Subgroups — Q₈×Q₁₆×D₅₃ #28264537

Order 13568.

1830 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈²×D₅₃	C₂	1	2	yes	maximal
2	Q₈×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₁₆×Dic₁₀₆	C₂	1	3	yes	maximal, complemented
4	#20287598	C₂	1	3	yes	maximal, complemented
5	Q₈×#11719925	C₂	1	2	yes	maximal, complemented
6	#20287596	C₂	1	6	yes	maximal, complemented
7	#500×D₅₃	C₂	1	3	yes	maximal
8	#477×D₅₃	C₂	1	6	yes	maximal
9	C₈×Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic
10	C₄×Q₁₆×D₅₃	C₂	1	3	yes	maximal
11	Q₈×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₅₃	C₂²	1	1	yes	characteristic
13	Q₈×Dic₁₀₆	—	2	2	no	—
14	Q₈×Dic₁₀₆	C₂²	1	6	yes	—
15	Q₈×Dic₁₀₆	C₂²	1	2	yes	—
16	#16811353	—	2	6	no	—
17	#16811353	C₂²	1	6	yes	—
18	C₄⋊Q₈×D₅₃	—	2	6	no	—
19	C₄⋊Q₈×D₅₃	C₂²	1	6	yes	—
20	C₄×Q₈×D₅₃	—	2	2	no	—
21	C₄×Q₈×D₅₃	C₂²	1	6	yes	—
22	C₄×Q₈×D₅₃	C₂²	1	1	yes	characteristic
23	#16811028	C₂²	1	3	yes	—
24	#16811027	C₂²	1	6	yes	—
25	#16811027	C₂²	1	6	yes	—
26	#16811026	C₂²	1	6	yes	—
27	#16810919	C₂²	1	6	yes	—
28	#16810918	C₂²	1	6	yes	—
29	#16810917	C₂²	1	12	yes	—
30	#16810916	C₂²	1	12	yes	—
31	#16810915	C₂²	1	6	yes	—
32	Q₈×#88322	C₂²	1	1	yes	characteristic
33	C₈×Dic₁₀₆	C₂²	1	3	yes	—
34	#16810747	C₂²	1	3	yes	—
35	C₄×Dic₂₁₂	C₂²	1	3	yes	—
36	Q₁₆×Dic₅₃	C₂²	1	3	yes	—
37	#16810717	C₂²	1	3	yes	—
38	C₄×#11719925	C₂²	1	6	yes	—
39	#16810715	C₂²	1	6	yes	—
40	#107×D₅₃	C₂²	1	3	yes	—
41	#107×D₅₃	C₂²	1	6	yes	—
42	C₄⋊C₈×D₅₃	C₂²	1	3	yes	—
43	#103×D₅₃	C₂²	1	6	yes	—
44	C₄×C₈×D₅₃	C₂²	1	3	yes	—
45	Q₈²×C₅₃	C₂²	1	2	yes	—
46	C₅₃×#500	C₂²	1	3	yes	—
47	C₅₃×#477	C₂²	1	6	yes	—
48	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
49	C₄×Q₁₆×C₅₃	C₂²	1	3	yes	—
50	C₂×Q₈×D₅₃	—	2	6	no	—
51	C₂×Q₈×D₅₃	D₄	1	1	yes	characteristic
52	C₂×Q₈×D₅₃	C₂³	1	2	yes	—
53	C₂×Dic₂₁₂	C₂³	1	1	yes	characteristic
54	C₂×#11719925	C₂³	1	2	yes	—
55	Q₁₆×D₅₃	Q₈	1	8	yes	direct product factor
56	C₂×C₈×D₅₃	C₂³	1	3	yes	—
57	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
58	#13133217	—	2	6	no	—
59	#13133217	C₂³	1	6	yes	—
60	#13133216	—	2	6	no	—
61	#13133216	—	2	6	no	—
62	#13133216	—	2	12	no	—
63	#13133216	—	2	6	no	—
