Subgroups — #28560681

Order 15104.

572 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21434679	C₂	1	1	yes	maximal, characteristic
2	#21433194	C₂	1	2	yes	maximal
3	#21422992	C₂	1	1	yes	maximal, characteristic, complemented
4	#21422965	C₂	1	2	yes	maximal, complemented
5	#21422964	C₂	1	2	yes	maximal, complemented
6	C₈×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₁₆×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#21422843	C₂	1	1	yes	maximal, characteristic
9	#21422515	C₂	1	2	yes	maximal
10	C₈×#11745659	C₂	1	1	yes	maximal, characteristic
11	C₅₉×#1591	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837956	C₂²	1	2	yes	—
13	#16837806	C₄	1	1	yes	characteristic
14	#16837780	C₄	1	1	yes	characteristic
15	#16837751	C₄	1	2	yes	—
16	#16837644	C₄	1	1	yes	characteristic
17	#16837642	C₄	1	2	yes	—
18	#16837640	C₄	1	1	yes	characteristic
19	#16837530	—	2	2	no	—
20	#16837529	C₂²	1	2	yes	—
21	Q₈×C₅₉⋊C₈	C₂²	1	2	yes	—
22	C₈×Dic₁₁₈	C₂²	1	2	yes	—
23	#16837526	—	2	2	no	—
24	C₄×Dic₂₃₆	C₂²	1	1	yes	characteristic
25	C₄×#11745672	C₂²	1	2	yes	—
26	#16837464	—	2	2	no	—
27	#16837462	—	2	2	no	—
28	#16837454	C₂²	1	2	yes	—
29	#16837453	C₂²	1	1	yes	characteristic
30	C₄×#11745659	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₅₉	C₂²	1	1	yes	characteristic
32	#16837155	C₂²	1	1	yes	characteristic
33	#16837152	C₂²	1	2	yes	—
34	#16837134	C₂²	1	2	yes	—
35	Q₈⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
36	#16837132	C₂²	1	2	yes	—
37	#16837124	C₂²	1	1	yes	characteristic
38	#16837123	C₂²	1	1	yes	characteristic
39	C₈×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
40	C₅₉×#446	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
42	C₅₉×#335	C₂²	1	1	yes	characteristic
43	C₅₉×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₈²×C₅₉	C₂²	1	1	yes	characteristic
45	C₂×Dic₂₃₆	C₂×C₄	1	1	yes	characteristic
46	C₂×#11745672	C₂×C₄	1	2	yes	—
47	C₂×#11745659	—	2	1	no	—
48	C₂×C₈×D₅₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₅₉	C₂×C₄	1	1	yes	characteristic
50	#14273726	C₂×C₄	1	2	yes	—
51	#14273725	C₂×C₄	1	2	yes	—
52	#14273717	—	2	2	no	—
53	#14273707	C₂×C₄	1	2	yes	—
54	#14273705	—	2	2	no	—
55	#14273703	C₂×C₄	1	2	yes	—
56	C₄×Dic₁₁₈	—	2	2	no	—
57	C₄×Dic₁₁₈	C₂³	1	2	yes	—
58	#14273684	—	2	2	no	—
59	C₄²×D₅₉	C₂³	1	1	yes	characteristic
60	#14273661	C₂×C₄	1	1	yes	characteristic
61	#14273660	C₂×C₄	1	2	yes	—
