Subgroups — C₅₉×#13067 #28425176

Order 15104.

536 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	C₂²×M₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₉×#466	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₉×#361	C₂	1	2	yes	maximal, complemented
4	C₅₉×#1810	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#1387	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₅₉	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
8	C₂×#121×C₅₉	—	2	1	no	—
9	C₂×M₃₂×C₅₉	—	2	1	no	—
10	C₂×M₃₂×C₅₉	—	2	4	no	—
11	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₅₉	—	2	1	no	—
13	C₅₉×#466	—	2	1	no	—
14	C₅₉×#466	—	2	1	no	—
15	C₅₉×#466	C₄	1	2	yes	—
16	C₅₉×#466	C₄	1	2	yes	—
17	C₂×#108×C₅₉	C₂²	1	2	yes	—
18	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×#102×C₅₉	—	2	1	no	—
20	C₅₉×#407	—	2	1	no	—
21	C₅₉×#361	—	2	4	no	—
22	C₅₉×#341	—	2	1	no	—
23	C₂²×D₄×C₅₉	—	2	1	no	—
24	C₂²×D₄×C₅₉	—	4	1	no	—
25	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
26	C₂×D₈×C₅₉	—	2	2	no	—
27	C₂×D₈×C₅₉	—	2	1	no	—
28	C₂×D₈×C₅₉	—	2	2	no	—
29	C₂×D₈×C₅₉	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₅₉	—	2	2	no	—
31	C₂²×C₈×C₅₉	—	2	2	no	—
32	C₂²×C₈×C₅₉	—	2	2	no	—
33	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
39	#121×C₅₉	—	2	2	no	—
40	#121×C₅₉	—	2	2	no	—
41	#121×C₅₉	—	4	1	no	—
42	#121×C₅₉	—	4	1	no	—
43	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
44	C₂×C₂²⋊C₄×C₅₉	—	4	1	no	—
45	M₃₂×C₅₉	—	2	4	no	—
46	M₃₂×C₅₉	—	2	2	no	—
47	M₃₂×C₅₉	D₄	1	4	yes	complemented
48	C₂×C₁₆×C₅₉	—	2	4	no	—
49	C₂×C₁₆×C₅₉	—	2	2	no	—
50	#108×C₅₉	—	2	2	no	—
51	#108×C₅₉	—	2	2	no	—
52	#107×C₅₉	—	2	1	no	—
53	#107×C₅₉	C₂×C₄	1	2	yes	—
54	#102×C₅₉	—	2	2	no	—
55	#102×C₅₉	—	2	2	no	—
56	C₂²⋊C₈×C₅₉	—	4	1	no	—
57	C₂²⋊C₈×C₅₉	—	4	1	no	—
58	C₂⁴×C₅₉	—	4	1	no	—
59	C₂⁴×C₅₉	SD₁₆	1	1	yes	characteristic
60	C₂×D₄×C₅₉	—	2	2	no	—
61	C₂×D₄×C₅₉	—	4	2	no	—
62	C₂×D₄×C₅₉	—	4	2	no	—
63	C₂×D₄×C₅₉	—	4	2	no	—
64	C₂×D₄×C₅₉	—	4	1	no	—
65	C₂×D₄×C₅₉	—	4	2	no	—
66	C₂×D₄×C₅₉	—	8	4	no	—
67	C₂²×C₄×C₅₉	—	2	2	no	—
68	C₂²×C₄×C₅₉	—	2	2	no	—
69	C₂²×C₄×C₅₉	—	4	1	no	—
70	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
75	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
76	D₈×C₅₉	—	2	4	no	—
77	D₈×C₅₉	—	4	1	no	—
78	D₈×C₅₉	—	4	1	no	—
79	M₁₆×C₅₉	—	4	2	no	—
80	M₁₆×C₅₉	—	4	2	no	—
81	C₂×C₈×C₅₉	—	2	2	no	—
82	C₂×C₈×C₅₉	—	2	2	no	—
83	C₂×C₈×C₅₉	—	2	2	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	—	2	1	no	—
88	C₂×C₈×C₅₉	—	4	2	no	—
89	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₅₉	#41	1	2	yes	—
95	C₄⋊C₄×C₅₉	—	2	2	no	—
96	C₄⋊C₄×C₅₉	—	4	1	no	—
97	C₂²⋊C₄×C₅₉	—	4	2	no	—
98	C₂²⋊C₄×C₅₉	—	4	2	no	—
99	C₁₆×C₅₉	—	2	4	no	—
100	C₂³×C₅₉	—	2	2	no	—
101	C₂³×C₅₉	—	2	2	no	—
102	C₂³×C₅₉	—	4	1	no	—
103	C₂³×C₅₉	—	4	2	no	—
104	C₂³×C₅₉	—	8	2	no	—
105	C₂³×C₅₉	—	8	2	no	—
106	C₂³×C₅₉	—	8	2	no	—
107	C₂³×C₅₉	#102	1	1	yes	characteristic
108	C₂³×C₅₉	#102	1	1	yes	characteristic
109	C₂³×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
110	D₄×C₅₉	—	4	2	no	—
111	D₄×C₅₉	—	4	2	no	—
112	D₄×C₅₉	—	8	4	no	—
113	D₄×C₅₉	—	8	4	no	—
114	D₄×C₅₉	—	8	2	no	—
115	C₂×C₄×C₅₉	—	2	2	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	—	2	1	no	—
