Subgroups — #21434662

Order 7552.

380 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837939	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837934	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837802	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837783	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837758	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837755	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837642	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837637	C₂	1	1	yes	maximal, characteristic
9	#16837632	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837628	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#11745672	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#11745660	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#11745669	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₈×D₅₉	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×SD₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11745672	C₂²	1	1	yes	characteristic
17	C₂×#11745660	C₂²	1	1	yes	characteristic
18	C₂×#11745669	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₅₉	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₅₉	C₂²	1	1	yes	characteristic
21	#14273722	—	2	1	no	—
22	#14273720	—	2	1	no	—
23	#14273718	C₂²	1	1	yes	characteristic
24	#14273718	C₂²	1	1	yes	characteristic
25	#14273716	C₂²	1	1	yes	characteristic
26	#14273714	C₂²	1	1	yes	characteristic
27	#14273712	—	2	1	no	—
28	#14273709	—	2	1	no	—
29	#14273707	C₂²	1	1	yes	characteristic
30	#14273703	C₂²	1	1	yes	characteristic
31	#14273702	C₂²	1	1	yes	characteristic
32	#14273698	C₂²	1	1	yes	characteristic
33	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
34	C₄×D₂₃₆	—	2	1	no	—
35	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
36	C₄×C₂²⋊D₅₉	—	2	1	no	—
37	#14273686	—	2	1	no	—
38	#14273684	—	2	1	no	—
39	C₄²×D₅₉	C₂²	1	1	yes	characteristic
40	#14273660	C₂²	1	1	yes	characteristic
41	#14273660	C₂²	1	1	yes	characteristic
42	#14273659	C₂²	1	1	yes	characteristic
43	#14273652	C₂²	1	1	yes	characteristic
44	#14273651	C₂²	1	1	yes	characteristic
45	#14273650	C₂²	1	1	yes	characteristic
46	#14273649	C₂²	1	1	yes	characteristic
47	#14273648	C₂²	1	1	yes	characteristic
48	#14273647	C₂²	1	1	yes	characteristic
49	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
50	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
51	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₅₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
55	#106×C₅₉	C₂²	1	1	yes	characteristic
56	#103×C₅₉	C₂²	1	1	yes	characteristic
57	#102×C₅₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	—	2	1	no	—
62	C₂×C₂²⋊D₅₉	—	2	1	no	—
63	C₂×D₂₃₆	—	2	1	no	—
64	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₅₉	—	2	1	no	—
66	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
67	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
68	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
69	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
70	SD₁₆×C₅₉	—	2	2	no	—
71	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₅₉	—	2	1	no	—
74	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
76	C₂²⋊C₄×C₅₉	—	2	1	no	—
77	C₄²×C₅₉	—	2	1	no	—
78	C₄²×C₅₉	C₂³	1	1	yes	characteristic
79	C₂²⋊Dic₅₉	—	2	1	no	—
80	C₂²⋊Dic₅₉	—	2	1	no	—
81	C₂²⋊Dic₅₉	—	2	1	no	—
82	#11745672	—	2	2	no	—
83	#11745669	—	2	2	no	—
84	(C₂×D₅₉)⋊C₄	—	2	1	no	—
