Subgroups — #28742224

Order 15104.

526 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21433071	C₂	1	1	yes	maximal, characteristic
2	#21426004	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₆×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#21425929	C₂	1	1	yes	maximal, characteristic, complemented
5	#21425921	C₂	1	2	yes	maximal, complemented
6	C₁₆×D₂₃₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₆×C₂²⋊D₅₉	C₂	1	2	yes	maximal, complemented
8	#21425901	C₂	1	2	yes	maximal, complemented
9	#21425899	C₂	1	1	yes	maximal, characteristic, complemented
10	#21425691	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837936	C₄	1	1	yes	characteristic
13	C₂×#14273666	C₂²	1	2	yes	—
14	C₂×C₁₆×D₅₉	C₂²	1	2	yes	—
15	#16837532	C₄	1	1	yes	characteristic
16	C₈×Dic₁₁₈	C₂²	1	1	yes	characteristic
17	#16837481	C₄	1	1	yes	characteristic
18	#16837473	C₄	1	2	yes	—
19	C₈×D₂₃₆	C₂²	1	1	yes	characteristic
20	C₈×C₂²⋊D₅₉	C₂²	1	2	yes	—
21	#16837461	C₄	1	2	yes	—
22	#16837459	C₄	1	1	yes	characteristic
23	#16837356	C₂²	1	2	yes	—
24	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	1	yes	characteristic
25	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	1	yes	characteristic
26	Dic₅₉⋊C₁₆	C₂²	1	2	yes	—
27	Dic₅₉⋊C₁₆	C₂²	1	2	yes	—
28	C₂²⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
29	(C₂×D₅₉)⋊C₁₆	C₂²	1	2	yes	—
30	(C₂×D₅₉)⋊C₁₆	C₂²	1	2	yes	—
31	#16837178	C₂²	1	2	yes	—
32	#16837177	C₂²	1	2	yes	—
33	C₁₆×Dic₅₉	C₂²	1	2	yes	—
34	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
35	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
37	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
39	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
41	C₂×#11745659	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₅₉	C₂³	1	2	yes	—
43	#14273721	C₈	1	2	yes	—
44	#14273719	C₈	1	1	yes	characteristic
45	#14273715	C₈	1	1	yes	characteristic
46	#14273710	C₈	1	2	yes	—
47	#14273706	C₈	1	1	yes	characteristic
48	#14273701	C₈	1	1	yes	characteristic
49	C₄×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
50	C₄×D₂₃₆	C₂×C₄	1	1	yes	characteristic
51	C₄×C₂²⋊D₅₉	C₂×C₄	1	2	yes	—
52	#14273684	C₂×C₄	1	2	yes	—
53	#14273666	—	2	4	no	—
54	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
55	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
56	C₁₆×D₅₉	—	2	4	no	—
57	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
58	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
59	Dic₅₉⋊C₈	C₂×C₄	1	2	yes	—
60	Dic₅₉⋊C₈	C₂×C₄	1	2	yes	—
61	C₂²⋊(C₅₉⋊C₈)	C₂×C₄	1	2	yes	—
62	(C₂×D₅₉)⋊C₈	C₂×C₄	1	2	yes	—
63	(C₂×D₅₉)⋊C₈	C₂×C₄	1	2	yes	—
64	#14273638	C₂×C₄	1	2	yes	—
65	#14273637	C₂×C₄	1	2	yes	—
66	C₈×Dic₅₉	C₂³	1	2	yes	—
67	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
70	C₂×C₁₆×C₅₉	—	2	2	no	—
71	C₂×C₁₆×C₅₉	—	2	2	no	—
72	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
