Subgroups — C₅₉×#21656 #28433765

Order 15104.

970 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₉×#490	C₂	1	1	yes	maximal, characteristic
2	C₄×#127×C₅₉	C₂	1	1	yes	maximal, characteristic
3	C₄×#125×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₉×#1992	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#1961	C₂	1	1	yes	maximal, characteristic
6	C₅₉×#1934	C₂	1	4	yes	maximal, complemented
7	C₅₉×#1836	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₉×#1762	C₂	1	4	yes	maximal, complemented
9	C₄×#97×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#127×C₅₉	C₂²	1	1	yes	characteristic
11	C₂×#125×C₅₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₅₉	—	2	2	no	—
13	C₂×C₄×D₄×C₅₉	C₂²	1	2	yes	—
14	C₅₉×#490	C₄	1	8	yes	complemented
15	C₂×#102×C₅₉	C₂²	1	4	yes	—
16	C₂×#102×C₅₉	C₂²	1	4	yes	—
17	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×#97×C₅₉	C₂²	1	2	yes	—
19	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
21	C₅₉×#390	—	2	2	no	—
22	C₅₉×#390	C₂²	1	2	yes	—
23	C₅₉×#389	C₂²	1	2	yes	—
24	C₅₉×#387	—	2	2	no	—
25	C₅₉×#384	C₂²	1	1	yes	characteristic
26	C₅₉×#382	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₅₉	—	2	2	no	—
28	C₄×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
29	C₄³×C₅₉	C₂²	1	1	yes	characteristic
30	C₅₉×#340	C₂²	1	4	yes	—
31	C₅₉×#340	C₂²	1	4	yes	—
32	C₂²×D₄×C₅₉	—	2	2	no	—
33	C₂²×D₄×C₅₉	C₂³	1	2	yes	—
34	C₂³×C₄×C₅₉	—	2	2	no	—
35	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
36	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
37	#127×C₅₉	—	2	2	no	—
38	#127×C₅₉	C₂×C₄	1	4	yes	—
39	#125×C₅₉	—	2	2	no	—
40	#125×C₅₉	C₂×C₄	1	4	yes	complemented
41	C₄×D₄×C₅₉	—	2	8	no	—
42	C₄×D₄×C₅₉	—	4	8	no	—
43	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
45	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₅₉	C₂³	1	2	yes	—
49	C₂×C₂²⋊C₄×C₅₉	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₅₉	—	2	4	no	—
51	C₂×C₄²×C₅₉	—	2	2	no	—
52	C₂×C₄²×C₅₉	D₄	1	2	yes	—
53	C₂×C₄²×C₅₉	D₄	1	2	yes	—
54	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
57	#102×C₅₉	—	2	8	no	—
58	#102×C₅₉	C₂×C₄	1	16	yes	complemented
59	#97×C₅₉	—	2	2	no	—
60	#97×C₅₉	—	2	4	no	—
61	#97×C₅₉	C₂×C₄	1	4	yes	complemented
62	C₄×C₈×C₅₉	—	2	4	no	—
63	#95×C₅₉	—	2	2	no	—
64	#95×C₅₉	—	2	2	no	—
65	#95×C₅₉	—	2	2	no	—
66	C₂⁴×C₅₉	—	2	2	no	—
67	C₂×D₄×C₅₉	—	2	8	no	—
68	C₂×D₄×C₅₉	—	2	4	no	—
69	C₂×D₄×C₅₉	—	2	4	no	—
70	C₂×D₄×C₅₉	—	2	2	no	—
71	C₂×D₄×C₅₉	—	4	4	no	—
72	C₂×D₄×C₅₉	—	4	4	no	—
73	C₂×D₄×C₅₉	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₅₉	—	2	2	no	—
75	C₂²×C₄×C₅₉	—	2	2	no	—
