Subgroups — #28482580

Order 15104.

446 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21433204	C₂	1	1	yes	maximal, characteristic
2	#21424213	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#14273655	C₂	1	1	yes	maximal, characteristic
4	#21422849	C₂	1	1	yes	maximal, characteristic, complemented
5	#21422484	C₂	1	1	yes	maximal, characteristic
6	#21421945	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1276	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11745685	C₂²	1	1	yes	characteristic
9	C₈×Dic₁₁₈	—	2	1	no	—
10	C₈×D₂₃₆	—	2	1	no	—
11	(C₂²⋊D₅₉)⋊C₈	—	2	2	no	—
12	#16837454	—	2	1	no	—
13	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×D₅₉	—	2	1	no	—
16	C₂×#14273655	C₄	1	1	yes	characteristic
17	C₂×#14273655	C₄	1	1	yes	characteristic
18	C₄×#11745664	C₂²	1	1	yes	characteristic
19	#16837162	C₄	1	1	yes	characteristic
20	#16837162	C₄	1	1	yes	characteristic
21	#16837157	—	2	2	no	—
22	#16837154	—	2	1	no	—
23	#16837152	—	2	1	no	—
24	#16837134	—	2	1	no	—
25	#16837131	—	2	1	no	—
26	C₅₉×#431	C₂²	1	1	yes	characteristic
27	C₅₉×#431	C₂²	1	1	yes	characteristic
28	C₄³×C₅₉	C₂²	1	1	yes	characteristic
29	C₂×#11745685	C₂×C₄	1	1	yes	characteristic
30	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
31	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₅₉⋊C₈	—	2	1	no	—
33	C₂²×C₅₉⋊C₈	D₄	1	2	yes	—
34	C₂×C₈×D₅₉	—	2	2	no	—
35	C₄×Dic₁₁₈	D₄	1	1	yes	characteristic
36	C₄×Dic₁₁₈	Q₈	1	1	yes	characteristic
37	C₄×D₂₃₆	D₄	1	1	yes	characteristic
38	C₄×D₂₃₆	Q₈	1	1	yes	characteristic
39	C₄×C₂²⋊D₅₉	—	2	1	no	—
40	C₄×C₂²⋊D₅₉	—	2	1	no	—
41	#14273686	C₂×C₄	1	1	yes	characteristic
42	#14273684	—	2	1	no	—
43	C₄²×D₅₉	—	2	1	no	—
44	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
45	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
46	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
47	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
48	Dic₅₉⋊C₈	—	2	2	no	—
49	#14273655	—	2	2	no	—
50	#14273655	C₈	1	4	yes	complemented
51	(C₂×D₅₉)⋊C₈	—	2	2	no	—
52	#14273638	—	2	1	no	—
53	C₄×C₅₉⋊C₈	—	2	1	no	—
54	C₈×Dic₅₉	—	2	2	no	—
55	C₂×C₄²×C₅₉	—	2	1	no	—
56	C₂×C₄²×C₅₉	—	2	1	no	—
57	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₅₉	—	2	1	no	—
61	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
62	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
63	C₂²⋊C₈×C₅₉	—	2	1	no	—
64	C₂²⋊C₈×C₅₉	—	2	1	no	—
65	C₄×C₈×C₅₉	—	2	1	no	—
66	C₄×C₈×C₅₉	—	2	1	no	—
67	C₂²×C₄×C₅₉	—	2	1	no	—
68	C₂²×C₄×C₅₉	—	2	1	no	—
69	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
72	C₂×C₂²⋊D₅₉	—	2	1	no	—
73	#11745685	—	2	1	no	—
74	#11745685	—	2	1	no	—
75	#11745685	C₂×C₈	1	2	yes	—
76	#11745685	M₁₆	1	2	yes	—
77	C₂×D₂₃₆	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×D₅₉	—	2	1	no	—
79	C₂×C₄×D₅₉	—	2	1	no	—
80	C₂×C₄×D₅₉	—	2	1	no	—
81	C₂×Dic₁₁₈	C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₅₉	—	2	1	no	—
83	C₂×C₈×C₅₉	—	2	1	no	—
