Subgroups — #27300550

Order 13568.

462 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810441	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16810402	C₂	1	1	yes	maximal, characteristic
4	C₈×#11719921	C₂	1	1	yes	maximal, characteristic
5	#20272793	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272792	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
9	C₂²×#11719908	C₂²	1	1	yes	characteristic
10	C₂²×#11719908	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×D₅₃	—	2	2	no	—
12	C₂×C₁₆×D₅₃	—	2	2	no	—
13	C₂×C₁₆×D₅₃	—	2	2	no	—
14	C₂×#13133129	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₄×#11719921	C₄	1	1	yes	characteristic
17	#16810441	—	2	2	no	—
18	#16810402	—	2	2	no	—
19	#16810378	C₄	1	1	yes	characteristic
20	#16810377	C₄	1	1	yes	characteristic
21	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×C₄⋊C₁₆	—	2	2	no	—
25	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
27	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×D₅₃	—	2	1	no	—
29	C₂×C₈×D₅₃	—	2	1	no	—
30	C₂×C₈×D₅₃	—	2	1	no	—
31	C₂×C₈×D₅₃	—	2	1	no	—
32	C₂×C₈×D₅₃	D₄	1	2	yes	—
33	C₂×C₈×D₅₃	Q₈	1	2	yes	—
34	C₂×#11719921	C₈	1	1	yes	characteristic
35	C₂×#11719921	C₈	1	1	yes	characteristic
36	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×#11719908	—	2	1	no	—
38	C₂×#11719908	—	2	1	no	—
39	C₂×#11719908	—	2	1	no	—
40	C₂×#11719908	—	2	1	no	—
41	C₂×#11719908	—	2	1	no	—
42	C₂×#11719908	D₄	1	2	yes	—
43	C₁₆×D₅₃	—	2	8	no	—
44	#13133129	—	2	2	no	—
45	C₈×Dic₅₃	—	2	1	no	—
46	C₈×Dic₅₃	—	2	1	no	—
47	#13133124	C₈	1	1	yes	characteristic
48	#13133124	C₈	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₁₆×C₅₃	—	2	1	no	—
53	C₂×C₁₆×C₅₃	—	2	1	no	—
54	C₂×C₁₆×C₅₃	—	2	1	no	—
55	C₂×C₁₆×C₅₃	—	2	1	no	—
56	C₂×C₁₆×C₅₃	—	2	1	no	—
57	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
58	C₄×C₈×C₅₃	—	2	1	no	—
59	C₄×C₈×C₅₃	—	2	1	no	—
60	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
65	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
66	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
67	C₂×C₄×D₅₃	—	2	1	no	—
68	C₂×C₄×D₅₃	—	2	1	no	—
69	C₂×C₄×D₅₃	—	2	1	no	—
70	C₂×C₄×D₅₃	—	2	1	no	—
71	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
72	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
73	C₂×C₈×C₅₃	—	2	1	no	—
74	C₂×C₈×C₅₃	—	2	1	no	—
75	C₂×C₈×C₅₃	—	2	1	no	—
76	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
81	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
82	C₄²×C₅₃	—	2	1	no	—
83	C₄²×C₅₃	—	2	1	no	—
84	#11719921	—	2	2	no	—
85	#11719921	C₁₆	1	4	yes	complemented
86	C₄×Dic₅₃	—	2	1	no	—
87	C₄×Dic₅₃	—	2	1	no	—
88	C₂×#88322	—	2	1	no	—
89	C₂×#88322	—	2	1	no	—
90	C₂×#88322	—	2	1	no	—
