Subgroups — C₅₉×#9858 #28421967

Order 15104.

428 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#3131	C₂	1	1	yes	maximal, characteristic
2	C₅₉×#1848	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₉×#1493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₉×#1493	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#1492	C₂	1	1	yes	maximal, characteristic
6	C₅₉×#1492	C₂	1	1	yes	maximal, characteristic
7	C₂×C₅₉×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₉×#533	C₂²	1	1	yes	characteristic
9	C₅₉×#532	C₂²	1	1	yes	characteristic
10	C₅₉×#488	—	2	2	no	—
11	C₅₉×#487	C₄	1	1	yes	characteristic
12	C₅₉×#487	C₄	1	1	yes	characteristic
13	C₅₉×#486	C₄	1	1	yes	characteristic
14	C₅₉×#486	C₄	1	1	yes	characteristic
15	C₂×#108×C₅₉	—	2	2	no	—
16	C₅₉×#423	C₂²	1	1	yes	characteristic
17	C₅₉×#423	C₂²	1	1	yes	characteristic
18	C₅₉×#416	—	2	2	no	—
19	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₅₉×Q₈⋊C₈	—	2	2	no	—
23	C₅₉×Q₈⋊C₈	—	2	2	no	—
24	C₅₉×D₄⋊C₈	—	2	2	no	—
25	C₅₉×D₄⋊C₈	—	2	2	no	—
26	C₅₉×#322	—	2	2	no	—
27	C₂×#41×C₅₉	—	2	1	no	—
28	C₂×M₁₆×C₅₉	—	2	1	no	—
29	C₂×M₁₆×C₅₉	—	2	1	no	—
30	C₂²×C₈×C₅₉	—	2	1	no	—
31	C₂²×C₈×C₅₉	—	2	1	no	—
32	C₂²×C₈×C₅₉	D₄	1	2	yes	—
33	C₄⋊Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
34	C₄⋊Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
35	C₄⋊Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
36	#127×C₅₉	C₂×C₄	1	1	yes	characteristic
37	#125×C₅₉	—	2	1	no	—
38	#124×C₅₉	—	2	1	no	—
39	C₂²⋊Q₈×C₅₉	—	2	1	no	—
40	C₂²⋊Q₈×C₅₉	—	2	1	no	—
41	#121×C₅₉	—	2	1	no	—
42	#121×C₅₉	—	2	1	no	—
43	C₄×Q₈×C₅₉	—	2	1	no	—
44	C₄×Q₈×C₅₉	—	2	1	no	—
45	C₄×D₄×C₅₉	—	2	1	no	—
46	C₄×D₄×C₅₉	—	2	1	no	—
47	#117×C₅₉	—	2	1	no	—
48	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
49	#108×C₅₉	—	4	4	no	—
50	C₄⋊C₈×C₅₉	—	2	1	no	—
51	C₄⋊C₈×C₅₉	—	2	1	no	—
52	C₄⋊C₈×C₅₉	—	2	1	no	—
53	C₄⋊C₈×C₅₉	—	2	1	no	—
54	#103×C₅₉	—	2	2	no	—
55	#103×C₅₉	—	2	2	no	—
56	#102×C₅₉	—	2	2	no	—
57	#102×C₅₉	—	2	2	no	—
58	C₄×C₈×C₅₉	—	2	1	no	—
59	C₄×C₈×C₅₉	—	2	1	no	—
60	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₅₉	D₄	1	2	yes	—
63	C₄×C₈×C₅₉	D₄	1	2	yes	—
64	C₄×C₈×C₅₉	D₄	1	2	yes	—
65	#41×C₅₉	—	4	2	no	—
66	#41×C₅₉	—	4	2	no	—
67	C₂×Q₈×C₅₉	—	2	1	no	—
68	C₂×Q₈×C₅₉	—	2	1	no	—
69	C₂×Q₈×C₅₉	—	2	1	no	—
70	C₂×D₄×C₅₉	—	2	1	no	—
71	C₂×D₄×C₅₉	—	2	1	no	—
72	C₂×D₄×C₅₉	—	2	1	no	—
73	C₂×D₄×C₅₉	—	4	1	no	—
74	C₂²×C₄×C₅₉	—	4	1	no	—
75	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
