Subgroups — #28596858

Order 15104.

722 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21433037	C₂	1	1	yes	maximal, characteristic, complemented
2	#21432814	C₂	1	1	yes	maximal, characteristic
3	#21425078	C₂	1	1	yes	maximal, characteristic, complemented
4	#21424929	C₂	1	1	yes	maximal, characteristic
5	#21424809	C₂	1	2	yes	maximal, complemented
6	#21424802	C₂	1	2	yes	maximal, complemented
7	#21424234	C₂	1	1	yes	maximal, characteristic, complemented
8	#21423943	C₂	1	2	yes	maximal, complemented
9	#21423936	C₂	1	2	yes	maximal, complemented
10	#21423782	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₉×#3131	C₂	1	1	yes	maximal, characteristic
12	C₂×#14273734	C₂²	1	2	yes	—
13	#16837533	—	2	4	no	—
14	#16837474	—	2	4	no	—
15	#16837429	C₂²	1	2	yes	—
16	#16837426	C₂²	1	1	yes	characteristic
17	#16837426	C₂²	1	1	yes	characteristic
18	C₂×#14273654	C₂²	1	2	yes	—
19	C₂×#14273654	C₂²	1	2	yes	—
20	#16837369	C₂²	1	2	yes	—
21	#16837358	C₂²	1	2	yes	—
22	C₂×#14273638	C₂²	1	1	yes	characteristic
23	C₂×#14273637	C₂²	1	1	yes	characteristic
24	C₂×#14273637	C₂²	1	1	yes	characteristic
25	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
26	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
27	#16837248	C₂²	1	1	yes	characteristic
28	#16837167	C₂²	1	2	yes	—
29	#16837167	C₂²	1	2	yes	—
30	#16837161	C₂²	1	2	yes	—
31	#16837161	C₂²	1	2	yes	—
32	C₅₉×#533	C₂²	1	1	yes	characteristic
33	C₅₉×#532	C₂²	1	1	yes	characteristic
34	C₅₉×#488	C₄	1	4	yes	complemented
35	C₅₉×#487	C₄	1	2	yes	complemented
36	C₅₉×#486	C₄	1	2	yes	complemented
37	C₅₉×#416	C₂²	1	2	yes	—
38	C₅₉×#416	C₂²	1	2	yes	—
39	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
40	#14273734	—	2	4	no	—
41	#14273734	—	2	4	no	—
42	C₂×#11745664	—	2	2	no	—
43	C₂×#11745664	C₂³	1	2	yes	—
44	C₂×#11745664	C₂³	1	2	yes	—
45	C₂²×C₅₉⋊C₈	—	2	2	no	—
46	C₂²×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
47	C₂²×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
48	C₂×C₄×Dic₅₉	C₂³	1	1	yes	characteristic
49	C₂×C₄×Dic₅₉	C₂³	1	1	yes	characteristic
50	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
51	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
52	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
53	#14273654	—	2	4	no	—
54	#14273654	—	2	4	no	—
55	C₂²⋊(C₅₉⋊C₈)	—	2	4	no	—
56	#14273638	—	2	2	no	—
57	#14273638	D₄	1	4	yes	—
58	#14273637	—	2	2	no	—
59	#14273637	—	2	2	no	—
60	C₄×C₅₉⋊C₈	—	2	2	no	—
