Subgroups — D₅₃×#2249 #27539487

Order 13568.

854 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×SD₃₂×D₅₃	C₂	1	2	yes	maximal, complemented
3	C₂×D₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#495×D₅₃	C₂	1	2	yes	maximal, complemented
5	#20276977	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276976	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276975	C₂	1	2	yes	maximal, complemented
8	#20276974	C₂	1	2	yes	maximal, complemented
9	#20276973	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2249	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₅₃	C₂²	1	2	yes	—
13	C₂×SD₁₆×D₅₃	—	2	2	no	—
14	C₂×D₈×D₅₃	C₂²	1	2	yes	—
15	#124×D₅₃	—	2	2	no	—
16	C₂×Dic₄₂₄	C₂²	1	1	yes	characteristic, complemented
17	C₂×#13133163	C₂²	1	2	yes	complemented
18	Q₃₂×D₅₃	—	2	4	no	—
19	C₂×#13133162	C₂²	1	2	yes	complemented
20	C₂×#13133161	C₂²	1	2	yes	complemented
21	C₂×#13133160	C₂²	1	2	yes	complemented
22	SD₃₂×D₅₃	—	2	8	no	—
23	C₂×D₈₄₈	C₂²	1	1	yes	characteristic
24	C₂×#13133158	C₂²	1	2	yes	—
25	D₁₆×D₅₃	—	2	4	no	—
26	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
28	#16811008	C₂²	1	2	yes	complemented
29	#16811007	C₂²	1	2	yes	complemented
30	#16811006	C₂²	1	2	yes	complemented
31	#16811005	—	2	2	no	—
32	#16811004	—	2	2	no	—
33	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
34	C₄×#11719908	C₂²	1	1	yes	characteristic, complemented
35	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
36	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
37	C₂×Q₃₂×C₅₃	C₂²	1	1	yes	characteristic, complemented
38	C₂×SD₃₂×C₅₃	C₂²	1	2	yes	complemented
39	C₂×D₁₆×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#495	C₂²	1	2	yes	complemented
41	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
42	C₂×Q₈×D₅₃	—	2	2	no	—
43	C₂×D₄×D₅₃	—	2	2	no	—
44	C₂×Dic₂₁₂	C₂³	1	2	yes	—
45	C₂×#11719925	—	2	2	no	—
46	Q₁₆×D₅₃	—	2	4	no	—
47	Q₁₆×D₅₃	—	2	4	no	—
48	C₂×#11719913	—	2	2	no	—
49	C₂×#11719923	—	2	2	no	—
50	C₂×#11719924	—	2	2	no	—
51	SD₁₆×D₅₃	—	4	8	no	—
52	C₂×D₄₂₄	C₂³	1	2	yes	—
53	C₂×#11719922	—	2	2	no	—
54	D₈×D₅₃	—	2	4	no	—
55	D₈×D₅₃	—	2	4	no	—
56	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
57	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
58	#13133206	—	2	2	no	—
59	#13133205	—	2	2	no	—
60	#13133204	—	2	2	no	—
61	#13133203	—	4	4	no	—
62	C₂²⋊C₄×D₅₃	—	4	4	no	—
63	C₄²×D₅₃	D₄	1	1	yes	characteristic
64	Dic₄₂₄	—	2	2	no	—
65	#13133163	—	2	4	no	—
66	#13133162	—	2	4	no	—
