Subgroups — #27372051

Order 13568.

710 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	#20285664	C₂	1	2	yes	maximal
2	#20285299	C₂	1	1	yes	maximal, characteristic
3	#20285256	C₂	1	2	yes	maximal
4	#500×D₅₃	C₂	1	1	yes	maximal, characteristic
5	#20274394	C₂	1	2	yes	maximal, complemented
6	#20274389	C₂	1	2	yes	maximal, complemented
7	#20274333	C₂	1	1	yes	maximal, characteristic, complemented
8	#20274332	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274309	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#11719912	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#1707	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811178	C₂²	1	2	yes	—
13	C₄⋊Q₈×D₅₃	C₂²	1	2	yes	—
14	#16811030	—	2	4	no	—
15	#16811028	C₂²	1	1	yes	characteristic
16	#16811028	C₂²	1	2	yes	—
17	#16811028	C₂²	1	1	yes	characteristic
18	#16811027	—	2	2	no	—
19	#16811026	C₂²	1	2	yes	—
20	#16811026	C₂²	1	2	yes	—
21	#16811019	C₂²	1	2	yes	—
22	#16811003	C₂²	1	2	yes	—
23	#16811002	C₂²	1	2	yes	—
24	#16810975	C₂²	1	2	yes	—
25	#16810973	C₂²	1	2	yes	—
26	#16810861	—	2	2	no	—
27	#16810832	—	2	4	no	—
28	#16810780	—	2	4	no	—
29	Q₁₆×Dic₅₃	—	2	2	no	—
30	#16810674	C₂²	1	1	yes	characteristic
31	#16810662	—	2	4	no	—
32	#107×D₅₃	—	2	2	no	—
33	C₄×#11719912	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
35	#16810345	C₂²	1	1	yes	characteristic
36	#16810344	C₂²	1	1	yes	characteristic
37	C₈×#88322	C₂²	1	1	yes	characteristic
38	C₅₃×#500	C₂²	1	1	yes	characteristic
39	C₅₃×#498	C₂²	1	2	yes	—
40	C₅₃×#494	C₂²	1	1	yes	characteristic
41	C₅₃×#487	C₂²	1	2	yes	—
42	C₈²×C₅₃	C₂²	1	1	yes	characteristic
43	C₂×Q₈×D₅₃	—	2	2	no	—
44	C₂×Dic₂₁₂	—	2	4	no	—
45	C₂×#11719925	—	2	4	no	—
46	C₂×#11719912	D₄	1	2	yes	—
47	C₂×C₈×D₅₃	D₄	1	2	yes	—
48	#13133217	C₂³	1	2	yes	—
49	#13133217	C₂³	1	2	yes	—
50	#13133216	—	2	2	no	—
51	#13133216	—	2	2	no	—
52	#13133215	—	2	2	no	—
53	#13133208	—	2	2	no	—
54	#13133197	—	2	2	no	—
55	#13133193	—	2	2	no	—
56	Q₈×Dic₅₃	—	2	2	no	—
57	#13133182	—	2	2	no	—
58	#13133175	—	2	2	no	—
59	C₄⋊C₄×D₅₃	—	2	2	no	—
60	C₄⋊C₄×D₅₃	—	2	2	no	—
61	C₄²×D₅₃	C₂³	1	1	yes	characteristic
62	#13133151	—	2	2	no	—
63	#13133151	—	2	4	no	—
64	#13133151	—	2	2	no	—
65	#13133150	—	2	4	no	—
66	#13133150	—	2	4	no	—
67	#13133148	—	2	4	no	—
68	#13133146	D₄	1	2	yes	—
69	#13133141	—	2	4	no	—
70	#13133140	—	2	4	no	—
71	#13133129	D₄	1	2	yes	—
72	#13133127	D₄	1	2	yes	—
73	C₄×#88322	C₂³	1	1	yes	characteristic
74	C₄×#88322	C₂³	1	1	yes	characteristic
75	C₄×#88322	C₂³	1	1	yes	characteristic
76	C₈×Dic₅₃	D₄	1	2	yes	—
