Subgroups — #28505150

Order 15104.

462 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₅₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×Dic₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×(C₂×D₅₉)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₈×(C₂×D₅₉)⋊C₄	C₂	1	1	yes	maximal, characteristic
6	#21422173	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₅₉	C₂²	1	1	yes	characteristic
9	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×D₅₉	—	2	2	no	—
12	C₂×C₁₆×D₅₉	—	2	2	no	—
13	C₂×C₁₆×D₅₉	—	2	2	no	—
14	C₂×(C₂×D₅₉)⋊C₈	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
16	C₄×(C₂×D₅₉)⋊C₄	C₄	1	1	yes	characteristic
17	C₄⋊(C₅₉⋊C₁₆)	—	2	2	no	—
18	Dic₅₉⋊C₁₆	—	2	2	no	—
19	(C₂×D₅₉)⋊C₁₆	—	2	2	no	—
20	#16837157	C₄	1	1	yes	characteristic
21	#16837156	C₄	1	1	yes	characteristic
22	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₂²×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×D₅₉	—	2	1	no	—
29	C₂×C₈×D₅₉	—	2	1	no	—
30	C₂×C₈×D₅₉	—	2	1	no	—
31	C₂×C₈×D₅₉	—	2	1	no	—
32	C₂×C₈×D₅₉	D₄	1	2	yes	—
33	C₂×C₈×D₅₉	Q₈	1	2	yes	—
34	C₂×(C₂×D₅₉)⋊C₄	C₈	1	1	yes	characteristic
35	C₂×(C₂×D₅₉)⋊C₄	C₈	1	1	yes	characteristic
36	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₅₉⋊C₁₆	—	2	1	no	—
38	C₂×C₅₉⋊C₁₆	—	2	1	no	—
39	C₂×C₅₉⋊C₁₆	—	2	1	no	—
40	C₂×C₅₉⋊C₁₆	—	2	1	no	—
41	C₂×C₅₉⋊C₁₆	—	2	1	no	—
42	C₂×C₅₉⋊C₁₆	D₄	1	2	yes	—
43	C₁₆×D₅₉	—	2	8	no	—
44	(C₂×D₅₉)⋊C₈	—	2	2	no	—
45	C₈×Dic₅₉	—	2	1	no	—
46	C₈×Dic₅₉	—	2	1	no	—
47	C₅₉⋊#95	C₈	1	1	yes	characteristic
48	C₅₉⋊#95	C₈	1	1	yes	characteristic
49	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₁₆×C₅₉	—	2	1	no	—
53	C₂×C₁₆×C₅₉	—	2	1	no	—
54	C₂×C₁₆×C₅₉	—	2	1	no	—
55	C₂×C₁₆×C₅₉	—	2	1	no	—
56	C₂×C₁₆×C₅₉	—	2	1	no	—
57	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
58	C₄×C₈×C₅₉	—	2	1	no	—
59	C₄×C₈×C₅₉	—	2	1	no	—
60	C₂³×D₅₉	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×Dic₅₉	C₂×C₈	1	1	yes	characteristic
65	C₂²×Dic₅₉	C₂×C₈	1	1	yes	characteristic
66	C₂²×Dic₅₉	C₂×C₈	1	1	yes	characteristic
67	C₂×C₄×D₅₉	—	2	1	no	—
68	C₂×C₄×D₅₉	—	2	1	no	—
69	C₂×C₄×D₅₉	—	2	1	no	—
70	C₂×C₄×D₅₉	—	2	1	no	—
71	C₂×C₄×D₅₉	C₄⋊C₄	1	2	yes	—
72	C₂×C₄×D₅₉	M₁₆	1	2	yes	—
73	C₂×C₈×C₅₉	—	2	1	no	—
74	C₂×C₈×C₅₉	—	2	1	no	—
75	C₂×C₈×C₅₉	—	2	1	no	—
76	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
81	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
82	C₄²×C₅₉	—	2	1	no	—
83	C₄²×C₅₉	—	2	1	no	—
84	(C₂×D₅₉)⋊C₄	—	2	2	no	—
85	(C₂×D₅₉)⋊C₄	C₁₆	1	4	yes	complemented
86	C₄×Dic₅₉	—	2	1	no	—
87	C₄×Dic₅₉	—	2	1	no	—
88	C₂×C₅₉⋊C₈	—	2	1	no	—
89	C₂×C₅₉⋊C₈	—	2	1	no	—
90	C₂×C₅₉⋊C₈	—	2	1	no	—
91	C₂×C₅₉⋊C₈	—	2	1	no	—
92	C₂×C₅₉⋊C₈	—	2	1	no	—
93	C₂×C₅₉⋊C₈	C₄⋊C₄	1	2	yes	—
94	C₈×D₅₉	—	2	2	no	—
