Subgroups — C₅₉×#8018 #28420127

Order 15104.

678 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	C₂×C₅₉×#432	C₂	1	1	yes	maximal, characteristic
2	C₅₉×#2285	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₉×#383	C₂	1	1	yes	maximal, characteristic
4	C₅₉×#1294	C₂	1	4	yes	maximal
5	C₂×#117×C₅₉	—	2	1	no	—
6	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
7	C₅₉×#432	—	2	2	no	—
8	C₅₉×#432	C₄	1	4	yes	complemented
9	C₂×C₄⋊C₈×C₅₉	—	2	1	no	—
10	C₂×C₂²⋊C₈×C₅₉	C₂²	1	2	yes	—
11	C₂×#97×C₅₉	—	2	1	no	—
12	C₅₉×#384	C₄	1	2	yes	complemented
13	C₅₉×#383	—	2	2	no	—
14	C₅₉×#383	C₄	1	4	yes	complemented
15	C₅₉×#382	—	2	1	no	—
16	C₅₉×#380	—	2	1	no	—
17	C₅₉×#380	C₂²	1	2	yes	—
18	C₄×C₄⋊C₄×C₅₉	—	2	1	no	—
19	C₄×C₄⋊C₄×C₅₉	C₄	1	2	yes	complemented
20	C₂×#95×C₅₉	—	2	1	no	—
21	C₂×#95×C₅₉	—	2	1	no	—
22	C₂×#95×C₅₉	C₂²	1	2	yes	—
23	C₂³×C₄×C₅₉	—	2	2	no	—
24	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₅₉	—	2	2	no	—
28	#117×C₅₉	—	4	4	no	—
29	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
31	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	4	yes	—
32	C₂×C₂²⋊C₄×C₅₉	—	2	2	no	—
33	C₂×C₄²×C₅₉	—	2	2	no	—
34	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₉	D₄	1	2	yes	—
41	C₂×C₄²×C₅₉	Q₈	1	2	yes	—
42	C₄⋊C₈×C₅₉	—	2	4	no	—
43	C₂²⋊C₈×C₅₉	—	2	2	no	—
44	C₂²⋊C₈×C₅₉	C₂×C₄	1	4	yes	—
45	#97×C₅₉	—	2	4	no	—
46	#95×C₅₉	—	2	4	no	—
47	#95×C₅₉	—	2	4	no	—
48	#95×C₅₉	—	2	2	no	—
49	#95×C₅₉	—	2	2	no	—
50	#95×C₅₉	—	2	2	no	—
51	#95×C₅₉	C₂×C₄	1	4	yes	—
52	C₂⁴×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₉	—	2	2	no	—
54	C₂²×C₄×C₅₉	—	2	2	no	—
55	C₂²×C₄×C₅₉	—	2	4	no	—
56	C₂²×C₄×C₅₉	—	2	4	no	—
57	C₂²×C₄×C₅₉	—	4	2	no	—
58	C₂²×C₄×C₅₉	—	4	2	no	—
59	C₂²×C₄×C₅₉	—	4	4	no	—
60	C₂²×C₄×C₅₉	—	4	4	no	—
61	C₂²×C₄×C₅₉	—	4	2	no	—
62	C₂²×C₄×C₅₉	—	4	2	no	—
63	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₉	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₈×C₅₉	—	2	4	no	—
78	C₂×C₈×C₅₉	—	4	4	no	—
79	C₄⋊C₄×C₅₉	—	4	4	no	—
80	C₄⋊C₄×C₅₉	—	4	4	no	—
81	C₄⋊C₄×C₅₉	—	4	4	no	—
82	C₂²⋊C₄×C₅₉	—	4	4	no	—
83	C₄²×C₅₉	—	4	4	no	—
84	C₄²×C₅₉	C₄²	1	2	yes	—
85	C₄²×C₅₉	C₄²	1	2	yes	—
86	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
88	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
