Subgroups — #27276811

Order 13568.

472 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	#20277192	C₂	1	1	yes	maximal, characteristic
2	#20274832	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810355	C₂	1	2	yes	maximal
4	#20272555	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	#16810529	C₄	1	1	yes	characteristic
9	#16810529	C₄	1	1	yes	characteristic
10	#16810521	C₄	1	1	yes	characteristic
11	#16810521	C₄	1	1	yes	characteristic
12	#16810507	—	2	1	no	—
13	#16810507	—	2	1	no	—
14	C₄×#11719919	—	2	1	no	—
15	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
16	#16810469	—	2	1	no	—
17	#16810378	—	2	1	no	—
18	#16810378	—	2	1	no	—
19	#16810355	—	2	4	no	—
20	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
22	C₄³×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#336	—	2	1	no	—
24	C₂²×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
25	C₂²×#88322	—	2	2	no	—
26	C₄×Dic₁₀₆	—	2	1	no	—
27	C₄×Dic₁₀₆	—	2	1	no	—
28	C₄×Dic₁₀₆	D₄	1	2	yes	—
29	C₄×Dic₁₀₆	Q₈	1	2	yes	—
30	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×#11719919	—	2	1	no	—
34	C₂×#11719919	—	2	1	no	—
35	C₂×#11719919	—	2	1	no	—
36	C₂×C₄×Dic₅₃	—	2	1	no	—
37	C₂×C₄×Dic₅₃	—	2	1	no	—
38	C₄⋊#88322	—	2	2	no	—
39	C₄⋊#88322	—	2	2	no	—
40	#13133124	—	2	1	no	—
41	#13133124	—	2	1	no	—
42	#13133124	—	2	1	no	—
43	#13133124	—	2	1	no	—
44	#13133124	—	2	1	no	—
45	#13133124	—	2	1	no	—
46	#13133124	—	2	1	no	—
47	C₂²×C₈×C₅₃	—	2	2	no	—
48	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₅₃	—	2	4	no	—
56	C₄×C₈×C₅₃	—	2	2	no	—
57	C₄×C₈×C₅₃	—	2	2	no	—
58	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×Dic₅₃	—	2	1	no	—
66	C₂²×Dic₅₃	—	2	1	no	—
67	C₂²×Dic₅₃	—	2	1	no	—
68	C₂²×Dic₅₃	—	2	1	no	—
69	C₂×Dic₁₀₆	—	2	2	no	—
70	C₂×Dic₁₀₆	—	2	2	no	—
71	C₂×Dic₁₀₆	—	2	1	no	—
72	C₂×Dic₁₀₆	C₄⋊C₄	1	2	yes	—
73	C₂×C₈×C₅₃	—	2	4	no	—
74	C₂×C₈×C₅₃	—	4	2	no	—
75	C₂×C₈×C₅₃	—	4	2	no	—
76	C₄²×C₅₃	—	2	2	no	—
77	C₄²×C₅₃	—	2	2	no	—
78	C₄²×C₅₃	—	2	2	no	—
79	C₄²×C₅₃	—	2	2	no	—
80	C₄²×C₅₃	—	2	2	no	—
81	C₄²×C₅₃	—	4	1	no	—
82	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
84	C₄²×C₅₃	#41	1	1	yes	characteristic
85	C₄²×C₅₃	#41	1	1	yes	characteristic
86	C₄⋊Dic₅₃	—	2	1	no	—
87	C₄⋊Dic₅₃	—	2	2	no	—
88	C₄⋊Dic₅₃	—	2	2	no	—
89	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
90	#11719919	—	2	2	no	—
91	#11719919	—	2	2	no	—
92	#11719919	—	4	2	no	—
93	#11719919	—	4	2	no	—
94	C₄×Dic₅₃	—	2	2	no	—
95	C₄×Dic₅₃	—	2	2	no	—
96	C₄×Dic₅₃	—	4	2	no	—
97	C₂×#88322	—	4	2	no	—
98	C₂×#88322	—	4	2	no	—
99	C₂×#88322	—	4	2	no	—
100	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	—	2	1	no	—
102	C₂×C₄×C₅₃	—	2	1	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	2	1	no	—
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	1	no	—
108	C₂×C₄×C₅₃	—	2	1	no	—
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	—	2	1	no	—
111	C₂×C₄×C₅₃	—	2	1	no	—
112	C₂×C₄×C₅₃	—	2	1	no	—
113	C₂×C₄×C₅₃	—	2	1	no	—
114	C₂×C₄×C₅₃	#104	1	2	yes	—
115	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
117	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
118	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₅₃	#131	1	2	yes	—
