Subgroups — #21434344

Order 7552.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838012	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837876	C₂	1	1	yes	maximal, characteristic
3	#16837743	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837705	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837701	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837606	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837606	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837555	C₂	1	1	yes	maximal, characteristic
9	#16837552	C₂	1	1	yes	maximal, characteristic
10	#16837505	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837501	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837489	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837487	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837460	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#485	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×D₂₃₆	C₂²	1	1	yes	characteristic
17	C₂×#11745660	C₂²	1	1	yes	characteristic
18	C₂×#11745660	C₂²	1	1	yes	characteristic
19	C₂×#11745671	C₂²	1	1	yes	characteristic
20	C₂×#11745671	C₂²	1	1	yes	characteristic
21	#14273725	C₂²	1	1	yes	characteristic, complemented
22	#14273720	—	2	1	no	—
23	#14273720	—	2	1	no	—
24	#14273718	—	2	1	no	—
25	#14273718	C₂²	1	1	yes	characteristic, complemented
26	#14273717	—	2	1	no	—
27	#14273706	C₂²	1	1	yes	characteristic
28	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic, complemented
29	#14273692	C₂²	1	1	yes	characteristic, complemented
30	C₄×D₂₃₆	—	2	1	no	—
31	C₄×D₂₃₆	—	2	1	no	—
32	#14273689	C₂²	1	1	yes	characteristic
33	#14273689	C₂²	1	1	yes	characteristic
34	#14273689	C₂²	1	1	yes	characteristic
35	#14273689	C₂²	1	1	yes	characteristic
36	#14273686	C₂²	1	1	yes	characteristic
37	C₂²⋊C₄×D₅₉	—	2	1	no	—
38	C₂²⋊C₄×D₅₉	—	2	1	no	—
39	#14273660	C₂²	1	1	yes	characteristic, complemented
40	#14273658	C₂²	1	1	yes	characteristic, complemented
41	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
42	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
43	#14273650	C₂²	1	1	yes	characteristic, complemented
44	#14273650	C₂²	1	1	yes	characteristic, complemented
45	#14273649	C₂²	1	1	yes	characteristic
46	#14273649	C₂²	1	1	yes	characteristic
47	#14273647	C₂²	1	1	yes	characteristic
48	#14273647	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
50	#14273637	C₂²	1	1	yes	characteristic, complemented
51	C₈×Dic₅₉	C₂²	1	1	yes	characteristic, complemented
52	C₂²⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
53	#117×C₅₉	C₂²	1	1	yes	characteristic
54	#107×C₅₉	C₂²	1	1	yes	characteristic, complemented
55	#106×C₅₉	C₂²	1	1	yes	characteristic, complemented
56	#103×C₅₉	C₂²	1	1	yes	characteristic, complemented
57	#103×C₅₉	C₂²	1	1	yes	characteristic, complemented
58	C₂²⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