75	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
76	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
77	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
78	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
80	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
81	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
82	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₂²⋊D₅₉	C₂×C₈	1	2	yes	—
84	C₂×D₂₃₆	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₅₉	C₂²×C₄	1	2	yes	—
86	C₂×Dic₁₁₈	C₂×C₈	1	1	yes	characteristic
87	C₂×C₈×C₅₉	—	2	2	no	—
88	C₂×C₈×C₅₉	—	2	1	no	—
89	C₂×C₈×C₅₉	—	2	1	no	—
90	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
94	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
95	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
96	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
98	C₂²⋊Dic₅₉	C₂×C₈	1	2	yes	—
99	(C₂×D₅₉)⋊C₄	C₂×C₈	1	2	yes	—
100	(C₂×D₅₉)⋊C₄	C₂×C₈	1	2	yes	—
101	(C₂×D₅₉)⋊C₄	C₂×C₈	1	2	yes	—
102	(C₂×D₅₉)⋊C₄	C₂×C₈	1	2	yes	—
103	C₄⋊Dic₅₉	C₂×C₈	1	1	yes	characteristic
104	C₄⋊Dic₅₉	C₂×C₈	1	1	yes	characteristic
105	Dic₅₉⋊C₄	C₂×C₈	1	2	yes	—
106	Dic₅₉⋊C₄	C₂×C₈	1	2	yes	—
107	Dic₅₉⋊C₄	C₂×C₈	1	2	yes	—
108	Dic₅₉⋊C₄	C₂×C₈	1	2	yes	—
109	C₄×Dic₅₉	C₂²×C₄	1	2	yes	—
110	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
111	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
112	#11745659	—	2	4	no	—
113	C₈×D₅₉	—	2	2	no	—
114	C₈×D₅₉	—	2	2	no	—
115	C₁₆×C₅₉	—	2	1	no	—
116	C₁₆×C₅₉	—	2	1	no	—
117	C₁₆×C₅₉	#41	1	2	yes	—
118	C₁₆×C₅₉	#41	1	2	yes	—
119	C₅₉⋊C₁₆	—	2	2	no	—
120	C₅₉⋊C₁₆	—	2	2	no	—
121	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
122	C₂²×D₅₉	C₂²×C₈	1	2	yes	—
123	C₂×C₄×C₅₉	—	2	2	no	—
124	C₂×C₄×C₅₉	—	2	1	no	—
125	C₂×C₄×C₅₉	—	2	1	no	—
126	C₂×C₄×C₅₉	—	2	2	no	—
127	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
133	C₂²⋊D₅₉	—	2	4	no	—
134	C₂×Dic₅₉	C₂²×C₈	1	2	yes	—
135	C₂×Dic₅₉	C₂²×C₈	1	2	yes	—
136	C₂×Dic₅₉	C₂²×C₈	1	2	yes	—
137	D₂₃₆	—	2	2	no	—
138	C₄×D₅₉	—	2	2	no	—
139	C₄×D₅₉	—	2	2	no	—
140	Dic₁₁₈	—	2	2	no	—
141	C₈×C₅₉	—	2	1	no	—
142	C₈×C₅₉	—	2	1	no	—
143	C₈×C₅₉	#131	1	2	yes	—
144	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
145	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
146	C₅₉⋊C₈	—	2	2	no	—
147	C₅₉⋊C₈	—	2	2	no	—
148	#33480	—	59	1	no	maximal, Sylow
149	C₂²×C₅₉	—	2	1	no	—
150	C₂²×C₅₉	—	2	1	no	—
151	C₂²×C₅₉	#504	1	2	yes	—
152	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
153	C₂×D₅₉	—	2	2	no	—
154	C₂×D₅₉	—	2	2	no	—
155	C₂×D₅₉	—	2	2	no	—
156	C₄×C₅₉	—	2	1	no	—
157	C₄×C₅₉	—	2	1	no	—
158	C₄×C₅₉	—	2	1	no	—
159	C₄×C₅₉	—	2	1	no	—
160	C₄×C₅₉	#504	1	2	yes	—
161	C₄×C₅₉	C₄×#41	1	1	yes	characteristic
162	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
163	Dic₅₉	—	2	2	no	—
164	Dic₅₉	—	2	2	no	—
165	Dic₅₉	—	2	2	no	—
166	#2966	—	59	1	no	—
167	#2183	—	59	1	no	—
168	Q₈×C₁₆	—	59	1	no	—
169	#2169	—	59	1	no	—