67	#13133161	—	2	4	no	—
68	#13133160	—	2	4	no	—
69	D₈₄₈	—	2	2	no	—
70	#13133158	—	2	4	no	—
71	C₂×#11719908	C₂³	1	1	yes	characteristic
72	C₂×#11719908	C₂³	1	1	yes	characteristic
73	C₁₆×D₅₃	D₄	1	4	yes	—
74	C₁₆×D₅₃	D₄	1	4	yes	—
75	C₄×#88322	D₄	1	1	yes	characteristic
76	C₈×Dic₅₃	D₄	1	1	yes	characteristic
77	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
78	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
79	C₂×SD₁₆×C₅₃	—	2	2	no	—
80	C₂×D₈×C₅₃	C₂³	1	2	yes	—
81	#124×C₅₃	—	2	2	no	—
82	Q₃₂×C₅₃	—	2	2	no	—
83	SD₃₂×C₅₃	—	2	4	no	—
84	D₁₆×C₅₃	—	2	2	no	—
85	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
86	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
88	C₂³×D₅₃	—	4	2	no	—
89	C₂×Q₈×C₅₃	—	2	2	no	—
90	C₂×D₄×C₅₃	—	2	2	no	—
91	C₂×C₂²⋊D₅₃	—	4	2	no	—
92	Q₈×D₅₃	—	4	4	no	—
93	Q₈×D₅₃	—	4	4	no	—
94	D₄×D₅₃	—	4	4	no	—
95	D₄×D₅₃	—	4	4	no	—
96	C₂×D₂₁₂	—	2	2	no	—
97	C₂×C₄×D₅₃	—	4	2	no	—
98	C₂×C₄×D₅₃	D₈	1	2	yes	—
99	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
100	C₂×Dic₁₀₆	—	2	2	no	—
101	C₂×Dic₁₀₆	—	4	2	no	—
102	Q₁₆×C₅₃	—	2	2	no	—
103	Q₁₆×C₅₃	—	2	2	no	—
104	SD₁₆×C₅₃	—	4	4	no	—
105	D₈×C₅₃	—	2	2	no	—
106	D₈×C₅₃	—	2	2	no	—
107	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
109	C₂²⋊C₄×C₅₃	—	4	4	no	—
110	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₂²⋊Dic₅₃	—	4	4	no	—
112	#11719925	—	4	4	no	—
113	#11719924	—	4	4	no	—
114	#11719923	—	4	4	no	—
115	#11719922	—	4	4	no	—
116	#11719921	—	4	4	no	—
117	#11719921	—	4	4	no	—
118	C₄×Dic₅₃	D₈	1	2	yes	—
119	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
120	C₂×#88322	C₂×D₄	1	1	yes	characteristic
121	C₂×#88322	C₂×D₄	1	1	yes	characteristic
122	Dic₂₁₂	—	2	2	no	—
123	Dic₂₁₂	—	2	2	no	—
124	D₄₂₄	—	2	2	no	—
125	D₄₂₄	—	2	2	no	—
126	#11719913	—	4	4	no	—
127	C₈×D₅₃	D₈	1	4	yes	—
128	C₈×D₅₃	C₂×D₄	1	2	yes	—
129	C₈×D₅₃	C₂×D₄	1	2	yes	—
130	C₁₆×C₅₃	C₂×D₄	1	2	yes	—
131	C₁₆×C₅₃	C₂×D₄	1	2	yes	—
132	#11719908	C₂×D₄	1	2	yes	—
133	#11719908	C₂×D₄	1	2	yes	—
134	C₂³×C₅₃	—	4	2	no	—
135	C₂²×D₅₃	—	4	2	no	—
136	C₂²×D₅₃	—	8	4	no	—
137	C₂²×D₅₃	—	8	8	no	—
138	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
139	Q₈×C₅₃	—	4	2	no	—
140	Q₈×C₅₃	—	4	2	no	—
141	D₄×C₅₃	—	4	2	no	—
142	D₄×C₅₃	—	4	2	no	—
143	C₂×C₄×C₅₃	—	4	2	no	—
144	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
146	C₂²⋊D₅₃	—	8	4	no	—
147	C₂×Dic₅₃	—	4	2	no	—
148	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
149	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
150	D₂₁₂	—	4	2	no	—
