Subgroups — C₅₉×#7581 #28419690

Order 15104.

424 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#2295	C₂	1	1	yes	maximal, characteristic
2	C₅₉×#1839	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#1744	C₂	1	1	yes	maximal, characteristic
4	C₅₉×#1272	C₂	1	2	yes	maximal
5	C₅₉×#1271	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈×C₅₉	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₅₉	—	2	1	no	—
8	C₅₉×#391	C₄	1	1	yes	characteristic
9	C₅₉×#391	C₂²	1	1	yes	characteristic
10	C₅₉×#389	C₄	1	1	yes	characteristic
11	C₅₉×#389	C₂²	1	1	yes	characteristic
12	C₅₉×#387	—	2	1	no	—
13	C₅₉×#387	—	2	1	no	—
14	C₄×C₄⋊C₄×C₅₉	—	2	1	no	—
15	C₄×C₄⋊C₄×C₅₉	C₄	1	1	yes	characteristic
16	C₅₉×#376	—	2	1	no	—
17	C₄³×C₅₉	C₂²	1	1	yes	characteristic
18	C₅₉×#336	—	2	1	no	—
19	C₅₉×#336	—	2	1	no	—
20	C₅₉×#336	—	2	1	no	—
21	C₅₉×#336	C₂²	1	2	yes	—
22	C₅₉×#336	C₂²	1	2	yes	—
23	C₂²×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₉	—	2	2	no	—
25	C₂²×C₈×C₅₉	—	2	2	no	—
26	C₄×Q₈×C₅₉	—	2	2	no	—
27	C₄×Q₈×C₅₉	C₈	1	4	yes	complemented
28	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
30	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
31	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₉	—	2	1	no	—
35	C₂×C₄²×C₅₉	—	2	1	no	—
36	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
43	C₄×C₈×C₅₉	—	2	4	no	—
44	#95×C₅₉	—	2	1	no	—
45	#95×C₅₉	—	2	1	no	—
46	#95×C₅₉	—	2	1	no	—
47	#95×C₅₉	—	2	1	no	—
48	#95×C₅₉	—	2	1	no	—
49	#95×C₅₉	—	2	1	no	—
50	#95×C₅₉	—	2	1	no	—
51	C₂×Q₈×C₅₉	—	2	1	no	—
52	C₂×Q₈×C₅₉	—	4	1	no	—
53	C₂×Q₈×C₅₉	—	4	1	no	—
54	C₂×Q₈×C₅₉	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₅₉	—	2	1	no	—
56	C₂²×C₄×C₅₉	—	2	1	no	—
57	C₂²×C₄×C₅₉	—	2	1	no	—
58	C₂²×C₄×C₅₉	—	2	1	no	—
59	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₅₉	—	2	4	no	—
67	C₂×C₈×C₅₉	—	4	2	no	—
68	C₂×C₈×C₅₉	—	4	2	no	—
69	C₂×C₈×C₅₉	—	4	2	no	—
70	C₂×C₈×C₅₉	—	4	2	no	—
71	C₂×C₈×C₅₉	—	4	2	no	—
72	C₄⋊C₄×C₅₉	—	2	4	no	—
73	C₄⋊C₄×C₅₉	—	2	1	no	—
74	C₄⋊C₄×C₅₉	—	4	1	no	—
75	C₄⋊C₄×C₅₉	—	4	1	no	—
76	C₄⋊C₄×C₅₉	—	4	2	no	—
77	C₄⋊C₄×C₅₉	—	4	2	no	—
78	C₄⋊C₄×C₅₉	C₂×C₈	1	2	yes	—
79	C₄²×C₅₉	—	2	4	no	—
80	C₄²×C₅₉	—	2	2	no	—
81	C₄²×C₅₉	—	2	2	no	—
82	C₄²×C₅₉	—	2	2	no	—
83	C₄²×C₅₉	—	4	1	no	—
84	C₄²×C₅₉	—	4	1	no	—
85	C₄²×C₅₉	—	4	1	no	—
86	C₄²×C₅₉	—	4	2	no	—
87	C₄²×C₅₉	C₂×C₈	1	2	yes	—
88	C₄²×C₅₉	M₁₆	1	2	yes	—
89	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	Q₈×C₅₉	—	4	2	no	—
91	Q₈×C₅₉	—	4	2	no	—
92	C₂×C₄×C₅₉	—	2	1	no	—
93	C₂×C₄×C₅₉	—	2	1	no	—
94	C₂×C₄×C₅₉	—	2	1	no	—
95	C₂×C₄×C₅₉	—	2	1	no	—
96	C₂×C₄×C₅₉	—	2	1	no	—
97	C₂×C₄×C₅₉	—	2	1	no	—
98	C₂×C₄×C₅₉	—	2	1	no	—
99	C₂×C₄×C₅₉	—	2	1	no	—
100	C₂×C₄×C₅₉	—	2	1	no	—
101	C₂×C₄×C₅₉	—	2	1	no	—
102	C₂×C₄×C₅₉	—	2	1	no	—
103	C₂×C₄×C₅₉	—	2	1	no	—
104	C₂×C₄×C₅₉	—	2	2	no	—
105	C₂×C₄×C₅₉	—	2	2	no	—
