Subgroups — #28560736

Order 15104.

598 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21433256	C₂	1	2	yes	maximal, complemented
2	#21422999	C₂	1	2	yes	maximal, complemented
3	#21422947	C₂	1	2	yes	maximal, complemented
4	#21422926	C₂	1	2	yes	maximal, complemented
5	SD₁₆⋊(C₅₉⋊C₈)	C₂	1	2	yes	maximal, complemented
6	C₂×#16837155	C₂	1	1	yes	maximal, characteristic, complemented
7	#21422509	C₂	1	2	yes	maximal, complemented
8	#21422385	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₉×#3168	C₂	1	1	yes	maximal, characteristic
10	#16837533	C₂²	1	2	yes	—
11	Q₈×C₅₉⋊C₈	C₂²	1	2	yes	—
12	#16837480	C₂²	1	2	yes	—
13	#16837474	—	2	2	no	—
14	D₄×C₅₉⋊C₈	—	2	2	no	—
15	D₄×C₅₉⋊C₈	C₂²	1	2	yes	—
16	#16837463	—	2	2	no	—
17	#16837455	C₂²	1	1	yes	characteristic
18	#16837429	—	2	2	no	—
19	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
20	C₄×#11745664	C₂²	1	1	yes	characteristic
21	#16837155	C₂²	1	1	yes	characteristic
22	#16837155	C₂²	1	2	yes	—
23	#16837155	C₂²	1	1	yes	characteristic
24	Q₈⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
25	Q₈⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
26	D₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
27	D₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
28	#16837124	C₂²	1	2	yes	—
29	#16837123	C₂²	1	2	yes	—
30	C₅₉×#529	C₂²	1	2	yes	—
31	C₅₉×#499	C₄	1	1	yes	characteristic
32	C₅₉×#495	C₄	1	1	yes	characteristic
33	C₅₉×#488	C₄	1	2	yes	—
34	C₅₉×#467	C₄	1	1	yes	characteristic
35	C₅₉×#466	C₄	1	1	yes	characteristic
36	C₅₉×#465	C₄	1	2	yes	—
37	C₄×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
38	C₄×SD₁₆×C₅₉	C₂²	1	2	yes	—
39	C₄×D₈×C₅₉	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
41	C₂×#11745664	—	2	1	no	—
42	C₂×#11745664	—	2	2	no	—
43	C₂²×C₅₉⋊C₈	—	2	2	no	—
44	C₂²×C₅₉⋊C₈	—	2	2	no	—
45	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
46	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
47	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
48	C₄⋊(C₅₉⋊C₈)	D₄	1	1	yes	characteristic
49	C₄⋊(C₅₉⋊C₈)	D₄	1	1	yes	characteristic
50	C₄⋊(C₅₉⋊C₈)	C₂³	1	2	yes	—
51	C₄⋊(C₅₉⋊C₈)	C₂³	1	2	yes	—
52	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
53	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
54	C₂²⋊(C₅₉⋊C₈)	D₄	1	2	yes	—
55	#14273638	—	2	1	no	—
56	#14273638	—	2	2	no	—
57	C₄×C₅₉⋊C₈	—	2	2	no	—
58	C₄×C₅₉⋊C₈	C₂³	1	2	yes	—
59	C₄×C₅₉⋊C₈	C₂³	1	2	yes	—
60	C₂×Q₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
61	C₂×SD₁₆×C₅₉	C₂×C₄	1	2	yes	—
62	C₂×D₈×C₅₉	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
65	#126×C₅₉	—	2	2	no	—
66	#125×C₅₉	C₂×C₄	1	2	yes	—
67	#124×C₅₉	C₂×C₄	1	2	yes	—
68	#123×C₅₉	—	2	2	no	—
69	C₂²⋊Q₈×C₅₉	C₂×C₄	1	2	yes	—
70	#121×C₅₉	C₂×C₄	1	2	yes	—
71	C₄×Q₈×C₅₉	C₂³	1	2	yes	—
72	C₄×D₄×C₅₉	—	2	2	no	—
73	C₄×D₄×C₅₉	C₂³	1	2	yes	—
74	#117×C₅₉	—	2	2	no	—
75	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