77	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
78	M₁₆×C₅₉	—	4	2	no	—
79	M₁₆×C₅₉	—	4	2	no	—
80	C₂×C₈×C₅₉	—	2	4	no	—
81	C₂×C₈×C₅₉	—	2	2	no	—
82	C₂×C₈×C₅₉	—	2	2	no	—
83	C₂×C₈×C₅₉	—	2	2	no	—
84	C₂×C₈×C₅₉	—	2	2	no	—
85	C₂×C₈×C₅₉	—	4	2	no	—
86	C₂×C₈×C₅₉	—	4	1	no	—
87	C₂×C₈×C₅₉	—	4	1	no	—
88	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
89	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
90	C₄⋊C₄×C₅₉	—	2	1	no	—
91	C₄⋊C₄×C₅₉	—	2	1	no	—
92	C₄⋊C₄×C₅₉	—	2	1	no	—
93	C₄⋊C₄×C₅₉	—	2	1	no	—
94	C₄⋊C₄×C₅₉	—	2	1	no	—
95	C₄⋊C₄×C₅₉	—	2	1	no	—
96	C₄⋊C₄×C₅₉	—	4	1	no	—
97	C₄⋊C₄×C₅₉	—	4	1	no	—
98	C₂²⋊C₄×C₅₉	—	4	1	no	—
99	C₂²⋊C₄×C₅₉	—	4	1	no	—
100	C₂²⋊C₄×C₅₉	—	4	1	no	—
101	C₄²×C₅₉	—	4	1	no	—
102	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
106	C₂³×C₅₉	—	4	1	no	—
107	C₂³×C₅₉	#124	1	1	yes	characteristic
108	Q₈×C₅₉	—	4	2	no	—
109	Q₈×C₅₉	—	4	2	no	—
110	Q₈×C₅₉	—	4	2	no	—
111	D₄×C₅₉	—	4	2	no	—
112	D₄×C₅₉	—	4	2	no	—
113	D₄×C₅₉	—	4	2	no	—
114	D₄×C₅₉	—	8	2	no	—
115	C₂×C₄×C₅₉	—	2	2	no	—
116	C₂×C₄×C₅₉	—	2	2	no	—
117	C₂×C₄×C₅₉	—	4	1	no	—
118	C₂×C₄×C₅₉	—	4	1	no	—
119	C₂×C₄×C₅₉	—	4	1	no	—
120	C₂×C₄×C₅₉	—	8	1	no	—
121	C₂×C₄×C₅₉	—	8	1	no	—
122	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
126	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
127	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
128	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
129	C₂×C₄×C₅₉	#137	1	2	yes	—
130	C₈×C₅₉	—	2	4	no	—
131	C₈×C₅₉	—	2	2	no	—
132	C₈×C₅₉	—	2	2	no	—
133	C₈×C₅₉	—	4	2	no	—
134	C₈×C₅₉	—	8	1	no	—
135	C₈×C₅₉	—	8	1	no	—
136	#9858	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂²×C₅₉	—	2	1	no	—
138	C₂²×C₅₉	—	2	1	no	—
139	C₂²×C₅₉	—	8	1	no	—
140	C₂²×C₅₉	—	8	1	no	—
141	C₂²×C₅₉	—	8	1	no	—
142	C₂²×C₅₉	#390	1	1	yes	characteristic
143	C₂²×C₅₉	#489	1	2	yes	—
144	C₄×C₅₉	—	2	1	no	—
145	C₄×C₅₉	—	2	1	no	—
146	C₄×C₅₉	—	8	1	no	—
147	C₄×C₅₉	—	8	1	no	—
148	C₄×C₅₉	—	8	1	no	—
149	C₄×C₅₉	#417	1	2	yes	—
150	C₄×C₅₉	#443	1	2	yes	—
151	C₄×C₅₉	#444	1	2	yes	—
152	C₄×C₅₉	#496	1	1	yes	characteristic
153	C₄×C₅₉	#497	1	1	yes	characteristic
154	#3131	C₂×C₅₉	1	1	yes	characteristic
155	#1848	C₂×C₅₉	1	1	yes	characteristic, complemented
156	#1493	C₂×C₅₉	1	1	yes	characteristic, complemented
157	#1493	C₂×C₅₉	1	1	yes	characteristic, complemented
158	#1492	C₂×C₅₉	1	1	yes	characteristic
159	#1492	C₂×C₅₉	1	1	yes	characteristic
160	C₂×#322	C₂×C₅₉	1	1	yes	characteristic, complemented
