Subgroups — #21436697

Order 7552.

360 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838104	C₂	1	1	yes	maximal, characteristic, complemented
2	#16838099	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837802	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837725	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837723	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837714	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837713	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837704	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837700	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837652	C₂	1	1	yes	maximal, characteristic, complemented
11	D₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837478	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837444	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837441	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#469	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11745659	C₂²	1	1	yes	characteristic
17	#14273720	—	2	1	no	—
18	#14273720	—	2	1	no	—
19	#14273718	C₂²	1	1	yes	characteristic
20	#14273718	C₂²	1	1	yes	characteristic
21	#14273711	—	2	1	no	—
22	#14273708	—	2	1	no	—
23	#14273699	—	2	1	no	—
24	#14273698	—	2	1	no	—
25	#14273698	C₂²	1	1	yes	characteristic
26	#14273698	C₂²	1	1	yes	characteristic
27	D₄×Dic₅₉	—	2	1	no	—
28	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
29	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊D₅₉	C₂²	1	1	yes	characteristic
31	(C₂²⋊D₅₉)⋊C₄	—	2	1	no	—
32	#14273686	C₂²	1	1	yes	characteristic
33	#14273685	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₅₉	C₂²	1	1	yes	characteristic
36	#14273661	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	#14273660	C₂²	1	1	yes	characteristic
41	#14273659	C₂²	1	1	yes	characteristic
42	#14273659	C₂²	1	1	yes	characteristic
43	#14273658	C₂²	1	1	yes	characteristic
44	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
45	#14273648	C₂²	1	1	yes	characteristic
46	#14273648	C₂²	1	1	yes	characteristic
47	#14273648	C₂²	1	1	yes	characteristic
48	#14273648	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₅₉)⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
52	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
54	#121×C₅₉	C₂²	1	1	yes	characteristic
55	#121×C₅₉	C₂²	1	1	yes	characteristic
56	#106×C₅₉	C₂²	1	1	yes	characteristic
57	#102×C₅₉	C₂²	1	1	yes	characteristic
58	#102×C₅₉	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₅₉	—	2	1	no	—
60	C₂×D₄×C₅₉	—	2	1	no	—
61	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₅₉	—	2	1	no	—
65	C₂×C₂²⋊D₅₉	—	2	1	no	—
66	C₂×C₂²⋊D₅₉	D₄	1	1	yes	characteristic
67	C₂²×Dic₅₉	—	2	1	no	—
68	C₂²×Dic₅₉	—	2	1	no	—
69	C₂²×Dic₅₉	—	2	1	no	—
70	C₂×C₄×D₅₉	—	2	1	no	—
71	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
72	D₈×C₅₉	—	2	2	no	—
73	M₁₆×C₅₉	—	2	2	no	—
74	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
75	C₂×C₈×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	C₂²⋊Dic₅₉	—	2	1	no	—
81	C₂²⋊Dic₅₉	—	2	1	no	—
82	C₂²⋊Dic₅₉	—	2	1	no	—
83	C₂²⋊Dic₅₉	—	2	1	no	—
84	C₂²⋊Dic₅₉	—	2	1	no	—
85	C₂²⋊Dic₅₉	D₄	1	1	yes	characteristic
86	(C₂×D₅₉)⋊C₄	—	2	1	no	—
87	(C₂×D₅₉)⋊C₄	D₄	1	1	yes	characteristic
88	C₄⋊Dic₅₉	—	2	1	no	—
