Subgroups — C₅₉×#33906 #28446015

Order 15104.

494 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#3339	C₂	1	2	yes	maximal, complemented
2	C₅₉×#2253	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₉×#2234	C₂	1	2	yes	maximal, complemented
4	C₅₉×#2233	C₂	1	2	yes	maximal, complemented
5	C₅₉×#2228	C₂	1	2	yes	maximal, complemented
6	C₅₉×#2215	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#2215	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₉×#2158	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅₉×#366	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₉×#553	—	2	2	no	—
13	C₂×D₁₆×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×#482	—	2	2	no	—
18	C₅₉×#481	—	2	2	no	—
19	C₅₉×#478	—	2	2	no	—
20	C₅₉×#466	C₂²	1	2	yes	—
21	C₅₉×#466	C₂²	1	2	yes	—
22	C₄×D₈×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×#357	C₂²	1	2	yes	—
34	C₅₉×#357	C₂²	1	2	yes	—
35	C₅₉×#357	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×D₈×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	#123×C₅₉	—	4	2	no	—
45	#121×C₅₉	—	2	2	no	—
46	#121×C₅₉	—	2	2	no	—
47	#121×C₅₉	—	4	2	no	—
48	C₄×D₄×C₅₉	—	2	2	no	—
49	C₄×D₄×C₅₉	—	4	2	no	—
50	C₄×D₄×C₅₉	D₄	1	1	yes	characteristic
51	#117×C₅₉	—	2	2	no	—
52	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
53	D₁₆×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	#102×C₅₉	—	2	2	no	—
68	#102×C₅₉	—	2	2	no	—
69	#102×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×D₄×C₅₉	—	2	2	no	—
74	C₂×D₄×C₅₉	—	4	2	no	—
75	C₂×D₄×C₅₉	—	4	2	no	—
76	C₂×D₄×C₅₉	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₅₉	—	4	2	no	—
78	C₂²×C₄×C₅₉	—	4	2	no	—
79	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
81	D₈×C₅₉	—	2	2	no	—
82	D₈×C₅₉	—	2	2	no	—
83	C₂×C₈×C₅₉	—	2	1	no	—
84	C₂×C₈×C₅₉	—	2	1	no	—
85	C₂×C₈×C₅₉	—	2	1	no	—
86	C₂×C₈×C₅₉	—	2	1	no	—
87	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₅₉	—	2	2	no	—
92	C₄⋊C₄×C₅₉	—	2	2	no	—
93	C₄⋊C₄×C₅₉	—	2	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	—	4	2	no	—
102	C₂²⋊C₄×C₅₉	D₈	1	1	yes	characteristic
103	C₂²⋊C₄×C₅₉	D₈	1	1	yes	characteristic
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	—	4	2	no	—
111	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
112	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
113	D₄×C₅₉	—	2	2	no	—
114	D₄×C₅₉	—	4	2	no	—
115	D₄×C₅₉	—	4	2	no	—
116	D₄×C₅₉	—	8	4	no	—
117	D₄×C₅₉	—	16	2	no	—
118	C₂×C₄×C₅₉	—	2	1	no	—
119	C₂×C₄×C₅₉	—	2	1	no	—
120	C₂×C₄×C₅₉	—	2	1	no	—
121	C₂×C₄×C₅₉	—	2	1	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	—	16	2	no	—
126	C₂×C₄×C₅₉	—	16	2	no	—
127	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
132	C₈×C₅₉	—	2	1	no	—
133	C₈×C₅₉	—	2	1	no	—
134	C₈×C₅₉	—	2	1	no	—
135	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
136	C₈×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
137	#33906	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₂²×C₅₉	—	2	1	no	—
139	C₂²×C₅₉	—	2	1	no	—
140	C₂²×C₅₉	—	2	1	no	—