170	#2168	—	59	2	no	—
171	D₄×C₁₆	—	59	2	no	—
172	D₄×C₁₆	—	59	1	no	—
173	#2164	—	59	2	no	—
174	#2163	—	59	1	no	—
175	#2108	—	59	2	no	—
176	C₂×C₄×C₁₆	D₅₉	1	1	yes	characteristic, complemented
177	C₂×C₅₉	—	2	2	no	—
178	C₂×C₅₉	C₈×#41	1	1	yes	characteristic
179	C₂×C₅₉	C₂×#504	1	1	yes	characteristic
180	C₂×C₅₉	C₂×#504	1	1	yes	characteristic
181	D₅₉	—	4	2	no	—
182	#529	—	59	1	no	—
183	C₂×M₃₂	—	59	2	no	—
184	C₂²×C₁₆	—	59	2	no	—
185	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
186	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
187	#446	—	59	1	no	—
188	C₈×Q₈	—	59	1	no	—
189	#436	—	59	1	no	—
190	#435	—	59	2	no	—
191	C₈×D₄	—	59	2	no	—
192	C₈×D₄	—	59	1	no	—
193	#433	—	59	2	no	—
194	#432	—	59	1	no	—
195	#405	—	59	2	no	—
196	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
197	C₄⋊C₁₆	—	59	2	no	—
198	C₄⋊C₁₆	—	59	2	no	—
199	C₄⋊C₁₆	—	59	1	no	—
200	C₄⋊C₁₆	—	59	1	no	—
201	C₂²⋊C₁₆	—	59	2	no	—
202	C₂²⋊C₁₆	—	59	2	no	—
203	C₂²⋊C₁₆	—	59	2	no	—
204	#346	—	59	2	no	—
205	#346	—	59	2	no	—
206	C₄×C₁₆	—	59	2	no	—
207	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
208	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
209	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
210	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
211	C₅₉	#33480	1	1	yes	characteristic, complemented, Sylow
212	C₂×M₁₆	—	59	2	no	—
213	C₂²×C₈	—	59	2	no	—
214	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
215	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
216	#126	—	59	2	no	—
217	#125	—	59	1	no	—
218	#124	—	59	1	no	—
219	#123	—	59	2	no	—
220	C₂²⋊Q₈	—	59	1	no	—
221	#121	—	59	1	no	—
222	C₄×Q₈	—	59	1	no	—
223	C₄×D₄	—	59	2	no	—
224	C₄×D₄	—	59	1	no	—
225	#117	—	59	2	no	—
226	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
227	M₃₂	—	118	4	no	—
228	C₂×C₁₆	—	2	2	no	—
229	C₂×C₁₆	—	2	2	no	—
230	C₂×C₁₆	—	59	2	no	—
231	C₂×C₁₆	—	59	2	no	—
232	C₂×C₁₆	—	118	4	no	—
233	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
234	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
235	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
236	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
237	C₄⋊C₈	—	59	2	no	—
238	C₄⋊C₈	—	59	2	no	—
239	C₄⋊C₈	—	59	1	no	—
240	C₄⋊C₈	—	59	1	no	—
241	C₂²⋊C₈	—	59	2	no	—
242	C₂²⋊C₈	—	59	2	no	—
243	C₂²⋊C₈	—	59	2	no	—
244	#97	—	59	2	no	—
245	#97	—	59	2	no	—
246	C₄×C₈	—	59	2	no	—
247	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
248	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
250	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
251	C₂×Q₈	—	59	1	no	—
252	C₂×D₄	—	59	2	no	—
253	C₂×D₄	—	59	1	no	—
254	C₂²×C₄	—	59	2	no	—
255	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
256	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
257	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
258	M₁₆	—	118	4	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	59	2	no	—
