Subgroups — #27299344

Order 13568.

498 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810705	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810437	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810435	C₂	1	1	yes	maximal, characteristic
4	M₃₂×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#108×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272823	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1387	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄₂₄	C₂²	1	1	yes	characteristic
9	C₂×#13133179	—	2	1	no	—
10	C₂×#13133157	C₂²	1	1	yes	characteristic
11	#16810705	—	2	1	no	—
12	#16810705	—	2	1	no	—
13	#16810705	C₄	1	2	yes	—
14	#16810705	C₄	1	2	yes	—
15	C₂×#13133153	C₂²	1	1	yes	characteristic
16	C₂×#13133137	—	2	1	no	—
17	M₁₆×Dic₅₃	—	2	1	no	—
18	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
19	#16810437	C₄	1	4	yes	complemented
20	#16810435	C₄	1	4	yes	complemented
21	#16810398	—	2	1	no	—
22	C₁₆×Dic₅₃	—	2	1	no	—
23	#16810387	—	2	1	no	—
24	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
27	C₂²×D₂₁₂	—	2	1	no	—
28	C₂²×C₄×D₅₃	—	4	1	no	—
29	C₂×D₄₂₄	—	2	1	no	—
30	C₂×D₄₂₄	—	2	1	no	—
31	C₂×D₄₂₄	—	2	1	no	—
32	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
33	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
34	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
35	C₂×#11719917	—	2	1	no	—
36	C₂²×#88322	D₄	1	1	yes	characteristic
37	#13133179	—	2	2	no	—
38	#13133179	—	2	2	no	—
39	#13133179	—	4	1	no	—
40	#13133179	—	4	1	no	—
41	C₂×#11719921	—	4	1	no	—
42	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
43	#13133157	—	2	1	no	—
44	#13133157	C₂×C₄	1	2	yes	complemented
45	C₂×#11719908	—	2	1	no	—
46	#13133153	—	2	1	no	—
47	#13133153	C₂×C₄	1	2	yes	complemented
48	#13133137	—	2	2	no	—
49	#13133137	—	2	2	no	—
50	#13133127	—	4	1	no	—
51	C₈×Dic₅₃	—	4	1	no	—
52	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
53	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
54	C₈×Dic₅₃	D₄	1	1	yes	characteristic
55	C₈×Dic₅₃	D₄	1	1	yes	characteristic
56	C₂×M₁₆×C₅₃	—	2	1	no	—
57	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
59	M₃₂×C₅₃	—	2	1	no	—
60	M₃₂×C₅₃	C₂×C₄	1	2	yes	complemented
61	C₂×C₁₆×C₅₃	—	2	1	no	—
62	#108×C₅₃	—	2	1	no	—
63	#108×C₅₃	C₂×C₄	1	2	yes	complemented
64	#107×C₅₃	—	2	1	no	—
65	#107×C₅₃	C₂×C₄	1	2	yes	—
66	C₂³×D₅₃	—	4	1	no	—
67	C₂²×C₄×C₅₃	—	4	1	no	—
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₅₃	SD₁₆	1	1	yes	characteristic
70	C₂²×Dic₅₃	SD₁₆	1	1	yes	characteristic
71	C₂×D₂₁₂	—	2	2	no	—
72	C₂×D₂₁₂	—	2	2	no	—
73	C₂×D₂₁₂	—	2	2	no	—
74	C₂×D₂₁₂	—	4	1	no	—
75	C₂×D₂₁₂	—	4	1	no	—
76	C₂×D₂₁₂	—	4	1	no	—
77	C₂×C₄×D₅₃	—	4	2	no	—
78	C₂×C₄×D₅₃	—	4	2	no	—
79	C₂×C₄×D₅₃	—	8	4	no	—
80	M₁₆×C₅₃	—	2	2	no	—
81	M₁₆×C₅₃	—	4	1	no	—