64	#13133216	C₂³	1	12	yes	—
65	#13133215	—	2	12	no	—
66	#13133215	—	2	6	no	—
67	Q₈×Dic₅₃	—	2	2	no	—
68	Q₈×Dic₅₃	D₄	1	1	yes	characteristic
69	Q₈×Dic₅₃	C₂³	1	6	yes	—
70	C₄×Dic₁₀₆	—	2	6	no	—
71	C₄×Dic₁₀₆	—	2	6	no	—
72	C₄×Dic₁₀₆	C₂³	1	3	yes	—
73	C₄×Dic₁₀₆	C₂³	1	6	yes	—
74	#13133182	—	2	6	no	—
75	#13133182	—	2	6	no	—
76	#13133182	D₄	1	3	yes	—
77	#13133182	C₂³	1	6	yes	—
78	C₄⋊C₄×D₅₃	—	2	6	no	—
79	C₄⋊C₄×D₅₃	—	2	6	no	—
80	C₄⋊C₄×D₅₃	D₄	1	3	yes	—
81	C₄⋊C₄×D₅₃	C₂³	1	6	yes	—
82	C₄²×D₅₃	—	2	6	no	—
83	C₄²×D₅₃	C₂³	1	3	yes	—
84	#13133151	C₂³	1	3	yes	—
85	#13133151	C₂³	1	6	yes	—
86	#13133150	C₂³	1	6	yes	—
87	#13133150	C₂³	1	12	yes	—
88	C₄⋊#88322	C₂³	1	3	yes	—
89	#13133146	C₂³	1	6	yes	—
90	#13133142	C₂³	1	6	yes	—
91	#13133141	C₂³	1	6	yes	—
92	#13133140	C₂³	1	6	yes	—
93	C₄×#88322	C₂³	1	3	yes	—
94	C₈×Dic₅₃	C₂³	1	3	yes	—
95	C₈×Dic₅₃	C₂³	1	3	yes	—
96	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
97	C₄⋊Q₈×C₅₃	—	2	6	no	—
98	C₄⋊Q₈×C₅₃	C₂³	1	6	yes	—
99	C₄×Q₈×C₅₃	—	2	2	no	—
100	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
101	C₄×Q₈×C₅₃	C₂³	1	6	yes	—
102	#107×C₅₃	C₂³	1	3	yes	—
103	#107×C₅₃	C₂³	1	6	yes	—
104	C₄⋊C₈×C₅₃	C₂³	1	3	yes	—
105	#103×C₅₃	C₂³	1	6	yes	—
106	C₄×C₈×C₅₃	C₂³	1	3	yes	—
107	C₂×Q₈×C₅₃	—	2	6	no	—
108	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
110	Q₈×D₅₃	—	2	4	no	—
111	Q₈×D₅₃	—	8	12	no	—
112	Q₈×D₅₃	Q₁₆	1	8	yes	direct product factor
113	Q₈×D₅₃	C₂×Q₈	1	8	yes	—
114	C₂×C₄×D₅₃	—	2	2	no	—
115	C₂×C₄×D₅₃	—	2	6	no	—
116	C₂×C₄×D₅₃	C₂×D₄	1	3	yes	—
117	C₂×C₄×D₅₃	C₂×D₄	1	3	yes	—
118	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
119	C₂×Dic₁₀₆	—	2	2	no	—
120	C₂×Dic₁₀₆	—	2	6	no	—
121	C₂×Dic₁₀₆	—	2	12	no	—
122	C₂×Dic₁₀₆	C₂×D₄	1	3	yes	—
123	C₂×Dic₁₀₆	C₂⁴	1	2	yes	—
124	Q₁₆×C₅₃	C₂×Q₈	1	4	yes	complemented
125	C₂×C₈×C₅₃	C₂⁴	1	3	yes	—
126	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
127	C₄⋊C₄×C₅₃	—	2	6	no	—
128	C₄⋊C₄×C₅₃	—	2	6	no	—
129	C₄⋊C₄×C₅₃	C₂×D₄	1	3	yes	—
130	C₄⋊C₄×C₅₃	C₂⁴	1	6	yes	—
131	C₄²×C₅₃	—	2	6	no	—
132	C₄²×C₅₃	C₂⁴	1	3	yes	—
133	#11719925	C₂×Q₈	1	8	yes	complemented
134	C₄⋊Dic₅₃	—	2	6	no	—
135	C₄⋊Dic₅₃	—	2	6	no	—
136	C₄⋊Dic₅₃	C₂×D₄	1	3	yes	—
