Subgroups — C₅₉×#17721 #28429830

Order 15104.

1172 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#3499	C₂	1	1	yes	maximal, characteristic
2	C₅₉×#2988	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₉×#2906	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₉×#1547	C₂	1	3	yes	maximal, complemented
5	C₅₉×#1546	C₂	1	3	yes	maximal, complemented
6	C₅₉×#1540	C₂	1	3	yes	maximal, complemented
7	C₂×C₅₉×#332	C₂	1	3	yes	maximal, complemented
8	C₂×2₋¹⁺⁴×C₅₉	—	2	1	no	—
9	C₂×#131×C₅₉	—	2	1	no	—
10	D₄×Q₈×C₅₉	—	2	1	no	—
11	D₄×Q₈×C₅₉	C₄	1	2	yes	—
12	C₅₉×#547	—	2	3	no	—
13	C₅₉×#547	C₄	1	6	yes	—
14	C₅₉×#536	—	2	3	no	—
15	C₅₉×#532	C₂²	1	3	yes	—
16	C₂×C₄⋊Q₈×C₅₉	C₂²	1	3	yes	—
17	C₄×#41×C₅₉	C₂²	1	1	yes	characteristic
18	C₈×Q₈×C₅₉	—	2	1	no	—
19	C₈×D₄×C₅₉	—	2	3	no	—
20	C₅₉×#431	C₂²	1	3	yes	—
21	C₅₉×#424	—	2	3	no	—
22	C₅₉×#423	C₂²	1	3	yes	—
23	C₂×C₄⋊C₈×C₅₉	C₂²	1	3	yes	—
24	C₂×C₄⋊C₈×C₅₉	C₂²	1	3	yes	—
25	C₅₉×#332	C₂²	1	3	yes	—
26	C₅₉×#332	C₂²	1	6	yes	—
27	C₅₉×#332	C₂²	1	3	yes	—
28	C₅₉×#331	C₂²	1	1	yes	characteristic
29	C₅₉×#331	C₂²	1	3	yes	—
30	2₋¹⁺⁴×C₅₉	—	4	4	no	—
31	2₋¹⁺⁴×C₅₉	—	4	4	no	—
32	C₂×#41×C₅₉	—	2	6	no	—
33	C₂×#41×C₅₉	—	2	3	no	—
34	C₂×#41×C₅₉	D₄	1	1	yes	characteristic
35	C₂²×Q₈×C₅₉	—	2	2	no	—
36	C₂²×Q₈×C₅₉	—	2	3	no	—
37	#131×C₅₉	—	4	4	no	—
38	C₂×M₁₆×C₅₉	—	2	3	no	—
39	C₂²×C₈×C₅₉	—	2	3	no	—
40	C₂²×C₈×C₅₉	—	2	3	no	—
41	C₄⋊Q₈×C₅₉	—	2	3	no	—
42	C₄⋊Q₈×C₅₉	—	2	3	no	—
43	C₄⋊Q₈×C₅₉	C₂×C₄	1	6	yes	—
44	C₄⋊Q₈×C₅₉	C₂³	1	3	yes	—
45	#127×C₅₉	C₂³	1	1	yes	characteristic
46	#124×C₅₉	—	2	3	no	—
47	#123×C₅₉	—	2	12	no	—
48	C₂²⋊Q₈×C₅₉	—	2	6	no	—
49	C₂²⋊Q₈×C₅₉	—	2	6	no	—
50	#121×C₅₉	—	2	3	no	—
51	#121×C₅₉	C₂×C₄	1	6	yes	—
52	C₄×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
53	C₄×Q₈×C₅₉	D₄	1	1	yes	characteristic
54	C₄×D₄×C₅₉	—	2	3	no	—
55	C₄×D₄×C₅₉	C₂×C₄	1	6	yes	—
56	C₄×D₄×C₅₉	C₂×C₄	1	3	yes	—
57	C₄×D₄×C₅₉	D₄	1	3	yes	—
58	#117×C₅₉	D₄	1	3	yes	—
59	C₂×C₄⋊C₄×C₅₉	—	2	3	no	—
60	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	6	yes	—
61	C₂×C₄²×C₅₉	C₂³	1	3	yes	—
62	C₄⋊C₈×C₅₉	—	2	3	no	—
63	C₄⋊C₈×C₅₉	—	2	3	no	—
64	C₄⋊C₈×C₅₉	—	2	3	no	—