89	C₄⋊Dic₅₉	—	2	1	no	—
90	C₄⋊Dic₅₉	—	2	1	no	—
91	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
99	C₄×Dic₅₉	—	2	1	no	—
100	C₄×Dic₅₉	—	2	1	no	—
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	C₂³×C₅₉	—	2	1	no	—
106	C₂³×C₅₉	—	2	1	no	—
107	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
108	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
109	D₄×C₅₉	—	2	1	no	—
110	D₄×C₅₉	—	2	1	no	—
111	D₄×C₅₉	—	2	1	no	—
112	D₄×C₅₉	—	2	1	no	—
113	D₄×C₅₉	—	4	2	no	—
114	D₄×C₅₉	—	8	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	—	2	1	no	—
119	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
121	C₂²⋊D₅₉	—	4	1	no	—
122	C₂²⋊D₅₉	—	4	2	no	—
123	C₂²⋊D₅₉	—	8	1	no	—
124	C₂×Dic₅₉	—	2	1	no	—
125	C₂×Dic₅₉	—	2	1	no	—
126	C₂×Dic₅₉	—	2	1	no	—
127	C₂×Dic₅₉	—	2	1	no	—
128	C₂×Dic₅₉	—	8	1	no	—
129	C₂×Dic₅₉	—	8	1	no	—
130	C₂×Dic₅₉	—	8	1	no	—
131	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
134	C₄×D₅₉	—	2	1	no	—
135	C₄×D₅₉	—	2	1	no	—
136	C₄×D₅₉	—	8	1	no	—
137	C₈×C₅₉	—	2	1	no	—
138	C₈×C₅₉	#41	1	2	yes	—
139	C₅₉⋊C₈	—	2	1	no	—
140	C₅₉⋊C₈	—	2	1	no	—
141	C₂²×C₅₉	—	2	1	no	—
142	C₂²×C₅₉	—	2	1	no	—
143	C₂²×C₅₉	—	2	1	no	—
144	C₂²×C₅₉	—	4	1	no	—
145	C₂²×C₅₉	—	4	1	no	—
146	C₂²×C₅₉	—	4	1	no	—
147	C₂²×C₅₉	—	4	1	no	—
148	C₂²×C₅₉	—	4	1	no	—
149	C₂²×C₅₉	—	4	1	no	—
150	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
151	C₂×D₅₉	—	2	1	no	—
152	C₂×D₅₉	—	2	1	no	—
153	C₂×D₅₉	—	2	1	no	—
154	C₄×C₅₉	—	2	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	2₋¹⁺⁴	1	1	yes	characteristic
159	Dic₅₉	—	2	1	no	—
160	Dic₅₉	—	2	1	no	—
161	Dic₅₉	—	2	1	no	—
162	Dic₅₉	—	4	1	no	—
163	Dic₅₉	—	4	1	no	—
164	Dic₅₉	—	4	1	no	—
165	Dic₅₉	—	4	1	no	—
166	#3355	—	59	1	no	maximal, Sylow
167	C₂×C₅₉	—	4	1	no	—
168	C₂×C₅₉	—	8	1	no	—
169	C₂×C₅₉	—	8	1	no	—
170	C₂×C₅₉	#547	1	1	yes	characteristic
171	C₂×C₅₉	#575	1	1	yes	characteristic
172	C₂×C₅₉	#576	1	1	yes	characteristic
173	D₅₉	—	4	1	no	—
174	#553	—	59	1	no	—
175	#553	—	59	1	no	—
176	#499	—	59	1	no	—
177	#482	—	59	1	no	—
178	#482	—	59	1	no	—
179	#480	—	59	1	no	—
180	#480	—	59	1	no	—
181	#478	—	59	1	no	—
182	#478	—	59	1	no	—
183	#469	—	59	1	no	—
184	#469	D₅₉	1	1	yes	characteristic, complemented
185	C₄×D₈	—	59	1	no	—
186	#436	—	59	1	no	—
187	#428	—	59	1	no	—
188	#428	—	59	1	no	—
189	C₅₉	#3355	1	1	yes	characteristic, complemented, Sylow
190	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
191	C₂×M₁₆	—	59	1	no	—
192	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
193	#126	—	118	1	no	—
194	#126	—	118	1	no	—
195	#125	—	59	1	no	—
196	#125	—	59	1	no	—
197	#123	—	118	1	no	—
198	#123	—	118	1	no	—
199	#121	—	59	1	no	—
200	#121	—	59	1	no	—
201	#121	—	118	1	no	—
202	#121	—	118	1	no	—
203	#121	C₂×D₅₉	1	1	yes	characteristic
204	#121	C₂×D₅₉	1	1	yes	characteristic
205	C₄×D₄	—	59	1	no	—
206	C₄×D₄	—	59	1	no	—
207	C₄×D₄	—	59	1	no	—
208	C₄×D₄	—	118	1	no	—
209	C₄×D₄	—	118	1	no	—
210	#117	—	59	1	no	—
211	#117	—	59	1	no	—
212	C₂×C₄⋊C₄	—	59	1	no	—
213	C₂×C₄⋊C₄	—	59	1	no	—
214	#107	—	59	1	no	—
215	#107	—	59	1	no	—
216	#107	—	59	1	no	—
217	#107	—	59	1	no	—
218	#107	—	59	1	no	—
219	#106	—	59	1	no	—
220	#106	—	59	1	no	—
221	#106	—	59	1	no	—
222	#106	C₂×D₅₉	1	1	yes	characteristic
223	C₄⋊C₈	—	59	1	no	—
224	#102	—	59	1	no	—
