Subgroups — C₅₉×#18043 #28430152

Order 15104.

644 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#2971	C₂	1	2	yes	maximal, complemented
2	C₄×#135×C₅₉	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#1594	C₂	1	2	yes	maximal, complemented
4	C₅₉×#1588	C₂	1	4	yes	maximal, complemented
5	C₅₉×#1579	C₂	1	2	yes	maximal, complemented
6	C₅₉×#1568	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1488	C₂	1	2	yes	maximal, complemented
8	C₅₉×#1451	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₅₉	C₄	1	1	yes	characteristic
10	C₄×#41×C₅₉	C₂²	1	2	yes	—
11	C₈×Q₈×C₅₉	C₂²	1	4	yes	—
12	C₄×Q₁₆×C₅₉	C₄	1	1	yes	characteristic
13	C₄×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	C₄×SD₁₆×C₅₉	C₄	1	2	yes	—
15	C₄×SD₁₆×C₅₉	C₂²	1	2	yes	—
16	C₄×D₈×C₅₉	C₄	1	1	yes	characteristic
17	C₄×D₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₅₉×#436	—	2	4	no	—
19	C₅₉×#436	C₂²	1	4	yes	—
20	C₅₉×#432	—	2	2	no	—
21	C₅₉×#432	C₂²	1	2	yes	—
22	C₅₉×#431	C₂²	1	1	yes	characteristic
23	C₅₉×#427	C₄	1	1	yes	characteristic
24	C₅₉×#416	C₄	1	2	yes	—
25	C₄×M₁₆×C₅₉	—	2	2	no	—
26	C₄×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
28	C₅₉×#335	C₂²	1	2	yes	—
29	C₅₉×#334	C₂²	1	2	yes	—
30	C₅₉×Q₈⋊C₈	C₂²	1	4	yes	—
31	C₅₉×D₄⋊C₈	C₂²	1	4	yes	—
32	C₅₉×#322	C₂²	1	2	yes	—
33	C₅₉×#322	C₂²	1	2	yes	—
34	C₂×#41×C₅₉	C₂×C₄	1	2	yes	—
35	#135×C₅₉	—	2	4	no	—
36	C₂×Q₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
37	C₂×SD₁₆×C₅₉	C₂×C₄	1	2	yes	—
38	C₂×D₈×C₅₉	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆×C₅₉	—	2	4	no	—
40	C₂×M₁₆×C₅₉	D₄	1	2	yes	—
41	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
43	C₄×Q₈×C₅₉	C₂×C₄	1	2	yes	—
44	C₄×Q₈×C₅₉	C₂³	1	2	yes	—
45	C₄×D₄×C₅₉	—	2	2	no	—
46	C₄×D₄×C₅₉	—	2	2	no	—
47	C₄×D₄×C₅₉	C₂×C₄	1	2	yes	—
48	C₄×D₄×C₅₉	C₂³	1	2	yes	—
49	#117×C₅₉	—	2	2	no	—
50	#117×C₅₉	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₅₉	—	2	2	no	—
52	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
53	#107×C₅₉	C₂×C₄	1	1	yes	characteristic
54	#107×C₅₉	C₂×C₄	1	1	yes	characteristic
55	#106×C₅₉	C₂×C₄	1	2	yes	—
56	C₄⋊C₈×C₅₉	—	2	4	no	—
57	C₄⋊C₈×C₅₉	—	2	1	no	—
58	C₄⋊C₈×C₅₉	C₂³	1	4	yes	—
59	#103×C₅₉	C₂×C₄	1	2	yes	—
60	#103×C₅₉	C₂×C₄	1	2	yes	—
61	#102×C₅₉	C₂×C₄	1	2	yes	—
62	#102×C₅₉	C₂×C₄	1	2	yes	—
63	C₂²⋊C₈×C₅₉	—	2	2	no	—
64	C₂²⋊C₈×C₅₉	—	2	1	no	—
65	C₂²⋊C₈×C₅₉	—	2	4	no	—
66	#97×C₅₉	—	2	2	no	—
67	#97×C₅₉	D₄	1	2	yes	—
68	C₄×C₈×C₅₉	—	2	4	no	—
69	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₅₉	C₂³	1	2	yes	—
72	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₅₉	C₂³	1	2	yes	—
75	#41×C₅₉	—	2	4	no	—
76	#41×C₅₉	—	4	2	no	—
77	C₂×Q₈×C₅₉	C₂²×C₄	1	2	yes	—
78	C₂×D₄×C₅₉	—	2	2	no	—
