Subgroups — Q₁₆×D₅₃⋊C₈ #28316355

Order 13568.

734 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₅₃⋊C₈	C₂	1	2	yes	maximal
2	#20290303	C₂	1	2	yes	maximal, complemented
3	#20290302	C₂	1	4	yes	maximal, complemented
4	Q₁₆×#88324	C₂	1	2	yes	maximal, complemented
5	D₅₃⋊#335	C₂	1	1	yes	maximal, characteristic
6	D₅₃⋊(Q₈⋊C₈)	C₂	1	2	yes	maximal
7	C₈×D₅₃⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₄×Q₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
9	#16811672	—	2	4	no	—
10	#16811672	C₂²	1	4	yes	—
11	Q₈×#88324	C₂²	1	4	yes	—
12	D₅₃⋊(C₄⋊C₈)	—	2	2	no	—
13	D₅₃⋊(C₄⋊C₈)	C₂²	1	2	yes	—
14	C₄×D₅₃⋊C₈	—	2	2	no	—
15	C₄×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
17	#16811487	C₂²	1	2	yes	—
18	#16811486	C₂²	1	2	yes	—
19	#16811473	C₂²	1	4	yes	—
20	Q₈⋊#88324	C₂²	1	4	yes	—
21	C₈×#88324	C₂²	1	2	yes	—
22	C₈×#88324	C₂²	1	2	yes	—
23	C₂×Q₁₆×D₅₃	C₄	1	1	yes	characteristic
24	C₄×Q₈×D₅₃	C₂²	1	2	yes	—
25	C₄×Dic₂₁₂	C₄	1	1	yes	characteristic
26	Q₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
27	#16810717	C₂²	1	1	yes	characteristic
28	C₄×#11719925	C₄	1	2	yes	—
29	#16810715	C₂²	1	2	yes	—
30	#107×D₅₃	C₄	1	1	yes	characteristic
31	#103×D₅₃	C₄	1	2	yes	—
32	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
33	C₄×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
34	C₂×D₅₃⋊C₈	—	2	2	no	—
35	C₂×D₅₃⋊C₈	—	2	1	no	—
36	C₂×D₅₃⋊C₈	D₄	1	1	yes	characteristic
37	C₂×D₅₃⋊C₈	D₄	1	1	yes	characteristic
38	#13133284	—	2	2	no	—
39	#13133284	—	2	4	no	—
40	C₄⋊#88324	—	2	4	no	—
41	C₄⋊#88324	C₂³	1	4	yes	—
42	C₄×#88324	—	2	1	no	—
43	C₄×#88324	—	2	4	no	—
44	C₄×#88324	—	2	4	no	—
45	C₄×#88324	D₄	1	2	yes	—
46	C₄×#88324	C₂³	1	2	yes	—
47	C₄×#88324	C₂³	1	2	yes	—
48	C₂×Q₈×D₅₃	C₂×C₄	1	2	yes	—
49	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
50	C₂×#11719925	C₂×C₄	1	2	yes	—
51	Q₁₆×D₅₃	C₈	1	8	yes	complemented
52	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
54	Q₈×Dic₅₃	C₂³	1	2	yes	—
55	C₄×Dic₁₀₆	—	2	2	no	—
56	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
57	#13133182	—	2	2	no	—
58	#13133182	C₂³	1	2	yes	—
59	C₄⋊C₄×D₅₃	—	2	2	no	—
60	C₄⋊C₄×D₅₃	C₂×C₄	1	2	yes	—
61	C₄²×D₅₃	—	2	2	no	—
62	C₄²×D₅₃	C₂³	1	1	yes	characteristic
63	#13133151	C₂×C₄	1	1	yes	characteristic
64	#13133150	C₂×C₄	1	2	yes	—
65	#13133142	C₂×C₄	1	2	yes	—
66	#13133141	C₂×C₄	1	2	yes	—
67	#13133140	C₂×C₄	1	2	yes	—
68	C₄×#88322	C₂×C₄	1	1	yes	characteristic
69	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
70	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
71	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
73	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
