Subgroups — #21435265

Order 7552.

334 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837937	C₂	1	1	yes	maximal, characteristic
2	#125×D₅₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837801	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837759	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837757	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837739	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837734	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837722	C₂	1	1	yes	maximal, characteristic
9	#16837717	C₂	1	1	yes	maximal, characteristic
10	#16837712	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837708	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837705	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837701	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837453	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#499	C₂	1	1	yes	maximal, characteristic, complemented
16	#14273722	—	2	1	no	—
17	#14273719	C₂²	1	1	yes	characteristic
18	#14273718	C₂²	1	1	yes	characteristic
19	#14273718	C₂²	1	1	yes	characteristic
20	#14273717	—	2	1	no	—
21	#14273717	—	2	1	no	—
22	#14273715	C₂²	1	1	yes	characteristic
23	#14273712	—	2	1	no	—
24	#14273707	C₂²	1	1	yes	characteristic
25	#14273702	C₂²	1	1	yes	characteristic
26	#14273692	C₂²	1	1	yes	characteristic
27	#14273689	—	2	1	no	—
28	#14273689	C₂²	1	1	yes	characteristic
29	#14273685	—	2	1	no	—
30	C₄⋊C₄×D₅₉	—	2	1	no	—
31	C₄⋊C₄×D₅₉	—	2	1	no	—
32	C₄⋊C₄×D₅₉	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₅₉	C₂²	1	1	yes	characteristic
34	C₄²×D₅₉	C₂²	1	1	yes	characteristic
35	#14273660	C₂²	1	1	yes	characteristic
36	#14273660	C₂²	1	1	yes	characteristic
37	#14273658	C₂²	1	1	yes	characteristic
38	#14273658	C₂²	1	1	yes	characteristic
39	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
40	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
41	#14273652	C₂²	1	1	yes	characteristic
42	#14273652	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	(C₂×D₅₉)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₅₉)⋊C₈	C₂²	1	1	yes	characteristic
51	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
52	#125×C₅₉	C₂²	1	1	yes	characteristic
53	#125×C₅₉	C₂²	1	1	yes	characteristic
54	#107×C₅₉	C₂²	1	1	yes	characteristic
55	#107×C₅₉	C₂²	1	1	yes	characteristic
56	#106×C₅₉	C₂²	1	1	yes	characteristic
57	#106×C₅₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
59	C₂×D₂₃₆	—	2	1	no	—
60	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
61	C₂×C₄×D₅₉	—	2	1	no	—
62	C₂×C₄×D₅₉	—	2	1	no	—
63	C₂×C₄×D₅₉	—	2	1	no	—
64	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
65	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
66	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
67	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₅₉	—	2	1	no	—
71	C₄⋊C₄×C₅₉	—	2	1	no	—
72	C₄⋊C₄×C₅₉	—	2	1	no	—
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	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
78	C₄²×C₅₉	C₂³	1	1	yes	characteristic
79	(C₂×D₅₉)⋊C₄	—	2	1	no	—
80	(C₂×D₅₉)⋊C₄	—	2	1	no	—
81	(C₂×D₅₉)⋊C₄	—	2	1	no	—
82	(C₂×D₅₉)⋊C₄	—	2	1	no	—
83	(C₂×D₅₉)⋊C₄	—	2	1	no	—
84	C₄⋊Dic₅₉	—	2	1	no	—
85	C₄⋊Dic₅₉	—	2	1	no	—
86	C₄⋊Dic₅₉	—	2	1	no	—
87	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
