Subgroups — #28479750

Order 15104.

400 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21433072	C₂	1	1	yes	maximal, characteristic
2	#21423848	C₂	1	1	yes	maximal, characteristic, complemented
3	#21422684	C₂	1	1	yes	maximal, characteristic
4	#21422377	C₂	1	1	yes	maximal, characteristic, complemented
5	#21421996	C₂	1	1	yes	maximal, characteristic, complemented
6	#21421989	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1500	C₂	1	1	yes	maximal, characteristic
8	#16837533	C₄	1	1	yes	characteristic, complemented
9	Q₈×C₅₉⋊C₈	C₄	1	1	yes	characteristic, complemented
10	#16837480	C₄	1	1	yes	characteristic, complemented
11	#16837474	—	2	1	no	—
12	D₄×C₅₉⋊C₈	—	2	1	no	—
13	D₄×C₅₉⋊C₈	C₄	1	1	yes	characteristic, complemented
14	#16837463	—	2	1	no	—
15	#16837459	C₂²	1	1	yes	characteristic
16	#16837358	—	2	1	no	—
17	C₄×#11745664	C₂²	1	1	yes	characteristic
18	C₂×#14273638	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
20	#16837157	—	2	1	no	—
21	#16837157	—	2	1	no	—
22	#16837144	C₄	1	2	yes	complemented
23	#16837142	C₄	1	2	yes	complemented
24	#16837124	—	2	1	no	—
25	C₈×C₅₉⋊C₈	—	2	1	no	—
26	C₅₉×#529	C₂²	1	1	yes	characteristic
27	C₅₉×#431	C₂²	1	1	yes	characteristic
28	C₅₉×#376	C₂²	1	1	yes	characteristic
29	C₅₉×#328	C₄	1	2	yes	—
30	C₅₉×#327	C₄	1	2	yes	—
31	C₂×#11745664	—	2	1	no	—
32	C₂×#11745664	—	2	1	no	—
33	C₂²×C₅₉⋊C₈	—	2	1	no	—
34	C₂²×C₅₉⋊C₈	—	2	1	no	—
35	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
36	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
37	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
38	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
39	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
40	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
41	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
42	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
43	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
44	#14273638	—	2	1	no	—
45	#14273638	—	2	1	no	—
46	#14273638	—	2	1	no	—
47	#14273638	C₂×C₄	1	2	yes	—
48	#14273638	C₂×C₄	1	2	yes	—
49	C₄×C₅₉⋊C₈	—	2	1	no	—
50	C₄×C₅₉⋊C₈	—	2	1	no	—
51	C₄×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₄×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
54	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
55	#126×C₅₉	—	2	1	no	—
56	#125×C₅₉	C₂×C₄	1	1	yes	characteristic
57	#124×C₅₉	C₂×C₄	1	1	yes	characteristic
58	#123×C₅₉	—	2	1	no	—
59	C₂²⋊Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
60	#121×C₅₉	C₂×C₄	1	1	yes	characteristic
61	C₄×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
62	C₄×D₄×C₅₉	—	2	1	no	—
