Subgroups — #28557733

Order 15104.

666 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	#21433209	C₂	1	2	yes	maximal
2	C₄×#14273733	C₂	1	1	yes	maximal, characteristic
3	#21422862	C₂	1	4	yes	maximal, complemented
4	#21422856	C₂	1	2	yes	maximal, complemented
5	#21422850	C₂	1	2	yes	maximal, complemented
6	#21422849	C₂	1	2	yes	maximal, complemented
7	C₂×#16837153	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₉×#1564	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14273733	C₄	1	1	yes	characteristic
10	C₂×#14273733	C₄	1	1	yes	characteristic
11	C₄×#11745685	C₂²	1	1	yes	characteristic
12	#16837532	—	2	2	no	—
13	#16837481	—	2	2	no	—
14	#16837470	—	2	4	no	—
15	#16837458	C₂²	1	4	yes	—
16	#16837424	C₂²	1	4	yes	—
17	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
18	#16837357	C₄	1	1	yes	characteristic
19	#16837357	C₄	1	1	yes	characteristic
20	#16837356	C₄	1	2	yes	—
21	#16837356	C₄	1	2	yes	—
22	C₄×#11745664	C₂²	1	1	yes	characteristic
23	C₄×#11745659	C₂²	1	2	yes	—
24	C₄×C₈×D₅₉	C₂²	1	2	yes	—
25	#16837154	C₂²	1	4	yes	—
26	#16837153	C₂²	1	2	yes	—
27	#16837153	C₂²	1	2	yes	—
28	#16837152	C₂²	1	4	yes	—
29	C₅₉×#431	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
31	C₅₉×#335	C₂²	1	2	yes	—
32	C₅₉×#334	C₂²	1	1	yes	characteristic
33	C₅₉×#334	C₂²	1	1	yes	characteristic
34	C₂×#11745685	C₂×C₄	1	1	yes	characteristic
35	#14273733	—	2	4	no	—
36	#14273733	—	2	4	no	—
37	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
38	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
39	C₂×#11745659	—	2	4	no	—
40	C₂×#11745659	C₂×C₄	1	2	yes	—
41	C₂×#11745659	C₂×C₄	1	2	yes	—
42	C₂²×C₅₉⋊C₈	—	2	2	no	—
43	C₂×C₈×D₅₉	C₂×C₄	1	2	yes	—
44	C₂×C₈×D₅₉	C₂×C₄	1	2	yes	—
45	C₄×Dic₁₁₈	D₄	1	1	yes	characteristic
46	C₄×Dic₁₁₈	Q₈	1	1	yes	characteristic
47	C₄×D₂₃₆	D₄	1	1	yes	characteristic
48	C₄×D₂₃₆	Q₈	1	1	yes	characteristic
49	C₄×C₂²⋊D₅₉	D₄	1	2	yes	—
50	C₄×C₂²⋊D₅₉	Q₈	1	2	yes	—
51	#14273686	C₂×C₄	1	1	yes	characteristic
52	#14273684	C₂×C₄	1	2	yes	—
53	C₄²×D₅₉	C₂³	1	2	yes	—
54	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
55	C₄⋊(C₅₉⋊C₈)	C₂³	1	4	yes	—
56	Dic₅₉⋊C₈	—	2	4	no	—
57	(C₂×D₅₉)⋊C₈	—	2	4	no	—
58	#14273638	C₂×C₄	1	2	yes	—
59	#14273637	—	2	4	no	—
60	#14273637	C₂×C₄	1	2	yes	—
61	#14273637	C₂×C₄	1	2	yes	—
62	C₄×C₅₉⋊C₈	C₂³	1	2	yes	—
63	C₈×Dic₅₉	C₂×C₄	1	2	yes	—
64	C₈×Dic₅₉	C₂×C₄	1	2	yes	—
65	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
68	C₄⋊C₈×C₅₉	C₂³	1	2	yes	—
69	C₄⋊C₈×C₅₉	C₂³	1	2	yes	—
70	C₂²⋊C₈×C₅₉	—	2	2	no	—
71	C₄×C₈×C₅₉	—	2	2	no	—
72	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₂²⋊D₅₉	C₄⋊C₄	1	2	yes	—
80	#11745685	—	2	2	no	—
81	#11745685	—	2	2	no	—
82	C₂×D₂₃₆	C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×D₅₉	C₂²×C₄	1	2	yes	—