62	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
63	C₄⋊(C₅₉⋊C₈)	C₂³	1	2	yes	—
64	Dic₅₉⋊C₈	—	2	2	no	—
65	Dic₅₉⋊C₈	—	2	2	no	—
66	Dic₅₉⋊C₈	D₄	1	1	yes	characteristic
67	Dic₅₉⋊C₈	D₄	1	1	yes	characteristic
68	#14273652	C₂×C₄	1	2	yes	—
69	#14273651	C₂×C₄	1	2	yes	—
70	#14273650	C₂×C₄	1	2	yes	—
71	(C₂×D₅₉)⋊C₈	—	2	2	no	—
72	(C₂×D₅₉)⋊C₈	D₄	1	1	yes	characteristic
73	(C₂×D₅₉)⋊C₈	D₄	1	1	yes	characteristic
74	#14273637	—	2	1	no	—
75	#14273637	—	2	2	no	—
76	C₄×C₅₉⋊C₈	—	2	2	no	—
77	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
78	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
79	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
80	C₈×Dic₅₉	—	2	2	no	—
81	C₈×Dic₅₉	—	2	2	no	—
82	C₈×Dic₅₉	C₂×C₄	1	1	yes	characteristic
83	C₈×Dic₅₉	C₂×C₄	1	1	yes	characteristic
84	C₂×Q₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈×C₅₉	C₂³	1	2	yes	—
86	#107×C₅₉	C₂×C₄	1	1	yes	characteristic
87	C₄⋊C₈×C₅₉	—	2	2	no	—
88	C₄⋊C₈×C₅₉	C₂³	1	2	yes	—
89	#103×C₅₉	C₂×C₄	1	2	yes	—
90	#97×C₅₉	—	2	2	no	—
91	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₅₉	C₂²×C₄	1	2	yes	—
95	C₂×C₄×D₅₉	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×D₅₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×D₅₉	#41	1	1	yes	characteristic
98	C₂×Dic₁₁₈	—	2	2	no	—
99	C₂×Dic₁₁₈	C₂²×C₄	1	2	yes	—
100	Q₁₆×C₅₉	—	2	2	no	—
101	C₂×C₈×C₅₉	—	2	1	no	—
102	C₂×C₈×C₅₉	—	2	2	no	—
103	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
107	C₄⋊C₄×C₅₉	—	2	2	no	—
108	C₄⋊C₄×C₅₉	C₂²×C₄	1	2	yes	—
109	C₄²×C₅₉	—	2	2	no	—
110	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
111	#11745672	—	4	2	no	—
112	(C₂×D₅₉)⋊C₄	—	2	2	no	—
113	(C₂×D₅₉)⋊C₄	—	2	2	no	—
114	C₄⋊Dic₅₉	—	2	2	no	—
115	C₄⋊Dic₅₉	C₂²×C₄	1	2	yes	—
116	Dic₅₉⋊C₄	—	2	2	no	—
117	Dic₅₉⋊C₄	—	2	2	no	—
118	Dic₅₉⋊C₄	—	2	2	no	—
119	Dic₅₉⋊C₄	—	2	2	no	—
120	Dic₅₉⋊C₄	—	2	2	no	—
121	C₄×Dic₅₉	—	2	2	no	—
122	C₄×Dic₅₉	C₂²×C₄	1	1	yes	characteristic
123	C₄×Dic₅₉	C₂×D₄	1	1	yes	characteristic
124	C₄×Dic₅₉	#41	1	1	yes	characteristic
125	C₂×C₅₉⋊C₈	—	2	1	no	—
126	C₂×C₅₉⋊C₈	—	2	2	no	—
127	C₂×C₅₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₅₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
130	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
131	Dic₂₃₆	—	2	2	no	—
132	#11745659	—	2	4	no	—
133	C₈×D₅₉	—	2	2	no	—
134	C₈×D₅₉	—	4	1	no	—
135	C₂²×D₅₉	C₄×D₄	1	1	yes	characteristic