76	#107×C₅₉	C₂×C₄	1	1	yes	characteristic
77	#107×C₅₉	C₂×C₄	1	1	yes	characteristic
78	#106×C₅₉	C₂×C₄	1	2	yes	—
79	#103×C₅₉	C₂×C₄	1	2	yes	—
80	#103×C₅₉	C₂×C₄	1	2	yes	—
81	#102×C₅₉	C₂×C₄	1	2	yes	—
82	#102×C₅₉	C₂×C₄	1	2	yes	—
83	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
84	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
85	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
87	C₂×Q₈×C₅₉	C₂²×C₄	1	2	yes	—
88	C₂×D₄×C₅₉	—	2	2	no	—
89	C₂×D₄×C₅₉	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₅₉	—	2	2	no	—
91	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
94	Q₁₆×C₅₉	—	4	1	no	—
95	SD₁₆×C₅₉	—	2	4	no	—
96	D₈×C₅₉	—	4	1	no	—
97	C₂×C₈×C₅₉	—	2	2	no	—
98	C₂×C₈×C₅₉	—	4	1	no	—
99	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
103	C₄⋊C₄×C₅₉	—	2	2	no	—
104	C₄⋊C₄×C₅₉	—	2	2	no	—
105	C₄⋊C₄×C₅₉	—	2	2	no	—
106	C₄⋊C₄×C₅₉	—	2	2	no	—
107	C₄⋊C₄×C₅₉	C₂²×C₄	1	2	yes	—
108	C₄⋊C₄×C₅₉	C₂²×C₄	1	2	yes	—
109	C₂²⋊C₄×C₅₉	—	2	2	no	—
110	C₂²⋊C₄×C₅₉	—	2	2	no	—
111	C₂²⋊C₄×C₅₉	—	2	2	no	—
112	C₂²⋊C₄×C₅₉	—	2	2	no	—
113	C₂²⋊C₄×C₅₉	—	2	2	no	—
114	C₄²×C₅₉	—	2	2	no	—
115	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₅₉	#41	1	1	yes	characteristic
118	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
119	#11745664	—	4	2	no	—
120	#11745664	—	8	2	no	—
121	C₂×C₅₉⋊C₈	—	2	2	no	—
122	C₂×C₅₉⋊C₈	—	2	2	no	—
123	C₂×C₅₉⋊C₈	—	2	2	no	—
124	C₂×C₅₉⋊C₈	—	8	2	no	—
125	C₂×C₅₉⋊C₈	—	8	2	no	—
126	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
127	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
128	C₂³×C₅₉	—	2	2	no	—
129	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
130	Q₈×C₅₉	—	2	2	no	—
131	Q₈×C₅₉	—	2	2	no	—
132	D₄×C₅₉	—	2	2	no	—
133	D₄×C₅₉	—	2	2	no	—
134	D₄×C₅₉	—	4	4	no	—
135	C₂×C₄×C₅₉	—	2	2	no	—
136	C₂×C₄×C₅₉	—	2	1	no	—
137	C₂×C₄×C₅₉	—	2	1	no	—
138	C₂×C₄×C₅₉	—	2	1	no	—
139	C₂×C₄×C₅₉	—	2	1	no	—
140	C₂×C₄×C₅₉	—	2	2	no	—
141	C₂×C₄×C₅₉	—	2	2	no	—
142	C₂×C₄×C₅₉	—	2	2	no	—
143	C₂×C₄×C₅₉	—	4	2	no	—
144	C₂×C₄×C₅₉	—	4	2	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	C₈×C₅₉	—	2	1	no	—
152	C₈×C₅₉	—	2	1	no	—
153	C₈×C₅₉	—	2	1	no	—
154	C₈×C₅₉	#131	1	2	yes	—
155	C₅₉⋊C₈	—	2	2	no	—
156	C₅₉⋊C₈	—	2	2	no	—
157	C₅₉⋊C₈	—	4	2	no	—
158	C₅₉⋊C₈	—	4	2	no	—
159	C₅₉⋊C₈	—	4	2	no	—
160	#18092	—	59	1	no	maximal, Sylow
161	C₂²×C₅₉	—	2	1	no	—
162	C₂²×C₅₉	—	2	1	no	—
163	C₂²×C₅₉	—	4	2	no	—
164	C₂²×C₅₉	—	4	2	no	—
165	C₂²×C₅₉	—	4	2	no	—
166	C₂²×C₅₉	#444	1	2	yes	—
167	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
168	C₄×C₅₉	—	2	1	no	—
169	C₄×C₅₉	—	2	1	no	—
170	C₄×C₅₉	—	2	1	no	—
171	C₄×C₅₉	—	2	1	no	—
172	C₄×C₅₉	—	4	2	no	—
173	C₄×C₅₉	—	4	2	no	—
174	C₄×C₅₉	—	4	2	no	—
175	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
176	C₄×C₅₉	#568	1	1	yes	characteristic
177	C₄×C₅₉	#576	1	1	yes	characteristic
178	C₄×C₅₉	#578	1	1	yes	characteristic
179	#3168	D₅₉	1	1	yes	characteristic
180	#2994	—	59	2	no	—
181	#1592	—	59	2	no	—
182	#1584	—	59	2	no	—
