Subgroups — C₅₉×#32667 #28444776

Order 15104.

666 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#2582	C₂	1	2	yes	maximal
2	C₅₉×#2096	C₂	1	2	yes	maximal
3	C₅₉×#2084	C₂	1	2	yes	maximal
4	C₅₉×#2066	C₂	1	2	yes	maximal
5	C₅₉×#1985	C₂	1	1	yes	maximal, characteristic
6	C₅₉×#1971	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#1944	C₂	1	1	yes	maximal, characteristic
8	C₅₉×#1878	C₂	1	2	yes	maximal
9	C₅₉×#1863	C₂	1	2	yes	maximal
10	C₂²×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₅₉	C₂²	1	2	yes	—
12	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
14	C₂×#106×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×#106×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×#106×C₅₉	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×#103×C₅₉	C₂²	1	2	yes	—
19	C₂×#103×C₅₉	C₂²	1	2	yes	—
20	C₂×#103×C₅₉	C₂²	1	2	yes	—
21	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
22	C₅₉×#398	—	2	2	no	—
23	C₅₉×#398	—	2	2	no	—
24	C₅₉×#391	—	2	2	no	—
25	C₅₉×#391	C₂²	1	2	yes	—
26	C₅₉×#391	C₂²	1	2	yes	—
27	C₅₉×#389	—	2	2	no	—
28	C₅₉×#389	C₂²	1	2	yes	—
29	C₅₉×#389	C₂²	1	2	yes	—
30	C₅₉×#384	C₂²	1	2	yes	—
31	C₅₉×#340	C₂²	1	2	yes	—
32	C₅₉×#340	C₂²	1	2	yes	—
33	C₅₉×#336	C₂²	1	2	yes	—
34	C₅₉×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₅₉	D₄	1	1	yes	characteristic
36	C₂²×Q₈×C₅₉	C₂³	1	2	yes	—
37	C₂×Q₁₆×C₅₉	—	4	1	no	—
38	C₂×Q₁₆×C₅₉	—	4	1	no	—
39	C₂×Q₁₆×C₅₉	Q₈	1	4	yes	complemented
40	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
41	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
43	C₄×Q₈×C₅₉	—	4	1	no	—
44	C₄×Q₈×C₅₉	—	4	1	no	—
45	C₄×Q₈×C₅₉	—	4	2	no	—
46	C₄×Q₈×C₅₉	—	4	2	no	—
47	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
48	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
49	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
50	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
51	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
52	C₂×C₄⋊C₄×C₅₉	D₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
56	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
57	C₂×C₄²×C₅₉	—	2	2	no	—
58	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
59	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
60	#107×C₅₉	—	4	1	no	—
61	#106×C₅₉	—	2	2	no	—
62	#106×C₅₉	—	4	2	no	—
63	#106×C₅₉	—	4	1	no	—
64	C₄⋊C₈×C₅₉	—	4	1	no	—
65	#103×C₅₉	—	2	4	no	—
66	#103×C₅₉	—	4	2	no	—
67	#103×C₅₉	—	4	2	no	—
68	#97×C₅₉	—	4	1	no	—
69	#95×C₅₉	—	2	2	no	—
70	#95×C₅₉	—	2	2	no	—
71	#95×C₅₉	—	2	2	no	—
72	#95×C₅₉	—	2	2	no	—
73	#95×C₅₉	—	2	2	no	—
74	#95×C₅₉	—	2	2	no	—
75	C₂×Q₈×C₅₉	—	2	4	no	—
76	C₂×Q₈×C₅₉	—	2	2	no	—
