Subgroups — #21434142

Order 7552.

454 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837931	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₄×D₅₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837736	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837719	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837703	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837702	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837614	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837551	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837534	C₂	1	1	yes	maximal, characteristic
10	SD₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837487	C₂	1	1	yes	maximal, characteristic, complemented
12	D₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837482	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837454	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#463	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₅₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₅₉	—	2	1	no	—
18	C₂×#11745671	C₂²	1	1	yes	characteristic
19	C₂×#11745669	C₂²	1	1	yes	characteristic
20	#14273720	—	2	1	no	—
21	#14273719	C₂²	1	1	yes	characteristic, complemented
22	#14273709	—	2	1	no	—
23	#14273703	C₂²	1	1	yes	characteristic
24	#14273698	C₂²	1	1	yes	characteristic
25	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic, complemented
26	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
27	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
28	D₄×Dic₅₉	C₂²	1	1	yes	characteristic, complemented
29	#14273689	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊D₅₉	—	2	1	no	—
31	(C₂²⋊D₅₉)⋊C₄	—	2	1	no	—
32	(C₂²⋊D₅₉)⋊C₄	—	2	1	no	—
33	#14273683	—	2	1	no	—
34	C₄⋊C₄×D₅₉	C₂²	1	1	yes	characteristic
35	C₂²⋊C₄×D₅₉	—	2	1	no	—
36	C₄²×D₅₉	C₂²	1	1	yes	characteristic
37	#14273661	C₂²	1	1	yes	characteristic, complemented
38	#14273660	C₂²	1	1	yes	characteristic, complemented
39	#14273659	C₂²	1	1	yes	characteristic, complemented
40	#14273658	C₂²	1	1	yes	characteristic, complemented
41	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic, complemented
42	#14273651	C₂²	1	1	yes	characteristic, complemented
43	#14273649	C₂²	1	1	yes	characteristic
44	#14273649	C₂²	1	1	yes	characteristic
45	#14273648	C₂²	1	1	yes	characteristic
46	#14273648	C₂²	1	1	yes	characteristic
47	#14273648	C₂²	1	1	yes	characteristic, complemented
48	(C₂×D₅₉)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×D₅₉)⋊C₈	C₂²	1	1	yes	characteristic
50	C₈×Dic₅₉	C₂²	1	1	yes	characteristic, complemented
51	C₈×Dic₅₉	C₂²	1	1	yes	characteristic, complemented
52	C₂×SD₁₆×C₅₉	C₂²	1	1	yes	characteristic
53	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
54	#124×C₅₉	C₂²	1	1	yes	characteristic, complemented
55	C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic, complemented
57	#103×C₅₉	C₂²	1	1	yes	characteristic, complemented
58	#102×C₅₉	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₅₉	—	2	1	no	—
60	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₅₉	—	2	1	no	—
64	C₂×C₂²⋊D₅₉	—	2	1	no	—
65	C₂×C₂²⋊D₅₉	—	2	1	no	—
