Subgroups — C₅₉×#10800 #28422909

Order 15104.

650 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₉×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₉×#338	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₉×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₉×H(3,ℤ/4)	C₂	1	2	yes	maximal
5	C₅₉×#1729	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₅₉	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
9	C₂×#117×C₅₉	C₂²	1	2	yes	—
10	C₂×#117×C₅₉	C₂²	1	1	yes	characteristic
11	C₅₉×#405	—	2	2	no	—
12	C₅₉×#405	—	2	2	no	—
13	C₂×#97×C₅₉	C₄	1	2	yes	complemented
14	C₂×C₄×C₈×C₅₉	C₄	1	2	yes	complemented
15	C₅₉×#338	—	2	2	no	—
16	C₅₉×H(3,ℤ/4)	—	2	2	no	—
17	C₅₉×H(3,ℤ/4)	—	2	4	no	—
18	C₅₉×#336	C₄	1	8	yes	complemented
19	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₅₉	—	2	2	no	—
21	C₂×M₁₆×C₅₉	—	2	2	no	—
22	C₂×M₁₆×C₅₉	—	2	2	no	—
23	C₂×M₁₆×C₅₉	—	2	2	no	—
24	C₂×M₁₆×C₅₉	—	2	2	no	—
25	C₂×M₁₆×C₅₉	—	2	2	no	—
26	C₂×M₁₆×C₅₉	—	2	2	no	—
27	C₂×M₁₆×C₅₉	—	2	2	no	—
28	C₂×M₁₆×C₅₉	D₄	1	2	yes	—
29	C₂×M₁₆×C₅₉	D₄	1	2	yes	—
30	C₂²×C₈×C₅₉	—	2	1	no	—
31	C₂²×C₈×C₅₉	—	2	1	no	—
32	C₂²×C₈×C₅₉	—	2	2	no	—
33	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
38	#117×C₅₉	—	2	1	no	—
39	#117×C₅₉	—	2	1	no	—
40	#117×C₅₉	—	2	2	no	—
41	#117×C₅₉	—	2	2	no	—
42	#117×C₅₉	—	2	2	no	—
43	#117×C₅₉	—	2	2	no	—
44	#117×C₅₉	D₄	1	2	yes	—
45	#117×C₅₉	Q₈	1	2	yes	—
46	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
47	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₅₉	—	2	1	no	—
49	C₂×C₂²⋊C₄×C₅₉	—	2	2	no	—
50	C₂×C₄²×C₅₉	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
52	#97×C₅₉	—	2	4	no	—
53	C₄×C₈×C₅₉	—	2	4	no	—
54	C₂⁴×C₅₉	#41	1	1	yes	characteristic
55	C₂²×C₄×C₅₉	—	2	2	no	—
56	C₂²×C₄×C₅₉	—	2	4	no	—
57	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
64	C₂²×C₄×C₅₉	#41	1	2	yes	—
65	C₂²×C₄×C₅₉	#41	1	2	yes	—
66	C₂²×C₄×C₅₉	#41	1	2	yes	—
67	M₁₆×C₅₉	—	2	4	no	—
68	M₁₆×C₅₉	—	2	4	no	—
69	M₁₆×C₅₉	—	4	2	no	—
70	M₁₆×C₅₉	—	4	2	no	—
71	M₁₆×C₅₉	—	4	2	no	—
72	M₁₆×C₅₉	—	4	2	no	—
73	C₂×C₈×C₅₉	—	2	1	no	—
74	C₂×C₈×C₅₉	—	2	1	no	—
75	C₂×C₈×C₅₉	—	2	1	no	—
76	C₂×C₈×C₅₉	—	2	1	no	—
77	C₂×C₈×C₅₉	—	2	2	no	—
78	C₂×C₈×C₅₉	—	2	2	no	—
79	C₂×C₈×C₅₉	—	2	4	no	—
80	C₂×C₈×C₅₉	—	2	4	no	—
81	C₂×C₈×C₅₉	—	2	4	no	—
82	C₂×C₈×C₅₉	—	2	4	no	—
83	C₂×C₈×C₅₉	—	2	4	no	—
