Subgroups — D₅₃×#2219 #27538540

Order 13568.

682 conjugacy classes of subgroups.

The subgroup lattice has 382 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×M₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#467×D₅₃	C₂	1	2	yes	maximal
4	#20276824	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276823	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276822	C₂	1	2	yes	maximal, complemented
7	#20276821	C₂	1	2	yes	maximal, complemented
8	#20276820	C₂	1	1	yes	maximal, characteristic, complemented
9	#367×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#358×D₅₃	C₂	1	2	yes	maximal, complemented
11	C₅₃×#2219	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₅₃	C₂²	1	2	yes	—
13	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
14	C₂²⋊Q₈×D₅₃	—	2	2	no	—
15	C₂×Dic₄₂₄	C₂²	1	1	yes	characteristic, complemented
16	C₂×#13133163	C₂²	1	2	yes	complemented
17	Q₃₂×D₅₃	—	2	4	no	—
18	C₂×#13133157	C₂²	1	1	yes	characteristic
19	C₂×#13133156	C₂²	1	1	yes	characteristic
20	C₂×#13133156	C₂²	1	1	yes	characteristic
21	M₃₂×D₅₃	—	2	4	no	—
22	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
24	#16810865	C₂²	1	2	yes	—
25	#16810864	C₂²	1	2	yes	—
26	#16810863	—	2	2	no	—
27	#16810862	—	2	2	no	—
28	#16810861	C₂²	1	2	yes	—
29	#107×D₅₃	C₂²	1	2	yes	—
30	#103×D₅₃	—	2	2	no	—
31	#16810451	C₂²	1	1	yes	characteristic, complemented
32	#16810450	C₂²	1	2	yes	complemented
33	#16810428	C₂²	1	2	yes	complemented
34	#16810427	C₂²	1	2	yes	complemented
35	#16810426	C₂²	1	2	yes	complemented
36	C₂×Q₃₂×C₅₃	C₂²	1	1	yes	characteristic, complemented
37	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×#467	C₂²	1	2	yes	—
39	C₅₃×#367	C₂²	1	1	yes	characteristic, complemented
40	C₅₃×#358	C₂²	1	2	yes	complemented
41	C₂×Q₈×D₅₃	—	2	2	no	—
42	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
43	C₂×Dic₂₁₂	C₂³	1	2	yes	—
44	C₂×#11719925	—	2	2	no	—
45	Q₁₆×D₅₃	—	2	4	no	—
46	Q₁₆×D₅₃	—	2	4	no	—
47	C₂²×#88322	D₄	1	1	yes	characteristic
48	C₂×C₈×D₅₃	—	2	2	no	—
49	C₂×C₈×D₅₃	—	2	2	no	—
50	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
51	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
52	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
53	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
54	#13133197	—	2	2	no	—
55	#13133196	—	2	2	no	—
56	#13133195	—	4	2	no	—
57	#13133194	—	4	2	no	—
58	#13133193	—	2	2	no	—
59	C₄⋊C₄×D₅₃	—	2	2	no	—
60	C₄⋊C₄×D₅₃	—	4	2	no	—
61	C₂²⋊C₄×D₅₃	—	4	2	no	—
62	Dic₄₂₄	—	2	2	no	—
63	#13133163	—	2	4	no	—
