Subgroups — #28560532

Order 15104.

592 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21433101	C₂	1	2	yes	maximal, complemented
2	#21423010	C₂	1	1	yes	maximal, characteristic, complemented
3	#21423005	C₂	1	1	yes	maximal, characteristic, complemented
4	#21422975	C₂	1	2	yes	maximal, complemented
5	#21422954	C₂	1	2	yes	maximal, complemented
6	#21422920	C₂	1	1	yes	maximal, characteristic, complemented
7	#21422915	C₂	1	1	yes	maximal, characteristic, complemented
8	#21422844	C₂	1	1	yes	maximal, characteristic, complemented
9	#21422504	C₂	1	2	yes	maximal, complemented
10	#21422385	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₉×#3168	C₂	1	1	yes	maximal, characteristic
12	#16837533	C₂²	1	2	yes	—
13	Q₈×C₅₉⋊C₈	C₂²	1	2	yes	—
14	#16837480	—	2	2	no	—
15	#16837480	C₂²	1	2	yes	—
16	#16837474	—	2	2	no	—
17	D₄×C₅₉⋊C₈	C₂²	1	2	yes	—
18	#16837463	—	2	2	no	—
19	#16837455	C₂²	1	2	yes	—
20	#16837455	C₂²	1	1	yes	characteristic
21	#16837358	—	2	2	no	—
22	C₄×#11745664	C₂²	1	1	yes	characteristic
23	#16837155	C₂²	1	2	yes	—
24	#16837153	C₂²	1	1	yes	characteristic
25	#16837153	C₂²	1	1	yes	characteristic
26	Q₈⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
27	Q₈⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
28	D₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
29	D₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
30	#16837124	C₂²	1	1	yes	characteristic
31	#16837124	C₂²	1	1	yes	characteristic
32	#16837123	C₂²	1	1	yes	characteristic
33	#16837123	C₂²	1	1	yes	characteristic
34	C₅₉×#529	C₂²	1	2	yes	—
35	C₅₉×#499	C₄	1	1	yes	characteristic
36	C₅₉×#495	C₄	1	1	yes	characteristic
37	C₅₉×#488	C₄	1	2	yes	—
38	C₅₉×#467	C₄	1	1	yes	characteristic
39	C₅₉×#466	C₄	1	1	yes	characteristic
40	C₅₉×#465	C₄	1	2	yes	—
41	C₄×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
42	C₄×SD₁₆×C₅₉	C₂²	1	2	yes	—
43	C₄×D₈×C₅₉	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
45	C₂×#11745664	—	2	2	no	—
46	C₂×#11745664	D₄	1	2	yes	—
47	C₂²×C₅₉⋊C₈	—	2	2	no	—
48	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
49	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
50	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
51	C₄⋊(C₅₉⋊C₈)	C₂³	1	2	yes	—
52	C₄⋊(C₅₉⋊C₈)	C₂³	1	2	yes	—
53	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
54	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
55	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
56	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
57	#14273638	—	2	2	no	—
58	#14273638	D₄	1	2	yes	—
59	C₄×C₅₉⋊C₈	—	2	2	no	—
60	C₄×C₅₉⋊C₈	—	2	2	no	—
61	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
62	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
63	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
64	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
65	C₂×Q₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
66	C₂×SD₁₆×C₅₉	C₂×C₄	1	2	yes	—
67	C₂×D₈×C₅₉	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
70	#126×C₅₉	—	2	2	no	—
71	#125×C₅₉	C₂×C₄	1	2	yes	—
72	#124×C₅₉	C₂×C₄	1	2	yes	—