85	(C₂×D₅₉)⋊C₄	—	2	1	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	—	2	1	no	—
93	C₄⋊Dic₅₉	C₂³	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	C₄×Dic₅₉	—	2	1	no	—
100	C₄×Dic₅₉	D₄	1	1	yes	characteristic
101	C₄×Dic₅₉	D₄	1	1	yes	characteristic
102	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
103	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
104	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
105	#11745660	—	2	2	no	—
106	C₈×D₅₉	—	2	2	no	—
107	C₈×D₅₉	—	2	2	no	—
108	C₂³×C₅₉	—	2	1	no	—
109	C₂²×D₅₉	—	2	1	no	—
110	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
111	Q₈×C₅₉	—	2	1	no	—
112	Q₈×C₅₉	—	2	1	no	—
113	D₄×C₅₉	—	2	1	no	—
114	D₄×C₅₉	—	2	1	no	—
115	C₂×C₄×C₅₉	—	2	1	no	—
116	C₂×C₄×C₅₉	—	2	1	no	—
117	C₂×C₄×C₅₉	—	2	1	no	—
118	C₂×C₄×C₅₉	—	4	2	no	—
119	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₅₉	—	4	2	no	—
123	C₂×Dic₅₉	—	2	1	no	—
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	D₂₃₆	—	2	1	no	—
130	D₂₃₆	—	2	1	no	—
131	D₂₃₆	—	4	2	no	—
132	C₄×D₅₉	—	2	2	no	—
133	C₄×D₅₉	—	2	2	no	—
134	C₄×D₅₉	—	2	1	no	—
135	C₄×D₅₉	—	2	1	no	—
136	C₄×D₅₉	—	4	2	no	—
137	Dic₁₁₈	—	2	1	no	—
138	Dic₁₁₈	—	2	1	no	—
139	C₈×C₅₉	—	2	1	no	—
140	C₈×C₅₉	#41	1	2	yes	—
141	C₅₉⋊C₈	—	2	1	no	—
142	C₅₉⋊C₈	#41	1	2	yes	—
143	C₂²×C₅₉	—	4	1	no	—
144	C₂²×C₅₉	—	4	1	no	—
145	C₂²×C₅₉	—	4	1	no	—
146	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
147	C₂×D₅₉	—	2	1	no	—
148	C₂×D₅₉	—	2	1	no	—
149	C₂×D₅₉	—	4	1	no	—
150	C₂×D₅₉	—	4	1	no	—
151	C₂×D₅₉	—	4	1	no	—
152	C₂×D₅₉	#135	1	2	yes	—
153	C₄×C₅₉	—	2	1	no	—
154	C₄×C₅₉	—	4	1	no	—
155	C₄×C₅₉	—	4	1	no	—
156	C₄×C₅₉	—	4	1	no	—
157	C₄×C₅₉	#135	1	2	yes	—
158	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
159	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
160	Dic₅₉	—	2	1	no	—
161	Dic₅₉	—	2	1	no	—
162	Dic₅₉	—	2	1	no	—
163	Dic₅₉	—	4	1	no	—
164	Dic₅₉	—	4	1	no	—
165	Dic₅₉	—	4	1	no	—
166	#3168	—	59	1	no	maximal, Sylow
167	C₂×C₅₉	—	8	1	no	—
168	C₂×C₅₉	#532	1	1	yes	characteristic
169	C₂×C₅₉	C₂×#135	1	1	yes	characteristic
170	C₂×C₅₉	C₂×#135	1	1	yes	characteristic
171	D₅₉	—	2	2	no	—
172	D₅₉	—	8	1	no	—
173	#529	—	59	1	no	—
174	#529	—	59	1	no	—
175	#499	—	59	1	no	—
176	#495	—	59	1	no	—
177	#488	—	59	1	no	—
178	#488	—	59	1	no	—
179	#467	—	59	1	no	—
180	#466	—	59	1	no	—
181	#465	—	59	1	no	—
182	#465	—	59	1	no	—
183	C₄×Q₁₆	—	59	1	no	—
184	C₄×SD₁₆	—	59	1	no	—
185	C₄×SD₁₆	D₅₉	1	1	yes	characteristic, complemented
186	C₄×D₈	—	59	1	no	—
187	C₂×C₄×C₈	—	59	1	no	—
188	C₅₉	#3168	1	1	yes	characteristic, complemented, Sylow
189	C₂×Q₁₆	—	59	1	no	—
190	C₂×SD₁₆	—	59	1	no	—
191	C₂×SD₁₆	C₂×D₅₉	1	1	yes	characteristic
192	C₂×D₈	—	59	1	no	—
193	C₂²×C₈	—	59	1	no	—
194	C₂²×C₈	—	59	1	no	—
195	#126	—	118	1	no	—
196	#126	—	118	1	no	—
197	#125	—	59	1	no	—
198	#125	—	59	1	no	—
199	#124	—	59	1	no	—
200	#124	—	59	1	no	—
201	#123	—	118	1	no	—
202	#123	—	118	1	no	—
203	C₂²⋊Q₈	—	59	1	no	—
204	C₂²⋊Q₈	—	59	1	no	—
205	#121	—	59	1	no	—
206	#121	—	59	1	no	—
207	C₄×Q₈	—	59	1	no	—
208	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
209	C₄×D₄	—	59	1	no	—
210	C₄×D₄	—	118	1	no	—
211	C₄×D₄	—	118	1	no	—
212	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
213	#117	—	118	1	no	—
214	#117	—	118	1	no	—
215	C₂×C₄²	—	59	1	no	—
216	#107	—	59	1	no	—
217	#107	—	59	1	no	—
218	#106	—	59	1	no	—
219	#106	C₂×D₅₉	1	1	yes	characteristic
220	#103	—	59	1	no	—
221	#103	—	59	1	no	—
222	#103	—	59	1	no	—
