Subgroups — #28482549

Order 15104.

482 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21433119	C₂	1	1	yes	maximal, characteristic
2	#21424889	C₂	1	1	yes	maximal, characteristic
3	C₄×#14273643	C₂	1	1	yes	maximal, characteristic
4	#21422846	C₂	1	1	yes	maximal, characteristic, complemented
5	#21421934	C₂	1	1	yes	maximal, characteristic, complemented
6	#21421932	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1565	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₂₃₆	C₂²	1	1	yes	characteristic
9	C₈×D₂₃₆	—	2	1	no	—
10	C₈×D₂₃₆	—	2	1	no	—
11	#16837467	—	2	1	no	—
12	#16837467	—	2	1	no	—
13	#16837455	C₂²	1	1	yes	characteristic
14	C₂×#14273643	C₄	1	1	yes	characteristic
15	C₂×#14273643	C₄	1	1	yes	characteristic
16	C₂×(C₂×D₅₉)⋊C₈	—	2	2	no	—
17	C₂²⋊C₈×D₅₉	—	2	2	no	—
18	C₄×#11745664	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
20	#16837170	C₄	1	1	yes	characteristic
21	#16837170	C₄	1	1	yes	characteristic
22	#16837154	—	2	1	no	—
23	#16837152	—	2	1	no	—
24	C₅₉×#431	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
26	C₄×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
27	C₅₉×#335	—	2	1	no	—
28	C₅₉×#334	—	2	1	no	—
29	C₂²×D₂₃₆	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×D₅₉	D₄	1	2	yes	—
31	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
32	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₅₉⋊C₈	—	2	1	no	—
34	C₂×C₈×D₅₉	—	2	2	no	—
35	C₂×C₈×D₅₉	—	2	2	no	—
36	C₄×D₂₃₆	—	2	1	no	—
37	C₄×D₂₃₆	—	2	1	no	—
38	C₄×D₂₃₆	—	2	1	no	—
39	C₄×D₂₃₆	—	2	1	no	—
40	C₂×C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
41	C₂×(C₂×D₅₉)⋊C₄	Q₈	1	2	yes	—
42	C₄⋊(C₅₉⋊C₈)	D₄	1	1	yes	characteristic
43	C₄⋊(C₅₉⋊C₈)	D₄	1	1	yes	characteristic
44	#14273643	—	2	2	no	—
45	#14273643	C₈	1	4	yes	complemented
46	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
47	(C₂×D₅₉)⋊C₈	—	2	2	no	—
48	(C₂×D₅₉)⋊C₈	—	2	2	no	—
49	(C₂×D₅₉)⋊C₈	—	4	4	no	—
50	#14273638	—	2	1	no	—
51	C₄×C₅₉⋊C₈	—	2	1	no	—
52	C₄×C₅₉⋊C₈	—	2	1	no	—
53	C₄×C₅₉⋊C₈	—	2	1	no	—
54	C₄×C₅₉⋊C₈	—	2	1	no	—
55	C₂×M₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
56	C₂×M₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₅₉	—	2	1	no	—
58	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₅₉	—	2	1	no	—
60	C₄⋊C₈×C₅₉	—	2	1	no	—
61	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
62	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
63	C₂²⋊C₈×C₅₉	—	2	1	no	—
64	#97×C₅₉	—	2	1	no	—
65	C₄×C₈×C₅₉	—	2	1	no	—
66	C₄×C₈×C₅₉	—	2	1	no	—
67	C₂³×D₅₉	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
71	C₂²×Dic₅₉	C₄⋊C₄	1	2	yes	—
72	C₂×D₂₃₆	—	2	1	no	—
73	C₂×D₂₃₆	—	2	1	no	—
74	C₂×D₂₃₆	—	2	1	no	—
75	C₂×D₂₃₆	—	2	1	no	—
76	C₂×D₂₃₆	C₂×C₈	1	2	yes	—
77	C₂×D₂₃₆	M₁₆	1	2	yes	—
78	C₂×C₄×D₅₉	—	2	2	no	—
79	C₂×C₄×D₅₉	—	2	2	no	—
