Subgroups — #28560474

Order 15104.

600 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21434666	C₂	1	1	yes	maximal, characteristic
2	#21433094	C₂	1	2	yes	maximal, complemented
3	#21422992	C₂	1	1	yes	maximal, characteristic, complemented
4	#21422980	C₂	1	1	yes	maximal, characteristic, complemented
5	#21422938	C₂	1	2	yes	maximal, complemented
6	#21422931	C₂	1	2	yes	maximal, complemented
7	#21422902	C₂	1	1	yes	maximal, characteristic, complemented
8	#21422890	C₂	1	1	yes	maximal, characteristic, complemented
9	#21422502	C₂	1	2	yes	maximal, complemented
10	C₈×#11745664	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₉×#1564	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837936	C₂²	1	2	yes	—
13	#16837804	C₄	1	1	yes	characteristic
14	#16837786	C₄	1	1	yes	characteristic
15	#16837757	C₄	1	2	yes	—
16	#16837643	C₄	1	1	yes	characteristic
17	#16837638	C₄	1	1	yes	characteristic
18	#16837631	C₄	1	2	yes	—
19	#16837529	C₂²	1	2	yes	—
20	#16837526	C₂²	1	2	yes	—
21	C₄×Dic₂₃₆	C₂²	1	1	yes	characteristic
22	C₄×#11745660	C₂²	1	2	yes	—
23	C₄×D₄₇₂	C₂²	1	1	yes	characteristic
24	#16837479	C₂²	1	2	yes	—
25	#16837477	—	2	2	no	—
26	#16837476	—	2	2	no	—
27	#16837467	C₂²	1	2	yes	—
28	#16837461	—	2	2	no	—
29	#16837455	C₂²	1	1	yes	characteristic
30	#16837356	—	2	2	no	—
31	C₄×#11745664	C₂²	1	1	yes	characteristic
32	#16837132	C₂²	1	2	yes	—
33	#16837132	C₂²	1	2	yes	—
34	#16837129	C₂²	1	2	yes	—
35	#16837129	C₂²	1	2	yes	—
36	#16837124	C₂²	1	1	yes	characteristic
37	#16837124	C₂²	1	1	yes	characteristic
38	C₈×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
39	C₈×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
40	C₅₉×#431	C₂²	1	2	yes	—
41	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
42	C₅₉×#335	C₂²	1	1	yes	characteristic
43	C₅₉×#335	C₂²	1	1	yes	characteristic
44	C₅₉×#334	C₂²	1	2	yes	—
45	C₂×Dic₂₃₆	C₂×C₄	1	1	yes	characteristic
46	C₂×#11745660	C₂×C₄	1	2	yes	—
47	C₂×D₄₇₂	C₂×C₄	1	1	yes	characteristic
48	C₂×#11745664	D₄	1	1	yes	characteristic
49	C₂×#11745664	D₄	1	1	yes	characteristic
50	C₂×#11745659	—	2	2	no	—
51	C₂×C₈×D₅₉	—	2	2	no	—
52	#14273721	—	2	2	no	—
53	#14273719	C₂×C₄	1	2	yes	—
54	#14273715	C₂×C₄	1	2	yes	—
55	#14273710	—	2	2	no	—
56	#14273706	C₂×C₄	1	2	yes	—
57	#14273701	C₂×C₄	1	2	yes	—
58	C₄×Dic₁₁₈	C₂³	1	2	yes	—
59	C₄×D₂₃₆	C₂³	1	2	yes	—
60	C₄×C₂²⋊D₅₉	—	2	2	no	—
61	#14273684	—	2	2	no	—
62	#14273661	C₂×C₄	1	1	yes	characteristic
63	#14273661	C₂×C₄	1	1	yes	characteristic
64	#14273659	C₂×C₄	1	2	yes	—
65	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
66	Dic₅₉⋊C₈	—	2	2	no	—
67	Dic₅₉⋊C₈	—	2	2	no	—
68	#14273652	C₂×C₄	1	2	yes	—
69	#14273652	C₂×C₄	1	2	yes	—
70	#14273649	C₂×C₄	1	2	yes	—
71	#14273649	C₂×C₄	1	2	yes	—
72	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
73	(C₂×D₅₉)⋊C₈	—	2	2	no	—
74	(C₂×D₅₉)⋊C₈	—	2	2	no	—
75	#14273638	D₄	1	1	yes	characteristic
76	#14273638	D₄	1	1	yes	characteristic
77	#14273637	—	2	2	no	—
78	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
79	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