106	C₂×C₄×C₅₉	—	2	1	no	—
107	C₂×C₄×C₅₉	—	2	1	no	—
108	C₂×C₄×C₅₉	—	2	1	no	—
109	C₂×C₄×C₅₉	—	4	1	no	—
110	C₂×C₄×C₅₉	—	4	1	no	—
111	C₂×C₄×C₅₉	—	4	1	no	—
112	C₂×C₄×C₅₉	—	4	1	no	—
113	C₂×C₄×C₅₉	—	4	1	no	—
114	C₂×C₄×C₅₉	—	4	1	no	—
115	C₂×C₄×C₅₉	—	4	1	no	—
116	C₂×C₄×C₅₉	—	4	1	no	—
117	C₂×C₄×C₅₉	—	4	1	no	—
118	C₂×C₄×C₅₉	—	4	1	no	—
119	C₂×C₄×C₅₉	C₂²⋊C₈	1	2	yes	—
120	C₂×C₄×C₅₉	C₂²⋊C₈	1	2	yes	—
121	C₂×C₄×C₅₉	#104	1	2	yes	—
122	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
124	C₂×C₄×C₅₉	#135	1	2	yes	—
125	C₂×C₄×C₅₉	#137	1	2	yes	—
126	C₈×C₅₉	—	4	4	no	—
127	C₈×C₅₉	—	8	2	no	—
128	#7581	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂²×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
130	C₂²×C₅₉	#410	1	1	yes	characteristic
131	C₂²×C₅₉	#413	1	1	yes	characteristic
132	C₂²×C₅₉	#416	1	1	yes	characteristic
133	C₂²×C₅₉	#419	1	1	yes	characteristic
134	C₂²×C₅₉	C₂×#104	1	1	yes	characteristic
135	C₂²×C₅₉	#421	1	1	yes	characteristic
136	C₄×C₅₉	—	2	1	no	—
137	C₄×C₅₉	—	2	1	no	—
138	C₄×C₅₉	—	2	2	no	—
139	C₄×C₅₉	—	2	2	no	—
140	C₄×C₅₉	—	2	2	no	—
141	C₄×C₅₉	—	2	2	no	—
142	C₄×C₅₉	—	2	2	no	—
143	C₄×C₅₉	—	4	1	no	—
144	C₄×C₅₉	—	4	2	no	—
145	C₄×C₅₉	—	4	2	no	—
146	C₄×C₅₉	—	8	1	no	—
147	C₄×C₅₉	—	8	1	no	—
148	#2295	C₂×C₅₉	1	1	yes	characteristic
149	#1839	C₂×C₅₉	1	1	yes	characteristic
150	#1744	C₂×C₅₉	1	1	yes	characteristic
151	#1272	C₂×C₅₉	1	2	yes	—
152	#1271	C₂×C₅₉	1	2	yes	—
153	C₂×C₅₉	#1458	1	1	yes	characteristic
154	C₂×C₅₉	#1476	1	1	yes	characteristic
155	C₂×C₅₉	#1479	1	1	yes	characteristic
156	C₂×C₅₉	#1506	1	1	yes	characteristic
157	C₂×C₅₉	#1521	1	1	yes	characteristic
158	C₂×C₅₉	#1532	1	1	yes	characteristic
159	C₂×C₅₉	#1554	1	1	yes	characteristic
160	C₂×C₄×Q₈	C₄×C₅₉	1	1	yes	characteristic
161	C₂×C₄×C₈	—	2	1	no	—
162	#391	C₄×C₅₉	1	1	yes	characteristic
163	#391	C₂²×C₅₉	1	1	yes	characteristic
164	#389	C₄×C₅₉	1	1	yes	characteristic
165	#389	C₂²×C₅₉	1	1	yes	characteristic
166	#387	—	2	1	no	—
167	#387	—	2	1	no	—
168	C₄×C₄⋊C₄	—	2	1	no	—
169	C₄×C₄⋊C₄	C₄×C₅₉	1	1	yes	characteristic
170	#376	—	2	1	no	—
171	C₄³	C₂²×C₅₉	1	1	yes	characteristic
172	#336	—	2	1	no	—
173	#336	—	2	1	no	—
174	#336	—	2	1	no	—
175	#336	C₂²×C₅₉	1	2	yes	—
176	#336	C₂²×C₅₉	1	2	yes	—
177	C₅₉	#7581	1	1	yes	characteristic, direct product factor, Sylow
178	C₂²×Q₈	C₂×C₄×C₅₉	1	1	yes	characteristic
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₄×Q₈	—	2	2	no	—
182	C₄×Q₈	C₈×C₅₉	1	4	yes	complemented
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	—	2	1	no	—
186	C₂×C₄⋊C₄	C₂×C₄×C₅₉	1	1	yes	characteristic
187	C₂×C₄⋊C₄	C₂×C₄×C₅₉	1	1	yes	characteristic
188	C₂×C₄⋊C₄	C₂×C₄×C₅₉	1	1	yes	characteristic
189	C₂×C₄²	—	2	1	no	—
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	C₂×C₄×C₅₉	1	1	yes	characteristic