80	C₂×C₄×D₅₉	—	4	2	no	—
81	C₂×C₄×D₅₉	—	4	2	no	—
82	M₁₆×C₅₉	—	2	1	no	—
83	M₁₆×C₅₉	—	2	1	no	—
84	M₁₆×C₅₉	—	2	1	no	—
85	M₁₆×C₅₉	C₂×C₈	1	2	yes	complemented
86	C₂×C₈×C₅₉	—	2	1	no	—
87	C₂×C₈×C₅₉	—	2	1	no	—
88	C₂×C₈×C₅₉	—	2	1	no	—
89	C₂×C₈×C₅₉	—	2	1	no	—
90	C₂×C₈×C₅₉	—	4	2	no	—
91	C₂×C₈×C₅₉	—	4	1	no	—
92	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₅₉	#41	1	1	yes	characteristic
95	C₄²×C₅₉	#41	1	1	yes	characteristic
96	(C₂×D₅₉)⋊C₄	—	2	2	no	—
97	(C₂×D₅₉)⋊C₄	—	2	2	no	—
98	C₄⋊Dic₅₉	—	2	1	no	—
99	C₄⋊Dic₅₉	—	2	1	no	—
100	#11745664	—	2	1	no	—
101	#11745664	—	2	1	no	—
102	#11745664	—	2	1	no	—
103	#11745664	C₂×C₈	1	2	yes	complemented
104	C₂×C₅₉⋊C₈	—	2	1	no	—
105	C₂×C₅₉⋊C₈	—	2	2	no	—
106	C₂×C₅₉⋊C₈	—	2	1	no	—
107	C₂×C₅₉⋊C₈	—	2	1	no	—
108	C₂×C₅₉⋊C₈	—	2	1	no	—
109	C₂×C₅₉⋊C₈	—	4	1	no	—
110	C₂×C₅₉⋊C₈	—	4	1	no	—
111	C₈×D₅₉	—	8	2	no	—
112	C₈×D₅₉	—	8	2	no	—
113	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×D₅₉	—	2	2	no	—
115	C₂²×D₅₉	—	2	2	no	—
116	C₂²×D₅₉	—	4	2	no	—
117	C₂²×D₅₉	—	4	2	no	—
118	C₂²×D₅₉	C₄⋊C₈	1	4	yes	—
119	C₂×C₄×C₅₉	—	2	1	no	—
120	C₂×C₄×C₅₉	—	2	1	no	—
121	C₂×C₄×C₅₉	—	4	1	no	—
122	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
128	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
130	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
131	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
132	C₂×Dic₅₉	—	2	2	no	—
133	C₂×Dic₅₉	—	4	2	no	—
134	D₂₃₆	—	2	2	no	—
135	D₂₃₆	—	2	2	no	—
136	D₂₃₆	—	2	2	no	—
137	D₂₃₆	—	2	2	no	—
138	C₄×D₅₉	—	4	2	no	—
139	C₄×D₅₉	—	4	2	no	—
140	C₄×D₅₉	—	8	2	no	—
141	C₈×C₅₉	—	2	2	no	—
142	C₈×C₅₉	—	4	1	no	—
143	C₈×C₅₉	—	4	1	no	—
144	C₈×C₅₉	—	4	1	no	—
145	C₈×C₅₉	—	8	1	no	—
146	C₅₉⋊C₈	—	2	2	no	—
147	C₅₉⋊C₈	—	4	2	no	—
148	C₅₉⋊C₈	—	4	1	no	—
149	C₅₉⋊C₈	—	4	1	no	—
150	C₅₉⋊C₈	—	4	1	no	—
151	#8835	—	59	1	no	maximal, Sylow
152	C₂²×C₅₉	—	2	1	no	—
153	C₂²×C₅₉	—	2	1	no	—
154	C₂²×C₅₉	#385	1	1	yes	characteristic
155	C₂²×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
156	C₂²×C₅₉	#424	1	1	yes	characteristic
157	C₂×D₅₉	—	2	4	no	—
158	C₂×D₅₉	—	4	2	no	—
159	C₂×D₅₉	—	4	2	no	—
160	C₂×D₅₉	—	4	4	no	—
161	C₂×D₅₉	—	8	2	no	—
162	C₄×C₅₉	—	2	1	no	—
163	C₄×C₅₉	—	2	1	no	—
164	C₄×C₅₉	—	2	1	no	—
165	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
166	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
167	C₄×C₅₉	#435	1	1	yes	characteristic
168	C₄×C₅₉	#435	1	1	yes	characteristic
169	C₄×C₅₉	#453	1	1	yes	characteristic
170	C₄×C₅₉	#455	1	1	yes	characteristic
171	Dic₅₉	—	4	2	no	—
172	#2972	—	59	1	no	—
173	#1954	—	59	1	no	—
174	C₄×#100	—	59	1	no	—
175	#1565	—	59	1	no	—
176	#1565	D₅₉	1	1	yes	characteristic, complemented
177	#1278	—	59	1	no	—
178	#1278	—	59	1	no	—
179	C₂×C₅₉	—	2	1	no	—
180	C₂×C₅₉	—	2	1	no	—
181	C₂×C₅₉	#1850	1	1	yes	characteristic
