Subgroups — #28479751

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	#21433071	C₂	1	1	yes	maximal, characteristic
2	#21423847	C₂	1	1	yes	maximal, characteristic, complemented
3	#21422685	C₂	1	1	yes	maximal, characteristic
4	#21422583	C₂	1	1	yes	maximal, characteristic
5	#21421996	C₂	1	1	yes	maximal, characteristic, complemented
6	#21421990	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×M₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837936	C₂²	1	1	yes	characteristic
9	#16837532	C₄	1	1	yes	characteristic, complemented
10	C₈×Dic₁₁₈	C₄	1	1	yes	characteristic, complemented
11	#16837481	C₄	1	1	yes	characteristic, complemented
12	#16837473	—	2	1	no	—
13	C₈×D₂₃₆	C₄	1	1	yes	characteristic, complemented
14	C₈×C₂²⋊D₅₉	—	2	1	no	—
15	#16837461	—	2	1	no	—
16	#16837459	C₂²	1	1	yes	characteristic
17	#16837356	—	2	1	no	—
18	C₂×#14273638	C₂²	1	1	yes	characteristic
19	#16837248	C₂²	1	1	yes	characteristic
20	#16837156	—	2	1	no	—
21	#16837156	—	2	1	no	—
22	#16837143	C₄	1	2	yes	complemented
23	#16837141	C₄	1	2	yes	complemented
24	#16837140	C₄	1	2	yes	—
25	#16837137	C₄	1	2	yes	—
26	C₅₉×#431	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
29	C₅₉×#322	—	2	1	no	—
30	C₈²×C₅₉	—	2	1	no	—
31	C₂×#11745659	—	2	1	no	—
32	C₂²×C₅₉⋊C₈	—	2	1	no	—
33	C₂×C₈×D₅₉	—	2	1	no	—
34	#14273721	—	2	1	no	—
35	#14273719	C₂×C₄	1	1	yes	characteristic
36	#14273715	C₂×C₄	1	1	yes	characteristic
37	#14273710	—	2	1	no	—
38	#14273706	C₂×C₄	1	1	yes	characteristic
39	#14273701	C₂×C₄	1	1	yes	characteristic
40	C₄×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
41	C₄×D₂₃₆	C₂×C₄	1	1	yes	characteristic
42	C₄×C₂²⋊D₅₉	—	2	1	no	—
43	#14273684	—	2	1	no	—
44	C₂×C₄×Dic₅₉	—	2	1	no	—
45	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
46	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
47	Dic₅₉⋊C₈	—	2	1	no	—
48	Dic₅₉⋊C₈	—	2	1	no	—
49	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
50	(C₂×D₅₉)⋊C₈	—	2	1	no	—
51	(C₂×D₅₉)⋊C₈	—	2	1	no	—
52	#14273638	—	2	1	no	—
53	#14273638	—	2	1	no	—
54	#14273638	C₂×C₄	1	2	yes	—
55	#14273637	—	2	1	no	—
56	C₈×Dic₅₉	—	2	1	no	—
57	C₅₉⋊#95	—	2	1	no	—
58	C₅₉⋊#95	C₂×C₄	1	2	yes	—
59	C₂×M₁₆×C₅₉	—	2	1	no	—
60	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
62	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
63	C₄⋊C₈×C₅₉	—	2	1	no	—
64	C₂²⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
65	#97×C₅₉	C₂×C₄	1	2	yes	—
66	C₄×C₈×C₅₉	—	2	1	no	—
67	C₄×C₈×C₅₉	—	2	1	no	—
68	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
71	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
75	C₂×C₂²⋊D₅₉	—	2	1	no	—
76	C₂²×Dic₅₉	—	2	2	no	—
77	C₂×D₂₃₆	C₄²	1	1	yes	characteristic
78	C₂×C₄×D₅₉	—	2	1	no	—
79	C₂×Dic₁₁₈	C₄²	1	1	yes	characteristic
80	M₁₆×C₅₉	—	2	4	no	—
81	C₂×C₈×C₅₉	—	2	2	no	—
82	C₂×C₈×C₅₉	—	2	2	no	—
83	C₂×C₈×C₅₉	—	4	1	no	—
84	C₂×C₈×C₅₉	—	4	1	no	—
85	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
89	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₅₉	#41	1	1	yes	characteristic
93	C₂²⋊Dic₅₉	—	2	1	no	—
94	(C₂×D₅₉)⋊C₄	—	2	1	no	—
95	(C₂×D₅₉)⋊C₄	—	2	1	no	—
96	(C₂×D₅₉)⋊C₄	—	2	1	no	—
97	(C₂×D₅₉)⋊C₄	—	2	1	no	—
98	C₄⋊Dic₅₉	C₄²	1	1	yes	characteristic
99	C₄⋊Dic₅₉	C₄²	1	1	yes	characteristic
100	Dic₅₉⋊C₄	—	2	1	no	—
101	Dic₅₉⋊C₄	—	2	1	no	—