64	#13133157	—	2	2	no	—
65	#13133156	—	2	2	no	—
66	#13133156	—	2	2	no	—
67	C₂×#11719908	C₂³	1	1	yes	characteristic
68	C₂×#11719908	C₂³	1	1	yes	characteristic
69	C₁₆×D₅₃	—	2	2	no	—
70	C₁₆×D₅₃	D₄	1	4	yes	—
71	#13133151	C₂³	1	2	yes	—
72	#13133150	—	2	2	no	—
73	#13133142	—	2	2	no	—
74	#13133141	—	2	2	no	—
75	#13133140	—	2	2	no	—
76	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
77	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₂²⋊Q₈×C₅₃	—	2	2	no	—
79	Q₃₂×C₅₃	—	2	2	no	—
80	M₃₂×C₅₃	—	2	2	no	—
81	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
82	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
83	#107×C₅₃	C₂³	1	2	yes	—
84	#103×C₅₃	—	2	2	no	—
85	C₂³×D₅₃	D₈	1	1	yes	characteristic
86	C₂×Q₈×C₅₃	—	2	2	no	—
87	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×Dic₅₃	D₈	1	1	yes	characteristic
89	Q₈×D₅₃	—	4	4	no	—
90	Q₈×D₅₃	—	4	4	no	—
91	C₂×C₄×D₅₃	—	2	2	no	—
92	C₂×C₄×D₅₃	—	2	2	no	—
93	C₂×C₄×D₅₃	—	4	2	no	—
94	C₂×C₄×D₅₃	—	4	2	no	—
95	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
96	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
97	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
99	C₂×Dic₁₀₆	—	2	2	no	—
100	C₂×Dic₁₀₆	—	4	2	no	—
101	Q₁₆×C₅₃	—	2	2	no	—
102	Q₁₆×C₅₃	—	2	2	no	—
103	C₂×C₈×C₅₃	—	2	1	no	—
104	C₂×C₈×C₅₃	—	2	1	no	—
105	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₄×C₅₃	—	2	2	no	—
108	C₄⋊C₄×C₅₃	—	4	2	no	—
109	C₂²⋊C₄×C₅₃	—	4	2	no	—
110	C₂²⋊Dic₅₃	—	4	2	no	—
111	#11719925	—	4	4	no	—
112	#11719921	—	4	2	no	—
113	#11719921	—	4	2	no	—
114	C₄⋊Dic₅₃	—	2	2	no	—
115	C₄⋊Dic₅₃	—	4	2	no	—
116	#11719919	—	4	2	no	—
117	#11719919	—	4	2	no	—
118	#11719919	—	4	2	no	—
119	C₂×#88322	—	2	2	no	—
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	C₈×D₅₃	—	2	2	no	—
125	C₈×D₅₃	—	2	1	no	—
126	C₈×D₅₃	—	2	1	no	—
127	C₈×D₅₃	—	2	2	no	—
128	C₈×D₅₃	C₂×D₄	1	2	yes	—
129	C₈×D₅₃	#41	1	2	yes	—
130	C₁₆×C₅₃	—	2	1	no	—
131	C₁₆×C₅₃	C₂×D₄	1	2	yes	—
132	#11719908	—	2	1	no	—
133	#11719908	C₂×D₄	1	2	yes	—
134	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
135	C₂²×D₅₃	—	2	2	no	—
136	C₂²×D₅₃	—	2	4	no	—
137	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
138	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
139	Q₈×C₅₃	—	4	2	no	—
140	Q₈×C₅₃	—	4	2	no	—
141	C₂×C₄×C₅₃	—	2	1	no	—
142	C₂×C₄×C₅₃	—	2	1	no	—
143	C₂×C₄×C₅₃	—	4	2	no	—
144	C₂×C₄×C₅₃	—	4	2	no	—
145	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×Dic₅₃	—	2	2	no	—
148	C₂×Dic₅₃	—	4	2	no	—
149	C₂×Dic₅₃	—	4	2	no	—
