Subgroups — C₅₉×#33927 #28446036

Order 15104.

486 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#3393	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₉×#3380	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₉×#2242	C₂	1	2	yes	maximal, complemented
4	C₅₉×#2241	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#2229	C₂	1	2	yes	maximal, complemented
6	C₅₉×#2226	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#2210	C₂	1	2	yes	maximal
8	C₅₉×#2208	C₂	1	1	yes	maximal, characteristic, complemented
9	Q₈×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×SD₃₂×C₅₉	C₂	1	2	yes	maximal, complemented
11	C₄×D₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
12	C₅₉×#556	—	2	1	no	—
13	C₅₉×#554	—	2	1	no	—
14	C₂×SD₃₂×C₅₉	C₂²	1	2	yes	—
15	C₂×D₁₆×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×#500	C₂²	1	1	yes	characteristic
17	C₅₉×#499	C₂²	1	2	yes	—
18	C₅₉×#495	C₂²	1	2	yes	—
19	C₅₉×#494	C₂²	1	1	yes	characteristic
20	C₅₉×#478	—	2	2	no	—
21	C₅₉×#476	—	2	1	no	—
22	C₅₉×#464	—	2	2	no	—
23	C₅₉×#461	—	2	1	no	—
24	C₈×Q₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₄×Q₁₆×C₅₉	C₂²	1	2	yes	—
26	C₄×D₈×C₅₉	C₂²	1	1	yes	characteristic
27	C₄×D₈×C₅₉	C₂²	1	1	yes	characteristic
28	C₄×D₈×C₅₉	C₂²	1	2	yes	—
29	C₅₉×#367	C₂²	1	2	yes	—
30	C₅₉×#366	C₂²	1	1	yes	characteristic
31	C₅₉×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
32	C₅₉×C₄⋊C₁₆	C₂²	1	2	yes	—
33	C₅₉×#358	C₂²	1	2	yes	—
34	C₅₉×#358	C₂²	1	2	yes	—
35	C₅₉×#358	C₂²	1	2	yes	—
36	C₅₉×#357	C₂²	1	2	yes	—
37	C₅₉×#357	C₂²	1	1	yes	characteristic
38	C₅₉×#357	C₂²	1	1	yes	characteristic
39	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₅₉	C₂²	1	2	yes	—
41	C₂×Q₁₆×C₅₉	—	2	1	no	—
42	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
43	C₂×SD₁₆×C₅₉	—	2	2	no	—
44	C₂×D₈×C₅₉	C₂³	1	1	yes	characteristic
45	C₂×D₈×C₅₉	C₂³	1	1	yes	characteristic
46	C₄⋊Q₈×C₅₉	—	2	1	no	—
47	C₄⋊Q₈×C₅₉	—	2	1	no	—
48	#126×C₅₉	—	4	2	no	—
49	#125×C₅₉	—	2	2	no	—
50	#124×C₅₉	—	2	2	no	—
51	C₂²⋊Q₈×C₅₉	—	4	1	no	—
52	C₂²⋊Q₈×C₅₉	—	4	1	no	—
53	C₂²⋊Q₈×C₅₉	—	4	2	no	—
54	C₄×Q₈×C₅₉	—	2	2	no	—
55	C₄×Q₈×C₅₉	D₄	1	1	yes	characteristic
56	C₄×D₄×C₅₉	—	2	1	no	—
57	C₄×D₄×C₅₉	—	2	1	no	—
58	C₄×D₄×C₅₉	—	2	2	no	—
59	#117×C₅₉	—	4	1	no	—
60	C₂×C₄⋊C₄×C₅₉	—	4	1	no	—
61	SD₃₂×C₅₉	—	2	4	no	—
62	D₁₆×C₅₉	—	2	2	no	—
63	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
64	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
65	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
66	#107×C₅₉	—	2	1	no	—
67	#107×C₅₉	C₂³	1	1	yes	characteristic
68	#107×C₅₉	C₂³	1	2	yes	—
69	#107×C₅₉	C₂³	1	1	yes	characteristic
70	#106×C₅₉	—	2	2	no	—
71	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
72	C₄⋊C₈×C₅₉	D₄	1	2	yes	—
73	#103×C₅₉	—	2	2	no	—
74	#102×C₅₉	—	2	1	no	—