73	#123×C₅₉	—	2	2	no	—
74	C₂²⋊Q₈×C₅₉	C₂×C₄	1	2	yes	—
75	#121×C₅₉	C₂×C₄	1	2	yes	—
76	C₄×Q₈×C₅₉	C₂³	1	2	yes	—
77	C₄×D₄×C₅₉	—	2	2	no	—
78	C₄×D₄×C₅₉	C₂³	1	2	yes	—
79	#117×C₅₉	—	2	2	no	—
80	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
81	#107×C₅₉	C₂×C₄	1	1	yes	characteristic
82	#107×C₅₉	C₂×C₄	1	1	yes	characteristic
83	#106×C₅₉	C₂×C₄	1	2	yes	—
84	#103×C₅₉	C₂×C₄	1	2	yes	—
85	#103×C₅₉	C₂×C₄	1	2	yes	—
86	#102×C₅₉	C₂×C₄	1	2	yes	—
87	#102×C₅₉	C₂×C₄	1	2	yes	—
88	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
89	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
90	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
92	C₂×Q₈×C₅₉	C₂²×C₄	1	2	yes	—
93	C₂×D₄×C₅₉	—	2	2	no	—
94	C₂×D₄×C₅₉	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₅₉	—	2	2	no	—
96	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
99	Q₁₆×C₅₉	—	2	2	no	—
100	SD₁₆×C₅₉	—	4	2	no	—
101	D₈×C₅₉	—	2	2	no	—
102	C₂×C₈×C₅₉	—	2	2	no	—
103	C₂×C₈×C₅₉	—	4	1	no	—
104	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
108	C₄⋊C₄×C₅₉	—	2	2	no	—
109	C₄⋊C₄×C₅₉	—	2	2	no	—
110	C₄⋊C₄×C₅₉	—	2	2	no	—
111	C₄⋊C₄×C₅₉	—	2	2	no	—
112	C₄⋊C₄×C₅₉	C₂²×C₄	1	2	yes	—
113	C₄⋊C₄×C₅₉	C₂²×C₄	1	2	yes	—
114	C₂²⋊C₄×C₅₉	—	2	2	no	—
115	C₂²⋊C₄×C₅₉	—	2	2	no	—
116	C₂²⋊C₄×C₅₉	—	2	2	no	—
117	C₂²⋊C₄×C₅₉	—	2	2	no	—
118	C₂²⋊C₄×C₅₉	—	2	2	no	—
119	C₄²×C₅₉	—	2	2	no	—
120	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
121	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₅₉	#41	1	1	yes	characteristic
123	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
124	#11745664	—	4	2	no	—
125	#11745664	—	4	4	no	—
126	C₂×C₅₉⋊C₈	—	2	2	no	—
127	C₂×C₅₉⋊C₈	—	2	2	no	—
128	C₂×C₅₉⋊C₈	—	2	1	no	—
129	C₂×C₅₉⋊C₈	—	2	1	no	—
130	C₂×C₅₉⋊C₈	—	4	4	no	—
131	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
132	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
133	C₂³×C₅₉	—	2	2	no	—
134	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
135	Q₈×C₅₉	—	2	2	no	—
136	Q₈×C₅₉	—	2	2	no	—
137	D₄×C₅₉	—	2	2	no	—
138	D₄×C₅₉	—	2	2	no	—
139	D₄×C₅₉	—	8	2	no	—
140	C₂×C₄×C₅₉	—	2	1	no	—
141	C₂×C₄×C₅₉	—	2	1	no	—
142	C₂×C₄×C₅₉	—	2	2	no	—
143	C₂×C₄×C₅₉	—	2	2	no	—
144	C₂×C₄×C₅₉	—	2	2	no	—
145	C₂×C₄×C₅₉	—	2	2	no	—
146	C₂×C₄×C₅₉	—	4	1	no	—
147	C₂×C₄×C₅₉	—	4	2	no	—
148	C₂×C₄×C₅₉	—	4	2	no	—
149	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
155	C₈×C₅₉	—	2	1	no	—
156	C₈×C₅₉	—	2	1	no	—
157	C₈×C₅₉	—	2	1	no	—
158	C₈×C₅₉	#131	1	2	yes	—
159	C₅₉⋊C₈	—	2	2	no	—
160	C₅₉⋊C₈	—	2	2	no	—
161	C₅₉⋊C₈	—	4	1	no	—
162	C₅₉⋊C₈	—	4	1	no	—
163	C₅₉⋊C₈	—	4	2	no	—
164	C₅₉⋊C₈	—	4	2	no	—
165	#18075	—	59	1	no	maximal, Sylow
166	C₂²×C₅₉	—	2	1	no	—
167	C₂²×C₅₉	—	2	1	no	—
168	C₂²×C₅₉	—	2	1	no	—
169	C₂²×C₅₉	—	4	2	no	—
170	C₂²×C₅₉	—	4	2	no	—
171	C₂²×C₅₉	—	4	2	no	—
172	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
173	C₄×C₅₉	—	2	1	no	—
174	C₄×C₅₉	—	2	1	no	—
175	C₄×C₅₉	—	2	1	no	—
176	C₄×C₅₉	—	2	1	no	—
177	C₄×C₅₉	—	4	2	no	—
178	C₄×C₅₉	—	4	2	no	—
179	C₄×C₅₉	—	4	2	no	—
180	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
181	C₄×C₅₉	#568	1	1	yes	characteristic