77	C₂×Q₈×C₅₉	—	2	2	no	—
78	C₂×Q₈×C₅₉	—	2	2	no	—
79	C₂×Q₈×C₅₉	—	2	2	no	—
80	C₂×Q₈×C₅₉	—	2	2	no	—
81	C₂×Q₈×C₅₉	—	4	1	no	—
82	C₂×Q₈×C₅₉	C₂×Q₈	1	4	yes	—
83	C₂²×C₄×C₅₉	—	2	2	no	—
84	C₂²×C₄×C₅₉	—	2	2	no	—
85	C₂²×C₄×C₅₉	—	2	2	no	—
86	C₂²×C₄×C₅₉	—	2	2	no	—
87	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
92	Q₁₆×C₅₉	—	4	4	no	—
93	C₂×C₈×C₅₉	—	2	1	no	—
94	C₂×C₈×C₅₉	—	2	1	no	—
95	C₂×C₈×C₅₉	—	2	1	no	—
96	C₂×C₈×C₅₉	—	2	1	no	—
97	C₂×C₈×C₅₉	—	2	2	no	—
98	C₂×C₈×C₅₉	—	2	2	no	—
99	C₂×C₈×C₅₉	—	2	2	no	—
100	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
101	C₂×C₈×C₅₉	#41	1	2	yes	—
102	C₄⋊C₄×C₅₉	—	2	2	no	—
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	—	2	2	no	—
108	C₄⋊C₄×C₅₉	—	4	2	no	—
109	C₄⋊C₄×C₅₉	—	4	1	no	—
110	C₄⋊C₄×C₅₉	—	4	4	no	—
111	C₄⋊C₄×C₅₉	—	4	4	no	—
112	C₄⋊C₄×C₅₉	—	8	2	no	—
113	C₄⋊C₄×C₅₉	—	8	2	no	—
114	C₄⋊C₄×C₅₉	—	8	2	no	—
115	C₄²×C₅₉	—	2	1	no	—
116	C₄²×C₅₉	—	2	1	no	—
117	C₄²×C₅₉	—	2	2	no	—
118	C₄²×C₅₉	—	2	2	no	—
119	C₄²×C₅₉	—	4	4	no	—
120	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
121	Q₈×C₅₉	—	2	4	no	—
122	Q₈×C₅₉	—	2	4	no	—
123	Q₈×C₅₉	—	4	4	no	—
124	Q₈×C₅₉	—	4	2	no	—
125	Q₈×C₅₉	—	4	2	no	—
126	C₂×C₄×C₅₉	—	2	2	no	—
127	C₂×C₄×C₅₉	—	2	2	no	—
128	C₂×C₄×C₅₉	—	2	2	no	—
129	C₂×C₄×C₅₉	—	2	1	no	—
130	C₂×C₄×C₅₉	—	2	1	no	—
131	C₂×C₄×C₅₉	—	2	4	no	—
132	C₂×C₄×C₅₉	—	2	2	no	—
133	C₂×C₄×C₅₉	—	2	2	no	—
134	C₂×C₄×C₅₉	—	2	2	no	—
135	C₂×C₄×C₅₉	—	2	2	no	—
136	C₂×C₄×C₅₉	—	4	2	no	—
137	C₂×C₄×C₅₉	—	4	2	no	—
138	C₂×C₄×C₅₉	—	4	2	no	—
139	C₂×C₄×C₅₉	—	4	2	no	—
140	C₂×C₄×C₅₉	—	4	2	no	—
141	C₂×C₄×C₅₉	—	4	2	no	—
142	C₂×C₄×C₅₉	—	4	2	no	—
143	C₂×C₄×C₅₉	—	4	2	no	—
144	C₂×C₄×C₅₉	—	4	2	no	—
145	C₂×C₄×C₅₉	—	4	2	no	—
146	C₂×C₄×C₅₉	—	4	2	no	—
147	C₂×C₄×C₅₉	#137	1	4	yes	—
148	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
149	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
151	C₂×C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂×C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
153	C₂×C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
154	C₈×C₅₉	—	2	2	no	—
155	C₈×C₅₉	—	2	2	no	—
156	C₈×C₅₉	—	4	2	no	—
157	#32667	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂²×C₅₉	#534	1	1	yes	characteristic
159	C₂²×C₅₉	D₄×Q₈	1	1	yes	characteristic
160	C₂²×C₅₉	#550	1	1	yes	characteristic
161	C₂²×C₅₉	C₂×#137	1	2	yes	—
162	C₂²×C₅₉	#577	1	2	yes	—
163	C₄×C₅₉	—	2	4	no	—
164	C₄×C₅₉	—	4	1	no	—
165	C₄×C₅₉	—	4	2	no	—
166	C₄×C₅₉	—	4	1	no	—
167	C₄×C₅₉	—	4	4	no	—
168	C₄×C₅₉	—	8	2	no	—
169	C₄×C₅₉	—	8	2	no	—
170	C₄×C₅₉	—	8	2	no	—
171	C₄×C₅₉	#539	1	1	yes	characteristic
172	C₄×C₅₉	#543	1	1	yes	characteristic
173	C₄×C₅₉	#554	1	1	yes	characteristic