59	C₂³×D₅₉	—	2	1	no	—
60	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
62	C₂×D₂₃₆	—	2	1	no	—
63	C₂×D₂₃₆	—	2	2	no	—
64	C₂×D₂₃₆	—	2	2	no	—
65	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
66	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₅₉	—	2	1	no	—
68	C₂×C₄×D₅₉	—	2	1	no	—
69	C₂×C₄×D₅₉	—	2	1	no	—
70	C₂×C₄×D₅₉	—	2	1	no	—
71	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₅₉	—	2	1	no	—
75	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
78	C₂²⋊C₄×C₅₉	—	2	1	no	—
79	C₂²⋊C₄×C₅₉	—	2	1	no	—
80	C₄²×C₅₉	—	2	1	no	—
81	C₂²⋊Dic₅₉	—	2	1	no	—
82	C₂²⋊Dic₅₉	D₄	1	1	yes	characteristic
83	C₂²⋊Dic₅₉	D₄	1	1	yes	characteristic
84	#11745671	—	2	2	no	—
85	#11745671	—	2	2	no	—
86	(C₂×D₅₉)⋊C₄	—	2	1	no	—
87	(C₂×D₅₉)⋊C₄	—	2	1	no	—
88	(C₂×D₅₉)⋊C₄	—	2	1	no	—
89	(C₂×D₅₉)⋊C₄	—	2	1	no	—
90	C₄⋊Dic₅₉	—	2	1	no	—
91	C₄⋊Dic₅₉	—	2	1	no	—
92	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
93	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
94	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
95	Dic₅₉⋊C₄	—	2	1	no	—
96	Dic₅₉⋊C₄	—	2	1	no	—
97	Dic₅₉⋊C₄	—	2	1	no	—
98	Dic₅₉⋊C₄	—	2	1	no	—
99	Dic₅₉⋊C₄	—	2	1	no	—
100	C₄×Dic₅₉	—	2	1	no	—
101	C₄×Dic₅₉	—	2	1	no	—
102	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
103	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
104	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
106	#11745660	—	2	2	no	—
107	#11745660	—	2	2	no	—
108	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₅₉	—	2	1	no	—
110	C₂²×D₅₉	—	2	1	no	—
111	C₂²×D₅₉	—	2	4	no	—
112	C₂²×D₅₉	—	8	1	no	—
113	C₂²×D₅₉	—	8	1	no	—
114	Q₈×C₅₉	—	2	1	no	—
115	Q₈×C₅₉	—	2	1	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	1	no	—
121	C₂×C₄×C₅₉	—	2	1	no	—
122	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
124	C₂×Dic₅₉	—	2	1	no	—
125	C₂×Dic₅₉	—	2	1	no	—
126	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
130	D₂₃₆	—	2	4	no	—
131	D₂₃₆	—	2	1	no	—
132	D₂₃₆	—	2	1	no	—
133	D₂₃₆	#41	1	2	yes	—
134	D₂₃₆	#41	1	2	yes	—
135	C₄×D₅₉	—	8	1	no	—
136	C₄×D₅₉	—	8	1	no	—
137	C₄×D₅₉	—	8	1	no	—
138	C₄×D₅₉	—	8	1	no	—
139	Dic₁₁₈	—	2	1	no	—
140	Dic₁₁₈	—	2	1	no	—
141	C₈×C₅₉	—	2	1	no	—
142	C₈×C₅₉	—	2	1	no	—
143	C₅₉⋊C₈	—	2	1	no	—
144	C₅₉⋊C₈	—	2	1	no	—
145	C₂²×C₅₉	—	2	1	no	—
146	C₂²×C₅₉	—	2	1	no	—
147	C₂²×C₅₉	—	2	1	no	—
148	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₅₉	—	2	2	no	—
150	C₂×D₅₉	—	2	2	no	—
151	C₂×D₅₉	—	4	1	no	—
152	C₂×D₅₉	—	4	1	no	—
153	C₂×D₅₉	—	4	1	no	—
154	C₂×D₅₉	—	4	1	no	—
155	C₂×D₅₉	—	8	4	no	—
156	C₄×C₅₉	—	2	1	no	—
157	C₄×C₅₉	—	4	1	no	—
158	C₄×C₅₉	—	4	1	no	—
159	C₄×C₅₉	—	4	1	no	—
160	C₄×C₅₉	—	4	1	no	—
161	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
162	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
163	Dic₅₉	—	2	1	no	—
164	Dic₅₉	—	2	1	no	—
165	Dic₅₉	—	2	1	no	—
166	Dic₅₉	—	2	1	no	—
167	Dic₅₉	—	4	1	no	—