76	C₂²×C₄×C₅₉	—	2	2	no	—
77	C₂²×C₄×C₅₉	—	4	8	no	—
78	C₂²×C₄×C₅₉	—	4	4	no	—
79	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
84	C₂×C₈×C₅₉	—	2	2	no	—
85	C₂×C₈×C₅₉	—	2	2	no	—
86	C₂×C₈×C₅₉	—	2	2	no	—
87	C₂×C₈×C₅₉	—	2	2	no	—
88	C₂×C₈×C₅₉	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₅₉	#41	1	4	yes	—
90	C₄⋊C₄×C₅₉	—	2	8	no	—
91	C₄⋊C₄×C₅₉	—	2	2	no	—
92	C₄⋊C₄×C₅₉	—	2	4	no	—
93	C₄⋊C₄×C₅₉	—	2	4	no	—
94	C₄⋊C₄×C₅₉	—	4	4	no	—
95	C₄⋊C₄×C₅₉	—	4	4	no	—
96	C₄⋊C₄×C₅₉	C₂²×C₄	1	4	yes	—
97	C₂²⋊C₄×C₅₉	—	4	4	no	—
98	C₂²⋊C₄×C₅₉	—	4	8	no	—
99	C₄²×C₅₉	—	2	4	no	—
100	C₄²×C₅₉	—	2	2	no	—
101	C₄²×C₅₉	—	4	2	no	—
102	C₄²×C₅₉	—	4	4	no	—
103	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
104	C₄²×C₅₉	#41	1	2	yes	—
105	C₄²×C₅₉	#41	1	4	yes	—
106	C₂³×C₅₉	—	2	4	no	—
107	C₂³×C₅₉	—	4	2	no	—
108	C₂³×C₅₉	—	4	2	no	—
109	C₂³×C₅₉	—	4	4	no	—
110	C₂³×C₅₉	—	4	4	no	—
111	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
112	D₄×C₅₉	—	2	8	no	—
113	D₄×C₅₉	—	4	16	no	—
114	D₄×C₅₉	—	4	4	no	—
115	C₂×C₄×C₅₉	—	2	2	no	—
116	C₂×C₄×C₅₉	—	2	2	no	—
117	C₂×C₄×C₅₉	—	2	2	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	4	no	—
122	C₂×C₄×C₅₉	—	4	2	no	—
123	C₂×C₄×C₅₉	—	4	2	no	—
124	C₂×C₄×C₅₉	—	4	2	no	—
125	C₂×C₄×C₅₉	—	4	2	no	—
126	C₂×C₄×C₅₉	—	4	4	no	—
127	C₂×C₄×C₅₉	—	4	2	no	—
128	C₂×C₄×C₅₉	—	4	2	no	—
129	C₂×C₄×C₅₉	—	8	8	no	—
130	C₂×C₄×C₅₉	C₄×D₄	1	4	yes	—
131	C₂×C₄×C₅₉	#124	1	2	yes	—
132	C₂×C₄×C₅₉	#124	1	2	yes	—
133	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	4	yes	—
134	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
136	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
137	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
138	C₈×C₅₉	—	2	4	no	—
139	C₈×C₅₉	—	2	4	no	—
140	#21656	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂²×C₅₉	—	4	8	no	—
142	C₂²×C₅₉	—	4	4	no	—
143	C₂²×C₅₉	—	8	2	no	—
144	C₂²×C₅₉	—	8	2	no	—
145	C₂²×C₅₉	—	8	4	no	—
146	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
147	C₂²×C₅₉	C₂×#124	1	1	yes	characteristic
148	C₂²×C₅₉	#532	1	1	yes	characteristic
149	C₂²×C₅₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
150	C₂²×C₅₉	#575	1	2	yes	—
151	C₄×C₅₉	—	2	2	no	—
152	C₄×C₅₉	—	2	2	no	—
153	C₄×C₅₉	—	2	4	no	—
154	C₄×C₅₉	—	2	4	no	—
155	C₄×C₅₉	—	4	4	no	—
156	C₄×C₅₉	—	8	2	no	—
157	C₄×C₅₉	—	8	2	no	—
158	C₄×C₅₉	C₄×#41	1	2	yes	—
159	C₄×C₅₉	#529	1	2	yes	—
160	C₂×#490	C₂×C₅₉	1	1	yes	characteristic
161	C₄×#127	C₂×C₅₉	1	1	yes	characteristic
162	C₄×#125	C₂×C₅₉	1	1	yes	characteristic, complemented
163	#1992	C₂×C₅₉	1	1	yes	characteristic, complemented
164	#1961	C₂×C₅₉	1	1	yes	characteristic