89	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
90	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
91	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
92	C₂³×C₅₉	—	2	2	no	—
93	C₂³×C₅₉	—	2	2	no	—
94	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂³×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
98	C₂³×C₅₉	#137	1	2	yes	—
99	C₂×C₄×C₅₉	—	2	2	no	—
100	C₂×C₄×C₅₉	—	2	2	no	—
101	C₂×C₄×C₅₉	—	2	4	no	—
102	C₂×C₄×C₅₉	—	2	1	no	—
103	C₂×C₄×C₅₉	—	2	1	no	—
104	C₂×C₄×C₅₉	—	2	1	no	—
105	C₂×C₄×C₅₉	—	2	1	no	—
106	C₂×C₄×C₅₉	—	2	2	no	—
107	C₂×C₄×C₅₉	—	2	2	no	—
108	C₂×C₄×C₅₉	—	2	4	no	—
109	C₂×C₄×C₅₉	—	4	2	no	—
110	C₂×C₄×C₅₉	—	4	2	no	—
111	C₂×C₄×C₅₉	—	4	1	no	—
112	C₂×C₄×C₅₉	—	4	1	no	—
113	C₂×C₄×C₅₉	—	4	1	no	—
114	C₂×C₄×C₅₉	—	4	1	no	—
115	C₂×C₄×C₅₉	—	4	4	no	—
116	C₂×C₄×C₅₉	—	4	4	no	—
117	C₂×C₄×C₅₉	—	4	4	no	—
118	C₂×C₄×C₅₉	—	4	4	no	—
119	C₂×C₄×C₅₉	—	4	4	no	—
120	C₂×C₄×C₅₉	—	4	4	no	—
121	C₂×C₄×C₅₉	—	8	4	no	—
122	C₂×C₄×C₅₉	—	8	4	no	—
123	C₂×C₄×C₅₉	#95	1	2	yes	—
124	C₂×C₄×C₅₉	#95	1	2	yes	—
125	C₂×C₄×C₅₉	#95	1	2	yes	—
126	C₂×C₄×C₅₉	#95	1	2	yes	—
127	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
128	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₈×C₅₉	—	4	4	no	—
132	#8018	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂²×C₅₉	—	2	1	no	—
134	C₂²×C₅₉	—	2	1	no	—
135	C₂²×C₅₉	—	2	1	no	—
136	C₂²×C₅₉	—	2	1	no	—
137	C₂²×C₅₉	—	2	4	no	—
138	C₂²×C₅₉	—	2	4	no	—
139	C₂²×C₅₉	C₂×#95	1	1	yes	characteristic
140	C₂²×C₅₉	#410	1	2	yes	—
141	C₂²×C₅₉	#417	1	2	yes	—
142	C₂²×C₅₉	#418	1	1	yes	characteristic
143	C₂²×C₅₉	#419	1	1	yes	characteristic
144	C₂²×C₅₉	#421	1	2	yes	—
145	C₂²×C₅₉	#428	1	2	yes	—
146	C₄×C₅₉	—	2	2	no	—
147	C₄×C₅₉	—	2	2	no	—
148	C₄×C₅₉	—	2	2	no	—
149	C₄×C₅₉	—	2	2	no	—
150	C₄×C₅₉	—	2	1	no	—
151	C₄×C₅₉	—	2	1	no	—
152	C₄×C₅₉	—	2	1	no	—
153	C₄×C₅₉	—	2	1	no	—
154	C₄×C₅₉	—	8	4	no	—
155	C₄×C₅₉	—	8	4	no	—
156	C₂×#432	C₂×C₅₉	1	1	yes	characteristic
157	#2285	C₂×C₅₉	1	1	yes	characteristic
158	C₂×#383	C₂×C₅₉	1	1	yes	characteristic
159	#1294	C₂×C₅₉	1	4	yes	—
160	C₂×C₅₉	—	2	2	no	—
161	C₂×C₅₉	—	2	2	no	—
162	C₂×C₅₉	#1514	1	2	yes	—
163	C₂×C₅₉	#1515	1	2	yes	—
164	C₂×C₅₉	#1733	1	1	yes	characteristic
165	C₂×C₅₉	#1736	1	1	yes	characteristic
166	C₂×C₅₉	#1740	1	1	yes	characteristic
167	C₂×#117	—	2	1	no	—