182	C₂×C₅₉	#1856	1	1	yes	characteristic
183	C₂×C₅₉	#1858	1	1	yes	characteristic
184	D₅₉	—	4	4	no	—
185	C₂×C₄×D₄	—	59	1	no	—
186	C₈×D₄	—	118	1	no	—
187	C₈×D₄	—	118	1	no	—
188	C₈×D₄	—	118	1	no	—
189	C₈×D₄	—	118	1	no	—
190	#431	—	59	1	no	—
191	#431	C₂×D₅₉	1	1	yes	characteristic
192	C₂×C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
193	C₂×#100	—	59	1	no	—
194	C₂×#100	—	59	1	no	—
195	C₂×C₂²⋊C₈	—	118	2	no	—
196	C₂×C₂²⋊C₈	—	118	2	no	—
197	C₄×M₁₆	—	59	1	no	—
198	C₄×M₁₆	C₂×D₅₉	1	1	yes	characteristic
199	C₂×C₄×C₈	—	59	1	no	—
200	#343	—	59	1	no	—
201	#343	—	59	1	no	—
202	#335	—	2	1	no	—
203	#335	—	118	1	no	—
204	#334	—	2	1	no	—
205	#334	—	118	1	no	—
206	C₅₉	#8835	1	1	yes	characteristic, complemented, Sylow
207	C₂²×D₄	—	59	1	no	—
208	C₂³×C₄	—	59	2	no	—
209	C₂×M₁₆	—	59	1	no	—
210	C₂×M₁₆	—	59	1	no	—
211	C₂×M₁₆	C₄×D₅₉	1	1	yes	characteristic
212	C₂×M₁₆	C₄×D₅₉	1	1	yes	characteristic
213	C₂²×C₈	—	2	1	no	—
214	C₂²×C₈	—	118	2	no	—
215	C₂²×C₈	—	118	2	no	—
216	C₂²×C₈	—	118	1	no	—
217	C₄×D₄	—	118	1	no	—
218	C₄×D₄	—	118	1	no	—
219	C₄×D₄	—	118	1	no	—
220	C₄×D₄	—	118	1	no	—
221	C₂×C₄⋊C₄	—	59	1	no	—
222	C₂×C₂²⋊C₄	—	59	2	no	—
223	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	59	1	no	—
227	C₄⋊C₈	—	59	1	no	—
228	C₄⋊C₈	C₂²⋊D₅₉	1	1	yes	characteristic
229	C₄⋊C₈	C₂²⋊D₅₉	1	1	yes	characteristic
230	#100	—	59	4	no	—
231	#100	—	118	2	no	—
232	C₂²⋊C₈	—	2	1	no	—
233	C₂²⋊C₈	—	118	2	no	—
234	C₂²⋊C₈	—	118	2	no	—
235	C₂²⋊C₈	—	118	1	no	—
236	C₂²⋊C₈	—	236	4	no	—
237	#97	—	2	1	no	—
238	#97	—	118	1	no	—
239	C₄×C₈	—	2	1	no	—
240	C₄×C₈	—	2	1	no	—
241	C₄×C₈	—	118	1	no	—
242	C₄×C₈	—	118	1	no	—
243	C₄×C₈	—	118	1	no	—
244	C₄×C₈	—	118	1	no	—
245	C₂⁴	—	59	2	no	—
246	C₂×D₄	—	59	2	no	—
247	C₂×D₄	—	59	2	no	—
248	C₂×D₄	—	118	1	no	—
249	C₂×D₄	—	118	1	no	—
250	C₂×D₄	—	118	1	no	—
251	C₂×D₄	—	118	1	no	—
252	C₂²×C₄	—	59	2	no	—
253	C₂²×C₄	—	118	2	no	—
254	C₂²×C₄	—	118	2	no	—
255	C₂²×C₄	—	236	2	no	—
256	C₂²×C₄	—	236	2	no	—
257	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
258	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
259	C₂²×C₄	Q₈×D₅₉	1	1	yes	characteristic
260	M₁₆	—	2	1	no	—
261	M₁₆	—	2	1	no	—
262	M₁₆	—	2	1	no	—
263	M₁₆	—	59	2	no	—
264	M₁₆	—	118	1	no	—
265	M₁₆	—	118	1	no	—
266	M₁₆	—	118	1	no	—
267	M₁₆	C₈×D₅₉	1	2	yes	complemented
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	4	2	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	118	1	no	—
275	C₂×C₈	—	118	2	no	—
276	C₂×C₈	—	118	1	no	—
277	C₂×C₈	—	118	1	no	—
278	C₂×C₈	—	118	1	no	—
279	C₂×C₈	—	472	2	no	—
280	C₂×C₈	—	472	2	no	—
281	C₂×C₈	—	236	1	no	—
282	C₂×C₈	—	236	1	no	—
283	C₄⋊C₄	—	118	1	no	—
284	C₄⋊C₄	—	118	1	no	—
285	C₂²⋊C₄	—	118	2	no	—
286	C₂²⋊C₄	—	118	2	no	—
287	C₄²	#11745685	1	1	yes	characteristic
288	C₄²	D₄×D₅₉	1	1	yes	characteristic