82	C₂×C₈×C₅₃	—	4	1	no	—
83	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
88	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
89	C₄⋊C₄×C₅₃	—	2	2	no	—
90	C₄⋊C₄×C₅₃	—	4	1	no	—
91	#11719921	—	4	2	no	—
92	#11719921	—	4	2	no	—
93	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
94	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
95	C₄×Dic₅₃	D₈	1	1	yes	characteristic
96	C₄×Dic₅₃	D₈	1	1	yes	characteristic
97	#11719917	—	2	2	no	—
98	#11719917	—	4	1	no	—
99	C₂×#88322	—	2	1	no	—
100	C₂×#88322	—	2	1	no	—
101	C₂×#88322	—	4	1	no	—
102	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
104	D₄₂₄	—	2	2	no	—
105	D₄₂₄	—	2	2	no	—
106	D₄₂₄	—	2	2	no	—
107	D₄₂₄	C₄²	1	4	yes	—
108	C₁₆×C₅₃	—	2	2	no	—
109	#11719908	—	2	2	no	—
110	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
111	C₂²×D₅₃	—	4	2	no	—
112	C₂²×D₅₃	—	4	2	no	—
113	C₂²×D₅₃	—	4	2	no	—
114	C₂²×D₅₃	—	8	2	no	—
115	C₂²×D₅₃	—	8	2	no	—
116	C₂×C₄×C₅₃	—	4	2	no	—
117	C₂×C₄×C₅₃	—	8	2	no	—
118	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
123	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
124	C₂×Dic₅₃	—	2	2	no	—
125	C₂×Dic₅₃	—	4	1	no	—
126	C₂×Dic₅₃	#102	1	1	yes	characteristic
127	C₂×Dic₅₃	#102	1	1	yes	characteristic
128	C₂×Dic₅₃	#102	1	1	yes	characteristic
129	C₂×Dic₅₃	#102	1	1	yes	characteristic
130	D₂₁₂	—	2	4	no	—
131	D₂₁₂	—	4	2	no	—
132	D₂₁₂	—	4	2	no	—
133	D₂₁₂	—	4	2	no	—
134	C₄×D₅₃	—	8	4	no	—
135	C₄×D₅₃	—	8	4	no	—
136	C₈×C₅₃	—	4	2	no	—
137	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
138	C₈×C₅₃	#117	1	1	yes	characteristic
139	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
140	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
141	#88322	—	2	1	no	—
142	#88322	—	2	1	no	—
143	#88322	—	4	2	no	—
144	C₄×#361	—	53	1	no	maximal, Sylow
145	C₂²×C₅₃	—	2	1	no	—
146	C₂²×C₅₃	—	2	1	no	—
147	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
148	C₂²×C₅₃	C₄×SD₁₆	1	1	yes	characteristic
149	C₂²×C₅₃	C₄×SD₁₆	1	1	yes	characteristic
150	C₂×D₅₃	—	4	4	no	—
151	C₂×D₅₃	—	8	4	no	—
152	C₂×D₅₃	—	8	4	no	—
153	C₂×D₅₃	—	8	2	no	—
154	C₂×D₅₃	—	8	2	no	—
155	C₄×C₅₃	—	8	2	no	—
156	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
157	C₄×C₅₃	#416	1	1	yes	characteristic
158	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
159	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
160	Dic₅₃	—	2	1	no	—
161	Dic₅₃	—	2	1	no	—
162	Dic₅₃	—	2	1	no	—
163	Dic₅₃	#361	1	2	yes	complemented
164	C₂×C₄×D₈	—	53	1	no	—
165	C₂×#361	—	53	1	no	—
166	C₂×#361	—	53	1	no	—
167	C₄×M₃₂	—	53	1	no	—
168	C₄×#108	—	53	1	no	—
169	#1387	—	53	1	no	—
170	#1387	D₅₃	1	1	yes	characteristic, complemented
171	C₂×C₅₃	—	2	1	no	—
172	C₂×C₅₃	—	2	1	no	—
173	C₂×C₅₃	C₄×#102	1	1	yes	characteristic
174	C₂×C₅₃	C₂×#361	1	1	yes	characteristic
