Subgroups — #28479747

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	#21423847	C₂	1	1	yes	maximal, characteristic, complemented
2	#21422684	C₂	1	1	yes	maximal, characteristic
3	#21422582	C₂	1	1	yes	maximal, characteristic
4	C₈×#11745664	C₂	1	1	yes	maximal, characteristic, complemented
5	#21421994	C₂	1	1	yes	maximal, characteristic, complemented
6	#21421988	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#2966	C₂	1	1	yes	maximal, characteristic
8	#16837455	C₂²	1	1	yes	characteristic
9	C₄×#11745664	C₂²	1	1	yes	characteristic
10	C₂×#14273638	C₂²	1	1	yes	characteristic
11	#16837248	C₂²	1	1	yes	characteristic
12	#16837156	—	2	1	no	—
13	#16837156	—	2	1	no	—
14	#16837144	C₄	1	2	yes	complemented
15	#16837142	C₄	1	2	yes	complemented
16	#16837139	C₄	1	2	yes	—
17	#16837136	C₄	1	2	yes	—
18	#16837124	—	2	1	no	—
19	C₈×C₅₉⋊C₈	—	2	1	no	—
20	C₅₉×#529	C₂²	1	1	yes	characteristic
21	C₅₉×#446	C₄	1	1	yes	characteristic, complemented
22	C₈×Q₈×C₅₉	C₄	1	1	yes	characteristic, complemented
23	C₅₉×#436	C₄	1	1	yes	characteristic, complemented
24	C₅₉×#435	—	2	1	no	—
25	C₈×D₄×C₅₉	—	2	1	no	—
26	C₈×D₄×C₅₉	C₄	1	1	yes	characteristic, complemented
27	C₅₉×#433	—	2	1	no	—
28	C₅₉×#432	C₂²	1	1	yes	characteristic
29	C₅₉×#405	—	2	1	no	—
30	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
31	C₂×#11745664	—	2	1	no	—
32	C₂²×C₅₉⋊C₈	—	2	1	no	—
33	C₂×C₄×Dic₅₉	—	2	1	no	—
34	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
35	C₂²⋊(C₅₉⋊C₈)	C₂×C₄	1	2	yes	—
36	#14273638	—	2	1	no	—
37	#14273638	C₂×C₄	1	2	yes	—
38	#14273638	C₂×C₄	1	2	yes	—
39	C₄×C₅₉⋊C₈	—	2	1	no	—
40	C₄×C₅₉⋊C₈	—	2	1	no	—
41	C₅₉⋊#95	—	2	1	no	—
42	C₅₉⋊#95	C₂×C₄	1	2	yes	—
43	C₂×M₁₆×C₅₉	—	2	1	no	—
44	C₂²×C₈×C₅₉	—	2	1	no	—
45	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
47	#126×C₅₉	—	2	1	no	—
48	#125×C₅₉	C₂×C₄	1	1	yes	characteristic
49	#124×C₅₉	C₂×C₄	1	1	yes	characteristic
50	#123×C₅₉	—	2	1	no	—
51	C₂²⋊Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
52	#121×C₅₉	C₂×C₄	1	1	yes	characteristic
53	C₄×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
54	C₄×D₄×C₅₉	—	2	1	no	—
55	C₄×D₄×C₅₉	C₂×C₄	1	1	yes	characteristic
56	#117×C₅₉	—	2	1	no	—
57	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₅₉	—	2	1	no	—
59	C₄⋊C₈×C₅₉	—	2	1	no	—
60	C₄⋊C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
61	C₄⋊C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
62	C₂²⋊C₈×C₅₉	—	2	1	no	—
63	C₂²⋊C₈×C₅₉	—	2	1	no	—
64	C₂²⋊C₈×C₅₉	—	2	1	no	—
65	#97×C₅₉	—	2	1	no	—
66	#97×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	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	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
79	C₂²×Dic₅₉	—	2	2	no	—
80	M₁₆×C₅₉	—	4	2	no	—
81	C₂×C₈×C₅₉	—	2	1	no	—
82	C₂×C₈×C₅₉	—	2	1	no	—
83	C₂×C₈×C₅₉	—	4	1	no	—
84	C₂×C₈×C₅₉	—	4	1	no	—
85	C₂×C₈×C₅₉	—	4	2	no	—
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	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
90	C₄⋊C₄×C₅₉	—	2	1	no	—
91	C₄⋊C₄×C₅₉	—	2	1	no	—
92	C₄⋊C₄×C₅₉	—	2	1	no	—
93	C₄⋊C₄×C₅₉	—	2	1	no	—
94	C₄⋊C₄×C₅₉	C₄²	1	1	yes	characteristic
95	C₄⋊C₄×C₅₉	C₄²	1	1	yes	characteristic
96	C₂²⋊C₄×C₅₉	—	2	1	no	—
97	C₂²⋊C₄×C₅₉	—	2	1	no	—
98	C₂²⋊C₄×C₅₉	—	2	1	no	—
99	C₂²⋊C₄×C₅₉	—	2	1	no	—
100	C₂²⋊C₄×C₅₉	—	2	1	no	—
101	C₄²×C₅₉	—	2	1	no	—
102	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₅₉	#41	1	1	yes	characteristic
106	C₄×Dic₅₉	—	4	1	no	—
107	C₄×Dic₅₉	—	4	1	no	—
108	#11745664	—	2	4	no	—
109	C₂×C₅₉⋊C₈	—	2	2	no	—
110	C₂×C₅₉⋊C₈	—	2	2	no	—
111	C₂×C₅₉⋊C₈	—	2	2	no	—
112	C₂×C₅₉⋊C₈	—	4	1	no	—
113	C₂×C₅₉⋊C₈	—	4	1	no	—
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	C₂×C₄²	1	1	yes	characteristic
129	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
131	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
134	C₂×Dic₅₉	—	4	2	no	—
135	C₂×Dic₅₉	—	4	2	no	—
136	C₂×Dic₅₉	—	4	2	no	—
137	C₈×C₅₉	—	2	1	no	—
138	C₈×C₅₉	—	2	1	no	—
139	C₈×C₅₉	—	2	1	no	—
140	C₈×C₅₉	—	2	1	no	—
141	C₈×C₅₉	—	4	1	no	—
142	C₈×C₅₉	—	4	1	no	—
143	C₅₉⋊C₈	—	2	4	no	—
144	C₅₉⋊C₈	—	4	2	no	—
145	C₅₉⋊C₈	—	4	2	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	#421	1	1	yes	characteristic
163	C₄×C₅₉	#421	1	1	yes	characteristic
164	Dic₅₉	—	8	2	no	—
165	#2966	D₅₉	1	1	yes	characteristic
166	#1746	—	59	1	no	—
167	#1524	—	59	1	no	—
168	#1500	—	59	1	no	—
