Subgroups — C₅₉×#33797 #28445906

Order 15104.

460 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#3170	C₂	1	1	yes	maximal, characteristic
2	C₅₉×#3160	C₂	1	1	yes	maximal, characteristic
3	C₅₉×#2254	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₉×#2252	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#2241	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₉×#2241	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#2237	C₂	1	4	yes	maximal
8	C₅₉×#2229	C₂	1	4	yes	maximal, complemented
9	C₅₉×#2162	C₂	1	1	yes	maximal, characteristic
10	C₅₉×#533	—	2	1	no	—
11	C₅₉×#533	—	2	1	no	—
12	C₅₉×#500	C₂²	1	1	yes	characteristic
13	C₅₉×#500	C₂²	1	1	yes	characteristic
14	C₅₉×#500	C₂²	1	1	yes	characteristic
15	C₅₉×#499	—	2	1	no	—
16	C₅₉×#499	C₂²	1	2	yes	—
17	C₅₉×#498	—	2	1	no	—
18	C₅₉×#494	C₂²	1	1	yes	characteristic
19	C₅₉×#487	—	2	1	no	—
20	C₅₉×#467	—	2	1	no	—
21	C₅₉×#467	C₂²	1	2	yes	—
22	C₄×Q₁₆×C₅₉	—	2	1	no	—
23	C₄×Q₁₆×C₅₉	C₂²	1	2	yes	—
24	C₅₉×#427	—	2	1	no	—
25	C₅₉×#427	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
27	C₅₉×#367	C₂²	1	4	yes	—
28	C₅₉×#366	C₂²	1	4	yes	—
29	C₅₉×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₅₉×#358	C₂²	1	4	yes	—
31	C₅₉×#358	C₂²	1	4	yes	—
32	C₅₉×C₂²⋊C₁₆	C₂²	1	2	yes	—
33	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
34	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
35	C₂×Q₁₆×C₅₉	—	2	1	no	—
36	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
37	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
39	C₄⋊Q₈×C₅₉	—	2	1	no	—
40	C₄⋊Q₈×C₅₉	—	2	1	no	—
41	C₄⋊Q₈×C₅₉	—	2	1	no	—
42	C₄⋊Q₈×C₅₉	—	2	1	no	—
43	#125×C₅₉	—	2	2	no	—
44	#125×C₅₉	—	4	1	no	—
45	C₂²⋊Q₈×C₅₉	—	2	2	no	—
46	C₂²⋊Q₈×C₅₉	—	4	1	no	—
47	C₂²⋊Q₈×C₅₉	—	4	1	no	—
48	C₂²⋊Q₈×C₅₉	—	4	1	no	—
49	C₄×Q₈×C₅₉	—	2	2	no	—
50	C₄×Q₈×C₅₉	—	4	1	no	—
51	C₄×Q₈×C₅₉	—	4	1	no	—
52	C₄×Q₈×C₅₉	—	4	1	no	—
53	#117×C₅₉	—	2	2	no	—
54	#117×C₅₉	—	4	1	no	—
55	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
56	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
57	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
58	#107×C₅₉	—	2	1	no	—
59	#107×C₅₉	—	2	1	no	—
60	#107×C₅₉	—	2	1	no	—
61	#107×C₅₉	C₂³	1	2	yes	—
62	#107×C₅₉	C₂³	1	2	yes	—
63	#107×C₅₉	C₂³	1	2	yes	—
64	#106×C₅₉	—	2	2	no	—
65	#106×C₅₉	—	2	2	no	—
66	#103×C₅₉	—	2	2	no	—
67	#103×C₅₉	—	2	2	no	—
68	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
69	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
70	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
72	C₂×Q₈×C₅₉	—	2	2	no	—
73	C₂×Q₈×C₅₉	—	4	1	no	—
74	C₂×Q₈×C₅₉	—	4	1	no	—
75	C₂×Q₈×C₅₉	—	4	1	no	—
76	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
79	Q₁₆×C₅₉	—	2	4	no	—
80	Q₁₆×C₅₉	—	4	2	no	—
81	C₂×C₈×C₅₉	—	2	2	no	—
82	C₂×C₈×C₅₉	—	2	2	no	—
83	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
87	C₄⋊C₄×C₅₉	—	2	2	no	—
88	C₄⋊C₄×C₅₉	—	2	2	no	—
89	C₄⋊C₄×C₅₉	—	2	2	no	—
90	C₄⋊C₄×C₅₉	—	4	1	no	—
91	C₄⋊C₄×C₅₉	—	4	1	no	—
92	C₄⋊C₄×C₅₉	—	4	2	no	—
93	C₄⋊C₄×C₅₉	—	4	2	no	—
94	C₄⋊C₄×C₅₉	—	4	2	no	—
95	C₄⋊C₄×C₅₉	—	4	2	no	—
96	C₄⋊C₄×C₅₉	—	4	1	no	—
97	C₄⋊C₄×C₅₉	—	4	1	no	—
98	C₄⋊C₄×C₅₉	—	4	1	no	—
99	C₂²⋊C₄×C₅₉	—	4	2	no	—
100	C₂²⋊C₄×C₅₉	—	4	2	no	—
101	C₄²×C₅₉	—	4	2	no	—