151	D₂₁₂	—	4	2	no	—
152	C₄×D₅₃	—	2	4	no	—
153	C₄×D₅₃	—	8	4	no	—
154	C₄×D₅₃	#127	1	2	yes	—
155	C₄×D₅₃	C₂×D₈	1	2	yes	—
156	Dic₁₀₆	—	4	2	no	—
157	Dic₁₀₆	—	4	2	no	—
158	Dic₁₀₆	—	8	4	no	—
159	C₈×C₅₃	C₂×D₈	1	2	yes	—
160	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
161	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
162	#88322	#127	1	2	yes	—
163	#88322	C₂×D₈	1	2	yes	—
164	C₂×#2249	—	53	1	no	maximal, Sylow
165	C₂²×C₅₃	—	8	2	no	—
166	C₂²×C₅₃	—	8	4	no	—
167	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
168	C₂×D₅₃	—	8	2	no	—
169	C₂×D₅₃	—	8	4	no	—
170	C₂×D₅₃	—	16	8	no	—
171	C₂×D₅₃	#493	1	2	yes	—
172	C₂×D₅₃	#508	1	4	yes	—
173	C₄×C₅₃	—	2	2	no	—
174	C₄×C₅₃	—	8	2	no	—
175	C₄×C₅₃	C₂×#127	1	1	yes	characteristic
176	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
177	Dic₅₃	—	2	2	no	—
178	Dic₅₃	—	8	2	no	—
179	Dic₅₃	#493	1	2	yes	—
180	C₂²×Q₃₂	—	53	1	no	—
181	C₂²×SD₃₂	—	53	2	no	—
182	C₂²×D₁₆	—	53	1	no	—
183	C₂×#495	—	53	2	no	—
184	#2249	—	53	1	no	—
185	#2249	—	53	2	no	—
186	#2249	—	53	1	no	—
187	#2249	—	53	2	no	—
188	#2249	—	53	1	no	—
189	#2249	D₅₃	1	1	yes	characteristic, direct product factor
190	C₂×C₄×C₁₆	—	53	1	no	—
191	C₂×C₅₃	—	16	2	no	—
192	C₂×C₅₃	C₂×#493	1	1	yes	characteristic
193	C₂×C₅₃	C₂×#508	1	2	yes	—
194	D₅₃	—	16	2	no	—
195	D₅₃	#2249	1	4	yes	direct product factor
196	C₂²×Q₁₆	—	53	2	no	—
197	C₂²×SD₁₆	—	106	2	no	—
198	C₂²×D₈	—	53	2	no	—
199	C₂×#124	—	106	2	no	—
200	C₂×Q₃₂	—	53	2	no	—
201	C₂×Q₃₂	—	53	1	no	—
202	C₂×Q₃₂	—	106	4	no	—
203	C₂×Q₃₂	C₂×D₅₃	1	1	yes	characteristic, complemented
204	C₂×SD₃₂	—	53	2	no	—
205	C₂×SD₃₂	—	53	2	no	—
206	C₂×SD₃₂	—	53	2	no	—
207	C₂×SD₃₂	—	106	8	no	—
208	C₂×SD₃₂	C₂×D₅₃	1	2	yes	complemented
209	C₂×D₁₆	—	53	2	no	—
210	C₂×D₁₆	—	53	1	no	—
211	C₂×D₁₆	—	106	4	no	—
212	C₂×D₁₆	C₂×D₅₃	1	1	yes	characteristic
213	C₂²×C₁₆	—	53	1	no	—
214	C₂²×C₁₆	—	53	1	no	—
215	#495	—	53	2	no	—
216	#495	—	53	2	no	—
217	#495	—	53	2	no	—
218	#495	—	106	2	no	—
219	#495	—	106	2	no	—
220	#495	C₂×D₅₃	1	2	yes	complemented
221	C₂×C₄×C₈	—	53	1	no	—
222	C₄×C₁₆	—	53	1	no	—
223	C₄×C₁₆	—	53	1	no	—
224	C₄×C₁₆	—	53	1	no	—
225	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
226	C₅₃	C₂×#2249	1	1	yes	characteristic, complemented, Sylow
227	C₂²×Q₈	—	106	2	no	—
228	C₂²×D₄	—	106	2	no	—
229	C₂×Q₁₆	—	53	2	no	—
230	C₂×Q₁₆	—	106	4	no	—
231	C₂×Q₁₆	—	106	4	no	—
232	C₂×Q₁₆	—	106	2	no	—
233	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
234	C₂×SD₁₆	—	2	2	no	—