75	#102×C₅₉	—	2	2	no	—
76	#102×C₅₉	—	2	1	no	—
77	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₅₉	C₂³	1	2	yes	—
79	C₂×Q₈×C₅₉	—	2	2	no	—
80	C₂×Q₈×C₅₉	—	4	1	no	—
81	C₂×Q₈×C₅₉	D₈	1	1	yes	characteristic
82	C₂×D₄×C₅₉	—	2	1	no	—
83	C₂×D₄×C₅₉	—	2	1	no	—
84	C₂²×C₄×C₅₉	—	4	1	no	—
85	C₂²×C₄×C₅₉	—	4	1	no	—
86	C₂²×C₄×C₅₉	—	4	2	no	—
87	Q₁₆×C₅₉	—	2	2	no	—
88	Q₁₆×C₅₉	—	2	2	no	—
89	Q₁₆×C₅₉	—	8	1	no	—
90	SD₁₆×C₅₉	—	8	2	no	—
91	D₈×C₅₉	—	2	1	no	—
92	D₈×C₅₉	—	2	1	no	—
93	D₈×C₅₉	—	2	1	no	—
94	D₈×C₅₉	#41	1	2	yes	—
95	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₅₉	—	2	1	no	—
99	C₄⋊C₄×C₅₉	—	2	1	no	—
100	C₄⋊C₄×C₅₉	—	2	2	no	—
101	C₄⋊C₄×C₅₉	—	4	2	no	—
102	C₄⋊C₄×C₅₉	—	4	1	no	—
103	C₄⋊C₄×C₅₉	—	4	1	no	—
104	C₄⋊C₄×C₅₉	—	4	2	no	—
105	C₄⋊C₄×C₅₉	—	4	2	no	—
106	C₄⋊C₄×C₅₉	—	4	1	no	—
107	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
108	C₄⋊C₄×C₅₉	D₈	1	2	yes	—
109	C₂²⋊C₄×C₅₉	—	4	2	no	—
110	C₂²⋊C₄×C₅₉	—	4	1	no	—
111	C₂²⋊C₄×C₅₉	—	4	2	no	—
112	C₂²⋊C₄×C₅₉	—	4	1	no	—
113	C₂²⋊C₄×C₅₉	—	4	2	no	—
114	C₄²×C₅₉	—	4	2	no	—
115	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₅₉	C₂×D₄	1	2	yes	—
117	C₁₆×C₅₉	—	2	1	no	—
118	C₁₆×C₅₉	—	2	2	no	—
119	C₁₆×C₅₉	—	2	1	no	—
120	C₂³×C₅₉	—	4	1	no	—
121	C₂³×C₅₉	—	4	1	no	—
122	Q₈×C₅₉	—	2	2	no	—
123	Q₈×C₅₉	—	4	2	no	—
124	Q₈×C₅₉	—	4	2	no	—
125	Q₈×C₅₉	—	16	1	no	—
126	D₄×C₅₉	—	2	2	no	—
127	D₄×C₅₉	—	4	1	no	—
128	D₄×C₅₉	—	4	1	no	—
129	D₄×C₅₉	—	4	1	no	—
130	C₂×C₄×C₅₉	—	4	1	no	—
131	C₂×C₄×C₅₉	—	4	2	no	—
132	C₂×C₄×C₅₉	—	4	1	no	—
133	C₂×C₄×C₅₉	—	4	2	no	—
134	C₂×C₄×C₅₉	—	8	2	no	—
135	C₂×C₄×C₅₉	—	8	2	no	—
136	C₂×C₄×C₅₉	—	16	2	no	—
137	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
138	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
142	C₈×C₅₉	—	2	2	no	—
143	C₈×C₅₉	—	2	1	no	—
144	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
145	C₈×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
146	#33927	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₂²×C₅₉	—	4	2	no	—
148	C₂²×C₅₉	—	8	1	no	—
149	C₂²×C₅₉	—	8	1	no	—
150	C₂²×C₅₉	—	8	1	no	—
151	C₂²×C₅₉	—	8	1	no	—
152	C₂²×C₅₉	—	8	1	no	—
153	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
154	C₄×C₅₉	—	2	2	no	—
155	C₄×C₅₉	—	2	2	no	—
156	C₄×C₅₉	—	2	1	no	—
157	C₄×C₅₉	—	8	2	no	—
158	C₄×C₅₉	—	8	1	no	—
159	C₄×C₅₉	—	8	1	no	—
160	C₄×C₅₉	—	8	2	no	—
161	C₄×C₅₉	#508	1	2	yes	—
162	C₄×C₅₉	#548	1	1	yes	characteristic
163	C₄×C₅₉	#577	1	1	yes	characteristic
164	#3393	C₂×C₅₉	1	1	yes	characteristic, complemented
165	#3380	C₂×C₅₉	1	1	yes	characteristic, complemented
166	#2242	C₂×C₅₉	1	2	yes	complemented
167	#2241	C₂×C₅₉	1	1	yes	characteristic, complemented
168	#2229	C₂×C₅₉	1	2	yes	complemented
169	#2226	C₂×C₅₉	1	1	yes	characteristic, complemented
170	#2210	C₂×C₅₉	1	2	yes	—
171	#2208	C₂×C₅₉	1	1	yes	characteristic, complemented