102	C₄²×C₅₉	—	4	2	no	—
103	C₄²×C₅₉	D₈	1	1	yes	characteristic
104	C₄²×C₅₉	D₈	1	1	yes	characteristic
105	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
107	C₁₆×C₅₉	—	2	2	no	—
108	C₁₆×C₅₉	—	2	2	no	—
109	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
110	Q₈×C₅₉	—	4	2	no	—
111	Q₈×C₅₉	—	4	2	no	—
112	Q₈×C₅₉	—	8	2	no	—
113	Q₈×C₅₉	—	8	2	no	—
114	Q₈×C₅₉	—	8	2	no	—
115	C₂×C₄×C₅₉	—	2	1	no	—
116	C₂×C₄×C₅₉	—	2	1	no	—
117	C₂×C₄×C₅₉	—	2	2	no	—
118	C₂×C₄×C₅₉	—	2	2	no	—
119	C₂×C₄×C₅₉	—	4	2	no	—
120	C₂×C₄×C₅₉	—	4	2	no	—
121	C₂×C₄×C₅₉	—	4	2	no	—
122	C₂×C₄×C₅₉	—	4	2	no	—
123	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
129	C₈×C₅₉	—	2	1	no	—
130	C₈×C₅₉	—	2	1	no	—
131	C₈×C₅₉	#137	1	2	yes	—
132	C₈×C₅₉	2₋¹⁺⁴	1	2	yes	—
133	#33797	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂²×C₅₉	—	2	1	no	—
135	C₂²×C₅₉	—	2	1	no	—
136	C₂²×C₅₉	—	2	1	no	—
137	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
138	C₄×C₅₉	—	2	1	no	—
139	C₄×C₅₉	—	2	1	no	—
140	C₄×C₅₉	—	2	1	no	—
141	C₄×C₅₉	—	8	2	no	—
142	C₄×C₅₉	—	8	2	no	—
143	C₄×C₅₉	—	8	2	no	—
144	C₄×C₅₉	—	8	2	no	—
145	C₄×C₅₉	#508	1	2	yes	—
146	C₄×C₅₉	#510	1	2	yes	—
147	C₄×C₅₉	#536	1	1	yes	characteristic
148	C₄×C₅₉	C₂×#137	1	1	yes	characteristic
149	#3170	C₂×C₅₉	1	1	yes	characteristic
150	#3160	C₂×C₅₉	1	1	yes	characteristic
151	#2254	C₂×C₅₉	1	1	yes	characteristic, complemented
152	#2252	C₂×C₅₉	1	1	yes	characteristic, complemented
153	#2241	C₂×C₅₉	1	1	yes	characteristic, complemented
154	#2241	C₂×C₅₉	1	1	yes	characteristic, complemented
155	#2237	C₂×C₅₉	1	4	yes	—
156	#2229	C₂×C₅₉	1	4	yes	complemented
157	#2162	C₂×C₅₉	1	1	yes	characteristic
158	C₂×C₅₉	—	4	1	no	—
159	C₂×C₅₉	#3226	1	1	yes	characteristic
160	C₂×C₅₉	C₂×#508	1	1	yes	characteristic
161	C₂×C₅₉	C₂×#510	1	1	yes	characteristic
162	#533	—	2	1	no	—
163	#533	—	2	1	no	—
164	#500	C₂²×C₅₉	1	1	yes	characteristic
165	#500	C₂²×C₅₉	1	1	yes	characteristic
166	#500	C₂²×C₅₉	1	1	yes	characteristic
167	#499	—	2	1	no	—
168	#499	C₂²×C₅₉	1	2	yes	—
169	#498	—	2	1	no	—
170	#494	C₂²×C₅₉	1	1	yes	characteristic
171	#487	—	2	1	no	—
172	#467	—	2	1	no	—
173	#467	C₂²×C₅₉	1	2	yes	—
174	C₄×Q₁₆	—	2	1	no	—
175	C₄×Q₁₆	C₂²×C₅₉	1	2	yes	—
176	#427	—	2	1	no	—
177	#427	C₂²×C₅₉	1	2	yes	—
178	C₂×C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic
179	#367	C₂²×C₅₉	1	4	yes	—
180	#366	C₂²×C₅₉	1	4	yes	—
181	C₄⋊C₁₆	C₂²×C₅₉	1	2	yes	—
182	#358	C₂²×C₅₉	1	4	yes	—
183	#358	C₂²×C₅₉	1	4	yes	—
184	C₂²⋊C₁₆	C₂²×C₅₉	1	2	yes	—
185	C₄×C₁₆	C₂²×C₅₉	1	1	yes	characteristic
186	C₄×C₁₆	C₂²×C₅₉	1	1	yes	characteristic
187	C₅₉	#33797	1	1	yes	characteristic, direct product factor, Sylow
188	C₂×Q₁₆	—	2	1	no	—
189	C₂×Q₁₆	C₂³×C₅₉	1	2	yes	—
190	C₂²×C₈	D₄×C₅₉	1	1	yes	characteristic
191	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
192	C₄⋊Q₈	—	2	1	no	—
193	C₄⋊Q₈	—	2	1	no	—
194	C₄⋊Q₈	—	2	1	no	—
195	C₄⋊Q₈	—	2	1	no	—
196	#125	—	2	2	no	—
197	#125	—	4	1	no	—
198	C₂²⋊Q₈	—	2	2	no	—
199	C₂²⋊Q₈	—	4	1	no	—
200	C₂²⋊Q₈	—	4	1	no	—
201	C₂²⋊Q₈	—	4	1	no	—
202	C₄×Q₈	—	2	2	no	—
203	C₄×Q₈	—	4	1	no	—
204	C₄×Q₈	—	4	1	no	—
205	C₄×Q₈	—	4	1	no	—
206	#117	—	2	2	no	—
207	#117	—	4	1	no	—
208	C₂×C₄²	D₄×C₅₉	1	1	yes	characteristic
