Subgroups — C₅₉×#9289 #28421398

Order 15104.

490 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#3206	C₂	1	1	yes	maximal, characteristic
2	C₅₉×#2932	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#1993	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₉×#1549	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#1517	C₂	1	1	yes	maximal, characteristic
6	C₅₉×#1286	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1275	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131×C₅₉	—	2	1	no	—
9	C₅₉×#532	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
11	C₅₉×#492	C₄	1	2	yes	complemented
12	C₅₉×#487	C₄	1	2	yes	complemented
13	C₅₉×#461	—	2	2	no	—
14	C₅₉×#448	—	2	2	no	—
15	C₅₉×#435	—	2	2	no	—
16	C₅₉×#435	—	2	2	no	—
17	C₅₉×#431	C₂²	1	1	yes	characteristic
18	C₅₉×#431	C₂²	1	1	yes	characteristic
19	C₅₉×#423	C₂²	1	1	yes	characteristic
20	C₅₉×#408	—	2	1	no	—
21	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
22	C₅₉×#382	C₂²	1	1	yes	characteristic
23	C₅₉×#340	—	2	1	no	—
24	C₅₉×#340	—	2	1	no	—
25	C₅₉×#332	—	2	2	no	—
26	C₅₉×#327	—	2	2	no	—
27	C₂×#41×C₅₉	—	2	1	no	—
28	C₂²×Q₈×C₅₉	—	2	1	no	—
29	C₂×SD₁₆×C₅₉	—	2	4	no	—
30	#131×C₅₉	—	4	2	no	—
31	#131×C₅₉	—	4	2	no	—
32	C₂×M₁₆×C₅₉	—	2	2	no	—
33	C₂×M₁₆×C₅₉	—	2	1	no	—
34	C₂²×C₈×C₅₉	—	2	2	no	—
35	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
37	C₄⋊Q₈×C₅₉	—	2	1	no	—
38	C₄⋊Q₈×C₅₉	—	2	1	no	—
39	C₄⋊Q₈×C₅₉	—	2	1	no	—
40	C₄⋊Q₈×C₅₉	C₂×C₄	1	2	yes	—
41	C₄⋊Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
42	#127×C₅₉	C₂×C₄	1	1	yes	characteristic
43	#124×C₅₉	—	2	1	no	—
44	#121×C₅₉	—	2	1	no	—
45	#121×C₅₉	D₄	1	2	yes	—
46	C₄×D₄×C₅₉	—	2	1	no	—
47	C₄×D₄×C₅₉	D₄	1	2	yes	—
48	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
49	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
50	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
51	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
52	C₄⋊C₈×C₅₉	—	2	1	no	—
53	C₄⋊C₈×C₅₉	—	2	1	no	—
54	C₄⋊C₈×C₅₉	—	2	1	no	—
55	C₄⋊C₈×C₅₉	D₄	1	2	yes	—
56	#103×C₅₉	—	2	4	no	—
57	C₂²⋊C₈×C₅₉	—	2	4	no	—
58	C₂²⋊C₈×C₅₉	—	2	1	no	—
59	C₂²⋊C₈×C₅₉	—	2	1	no	—
60	#97×C₅₉	—	2	1	no	—
61	#97×C₅₉	—	2	1	no	—
62	C₄×C₈×C₅₉	—	2	1	no	—
63	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
64	#95×C₅₉	—	2	1	no	—
65	#95×C₅₉	—	2	1	no	—
66	#41×C₅₉	—	4	1	no	—
67	#41×C₅₉	—	4	1	no	—
68	#41×C₅₉	—	4	1	no	—
69	#41×C₅₉	—	4	1	no	—
70	C₂×Q₈×C₅₉	—	2	1	no	—
71	C₂×Q₈×C₅₉	—	2	2	no	—
72	C₂×Q₈×C₅₉	—	4	1	no	—
73	C₂×Q₈×C₅₉	—	8	2	no	—
74	C₂×D₄×C₅₉	—	2	2	no	—
75	C₂×D₄×C₅₉	—	2	1	no	—
76	C₂×D₄×C₅₉	—	2	1	no	—
77	C₂×D₄×C₅₉	C₂²⋊C₄	1	2	yes	—
78	C₂²×C₄×C₅₉	—	2	2	no	—
79	C₂²×C₄×C₅₉	—	4	1	no	—