174	C₄×C₅₉	#562	1	1	yes	characteristic
175	#2582	C₂×C₅₉	1	2	yes	—
176	#2096	C₂×C₅₉	1	2	yes	—
177	#2084	C₂×C₅₉	1	2	yes	—
178	#2066	C₂×C₅₉	1	2	yes	—
179	#1985	C₂×C₅₉	1	1	yes	characteristic
180	#1971	C₂×C₅₉	1	1	yes	characteristic
181	#1944	C₂×C₅₉	1	1	yes	characteristic
182	#1878	C₂×C₅₉	1	2	yes	—
183	#1863	C₂×C₅₉	1	2	yes	—
184	C₂×C₅₉	#2732	1	1	yes	characteristic
185	C₂×C₅₉	#3195	1	1	yes	characteristic
186	C₂×C₅₉	#3204	1	1	yes	characteristic
187	C₂×C₅₉	#3387	1	2	yes	—
188	C₂×C₅₉	#3400	1	2	yes	—
189	C₂²×Q₁₆	C₂²×C₅₉	1	1	yes	characteristic
190	C₂×C₄×Q₈	C₂²×C₅₉	1	2	yes	—
191	C₂×C₄×Q₈	C₂²×C₅₉	1	1	yes	characteristic
192	C₂×#107	C₂²×C₅₉	1	1	yes	characteristic
193	C₂×#106	C₂²×C₅₉	1	1	yes	characteristic
194	C₂×#106	C₂²×C₅₉	1	2	yes	—
195	C₂×#106	C₂²×C₅₉	1	1	yes	characteristic
196	C₂×C₄⋊C₈	C₂²×C₅₉	1	1	yes	characteristic
197	C₂×#103	C₂²×C₅₉	1	2	yes	—
198	C₂×#103	C₂²×C₅₉	1	2	yes	—
199	C₂×#103	C₂²×C₅₉	1	2	yes	—
200	C₂×#97	C₂²×C₅₉	1	1	yes	characteristic
201	#398	—	2	2	no	—
202	#398	—	2	2	no	—
203	#391	—	2	2	no	—
204	#391	C₂²×C₅₉	1	2	yes	—
205	#391	C₂²×C₅₉	1	2	yes	—
206	#389	—	2	2	no	—
207	#389	C₂²×C₅₉	1	2	yes	—
208	#389	C₂²×C₅₉	1	2	yes	—
209	#384	C₂²×C₅₉	1	2	yes	—
210	#340	C₂²×C₅₉	1	2	yes	—
211	#340	C₂²×C₅₉	1	2	yes	—
212	#336	C₂²×C₅₉	1	2	yes	—
213	#336	C₂²×C₅₉	1	2	yes	—
214	C₅₉	#32667	1	1	yes	characteristic, direct product factor, Sylow
215	C₂²×Q₈	D₄×C₅₉	1	1	yes	characteristic
216	C₂²×Q₈	C₂³×C₅₉	1	2	yes	—
217	C₂×Q₁₆	—	4	1	no	—
218	C₂×Q₁₆	—	4	1	no	—
219	C₂×Q₁₆	Q₈×C₅₉	1	4	yes	complemented
220	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
221	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
222	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
223	C₄×Q₈	—	4	1	no	—
224	C₄×Q₈	—	4	1	no	—
225	C₄×Q₈	—	4	2	no	—
226	C₄×Q₈	—	4	2	no	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	—	2	2	no	—
229	C₂×C₄⋊C₄	—	2	2	no	—
230	C₂×C₄⋊C₄	—	2	2	no	—
231	C₂×C₄⋊C₄	—	2	2	no	—
232	C₂×C₄⋊C₄	D₄×C₅₉	1	2	yes	—
233	C₂×C₄⋊C₄	D₄×C₅₉	1	1	yes	characteristic
234	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
235	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
236	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
237	C₂×C₄²	—	2	2	no	—
238	C₂×C₄²	C₂³×C₅₉	1	2	yes	—
239	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
240	#107	—	4	1	no	—
241	#106	—	2	2	no	—
242	#106	—	4	2	no	—
243	#106	—	4	1	no	—
244	C₄⋊C₈	—	4	1	no	—
245	#103	—	2	4	no	—
246	#103	—	4	2	no	—
247	#103	—	4	2	no	—
248	#97	—	4	1	no	—
249	#95	—	2	2	no	—
250	#95	—	2	2	no	—
251	#95	—	2	2	no	—
252	#95	—	2	2	no	—
253	#95	—	2	2	no	—
254	#95	—	2	2	no	—