263	C₂×C₈	—	59	2	no	—
264	C₂×C₈	—	118	2	no	—
265	C₂×C₈	—	118	2	no	—
266	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
269	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
270	C₄⋊C₄	—	59	2	no	—
271	C₄⋊C₄	—	59	2	no	—
272	C₄⋊C₄	—	59	1	no	—
273	C₄⋊C₄	—	59	2	no	—
274	C₄⋊C₄	—	59	1	no	—
275	C₄⋊C₄	—	59	2	no	—
276	C₂²⋊C₄	—	59	2	no	—
277	C₂²⋊C₄	—	59	2	no	—
278	C₂²⋊C₄	—	59	2	no	—
279	C₂²⋊C₄	—	59	2	no	—
280	C₂²⋊C₄	—	59	2	no	—
281	C₄²	—	59	2	no	—
282	C₄²	C₈×D₅₉	1	1	yes	characteristic
283	C₄²	C₈×D₅₉	1	1	yes	characteristic
284	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
285	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
286	C₁₆	—	2	1	no	—
287	C₁₆	—	2	1	no	—
288	C₁₆	—	118	2	no	—
289	C₁₆	—	118	2	no	—
290	C₁₆	#11745685	1	2	yes	—
291	C₁₆	#11745685	1	2	yes	—
292	C₂³	—	59	2	no	—
293	C₂³	C₂×C₈×D₅₉	1	1	yes	characteristic
294	Q₈	—	118	2	no	—
295	D₄	—	118	4	no	—
296	D₄	—	118	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	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	C₂×C₈×D₅₉	1	1	yes	characteristic
307	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
308	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
309	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
310	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
311	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	118	2	no	—
315	C₈	—	118	2	no	—
316	C₈	#14273733	1	2	yes	—
317	C₈	C₂×#11745685	1	1	yes	characteristic
318	C₈	C₂×#11745685	1	1	yes	characteristic
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	118	2	no	—
322	C₂²	—	118	2	no	—
323	C₂²	—	118	2	no	—
324	C₂²	#16837818	1	2	yes	—
325	C₂²	C₂²×C₈×D₅₉	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	118	2	no	—
331	C₄	—	118	2	no	—
332	C₄	—	118	2	no	—
333	C₄	#16837818	1	2	yes	—
334	C₄	C₄×#11745685	1	1	yes	characteristic
335	C₄	C₂×#14273733	1	1	yes	characteristic
336	C₂	—	2	2	no	—
337	C₂	—	236	2	no	—
338	C₂	C₈×#11745685	1	1	yes	characteristic
339	C₂	C₂×#16837818	1	1	yes	characteristic
340	C₂	C₂×#16837818	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₄×C₁₆×C₅₉	C₂
prime radical	#14273701	C₈
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₅₉
hypocenter	C₅₉	#33480
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #21426004	C₂	semidirect
3. C₁₆×Dic₁₁₈	C₂	semidirect
4. #21425929	C₂	semidirect
5. #21425921	C₂	semidirect
6. C₁₆×D₂₃₆	C₂	semidirect
7. C₁₆×C₂²⋊D₅₉	C₂	semidirect
8. #21425901	C₂	semidirect
9. #21425899	C₂	semidirect
10. #21425691	C₂	semidirect
11. C₂×C₄×C₁₆×C₅₉	C₂	semidirect
176. C₂×C₄×C₁₆	D₅₉	semidirect
211. C₅₉	#33480	semidirect

Maximal subgroups

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

subgroup	index
#21433071	2
#21426004	2
C₁₆×Dic₁₁₈	2
#21425929	2
#21425921	2
C₁₆×D₂₃₆	2
C₁₆×C₂²⋊D₅₉	2
#21425901	2
#21425899	2
#21425691	2
C₂×C₄×C₁₆×C₅₉	2
#33480	59