Subgroups — #27539465

Order 13568.

622 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#20285144	C₂	1	1	yes	maximal, characteristic
3	#20276953	C₂	1	4	yes	maximal, complemented
4	#20276942	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276940	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276939	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276426	C₂	1	4	yes	maximal, complemented
8	C₄×#13133156	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#2241	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13133240	—	2	1	no	—
11	C₂×#13133240	C₂²	1	2	yes	—
12	#16811174	—	2	1	no	—
13	C₄⋊Q₈×D₅₃	—	2	1	no	—
14	C₂×#13133156	C₂²	1	2	yes	—
15	#16811028	C₂²	1	1	yes	characteristic
16	#16811027	—	2	1	no	—
17	#16811027	C₂²	1	2	yes	—
18	#16811026	—	2	1	no	—
19	#16811008	—	2	1	no	—
20	#16811008	C₂²	1	2	yes	—
21	#16811000	C₂²	1	1	yes	characteristic
22	#16810997	—	2	1	no	—
23	#107×D₅₃	—	2	1	no	—
24	#107×D₅₃	C₂²	1	2	yes	—
25	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
26	#16810427	C₂²	1	4	yes	—
27	#16810425	C₂²	1	4	yes	—
28	#16810423	C₂²	1	4	yes	—
29	#16810398	C₂²	1	2	yes	—
30	C₄×#11719908	C₂²	1	1	yes	characteristic
31	C₅₃×#500	C₂²	1	1	yes	characteristic
32	C₅₃×#494	C₂²	1	1	yes	characteristic
33	C₅₃×#358	C₂²	1	4	yes	—
34	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×#11719949	—	2	2	no	—
36	C₂×Q₈×D₅₃	—	4	1	no	—
37	#13133240	—	2	8	no	—
38	#13133240	—	4	4	no	—
39	C₂×#11719924	—	2	2	no	—
40	C₂×#11719924	—	2	2	no	—
41	C₂×D₄₂₄	—	2	1	no	—
42	C₂×D₄₂₄	C₂³	1	2	yes	—
43	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
44	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
45	#13133217	—	2	1	no	—
46	#13133216	—	2	2	no	—
47	#13133216	—	4	1	no	—
48	#13133215	—	4	1	no	—
49	#13133214	—	2	1	no	—
50	#13133206	—	2	2	no	—
51	#13133192	—	4	1	no	—
52	#13133192	—	4	1	no	—
53	#13133179	—	4	1	no	—
54	#13133179	—	4	1	no	—
55	C₄⋊C₄×D₅₃	—	2	2	no	—
56	C₄⋊C₄×D₅₃	—	4	1	no	—
57	C₄²×D₅₃	D₄	1	1	yes	characteristic
58	#13133156	—	2	4	no	—
59	C₂×#11719908	C₂³	1	2	yes	—
60	#13133151	—	2	1	no	—
61	#13133151	C₂³	1	2	yes	—
62	#13133150	—	2	2	no	—
63	#13133150	—	2	2	no	—
64	C₄×#88322	D₄	1	1	yes	characteristic
65	C₈×Dic₅₃	D₄	1	1	yes	characteristic
66	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
67	C₂×Q₁₆×C₅₃	—	2	1	no	—
68	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₅₃	—	2	1	no	—
70	C₄⋊Q₈×C₅₃	—	2	1	no	—
71	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
72	#107×C₅₃	—	2	1	no	—
73	#107×C₅₃	C₂³	1	2	yes	—
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₂×Q₈×C₅₃	—	2	2	no	—
76	C₂×Q₈×C₅₃	—	4	1	no	—
77	#11719949	—	4	4	no	—