84	C₂×C₈×C₅₉	—	4	1	no	—
85	C₄²×C₅₉	—	2	1	no	—
86	C₄²×C₅₉	—	2	1	no	—
87	C₄²×C₅₉	—	2	1	no	—
88	C₄²×C₅₉	—	4	1	no	—
89	C₄²×C₅₉	—	4	1	no	—
90	C₄²×C₅₉	—	4	2	no	—
91	C₄²×C₅₉	C₂×C₈	1	2	yes	complemented
92	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
95	C₄²×C₅₉	#41	1	1	yes	characteristic
96	C₂²⋊Dic₅₉	—	2	1	no	—
97	(C₂×D₅₉)⋊C₄	—	2	1	no	—
98	(C₂×D₅₉)⋊C₄	—	2	1	no	—
99	C₄⋊Dic₅₉	C₄⋊C₄	1	1	yes	characteristic
100	C₄⋊Dic₅₉	C₄⋊C₄	1	1	yes	characteristic
101	Dic₅₉⋊C₄	—	2	1	no	—
102	Dic₅₉⋊C₄	—	2	1	no	—
103	C₄×Dic₅₉	—	2	1	no	—
104	C₄×Dic₅₉	—	2	1	no	—
105	C₄×Dic₅₉	—	2	1	no	—
106	#11745664	—	2	1	no	—
107	#11745664	—	2	1	no	—
108	#11745664	—	2	1	no	—
109	#11745664	C₂×C₈	1	2	yes	complemented
110	C₂×C₅₉⋊C₈	—	2	1	no	—
111	C₂×C₅₉⋊C₈	—	2	1	no	—
112	C₂×C₅₉⋊C₈	—	2	2	no	—
113	C₂×C₅₉⋊C₈	—	4	2	no	—
114	C₂×C₅₉⋊C₈	—	4	2	no	—
115	C₂×C₅₉⋊C₈	—	4	1	no	—
116	C₈×D₅₉	—	8	2	no	—
117	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
118	C₂²×D₅₉	—	2	1	no	—
119	C₂×C₄×C₅₉	—	2	1	no	—
120	C₂×C₄×C₅₉	—	2	1	no	—
121	C₂×C₄×C₅₉	—	2	2	no	—
122	C₂×C₄×C₅₉	—	2	2	no	—
123	C₂×C₄×C₅₉	—	4	1	no	—
124	C₂×C₄×C₅₉	—	4	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×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
131	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
133	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
134	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
136	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
137	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
138	C₂²⋊D₅₉	—	2	2	no	—
139	C₂²⋊D₅₉	—	2	2	no	—
140	C₂×Dic₅₉	—	2	1	no	—
141	C₂×Dic₅₉	—	2	1	no	—
142	C₂×Dic₅₉	—	2	1	no	—
143	D₂₃₆	—	2	1	no	—
144	D₂₃₆	C₄⋊C₈	1	2	yes	—
145	C₄×D₅₉	—	2	2	no	—
146	C₄×D₅₉	—	2	2	no	—
147	C₄×D₅₉	—	4	1	no	—
148	C₄×D₅₉	—	4	1	no	—
149	C₄×D₅₉	—	8	1	no	—
150	Dic₁₁₈	—	2	1	no	—
151	Dic₁₁₈	C₄⋊C₈	1	2	yes	—
152	C₈×C₅₉	—	4	1	no	—
153	C₈×C₅₉	—	4	1	no	—
154	C₈×C₅₉	—	8	1	no	—
155	C₅₉⋊C₈	—	2	2	no	—
156	C₅₉⋊C₈	—	4	1	no	—
157	C₅₉⋊C₈	—	4	2	no	—
158	C₅₉⋊C₈	—	8	1	no	—
159	#8840	—	59	1	no	maximal, Sylow
160	C₂²×C₅₉	—	2	1	no	—
161	C₂²×C₅₉	—	2	1	no	—
162	C₂²×C₅₉	#385	1	1	yes	characteristic
163	C₂²×C₅₉	C₈×D₄	1	1	yes	characteristic
164	C₂²×C₅₉	#435	1	1	yes	characteristic
165	C₂×D₅₉	—	2	2	no	—
166	C₂×D₅₉	—	4	1	no	—
167	C₂×D₅₉	—	4	1	no	—
168	C₄×C₅₉	—	2	1	no	—
169	C₄×C₅₉	—	2	1	no	—
170	C₄×C₅₉	—	2	1	no	—
171	C₄×C₅₉	—	4	2	no	—
172	C₄×C₅₉	—	4	2	no	—
173	C₄×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
174	C₄×C₅₉	#424	1	1	yes	characteristic
175	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
176	C₄×C₅₉	#435	1	1	yes	characteristic
177	C₄×C₅₉	#454	1	1	yes	characteristic
178	C₄×C₅₉	#455	1	1	yes	characteristic
179	Dic₅₉	—	2	2	no	—
180	Dic₅₉	—	4	1	no	—
181	Dic₅₉	—	4	1	no	—
182	#2988	—	59	1	no	—
183	#1831	—	59	1	no	—
184	C₄×#104	—	59	1	no	—
185	#1565	—	59	1	no	—
186	#1477	—	59	1	no	—
187	#1283	—	59	1	no	—
188	#1276	D₅₉	1	1	yes	characteristic, complemented