122	C₂×C₄×C₅₃	#131	1	2	yes	—
123	C₂×C₄×C₅₃	#135	1	2	yes	—
124	C₂×C₄×C₅₃	#137	1	2	yes	—
125	C₂×Dic₅₃	—	2	2	no	—
126	C₂×Dic₅₃	—	2	2	no	—
127	C₂×Dic₅₃	—	4	1	no	—
128	C₂×Dic₅₃	—	4	1	no	—
129	C₂×Dic₅₃	—	4	1	no	—
130	C₂×Dic₅₃	—	4	1	no	—
131	C₂×Dic₅₃	—	4	1	no	—
132	C₂×Dic₅₃	—	4	1	no	—
133	C₂×Dic₅₃	—	4	1	no	—
134	C₂×Dic₅₃	—	4	1	no	—
135	C₂×Dic₅₃	—	4	1	no	—
136	C₂×Dic₅₃	—	4	1	no	—
137	Dic₁₀₆	—	2	4	no	—
138	Dic₁₀₆	—	4	2	no	—
139	C₈×C₅₃	—	4	4	no	—
140	#88322	—	8	2	no	—
141	#8570	—	53	1	no	maximal, Sylow
142	C₂²×C₅₃	#380	1	1	yes	characteristic
143	C₂²×C₅₃	#408	1	1	yes	characteristic
144	C₂²×C₅₃	#408	1	1	yes	characteristic
145	C₂²×C₅₃	#416	1	1	yes	characteristic
146	C₂²×C₅₃	#419	1	1	yes	characteristic
147	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
148	C₂²×C₅₃	#421	1	1	yes	characteristic
149	C₄×C₅₃	—	2	1	no	—
150	C₄×C₅₃	—	2	1	no	—
151	C₄×C₅₃	—	2	2	no	—
152	C₄×C₅₃	—	4	1	no	—
153	C₄×C₅₃	—	4	1	no	—
154	C₄×C₅₃	—	4	1	no	—
155	C₄×C₅₃	#424	1	2	yes	—
156	C₄×C₅₃	#433	1	2	yes	—
157	C₄×C₅₃	#464	1	2	yes	—
158	C₄×C₅₃	#479	1	2	yes	—
159	Dic₅₃	—	4	2	no	—
160	Dic₅₃	—	4	2	no	—
161	Dic₅₃	—	8	1	no	—
162	Dic₅₃	—	8	1	no	—
163	#2295	—	53	1	no	—
164	#1831	—	53	1	no	—
165	C₄×C₄⋊C₈	D₅₃	1	1	yes	characteristic
166	C₂×Q₈⋊C₈	—	53	2	no	—
167	#1279	—	53	2	no	—
168	C₂×C₅₃	#1482	1	1	yes	characteristic
169	C₂×C₅₃	#1485	1	1	yes	characteristic
170	C₂×C₅₃	#1494	1	1	yes	characteristic
171	C₂×C₅₃	#1497	1	1	yes	characteristic
172	C₂×C₅₃	#1798	1	1	yes	characteristic
173	C₂×C₅₃	#1805	1	1	yes	characteristic
174	C₂×C₅₃	#1806	1	1	yes	characteristic
175	C₂×C₄×Q₈	—	53	1	no	—
176	C₂×C₄⋊C₈	—	2	1	no	—
177	C₂×C₄⋊C₈	—	53	2	no	—
178	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
179	#391	—	53	1	no	—
180	#391	—	53	1	no	—
181	#389	—	53	1	no	—
182	#389	—	53	1	no	—
183	#387	—	106	1	no	—
184	#387	—	106	1	no	—
185	C₄×C₄⋊C₄	—	53	1	no	—
186	C₄×C₄⋊C₄	—	106	1	no	—
187	#376	—	106	1	no	—
188	C₄³	C₂×D₅₃	1	1	yes	characteristic
189	#336	—	2	1	no	—
190	#336	—	106	1	no	—
191	#336	—	106	1	no	—
192	Q₈⋊C₈	—	106	4	no	—
193	C₅₃	#8570	1	1	yes	characteristic, complemented, Sylow
194	C₂²×Q₈	—	53	1	no	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	—	106	2	no	—
197	C₄×Q₈	—	53	2	no	—
198	C₄×Q₈	—	53	2	no	—
199	C₄×Q₈	—	106	1	no	—
200	C₄×Q₈	—	106	1	no	—
201	C₂×C₄⋊C₄	—	53	1	no	—
202	C₂×C₄⋊C₄	—	53	1	no	—
203	C₂×C₄⋊C₄	—	53	1	no	—
204	C₂×C₄⋊C₄	—	106	1	no	—
205	C₂×C₄⋊C₄	—	106	1	no	—
206	C₂×C₄⋊C₄	—	106	1	no	—
207	C₂×C₄²	—	106	1	no	—
208	C₂×C₄²	—	106	1	no	—
209	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
210	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
211	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
212	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
213	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
214	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
215	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
216	C₄⋊C₈	—	2	4	no	—
217	C₄⋊C₈	—	106	2	no	—
218	C₄⋊C₈	—	106	2	no	—
219	C₄×C₈	—	2	2	no	—
220	C₄×C₈	—	2	2	no	—
221	#95	—	106	1	no	—
222	#95	—	106	1	no	—
223	#95	—	106	1	no	—
224	#95	—	106	1	no	—
