Subgroups — C₅₉×#33903 #28446012

Order 15104.

470 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#3336	C₂	1	2	yes	maximal
2	C₅₉×#2253	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₉×#2237	C₂	1	2	yes	maximal, complemented
4	C₅₉×#2235	C₂	1	2	yes	maximal, complemented
5	C₅₉×#2229	C₂	1	2	yes	maximal, complemented
6	C₅₉×#2216	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#2216	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×Q₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
10	C₅₉×#2158	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅₉×#366	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₉×#552	—	2	2	no	—
13	C₂×Q₃₂×C₅₉	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×#499	C₂²	1	2	yes	—
17	C₅₉×#485	—	2	2	no	—
18	C₅₉×#483	—	2	2	no	—
19	C₅₉×#479	—	2	2	no	—
20	C₅₉×#467	C₂²	1	2	yes	—
21	C₅₉×#467	C₂²	1	2	yes	—
22	C₄×Q₁₆×C₅₉	C₂²	1	2	yes	—
23	C₈×D₄×C₅₉	C₂²	1	1	yes	characteristic
24	C₅₉×#427	C₂²	1	2	yes	—
25	C₂×#107×C₅₉	C₂²	1	2	yes	—
26	C₅₉×#367	C₂²	1	2	yes	—
27	C₅₉×#367	C₂²	1	2	yes	—
28	C₅₉×#366	C₂²	1	2	yes	—
29	C₅₉×#366	C₂²	1	1	yes	characteristic
30	C₅₉×#366	C₂²	1	1	yes	characteristic
31	C₅₉×#366	C₂²	1	1	yes	characteristic
32	C₅₉×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₅₉×#358	C₂²	1	2	yes	—
34	C₅₉×#358	C₂²	1	2	yes	—
35	C₅₉×#358	C₂²	1	2	yes	—
36	C₅₉×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
37	C₅₉×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
39	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
40	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
42	#126×C₅₉	—	4	2	no	—
43	#125×C₅₉	—	2	2	no	—
44	#125×C₅₉	—	4	2	no	—
45	#123×C₅₉	—	4	2	no	—
46	C₂²⋊Q₈×C₅₉	—	2	2	no	—
47	C₂²⋊Q₈×C₅₉	—	2	2	no	—
48	C₄×Q₈×C₅₉	—	2	2	no	—
49	C₄×D₄×C₅₉	D₄	1	1	yes	characteristic
50	#117×C₅₉	—	2	2	no	—
51	#117×C₅₉	—	4	2	no	—
52	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
53	Q₃₂×C₅₉	—	2	2	no	—
54	C₂×C₁₆×C₅₉	—	2	2	no	—
55	C₂×C₁₆×C₅₉	—	2	2	no	—
56	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
57	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
58	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
59	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
60	#107×C₅₉	—	2	2	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	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
67	#103×C₅₉	—	2	2	no	—
68	#103×C₅₉	—	2	2	no	—
69	#103×C₅₉	—	2	2	no	—
70	C₂²⋊C₈×C₅₉	D₄	1	1	yes	characteristic
71	C₂²⋊C₈×C₅₉	D₄	1	1	yes	characteristic
72	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
73	C₂×Q₈×C₅₉	—	2	2	no	—
74	C₂×D₄×C₅₉	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₅₉	—	4	2	no	—
76	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
78	Q₁₆×C₅₉	—	2	2	no	—
79	Q₁₆×C₅₉	—	2	2	no	—
80	C₂×C₈×C₅₉	—	2	1	no	—
81	C₂×C₈×C₅₉	—	2	1	no	—
82	C₂×C₈×C₅₉	—	2	1	no	—
83	C₂×C₈×C₅₉	—	2	1	no	—
84	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₅₉	—	2	2	no	—
89	C₄⋊C₄×C₅₉	—	2	2	no	—
90	C₄⋊C₄×C₅₉	—	2	2	no	—
91	C₄⋊C₄×C₅₉	—	4	2	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	2	no	—
97	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
98	C₂²⋊C₄×C₅₉	—	4	2	no	—
99	C₂²⋊C₄×C₅₉	—	4	2	no	—
100	C₂²⋊C₄×C₅₉	—	4	2	no	—
101	C₂²⋊C₄×C₅₉	D₈	1	1	yes	characteristic
102	C₂²⋊C₄×C₅₉	D₈	1	1	yes	characteristic
103	C₄²×C₅₉	—	4	2	no	—
104	C₄²×C₅₉	—	4	2	no	—
105	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
