Subgroups — #27276810

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	#20274832	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#88322	C₂	1	2	yes	maximal
4	#20272556	C₂	1	2	yes	maximal
5	C₅₃×#2295	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	—	2	1	no	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	C₂×C₄×#88322	C₂²	1	2	yes	—
9	#16810378	—	2	1	no	—
10	#16810378	—	2	1	no	—
11	#16810378	—	2	1	no	—
12	Q₈⋊#88322	—	2	4	no	—
13	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#391	C₄	1	1	yes	characteristic
15	C₅₃×#391	C₄	1	1	yes	characteristic
16	C₅₃×#389	C₄	1	1	yes	characteristic
17	C₅₃×#389	C₄	1	1	yes	characteristic
18	C₅₃×#387	—	2	1	no	—
19	C₅₃×#387	—	2	1	no	—
20	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
21	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#376	—	2	1	no	—
23	C₄³×C₅₃	C₂²	1	1	yes	characteristic
24	C₂²×#88322	—	2	2	no	—
25	C₂²×#88322	—	2	2	no	—
26	C₄⋊#88322	—	2	4	no	—
27	C₄⋊#88322	—	2	2	no	—
28	C₄⋊#88322	—	2	2	no	—
29	C₄×#88322	—	2	2	no	—
30	C₄×#88322	—	2	2	no	—
31	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄×Q₈×C₅₃	—	2	1	no	—
33	C₄×Q₈×C₅₃	—	2	1	no	—
34	C₄×Q₈×C₅₃	D₄	1	2	yes	—
35	C₄×Q₈×C₅₃	Q₈	1	2	yes	—
36	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
38	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
39	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	—	2	1	no	—
43	C₂×C₄²×C₅₃	—	2	1	no	—
44	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
51	#95×C₅₃	—	2	1	no	—
52	#95×C₅₃	—	2	1	no	—
53	#95×C₅₃	—	2	1	no	—
54	#95×C₅₃	—	2	1	no	—
55	#95×C₅₃	—	2	1	no	—
56	#95×C₅₃	—	2	1	no	—
57	#95×C₅₃	—	2	1	no	—
58	C₂×Q₈×C₅₃	—	2	1	no	—
59	C₂×Q₈×C₅₃	—	2	2	no	—
60	C₂×Q₈×C₅₃	—	2	2	no	—
61	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₅₃	—	2	1	no	—
63	C₂²×C₄×C₅₃	—	2	1	no	—
64	C₂²×C₄×C₅₃	—	2	1	no	—
65	C₂²×C₄×C₅₃	—	2	1	no	—
66	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₄⋊C₄×C₅₃	—	2	1	no	—
74	C₄⋊C₄×C₅₃	—	2	2	no	—
75	C₄⋊C₄×C₅₃	—	2	2	no	—
76	C₄⋊C₄×C₅₃	—	2	2	no	—
77	C₄⋊C₄×C₅₃	—	2	2	no	—
78	C₄⋊C₄×C₅₃	—	4	2	no	—
79	C₄⋊C₄×C₅₃	—	4	2	no	—
80	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
81	C₄²×C₅₃	—	2	2	no	—
82	C₄²×C₅₃	—	2	2	no	—
83	C₄²×C₅₃	—	2	2	no	—
84	C₄²×C₅₃	—	2	2	no	—
85	C₄²×C₅₃	—	2	2	no	—
86	C₄²×C₅₃	—	2	2	no	—
87	C₄²×C₅₃	—	2	2	no	—
88	C₄²×C₅₃	—	4	1	no	—
89	C₄²×C₅₃	—	4	2	no	—
90	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
92	C₄²×C₅₃	#41	1	1	yes	characteristic
93	C₄²×C₅₃	#41	1	1	yes	characteristic
94	C₂×#88322	—	2	4	no	—
95	C₂×#88322	—	4	2	no	—
96	C₂×#88322	—	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	Q₈×C₅₃	—	2	4	no	—
102	Q₈×C₅₃	—	4	2	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	2	no	—
108	C₂×C₄×C₅₃	—	2	2	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	—	2	1	no	—
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	—	2	1	no	—
117	C₂×C₄×C₅₃	—	2	1	no	—
118	C₂×C₄×C₅₃	—	4	1	no	—
119	C₂×C₄×C₅₃	—	4	1	no	—
120	C₂×C₄×C₅₃	—	4	1	no	—
121	C₂×C₄×C₅₃	—	4	1	no	—
122	C₂×C₄×C₅₃	—	4	1	no	—
123	C₂×C₄×C₅₃	—	4	1	no	—
124	C₂×C₄×C₅₃	—	4	1	no	—
125	C₂×C₄×C₅₃	—	4	1	no	—
126	C₂×C₄×C₅₃	—	4	1	no	—
127	C₂×C₄×C₅₃	—	4	1	no	—
128	C₂×C₄×C₅₃	#104	1	2	yes	—
129	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
131	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
132	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×C₄×C₅₃	#131	1	2	yes	—
136	C₂×C₄×C₅₃	#131	1	2	yes	—
137	C₂×C₄×C₅₃	#135	1	2	yes	—
138	C₂×C₄×C₅₃	#137	1	2	yes	—
139	#88322	—	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	2	no	—
153	C₄×C₅₃	—	4	2	no	—
154	C₄×C₅₃	—	4	1	no	—
155	C₄×C₅₃	—	4	1	no	—
156	C₄×C₅₃	—	4	1	no	—
157	C₄×C₅₃	—	8	1	no	—
158	C₄×C₅₃	—	8	1	no	—
159	C₄×C₅₃	#424	1	2	yes	—
160	C₄×C₅₃	#433	1	2	yes	—
161	C₄×C₅₃	#464	1	2	yes	—
162	C₄×C₅₃	#479	1	2	yes	—
163	#2295	D₅₃	1	1	yes	characteristic
164	#1831	—	53	1	no	—
165	C₄×C₄⋊C₈	—	53	1	no	—
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	C₂×D₅₃	1	1	yes	characteristic
176	C₂×C₄⋊C₈	—	53	2	no	—
177	C₂×C₄⋊C₈	—	106	1	no	—