223	#103	C₂×D₅₉	1	1	yes	characteristic
224	#102	—	59	1	no	—
225	#102	—	59	1	no	—
226	#102	—	59	1	no	—
227	#102	C₂×D₅₉	1	1	yes	characteristic
228	C₄×C₈	—	59	1	no	—
229	C₄×C₈	—	59	1	no	—
230	C₄×C₈	—	59	1	no	—
231	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
232	C₂×Q₈	—	59	1	no	—
233	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
234	C₂×D₄	—	59	1	no	—
235	C₂×D₄	—	118	1	no	—
236	C₂×D₄	—	118	1	no	—
237	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	59	1	no	—
240	C₂²×C₄	—	59	1	no	—
241	C₂²×C₄	—	59	1	no	—
242	C₂²×C₄	—	118	1	no	—
243	Q₁₆	—	118	2	no	—
244	SD₁₆	—	2	2	no	—
245	SD₁₆	—	118	2	no	—
246	D₈	—	118	2	no	—
247	C₂×C₈	—	59	1	no	—
248	C₂×C₈	—	59	1	no	—
249	C₂×C₈	—	118	2	no	—
250	C₂×C₈	—	118	2	no	—
251	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	59	1	no	—
255	C₄⋊C₄	—	59	1	no	—
256	C₄⋊C₄	—	118	1	no	—
257	C₄⋊C₄	—	118	1	no	—
258	C₄⋊C₄	—	118	1	no	—
259	C₄⋊C₄	—	118	1	no	—
260	C₄⋊C₄	—	118	1	no	—
261	C₄⋊C₄	—	118	1	no	—
262	C₄⋊C₄	—	118	1	no	—
263	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	118	1	no	—
267	C₂²⋊C₄	—	118	1	no	—
268	C₂²⋊C₄	—	118	1	no	—
269	C₂²⋊C₄	—	118	1	no	—
270	C₂²⋊C₄	—	118	1	no	—
271	C₂²⋊C₄	—	118	1	no	—
272	C₂²⋊C₄	—	118	1	no	—
273	C₂²⋊C₄	—	118	1	no	—
274	C₂²⋊C₄	—	118	1	no	—
275	C₄²	—	2	1	no	—
276	C₄²	—	59	1	no	—
277	C₄²	—	59	1	no	—
278	C₄²	—	59	1	no	—
279	C₄²	—	118	1	no	—
280	C₄²	C₂²×D₅₉	1	1	yes	characteristic
281	C₂³	—	2	1	no	—
282	C₂³	—	59	1	no	—
283	C₂³	—	118	1	no	—
284	Q₈	—	2	1	no	—
285	Q₈	—	2	1	no	—
286	Q₈	—	118	1	no	—
287	Q₈	—	118	1	no	—
288	D₄	—	2	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	118	1	no	—
291	D₄	—	118	1	no	—
292	D₄	—	236	2	no	—
293	D₄	—	236	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	59	1	no	—
299	C₂×C₄	—	59	1	no	—
300	C₂×C₄	—	59	1	no	—
301	C₂×C₄	—	118	2	no	—
302	C₂×C₄	—	118	2	no	—
303	C₂×C₄	—	118	1	no	—
304	C₂×C₄	—	118	1	no	—
305	C₂×C₄	—	118	1	no	—
306	C₂×C₄	—	118	1	no	—
307	C₂×C₄	—	118	1	no	—
308	C₂×C₄	—	236	2	no	—
309	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
310	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	59	2	no	—
314	C₈	—	118	1	no	—
315	C₈	#11745685	1	2	yes	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	59	2	no	—
320	C₂²	—	118	1	no	—
321	C₂²	—	118	1	no	—
322	C₂²	—	236	1	no	—
323	C₂²	—	236	1	no	—
324	C₂²	—	236	1	no	—
325	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	118	1	no	—
331	C₄	—	118	1	no	—
332	C₄	—	118	1	no	—
333	C₄	—	236	1	no	—
334	C₄	—	236	1	no	—
335	C₄	—	236	1	no	—
336	C₄	#14273748	1	2	yes	—
337	C₄	#41×D₅₉	1	1	yes	characteristic
338	C₄	C₂×#11745685	1	1	yes	characteristic
339	C₂	—	8	1	no	—
340	C₂	—	118	2	no	—
341	C₂	—	472	1	no	—
342	C₂	#16837949	1	1	yes	characteristic
343	C₂	#135×D₅₉	1	1	yes	characteristic
344	C₂	C₂×#14273748	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₄×SD₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	C₄×SD₁₆	D₅₉
hypocenter	C₅₉	#3168
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #16837939	C₂	semidirect
2. #16837934	C₂	semidirect
3. #16837802	C₂	semidirect
4. #16837783	C₂	semidirect
5. #16837758	C₂	semidirect
6. #16837755	C₂	semidirect
7. #16837642	C₂	semidirect
9. #16837632	C₂	semidirect
10. #16837628	C₂	semidirect
11. C₄×#11745672	C₂	semidirect
12. C₄×#11745660	C₂	semidirect
13. C₄×#11745669	C₂	semidirect
14. C₄×C₈×D₅₉	C₂	semidirect
15. C₄×SD₁₆×C₅₉	C₂	semidirect
185. C₄×SD₁₆	D₅₉	semidirect
188. C₅₉	#3168	semidirect