168	Dic₅₉	—	4	1	no	—
169	#3123	—	59	1	no	maximal, Sylow
170	C₂×C₅₉	—	4	1	no	—
171	C₂×C₅₉	#525	1	1	yes	characteristic
172	C₂×C₅₉	#577	1	1	yes	characteristic
173	C₂×C₅₉	#577	1	1	yes	characteristic
174	D₅₉	—	4	2	no	—
175	D₅₉	—	4	2	no	—
176	#541	—	59	1	no	—
177	#518	—	59	1	no	—
178	#485	—	59	1	no	—
179	#485	D₅₉	1	1	yes	characteristic, complemented
180	#478	—	59	1	no	—
181	#478	—	59	1	no	—
182	#461	—	59	1	no	—
183	#461	—	59	1	no	—
184	#450	—	59	1	no	—
185	#450	—	59	1	no	—
186	#440	—	59	1	no	—
187	#440	—	59	1	no	—
188	C₄×SD₁₆	—	59	1	no	—
189	C₄×SD₁₆	—	59	1	no	—
190	#433	—	59	1	no	—
191	C₅₉	#3123	1	1	yes	characteristic, complemented, Sylow
192	C₂²×D₄	—	59	1	no	—
193	C₂×SD₁₆	—	59	1	no	—
194	C₂×SD₁₆	—	59	1	no	—
195	C₂×SD₁₆	—	59	1	no	—
196	C₂×SD₁₆	—	59	1	no	—
197	C₄⋊Q₈	—	59	1	no	—
198	#126	—	118	1	no	—
199	#126	—	118	1	no	—
200	#125	—	59	1	no	—
201	#125	—	118	1	no	—
202	#125	—	118	1	no	—
203	C₂²⋊Q₈	—	59	1	no	—
204	C₂²⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
205	C₄×Q₈	—	59	1	no	—
206	C₄×Q₈	—	59	1	no	—
207	C₄×D₄	—	59	1	no	—
208	C₄×D₄	—	59	1	no	—
209	C₄×D₄	—	59	1	no	—
210	C₄×D₄	—	59	1	no	—
211	C₄×D₄	—	118	1	no	—
212	C₄×D₄	—	118	1	no	—
213	#117	—	59	1	no	—
214	#117	C₂×D₅₉	1	1	yes	characteristic
215	C₂×C₂²⋊C₄	—	118	1	no	—
216	C₂×C₂²⋊C₄	—	118	1	no	—
217	#107	—	59	1	no	—
218	#107	C₂×D₅₉	1	1	yes	characteristic, complemented
219	#106	—	59	1	no	—
220	#106	C₂×D₅₉	1	1	yes	characteristic, complemented
221	C₄⋊C₈	—	59	1	no	—
222	C₄⋊C₈	—	59	1	no	—
223	#103	—	59	1	no	—
224	#103	—	59	1	no	—
225	#103	C₂×D₅₉	1	1	yes	characteristic, complemented
226	#103	C₂×D₅₉	1	1	yes	characteristic, complemented
227	#102	—	59	1	no	—
228	#102	—	59	1	no	—
229	#102	—	59	1	no	—
230	#102	—	59	1	no	—
231	C₂²⋊C₈	—	59	1	no	—
232	C₂²⋊C₈	C₂×D₅₉	1	1	yes	characteristic
233	#97	—	59	1	no	—
234	C₄×C₈	—	59	1	no	—
235	C₂⁴	—	118	1	no	—
236	C₂×Q₈	—	59	1	no	—
237	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
238	C₂×D₄	—	59	1	no	—
239	C₂×D₄	—	59	1	no	—
240	C₂×D₄	—	118	2	no	—
241	C₂×D₄	—	118	2	no	—
242	C₂×D₄	—	118	1	no	—
243	C₂²×C₄	—	118	1	no	—
244	C₂²×C₄	—	118	1	no	—
245	C₂²×C₄	—	118	1	no	—
246	C₂²×C₄	—	118	1	no	—
247	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
248	SD₁₆	—	118	2	no	—
249	SD₁₆	—	118	2	no	—
250	SD₁₆	—	118	2	no	—
251	SD₁₆	—	118	2	no	—
252	C₂×C₈	—	59	1	no	—
253	C₂×C₈	—	59	1	no	—
254	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
255	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	59	1	no	—
258	C₄⋊C₄	—	59	1	no	—
259	C₄⋊C₄	—	59	1	no	—
260	C₄⋊C₄	—	118	1	no	—
261	C₄⋊C₄	—	118	1	no	—
262	C₄⋊C₄	—	118	1	no	—
263	C₄⋊C₄	—	118	1	no	—
264	C₄⋊C₄	—	118	1	no	—
265	C₄⋊C₄	—	118	1	no	—
266	C₄⋊C₄	—	118	1	no	—
267	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
268	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
269	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	2	1	no	—