66	C₂²×Dic₅₉	—	2	1	no	—
67	C₂²×Dic₅₉	—	2	1	no	—
68	D₄×D₅₉	—	2	2	no	—
69	D₄×D₅₉	—	2	2	no	—
70	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
71	C₂×C₄×D₅₉	—	2	1	no	—
72	C₂×C₄×D₅₉	—	2	1	no	—
73	C₂×C₄×D₅₉	—	2	1	no	—
74	C₂×C₄×D₅₉	—	4	4	no	—
75	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
76	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
77	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
78	SD₁₆×C₅₉	—	2	2	no	—
79	D₈×C₅₉	—	2	2	no	—
80	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
81	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
83	C₂²⋊C₄×C₅₉	—	2	1	no	—
84	C₂²⋊C₄×C₅₉	—	2	1	no	—
85	C₂²⋊C₄×C₅₉	—	2	1	no	—
86	C₄²×C₅₉	C₂³	1	1	yes	characteristic
87	C₂²⋊Dic₅₉	—	2	1	no	—
88	C₂²⋊Dic₅₉	—	2	1	no	—
89	C₂²⋊Dic₅₉	—	2	1	no	—
90	#11745671	—	2	2	no	—
91	#11745669	—	2	2	no	—
92	(C₂×D₅₉)⋊C₄	—	2	1	no	—
93	(C₂×D₅₉)⋊C₄	—	2	1	no	—
94	(C₂×D₅₉)⋊C₄	—	2	1	no	—
95	C₄⋊Dic₅₉	—	2	1	no	—
96	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
97	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
98	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
99	Dic₅₉⋊C₄	—	2	1	no	—
100	Dic₅₉⋊C₄	—	2	1	no	—
101	Dic₅₉⋊C₄	—	2	1	no	—
102	Dic₅₉⋊C₄	—	2	1	no	—
103	C₄×Dic₅₉	—	2	1	no	—
104	C₄×Dic₅₉	D₄	1	1	yes	characteristic
105	C₄×Dic₅₉	D₄	1	1	yes	characteristic
106	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
107	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
108	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
109	C₂³×C₅₉	—	2	1	no	—
110	C₂³×C₅₉	—	2	1	no	—
111	C₂²×D₅₉	—	2	4	no	—
112	C₂²×D₅₉	—	2	1	no	—
113	C₂²×D₅₉	—	4	2	no	—
114	C₂²×D₅₉	—	4	2	no	—
115	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
116	Q₈×C₅₉	—	2	1	no	—
117	Q₈×C₅₉	—	2	1	no	—
118	D₄×C₅₉	—	2	1	no	—
119	D₄×C₅₉	—	2	1	no	—
120	D₄×C₅₉	—	2	1	no	—
121	D₄×C₅₉	#41	1	2	yes	—
122	C₂×C₄×C₅₉	—	2	1	no	—
123	C₂×C₄×C₅₉	—	2	1	no	—
124	C₂×C₄×C₅₉	—	4	2	no	—
125	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
128	C₂²⋊D₅₉	—	2	4	no	—
129	C₂²⋊D₅₉	—	4	2	no	—
130	C₂×Dic₅₉	—	2	1	no	—
131	C₂×Dic₅₉	—	2	1	no	—
132	C₂×Dic₅₉	—	2	1	no	—
133	C₂×Dic₅₉	—	4	2	no	—
134	C₂×Dic₅₉	—	4	2	no	—
135	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
138	D₂₃₆	—	2	1	no	—
139	D₂₃₆	#41	1	2	yes	—
140	C₄×D₅₉	—	2	2	no	—
141	C₄×D₅₉	—	2	1	no	—
142	C₄×D₅₉	—	2	1	no	—
143	C₄×D₅₉	—	2	2	no	—
144	C₄×D₅₉	—	4	2	no	—
145	C₄×D₅₉	—	4	2	no	—
146	C₄×D₅₉	—	4	2	no	—
147	C₈×C₅₉	—	2	1	no	—
148	C₈×C₅₉	—	2	1	no	—
149	C₅₉⋊C₈	—	2	1	no	—
150	C₅₉⋊C₈	—	2	1	no	—
151	C₂²×C₅₉	—	2	2	no	—
152	C₂²×C₅₉	—	4	1	no	—
153	C₂²×C₅₉	—	4	1	no	—
154	C₂²×C₅₉	—	4	1	no	—
155	C₂²×C₅₉	—	4	1	no	—
156	C₂²×C₅₉	—	4	1	no	—
157	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
158	C₂×D₅₉	—	2	1	no	—
159	C₂×D₅₉	—	2	1	no	—
160	C₂×D₅₉	—	2	2	no	—
161	C₂×D₅₉	—	4	8	no	—
162	C₂×D₅₉	—	4	1	no	—
163	C₂×D₅₉	—	4	1	no	—
164	C₂×D₅₉	#135	1	2	yes	—
165	C₄×C₅₉	—	2	1	no	—
166	C₄×C₅₉	—	2	1	no	—
167	C₄×C₅₉	—	4	1	no	—
168	C₄×C₅₉	—	4	1	no	—
169	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
170	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
171	Dic₅₉	—	2	1	no	—
172	Dic₅₉	—	2	1	no	—
173	Dic₅₉	—	4	1	no	—
