Subgroups — #28506854

Order 15104.

442 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21432806	C₂	1	1	yes	maximal, characteristic
2	#21425851	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16837187	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16837186	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16837185	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16837177	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₉×#350	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14273733	C₂²	1	1	yes	characteristic
9	#16837948	C₄	1	1	yes	characteristic
10	C₂×#14273667	C₂²	1	1	yes	characteristic
11	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
12	#16837478	—	2	2	no	—
13	C₈×C₂²⋊D₅₉	—	2	2	no	—
14	#16837425	C₄	1	1	yes	characteristic
15	#16837370	C₄	1	1	yes	characteristic
16	#16837369	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
18	Dic₅₉⋊C₁₆	—	2	2	no	—
19	#16837187	—	2	2	no	—
20	#16837186	—	2	2	no	—
21	#16837185	—	2	2	no	—
22	#16837177	—	2	2	no	—
23	C₂×#131×C₅₉	C₂²	1	1	yes	characteristic
24	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
26	C₅₉×#350	—	2	2	no	—
27	C₂×#11745685	C₂×C₄	1	1	yes	characteristic
28	#14273733	—	2	1	no	—
29	#14273733	—	2	1	no	—
30	#14273733	—	2	1	no	—
31	#14273733	—	2	1	no	—
32	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
33	C₂×#11745659	—	2	1	no	—
34	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₅₉	—	2	1	no	—
36	#14273725	C₈	1	1	yes	characteristic
37	#14273723	C₈	1	1	yes	characteristic
38	#14273716	C₈	1	1	yes	characteristic
39	#14273714	C₈	1	1	yes	characteristic
40	#14273699	—	2	2	no	—
41	C₄×C₂²⋊D₅₉	—	2	2	no	—
42	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
43	#14273667	—	2	1	no	—
44	#14273667	—	2	1	no	—
45	C₂×C₅₉⋊C₁₆	—	2	1	no	—
46	C₂×C₅₉⋊C₁₆	—	2	1	no	—
47	C₂×C₅₉⋊C₁₆	—	2	1	no	—
48	C₂×C₅₉⋊C₁₆	D₄	1	2	yes	complemented
49	Dic₅₉⋊C₈	—	2	2	no	—
50	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
51	(C₂×D₅₉)⋊C₈	—	2	2	no	—
52	C₈×Dic₅₉	D₄	1	2	yes	—
53	C₈×Dic₅₉	D₄	1	2	yes	—
54	C₂×#41×C₅₉	C₂×C₄	1	1	yes	characteristic
55	#131×C₅₉	—	2	1	no	—
56	#131×C₅₉	—	2	1	no	—
57	#131×C₅₉	—	2	1	no	—
58	#131×C₅₉	—	2	1	no	—
59	C₂×M₁₆×C₅₉	—	2	1	no	—
60	C₂×M₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₈×C₅₉	—	2	1	no	—
62	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
63	M₃₂×C₅₉	—	2	1	no	—
64	M₃₂×C₅₉	—	2	1	no	—
65	C₂×C₁₆×C₅₉	—	2	1	no	—
66	C₂×C₁₆×C₅₉	—	2	1	no	—
67	C₂×C₁₆×C₅₉	—	2	1	no	—
68	C₂×C₁₆×C₅₉	D₄	1	2	yes	complemented
69	#41×C₅₉	—	2	1	no	—
70	#41×C₅₉	—	2	1	no	—
71	#41×C₅₉	—	2	1	no	—
72	#41×C₅₉	—	2	1	no	—
73	C₂×Q₈×C₅₉	C₂×C₈	1	1	yes	characteristic
74	C₂×D₄×C₅₉	—	2	1	no	—
75	C₂×D₄×C₅₉	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄×C₅₉	—	2	1	no	—
77	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₂²⋊D₅₉	—	2	1	no	—
79	C₂×C₂²⋊D₅₉	—	2	2	no	—
80	C₂²×Dic₅₉	M₁₆	1	2	yes	—
81	#11745685	—	2	1	no	—
