Subgroups — #28560720

Order 15104.

564 conjugacy classes of subgroups.

The subgroup lattice has 362 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	#21423004	C₂	1	1	yes	maximal, characteristic, complemented
4	#21422944	C₂	1	2	yes	maximal, complemented
5	#21422943	C₂	1	2	yes	maximal, complemented
6	#21422916	C₂	1	1	yes	maximal, characteristic, complemented
7	#21422915	C₂	1	1	yes	maximal, characteristic, complemented
8	#21422841	C₂	1	1	yes	maximal, characteristic
9	#21422515	C₂	1	2	yes	maximal
10	#21422382	C₂	1	1	yes	maximal, characteristic
11	C₅₉×#1593	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	#16837154	C₂²	1	2	yes	—
33	#16837153	C₂²	1	1	yes	characteristic
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	#16837123	C₂²	1	1	yes	characteristic
40	C₅₉×#446	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
42	C₅₉×#334	C₂²	1	1	yes	characteristic
43	C₅₉×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₅₉×#322	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	2	yes	—
67	#14273652	C₂×C₄	1	2	yes	—
68	#14273651	C₂×C₄	1	2	yes	—
69	#14273650	C₂×C₄	1	2	yes	—
70	(C₂×D₅₉)⋊C₈	—	2	2	no	—
71	(C₂×D₅₉)⋊C₈	D₄	1	2	yes	—
72	#14273637	—	2	1	no	—
73	#14273637	—	2	2	no	—
74	C₄×C₅₉⋊C₈	—	2	2	no	—
75	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
76	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
77	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
78	C₈×Dic₅₉	—	2	2	no	—
79	C₈×Dic₅₉	—	2	2	no	—
80	C₈×Dic₅₉	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₅₉	C₂×C₄	1	1	yes	characteristic
82	C₂×Q₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
83	C₄×Q₈×C₅₉	C₂³	1	2	yes	—
84	#107×C₅₉	C₂×C₄	1	1	yes	characteristic
85	C₄⋊C₈×C₅₉	—	2	2	no	—
86	C₄⋊C₈×C₅₉	C₂³	1	2	yes	—
87	#103×C₅₉	C₂×C₄	1	2	yes	—
88	#97×C₅₉	—	2	2	no	—
89	C₄×C₈×C₅₉	C₂³	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×C₄×D₅₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×D₅₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×D₅₉	#41	1	1	yes	characteristic
96	C₂×Dic₁₁₈	—	2	2	no	—
97	C₂×Dic₁₁₈	C₂²×C₄	1	2	yes	—
98	Q₁₆×C₅₉	—	2	2	no	—
99	C₂×C₈×C₅₉	—	2	2	no	—
100	C₂×C₈×C₅₉	—	2	1	no	—
101	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
104	C₄⋊C₄×C₅₉	—	2	2	no	—
105	C₄⋊C₄×C₅₉	C₂²×C₄	1	2	yes	—
106	C₄²×C₅₉	—	2	2	no	—
107	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
108	#11745672	—	4	2	no	—
109	(C₂×D₅₉)⋊C₄	—	2	2	no	—
110	(C₂×D₅₉)⋊C₄	—	2	2	no	—
