Subgroups — C₅₉×#33280 #28445389

Order 15104.

798 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#2864	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₉×#2858	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#1993	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₉×#1993	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#1961	C₂	1	1	yes	maximal, characteristic
6	C₅₉×#1961	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#1936	C₂	1	4	yes	maximal, complemented
8	C₅₉×#1934	C₂	1	4	yes	maximal, complemented
9	C₄×#97×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
11	C₂×#127×C₅₉	C₂²	1	1	yes	characteristic
12	C₂×#124×C₅₉	—	2	2	no	—
13	C₂×#102×C₅₉	C₂²	1	4	yes	—
14	C₂×#102×C₅₉	C₂²	1	4	yes	—
15	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
16	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×#97×C₅₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
19	C₅₉×#390	—	2	2	no	—
20	C₅₉×#390	C₂²	1	2	yes	—
21	C₅₉×#390	C₂²	1	2	yes	—
22	C₅₉×#389	C₂²	1	2	yes	—
23	C₅₉×#389	C₂²	1	2	yes	—
24	C₅₉×#387	—	2	2	no	—
25	C₅₉×#387	—	2	2	no	—
26	C₅₉×#382	C₂²	1	1	yes	characteristic
27	C₅₉×#382	C₂²	1	1	yes	characteristic
28	C₄³×C₅₉	C₂²	1	1	yes	characteristic
29	C₅₉×#340	C₂²	1	4	yes	—
30	C₅₉×#340	C₂²	1	4	yes	—
31	C₂²×Q₈×C₅₉	—	2	1	no	—
32	C₂²×D₄×C₅₉	—	2	1	no	—
33	C₂²×D₄×C₅₉	—	2	1	no	—
34	C₂²×D₄×C₅₉	C₂³	1	2	yes	—
35	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
36	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
37	C₄⋊Q₈×C₅₉	—	2	2	no	—
38	C₄⋊Q₈×C₅₉	—	2	2	no	—
39	#127×C₅₉	—	2	2	no	—
40	#127×C₅₉	—	2	2	no	—
41	#124×C₅₉	—	4	8	no	—
42	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
43	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
45	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
48	C₂×C₂²⋊C₄×C₅₉	—	2	4	no	—
49	C₂×C₂²⋊C₄×C₅₉	—	2	4	no	—
50	C₂×C₄²×C₅₉	D₄	1	2	yes	—
51	C₂×C₄²×C₅₉	D₄	1	2	yes	—
52	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
55	#102×C₅₉	—	2	8	no	—
56	#102×C₅₉	—	2	8	no	—
57	#97×C₅₉	—	2	4	no	—
58	#97×C₅₉	—	2	2	no	—
59	#97×C₅₉	—	2	2	no	—
60	C₄×C₈×C₅₉	—	2	4	no	—
61	#95×C₅₉	—	2	1	no	—
62	#95×C₅₉	—	2	1	no	—
63	#95×C₅₉	—	2	1	no	—
64	#95×C₅₉	—	2	1	no	—
65	#95×C₅₉	—	2	2	no	—
66	#95×C₅₉	—	2	2	no	—
67	C₂⁴×C₅₉	—	2	2	no	—
68	C₂×Q₈×C₅₉	—	4	4	no	—
69	C₂×Q₈×C₅₉	—	4	2	no	—
70	C₂×D₄×C₅₉	—	2	4	no	—
71	C₂×D₄×C₅₉	—	2	4	no	—
72	C₂×D₄×C₅₉	—	2	2	no	—
73	C₂×D₄×C₅₉	—	2	2	no	—
74	C₂×D₄×C₅₉	—	4	4	no	—
75	C₂×D₄×C₅₉	—	4	4	no	—
76	C₂×D₄×C₅₉	—	4	2	no	—
