Subgroups — D₅₃×#1335 #27293873

Order 13568.

414 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#443×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#364×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈²×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272689	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272688	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272687	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1335	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₅₃	C₄	1	1	yes	characteristic
9	#131×D₅₃	—	2	1	no	—
10	M₃₂×D₅₃	—	2	1	no	—
11	#16810741	C₂²	1	1	yes	characteristic
12	#16810740	C₄	1	1	yes	characteristic
13	#16810739	C₄	1	1	yes	characteristic
14	#16810738	—	2	1	no	—
15	#16810737	—	2	1	no	—
16	#108×D₅₃	C₄	1	1	yes	characteristic
17	#104×D₅₃	—	2	1	no	—
18	C₄×C₈×D₅₃	—	2	1	no	—
19	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
20	#16810444	C₂²	1	1	yes	characteristic, complemented
21	#16810443	—	2	1	no	—
22	C₈×#88322	—	2	1	no	—
23	C₈×#88322	C₂²	1	1	yes	characteristic, complemented
24	C₅₃×#443	C₂²	1	1	yes	characteristic
25	C₅₃×#364	C₂²	1	1	yes	characteristic, complemented
26	C₈²×C₅₃	C₂²	1	1	yes	characteristic, complemented
27	#41×D₅₃	—	2	1	no	—
28	#13133240	C₈	1	1	yes	characteristic, complemented
29	#13133239	C₂×C₄	1	1	yes	characteristic
30	#13133238	—	2	1	no	—
31	#13133237	—	2	1	no	—
32	#13133236	C₈	1	1	yes	characteristic, complemented
33	Q₁₆×D₅₃	C₈	1	1	yes	characteristic, complemented
34	SD₁₆×D₅₃	—	2	1	no	—
35	D₈×D₅₃	C₈	1	1	yes	characteristic, complemented
36	#13133225	—	2	1	no	—
37	#13133225	—	4	1	no	—
38	#13133224	—	2	1	no	—
39	#13133223	—	2	1	no	—
40	#13133223	—	4	1	no	—
41	M₁₆×D₅₃	—	2	1	no	—
42	M₁₆×D₅₃	—	4	1	no	—
43	C₂×C₈×D₅₃	—	2	2	no	—
44	C₂×C₈×D₅₃	—	4	1	no	—
45	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
47	C₄²×D₅₃	D₄	1	1	yes	characteristic
48	#13133157	—	2	1	no	—
49	#13133156	—	4	1	no	—
50	C₁₆×D₅₃	—	4	1	no	—
51	#13133153	C₂×C₄	1	1	yes	characteristic
52	#13133152	—	2	1	no	—
53	#13133145	—	2	1	no	—
54	#13133144	—	2	1	no	—
55	#13133143	—	2	1	no	—
56	C₄×#88322	—	2	1	no	—
57	C₄×#88322	D₄	1	1	yes	characteristic
58	C₈×Dic₅₃	—	2	2	no	—
59	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
60	C₈×Dic₅₃	D₄	1	1	yes	characteristic
61	#135×C₅₃	C₂×C₄	1	1	yes	characteristic
62	#131×C₅₃	—	2	1	no	—
63	M₃₂×C₅₃	—	2	1	no	—
64	#108×C₅₃	C₂×C₄	1	1	yes	characteristic
65	#104×C₅₃	—	2	1	no	—
66	C₄×C₈×C₅₃	—	2	1	no	—
67	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	#41×C₅₃	—	2	1	no	—
69	#11719949	—	2	1	no	—
70	Q₈×D₅₃	—	2	1	no	—
71	#11719947	—	2	1	no	—
72	#11719947	—	4	1	no	—
73	D₄×D₅₃	—	2	1	no	—
74	D₄×D₅₃	—	4	1	no	—
75	#11719945	—	2	1	no	—
76	#11719945	—	4	1	no	—
77	C₂×C₄×D₅₃	—	2	1	no	—
78	C₂×C₄×D₅₃	—	4	1	no	—
79	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
80	Q₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic, complemented
81	SD₁₆×C₅₃	—	2	1	no	—
82	D₈×C₅₃	C₂×C₈	1	1	yes	characteristic, complemented
83	M₁₆×C₅₃	—	2	1	no	—
84	M₁₆×C₅₃	—	4	1	no	—
85	C₂×C₈×C₅₃	—	2	2	no	—
86	C₂×C₈×C₅₃	—	4	1	no	—
87	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
90	#11719925	—	2	1	no	—
91	#11719924	—	2	1	no	—
92	#11719923	—	2	1	no	—
93	#11719922	—	2	1	no	—
94	C₄×Dic₅₃	—	2	1	no	—
95	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
96	#11719917	—	2	1	no	—
97	#11719917	—	4	1	no	—
98	C₂×#88322	—	2	2	no	—
99	C₂×#88322	—	4	1	no	—
100	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
102	Dic₂₁₂	C₂×C₈	1	1	yes	characteristic, complemented
103	D₄₂₄	C₂×C₈	1	1	yes	characteristic, complemented
104	#11719913	—	2	1	no	—
105	#11719912	—	4	1	no	—
106	#11719912	—	4	1	no	—
