Subgroups — #21433915

Order 7552.

384 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#125×D₅₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#11745689	C₂	1	1	yes	maximal, characteristic
3	#16837711	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837709	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837707	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837705	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837703	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837701	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837699	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837443	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837441	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837424	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837402	C₂	1	1	yes	maximal, characteristic
14	#16837402	C₂	1	1	yes	maximal, characteristic
15	C₅₉×#479	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11745689	C₂²	1	1	yes	characteristic
17	C₂×#11745659	C₂²	1	1	yes	characteristic
18	C₂×#11745659	C₂²	1	1	yes	characteristic
19	#14273719	C₂²	1	1	yes	characteristic
20	#14273718	C₂²	1	1	yes	characteristic
21	#14273718	C₂²	1	1	yes	characteristic
22	#14273717	—	2	1	no	—
23	#14273717	—	2	1	no	—
24	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
25	C₄×D₂₃₆	—	2	1	no	—
26	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
27	#14273689	—	2	1	no	—
28	#14273689	C₂²	1	1	yes	characteristic
29	#14273685	—	2	1	no	—
30	#14273685	C₂²	1	1	yes	characteristic
31	C₄⋊C₄×D₅₉	—	2	1	no	—
32	C₄⋊C₄×D₅₉	—	2	1	no	—
33	C₄⋊C₄×D₅₉	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₅₉	C₂²	1	1	yes	characteristic
35	C₄²×D₅₉	—	2	1	no	—
36	C₄²×D₅₉	C₂²	1	1	yes	characteristic
37	#14273661	C₂²	1	1	yes	characteristic
38	#14273660	C₂²	1	1	yes	characteristic
39	#14273660	C₂²	1	1	yes	characteristic
40	#14273659	C₂²	1	1	yes	characteristic
41	#14273658	C₂²	1	1	yes	characteristic
42	#14273658	C₂²	1	1	yes	characteristic
43	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
44	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
46	#14273651	C₂²	1	1	yes	characteristic
47	#14273651	C₂²	1	1	yes	characteristic
48	#14273649	C₂²	1	1	yes	characteristic
49	#14273649	C₂²	1	1	yes	characteristic
50	#14273647	C₂²	1	1	yes	characteristic
51	#14273647	C₂²	1	1	yes	characteristic
52	#125×C₅₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
54	#107×C₅₉	C₂²	1	1	yes	characteristic
55	#106×C₅₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
57	#103×C₅₉	C₂²	1	1	yes	characteristic
58	#103×C₅₉	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
60	#11745689	—	2	2	no	—
61	#11745689	Q₈	1	4	yes	—
62	C₂×D₂₃₆	—	2	1	no	—
63	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₅₉	—	2	1	no	—
65	C₂×C₄×D₅₉	—	2	1	no	—
66	C₂×C₄×D₅₉	—	2	1	no	—
67	C₂×C₄×D₅₉	—	2	1	no	—
68	C₂×C₄×D₅₉	—	2	1	no	—
69	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
70	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
71	C₂×C₄×D₅₉	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	—	2	1	no	—
76	C₄⋊C₄×C₅₉	—	2	1	no	—
77	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
80	C₄²×C₅₉	—	2	1	no	—
81	C₄²×C₅₉	C₂³	1	1	yes	characteristic
82	(C₂×D₅₉)⋊C₄	—	2	1	no	—
83	(C₂×D₅₉)⋊C₄	—	2	1	no	—
84	(C₂×D₅₉)⋊C₄	—	2	1	no	—
85	C₄⋊Dic₅₉	—	2	1	no	—
86	C₄⋊Dic₅₉	—	2	1	no	—