78	#11719949	—	4	4	no	—
79	Q₈×D₅₃	—	8	4	no	—
80	C₂×D₂₁₂	—	2	2	no	—
81	C₂×D₂₁₂	—	4	2	no	—
82	C₂×D₂₁₂	—	4	1	no	—
83	C₂×D₂₁₂	—	4	1	no	—
84	C₂×C₄×D₅₃	—	4	2	no	—
85	C₂×C₄×D₅₃	—	4	2	no	—
86	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
87	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
88	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂×Dic₁₀₆	—	4	1	no	—
90	C₂×Dic₁₀₆	—	4	2	no	—
91	Q₁₆×C₅₃	—	2	4	no	—
92	Q₁₆×C₅₃	—	4	2	no	—
93	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₅₃	—	2	2	no	—
96	C₄⋊C₄×C₅₃	—	4	1	no	—
97	C₄⋊C₄×C₅₃	—	4	1	no	—
98	C₄⋊C₄×C₅₃	—	4	1	no	—
99	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
100	#11719924	—	4	4	no	—
101	#11719924	—	4	4	no	—
102	#11719921	—	4	2	no	—
103	#11719921	—	4	2	no	—
104	C₄⋊Dic₅₃	—	2	2	no	—
105	C₄⋊Dic₅₃	—	4	1	no	—
106	#11719919	—	4	2	no	—
107	#11719919	—	4	2	no	—
108	C₄×Dic₅₃	D₈	1	1	yes	characteristic
109	C₄×Dic₅₃	D₈	1	1	yes	characteristic
110	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
111	C₂×#88322	C₂×D₄	1	1	yes	characteristic
112	C₂×#88322	C₂×D₄	1	1	yes	characteristic
113	D₄₂₄	—	2	4	no	—
114	D₄₂₄	—	4	2	no	—
115	C₈×D₅₃	SD₁₆	1	4	yes	—
116	C₈×D₅₃	#41	1	4	yes	—
117	C₁₆×C₅₃	—	2	2	no	—
118	#11719908	—	2	2	no	—
119	C₂²×D₅₃	—	4	2	no	—
120	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
121	Q₈×C₅₃	—	4	2	no	—
122	Q₈×C₅₃	—	4	2	no	—
123	Q₈×C₅₃	—	8	2	no	—
124	C₂×C₄×C₅₃	—	4	2	no	—
125	C₂×C₄×C₅₃	—	4	2	no	—
126	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×Dic₅₃	—	4	2	no	—
130	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
132	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
133	D₂₁₂	—	4	2	no	—
134	D₂₁₂	—	4	2	no	—
135	D₂₁₂	—	8	4	no	—
136	D₂₁₂	—	8	2	no	—
137	D₂₁₂	—	8	2	no	—
138	C₄×D₅₃	—	2	2	no	—
139	C₄×D₅₃	—	2	2	no	—
140	C₄×D₅₃	—	8	4	no	—
141	C₄×D₅₃	—	8	4	no	—
142	C₄×D₅₃	#124	1	2	yes	—
143	C₄×D₅₃	C₂×SD₁₆	1	2	yes	—
144	Dic₁₀₆	—	8	4	no	—
145	Dic₁₀₆	—	8	2	no	—
146	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
147	C₈×C₅₃	C₂×#41	1	2	yes	—
148	#88322	#124	1	2	yes	—
149	#88322	C₂×SD₁₆	1	2	yes	—
150	#33707	—	53	1	no	maximal, Sylow
151	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
152	C₂×D₅₃	—	2	1	no	—
153	C₂×D₅₃	—	2	1	no	—
154	C₂×D₅₃	—	8	2	no	—
155	C₂×D₅₃	—	8	2	no	—
156	C₂×D₅₃	—	8	2	no	—
157	C₂×D₅₃	#486	1	2	yes	—
158	C₄×C₅₃	—	8	2	no	—
159	C₄×C₅₃	—	8	2	no	—
160	C₄×C₅₃	#509	1	2	yes	—
161	C₄×C₅₃	#510	1	2	yes	—
162	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
163	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
164	Dic₅₃	—	2	1	no	—
165	Dic₅₃	—	2	1	no	—
166	Dic₅₃	—	8	2	no	—
167	Dic₅₃	#486	1	2	yes	—
168	C₂×#500	—	53	1	no	—
169	#3147	—	53	1	no	—