183	#1581	—	59	2	no	—
184	SD₁₆⋊C₈	—	59	2	no	—
185	C₂×#335	—	59	1	no	—
186	#1483	—	59	2	no	—
187	#1451	—	59	1	no	—
188	C₂×C₅₉	—	2	2	no	—
189	C₂×C₅₉	—	8	2	no	—
190	C₂×C₅₉	#2935	1	1	yes	characteristic
191	C₂×C₅₉	#2951	1	1	yes	characteristic
192	C₂×C₅₉	C₂×#444	1	1	yes	characteristic
193	#529	C₂×D₅₉	1	2	yes	—
194	#499	Dic₅₉	1	1	yes	characteristic
195	#495	Dic₅₉	1	1	yes	characteristic
196	#488	Dic₅₉	1	2	yes	—
197	#467	Dic₅₉	1	1	yes	characteristic
198	#466	Dic₅₉	1	1	yes	characteristic
199	#465	Dic₅₉	1	2	yes	—
200	#446	—	59	2	no	—
201	C₈×Q₈	—	59	2	no	—
202	C₄×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
203	C₄×SD₁₆	C₂×D₅₉	1	2	yes	—
204	C₄×D₈	C₂×D₅₉	1	1	yes	characteristic
205	#436	—	59	2	no	—
206	#435	—	118	2	no	—
207	C₈×D₄	—	59	2	no	—
208	C₈×D₄	—	118	2	no	—
209	#433	—	118	2	no	—
210	#431	—	59	1	no	—
211	#424	—	118	2	no	—
212	C₂×C₄⋊C₈	—	59	2	no	—
213	C₄×M₁₆	—	59	1	no	—
214	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
215	#335	—	59	1	no	—
216	#335	—	59	2	no	—
217	#335	—	59	1	no	—
218	Q₈⋊C₈	—	59	2	no	—
219	Q₈⋊C₈	—	59	2	no	—
220	D₄⋊C₈	—	59	2	no	—
221	D₄⋊C₈	—	59	2	no	—
222	#322	—	59	2	no	—
223	#322	—	59	2	no	—
224	C₅₉	#18092	1	1	yes	characteristic, complemented, Sylow
225	C₂×Q₁₆	C₂×Dic₅₉	1	1	yes	characteristic
226	C₂×SD₁₆	C₂×Dic₅₉	1	2	yes	—
227	C₂×D₈	C₂×Dic₅₉	1	1	yes	characteristic
228	C₂×M₁₆	—	118	2	no	—
229	C₂×M₁₆	—	118	1	no	—
230	C₂²×C₈	—	118	2	no	—
231	C₂²×C₈	—	118	2	no	—
232	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
233	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
234	#126	—	2	2	no	—
235	#125	C₂×Dic₅₉	1	2	yes	—
236	#124	C₂×Dic₅₉	1	2	yes	—
237	#123	—	2	2	no	—
238	C₂²⋊Q₈	C₂×Dic₅₉	1	2	yes	—
239	#121	C₂×Dic₅₉	1	2	yes	—
240	C₄×Q₈	C₂²×D₅₉	1	2	yes	—
241	C₄×D₄	—	2	2	no	—
242	C₄×D₄	C₂²×D₅₉	1	2	yes	—
243	#117	—	2	2	no	—
244	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
245	#107	C₂×Dic₅₉	1	1	yes	characteristic
246	#107	C₂×Dic₅₉	1	1	yes	characteristic
247	#106	C₂×Dic₅₉	1	2	yes	—
248	C₄⋊C₈	—	59	1	no	—
249	C₄⋊C₈	—	59	1	no	—
250	C₄⋊C₈	—	59	2	no	—
251	C₄⋊C₈	—	59	2	no	—
252	C₄⋊C₈	—	118	2	no	—
253	C₄⋊C₈	—	118	2	no	—
254	C₄⋊C₈	—	118	2	no	—
255	#103	C₂×Dic₅₉	1	2	yes	—
256	#103	C₂×Dic₅₉	1	2	yes	—
257	#102	C₂×Dic₅₉	1	2	yes	—
258	#102	C₂×Dic₅₉	1	2	yes	—
259	C₂²⋊C₈	—	59	2	no	—
260	C₂²⋊C₈	—	118	2	no	—
261	C₂²⋊C₈	—	118	2	no	—
262	#97	—	118	2	no	—
263	#97	—	118	1	no	—
264	C₄×C₈	—	59	2	no	—
265	C₄×C₈	—	59	2	no	—
266	C₄×C₈	—	118	2	no	—
267	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
268	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
269	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
270	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
271	C₂×Q₈	C₂²×Dic₅₉	1	2	yes	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	C₂²×Dic₅₉	1	2	yes	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
276	C₂²×C₄	#11745687	1	1	yes	characteristic
277	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