61	C₈×Dic₅₉	—	2	2	no	—
62	C₈×Dic₅₉	—	2	2	no	—
63	C₅₉⋊#95	—	2	2	no	—
64	C₂×#41×C₅₉	C₂³	1	2	yes	—
65	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
67	C₄⋊Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
68	C₄⋊Q₈×C₅₉	C₂×C₄	1	2	yes	—
69	#127×C₅₉	C₂×C₄	1	1	yes	characteristic
70	#125×C₅₉	C₂×C₄	1	2	yes	—
71	#124×C₅₉	C₂×C₄	1	2	yes	—
72	C₂²⋊Q₈×C₅₉	—	2	2	no	—
73	#121×C₅₉	—	2	2	no	—
74	C₄×Q₈×C₅₉	—	2	2	no	—
75	C₄×D₄×C₅₉	—	2	2	no	—
76	#117×C₅₉	C₂³	1	2	yes	—
77	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
78	#103×C₅₉	C₂×C₄	1	4	yes	—
79	#103×C₅₉	C₂×C₄	1	4	yes	—
80	#102×C₅₉	C₂×C₄	1	4	yes	—
81	#102×C₅₉	C₂×C₄	1	4	yes	—
82	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
83	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
84	#41×C₅₉	—	2	2	no	—
85	#41×C₅₉	—	2	2	no	—
86	#41×C₅₉	—	2	2	no	—
87	#41×C₅₉	—	2	2	no	—
88	C₂×Q₈×C₅₉	—	2	2	no	—
89	C₂×Q₈×C₅₉	C₂²×C₄	1	2	yes	—
90	C₂×D₄×C₅₉	—	2	2	no	—
91	C₂×D₄×C₅₉	—	2	2	no	—
92	C₂×D₄×C₅₉	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₅₉	—	2	2	no	—
94	C₂²×C₄×C₅₉	#41	1	2	yes	—
95	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
96	C₂²×Dic₅₉	—	2	1	no	—
97	C₂²×Dic₅₉	—	2	1	no	—
98	C₂×C₈×C₅₉	—	2	1	no	—
99	C₂×C₈×C₅₉	—	2	1	no	—
100	C₂×C₈×C₅₉	—	2	1	no	—
101	C₂×C₈×C₅₉	—	2	1	no	—
102	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
104	C₄⋊C₄×C₅₉	—	2	2	no	—
105	C₄⋊C₄×C₅₉	—	2	4	no	—
106	C₄⋊C₄×C₅₉	—	2	2	no	—
107	C₄⋊C₄×C₅₉	C₂²×C₄	1	4	yes	—
108	C₂²⋊C₄×C₅₉	—	2	2	no	—
109	C₂²⋊C₄×C₅₉	—	2	4	no	—
110	C₄²×C₅₉	—	2	2	no	—
111	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₅₉	C₂×D₄	1	2	yes	—
114	C₄×Dic₅₉	—	2	1	no	—
115	C₄×Dic₅₉	—	2	1	no	—
116	C₄×Dic₅₉	#41	1	2	yes	—
117	C₄×Dic₅₉	#41	1	2	yes	—
118	#11745664	—	2	2	no	—
119	#11745664	—	2	2	no	—
120	#11745664	—	2	2	no	—
121	#11745664	—	2	2	no	—
122	#11745664	—	4	4	no	—
123	C₂×C₅₉⋊C₈	—	2	1	no	—
124	C₂×C₅₉⋊C₈	—	2	1	no	—
125	C₂×C₅₉⋊C₈	—	2	2	no	—
126	C₂×C₅₉⋊C₈	—	2	2	no	—
127	C₂×C₅₉⋊C₈	—	4	4	no	—
128	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
129	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
130	C₂×C₅₉⋊C₈	#41	1	2	yes	—
131	C₂×C₅₉⋊C₈	#41	1	2	yes	—
132	C₂³×C₅₉	—	2	2	no	—
133	C₂³×C₅₉	C₂×#41	1	1	yes	characteristic
134	Q₈×C₅₉	—	2	2	no	—
135	Q₈×C₅₉	—	2	2	no	—
136	Q₈×C₅₉	—	8	2	no	—
137	D₄×C₅₉	—	2	2	no	—
138	D₄×C₅₉	—	2	2	no	—
139	D₄×C₅₉	—	4	2	no	—
140	D₄×C₅₉	—	4	2	no	—
141	D₄×C₅₉	—	8	2	no	—
142	C₂×C₄×C₅₉	—	2	4	no	—
143	C₂×C₄×C₅₉	—	2	2	no	—
144	C₂×C₄×C₅₉	—	4	2	no	—