82	#11745685	—	2	1	no	—
83	#11745685	—	2	1	no	—
84	#11745685	—	2	1	no	—
85	C₂×D₂₃₆	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄×D₅₉	—	2	1	no	—
87	C₂×Dic₁₁₈	C₂×C₈	1	1	yes	characteristic
88	M₁₆×C₅₉	—	2	1	no	—
89	M₁₆×C₅₉	—	2	1	no	—
90	M₁₆×C₅₉	—	4	1	no	—
91	M₁₆×C₅₉	—	4	1	no	—
92	C₂×C₈×C₅₉	—	2	1	no	—
93	C₂×C₈×C₅₉	—	4	1	no	—
94	C₂×C₈×C₅₉	—	4	1	no	—
95	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
100	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
101	C₂²⋊Dic₅₉	—	2	2	no	—
102	(C₂×D₅₉)⋊C₄	—	2	2	no	—
103	Dic₅₉⋊C₄	—	2	2	no	—
104	C₄×Dic₅₉	C₂²⋊C₄	1	2	yes	—
105	C₄×Dic₅₉	C₂×C₈	1	1	yes	characteristic
106	C₄×Dic₅₉	C₂×C₈	1	1	yes	characteristic
107	#11745664	—	2	1	no	—
108	#11745664	—	2	1	no	—
109	C₂×C₅₉⋊C₈	—	2	1	no	—
110	C₂×C₅₉⋊C₈	—	2	1	no	—
111	C₂×C₅₉⋊C₈	—	2	1	no	—
112	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	2	yes	—
113	#11745659	—	4	1	no	—
114	#11745659	—	4	1	no	—
115	C₈×D₅₉	—	4	1	no	—
116	C₈×D₅₉	—	4	1	no	—
117	C₁₆×C₅₉	—	2	2	no	—
118	C₁₆×C₅₉	—	4	1	no	—
119	C₅₉⋊C₁₆	—	2	2	no	—
120	C₅₉⋊C₁₆	—	4	1	no	—
121	C₂³×C₅₉	—	2	1	no	—
122	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
123	C₂²×D₅₉	—	2	1	no	—
124	Q₈×C₅₉	—	2	1	no	—
125	Q₈×C₅₉	—	2	1	no	—
126	D₄×C₅₉	—	2	1	no	—
127	D₄×C₅₉	—	2	1	no	—
128	D₄×C₅₉	—	4	1	no	—
129	D₄×C₅₉	—	4	1	no	—
130	C₂×C₄×C₅₉	—	2	1	no	—
131	C₂×C₄×C₅₉	—	4	1	no	—
132	C₂×C₄×C₅₉	—	4	1	no	—
133	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	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	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
138	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
139	C₂²⋊D₅₉	—	4	4	no	—
140	C₂²⋊D₅₉	—	4	1	no	—
141	C₂²⋊D₅₉	—	4	1	no	—
142	C₂×Dic₅₉	—	2	1	no	—
143	C₂×Dic₅₉	—	4	2	no	—
144	C₂×Dic₅₉	C₂²⋊C₈	1	2	yes	—
145	C₂×Dic₅₉	C₂²⋊C₈	1	2	yes	—
146	D₂₃₆	—	2	1	no	—
147	D₂₃₆	—	2	1	no	—
148	C₄×D₅₉	—	4	1	no	—
149	C₄×D₅₉	—	4	1	no	—
150	Dic₁₁₈	—	2	1	no	—
151	Dic₁₁₈	—	2	1	no	—
152	C₈×C₅₉	—	4	1	no	—
153	C₈×C₅₉	#121	1	1	yes	characteristic
154	C₈×C₅₉	#121	1	1	yes	characteristic
155	C₈×C₅₉	#124	1	1	yes	characteristic
156	C₈×C₅₉	#127	1	1	yes	characteristic
157	C₅₉⋊C₈	—	2	2	no	—
158	C₅₉⋊C₈	—	4	1	no	—
159	#13192	—	59	1	no	maximal, Sylow
160	C₂²×C₅₉	—	2	2	no	—
161	C₂²×C₅₉	—	4	2	no	—
162	C₂²×C₅₉	—	4	1	no	—
163	C₂²×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
164	C₂²×C₅₉	#436	1	1	yes	characteristic
165	C₂²×C₅₉	#436	1	1	yes	characteristic
166	C₂×D₅₉	—	4	2	no	—
167	C₂×D₅₉	—	4	1	no	—
168	C₄×C₅₉	—	4	1	no	—
169	C₄×C₅₉	#390	1	1	yes	characteristic
170	C₄×C₅₉	#408	1	1	yes	characteristic
171	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
172	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
173	Dic₅₉	—	2	2	no	—
174	Dic₅₉	—	2	2	no	—
175	Dic₅₉	—	4	1	no	—
176	#2931	—	59	1	no	—
177	#2150	—	59	1	no	—
178	C₂×#350	—	59	1	no	—
179	C₂×#350	—	59	1	no	—
180	C₂×#350	—	59	1	no	—
181	C₂×#350	D₅₉	1	1	yes	characteristic, complemented