79	C₂×D₄×C₅₉	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₅₉	—	2	2	no	—
81	C₂²×C₄×C₅₉	—	2	2	no	—
82	C₂²×C₄×C₅₉	—	2	2	no	—
83	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
86	Q₁₆×C₅₉	M₁₆	1	4	yes	complemented
87	SD₁₆×C₅₉	—	2	4	no	—
88	D₈×C₅₉	—	2	2	no	—
89	M₁₆×C₅₉	—	4	2	no	—
90	M₁₆×C₅₉	—	4	8	no	—
91	C₂×C₈×C₅₉	—	2	2	no	—
92	C₂×C₈×C₅₉	—	2	4	no	—
93	C₂×C₈×C₅₉	—	2	2	no	—
94	C₂×C₈×C₅₉	—	4	1	no	—
95	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
100	C₄⋊C₄×C₅₉	—	2	2	no	—
101	C₄⋊C₄×C₅₉	—	2	2	no	—
102	C₄⋊C₄×C₅₉	C₂²×C₄	1	2	yes	—
103	C₄⋊C₄×C₅₉	C₂²×C₄	1	2	yes	—
104	C₂²⋊C₄×C₅₉	—	2	2	no	—
105	C₂²⋊C₄×C₅₉	—	2	2	no	—
106	C₂²⋊C₄×C₅₉	—	2	2	no	—
107	C₄²×C₅₉	—	2	2	no	—
108	C₄²×C₅₉	—	2	2	no	—
109	C₄²×C₅₉	—	2	2	no	—
110	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₅₉	#41	1	1	yes	characteristic
113	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
114	C₂³×C₅₉	—	2	2	no	—
115	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
116	Q₈×C₅₉	—	2	2	no	—
117	Q₈×C₅₉	C₂×M₁₆	1	4	yes	—
118	D₄×C₅₉	—	2	2	no	—
119	D₄×C₅₉	—	2	2	no	—
120	D₄×C₅₉	—	4	4	no	—
121	C₂×C₄×C₅₉	—	2	2	no	—
122	C₂×C₄×C₅₉	—	2	1	no	—
123	C₂×C₄×C₅₉	—	2	1	no	—
124	C₂×C₄×C₅₉	—	2	2	no	—
125	C₂×C₄×C₅₉	—	2	2	no	—
126	C₂×C₄×C₅₉	—	2	2	no	—
127	C₂×C₄×C₅₉	—	4	1	no	—
128	C₂×C₄×C₅₉	—	4	4	no	—
129	C₂×C₄×C₅₉	—	4	4	no	—
130	C₂×C₄×C₅₉	—	8	2	no	—
131	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
137	C₈×C₅₉	—	2	1	no	—
138	C₈×C₅₉	—	2	1	no	—
139	C₈×C₅₉	—	2	1	no	—
140	C₈×C₅₉	—	2	4	no	—
141	C₈×C₅₉	—	4	2	no	—
142	C₈×C₅₉	—	4	4	no	—
143	C₈×C₅₉	C₂×M₁₆	1	2	yes	—
144	#18043	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂²×C₅₉	—	2	1	no	—
146	C₂²×C₅₉	—	2	1	no	—
147	C₂²×C₅₉	—	2	1	no	—
148	C₂²×C₅₉	—	4	2	no	—
149	C₂²×C₅₉	—	4	2	no	—
150	C₂²×C₅₉	—	4	2	no	—
151	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
152	C₄×C₅₉	—	2	1	no	—
153	C₄×C₅₉	—	2	4	no	—
154	C₄×C₅₉	—	2	1	no	—
155	C₄×C₅₉	—	2	1	no	—
156	C₄×C₅₉	—	2	1	no	—
157	C₄×C₅₉	—	4	2	no	—
158	C₄×C₅₉	—	4	2	no	—
159	C₄×C₅₉	C₂²×M₁₆	1	1	yes	characteristic
160	C₄×C₅₉	#568	1	1	yes	characteristic
161	C₄×C₅₉	#575	1	2	yes	—
162	#2971	C₂×C₅₉	1	2	yes	complemented
163	C₄×#135	C₂×C₅₉	1	1	yes	characteristic
164	#1594	C₂×C₅₉	1	2	yes	complemented
165	#1588	C₂×C₅₉	1	4	yes	complemented
166	#1579	C₂×C₅₉	1	2	yes	complemented
167	#1568	C₂×C₅₉	1	1	yes	characteristic, complemented
168	#1488	C₂×C₅₉	1	2	yes	complemented
169	#1451	C₂×C₅₉	1	1	yes	characteristic, complemented
170	C₂×C₅₉	—	4	1	no	—
171	C₂×C₅₉	—	8	2	no	—
172	C₂×C₅₉	D₄×M₁₆	1	1	yes	characteristic
173	C₂×C₅₉	#2943	1	1	yes	characteristic
174	C₂×C₅₉	#2957	1	1	yes	characteristic
175	C₂×#135	C₄×C₅₉	1	1	yes	characteristic