235	C₂×SD₁₆	—	106	2	no	—
236	C₂×SD₁₆	—	106	2	no	—
237	C₂×SD₁₆	—	106	2	no	—
238	C₂×SD₁₆	—	212	8	no	—
239	C₂×D₈	—	53	2	no	—
240	C₂×D₈	—	106	4	no	—
241	C₂×D₈	—	106	4	no	—
242	C₂×D₈	—	106	2	no	—
243	C₂×D₈	C₂²×D₅₃	1	2	yes	—
244	C₂²×C₈	—	53	1	no	—
245	C₂²×C₈	—	53	1	no	—
246	#124	—	2	2	no	—
247	#124	—	106	2	no	—
248	#124	—	106	2	no	—
249	#124	—	106	2	no	—
250	#124	—	212	4	no	—
251	C₂×C₂²⋊C₄	—	212	4	no	—
252	C₂×C₄²	—	53	1	no	—
253	Q₃₂	—	2	2	no	—
254	Q₃₂	—	106	4	no	—
255	Q₃₂	—	106	2	no	—
256	SD₃₂	—	2	4	no	—
257	SD₃₂	—	106	4	no	—
258	SD₃₂	—	106	4	no	—
259	SD₃₂	—	106	4	no	—
260	D₁₆	—	2	2	no	—
261	D₁₆	—	106	4	no	—
262	D₁₆	—	106	2	no	—
263	C₂×C₁₆	—	53	1	no	—
264	C₂×C₁₆	—	53	1	no	—
265	C₂×C₁₆	—	53	4	no	—
266	C₂×C₁₆	—	53	4	no	—
267	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
268	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
269	C₄×C₈	—	53	1	no	—
270	C₄×C₈	—	53	1	no	—
271	C₄×C₈	—	53	1	no	—
272	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
273	C₂⁴	—	212	2	no	—
274	C₂×Q₈	—	2	2	no	—
275	C₂×Q₈	—	106	2	no	—
276	C₂×Q₈	—	212	2	no	—
277	C₂×Q₈	—	212	4	no	—
278	C₂×Q₈	—	212	4	no	—
279	C₂×D₄	—	2	2	no	—
280	C₂×D₄	—	106	2	no	—
281	C₂×D₄	—	212	4	no	—
282	C₂×D₄	—	212	4	no	—
283	C₂×D₄	—	212	2	no	—
284	C₂²×C₄	—	53	1	no	—
285	C₂²×C₄	—	53	2	no	—
286	C₂²×C₄	—	212	2	no	—
287	Q₁₆	—	2	2	no	—
288	Q₁₆	—	2	2	no	—
289	Q₁₆	—	106	2	no	—
290	Q₁₆	—	106	2	no	—
291	Q₁₆	—	212	4	no	—
292	SD₁₆	—	4	4	no	—
293	SD₁₆	—	212	4	no	—
294	SD₁₆	—	212	4	no	—
295	SD₁₆	—	212	4	no	—
296	D₈	—	2	2	no	—
297	D₈	—	2	2	no	—
298	D₈	—	106	2	no	—
299	D₈	—	106	2	no	—
300	D₈	—	212	4	no	—
301	C₂×C₈	—	53	1	no	—
302	C₂×C₈	—	53	1	no	—
303	C₂×C₈	—	53	4	no	—
304	C₂×C₈	—	53	2	no	—
305	C₂×C₈	—	53	2	no	—
306	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
307	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
308	C₂²⋊C₄	—	4	4	no	—
309	C₂²⋊C₄	—	212	4	no	—
310	C₂²⋊C₄	—	212	4	no	—
311	C₂²⋊C₄	—	212	4	no	—
312	C₄²	—	53	2	no	—
313	C₄²	—	53	1	no	—
314	C₄²	D₄×D₅₃	1	1	yes	characteristic
315	C₁₆	—	53	2	no	—
316	C₁₆	—	53	2	no	—
317	C₁₆	D₄×D₅₃	1	2	yes	—
318	C₁₆	D₄×D₅₃	1	2	yes	—
319	C₂³	—	4	2	no	—
320	C₂³	—	53	1	no	—
321	C₂³	—	424	4	no	—
322	C₂³	—	424	8	no	—
323	C₂³	—	212	2	no	—
324	Q₈	—	4	2	no	—
325	Q₈	—	4	2	no	—
326	Q₈	—	424	4	no	—
327	Q₈	—	212	2	no	—
328	Q₈	—	212	2	no	—
329	D₄	—	4	2	no	—
330	D₄	—	4	2	no	—
331	D₄	—	424	4	no	—
332	D₄	—	212	2	no	—
333	D₄	—	212	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	53	1	no	—
336	C₂×C₄	—	53	2	no	—