182	C₂×#346	—	59	1	no	—
183	C₂×C₅₉	—	2	1	no	—
184	C₂×C₅₉	—	2	1	no	—
185	C₂×C₅₉	—	8	1	no	—
186	C₂×C₅₉	#1954	1	1	yes	characteristic
187	C₂×C₅₉	#2117	1	2	yes	—
188	D₅₉	—	8	1	no	—
189	C₂×#131	—	59	1	no	—
190	C₂×#131	C₂×D₅₉	1	1	yes	characteristic
191	#532	—	59	1	no	—
192	C₂×M₃₂	—	59	1	no	—
193	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
194	C₂²×C₁₆	—	59	1	no	—
195	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
196	#436	—	118	2	no	—
197	C₈×D₄	—	118	2	no	—
198	#423	—	59	1	no	—
199	#408	—	59	1	no	—
200	#408	—	59	1	no	—
201	C₂×C₄×C₈	—	59	1	no	—
202	C₄⋊C₁₆	—	118	2	no	—
203	#350	—	2	2	no	—
204	#350	—	118	2	no	—
205	#350	—	118	2	no	—
206	#350	—	118	2	no	—
207	#346	—	118	2	no	—
208	C₅₉	#13192	1	1	yes	characteristic, complemented, Sylow
209	C₂×#41	—	59	1	no	—
210	C₂×#41	C₄×D₅₉	1	1	yes	characteristic
211	#131	—	2	1	no	—
212	#131	—	2	1	no	—
213	#131	—	2	1	no	—
214	#131	—	2	1	no	—
215	#131	—	118	1	no	—
216	#131	—	118	1	no	—
217	#131	—	118	1	no	—
218	#131	—	118	1	no	—
219	C₂×M₁₆	—	2	1	no	—
220	C₂×M₁₆	—	59	1	no	—
221	C₂×M₁₆	—	118	1	no	—
222	C₂×M₁₆	C₄×D₅₉	1	1	yes	characteristic
223	C₂²×C₈	—	2	1	no	—
224	C₂²×C₈	—	59	1	no	—
225	C₂²×C₈	—	118	1	no	—
226	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
227	C₄⋊Q₈	—	59	1	no	—
228	#127	—	59	1	no	—
229	#124	—	59	1	no	—
230	#124	—	59	1	no	—
231	#121	—	118	2	no	—
232	C₄×D₄	—	118	2	no	—
233	C₂×C₄²	—	59	1	no	—
234	M₃₂	—	2	1	no	—
235	M₃₂	—	2	1	no	—
236	M₃₂	—	118	1	no	—
237	M₃₂	—	118	1	no	—
238	C₂×C₁₆	—	2	1	no	—
239	C₂×C₁₆	—	2	1	no	—
240	C₂×C₁₆	—	2	1	no	—
241	C₂×C₁₆	—	59	2	no	—
242	C₂×C₁₆	—	118	1	no	—
243	C₂×C₁₆	—	118	1	no	—
244	C₂×C₁₆	—	118	1	no	—
245	C₂×C₁₆	C₂²⋊D₅₉	1	2	yes	complemented
246	C₄⋊C₈	—	118	2	no	—
247	C₂²⋊C₈	—	118	2	no	—
248	C₂²⋊C₈	—	118	2	no	—
249	C₄×C₈	—	59	2	no	—
250	C₄×C₈	—	59	2	no	—
251	#41	—	2	1	no	—
252	#41	—	2	1	no	—
253	#41	—	2	1	no	—
254	#41	—	2	1	no	—
255	#41	—	118	1	no	—
256	#41	—	118	1	no	—
257	#41	—	118	1	no	—
258	#41	—	118	1	no	—
259	C₂×Q₈	—	59	1	no	—
260	C₂×Q₈	C₈×D₅₉	1	1	yes	characteristic
261	C₂×D₄	—	2	1	no	—
262	C₂×D₄	—	59	1	no	—
263	C₂×D₄	—	118	2	no	—
264	C₂×D₄	—	118	1	no	—
265	C₂×D₄	C₈×D₅₉	1	1	yes	characteristic
266	C₂²×C₄	—	2	1	no	—
267	C₂²×C₄	—	59	2	no	—
268	C₂²×C₄	—	118	1	no	—
269	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
270	M₁₆	—	2	1	no	—
271	M₁₆	—	2	1	no	—
272	M₁₆	—	4	1	no	—
273	M₁₆	—	4	1	no	—
274	M₁₆	—	118	1	no	—
275	M₁₆	—	118	1	no	—
276	M₁₆	—	236	1	no	—
277	M₁₆	—	236	1	no	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	4	1	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	59	2	no	—
282	C₂×C₈	—	118	1	no	—
283	C₂×C₈	—	118	1	no	—
284	C₂×C₈	—	118	1	no	—
285	C₂×C₈	—	236	1	no	—
286	C₂×C₈	—	236	1	no	—
287	C₂×C₈	#11745685	1	1	yes	characteristic
288	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
289	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
290	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
