Subgroups — #28559239

Order 15104.

666 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21434651	C₂	1	1	yes	maximal, characteristic
2	#21433030	C₂	1	1	yes	maximal, characteristic, complemented
3	#21422998	C₂	1	4	yes	maximal, complemented
4	#21422908	C₂	1	4	yes	maximal, complemented
5	#21422854	C₂	1	1	yes	maximal, characteristic, complemented
6	#21422550	C₂	1	1	yes	maximal, characteristic, complemented
7	#21422547	C₂	1	1	yes	maximal, characteristic
8	C₂×#16837124	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₉×#2931	C₂	1	1	yes	maximal, characteristic
10	#16837959	C₂²	1	1	yes	characteristic
11	#16837798	C₄	1	1	yes	characteristic
12	#16837774	C₄	1	1	yes	characteristic
13	#16837754	C₄	1	1	yes	characteristic
14	#16837748	C₄	1	1	yes	characteristic
15	#16837629	C₄	1	4	yes	—
16	#16837532	—	2	1	no	—
17	#16837532	C₂²	1	2	yes	—
18	C₈×Dic₁₁₈	—	2	1	no	—
19	C₈×Dic₁₁₈	C₂²	1	2	yes	—
20	C₄×#11745670	C₂²	1	4	yes	—
21	#16837428	—	2	1	no	—
22	#16837426	C₂²	1	1	yes	characteristic
23	#16837357	—	2	1	no	—
24	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
26	#16837154	C₂²	1	4	yes	—
27	#16837132	C₂²	1	4	yes	—
28	D₄⋊(C₅₉⋊C₈)	C₂²	1	4	yes	—
29	#16837124	C₂²	1	2	yes	—
30	#16837124	C₂²	1	2	yes	—
31	C₂×#131×C₅₉	—	2	1	no	—
32	C₅₉×#532	C₂²	1	1	yes	characteristic
33	C₅₉×#436	—	2	1	no	—
34	C₅₉×#436	C₂²	1	2	yes	—
35	C₈×D₄×C₅₉	—	2	1	no	—
36	C₈×D₄×C₅₉	C₂²	1	2	yes	—
37	C₅₉×#423	C₂²	1	1	yes	characteristic
38	C₅₉×#408	—	2	1	no	—
39	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
40	C₂×#11745670	C₂×C₄	1	4	yes	—
41	C₂×#11745664	—	2	1	no	—
42	C₂²×C₅₉⋊C₈	—	2	1	no	—
43	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
44	#14273727	C₂×C₄	1	1	yes	characteristic
45	#14273727	C₂×C₄	1	1	yes	characteristic
46	#14273719	—	2	1	no	—
47	#14273706	—	2	1	no	—
48	#14273706	C₂×C₄	1	2	yes	—
49	C₄×Dic₁₁₈	—	2	1	no	—
50	C₄×Dic₁₁₈	C₂³	1	2	yes	—
51	#14273686	—	2	1	no	—
52	#14273658	C₂×C₄	1	4	yes	—
53	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
54	C₄⋊(C₅₉⋊C₈)	C₂³	1	2	yes	—
55	Dic₅₉⋊C₈	—	2	2	no	—
56	Dic₅₉⋊C₈	—	2	2	no	—
57	#14273650	C₂×C₄	1	4	yes	—
58	#14273648	C₂×C₄	1	4	yes	—
59	#14273637	—	2	2	no	—
60	C₄×C₅₉⋊C₈	—	2	1	no	—
61	C₄×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
62	C₄×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
63	C₄×C₅₉⋊C₈	C₂³	1	2	yes	—
64	C₄×C₅₉⋊C₈	C₂³	1	2	yes	—
65	C₈×Dic₅₉	—	2	2	no	—
66	C₂×#41×C₅₉	—	2	1	no	—
67	#131×C₅₉	—	4	4	no	—
68	C₂×M₁₆×C₅₉	—	2	1	no	—
69	C₂×M₁₆×C₅₉	—	2	2	no	—
70	C₂²×C₈×C₅₉	—	2	2	no	—
71	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
72	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
73	C₄⋊Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
74	#127×C₅₉	C₂×C₄	1	1	yes	characteristic
75	#124×C₅₉	—	2	1	no	—
76	#121×C₅₉	—	2	1	no	—
77	#121×C₅₉	C₂×C₄	1	2	yes	—
78	C₄×D₄×C₅₉	—	2	1	no	—
79	C₄×D₄×C₅₉	C₂³	1	2	yes	—
80	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
81	C₄⋊C₈×C₅₉	—	2	1	no	—
82	C₄⋊C₈×C₅₉	C₂³	1	2	yes	—
