Subgroups — #28559365

Order 15104.

560 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21434678	C₂	1	1	yes	maximal, characteristic
2	#21433190	C₂	1	1	yes	maximal, characteristic
3	#21423003	C₂	1	2	yes	maximal, complemented
4	#21422991	C₂	1	2	yes	maximal, complemented
5	#21422913	C₂	1	2	yes	maximal, complemented
6	C₈×#11745672	C₂	1	2	yes	maximal, complemented
7	#21422857	C₂	1	1	yes	maximal, characteristic
8	#21422569	C₂	1	1	yes	maximal, characteristic
9	#21422568	C₂	1	1	yes	maximal, characteristic
10	#21422380	C₂	1	1	yes	maximal, characteristic
11	C₅₉×#2985	C₂	1	1	yes	maximal, characteristic
12	#16837959	C₂²	1	1	yes	characteristic
13	#16837805	C₄	1	1	yes	characteristic
14	#16837781	C₄	1	1	yes	characteristic
15	#16837754	C₄	1	1	yes	characteristic
16	#16837752	C₄	1	1	yes	characteristic
17	#16837641	C₄	1	2	yes	—
18	#16837641	C₄	1	2	yes	—
19	#16837532	C₂²	1	2	yes	—
20	#16837531	—	2	2	no	—
21	C₈×Dic₁₁₈	C₂²	1	2	yes	—
22	C₄×#11745672	C₂²	1	2	yes	—
23	C₄×#11745672	C₂²	1	2	yes	—
24	#16837460	—	2	2	no	—
25	#16837459	C₂²	1	1	yes	characteristic
26	#16837455	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
28	#16837154	C₂²	1	2	yes	—
29	#16837152	C₂²	1	2	yes	—
30	Q₈⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
31	Q₈⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
32	#16837132	C₂²	1	2	yes	—
33	#16837132	C₂²	1	2	yes	—
34	#16837124	C₂²	1	1	yes	characteristic
35	#16837124	C₂²	1	1	yes	characteristic
36	C₈×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
37	C₈×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
38	C₅₉×#533	C₂²	1	1	yes	characteristic
39	C₅₉×#446	—	2	2	no	—
40	C₅₉×#446	C₂²	1	2	yes	—
41	C₈×Q₈×C₅₉	C₂²	1	2	yes	—
42	C₅₉×#433	—	2	2	no	—
43	C₅₉×#432	C₂²	1	1	yes	characteristic
44	C₅₉×#431	C₂²	1	1	yes	characteristic
45	C₂×#11745672	C₂×C₄	1	2	yes	—
46	C₂×#11745672	C₂×C₄	1	2	yes	—
47	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
49	#14273727	C₂×C₄	1	1	yes	characteristic
50	#14273727	C₂×C₄	1	1	yes	characteristic
51	#14273719	—	2	1	no	—
52	#14273706	—	2	1	no	—
53	#14273706	C₂×C₄	1	2	yes	—
54	C₄×Dic₁₁₈	—	2	1	no	—
55	C₄×Dic₁₁₈	C₂³	1	2	yes	—
56	#14273686	—	2	1	no	—
57	#14273660	C₂×C₄	1	2	yes	—
58	#14273660	C₂×C₄	1	2	yes	—
59	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
60	C₄⋊(C₅₉⋊C₈)	C₂³	1	2	yes	—
61	Dic₅₉⋊C₈	—	2	2	no	—
62	Dic₅₉⋊C₈	—	2	2	no	—
63	#14273651	C₂×C₄	1	2	yes	—
64	#14273651	C₂×C₄	1	2	yes	—
65	#14273650	C₂×C₄	1	2	yes	—
66	#14273650	C₂×C₄	1	2	yes	—
67	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
68	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
69	#14273638	—	2	1	no	—
70	#14273637	—	2	2	no	—
71	C₄×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
72	C₄×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
73	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
74	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
75	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
76	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