136	Q₈×C₅₉	—	2	2	no	—
137	Q₈×C₅₉	—	2	2	no	—
138	C₂×C₄×C₅₉	—	2	2	no	—
139	C₂×C₄×C₅₉	—	2	2	no	—
140	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
143	C₂×Dic₅₉	—	2	2	no	—
144	C₂×Dic₅₉	—	2	2	no	—
145	C₂×Dic₅₉	C₄×D₄	1	1	yes	characteristic
146	C₂×Dic₅₉	C₄×D₄	1	1	yes	characteristic
147	C₂×Dic₅₉	C₄×D₄	1	1	yes	characteristic
148	C₄×D₅₉	—	2	1	no	—
149	C₄×D₅₉	—	2	1	no	—
150	C₄×D₅₉	—	2	1	no	—
151	C₄×D₅₉	—	2	1	no	—
152	C₄×D₅₉	—	2	2	no	—
153	Dic₁₁₈	—	2	2	no	—
154	Dic₁₁₈	—	2	2	no	—
155	Dic₁₁₈	—	4	4	no	—
156	C₈×C₅₉	—	2	2	no	—
157	C₈×C₅₉	—	2	1	no	—
158	C₈×C₅₉	—	2	1	no	—
159	C₈×C₅₉	—	2	1	no	—
160	C₈×C₅₉	—	4	2	no	—
161	C₈×C₅₉	#131	1	2	yes	—
162	C₅₉⋊C₈	—	2	2	no	—
163	C₅₉⋊C₈	—	2	1	no	—
164	C₅₉⋊C₈	—	2	1	no	—
165	C₅₉⋊C₈	—	2	1	no	—
166	C₅₉⋊C₈	—	2	1	no	—
167	C₅₉⋊C₈	—	4	2	no	—
168	#18086	—	59	1	no	maximal, Sylow
169	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂×D₅₉	—	2	1	no	—
171	C₂×D₅₉	—	2	1	no	—
172	C₂×D₅₉	—	2	1	no	—
173	C₄×C₅₉	—	2	1	no	—
174	C₄×C₅₉	—	4	2	no	—
175	C₄×C₅₉	—	4	2	no	—
176	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
177	C₄×C₅₉	#568	1	1	yes	characteristic
178	C₄×C₅₉	#578	1	1	yes	characteristic
179	C₄×C₅₉	#578	1	1	yes	characteristic
180	Dic₅₉	—	2	1	no	—
181	Dic₅₉	—	2	1	no	—
182	Dic₅₉	—	4	2	no	—
183	Dic₅₉	—	4	2	no	—
184	Dic₅₉	#443	1	2	yes	—
185	#3170	—	59	1	no	—
186	#2986	—	59	2	no	—
187	#1591	—	59	1	no	—
188	#1591	D₅₉	1	1	yes	characteristic, complemented
189	#1587	—	59	2	no	—
190	#1587	—	59	2	no	—
191	C₈×Q₁₆	—	59	1	no	—
192	C₈×Q₁₆	—	59	1	no	—
193	#1564	—	59	1	no	—
194	#1485	—	59	2	no	—
195	C₈×M₁₆	—	59	1	no	—
196	C₂×C₅₉	#2935	1	1	yes	characteristic
197	C₂×C₅₉	#2950	1	1	yes	characteristic
198	C₂×C₅₉	C₂×#443	1	1	yes	characteristic
199	D₅₉	—	4	1	no	—
200	#533	—	59	2	no	—
201	#500	—	59	1	no	—
202	#494	—	59	1	no	—
203	#487	—	59	2	no	—
204	#467	—	59	1	no	—
205	#467	—	59	1	no	—
206	#467	—	59	2	no	—
207	#446	—	59	2	no	—
208	#446	—	118	2	no	—
209	#446	C₂×D₅₉	1	2	yes	—
210	C₈×Q₈	—	59	2	no	—
211	C₈×Q₈	—	59	2	no	—
212	C₈×Q₈	—	118	2	no	—
213	C₄×Q₁₆	—	59	2	no	—
214	C₄×Q₁₆	—	59	1	no	—
215	C₄×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
216	#433	—	118	2	no	—
217	#433	—	118	2	no	—
218	#431	—	59	1	no	—
219	#431	—	59	2	no	—
220	C₄×M₁₆	—	59	1	no	—
221	C₂×C₄×C₈	—	59	1	no	—
222	#335	—	59	1	no	—
223	#335	C₂×D₅₉	1	1	yes	characteristic
224	#334	—	59	2	no	—
225	Q₈⋊C₈	—	59	2	no	—