84	C₂×C₈×C₅₉	C₄²	1	2	yes	—
85	C₂×C₈×C₅₉	C₄²	1	2	yes	—
86	C₂×C₈×C₅₉	C₂²⋊C₄	1	4	yes	—
87	C₄⋊C₄×C₅₉	—	2	2	no	—
88	C₄⋊C₄×C₅₉	—	2	2	no	—
89	C₄⋊C₄×C₅₉	—	4	2	no	—
90	C₄⋊C₄×C₅₉	—	4	2	no	—
91	C₂²⋊C₄×C₅₉	—	2	2	no	—
92	C₂²⋊C₄×C₅₉	—	2	2	no	—
93	C₂²⋊C₄×C₅₉	—	4	2	no	—
94	C₂²⋊C₄×C₅₉	—	4	2	no	—
95	C₄²×C₅₉	—	2	2	no	—
96	C₄²×C₅₉	—	2	4	no	—
97	C₄²×C₅₉	—	2	4	no	—
98	C₄²×C₅₉	C₄⋊C₄	1	4	yes	—
99	C₂³×C₅₉	—	4	2	no	—
100	C₂³×C₅₉	#117	1	1	yes	characteristic
101	C₂³×C₅₉	#117	1	2	yes	—
102	C₂³×C₅₉	#121	1	1	yes	characteristic
103	C₂³×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂³×C₅₉	#126	1	2	yes	—
105	C₂×C₄×C₅₉	—	2	4	no	—
106	C₂×C₄×C₅₉	—	4	4	no	—
107	C₂×C₄×C₅₉	—	4	8	no	—
108	C₂×C₄×C₅₉	—	4	4	no	—
109	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
113	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
114	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
115	C₂×C₄×C₅₉	#117	1	2	yes	—
116	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
120	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
121	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₅₉	#123	1	2	yes	—
123	C₂×C₄×C₅₉	#123	1	2	yes	—
124	C₂×C₄×C₅₉	#124	1	2	yes	—
125	C₂×C₄×C₅₉	#125	1	2	yes	—
126	C₂×C₄×C₅₉	#126	1	2	yes	—
127	C₈×C₅₉	—	2	2	no	—
128	C₈×C₅₉	—	2	2	no	—
129	C₈×C₅₉	—	2	4	no	—
130	C₈×C₅₉	—	4	4	no	—
131	#10800	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂²×C₅₉	—	2	2	no	—
133	C₂²×C₅₉	—	2	4	no	—
134	C₂²×C₅₉	—	4	1	no	—
135	C₂²×C₅₉	—	4	1	no	—
136	C₂²×C₅₉	—	4	2	no	—
137	C₂²×C₅₉	#376	1	1	yes	characteristic
138	C₂²×C₅₉	#380	1	1	yes	characteristic
139	C₂²×C₅₉	#383	1	1	yes	characteristic
140	C₂²×C₅₉	#387	1	2	yes	—
141	C₂²×C₅₉	#388	1	2	yes	—
142	C₄×C₅₉	—	4	4	no	—
143	C₄×C₅₉	—	8	4	no	—
144	C₄×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
145	C₄×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
146	C₄×C₅₉	#381	1	1	yes	characteristic
147	C₄×C₅₉	#382	1	1	yes	characteristic
148	C₄×C₅₉	#387	1	2	yes	—
149	C₄×C₅₉	#388	1	2	yes	—
150	C₂×#405	C₂×C₅₉	1	1	yes	characteristic
151	C₂×#338	C₂×C₅₉	1	1	yes	characteristic
152	C₂×H(3,ℤ/4)	C₂×C₅₉	1	1	yes	characteristic
153	C₂×H(3,ℤ/4)	C₂×C₅₉	1	2	yes	—
154	#1729	C₂×C₅₉	1	2	yes	—
155	C₂×C₅₉	—	2	1	no	—
156	C₂×C₅₉	—	2	2	no	—
157	C₂×C₅₉	—	2	1	no	—
158	C₂×C₅₉	—	2	2	no	—
159	C₂×C₅₉	#1434	1	1	yes	characteristic
160	C₂×C₅₉	#1747	1	2	yes	—