120	C₂×C₄×C₅₉	—	2	2	no	—
121	C₂×C₄×C₅₉	—	2	2	no	—
122	C₂×C₄×C₅₉	—	4	2	no	—
123	C₂×C₄×C₅₉	—	8	2	no	—
124	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
125	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₉	#121	1	2	yes	—
128	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₅₉	#135	1	2	yes	—
130	C₈×C₅₉	—	2	1	no	—
131	C₈×C₅₉	—	2	1	no	—
132	C₈×C₅₉	—	8	2	no	—
133	C₈×C₅₉	C₄×D₄	1	2	yes	—
134	C₈×C₅₉	#120	1	1	yes	characteristic
135	C₈×C₅₉	#123	1	1	yes	characteristic
136	#13067	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂²×C₅₉	—	2	1	no	—
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	—	2	1	no	—
144	C₂²×C₅₉	—	4	4	no	—
145	C₂²×C₅₉	—	8	2	no	—
146	C₂²×C₅₉	—	8	2	no	—
147	C₂²×C₅₉	—	8	2	no	—
148	C₂²×C₅₉	—	16	4	no	—
149	C₂²×C₅₉	#361	1	2	yes	complemented
150	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
151	C₂²×C₅₉	#465	1	2	yes	—
152	C₄×C₅₉	—	2	1	no	—
153	C₄×C₅₉	—	2	1	no	—
154	C₄×C₅₉	—	8	2	no	—
155	C₄×C₅₉	#386	1	1	yes	characteristic
156	C₄×C₅₉	#416	1	1	yes	characteristic
157	C₄×C₅₉	#466	1	2	yes	—
158	C₂²×M₃₂	C₂×C₅₉	1	1	yes	characteristic, complemented
159	C₂×#466	C₂×C₅₉	1	1	yes	characteristic
160	C₂×#361	C₂×C₅₉	1	2	yes	complemented
161	#1810	C₂×C₅₉	1	1	yes	characteristic, complemented
162	#1387	C₂×C₅₉	1	2	yes	complemented
163	C₂×C₅₉	—	2	2	no	—
164	C₂×C₅₉	—	2	2	no	—
165	C₂×C₅₉	—	4	1	no	—
166	C₂×C₅₉	—	16	2	no	—
167	C₂×C₅₉	#1893	1	1	yes	characteristic
168	C₂×C₅₉	C₂×#361	1	1	yes	characteristic
169	C₂×C₅₉	#2148	1	1	yes	characteristic
170	C₂²×D₈	C₂²×C₅₉	1	1	yes	characteristic
171	C₂³×C₈	C₂²×C₅₉	1	1	yes	characteristic
172	C₂×#121	—	2	1	no	—
173	C₂×M₃₂	—	2	1	no	—
174	C₂×M₃₂	—	2	4	no	—
175	C₂×M₃₂	C₂²×C₅₉	1	2	yes	—
176	C₂²×C₁₆	—	2	1	no	—
177	#466	—	2	1	no	—
178	#466	—	2	1	no	—
179	#466	C₄×C₅₉	1	2	yes	—
180	#466	C₄×C₅₉	1	2	yes	—
181	C₂×#108	C₂²×C₅₉	1	2	yes	—
182	C₂×#107	C₂²×C₅₉	1	1	yes	characteristic
183	C₂×#102	—	2	1	no	—
184	#407	—	2	1	no	—
185	#361	—	2	4	no	—
186	#341	—	2	1	no	—
187	C₅₉	#13067	1	1	yes	characteristic, direct product factor, Sylow
188	C₂²×D₄	—	2	1	no	—
189	C₂²×D₄	—	4	1	no	—
190	C₂³×C₄	D₄×C₅₉	1	1	yes	characteristic
191	C₂×D₈	—	2	2	no	—
192	C₂×D₈	—	2	1	no	—
193	C₂×D₈	—	2	2	no	—
194	C₂×D₈	C₂×C₄×C₅₉	1	2	yes	—
195	C₂×M₁₆	—	2	2	no	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	C₂×C₄×C₅₉	1	1	yes	characteristic
199	C₂²×C₈	C₂×C₄×C₅₉	1	1	yes	characteristic
200	C₂²×C₈	D₄×C₅₉	1	1	yes	characteristic
201	C₂²×C₈	D₄×C₅₉	1	1	yes	characteristic
202	C₂²×C₈	D₄×C₅₉	1	1	yes	characteristic
203	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
204	#121	—	2	2	no	—
205	#121	—	2	2	no	—
206	#121	—	4	1	no	—
207	#121	—	4	1	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₂²⋊C₄	—	4	1	no	—
210	M₃₂	—	2	4	no	—
211	M₃₂	—	2	2	no	—
212	M₃₂	D₄×C₅₉	1	4	yes	complemented
213	C₂×C₁₆	—	2	4	no	—
214	C₂×C₁₆	—	2	2	no	—
215	#108	—	2	2	no	—
216	#108	—	2	2	no	—
217	#107	—	2	1	no	—
218	#107	C₂×C₄×C₅₉	1	2	yes	—
219	#102	—	2	2	no	—
220	#102	—	2	2	no	—
221	C₂²⋊C₈	—	4	1	no	—
222	C₂²⋊C₈	—	4	1	no	—
223	C₂⁴	—	4	1	no	—
224	C₂⁴	SD₁₆×C₅₉	1	1	yes	characteristic
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂×D₄	—	4	1	no	—
230	C₂×D₄	—	4	2	no	—