95	C₈×D₅₉	—	2	2	no	—
96	C₈×D₅₉	—	2	2	no	—
97	C₈×D₅₉	—	2	2	no	—
98	C₁₆×C₅₉	—	2	2	no	—
99	C₁₆×C₅₉	—	4	1	no	—
100	C₅₉⋊C₁₆	—	2	2	no	—
101	C₅₉⋊C₁₆	—	4	1	no	—
102	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
103	C₂²×D₅₉	—	2	1	no	—
104	C₂²×D₅₉	—	2	1	no	—
105	C₂²×D₅₉	—	2	1	no	—
106	C₂²×D₅₉	—	2	1	no	—
107	C₂²×D₅₉	C₄⋊C₈	1	2	yes	—
108	C₂²×D₅₉	C₂×C₁₆	1	2	yes	—
109	C₂²×D₅₉	M₃₂	1	2	yes	—
110	C₂×C₄×C₅₉	—	2	1	no	—
111	C₂×C₄×C₅₉	—	2	1	no	—
112	C₂×C₄×C₅₉	—	2	1	no	—
113	C₂×C₄×C₅₉	—	2	1	no	—
114	C₂×C₄×C₅₉	—	2	1	no	—
115	C₂×C₄×C₅₉	C₂×C₁₆	1	2	yes	complemented
116	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
121	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
122	C₂×Dic₅₉	—	2	1	no	—
123	C₂×Dic₅₉	—	2	1	no	—
124	C₂×Dic₅₉	—	2	1	no	—
125	C₂×Dic₅₉	—	2	1	no	—
126	C₂×Dic₅₉	—	2	1	no	—
127	C₂×Dic₅₉	—	2	1	no	—
128	C₂×Dic₅₉	—	2	1	no	—
129	C₂×Dic₅₉	C₄⋊C₈	1	2	yes	—
130	C₂×Dic₅₉	C₂×C₁₆	1	2	yes	complemented
131	C₄×D₅₉	—	2	2	no	—
132	C₄×D₅₉	—	2	2	no	—
133	C₄×D₅₉	—	2	2	no	—
134	C₄×D₅₉	—	2	2	no	—
135	C₈×C₅₉	—	4	1	no	—
136	C₈×C₅₉	#120	1	1	yes	characteristic
137	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
138	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
139	C₈×C₅₉	#123	1	1	yes	characteristic
140	C₅₉⋊C₈	—	2	2	no	—
141	C₅₉⋊C₈	—	4	1	no	—
142	#12912	—	59	1	no	maximal, Sylow
143	C₂²×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
144	C₂²×C₅₉	C₈×D₄	1	1	yes	characteristic
145	C₂²×C₅₉	C₈×D₄	1	1	yes	characteristic
146	C₂²×C₅₉	C₂²×C₁₆	1	1	yes	characteristic
147	C₂²×C₅₉	C₂×M₃₂	1	1	yes	characteristic
148	C₂²×C₅₉	#504	1	1	yes	characteristic
149	C₂²×C₅₉	#504	1	1	yes	characteristic
150	C₂×D₅₉	—	2	2	no	—
151	C₂×D₅₉	—	2	2	no	—
152	C₂×D₅₉	—	2	2	no	—
153	C₂×D₅₉	—	2	2	no	—
154	C₂×D₅₉	—	2	2	no	—
155	C₂×D₅₉	—	2	2	no	—
156	C₂×D₅₉	C₄⋊C₁₆	1	4	yes	complemented
157	C₄×C₅₉	—	2	2	no	—
158	C₄×C₅₉	—	4	1	no	—
159	C₄×C₅₉	#385	1	1	yes	characteristic
160	C₄×C₅₉	#424	1	1	yes	characteristic
161	C₄×C₅₉	#435	1	1	yes	characteristic
162	C₄×C₅₉	#435	1	1	yes	characteristic
163	Dic₅₉	—	2	2	no	—
164	Dic₅₉	—	2	2	no	—
165	Dic₅₉	—	4	1	no	—
166	C₂³×C₁₆	—	59	1	no	—
167	C₂×C₄⋊C₁₆	—	59	1	no	—
168	C₂×C₄⋊C₁₆	—	59	1	no	—
169	C₂×C₂²⋊C₁₆	—	59	1	no	—
170	C₈×C₂²⋊C₄	—	59	1	no	—
171	#1374	—	59	1	no	—
172	#1374	D₅₉	1	1	yes	characteristic, complemented
173	C₂×C₅₉	#1850	1	1	yes	characteristic
174	C₂×C₅₉	C₂×C₄⋊C₁₆	1	1	yes	characteristic
175	C₂×C₅₉	#2151	1	1	yes	characteristic
176	C₂×C₅₉	D₄×C₁₆	1	1	yes	characteristic
177	C₂×C₅₉	D₄×C₁₆	1	1	yes	characteristic
178	C₂×C₅₉	#2168	1	1	yes	characteristic
179	C₂×C₅₉	#2168	1	1	yes	characteristic
180	D₅₉	—	2	4	no	—
181	C₂³×C₈	—	59	1	no	—
182	C₂²×C₁₆	—	59	1	no	—
183	C₂²×C₁₆	—	59	1	no	—
184	C₂²×C₁₆	—	118	2	no	—
185	C₂²×C₁₆	—	118	2	no	—