111	C₄⋊Dic₅₉	—	2	2	no	—
112	C₄⋊Dic₅₉	C₂²×C₄	1	2	yes	—
113	Dic₅₉⋊C₄	—	2	2	no	—
114	Dic₅₉⋊C₄	—	2	2	no	—
115	Dic₅₉⋊C₄	—	2	2	no	—
116	Dic₅₉⋊C₄	—	2	2	no	—
117	Dic₅₉⋊C₄	—	2	2	no	—
118	C₄×Dic₅₉	—	2	2	no	—
119	C₄×Dic₅₉	C₂²×C₄	1	1	yes	characteristic
120	C₄×Dic₅₉	C₂×D₄	1	1	yes	characteristic
121	C₄×Dic₅₉	#41	1	1	yes	characteristic
122	C₂×C₅₉⋊C₈	—	2	2	no	—
123	C₂×C₅₉⋊C₈	—	2	1	no	—
124	C₂×C₅₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₅₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
127	Dic₂₃₆	—	2	2	no	—
128	#11745659	—	4	2	no	—
129	C₈×D₅₉	—	2	2	no	—
130	C₈×D₅₉	—	4	1	no	—
131	C₂²×D₅₉	C₄×D₄	1	1	yes	characteristic
132	Q₈×C₅₉	—	2	2	no	—
133	Q₈×C₅₉	—	2	2	no	—
134	C₂×C₄×C₅₉	—	2	2	no	—
135	C₂×C₄×C₅₉	—	2	2	no	—
136	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
139	C₂×Dic₅₉	—	2	2	no	—
140	C₂×Dic₅₉	—	2	2	no	—
141	C₂×Dic₅₉	C₄×D₄	1	1	yes	characteristic
142	C₂×Dic₅₉	C₄×D₄	1	1	yes	characteristic
143	C₂×Dic₅₉	C₄×D₄	1	1	yes	characteristic
144	C₄×D₅₉	—	2	1	no	—
145	C₄×D₅₉	—	2	1	no	—
146	C₄×D₅₉	—	2	2	no	—
147	C₄×D₅₉	—	2	2	no	—
148	Dic₁₁₈	—	2	2	no	—
149	Dic₁₁₈	—	2	2	no	—
150	Dic₁₁₈	—	4	4	no	—
151	C₈×C₅₉	—	2	1	no	—
152	C₈×C₅₉	—	2	2	no	—
153	C₈×C₅₉	—	4	1	no	—
154	C₈×C₅₉	—	4	2	no	—
155	C₈×C₅₉	#131	1	2	yes	—
156	C₅₉⋊C₈	—	2	1	no	—
157	C₅₉⋊C₈	—	2	1	no	—
158	C₅₉⋊C₈	—	2	2	no	—
159	C₅₉⋊C₈	—	4	1	no	—
160	C₅₉⋊C₈	—	4	2	no	—
161	#18090	—	59	1	no	maximal, Sylow
162	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
163	C₂×D₅₉	—	2	1	no	—
164	C₂×D₅₉	—	2	1	no	—
165	C₂×D₅₉	—	2	1	no	—
166	C₄×C₅₉	—	2	1	no	—
167	C₄×C₅₉	—	4	2	no	—
168	C₄×C₅₉	—	4	2	no	—
169	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
170	C₄×C₅₉	#568	1	1	yes	characteristic
171	C₄×C₅₉	#578	1	2	yes	—
172	Dic₅₉	—	2	1	no	—
173	Dic₅₉	—	2	1	no	—
174	Dic₅₉	—	4	2	no	—
175	Dic₅₉	—	4	2	no	—
176	Dic₅₉	#444	1	2	yes	—
177	#3170	—	59	1	no	—
178	#2986	—	59	2	no	—
179	#1593	—	59	1	no	—
180	#1593	D₅₉	1	1	yes	characteristic, complemented
181	#1584	—	59	2	no	—
182	#1584	—	59	2	no	—
183	#1579	—	59	1	no	—
184	#1579	—	59	1	no	—
185	#1564	—	59	1	no	—
186	#1485	—	59	2	no	—
187	#1451	—	59	1	no	—
188	C₂×C₅₉	#2935	1	1	yes	characteristic
189	C₂×C₅₉	#2950	1	1	yes	characteristic
190	C₂×C₅₉	C₂×#444	1	1	yes	characteristic
191	D₅₉	—	4	1	no	—
192	#533	—	59	2	no	—
193	#500	—	59	1	no	—
194	#494	—	59	1	no	—
195	#487	—	59	2	no	—
196	#467	—	59	1	no	—
197	#467	—	59	1	no	—
198	#467	—	59	2	no	—
199	#446	—	59	2	no	—
200	#446	—	118	2	no	—