80	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
81	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
82	C₈×Dic₅₉	—	2	2	no	—
83	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
84	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₅₉	C₂³	1	2	yes	—
87	C₄⋊C₈×C₅₉	C₂³	1	2	yes	—
88	C₂²⋊C₈×C₅₉	—	2	2	no	—
89	C₄×C₈×C₅₉	—	2	2	no	—
90	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
91	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
92	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
97	C₂×C₂²⋊D₅₉	—	2	2	no	—
98	C₂×D₂₃₆	C₂²×C₄	1	2	yes	—
99	C₂×C₄×D₅₉	—	2	2	no	—
100	C₂×Dic₁₁₈	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₅₉	—	2	2	no	—
102	C₂×C₈×C₅₉	—	2	2	no	—
103	C₂×C₈×C₅₉	—	2	2	no	—
104	C₂×C₈×C₅₉	—	4	1	no	—
105	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₅₉	#41	1	1	yes	characteristic
112	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
113	C₂²⋊Dic₅₉	—	2	2	no	—
114	(C₂×D₅₉)⋊C₄	—	2	2	no	—
115	(C₂×D₅₉)⋊C₄	—	2	2	no	—
116	(C₂×D₅₉)⋊C₄	—	2	2	no	—
117	(C₂×D₅₉)⋊C₄	—	2	2	no	—
118	C₄⋊Dic₅₉	C₂²×C₄	1	2	yes	—
119	C₄⋊Dic₅₉	C₂²×C₄	1	2	yes	—
120	Dic₅₉⋊C₄	—	2	2	no	—
121	Dic₅₉⋊C₄	—	2	2	no	—
122	Dic₅₉⋊C₄	—	2	2	no	—
123	Dic₅₉⋊C₄	—	2	2	no	—
124	C₄×Dic₅₉	—	2	2	no	—
125	#11745664	—	2	2	no	—
126	#11745664	—	2	2	no	—
127	C₂×C₅₉⋊C₈	—	2	1	no	—
128	C₂×C₅₉⋊C₈	—	2	1	no	—
129	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
130	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
131	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
132	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
133	Dic₂₃₆	—	2	2	no	—
134	D₄₇₂	—	2	2	no	—
135	#11745660	—	4	2	no	—
136	#11745659	—	4	4	no	—
137	C₈×D₅₉	—	4	4	no	—
138	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
139	C₂²×D₅₉	—	2	2	no	—
140	C₂×C₄×C₅₉	—	2	1	no	—
141	C₂×C₄×C₅₉	—	2	1	no	—
142	C₂×C₄×C₅₉	—	2	1	no	—
143	C₂×C₄×C₅₉	—	2	1	no	—
144	C₂×C₄×C₅₉	—	4	1	no	—
145	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
151	C₂²⋊D₅₉	—	8	2	no	—
152	C₂×Dic₅₉	—	2	2	no	—
153	C₂×Dic₅₉	—	2	2	no	—
154	C₂×Dic₅₉	—	2	2	no	—
155	D₂₃₆	—	2	2	no	—
156	D₂₃₆	—	2	2	no	—
157	C₄×D₅₉	—	4	2	no	—
158	C₄×D₅₉	—	4	2	no	—
159	Dic₁₁₈	—	2	2	no	—
160	Dic₁₁₈	—	2	2	no	—
161	C₈×C₅₉	—	2	1	no	—
162	C₈×C₅₉	—	2	1	no	—
163	C₈×C₅₉	—	2	1	no	—
164	C₈×C₅₉	—	4	2	no	—
165	C₈×C₅₉	—	4	2	no	—
166	C₈×C₅₉	#131	1	2	yes	—
167	C₅₉⋊C₈	—	2	1	no	—
168	C₅₉⋊C₈	—	2	1	no	—
169	C₅₉⋊C₈	—	4	1	no	—
170	C₅₉⋊C₈	—	4	1	no	—
171	C₅₉⋊C₈	#135	1	2	yes	—
172	C₅₉⋊C₈	#135	1	2	yes	—
173	#18069	—	59	1	no	maximal, Sylow
174	C₂²×C₅₉	—	2	1	no	—
175	C₂²×C₅₉	—	2	1	no	—
176	C₂²×C₅₉	—	2	1	no	—
177	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
178	C₂×D₅₉	—	4	2	no	—
179	C₂×D₅₉	—	4	2	no	—
180	C₂×D₅₉	—	4	2	no	—
181	C₄×C₅₉	—	2	1	no	—
182	C₄×C₅₉	—	2	1	no	—
183	C₄×C₅₉	—	2	1	no	—
184	C₄×C₅₉	—	2	1	no	—
185	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
186	C₄×C₅₉	#568	1	1	yes	characteristic
187	C₄×C₅₉	C₂×#135	1	1	yes	characteristic
188	C₄×C₅₉	C₂×#135	1	1	yes	characteristic
189	Dic₅₉	—	4	2	no	—
190	Dic₅₉	—	4	2	no	—
191	Dic₅₉	—	4	2	no	—
192	#3168	—	59	1	no	—
193	#2969	—	59	2	no	—