175	C₂×C₅₃	#2148	1	1	yes	characteristic
176	D₅₃	—	8	4	no	—
177	C₂²×D₈	—	53	1	no	—
178	C₂×C₄×D₄	—	106	1	no	—
179	C₂×M₃₂	—	53	1	no	—
180	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
181	C₄×D₈	—	53	2	no	—
182	C₄×D₈	—	53	2	no	—
183	C₄×D₈	—	106	1	no	—
184	C₄×D₈	—	106	1	no	—
185	C₂×#108	—	53	1	no	—
186	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
187	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
188	C₂×#102	—	106	1	no	—
189	C₄×M₁₆	—	106	1	no	—
190	C₂×C₄×C₈	—	53	1	no	—
191	#361	—	53	4	no	—
192	#361	—	53	4	no	—
193	#346	—	106	1	no	—
194	C₄×C₁₆	—	106	1	no	—
195	#341	—	106	1	no	—
196	C₅₃	C₄×#361	1	1	yes	characteristic, complemented, Sylow
197	C₂²×D₄	—	106	1	no	—
198	C₂³×C₄	—	212	1	no	—
199	C₂×D₈	—	53	2	no	—
200	C₂×D₈	—	53	2	no	—
201	C₂×D₈	—	53	2	no	—
202	C₂×D₈	—	106	1	no	—
203	C₂×D₈	—	106	1	no	—
204	C₂×D₈	—	106	1	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	—	106	1	no	—
207	C₂²×C₈	—	53	1	no	—
208	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
209	C₄×D₄	—	106	2	no	—
210	C₄×D₄	—	106	2	no	—
211	C₄×D₄	—	212	1	no	—
212	C₄×D₄	—	212	1	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₂²⋊C₄	—	212	1	no	—
215	C₂×C₄²	—	53	1	no	—
216	M₃₂	—	2	1	no	—
217	M₃₂	—	53	2	no	—
218	M₃₂	—	106	1	no	—
219	M₃₂	C₄×D₅₃	1	2	yes	complemented
220	C₂×C₁₆	—	2	1	no	—
221	C₂×C₁₆	—	106	1	no	—
222	#108	—	2	1	no	—
223	#108	—	53	2	no	—
224	#108	—	106	1	no	—
225	#108	C₄×D₅₃	1	2	yes	complemented
226	#107	—	2	1	no	—
227	#107	C₄×D₅₃	1	2	yes	—
228	#102	—	106	2	no	—
229	#102	—	106	2	no	—
230	#97	—	212	1	no	—
231	C₄×C₈	—	53	1	no	—
232	C₄×C₈	—	53	1	no	—
233	C₄×C₈	—	53	1	no	—
234	C₄×C₈	—	53	1	no	—
235	C₄×C₈	—	212	1	no	—
236	C₂⁴	—	212	1	no	—
237	C₂×D₄	—	106	2	no	—
238	C₂×D₄	—	106	2	no	—
239	C₂×D₄	—	106	2	no	—
240	C₂×D₄	—	212	1	no	—
241	C₂×D₄	—	212	1	no	—
242	C₂×D₄	—	212	1	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	—	53	1	no	—
245	C₂²×C₄	—	53	1	no	—
246	C₂²×C₄	—	424	4	no	—
247	C₂²×C₄	—	212	2	no	—
248	C₂²×C₄	—	212	2	no	—
249	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
250	D₈	—	53	4	no	—
251	D₈	—	106	2	no	—
252	D₈	—	106	2	no	—
253	D₈	—	106	2	no	—
254	M₁₆	—	2	2	no	—
255	M₁₆	—	4	1	no	—
256	M₁₆	—	106	2	no	—
257	M₁₆	—	212	1	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	53	1	no	—
260	C₂×C₈	—	53	1	no	—
261	C₂×C₈	—	106	1	no	—
262	C₂×C₈	—	106	1	no	—
263	C₂×C₈	—	212	1	no	—
264	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
265	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
267	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
268	C₂×C₈	#11719947	1	1	yes	characteristic
269	C₂×C₈	#11719949	1	1	yes	characteristic
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₂²⋊C₄	—	212	2	no	—
273	C₂²⋊C₄	—	212	2	no	—
274	C₄²	—	53	1	no	—
275	C₄²	—	53	1	no	—
276	C₄²	—	53	1	no	—