182	C₄×C₅₉	#575	1	2	yes	—
183	#3168	D₅₉	1	1	yes	characteristic
184	#2969	—	59	2	no	—
185	#1594	—	59	1	no	—
186	#1593	—	59	1	no	—
187	#1588	—	59	2	no	—
188	#1585	—	59	2	no	—
189	#1580	—	59	1	no	—
190	#1579	—	59	1	no	—
191	#1564	—	59	1	no	—
192	#1482	—	59	2	no	—
193	#1451	—	59	1	no	—
194	C₂×C₅₉	—	4	1	no	—
195	C₂×C₅₉	—	8	2	no	—
196	C₂×C₅₉	#2935	1	1	yes	characteristic
197	C₂×C₅₉	#2943	1	1	yes	characteristic
198	C₂×C₅₉	#2957	1	1	yes	characteristic
199	#529	C₂×D₅₉	1	2	yes	—
200	#499	Dic₅₉	1	1	yes	characteristic
201	#495	Dic₅₉	1	1	yes	characteristic
202	#488	Dic₅₉	1	2	yes	—
203	#467	Dic₅₉	1	1	yes	characteristic
204	#466	Dic₅₉	1	1	yes	characteristic
205	#465	Dic₅₉	1	2	yes	—
206	#446	—	59	2	no	—
207	C₈×Q₈	—	59	2	no	—
208	C₄×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
209	C₄×SD₁₆	C₂×D₅₉	1	2	yes	—
210	C₄×D₈	C₂×D₅₉	1	1	yes	characteristic
211	#436	—	59	2	no	—
212	#436	—	118	2	no	—
213	#435	—	118	2	no	—
214	C₈×D₄	—	59	2	no	—
215	#433	—	118	2	no	—
216	#431	—	59	1	no	—
217	#431	—	59	2	no	—
218	#405	—	118	2	no	—
219	C₄×M₁₆	—	59	1	no	—
220	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
221	#335	—	59	2	no	—
222	#334	—	59	1	no	—
223	#334	—	59	1	no	—
224	Q₈⋊C₈	—	59	2	no	—
225	Q₈⋊C₈	—	59	2	no	—
226	D₄⋊C₈	—	59	2	no	—
227	D₄⋊C₈	—	59	2	no	—
228	#322	—	59	1	no	—
229	#322	—	59	1	no	—
230	#322	—	59	1	no	—
231	#322	—	59	1	no	—
232	C₅₉	#18075	1	1	yes	characteristic, complemented, Sylow
233	C₂×Q₁₆	C₂×Dic₅₉	1	1	yes	characteristic
234	C₂×SD₁₆	C₂×Dic₅₉	1	2	yes	—
235	C₂×D₈	C₂×Dic₅₉	1	1	yes	characteristic
236	C₂×M₁₆	—	59	2	no	—
237	C₂×M₁₆	—	118	2	no	—
238	C₂²×C₈	—	118	2	no	—
239	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
240	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
241	#126	—	2	2	no	—
242	#125	C₂×Dic₅₉	1	2	yes	—
243	#124	C₂×Dic₅₉	1	2	yes	—
244	#123	—	2	2	no	—
245	C₂²⋊Q₈	C₂×Dic₅₉	1	2	yes	—
246	#121	C₂×Dic₅₉	1	2	yes	—
247	C₄×Q₈	C₂²×D₅₉	1	2	yes	—
248	C₄×D₄	—	2	2	no	—
249	C₄×D₄	C₂²×D₅₉	1	2	yes	—
250	#117	—	2	2	no	—
251	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
252	#107	C₂×Dic₅₉	1	1	yes	characteristic
253	#107	C₂×Dic₅₉	1	1	yes	characteristic
254	#106	C₂×Dic₅₉	1	2	yes	—
255	C₄⋊C₈	—	59	2	no	—
256	C₄⋊C₈	—	59	2	no	—
257	C₄⋊C₈	—	118	2	no	—
258	C₄⋊C₈	—	118	2	no	—
259	C₄⋊C₈	—	118	1	no	—
260	#103	C₂×Dic₅₉	1	2	yes	—
261	#103	C₂×Dic₅₉	1	2	yes	—
262	#102	C₂×Dic₅₉	1	2	yes	—
263	#102	C₂×Dic₅₉	1	2	yes	—
264	C₂²⋊C₈	—	118	2	no	—
265	C₂²⋊C₈	—	118	2	no	—
266	C₂²⋊C₈	—	118	2	no	—
267	C₂²⋊C₈	—	118	1	no	—
268	#97	—	59	2	no	—
269	#97	—	118	2	no	—
270	C₄×C₈	—	59	1	no	—
271	C₄×C₈	—	59	1	no	—
272	C₄×C₈	—	59	1	no	—
273	C₄×C₈	—	59	1	no	—
274	C₄×C₈	—	118	2	no	—
275	C₄×C₈	—	118	2	no	—
276	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
277	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
278	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
280	C₂×Q₈	C₂²×Dic₅₉	1	2	yes	—
281	C₂×D₄	—	2	2	no	—
282	C₂×D₄	C₂²×Dic₅₉	1	2	yes	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
285	C₂²×C₄	#11745687	1	1	yes	characteristic