150	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
151	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
152	C₄×D₅₃	—	2	2	no	—
153	C₄×D₅₃	—	2	1	no	—
154	C₄×D₅₃	—	2	1	no	—
155	C₄×D₅₃	—	2	2	no	—
156	C₄×D₅₃	—	8	4	no	—
157	C₄×D₅₃	—	8	4	no	—
158	C₄×D₅₃	#121	1	2	yes	—
159	C₄×D₅₃	#135	1	2	yes	—
160	Dic₁₀₆	—	4	2	no	—
161	Dic₁₀₆	—	4	2	no	—
162	Dic₁₀₆	—	8	4	no	—
163	C₈×C₅₃	—	2	1	no	—
164	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
165	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
166	#88322	—	2	1	no	—
167	#88322	#121	1	2	yes	—
168	C₂×#2219	—	53	1	no	maximal, Sylow
169	C₂²×C₅₃	—	2	2	no	—
170	C₂²×C₅₃	—	2	1	no	—
171	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
172	C₂×D₅₃	—	2	2	no	—
173	C₂×D₅₃	—	2	1	no	—
174	C₂×D₅₃	—	4	4	no	—
175	C₂×D₅₃	#466	1	2	yes	—
176	C₂×D₅₃	#510	1	4	yes	—
177	C₄×C₅₃	—	2	1	no	—
178	C₄×C₅₃	—	8	2	no	—
179	C₄×C₅₃	—	8	2	no	—
180	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
181	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
182	Dic₅₃	—	2	1	no	—
183	Dic₅₃	—	8	2	no	—
184	Dic₅₃	—	8	2	no	—
185	Dic₅₃	#466	1	2	yes	—
186	C₂²×Q₃₂	—	53	1	no	—
187	C₂²×M₃₂	—	53	1	no	—
188	C₂×#467	—	53	2	no	—
189	#2219	—	53	1	no	—
190	#2219	—	53	1	no	—
191	#2219	—	53	2	no	—
192	#2219	—	53	1	no	—
193	#2219	—	53	2	no	—
194	#2219	D₅₃	1	1	yes	characteristic, direct product factor
195	C₂×#367	—	53	1	no	—
196	C₂×#358	—	53	2	no	—
197	C₂×C₅₃	—	4	1	no	—
198	C₂×C₅₃	C₂×#466	1	1	yes	characteristic
199	C₂×C₅₃	C₂×#510	1	2	yes	—
200	D₅₃	—	4	1	no	—
201	D₅₃	#2219	1	4	yes	direct product factor
202	C₂²×Q₁₆	—	53	2	no	—
203	C₂³×C₈	—	53	1	no	—
204	C₂×C₂²⋊Q₈	—	106	2	no	—
205	C₂×Q₃₂	—	53	2	no	—
206	C₂×Q₃₂	—	53	1	no	—
207	C₂×Q₃₂	—	106	4	no	—
208	C₂×Q₃₂	C₂×D₅₃	1	1	yes	characteristic, complemented
209	C₂×M₃₂	—	53	1	no	—
210	C₂×M₃₂	—	53	1	no	—
211	C₂×M₃₂	—	53	1	no	—
212	C₂×M₃₂	—	106	4	no	—
213	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
214	C₂²×C₁₆	—	53	1	no	—
215	C₂²×C₁₆	—	53	1	no	—
216	#467	—	53	2	no	—
217	#467	—	53	2	no	—
218	#467	—	53	2	no	—
219	#467	—	106	2	no	—
220	#467	—	106	2	no	—
221	#467	C₂×D₅₃	1	2	yes	—
222	C₂×#107	—	53	2	no	—
223	C₂×#103	—	106	2	no	—
224	#367	—	53	1	no	—
225	#367	—	53	2	no	—
226	#367	C₂×D₅₃	1	1	yes	characteristic, complemented
227	#358	—	53	2	no	—
228	#358	—	53	2	no	—
229	#358	—	53	2	no	—
230	#358	C₂×D₅₃	1	2	yes	complemented
231	C₅₃	C₂×#2219	1	1	yes	characteristic, complemented, Sylow
232	C₂²×Q₈	—	106	2	no	—
233	C₂³×C₄	—	53	1	no	—
234	C₂×Q₁₆	—	53	2	no	—
235	C₂×Q₁₆	—	106	4	no	—