231	C₂×D₄	—	8	4	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
236	C₂²×C₄	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
237	C₂²×C₄	D₈×C₅₉	1	1	yes	characteristic
238	C₂²×C₄	D₈×C₅₉	1	1	yes	characteristic
239	C₂²×C₄	SD₁₆×C₅₉	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
241	D₈	—	2	4	no	—
242	D₈	—	4	1	no	—
243	D₈	—	4	1	no	—
244	M₁₆	—	4	2	no	—
245	M₁₆	—	4	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
255	C₂×C₈	C₂²⋊C₄×C₅₉	1	1	yes	characteristic
256	C₂×C₈	C₂²×C₄×C₅₉	1	1	yes	characteristic
257	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
258	C₂×C₈	C₂×D₄×C₅₉	1	1	yes	characteristic
259	C₂×C₈	#41×C₅₉	1	2	yes	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₁₆	—	2	4	no	—
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	4	1	no	—
268	C₂³	—	4	2	no	—
269	C₂³	—	8	2	no	—
270	C₂³	—	8	2	no	—
271	C₂³	—	8	2	no	—
272	C₂³	#102×C₅₉	1	1	yes	characteristic
273	C₂³	#102×C₅₉	1	1	yes	characteristic
274	C₂³	C₂×SD₁₆×C₅₉	1	1	yes	characteristic
275	D₄	—	4	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	8	4	no	—
278	D₄	—	8	4	no	—
279	D₄	—	8	2	no	—
280	C₂×C₄	—	2	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	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	8	2	no	—
289	C₂×C₄	#102×C₅₉	1	1	yes	characteristic
290	C₂×C₄	#102×C₅₉	1	1	yes	characteristic
291	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
292	C₂×C₄	#121×C₅₉	1	2	yes	—
293	C₂×C₄	C₂×D₈×C₅₉	1	1	yes	characteristic
294	C₂×C₄	#135×C₅₉	1	2	yes	—
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	—	8	2	no	—
298	C₈	C₄×D₄×C₅₉	1	2	yes	—
299	C₈	#120×C₅₉	1	1	yes	characteristic
300	C₈	#123×C₅₉	1	1	yes	characteristic
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	4	4	no	—
309	C₂²	—	8	2	no	—
310	C₂²	—	8	2	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	16	4	no	—
313	C₂²	C₅₉×#361	1	2	yes	complemented
314	C₂²	C₂×#102×C₅₉	1	1	yes	characteristic
315	C₂²	C₅₉×#465	1	2	yes	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	8	2	no	—
319	C₄	C₅₉×#386	1	1	yes	characteristic
320	C₄	C₅₉×#416	1	1	yes	characteristic
321	C₄	C₅₉×#466	1	2	yes	—
322	C₂	—	2	2	no	—
323	C₂	—	2	2	no	—
324	C₂	—	4	1	no	—
325	C₂	—	16	2	no	—
326	C₂	C₅₉×#1893	1	1	yes	characteristic
327	C₂	C₂×C₅₉×#361	1	1	yes	characteristic
328	C₂	C₅₉×#2148	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅₉
Frattini, cosocle	C₂²×C₈	C₂³×C₅₉
prime radical	C₂×C₅₉×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₂³×C₄×C₅₉ ◁ C₂³×C₈×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×M₃₂×C₅₉	C₂	semidirect
3. C₂×C₅₉×#361	C₂	semidirect
4. C₅₉×#1810	C₂	semidirect
5. C₅₉×#1387	C₂	semidirect
47. M₃₂×C₅₉	D₄	semidirect
136. #13067	C₅₉	direct (N × H)
149. C₂²×C₅₉	#361	semidirect
158. C₂²×M₃₂	C₂×C₅₉	semidirect
160. C₂×#361	C₂×C₅₉	semidirect
161. #1810	C₂×C₅₉	semidirect
162. #1387	C₂×C₅₉	semidirect
187. C₅₉	#13067	direct (N × H)
212. M₃₂	D₄×C₅₉	semidirect
313. C₂²	C₅₉×#361	semidirect

Maximal subgroups

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

subgroup	index
C₂²×M₃₂×C₅₉	2
C₂×C₅₉×#466	2
C₂×C₅₉×#361	2
C₅₉×#1810	2
C₅₉×#1387	2
#13067	59

Subgroups — C59×#13067 #28425176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#13067 #28425176