225	#102	—	59	1	no	—
226	#102	—	59	1	no	—
227	#102	—	59	1	no	—
228	#102	C₂×D₅₉	1	1	yes	characteristic
229	#102	C₂×D₅₉	1	1	yes	characteristic
230	C₂²⋊C₈	—	59	1	no	—
231	C₂²⋊C₈	—	59	1	no	—
232	C₄×C₈	—	59	1	no	—
233	C₂×D₄	—	2	1	no	—
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	—	59	1	no	—
236	C₂×D₄	—	118	1	no	—
237	C₂×D₄	—	118	1	no	—
238	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
239	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
240	C₂²×C₄	—	59	1	no	—
241	C₂²×C₄	—	118	1	no	—
242	C₂²×C₄	—	118	1	no	—
243	C₂²×C₄	—	118	1	no	—
244	C₂²×C₄	—	118	1	no	—
245	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
246	D₈	—	2	2	no	—
247	M₁₆	—	2	2	no	—
248	M₁₆	—	118	2	no	—
249	C₂×C₈	—	59	1	no	—
250	C₂×C₈	—	59	1	no	—
251	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
253	C₄⋊C₄	—	59	1	no	—
254	C₄⋊C₄	—	59	1	no	—
255	C₄⋊C₄	—	59	1	no	—
256	C₄⋊C₄	—	59	1	no	—
257	C₄⋊C₄	—	59	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	—	118	1	no	—
264	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
265	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	59	1	no	—
269	C₂²⋊C₄	—	59	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	C₂²⋊C₄	—	118	1	no	—
276	C₄²	—	59	1	no	—
277	C₄²	—	118	1	no	—
278	C₄²	—	118	1	no	—
279	C₂³	—	2	1	no	—
280	C₂³	—	2	1	no	—
281	C₂³	—	59	1	no	—
282	C₂³	D₄×D₅₉	1	1	yes	characteristic
283	D₄	—	2	1	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	2	1	no	—
287	D₄	—	4	2	no	—
288	D₄	—	8	1	no	—
289	D₄	—	472	1	no	—
290	D₄	—	236	1	no	—
291	D₄	—	236	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	59	1	no	—
297	C₂×C₄	—	59	1	no	—
298	C₂×C₄	—	59	1	no	—
299	C₂×C₄	—	118	1	no	—
300	C₂×C₄	—	118	1	no	—
301	C₂×C₄	—	118	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	—	472	1	no	—
307	C₂×C₄	—	472	1	no	—
308	C₂×C₄	—	472	1	no	—
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	—	118	1	no	—
313	C₈	—	118	1	no	—
314	C₈	#11745687	1	2	yes	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	118	1	no	—
325	C₂²	—	118	1	no	—
326	C₂²	—	118	1	no	—
327	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	118	1	no	—
332	C₄	—	118	1	no	—
333	C₄	—	118	1	no	—
334	C₄	—	236	1	no	—
335	C₄	—	236	1	no	—
336	C₄	—	236	1	no	—
337	C₄	—	236	1	no	—
338	C₄	C₂×#11745687	1	1	yes	characteristic
339	C₄	#14273774	1	1	yes	characteristic
340	C₂	—	4	1	no	—
341	C₂	—	8	1	no	—
342	C₂	—	8	1	no	—
343	C₂	—	236	1	no	—
344	C₂	#16838052	1	1	yes	characteristic
345	C₂	#16838198	1	1	yes	characteristic
346	C₂	#16838206	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#469	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#469	D₅₉
hypocenter	C₅₉	#3355
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #469

Complements

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

N	H ≅ G/N	type
1. #16838104	C₂	semidirect
2. #16838099	C₂	semidirect
3. #16837802	C₂	semidirect
4. #16837725	C₂	semidirect
5. #16837723	C₂	semidirect
6. #16837714	C₂	semidirect
7. #16837713	C₂	semidirect
8. #16837704	C₂	semidirect
9. #16837700	C₂	semidirect
10. #16837652	C₂	semidirect
11. D₈×Dic₅₉	C₂	semidirect
12. #16837478	C₂	semidirect
13. #16837444	C₂	semidirect
14. #16837441	C₂	semidirect
15. C₅₉×#469	C₂	semidirect
184. #469	D₅₉	semidirect
189. C₅₉	#3355	semidirect