84	C₂×C₄×D₅₉	C₂×D₄	1	2	yes	—
85	C₂×C₄×D₅₉	C₂×Q₈	1	2	yes	—
86	C₂×Dic₁₁₈	C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₅₉	—	2	2	no	—
88	C₂×C₈×C₅₉	—	2	2	no	—
89	C₂×C₈×C₅₉	—	2	2	no	—
90	C₂×C₈×C₅₉	M₁₆	1	4	yes	complemented
91	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
98	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
99	C₂²⋊Dic₅₉	C₄⋊C₄	1	2	yes	—
100	(C₂×D₅₉)⋊C₄	C₄⋊C₄	1	2	yes	—
101	(C₂×D₅₉)⋊C₄	C₄⋊C₄	1	2	yes	—
102	C₄⋊Dic₅₉	C₄⋊C₄	1	1	yes	characteristic
103	C₄⋊Dic₅₉	C₄⋊C₄	1	1	yes	characteristic
104	Dic₅₉⋊C₄	C₄⋊C₄	1	2	yes	—
105	Dic₅₉⋊C₄	C₄⋊C₄	1	2	yes	—
106	C₄×Dic₅₉	C₂²×C₄	1	2	yes	—
107	C₄×Dic₅₉	C₂×D₄	1	2	yes	—
108	C₄×Dic₅₉	C₂×Q₈	1	2	yes	—
109	#11745664	—	2	2	no	—
110	#11745664	M₁₆	1	4	yes	complemented
111	C₂×C₅₉⋊C₈	—	2	4	no	—
112	C₂×C₅₉⋊C₈	—	8	2	no	—
113	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
114	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
115	#11745659	—	2	4	no	—
116	#11745659	—	2	4	no	—
117	#11745659	—	8	4	no	—
118	C₈×D₅₉	—	2	4	no	—
119	C₈×D₅₉	—	2	4	no	—
120	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×D₅₉	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₅₉	—	2	2	no	—
123	C₂×C₄×C₅₉	—	4	1	no	—
124	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
128	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
129	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
132	C₂²⋊D₅₉	—	2	4	no	—
133	C₂×Dic₅₉	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×Dic₅₉	C₂×C₄⋊C₄	1	2	yes	—
135	C₂×Dic₅₉	C₂×C₄⋊C₄	1	2	yes	—
136	D₂₃₆	—	2	2	no	—
137	C₄×D₅₉	—	2	2	no	—
138	C₄×D₅₉	—	2	2	no	—
139	C₄×D₅₉	—	2	4	no	—
140	C₄×D₅₉	—	4	2	no	—
141	Dic₁₁₈	—	2	2	no	—
142	C₈×C₅₉	—	2	1	no	—
143	C₈×C₅₉	—	2	1	no	—
144	C₈×C₅₉	—	4	2	no	—
145	C₈×C₅₉	—	4	2	no	—
146	C₈×C₅₉	C₂×M₁₆	1	2	yes	—
147	C₈×C₅₉	C₂×M₁₆	1	2	yes	—
148	C₅₉⋊C₈	—	2	2	no	—
149	C₅₉⋊C₈	—	4	4	no	—
150	C₅₉⋊C₈	C₂×M₁₆	1	4	yes	—
151	#17798	—	59	1	no	maximal, Sylow
152	C₂²×C₅₉	—	2	1	no	—
153	C₂²×C₅₉	—	2	1	no	—
154	C₂²×C₅₉	#423	1	2	yes	—
155	C₂²×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
156	C₂×D₅₉	—	2	2	no	—
157	C₂×D₅₉	—	2	2	no	—
158	C₂×D₅₉	—	2	2	no	—
159	C₄×C₅₉	—	2	1	no	—
160	C₄×C₅₉	—	2	1	no	—
161	C₄×C₅₉	—	2	1	no	—
162	C₄×C₅₉	#423	1	2	yes	—
163	C₄×C₅₉	C₂²×M₁₆	1	1	yes	characteristic
164	C₄×C₅₉	C₂²×M₁₆	1	1	yes	characteristic
165	C₄×C₅₉	#577	1	2	yes	—
166	Dic₅₉	—	2	2	no	—
167	Dic₅₉	—	2	2	no	—
168	Dic₅₉	—	2	2	no	—
169	#2989	—	59	2	no	—
170	C₄×#131	—	59	1	no	—
171	#1569	—	59	4	no	—
172	#1568	—	59	2	no	—
173	#1566	—	59	2	no	—
174	#1565	—	59	2	no	—
175	#1564	D₅₉	1	1	yes	characteristic, complemented
176	C₂×#334	—	59	1	no	—
177	C₂×C₅₉	—	2	2	no	—
178	C₂×C₅₉	C₂×#423	1	1	yes	characteristic