170	#2243	—	53	4	no	—
171	#2241	—	53	1	no	—
172	#2241	D₅₃	1	1	yes	characteristic, complemented
173	#2240	—	53	1	no	—
174	#2240	—	53	1	no	—
175	#2139	—	53	4	no	—
176	C₄×M₃₂	—	53	1	no	—
177	C₂×C₅₃	C₂×#486	1	1	yes	characteristic
178	C₂×C₅₃	C₂×#509	1	1	yes	characteristic
179	C₂×C₅₃	C₂×#510	1	1	yes	characteristic
180	D₅₃	—	2	2	no	—
181	D₅₃	—	16	2	no	—
182	C₂×#135	—	53	2	no	—
183	C₂×#135	—	106	1	no	—
184	#532	—	106	1	no	—
185	C₂×C₄⋊Q₈	—	106	1	no	—
186	C₂×M₃₂	—	53	2	no	—
187	#500	—	53	2	no	—
188	#500	—	53	1	no	—
189	#500	—	106	1	no	—
190	#500	—	106	1	no	—
191	#500	C₂×D₅₃	1	1	yes	characteristic
192	#495	—	53	2	no	—
193	#495	—	106	1	no	—
194	#494	C₂×D₅₃	1	1	yes	characteristic
195	#493	—	53	1	no	—
196	#492	—	106	1	no	—
197	C₂×#107	—	53	2	no	—
198	C₂×#107	—	106	1	no	—
199	C₂×C₄×C₈	—	53	1	no	—
200	#358	—	53	4	no	—
201	#358	C₂×D₅₃	1	4	yes	—
202	#357	—	53	4	no	—
203	#357	—	53	4	no	—
204	#346	—	53	2	no	—
205	C₄×C₁₆	—	53	1	no	—
206	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
207	C₅₃	#33707	1	1	yes	characteristic, complemented, Sylow
208	C₂×#41	—	106	2	no	—
209	C₂²×Q₈	—	212	1	no	—
210	#135	—	106	8	no	—
211	#135	—	212	4	no	—
212	C₂×Q₁₆	—	2	1	no	—
213	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
214	C₂×SD₁₆	—	106	2	no	—
215	C₂×SD₁₆	—	106	2	no	—
216	C₂×D₈	—	53	2	no	—
217	C₂×D₈	—	106	1	no	—
218	C₂²×C₈	—	53	1	no	—
219	C₂²×C₈	—	53	1	no	—
220	C₄⋊Q₈	—	2	1	no	—
221	C₄⋊Q₈	—	2	1	no	—
222	C₄⋊Q₈	—	106	2	no	—
223	C₄⋊Q₈	—	106	1	no	—
224	C₄⋊Q₈	—	212	1	no	—
225	C₄⋊Q₈	—	212	1	no	—
226	#127	—	106	1	no	—
227	#124	—	106	2	no	—
228	#121	—	212	1	no	—
229	#121	—	212	1	no	—
230	C₄×D₄	—	212	1	no	—
231	C₄×D₄	—	212	1	no	—
232	C₂×C₄⋊C₄	—	106	2	no	—
233	C₂×C₄⋊C₄	—	212	1	no	—
234	C₂×C₄²	—	53	1	no	—
235	M₃₂	—	106	4	no	—
236	C₂×C₁₆	—	53	2	no	—
237	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
238	#107	—	2	1	no	—
239	#107	—	53	2	no	—
240	#107	—	106	2	no	—
241	#107	—	106	2	no	—
242	#107	—	106	1	no	—
243	#107	C₂²×D₅₃	1	2	yes	—
244	C₄×C₈	—	53	1	no	—
245	C₄×C₈	—	53	1	no	—
246	C₄×C₈	—	53	1	no	—
247	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
248	#41	—	212	4	no	—
249	#41	—	212	4	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	4	1	no	—
252	C₂×Q₈	—	424	4	no	—
253	C₂×Q₈	—	212	2	no	—
254	C₂×Q₈	—	212	1	no	—
255	C₂×D₄	—	106	2	no	—
256	C₂×D₄	—	212	2	no	—
257	C₂×D₄	—	212	1	no	—
258	C₂×D₄	—	212	1	no	—
259	C₂²×C₄	—	53	1	no	—
260	C₂²×C₄	—	53	1	no	—
261	C₂²×C₄	—	53	1	no	—
262	C₂²×C₄	—	212	2	no	—
263	C₂²×C₄	—	212	2	no	—
264	Q₁₆	—	2	4	no	—
265	Q₁₆	—	4	2	no	—
266	SD₁₆	—	212	4	no	—
267	SD₁₆	—	212	4	no	—
268	D₈	—	106	4	no	—
269	D₈	—	212	2	no	—
270	C₂×C₈	—	53	1	no	—