176	C₄×#41	C₂²×C₅₉	1	2	yes	—
177	C₈×Q₈	C₂²×C₅₉	1	4	yes	—
178	C₄×Q₁₆	C₄×C₅₉	1	1	yes	characteristic
179	C₄×Q₁₆	C₂²×C₅₉	1	1	yes	characteristic
180	C₄×SD₁₆	C₄×C₅₉	1	2	yes	—
181	C₄×SD₁₆	C₂²×C₅₉	1	2	yes	—
182	C₄×D₈	C₄×C₅₉	1	1	yes	characteristic
183	C₄×D₈	C₂²×C₅₉	1	1	yes	characteristic
184	#436	—	2	4	no	—
185	#436	C₂²×C₅₉	1	4	yes	—
186	#432	—	2	2	no	—
187	#432	C₂²×C₅₉	1	2	yes	—
188	#431	C₂²×C₅₉	1	1	yes	characteristic
189	#427	C₄×C₅₉	1	1	yes	characteristic
190	#416	C₄×C₅₉	1	2	yes	—
191	C₄×M₁₆	—	2	2	no	—
192	C₄×M₁₆	C₂²×C₅₉	1	1	yes	characteristic
193	C₂×C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic
194	#335	C₂²×C₅₉	1	2	yes	—
195	#334	C₂²×C₅₉	1	2	yes	—
196	Q₈⋊C₈	C₂²×C₅₉	1	4	yes	—
197	D₄⋊C₈	C₂²×C₅₉	1	4	yes	—
198	#322	C₂²×C₅₉	1	2	yes	—
199	#322	C₂²×C₅₉	1	2	yes	—
200	C₅₉	#18043	1	1	yes	characteristic, direct product factor, Sylow
201	C₂×#41	C₂×C₄×C₅₉	1	2	yes	—
202	#135	—	2	4	no	—
203	C₂×Q₁₆	C₂×C₄×C₅₉	1	1	yes	characteristic
204	C₂×SD₁₆	C₂×C₄×C₅₉	1	2	yes	—
205	C₂×D₈	C₂×C₄×C₅₉	1	1	yes	characteristic
206	C₂×M₁₆	—	2	4	no	—
207	C₂×M₁₆	D₄×C₅₉	1	2	yes	—
208	C₂²×C₈	C₂×C₄×C₅₉	1	1	yes	characteristic
209	C₂²×C₈	C₂×C₄×C₅₉	1	1	yes	characteristic
210	C₄×Q₈	C₂×C₄×C₅₉	1	2	yes	—
211	C₄×Q₈	C₂³×C₅₉	1	2	yes	—
212	C₄×D₄	—	2	2	no	—
213	C₄×D₄	—	2	2	no	—
214	C₄×D₄	C₂×C₄×C₅₉	1	2	yes	—
215	C₄×D₄	C₂³×C₅₉	1	2	yes	—
216	#117	—	2	2	no	—
217	#117	C₂×C₄×C₅₉	1	2	yes	—
218	C₂×C₄²	—	2	2	no	—
219	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
220	#107	C₂×C₄×C₅₉	1	1	yes	characteristic
221	#107	C₂×C₄×C₅₉	1	1	yes	characteristic
222	#106	C₂×C₄×C₅₉	1	2	yes	—
223	C₄⋊C₈	—	2	4	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	C₂³×C₅₉	1	4	yes	—
226	#103	C₂×C₄×C₅₉	1	2	yes	—
227	#103	C₂×C₄×C₅₉	1	2	yes	—
228	#102	C₂×C₄×C₅₉	1	2	yes	—
229	#102	C₂×C₄×C₅₉	1	2	yes	—
230	C₂²⋊C₈	—	2	2	no	—
231	C₂²⋊C₈	—	2	1	no	—
232	C₂²⋊C₈	—	2	4	no	—
233	#97	—	2	2	no	—
234	#97	D₄×C₅₉	1	2	yes	—
235	C₄×C₈	—	2	4	no	—
236	C₄×C₈	C₂×C₄×C₅₉	1	1	yes	characteristic
237	C₄×C₈	C₂×C₄×C₅₉	1	1	yes	characteristic
238	C₄×C₈	C₂³×C₅₉	1	2	yes	—
239	C₄×C₈	C₂³×C₅₉	1	1	yes	characteristic
240	C₄×C₈	C₂³×C₅₉	1	1	yes	characteristic
241	C₄×C₈	C₂³×C₅₉	1	2	yes	—
242	#41	—	2	4	no	—
243	#41	—	4	2	no	—
244	C₂×Q₈	C₂²×C₄×C₅₉	1	2	yes	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	C₂²×C₄×C₅₉	1	2	yes	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	C₂²×C₄×C₅₉	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
252	C₂²×C₄	#41×C₅₉	1	1	yes	characteristic
253	Q₁₆	M₁₆×C₅₉	1	4	yes	complemented
254	SD₁₆	—	2	4	no	—
255	D₈	—	2	2	no	—
256	M₁₆	—	4	2	no	—
257	M₁₆	—	4	8	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	C₂²×C₄×C₅₉	1	1	yes	characteristic