63	C₄×D₄×C₅₉	C₂×C₄	1	1	yes	characteristic
64	#117×C₅₉	—	2	1	no	—
65	C₂×C₄²×C₅₉	—	2	1	no	—
66	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₅₉	—	2	1	no	—
68	C₂²⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₅₉	—	2	1	no	—
70	#95×C₅₉	—	2	1	no	—
71	#95×C₅₉	C₂×C₄	1	2	yes	—
72	C₂×Q₈×C₅₉	C₄²	1	1	yes	characteristic
73	C₂×D₄×C₅₉	—	2	1	no	—
74	C₂×D₄×C₅₉	C₄²	1	1	yes	characteristic
75	C₂²×C₄×C₅₉	—	2	1	no	—
76	C₂²×C₄×C₅₉	—	2	2	no	—
77	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
80	C₂×C₈×C₅₉	—	2	2	no	—
81	C₄⋊C₄×C₅₉	—	2	1	no	—
82	C₄⋊C₄×C₅₉	—	2	1	no	—
83	C₄⋊C₄×C₅₉	—	2	1	no	—
84	C₄⋊C₄×C₅₉	—	2	1	no	—
85	C₄⋊C₄×C₅₉	C₄²	1	1	yes	characteristic
86	C₄⋊C₄×C₅₉	C₄²	1	1	yes	characteristic
87	C₂²⋊C₄×C₅₉	—	2	1	no	—
88	C₂²⋊C₄×C₅₉	—	2	1	no	—
89	C₂²⋊C₄×C₅₉	—	2	1	no	—
90	C₂²⋊C₄×C₅₉	—	2	1	no	—
91	C₂²⋊C₄×C₅₉	—	2	1	no	—
92	C₄²×C₅₉	—	2	1	no	—
93	C₄²×C₅₉	—	4	1	no	—
94	C₄²×C₅₉	—	4	1	no	—
95	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₅₉	#41	1	1	yes	characteristic
99	#11745664	—	2	4	no	—
100	#11745664	—	4	2	no	—
101	C₂×C₅₉⋊C₈	—	2	2	no	—
102	C₂×C₅₉⋊C₈	—	2	1	no	—
103	C₂×C₅₉⋊C₈	—	2	1	no	—
104	C₂×C₅₉⋊C₈	—	2	2	no	—
105	C₂×C₅₉⋊C₈	—	4	1	no	—
106	C₂×C₅₉⋊C₈	—	4	1	no	—
107	C₂×C₅₉⋊C₈	—	4	2	no	—
108	C₂×C₅₉⋊C₈	—	4	1	no	—
109	C₂×C₅₉⋊C₈	—	4	1	no	—
110	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
114	C₂³×C₅₉	—	2	1	no	—
115	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
116	Q₈×C₅₉	—	4	1	no	—
117	D₄×C₅₉	—	4	1	no	—
118	D₄×C₅₉	—	4	2	no	—
119	C₂×C₄×C₅₉	—	2	2	no	—
120	C₂×C₄×C₅₉	—	2	1	no	—
121	C₂×C₄×C₅₉	—	2	1	no	—
122	C₂×C₄×C₅₉	—	2	1	no	—
123	C₂×C₄×C₅₉	—	2	1	no	—
124	C₂×C₄×C₅₉	—	2	1	no	—
125	C₂×C₄×C₅₉	—	4	1	no	—
126	C₂×C₄×C₅₉	—	4	1	no	—
127	C₂×C₄×C₅₉	—	4	1	no	—
128	C₂×C₄×C₅₉	—	4	2	no	—
129	C₂×C₄×C₅₉	—	4	2	no	—
130	C₂×C₄×C₅₉	—	4	2	no	—
131	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
132	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
134	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
137	C₈×C₅₉	—	4	2	no	—
138	C₅₉⋊C₈	—	2	4	no	—
139	C₅₉⋊C₈	—	2	1	no	—
140	C₅₉⋊C₈	—	2	1	no	—
141	C₅₉⋊C₈	—	2	1	no	—
142	C₅₉⋊C₈	—	2	1	no	—
143	C₅₉⋊C₈	—	4	2	no	—
144	C₅₉⋊C₈	—	4	1	no	—
145	C₅₉⋊C₈	—	4	1	no	—
146	#8250	—	59	1	no	maximal, Sylow
147	C₂²×C₅₉	—	2	1	no	—
148	C₂²×C₅₉	—	2	1	no	—
149	C₂²×C₅₉	—	4	1	no	—
150	C₂²×C₅₉	—	4	1	no	—
151	C₂²×C₅₉	—	4	1	no	—
152	C₂²×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
153	C₂²×C₅₉	#443	1	2	yes	—
154	C₄×C₅₉	—	2	1	no	—
155	C₄×C₅₉	—	2	1	no	—
156	C₄×C₅₉	—	2	1	no	—
157	C₄×C₅₉	—	2	1	no	—
158	C₄×C₅₉	—	2	1	no	—
159	C₄×C₅₉	—	4	1	no	—
160	C₄×C₅₉	—	4	1	no	—
161	C₄×C₅₉	—	4	1	no	—
