₈	C₂²×C₅₉	1	1	yes	characteristic
339	#416	C₂²×C₅₉	1	8	yes	complemented
340	C₂×C₂²⋊C₈	C₂²×C₅₉	1	2	yes	—
341	C₂×C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic
342	C₅₉	#62249	1	1	yes	characteristic, direct product factor, Sylow
343	C₂×#41	—	2	2	no	—
344	C₂×#41	C₂³×C₅₉	1	8	yes	—
345	C₂²×D₄	D₄×C₅₉	1	1	yes	characteristic
346	C₂³×C₄	C₂³×C₅₉	1	2	yes	—
347	#135	—	4	8	no	—
348	C₂×Q₁₆	C₂³×C₅₉	1	2	yes	—
349	C₂×Q₁₆	C₂³×C₅₉	1	2	yes	—
350	C₂×SD₁₆	C₂³×C₅₉	1	8	yes	—
351	C₂×D₈	C₂³×C₅₉	1	2	yes	—
352	C₂×D₈	C₂³×C₅₉	1	2	yes	—
353	C₂²×C₈	—	4	4	no	—
354	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
355	C₂²×C₈	C₂³×C₅₉	1	4	yes	—
356	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
357	C₂²×C₈	C₂³×C₅₉	1	4	yes	—
358	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
359	C₄⋊Q₈	C₂³×C₅₉	1	2	yes	—
360	C₄⋊Q₈	C₂³×C₅₉	1	2	yes	—
361	C₄⋊Q₈	C₂³×C₅₉	1	2	yes	—
362	#127	—	2	4	no	—
363	#127	C₂³×C₅₉	1	2	yes	—
364	#126	—	2	8	no	—
365	#125	—	2	2	no	—
366	#124	—	2	4	no	—
367	#124	—	2	8	no	—
368	#124	—	2	2	no	—
369	#123	—	2	8	no	—
370	#123	—	2	8	no	—
371	C₂²⋊Q₈	—	2	2	no	—
372	C₂²⋊Q₈	C₂³×C₅₉	1	4	yes	—
373	C₂²⋊Q₈	C₂³×C₅₉	1	8	yes	—
374	#121	—	2	4	no	—
375	#121	—	2	4	no	—
376	#121	—	2	2	no	—
377	#121	—	2	8	no	—
378	#121	C₂³×C₅₉	1	4	yes	—
379	#121	C₂³×C₅₉	1	8	yes	—
380	#120	—	2	8	no	—
381	C₄×Q₈	—	2	4	no	—
382	C₄×Q₈	—	2	2	no	—
383	C₄×Q₈	C₂³×C₅₉	1	4	yes	—
384	C₄×D₄	—	2	8	no	—
385	C₄×D₄	—	2	8	no	—
386	C₄×D₄	—	2	4	no	—
387	C₄×D₄	—	2	2	no	—
388	C₄×D₄	D₄×C₅₉	1	2	yes	—
389	C₄×D₄	D₄×C₅₉	1	2	yes	—
390	C₄×D₄	C₂³×C₅₉	1	4	yes	—
391	C₄×D₄	C₂³×C₅₉	1	2	yes	—
392	C₄×D₄	C₂³×C₅₉	1	2	yes	—
393	#117	—	2	2	no	—
394	#117	—	2	4	no	—
395	#117	C₂³×C₅₉	1	8	yes	—
396	C₂×C₄⋊C₄	—	2	4	no	—
397	C₂×C₄⋊C₄	D₄×C₅₉	1	1	yes	characteristic
398	C₂×C₂²⋊C₄	D₄×C₅₉	1	2	yes	—
399	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
400	#107	C₂³×C₅₉	1	2	yes	—
401	#107	C₂³×C₅₉	1	2	yes	—
402	#107	C₂³×C₅₉	1	2	yes	—
403	#107	C₂³×C₅₉	1	2	yes	—
404	#106	C₂³×C₅₉	1	8	yes	—
405	C₄⋊C₈	C₂³×C₅₉	1	2	yes	—
406	C₄⋊C₈	C₂³×C₅₉	1	2	yes	—
407	#103	C₂³×C₅₉	1	4	yes	—
408	#103	C₂³×C₅₉	1	8	yes	—
409	#103	C₂³×C₅₉	1	4	yes	—
410	#102	C₂³×C₅₉	1	4	yes	—
411	#102	C₂³×C₅₉	1	8	yes	—
412	#102	C₂³×C₅₉	1	4	yes	—