145	C₂×C₄×C₅₉	—	4	2	no	—
146	C₂×C₄×C₅₉	—	4	2	no	—
147	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
148	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
149	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
154	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
155	C₂×Dic₅₉	—	2	2	no	—
156	C₂×Dic₅₉	—	2	2	no	—
157	C₂×Dic₅₉	—	4	2	no	—
158	C₂×Dic₅₉	—	4	2	no	—
159	C₈×C₅₉	—	2	2	no	—
160	C₈×C₅₉	—	2	2	no	—
161	C₅₉⋊C₈	—	2	2	no	—
162	C₅₉⋊C₈	—	4	2	no	—
163	C₅₉⋊C₈	—	4	2	no	—
164	C₅₉⋊C₈	#124	1	4	yes	—
165	#21659	—	59	1	no	maximal, Sylow
166	C₂²×C₅₉	—	2	2	no	—
167	C₂²×C₅₉	—	4	4	no	—
168	C₂²×C₅₉	—	4	2	no	—
169	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂²×C₅₉	#529	1	1	yes	characteristic
171	C₂²×C₅₉	#529	1	1	yes	characteristic
172	C₄×C₅₉	—	2	2	no	—
173	C₄×C₅₉	—	2	2	no	—
174	C₄×C₅₉	—	4	4	no	—
175	C₄×C₅₉	—	4	2	no	—
176	C₄×C₅₉	C₄×#41	1	1	yes	characteristic
177	C₄×C₅₉	C₄×#41	1	1	yes	characteristic
178	C₄×C₅₉	C₂×#124	1	1	yes	characteristic
179	C₄×C₅₉	#532	1	1	yes	characteristic
180	Dic₅₉	—	4	2	no	—
181	Dic₅₉	—	4	2	no	—
182	#3131	D₅₉	1	1	yes	characteristic
183	#2961	—	59	1	no	—
184	#2932	—	59	1	no	—
185	#1995	—	59	1	no	—
186	#1963	—	59	1	no	—
187	#1938	—	59	2	no	—
188	#1937	—	59	2	no	—
189	#1837	—	59	1	no	—
190	#1765	—	59	2	no	—
191	#1764	—	59	2	no	—
192	#1726	—	59	1	no	—
193	C₂×C₅₉	—	2	1	no	—
194	C₂×C₅₉	—	2	1	no	—
195	C₂×C₅₉	—	8	2	no	—
196	C₂×C₅₉	C₄×#124	1	1	yes	characteristic
197	C₂×C₅₉	#2957	1	2	yes	—
198	C₂×#131	—	59	2	no	—
199	#533	C₂×D₅₉	1	1	yes	characteristic
200	#532	C₂×D₅₉	1	1	yes	characteristic
201	#488	Dic₅₉	1	4	yes	complemented
202	#487	Dic₅₉	1	2	yes	complemented
203	#486	Dic₅₉	1	2	yes	complemented
204	#446	—	118	4	no	—
205	#435	—	118	4	no	—
206	#424	—	59	2	no	—
207	#423	—	59	1	no	—
208	#423	—	59	1	no	—
209	C₂×#104	—	59	2	no	—
210	C₂×#104	—	59	2	no	—
211	#416	C₂×D₅₉	1	2	yes	—
212	#416	C₂×D₅₉	1	2	yes	—
213	#408	—	59	2	no	—
214	#405	—	59	2	no	—
215	C₂×#97	—	59	1	no	—
216	C₂×#97	—	59	1	no	—
217	C₂×#97	—	59	1	no	—
218	C₂×C₄×C₈	—	59	1	no	—
219	C₂×C₄×C₈	—	59	1	no	—
220	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
221	#376	—	59	1	no	—
222	#341	—	59	2	no	—
223	#341	—	59	2	no	—
224	#339	—	59	2	no	—
225	#339	—	59	2	no	—
226	C₅₉	#21659	1	1	yes	characteristic, complemented, Sylow
227	C₂×#41	C₂²×D₅₉	1	2	yes	—
228	#131	—	118	4	no	—
229	#131	—	118	4	no	—
230	C₂×M₁₆	—	59	2	no	—
231	C₂×M₁₆	—	59	2	no	—
232	C₂×M₁₆	—	118	2	no	—