189	C₂×C₅₉	—	2	1	no	—
190	C₂×C₅₉	—	2	1	no	—
191	C₂×C₅₉	#1850	1	1	yes	characteristic
192	C₂×C₅₉	#1860	1	1	yes	characteristic
193	C₂×C₅₉	#1861	1	1	yes	characteristic
194	D₅₉	—	4	2	no	—
195	C₄×#41	—	59	1	no	—
196	C₈×Q₈	—	118	1	no	—
197	C₈×D₄	—	118	1	no	—
198	C₈×D₄	—	118	2	no	—
199	#431	—	118	1	no	—
200	#431	C₂×D₅₉	1	1	yes	characteristic
201	#431	C₂×D₅₉	1	1	yes	characteristic
202	C₂×C₄⋊C₈	—	59	1	no	—
203	C₂×C₄⋊C₈	—	59	1	no	—
204	C₂×C₄⋊C₈	—	118	1	no	—
205	C₂×#104	—	59	1	no	—
206	C₂×#104	—	59	1	no	—
207	C₄×M₁₆	—	59	1	no	—
208	C₄³	C₂×D₅₉	1	1	yes	characteristic
209	#339	—	59	1	no	—
210	#339	—	59	1	no	—
211	#336	—	118	2	no	—
212	#335	—	118	1	no	—
213	#334	—	118	1	no	—
214	Q₈⋊C₈	—	118	1	no	—
215	D₄⋊C₈	—	118	1	no	—
216	C₅₉	#8840	1	1	yes	characteristic, complemented, Sylow
217	C₂×#41	—	59	1	no	—
218	C₂×M₁₆	—	59	1	no	—
219	C₂×M₁₆	—	59	1	no	—
220	C₂²×C₈	—	59	2	no	—
221	C₂²×C₈	—	118	2	no	—
222	C₂²×C₈	—	118	1	no	—
223	C₄×Q₈	—	59	1	no	—
224	C₄×Q₈	—	59	1	no	—
225	C₄×D₄	—	59	1	no	—
226	C₄×D₄	—	59	1	no	—
227	C₄×D₄	—	118	1	no	—
228	C₄×D₄	—	118	1	no	—
229	#117	—	59	1	no	—
230	#117	—	118	1	no	—
231	C₂×C₄²	—	2	1	no	—
232	C₂×C₄²	—	2	1	no	—
233	C₂×C₄²	—	118	1	no	—
234	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
235	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
236	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
237	C₄⋊C₈	—	2	1	no	—
238	C₄⋊C₈	—	118	2	no	—
239	C₄⋊C₈	—	118	1	no	—
240	C₄⋊C₈	—	118	1	no	—
241	C₄⋊C₈	—	118	1	no	—
242	C₄⋊C₈	—	118	1	no	—
243	C₄⋊C₈	D₂₃₆	1	1	yes	characteristic
244	C₄⋊C₈	D₂₃₆	1	1	yes	characteristic
245	#104	—	59	4	no	—
246	#104	—	118	2	no	—
247	C₂²⋊C₈	—	2	1	no	—
248	C₂²⋊C₈	—	2	1	no	—
249	C₂²⋊C₈	—	118	2	no	—
250	#97	—	118	1	no	—
251	C₄×C₈	—	2	1	no	—
252	C₄×C₈	—	2	1	no	—
253	C₄×C₈	—	118	2	no	—
254	C₄×C₈	—	118	1	no	—
255	#41	—	59	2	no	—
256	#41	—	59	2	no	—
257	#41	—	118	1	no	—
258	#41	—	118	1	no	—
259	C₂×Q₈	—	59	1	no	—
260	C₂×D₄	—	59	1	no	—
261	C₂×D₄	—	118	1	no	—
262	C₂²×C₄	—	2	1	no	—
263	C₂²×C₄	—	2	1	no	—
264	C₂²×C₄	—	118	1	no	—
265	C₂²×C₄	—	118	1	no	—
266	C₂²×C₄	—	118	1	no	—
267	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
268	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
269	C₂²×C₄	#11745687	1	1	yes	characteristic
270	M₁₆	—	59	2	no	—
271	M₁₆	—	118	1	no	—
272	M₁₆	—	118	1	no	—
273	M₁₆	—	118	1	no	—
274	C₂×C₈	—	2	1	no	—
275	C₂×C₈	—	2	1	no	—
276	C₂×C₈	—	4	1	no	—
277	C₂×C₈	—	118	1	no	—
278	C₂×C₈	—	118	1	no	—
279	C₂×C₈	—	118	2	no	—
280	C₂×C₈	—	472	2	no	—
281	C₂×C₈	—	236	2	no	—
282	C₂×C₈	—	236	1	no	—
283	C₂×C₈	—	236	2	no	—
284	C₄⋊C₄	—	59	1	no	—
285	C₄⋊C₄	—	59	1	no	—
286	C₄⋊C₄	—	118	1	no	—
287	C₄⋊C₄	—	118	1	no	—
288	C₂²⋊C₄	—	118	1	no	—
289	C₂²⋊C₄	—	118	1	no	—
290	C₂²⋊C₄	—	118	1	no	—
291	C₄²	—	2	1	no	—
292	C₄²	—	2	1	no	—
293	C₄²	—	2	1	no	—
294	C₄²	—	4	1	no	—
295	C₄²	—	4	1	no	—
296	C₄²	—	4	2	no	—
297	C₄²	—	118	1	no	—
298	C₄²	—	118	1	no	—
299	C₄²	—	118	1	no	—
