Subgroups — #21434462

Order 7552.

372 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838032	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837977	C₂	1	1	yes	maximal, characteristic
3	#16837769	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837763	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837762	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837759	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837758	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837752	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837751	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837409	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837402	C₂	1	1	yes	maximal, characteristic
12	#16837397	C₂	1	1	yes	maximal, characteristic
13	#103×D₅₉	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837359	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#491	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11745689	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₅₉	C₂²	1	1	yes	characteristic
18	C₂×#11745659	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₅₉	C₂²	1	1	yes	characteristic
20	#14273727	C₂²	1	1	yes	characteristic
21	#14273726	—	2	1	no	—
22	#14273726	C₂²	1	1	yes	characteristic
23	#14273725	C₂²	1	1	yes	characteristic
24	#14273718	C₂²	1	1	yes	characteristic
25	#14273717	—	2	1	no	—
26	#14273716	C₂²	1	1	yes	characteristic
27	#14273715	C₂²	1	1	yes	characteristic
28	#14273712	—	2	1	no	—
29	#14273712	—	2	1	no	—
30	#14273707	—	2	1	no	—
31	#14273705	—	2	1	no	—
32	#14273705	—	2	1	no	—
33	#14273705	—	2	1	no	—
34	#14273685	C₂²	1	1	yes	characteristic
35	#14273684	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₅₉	C₂²	1	1	yes	characteristic
37	#14273652	C₂²	1	1	yes	characteristic
38	#14273652	C₂²	1	1	yes	characteristic
39	#14273651	C₂²	1	1	yes	characteristic
40	#14273651	C₂²	1	1	yes	characteristic
41	#14273651	C₂²	1	1	yes	characteristic
42	#14273651	C₂²	1	1	yes	characteristic
43	#14273650	C₂²	1	1	yes	characteristic
44	#14273650	C₂²	1	1	yes	characteristic
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	#14273637	C₂²	1	1	yes	characteristic
50	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
53	#125×C₅₉	C₂²	1	1	yes	characteristic
54	#103×C₅₉	C₂²	1	1	yes	characteristic
55	#103×C₅₉	C₂²	1	1	yes	characteristic
56	#103×C₅₉	C₂²	1	1	yes	characteristic
57	#103×C₅₉	C₂²	1	1	yes	characteristic
58	#97×C₅₉	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
61	#11745689	—	2	2	no	—
62	#11745689	—	2	2	no	—
63	Q₈×D₅₉	—	2	2	no	—
64	Q₈×D₅₉	—	2	2	no	—
65	C₂×D₂₃₆	—	2	1	no	—
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	C₂³	1	1	yes	characteristic
71	C₂×Dic₁₁₈	—	2	1	no	—
72	C₂×Dic₁₁₈	—	2	1	no	—
73	C₂×Dic₁₁₈	—	2	1	no	—
74	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₅₉	—	2	1	no	—
78	C₄⋊C₄×C₅₉	—	2	1	no	—
79	C₄⋊C₄×C₅₉	—	2	1	no	—
80	C₄⋊C₄×C₅₉	—	2	1	no	—
81	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
83	C₄²×C₅₉	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
90	(C₂×D₅₉)⋊C₄	D₄	1	1	yes	characteristic