77	C₈×Dic₅₉	—	2	2	no	—
78	C₄⋊Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
79	C₄⋊Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
80	#125×C₅₉	—	2	1	no	—
81	C₂²⋊Q₈×C₅₉	—	2	1	no	—
82	C₂²⋊Q₈×C₅₉	C₂×C₄	1	2	yes	—
83	C₄×Q₈×C₅₉	—	2	1	no	—
84	C₄×Q₈×C₅₉	C₂³	1	2	yes	—
85	#117×C₅₉	—	2	1	no	—
86	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₅₉	—	2	2	no	—
88	C₄⋊C₈×C₅₉	—	2	2	no	—
89	C₄⋊C₈×C₅₉	D₄	1	2	yes	—
90	C₄⋊C₈×C₅₉	C₂³	1	2	yes	—
91	C₂²⋊C₈×C₅₉	—	2	1	no	—
92	C₂²⋊C₈×C₅₉	D₄	1	2	yes	—
93	#97×C₅₉	—	2	2	no	—
94	#97×C₅₉	—	2	1	no	—
95	C₄×C₈×C₅₉	—	2	2	no	—
96	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
97	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
98	C₂×Q₈×C₅₉	—	2	1	no	—
99	C₂×Q₈×C₅₉	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
103	C₂×Dic₁₁₈	—	2	1	no	—
104	C₂×Dic₁₁₈	C₂²×C₄	1	2	yes	—
105	C₂×C₈×C₅₉	—	2	2	no	—
106	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
108	C₄⋊C₄×C₅₉	—	2	1	no	—
109	C₄⋊C₄×C₅₉	—	2	1	no	—
110	C₄⋊C₄×C₅₉	—	2	1	no	—
111	C₄⋊C₄×C₅₉	—	2	2	no	—
112	C₄⋊C₄×C₅₉	—	2	2	no	—
113	C₄⋊C₄×C₅₉	C₂²×C₄	1	2	yes	—
114	C₂²⋊C₄×C₅₉	—	2	2	no	—
115	C₄²×C₅₉	—	2	2	no	—
116	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
117	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₅₉	#41	1	1	yes	characteristic
119	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
120	C₂²⋊Dic₅₉	—	2	2	no	—
121	#11745672	—	2	4	no	—
122	#11745672	—	2	4	no	—
123	C₄⋊Dic₅₉	—	2	1	no	—
124	C₄⋊Dic₅₉	—	2	1	no	—
125	C₄⋊Dic₅₉	—	2	1	no	—
126	C₄⋊Dic₅₉	C₂²×C₄	1	2	yes	—
127	Dic₅₉⋊C₄	—	2	2	no	—
128	Dic₅₉⋊C₄	—	2	2	no	—
129	C₄×Dic₅₉	—	2	2	no	—
130	C₂×C₅₉⋊C₈	—	2	2	no	—
131	C₂×C₅₉⋊C₈	—	2	2	no	—
132	C₂×C₅₉⋊C₈	—	2	2	no	—
133	C₂×C₅₉⋊C₈	—	2	1	no	—
134	C₂×C₅₉⋊C₈	—	2	1	no	—
135	C₂×C₅₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₅₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
137	C₂×C₅₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
138	C₂×C₅₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
139	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
140	Q₈×C₅₉	—	2	2	no	—
141	Q₈×C₅₉	—	2	2	no	—
142	Q₈×C₅₉	—	8	1	no	—
143	C₂×C₄×C₅₉	—	2	1	no	—
144	C₂×C₄×C₅₉	—	2	1	no	—
145	C₂×C₄×C₅₉	—	2	2	no	—
146	C₂×C₄×C₅₉	—	2	2	no	—
147	C₂×C₄×C₅₉	—	2	2	no	—
148	C₂×C₄×C₅₉	—	4	1	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	C₂×Dic₅₉	—	2	2	no	—
156	C₂×Dic₅₉	—	2	2	no	—
157	Dic₁₁₈	—	2	2	no	—
158	Dic₁₁₈	—	2	2	no	—
159	Dic₁₁₈	—	8	1	no	—
160	C₈×C₅₉	—	2	2	no	—
161	C₈×C₅₉	—	2	2	no	—
162	C₈×C₅₉	—	4	2	no	—
163	C₅₉⋊C₈	—	2	1	no	—
164	C₅₉⋊C₈	—	2	1	no	—
165	C₅₉⋊C₈	—	4	1	no	—
166	C₅₉⋊C₈	—	4	1	no	—
167	C₅₉⋊C₈	—	4	2	no	—
168	C₅₉⋊C₈	#131	1	2	yes	—
169	C₅₉⋊C₈	#131	1	2	yes	—
170	#17972	—	59	1	no	maximal, Sylow
171	C₂²×C₅₉	—	2	1	no	—
172	C₂²×C₅₉	—	2	1	no	—
173	C₂²×C₅₉	—	2	1	no	—
174	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
175	C₄×C₅₉	—	2	1	no	—
176	C₄×C₅₉	—	2	1	no	—
177	C₄×C₅₉	—	2	1	no	—
178	C₄×C₅₉	—	2	1	no	—
179	C₄×C₅₉	—	4	2	no	—
180	C₄×C₅₉	—	4	2	no	—