179	C₂×C₅₉	#2912	1	1	yes	characteristic
180	C₂×C₅₉	#2916	1	1	yes	characteristic
181	D₅₉	—	4	2	no	—
182	C₂×#131	—	59	1	no	—
183	C₂×#131	—	59	1	no	—
184	C₄×#41	—	59	1	no	—
185	#446	—	118	2	no	—
186	#436	—	118	2	no	—
187	#435	—	118	4	no	—
188	#432	—	59	4	no	—
189	#431	C₂×D₅₉	1	2	yes	—
190	#423	—	59	4	no	—
191	C₂×C₄⋊C₈	—	59	2	no	—
192	#405	—	59	1	no	—
193	#405	—	59	1	no	—
194	#405	—	59	2	no	—
195	#405	—	59	2	no	—
196	C₄×M₁₆	—	59	1	no	—
197	C₄×M₁₆	—	59	2	no	—
198	C₂×C₄×C₈	—	59	2	no	—
199	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
200	#335	—	59	4	no	—
201	#335	C₂×D₅₉	1	2	yes	—
202	#334	—	59	4	no	—
203	#334	—	59	2	no	—
204	#334	—	59	2	no	—
205	#334	C₂×D₅₉	1	1	yes	characteristic
206	#334	C₂×D₅₉	1	1	yes	characteristic
207	C₅₉	#17798	1	1	yes	characteristic, complemented, Sylow
208	C₂×#41	—	59	1	no	—
209	#131	—	118	4	no	—
210	#131	—	118	4	no	—
211	C₂×M₁₆	—	59	1	no	—
212	C₂×M₁₆	—	59	1	no	—
213	C₂×M₁₆	—	59	2	no	—
214	C₂×M₁₆	—	59	2	no	—
215	C₂×M₁₆	—	118	4	no	—
216	C₂²×C₈	—	59	2	no	—
217	C₂²×C₈	—	59	2	no	—
218	C₂²×C₈	—	118	2	no	—
219	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
220	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
221	C₄×Q₈	—	59	1	no	—
222	C₄×Q₈	—	59	1	no	—
223	C₄×D₄	—	59	2	no	—
224	C₄×D₄	—	59	2	no	—
225	C₄×D₄	—	59	1	no	—
226	C₄×D₄	—	59	1	no	—
227	#117	—	59	1	no	—
228	#117	—	59	2	no	—
229	C₂×C₄²	—	59	2	no	—
230	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
231	C₄⋊C₈	—	59	4	no	—
232	C₄⋊C₈	—	118	4	no	—
233	C₄⋊C₈	—	118	2	no	—
234	C₄⋊C₈	C₂²×D₅₉	1	2	yes	—
235	C₄⋊C₈	C₂²×D₅₉	1	2	yes	—
236	C₂²⋊C₈	—	2	2	no	—
237	C₂²⋊C₈	—	118	4	no	—
238	#97	—	59	2	no	—
239	#97	—	59	2	no	—
240	#97	—	59	2	no	—
241	#97	—	118	4	no	—
242	C₄×C₈	—	2	2	no	—
243	C₄×C₈	—	59	2	no	—
244	C₄×C₈	—	59	2	no	—
245	C₄×C₈	—	59	2	no	—
246	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
247	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
250	#41	—	118	2	no	—
251	#41	—	118	2	no	—
252	C₂×Q₈	—	59	1	no	—
253	C₂×D₄	—	59	2	no	—
254	C₂×D₄	—	59	1	no	—
255	C₂²×C₄	—	59	2	no	—
256	C₂²×C₄	—	59	2	no	—
257	C₂²×C₄	—	59	2	no	—
258	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
259	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
260	C₂²×C₄	Q₈×D₅₉	1	1	yes	characteristic
261	M₁₆	—	59	4	no	—
262	M₁₆	—	118	4	no	—
263	M₁₆	—	118	2	no	—
264	M₁₆	—	118	4	no	—
265	M₁₆	—	472	4	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	59	2	no	—
270	C₂×C₈	—	59	2	no	—
271	C₂×C₈	—	118	4	no	—
272	C₂×C₈	—	118	4	no	—
273	C₂×C₈	—	118	4	no	—
274	C₂×C₈	—	472	2	no	—
275	C₂×C₈	#11745659	1	4	yes	complemented
276	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
277	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
278	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