87	C₄⋊Dic₅₉	—	2	1	no	—
88	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
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	Dic₅₉⋊C₄	—	2	1	no	—
97	C₄×Dic₅₉	—	2	1	no	—
98	C₄×Dic₅₉	—	2	1	no	—
99	C₄×Dic₅₉	D₄	1	1	yes	characteristic
100	C₄×Dic₅₉	D₄	1	1	yes	characteristic
101	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
102	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
103	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
104	#11745659	—	2	2	no	—
105	#11745659	—	2	2	no	—
106	C₂²×D₅₉	—	2	1	no	—
107	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
108	Q₈×C₅₉	—	2	1	no	—
109	Q₈×C₅₉	C₂×Q₈	1	2	yes	—
110	C₂×C₄×C₅₉	—	2	1	no	—
111	C₂×C₄×C₅₉	—	2	1	no	—
112	C₂×C₄×C₅₉	—	2	1	no	—
113	C₂×C₄×C₅₉	—	2	1	no	—
114	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
117	C₂×Dic₅₉	—	2	1	no	—
118	C₂×Dic₅₉	—	2	1	no	—
119	C₂×Dic₅₉	—	2	1	no	—
120	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
123	D₂₃₆	—	2	4	no	—
124	D₂₃₆	—	2	1	no	—
125	D₂₃₆	C₂×Q₈	1	2	yes	—
126	C₄×D₅₉	—	2	4	no	—
127	C₄×D₅₉	—	4	1	no	—
128	C₄×D₅₉	—	4	1	no	—
129	C₄×D₅₉	—	4	2	no	—
130	C₄×D₅₉	—	4	2	no	—
131	C₄×D₅₉	—	4	2	no	—
132	C₄×D₅₉	—	4	2	no	—
133	C₄×D₅₉	C₂×Q₈	1	2	yes	—
134	C₄×D₅₉	#41	1	2	yes	—
135	C₈×C₅₉	—	2	1	no	—
136	C₈×C₅₉	—	2	1	no	—
137	C₅₉⋊C₈	—	2	1	no	—
138	C₅₉⋊C₈	—	2	1	no	—
139	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
140	C₂×D₅₉	—	2	1	no	—
141	C₂×D₅₉	—	2	1	no	—
142	C₂×D₅₉	—	2	2	no	—
143	C₂×D₅₉	—	4	1	no	—
144	C₂×D₅₉	—	4	1	no	—
145	C₂×D₅₉	C₂²⋊Q₈	1	2	yes	—
146	C₄×C₅₉	—	2	2	no	—
147	C₄×C₅₉	—	2	1	no	—
148	C₄×C₅₉	—	2	1	no	—
149	C₄×C₅₉	—	4	1	no	—
150	C₄×C₅₉	—	4	1	no	—
151	C₄×C₅₉	—	4	1	no	—
152	C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
153	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
154	Dic₅₉	—	2	1	no	—
155	Dic₅₉	—	2	1	no	—
156	Dic₅₉	—	4	1	no	—
157	Dic₅₉	—	4	1	no	—
158	Dic₅₉	—	4	1	no	—
159	Dic₅₉	C₂²⋊Q₈	1	2	yes	—
160	#3080	—	59	1	no	maximal, Sylow
161	C₂×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
162	C₂×C₅₉	#575	1	1	yes	characteristic
163	C₂×C₅₉	#575	1	1	yes	characteristic
164	D₅₉	—	2	2	no	—
165	D₅₉	—	4	2	no	—
166	C₂×#125	—	59	1	no	—
167	C₄×#41	—	59	1	no	—
168	#479	—	59	1	no	—
169	#479	—	59	1	no	—
170	#479	—	59	1	no	—
171	#479	D₅₉	1	1	yes	characteristic, complemented
172	#478	—	59	1	no	—
173	#478	—	59	1	no	—
174	#478	—	59	1	no	—
175	#478	—	59	1	no	—
176	#428	—	59	1	no	—
177	#428	—	59	1	no	—
178	#423	—	59	1	no	—
179	#417	—	59	1	no	—
180	#417	—	59	1	no	—
181	C₅₉	#3080	1	1	yes	characteristic, complemented, Sylow
182	C₂×#41	—	59	1	no	—
183	C₂×M₁₆	—	59	1	no	—
184	C₂×M₁₆	—	59	1	no	—
185	#125	—	59	1	no	—
186	#125	—	59	1	no	—
187	#125	—	59	1	no	—
188	#125	—	118	1	no	—
189	#125	—	118	1	no	—
190	#125	C₂×D₅₉	1	1	yes	characteristic
191	C₄×Q₈	—	59	1	no	—
192	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
193	C₄×D₄	—	59	1	no	—
194	C₄×D₄	—	59	1	no	—
195	C₄×D₄	—	118	1	no	—
196	C₄×D₄	—	118	1	no	—
197	#117	—	59	1	no	—
198	#117	—	118	1	no	—
199	C₂×C₄⋊C₄	—	59	1	no	—
200	C₂×C₄⋊C₄	—	59	1	no	—
201	C₂×C₄⋊C₄	—	118	1	no	—
202	C₂×C₄⋊C₄	—	118	1	no	—
203	C₂×C₄²	—	59	1	no	—
204	C₂×C₄²	—	118	1	no	—
205	#107	—	59	1	no	—
206	#107	—	59	1	no	—
207	#107	—	59	1	no	—
208	#107	C₂×D₅₉	1	1	yes	characteristic
209	#106	—	59	1	no	—
210	#106	—	59	1	no	—
211	#106	—	59	1	no	—
212	#106	C₂×D₅₉	1	1	yes	characteristic
213	C₄⋊C₈	—	59	1	no	—
214	C₄⋊C₈	—	59	1	no	—