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	Dic₅₉	1	1	yes	characteristic, complemented
179	C₈×Q₈	Dic₅₉	1	1	yes	characteristic, complemented
180	#436	Dic₅₉	1	1	yes	characteristic, complemented
181	#435	—	2	1	no	—
182	C₈×D₄	—	2	1	no	—
183	C₈×D₄	Dic₅₉	1	1	yes	characteristic, complemented
184	#433	—	2	1	no	—
185	#432	C₂×D₅₉	1	1	yes	characteristic
186	#431	—	59	1	no	—
187	#405	—	2	1	no	—
188	C₄×M₁₆	—	59	1	no	—
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	—	118	1	no	—
199	C₈²	—	118	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	—	2	1	no	—
204	C₂²×C₈	—	118	1	no	—
205	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
206	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
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	C₄×D₅₉	1	1	yes	characteristic
212	#121	C₄×D₅₉	1	1	yes	characteristic
213	C₄×Q₈	C₂×Dic₅₉	1	1	yes	characteristic
214	C₄×D₄	—	2	1	no	—
215	C₄×D₄	C₂×Dic₅₉	1	1	yes	characteristic
216	#117	—	2	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	—	2	1	no	—
221	C₄⋊C₈	—	118	1	no	—
222	C₄⋊C₈	C₂×Dic₅₉	1	1	yes	characteristic
223	C₄⋊C₈	C₂×Dic₅₉	1	1	yes	characteristic
224	C₂²⋊C₈	—	2	1	no	—
225	C₂²⋊C₈	—	2	1	no	—
226	C₂²⋊C₈	—	2	1	no	—
227	C₂²⋊C₈	—	59	2	no	—
228	#97	—	2	1	no	—
229	#97	—	2	1	no	—
230	#97	—	59	2	no	—
231	#97	—	59	2	no	—
232	#97	—	118	1	no	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	—	118	1	no	—
235	C₄×C₈	—	118	1	no	—
236	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
237	C₄×C₈	C₂×Dic₅₉	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	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	—	118	2	no	—
247	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
248	C₂²×C₄	#11745685	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
250	M₁₆	—	4	2	no	—
251	M₁₆	—	118	4	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	118	2	no	—
258	C₂×C₈	—	118	2	no	—
259	C₂×C₈	—	118	2	no	—
260	C₂×C₈	—	236	1	no	—
261	C₂×C₈	—	236	1	no	—
262	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
263	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
264	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
265	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
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	—	236	1	no	—
279	C₄²	—	236	1	no	—
280	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
281	C₄²	#11745685	1	1	yes	characteristic
282	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
283	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
284	C₂³	—	2	1	no	—
285	C₂³	C₄×C₂²⋊D₅₉	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	—	236	2	no	—
299	C₂×C₄	—	236	2	no	—
300	C₂×C₄	—	236	2	no	—
301	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
302	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
303	C₂×C₄	#14273686	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×C₂²⋊D₅₉	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	4	1	no	—
312	C₈	—	4	1	no	—
313	C₈	—	118	4	no	—
314	C₈	—	236	2	no	—
315	C₈	—	236	2	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	C₄×C₂²⋊Dic₅₉	1	1	yes	characteristic
322	C₂²	#16837517	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	—	472	2	no	—
332	C₄	#16837419	1	1	yes	characteristic
333	C₄	#16837419	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	8	1	no	—
336	C₂	#21423896	1	1	yes	characteristic
337	C₂	#21423906	1	1	yes	characteristic
338	C₂	#21423931	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₅₉
commutator	C₂×C₄×C₅₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#2966	C₂
socle	C₂²×C₅₉	C₄×C₂²⋊C₄
prime radical	#121×C₅₉	C₂×C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#2966	D₅₉
hypocenter	C₅₉	#8250
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2966

Complements

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

N	H ≅ G/N	type
1. #21423847	C₂	semidirect
4. C₈×#11745664	C₂	semidirect
5. #21421994	C₂	semidirect
6. #21421988	C₂	semidirect
14. #16837144	C₄	semidirect
15. #16837142	C₄	semidirect
21. C₅₉×#446	C₄	semidirect
22. C₈×Q₈×C₅₉	C₄	semidirect
23. C₅₉×#436	C₄	semidirect
26. C₈×D₄×C₅₉	C₄	semidirect
178. #446	Dic₅₉	semidirect
179. C₈×Q₈	Dic₅₉	semidirect
180. #436	Dic₅₉	semidirect
183. C₈×D₄	Dic₅₉	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
#21423847	2
#21422684	2
#21422582	2
C₈×#11745664	2
#21421994	2
#21421988	2
C₅₉×#2966	2
#8250	59