65	C₄⋊C₈×C₅₉	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₅₉	C₂³	1	3	yes	—
67	C₄⋊C₈×C₅₉	C₂³	1	3	yes	—
68	C₄⋊C₈×C₅₉	C₂³	1	1	yes	characteristic
69	C₂²⋊C₈×C₅₉	—	2	3	no	—
70	C₄×C₈×C₅₉	—	2	3	no	—
71	#41×C₅₉	—	4	24	no	—
72	#41×C₅₉	—	4	6	no	—
73	#41×C₅₉	—	4	6	no	—
74	#41×C₅₉	D₈	1	4	yes	—
75	#41×C₅₉	SD₁₆	1	4	yes	—
76	C₂×Q₈×C₅₉	—	2	6	no	—
77	C₂×Q₈×C₅₉	—	2	3	no	—
78	C₂×Q₈×C₅₉	—	4	8	no	—
79	C₂×Q₈×C₅₉	—	4	6	no	—
80	C₂×Q₈×C₅₉	—	4	6	no	—
81	C₂×Q₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×D₄×C₅₉	—	2	6	no	—
83	C₂×D₄×C₅₉	—	2	1	no	—
84	C₂×D₄×C₅₉	—	2	1	no	—
85	C₂×D₄×C₅₉	C₂²⋊C₄	1	3	yes	—
86	C₂×D₄×C₅₉	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₅₉	—	2	6	no	—
88	C₂²×C₄×C₅₉	—	2	6	no	—
89	C₂²×C₄×C₅₉	C₂²×C₄	1	3	yes	—
90	C₂²×C₄×C₅₉	C₂×D₄	1	3	yes	—
91	C₂²×C₄×C₅₉	C₂×D₄	1	3	yes	—
92	M₁₆×C₅₉	—	4	6	no	—
93	C₂×C₈×C₅₉	—	2	1	no	—
94	C₂×C₈×C₅₉	—	2	3	no	—
95	C₂×C₈×C₅₉	—	2	1	no	—
96	C₂×C₈×C₅₉	—	2	3	no	—
97	C₂×C₈×C₅₉	—	4	6	no	—
98	C₂×C₈×C₅₉	—	8	3	no	—
99	C₄⋊C₄×C₅₉	—	2	6	no	—
100	C₄⋊C₄×C₅₉	—	2	6	no	—
101	C₄⋊C₄×C₅₉	—	2	3	no	—
102	C₄⋊C₄×C₅₉	C₂²⋊C₄	1	3	yes	—
103	C₄⋊C₄×C₅₉	C₂²⋊C₄	1	3	yes	—
104	C₄⋊C₄×C₅₉	C₂²×C₄	1	6	yes	—
105	C₂²⋊C₄×C₅₉	—	2	6	no	—
106	C₂²⋊C₄×C₅₉	—	2	6	no	—
107	C₂²⋊C₄×C₅₉	C₂²⋊C₄	1	6	yes	—
108	C₄²×C₅₉	C₂²×C₄	1	3	yes	—
109	C₄²×C₅₉	C₂×D₄	1	3	yes	—
110	C₄²×C₅₉	C₂×D₄	1	3	yes	—
111	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
112	C₂³×C₅₉	—	2	2	no	—
113	C₂³×C₅₉	C₂×C₂²⋊C₄	1	3	yes	—
114	Q₈×C₅₉	—	4	3	no	—
115	Q₈×C₅₉	—	4	3	no	—
116	Q₈×C₅₉	—	4	12	no	—
117	Q₈×C₅₉	#102	1	4	yes	—
118	D₄×C₅₉	—	4	2	no	—
119	D₄×C₅₉	—	4	12	no	—
120	D₄×C₅₉	—	4	1	no	—
121	D₄×C₅₉	—	4	1	no	—
122	D₄×C₅₉	—	4	1	no	—
123	D₄×C₅₉	—	4	1	no	—
124	D₄×C₅₉	#102	1	12	yes	—
125	C₂×C₄×C₅₉	—	2	3	no	—
126	C₂×C₄×C₅₉	—	2	3	no	—
127	C₂×C₄×C₅₉	—	2	6	no	—
128	C₂×C₄×C₅₉	—	4	3	no	—
129	C₂×C₄×C₅₉	—	4	12	no	—
130	C₂×C₄×C₅₉	—	4	12	no	—
131	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	3	yes	—
132	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	3	yes	—
133	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	3	yes	—
134	C₂×C₄×C₅₉	C₂×D₈	1	6	yes	—
135	C₂×C₄×C₅₉	C₂×SD₁₆	1	6	yes	—