165	#1934	C₂×C₅₉	1	4	yes	complemented
166	#1836	C₂×C₅₉	1	1	yes	characteristic, complemented
167	#1762	C₂×C₅₉	1	4	yes	complemented
168	C₄×#97	C₂×C₅₉	1	1	yes	characteristic, complemented
169	C₂×C₅₉	—	8	4	no	—
170	C₂×C₅₉	C₄×#124	1	1	yes	characteristic
171	C₂×C₅₉	C₄×AGL(1,ℤ/8)	1	2	yes	—
172	C₂×C₅₉	#3138	1	1	yes	characteristic
173	C₂×C₅₉	#3142	1	1	yes	characteristic
174	C₂×C₅₉	#3180	1	2	yes	—
175	C₂×#127	C₂²×C₅₉	1	1	yes	characteristic
176	C₂×#125	C₂²×C₅₉	1	1	yes	characteristic
177	C₂×C₄×D₄	—	2	2	no	—
178	C₂×C₄×D₄	C₂²×C₅₉	1	2	yes	—
179	#490	C₄×C₅₉	1	8	yes	complemented
180	C₂×#102	C₂²×C₅₉	1	4	yes	—
181	C₂×#102	C₂²×C₅₉	1	4	yes	—
182	C₂×#97	C₂²×C₅₉	1	1	yes	characteristic
183	C₂×#97	C₂²×C₅₉	1	2	yes	—
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	#389	C₂²×C₅₉	1	2	yes	—
189	#387	—	2	2	no	—
190	#384	C₂²×C₅₉	1	1	yes	characteristic
191	#382	C₂²×C₅₉	1	1	yes	characteristic
192	C₄×C₄⋊C₄	—	2	2	no	—
193	C₄×C₄⋊C₄	C₂²×C₅₉	1	2	yes	—
194	C₄³	C₂²×C₅₉	1	1	yes	characteristic
195	#340	C₂²×C₅₉	1	4	yes	—
196	#340	C₂²×C₅₉	1	4	yes	—
197	C₅₉	#21656	1	1	yes	characteristic, direct product factor, Sylow
198	C₂²×D₄	—	2	2	no	—
199	C₂²×D₄	C₂³×C₅₉	1	2	yes	—
200	C₂³×C₄	—	2	2	no	—
201	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
202	C₂²×C₈	C₂³×C₅₉	1	2	yes	—
203	#127	—	2	2	no	—
204	#127	C₂×C₄×C₅₉	1	4	yes	—
205	#125	—	2	2	no	—
206	#125	C₂×C₄×C₅₉	1	4	yes	complemented
207	C₄×D₄	—	2	8	no	—
208	C₄×D₄	—	4	8	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
213	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
214	C₂×C₄⋊C₄	C₂³×C₅₉	1	2	yes	—
215	C₂×C₂²⋊C₄	—	2	4	no	—
216	C₂×C₂²⋊C₄	—	2	2	no	—
217	C₂×C₄²	—	2	2	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	C₂×C₄×C₅₉	1	16	yes	complemented
225	#97	—	2	2	no	—
226	#97	—	2	4	no	—
227	#97	C₂×C₄×C₅₉	1	4	yes	complemented
228	C₄×C₈	—	2	4	no	—
229	#95	—	2	2	no	—
230	#95	—	2	2	no	—
231	#95	—	2	2	no	—
232	C₂⁴	—	2	2	no	—
233	C₂×D₄	—	2	8	no	—
234	C₂×D₄	—	2	4	no	—
235	C₂×D₄	—	2	4	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	4	4	no	—
238	C₂×D₄	—	4	4	no	—
239	C₂×D₄	C₂²×C₄×C₅₉	1	4	yes	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	4	8	no	—
244	C₂²×C₄	—	4	4	no	—
245	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
246	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₅₉	1	2	yes	—
249	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	C₂²×C₄×C₅₉	1	4	yes	—
255	C₂×C₈	#41×C₅₉	1	4	yes	—
256	C₄⋊C₄	—	2	8	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	4	4	no	—
261	C₄⋊C₄	—	4	4	no	—
262	C₄⋊C₄	C₂²×C₄×C₅₉	1	4	yes	—
263	C₂²⋊C₄	—	4	4	no	—
264	C₂²⋊C₄	—	4	8	no	—