80	C₂²×C₄×C₅₉	—	4	1	no	—
81	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
84	SD₁₆×C₅₉	—	8	4	no	—
85	M₁₆×C₅₉	—	4	4	no	—
86	M₁₆×C₅₉	—	4	1	no	—
87	M₁₆×C₅₉	—	4	1	no	—
88	C₂×C₈×C₅₉	—	2	1	no	—
89	C₂×C₈×C₅₉	—	2	1	no	—
90	C₂×C₈×C₅₉	—	2	2	no	—
91	C₂×C₈×C₅₉	—	2	1	no	—
92	C₂×C₈×C₅₉	—	2	2	no	—
93	C₂×C₈×C₅₉	—	2	1	no	—
94	C₂×C₈×C₅₉	—	2	1	no	—
95	C₂×C₈×C₅₉	—	4	4	no	—
96	C₂×C₈×C₅₉	—	4	1	no	—
97	C₂×C₈×C₅₉	SD₁₆	1	2	yes	complemented
98	C₄⋊C₄×C₅₉	—	2	1	no	—
99	C₄⋊C₄×C₅₉	—	2	2	no	—
100	C₄⋊C₄×C₅₉	—	2	2	no	—
101	C₄⋊C₄×C₅₉	—	4	1	no	—
102	C₄⋊C₄×C₅₉	—	4	1	no	—
103	C₄⋊C₄×C₅₉	—	4	1	no	—
104	C₄⋊C₄×C₅₉	—	4	1	no	—
105	C₄⋊C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
106	C₂²⋊C₄×C₅₉	—	2	2	no	—
107	C₂²⋊C₄×C₅₉	—	2	2	no	—
108	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₅₉	#41	1	1	yes	characteristic
112	C₂³×C₅₉	—	2	2	no	—
113	C₂³×C₅₉	#121	1	1	yes	characteristic
114	Q₈×C₅₉	—	4	4	no	—
115	Q₈×C₅₉	—	4	1	no	—
116	Q₈×C₅₉	—	4	1	no	—
117	Q₈×C₅₉	—	8	2	no	—
118	D₄×C₅₉	—	4	2	no	—
119	D₄×C₅₉	—	4	2	no	—
120	D₄×C₅₉	—	4	2	no	—
121	D₄×C₅₉	—	4	1	no	—
122	D₄×C₅₉	—	4	1	no	—
123	D₄×C₅₉	—	8	1	no	—
124	C₂×C₄×C₅₉	—	2	2	no	—
125	C₂×C₄×C₅₉	—	4	1	no	—
126	C₂×C₄×C₅₉	—	4	1	no	—
127	C₂×C₄×C₅₉	—	4	2	no	—
128	C₂×C₄×C₅₉	—	4	2	no	—
129	C₂×C₄×C₅₉	—	4	2	no	—
130	C₂×C₄×C₅₉	—	8	2	no	—
131	C₂×C₄×C₅₉	—	8	1	no	—
132	C₂×C₄×C₅₉	—	8	1	no	—
133	C₂×C₄×C₅₉	—	8	1	no	—
134	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
138	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
139	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
140	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
141	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
142	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
144	C₈×C₅₉	—	2	2	no	—
145	C₈×C₅₉	—	4	2	no	—
146	C₈×C₅₉	—	4	2	no	—
147	C₈×C₅₉	—	4	1	no	—
148	C₈×C₅₉	—	8	1	no	—
149	#9289	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂²×C₅₉	—	2	1	no	—
151	C₂²×C₅₉	—	2	1	no	—
152	C₂²×C₅₉	—	4	2	no	—
153	C₂²×C₅₉	—	4	2	no	—
154	C₂²×C₅₉	—	4	2	no	—
155	C₂²×C₅₉	#386	1	1	yes	characteristic
156	C₂²×C₅₉	#461	1	1	yes	characteristic
157	C₂²×C₅₉	#463	1	1	yes	characteristic
158	C₄×C₅₉	—	2	1	no	—
159	C₄×C₅₉	—	2	1	no	—
160	C₄×C₅₉	—	2	1	no	—
161	C₄×C₅₉	—	2	1	no	—
162	C₄×C₅₉	—	4	2	no	—
163	C₄×C₅₉	—	8	2	no	—
164	C₄×C₅₉	—	16	1	no	—
165	C₄×C₅₉	C₄×SD₁₆	1	1	yes	characteristic
166	C₄×C₅₉	#440	1	1	yes	characteristic
167	C₄×C₅₉	#450	1	1	yes	characteristic
168	C₄×C₅₉	#452	1	1	yes	characteristic
169	#3206	C₂×C₅₉	1	1	yes	characteristic
170	#2932	C₂×C₅₉	1	1	yes	characteristic
171	#1993	C₂×C₅₉	1	1	yes	characteristic, complemented
172	#1549	C₂×C₅₉	1	1	yes	characteristic, complemented