161	C₂²×M₁₆	C₂²×C₅₉	1	2	yes	—
162	C₂²×M₁₆	C₂²×C₅₉	1	1	yes	characteristic
163	C₂³×C₈	C₂²×C₅₉	1	1	yes	characteristic
164	C₂×#117	C₂²×C₅₉	1	2	yes	—
165	C₂×#117	C₂²×C₅₉	1	1	yes	characteristic
166	#405	—	2	2	no	—
167	#405	—	2	2	no	—
168	C₂×#97	C₄×C₅₉	1	2	yes	complemented
169	C₂×C₄×C₈	C₄×C₅₉	1	2	yes	complemented
170	#338	—	2	2	no	—
171	H(3,ℤ/4)	—	2	2	no	—
172	H(3,ℤ/4)	—	2	4	no	—
173	#336	C₄×C₅₉	1	8	yes	complemented
174	C₅₉	#10800	1	1	yes	characteristic, direct product factor, Sylow
175	C₂³×C₄	C₂³×C₅₉	1	1	yes	characteristic
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	D₄×C₅₉	1	2	yes	—
185	C₂×M₁₆	D₄×C₅₉	1	2	yes	—
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	—	2	1	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	C₂×C₄×C₅₉	1	4	yes	—
190	C₂²×C₈	C₂×C₄×C₅₉	1	4	yes	—
191	C₂²×C₈	C₂×C₄×C₅₉	1	1	yes	characteristic
192	C₂²×C₈	C₂×C₄×C₅₉	1	1	yes	characteristic
193	C₂²×C₈	C₂×C₄×C₅₉	1	2	yes	—
194	#117	—	2	1	no	—
195	#117	—	2	1	no	—
196	#117	—	2	2	no	—
197	#117	—	2	2	no	—
198	#117	—	2	2	no	—
199	#117	—	2	2	no	—
200	#117	D₄×C₅₉	1	2	yes	—
201	#117	Q₈×C₅₉	1	2	yes	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₂²⋊C₄	—	2	1	no	—
205	C₂×C₂²⋊C₄	—	2	2	no	—
206	C₂×C₄²	C₂×C₄×C₅₉	1	4	yes	—
207	C₂×C₄²	C₂×C₄×C₅₉	1	2	yes	—
208	#97	—	2	4	no	—
209	C₄×C₈	—	2	4	no	—
210	C₂⁴	#41×C₅₉	1	1	yes	characteristic
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	C₂²×C₄×C₅₉	1	1	yes	characteristic
214	C₂²×C₄	C₂²×C₄×C₅₉	1	2	yes	—
215	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
218	C₂²×C₄	C₂×Q₈×C₅₉	1	1	yes	characteristic
219	C₂²×C₄	#41×C₅₉	1	1	yes	characteristic
220	C₂²×C₄	#41×C₅₉	1	2	yes	—
221	C₂²×C₄	#41×C₅₉	1	2	yes	—
222	C₂²×C₄	#41×C₅₉	1	2	yes	—
223	M₁₆	—	2	4	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	4	2	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	4	2	no	—
228	M₁₆	—	4	2	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	C₄²×C₅₉	1	2	yes	—
241	C₂×C₈	C₄²×C₅₉	1	2	yes	—
242	C₂×C₈	C₂²⋊C₄×C₅₉	1	4	yes	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₂²⋊C₄	—	2	2	no	—
248	C₂²⋊C₄	—	2	2	no	—
249	C₂²⋊C₄	—	4	2	no	—
250	C₂²⋊C₄	—	4	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	4	no	—
253	C₄²	—	2	4	no	—
254	C₄²	C₄⋊C₄×C₅₉	1	4	yes	—
255	C₂³	—	4	2	no	—
256	C₂³	#117×C₅₉	1	1	yes	characteristic
257	C₂³	#117×C₅₉	1	2	yes	—