136	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
139	C₈×C₅₉	—	4	1	no	—
140	C₈×C₅₉	—	4	3	no	—
141	C₈×C₅₉	—	4	3	no	—
142	C₈×C₅₉	—	4	1	no	—
143	#17721	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂²×C₅₉	—	2	3	no	—
145	C₂²×C₅₉	—	2	3	no	—
146	C₂²×C₅₉	—	4	2	no	—
147	C₂²×C₅₉	—	4	2	no	—
148	C₂²×C₅₉	—	4	2	no	—
149	C₂²×C₅₉	C₂×#102	1	6	yes	—
150	C₂²×C₅₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
151	C₄×C₅₉	—	2	1	no	—
152	C₄×C₅₉	—	2	3	no	—
153	C₄×C₅₉	—	2	3	no	—
154	C₄×C₅₉	—	4	6	no	—
155	C₄×C₅₉	C₂×#102	1	6	yes	—
156	C₄×C₅₉	C₂²×D₈	1	1	yes	characteristic
157	C₄×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
158	C₄×C₅₉	#577	1	1	yes	characteristic
159	C₄×C₅₉	#578	1	1	yes	characteristic
160	#3499	C₂×C₅₉	1	1	yes	characteristic
161	#2988	C₂×C₅₉	1	1	yes	characteristic, complemented
162	#2906	C₂×C₅₉	1	1	yes	characteristic, complemented
163	#1547	C₂×C₅₉	1	3	yes	complemented
164	#1546	C₂×C₅₉	1	3	yes	complemented
165	#1540	C₂×C₅₉	1	3	yes	complemented
166	C₂×#332	C₂×C₅₉	1	3	yes	complemented
167	C₂×C₅₉	—	2	6	no	—
168	C₂×C₅₉	—	8	2	no	—
169	C₂×C₅₉	#2889	1	1	yes	characteristic
170	C₂×C₅₉	C₂²×#102	1	1	yes	characteristic
171	C₂×C₅₉	#2898	1	1	yes	characteristic
172	C₂×2₋¹⁺⁴	—	2	1	no	—
173	C₂×#131	—	2	1	no	—
174	D₄×Q₈	—	2	1	no	—
175	D₄×Q₈	C₄×C₅₉	1	2	yes	—
176	#547	—	2	3	no	—
177	#547	C₄×C₅₉	1	6	yes	—
178	#536	—	2	3	no	—
179	#532	C₂²×C₅₉	1	3	yes	—
180	C₂×C₄⋊Q₈	C₂²×C₅₉	1	3	yes	—
181	C₄×#41	C₂²×C₅₉	1	1	yes	characteristic
182	C₈×Q₈	—	2	1	no	—
183	C₈×D₄	—	2	3	no	—
184	#431	C₂²×C₅₉	1	3	yes	—
185	#424	—	2	3	no	—
186	#423	C₂²×C₅₉	1	3	yes	—
187	C₂×C₄⋊C₈	C₂²×C₅₉	1	3	yes	—
188	C₂×C₄⋊C₈	C₂²×C₅₉	1	3	yes	—
189	#332	C₂²×C₅₉	1	3	yes	—
190	#332	C₂²×C₅₉	1	6	yes	—
191	#332	C₂²×C₅₉	1	3	yes	—
192	#331	C₂²×C₅₉	1	1	yes	characteristic
193	#331	C₂²×C₅₉	1	3	yes	—
194	C₅₉	#17721	1	1	yes	characteristic, direct product factor, Sylow
195	2₋¹⁺⁴	—	4	4	no	—
196	2₋¹⁺⁴	—	4	4	no	—
197	C₂×#41	—	2	6	no	—
198	C₂×#41	—	2	3	no	—
199	C₂×#41	D₄×C₅₉	1	1	yes	characteristic
200	C₂²×Q₈	—	2	2	no	—
201	C₂²×Q₈	—	2	3	no	—
202	#131	—	4	4	no	—
203	C₂×M₁₆	—	2	3	no	—
204	C₂²×C₈	—	2	3	no	—
205	C₂²×C₈	—	2	3	no	—