236	C₂×Q₁₆	—	106	4	no	—
237	C₂×Q₁₆	—	106	2	no	—
238	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
239	C₂²×C₈	—	53	1	no	—
240	C₂²×C₈	—	53	1	no	—
241	C₂²×C₈	—	53	1	no	—
242	C₂²×C₈	—	53	1	no	—
243	C₂²×C₈	—	53	1	no	—
244	C₂²×C₈	—	106	2	no	—
245	C₂²×C₈	—	106	2	no	—
246	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
247	C₂²⋊Q₈	—	2	2	no	—
248	C₂²⋊Q₈	—	106	2	no	—
249	C₂²⋊Q₈	—	106	2	no	—
250	C₂²⋊Q₈	—	106	2	no	—
251	C₂²⋊Q₈	—	212	2	no	—
252	C₂²⋊Q₈	—	212	2	no	—
253	C₂×C₄⋊C₄	—	106	2	no	—
254	C₂×C₄⋊C₄	—	212	2	no	—
255	C₂×C₂²⋊C₄	—	212	2	no	—
256	Q₃₂	—	2	2	no	—
257	Q₃₂	—	106	4	no	—
258	Q₃₂	—	106	2	no	—
259	M₃₂	—	2	2	no	—
260	M₃₂	—	106	2	no	—
261	M₃₂	—	106	2	no	—
262	M₃₂	—	106	2	no	—
263	C₂×C₁₆	—	53	1	no	—
264	C₂×C₁₆	—	53	4	no	—
265	C₂×C₁₆	—	53	1	no	—
266	C₂×C₁₆	—	106	2	no	—
267	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
268	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
269	#107	—	53	2	no	—
270	#107	—	106	2	no	—
271	#107	C₂²×D₅₃	1	2	yes	—
272	#103	—	2	2	no	—
273	#103	—	106	2	no	—
274	#103	—	106	2	no	—
275	#103	—	106	2	no	—
276	C₂⁴	—	53	1	no	—
277	C₂×Q₈	—	2	2	no	—
278	C₂×Q₈	—	106	2	no	—
279	C₂×Q₈	—	212	2	no	—
280	C₂×Q₈	—	212	4	no	—
281	C₂×Q₈	—	212	4	no	—
282	C₂²×C₄	—	53	1	no	—
283	C₂²×C₄	—	53	1	no	—
284	C₂²×C₄	—	53	1	no	—
285	C₂²×C₄	—	53	1	no	—
286	C₂²×C₄	—	53	1	no	—
287	C₂²×C₄	—	106	2	no	—
288	C₂²×C₄	—	106	2	no	—
289	C₂²×C₄	—	212	2	no	—
290	C₂²×C₄	—	212	2	no	—
291	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
292	Q₁₆	—	2	2	no	—
293	Q₁₆	—	2	2	no	—
294	Q₁₆	—	106	2	no	—
295	Q₁₆	—	106	2	no	—
296	Q₁₆	—	212	4	no	—
297	C₂×C₈	—	2	1	no	—
298	C₂×C₈	—	2	1	no	—
299	C₂×C₈	—	53	1	no	—
300	C₂×C₈	—	53	2	no	—
301	C₂×C₈	—	53	2	no	—
302	C₂×C₈	—	53	1	no	—
303	C₂×C₈	—	106	2	no	—
304	C₂×C₈	—	106	2	no	—
305	C₂×C₈	—	106	2	no	—
306	C₂×C₈	—	106	1	no	—
307	C₂×C₈	—	106	1	no	—
308	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
309	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	4	2	no	—
312	C₄⋊C₄	—	106	2	no	—
313	C₄⋊C₄	—	212	2	no	—
314	C₄⋊C₄	—	212	2	no	—
315	C₄⋊C₄	—	212	2	no	—
316	C₄⋊C₄	—	212	2	no	—
317	C₂²⋊C₄	—	4	2	no	—
318	C₂²⋊C₄	—	212	2	no	—
319	C₂²⋊C₄	—	212	2	no	—
320	C₂²⋊C₄	—	212	2	no	—
321	C₁₆	—	2	1	no	—
322	C₁₆	—	53	2	no	—
323	C₁₆	—	106	1	no	—
324	C₁₆	D₄×D₅₃	1	2	yes	—
325	C₂³	—	53	1	no	—
326	C₂³	—	53	1	no	—
327	C₂³	—	106	2	no	—
328	C₂³	—	106	4	no	—
329	C₂³	D₈×D₅₃	1	1	yes	characteristic
330	Q₈	—	4	2	no	—
331	Q₈	—	4	2	no	—
332	Q₈	—	424	4	no	—
333	Q₈	—	212	2	no	—