107	#11719912	—	4	1	no	—
108	C₈×D₅₃	—	2	8	no	—
109	C₈×D₅₃	—	4	1	no	—
110	C₈×D₅₃	—	4	1	no	—
111	C₈×D₅₃	—	4	1	no	—
112	C₈×D₅₃	C₂×C₈	1	1	yes	characteristic
113	C₈×D₅₃	C₂×C₈	1	1	yes	characteristic
114	C₈×D₅₃	M₁₆	1	1	yes	characteristic
115	C₈×D₅₃	M₁₆	1	1	yes	characteristic
116	C₈×D₅₃	D₈	1	1	yes	characteristic
117	C₈×D₅₃	D₈	1	1	yes	characteristic
118	C₈×D₅₃	SD₁₆	1	1	yes	characteristic
119	C₈×D₅₃	SD₁₆	1	1	yes	characteristic
120	C₁₆×C₅₃	—	4	1	no	—
121	#11719908	—	4	1	no	—
122	C₂²×D₅₃	—	4	1	no	—
123	C₂²×D₅₃	#104	1	1	yes	characteristic
124	Q₈×C₅₃	—	2	1	no	—
125	D₄×C₅₃	—	2	1	no	—
126	D₄×C₅₃	—	4	1	no	—
127	C₂×C₄×C₅₃	—	2	1	no	—
128	C₂×C₄×C₅₃	—	4	1	no	—
129	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²⋊D₅₃	—	4	1	no	—
131	C₂²⋊D₅₃	—	4	1	no	—
132	C₂²⋊D₅₃	—	4	1	no	—
133	C₂×Dic₅₃	—	2	1	no	—
134	C₂×Dic₅₃	—	4	1	no	—
135	C₂×Dic₅₃	#104	1	1	yes	characteristic
136	D₂₁₂	—	2	1	no	—
137	D₂₁₂	—	4	1	no	—
138	C₄×D₅₃	—	2	4	no	—
139	C₄×D₅₃	—	4	1	no	—
140	C₄×D₅₃	—	4	1	no	—
141	C₄×D₅₃	—	4	1	no	—
142	C₄×D₅₃	C₂²⋊C₈	1	1	yes	characteristic
143	C₄×D₅₃	C₂²⋊C₈	1	1	yes	characteristic
144	C₄×D₅₃	#102	1	1	yes	characteristic
145	C₄×D₅₃	#102	1	1	yes	characteristic
146	Dic₁₀₆	—	2	1	no	—
147	Dic₁₀₆	—	4	1	no	—
148	C₈×C₅₃	—	2	4	no	—
149	C₈×C₅₃	—	4	1	no	—
150	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
151	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
152	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
153	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
154	#88322	—	2	4	no	—
155	#88322	—	4	1	no	—
156	#88322	C₂²⋊C₈	1	1	yes	characteristic
157	#88322	C₂²⋊C₈	1	1	yes	characteristic
158	#88322	#102	1	1	yes	characteristic
159	#88322	#102	1	1	yes	characteristic
160	C₂×#1335	—	53	1	no	maximal, Sylow
161	C₂²×C₅₃	—	4	1	no	—
162	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
163	C₂×D₅₃	—	2	2	no	—
164	C₂×D₅₃	—	4	1	no	—
165	C₂×D₅₃	—	8	2	no	—
166	C₂×D₅₃	D₄⋊C₈	1	1	yes	characteristic
167	C₂×D₅₃	D₄⋊C₈	1	1	yes	characteristic
168	C₄×C₅₃	—	2	2	no	—
169	C₄×C₅₃	—	4	1	no	—
170	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
171	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
172	Dic₅₃	—	2	2	no	—
173	Dic₅₃	—	4	1	no	—
174	Dic₅₃	D₄⋊C₈	1	1	yes	characteristic
175	Dic₅₃	D₄⋊C₈	1	1	yes	characteristic
176	C₂×#443	—	53	1	no	—
177	C₂×#364	—	53	1	no	—
178	C₂×C₈²	—	53	1	no	—
179	#1335	—	53	1	no	—
180	#1335	—	53	1	no	—
181	#1335	—	53	1	no	—
182	#1335	D₅₃	1	1	yes	characteristic, direct product factor
183	C₂×C₅₃	—	2	1	no	—
184	C₂×C₅₃	—	8	1	no	—
185	C₂×C₅₃	C₂×D₄⋊C₈	1	1	yes	characteristic
186	D₅₃	—	2	1	no	—
187	D₅₃	—	8	1	no	—
188	D₅₃	#1335	1	2	yes	direct product factor
189	C₂×#135	—	53	1	no	—
190	C₂×#131	—	106	1	no	—
191	C₂×M₃₂	—	106	1	no	—
192	#443	—	53	1	no	—
193	#443	—	53	1	no	—
194	#443	—	53	1	no	—
195	#443	—	106	1	no	—
196	#443	—	106	1	no	—
197	#443	C₂×D₅₃	1	1	yes	characteristic
198	C₂×#108	—	53	1	no	—
199	C₂×#104	—	106	1	no	—
200	C₂×C₄×C₈	—	53	1	no	—
201	C₂×C₄×C₈	—	106	1	no	—
202	#364	—	53	1	no	—
203	#364	—	106	1	no	—
204	#364	C₂×D₅₃	1	1	yes	characteristic, complemented
205	C₈²	—	53	1	no	—
206	C₈²	—	106	1	no	—
207	C₈²	C₂×D₅₃	1	1	yes	characteristic, complemented
208	C₅₃	C₂×#1335	1	1	yes	characteristic, complemented, Sylow
209	C₂×#41	—	106	1	no	—
210	#135	—	53	1	no	—
211	#135	—	53	1	no	—
212	#135	—	53	1	no	—
213	#135	—	106	1	no	—
214	#135	—	106	1	no	—
215	#135	C₄×D₅₃	1	1	yes	characteristic
216	C₂×Q₁₆	—	53	1	no	—
217	C₂×SD₁₆	—	106	1	no	—
218	C₂×D₈	—	53	1	no	—
219	#131	—	2	1	no	—
220	#131	—	106	1	no	—
221	#131	—	106	1	no	—
222	#131	—	106	1	no	—
223	#131	—	212	1	no	—
224	#131	—	212	1	no	—
225	C₂×M₁₆	—	106	1	no	—
226	C₂×M₁₆	—	212	1	no	—