172	Q₈×C₁₆	C₂×C₅₉	1	1	yes	characteristic, complemented
173	C₄×SD₃₂	C₂×C₅₉	1	2	yes	complemented
174	C₄×D₁₆	C₂×C₅₉	1	1	yes	characteristic
175	C₂×C₅₉	—	8	2	no	—
176	C₂×C₅₉	—	16	1	no	—
177	C₂×C₅₉	#3358	1	1	yes	characteristic
178	C₂×C₅₉	C₂×#508	1	1	yes	characteristic
179	C₂×C₅₉	#3416	1	1	yes	characteristic
180	#556	—	2	1	no	—
181	#554	—	2	1	no	—
182	C₂×SD₃₂	C₂²×C₅₉	1	2	yes	—
183	C₂×D₁₆	C₂²×C₅₉	1	1	yes	characteristic
184	#500	C₂²×C₅₉	1	1	yes	characteristic
185	#499	C₂²×C₅₉	1	2	yes	—
186	#495	C₂²×C₅₉	1	2	yes	—
187	#494	C₂²×C₅₉	1	1	yes	characteristic
188	#478	—	2	2	no	—
189	#476	—	2	1	no	—
190	#464	—	2	2	no	—
191	#461	—	2	1	no	—
192	C₈×Q₈	C₂²×C₅₉	1	1	yes	characteristic
193	C₄×Q₁₆	C₂²×C₅₉	1	2	yes	—
194	C₄×D₈	C₂²×C₅₉	1	1	yes	characteristic
195	C₄×D₈	C₂²×C₅₉	1	2	yes	—
196	C₄×D₈	C₂²×C₅₉	1	1	yes	characteristic
197	#367	C₂²×C₅₉	1	2	yes	—
198	#366	C₂²×C₅₉	1	1	yes	characteristic
199	C₄⋊C₁₆	C₂²×C₅₉	1	1	yes	characteristic
200	C₄⋊C₁₆	C₂²×C₅₉	1	2	yes	—
201	#358	C₂²×C₅₉	1	2	yes	—
202	#358	C₂²×C₅₉	1	2	yes	—
203	#358	C₂²×C₅₉	1	2	yes	—
204	#357	C₂²×C₅₉	1	1	yes	characteristic
205	#357	C₂²×C₅₉	1	1	yes	characteristic
206	#357	C₂²×C₅₉	1	2	yes	—
207	C₄×C₁₆	C₂²×C₅₉	1	1	yes	characteristic
208	C₄×C₁₆	C₂²×C₅₉	1	2	yes	—
209	C₅₉	#33927	1	1	yes	characteristic, direct product factor, Sylow
210	C₂×Q₁₆	—	2	1	no	—
211	C₂×Q₁₆	C₂³×C₅₉	1	2	yes	—
212	C₂×SD₁₆	—	2	2	no	—
213	C₂×D₈	C₂³×C₅₉	1	1	yes	characteristic
214	C₂×D₈	C₂³×C₅₉	1	1	yes	characteristic
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	#126	—	4	2	no	—
218	#125	—	2	2	no	—
219	#124	—	2	2	no	—
220	C₂²⋊Q₈	—	4	1	no	—
221	C₂²⋊Q₈	—	4	1	no	—
222	C₂²⋊Q₈	—	4	2	no	—
223	C₄×Q₈	—	2	2	no	—
224	C₄×Q₈	D₄×C₅₉	1	1	yes	characteristic
225	C₄×D₄	—	2	1	no	—
226	C₄×D₄	—	2	2	no	—
227	C₄×D₄	—	2	1	no	—
228	#117	—	4	1	no	—
229	C₂×C₄⋊C₄	—	4	1	no	—
230	SD₃₂	—	2	4	no	—
231	D₁₆	—	2	2	no	—
232	C₂×C₁₆	C₂³×C₅₉	1	1	yes	characteristic
233	C₂×C₁₆	C₂³×C₅₉	1	2	yes	—
234	C₂×C₁₆	C₂³×C₅₉	1	1	yes	characteristic
235	#107	—	2	1	no	—
236	#107	C₂³×C₅₉	1	1	yes	characteristic
237	#107	C₂³×C₅₉	1	2	yes	—
238	#107	C₂³×C₅₉	1	1	yes	characteristic
239	#106	—	2	2	no	—
240	C₄⋊C₈	D₄×C₅₉	1	1	yes	characteristic
241	C₄⋊C₈	D₄×C₅₉	1	2	yes	—
242	#103	—	2	2	no	—
243	#102	—	2	1	no	—
244	#102	—	2	2	no	—
245	#102	—	2	1	no	—
246	C₄×C₈	C₂³×C₅₉	1	1	yes	characteristic
247	C₄×C₈	C₂³×C₅₉	1	2	yes	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×Q₈	D₈×C₅₉	1	1	yes	characteristic
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	2	1	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	4	2	no	—
256	Q₁₆	—	2	2	no	—
257	Q₁₆	—	2	2	no	—
258	Q₁₆	—	8	1	no	—
259	SD₁₆	—	8	2	no	—
260	D₈	—	2	1	no	—
261	D₈	—	2	1	no	—
262	D₈	—	2	1	no	—
263	D₈	#41×C₅₉	1	2	yes	—
264	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