74	#103×C₅₃	C₂×C₄	1	2	yes	—
75	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
76	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
77	Q₈×D₅₃	—	2	4	no	—
78	Q₈×D₅₃	C₂×C₈	1	8	yes	—
79	C₂×C₄×D₅₃	—	2	2	no	—
80	C₂×C₄×D₅₃	—	2	2	no	—
81	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
84	C₂×Dic₁₀₆	—	2	2	no	—
85	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
86	C₂×#88324	—	2	2	no	—
87	C₂×#88324	—	2	4	no	—
88	C₂×#88324	C₂×D₄	1	2	yes	—
89	C₂×#88324	C₂×D₄	1	2	yes	—
90	D₅₃⋊C₈	—	2	4	no	—
91	D₅₃⋊C₈	—	2	2	no	—
92	D₅₃⋊C₈	—	4	4	no	—
93	D₅₃⋊C₈	Q₁₆	1	4	yes	direct product factor
94	Q₁₆×C₅₃	C₂×C₈	1	4	yes	complemented
95	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₄⋊C₄×C₅₃	—	2	2	no	—
98	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
99	C₄²×C₅₃	—	2	2	no	—
100	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	#11719925	C₂×C₈	1	8	yes	complemented
102	C₄⋊Dic₅₃	—	2	2	no	—
103	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
104	#11719919	—	2	2	no	—
105	#11719919	—	2	2	no	—
106	#11719919	—	2	2	no	—
107	C₄×Dic₅₃	—	2	2	no	—
108	C₄×Dic₅₃	—	2	2	no	—
109	C₄×Dic₅₃	—	2	2	no	—
110	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
111	C₄×Dic₅₃	#41	1	1	yes	characteristic
112	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
113	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
114	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
115	Dic₂₁₂	C₂×C₈	1	4	yes	complemented
116	C₈×D₅₃	—	2	2	no	—
117	C₈×D₅₃	C₂×C₈	1	4	yes	—
118	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
119	Q₈×C₅₃	—	2	2	no	—
120	Q₈×C₅₃	C₂²×C₈	1	4	yes	—
121	C₂×C₄×C₅₃	—	2	2	no	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×Dic₅₃	—	2	2	no	—
127	C₂×Dic₅₃	—	2	2	no	—
128	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
129	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×Dic₅₃	C₂×#41	1	1	yes	characteristic
131	C₄×D₅₃	—	2	8	no	—
132	C₄×D₅₃	—	2	2	no	—
133	C₄×D₅₃	—	4	4	no	—
134	C₄×D₅₃	C₂²×C₈	1	2	yes	—
135	C₄×D₅₃	#131	1	2	yes	—
136	C₄×D₅₃	C₂×Q₁₆	1	2	yes	—
137	C₄×D₅₃	#135	1	2	yes	—
138	Dic₁₀₆	—	2	2	no	—
139	Dic₁₀₆	—	2	8	no	—
140	Dic₁₀₆	C₂²×C₈	1	4	yes	—
141	#88324	—	2	4	no	—
142	#88324	—	2	2	no	—
143	#88324	—	4	4	no	—
144	#88324	C₂×Q₁₆	1	4	yes	complemented
145	C₈×C₅₃	—	2	1	no	—
146	C₈×C₅₃	C₂²×C₈	1	2	yes	—
147	#88322	—	2	1	no	—
148	#88322	C₂²×C₈	1	2	yes	—
149	C₂×C₈×Q₁₆	—	53	1	no	maximal, Sylow
150	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
151	C₂×D₅₃	C₈×D₄	1	2	yes	—
152	C₂×D₅₃	C₄×Q₁₆	1	2	yes	—
153	C₂×D₅₃	#443	1	2	yes	—
154	C₄×C₅₃	—	2	4	no	—
155	C₄×C₅₃	—	2	1	no	—
156	C₄×C₅₃	—	4	2	no	—