413	C₂²⋊C₈	C₂³×C₅₉	1	4	yes	—
414	C₂²⋊C₈	C₂³×C₅₉	1	4	yes	—
415	C₄×C₈	C₂³×C₅₉	1	1	yes	characteristic
416	C₄×C₈	C₂³×C₅₉	1	1	yes	characteristic
417	C₄×C₈	C₂³×C₅₉	1	1	yes	characteristic
418	C₄×C₈	C₂³×C₅₉	1	1	yes	characteristic
419	C₂⁴	C₂×D₄×C₅₉	1	2	yes	—
420	#41	—	4	8	no	—
421	#41	—	4	8	no	—
422	#41	—	4	8	no	—
423	C₂×Q₈	—	2	2	no	—
424	C₂×Q₈	—	2	2	no	—
425	C₂×Q₈	C₂⁴×C₅₉	1	4	yes	—
426	C₂×D₄	—	2	4	no	—
427	C₂×D₄	—	2	8	no	—
428	C₂×D₄	—	2	8	no	—
429	C₂×D₄	—	2	4	no	—
430	C₂×D₄	—	2	2	no	—
431	C₂×D₄	—	2	2	no	—
432	C₂×D₄	C₂×D₄×C₅₉	1	2	yes	—
433	C₂×D₄	C₂×D₄×C₅₉	1	2	yes	—
434	C₂×D₄	C₂⁴×C₅₉	1	4	yes	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	2	8	no	—
437	C₂²×C₄	—	4	2	no	—
438	C₂²×C₄	—	4	2	no	—
439	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
440	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
441	C₂²×C₄	C₂×D₄×C₅₉	1	4	yes	—
442	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
443	C₂²×C₄	C₂⁴×C₅₉	1	4	yes	—
444	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
445	Q₁₆	—	2	4	no	—
446	Q₁₆	—	4	2	no	—
447	SD₁₆	—	4	8	no	—
448	D₈	—	2	4	no	—
449	D₈	—	4	2	no	—
450	C₂×C₈	—	2	2	no	—
451	C₂×C₈	—	2	8	no	—
452	C₂×C₈	—	4	4	no	—
453	C₂×C₈	—	4	2	no	—
454	C₂×C₈	—	4	1	no	—
455	C₂×C₈	C₂⁴×C₅₉	1	1	yes	characteristic
456	C₂×C₈	C₂⁴×C₅₉	1	1	yes	characteristic
457	C₂×C₈	C₂⁴×C₅₉	1	1	yes	characteristic
458	C₂×C₈	C₂⁴×C₅₉	1	4	yes	—
459	C₂×C₈	C₂⁴×C₅₉	1	1	yes	characteristic
460	C₄⋊C₄	—	2	2	no	—
461	C₄⋊C₄	—	2	8	no	—
462	C₄⋊C₄	—	2	2	no	—
463	C₄⋊C₄	—	2	4	no	—
464	C₄⋊C₄	—	2	2	no	—
465	C₄⋊C₄	—	2	2	no	—
466	C₄⋊C₄	—	2	4	no	—
467	C₄⋊C₄	C₂×D₄×C₅₉	1	2	yes	—
468	C₄⋊C₄	C₂×D₄×C₅₉	1	2	yes	—
469	C₄⋊C₄	C₂⁴×C₅₉	1	4	yes	—
470	C₄⋊C₄	C₂⁴×C₅₉	1	4	yes	—
471	C₂²⋊C₄	—	2	4	no	—
472	C₂²⋊C₄	—	2	8	no	—
473	C₂²⋊C₄	—	2	8	no	—
474	C₂²⋊C₄	—	2	8	no	—
475	C₂²⋊C₄	—	2	4	no	—
476	C₂²⋊C₄	—	2	4	no	—
477	C₂²⋊C₄	C₂×D₄×C₅₉	1	4	yes	—
478	C₂²⋊C₄	C₂×D₄×C₅₉	1	4	yes	—
479	C₄²	—	2	8	no	—
480	C₄²	—	2	4	no	—
481	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
482	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
483	C₄²	C₂⁴×C₅₉	1	1	yes	characteristic
484	C₄²	C₂⁴×C₅₉	1	1	yes	characteristic
485	C₂³	—	2	4	no	—
486	C₂³	—	2	4	no	—
487	C₂³	—	4	2	no	—
488	C₂³	—	4	2	no	—
489	C₂³	C₂²×D₄×C₅₉	1	4	yes	—