265	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
266	C₂×C₈	C₂⁴×C₅₉	1	1	yes	characteristic
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	D₈×C₅₉	1	1	yes	characteristic
277	C₄⋊C₄	D₈×C₅₉	1	2	yes	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₂²⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	1	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₄²	—	4	2	no	—
284	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
285	C₄²	C₂×D₄×C₅₉	1	2	yes	—
286	C₁₆	—	2	1	no	—
287	C₁₆	—	2	2	no	—
288	C₁₆	—	2	1	no	—
289	C₂³	—	4	1	no	—
290	C₂³	—	4	1	no	—
291	Q₈	—	2	2	no	—
292	Q₈	—	4	2	no	—
293	Q₈	—	4	2	no	—
294	Q₈	—	16	1	no	—
295	D₄	—	2	2	no	—
296	D₄	—	4	1	no	—
297	D₄	—	4	1	no	—
298	D₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	8	2	no	—
304	C₂×C₄	—	8	2	no	—
305	C₂×C₄	—	16	2	no	—
306	C₂×C₄	C₂×D₈×C₅₉	1	2	yes	—
307	C₂×C₄	C₂×D₈×C₅₉	1	2	yes	—
308	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
309	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
310	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	2	1	no	—
313	C₈	C₂×#41×C₅₉	1	1	yes	characteristic
314	C₈	2₋¹⁺⁴×C₅₉	1	1	yes	characteristic
315	C₂²	—	4	2	no	—
316	C₂²	—	8	1	no	—
317	C₂²	—	8	1	no	—
318	C₂²	—	8	1	no	—
319	C₂²	—	8	1	no	—
320	C₂²	—	8	1	no	—
321	C₂²	C₂²×D₈×C₅₉	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	1	no	—
325	C₄	—	8	2	no	—
326	C₄	—	8	1	no	—
327	C₄	—	8	1	no	—
328	C₄	—	8	2	no	—
329	C₄	C₅₉×#508	1	2	yes	—
330	C₄	C₅₉×#548	1	1	yes	characteristic
331	C₄	C₅₉×#577	1	1	yes	characteristic
332	C₂	—	8	2	no	—
333	C₂	—	16	1	no	—
334	C₂	C₅₉×#3358	1	1	yes	characteristic
335	C₂	C₂×C₅₉×#508	1	1	yes	characteristic
336	C₂	C₅₉×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₉
prime radical	C₂×D₁₆×C₅₉	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₄×Q₈×C₅₉ ◁ C₈×Q₈×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₉×#3393	C₂	semidirect
2. C₅₉×#3380	C₂	semidirect
3. C₅₉×#2242	C₂	semidirect
4. C₅₉×#2241	C₂	semidirect
5. C₅₉×#2229	C₂	semidirect
6. C₅₉×#2226	C₂	semidirect
8. C₅₉×#2208	C₂	semidirect
9. Q₈×C₁₆×C₅₉	C₂	semidirect
10. C₄×SD₃₂×C₅₉	C₂	semidirect
146. #33927	C₅₉	direct (N × H)
164. #3393	C₂×C₅₉	semidirect
165. #3380	C₂×C₅₉	semidirect
166. #2242	C₂×C₅₉	semidirect
167. #2241	C₂×C₅₉	semidirect
168. #2229	C₂×C₅₉	semidirect
169. #2226	C₂×C₅₉	semidirect
171. #2208	C₂×C₅₉	semidirect
172. Q₈×C₁₆	C₂×C₅₉	semidirect
173. C₄×SD₃₂	C₂×C₅₉	semidirect
209. C₅₉	#33927	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₉×#3393	2
C₅₉×#3380	2
C₅₉×#2242	2
C₅₉×#2241	2
C₅₉×#2229	2
C₅₉×#2226	2
C₅₉×#2210	2
C₅₉×#2208	2
Q₈×C₁₆×C₅₉	2
C₄×SD₃₂×C₅₉	2
C₄×D₁₆×C₅₉	2
#33927	59

Subgroups — C59×#33927 #28446036

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#33927 #28446036