157	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
158	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
159	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
160	Dic₅₃	—	2	4	no	—
161	Dic₅₃	—	2	1	no	—
162	Dic₅₃	—	4	2	no	—
163	Dic₅₃	C₈×D₄	1	2	yes	—
164	Dic₅₃	C₂²×Q₁₆	1	1	yes	characteristic
165	Dic₅₃	C₂×#135	1	1	yes	characteristic
166	C₂×C₈×Q₈	—	53	2	no	—
167	C₂×C₄×Q₁₆	—	53	1	no	—
168	C₈×Q₁₆	—	53	2	no	—
169	C₈×Q₁₆	—	53	4	no	—
170	C₈×Q₁₆	—	53	2	no	—
171	C₂×#335	—	53	1	no	—
172	C₂×Q₈⋊C₈	—	53	2	no	—
173	C₂×C₈²	—	53	1	no	—
174	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
175	C₂×C₅₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
176	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
177	D₅₃	C₈×Q₁₆	1	4	yes	complemented
178	C₂²×Q₁₆	—	53	1	no	—
179	C₂×C₄×Q₈	—	53	2	no	—
180	C₈×Q₈	—	53	4	no	—
181	C₈×Q₈	—	53	4	no	—
182	C₈×Q₈	—	106	4	no	—
183	C₄×Q₁₆	—	53	1	no	—
184	C₄×Q₁₆	—	53	2	no	—
185	C₄×Q₁₆	—	53	1	no	—
186	C₄×Q₁₆	—	53	2	no	—
187	C₄×Q₁₆	—	53	1	no	—
188	C₄×Q₁₆	#6157	1	1	yes	characteristic
189	C₂×#107	—	53	1	no	—
190	C₂×C₄⋊C₈	—	53	2	no	—
191	C₂×C₄⋊C₈	—	106	2	no	—
192	C₂×#103	—	53	2	no	—
193	C₂×C₄×C₈	—	53	1	no	—
194	C₂×C₄×C₈	—	53	1	no	—
195	C₂×C₄×C₈	—	53	1	no	—
196	C₂×C₄×C₈	—	106	2	no	—
197	#335	—	53	2	no	—
198	#335	—	53	2	no	—
199	Q₈⋊C₈	—	53	4	no	—
200	Q₈⋊C₈	—	53	4	no	—
201	C₈²	—	53	2	no	—
202	C₈²	—	53	2	no	—
203	C₅₃	C₂×C₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
204	C₂²×Q₈	—	53	2	no	—
205	C₂×Q₁₆	—	53	2	no	—
206	C₂×Q₁₆	—	53	8	no	—
207	C₂×Q₁₆	—	53	1	no	—
208	C₂×Q₁₆	C₂×#6157	1	1	yes	characteristic
209	C₂²×C₈	—	53	1	no	—
210	C₂²×C₈	—	53	1	no	—
211	C₂²×C₈	—	53	1	no	—
212	C₂²×C₈	—	53	1	no	—
213	C₂²×C₈	—	106	1	no	—
214	C₂²×C₈	—	106	2	no	—
215	C₄×Q₈	—	53	2	no	—
216	C₄×Q₈	—	53	2	no	—
217	C₄×Q₈	—	53	2	no	—
218	C₄×Q₈	—	106	2	no	—
219	C₄×Q₈	—	106	2	no	—
220	C₄×Q₈	C₂×#6157	1	2	yes	—
221	C₂×C₄⋊C₄	—	53	2	no	—
222	C₂×C₄⋊C₄	—	106	2	no	—
223	C₂×C₄²	—	53	1	no	—
224	C₂×C₄²	—	106	2	no	—
225	#107	—	53	1	no	—
226	#107	—	53	2	no	—
227	#107	C₂×#6157	1	1	yes	characteristic
228	C₄⋊C₈	—	53	4	no	—
229	C₄⋊C₈	—	106	2	no	—
230	C₄⋊C₈	—	106	4	no	—
231	C₄⋊C₈	—	106	4	no	—
232	#103	—	53	2	no	—
233	#103	—	53	2	no	—
234	#103	—	53	2	no	—
235	#103	C₂×#6157	1	2	yes	—
236	C₄×C₈	—	53	2	no	—
237	C₄×C₈	—	53	2	no	—
238	C₄×C₈	—	53	1	no	—
239	C₄×C₈	—	53	1	no	—
240	C₄×C₈	—	53	2	no	—
241	C₄×C₈	—	53	1	no	—
242	C₄×C₈	—	106	1	no	—
243	C₄×C₈	—	106	4	no	—
244	C₄×C₈	—	106	4	no	—
245	C₄×C₈	C₂×#6157	1	1	yes	characteristic
246	C₂×Q₈	—	53	8	no	—
247	C₂×Q₈	—	53	2	no	—
248	C₂×Q₈	—	106	2	no	—
249	C₂×Q₈	—	106	4	no	—
250	C₂×Q₈	C₂²×#6157	1	2	yes	—