201	#446	C₂×D₅₉	1	2	yes	—
202	C₈×Q₈	—	59	2	no	—
203	C₈×Q₈	—	59	2	no	—
204	C₈×Q₈	—	118	2	no	—
205	C₄×Q₁₆	—	59	2	no	—
206	C₄×Q₁₆	—	59	1	no	—
207	C₄×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
208	#433	—	118	2	no	—
209	#433	—	118	2	no	—
210	#431	—	59	1	no	—
211	#431	—	59	2	no	—
212	C₄×M₁₆	—	59	1	no	—
213	C₂×C₄×C₈	—	59	1	no	—
214	#335	—	59	2	no	—
215	#334	—	59	1	no	—
216	#334	C₂×D₅₉	1	1	yes	characteristic
217	Q₈⋊C₈	—	59	2	no	—
218	Q₈⋊C₈	—	59	2	no	—
219	Q₈⋊C₈	—	59	2	no	—
220	Q₈⋊C₈	C₂×D₅₉	1	2	yes	—
221	#322	—	59	1	no	—
222	#322	—	59	1	no	—
223	#322	—	59	1	no	—
224	#322	C₂×D₅₉	1	1	yes	characteristic
225	C₅₉	#18090	1	1	yes	characteristic, complemented, Sylow
226	C₂×Q₁₆	—	59	2	no	—
227	C₂×Q₁₆	—	59	1	no	—
228	C₂×Q₁₆	C₄×D₅₉	1	1	yes	characteristic
229	C₂×M₁₆	—	118	1	no	—
230	C₂²×C₈	—	59	1	no	—
231	C₂²×C₈	—	59	1	no	—
232	C₄⋊Q₈	—	59	2	no	—
233	C₄⋊Q₈	—	59	2	no	—
234	#125	—	118	2	no	—
235	C₂²⋊Q₈	—	59	2	no	—
236	C₂²⋊Q₈	—	59	2	no	—
237	C₂²⋊Q₈	—	118	2	no	—
238	C₄×Q₈	—	59	2	no	—
239	C₄×Q₈	—	118	2	no	—
240	C₄×Q₈	C₂²×D₅₉	1	2	yes	—
241	#117	—	118	2	no	—
242	C₂×C₄²	—	59	1	no	—
243	#107	—	59	1	no	—
244	#107	—	59	2	no	—
245	#107	C₄×D₅₉	1	1	yes	characteristic
246	C₄⋊C₈	—	2	2	no	—
247	C₄⋊C₈	—	59	2	no	—
248	C₄⋊C₈	—	59	2	no	—
249	C₄⋊C₈	—	118	2	no	—
250	C₄⋊C₈	—	118	2	no	—
251	C₄⋊C₈	—	118	2	no	—
252	C₄⋊C₈	C₂²×D₅₉	1	2	yes	—
253	#103	—	59	2	no	—
254	#103	—	59	2	no	—
255	#103	—	59	2	no	—
256	#103	C₄×D₅₉	1	2	yes	—
257	C₂²⋊C₈	—	59	2	no	—
258	C₂²⋊C₈	—	118	2	no	—
259	#97	—	2	2	no	—
260	#97	—	118	1	no	—
261	#97	—	118	2	no	—
262	C₄×C₈	—	59	1	no	—
263	C₄×C₈	—	59	1	no	—
264	C₄×C₈	—	59	1	no	—
265	C₄×C₈	—	59	1	no	—
266	C₄×C₈	—	59	1	no	—
267	C₄×C₈	—	118	2	no	—
268	C₄×C₈	—	118	2	no	—
269	C₄×C₈	—	118	2	no	—
270	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
271	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
272	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
273	C₂×Q₈	—	59	2	no	—
274	C₂×Q₈	—	118	2	no	—
275	C₂×Q₈	C₂×C₄×D₅₉	1	2	yes	—
276	C₂²×C₄	—	59	1	no	—
277	C₂²×C₄	—	59	1	no	—
278	C₂²×C₄	—	59	1	no	—
279	Q₁₆	—	2	2	no	—
280	Q₁₆	—	118	2	no	—
281	Q₁₆	—	236	2	no	—
282	M₁₆	—	236	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	59	1	no	—
286	C₂×C₈	—	59	1	no	—
287	C₂×C₈	—	59	2	no	—
288	C₂×C₈	—	118	2	no	—
289	C₂×C₈	—	118	2	no	—
290	C₂×C₈	—	118	1	no	—
291	C₂×C₈	—	236	1	no	—
292	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
293	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