91	C₂×#88322	—	2	1	no	—
92	C₂×#88322	—	2	1	no	—
93	C₂×#88322	C₄⋊C₄	1	2	yes	—
94	C₈×D₅₃	—	2	2	no	—
95	C₈×D₅₃	—	2	2	no	—
96	C₈×D₅₃	—	2	2	no	—
97	C₈×D₅₃	—	2	2	no	—
98	C₁₆×C₅₃	—	2	2	no	—
99	C₁₆×C₅₃	—	4	1	no	—
100	#11719908	—	2	2	no	—
101	#11719908	—	4	1	no	—
102	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
103	C₂²×D₅₃	—	2	1	no	—
104	C₂²×D₅₃	—	2	1	no	—
105	C₂²×D₅₃	—	2	1	no	—
106	C₂²×D₅₃	—	2	1	no	—
107	C₂²×D₅₃	C₄⋊C₈	1	2	yes	—
108	C₂²×D₅₃	C₂×C₁₆	1	2	yes	—
109	C₂²×D₅₃	M₃₂	1	2	yes	—
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	C₂×C₁₆	1	2	yes	complemented
116	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
121	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
122	C₂×Dic₅₃	—	2	1	no	—
123	C₂×Dic₅₃	—	2	1	no	—
124	C₂×Dic₅₃	—	2	1	no	—
125	C₂×Dic₅₃	—	2	1	no	—
126	C₂×Dic₅₃	—	2	1	no	—
127	C₂×Dic₅₃	—	2	1	no	—
128	C₂×Dic₅₃	—	2	1	no	—
129	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
130	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	complemented
131	C₄×D₅₃	—	2	2	no	—
132	C₄×D₅₃	—	2	2	no	—
133	C₄×D₅₃	—	2	2	no	—
134	C₄×D₅₃	—	2	2	no	—
135	C₈×C₅₃	—	4	1	no	—
136	C₈×C₅₃	#120	1	1	yes	characteristic
137	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
138	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
139	C₈×C₅₃	#123	1	1	yes	characteristic
140	#88322	—	2	2	no	—
141	#88322	—	4	1	no	—
142	#12912	—	53	1	no	maximal, Sylow
143	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
144	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
145	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
146	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
147	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
148	C₂²×C₅₃	#504	1	1	yes	characteristic
149	C₂²×C₅₃	#504	1	1	yes	characteristic
150	C₂×D₅₃	—	2	2	no	—
151	C₂×D₅₃	—	2	2	no	—
152	C₂×D₅₃	—	2	2	no	—
153	C₂×D₅₃	—	2	2	no	—
154	C₂×D₅₃	—	2	2	no	—
155	C₂×D₅₃	—	2	2	no	—
156	C₂×D₅₃	C₄⋊C₁₆	1	4	yes	complemented
157	C₄×C₅₃	—	2	2	no	—
158	C₄×C₅₃	—	4	1	no	—
159	C₄×C₅₃	#385	1	1	yes	characteristic
160	C₄×C₅₃	#424	1	1	yes	characteristic
161	C₄×C₅₃	#435	1	1	yes	characteristic
162	C₄×C₅₃	#435	1	1	yes	characteristic
163	Dic₅₃	—	2	2	no	—
164	Dic₅₃	—	2	2	no	—
165	Dic₅₃	—	4	1	no	—
166	C₂³×C₁₆	—	53	1	no	—
167	C₂×C₄⋊C₁₆	—	53	1	no	—
168	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic, complemented
169	C₂×C₂²⋊C₁₆	—	53	1	no	—
170	C₈×C₂²⋊C₄	—	53	1	no	—
171	#1374	—	53	1	no	—
172	#1374	—	53	1	no	—
173	C₂×C₅₃	#1850	1	1	yes	characteristic
174	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
175	C₂×C₅₃	#2151	1	1	yes	characteristic
176	C₂×C₅₃	D₄×C₁₆	1	1	yes	characteristic
177	C₂×C₅₃	D₄×C₁₆	1	1	yes	characteristic
178	C₂×C₅₃	#2168	1	1	yes	characteristic