186	C₂²×C₁₆	—	118	2	no	—
187	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
188	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
189	C₂×C₂²⋊C₈	—	59	1	no	—
190	C₂×C₄×C₈	—	59	1	no	—
191	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
192	C₄×C₂²⋊C₄	—	59	1	no	—
193	C₄⋊C₁₆	—	118	2	no	—
194	C₄⋊C₁₆	—	118	2	no	—
195	C₂²⋊C₁₆	—	118	2	no	—
196	#336	—	59	1	no	—
197	#336	—	59	1	no	—
198	C₅₉	#12912	1	1	yes	characteristic, complemented, Sylow
199	C₂³×C₄	—	59	1	no	—
200	C₂²×C₈	—	59	1	no	—
201	C₂²×C₈	—	59	2	no	—
202	C₂²×C₈	—	59	2	no	—
203	C₂²×C₈	—	59	1	no	—
204	C₂²×C₈	—	118	1	no	—
205	C₂²×C₈	—	118	1	no	—
206	C₂²×C₈	—	118	1	no	—
207	C₂²×C₈	—	118	1	no	—
208	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
209	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
210	C₂×C₂²⋊C₄	—	59	1	no	—
211	C₂×C₂²⋊C₄	—	59	1	no	—
212	C₂×C₄²	—	59	1	no	—
213	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
214	C₂×C₁₆	—	2	1	no	—
215	C₂×C₁₆	—	2	1	no	—
216	C₂×C₁₆	—	2	1	no	—
217	C₂×C₁₆	—	2	1	no	—
218	C₂×C₁₆	—	2	1	no	—
219	C₂×C₁₆	—	59	2	no	—
220	C₂×C₁₆	—	118	8	no	—
221	C₂×C₁₆	—	118	1	no	—
222	C₂×C₁₆	—	118	1	no	—
223	C₂×C₁₆	—	118	1	no	—
224	C₂×C₁₆	—	118	1	no	—
225	C₂×C₁₆	—	118	1	no	—
226	C₂×C₁₆	D₂₃₆	1	2	yes	—
227	C₂²⋊C₈	—	118	2	no	—
228	C₄×C₈	—	2	1	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	118	1	no	—
231	C₄×C₈	—	118	1	no	—
232	#95	—	59	1	no	—
233	#95	—	59	1	no	—
234	C₂⁴	—	59	1	no	—
235	C₂²×C₄	—	59	1	no	—
236	C₂²×C₄	—	59	1	no	—
237	C₂²×C₄	—	59	2	no	—
238	C₂²×C₄	—	59	2	no	—
239	C₂²×C₄	—	59	1	no	—
240	C₂²×C₄	—	118	1	no	—
241	C₂²×C₄	—	118	1	no	—
242	C₂²×C₄	—	118	1	no	—
243	C₂²×C₄	—	118	1	no	—
244	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
245	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	59	2	no	—
251	C₂×C₈	—	118	1	no	—
252	C₂×C₈	—	118	1	no	—
253	C₂×C₈	—	118	2	no	—
254	C₂×C₈	—	118	2	no	—
255	C₂×C₈	—	118	2	no	—
256	C₂×C₈	—	118	2	no	—
257	C₂×C₈	—	118	1	no	—
258	C₂×C₈	—	118	1	no	—
259	C₂×C₈	—	118	1	no	—
260	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
261	C₂×C₈	#11745685	1	1	yes	characteristic
262	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
263	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
264	C₂×C₈	#11745687	1	1	yes	characteristic
265	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
266	C₂²⋊C₄	—	59	4	no	—
267	C₂²⋊C₄	—	118	2	no	—
268	C₄²	—	2	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	118	1	no	—
271	C₄²	—	118	1	no	—
272	C₁₆	—	2	2	no	—
273	C₁₆	—	4	1	no	—
274	C₁₆	—	118	2	no	—
275	C₁₆	—	236	1	no	—
276	C₂³	—	59	2	no	—
277	C₂³	—	59	2	no	—
278	C₂³	—	59	2	no	—
279	C₂³	—	118	1	no	—
280	C₂³	—	118	1	no	—
281	C₂³	—	118	1	no	—
282	C₂³	—	118	1	no	—