141	C₂²×C₅₉	—	2	1	no	—
142	C₂²×C₅₉	—	2	1	no	—
143	C₂²×C₅₉	—	8	2	no	—
144	C₂²×C₅₉	—	8	2	no	—
145	C₂²×C₅₉	—	8	2	no	—
146	C₂²×C₅₉	#508	1	2	yes	—
147	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
148	C₄×C₅₉	—	2	1	no	—
149	C₄×C₅₉	—	2	1	no	—
150	C₄×C₅₉	—	2	1	no	—
151	C₄×C₅₉	—	2	1	no	—
152	C₄×C₅₉	—	8	2	no	—
153	C₄×C₅₉	—	8	2	no	—
154	C₄×C₅₉	—	8	2	no	—
155	C₄×C₅₉	#547	1	1	yes	characteristic
156	C₄×C₅₉	#578	1	1	yes	characteristic
157	#3339	C₂×C₅₉	1	2	yes	complemented
158	#2253	C₂×C₅₉	1	1	yes	characteristic, complemented
159	#2234	C₂×C₅₉	1	2	yes	complemented
160	#2233	C₂×C₅₉	1	2	yes	complemented
161	#2228	C₂×C₅₉	1	2	yes	complemented
162	#2215	C₂×C₅₉	1	1	yes	characteristic, complemented
163	#2215	C₂×C₅₉	1	1	yes	characteristic, complemented
164	C₄×D₁₆	C₂×C₅₉	1	1	yes	characteristic, complemented
165	D₄×C₁₆	C₂×C₅₉	1	1	yes	characteristic, complemented
166	#2158	C₂×C₅₉	1	1	yes	characteristic, complemented
167	C₂×#366	C₂×C₅₉	1	1	yes	characteristic, complemented
168	C₂×C₅₉	—	2	2	no	—
169	C₂×C₅₉	—	4	1	no	—
170	C₂×C₅₉	—	16	2	no	—
171	C₂×C₅₉	#3334	1	1	yes	characteristic
172	C₂×C₅₉	C₂×#508	1	1	yes	characteristic
173	C₂×C₅₉	#3415	1	1	yes	characteristic
174	#553	—	2	2	no	—
175	C₂×D₁₆	C₂²×C₅₉	1	1	yes	characteristic
176	C₂²×C₁₆	C₂²×C₅₉	1	1	yes	characteristic
177	C₂²×C₁₆	C₂²×C₅₉	1	1	yes	characteristic
178	#499	C₂²×C₅₉	1	2	yes	—
179	#482	—	2	2	no	—
180	#481	—	2	2	no	—
181	#478	—	2	2	no	—
182	#466	C₂²×C₅₉	1	2	yes	—
183	#466	C₂²×C₅₉	1	2	yes	—
184	C₄×D₈	C₂²×C₅₉	1	2	yes	—
185	C₈×D₄	C₂²×C₅₉	1	1	yes	characteristic
186	#427	C₂²×C₅₉	1	2	yes	—
187	C₂×#107	C₂²×C₅₉	1	2	yes	—
188	#367	C₂²×C₅₉	1	2	yes	—
189	#367	C₂²×C₅₉	1	2	yes	—
190	#366	C₂²×C₅₉	1	1	yes	characteristic
191	#366	C₂²×C₅₉	1	1	yes	characteristic
192	#366	C₂²×C₅₉	1	2	yes	—
193	#366	C₂²×C₅₉	1	1	yes	characteristic
194	C₄⋊C₁₆	C₂²×C₅₉	1	1	yes	characteristic
195	#357	C₂²×C₅₉	1	2	yes	—
196	#357	C₂²×C₅₉	1	2	yes	—
197	#357	C₂²×C₅₉	1	2	yes	—
198	C₂²⋊C₁₆	C₂²×C₅₉	1	1	yes	characteristic
199	C₂²⋊C₁₆	C₂²×C₅₉	1	1	yes	characteristic
200	C₄×C₁₆	C₂²×C₅₉	1	1	yes	characteristic
201	C₅₉	#33906	1	1	yes	characteristic, direct product factor, Sylow
202	C₂×D₈	C₂³×C₅₉	1	2	yes	—
203	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
205	#126	—	4	2	no	—
206	#125	—	2	2	no	—
207	#123	—	4	2	no	—
208	#121	—	2	2	no	—
209	#121	—	2	2	no	—
210	#121	—	4	2	no	—
211	C₄×D₄	—	2	2	no	—
212	C₄×D₄	—	4	2	no	—
213	C₄×D₄	D₄×C₅₉	1	1	yes	characteristic
214	#117	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	D₁₆	—	2	2	no	—
217	C₂×C₁₆	—	2	2	no	—
218	C₂×C₁₆	—	2	2	no	—
219	C₂×C₁₆	C₂³×C₅₉	1	1	yes	characteristic
220	C₂×C₁₆	C₂³×C₅₉	1	1	yes	characteristic
221	C₂×C₁₆	C₂³×C₅₉	1	1	yes	characteristic
222	C₂×C₁₆	C₂³×C₅₉	1	1	yes	characteristic
223	#107	—	2	2	no	—
224	#107	C₂³×C₅₉	1	2	yes	—
225	#107	C₂³×C₅₉	1	2	yes	—
226	#107	C₂³×C₅₉	1	2	yes	—
227	#106	—	2	2	no	—
228	#106	—	2	2	no	—
229	C₄⋊C₈	D₄×C₅₉	1	1	yes	characteristic
230	#102	—	2	2	no	—
231	#102	—	2	2	no	—
232	#102	—	2	2	no	—