272	C₂²⋊C₄	—	59	1	no	—
273	C₂²⋊C₄	—	59	1	no	—
274	C₂²⋊C₄	—	118	1	no	—
275	C₂²⋊C₄	—	118	1	no	—
276	C₂²⋊C₄	—	118	1	no	—
277	C₂²⋊C₄	—	118	1	no	—
278	C₂²⋊C₄	—	118	1	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	59	1	no	—
281	C₄²	—	59	1	no	—
282	C₄²	—	118	1	no	—
283	C₄²	—	118	1	no	—
284	C₂³	—	118	4	no	—
285	C₂³	—	118	1	no	—
286	C₂³	—	118	1	no	—
287	C₂³	—	472	1	no	—
288	C₂³	—	472	1	no	—
289	C₂³	D₄×D₅₉	1	1	yes	characteristic
290	Q₈	—	2	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	118	1	no	—
293	Q₈	—	118	1	no	—
294	D₄	—	59	2	no	—
295	D₄	—	59	2	no	—
296	D₄	—	118	4	no	—
297	D₄	—	118	1	no	—
298	D₄	—	118	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	59	1	no	—
306	C₂×C₄	—	59	1	no	—
307	C₂×C₄	—	59	1	no	—
308	C₂×C₄	—	59	1	no	—
309	C₂×C₄	—	118	1	no	—
310	C₂×C₄	—	118	1	no	—
311	C₂×C₄	—	472	1	no	—
312	C₂×C₄	—	472	1	no	—
313	C₂×C₄	—	472	1	no	—
314	C₂×C₄	—	472	1	no	—
315	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	118	1	no	—
320	C₈	—	118	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	118	2	no	—
325	C₂²	—	118	2	no	—
326	C₂²	—	472	4	no	—
327	C₂²	—	236	1	no	—
328	C₂²	—	236	1	no	—
329	C₂²	—	236	1	no	—
330	C₂²	—	236	1	no	—
331	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	118	1	no	—
338	C₄	—	118	1	no	—
339	C₄	—	118	1	no	—
340	C₄	—	118	1	no	—
341	C₄	—	236	1	no	—
342	C₄	—	236	1	no	—
343	C₄	#41×D₅₉	1	1	yes	characteristic
344	C₄	#41×D₅₉	1	1	yes	characteristic
345	C₂	—	4	1	no	—
346	C₂	—	236	2	no	—
347	C₂	—	236	2	no	—
348	C₂	#16837910	1	1	yes	characteristic
349	C₂	#16838209	1	1	yes	characteristic
350	C₂	#16838211	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#485	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#485	D₅₉
hypocenter	C₅₉	#3123
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #485

Complements

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

N	H ≅ G/N	type
1. #16838012	C₂	semidirect
3. #16837743	C₂	semidirect
4. #16837705	C₂	semidirect
5. #16837701	C₂	semidirect
6. #16837606	C₂	semidirect
7. #16837606	C₂	semidirect
10. #16837505	C₂	semidirect
11. #16837501	C₂	semidirect
12. #16837489	C₂	semidirect
13. #16837487	C₂	semidirect
14. #16837460	C₂	semidirect
15. C₅₉×#485	C₂	semidirect
21. #14273725	C₂²	semidirect
25. #14273718	C₂²	semidirect
28. Q₈×Dic₅₉	C₂²	semidirect
29. #14273692	C₂²	semidirect
39. #14273660	C₂²	semidirect
40. #14273658	C₂²	semidirect
41. Dic₅₉⋊C₈	C₂²	semidirect
42. Dic₅₉⋊C₈	C₂²	semidirect
43. #14273650	C₂²	semidirect
44. #14273650	C₂²	semidirect
50. #14273637	C₂²	semidirect
51. C₈×Dic₅₉	C₂²	semidirect
54. #107×C₅₉	C₂²	semidirect
55. #106×C₅₉	C₂²	semidirect
56. #103×C₅₉	C₂²	semidirect
57. #103×C₅₉	C₂²	semidirect
179. #485	D₅₉	semidirect
191. C₅₉	#3123	semidirect
218. #107	C₂×D₅₉	semidirect
220. #106	C₂×D₅₉	semidirect
225. #103	C₂×D₅₉	semidirect
226. #103	C₂×D₅₉	semidirect