258	C₂³	#121×C₅₉	1	1	yes	characteristic
259	C₂³	C₂²⋊Q₈×C₅₉	1	1	yes	characteristic
260	C₂³	#126×C₅₉	1	2	yes	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	8	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	C₂×C₄²×C₅₉	1	1	yes	characteristic
266	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
267	C₂×C₄	C₂×C₄⋊C₄×C₅₉	1	1	yes	characteristic
268	C₂×C₄	#117×C₅₉	1	1	yes	characteristic
269	C₂×C₄	#117×C₅₉	1	1	yes	characteristic
270	C₂×C₄	#117×C₅₉	1	1	yes	characteristic
271	C₂×C₄	#117×C₅₉	1	2	yes	—
272	C₂×C₄	C₄×D₄×C₅₉	1	2	yes	—
273	C₂×C₄	C₄×D₄×C₅₉	1	2	yes	—
274	C₂×C₄	C₄×D₄×C₅₉	1	2	yes	—
275	C₂×C₄	C₄×Q₈×C₅₉	1	2	yes	—
276	C₂×C₄	#121×C₅₉	1	1	yes	characteristic
277	C₂×C₄	C₂²⋊Q₈×C₅₉	1	1	yes	characteristic
278	C₂×C₄	#123×C₅₉	1	2	yes	—
279	C₂×C₄	#123×C₅₉	1	2	yes	—
280	C₂×C₄	#124×C₅₉	1	2	yes	—
281	C₂×C₄	#125×C₅₉	1	2	yes	—
282	C₂×C₄	#126×C₅₉	1	2	yes	—
283	C₈	—	2	2	no	—
284	C₈	—	2	2	no	—
285	C₈	—	2	4	no	—
286	C₈	—	4	4	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	2	no	—
292	C₂²	C₅₉×#376	1	1	yes	characteristic
293	C₂²	C₅₉×#380	1	1	yes	characteristic
294	C₂²	C₅₉×#383	1	1	yes	characteristic
295	C₂²	C₅₉×#387	1	2	yes	—
296	C₂²	C₅₉×#388	1	2	yes	—
297	C₄	—	4	4	no	—
298	C₄	—	8	4	no	—
299	C₄	C₄×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
300	C₄	C₄×C₄⋊C₄×C₅₉	1	1	yes	characteristic
301	C₄	C₅₉×#381	1	1	yes	characteristic
302	C₄	C₅₉×#382	1	1	yes	characteristic
303	C₄	C₅₉×#387	1	2	yes	—
304	C₄	C₅₉×#388	1	2	yes	—
305	C₂	—	2	1	no	—
306	C₂	—	2	2	no	—
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	C₅₉×#1434	1	1	yes	characteristic
310	C₂	C₅₉×#1747	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₉	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₅₉
Frattini, cosocle	C₂³×C₄	C₂³×C₅₉
socle	C₂²×C₅₉	#376
prime radical	C₂⁴×C₅₉	#41
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₂×#97×C₅₉	C₄	semidirect
14. C₂×C₄×C₈×C₅₉	C₄	semidirect
18. C₅₉×#336	C₄	semidirect
131. #10800	C₅₉	direct (N × H)
168. C₂×#97	C₄×C₅₉	semidirect
169. C₂×C₄×C₈	C₄×C₅₉	semidirect
173. #336	C₄×C₅₉	semidirect
174. C₅₉	#10800	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₉×#405	2
C₂×C₅₉×#338	2
C₂×C₅₉×H(3,ℤ/4)	2
C₅₉×#1729	2
#10800	59

Subgroups — C59×#10800 #28422909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#10800 #28422909