179	C₂×C₅₃	#2168	1	1	yes	characteristic
180	D₅₃	—	2	4	no	—
181	C₂³×C₈	—	53	1	no	—
182	C₂²×C₁₆	—	53	1	no	—
183	C₂²×C₁₆	—	53	1	no	—
184	C₂²×C₁₆	—	106	2	no	—
185	C₂²×C₁₆	—	106	2	no	—
186	C₂²×C₁₆	—	106	2	no	—
187	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
188	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
189	C₂×C₂²⋊C₈	—	53	1	no	—
190	C₂×C₄×C₈	—	53	1	no	—
191	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
192	C₄×C₂²⋊C₄	—	53	1	no	—
193	C₄⋊C₁₆	—	2	2	no	—
194	C₄⋊C₁₆	—	106	2	no	—
195	C₂²⋊C₁₆	—	106	2	no	—
196	#336	—	53	1	no	—
197	#336	—	53	1	no	—
198	C₅₃	#12912	1	1	yes	characteristic, complemented, Sylow
199	C₂³×C₄	—	53	1	no	—
200	C₂²×C₈	—	53	1	no	—
201	C₂²×C₈	—	53	2	no	—
202	C₂²×C₈	—	53	2	no	—
203	C₂²×C₈	—	53	1	no	—
204	C₂²×C₈	—	106	1	no	—
205	C₂²×C₈	—	106	1	no	—
206	C₂²×C₈	—	106	1	no	—
207	C₂²×C₈	—	106	1	no	—
208	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
209	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
210	C₂×C₂²⋊C₄	—	53	1	no	—
211	C₂×C₂²⋊C₄	—	53	1	no	—
212	C₂×C₄²	—	53	1	no	—
213	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
214	C₂×C₁₆	—	2	1	no	—
215	C₂×C₁₆	—	2	1	no	—
216	C₂×C₁₆	—	2	1	no	—
217	C₂×C₁₆	—	2	1	no	—
218	C₂×C₁₆	—	2	1	no	—
219	C₂×C₁₆	—	53	2	no	—
220	C₂×C₁₆	—	106	8	no	—
221	C₂×C₁₆	—	106	1	no	—
222	C₂×C₁₆	—	106	1	no	—
223	C₂×C₁₆	—	106	1	no	—
224	C₂×C₁₆	—	106	1	no	—
225	C₂×C₁₆	—	106	1	no	—
226	C₂×C₁₆	C₂²⋊D₅₃	1	2	yes	—
227	C₂²⋊C₈	—	106	2	no	—
228	C₄×C₈	—	2	1	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	106	1	no	—
231	C₄×C₈	—	106	1	no	—
232	#95	—	53	1	no	—
233	#95	—	53	1	no	—
234	C₂⁴	—	53	1	no	—
235	C₂²×C₄	—	53	1	no	—
236	C₂²×C₄	—	53	1	no	—
237	C₂²×C₄	—	53	2	no	—
238	C₂²×C₄	—	53	2	no	—
239	C₂²×C₄	—	53	1	no	—
240	C₂²×C₄	—	106	1	no	—
241	C₂²×C₄	—	106	1	no	—
242	C₂²×C₄	—	106	1	no	—
243	C₂²×C₄	—	106	1	no	—
244	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
245	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	53	2	no	—
251	C₂×C₈	—	106	1	no	—
252	C₂×C₈	—	106	1	no	—
253	C₂×C₈	—	106	2	no	—
254	C₂×C₈	—	106	2	no	—
255	C₂×C₈	—	106	2	no	—
256	C₂×C₈	—	106	2	no	—
257	C₂×C₈	—	106	1	no	—
258	C₂×C₈	—	106	1	no	—
259	C₂×C₈	—	106	1	no	—
260	C₂×C₈	#11719945	1	1	yes	characteristic
261	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
262	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
263	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
264	C₂×C₈	#11719949	1	1	yes	characteristic
265	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
266	C₂²⋊C₄	—	53	4	no	—
267	C₂²⋊C₄	—	106	2	no	—
268	C₄²	—	2	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	106	1	no	—
271	C₄²	—	106	1	no	—
272	C₁₆	—	2	2	no	—
273	C₁₆	—	4	1	no	—