83	C₂²⋊C₈×C₅₉	—	2	2	no	—
84	C₂²⋊C₈×C₅₉	—	2	2	no	—
85	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
86	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
87	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
89	#41×C₅₉	—	4	2	no	—
90	#41×C₅₉	—	4	2	no	—
91	C₂×Q₈×C₅₉	—	2	1	no	—
92	C₂×D₄×C₅₉	—	2	2	no	—
93	C₂×D₄×C₅₉	—	2	1	no	—
94	C₂×D₄×C₅₉	—	2	1	no	—
95	C₂×D₄×C₅₉	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₅₉	—	2	2	no	—
97	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
100	C₂×Dic₁₁₈	—	2	1	no	—
101	C₂×Dic₁₁₈	C₂²×C₄	1	2	yes	—
102	M₁₆×C₅₉	—	4	4	no	—
103	M₁₆×C₅₉	—	4	2	no	—
104	C₂×C₈×C₅₉	—	2	2	no	—
105	C₂×C₈×C₅₉	—	2	2	no	—
106	C₂×C₈×C₅₉	—	2	2	no	—
107	C₂×C₈×C₅₉	—	4	4	no	—
108	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
112	C₄⋊C₄×C₅₉	—	2	1	no	—
113	C₄⋊C₄×C₅₉	C₂²×C₄	1	2	yes	—
114	C₂²⋊C₄×C₅₉	—	2	2	no	—
115	C₂²⋊C₄×C₅₉	—	2	2	no	—
116	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
117	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₅₉	#41	1	1	yes	characteristic
119	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
120	C₂²⋊Dic₅₉	—	2	2	no	—
121	#11745670	—	2	8	no	—
122	C₄⋊Dic₅₉	—	2	1	no	—
123	C₄⋊Dic₅₉	—	2	1	no	—
124	C₄⋊Dic₅₉	—	2	1	no	—
125	C₄⋊Dic₅₉	C₂²×C₄	1	2	yes	—
126	Dic₅₉⋊C₄	—	2	2	no	—
127	Dic₅₉⋊C₄	—	2	2	no	—
128	C₄×Dic₅₉	—	2	2	no	—
129	#11745664	—	4	2	no	—
130	C₂×C₅₉⋊C₈	—	2	4	no	—
131	C₂×C₅₉⋊C₈	—	2	2	no	—
132	C₂×C₅₉⋊C₈	—	2	2	no	—
133	C₂×C₅₉⋊C₈	—	4	2	no	—
134	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
135	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
136	C₂³×C₅₉	—	2	2	no	—
137	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
138	Q₈×C₅₉	—	4	2	no	—
139	D₄×C₅₉	—	2	4	no	—
140	D₄×C₅₉	—	4	4	no	—
141	D₄×C₅₉	—	4	2	no	—
142	D₄×C₅₉	—	4	2	no	—
143	C₂×C₄×C₅₉	—	2	2	no	—
144	C₂×C₄×C₅₉	—	2	2	no	—
145	C₂×C₄×C₅₉	—	2	2	no	—
146	C₂×C₄×C₅₉	—	4	2	no	—
147	C₂×C₄×C₅₉	—	4	2	no	—
148	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
152	C₂×C₄×C₅₉	#137	1	2	yes	—
153	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
154	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
156	C₂×Dic₅₉	—	2	2	no	—
157	C₂×Dic₅₉	—	2	2	no	—
158	Dic₁₁₈	—	2	4	no	—
159	Dic₁₁₈	—	4	2	no	—
160	C₈×C₅₉	—	2	1	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	C₅₉⋊C₈	—	2	2	no	—
166	C₅₉⋊C₈	—	4	2	no	—
167	C₅₉⋊C₈	—	4	2	no	—
168	C₅₉⋊C₈	#131	1	4	yes	complemented
169	#17960	—	59	1	no	maximal, Sylow
170	C₂²×C₅₉	—	2	1	no	—
171	C₂²×C₅₉	—	2	1	no	—
172	C₂²×C₅₉	—	4	2	no	—
173	C₂²×C₅₉	—	4	2	no	—
174	C₂²×C₅₉	—	4	2	no	—
175	C₂²×C₅₉	#440	1	2	yes	—
176	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
177	C₄×C₅₉	—	2	1	no	—
178	C₄×C₅₉	—	2	1	no	—
179	C₄×C₅₉	—	4	2	no	—
180	C₄×C₅₉	#440	1	2	yes	—
181	C₄×C₅₉	#444	1	2	yes	—
182	C₄×C₅₉	C₂×#131	1	2	yes	—
183	C₄×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
184	C₄×C₅₉	C₂×#137	1	1	yes	characteristic
185	Dic₅₉	—	4	2	no	—
186	Dic₅₉	—	4	2	no	—
187	#3167	—	59	1	no	—
188	#2960	—	59	1	no	—
189	#2931	D₅₉	1	1	yes	characteristic