227	C₂²×C₈	—	53	1	no	—
228	C₂²×C₈	—	53	1	no	—
229	C₂²×C₈	—	106	2	no	—
230	C₂²×C₈	—	212	1	no	—
231	C₂×C₄²	—	53	1	no	—
232	M₃₂	—	2	1	no	—
233	M₃₂	—	106	1	no	—
234	M₃₂	—	212	1	no	—
235	C₂×C₁₆	—	212	1	no	—
236	#108	—	53	1	no	—
237	#108	—	106	1	no	—
238	#108	C₄×D₅₃	1	1	yes	characteristic
239	#104	—	2	1	no	—
240	#104	—	106	1	no	—
241	#104	—	106	1	no	—
242	#104	—	106	1	no	—
243	C₄×C₈	—	2	1	no	—
244	C₄×C₈	—	53	1	no	—
245	C₄×C₈	—	53	1	no	—
246	C₄×C₈	—	53	1	no	—
247	C₄×C₈	—	106	2	no	—
248	C₄×C₈	—	106	1	no	—
249	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
250	#41	—	2	1	no	—
251	#41	—	106	1	no	—
252	#41	—	106	1	no	—
253	#41	—	106	1	no	—
254	#41	—	212	1	no	—
255	#41	—	212	1	no	—
256	C₂×Q₈	—	106	1	no	—
257	C₂×D₄	—	106	1	no	—
258	C₂×D₄	—	212	1	no	—
259	C₂²×C₄	—	53	1	no	—
260	C₂²×C₄	—	106	1	no	—
261	C₂²×C₄	—	212	1	no	—
262	Q₁₆	—	53	1	no	—
263	Q₁₆	—	106	1	no	—
264	Q₁₆	C₈×D₅₃	1	1	yes	characteristic, complemented
265	SD₁₆	—	2	1	no	—
266	SD₁₆	—	106	1	no	—
267	SD₁₆	—	106	1	no	—
268	SD₁₆	—	106	1	no	—
269	D₈	—	53	1	no	—
270	D₈	—	106	1	no	—
271	D₈	C₈×D₅₃	1	1	yes	characteristic, complemented
272	M₁₆	—	2	1	no	—
273	M₁₆	—	4	1	no	—
274	M₁₆	—	106	1	no	—
275	M₁₆	—	212	1	no	—
276	M₁₆	—	212	1	no	—
277	M₁₆	—	212	1	no	—
278	M₁₆	—	212	1	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	53	1	no	—
282	C₂×C₈	—	53	1	no	—
283	C₂×C₈	—	53	1	no	—
284	C₂×C₈	—	53	1	no	—
285	C₂×C₈	—	53	1	no	—
286	C₂×C₈	—	53	1	no	—
287	C₂×C₈	—	53	1	no	—
288	C₂×C₈	—	53	1	no	—
289	C₂×C₈	—	53	1	no	—
290	C₂×C₈	—	53	1	no	—
291	C₂×C₈	—	106	8	no	—
292	C₂×C₈	—	106	2	no	—
293	C₂×C₈	—	212	1	no	—
294	C₂×C₈	—	212	1	no	—
295	C₂×C₈	—	212	1	no	—
296	C₂×C₈	—	212	1	no	—
297	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
298	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
299	C₄²	—	53	1	no	—
300	C₄²	—	106	1	no	—
301	C₄²	D₄×D₅₃	1	1	yes	characteristic
302	C₁₆	—	4	1	no	—
303	C₁₆	—	212	1	no	—
304	C₂³	—	53	1	no	—
305	C₂³	—	212	1	no	—
306	Q₈	—	2	1	no	—
307	Q₈	—	106	1	no	—
308	Q₈	—	212	1	no	—
309	D₄	—	2	1	no	—
310	D₄	—	4	1	no	—
311	D₄	—	106	1	no	—
312	D₄	—	212	1	no	—
313	D₄	—	212	1	no	—
314	D₄	—	212	1	no	—
315	D₄	—	212	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	53	1	no	—
319	C₂×C₄	—	53	1	no	—
320	C₂×C₄	—	53	1	no	—
321	C₂×C₄	—	53	1	no	—
322	C₂×C₄	—	53	1	no	—
323	C₂×C₄	—	106	4	no	—
324	C₂×C₄	—	106	1	no	—
325	C₂×C₄	—	212	1	no	—
326	C₂×C₄	—	212	1	no	—
327	C₂×C₄	—	212	1	no	—
328	C₂×C₄	—	212	1	no	—
329	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
330	C₈	—	2	4	no	—
331	C₈	—	4	1	no	—
332	C₈	—	53	1	no	—
333	C₈	—	53	1	no	—
334	C₈	—	53	1	no	—
335	C₈	—	53	1	no	—
336	C₈	—	106	4	no	—
337	C₈	—	212	1	no	—
338	C₈	C₂×C₈×D₅₃	1	1	yes	characteristic
339	C₈	M₁₆×D₅₃	1	1	yes	characteristic
340	C₈	D₈×D₅₃	1	1	yes	characteristic
341	C₈	SD₁₆×D₅₃	1	1	yes	characteristic
342	C₂²	—	4	1	no	—
343	C₂²	—	53	1	no	—
344	C₂²	—	53	1	no	—
345	C₂²	—	106	2	no	—
346	C₂²	—	424	2	no	—
347	C₂²	—	212	1	no	—
348	C₂²	#104×D₅₃	1	1	yes	characteristic
349	C₄	—	2	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	53	1	no	—
352	C₄	—	53	1	no	—
353	C₄	—	106	2	no	—
354	C₄	—	212	1	no	—
355	C₄	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
356	C₄	#102×D₅₃	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	8	1	no	—
359	C₂	—	53	2	no	—
360	C₂	—	106	1	no	—
361	C₂	—	424	1	no	—
362	C₂	D₄⋊C₈×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₈×D₅₃
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1335	C₂
socle	C₂×C₅₃	C₂×D₄⋊C₈
prime radical	#135×D₅₃	C₄
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#1335	D₅₃
hypocenter	C₅₃	C₂×#1335
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1335