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	Dic₅₉⋊C₄	—	2	1	no	—
100	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
101	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
102	C₄×Dic₅₉	—	2	1	no	—
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	#11745659	—	2	2	no	—
107	C₈×D₅₉	—	2	2	no	—
108	C₂²×D₅₉	—	2	1	no	—
109	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₅₉	—	2	1	no	—
111	Q₈×C₅₉	—	2	1	no	—
112	Q₈×C₅₉	—	2	1	no	—
113	Q₈×C₅₉	—	2	1	no	—
114	C₂×C₄×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	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
121	C₂×Dic₅₉	—	2	1	no	—
122	C₂×Dic₅₉	—	2	1	no	—
123	C₂×Dic₅₉	—	2	1	no	—
124	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
127	D₂₃₆	—	2	1	no	—
128	D₂₃₆	—	2	1	no	—
129	D₂₃₆	—	4	2	no	—
130	C₄×D₅₉	—	4	2	no	—
131	C₄×D₅₉	—	4	2	no	—
132	C₄×D₅₉	—	8	1	no	—
133	C₄×D₅₉	#41	1	2	yes	—
134	C₄×D₅₉	#41	1	2	yes	—
135	Dic₁₁₈	—	2	1	no	—
136	Dic₁₁₈	—	2	1	no	—
137	Dic₁₁₈	—	4	2	no	—
138	Dic₁₁₈	—	8	1	no	—
139	Dic₁₁₈	—	8	1	no	—
140	C₈×C₅₉	—	2	1	no	—
141	C₈×C₅₉	—	2	1	no	—
142	C₅₉⋊C₈	—	2	1	no	—
143	C₅₉⋊C₈	—	2	1	no	—
144	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₅₉	—	2	1	no	—
146	C₂×D₅₉	—	2	1	no	—
147	C₂×D₅₉	—	4	1	no	—
148	C₂×D₅₉	—	4	1	no	—
149	C₂×D₅₉	—	4	1	no	—
150	C₂×D₅₉	#124	1	2	yes	—
151	C₄×C₅₉	—	2	1	no	—
152	C₄×C₅₉	—	2	1	no	—
153	C₄×C₅₉	—	4	1	no	—
154	C₄×C₅₉	—	4	1	no	—
155	C₄×C₅₉	—	4	1	no	—
156	C₄×C₅₉	—	4	1	no	—
157	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
158	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
159	Dic₅₉	—	2	1	no	—
160	Dic₅₉	—	2	1	no	—
161	Dic₅₉	—	4	1	no	—
162	Dic₅₉	—	4	1	no	—
163	Dic₅₉	—	4	1	no	—
164	Dic₅₉	#124	1	2	yes	—
165	#3137	—	59	1	no	maximal, Sylow
166	C₂×C₅₉	C₂×#124	1	1	yes	characteristic
167	C₂×C₅₉	#577	1	1	yes	characteristic
168	C₂×C₅₉	#578	1	1	yes	characteristic
169	D₅₉	—	2	2	no	—
170	D₅₉	—	8	1	no	—
171	#544	—	59	1	no	—
172	#536	—	59	1	no	—
173	#491	—	59	1	no	—
174	#491	D₅₉	1	1	yes	characteristic, complemented
175	#489	—	59	1	no	—
176	#489	—	59	1	no	—
177	#488	—	59	1	no	—
178	#488	—	59	1	no	—
179	#487	—	59	1	no	—
180	#487	—	59	1	no	—
181	#419	—	59	1	no	—
182	#417	—	59	1	no	—
183	#416	—	59	1	no	—
184	C₂×#103	—	59	1	no	—
185	#405	—	59	1	no	—
186	C₅₉	#3137	1	1	yes	characteristic, complemented, Sylow
187	C₂×#41	—	59	1	no	—
188	C₂²×Q₈	—	59	1	no	—
189	C₂×M₁₆	—	59	1	no	—
190	C₂²×C₈	—	59	1	no	—
191	C₄⋊Q₈	—	59	1	no	—
192	C₄⋊Q₈	—	59	1	no	—
193	C₄⋊Q₈	—	59	1	no	—
194	C₄⋊Q₈	—	118	1	no	—
195	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
196	#125	—	59	1	no	—
197	#125	—	118	1	no	—
198	#125	C₂×D₅₉	1	1	yes	characteristic
199	#124	—	59	1	no	—
200	#124	—	59	1	no	—
201	#123	—	118	1	no	—
202	#123	—	118	1	no	—
203	C₂²⋊Q₈	—	118	1	no	—
204	C₂²⋊Q₈	—	118	1	no	—
205	C₂²⋊Q₈	—	118	1	no	—
206	C₂²⋊Q₈	—	118	1	no	—
207	#117	—	59	1	no	—
208	#117	—	59	1	no	—
209	C₂×C₄⋊C₄	—	59	1	no	—
210	#103	—	59	1	no	—
211	#103	—	59	1	no	—
212	#103	—	59	1	no	—
213	#103	—	59	1	no	—
214	#103	—	59	1	no	—
215	#103	—	59	1	no	—
216	#103	—	59	1	no	—
217	#103	—	59	1	no	—
218	#103	C₂×D₅₉	1	1	yes	characteristic