88	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
89	Dic₅₉⋊C₄	—	2	1	no	—
90	Dic₅₉⋊C₄	—	2	1	no	—
91	Dic₅₉⋊C₄	—	2	1	no	—
92	Dic₅₉⋊C₄	—	2	1	no	—
93	Dic₅₉⋊C₄	—	2	1	no	—
94	Dic₅₉⋊C₄	—	2	1	no	—
95	Dic₅₉⋊C₄	—	2	1	no	—
96	C₄×Dic₅₉	—	2	1	no	—
97	C₄×Dic₅₉	D₄	1	1	yes	characteristic
98	C₄×Dic₅₉	D₄	1	1	yes	characteristic
99	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
100	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
101	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
102	C₂²×D₅₉	—	2	1	no	—
103	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₅₉	—	2	1	no	—
105	C₂×C₄×C₅₉	—	2	1	no	—
106	C₂×C₄×C₅₉	—	2	1	no	—
107	C₂×C₄×C₅₉	—	2	1	no	—
108	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
111	C₂×Dic₅₉	—	2	1	no	—
112	C₂×Dic₅₉	—	2	1	no	—
113	C₂×Dic₅₉	—	2	1	no	—
114	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
115	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
116	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
117	D₂₃₆	—	2	1	no	—
118	D₂₃₆	—	2	1	no	—
119	D₂₃₆	—	4	2	no	—
120	C₄×D₅₉	—	2	2	no	—
121	C₄×D₅₉	—	2	1	no	—
122	C₄×D₅₉	—	2	1	no	—
123	C₄×D₅₉	—	4	1	no	—
124	C₄×D₅₉	—	4	2	no	—
125	C₄×D₅₉	—	4	2	no	—
126	C₄×D₅₉	—	4	2	no	—
127	Dic₁₁₈	—	2	1	no	—
128	Dic₁₁₈	—	2	1	no	—
129	C₈×C₅₉	—	2	1	no	—
130	C₈×C₅₉	—	2	1	no	—
131	C₅₉⋊C₈	—	2	1	no	—
132	C₅₉⋊C₈	—	2	1	no	—
133	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
134	C₂×D₅₉	—	2	1	no	—
135	C₂×D₅₉	—	2	1	no	—
136	C₂×D₅₉	—	4	1	no	—
137	C₂×D₅₉	—	4	1	no	—
138	C₂×D₅₉	—	4	1	no	—
139	C₂×D₅₉	#135	1	2	yes	—
140	C₄×C₅₉	—	2	1	no	—
141	C₄×C₅₉	—	2	1	no	—
142	C₄×C₅₉	—	4	1	no	—
143	C₄×C₅₉	—	4	1	no	—
144	C₄×C₅₉	—	4	1	no	—
145	C₄×C₅₉	—	4	1	no	—
146	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
147	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
148	Dic₅₉	—	2	1	no	—
149	Dic₅₉	—	2	1	no	—
150	Dic₅₉	—	2	1	no	—
151	Dic₅₉	—	4	1	no	—
152	Dic₅₉	—	4	1	no	—
153	Dic₅₉	—	4	1	no	—
154	#3232	—	59	1	no	maximal, Sylow
155	C₂×C₅₉	#536	1	1	yes	characteristic
156	C₂×C₅₉	C₂×#135	1	1	yes	characteristic
157	C₂×C₅₉	#575	1	1	yes	characteristic
158	D₅₉	—	2	2	no	—
159	D₅₉	—	8	1	no	—
160	#529	—	59	1	no	—
161	C₂×#125	—	59	1	no	—
162	#499	—	59	1	no	—
163	#499	D₅₉	1	1	yes	characteristic, complemented
164	#488	—	59	1	no	—
165	#488	—	59	1	no	—
166	#484	—	59	1	no	—
167	#483	—	59	1	no	—
168	#481	—	59	1	no	—
169	#480	—	59	1	no	—
170	#479	—	59	1	no	—
171	#479	—	59	1	no	—
172	#478	—	59	1	no	—
173	#478	—	59	1	no	—
174	#431	—	59	1	no	—
175	C₅₉	#3232	1	1	yes	characteristic, complemented, Sylow
176	#126	—	118	1	no	—
177	#125	—	59	1	no	—
178	#125	—	59	1	no	—
179	#125	—	59	1	no	—
180	#125	—	118	1	no	—
181	#125	—	118	1	no	—
182	#125	C₂×D₅₉	1	1	yes	characteristic
183	#125	C₂×D₅₉	1	1	yes	characteristic
184	#124	—	59	1	no	—
185	#123	—	118	1	no	—
186	C₂²⋊Q₈	—	59	1	no	—
187	#121	—	59	1	no	—
188	C₄×Q₈	—	59	1	no	—
189	C₄×D₄	—	59	1	no	—
190	C₄×D₄	—	118	1	no	—
191	#117	—	118	1	no	—
192	C₂×C₄⋊C₄	—	59	1	no	—
193	C₂×C₄⋊C₄	—	59	1	no	—
194	C₂×C₄⋊C₄	—	118	1	no	—
195	C₂×C₄⋊C₄	—	118	1	no	—
196	C₂×C₄²	—	59	1	no	—
197	#107	—	59	1	no	—
198	#107	—	59	1	no	—
199	#107	C₂×D₅₉	1	1	yes	characteristic
200	#107	C₂×D₅₉	1	1	yes	characteristic
201	#106	—	59	1	no	—
202	#106	—	59	1	no	—
203	#106	C₂×D₅₉	1	1	yes	characteristic
204	#106	C₂×D₅₉	1	1	yes	characteristic