283	C₂³	C₂×C₈×D₅₉	1	1	yes	characteristic
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	59	2	no	—
290	C₂×C₄	—	59	2	no	—
291	C₂×C₄	—	118	1	no	—
292	C₂×C₄	—	118	1	no	—
293	C₂×C₄	—	118	1	no	—
294	C₂×C₄	—	118	1	no	—
295	C₂×C₄	—	118	2	no	—
296	C₂×C₄	—	118	2	no	—
297	C₂×C₄	—	118	2	no	—
298	C₂×C₄	—	118	2	no	—
299	C₂×C₄	—	118	1	no	—
300	C₂×C₄	—	118	1	no	—
301	C₂×C₄	—	118	1	no	—
302	C₂×C₄	C₁₆×D₅₉	1	2	yes	complemented
303	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
304	C₂×C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
305	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
306	C₂×C₄	M₁₆×D₅₉	1	1	yes	characteristic
307	C₂×C₄	#14273733	1	1	yes	characteristic
308	C₂×C₄	#14273735	1	1	yes	characteristic
309	C₈	—	4	1	no	—
310	C₈	—	118	2	no	—
311	C₈	—	236	1	no	—
312	C₈	#14273697	1	1	yes	characteristic
313	C₈	#14273705	1	1	yes	characteristic
314	C₈	#14273707	1	1	yes	characteristic
315	C₈	#14273709	1	1	yes	characteristic
316	C₂²	—	59	4	no	—
317	C₂²	—	118	2	no	—
318	C₂²	—	118	2	no	—
319	C₂²	—	118	2	no	—
320	C₂²	—	118	2	no	—
321	C₂²	—	118	2	no	—
322	C₂²	—	118	2	no	—
323	C₂²	C₄⋊C₈×D₅₉	1	1	yes	characteristic
324	C₂²	(C₂²⋊D₅₉)⋊C₈	1	1	yes	characteristic
325	C₂²	C₈×D₂₃₆	1	1	yes	characteristic
326	C₂²	C₂×C₁₆×D₅₉	1	1	yes	characteristic
327	C₂²	M₃₂×D₅₉	1	1	yes	characteristic
328	C₂²	#16837818	1	1	yes	characteristic
329	C₂²	#16837820	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	4	1	no	—
332	C₄	—	118	2	no	—
333	C₄	—	118	2	no	—
334	C₄	—	236	1	no	—
335	C₄	#16837276	1	1	yes	characteristic
336	C₄	#16837427	1	1	yes	characteristic
337	C₄	#16837475	1	1	yes	characteristic
338	C₄	#16837476	1	1	yes	characteristic
339	C₂	—	118	4	no	—
340	C₂	#21424275	1	1	yes	characteristic
341	C₂	C₄⋊C₁₆×D₅₉	1	1	yes	characteristic
342	C₂	#21425853	1	1	yes	characteristic
343	C₂	(C₂²⋊D₅₉)⋊C₁₆	1	1	yes	characteristic
344	C₂	C₁₆×D₂₃₆	1	1	yes	characteristic
345	C₂	#21425923	1	1	yes	characteristic
346	C₂	#21425924	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₉
commutator	C₂²×C₅₉	C₂²×C₁₆
Fitting	C₅₉×#1374	C₂
socle	C₂³×C₅₉	C₂²×C₈
prime radical	C₂³×D₅₉	C₂×C₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1374	D₅₉
hypocenter	C₅₉	#12912
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	semidirect
3. C₂×Dic₅₉⋊C₁₆	C₂	semidirect
6. #21422173	C₂	semidirect
7. C₅₉×#1374	C₂	semidirect
85. (C₂×D₅₉)⋊C₄	C₁₆	semidirect
115. C₂×C₄×C₅₉	C₂×C₁₆	semidirect
130. C₂×Dic₅₉	C₂×C₁₆	semidirect
156. C₂×D₅₉	C₄⋊C₁₆	semidirect
172. #1374	D₅₉	semidirect
198. C₅₉	#12912	semidirect
302. C₂×C₄	C₁₆×D₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×D₅₉	2
C₂×C₄⋊(C₅₉⋊C₁₆)	2
C₂×Dic₅₉⋊C₁₆	2
C₂×(C₂×D₅₉)⋊C₁₆	2
C₈×(C₂×D₅₉)⋊C₄	2
#21422173	2
C₅₉×#1374	2
#12912	59