174	Dic₅₉	—	4	1	no	—
175	Dic₅₉	—	4	1	no	—
176	Dic₅₉	AGL(1,ℤ/8)	1	2	yes	—
177	#3105	—	59	1	no	maximal, Sylow
178	C₂×C₅₉	—	4	2	no	—
179	C₂×C₅₉	—	8	1	no	—
180	C₂×C₅₉	#525	1	1	yes	characteristic
181	C₂×C₅₉	C₂×#135	1	1	yes	characteristic
182	C₂×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
183	D₅₉	—	2	2	no	—
184	D₅₉	—	4	2	no	—
185	#529	—	59	1	no	—
186	C₂×C₄×D₄	—	59	1	no	—
187	#484	—	59	1	no	—
188	#481	—	59	1	no	—
189	#478	—	59	1	no	—
190	#478	—	59	1	no	—
191	#463	—	59	1	no	—
192	#463	D₅₉	1	1	yes	characteristic, complemented
193	#450	—	59	1	no	—
194	#447	—	59	1	no	—
195	C₄×SD₁₆	—	59	1	no	—
196	C₄×SD₁₆	—	59	1	no	—
197	C₄×D₈	—	59	1	no	—
198	C₄×D₈	—	59	1	no	—
199	#431	—	59	1	no	—
200	C₅₉	#3105	1	1	yes	characteristic, complemented, Sylow
201	C₂²×D₄	—	59	1	no	—
202	C₂³×C₄	—	118	1	no	—
203	C₂×SD₁₆	—	59	1	no	—
204	C₂×SD₁₆	C₂×D₅₉	1	1	yes	characteristic
205	C₂×D₈	—	59	1	no	—
206	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
207	#126	—	118	1	no	—
208	#125	—	59	1	no	—
209	#124	C₂×D₅₉	1	1	yes	characteristic, complemented
210	#123	—	118	1	no	—
211	C₂²⋊Q₈	—	59	1	no	—
212	#121	—	59	1	no	—
213	C₄×Q₈	—	59	1	no	—
214	C₄×D₄	—	59	1	no	—
215	C₄×D₄	—	59	1	no	—
216	C₄×D₄	—	59	1	no	—
217	C₄×D₄	—	59	1	no	—
218	C₄×D₄	—	118	1	no	—
219	C₄×D₄	—	118	1	no	—
220	C₄×D₄	—	118	1	no	—
221	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
222	#117	—	118	1	no	—
223	C₂×C₄⋊C₄	—	59	1	no	—
224	C₂×C₂²⋊C₄	—	118	1	no	—
225	C₂×C₄²	—	59	1	no	—
226	#107	—	59	1	no	—
227	#107	—	59	1	no	—
228	#106	—	59	1	no	—
229	#106	—	59	1	no	—
230	C₄⋊C₈	—	59	1	no	—
231	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic, complemented
232	#103	—	59	1	no	—
233	#103	C₂×D₅₉	1	1	yes	characteristic, complemented
234	#102	—	59	1	no	—
235	#102	—	59	1	no	—
236	#102	—	59	1	no	—
237	#102	—	59	1	no	—
238	#102	—	59	1	no	—
239	#102	C₂×D₅₉	1	1	yes	characteristic, complemented
240	C₂²⋊C₈	—	59	1	no	—
241	C₂²⋊C₈	—	59	1	no	—
242	C₄×C₈	—	59	1	no	—
243	C₄×C₈	—	59	1	no	—
244	C₂⁴	—	118	1	no	—
245	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
246	C₂×D₄	—	59	1	no	—
247	C₂×D₄	—	118	1	no	—
248	C₂×D₄	—	118	2	no	—
249	C₂×D₄	—	118	2	no	—
250	C₂×D₄	—	118	1	no	—
251	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
252	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	59	1	no	—
255	C₂²×C₄	—	59	1	no	—
256	C₂²×C₄	—	59	1	no	—
257	C₂²×C₄	—	118	1	no	—
258	C₂²×C₄	—	118	1	no	—
259	C₂²×C₄	—	118	1	no	—
260	C₂²×C₄	—	118	1	no	—
261	C₂²×C₄	—	118	1	no	—
262	C₂²×C₄	—	236	4	no	—
263	SD₁₆	—	2	2	no	—
264	SD₁₆	—	118	2	no	—
265	D₈	—	2	2	no	—
266	D₈	—	118	2	no	—
267	C₂×C₈	—	59	1	no	—
268	C₂×C₈	—	59	1	no	—
269	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
270	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
271	C₄⋊C₄	—	59	1	no	—
272	C₄⋊C₄	—	59	1	no	—
273	C₄⋊C₄	—	59	1	no	—
274	C₄⋊C₄	—	118	1	no	—
275	C₄⋊C₄	—	118	1	no	—
276	C₄⋊C₄	—	118	1	no	—
277	C₄⋊C₄	—	118	1	no	—
278	C₄⋊C₄	—	118	1	no	—
279	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	2	1	no	—
282	C₂²⋊C₄	—	2	1	no	—
283	C₂²⋊C₄	—	118	1	no	—
284	C₂²⋊C₄	—	118	1	no	—
285	C₂²⋊C₄	—	118	1	no	—
286	C₂²⋊C₄	—	118	1	no	—