161	C₂×C₅₉	—	2	2	no	—
162	C₂×C₅₉	—	16	1	no	—
163	C₂×C₅₉	#1959	1	1	yes	characteristic
164	C₂×C₅₉	#1974	1	1	yes	characteristic
165	C₂×C₅₉	#1975	1	1	yes	characteristic
166	#533	C₂²×C₅₉	1	1	yes	characteristic
167	#532	C₂²×C₅₉	1	1	yes	characteristic
168	#488	—	2	2	no	—
169	#487	C₄×C₅₉	1	1	yes	characteristic
170	#487	C₄×C₅₉	1	1	yes	characteristic
171	#486	C₄×C₅₉	1	1	yes	characteristic
172	#486	C₄×C₅₉	1	1	yes	characteristic
173	C₂×#108	—	2	2	no	—
174	#423	C₂²×C₅₉	1	1	yes	characteristic
175	#423	C₂²×C₅₉	1	1	yes	characteristic
176	#416	—	2	2	no	—
177	C₂×C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic
178	C₂×C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic
179	C₂×C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic
180	Q₈⋊C₈	—	2	2	no	—
181	Q₈⋊C₈	—	2	2	no	—
182	D₄⋊C₈	—	2	2	no	—
183	D₄⋊C₈	—	2	2	no	—
184	#322	—	2	2	no	—
185	C₅₉	#9858	1	1	yes	characteristic, direct product factor, Sylow
186	C₂×#41	—	2	1	no	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	—	2	1	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	D₄×C₅₉	1	2	yes	—
192	C₄⋊Q₈	C₂×C₄×C₅₉	1	1	yes	characteristic
193	C₄⋊Q₈	C₂×C₄×C₅₉	1	1	yes	characteristic
194	C₄⋊Q₈	C₂×C₄×C₅₉	1	1	yes	characteristic
195	#127	C₂×C₄×C₅₉	1	1	yes	characteristic
196	#125	—	2	1	no	—
197	#124	—	2	1	no	—
198	C₂²⋊Q₈	—	2	1	no	—
199	C₂²⋊Q₈	—	2	1	no	—
200	#121	—	2	1	no	—
201	#121	—	2	1	no	—
202	C₄×Q₈	—	2	1	no	—
203	C₄×Q₈	—	2	1	no	—
204	C₄×D₄	—	2	1	no	—
205	C₄×D₄	—	2	1	no	—
206	#117	—	2	1	no	—
207	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
208	#108	—	4	4	no	—
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	2	1	no	—
211	C₄⋊C₈	—	2	1	no	—
212	C₄⋊C₈	—	2	1	no	—
213	#103	—	2	2	no	—
214	#103	—	2	2	no	—
215	#102	—	2	2	no	—
216	#102	—	2	2	no	—
217	C₄×C₈	—	2	1	no	—
218	C₄×C₈	—	2	1	no	—
219	C₄×C₈	C₂×C₄×C₅₉	1	1	yes	characteristic
220	C₄×C₈	C₂×C₄×C₅₉	1	1	yes	characteristic
221	C₄×C₈	D₄×C₅₉	1	2	yes	—
222	C₄×C₈	D₄×C₅₉	1	2	yes	—
223	C₄×C₈	D₄×C₅₉	1	2	yes	—
224	#41	—	4	2	no	—
225	#41	—	4	2	no	—
226	C₂×Q₈	—	2	1	no	—
227	C₂×Q₈	—	2	1	no	—
228	C₂×Q₈	—	2	1	no	—
229	C₂×D₄	—	2	1	no	—
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	—	2	1	no	—
232	C₂×D₄	—	4	1	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
235	C₂²×C₄	#41×C₅₉	1	1	yes	characteristic
236	C₂²×C₄	#41×C₅₉	1	1	yes	characteristic
237	M₁₆	—	4	2	no	—
238	M₁₆	—	4	2	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	C₂²⋊C₄×C₅₉	1	2	yes	—