190	#1592	—	59	4	no	—
191	SD₁₆⋊C₈	—	59	4	no	—
192	#1567	—	59	1	no	—
193	#1493	—	59	1	no	—
194	#1492	—	59	1	no	—
195	C₂×#322	—	59	1	no	—
196	C₂×C₅₉	—	2	2	no	—
197	C₂×C₅₉	—	8	2	no	—
198	C₂×C₅₉	#2931	1	1	yes	characteristic
199	C₂×C₅₉	C₂×#440	1	1	yes	characteristic
200	C₂×C₅₉	C₂×#444	1	1	yes	characteristic
201	C₂×#131	—	2	1	no	—
202	#533	—	59	1	no	—
203	#532	C₂×D₅₉	1	1	yes	characteristic
204	#498	—	59	1	no	—
205	#492	—	59	1	no	—
206	#487	—	59	1	no	—
207	#486	—	59	1	no	—
208	#465	—	59	4	no	—
209	#446	—	59	2	no	—
210	#446	—	118	1	no	—
211	C₈×Q₈	—	59	2	no	—
212	C₈×Q₈	—	118	1	no	—
213	C₄×SD₁₆	—	59	4	no	—
214	#436	—	2	1	no	—
215	#436	C₂×D₅₉	1	2	yes	—
216	C₈×D₄	—	2	1	no	—
217	C₈×D₄	C₂×D₅₉	1	2	yes	—
218	#424	—	118	1	no	—
219	#423	—	59	1	no	—
220	#423	C₂×D₅₉	1	1	yes	characteristic
221	#408	—	2	1	no	—
222	#405	—	118	1	no	—
223	C₂×C₄×C₈	—	59	1	no	—
224	C₂×C₄×C₈	—	59	1	no	—
225	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
226	#335	—	59	4	no	—
227	Q₈⋊C₈	—	59	4	no	—
228	D₄⋊C₈	—	59	4	no	—
229	#322	—	59	2	no	—
230	#322	—	59	2	no	—
231	C₅₉	#17960	1	1	yes	characteristic, complemented, Sylow
232	C₂×#41	—	2	1	no	—
233	C₂×SD₁₆	—	59	4	no	—
234	#131	—	4	4	no	—
235	C₂×M₁₆	—	2	1	no	—
236	C₂×M₁₆	—	2	2	no	—
237	C₂×M₁₆	—	118	1	no	—
238	C₂²×C₈	—	2	2	no	—
239	C₂²×C₈	—	59	2	no	—
240	C₂²×C₈	—	118	1	no	—
241	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
242	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
243	C₄⋊Q₈	—	59	1	no	—
244	C₄⋊Q₈	—	59	1	no	—
245	C₄⋊Q₈	C₄×D₅₉	1	1	yes	characteristic
246	#127	C₄×D₅₉	1	1	yes	characteristic
247	#125	—	118	1	no	—
248	#124	—	2	1	no	—
249	C₂²⋊Q₈	—	59	2	no	—
250	C₂²⋊Q₈	—	118	1	no	—
251	#121	—	2	1	no	—
252	#121	C₄×D₅₉	1	2	yes	—
253	C₄×Q₈	—	59	2	no	—
254	C₄×Q₈	—	118	1	no	—
255	C₄×D₄	—	2	1	no	—
256	C₄×D₄	C₂²×D₅₉	1	2	yes	—
257	#117	—	118	1	no	—
258	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
259	#106	—	59	4	no	—
260	C₄⋊C₈	—	2	1	no	—
261	C₄⋊C₈	—	59	2	no	—
262	C₄⋊C₈	—	118	2	no	—
263	C₄⋊C₈	—	118	2	no	—
264	C₄⋊C₈	—	118	1	no	—
265	C₄⋊C₈	C₂²×D₅₉	1	2	yes	—
266	#103	—	59	4	no	—
267	#102	—	59	4	no	—
268	C₂²⋊C₈	—	2	2	no	—
269	C₂²⋊C₈	—	2	2	no	—
270	#97	—	118	2	no	—
271	C₄×C₈	—	59	1	no	—
272	C₄×C₈	—	59	1	no	—
273	C₄×C₈	—	59	2	no	—
274	C₄×C₈	—	59	2	no	—
275	C₄×C₈	—	118	2	no	—
276	C₄×C₈	—	118	1	no	—
277	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
278	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
281	#41	—	4	2	no	—
282	#41	—	4	2	no	—
283	C₂×Q₈	—	2	1	no	—
284	C₂×Q₈	—	59	2	no	—
285	C₂×Q₈	—	118	1	no	—
286	C₂×D₄	—	2	2	no	—
287	C₂×D₄	—	2	1	no	—
288	C₂×D₄	—	2	1	no	—
289	C₂×D₄	C₂×C₄×D₅₉	1	2	yes	—
290	C₂²×C₄	—	2	2	no	—
291	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
292	C₂²×C₄	#11745685	1	1	yes	characteristic
293	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
294	SD₁₆	—	118	8	no	—
295	M₁₆	—	4	4	no	—
296	M₁₆	—	4	2	no	—
297	M₁₆	—	236	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	4	4	no	—
302	C₂×C₈	—	59	2	no	—