77	C₂×Q₁₆×C₅₃	—	2	2	no	—
78	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
79	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
80	#107×C₅₃	—	2	2	no	—
81	#106×C₅₃	—	2	4	no	—
82	#103×C₅₃	—	2	4	no	—
83	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₂×Q₈×C₅₃	—	2	2	no	—
87	C₂×Q₈×C₅₃	—	2	2	no	—
88	Q₈×D₅₃	—	8	4	no	—
89	C₂×C₄×D₅₃	—	4	2	no	—
90	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂×Dic₁₀₆	—	2	2	no	—
94	C₂×Dic₁₀₆	—	2	2	no	—
95	C₂×Dic₁₀₆	—	4	2	no	—
96	Q₁₆×C₅₃	—	4	4	no	—
97	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
100	C₄⋊C₄×C₅₃	—	2	2	no	—
101	C₄⋊C₄×C₅₃	—	2	2	no	—
102	C₄⋊C₄×C₅₃	—	2	2	no	—
103	C₄⋊C₄×C₅₃	—	2	2	no	—
104	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
105	#11719925	—	8	4	no	—
106	#11719921	—	4	2	no	—
107	C₄⋊Dic₅₃	—	2	2	no	—
108	C₄⋊Dic₅₃	—	2	2	no	—
109	C₄⋊Dic₅₃	—	2	2	no	—
110	C₄⋊Dic₅₃	—	2	2	no	—
111	C₄⋊Dic₅₃	—	4	2	no	—
112	#11719919	—	4	2	no	—
113	#11719919	—	4	2	no	—
114	#11719919	—	4	2	no	—
115	C₄×Dic₅₃	—	4	2	no	—
116	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
117	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
118	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
119	C₂×#88322	C₂×D₄	1	2	yes	—
120	C₂×#88322	C₂×D₄	1	2	yes	—
121	C₂×#88322	C₂×D₄	1	2	yes	—
122	Dic₂₁₂	—	8	4	no	—
123	#11719912	Q₁₆	1	8	yes	complemented
124	C₈×D₅₃	—	2	4	no	—
125	C₂²×D₅₃	#127	1	1	yes	characteristic
126	Q₈×C₅₃	—	4	4	no	—
127	Q₈×C₅₃	—	4	4	no	—
128	C₂×C₄×C₅₃	—	4	2	no	—
129	C₂×C₄×C₅₃	—	4	2	no	—
130	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×Dic₅₃	—	4	2	no	—
134	C₂×Dic₅₃	—	4	2	no	—
135	C₂×Dic₅₃	#127	1	1	yes	characteristic
136	C₂×Dic₅₃	#127	1	1	yes	characteristic
137	C₂×Dic₅₃	#127	1	1	yes	characteristic
138	C₄×D₅₃	—	2	2	no	—
139	C₄×D₅₃	—	2	2	no	—
140	C₄×D₅₃	—	8	4	no	—
141	C₄×D₅₃	C₂×Q₁₆	1	4	yes	—
142	Dic₁₀₆	—	4	4	no	—
143	Dic₁₀₆	—	4	4	no	—
144	Dic₁₀₆	—	8	4	no	—
145	C₈×C₅₃	—	2	2	no	—
146	C₈×C₅₃	C₂×Q₁₆	1	4	yes	—
147	C₈×C₅₃	#137	1	4	yes	—
148	#88322	—	2	2	no	—
149	#88322	—	2	2	no	—
150	#88322	C₂×Q₁₆	1	4	yes	—
151	#19633	—	53	1	no	maximal, Sylow
152	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
153	C₂×D₅₃	—	2	1	no	—
154	C₂×D₅₃	—	2	1	no	—
155	C₂×D₅₃	#494	1	2	yes	—
156	C₄×C₅₃	—	8	2	no	—
157	C₄×C₅₃	—	8	2	no	—
158	C₄×C₅₃	C₂²×Q₁₆	1	2	yes	—
159	C₄×C₅₃	C₂×#137	1	2	yes	—
160	C₄×C₅₃	#576	1	2	yes	—
161	Dic₅₃	—	2	1	no	—
162	Dic₅₃	—	2	1	no	—
163	Dic₅₃	—	8	2	no	—
164	Dic₅₃	—	8	2	no	—
165	Dic₅₃	#494	1	2	yes	—
166	#3209	—	53	2	no	—
167	#3170	—	53	1	no	—
168	#3165	—	53	2	no	—
169	C₂×#500	—	53	1	no	—