248	C₂×C₈	C₂²⋊C₄×C₅₉	1	2	yes	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₂²⋊C₄	—	4	1	no	—
258	C₂²⋊C₄	—	4	1	no	—
259	C₂²⋊C₄	—	4	1	no	—
260	C₄²	—	4	1	no	—
261	C₄²	C₂²×C₄×C₅₉	1	1	yes	characteristic
262	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
263	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
264	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
265	C₂³	—	4	1	no	—
266	C₂³	#124×C₅₉	1	1	yes	characteristic
267	Q₈	—	4	2	no	—
268	Q₈	—	4	2	no	—
269	Q₈	—	4	2	no	—
270	D₄	—	4	2	no	—
271	D₄	—	4	2	no	—
272	D₄	—	4	2	no	—
273	D₄	—	8	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	8	1	no	—
280	C₂×C₄	—	8	1	no	—
281	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
282	C₂×C₄	C₄×D₄×C₅₉	1	1	yes	characteristic
283	C₂×C₄	C₄×D₄×C₅₉	1	1	yes	characteristic
284	C₂×C₄	#121×C₅₉	1	1	yes	characteristic
285	C₂×C₄	#121×C₅₉	1	1	yes	characteristic
286	C₂×C₄	#127×C₅₉	1	1	yes	characteristic
287	C₂×C₄	AGL(1,ℤ/8)×C₅₉	1	2	yes	—
288	C₂×C₄	#137×C₅₉	1	2	yes	—
289	C₈	—	2	4	no	—
290	C₈	—	2	2	no	—
291	C₈	—	2	2	no	—
292	C₈	—	4	2	no	—
293	C₈	—	8	1	no	—
294	C₈	—	8	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	8	1	no	—
298	C₂²	—	8	1	no	—
299	C₂²	—	8	1	no	—
300	C₂²	C₅₉×#390	1	1	yes	characteristic
301	C₂²	C₅₉×#489	1	2	yes	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	8	1	no	—
305	C₄	—	8	1	no	—
306	C₄	—	8	1	no	—
307	C₄	C₅₉×#417	1	2	yes	—
308	C₄	C₅₉×#443	1	2	yes	—
309	C₄	C₅₉×#444	1	2	yes	—
310	C₄	C₅₉×#496	1	1	yes	characteristic
311	C₄	C₅₉×#497	1	1	yes	characteristic
312	C₂	—	2	2	no	—
313	C₂	—	16	1	no	—
314	C₂	C₅₉×#1959	1	1	yes	characteristic
315	C₂	C₅₉×#1974	1	1	yes	characteristic
316	C₂	C₅₉×#1975	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₉	#127
commutator	C₄²	C₂²×C₄×C₅₉
Frattini, cosocle	C₂×C₄²	C₂³×C₅₉
socle	C₂²×C₅₉	#390
prime radical	C₅₉×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₉ ◁ C₂×C₄²×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₉×#1848	C₂	semidirect
3. C₅₉×#1493	C₂	semidirect
4. C₅₉×#1493	C₂	semidirect
7. C₂×C₅₉×#322	C₂	semidirect
136. #9858	C₅₉	direct (N × H)
155. #1848	C₂×C₅₉	semidirect
156. #1493	C₂×C₅₉	semidirect
157. #1493	C₂×C₅₉	semidirect
160. C₂×#322	C₂×C₅₉	semidirect
185. C₅₉	#9858	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×#3131	2
C₅₉×#1848	2
C₅₉×#1493	2
C₅₉×#1492	2
C₂×C₅₉×#322	2
#9858	59

Subgroups — C59×#9858 #28421967

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#9858 #28421967