77	C₂×D₄×C₅₉	—	4	2	no	—
78	C₂²×C₄×C₅₉	—	2	2	no	—
79	C₂²×C₄×C₅₉	—	2	2	no	—
80	C₂²×C₄×C₅₉	—	2	2	no	—
81	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
86	C₂×C₈×C₅₉	—	2	2	no	—
87	C₂×C₈×C₅₉	—	2	2	no	—
88	C₂×C₈×C₅₉	—	2	2	no	—
89	C₂×C₈×C₅₉	—	2	2	no	—
90	C₂×C₈×C₅₉	#41	1	4	yes	—
91	C₂×C₈×C₅₉	#41	1	4	yes	—
92	C₄⋊C₄×C₅₉	—	2	4	no	—
93	C₄⋊C₄×C₅₉	—	2	4	no	—
94	C₄⋊C₄×C₅₉	—	2	2	no	—
95	C₄⋊C₄×C₅₉	—	2	2	no	—
96	C₄⋊C₄×C₅₉	—	4	4	no	—
97	C₄⋊C₄×C₅₉	—	4	2	no	—
98	C₄⋊C₄×C₅₉	—	4	4	no	—
99	C₂²⋊C₄×C₅₉	—	4	8	no	—
100	C₂²⋊C₄×C₅₉	—	4	8	no	—
101	C₄²×C₅₉	—	2	4	no	—
102	C₄²×C₅₉	—	2	2	no	—
103	C₄²×C₅₉	—	2	2	no	—
104	C₄²×C₅₉	—	4	2	no	—
105	C₄²×C₅₉	#41	1	2	yes	—
106	C₄²×C₅₉	#41	1	2	yes	—
107	C₂³×C₅₉	—	4	2	no	—
108	C₂³×C₅₉	—	4	2	no	—
109	C₂³×C₅₉	—	4	2	no	—
110	C₂³×C₅₉	—	4	4	no	—
111	C₂³×C₅₉	—	4	4	no	—
112	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
113	Q₈×C₅₉	—	8	4	no	—
114	D₄×C₅₉	—	4	4	no	—
115	D₄×C₅₉	—	4	4	no	—
116	D₄×C₅₉	—	8	4	no	—
117	D₄×C₅₉	—	8	4	no	—
118	C₂×C₄×C₅₉	—	2	2	no	—
119	C₂×C₄×C₅₉	—	2	2	no	—
120	C₂×C₄×C₅₉	—	2	2	no	—
121	C₂×C₄×C₅₉	—	2	2	no	—
122	C₂×C₄×C₅₉	—	2	2	no	—
123	C₂×C₄×C₅₉	—	2	2	no	—
124	C₂×C₄×C₅₉	—	2	2	no	—
125	C₂×C₄×C₅₉	—	4	2	no	—
126	C₂×C₄×C₅₉	—	4	2	no	—
127	C₂×C₄×C₅₉	—	4	2	no	—
128	C₂×C₄×C₅₉	—	4	4	no	—
129	C₂×C₄×C₅₉	—	4	2	no	—
130	C₂×C₄×C₅₉	—	4	2	no	—
131	C₂×C₄×C₅₉	—	4	2	no	—
132	C₂×C₄×C₅₉	—	4	2	no	—
133	C₂×C₄×C₅₉	#124	1	2	yes	—
134	C₂×C₄×C₅₉	#124	1	2	yes	—
135	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	4	yes	—
136	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
139	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
140	C₈×C₅₉	—	2	4	no	—
141	C₈×C₅₉	—	2	4	no	—
142	#33280	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂²×C₅₉	—	8	2	no	—
144	C₂²×C₅₉	—	8	2	no	—
145	C₂²×C₅₉	—	8	2	no	—
146	C₂²×C₅₉	—	8	4	no	—
147	C₂²×C₅₉	—	8	4	no	—
148	C₂²×C₅₉	C₂×#124	1	1	yes	characteristic
149	C₂²×C₅₉	#532	1	1	yes	characteristic
150	C₂²×C₅₉	#532	1	1	yes	characteristic
151	C₂²×C₅₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
152	C₂²×C₅₉	#575	1	2	yes	—
153	C₄×C₅₉	—	2	2	no	—
154	C₄×C₅₉	—	2	2	no	—
155	C₄×C₅₉	—	2	4	no	—
156	C₄×C₅₉	—	4	2	no	—
157	C₄×C₅₉	—	8	2	no	—
158	C₄×C₅₉	—	8	2	no	—
159	C₄×C₅₉	—	8	2	no	—
160	C₄×C₅₉	#529	1	2	yes	—
161	C₄×C₅₉	#529	1	2	yes	—
162	#2864	C₂×C₅₉	1	1	yes	characteristic, complemented
163	#2858	C₂×C₅₉	1	1	yes	characteristic
164	#1993	C₂×C₅₉	1	1	yes	characteristic, complemented
165	#1993	C₂×C₅₉	1	1	yes	characteristic, complemented
166	#1961	C₂×C₅₉	1	1	yes	characteristic