303	C₂×C₈	—	59	2	no	—
304	C₂×C₈	—	118	4	no	—
305	C₂×C₈	—	118	2	no	—
306	C₂×C₈	—	118	2	no	—
307	C₂×C₈	—	236	2	no	—
308	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
309	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
310	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
311	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	59	2	no	—
314	C₄⋊C₄	—	118	1	no	—
315	C₄⋊C₄	—	118	1	no	—
316	C₄⋊C₄	—	118	2	no	—
317	C₄⋊C₄	—	118	1	no	—
318	C₄⋊C₄	—	118	2	no	—
319	C₄⋊C₄	C₂×C₄×D₅₉	1	2	yes	—
320	C₂²⋊C₄	—	2	2	no	—
321	C₂²⋊C₄	—	2	2	no	—
322	C₂²⋊C₄	—	118	2	no	—
323	C₄²	—	118	2	no	—
324	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
325	C₄²	#11745685	1	1	yes	characteristic
326	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
327	C₄²	C₂³×D₅₉	1	1	yes	characteristic
328	C₂³	—	2	2	no	—
329	C₂³	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
330	Q₈	—	4	2	no	—
331	Q₈	—	118	4	no	—
332	Q₈	—	236	2	no	—
333	D₄	—	2	4	no	—
334	D₄	—	4	4	no	—
335	D₄	—	4	2	no	—
336	D₄	—	4	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	4	2	no	—
342	C₂×C₄	—	118	2	no	—
343	C₂×C₄	—	118	2	no	—
344	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
345	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
346	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
347	C₂×C₄	#14273753	1	2	yes	—
348	C₂×C₄	#14273757	1	2	yes	—
349	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
350	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
351	C₂×C₄	C₂×#11745685	1	1	yes	characteristic
352	C₈	—	2	1	no	—
353	C₈	—	2	1	no	—
354	C₈	—	2	1	no	—
355	C₈	—	2	1	no	—
356	C₈	—	4	2	no	—
357	C₈	—	59	4	no	—
358	C₈	—	118	2	no	—
359	C₈	—	236	2	no	—
360	C₈	—	236	2	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	4	2	no	—
364	C₂²	—	4	2	no	—
365	C₂²	—	4	2	no	—
366	C₂²	#16837502	1	2	yes	—
367	C₂²	C₂×C₄×C₂²⋊D₅₉	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	4	2	no	—
371	C₄	—	236	2	no	—
372	C₄	—	236	2	no	—
373	C₄	#16837502	1	2	yes	—
374	C₄	#16837522	1	2	yes	—
375	C₄	#131×D₅₉	1	2	yes	—
376	C₄	C₂×#14273753	1	1	yes	characteristic
377	C₄	C₂×#14273757	1	1	yes	characteristic
378	C₂	—	2	2	no	—
379	C₂	—	8	2	no	—
380	C₂	#21432806	1	1	yes	characteristic
381	C₂	C₂×#16837502	1	1	yes	characteristic
382	C₂	C₂×#16837522	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₄²	C₂³×D₅₉
Fitting	C₅₉×#2931	C₂
socle	C₂²×C₅₉	C₂×C₄×D₄
prime radical	C₅₉×#532	C₂²
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	#2931	D₅₉
hypocenter	C₅₉	#17960
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
2. #21433030	C₂	semidirect
3. #21422998	C₂	semidirect
4. #21422908	C₂	semidirect
5. #21422854	C₂	semidirect
6. #21422550	C₂	semidirect
8. C₂×#16837124	C₂	semidirect
168. C₅₉⋊C₈	#131	semidirect
231. C₅₉	#17960	semidirect

Maximal subgroups

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

subgroup	index
#21434651	2
#21433030	2
#21422998	2
#21422908	2
#21422854	2
#21422550	2
#21422547	2
C₂×#16837124	2
C₅₉×#2931	2
#17960	59