490	C₂³	C₂²×D₄×C₅₉	1	1	yes	characteristic
491	Q₈	—	2	4	no	—
492	Q₈	—	2	4	no	—
493	Q₈	—	4	4	no	—
494	Q₈	—	8	2	no	—
495	D₄	—	2	4	no	—
496	D₄	—	2	4	no	—
497	D₄	—	2	4	no	—
498	D₄	—	2	4	no	—
499	D₄	—	4	8	no	—
500	D₄	—	4	4	no	—
501	D₄	—	8	2	no	—
502	D₄	—	8	8	no	—
503	D₄	—	8	8	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	4	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	4	4	no	—
512	C₂×C₄	—	4	2	no	—
513	C₂×C₄	—	4	1	no	—
514	C₂×C₄	—	4	8	no	—
515	C₂×C₄	—	8	8	no	—
516	C₂×C₄	#135×C₅₉	1	4	yes	complemented
517	C₂×C₄	C₂²×D₄×C₅₉	1	4	yes	—
518	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
519	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
520	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
521	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
522	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
523	C₂×C₄	C₂⁵×C₅₉	1	1	yes	characteristic
524	C₈	—	2	1	no	—
525	C₈	—	2	4	no	—
526	C₈	—	2	1	no	—
527	C₈	—	2	1	no	—
528	C₈	H(5,𝔽₂)×C₅₉	1	2	yes	—
529	C₂²	—	2	1	no	—
530	C₂²	—	2	1	no	—
531	C₂²	—	2	4	no	—
532	C₂²	—	2	4	no	—
533	C₂²	—	2	4	no	—
534	C₂²	—	4	8	no	—
535	C₂²	—	4	4	no	—
536	C₂²	—	4	4	no	—
537	C₂²	—	4	4	no	—
538	C₂²	C₂×#135×C₅₉	1	2	yes	—
539	C₂²	C₂³×D₄×C₅₉	1	1	yes	characteristic
540	C₄	—	2	1	no	—
541	C₄	—	2	1	no	—
542	C₄	—	2	1	no	—
543	C₄	—	2	4	no	—
544	C₄	—	4	4	no	—
545	C₄	—	4	4	no	—
546	C₄	—	4	4	no	—
547	C₄	C₂×#135×C₅₉	1	2	yes	—
548	C₄	C₂×#135×C₅₉	1	2	yes	—
549	C₄	C₂×H(5,𝔽₂)×C₅₉	1	1	yes	characteristic
550	C₄	C₅₉×#585	1	1	yes	characteristic
551	C₂	—	2	2	no	—
552	C₂	—	4	4	no	—
553	C₂	—	8	4	no	—
554	C₂	C₅₉×#3496	1	1	yes	characteristic
555	C₂	C₂²×#135×C₅₉	1	1	yes	characteristic
556	C₂	C₅₉×#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₂×C₄×D₄×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₉×#3506	C₂	semidirect
2. C₅₉×#3342	C₂	semidirect
3. C₅₉×#3341	C₂	semidirect
4. C₅₉×#3304	C₂	semidirect
5. C₅₉×#3303	C₂	semidirect
6. C₅₉×#3287	C₂	semidirect
7. C₅₉×#3283	C₂	semidirect
8. C₅₉×#3246	C₂	semidirect
9. C₅₉×#3191	C₂	semidirect
10. C₅₉×#3170	C₂	semidirect
11. C₅₉×#3169	C₂	semidirect
12. C₅₉×#3050	C₂	semidirect
13. C₂×C₈×D₄×C₅₉	C₂	semidirect
14. C₂×#135×C₅₉	C₂²	semidirect