167	#1961	C₂×C₅₉	1	1	yes	characteristic
168	#1936	C₂×C₅₉	1	4	yes	complemented
169	#1934	C₂×C₅₉	1	4	yes	complemented
170	C₄×#97	C₂×C₅₉	1	1	yes	characteristic, complemented
171	C₂×C₅₉	—	16	2	no	—
172	C₂×C₅₉	#2861	1	1	yes	characteristic
173	C₂×C₅₉	#3138	1	1	yes	characteristic
174	C₂×C₅₉	#3140	1	1	yes	characteristic
175	C₂×C₅₉	#3171	1	2	yes	—
176	C₂×C₅₉	#3180	1	2	yes	—
177	C₂×C₄⋊Q₈	C₂²×C₅₉	1	1	yes	characteristic
178	C₂×#127	C₂²×C₅₉	1	1	yes	characteristic
179	C₂×#124	—	2	2	no	—
180	C₂×#102	C₂²×C₅₉	1	4	yes	—
181	C₂×#102	C₂²×C₅₉	1	4	yes	—
182	C₂×#97	C₂²×C₅₉	1	2	yes	—
183	C₂×#97	C₂²×C₅₉	1	1	yes	characteristic
184	C₂×#97	C₂²×C₅₉	1	1	yes	characteristic
185	C₂×C₄×C₈	C₂²×C₅₉	1	2	yes	—
186	#390	—	2	2	no	—
187	#390	C₂²×C₅₉	1	2	yes	—
188	#390	C₂²×C₅₉	1	2	yes	—
189	#389	C₂²×C₅₉	1	2	yes	—
190	#389	C₂²×C₅₉	1	2	yes	—
191	#387	—	2	2	no	—
192	#387	—	2	2	no	—
193	#382	C₂²×C₅₉	1	1	yes	characteristic
194	#382	C₂²×C₅₉	1	1	yes	characteristic
195	C₄³	C₂²×C₅₉	1	1	yes	characteristic
196	#340	C₂²×C₅₉	1	4	yes	—
197	#340	C₂²×C₅₉	1	4	yes	—
198	C₅₉	#33280	1	1	yes	characteristic, direct product factor, Sylow
199	C₂²×Q₈	—	2	1	no	—
200	C₂²×D₄	—	2	1	no	—
201	C₂²×D₄	—	2	1	no	—
202	C₂²×D₄	C₂³×C₅₉	1	2	yes	—
203	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
204	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
205	C₄⋊Q₈	—	2	2	no	—
206	C₄⋊Q₈	—	2	2	no	—
207	#127	—	2	2	no	—
208	#127	—	2	2	no	—
209	#124	—	4	8	no	—
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
214	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
215	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
216	C₂×C₂²⋊C₄	—	2	4	no	—
217	C₂×C₂²⋊C₄	—	2	4	no	—
218	C₂×C₄²	D₄×C₅₉	1	2	yes	—
219	C₂×C₄²	D₄×C₅₉	1	2	yes	—
220	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
221	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
222	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
223	#102	—	2	8	no	—
224	#102	—	2	8	no	—
225	#97	—	2	4	no	—
226	#97	—	2	2	no	—
227	#97	—	2	2	no	—
228	C₄×C₈	—	2	4	no	—
229	#95	—	2	1	no	—
230	#95	—	2	1	no	—
231	#95	—	2	1	no	—
232	#95	—	2	1	no	—
233	#95	—	2	2	no	—
234	#95	—	2	2	no	—
235	C₂⁴	—	2	2	no	—
236	C₂×Q₈	—	4	4	no	—
237	C₂×Q₈	—	4	2	no	—
238	C₂×D₄	—	2	4	no	—
239	C₂×D₄	—	2	4	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	4	4	no	—
243	C₂×D₄	—	4	4	no	—
244	C₂×D₄	—	4	2	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
250	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