291	C₂×C₈	#11745687	1	1	yes	characteristic
292	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
293	C₄⋊C₄	—	118	2	no	—
294	C₂²⋊C₄	—	118	2	no	—
295	C₂²⋊C₄	—	118	2	no	—
296	C₄²	—	59	2	no	—
297	C₄²	—	59	1	no	—
298	C₄²	—	59	1	no	—
299	C₁₆	—	2	2	no	—
300	C₁₆	—	4	1	no	—
301	C₁₆	—	118	2	no	—
302	C₁₆	—	236	1	no	—
303	C₂³	—	2	1	no	—
304	C₂³	—	118	1	no	—
305	C₂³	M₁₆×D₅₉	1	1	yes	characteristic
306	Q₈	—	2	1	no	—
307	Q₈	—	2	1	no	—
308	Q₈	—	118	1	no	—
309	Q₈	—	118	1	no	—
310	D₄	—	2	1	no	—
311	D₄	—	2	1	no	—
312	D₄	—	4	1	no	—
313	D₄	—	4	1	no	—
314	D₄	—	118	1	no	—
315	D₄	—	118	1	no	—
316	D₄	—	236	4	no	—
317	D₄	—	236	1	no	—
318	D₄	—	236	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	59	2	no	—
323	C₂×C₄	—	59	2	no	—
324	C₂×C₄	—	118	1	no	—
325	C₂×C₄	—	236	2	no	—
326	C₂×C₄	—	236	1	no	—
327	C₂×C₄	—	236	1	no	—
328	C₂×C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
329	C₂×C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
330	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
331	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
332	C₂×C₄	#14273733	1	1	yes	characteristic
333	C₂×C₄	#14273735	1	1	yes	characteristic
334	C₈	—	4	1	no	—
335	C₈	—	118	2	no	—
336	C₈	—	236	1	no	—
337	C₈	#14273699	1	1	yes	characteristic
338	C₈	#14273700	1	1	yes	characteristic
339	C₈	#14273713	1	1	yes	characteristic
340	C₈	#14273723	1	1	yes	characteristic
341	C₂²	—	2	2	no	—
342	C₂²	—	4	2	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	236	2	no	—
345	C₂²	—	236	1	no	—
346	C₂²	C₂²⋊C₈×D₅₉	1	1	yes	characteristic
347	C₂²	#16837477	1	1	yes	characteristic
348	C₂²	#16837478	1	1	yes	characteristic
349	C₄	—	4	1	no	—
350	C₄	—	118	2	no	—
351	C₄	—	118	2	no	—
352	C₄	—	236	1	no	—
353	C₄	#16837302	1	1	yes	characteristic
354	C₄	#16837367	1	1	yes	characteristic
355	C₄	(C₂²⋊D₅₉)⋊C₈	1	1	yes	characteristic
356	C₄	C₈×C₂²⋊D₅₉	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	8	1	no	—
360	C₂	—	472	1	no	—
361	C₂	#21424888	1	1	yes	characteristic
362	C₂	#21425722	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#350	C₂
socle	C₂²×C₅₉	C₂×C₂²⋊C₈
prime radical	#16837948	C₄
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#350	D₅₉
hypocenter	C₅₉	#13192
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#350

Complements

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

N	H ≅ G/N	type
2. #21425851	C₂	semidirect
3. C₂×#16837187	C₂	semidirect
4. C₂×#16837186	C₂	semidirect
5. C₂×#16837185	C₂	semidirect
6. C₂×#16837177	C₂	semidirect
7. C₂×C₅₉×#350	C₂	semidirect
48. C₂×C₅₉⋊C₁₆	D₄	semidirect
68. C₂×C₁₆×C₅₉	D₄	semidirect
181. C₂×#350	D₅₉	semidirect
208. C₅₉	#13192	semidirect
245. C₂×C₁₆	C₂²⋊D₅₉	semidirect

Maximal subgroups

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

subgroup	index
#21432806	2
#21425851	2
C₂×#16837187	2
C₂×#16837186	2
C₂×#16837185	2
C₂×#16837177	2
C₂×C₅₉×#350	2
#13192	59