170	#1723	—	53	2	no	—
171	#1723	—	53	2	no	—
172	#1713	—	53	1	no	—
173	#1713	—	53	1	no	—
174	#1707	—	53	1	no	—
175	#1707	D₅₃	1	1	yes	characteristic, complemented
176	C₈×M₁₆	—	53	1	no	—
177	C₂×C₅₃	C₂×#494	1	1	yes	characteristic
178	C₂×C₅₃	#3152	1	1	yes	characteristic
179	C₂×C₅₃	#3155	1	1	yes	characteristic
180	D₅₃	—	2	2	no	—
181	#533	—	53	2	no	—
182	C₂×C₄⋊Q₈	—	53	2	no	—
183	#501	—	106	4	no	—
184	#500	—	53	2	no	—
185	#500	—	53	2	no	—
186	#500	—	53	1	no	—
187	#500	—	53	2	no	—
188	#500	—	53	1	no	—
189	#500	—	106	2	no	—
190	#500	C₂×D₅₃	1	1	yes	characteristic
191	#498	—	53	2	no	—
192	#498	C₂×D₅₃	1	2	yes	—
193	#494	—	53	2	no	—
194	#494	—	53	2	no	—
195	#494	C₂×D₅₃	1	1	yes	characteristic
196	#487	—	53	2	no	—
197	#487	—	53	2	no	—
198	#487	C₂×D₅₃	1	2	yes	—
199	#467	—	106	2	no	—
200	#462	—	106	4	no	—
201	#451	—	106	4	no	—
202	C₄×Q₁₆	—	106	2	no	—
203	#431	—	53	1	no	—
204	#428	—	106	4	no	—
205	C₂×#107	—	106	2	no	—
206	C₄×M₁₆	—	53	1	no	—
207	C₂×C₄×C₈	—	53	1	no	—
208	#322	—	53	1	no	—
209	#322	—	53	1	no	—
210	C₈²	—	53	1	no	—
211	C₈²	C₂×D₅₃	1	1	yes	characteristic
212	C₅₃	#19633	1	1	yes	characteristic, complemented, Sylow
213	C₂²×Q₈	—	106	2	no	—
214	C₂×Q₁₆	—	2	2	no	—
215	C₂×Q₁₆	—	106	4	no	—
216	C₂×Q₁₆	—	106	4	no	—
217	C₂×M₁₆	—	53	2	no	—
218	C₂²×C₈	—	53	2	no	—
219	C₄⋊Q₈	—	53	2	no	—
220	C₄⋊Q₈	—	53	2	no	—
221	C₄⋊Q₈	—	106	2	no	—
222	C₄⋊Q₈	—	106	2	no	—
223	C₄⋊Q₈	—	106	2	no	—
224	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
225	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
226	#125	—	106	2	no	—
227	C₂²⋊Q₈	—	106	2	no	—
228	C₂²⋊Q₈	—	106	2	no	—
229	C₄×Q₈	—	106	2	no	—
230	C₄×Q₈	—	106	2	no	—
231	#117	—	106	2	no	—
232	C₂×C₄⋊C₄	—	106	2	no	—
233	C₂×C₄⋊C₄	—	106	2	no	—
234	C₂×C₄²	—	53	1	no	—
235	#107	—	2	2	no	—
236	#107	—	106	4	no	—
237	#107	—	106	4	no	—
238	#107	—	106	2	no	—
239	#107	—	106	4	no	—
240	#107	—	106	2	no	—
241	#106	—	2	4	no	—
242	#106	—	106	4	no	—
243	C₄⋊C₈	—	53	2	no	—
244	#103	—	2	4	no	—
245	#103	—	106	4	no	—
246	#103	—	106	4	no	—
247	C₂²⋊C₈	—	53	2	no	—
248	#97	—	53	2	no	—
249	C₄×C₈	—	53	2	no	—
250	C₄×C₈	—	53	1	no	—
251	C₄×C₈	—	53	1	no	—
252	C₄×C₈	—	53	1	no	—
253	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
254	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
255	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	—	106	2	no	—
259	C₂×Q₈	—	106	2	no	—
260	C₂×Q₈	—	424	4	no	—
261	C₂×Q₈	—	212	2	no	—
262	C₂²×C₄	—	53	1	no	—
263	C₂²×C₄	—	53	1	no	—
264	C₂²×C₄	—	53	1	no	—
265	C₂²×C₄	—	212	2	no	—
266	Q₁₆	—	4	4	no	—
267	Q₁₆	—	424	4	no	—