137	C₄⋊Dic₅₃	C₂⁴	1	6	yes	—
138	#11719919	—	2	6	no	—
139	#11719919	—	2	12	no	—
140	#11719919	—	2	6	no	—
141	#11719919	—	2	6	no	—
142	#11719919	C₂×D₄	1	6	yes	—
143	C₄×Dic₅₃	—	2	6	no	—
144	C₄×Dic₅₃	—	2	6	no	—
145	C₄×Dic₅₃	—	2	6	no	—
146	C₄×Dic₅₃	C₂×D₄	1	3	yes	—
147	C₄×Dic₅₃	C₂×D₄	1	3	yes	—
148	C₄×Dic₅₃	C₂⁴	1	3	yes	—
149	C₂×#88322	C₂⁴	1	3	yes	—
150	C₂×#88322	C₂⁴	1	1	yes	characteristic
151	Dic₂₁₂	C₂×Q₈	1	4	yes	complemented
152	C₈×D₅₃	—	2	6	no	—
153	C₈×D₅₃	C₂×Q₈	1	4	yes	—
154	C₂²×D₅₃	C₂²×D₄	1	1	yes	characteristic
155	Q₈×C₅₃	—	2	2	no	—
156	Q₈×C₅₃	—	8	6	no	—
157	Q₈×C₅₃	C₂×Q₁₆	1	4	yes	complemented
158	Q₈×C₅₃	C₂²×Q₈	1	4	yes	—
159	C₂×C₄×C₅₃	—	2	2	no	—
160	C₂×C₄×C₅₃	—	2	6	no	—
161	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
162	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
163	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
164	C₂×Dic₅₃	—	2	2	no	—
165	C₂×Dic₅₃	—	2	6	no	—
166	C₂×Dic₅₃	C₂²×D₄	1	3	yes	—
167	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
168	C₂×Dic₅₃	C₂²×D₄	1	3	yes	—
169	C₄×D₅₃	—	2	8	no	—
170	C₄×D₅₃	—	2	6	no	—
171	C₄×D₅₃	—	4	12	no	—
172	C₄×D₅₃	C₂×Q₁₆	1	12	yes	—
173	C₄×D₅₃	C₂²×Q₈	1	2	yes	—
174	C₄×D₅₃	2₋¹⁺⁴	1	2	yes	—
175	Dic₁₀₆	—	2	2	no	—
176	Dic₁₀₆	—	2	8	no	—
177	Dic₁₀₆	—	8	12	no	—
178	Dic₁₀₆	—	8	6	no	—
179	Dic₁₀₆	C₂×Q₁₆	1	12	yes	complemented
180	Dic₁₀₆	C₂²×Q₈	1	4	yes	—
181	C₈×C₅₃	—	2	3	no	—
182	C₈×C₅₃	C₂²×Q₈	1	2	yes	—
183	#88322	—	2	3	no	—
184	#88322	C₂²×Q₈	1	2	yes	—
185	C₂×Q₈×Q₁₆	—	53	1	no	maximal, Sylow
186	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
187	C₂×D₅₃	D₄×Q₈	1	2	yes	—
188	C₂×D₅₃	C₂²×Q₁₆	1	2	yes	—
189	C₂×D₅₃	#576	1	2	yes	—
190	C₄×C₅₃	—	2	4	no	—
191	C₄×C₅₃	—	2	3	no	—
192	C₄×C₅₃	—	4	6	no	—
193	C₄×C₅₃	C₂²×Q₁₆	1	6	yes	—
194	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
195	C₄×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
196	Dic₅₃	—	2	4	no	—
197	Dic₅₃	—	2	3	no	—
198	Dic₅₃	—	4	6	no	—
199	Dic₅₃	D₄×Q₈	1	2	yes	—
200	Dic₅₃	C₂²×Q₁₆	1	6	yes	—
201	C₂×Q₈²	—	53	2	no	—
202	Q₈×Q₁₆	—	53	3	no	—
203	Q₈×Q₁₆	—	53	6	no	—
204	Q₈×Q₁₆	—	53	1	no	—
205	Q₈×Q₁₆	—	53	3	no	—
206	Q₈×Q₁₆	—	53	2	no	—