289	C₄²	#11745689	1	1	yes	characteristic
290	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
291	C₂³	—	59	4	no	—
292	C₂³	—	118	2	no	—
293	C₂³	—	118	2	no	—
294	C₂³	—	236	2	no	—
295	C₂³	—	236	2	no	—
296	C₂³	C₄⋊C₄×D₅₉	1	1	yes	characteristic
297	D₄	—	118	2	no	—
298	D₄	—	118	2	no	—
299	D₄	—	118	2	no	—
300	D₄	—	118	2	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	118	2	no	—
305	C₂×C₄	—	472	2	no	—
306	C₂×C₄	—	236	2	no	—
307	C₂×C₄	—	236	2	no	—
308	C₂×C₄	—	236	2	no	—
309	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
310	C₂×C₄	#14273689	1	1	yes	characteristic
311	C₂×C₄	#14273695	1	1	yes	characteristic
312	C₂×C₄	#14273703	1	1	yes	characteristic
313	C₂×C₄	#14273705	1	1	yes	characteristic
314	C₂×C₄	#14273712	1	1	yes	characteristic
315	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
316	C₂×C₄	M₁₆×D₅₉	1	1	yes	characteristic
317	C₂×C₄	#14273733	1	1	yes	characteristic
318	C₂×C₄	#14273735	1	1	yes	characteristic
319	C₈	—	2	2	no	—
320	C₈	—	4	1	no	—
321	C₈	—	4	1	no	—
322	C₈	—	4	1	no	—
323	C₈	—	8	1	no	—
324	C₈	—	118	2	no	—
325	C₈	—	236	2	no	—
326	C₈	—	236	1	no	—
327	C₈	—	236	1	no	—
328	C₈	—	236	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	118	4	no	—
332	C₂²	—	472	2	no	—
333	C₂²	—	236	4	no	—
334	C₂²	—	236	2	no	—
335	C₂²	—	236	2	no	—
336	C₂²	#16837273	1	1	yes	characteristic
337	C₂²	C₄⋊C₈×D₅₉	1	1	yes	characteristic
338	C₂²	#16837427	1	1	yes	characteristic
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	236	2	no	—
343	C₄	C₈×C₂²⋊D₅₉	1	1	yes	characteristic
344	C₄	#16837467	1	1	yes	characteristic
345	C₄	#16837471	1	1	yes	characteristic
346	C₄	#16837472	1	1	yes	characteristic
347	C₄	#16837568	1	1	yes	characteristic
348	C₄	#16837578	1	1	yes	characteristic
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	236	4	no	—
352	C₂	#21424272	1	1	yes	characteristic
353	C₂	#21424304	1	1	yes	characteristic
354	C₂	#21424314	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14273695
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1565	C₂
socle	C₂²×C₅₉	#385
prime radical	C₂²×D₂₃₆	C₂×C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1565	D₅₉
hypocenter	C₅₉	#8835
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1565

Complements

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

N	H ≅ G/N	type
4. #21422846	C₂	semidirect
5. #21421934	C₂	semidirect
6. #21421932	C₂	semidirect
7. C₅₉×#1565	C₂	semidirect
45. #14273643	C₈	semidirect
85. M₁₆×C₅₉	C₂×C₈	semidirect
103. #11745664	C₂×C₈	semidirect
176. #1565	D₅₉	semidirect
206. C₅₉	#8835	semidirect
267. M₁₆	C₈×D₅₉	semidirect

Maximal subgroups

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

subgroup	index
#21433119	2
#21424889	2
C₄×#14273643	2
#21422846	2
#21421934	2
#21421932	2
C₅₉×#1565	2
#8835	59