268	Q₁₆	—	424	4	no	—
269	M₁₆	—	53	8	no	—
270	C₂×C₈	—	53	2	no	—
271	C₂×C₈	—	53	2	no	—
272	C₂×C₈	—	53	2	no	—
273	C₂×C₈	—	106	4	no	—
274	C₂×C₈	D₄×D₅₃	1	2	yes	—
275	C₂×C₈	D₄×D₅₃	1	2	yes	—
276	C₂×C₈	D₄×D₅₃	1	2	yes	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	106	2	no	—
282	C₄⋊C₄	—	106	2	no	—
283	C₄⋊C₄	—	106	2	no	—
284	C₄⋊C₄	—	106	2	no	—
285	C₄⋊C₄	—	212	2	no	—
286	C₄⋊C₄	—	212	2	no	—
287	C₄⋊C₄	—	212	2	no	—
288	C₄⋊C₄	—	212	2	no	—
289	C₂²⋊C₄	—	212	2	no	—
290	C₄²	—	53	1	no	—
291	C₄²	—	53	1	no	—
292	C₄²	—	53	1	no	—
293	C₄²	—	212	2	no	—
294	C₄²	C₂³×D₅₃	1	1	yes	characteristic
295	C₂³	—	53	1	no	—
296	Q₈	—	4	4	no	—
297	Q₈	—	4	4	no	—
298	Q₈	—	424	4	no	—
299	Q₈	—	212	4	no	—
300	Q₈	—	212	4	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	53	1	no	—
304	C₂×C₄	—	53	4	no	—
305	C₂×C₄	—	53	1	no	—
306	C₂×C₄	—	53	1	no	—
307	C₂×C₄	—	106	2	no	—
308	C₂×C₄	—	106	2	no	—
309	C₂×C₄	—	424	4	no	—
310	C₂×C₄	—	212	2	no	—
311	C₂×C₄	—	212	2	no	—
312	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
313	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
314	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
315	C₈	—	2	2	no	—
316	C₈	—	53	4	no	—
317	C₈	—	106	2	no	—
318	C₈	—	106	2	no	—
319	C₈	Q₁₆×D₅₃	1	4	yes	—
320	C₈	#13133246	1	4	yes	—
321	C₂²	—	53	2	no	—
322	C₂²	—	106	1	no	—
323	C₂²	—	106	1	no	—
324	C₂²	#127×D₅₃	1	1	yes	characteristic
325	C₄	—	8	2	no	—
326	C₄	—	8	2	no	—
327	C₄	—	53	2	no	—
328	C₄	—	106	1	no	—
329	C₄	—	106	1	no	—
330	C₄	—	424	2	no	—
331	C₄	—	424	2	no	—
332	C₄	C₂×Q₁₆×D₅₃	1	2	yes	—
333	C₄	C₂×#13133246	1	2	yes	—
334	C₄	#16811425	1	2	yes	—
335	C₂	—	106	2	no	—
336	C₂	#494×D₅₃	1	1	yes	characteristic
337	C₂	#20285169	1	1	yes	characteristic
338	C₂	#20285197	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1707	C₂
socle	C₂²×C₅₃	C₂×#127
prime radical	C₂²×D₅₃	#127
hypercenter	#1707	D₅₃
hypocenter	C₅₃	#19633
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1707

Complements

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

N	H ≅ G/N	type
5. #20274394	C₂	semidirect
6. #20274389	C₂	semidirect
7. #20274333	C₂	semidirect
8. #20274332	C₂	semidirect
9. #20274309	C₂	semidirect
11. C₅₃×#1707	C₂	semidirect
123. #11719912	Q₁₆	semidirect
175. #1707	D₅₃	semidirect
212. C₅₃	#19633	semidirect

Maximal subgroups

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

subgroup	index
#20285664	2
#20285299	2
#20285256	2
#500×D₅₃	2
#20274394	2
#20274389	2
#20274333	2
#20274332	2
#20274309	2
C₈×#11719912	2
C₅₃×#1707	2
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