207	Q₈×Q₁₆	D₅₃	1	1	yes	characteristic, direct product factor
208	C₂×#500	—	53	3	no	—
209	C₂×#477	—	53	6	no	—
210	C₂×C₈×Q₈	—	53	1	no	—
211	C₂×C₄×Q₁₆	—	53	3	no	—
212	C₂×C₅₃	C₂×D₄×Q₈	1	1	yes	characteristic
213	C₂×C₅₃	C₂³×Q₁₆	1	1	yes	characteristic
214	C₂×C₅₃	C₂×#576	1	1	yes	characteristic
215	D₅₃	Q₈×Q₁₆	1	4	yes	direct product factor
216	C₂²×Q₁₆	—	53	1	no	—
217	Q₈²	—	53	6	no	—
218	Q₈²	—	53	6	no	—
219	Q₈²	—	53	2	no	—
220	Q₈²	—	106	2	no	—
221	Q₈²	—	106	6	no	—
222	Q₈²	C₂×D₅₃	1	2	yes	—
223	C₂×C₄⋊Q₈	—	53	6	no	—
224	C₂×C₄⋊Q₈	—	106	6	no	—
225	C₂×C₄×Q₈	—	53	1	no	—
226	C₂×C₄×Q₈	—	53	6	no	—
227	C₂×C₄×Q₈	—	106	2	no	—
228	#500	—	53	6	no	—
229	#500	—	53	6	no	—
230	#500	—	53	6	no	—
231	#500	—	53	3	no	—
232	#500	C₂×D₅₃	1	3	yes	—
233	#477	—	53	6	no	—
234	#477	—	53	6	no	—
235	#477	—	53	12	no	—
236	#477	—	53	6	no	—
237	#477	—	53	12	no	—
238	#477	C₂×D₅₃	1	6	yes	—
239	C₈×Q₈	—	53	1	no	—
240	C₈×Q₈	—	53	3	no	—
241	C₈×Q₈	—	53	3	no	—
242	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
243	C₄×Q₁₆	—	53	3	no	—
244	C₄×Q₁₆	—	53	6	no	—
245	C₄×Q₁₆	—	53	3	no	—
246	C₄×Q₁₆	—	53	6	no	—
247	C₄×Q₁₆	—	53	3	no	—
248	C₄×Q₁₆	C₂×D₅₃	1	3	yes	—
249	C₂×#107	—	53	6	no	—
250	C₂×#107	—	53	3	no	—
251	C₂×C₄⋊C₈	—	53	3	no	—
252	C₂×#103	—	53	6	no	—
253	C₂×C₄×C₈	—	53	3	no	—
254	C₅₃	C₂×Q₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
255	C₂²×Q₈	—	53	1	no	—
256	C₂²×Q₈	—	53	2	no	—
257	C₂²×Q₈	—	106	6	no	—
258	C₂×Q₁₆	—	53	2	no	—
259	C₂×Q₁₆	—	53	8	no	—
260	C₂×Q₁₆	—	53	1	no	—
261	C₂×Q₁₆	C₂²×D₅₃	1	1	yes	characteristic
262	C₂²×C₈	—	53	1	no	—
263	C₂²×C₈	—	53	3	no	—
264	C₄⋊Q₈	—	2	6	no	—
265	C₄⋊Q₈	—	53	12	no	—
266	C₄⋊Q₈	—	53	6	no	—
267	C₄⋊Q₈	—	106	6	no	—
268	C₄⋊Q₈	—	106	6	no	—
269	C₄⋊Q₈	—	106	6	no	—
270	C₄⋊Q₈	—	106	12	no	—
271	C₄⋊Q₈	—	106	6	no	—
272	C₄⋊Q₈	—	106	12	no	—
273	C₄⋊Q₈	—	106	6	no	—
274	C₄⋊Q₈	C₂²×D₅₃	1	6	yes	—
275	C₄×Q₈	—	2	2	no	—
276	C₄×Q₈	—	53	1	no	—
277	C₄×Q₈	—	53	3	no	—
278	C₄×Q₈	—	53	3	no	—
279	C₄×Q₈	—	53	6	no	—
280	C₄×Q₈	—	53	6	no	—
281	C₄×Q₈	—	53	6	no	—
282	C₄×Q₈	—	106	6	no	—
283	C₄×Q₈	—	106	6	no	—
284	C₄×Q₈	—	106	6	no	—
285	C₄×Q₈	—	106	6	no	—
286	C₄×Q₈	—	106	2	no	—
287	C₄×Q₈	C₂²×D₅₃	1	6	yes	—