205	C₄⋊C₈	—	59	1	no	—
206	C₄⋊C₈	—	59	1	no	—
207	#103	—	59	1	no	—
208	#103	—	59	1	no	—
209	#103	—	59	1	no	—
210	#103	—	59	1	no	—
211	#102	—	59	1	no	—
212	#102	—	59	1	no	—
213	#102	—	59	1	no	—
214	#102	—	59	1	no	—
215	C₂²⋊C₈	—	59	1	no	—
216	C₂²⋊C₈	—	59	1	no	—
217	C₄×C₈	—	59	1	no	—
218	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
219	C₂×Q₈	—	59	1	no	—
220	C₂×D₄	—	59	1	no	—
221	C₂×D₄	—	118	1	no	—
222	C₂²×C₄	—	59	1	no	—
223	C₂²×C₄	—	59	1	no	—
224	C₂²×C₄	—	59	1	no	—
225	C₂²×C₄	—	118	1	no	—
226	C₂²×C₄	—	118	1	no	—
227	C₂²×C₄	—	118	1	no	—
228	C₂×C₈	—	59	1	no	—
229	C₂×C₈	—	59	1	no	—
230	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
231	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	59	1	no	—
237	C₄⋊C₄	—	59	1	no	—
238	C₄⋊C₄	—	118	1	no	—
239	C₄⋊C₄	—	118	1	no	—
240	C₄⋊C₄	—	118	1	no	—
241	C₄⋊C₄	—	118	1	no	—
242	C₄⋊C₄	—	118	1	no	—
243	C₄⋊C₄	—	118	1	no	—
244	C₄⋊C₄	—	118	1	no	—
245	C₄⋊C₄	—	118	1	no	—
246	C₄⋊C₄	—	118	1	no	—
247	C₄⋊C₄	—	118	1	no	—
248	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
250	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
251	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
252	C₂²⋊C₄	—	118	1	no	—
253	C₂²⋊C₄	—	118	1	no	—
254	C₂²⋊C₄	—	118	1	no	—
255	C₂²⋊C₄	—	118	1	no	—
256	C₂²⋊C₄	—	118	1	no	—
257	C₄²	—	59	1	no	—
258	C₄²	—	59	1	no	—
259	C₄²	—	59	1	no	—
260	C₄²	—	118	1	no	—
261	C₄²	C₂²×D₅₉	1	1	yes	characteristic
262	C₂³	—	59	1	no	—
263	C₂³	—	118	1	no	—
264	Q₈	—	118	1	no	—
265	Q₈	—	118	1	no	—
266	D₄	—	118	1	no	—
267	D₄	—	118	1	no	—
268	D₄	—	236	2	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	59	1	no	—
274	C₂×C₄	—	59	1	no	—
275	C₂×C₄	—	59	1	no	—
276	C₂×C₄	—	118	2	no	—
277	C₂×C₄	—	118	1	no	—
278	C₂×C₄	—	118	1	no	—
279	C₂×C₄	—	118	1	no	—
280	C₂×C₄	—	118	1	no	—
281	C₂×C₄	—	118	1	no	—
282	C₂×C₄	—	236	1	no	—
283	C₂×C₄	—	236	2	no	—
284	C₂×C₄	—	236	2	no	—
285	C₂×C₄	—	236	2	no	—
286	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
287	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
288	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	118	1	no	—
292	C₈	—	118	1	no	—
293	C₂²	—	59	2	no	—
294	C₂²	—	118	1	no	—
295	C₂²	—	118	1	no	—
296	C₂²	—	236	1	no	—
297	C₂²	—	236	1	no	—
298	C₂²	—	236	1	no	—
299	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	118	1	no	—
307	C₄	—	118	1	no	—
308	C₄	—	118	1	no	—
309	C₄	—	236	1	no	—
310	C₄	—	236	1	no	—
311	C₄	—	236	1	no	—
312	C₄	#14273775	1	1	yes	characteristic
313	C₄	#14273775	1	1	yes	characteristic
314	C₂	—	118	2	no	—
315	C₂	—	472	1	no	—
316	C₂	#16837977	1	1	yes	characteristic
317	C₂	#135×D₅₉	1	1	yes	characteristic
318	C₂	#16838199	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#499	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#499	D₅₉
hypocenter	C₅₉	#3232
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

Complements

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

N	H ≅ G/N	type
2. #125×D₅₉	C₂	semidirect
3. #16837801	C₂	semidirect
4. #16837759	C₂	semidirect
5. #16837757	C₂	semidirect
6. #16837739	C₂	semidirect
7. #16837734	C₂	semidirect
10. #16837712	C₂	semidirect
11. #16837708	C₂	semidirect
12. #16837705	C₂	semidirect
13. #16837701	C₂	semidirect
14. #16837453	C₂	semidirect
15. C₅₉×#499	C₂	semidirect
163. #499	D₅₉	semidirect
175. C₅₉	#3232	semidirect