226	Q₈⋊C₈	—	59	2	no	—
227	Q₈⋊C₈	—	59	2	no	—
228	Q₈⋊C₈	C₂×D₅₉	1	2	yes	—
229	#322	—	59	1	no	—
230	#322	—	59	1	no	—
231	C₈²	—	59	1	no	—
232	C₈²	C₂×D₅₉	1	1	yes	characteristic
233	C₅₉	#18086	1	1	yes	characteristic, complemented, Sylow
234	C₂×Q₁₆	—	59	2	no	—
235	C₂×Q₁₆	—	59	1	no	—
236	C₂×Q₁₆	C₄×D₅₉	1	1	yes	characteristic
237	C₂×M₁₆	—	118	1	no	—
238	C₂²×C₈	—	59	1	no	—
239	C₂²×C₈	—	59	1	no	—
240	C₄⋊Q₈	—	59	2	no	—
241	C₄⋊Q₈	—	59	2	no	—
242	#125	—	118	2	no	—
243	C₂²⋊Q₈	—	59	2	no	—
244	C₂²⋊Q₈	—	59	2	no	—
245	C₂²⋊Q₈	—	118	2	no	—
246	C₄×Q₈	—	59	2	no	—
247	C₄×Q₈	—	118	2	no	—
248	C₄×Q₈	C₂²×D₅₉	1	2	yes	—
249	#117	—	118	2	no	—
250	C₂×C₄²	—	59	1	no	—
251	#107	—	59	1	no	—
252	#107	—	59	2	no	—
253	#107	C₄×D₅₉	1	1	yes	characteristic
254	C₄⋊C₈	—	2	2	no	—
255	C₄⋊C₈	—	59	1	no	—
256	C₄⋊C₈	—	59	1	no	—
257	C₄⋊C₈	—	59	2	no	—
258	C₄⋊C₈	—	118	2	no	—
259	C₄⋊C₈	—	118	2	no	—
260	C₄⋊C₈	—	118	2	no	—
261	C₄⋊C₈	C₂²×D₅₉	1	2	yes	—
262	#103	—	59	2	no	—
263	#103	—	59	2	no	—
264	#103	—	59	2	no	—
265	#103	C₄×D₅₉	1	2	yes	—
266	C₂²⋊C₈	—	59	1	no	—
267	C₂²⋊C₈	—	59	1	no	—
268	C₂²⋊C₈	—	118	2	no	—
269	#97	—	2	2	no	—
270	#97	—	118	1	no	—
271	#97	—	118	2	no	—
272	C₄×C₈	—	59	1	no	—
273	C₄×C₈	—	59	1	no	—
274	C₄×C₈	—	59	1	no	—
275	C₄×C₈	—	59	1	no	—
276	C₄×C₈	—	59	1	no	—
277	C₄×C₈	—	118	2	no	—
278	C₄×C₈	—	118	2	no	—
279	C₄×C₈	—	118	2	no	—
280	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
281	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
282	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
283	C₂×Q₈	—	59	2	no	—
284	C₂×Q₈	—	118	2	no	—
285	C₂×Q₈	C₂×C₄×D₅₉	1	2	yes	—
286	C₂²×C₄	—	59	1	no	—
287	C₂²×C₄	—	59	1	no	—
288	C₂²×C₄	—	59	1	no	—
289	Q₁₆	—	2	2	no	—
290	Q₁₆	—	118	2	no	—
291	Q₁₆	—	236	2	no	—
292	M₁₆	—	118	4	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	59	1	no	—
296	C₂×C₈	—	59	1	no	—
297	C₂×C₈	—	59	1	no	—
298	C₂×C₈	—	59	1	no	—
299	C₂×C₈	—	118	1	no	—
300	C₂×C₈	—	118	2	no	—
301	C₂×C₈	—	118	2	no	—
302	C₂×C₈	—	236	1	no	—
303	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
304	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
305	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
306	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	59	2	no	—
309	C₄⋊C₄	—	118	2	no	—
310	C₄⋊C₄	—	118	2	no	—
311	C₄⋊C₄	—	118	2	no	—
312	C₄⋊C₄	—	118	2	no	—
313	C₄⋊C₄	—	118	2	no	—
314	C₄⋊C₄	—	118	2	no	—
315	C₄⋊C₄	C₂×C₄×D₅₉	1	2	yes	—