194	#1591	—	59	1	no	—
195	#1589	—	59	1	no	—
196	#1583	—	59	2	no	—
197	#1582	—	59	2	no	—
198	C₈×Q₁₆	—	59	1	no	—
199	C₈×D₈	—	59	1	no	—
200	#1564	D₅₉	1	1	yes	characteristic, complemented
201	#1482	—	59	2	no	—
202	C₈×M₁₆	—	59	1	no	—
203	C₂×C₅₉	—	4	1	no	—
204	C₂×C₅₉	#2935	1	1	yes	characteristic
205	C₂×C₅₉	C₄×#135	1	1	yes	characteristic
206	C₂×C₅₉	#2957	1	1	yes	characteristic
207	D₅₉	—	8	2	no	—
208	#529	—	59	2	no	—
209	#499	—	59	1	no	—
210	#495	—	59	1	no	—
211	#488	—	59	2	no	—
212	#467	—	59	1	no	—
213	#466	—	59	1	no	—
214	#465	—	59	2	no	—
215	#446	—	59	2	no	—
216	C₈×Q₈	—	59	2	no	—
217	C₄×Q₁₆	—	59	1	no	—
218	C₄×SD₁₆	—	59	2	no	—
219	C₄×D₈	—	59	1	no	—
220	#436	—	59	2	no	—
221	#436	—	118	2	no	—
222	#435	—	118	2	no	—
223	C₈×D₄	—	59	2	no	—
224	#433	—	118	2	no	—
225	#431	—	59	1	no	—
226	#431	C₂×D₅₉	1	2	yes	—
227	#405	—	118	2	no	—
228	C₄×M₁₆	—	59	1	no	—
229	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
230	#335	C₂×D₅₉	1	1	yes	characteristic
231	#335	C₂×D₅₉	1	1	yes	characteristic
232	#334	C₂×D₅₉	1	2	yes	—
233	Q₈⋊C₈	—	59	2	no	—
234	Q₈⋊C₈	—	59	2	no	—
235	D₄⋊C₈	—	59	2	no	—
236	D₄⋊C₈	—	59	2	no	—
237	#322	—	59	1	no	—
238	#322	—	59	1	no	—
239	C₈²	—	59	1	no	—
240	C₈²	—	59	1	no	—
241	C₅₉	#18069	1	1	yes	characteristic, complemented, Sylow
242	C₂×Q₁₆	—	59	1	no	—
243	C₂×SD₁₆	—	59	2	no	—
244	C₂×D₈	—	59	1	no	—
245	C₂×M₁₆	—	59	1	no	—
246	C₂×M₁₆	—	59	1	no	—
247	C₂×M₁₆	—	118	2	no	—
248	C₂²×C₈	—	118	2	no	—
249	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
250	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
251	#126	—	118	2	no	—
252	#125	—	59	2	no	—
253	#124	—	59	2	no	—
254	#123	—	118	2	no	—
255	C₂²⋊Q₈	—	59	2	no	—
256	#121	—	59	2	no	—
257	C₄×Q₈	—	59	2	no	—
258	C₄×D₄	—	59	2	no	—
259	C₄×D₄	—	118	2	no	—
260	#117	—	118	2	no	—
261	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
262	#107	—	59	1	no	—
263	#107	—	59	1	no	—
264	#106	—	59	2	no	—
265	C₄⋊C₈	—	118	2	no	—
266	C₄⋊C₈	—	118	2	no	—
267	C₄⋊C₈	—	118	1	no	—
268	C₄⋊C₈	C₂²×D₅₉	1	2	yes	—
269	C₄⋊C₈	C₂²×D₅₉	1	2	yes	—
270	#103	—	59	2	no	—
271	#103	—	59	2	no	—
272	#102	—	59	2	no	—
273	#102	—	59	2	no	—
274	C₂²⋊C₈	—	2	2	no	—
275	C₂²⋊C₈	—	118	2	no	—
276	C₂²⋊C₈	—	118	2	no	—
277	C₂²⋊C₈	—	118	1	no	—
278	#97	—	59	1	no	—
279	#97	—	59	1	no	—
280	#97	—	118	2	no	—
281	C₄×C₈	—	2	2	no	—
282	C₄×C₈	—	59	1	no	—
283	C₄×C₈	—	59	1	no	—
284	C₄×C₈	—	59	1	no	—
285	C₄×C₈	—	59	1	no	—
286	C₄×C₈	—	118	2	no	—
287	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
288	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
290	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
291	C₂×Q₈	—	59	2	no	—
292	C₂×D₄	—	59	2	no	—
293	C₂×D₄	—	118	2	no	—
294	C₂²×C₄	—	118	2	no	—
295	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
296	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
297	C₂²×C₄	#11745689	1	1	yes	characteristic
298	Q₁₆	—	118	2	no	—
299	SD₁₆	—	236	2	no	—
300	D₈	—	118	2	no	—
301	M₁₆	—	118	2	no	—
302	M₁₆	—	118	2	no	—
303	M₁₆	—	236	4	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	4	1	no	—
308	C₂×C₈	—	59	1	no	—