255	C₂×Q₈	—	2	4	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	—	2	2	no	—
259	C₂×Q₈	—	2	2	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	4	1	no	—
262	C₂×Q₈	C₂×Q₈×C₅₉	1	4	yes	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
270	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
271	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
272	Q₁₆	—	4	4	no	—
273	C₂×C₈	—	2	1	no	—
274	C₂×C₈	—	2	1	no	—
275	C₂×C₈	—	2	1	no	—
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	C₂×Q₈×C₅₉	1	2	yes	—
281	C₂×C₈	#41×C₅₉	1	2	yes	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	4	2	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	—	4	4	no	—
291	C₄⋊C₄	—	4	4	no	—
292	C₄⋊C₄	—	8	2	no	—
293	C₄⋊C₄	—	8	2	no	—
294	C₄⋊C₄	—	8	2	no	—
295	C₄²	—	2	1	no	—
296	C₄²	—	2	1	no	—
297	C₄²	—	2	2	no	—
298	C₄²	—	2	2	no	—
299	C₄²	—	4	4	no	—
300	C₂³	C₂²×D₄×C₅₉	1	1	yes	characteristic
301	Q₈	—	2	4	no	—
302	Q₈	—	2	4	no	—
303	Q₈	—	4	4	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	4	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	#137×C₅₉	1	4	yes	—
328	C₂×C₄	C₂²×Q₈×C₅₉	1	1	yes	characteristic
329	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
330	C₂×C₄	H(5,𝔽₂)×C₅₉	1	1	yes	characteristic
331	C₂×C₄	H(5,𝔽₂)×C₅₉	1	1	yes	characteristic
332	C₂×C₄	H(5,𝔽₂)×C₅₉	1	1	yes	characteristic
333	C₂×C₄	2₋¹⁺⁴×C₅₉	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	2	2	no	—
336	C₈	—	4	2	no	—
337	C₂²	C₅₉×#534	1	1	yes	characteristic
338	C₂²	D₄×Q₈×C₅₉	1	1	yes	characteristic
339	C₂²	C₅₉×#550	1	1	yes	characteristic
340	C₂²	C₂×#137×C₅₉	1	2	yes	—
341	C₂²	C₅₉×#577	1	2	yes	—
342	C₄	—	2	4	no	—
343	C₄	—	4	1	no	—
344	C₄	—	4	2	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	4	no	—
347	C₄	—	8	2	no	—
348	C₄	—	8	2	no	—
349	C₄	—	8	2	no	—
350	C₄	C₅₉×#539	1	1	yes	characteristic
351	C₄	C₅₉×#543	1	1	yes	characteristic
352	C₄	C₅₉×#554	1	1	yes	characteristic
353	C₄	C₅₉×#562	1	1	yes	characteristic
354	C₂	C₅₉×#2732	1	1	yes	characteristic
355	C₂	C₅₉×#3195	1	1	yes	characteristic
356	C₂	C₅₉×#3204	1	1	yes	characteristic
357	C₂	C₅₉×#3387	1	2	yes	—
358	C₂	C₅₉×#3400	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₉ ◁ C₂×C₄×Q₈×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
39. C₂×Q₁₆×C₅₉	Q₈	semidirect
157. #32667	C₅₉	direct (N × H)
214. C₅₉	#32667	direct (N × H)
219. C₂×Q₁₆	Q₈×C₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₅₉×#2582	2
C₅₉×#2096	2
C₅₉×#2084	2
C₅₉×#2066	2
C₅₉×#1985	2
C₅₉×#1971	2
C₅₉×#1944	2
C₅₉×#1878	2
C₅₉×#1863	2
#32667	59

Subgroups — C59×#32667 #28444776

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#32667 #28444776