173	#1517	C₂×C₅₉	1	1	yes	characteristic
174	#1286	C₂×C₅₉	1	1	yes	characteristic, complemented
175	#1275	C₂×C₅₉	1	1	yes	characteristic, complemented
176	C₂×C₅₉	—	2	1	no	—
177	C₂×C₅₉	—	2	1	no	—
178	C₂×C₅₉	—	8	2	no	—
179	C₂×C₅₉	#1877	1	1	yes	characteristic
180	C₂×C₅₉	#1902	1	1	yes	characteristic
181	C₂×C₅₉	#1910	1	1	yes	characteristic
182	C₂×#131	—	2	1	no	—
183	#532	C₂²×C₅₉	1	1	yes	characteristic
184	C₂×C₄⋊Q₈	C₂²×C₅₉	1	1	yes	characteristic
185	#492	C₄×C₅₉	1	2	yes	complemented
186	#487	C₄×C₅₉	1	2	yes	complemented
187	#461	—	2	2	no	—
188	#448	—	2	2	no	—
189	#435	—	2	2	no	—
190	#435	—	2	2	no	—
191	#431	C₂²×C₅₉	1	1	yes	characteristic
192	#431	C₂²×C₅₉	1	1	yes	characteristic
193	#423	C₂²×C₅₉	1	1	yes	characteristic
194	#408	—	2	1	no	—
195	C₂×#97	C₂²×C₅₉	1	1	yes	characteristic
196	#382	C₂²×C₅₉	1	1	yes	characteristic
197	#340	—	2	1	no	—
198	#340	—	2	1	no	—
199	#332	—	2	2	no	—
200	#327	—	2	2	no	—
201	C₅₉	#9289	1	1	yes	characteristic, direct product factor, Sylow
202	C₂×#41	—	2	1	no	—
203	C₂²×Q₈	—	2	1	no	—
204	C₂×SD₁₆	—	2	4	no	—
205	#131	—	4	2	no	—
206	#131	—	4	2	no	—
207	C₂×M₁₆	—	2	2	no	—
208	C₂×M₁₆	—	2	1	no	—
209	C₂²×C₈	—	2	2	no	—
210	C₂²×C₈	D₄×C₅₉	1	1	yes	characteristic
211	C₂²×C₈	D₄×C₅₉	1	1	yes	characteristic
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	C₂×C₄×C₅₉	1	2	yes	—
216	C₄⋊Q₈	C₂×C₄×C₅₉	1	1	yes	characteristic
217	#127	C₂×C₄×C₅₉	1	1	yes	characteristic
218	#124	—	2	1	no	—
219	#121	—	2	1	no	—
220	#121	D₄×C₅₉	1	2	yes	—
221	C₄×D₄	—	2	1	no	—
222	C₄×D₄	D₄×C₅₉	1	2	yes	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	—	2	1	no	—
225	C₂×C₄⋊C₄	—	2	1	no	—
226	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
227	C₄⋊C₈	—	2	1	no	—
228	C₄⋊C₈	—	2	1	no	—
229	C₄⋊C₈	—	2	1	no	—
230	C₄⋊C₈	D₄×C₅₉	1	2	yes	—
231	#103	—	2	4	no	—
232	C₂²⋊C₈	—	2	4	no	—
233	C₂²⋊C₈	—	2	1	no	—
234	C₂²⋊C₈	—	2	1	no	—
235	#97	—	2	1	no	—
236	#97	—	2	1	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	C₂×C₄×C₅₉	1	2	yes	—
239	#95	—	2	1	no	—
240	#95	—	2	1	no	—
241	#41	—	4	1	no	—
242	#41	—	4	1	no	—
243	#41	—	4	1	no	—
244	#41	—	4	1	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	8	2	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	C₂²⋊C₄×C₅₉	1	2	yes	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
258	C₂²×C₄	#41×C₅₉	1	1	yes	characteristic
259	SD₁₆	—	8	4	no	—
260	M₁₆	—	4	4	no	—
261	M₁₆	—	4	1	no	—
262	M₁₆	—	4	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	4	4	no	—
271	C₂×C₈	—	4	1	no	—
272	C₂×C₈	SD₁₆×C₅₉	1	2	yes	complemented
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	C₂²⋊C₄×C₅₉	1	2	yes	—
281	C₂²⋊C₄	—	2	2	no	—
282	C₂²⋊C₄	—	2	2	no	—