334	Q₈	—	212	2	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	53	1	no	—
340	C₂×C₄	—	53	2	no	—
341	C₂×C₄	—	53	2	no	—
342	C₂×C₄	—	53	1	no	—
343	C₂×C₄	—	106	2	no	—
344	C₂×C₄	—	106	2	no	—
345	C₂×C₄	—	106	2	no	—
346	C₂×C₄	—	106	1	no	—
347	C₂×C₄	—	106	1	no	—
348	C₂×C₄	—	424	4	no	—
349	C₂×C₄	—	424	4	no	—
350	C₂×C₄	—	212	2	no	—
351	C₂×C₄	—	212	2	no	—
352	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
353	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
354	C₈	—	2	1	no	—
355	C₈	—	53	2	no	—
356	C₈	—	106	1	no	—
357	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
358	C₈	#41×D₅₃	1	1	yes	characteristic
359	C₂²	—	2	2	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	53	4	no	—
362	C₂²	—	53	2	no	—
363	C₂²	—	106	2	no	—
364	C₂²	—	106	1	no	—
365	C₂²	—	212	4	no	—
366	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
367	C₄	—	2	1	no	—
368	C₄	—	8	2	no	—
369	C₄	—	8	2	no	—
370	C₄	—	53	2	no	—
371	C₄	—	106	1	no	—
372	C₄	—	424	2	no	—
373	C₄	—	424	2	no	—
374	C₄	#121×D₅₃	1	1	yes	characteristic
375	C₄	#135×D₅₃	1	1	yes	characteristic
376	C₂	—	4	1	no	—
377	C₂	—	53	4	no	—
378	C₂	—	212	1	no	—
379	C₂	#466×D₅₃	1	1	yes	characteristic
380	C₂	#510×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×#2219	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂³×D₅₃	D₈
hypercenter	#2219	D₅₃
hypocenter	C₅₃	C₂×#2219
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2219

Complements

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

N	H ≅ G/N	type
1. C₂×Q₃₂×D₅₃	C₂	semidirect
4. #20276824	C₂	semidirect
5. #20276823	C₂	semidirect
6. #20276822	C₂	semidirect
7. #20276821	C₂	semidirect
8. #20276820	C₂	semidirect
9. #367×D₅₃	C₂	semidirect
10. #358×D₅₃	C₂	semidirect
11. C₅₃×#2219	C₂	semidirect
15. C₂×Dic₄₂₄	C₂²	semidirect
16. C₂×#13133163	C₂²	semidirect
31. #16810451	C₂²	semidirect
32. #16810450	C₂²	semidirect
33. #16810428	C₂²	semidirect
34. #16810427	C₂²	semidirect
35. #16810426	C₂²	semidirect
36. C₂×Q₃₂×C₅₃	C₂²	semidirect
39. C₅₃×#367	C₂²	semidirect
40. C₅₃×#358	C₂²	semidirect
194. #2219	D₅₃	direct (N × H)
201. D₅₃	#2219	direct (N × H)
208. C₂×Q₃₂	C₂×D₅₃	semidirect
226. #367	C₂×D₅₃	semidirect
230. #358	C₂×D₅₃	semidirect
231. C₅₃	C₂×#2219	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×M₃₂×D₅₃	2
#467×D₅₃	2
#20276824	2
#20276823	2
#20276822	2
#20276821	2
#20276820	2
#367×D₅₃	2
#358×D₅₃	2
C₅₃×#2219	2
C₂×#2219	53

Subgroups — D53×#2219 #27538540

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#2219 #27538540