181	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
182	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
183	C₄×C₅₉	#578	1	2	yes	—
184	Dic₅₉	—	4	2	no	—
185	Dic₅₉	—	4	2	no	—
186	#3170	—	59	1	no	—
187	#2985	—	59	1	no	—
188	#2985	D₅₉	1	1	yes	characteristic
189	#1593	—	59	2	no	—
190	#1591	—	59	2	no	—
191	#1579	—	59	2	no	—
192	C₈×Q₁₆	—	59	2	no	—
193	#1568	—	59	1	no	—
194	#1497	—	59	1	no	—
195	#1497	—	59	1	no	—
196	#1450	—	59	1	no	—
197	C₂×C₅₉	—	4	1	no	—
198	C₂×C₅₉	#2931	1	1	yes	characteristic
199	C₂×C₅₉	#2950	1	1	yes	characteristic
200	C₂×C₅₉	#2957	1	1	yes	characteristic
201	#533	—	59	1	no	—
202	#533	C₂×D₅₉	1	1	yes	characteristic
203	#500	—	59	1	no	—
204	#494	—	59	1	no	—
205	#487	—	59	1	no	—
206	#487	—	59	1	no	—
207	#467	—	59	2	no	—
208	#467	—	59	2	no	—
209	#446	—	2	2	no	—
210	#446	—	59	2	no	—
211	#446	—	118	2	no	—
212	#446	C₂×D₅₉	1	2	yes	—
213	C₈×Q₈	—	59	2	no	—
214	C₈×Q₈	C₂×D₅₉	1	2	yes	—
215	C₄×Q₁₆	—	59	2	no	—
216	C₄×Q₁₆	—	59	2	no	—
217	#433	—	2	2	no	—
218	#433	—	118	2	no	—
219	#432	—	59	1	no	—
220	#432	C₂×D₅₉	1	1	yes	characteristic
221	#431	—	59	1	no	—
222	#431	C₂×D₅₉	1	1	yes	characteristic
223	C₂×C₄×C₈	—	59	1	no	—
224	#335	—	59	2	no	—
225	#334	—	59	2	no	—
226	Q₈⋊C₈	—	59	2	no	—
227	Q₈⋊C₈	—	59	2	no	—
228	Q₈⋊C₈	—	59	2	no	—
229	Q₈⋊C₈	—	59	2	no	—
230	#322	—	59	1	no	—
231	#322	—	59	1	no	—
232	C₈²	—	59	1	no	—
233	C₈²	—	59	1	no	—
234	C₅₉	#17972	1	1	yes	characteristic, complemented, Sylow
235	C₂×Q₁₆	—	59	2	no	—
236	C₂×Q₁₆	—	59	2	no	—
237	C₂²×C₈	—	59	1	no	—
238	C₂²×C₈	—	59	1	no	—
239	C₄⋊Q₈	—	59	1	no	—
240	C₄⋊Q₈	—	59	1	no	—
241	C₄⋊Q₈	C₄×D₅₉	1	1	yes	characteristic
242	C₄⋊Q₈	C₄×D₅₉	1	1	yes	characteristic
243	#125	—	2	1	no	—
244	#125	—	118	1	no	—
245	C₂²⋊Q₈	—	2	1	no	—
246	C₂²⋊Q₈	—	59	2	no	—
247	C₂²⋊Q₈	—	118	1	no	—
248	C₂²⋊Q₈	C₄×D₅₉	1	2	yes	—
249	C₄×Q₈	—	2	1	no	—
250	C₄×Q₈	—	59	2	no	—
251	C₄×Q₈	—	118	1	no	—
252	C₄×Q₈	C₂²×D₅₉	1	2	yes	—
253	#117	—	2	1	no	—
254	#117	—	118	1	no	—
255	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
256	#107	—	59	2	no	—
257	#107	—	59	2	no	—
258	C₄⋊C₈	—	2	2	no	—
259	C₄⋊C₈	—	2	2	no	—
260	C₄⋊C₈	—	59	2	no	—
261	C₄⋊C₈	—	118	2	no	—
262	C₄⋊C₈	—	118	2	no	—
263	C₄⋊C₈	—	118	1	no	—
264	C₄⋊C₈	C₂²⋊D₅₉	1	2	yes	—
265	C₄⋊C₈	C₂²×D₅₉	1	2	yes	—
266	#103	—	59	2	no	—
267	#103	—	59	2	no	—
268	#103	—	59	2	no	—
269	#103	—	59	2	no	—
270	C₂²⋊C₈	—	2	1	no	—
271	C₂²⋊C₈	—	118	1	no	—
272	C₂²⋊C₈	—	118	1	no	—
273	C₂²⋊C₈	C₂²⋊D₅₉	1	2	yes	—
274	#97	—	2	2	no	—
275	#97	—	2	1	no	—
276	#97	—	118	2	no	—
277	#97	—	118	1	no	—
278	C₄×C₈	—	2	2	no	—
279	C₄×C₈	—	59	1	no	—
280	C₄×C₈	—	59	1	no	—
281	C₄×C₈	—	59	1	no	—
282	C₄×C₈	—	59	1	no	—
283	C₄×C₈	—	59	1	no	—
284	C₄×C₈	—	59	1	no	—
285	C₄×C₈	—	118	2	no	—
286	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
287	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
288	C₂×Q₈	—	2	1	no	—
289	C₂×Q₈	—	59	2	no	—
290	C₂×Q₈	—	118	1	no	—
291	C₂×Q₈	C₂×C₄×D₅₉	1	2	yes	—
292	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
293	C₂²×C₄	#11745685	1	1	yes	characteristic