106	C₁₆×C₅₉	—	2	1	no	—
107	C₁₆×C₅₉	—	2	1	no	—
108	C₁₆×C₅₉	—	2	1	no	—
109	C₁₆×C₅₉	#41	1	2	yes	—
110	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
111	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
112	Q₈×C₅₉	—	4	2	no	—
113	Q₈×C₅₉	—	4	2	no	—
114	D₄×C₅₉	—	2	2	no	—
115	C₂×C₄×C₅₉	—	2	1	no	—
116	C₂×C₄×C₅₉	—	2	1	no	—
117	C₂×C₄×C₅₉	—	2	1	no	—
118	C₂×C₄×C₅₉	—	2	1	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	—	8	4	no	—
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	—	2	1	no	—
132	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
133	C₈×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
134	#33903	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂²×C₅₉	—	2	1	no	—
136	C₂²×C₅₉	—	2	1	no	—
137	C₂²×C₅₉	—	2	1	no	—
138	C₂²×C₅₉	—	2	1	no	—
139	C₂²×C₅₉	—	2	1	no	—
140	C₂²×C₅₉	#508	1	2	yes	—
141	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
142	C₄×C₅₉	—	2	1	no	—
143	C₄×C₅₉	—	2	1	no	—
144	C₄×C₅₉	—	2	1	no	—
145	C₄×C₅₉	—	2	1	no	—
146	C₄×C₅₉	—	8	2	no	—
147	C₄×C₅₉	—	8	2	no	—
148	C₄×C₅₉	—	8	2	no	—
149	C₄×C₅₉	—	8	2	no	—
150	C₄×C₅₉	#547	1	1	yes	characteristic
151	C₄×C₅₉	#578	1	1	yes	characteristic
152	#3336	C₂×C₅₉	1	2	yes	—
153	#2253	C₂×C₅₉	1	1	yes	characteristic, complemented
154	#2237	C₂×C₅₉	1	2	yes	complemented
155	#2235	C₂×C₅₉	1	2	yes	complemented
156	#2229	C₂×C₅₉	1	2	yes	complemented
157	#2216	C₂×C₅₉	1	1	yes	characteristic, complemented
158	#2216	C₂×C₅₉	1	1	yes	characteristic, complemented
159	C₄×Q₃₂	C₂×C₅₉	1	1	yes	characteristic, complemented
160	D₄×C₁₆	C₂×C₅₉	1	1	yes	characteristic
161	#2158	C₂×C₅₉	1	1	yes	characteristic, complemented
162	C₂×#366	C₂×C₅₉	1	1	yes	characteristic, complemented
163	C₂×C₅₉	—	2	2	no	—
164	C₂×C₅₉	—	4	1	no	—
165	C₂×C₅₉	#3334	1	1	yes	characteristic
166	C₂×C₅₉	C₂×#508	1	1	yes	characteristic
167	C₂×C₅₉	#3417	1	1	yes	characteristic
168	#552	—	2	2	no	—
169	C₂×Q₃₂	C₂²×C₅₉	1	1	yes	characteristic
170	C₂²×C₁₆	C₂²×C₅₉	1	1	yes	characteristic
171	C₂²×C₁₆	C₂²×C₅₉	1	1	yes	characteristic
172	#499	C₂²×C₅₉	1	2	yes	—
173	#485	—	2	2	no	—
174	#483	—	2	2	no	—
175	#479	—	2	2	no	—
176	#467	C₂²×C₅₉	1	2	yes	—
177	#467	C₂²×C₅₉	1	2	yes	—
178	C₄×Q₁₆	C₂²×C₅₉	1	2	yes	—
179	C₈×D₄	C₂²×C₅₉	1	1	yes	characteristic
180	#427	C₂²×C₅₉	1	2	yes	—
181	C₂×#107	C₂²×C₅₉	1	2	yes	—
182	#367	C₂²×C₅₉	1	2	yes	—
183	#367	C₂²×C₅₉	1	2	yes	—
184	#366	C₂²×C₅₉	1	1	yes	characteristic
185	#366	C₂²×C₅₉	1	1	yes	characteristic
186	#366	C₂²×C₅₉	1	2	yes	—
187	#366	C₂²×C₅₉	1	1	yes	characteristic
188	C₄⋊C₁₆	C₂²×C₅₉	1	1	yes	characteristic
189	#358	C₂²×C₅₉	1	2	yes	—
190	#358	C₂²×C₅₉	1	2	yes	—
191	#358	C₂²×C₅₉	1	2	yes	—
192	C₂²⋊C₁₆	C₂²×C₅₉	1	1	yes	characteristic
193	C₂²⋊C₁₆	C₂²×C₅₉	1	1	yes	characteristic
194	C₄×C₁₆	C₂²×C₅₉	1	1	yes	characteristic
195	C₅₉	#33903	1	1	yes	characteristic, direct product factor, Sylow
196	C₂×Q₁₆	C₂³×C₅₉	1	2	yes	—
197	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
199	#126	—	4	2	no	—
200	#125	—	2	2	no	—
201	#125	—	4	2	no	—
202	#123	—	4	2	no	—
203	C₂²⋊Q₈	—	2	2	no	—
204	C₂²⋊Q₈	—	2	2	no	—
205	C₄×Q₈	—	2	2	no	—
206	C₄×D₄	D₄×C₅₉	1	1	yes	characteristic
207	#117	—	2	2	no	—
208	#117	—	4	2	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	Q₃₂	—	2	2	no	—
211	C₂×C₁₆	—	2	2	no	—
212	C₂×C₁₆	—	2	2	no	—
213	C₂×C₁₆	C₂³×C₅₉	1	1	yes	characteristic
214	C₂×C₁₆	C₂³×C₅₉	1	1	yes	characteristic