168	C₂²×C₄²	C₂²×C₅₉	1	1	yes	characteristic
169	#432	—	2	2	no	—
170	#432	C₄×C₅₉	1	4	yes	complemented
171	C₂×C₄⋊C₈	—	2	1	no	—
172	C₂×C₂²⋊C₈	C₂²×C₅₉	1	2	yes	—
173	C₂×#97	—	2	1	no	—
174	#384	C₄×C₅₉	1	2	yes	complemented
175	#383	—	2	2	no	—
176	#383	C₄×C₅₉	1	4	yes	complemented
177	#382	—	2	1	no	—
178	#380	—	2	1	no	—
179	#380	C₂²×C₅₉	1	2	yes	—
180	C₄×C₄⋊C₄	—	2	1	no	—
181	C₄×C₄⋊C₄	C₄×C₅₉	1	2	yes	complemented
182	C₂×#95	—	2	1	no	—
183	C₂×#95	—	2	1	no	—
184	C₂×#95	C₂²×C₅₉	1	2	yes	—
185	C₅₉	#8018	1	1	yes	characteristic, direct product factor, Sylow
186	C₂³×C₄	—	2	2	no	—
187	C₂³×C₄	D₄×C₅₉	1	1	yes	characteristic
188	C₂³×C₄	D₄×C₅₉	1	1	yes	characteristic
189	C₂³×C₄	C₂³×C₅₉	1	1	yes	characteristic
190	C₂²×C₈	—	2	2	no	—
191	#117	—	4	4	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	C₂×C₄×C₅₉	1	4	yes	—
195	C₂×C₂²⋊C₄	—	2	2	no	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	C₂×C₄×C₅₉	1	2	yes	—
198	C₂×C₄²	C₂×C₄×C₅₉	1	1	yes	characteristic
199	C₂×C₄²	C₂×C₄×C₅₉	1	2	yes	—
200	C₂×C₄²	C₂×C₄×C₅₉	1	1	yes	characteristic
201	C₂×C₄²	D₄×C₅₉	1	1	yes	characteristic
202	C₂×C₄²	D₄×C₅₉	1	1	yes	characteristic
203	C₂×C₄²	D₄×C₅₉	1	2	yes	—
204	C₂×C₄²	Q₈×C₅₉	1	2	yes	—
205	C₄⋊C₈	—	2	4	no	—
206	C₂²⋊C₈	—	2	2	no	—
207	C₂²⋊C₈	C₂×C₄×C₅₉	1	4	yes	—
208	#97	—	2	4	no	—
209	#95	—	2	4	no	—
210	#95	—	2	4	no	—
211	#95	—	2	2	no	—
212	#95	—	2	2	no	—
213	#95	—	2	2	no	—
214	#95	C₂×C₄×C₅₉	1	4	yes	—
215	C₂⁴	C₂×D₄×C₅₉	1	1	yes	characteristic
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	—	4	4	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	C₂²⋊C₄×C₅₉	1	2	yes	—
227	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
228	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
229	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
230	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
231	C₂²×C₄	C₂²⋊C₄×C₅₉	1	2	yes	—
232	C₂²×C₄	C₄⋊C₄×C₅₉	1	2	yes	—
233	C₂²×C₄	C₄⋊C₄×C₅₉	1	2	yes	—
234	C₂²×C₄	C₂²×C₄×C₅₉	1	1	yes	characteristic
235	C₂²×C₄	C₂²×C₄×C₅₉	1	1	yes	characteristic
236	C₂²×C₄	C₂²×C₄×C₅₉	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
239	C₂²×C₄	C₂×Q₈×C₅₉	1	1	yes	characteristic
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	4	4	no	—
242	C₄⋊C₄	—	4	4	no	—
243	C₄⋊C₄	—	4	4	no	—
244	C₄⋊C₄	—	4	4	no	—