286	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
287	Q₁₆	—	2	2	no	—
288	SD₁₆	—	4	2	no	—
289	D₈	—	2	2	no	—
290	M₁₆	—	236	2	no	—
291	M₁₆	—	236	4	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	4	1	no	—
294	C₂×C₈	—	59	2	no	—
295	C₂×C₈	—	59	2	no	—
296	C₂×C₈	—	118	2	no	—
297	C₂×C₈	—	118	2	no	—
298	C₂×C₈	—	118	1	no	—
299	C₂×C₈	—	118	1	no	—
300	C₂×C₈	—	236	4	no	—
301	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
302	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
303	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
304	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	C₂²×Dic₅₉	1	2	yes	—
310	C₄⋊C₄	C₂²×Dic₅₉	1	2	yes	—
311	C₂²⋊C₄	—	2	2	no	—
312	C₂²⋊C₄	—	2	2	no	—
313	C₂²⋊C₄	—	2	2	no	—
314	C₂²⋊C₄	—	2	2	no	—
315	C₂²⋊C₄	—	2	2	no	—
316	C₄²	—	2	2	no	—
317	C₄²	D₄×D₅₉	1	1	yes	characteristic
318	C₄²	#11745687	1	1	yes	characteristic
319	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
320	C₄²	C₂³×D₅₉	1	1	yes	characteristic
321	C₂³	—	2	2	no	—
322	C₂³	D₄×Dic₅₉	1	1	yes	characteristic
323	Q₈	—	2	2	no	—
324	Q₈	—	2	2	no	—
325	D₄	—	2	2	no	—
326	D₄	—	2	2	no	—
327	D₄	—	8	2	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
338	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
339	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
340	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
341	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
342	C₂×C₄	C₂×#11745687	1	1	yes	characteristic
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	2	1	no	—
346	C₈	—	118	2	no	—
347	C₈	—	118	2	no	—
348	C₈	—	236	1	no	—
349	C₈	—	236	1	no	—
350	C₈	—	236	2	no	—
351	C₈	—	236	2	no	—
352	C₈	#14273734	1	2	yes	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	4	2	no	—
357	C₂²	—	4	2	no	—
358	C₂²	—	4	2	no	—
359	C₂²	C₂×D₄×Dic₅₉	1	1	yes	characteristic
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	4	2	no	—
365	C₄	—	4	2	no	—
366	C₄	—	4	2	no	—
367	C₄	C₂×#14273734	1	1	yes	characteristic
368	C₄	#16838169	1	1	yes	characteristic
369	C₄	#16838199	1	2	yes	—
370	C₂	—	4	1	no	—
371	C₂	—	8	2	no	—
372	C₂	#21432840	1	1	yes	characteristic
373	C₂	#21432897	1	1	yes	characteristic
374	C₂	#21433013	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×#3168	C₂
socle	C₂²×C₅₉	C₂×C₄×D₄
prime radical	C₅₉×#466	C₄
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	#3168	D₅₉
hypocenter	C₅₉	#18075
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3168

Complements

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

N	H ≅ G/N	type
1. #21433101	C₂	semidirect
2. #21423010	C₂	semidirect
3. #21423005	C₂	semidirect
4. #21422975	C₂	semidirect
5. #21422954	C₂	semidirect
6. #21422920	C₂	semidirect
7. #21422915	C₂	semidirect
8. #21422844	C₂	semidirect
9. #21422504	C₂	semidirect
10. #21422385	C₂	semidirect
232. C₅₉	#18075	semidirect

Maximal subgroups

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

subgroup	index
#21433101	2
#21423010	2
#21423005	2
#21422975	2
#21422954	2
#21422920	2
#21422915	2
#21422844	2
#21422504	2
#21422385	2
C₅₉×#3168	2
#18075	59