162	C₄×C₅₉	—	8	2	no	—
163	C₄×C₅₉	#421	1	1	yes	characteristic
164	C₄×C₅₉	#421	1	1	yes	characteristic
165	#2966	—	59	1	no	—
166	#1746	—	59	1	no	—
167	#1524	—	59	1	no	—
168	#1500	D₅₉	1	1	yes	characteristic
169	C₈×M₁₆	—	59	1	no	—
170	#1302	—	59	1	no	—
171	#1300	—	59	1	no	—
172	C₂×C₅₉	—	2	2	no	—
173	C₂×C₅₉	—	8	1	no	—
174	C₂×C₅₉	#1757	1	1	yes	characteristic
175	C₂×C₅₉	#1759	1	1	yes	characteristic
176	C₂×C₅₉	#1764	1	1	yes	characteristic
177	#529	C₂×D₅₉	1	1	yes	characteristic
178	#446	—	59	1	no	—
179	C₈×Q₈	—	59	1	no	—
180	#436	—	59	1	no	—
181	#435	—	118	1	no	—
182	C₈×D₄	—	59	1	no	—
183	C₈×D₄	—	118	1	no	—
184	#433	—	118	1	no	—
185	#432	—	59	1	no	—
186	#431	C₂×D₅₉	1	1	yes	characteristic
187	#405	—	118	1	no	—
188	C₄×M₁₆	—	59	1	no	—
189	C₂×#97	—	59	1	no	—
190	C₂×C₄×C₈	—	59	1	no	—
191	#376	C₂×D₅₉	1	1	yes	characteristic
192	#336	—	118	1	no	—
193	#336	—	118	1	no	—
194	#330	—	59	2	no	—
195	#329	—	59	2	no	—
196	#328	Dic₅₉	1	2	yes	—
197	#327	Dic₅₉	1	2	yes	—
198	#322	—	118	1	no	—
199	C₈²	—	118	1	no	—
200	C₅₉	#8250	1	1	yes	characteristic, complemented, Sylow
201	C₂×M₁₆	—	118	1	no	—
202	C₂×M₁₆	—	118	1	no	—
203	C₂²×C₈	—	59	1	no	—
204	C₂²×C₈	—	59	1	no	—
205	C₂²×C₈	—	118	1	no	—
206	C₂²×C₈	—	118	1	no	—
207	#126	—	2	1	no	—
208	#125	C₄×D₅₉	1	1	yes	characteristic
209	#124	C₄×D₅₉	1	1	yes	characteristic
210	#123	—	2	1	no	—
211	C₂²⋊Q₈	C₂×Dic₅₉	1	1	yes	characteristic
212	#121	C₂×Dic₅₉	1	1	yes	characteristic
213	C₄×Q₈	C₄×D₅₉	1	1	yes	characteristic
214	C₄×D₄	—	2	1	no	—
215	C₄×D₄	C₄×D₅₉	1	1	yes	characteristic
216	#117	—	2	1	no	—
217	C₂×C₄²	—	2	1	no	—
218	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	59	1	no	—
221	C₄⋊C₈	—	59	1	no	—
222	C₄⋊C₈	—	118	1	no	—
223	C₄⋊C₈	—	118	1	no	—
224	C₂²⋊C₈	—	118	1	no	—
225	C₂²⋊C₈	—	118	1	no	—
226	C₂²⋊C₈	—	118	1	no	—
227	C₂²⋊C₈	C₂×Dic₅₉	1	2	yes	—
228	#97	—	59	2	no	—
229	#97	—	59	2	no	—
230	#97	—	118	1	no	—
231	#97	—	118	1	no	—
232	#97	—	118	1	no	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	—	59	1	no	—
235	C₄×C₈	—	59	1	no	—
236	C₄×C₈	—	59	1	no	—
237	C₄×C₈	—	59	1	no	—
238	C₄×C₈	—	118	1	no	—
239	C₄×C₈	—	118	1	no	—
240	#95	—	2	1	no	—
241	#95	C₂×Dic₅₉	1	2	yes	—
242	C₂×Q₈	C₄×Dic₅₉	1	1	yes	characteristic
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	C₄×Dic₅₉	1	1	yes	characteristic
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
248	C₂²×C₄	#11745687	1	1	yes	characteristic
249	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
250	M₁₆	—	118	4	no	—
251	M₁₆	—	236	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	59	1	no	—
254	C₂×C₈	—	59	1	no	—
255	C₂×C₈	—	59	1	no	—
256	C₂×C₈	—	59	1	no	—