219	#103	C₂×D₅₉	1	1	yes	characteristic
220	#103	C₂×D₅₉	1	1	yes	characteristic
221	#103	C₂×D₅₉	1	1	yes	characteristic
222	#102	—	59	1	no	—
223	#102	—	59	1	no	—
224	#102	—	59	1	no	—
225	#102	—	59	1	no	—
226	#97	—	59	1	no	—
227	#97	C₂×D₅₉	1	1	yes	characteristic
228	C₄×C₈	—	59	1	no	—
229	C₄×C₈	—	59	1	no	—
230	#41	—	118	2	no	—
231	#41	—	118	2	no	—
232	C₂×Q₈	—	59	1	no	—
233	C₂×Q₈	—	118	1	no	—
234	C₂×Q₈	—	118	1	no	—
235	C₂×Q₈	—	118	2	no	—
236	C₂×Q₈	—	118	2	no	—
237	C₂×Q₈	—	118	1	no	—
238	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
239	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
240	C₂×D₄	—	59	1	no	—
241	C₂×D₄	—	118	1	no	—
242	C₂²×C₄	—	59	1	no	—
243	C₂²×C₄	—	118	1	no	—
244	C₂²×C₄	—	118	1	no	—
245	C₂²×C₄	—	118	1	no	—
246	M₁₆	—	118	2	no	—
247	C₂×C₈	—	59	1	no	—
248	C₂×C₈	—	59	1	no	—
249	C₂×C₈	—	118	2	no	—
250	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
251	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	59	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	C₂²⋊C₄	—	59	1	no	—
270	C₂²⋊C₄	—	59	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	C₂²⋊C₄	—	118	1	no	—
276	C₄²	—	59	1	no	—
277	C₄²	—	118	1	no	—
278	C₄²	C₂²×D₅₉	1	1	yes	characteristic
279	C₂³	—	59	1	no	—
280	C₂³	—	118	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	2	1	no	—
285	Q₈	—	118	1	no	—
286	Q₈	—	118	1	no	—
287	Q₈	—	472	1	no	—
288	Q₈	—	472	1	no	—
289	Q₈	—	236	2	no	—
290	D₄	—	118	1	no	—
291	D₄	—	118	1	no	—
292	D₄	—	236	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	59	1	no	—
298	C₂×C₄	—	59	2	no	—
299	C₂×C₄	—	59	2	no	—
300	C₂×C₄	—	59	1	no	—
301	C₂×C₄	—	59	1	no	—
302	C₂×C₄	—	118	1	no	—
303	C₂×C₄	—	118	1	no	—
304	C₂×C₄	—	118	1	no	—
305	C₂×C₄	—	472	1	no	—
306	C₂×C₄	—	236	2	no	—
307	C₂×C₄	—	236	2	no	—
308	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
309	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	118	1	no	—
314	C₈	—	118	1	no	—
315	C₂²	—	59	2	no	—
316	C₂²	—	118	1	no	—
317	C₂²	—	118	1	no	—
318	C₂²	—	236	1	no	—
319	C₂²	—	236	1	no	—
320	C₂²	—	236	1	no	—
321	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	59	2	no	—
329	C₄	—	118	1	no	—
330	C₄	—	118	1	no	—
331	C₄	—	236	1	no	—
332	C₄	—	236	1	no	—
333	C₄	—	236	1	no	—
334	C₄	#41×D₅₉	1	1	yes	characteristic
335	C₄	#41×D₅₉	1	1	yes	characteristic
336	C₂	—	118	2	no	—
337	C₂	—	472	1	no	—
338	C₂	#124×D₅₉	1	1	yes	characteristic
339	C₂	#16838211	1	1	yes	characteristic
340	C₂	#16838218	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#491	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#491	D₅₉
hypocenter	C₅₉	#3137
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #491

Complements

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

N	H ≅ G/N	type
1. #16838032	C₂	semidirect
3. #16837769	C₂	semidirect
4. #16837763	C₂	semidirect
5. #16837762	C₂	semidirect
6. #16837759	C₂	semidirect
7. #16837758	C₂	semidirect
8. #16837752	C₂	semidirect
9. #16837751	C₂	semidirect
10. #16837409	C₂	semidirect
13. #103×D₅₉	C₂	semidirect
14. #16837359	C₂	semidirect
15. C₅₉×#491	C₂	semidirect
174. #491	D₅₉	semidirect
186. C₅₉	#3137	semidirect