102	Dic₅₉⋊C₄	—	2	1	no	—
103	Dic₅₉⋊C₄	—	2	1	no	—
104	C₄×Dic₅₉	—	2	1	no	—
105	C₄×Dic₅₉	—	4	1	no	—
106	C₄×Dic₅₉	—	4	1	no	—
107	C₂×C₅₉⋊C₈	—	2	1	no	—
108	C₂×C₅₉⋊C₈	—	2	1	no	—
109	C₂×C₅₉⋊C₈	—	2	2	no	—
110	C₂×C₅₉⋊C₈	—	4	1	no	—
111	C₂×C₅₉⋊C₈	—	4	1	no	—
112	#11745659	—	4	2	no	—
113	C₈×D₅₉	—	4	2	no	—
114	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
115	C₂²×D₅₉	—	2	1	no	—
116	C₂×C₄×C₅₉	—	2	2	no	—
117	C₂×C₄×C₅₉	—	2	1	no	—
118	C₂×C₄×C₅₉	—	2	1	no	—
119	C₂×C₄×C₅₉	—	4	1	no	—
120	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
121	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
126	C₂²⋊D₅₉	—	4	2	no	—
127	C₂×Dic₅₉	—	2	1	no	—
128	C₂×Dic₅₉	—	2	1	no	—
129	C₂×Dic₅₉	—	2	1	no	—
130	C₂×Dic₅₉	—	4	2	no	—
131	C₂×Dic₅₉	—	4	2	no	—
132	C₂×Dic₅₉	—	4	2	no	—
133	D₂₃₆	—	4	1	no	—
134	C₄×D₅₉	—	4	1	no	—
135	C₄×D₅₉	—	4	1	no	—
136	Dic₁₁₈	—	4	1	no	—
137	C₈×C₅₉	—	2	4	no	—
138	C₈×C₅₉	—	2	1	no	—
139	C₈×C₅₉	—	2	1	no	—
140	C₈×C₅₉	—	2	1	no	—
141	C₈×C₅₉	—	2	1	no	—
142	C₈×C₅₉	—	4	2	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	C₄×C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₅₉	#443	1	2	yes	—
151	C₂×D₅₉	—	4	1	no	—
152	C₂×D₅₉	—	4	1	no	—
153	C₂×D₅₉	—	4	1	no	—
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	#421	1	1	yes	characteristic
160	C₄×C₅₉	#421	1	1	yes	characteristic
161	Dic₅₉	—	4	1	no	—
162	Dic₅₉	—	4	1	no	—
163	Dic₅₉	—	4	1	no	—
164	Dic₅₉	—	8	2	no	—
165	#2966	—	59	1	no	—
166	#1746	—	59	1	no	—
167	#1524	—	59	1	no	—
168	#1500	—	59	1	no	—
169	C₈×M₁₆	D₅₉	1	1	yes	characteristic, complemented
170	#1302	—	59	1	no	—
171	#1300	—	59	1	no	—
172	C₂×C₅₉	—	2	2	no	—
173	C₂×C₅₉	#1757	1	1	yes	characteristic
174	C₂×C₅₉	#1759	1	1	yes	characteristic
175	C₂×C₅₉	#1764	1	1	yes	characteristic
176	D₅₉	—	8	1	no	—
177	#529	—	59	1	no	—
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	C₂×D₅₉	1	1	yes	characteristic
189	C₂×#97	—	59	1	no	—
190	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
191	#376	—	59	1	no	—
192	#336	—	118	1	no	—
193	#336	—	118	1	no	—
194	#330	—	59	2	no	—
195	#329	—	59	2	no	—
196	#328	—	59	2	no	—
197	#327	—	59	2	no	—
198	#322	—	2	1	no	—
199	C₈²	—	2	1	no	—
200	C₅₉	#8250	1	1	yes	characteristic, complemented, Sylow
201	C₂×M₁₆	—	2	1	no	—
202	C₂×M₁₆	—	118	1	no	—
203	C₂²×C₈	—	118	1	no	—
204	C₂²×C₈	—	118	1	no	—
205	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
206	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
207	#126	—	118	1	no	—
208	#125	—	59	1	no	—
209	#124	—	59	1	no	—
210	#123	—	118	1	no	—
211	C₂²⋊Q₈	—	59	1	no	—
212	#121	—	59	1	no	—
213	C₄×Q₈	—	59	1	no	—
214	C₄×D₄	—	59	1	no	—
215	C₄×D₄	—	118	1	no	—
216	#117	—	118	1	no	—
217	C₂×C₄²	—	118	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	C₄×D₅₉	1	2	yes	—
228	#97	—	59	2	no	—
229	#97	—	118	1	no	—
230	#97	—	118	1	no	—
231	#97	—	118	1	no	—
232	#97	C₄×D₅₉	1	2	yes	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	—	2	1	no	—
235	C₄×C₈	—	118	1	no	—
236	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
237	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
238	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
239	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
240	#95	—	59	2	no	—
241	#95	—	118	1	no	—
242	C₂×Q₈	—	59	1	no	—
243	C₂×D₄	—	59	1	no	—
244	C₂×D₄	—	118	1	no	—