233	C₂²×C₈	—	59	1	no	—
234	C₂²×C₈	—	59	1	no	—
235	C₂²×C₈	—	118	2	no	—
236	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
237	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
238	C₄⋊Q₈	C₂×Dic₅₉	1	1	yes	characteristic
239	C₄⋊Q₈	C₂×Dic₅₉	1	2	yes	—
240	#127	C₂×Dic₅₉	1	1	yes	characteristic
241	#125	C₂×Dic₅₉	1	2	yes	—
242	#124	C₂×Dic₅₉	1	2	yes	—
243	C₂²⋊Q₈	—	2	2	no	—
244	#121	—	2	2	no	—
245	C₄×Q₈	—	2	2	no	—
246	C₄×D₄	—	2	2	no	—
247	#117	C₂²×D₅₉	1	2	yes	—
248	C₂×C₄²	—	59	1	no	—
249	C₂×C₄²	—	59	1	no	—
250	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
251	C₄⋊C₈	—	118	4	no	—
252	C₄⋊C₈	—	118	2	no	—
253	C₄⋊C₈	—	118	2	no	—
254	#104	—	118	4	no	—
255	#104	—	118	4	no	—
256	#103	C₂×Dic₅₉	1	4	yes	—
257	#103	C₂×Dic₅₉	1	4	yes	—
258	#102	C₂×Dic₅₉	1	4	yes	—
259	#102	C₂×Dic₅₉	1	4	yes	—
260	C₂²⋊C₈	—	118	4	no	—
261	#97	—	59	4	no	—
262	#97	—	118	2	no	—
263	#97	—	118	2	no	—
264	#97	—	118	2	no	—
265	C₄×C₈	—	118	2	no	—
266	C₄×C₈	—	118	2	no	—
267	C₄×C₈	—	118	2	no	—
268	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
269	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
270	#95	—	118	2	no	—
271	#41	—	2	2	no	—
272	#41	—	2	2	no	—
273	#41	—	2	2	no	—
274	#41	—	2	2	no	—
275	C₂×Q₈	—	2	2	no	—
276	C₂×Q₈	C₂²×Dic₅₉	1	2	yes	—
277	C₂×D₄	—	2	2	no	—
278	C₂×D₄	—	2	2	no	—
279	C₂×D₄	C₂²×Dic₅₉	1	2	yes	—
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	118	1	no	—
282	C₂²×C₄	—	118	1	no	—
283	C₂²×C₄	#11745687	1	2	yes	—
284	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
285	M₁₆	—	118	2	no	—
286	M₁₆	—	118	2	no	—
287	M₁₆	—	118	2	no	—
288	M₁₆	—	118	2	no	—
289	M₁₆	—	236	4	no	—
290	C₂×C₈	—	2	1	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	59	2	no	—
295	C₂×C₈	—	59	2	no	—
296	C₂×C₈	—	59	2	no	—
297	C₂×C₈	—	59	2	no	—
298	C₂×C₈	—	118	1	no	—
299	C₂×C₈	—	118	1	no	—
300	C₂×C₈	—	118	2	no	—
301	C₂×C₈	—	118	2	no	—
302	C₂×C₈	—	236	4	no	—
303	C₂×C₈	C₂²×Dic₅₉	1	2	yes	—
304	C₂×C₈	C₂²×Dic₅₉	1	2	yes	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	2	4	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	C₂²×Dic₅₉	1	4	yes	—
309	C₂²⋊C₄	—	2	2	no	—
310	C₂²⋊C₄	—	2	4	no	—
311	C₄²	—	2	2	no	—
312	C₄²	—	59	2	no	—
313	C₄²	—	59	2	no	—
314	C₄²	—	118	1	no	—
315	C₄²	—	118	1	no	—
316	C₄²	D₄×D₅₉	1	2	yes	—
317	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
318	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
319	C₂³	—	2	2	no	—
320	C₂³	C₂×#11745687	1	1	yes	characteristic