278	Q₁₆	—	4	1	no	—
279	SD₁₆	—	2	4	no	—
280	D₈	—	4	1	no	—
281	M₁₆	—	472	2	no	—
282	M₁₆	—	236	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	59	2	no	—
286	C₂×C₈	—	59	2	no	—
287	C₂×C₈	—	118	2	no	—
288	C₂×C₈	—	118	2	no	—
289	C₂×C₈	—	118	2	no	—
290	C₂×C₈	—	472	2	no	—
291	C₂×C₈	—	472	2	no	—
292	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
293	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
294	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
295	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	C₂²×Dic₅₉	1	2	yes	—
301	C₄⋊C₄	C₂²×Dic₅₉	1	2	yes	—
302	C₂²⋊C₄	—	2	2	no	—
303	C₂²⋊C₄	—	2	2	no	—
304	C₂²⋊C₄	—	2	2	no	—
305	C₂²⋊C₄	—	2	2	no	—
306	C₂²⋊C₄	—	2	2	no	—
307	C₄²	—	2	2	no	—
308	C₄²	D₄×D₅₉	1	1	yes	characteristic
309	C₄²	#11745687	1	1	yes	characteristic
310	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
311	C₄²	C₂³×D₅₉	1	1	yes	characteristic
312	C₂³	—	2	2	no	—
313	C₂³	D₄×Dic₅₉	1	1	yes	characteristic
314	Q₈	—	2	2	no	—
315	Q₈	—	2	2	no	—
316	D₄	—	2	2	no	—
317	D₄	—	2	2	no	—
318	D₄	—	4	4	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
330	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
331	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
332	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
334	C₂×C₄	C₂×#11745687	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	—	118	2	no	—
339	C₈	—	118	2	no	—
340	C₈	—	236	2	no	—
341	C₈	—	236	2	no	—
342	C₈	—	236	2	no	—
343	C₈	#14273734	1	2	yes	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	4	2	no	—
347	C₂²	—	4	2	no	—
348	C₂²	—	4	2	no	—
349	C₂²	#16837524	1	2	yes	—
350	C₂²	C₂×D₄×Dic₅₉	1	1	yes	characteristic
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	4	2	no	—
356	C₄	—	4	2	no	—
357	C₄	—	4	2	no	—
358	C₄	C₂×#14273734	1	1	yes	characteristic
359	C₄	#16838169	1	1	yes	characteristic
360	C₄	#16838204	1	1	yes	characteristic
361	C₄	#16838216	1	1	yes	characteristic
362	C₂	—	2	2	no	—
363	C₂	—	8	2	no	—
364	C₂	#21432840	1	1	yes	characteristic
365	C₂	#21432972	1	1	yes	characteristic
366	C₂	C₂×#16837524	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×#3168	C₂
socle	C₂²×C₅₉	C₂×C₄×D₄
prime radical	C₅₉×#466	C₄
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	#3168	D₅₉
hypocenter	C₅₉	#18092
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3168

Complements

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

N	H ≅ G/N	type
1. #21433256	C₂	semidirect
2. #21422999	C₂	semidirect
3. #21422947	C₂	semidirect
4. #21422926	C₂	semidirect
5. SD₁₆⋊(C₅₉⋊C₈)	C₂	semidirect
6. C₂×#16837155	C₂	semidirect
7. #21422509	C₂	semidirect
8. #21422385	C₂	semidirect
224. C₅₉	#18092	semidirect

Maximal subgroups

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

subgroup	index
#21433256	2
#21422999	2
#21422947	2
#21422926	2
SD₁₆⋊(C₅₉⋊C₈)	2
C₂×#16837155	2
#21422509	2
#21422385	2
C₅₉×#3168	2
#18092	59