274	C₁₆	—	106	2	no	—
275	C₁₆	—	212	1	no	—
276	C₂³	—	53	2	no	—
277	C₂³	—	53	2	no	—
278	C₂³	—	53	2	no	—
279	C₂³	—	106	1	no	—
280	C₂³	—	106	1	no	—
281	C₂³	—	106	1	no	—
282	C₂³	—	106	1	no	—
283	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	53	2	no	—
290	C₂×C₄	—	53	2	no	—
291	C₂×C₄	—	106	1	no	—
292	C₂×C₄	—	106	1	no	—
293	C₂×C₄	—	106	1	no	—
294	C₂×C₄	—	106	1	no	—
295	C₂×C₄	—	106	2	no	—
296	C₂×C₄	—	106	2	no	—
297	C₂×C₄	—	106	2	no	—
298	C₂×C₄	—	106	2	no	—
299	C₂×C₄	—	106	1	no	—
300	C₂×C₄	—	106	1	no	—
301	C₂×C₄	—	106	1	no	—
302	C₂×C₄	C₁₆×D₅₃	1	2	yes	complemented
303	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
304	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
305	C₂×C₄	#13133179	1	1	yes	characteristic
306	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
307	C₂×C₄	#13133223	1	1	yes	characteristic
308	C₂×C₄	#13133225	1	1	yes	characteristic
309	C₈	—	4	1	no	—
310	C₈	—	106	2	no	—
311	C₈	—	212	1	no	—
312	C₈	#13133185	1	1	yes	characteristic
313	C₈	#13133193	1	1	yes	characteristic
314	C₈	#13133195	1	1	yes	characteristic
315	C₈	#13133202	1	1	yes	characteristic
316	C₂²	—	53	4	no	—
317	C₂²	—	106	2	no	—
318	C₂²	—	106	2	no	—
319	C₂²	—	106	2	no	—
320	C₂²	—	106	2	no	—
321	C₂²	—	106	2	no	—
322	C₂²	—	106	2	no	—
323	C₂²	C₄⋊C₈×D₅₃	1	1	yes	characteristic
324	C₂²	C₈×C₂²⋊D₅₃	1	1	yes	characteristic
325	C₂²	#16810688	1	1	yes	characteristic
326	C₂²	C₂×C₁₆×D₅₃	1	1	yes	characteristic
327	C₂²	M₃₂×D₅₃	1	1	yes	characteristic
328	C₂²	#16811039	1	1	yes	characteristic
329	C₂²	#16811041	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	4	1	no	—
332	C₄	—	106	2	no	—
333	C₄	—	106	2	no	—
334	C₄	—	212	1	no	—
335	C₄	#16810494	1	1	yes	characteristic
336	C₄	#16810648	1	1	yes	characteristic
337	C₄	#16810692	1	1	yes	characteristic
338	C₄	#16810693	1	1	yes	characteristic
339	C₂	—	106	4	no	—
340	C₂	#20274891	1	1	yes	characteristic
341	C₂	C₄⋊C₁₆×D₅₃	1	1	yes	characteristic
342	C₂	#20276472	1	1	yes	characteristic
343	C₂	C₁₆×C₂²⋊D₅₃	1	1	yes	characteristic
344	C₂	#20276534	1	1	yes	characteristic
345	C₂	#20276538	1	1	yes	characteristic
346	C₂	#20276539	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#12912
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#16810441	C₂	semidirect
5. #20272793	C₂	semidirect
6. #20272792	C₂	semidirect
7. C₂×C₅₃×C₄⋊C₁₆	C₂	semidirect
85. #11719921	C₁₆	semidirect
115. C₂×C₄×C₅₃	C₂×C₁₆	semidirect
130. C₂×Dic₅₃	C₂×C₁₆	semidirect
156. C₂×D₅₃	C₄⋊C₁₆	semidirect
168. C₂×C₄⋊C₁₆	D₅₃	semidirect
198. C₅₃	#12912	semidirect
302. C₂×C₄	C₁₆×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×D₅₃	2
C₂×#16810441	2
C₂×#16810402	2
C₈×#11719921	2
#20272793	2
#20272792	2
C₂×C₅₃×C₄⋊C₁₆	2
#12912	53