321	Q₈	—	2	2	no	—
322	Q₈	—	2	2	no	—
323	Q₈	—	8	2	no	—
324	D₄	—	2	2	no	—
325	D₄	—	2	2	no	—
326	D₄	—	4	2	no	—
327	D₄	—	4	2	no	—
328	D₄	—	8	2	no	—
329	C₂×C₄	—	2	4	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	118	2	no	—
335	C₂×C₄	—	118	2	no	—
336	C₂×C₄	—	236	2	no	—
337	C₂×C₄	—	236	2	no	—
338	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
339	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
340	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
341	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
342	C₂×C₄	#41×D₅₉	1	1	yes	characteristic
343	C₂×C₄	#41×D₅₉	1	1	yes	characteristic
344	C₂×C₄	#41×D₅₉	1	1	yes	characteristic
345	C₂×C₄	#41×D₅₉	1	1	yes	characteristic
346	C₈	—	2	2	no	—
347	C₈	—	2	2	no	—
348	C₈	—	59	4	no	—
349	C₈	—	118	2	no	—
350	C₈	—	236	2	no	—
351	C₈	—	236	2	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	4	4	no	—
354	C₂²	—	4	2	no	—
355	C₂²	C₂×D₄×Dic₅₉	1	1	yes	characteristic
356	C₂²	#16837931	1	1	yes	characteristic
357	C₂²	#16837931	1	1	yes	characteristic
358	C₄	—	2	2	no	—
359	C₄	—	2	2	no	—
360	C₄	—	4	4	no	—
361	C₄	—	4	2	no	—
362	C₄	—	236	2	no	—
363	C₄	—	236	2	no	—
364	C₄	#41×Dic₅₉	1	1	yes	characteristic
365	C₄	#41×Dic₅₉	1	1	yes	characteristic
366	C₄	#124×D₅₉	1	1	yes	characteristic
367	C₄	#16837951	1	1	yes	characteristic
368	C₂	—	2	1	no	—
369	C₂	—	2	1	no	—
370	C₂	—	8	2	no	—
371	C₂	#124×Dic₅₉	1	1	yes	characteristic
372	C₂	#21433013	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₅₉×#3131	C₂
socle	C₂²×C₅₉	C₂×C₄×D₄
prime radical	C₅₉×#532	C₂²
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	#3131	D₅₉
hypocenter	C₅₉	#21659
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3131

Complements

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

N	H ≅ G/N	type
1. #21433037	C₂	semidirect
3. #21425078	C₂	semidirect
5. #21424809	C₂	semidirect
6. #21424802	C₂	semidirect
7. #21424234	C₂	semidirect
8. #21423943	C₂	semidirect
9. #21423936	C₂	semidirect
10. #21423782	C₂	semidirect
34. C₅₉×#488	C₄	semidirect
35. C₅₉×#487	C₄	semidirect
36. C₅₉×#486	C₄	semidirect
201. #488	Dic₅₉	semidirect
202. #487	Dic₅₉	semidirect
203. #486	Dic₅₉	semidirect
226. C₅₉	#21659	semidirect

Maximal subgroups

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

subgroup	index
#21433037	2
#21432814	2
#21425078	2
#21424929	2
#21424809	2
#21424802	2
#21424234	2
#21423943	2
#21423936	2
#21423782	2
C₅₉×#3131	2
#21659	59