29. C₅₉×#500	C₂²	semidirect
30. C₅₉×#500	C₂²	semidirect
31. C₅₉×#494	C₂²	semidirect
32. C₅₉×#493	C₂²	semidirect
33. C₅₉×#487	C₂²	semidirect
34. C₅₉×#486	C₂²	semidirect
35. C₅₉×#485	C₂²	semidirect
36. C₅₉×#482	C₂²	semidirect
37. C₅₉×#476	C₂²	semidirect
38. C₅₉×#475	C₂²	semidirect
41. C₅₉×#467	C₂²	semidirect
44. C₅₉×#466	C₂²	semidirect
46. C₅₉×#465	C₂²	semidirect
47. C₅₉×#461	C₂²	semidirect
48. C₅₉×#460	C₂²	semidirect
49. C₅₉×#452	C₂²	semidirect
50. C₅₉×#449	C₂²	semidirect
51. C₄×Q₁₆×C₅₉	C₂²	semidirect
52. C₄×Q₁₆×C₅₉	C₂²	semidirect
53. C₄×D₈×C₅₉	C₂²	semidirect
54. C₄×D₈×C₅₉	C₂²	semidirect
59. C₅₉×#427	C₂²	semidirect
61. C₅₉×#416	C₂²	semidirect
237. C₂×C₄×C₅₉	#135	semidirect
250. #62249	C₅₉	direct (N × H)
273. #3506	C₂×C₅₉	semidirect
274. #3342	C₂×C₅₉	semidirect
275. #3341	C₂×C₅₉	semidirect
276. #3304	C₂×C₅₉	semidirect
277. #3303	C₂×C₅₉	semidirect
278. #3287	C₂×C₅₉	semidirect
279. #3283	C₂×C₅₉	semidirect
280. #3246	C₂×C₅₉	semidirect
281. #3191	C₂×C₅₉	semidirect
282. #3170	C₂×C₅₉	semidirect
283. #3169	C₂×C₅₉	semidirect
284. #3050	C₂×C₅₉	semidirect
285. C₂×C₈×D₄	C₂×C₅₉	semidirect
292. C₂×#135	C₂²×C₅₉	semidirect
307. #500	C₂²×C₅₉	semidirect
308. #500	C₂²×C₅₉	semidirect
309. #494	C₂²×C₅₉	semidirect
310. #493	C₂²×C₅₉	semidirect
311. #487	C₂²×C₅₉	semidirect
312. #486	C₂²×C₅₉	semidirect
313. #485	C₂²×C₅₉	semidirect
314. #482	C₂²×C₅₉	semidirect
315. #476	C₂²×C₅₉	semidirect
316. #475	C₂²×C₅₉	semidirect
319. #467	C₂²×C₅₉	semidirect
322. #466	C₂²×C₅₉	semidirect
324. #465	C₂²×C₅₉	semidirect
325. #461	C₂²×C₅₉	semidirect
326. #460	C₂²×C₅₉	semidirect
327. #452	C₂²×C₅₉	semidirect
328. #449	C₂²×C₅₉	semidirect
329. C₄×Q₁₆	C₂²×C₅₉	semidirect
330. C₄×Q₁₆	C₂²×C₅₉	semidirect
331. C₄×D₈	C₂²×C₅₉	semidirect
332. C₄×D₈	C₂²×C₅₉	semidirect
337. #427	C₂²×C₅₉	semidirect
339. #416	C₂²×C₅₉	semidirect
342. C₅₉	#62249	direct (N × H)
516. C₂×C₄	#135×C₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₅₉×#3506	2
C₅₉×#3342	2
C₅₉×#3341	2
C₅₉×#3304	2
C₅₉×#3303	2
C₅₉×#3287	2
C₅₉×#3283	2
C₅₉×#3246	2
C₅₉×#3191	2
C₅₉×#3170	2
C₅₉×#3169	2
C₅₉×#3050	2
C₂×C₈×D₄×C₅₉	2
#62249	59

Subgroups — C59×#62249 #28474358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#62249 #28474358