206	C₄⋊Q₈	—	2	3	no	—
207	C₄⋊Q₈	—	2	3	no	—
208	C₄⋊Q₈	C₂×C₄×C₅₉	1	6	yes	—
209	C₄⋊Q₈	C₂³×C₅₉	1	3	yes	—
210	#127	C₂³×C₅₉	1	1	yes	characteristic
211	#124	—	2	3	no	—
212	#123	—	2	12	no	—
213	C₂²⋊Q₈	—	2	6	no	—
214	C₂²⋊Q₈	—	2	6	no	—
215	#121	—	2	3	no	—
216	#121	C₂×C₄×C₅₉	1	6	yes	—
217	C₄×Q₈	C₂×C₄×C₅₉	1	1	yes	characteristic
218	C₄×Q₈	D₄×C₅₉	1	1	yes	characteristic
219	C₄×D₄	—	2	3	no	—
220	C₄×D₄	C₂×C₄×C₅₉	1	6	yes	—
221	C₄×D₄	C₂×C₄×C₅₉	1	3	yes	—
222	C₄×D₄	D₄×C₅₉	1	3	yes	—
223	#117	D₄×C₅₉	1	3	yes	—
224	C₂×C₄⋊C₄	—	2	3	no	—
225	C₂×C₄⋊C₄	C₂×C₄×C₅₉	1	6	yes	—
226	C₂×C₄²	C₂³×C₅₉	1	3	yes	—
227	C₄⋊C₈	—	2	3	no	—
228	C₄⋊C₈	—	2	3	no	—
229	C₄⋊C₈	—	2	3	no	—
230	C₄⋊C₈	C₂³×C₅₉	1	1	yes	characteristic
231	C₄⋊C₈	C₂³×C₅₉	1	3	yes	—
232	C₄⋊C₈	C₂³×C₅₉	1	3	yes	—
233	C₄⋊C₈	C₂³×C₅₉	1	1	yes	characteristic
234	C₂²⋊C₈	—	2	3	no	—
235	C₄×C₈	—	2	3	no	—
236	#41	—	4	24	no	—
237	#41	—	4	6	no	—
238	#41	—	4	6	no	—
239	#41	D₈×C₅₉	1	4	yes	—
240	#41	SD₁₆×C₅₉	1	4	yes	—
241	C₂×Q₈	—	2	6	no	—
242	C₂×Q₈	—	2	3	no	—
243	C₂×Q₈	—	4	8	no	—
244	C₂×Q₈	—	4	6	no	—
245	C₂×Q₈	—	4	6	no	—
246	C₂×Q₈	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
247	C₂×D₄	—	2	6	no	—
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	C₂²⋊C₄×C₅₉	1	3	yes	—
251	C₂×D₄	C₂²×C₄×C₅₉	1	2	yes	—
252	C₂²×C₄	—	2	6	no	—
253	C₂²×C₄	—	2	6	no	—
254	C₂²×C₄	C₂²×C₄×C₅₉	1	3	yes	—
255	C₂²×C₄	C₂×D₄×C₅₉	1	3	yes	—
256	C₂²×C₄	C₂×D₄×C₅₉	1	3	yes	—
257	M₁₆	—	4	6	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	3	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	3	no	—
262	C₂×C₈	—	4	6	no	—
263	C₂×C₈	—	8	3	no	—
264	C₄⋊C₄	—	2	6	no	—
265	C₄⋊C₄	—	2	6	no	—
266	C₄⋊C₄	—	2	3	no	—
267	C₄⋊C₄	C₂²⋊C₄×C₅₉	1	3	yes	—
268	C₄⋊C₄	C₂²⋊C₄×C₅₉	1	3	yes	—
269	C₄⋊C₄	C₂²×C₄×C₅₉	1	6	yes	—
270	C₂²⋊C₄	—	2	6	no	—
271	C₂²⋊C₄	—	2	6	no	—
272	C₂²⋊C₄	C₂²⋊C₄×C₅₉	1	6	yes	—
273	C₄²	C₂²×C₄×C₅₉	1	3	yes	—
274	C₄²	C₂×D₄×C₅₉	1	3	yes	—
275	C₄²	C₂×D₄×C₅₉	1	3	yes	—
276	C₄²	C₂⁴×C₅₉	1	1	yes	characteristic
277	C₂³	—	2	2	no	—
278	C₂³	C₂×C₂²⋊C₄×C₅₉	1	3	yes	—
279	Q₈	—	4	12	no	—
280	Q₈	—	4	3	no	—
281	Q₈	—	4	3	no	—
282	Q₈	#102×C₅₉	1	4	yes	—
283	D₄	—	4	2	no	—
284	D₄	—	4	12	no	—