300	C₄²	C₈×D₅₉	1	2	yes	complemented
301	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
302	C₄²	#11745685	1	1	yes	characteristic
303	C₄²	D₄×D₅₉	1	1	yes	characteristic
304	C₄²	Q₈×D₅₉	1	1	yes	characteristic
305	C₂³	—	118	1	no	—
306	C₂³	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
307	Q₈	—	59	2	no	—
308	Q₈	—	118	1	no	—
309	D₄	—	59	2	no	—
310	D₄	—	118	2	no	—
311	D₄	—	118	2	no	—
312	D₄	—	118	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	118	1	no	—
323	C₂×C₄	—	118	2	no	—
324	C₂×C₄	—	118	2	no	—
325	C₂×C₄	—	118	1	no	—
326	C₂×C₄	—	118	1	no	—
327	C₂×C₄	—	472	1	no	—
328	C₂×C₄	—	236	1	no	—
329	C₂×C₄	—	236	1	no	—
330	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
331	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
332	C₂×C₄	#14273697	1	1	yes	characteristic
333	C₂×C₄	#14273705	1	1	yes	characteristic
334	C₂×C₄	#14273707	1	1	yes	characteristic
335	C₂×C₄	#14273709	1	1	yes	characteristic
336	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
337	C₂×C₄	M₁₆×D₅₉	1	1	yes	characteristic
338	C₂×C₄	#14273733	1	1	yes	characteristic
339	C₂×C₄	#14273735	1	1	yes	characteristic
340	C₈	—	4	1	no	—
341	C₈	—	4	1	no	—
342	C₈	—	8	1	no	—
343	C₈	—	118	2	no	—
344	C₈	—	472	1	no	—
345	C₈	—	236	1	no	—
346	C₈	—	236	2	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	118	2	no	—
350	C₂²	—	236	1	no	—
351	C₂²	—	236	1	no	—
352	C₂²	#16837276	1	1	yes	characteristic
353	C₂²	(C₂²⋊D₅₉)⋊C₈	1	1	yes	characteristic
354	C₂²	#16837476	1	1	yes	characteristic
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	4	2	no	—
359	C₄	—	4	2	no	—
360	C₄	—	118	2	no	—
361	C₄	—	236	1	no	—
362	C₄	—	236	1	no	—
363	C₄	C₄⋊C₈×D₅₉	1	1	yes	characteristic
364	C₄	#16837427	1	1	yes	characteristic
365	C₄	C₈×D₂₃₆	1	1	yes	characteristic
366	C₄	#16837475	1	1	yes	characteristic
367	C₄	#16837576	1	1	yes	characteristic
368	C₄	#16837581	1	1	yes	characteristic
369	C₂	—	2	1	no	—
370	C₂	—	2	1	no	—
371	C₂	—	236	2	no	—
372	C₂	#21424275	1	1	yes	characteristic
373	C₂	#21424330	1	1	yes	characteristic
374	C₂	#21424337	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14273697
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1276	C₂
socle	C₂²×C₅₉	#385
prime radical	C₂×#11745685	C₂×C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1276	D₅₉
hypocenter	C₅₉	#8840
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1276

Complements

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

N	H ≅ G/N	type
2. #21424213	C₂	semidirect
4. #21422849	C₂	semidirect
6. #21421945	C₂	semidirect
7. C₅₉×#1276	C₂	semidirect
50. #14273655	C₈	semidirect
91. C₄²×C₅₉	C₂×C₈	semidirect
109. #11745664	C₂×C₈	semidirect
188. #1276	D₅₉	semidirect
216. C₅₉	#8840	semidirect
300. C₄²	C₈×D₅₉	semidirect

Maximal subgroups

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

subgroup	index
#21433204	2
#21424213	2
C₄×#14273655	2
#21422849	2
#21422484	2
#21421945	2
C₅₉×#1276	2
#8840	59