257	C₂×C₈	—	118	2	no	—
258	C₂×C₈	—	118	1	no	—
259	C₂×C₈	—	118	1	no	—
260	C₂×C₈	—	118	2	no	—
261	C₂×C₈	—	236	1	no	—
262	C₂×C₈	—	236	1	no	—
263	C₂×C₈	—	236	2	no	—
264	C₂×C₈	—	236	1	no	—
265	C₂×C₈	—	236	1	no	—
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	C₄×Dic₅₉	1	1	yes	characteristic
271	C₄⋊C₄	C₄×Dic₅₉	1	1	yes	characteristic
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	2	1	no	—
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₄²	—	2	1	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	4	1	no	—
280	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
281	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
282	C₄²	D₄×D₅₉	1	1	yes	characteristic
283	C₄²	#11745687	1	1	yes	characteristic
284	C₂³	—	2	1	no	—
285	C₂³	D₄×Dic₅₉	1	1	yes	characteristic
286	Q₈	—	4	1	no	—
287	D₄	—	4	1	no	—
288	D₄	—	4	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
302	C₂×C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
303	C₂×C₄	#14273683	1	1	yes	characteristic
304	C₂×C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
305	C₂×C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
306	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
307	C₈	—	4	2	no	—
308	C₈	—	118	4	no	—
309	C₈	—	118	1	no	—
310	C₈	—	118	1	no	—
311	C₈	—	118	1	no	—
312	C₈	—	118	1	no	—
313	C₈	—	236	2	no	—
314	C₈	—	236	1	no	—
315	C₈	—	236	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	C₂²⋊C₄×Dic₅₉	1	1	yes	characteristic
322	C₂²	#16837519	1	2	yes	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	8	2	no	—
332	C₄	#16837417	1	1	yes	characteristic
333	C₄	#16837417	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	8	1	no	—
336	C₂	#21423898	1	1	yes	characteristic
337	C₂	#21423908	1	1	yes	characteristic
338	C₂	#21423936	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1500	C₂
socle	C₂²×C₅₉	C₄×C₂²⋊C₄
prime radical	#121×C₅₉	C₂×C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1500	D₅₉
hypocenter	C₅₉	#8250
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1500

Complements

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

N	H ≅ G/N	type
2. #21423848	C₂	semidirect
4. #21422377	C₂	semidirect
5. #21421996	C₂	semidirect
6. #21421989	C₂	semidirect
8. #16837533	C₄	semidirect
9. Q₈×C₅₉⋊C₈	C₄	semidirect
10. #16837480	C₄	semidirect
13. D₄×C₅₉⋊C₈	C₄	semidirect
22. #16837144	C₄	semidirect
23. #16837142	C₄	semidirect
200. C₅₉	#8250	semidirect

Maximal subgroups

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

subgroup	index
#21433072	2
#21423848	2
#21422684	2
#21422377	2
#21421996	2
#21421989	2
C₅₉×#1500	2
#8250	59