279	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
280	C₄⋊C₄	—	59	2	no	—
281	C₄⋊C₄	—	59	2	no	—
282	C₄⋊C₄	—	59	1	no	—
283	C₄⋊C₄	—	59	1	no	—
284	C₂²⋊C₄	—	59	2	no	—
285	C₂²⋊C₄	—	59	2	no	—
286	C₂²⋊C₄	—	59	2	no	—
287	C₄²	—	59	2	no	—
288	C₄²	—	59	2	no	—
289	C₄²	—	59	2	no	—
290	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
291	C₄²	D₄×D₅₉	1	1	yes	characteristic
292	C₄²	Q₈×D₅₉	1	1	yes	characteristic
293	C₄²	C₂³×D₅₉	1	1	yes	characteristic
294	C₂³	—	59	2	no	—
295	C₂³	C₄⋊C₄×D₅₉	1	1	yes	characteristic
296	Q₈	—	118	2	no	—
297	D₄	—	118	4	no	—
298	D₄	—	118	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	59	2	no	—
302	C₂×C₄	—	59	2	no	—
303	C₂×C₄	—	59	2	no	—
304	C₂×C₄	—	118	2	no	—
305	C₂×C₄	—	118	2	no	—
306	C₂×C₄	—	118	4	no	—
307	C₂×C₄	—	236	2	no	—
308	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
309	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
310	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
311	C₂×C₄	M₁₆×D₅₉	1	2	yes	—
312	C₂×C₄	C₂×#11745659	1	2	yes	—
313	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
314	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
315	C₂×C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	4	2	no	—
319	C₈	—	4	2	no	—
320	C₈	—	59	4	no	—
321	C₈	—	118	2	no	—
322	C₈	—	236	4	no	—
323	C₈	C₂×#11745659	1	2	yes	—
324	C₈	C₂×#11745659	1	2	yes	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	118	2	no	—
328	C₂²	—	118	2	no	—
329	C₂²	—	118	2	no	—
330	C₂²	#16837424	1	2	yes	—
331	C₂²	C₂×C₄⋊C₄×D₅₉	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	118	2	no	—
336	C₄	—	118	2	no	—
337	C₄	—	118	2	no	—
338	C₄	#16837424	1	2	yes	—
339	C₄	C₂×M₁₆×D₅₉	1	1	yes	characteristic
340	C₄	C₂²×#11745659	1	1	yes	characteristic
341	C₄	#16838209	1	2	yes	—
342	C₂	—	2	2	no	—
343	C₂	—	236	2	no	—
344	C₂	C₂×#16837424	1	1	yes	characteristic
345	C₂	#21432688	1	1	yes	characteristic
346	C₂	#21432704	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×#1564	C₂
socle	C₂²×C₅₉	C₂²×C₄⋊C₄
prime radical	C₂×#11745685	C₂×C₄
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	#1564	D₅₉
hypocenter	C₅₉	#17798
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
3. #21422862	C₂	semidirect
4. #21422856	C₂	semidirect
5. #21422850	C₂	semidirect
6. #21422849	C₂	semidirect
7. C₂×#16837153	C₂	semidirect
8. C₅₉×#1564	C₂	semidirect
90. C₂×C₈×C₅₉	M₁₆	semidirect
110. #11745664	M₁₆	semidirect
175. #1564	D₅₉	semidirect
207. C₅₉	#17798	semidirect
275. C₂×C₈	#11745659	semidirect

Maximal subgroups

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

subgroup	index
#21433209	2
C₄×#14273733	2
#21422862	2
#21422856	2
#21422850	2
#21422849	2
C₂×#16837153	2
C₅₉×#1564	2
#17798	59