288	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
289	C₂×C₄⋊C₄	—	53	3	no	—
290	C₂×C₄⋊C₄	—	53	6	no	—
291	C₂×C₄⋊C₄	—	106	6	no	—
292	C₂×C₄⋊C₄	—	106	6	no	—
293	C₂×C₄²	—	53	3	no	—
294	C₂×C₄²	—	106	6	no	—
295	#107	—	53	3	no	—
296	#107	—	53	6	no	—
297	#107	—	53	6	no	—
298	#107	—	53	12	no	—
299	#107	C₂²×D₅₃	1	3	yes	—
300	#107	C₂²×D₅₃	1	6	yes	—
301	C₄⋊C₈	—	53	6	no	—
302	C₄⋊C₈	—	53	3	no	—
303	C₄⋊C₈	C₂²×D₅₃	1	3	yes	—
304	#103	—	53	6	no	—
305	#103	—	53	6	no	—
306	#103	—	53	6	no	—
307	#103	C₂²×D₅₃	1	6	yes	—
308	C₄×C₈	—	53	3	no	—
309	C₄×C₈	—	53	3	no	—
310	C₄×C₈	—	53	3	no	—
311	C₄×C₈	C₂²×D₅₃	1	3	yes	—
312	C₂×Q₈	—	2	6	no	—
313	C₂×Q₈	—	53	8	no	—
314	C₂×Q₈	—	53	3	no	—
315	C₂×Q₈	—	53	8	no	—
316	C₂×Q₈	—	53	2	no	—
317	C₂×Q₈	—	106	2	no	—
318	C₂×Q₈	—	106	12	no	—
319	C₂×Q₈	—	106	4	no	—
320	C₂×Q₈	—	106	6	no	—
321	C₂×Q₈	—	424	12	no	—
322	C₂×Q₈	D₄×D₅₃	1	1	yes	characteristic
323	C₂×Q₈	C₂³×D₅₃	1	2	yes	—
324	C₂²×C₄	—	53	3	no	—
325	C₂²×C₄	—	53	1	no	—
326	C₂²×C₄	—	53	3	no	—
327	C₂²×C₄	—	106	2	no	—
328	C₂²×C₄	—	106	6	no	—
329	Q₁₆	—	53	8	no	—
330	Q₁₆	—	53	4	no	—
331	Q₁₆	Q₈×D₅₃	1	4	yes	direct product factor
332	C₂×C₈	—	53	1	no	—
333	C₂×C₈	—	53	4	no	—
334	C₂×C₈	—	53	3	no	—
335	C₂×C₈	—	106	6	no	—
336	C₂×C₈	C₂³×D₅₃	1	3	yes	—
337	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
338	C₄⋊C₄	—	2	6	no	—
339	C₄⋊C₄	—	2	6	no	—
340	C₄⋊C₄	—	53	6	no	—
341	C₄⋊C₄	—	53	3	no	—
342	C₄⋊C₄	—	53	6	no	—
343	C₄⋊C₄	—	106	6	no	—
344	C₄⋊C₄	—	106	6	no	—
345	C₄⋊C₄	—	106	6	no	—
346	C₄⋊C₄	—	106	12	no	—
347	C₄⋊C₄	—	106	6	no	—
348	C₄⋊C₄	—	106	6	no	—
349	C₄⋊C₄	D₄×D₅₃	1	3	yes	—
350	C₄⋊C₄	C₂³×D₅₃	1	6	yes	—
351	C₄²	—	2	6	no	—
352	C₄²	—	53	3	no	—
353	C₄²	—	53	3	no	—
354	C₄²	—	53	3	no	—
355	C₄²	—	106	6	no	—
356	C₄²	—	106	6	no	—
357	C₄²	—	106	6	no	—
358	C₄²	C₂³×D₅₃	1	3	yes	—
359	C₂³	—	53	1	no	—
360	Q₈	—	2	2	no	—
361	Q₈	—	8	6	no	—
362	Q₈	—	53	12	no	—
363	Q₈	—	53	4	no	—
364	Q₈	—	106	8	no	—
365	Q₈	—	106	2	no	—
366	Q₈	—	424	6	no	—
367	Q₈	—	424	12	no	—
368	Q₈	Q₁₆×D₅₃	1	4	yes	direct product factor
369	Q₈	C₂×Q₈×D₅₃	1	4	yes	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	6	no	—
372	C₂×C₄	—	53	1	no	—
373	C₂×C₄	—	53	3	no	—