245	C₂²×C₄	—	118	1	no	—
246	C₂²×C₄	—	118	2	no	—
247	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
249	C₂²×C₄	#11745685	1	1	yes	characteristic
250	M₁₆	—	2	4	no	—
251	M₁₆	—	236	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	118	1	no	—
257	C₂×C₈	—	118	1	no	—
258	C₂×C₈	—	118	2	no	—
259	C₂×C₈	—	236	2	no	—
260	C₂×C₈	—	236	1	no	—
261	C₂×C₈	—	236	1	no	—
262	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
263	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
264	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
265	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
266	C₄⋊C₄	—	59	1	no	—
267	C₄⋊C₄	—	59	1	no	—
268	C₄⋊C₄	—	118	1	no	—
269	C₄⋊C₄	—	118	1	no	—
270	C₄⋊C₄	—	118	1	no	—
271	C₄⋊C₄	—	118	1	no	—
272	C₂²⋊C₄	—	118	1	no	—
273	C₂²⋊C₄	—	118	1	no	—
274	C₂²⋊C₄	—	118	1	no	—
275	C₂²⋊C₄	—	118	1	no	—
276	C₂²⋊C₄	—	118	1	no	—
277	C₄²	—	118	1	no	—
278	C₄²	—	236	1	no	—
279	C₄²	—	236	1	no	—
280	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
281	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
282	C₄²	#11745685	1	1	yes	characteristic
283	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
284	C₂³	—	118	1	no	—
285	C₂³	C₄×D₂₃₆	1	1	yes	characteristic
286	Q₈	—	236	1	no	—
287	D₄	—	236	1	no	—
288	D₄	—	236	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	118	1	no	—
294	C₂×C₄	—	118	1	no	—
295	C₂×C₄	—	118	1	no	—
296	C₂×C₄	—	236	2	no	—
297	C₂×C₄	—	236	2	no	—
298	C₂×C₄	—	236	2	no	—
299	C₂×C₄	—	236	1	no	—
300	C₂×C₄	—	236	1	no	—
301	C₂×C₄	C₄²×D₅₉	1	1	yes	characteristic
302	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
303	C₂×C₄	#14273684	1	1	yes	characteristic
304	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
305	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
306	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
307	C₈	—	2	4	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	4	2	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	—	236	1	no	—
319	C₂²	—	236	1	no	—
320	C₂²	—	236	1	no	—
321	C₂²	C₄×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
322	C₂²	#16837520	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	—	472	2	no	—
329	C₄	—	236	1	no	—
330	C₄	—	236	1	no	—
331	C₄	—	236	1	no	—
332	C₄	#16837418	1	1	yes	characteristic
333	C₄	#16837418	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	472	1	no	—
336	C₂	#21423897	1	1	yes	characteristic
337	C₂	#21423909	1	1	yes	characteristic
338	C₂	#21423933	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄
commutator	C₂×C₄×C₅₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₈×M₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₄×C₂²⋊C₄
prime radical	#14273701	C₂×C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₈×M₁₆	D₅₉
hypocenter	C₅₉	#8250
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
2. #21423847	C₂	semidirect
5. #21421996	C₂	semidirect
6. #21421990	C₂	semidirect
7. C₈×M₁₆×C₅₉	C₂	semidirect
9. #16837532	C₄	semidirect
10. C₈×Dic₁₁₈	C₄	semidirect
11. #16837481	C₄	semidirect
13. C₈×D₂₃₆	C₄	semidirect
22. #16837143	C₄	semidirect
23. #16837141	C₄	semidirect
169. C₈×M₁₆	D₅₉	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
#21433071	2
#21423847	2
#21422685	2
#21422583	2
#21421996	2
#21421990	2
C₈×M₁₆×C₅₉	2
#8250	59