287	C₂²⋊C₄	—	118	1	no	—
288	C₂²⋊C₄	—	118	1	no	—
289	C₄²	—	59	1	no	—
290	C₄²	—	59	1	no	—
291	C₄²	—	59	1	no	—
292	C₄²	—	118	1	no	—
293	C₄²	C₂²×D₅₉	1	1	yes	characteristic
294	C₂³	—	2	1	no	—
295	C₂³	—	2	1	no	—
296	C₂³	—	59	1	no	—
297	C₂³	—	118	4	no	—
298	C₂³	—	118	1	no	—
299	C₂³	—	236	2	no	—
300	C₂³	—	236	2	no	—
301	Q₈	—	2	1	no	—
302	Q₈	—	2	1	no	—
303	D₄	—	2	1	no	—
304	D₄	—	2	1	no	—
305	D₄	—	2	1	no	—
306	D₄	—	59	2	no	—
307	D₄	—	118	4	no	—
308	D₄	—	118	1	no	—
309	D₄	—	236	2	no	—
310	D₄	#11745687	1	2	yes	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	59	1	no	—
315	C₂×C₄	—	59	1	no	—
316	C₂×C₄	—	59	1	no	—
317	C₂×C₄	—	118	2	no	—
318	C₂×C₄	—	118	2	no	—
319	C₂×C₄	—	118	1	no	—
320	C₂×C₄	—	118	1	no	—
321	C₂×C₄	—	118	1	no	—
322	C₂×C₄	—	118	1	no	—
323	C₂×C₄	—	118	1	no	—
324	C₂×C₄	—	236	2	no	—
325	C₂×C₄	—	236	2	no	—
326	C₂×C₄	—	236	2	no	—
327	C₂×C₄	—	236	2	no	—
328	C₂×C₄	—	236	2	no	—
329	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
330	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
331	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	118	1	no	—
335	C₈	—	118	1	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	59	2	no	—
343	C₂²	—	118	1	no	—
344	C₂²	—	118	1	no	—
345	C₂²	—	118	2	no	—
346	C₂²	—	236	8	no	—
347	C₂²	—	236	1	no	—
348	C₂²	—	236	1	no	—
349	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	4	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	59	2	no	—
355	C₄	—	118	1	no	—
356	C₄	—	118	1	no	—
357	C₄	—	236	1	no	—
358	C₄	—	236	1	no	—
359	C₄	—	236	1	no	—
360	C₄	#41×D₅₉	1	1	yes	characteristic
361	C₄	C₂×#11745687	1	1	yes	characteristic
362	C₂	—	4	2	no	—
363	C₂	—	8	1	no	—
364	C₂	—	118	2	no	—
365	C₂	—	236	2	no	—
366	C₂	#16837912	1	1	yes	characteristic
367	C₂	#135×D₅₉	1	1	yes	characteristic
368	C₂	AGL(1,ℤ/8)×D₅₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#463	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#463	D₅₉
hypocenter	C₅₉	#3105
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #463

Complements

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

N	H ≅ G/N	type
1. #16837931	C₂	semidirect
2. C₄×D₄×D₅₉	C₂	semidirect
3. #16837736	C₂	semidirect
4. #16837719	C₂	semidirect
5. #16837703	C₂	semidirect
6. #16837702	C₂	semidirect
7. #16837614	C₂	semidirect
8. #16837551	C₂	semidirect
10. SD₁₆×Dic₅₉	C₂	semidirect
11. #16837487	C₂	semidirect
12. D₈×Dic₅₉	C₂	semidirect
13. #16837482	C₂	semidirect
14. #16837454	C₂	semidirect
15. C₅₉×#463	C₂	semidirect
21. #14273719	C₂²	semidirect
25. Q₈×Dic₅₉	C₂²	semidirect
28. D₄×Dic₅₉	C₂²	semidirect
37. #14273661	C₂²	semidirect
38. #14273660	C₂²	semidirect
39. #14273659	C₂²	semidirect
40. #14273658	C₂²	semidirect
41. C₄⋊(C₅₉⋊C₈)	C₂²	semidirect
42. #14273651	C₂²	semidirect
47. #14273648	C₂²	semidirect
50. C₈×Dic₅₉	C₂²	semidirect
51. C₈×Dic₅₉	C₂²	semidirect
54. #124×C₅₉	C₂²	semidirect
56. C₄⋊C₈×C₅₉	C₂²	semidirect
57. #103×C₅₉	C₂²	semidirect
58. #102×C₅₉	C₂²	semidirect
192. #463	D₅₉	semidirect
200. C₅₉	#3105	semidirect
209. #124	C₂×D₅₉	semidirect
231. C₄⋊C₈	C₂×D₅₉	semidirect
233. #103	C₂×D₅₉	semidirect
239. #102	C₂×D₅₉	semidirect