283	C₄²	C₂²×C₄×C₅₉	1	1	yes	characteristic
284	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
285	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
286	C₄²	#41×C₅₉	1	1	yes	characteristic
287	C₂³	—	2	2	no	—
288	C₂³	#121×C₅₉	1	1	yes	characteristic
289	Q₈	—	4	4	no	—
290	Q₈	—	4	1	no	—
291	Q₈	—	4	1	no	—
292	Q₈	—	8	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	2	no	—
295	D₄	—	4	2	no	—
296	D₄	—	4	1	no	—
297	D₄	—	4	1	no	—
298	D₄	—	8	1	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	8	2	no	—
306	C₂×C₄	—	8	1	no	—
307	C₂×C₄	—	8	1	no	—
308	C₂×C₄	—	8	1	no	—
309	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
310	C₂×C₄	C₄×D₄×C₅₉	1	1	yes	characteristic
311	C₂×C₄	C₄×D₄×C₅₉	1	1	yes	characteristic
312	C₂×C₄	#120×C₅₉	1	1	yes	characteristic
313	C₂×C₄	#121×C₅₉	1	1	yes	characteristic
314	C₂×C₄	#123×C₅₉	1	1	yes	characteristic
315	C₂×C₄	C₂×SD₁₆×C₅₉	1	1	yes	characteristic
316	C₂×C₄	#135×C₅₉	1	1	yes	characteristic
317	C₂×C₄	AGL(1,ℤ/8)×C₅₉	1	1	yes	characteristic
318	C₂×C₄	#137×C₅₉	1	1	yes	characteristic
319	C₈	—	2	2	no	—
320	C₈	—	4	2	no	—
321	C₈	—	4	2	no	—
322	C₈	—	4	1	no	—
323	C₈	—	8	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	4	2	no	—
327	C₂²	—	4	2	no	—
328	C₂²	—	4	2	no	—
329	C₂²	C₅₉×#386	1	1	yes	characteristic
330	C₂²	C₅₉×#461	1	1	yes	characteristic
331	C₂²	C₅₉×#463	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	4	2	no	—
337	C₄	—	8	2	no	—
338	C₄	—	16	1	no	—
339	C₄	C₄×SD₁₆×C₅₉	1	1	yes	characteristic
340	C₄	C₅₉×#440	1	1	yes	characteristic
341	C₄	C₅₉×#450	1	1	yes	characteristic
342	C₄	C₅₉×#452	1	1	yes	characteristic
343	C₂	—	2	1	no	—
344	C₂	—	2	1	no	—
345	C₂	—	8	2	no	—
346	C₂	C₅₉×#1877	1	1	yes	characteristic
347	C₂	C₅₉×#1902	1	1	yes	characteristic
348	C₂	C₅₉×#1910	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₉	#386
commutator	C₄²	C₂²×C₄×C₅₉
Frattini, cosocle	C₂×C₄²	C₂³×C₅₉
prime radical	C₅₉×#532	C₂²
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
3. C₅₉×#1993	C₂	semidirect
4. C₅₉×#1549	C₂	semidirect
6. C₅₉×#1286	C₂	semidirect
7. C₅₉×#1275	C₂	semidirect
11. C₅₉×#492	C₄	semidirect
12. C₅₉×#487	C₄	semidirect
97. C₂×C₈×C₅₉	SD₁₆	semidirect
149. #9289	C₅₉	direct (N × H)
171. #1993	C₂×C₅₉	semidirect
172. #1549	C₂×C₅₉	semidirect
174. #1286	C₂×C₅₉	semidirect
175. #1275	C₂×C₅₉	semidirect
185. #492	C₄×C₅₉	semidirect
186. #487	C₄×C₅₉	semidirect
201. C₅₉	#9289	direct (N × H)
272. C₂×C₈	SD₁₆×C₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₅₉×#3206	2
C₅₉×#2932	2
C₅₉×#1993	2
C₅₉×#1549	2
C₅₉×#1517	2
C₅₉×#1286	2
C₅₉×#1275	2
#9289	59

Subgroups — C59×#9289 #28421398

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#9289 #28421398