215	C₂×C₁₆	C₂³×C₅₉	1	1	yes	characteristic
216	C₂×C₁₆	C₂³×C₅₉	1	1	yes	characteristic
217	#107	—	2	2	no	—
218	#107	C₂³×C₅₉	1	2	yes	—
219	#107	C₂³×C₅₉	1	2	yes	—
220	#107	C₂³×C₅₉	1	2	yes	—
221	#106	—	2	2	no	—
222	#106	—	2	2	no	—
223	C₄⋊C₈	D₄×C₅₉	1	1	yes	characteristic
224	#103	—	2	2	no	—
225	#103	—	2	2	no	—
226	#103	—	2	2	no	—
227	C₂²⋊C₈	D₄×C₅₉	1	1	yes	characteristic
228	C₂²⋊C₈	D₄×C₅₉	1	1	yes	characteristic
229	C₄×C₈	C₂³×C₅₉	1	1	yes	characteristic
230	C₂×Q₈	—	2	2	no	—
231	C₂×D₄	D₈×C₅₉	1	1	yes	characteristic
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
235	Q₁₆	—	2	2	no	—
236	Q₁₆	—	2	2	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
242	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
243	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
244	C₂×C₈	C₂⁴×C₅₉	1	1	yes	characteristic
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	D₈×C₅₉	1	1	yes	characteristic
255	C₂²⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₂²⋊C₄	D₈×C₅₉	1	1	yes	characteristic
259	C₂²⋊C₄	D₈×C₅₉	1	1	yes	characteristic
260	C₄²	—	4	2	no	—
261	C₄²	—	4	2	no	—
262	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
263	C₁₆	—	2	1	no	—
264	C₁₆	—	2	1	no	—
265	C₁₆	—	2	1	no	—
266	C₁₆	#41×C₅₉	1	2	yes	—
267	C₂³	C₂×D₈×C₅₉	1	1	yes	characteristic
268	C₂³	C₂×D₈×C₅₉	1	1	yes	characteristic
269	Q₈	—	4	2	no	—
270	Q₈	—	4	2	no	—
271	D₄	—	2	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	8	4	no	—
281	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
282	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
283	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
284	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
285	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
286	C₈	—	2	1	no	—
287	C₈	—	2	1	no	—
288	C₈	—	2	1	no	—
289	C₈	C₂×#41×C₅₉	1	1	yes	characteristic
290	C₈	2₋¹⁺⁴×C₅₉	1	1	yes	characteristic
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	C₅₉×#508	1	2	yes	—
297	C₂²	C₂²×D₈×C₅₉	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	8	2	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	—	8	2	no	—
306	C₄	C₅₉×#547	1	1	yes	characteristic
307	C₄	C₅₉×#578	1	1	yes	characteristic
308	C₂	—	2	2	no	—
309	C₂	—	4	1	no	—
310	C₂	C₅₉×#3334	1	1	yes	characteristic
311	C₂	C₂×C₅₉×#508	1	1	yes	characteristic
312	C₂	C₅₉×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₉
prime radical	C₂×D₄×C₅₉	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₄×D₄×C₅₉ ◁ C₈×D₄×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₉×#2253	C₂	semidirect
3. C₅₉×#2237	C₂	semidirect
4. C₅₉×#2235	C₂	semidirect
5. C₅₉×#2229	C₂	semidirect
6. C₅₉×#2216	C₂	semidirect
7. C₅₉×#2216	C₂	semidirect
8. C₄×Q₃₂×C₅₉	C₂	semidirect
10. C₅₉×#2158	C₂	semidirect
11. C₂×C₅₉×#366	C₂	semidirect
134. #33903	C₅₉	direct (N × H)
153. #2253	C₂×C₅₉	semidirect
154. #2237	C₂×C₅₉	semidirect
155. #2235	C₂×C₅₉	semidirect
156. #2229	C₂×C₅₉	semidirect
157. #2216	C₂×C₅₉	semidirect
158. #2216	C₂×C₅₉	semidirect
159. C₄×Q₃₂	C₂×C₅₉	semidirect
161. #2158	C₂×C₅₉	semidirect
162. C₂×#366	C₂×C₅₉	semidirect
195. C₅₉	#33903	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×#3336	2
C₅₉×#2253	2
C₅₉×#2237	2
C₅₉×#2235	2
C₅₉×#2229	2
C₅₉×#2216	2
C₄×Q₃₂×C₅₉	2
D₄×C₁₆×C₅₉	2
C₅₉×#2158	2
C₂×C₅₉×#366	2
#33903	59

Subgroups — C59×#33903 #28446012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#33903 #28446012