Complements

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

N	H ≅ G/N	type
2. #364×D₅₃	C₂	semidirect
3. C₈²×D₅₃	C₂	semidirect
4. #20272689	C₂	semidirect
5. #20272688	C₂	semidirect
6. #20272687	C₂	semidirect
7. C₅₃×#1335	C₂	semidirect
20. #16810444	C₂²	semidirect
23. C₈×#88322	C₂²	semidirect
25. C₅₃×#364	C₂²	semidirect
26. C₈²×C₅₃	C₂²	semidirect
28. #13133240	C₈	semidirect
32. #13133236	C₈	semidirect
33. Q₁₆×D₅₃	C₈	semidirect
35. D₈×D₅₃	C₈	semidirect
80. Q₁₆×C₅₃	C₂×C₈	semidirect
82. D₈×C₅₃	C₂×C₈	semidirect
102. Dic₂₁₂	C₂×C₈	semidirect
103. D₄₂₄	C₂×C₈	semidirect
182. #1335	D₅₃	direct (N × H)
188. D₅₃	#1335	direct (N × H)
204. #364	C₂×D₅₃	semidirect
207. C₈²	C₂×D₅₃	semidirect
208. C₅₃	C₂×#1335	semidirect
264. Q₁₆	C₈×D₅₃	semidirect
271. D₈	C₈×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#443×D₅₃	2
#364×D₅₃	2
C₈²×D₅₃	2
#20272689	2
#20272688	2
#20272687	2
C₅₃×#1335	2
C₂×#1335	53

Subgroups — D53×#1335 #27293873

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#1335 #27293873