215	C₄⋊C₈	—	59	1	no	—
216	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
217	#103	—	59	1	no	—
218	#103	—	59	1	no	—
219	#103	C₂×D₅₉	1	1	yes	characteristic
220	#103	C₂×D₅₉	1	1	yes	characteristic
221	#102	—	59	1	no	—
222	#102	—	59	1	no	—
223	#102	—	59	1	no	—
224	#102	—	59	1	no	—
225	#41	—	59	4	no	—
226	#41	—	118	2	no	—
227	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
228	C₂×D₄	—	59	1	no	—
229	C₂×D₄	—	118	1	no	—
230	C₂²×C₄	—	59	1	no	—
231	C₂²×C₄	—	59	1	no	—
232	C₂²×C₄	—	59	1	no	—
233	C₂²×C₄	—	118	1	no	—
234	C₂²×C₄	—	118	1	no	—
235	C₂²×C₄	—	118	1	no	—
236	C₂²×C₄	—	118	1	no	—
237	C₂²×C₄	—	118	1	no	—
238	M₁₆	—	118	2	no	—
239	M₁₆	—	118	2	no	—
240	C₂×C₈	—	59	1	no	—
241	C₂×C₈	—	59	1	no	—
242	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	59	1	no	—
248	C₄⋊C₄	—	59	1	no	—
249	C₄⋊C₄	—	59	1	no	—
250	C₄⋊C₄	—	118	1	no	—
251	C₄⋊C₄	—	118	1	no	—
252	C₄⋊C₄	—	118	1	no	—
253	C₄⋊C₄	—	118	1	no	—
254	C₄⋊C₄	—	118	1	no	—
255	C₄⋊C₄	—	118	1	no	—
256	C₄⋊C₄	—	118	1	no	—
257	C₄⋊C₄	—	118	1	no	—
258	C₄⋊C₄	—	118	1	no	—
259	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
260	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
261	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
262	C₂²⋊C₄	—	118	1	no	—
263	C₂²⋊C₄	—	118	1	no	—
264	C₂²⋊C₄	—	118	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	59	1	no	—
267	C₄²	—	59	1	no	—
268	C₄²	—	59	1	no	—
269	C₄²	—	118	1	no	—
270	C₄²	—	118	1	no	—
271	C₄²	C₂²×D₅₉	1	1	yes	characteristic
272	C₂³	—	59	1	no	—
273	C₂³	—	118	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	Q₈×D₅₉	1	2	yes	—
276	D₄	—	59	2	no	—
277	D₄	—	118	4	no	—
278	D₄	—	118	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	59	1	no	—
284	C₂×C₄	—	59	2	no	—
285	C₂×C₄	—	59	2	no	—
286	C₂×C₄	—	59	1	no	—
287	C₂×C₄	—	59	1	no	—
288	C₂×C₄	—	118	4	no	—
289	C₂×C₄	—	118	1	no	—
290	C₂×C₄	—	118	1	no	—
291	C₂×C₄	—	118	1	no	—
292	C₂×C₄	—	236	1	no	—
293	C₂×C₄	—	236	1	no	—
294	C₂×C₄	—	236	2	no	—
295	C₂×C₄	—	236	2	no	—
296	C₂×C₄	—	236	2	no	—
297	C₂×C₄	—	236	2	no	—
298	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
299	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
300	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	118	1	no	—
304	C₈	—	118	1	no	—
305	C₂²	—	59	2	no	—
306	C₂²	—	118	1	no	—
307	C₂²	—	118	1	no	—
308	C₂²	—	118	2	no	—
309	C₂²	—	236	1	no	—
310	C₂²	—	236	1	no	—
311	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	59	2	no	—
319	C₄	—	118	1	no	—
320	C₄	—	118	1	no	—
321	C₄	—	236	1	no	—
322	C₄	—	236	1	no	—
323	C₄	—	236	1	no	—
324	C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
325	C₄	#41×D₅₉	1	1	yes	characteristic
326	C₂	—	118	2	no	—
327	C₂	—	236	2	no	—
328	C₂	C₂²⋊Q₈×D₅₉	1	1	yes	characteristic
329	C₂	#16838199	1	1	yes	characteristic
330	C₂	#16838201	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#479	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#479	D₅₉
hypocenter	C₅₉	#3080
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
1. #125×D₅₉	C₂	semidirect
3. #16837711	C₂	semidirect
4. #16837709	C₂	semidirect
5. #16837707	C₂	semidirect
6. #16837705	C₂	semidirect
7. #16837703	C₂	semidirect
8. #16837701	C₂	semidirect
9. #16837699	C₂	semidirect
10. #16837443	C₂	semidirect
11. #16837441	C₂	semidirect
12. #16837424	C₂	semidirect
15. C₅₉×#479	C₂	semidirect
171. #479	D₅₉	semidirect
181. C₅₉	#3080	semidirect