374	C₂×C₄	—	53	12	no	—
375	C₂×C₄	—	53	2	no	—
376	C₂×C₄	—	53	2	no	—
377	C₂×C₄	—	53	3	no	—
378	C₂×C₄	—	106	6	no	—
379	C₂×C₄	—	106	8	no	—
380	C₂×C₄	—	106	2	no	—
381	C₂×C₄	—	106	6	no	—
382	C₂×C₄	—	212	12	no	—
383	C₂×C₄	C₂×D₄×D₅₃	1	3	yes	—
384	C₂×C₄	C₂×D₄×D₅₃	1	3	yes	—
385	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
386	C₈	—	2	3	no	—
387	C₈	—	53	2	no	—
388	C₈	—	106	3	no	—
389	C₈	C₂×Q₈×D₅₃	1	2	yes	—
390	C₂²	—	53	2	no	—
391	C₂²	—	53	2	no	—
392	C₂²	—	53	2	no	—
393	C₂²	C₂²×D₄×D₅₃	1	1	yes	characteristic
394	C₄	—	2	3	no	—
395	C₄	—	2	4	no	—
396	C₄	—	4	6	no	—
397	C₄	—	53	6	no	—
398	C₄	—	53	2	no	—
399	C₄	—	106	3	no	—
400	C₄	—	106	4	no	—
401	C₄	—	212	6	no	—
402	C₄	C₂×Q₁₆×D₅₃	1	6	yes	—
403	C₄	C₂²×Q₈×D₅₃	1	1	yes	characteristic
404	C₄	2₋¹⁺⁴×D₅₃	1	1	yes	characteristic
405	C₂	—	53	4	no	—
406	C₂	D₄×Q₈×D₅₃	1	1	yes	characteristic
407	C₂	C₂²×Q₁₆×D₅₃	1	1	yes	characteristic
408	C₂	#576×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄×D₅₃
commutator, cosocle	C₂×C₄×C₅₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	Q₈×Q₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂³×D₄
prime radical	C₂²×D₅₃	C₂²×D₄
hypercenter	Q₈×Q₁₆	D₅₃
hypocenter	C₅₃	C₂×Q₈×Q₁₆
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈×Q₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×Dic₂₁₂	C₂	semidirect
3. Q₁₆×Dic₁₀₆	C₂	semidirect
4. #20287598	C₂	semidirect
5. Q₈×#11719925	C₂	semidirect
6. #20287596	C₂	semidirect
11. Q₈×Q₁₆×C₅₃	C₂	semidirect
55. Q₁₆×D₅₃	Q₈	direct (N × H)
112. Q₈×D₅₃	Q₁₆	direct (N × H)
124. Q₁₆×C₅₃	C₂×Q₈	semidirect
133. #11719925	C₂×Q₈	semidirect
151. Dic₂₁₂	C₂×Q₈	semidirect
157. Q₈×C₅₃	C₂×Q₁₆	semidirect
179. Dic₁₀₆	C₂×Q₁₆	semidirect
207. Q₈×Q₁₆	D₅₃	direct (N × H)
215. D₅₃	Q₈×Q₁₆	direct (N × H)
254. C₅₃	C₂×Q₈×Q₁₆	semidirect
331. Q₁₆	Q₈×D₅₃	direct (N × H)
368. Q₈	Q₁₆×D₅₃	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈²×D₅₃	2
Q₈×Dic₂₁₂	2
Q₁₆×Dic₁₀₆	2
#20287598	2
Q₈×#11719925	2
#20287596	2
#500×D₅₃	2
#477×D₅₃	2
C₈×Q₈×D₅₃	2
C₄×Q₁₆×D₅₃	2
Q₈×Q₁₆×C₅₃	2
C₂×Q₈×Q₁₆	53

Subgroups — Q8×Q16×D53 #28264537

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×Q₁₆×D₅₃ #28264537