294	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
295	Q₁₆	—	118	4	no	—
296	Q₁₆	—	118	4	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	59	1	no	—
299	C₂×C₈	—	59	1	no	—
300	C₂×C₈	—	59	1	no	—
301	C₂×C₈	—	59	1	no	—
302	C₂×C₈	—	118	2	no	—
303	C₂×C₈	—	118	2	no	—
304	C₂×C₈	—	118	2	no	—
305	C₂×C₈	—	118	1	no	—
306	C₂×C₈	—	118	1	no	—
307	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
308	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	2	1	no	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	59	2	no	—
315	C₄⋊C₄	—	118	1	no	—
316	C₄⋊C₄	—	118	1	no	—
317	C₄⋊C₄	—	118	2	no	—
318	C₄⋊C₄	—	118	1	no	—
319	C₄⋊C₄	—	118	2	no	—
320	C₄⋊C₄	C₂×C₄×D₅₉	1	2	yes	—
321	C₂²⋊C₄	—	2	2	no	—
322	C₂²⋊C₄	—	118	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	—	118	2	no	—
325	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
326	C₄²	#11745685	1	1	yes	characteristic
327	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
328	C₄²	C₂³×D₅₉	1	1	yes	characteristic
329	C₂³	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
330	Q₈	—	2	2	no	—
331	Q₈	—	2	2	no	—
332	Q₈	—	8	1	no	—
333	Q₈	—	118	2	no	—
334	Q₈	—	118	2	no	—
335	Q₈	—	472	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	4	1	no	—
342	C₂×C₄	—	118	2	no	—
343	C₂×C₄	—	118	2	no	—
344	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
345	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
346	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
347	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
348	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
349	C₂×C₄	C₂×#11745685	1	1	yes	characteristic
350	C₈	—	2	2	no	—
351	C₈	—	2	2	no	—
352	C₈	—	4	2	no	—
353	C₈	—	59	2	no	—
354	C₈	—	59	2	no	—
355	C₈	—	118	1	no	—
356	C₈	—	118	1	no	—
357	C₈	—	236	1	no	—
358	C₈	—	236	1	no	—
359	C₈	—	236	2	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	C₂×C₄×C₂²⋊D₅₉	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	4	2	no	—
369	C₄	—	4	2	no	—
370	C₄	—	236	2	no	—
371	C₄	—	236	2	no	—
372	C₄	#131×D₅₉	1	1	yes	characteristic
373	C₄	#131×D₅₉	1	1	yes	characteristic
374	C₄	#16838217	1	2	yes	—
375	C₂	—	4	1	no	—
376	C₂	#21432806	1	1	yes	characteristic
377	C₂	#21432962	1	1	yes	characteristic
378	C₂	#21433011	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₄²	C₂³×D₅₉
Fitting	C₅₉×#2985	C₂
socle	C₂²×C₅₉	C₂×C₄×D₄
prime radical	C₂³×C₅₉	C₄×D₄
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	#2985	D₅₉
hypocenter	C₅₉	#17972
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #2985

Complements

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

N	H ≅ G/N	type
3. #21423003	C₂	semidirect
4. #21422991	C₂	semidirect
5. #21422913	C₂	semidirect
6. C₈×#11745672	C₂	semidirect
234. C₅₉	#17972	semidirect

Maximal subgroups

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

subgroup	index
#21434678	2
#21433190	2
#21423003	2
#21422991	2
#21422913	2
C₈×#11745672	2
#21422857	2
#21422569	2
#21422568	2
#21422380	2
C₅₉×#2985	2
#17972	59