263	C₂×C₈	C₂²×C₄×C₅₉	1	1	yes	characteristic
264	C₂×C₈	C₂²×C₄×C₅₉	1	1	yes	characteristic
265	C₂×C₈	C₂²×C₄×C₅₉	1	1	yes	characteristic
266	C₂×C₈	C₂×D₄×C₅₉	1	4	yes	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	C₂²×C₄×C₅₉	1	2	yes	—
270	C₄⋊C₄	C₂²×C₄×C₅₉	1	2	yes	—
271	C₂²⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	2	2	no	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	C₂²×C₄×C₅₉	1	1	yes	characteristic
278	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
279	C₄²	#41×C₅₉	1	1	yes	characteristic
280	C₄²	C₂⁴×C₅₉	1	1	yes	characteristic
281	C₂³	—	2	2	no	—
282	C₂³	C₄×D₄×C₅₉	1	1	yes	characteristic
283	Q₈	—	2	2	no	—
284	Q₈	C₂×M₁₆×C₅₉	1	4	yes	—
285	D₄	—	2	2	no	—
286	D₄	—	2	2	no	—
287	D₄	—	4	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	8	2	no	—
298	C₂×C₄	C₄×D₄×C₅₉	1	1	yes	characteristic
299	C₂×C₄	C₄×D₄×C₅₉	1	1	yes	characteristic
300	C₂×C₄	C₄×D₄×C₅₉	1	1	yes	characteristic
301	C₂×C₄	C₂³×C₄×C₅₉	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
303	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	2	4	no	—
308	C₈	—	4	2	no	—
309	C₈	—	4	4	no	—
310	C₈	C₂×M₁₆×C₅₉	1	2	yes	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	C₂×C₄×D₄×C₅₉	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	4	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	2	no	—
325	C₄	C₂²×M₁₆×C₅₉	1	1	yes	characteristic
326	C₄	C₅₉×#568	1	1	yes	characteristic
327	C₄	C₅₉×#575	1	2	yes	—
328	C₂	—	4	1	no	—
329	C₂	—	8	2	no	—
330	C₂	D₄×M₁₆×C₅₉	1	1	yes	characteristic
331	C₂	C₅₉×#2943	1	1	yes	characteristic
332	C₂	C₅₉×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₉	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₉
Frattini, cosocle	C₄²	C₂⁴×C₅₉
socle	C₂²×C₅₉	C₂×C₄×D₄
prime radical	C₂×#135×C₅₉	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₉ ◁ C₄×M₁₆×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₉×#2971	C₂	semidirect
3. C₅₉×#1594	C₂	semidirect
4. C₅₉×#1588	C₂	semidirect
5. C₅₉×#1579	C₂	semidirect
6. C₅₉×#1568	C₂	semidirect
7. C₅₉×#1488	C₂	semidirect
8. C₅₉×#1451	C₂	semidirect
86. Q₁₆×C₅₉	M₁₆	semidirect
144. #18043	C₅₉	direct (N × H)
162. #2971	C₂×C₅₉	semidirect
164. #1594	C₂×C₅₉	semidirect
165. #1588	C₂×C₅₉	semidirect
166. #1579	C₂×C₅₉	semidirect
167. #1568	C₂×C₅₉	semidirect
168. #1488	C₂×C₅₉	semidirect
169. #1451	C₂×C₅₉	semidirect
200. C₅₉	#18043	direct (N × H)
253. Q₁₆	M₁₆×C₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₅₉×#2971	2
C₄×#135×C₅₉	2
C₅₉×#1594	2
C₅₉×#1588	2
C₅₉×#1579	2
C₅₉×#1568	2
C₅₉×#1488	2
C₅₉×#1451	2
#18043	59

Subgroups — C59×#18043 #28430152

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#18043 #28430152