309	C₂×C₈	—	59	1	no	—
310	C₂×C₈	—	59	1	no	—
311	C₂×C₈	—	59	1	no	—
312	C₂×C₈	—	118	1	no	—
313	C₂×C₈	—	118	1	no	—
314	C₂×C₈	—	236	4	no	—
315	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
316	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
317	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
318	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
319	C₄⋊C₄	—	59	2	no	—
320	C₄⋊C₄	—	59	2	no	—
321	C₄⋊C₄	—	118	2	no	—
322	C₄⋊C₄	—	118	2	no	—
323	C₄⋊C₄	—	118	2	no	—
324	C₄⋊C₄	—	118	2	no	—
325	C₂²⋊C₄	—	118	2	no	—
326	C₂²⋊C₄	—	118	2	no	—
327	C₂²⋊C₄	—	118	2	no	—
328	C₂²⋊C₄	—	118	2	no	—
329	C₂²⋊C₄	—	118	2	no	—
330	C₄²	—	118	2	no	—
331	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
332	C₄²	D₄×D₅₉	1	1	yes	characteristic
333	C₄²	#11745689	1	1	yes	characteristic
334	C₄²	C₂³×D₅₉	1	1	yes	characteristic
335	C₂³	—	118	2	no	—
336	C₂³	#14273689	1	1	yes	characteristic
337	Q₈	—	118	2	no	—
338	Q₈	—	118	2	no	—
339	D₄	—	118	2	no	—
340	D₄	—	118	2	no	—
341	D₄	—	472	2	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	118	2	no	—
348	C₂×C₄	—	118	2	no	—
349	C₂×C₄	—	118	2	no	—
350	C₂×C₄	—	236	2	no	—
351	C₂×C₄	—	236	2	no	—
352	C₂×C₄	#14273689	1	1	yes	characteristic
353	C₂×C₄	#14273689	1	1	yes	characteristic
354	C₂×C₄	#14273689	1	1	yes	characteristic
355	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
356	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
357	C₂×C₄	C₂×#11745689	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	4	2	no	—
362	C₈	—	4	2	no	—
363	C₈	—	59	2	no	—
364	C₈	—	59	2	no	—
365	C₈	—	118	1	no	—
366	C₈	—	118	1	no	—
367	C₈	—	236	1	no	—
368	C₈	—	236	1	no	—
369	C₈	#14273733	1	2	yes	—
370	C₂²	—	2	1	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	2	1	no	—
373	C₂²	—	236	2	no	—
374	C₂²	—	236	2	no	—
375	C₂²	—	236	2	no	—
376	C₂²	C₂×#14273689	1	1	yes	characteristic
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	236	2	no	—
382	C₄	—	236	2	no	—
383	C₄	—	236	2	no	—
384	C₄	C₂×#14273733	1	1	yes	characteristic
385	C₄	#16838168	1	1	yes	characteristic
386	C₄	#135×D₅₉	1	1	yes	characteristic
387	C₄	#135×D₅₉	1	1	yes	characteristic
388	C₂	—	4	1	no	—
389	C₂	—	472	2	no	—
390	C₂	#21432839	1	1	yes	characteristic
391	C₂	#21432867	1	1	yes	characteristic
392	C₂	#21433012	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	C₂×C₄×D₄
prime radical	#16837638	C₄
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	#1564	D₅₉
hypocenter	C₅₉	#18069
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
2. #21433094	C₂	semidirect
3. #21422992	C₂	semidirect
4. #21422980	C₂	semidirect
5. #21422938	C₂	semidirect
6. #21422931	C₂	semidirect
7. #21422902	C₂	semidirect
8. #21422890	C₂	semidirect
9. #21422502	C₂	semidirect
10. C₈×#11745664	C₂	semidirect
11. C₅₉×#1564	C₂	semidirect
200. #1564	D₅₉	semidirect
241. C₅₉	#18069	semidirect

Maximal subgroups

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

subgroup	index
#21434666	2
#21433094	2
#21422992	2
#21422980	2
#21422938	2
#21422931	2
#21422902	2
#21422890	2
#21422502	2
C₈×#11745664	2
C₅₉×#1564	2
#18069	59