233	C₂²⋊C₈	D₄×C₅₉	1	1	yes	characteristic
234	C₂²⋊C₈	D₄×C₅₉	1	1	yes	characteristic
235	C₄×C₈	C₂³×C₅₉	1	1	yes	characteristic
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂×D₄	—	4	2	no	—
239	C₂×D₄	D₈×C₅₉	1	1	yes	characteristic
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
244	D₈	—	2	2	no	—
245	D₈	—	2	2	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
251	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
252	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
253	C₂×C₈	C₂⁴×C₅₉	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	D₈×C₅₉	1	1	yes	characteristic
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	D₈×C₅₉	1	1	yes	characteristic
266	C₂²⋊C₄	D₈×C₅₉	1	1	yes	characteristic
267	C₄²	—	4	2	no	—
268	C₄²	C₂×D₄×C₅₉	1	1	yes	characteristic
269	C₁₆	—	2	1	no	—
270	C₁₆	—	2	1	no	—
271	C₁₆	—	2	1	no	—
272	C₁₆	#41×C₅₉	1	2	yes	—
273	C₂³	—	4	2	no	—
274	C₂³	C₂×D₈×C₅₉	1	1	yes	characteristic
275	C₂³	C₂×D₈×C₅₉	1	1	yes	characteristic
276	D₄	—	2	2	no	—
277	D₄	—	4	2	no	—
278	D₄	—	4	2	no	—
279	D₄	—	8	4	no	—
280	D₄	—	16	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	16	2	no	—
289	C₂×C₄	—	16	2	no	—
290	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
291	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
292	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
293	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	C₂×#41×C₅₉	1	1	yes	characteristic
299	C₈	2₋¹⁺⁴×C₅₉	1	1	yes	characteristic
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	—	8	2	no	—
308	C₂²	C₅₉×#508	1	2	yes	—
309	C₂²	C₂²×D₈×C₅₉	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	8	2	no	—
315	C₄	—	8	2	no	—
316	C₄	—	8	2	no	—
317	C₄	C₅₉×#547	1	1	yes	characteristic
318	C₄	C₅₉×#578	1	1	yes	characteristic
319	C₂	—	2	2	no	—
320	C₂	—	4	1	no	—
321	C₂	—	16	2	no	—
322	C₂	C₅₉×#3334	1	1	yes	characteristic
323	C₂	C₂×C₅₉×#508	1	1	yes	characteristic
324	C₂	C₅₉×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₉
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
1. C₅₉×#3339	C₂	semidirect
2. C₅₉×#2253	C₂	semidirect
3. C₅₉×#2234	C₂	semidirect
4. C₅₉×#2233	C₂	semidirect
5. C₅₉×#2228	C₂	semidirect
6. C₅₉×#2215	C₂	semidirect
7. C₅₉×#2215	C₂	semidirect
8. C₄×D₁₆×C₅₉	C₂	semidirect
9. D₄×C₁₆×C₅₉	C₂	semidirect
10. C₅₉×#2158	C₂	semidirect
11. C₂×C₅₉×#366	C₂	semidirect
137. #33906	C₅₉	direct (N × H)
157. #3339	C₂×C₅₉	semidirect
158. #2253	C₂×C₅₉	semidirect
159. #2234	C₂×C₅₉	semidirect
160. #2233	C₂×C₅₉	semidirect
161. #2228	C₂×C₅₉	semidirect
162. #2215	C₂×C₅₉	semidirect
163. #2215	C₂×C₅₉	semidirect
164. C₄×D₁₆	C₂×C₅₉	semidirect
165. D₄×C₁₆	C₂×C₅₉	semidirect
166. #2158	C₂×C₅₉	semidirect
167. C₂×#366	C₂×C₅₉	semidirect
201. C₅₉	#33906	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×#3339	2
C₅₉×#2253	2
C₅₉×#2234	2
C₅₉×#2233	2
C₅₉×#2228	2
C₅₉×#2215	2
C₄×D₁₆×C₅₉	2
D₄×C₁₆×C₅₉	2
C₅₉×#2158	2
C₂×C₅₉×#366	2
#33906	59

Subgroups — C59×#33906 #28446015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#33906 #28446015