253	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	#41×C₅₉	1	4	yes	—
259	C₂×C₈	#41×C₅₉	1	4	yes	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	4	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	4	no	—
267	C₂²⋊C₄	—	4	8	no	—
268	C₂²⋊C₄	—	4	8	no	—
269	C₄²	—	2	4	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	4	2	no	—
273	C₄²	#41×C₅₉	1	2	yes	—
274	C₄²	#41×C₅₉	1	2	yes	—
275	C₂³	—	4	2	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	4	2	no	—
278	C₂³	—	4	4	no	—
279	C₂³	—	4	4	no	—
280	C₂³	C₂²×D₄×C₅₉	1	1	yes	characteristic
281	Q₈	—	8	4	no	—
282	D₄	—	4	4	no	—
283	D₄	—	4	4	no	—
284	D₄	—	8	4	no	—
285	D₄	—	8	4	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	#124×C₅₉	1	2	yes	—
302	C₂×C₄	#124×C₅₉	1	2	yes	—
303	C₂×C₄	AGL(1,ℤ/8)×C₅₉	1	4	yes	—
304	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
305	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
306	C₂×C₄	C₂×#41×C₅₉	1	2	yes	—
307	C₂×C₄	C₂×#41×C₅₉	1	2	yes	—
308	C₈	—	2	4	no	—
309	C₈	—	2	4	no	—
310	C₂²	—	8	2	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	4	no	—
314	C₂²	—	8	4	no	—
315	C₂²	C₂×#124×C₅₉	1	1	yes	characteristic
316	C₂²	C₅₉×#532	1	1	yes	characteristic
317	C₂²	C₅₉×#532	1	1	yes	characteristic
318	C₂²	C₂×AGL(1,ℤ/8)×C₅₉	1	2	yes	—
319	C₂²	C₅₉×#575	1	2	yes	—
320	C₄	—	2	2	no	—
321	C₄	—	2	2	no	—
322	C₄	—	2	4	no	—
323	C₄	—	4	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	—	8	2	no	—
327	C₄	C₅₉×#529	1	2	yes	—
328	C₄	C₅₉×#529	1	2	yes	—
329	C₂	—	16	2	no	—
330	C₂	C₅₉×#2861	1	1	yes	characteristic
331	C₂	C₅₉×#3138	1	1	yes	characteristic
332	C₂	C₅₉×#3140	1	1	yes	characteristic
333	C₂	C₅₉×#3171	1	2	yes	—
334	C₂	C₅₉×#3180	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₉
prime radical	C₂×#127×C₅₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₉ ◁ C₄³×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₉×#2864	C₂	semidirect
3. C₅₉×#1993	C₂	semidirect
4. C₅₉×#1993	C₂	semidirect
7. C₅₉×#1936	C₂	semidirect
8. C₅₉×#1934	C₂	semidirect
9. C₄×#97×C₅₉	C₂	semidirect
142. #33280	C₅₉	direct (N × H)
162. #2864	C₂×C₅₉	semidirect
164. #1993	C₂×C₅₉	semidirect
165. #1993	C₂×C₅₉	semidirect
168. #1936	C₂×C₅₉	semidirect
169. #1934	C₂×C₅₉	semidirect
170. C₄×#97	C₂×C₅₉	semidirect
198. C₅₉	#33280	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×#2864	2
C₅₉×#2858	2
C₅₉×#1993	2
C₅₉×#1961	2
C₅₉×#1936	2
C₅₉×#1934	2
C₄×#97×C₅₉	2
#33280	59

Subgroups — C59×#33280 #28445389

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#33280 #28445389