271	C₂×C₈	—	53	1	no	—
272	C₂×C₈	—	53	4	no	—
273	C₂×C₈	—	53	4	no	—
274	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
275	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	106	2	no	—
281	C₄⋊C₄	—	212	2	no	—
282	C₄⋊C₄	—	212	2	no	—
283	C₄⋊C₄	—	212	1	no	—
284	C₂²⋊C₄	—	212	2	no	—
285	C₂²⋊C₄	—	212	2	no	—
286	C₄²	—	53	1	no	—
287	C₄²	—	53	1	no	—
288	C₄²	—	53	1	no	—
289	C₄²	D₄×D₅₃	1	1	yes	characteristic
290	C₁₆	—	2	2	no	—
291	C₁₆	—	106	2	no	—
292	C₂³	—	53	1	no	—
293	C₂³	—	212	2	no	—
294	Q₈	—	4	2	no	—
295	Q₈	—	4	2	no	—
296	Q₈	—	8	2	no	—
297	Q₈	—	424	4	no	—
298	Q₈	—	424	2	no	—
299	D₄	—	424	4	no	—
300	D₄	—	424	2	no	—
301	D₄	—	424	2	no	—
302	D₄	—	212	2	no	—
303	D₄	—	212	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	53	1	no	—
307	C₂×C₄	—	53	2	no	—
308	C₂×C₄	—	53	2	no	—
309	C₂×C₄	—	53	1	no	—
310	C₂×C₄	—	53	1	no	—
311	C₂×C₄	—	106	2	no	—
312	C₂×C₄	—	106	2	no	—
313	C₂×C₄	—	424	4	no	—
314	C₂×C₄	—	424	4	no	—
315	C₂×C₄	—	212	2	no	—
316	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
317	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
318	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
319	C₈	—	53	2	no	—
320	C₈	—	53	2	no	—
321	C₈	SD₁₆×D₅₃	1	2	yes	—
322	C₈	#41×D₅₃	1	2	yes	—
323	C₂²	—	53	2	no	—
324	C₂²	—	106	1	no	—
325	C₂²	—	106	1	no	—
326	C₂²	—	424	2	no	—
327	C₂²	—	424	2	no	—
328	C₂²	—	424	2	no	—
329	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
330	C₄	—	8	2	no	—
331	C₄	—	8	2	no	—
332	C₄	—	53	2	no	—
333	C₄	—	106	1	no	—
334	C₄	—	106	1	no	—
335	C₄	—	424	2	no	—
336	C₄	#16811060	1	2	yes	—
337	C₄	#16811065	1	2	yes	—
338	C₄	#124×D₅₃	1	1	yes	characteristic
339	C₄	C₂×SD₁₆×D₅₃	1	1	yes	characteristic
340	C₂	—	848	2	no	—
341	C₂	—	106	2	no	—
342	C₂	#486×D₅₃	1	1	yes	characteristic
343	C₂	C₂×#16811060	1	1	yes	characteristic
344	C₂	C₂×#16811065	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2241	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#20285144	C₂
hypercenter	#2241	D₅₃
hypocenter	C₅₃	#33707
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2241

Complements

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

N	H ≅ G/N	type
1. #500×D₅₃	C₂	semidirect
3. #20276953	C₂	semidirect
4. #20276942	C₂	semidirect
5. #20276940	C₂	semidirect
6. #20276939	C₂	semidirect
7. #20276426	C₂	semidirect
8. C₄×#13133156	C₂	semidirect
9. C₅₃×#2241	C₂	semidirect
172. #2241	D₅₃	semidirect
207. C₅₃	#33707	semidirect

Maximal subgroups

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

subgroup	index
#500×D₅₃	2
#20285144	2
#20276953	2
#20276942	2
#20276940	2
#20276939	2
#20276426	2
C₄×#13133156	2
C₅₃×#2241	2
#33707	53