265	C₄²	—	2	4	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	4	2	no	—
268	C₄²	—	4	4	no	—
269	C₄²	C₂²×C₄×C₅₉	1	2	yes	—
270	C₄²	#41×C₅₉	1	2	yes	—
271	C₄²	#41×C₅₉	1	4	yes	—
272	C₂³	—	2	4	no	—
273	C₂³	—	4	2	no	—
274	C₂³	—	4	2	no	—
275	C₂³	—	4	4	no	—
276	C₂³	—	4	4	no	—
277	C₂³	C₂²×D₄×C₅₉	1	1	yes	characteristic
278	D₄	—	2	8	no	—
279	D₄	—	4	16	no	—
280	D₄	—	4	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	8	8	no	—
296	C₂×C₄	C₄×D₄×C₅₉	1	4	yes	—
297	C₂×C₄	#124×C₅₉	1	2	yes	—
298	C₂×C₄	#124×C₅₉	1	2	yes	—
299	C₂×C₄	AGL(1,ℤ/8)×C₅₉	1	4	yes	—
300	C₂×C₄	C₂³×C₄×C₅₉	1	1	yes	characteristic
301	C₂×C₄	C₂×#41×C₅₉	1	2	yes	—
302	C₂×C₄	C₂×#41×C₅₉	1	2	yes	—
303	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
304	C₈	—	2	4	no	—
305	C₈	—	2	4	no	—
306	C₂²	—	4	4	no	—
307	C₂²	—	4	8	no	—
308	C₂²	—	8	2	no	—
309	C₂²	—	8	2	no	—
310	C₂²	—	8	4	no	—
311	C₂²	C₂×C₄×D₄×C₅₉	1	1	yes	characteristic
312	C₂²	C₂×#124×C₅₉	1	1	yes	characteristic
313	C₂²	C₅₉×#532	1	1	yes	characteristic
314	C₂²	C₂×AGL(1,ℤ/8)×C₅₉	1	2	yes	—
315	C₂²	C₅₉×#575	1	2	yes	—
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	4	no	—
319	C₄	—	2	4	no	—
320	C₄	—	4	4	no	—
321	C₄	—	8	2	no	—
322	C₄	—	8	2	no	—
323	C₄	C₄×#41×C₅₉	1	2	yes	—
324	C₄	C₅₉×#529	1	2	yes	—
325	C₂	—	8	4	no	—
326	C₂	C₄×#124×C₅₉	1	1	yes	characteristic
327	C₂	C₄×AGL(1,ℤ/8)×C₅₉	1	2	yes	—
328	C₂	C₅₉×#3138	1	1	yes	characteristic
329	C₂	C₅₉×#3142	1	1	yes	characteristic
330	C₂	C₅₉×#3180	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₉	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₉
prime radical	C₂×#127×C₅₉	C₂²
LCS	G ▷ C₂×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
3. C₄×#125×C₅₉	C₂	semidirect
4. C₅₉×#1992	C₂	semidirect
6. C₅₉×#1934	C₂	semidirect
7. C₅₉×#1836	C₂	semidirect
8. C₅₉×#1762	C₂	semidirect
9. C₄×#97×C₅₉	C₂	semidirect
14. C₅₉×#490	C₄	semidirect
40. #125×C₅₉	C₂×C₄	semidirect
58. #102×C₅₉	C₂×C₄	semidirect
61. #97×C₅₉	C₂×C₄	semidirect
140. #21656	C₅₉	direct (N × H)
162. C₄×#125	C₂×C₅₉	semidirect
163. #1992	C₂×C₅₉	semidirect
165. #1934	C₂×C₅₉	semidirect
166. #1836	C₂×C₅₉	semidirect
167. #1762	C₂×C₅₉	semidirect
168. C₄×#97	C₂×C₅₉	semidirect
179. #490	C₄×C₅₉	semidirect
197. C₅₉	#21656	direct (N × H)
206. #125	C₂×C₄×C₅₉	semidirect
224. #102	C₂×C₄×C₅₉	semidirect
227. #97	C₂×C₄×C₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₉×#490	2
C₄×#127×C₅₉	2
C₄×#125×C₅₉	2
C₅₉×#1992	2
C₅₉×#1961	2
C₅₉×#1934	2
C₅₉×#1836	2
C₅₉×#1762	2
C₄×#97×C₅₉	2
#21656	59

Subgroups — C59×#21656 #28433765

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#21656 #28433765