337	C₂×C₄	—	53	2	no	—
338	C₂×C₄	—	53	2	no	—
339	C₂×C₄	—	106	4	no	—
340	C₂×C₄	—	424	4	no	—
341	C₂×C₄	—	212	2	no	—
342	C₂×C₄	D₈×D₅₃	1	2	yes	—
343	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
344	C₈	—	53	2	no	—
345	C₈	—	53	2	no	—
346	C₈	D₈×D₅₃	1	2	yes	—
347	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
348	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
349	C₂²	—	8	4	no	—
350	C₂²	—	8	2	no	—
351	C₂²	—	53	4	no	—
352	C₂²	—	53	2	no	—
353	C₂²	—	848	8	no	—
354	C₂²	—	424	4	no	—
355	C₂²	—	424	2	no	—
356	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
357	C₄	—	2	2	no	—
358	C₄	—	8	2	no	—
359	C₄	—	53	2	no	—
360	C₄	—	106	2	no	—
361	C₄	—	424	2	no	—
362	C₄	#127×D₅₃	1	1	yes	characteristic
363	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
364	C₂	—	16	2	no	—
365	C₂	—	53	4	no	—
366	C₂	—	848	2	no	—
367	C₂	#493×D₅₃	1	1	yes	characteristic
368	C₂	#508×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2249	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×D₁₆×D₅₃	C₂
hypercenter	#2249	D₅₃
hypocenter	C₅₃	C₂×#2249
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2249

Complements

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

N	H ≅ G/N	type
1. C₂×Q₃₂×D₅₃	C₂	semidirect
2. C₂×SD₃₂×D₅₃	C₂	semidirect
4. #495×D₅₃	C₂	semidirect
5. #20276977	C₂	semidirect
6. #20276976	C₂	semidirect
7. #20276975	C₂	semidirect
8. #20276974	C₂	semidirect
9. #20276973	C₂	semidirect
10. C₄×C₁₆×D₅₃	C₂	semidirect
11. C₅₃×#2249	C₂	semidirect
16. C₂×Dic₄₂₄	C₂²	semidirect
17. C₂×#13133163	C₂²	semidirect
19. C₂×#13133162	C₂²	semidirect
20. C₂×#13133161	C₂²	semidirect
21. C₂×#13133160	C₂²	semidirect
28. #16811008	C₂²	semidirect
29. #16811007	C₂²	semidirect
30. #16811006	C₂²	semidirect
34. C₄×#11719908	C₂²	semidirect
35. C₁₆×Dic₅₃	C₂²	semidirect
36. C₁₆×Dic₅₃	C₂²	semidirect
37. C₂×Q₃₂×C₅₃	C₂²	semidirect
38. C₂×SD₃₂×C₅₃	C₂²	semidirect
40. C₅₃×#495	C₂²	semidirect
41. C₄×C₁₆×C₅₃	C₂²	semidirect
189. #2249	D₅₃	direct (N × H)
195. D₅₃	#2249	direct (N × H)
203. C₂×Q₃₂	C₂×D₅₃	semidirect
208. C₂×SD₃₂	C₂×D₅₃	semidirect
220. #495	C₂×D₅₃	semidirect
225. C₄×C₁₆	C₂×D₅₃	semidirect
226. C₅₃	C₂×#2249	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₃₂×D₅₃	2
C₂×SD₃₂×D₅₃	2
C₂×D₁₆×D₅₃	2
#495×D₅₃	2
#20276977	2
#20276976	2
#20276975	2
#20276974	2
#20276973	2
C₄×C₁₆×D₅₃	2
C₅₃×#2249	2
C₂×#2249	53

Subgroups — D53×#2249 #27539487

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#2249 #27539487