209	C₂×C₁₆	C₂³×C₅₉	1	2	yes	—
210	C₂×C₁₆	C₂³×C₅₉	1	2	yes	—
211	#107	—	2	1	no	—
212	#107	—	2	1	no	—
213	#107	—	2	1	no	—
214	#107	C₂³×C₅₉	1	2	yes	—
215	#107	C₂³×C₅₉	1	2	yes	—
216	#107	C₂³×C₅₉	1	2	yes	—
217	#106	—	2	2	no	—
218	#106	—	2	2	no	—
219	#103	—	2	2	no	—
220	#103	—	2	2	no	—
221	C₄×C₈	D₄×C₅₉	1	1	yes	characteristic
222	C₄×C₈	D₄×C₅₉	1	1	yes	characteristic
223	C₄×C₈	C₂³×C₅₉	1	1	yes	characteristic
224	C₄×C₈	C₂³×C₅₉	1	1	yes	characteristic
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	4	1	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	—	4	1	no	—
229	C₂²×C₄	D₈×C₅₉	1	1	yes	characteristic
230	C₂²×C₄	D₈×C₅₉	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
232	Q₁₆	—	2	4	no	—
233	Q₁₆	—	4	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
237	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
238	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
239	C₂×C₈	C₂⁴×C₅₉	1	1	yes	characteristic
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	4	1	no	—
244	C₄⋊C₄	—	4	1	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	1	no	—
250	C₄⋊C₄	—	4	1	no	—
251	C₄⋊C₄	—	4	1	no	—
252	C₂²⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	4	2	no	—
254	C₄²	—	4	2	no	—
255	C₄²	—	4	2	no	—
256	C₄²	D₈×C₅₉	1	1	yes	characteristic
257	C₄²	D₈×C₅₉	1	1	yes	characteristic
258	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
259	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
260	C₁₆	—	2	2	no	—
261	C₁₆	—	2	2	no	—
262	C₂³	C₂×D₈×C₅₉	1	1	yes	characteristic
263	Q₈	—	4	2	no	—
264	Q₈	—	4	2	no	—
265	Q₈	—	8	2	no	—
266	Q₈	—	8	2	no	—
267	Q₈	—	8	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
277	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
278	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
279	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
280	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	#137×C₅₉	1	2	yes	—
285	C₈	2₋¹⁺⁴×C₅₉	1	2	yes	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	C₂²×D₈×C₅₉	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	8	2	no	—
294	C₄	—	8	2	no	—
295	C₄	—	8	2	no	—
296	C₄	—	8	2	no	—
297	C₄	C₅₉×#508	1	2	yes	—
298	C₄	C₅₉×#510	1	2	yes	—
299	C₄	C₅₉×#536	1	1	yes	characteristic
300	C₄	C₂×#137×C₅₉	1	1	yes	characteristic
301	C₂	—	4	1	no	—
302	C₂	C₅₉×#3226	1	1	yes	characteristic
303	C₂	C₂×C₅₉×#508	1	1	yes	characteristic
304	C₂	C₂×C₅₉×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₉
prime radical	C₂³×C₅₉	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₂×C₄²×C₅₉ ◁ C₂×C₄×C₈×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₉×#2254	C₂	semidirect
4. C₅₉×#2252	C₂	semidirect
5. C₅₉×#2241	C₂	semidirect
6. C₅₉×#2241	C₂	semidirect
8. C₅₉×#2229	C₂	semidirect
133. #33797	C₅₉	direct (N × H)
151. #2254	C₂×C₅₉	semidirect
152. #2252	C₂×C₅₉	semidirect
153. #2241	C₂×C₅₉	semidirect
154. #2241	C₂×C₅₉	semidirect
156. #2229	C₂×C₅₉	semidirect
187. C₅₉	#33797	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×#3170	2
C₅₉×#3160	2
C₅₉×#2254	2
C₅₉×#2252	2
C₅₉×#2241	2
C₅₉×#2237	2
C₅₉×#2229	2
C₅₉×#2162	2
#33797	59

Subgroups — C59×#33797 #28445906

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#33797 #28445906