294	C₂×C₈	D₄×D₅₉	1	2	yes	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	59	2	no	—
297	C₄⋊C₄	—	118	2	no	—
298	C₄⋊C₄	—	118	2	no	—
299	C₄⋊C₄	—	118	2	no	—
300	C₄⋊C₄	—	118	2	no	—
301	C₄⋊C₄	—	118	2	no	—
302	C₄⋊C₄	—	118	2	no	—
303	C₄⋊C₄	C₂×C₄×D₅₉	1	2	yes	—
304	C₂²⋊C₄	—	118	2	no	—
305	C₂²⋊C₄	—	118	2	no	—
306	C₄²	—	2	2	no	—
307	C₄²	—	59	1	no	—
308	C₄²	—	59	1	no	—
309	C₄²	—	59	1	no	—
310	C₄²	—	118	2	no	—
311	C₄²	C₂³×D₅₉	1	1	yes	characteristic
312	C₂³	—	59	1	no	—
313	Q₈	—	2	2	no	—
314	Q₈	—	2	2	no	—
315	Q₈	—	118	2	no	—
316	Q₈	—	118	2	no	—
317	Q₈	—	236	4	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	59	1	no	—
321	C₂×C₄	—	59	1	no	—
322	C₂×C₄	—	59	1	no	—
323	C₂×C₄	—	118	2	no	—
324	C₂×C₄	—	118	1	no	—
325	C₂×C₄	—	118	1	no	—
326	C₂×C₄	—	118	2	no	—
327	C₂×C₄	—	118	2	no	—
328	C₂×C₄	—	118	2	no	—
329	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
330	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
331	C₂×C₄	#41×D₅₉	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	2	no	—
334	C₈	—	4	1	no	—
335	C₈	—	4	2	no	—
336	C₈	—	118	1	no	—
337	C₈	—	118	1	no	—
338	C₈	—	118	2	no	—
339	C₈	—	236	1	no	—
340	C₈	—	236	2	no	—
341	C₈	#14273735	1	2	yes	—
342	C₂²	—	118	1	no	—
343	C₂²	—	118	1	no	—
344	C₂²	—	118	1	no	—
345	C₂²	C₄×D₄×D₅₉	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	4	2	no	—
348	C₄	—	4	2	no	—
349	C₄	—	59	2	no	—
350	C₄	—	118	1	no	—
351	C₄	—	118	1	no	—
352	C₄	—	236	2	no	—
353	C₄	—	236	2	no	—
354	C₄	C₂×#14273735	1	1	yes	characteristic
355	C₄	#16838167	1	1	yes	characteristic
356	C₄	#16838218	1	2	yes	—
357	C₂	—	236	1	no	—
358	C₂	#21432841	1	1	yes	characteristic
359	C₂	#21432961	1	1	yes	characteristic
360	C₂	#444×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×#1593	C₂
socle	C₂²×C₅₉	C₂×C₄×D₄
prime radical	C₂²×D₅₉	C₄×D₄
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	#1593	D₅₉
hypocenter	C₅₉	#18090
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1593

Complements

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

N	H ≅ G/N	type
3. #21423004	C₂	semidirect
4. #21422944	C₂	semidirect
5. #21422943	C₂	semidirect
6. #21422916	C₂	semidirect
7. #21422915	C₂	semidirect
11. C₅₉×#1593	C₂	semidirect
180. #1593	D₅₉	semidirect
225. C₅₉	#18090	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
#21423004	2
#21422944	2
#21422943	2
#21422916	2
#21422915	2
#21422841	2
#21422515	2
#21422382	2
C₅₉×#1593	2
#18090	59