285	D₄	—	4	1	no	—
286	D₄	—	4	1	no	—
287	D₄	—	4	1	no	—
288	D₄	—	4	1	no	—
289	D₄	#102×C₅₉	1	12	yes	—
290	C₂×C₄	—	2	3	no	—
291	C₂×C₄	—	2	3	no	—
292	C₂×C₄	—	2	6	no	—
293	C₂×C₄	—	4	3	no	—
294	C₂×C₄	—	4	12	no	—
295	C₂×C₄	—	4	12	no	—
296	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	3	yes	—
297	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	3	yes	—
298	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	3	yes	—
299	C₂×C₄	C₂×D₈×C₅₉	1	6	yes	—
300	C₂×C₄	C₂×SD₁₆×C₅₉	1	6	yes	—
301	C₂×C₄	C₂³×C₄×C₅₉	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
304	C₈	—	4	1	no	—
305	C₈	—	4	3	no	—
306	C₈	—	4	3	no	—
307	C₈	—	4	1	no	—
308	C₂²	—	2	3	no	—
309	C₂²	—	2	3	no	—
310	C₂²	—	4	2	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	4	2	no	—
313	C₂²	C₂×#102×C₅₉	1	6	yes	—
314	C₂²	C₂²×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	2	3	no	—
317	C₄	—	2	3	no	—
318	C₄	—	4	6	no	—
319	C₄	C₂×#102×C₅₉	1	6	yes	—
320	C₄	C₂²×D₈×C₅₉	1	1	yes	characteristic
321	C₄	C₂²×SD₁₆×C₅₉	1	1	yes	characteristic
322	C₄	C₅₉×#577	1	1	yes	characteristic
323	C₄	C₅₉×#578	1	1	yes	characteristic
324	C₂	—	2	6	no	—
325	C₂	—	8	2	no	—
326	C₂	C₅₉×#2889	1	1	yes	characteristic
327	C₂	C₂²×#102×C₅₉	1	1	yes	characteristic
328	C₂	C₅₉×#2898	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₅₉
Frattini, cosocle	C₄²	C₂⁴×C₅₉
prime radical	C₅₉×#3499	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₄×#41×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₉×#2988	C₂	semidirect
3. C₅₉×#2906	C₂	semidirect
4. C₅₉×#1547	C₂	semidirect
5. C₅₉×#1546	C₂	semidirect
6. C₅₉×#1540	C₂	semidirect
7. C₂×C₅₉×#332	C₂	semidirect
143. #17721	C₅₉	direct (N × H)
161. #2988	C₂×C₅₉	semidirect
162. #2906	C₂×C₅₉	semidirect
163. #1547	C₂×C₅₉	semidirect
164. #1546	C₂×C₅₉	semidirect
165. #1540	C₂×C₅₉	semidirect
166. C₂×#332	C₂×C₅₉	semidirect
194. C₅₉	#17721	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×#3499	2
C₅₉×#2988	2
C₅₉×#2906	2
C₅₉×#1547	2
C₅₉×#1546	2
C₅₉×#1540	2
C₂×C₅₉×#332	2
#17721	59

Subgroups — C59×#17721 #28429830

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#17721 #28429830