192	C₂×C₄²	C₂×C₄×C₅₉	1	1	yes	characteristic
193	C₂×C₄²	D₄×C₅₉	1	1	yes	characteristic
194	C₂×C₄²	D₄×C₅₉	1	1	yes	characteristic
195	C₂×C₄²	D₄×C₅₉	1	1	yes	characteristic
196	C₂×C₄²	D₄×C₅₉	1	1	yes	characteristic
197	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
198	C₄×C₈	—	2	4	no	—
199	#95	—	2	1	no	—
200	#95	—	2	1	no	—
201	#95	—	2	1	no	—
202	#95	—	2	1	no	—
203	#95	—	2	1	no	—
204	#95	—	2	1	no	—
205	#95	—	2	1	no	—
206	C₂×Q₈	—	2	1	no	—
207	C₂×Q₈	—	4	1	no	—
208	C₂×Q₈	—	4	1	no	—
209	C₂×Q₈	C₂×C₈×C₅₉	1	2	yes	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
215	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
216	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
217	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
218	C₂²×C₄	C₂²×C₄×C₅₉	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	4	1	no	—
230	C₄⋊C₄	—	4	1	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	C₂×C₈×C₅₉	1	2	yes	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	4	1	no	—
239	C₄²	—	4	1	no	—
240	C₄²	—	4	1	no	—
241	C₄²	—	4	2	no	—
242	C₄²	C₂×C₈×C₅₉	1	2	yes	—
243	C₄²	M₁₆×C₅₉	1	2	yes	—
244	C₂³	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
245	Q₈	—	4	2	no	—
246	Q₈	—	4	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	C₂²⋊C₈×C₅₉	1	2	yes	—
275	C₂×C₄	C₂²⋊C₈×C₅₉	1	2	yes	—
276	C₂×C₄	#104×C₅₉	1	2	yes	—
277	C₂×C₄	C₂²×C₈×C₅₉	1	1	yes	characteristic
278	C₂×C₄	C₂×M₁₆×C₅₉	1	1	yes	characteristic
279	C₂×C₄	#135×C₅₉	1	2	yes	—
280	C₂×C₄	#137×C₅₉	1	2	yes	—
281	C₈	—	4	4	no	—
282	C₈	—	8	2	no	—
283	C₂²	C₂×C₂²⋊C₈×C₅₉	1	1	yes	characteristic
284	C₂²	C₅₉×#410	1	1	yes	characteristic
285	C₂²	C₅₉×#413	1	1	yes	characteristic
286	C₂²	C₅₉×#416	1	1	yes	characteristic
287	C₂²	C₅₉×#419	1	1	yes	characteristic
288	C₂²	C₂×#104×C₅₉	1	1	yes	characteristic
289	C₂²	C₅₉×#421	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	4	2	no	—
298	C₄	—	4	2	no	—
299	C₄	—	4	1	no	—
300	C₄	—	8	1	no	—
301	C₄	—	8	1	no	—
302	C₂	C₅₉×#1458	1	1	yes	characteristic
303	C₂	C₅₉×#1476	1	1	yes	characteristic
304	C₂	C₅₉×#1479	1	1	yes	characteristic
305	C₂	C₅₉×#1506	1	1	yes	characteristic
306	C₂	C₅₉×#1521	1	1	yes	characteristic
307	C₂	C₅₉×#1532	1	1	yes	characteristic
308	C₂	C₅₉×#1554	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₉	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₅₉
Frattini, cosocle	C₂×C₄²	C₂³×C₅₉
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₉ ◁ C₄³×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
27. C₄×Q₈×C₅₉	C₈	semidirect
128. #7581	C₅₉	direct (N × H)
177. C₅₉	#7581	direct (N × H)
182. C₄×Q₈	C₈×C₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₅₉×#2295	2
C₅₉×#1839	2
C₅₉×#1744	2
C₅₉×#1272	2
C₅₉×#1271	2
#7581	59

Subgroups — C59×#7581 #28419690

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#7581 #28419690