178	C₂×C₄×C₈	—	53	2	no	—
179	#391	Dic₅₃	1	1	yes	characteristic
180	#391	Dic₅₃	1	1	yes	characteristic
181	#389	Dic₅₃	1	1	yes	characteristic
182	#389	Dic₅₃	1	1	yes	characteristic
183	#387	—	2	1	no	—
184	#387	—	2	1	no	—
185	C₄×C₄⋊C₄	—	2	1	no	—
186	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
187	#376	—	2	1	no	—
188	C₄³	C₂×D₅₃	1	1	yes	characteristic
189	#336	—	106	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	C₂×Dic₅₃	1	1	yes	characteristic
195	C₂²×C₈	—	106	2	no	—
196	C₂²×C₈	—	106	2	no	—
197	C₄×Q₈	—	2	1	no	—
198	C₄×Q₈	—	2	1	no	—
199	C₄×Q₈	Dic₁₀₆	1	2	yes	—
200	C₄×Q₈	D₂₁₂	1	2	yes	—
201	C₂×C₄⋊C₄	—	2	1	no	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
205	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
206	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	—	2	1	no	—
209	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
210	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
211	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
212	C₂×C₄²	C₂²⋊D₅₃	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	—	106	4	no	—
217	C₄⋊C₈	—	106	2	no	—
218	C₄⋊C₈	—	106	2	no	—
219	C₄×C₈	—	106	2	no	—
220	C₄×C₈	—	106	2	no	—
221	#95	—	2	1	no	—
222	#95	—	2	1	no	—
223	#95	—	2	1	no	—
224	#95	—	2	1	no	—
225	#95	—	2	1	no	—
226	#95	—	2	1	no	—
227	#95	—	2	1	no	—
228	C₂×Q₈	—	2	1	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	C₄⋊Dic₅₃	1	2	yes	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
237	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
238	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
239	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
240	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
243	C₂×C₈	—	106	4	no	—
244	C₂×C₈	—	212	2	no	—
245	C₂×C₈	—	212	2	no	—
246	C₂×C₈	—	212	2	no	—
247	C₂×C₈	—	212	2	no	—
248	C₂×C₈	—	212	2	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
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	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	4	1	no	—
265	C₄²	—	4	2	no	—
266	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
267	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
268	C₄²	#11719945	1	1	yes	characteristic
269	C₄²	#11719945	1	1	yes	characteristic
270	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
271	Q₈	—	2	4	no	—
272	Q₈	—	4	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	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	#13133144	1	2	yes	—
299	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
300	C₂×C₄	#13133176	1	1	yes	characteristic
301	C₂×C₄	#13133191	1	1	yes	characteristic
302	C₂×C₄	#13133191	1	1	yes	characteristic
303	C₂×C₄	#13133196	1	1	yes	characteristic
304	C₂×C₄	#13133196	1	1	yes	characteristic
305	C₂×C₄	#13133224	1	2	yes	—
306	C₂×C₄	#13133224	1	2	yes	—
307	C₂×C₄	#13133237	1	2	yes	—
308	C₂×C₄	#13133248	1	2	yes	—
309	C₈	—	424	2	no	—
310	C₈	—	212	4	no	—
311	C₂²	#16810483	1	1	yes	characteristic
312	C₂²	#16810590	1	1	yes	characteristic
313	C₂²	#16810590	1	1	yes	characteristic
314	C₂²	#16810619	1	1	yes	characteristic
315	C₂²	#16810632	1	1	yes	characteristic
316	C₂²	C₂×#13133144	1	1	yes	characteristic
317	C₂²	#16810640	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	4	2	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	8	1	no	—
327	C₄	—	8	1	no	—
328	C₄	#16810650	1	2	yes	—
329	C₄	#16810684	1	2	yes	—
330	C₄	#16810845	1	2	yes	—
331	C₄	#16810930	1	2	yes	—
332	C₂	#20273123	1	1	yes	characteristic
333	C₂	#20273136	1	1	yes	characteristic
334	C₂	#20273175	1	1	yes	characteristic
335	C₂	#20273188	1	1	yes	characteristic
336	C₂	#20274699	1	1	yes	characteristic
337	C₂	#20274730	1	1	yes	characteristic
338	C₂	#20274735	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2295	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2295	D₅₃
hypocenter	C₅₃	#8570
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #2295

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
#20274832	2
C₄×C₄⋊#88322	2
C₂×Q₈⋊#88322	2
#20272556	2
C₅₃×#2295	2
#8570	53