245	C₂²⋊C₄	—	4	4	no	—
246	C₄²	—	4	4	no	—
247	C₄²	C₄²×C₅₉	1	2	yes	—
248	C₄²	C₄²×C₅₉	1	2	yes	—
249	C₄²	C₂²⋊C₄×C₅₉	1	2	yes	—
250	C₄²	C₂²⋊C₄×C₅₉	1	2	yes	—
251	C₄²	C₄⋊C₄×C₅₉	1	2	yes	—
252	C₄²	C₄⋊C₄×C₅₉	1	2	yes	—
253	C₄²	C₄⋊C₄×C₅₉	1	2	yes	—
254	C₄²	C₄⋊C₄×C₅₉	1	2	yes	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
258	C₂³	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
259	C₂³	C₂×C₄⋊C₄×C₅₉	1	1	yes	characteristic
260	C₂³	AGL(1,ℤ/8)×C₅₉	1	2	yes	—
261	C₂³	#137×C₅₉	1	2	yes	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	8	4	no	—
285	C₂×C₄	—	8	4	no	—
286	C₂×C₄	#95×C₅₉	1	2	yes	—
287	C₂×C₄	#95×C₅₉	1	2	yes	—
288	C₂×C₄	#95×C₅₉	1	2	yes	—
289	C₂×C₄	#95×C₅₉	1	2	yes	—
290	C₂×C₄	C₂×C₄²×C₅₉	1	1	yes	characteristic
291	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
292	C₂×C₄	C₂×C₄⋊C₄×C₅₉	1	1	yes	characteristic
293	C₂×C₄	C₂×C₄⋊C₄×C₅₉	1	1	yes	characteristic
294	C₈	—	4	4	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	4	no	—
301	C₂²	C₂×#95×C₅₉	1	1	yes	characteristic
302	C₂²	C₅₉×#410	1	2	yes	—
303	C₂²	C₅₉×#417	1	2	yes	—
304	C₂²	C₅₉×#418	1	1	yes	characteristic
305	C₂²	C₅₉×#419	1	1	yes	characteristic
306	C₂²	C₅₉×#421	1	2	yes	—
307	C₂²	C₅₉×#428	1	2	yes	—
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	8	4	no	—
317	C₄	—	8	4	no	—
318	C₂	—	2	2	no	—
319	C₂	—	2	2	no	—
320	C₂	C₅₉×#1514	1	2	yes	—
321	C₂	C₅₉×#1515	1	2	yes	—
322	C₂	C₅₉×#1733	1	1	yes	characteristic
323	C₂	C₅₉×#1736	1	1	yes	characteristic
324	C₂	C₅₉×#1740	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₉	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₅₉
Frattini, cosocle	C₂³×C₄	C₂³×C₅₉
prime radical	C₂⁴×C₅₉	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₉ ◁ C₂²×C₄²×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₅₉×#432	C₄	semidirect
12. C₅₉×#384	C₄	semidirect
14. C₅₉×#383	C₄	semidirect
19. C₄×C₄⋊C₄×C₅₉	C₄	semidirect
132. #8018	C₅₉	direct (N × H)
170. #432	C₄×C₅₉	semidirect
174. #384	C₄×C₅₉	semidirect
176. #383	C₄×C₅₉	semidirect
181. C₄×C₄⋊C₄	C₄×C₅₉	semidirect
185. C₅₉	#8018	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×#432	2
C₅₉×#2285	2
C₂×C₅₉×#383	2
C₅₉×#1294	2
#8018	59

Subgroups — C59×#8018 #28420127

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#8018 #28420127