225	#95	—	106	1	no	—
226	#95	—	106	1	no	—
227	#95	—	106	1	no	—
228	C₂×Q₈	—	53	2	no	—
229	C₂×Q₈	—	106	1	no	—
230	C₂×Q₈	—	106	2	no	—
231	C₂×Q₈	—	106	2	no	—
232	C₂²×C₄	—	106	1	no	—
233	C₂²×C₄	—	106	1	no	—
234	C₂²×C₄	—	106	1	no	—
235	C₂²×C₄	—	106	1	no	—
236	C₂²×C₄	#11719921	1	1	yes	characteristic
237	C₂²×C₄	#11719921	1	1	yes	characteristic
238	C₂²×C₄	#11719921	1	1	yes	characteristic
239	C₂²×C₄	#11719921	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	212	2	no	—
247	C₂×C₈	—	212	2	no	—
248	C₂×C₈	—	212	2	no	—
249	C₄⋊C₄	—	53	2	no	—
250	C₄⋊C₄	—	106	1	no	—
251	C₄⋊C₄	—	106	2	no	—
252	C₄⋊C₄	—	106	2	no	—
253	C₄⋊C₄	—	106	2	no	—
254	C₄⋊C₄	—	106	2	no	—
255	C₄⋊C₄	—	212	2	no	—
256	C₄⋊C₄	—	212	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	4	1	no	—
263	C₄²	—	106	2	no	—
264	C₄²	—	106	2	no	—
265	C₄²	—	212	2	no	—
266	C₄²	D₄×D₅₃	1	1	yes	characteristic
267	C₄²	#11719947	1	1	yes	characteristic
268	C₄²	Q₈×D₅₃	1	1	yes	characteristic
269	C₄²	#11719949	1	1	yes	characteristic
270	C₂³	C₂×#11719921	1	1	yes	characteristic
271	Q₈	—	106	4	no	—
272	Q₈	—	212	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	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	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	106	2	no	—
287	C₂×C₄	—	106	2	no	—
288	C₂×C₄	—	212	1	no	—
289	C₂×C₄	—	212	1	no	—
290	C₂×C₄	—	212	1	no	—
291	C₂×C₄	—	212	1	no	—
292	C₂×C₄	—	212	1	no	—
293	C₂×C₄	—	212	1	no	—
294	C₂×C₄	—	212	1	no	—
295	C₂×C₄	—	212	1	no	—
296	C₂×C₄	—	212	1	no	—
297	C₂×C₄	—	212	1	no	—
298	C₂×C₄	#13133145	1	2	yes	—
299	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
300	C₂×C₄	#13133175	1	1	yes	characteristic
301	C₂×C₄	#13133188	1	1	yes	characteristic
302	C₂×C₄	#13133192	1	1	yes	characteristic
303	C₂×C₄	#13133193	1	1	yes	characteristic
304	C₂×C₄	#13133197	1	1	yes	characteristic
305	C₂×C₄	#13133223	1	2	yes	—
306	C₂×C₄	#13133225	1	2	yes	—
307	C₂×C₄	#13133239	1	2	yes	—
308	C₂×C₄	#13133250	1	2	yes	—
309	C₈	—	4	4	no	—
310	C₈	—	424	2	no	—
311	C₂²	#16810480	1	1	yes	characteristic
312	C₂²	#16810589	1	1	yes	characteristic
313	C₂²	#16810591	1	1	yes	characteristic
314	C₂²	#16810620	1	1	yes	characteristic
315	C₂²	#16810633	1	1	yes	characteristic
316	C₂²	C₂×#13133145	1	1	yes	characteristic
317	C₂²	#16810641	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	424	1	no	—
325	C₄	—	424	1	no	—
326	C₄	—	212	2	no	—
327	C₄	—	212	2	no	—
328	C₄	#16810648	1	2	yes	—
329	C₄	#16810683	1	2	yes	—
330	C₄	#16810846	1	2	yes	—
331	C₄	#16810931	1	2	yes	—
332	C₂	#20273124	1	1	yes	characteristic
333	C₂	#20273137	1	1	yes	characteristic
334	C₂	#20273176	1	1	yes	characteristic
335	C₂	#20273189	1	1	yes	characteristic
336	C₂	#20274696	1	1	yes	characteristic
337	C₂	#20274731	1	1	yes	characteristic
338	C₂	#20274736	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×C₄⋊C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#8570
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
193. C₅₃	#8570	semidirect

Maximal subgroups

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

subgroup	index
#20277192	2
#20274832	2
C₂×#16810355	2
#20272555	2
C₄×C₄⋊C₈×C₅₃	2
#8570	53