277	C₄²	—	53	1	no	—
278	C₁₆	—	2	2	no	—
279	C₁₆	—	106	2	no	—
280	C₂³	—	424	2	no	—
281	C₂³	—	424	2	no	—
282	C₂³	—	212	2	no	—
283	C₂³	—	212	2	no	—
284	C₂³	—	212	2	no	—
285	C₂³	SD₁₆×D₅₃	1	1	yes	characteristic
286	D₄	—	106	4	no	—
287	D₄	—	212	2	no	—
288	D₄	—	212	2	no	—
289	D₄	—	212	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	8	2	no	—
292	C₂×C₄	—	53	1	no	—
293	C₂×C₄	—	53	1	no	—
294	C₂×C₄	—	53	1	no	—
295	C₂×C₄	—	53	1	no	—
296	C₂×C₄	—	106	2	no	—
297	C₂×C₄	—	424	4	no	—
298	C₂×C₄	—	424	4	no	—
299	C₂×C₄	—	212	1	no	—
300	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
301	C₂×C₄	#13133177	1	1	yes	characteristic
302	C₂×C₄	#13133179	1	1	yes	characteristic
303	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
304	C₂×C₄	#13133238	1	1	yes	characteristic
305	C₂×C₄	#13133240	1	1	yes	characteristic
306	C₈	—	4	2	no	—
307	C₈	—	106	1	no	—
308	C₈	—	106	1	no	—
309	C₈	—	212	2	no	—
310	C₈	C₄²×D₅₃	1	1	yes	characteristic
311	C₈	#13133175	1	1	yes	characteristic
312	C₈	#13133177	1	1	yes	characteristic
313	C₈	#13133179	1	1	yes	characteristic
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	424	4	no	—
317	C₂²	—	424	4	no	—
318	C₂²	—	424	2	no	—
319	C₂²	—	424	2	no	—
320	C₂²	—	212	4	no	—
321	C₂²	#102×D₅₃	1	1	yes	characteristic
322	C₂²	#16810708	1	1	yes	characteristic
323	C₂²	#16810710	1	1	yes	characteristic
324	C₄	—	8	2	no	—
325	C₄	—	53	2	no	—
326	C₄	—	106	1	no	—
327	C₄	—	106	1	no	—
328	C₄	—	106	1	no	—
329	C₄	#16810470	1	1	yes	characteristic
330	C₄	#16810618	1	1	yes	characteristic
331	C₄	#16810703	1	1	yes	characteristic
332	C₄	#16810705	1	1	yes	characteristic
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	424	4	no	—
336	C₂	#20274529	1	1	yes	characteristic
337	C₂	#361×D₅₃	1	1	yes	characteristic
338	C₂	#20276463	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#102×D₅₃
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1387	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂²×D₄₂₄	C₂²
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1387	D₅₃
hypocenter	C₅₃	C₄×#361
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1387

Complements

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

N	H ≅ G/N	type
4. M₃₂×Dic₅₃	C₂	semidirect
5. #108×Dic₅₃	C₂	semidirect
6. #20272823	C₂	semidirect
7. C₅₃×#1387	C₂	semidirect
19. #16810437	C₄	semidirect
20. #16810435	C₄	semidirect
44. #13133157	C₂×C₄	semidirect
47. #13133153	C₂×C₄	semidirect
60. M₃₂×C₅₃	C₂×C₄	semidirect
63. #108×C₅₃	C₂×C₄	semidirect
163. Dic₅₃	#361	semidirect
170. #1387	D₅₃	semidirect
196. C₅₃	C₄×#361	semidirect
219. M₃₂	C₄×D₅₃	semidirect
225. #108	C₄×D₅₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16810705	2
C₂×#16810437	2
C₂×#16810435	2
M₃₂×Dic₅₃	2
#108×Dic₅₃	2
#20272823	2
C₅₃×#1387	2
C₄×#361	53