316	C₂²⋊C₄	—	118	2	no	—
317	C₂²⋊C₄	—	118	2	no	—
318	C₄²	—	2	2	no	—
319	C₄²	—	59	1	no	—
320	C₄²	—	59	1	no	—
321	C₄²	—	59	1	no	—
322	C₄²	—	118	2	no	—
323	C₄²	C₂³×D₅₉	1	1	yes	characteristic
324	C₂³	—	59	1	no	—
325	Q₈	—	2	2	no	—
326	Q₈	—	2	2	no	—
327	Q₈	—	118	2	no	—
328	Q₈	—	118	2	no	—
329	Q₈	—	236	4	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	59	1	no	—
333	C₂×C₄	—	59	1	no	—
334	C₂×C₄	—	59	1	no	—
335	C₂×C₄	—	118	2	no	—
336	C₂×C₄	—	118	1	no	—
337	C₂×C₄	—	118	1	no	—
338	C₂×C₄	—	118	1	no	—
339	C₂×C₄	—	118	1	no	—
340	C₂×C₄	—	118	2	no	—
341	C₂×C₄	—	118	2	no	—
342	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
343	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
344	C₂×C₄	#41×D₅₉	1	1	yes	characteristic
345	C₈	—	2	2	no	—
346	C₈	—	2	1	no	—
347	C₈	—	2	1	no	—
348	C₈	—	2	1	no	—
349	C₈	—	4	2	no	—
350	C₈	—	118	2	no	—
351	C₈	—	118	1	no	—
352	C₈	—	118	1	no	—
353	C₈	—	118	1	no	—
354	C₈	—	118	1	no	—
355	C₈	—	236	2	no	—
356	C₈	#14273735	1	2	yes	—
357	C₂²	—	118	1	no	—
358	C₂²	—	118	1	no	—
359	C₂²	—	118	1	no	—
360	C₂²	C₄×D₄×D₅₉	1	1	yes	characteristic
361	C₄	—	2	1	no	—
362	C₄	—	4	2	no	—
363	C₄	—	4	2	no	—
364	C₄	—	59	2	no	—
365	C₄	—	118	1	no	—
366	C₄	—	118	1	no	—
367	C₄	—	236	2	no	—
368	C₄	—	236	2	no	—
369	C₄	C₂×#14273735	1	1	yes	characteristic
370	C₄	#16838167	1	1	yes	characteristic
371	C₄	#16838215	1	1	yes	characteristic
372	C₄	#16838216	1	1	yes	characteristic
373	C₂	—	236	1	no	—
374	C₂	#21432841	1	1	yes	characteristic
375	C₂	#21432959	1	1	yes	characteristic
376	C₂	#443×D₅₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₄²	C₂³×D₅₉
Fitting	C₅₉×#1591	C₂
socle	C₂²×C₅₉	C₂×C₄×D₄
prime radical	C₂²×D₅₉	C₄×D₄
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	#1591	D₅₉
hypocenter	C₅₉	#18086
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1591

Complements

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

N	H ≅ G/N	type
3. #21422992	C₂	semidirect
4. #21422965	C₂	semidirect
5. #21422964	C₂	semidirect
6. C₈×Dic₂₃₆	C₂	semidirect
7. Q₁₆×C₅₉⋊C₈	C₂	semidirect
11. C₅₉×#1591	C₂	semidirect
188. #1591	D₅₉	semidirect
233. C₅₉	#18086	semidirect

Maximal subgroups

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

subgroup	index
#21434679	2
#21433194	2
#21422992	2
#21422965	2
#21422964	2
C₈×Dic₂₃₆	2
Q₁₆×C₅₉⋊C₈	2
#21422843	2
#21422515	2
C₈×#11745659	2
C₅₉×#1591	2
#18086	59