251	C₂²×C₄	—	53	1	no	—
252	C₂²×C₄	—	53	1	no	—
253	C₂²×C₄	—	53	1	no	—
254	C₂²×C₄	—	106	2	no	—
255	C₂²×C₄	—	106	2	no	—
256	Q₁₆	—	53	8	no	—
257	Q₁₆	—	53	4	no	—
258	Q₁₆	D₅₃⋊C₈	1	4	yes	direct product factor
259	C₂×C₈	—	53	2	no	—
260	C₂×C₈	—	53	4	no	—
261	C₂×C₈	—	53	1	no	—
262	C₂×C₈	—	53	4	no	—
263	C₂×C₈	—	53	2	no	—
264	C₂×C₈	—	53	1	no	—
265	C₂×C₈	—	106	2	no	—
266	C₂×C₈	—	106	2	no	—
267	C₂×C₈	—	106	2	no	—
268	C₂×C₈	—	106	4	no	—
269	C₂×C₈	—	106	4	no	—
270	C₂×C₈	—	212	4	no	—
271	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
272	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	53	2	no	—
275	C₄⋊C₄	—	106	2	no	—
276	C₄⋊C₄	—	106	2	no	—
277	C₄⋊C₄	—	106	2	no	—
278	C₄⋊C₄	—	106	2	no	—
279	C₄⋊C₄	C₂²×#6157	1	2	yes	—
280	C₄²	—	2	2	no	—
281	C₄²	—	53	1	no	—
282	C₄²	—	53	1	no	—
283	C₄²	—	53	1	no	—
284	C₄²	—	106	2	no	—
285	C₄²	—	106	2	no	—
286	C₄²	—	106	2	no	—
287	C₄²	C₂²×#6157	1	1	yes	characteristic
288	C₂³	—	53	1	no	—
289	Q₈	—	2	2	no	—
290	Q₈	—	53	4	no	—
291	Q₈	—	106	8	no	—
292	Q₈	—	106	2	no	—
293	Q₈	C₂×D₅₃⋊C₈	1	4	yes	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	53	1	no	—
297	C₂×C₄	—	53	1	no	—
298	C₂×C₄	—	53	2	no	—
299	C₂×C₄	—	53	2	no	—
300	C₂×C₄	—	53	2	no	—
301	C₂×C₄	—	53	2	no	—
302	C₂×C₄	—	53	1	no	—
303	C₂×C₄	—	106	2	no	—
304	C₂×C₄	—	106	8	no	—
305	C₂×C₄	—	106	2	no	—
306	C₂×C₄	—	106	2	no	—
307	C₂×C₄	—	212	4	no	—
308	C₂×C₄	D₄×#6157	1	1	yes	characteristic
309	C₂×C₄	D₄×#6157	1	1	yes	characteristic
310	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	53	4	no	—
313	C₈	—	53	2	no	—
314	C₈	—	106	2	no	—
315	C₈	—	106	4	no	—
316	C₈	—	106	1	no	—
317	C₈	—	212	4	no	—
318	C₈	C₂×D₅₃⋊C₈	1	2	yes	—
319	C₂²	—	53	2	no	—
320	C₂²	—	53	2	no	—
321	C₂²	—	53	2	no	—
322	C₂²	C₂×D₄×#6157	1	1	yes	characteristic
323	C₄	—	2	4	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	2	no	—
326	C₄	—	53	2	no	—
327	C₄	—	53	1	no	—
328	C₄	—	53	1	no	—
329	C₄	—	106	4	no	—
330	C₄	—	106	1	no	—
331	C₄	—	212	2	no	—
332	C₄	Q₁₆×#6157	1	2	yes	—
333	C₄	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
334	C₄	D₅₃⋊#131	1	1	yes	characteristic
335	C₂	—	53	4	no	—
336	C₂	D₄×D₅₃⋊C₈	1	1	yes	characteristic
337	C₂	C₂×Q₁₆×#6157	1	1	yes	characteristic
338	C₂	D₅₃⋊#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6157
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₄×Q₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂²×D₅₃	C₄×D₄
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	C₄×Q₁₆	#6157
hypocenter	C₅₃	C₂×C₈×Q₁₆
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #20290303	C₂	semidirect
3. #20290302	C₂	semidirect
4. Q₁₆×#88324	C₂	semidirect
51. Q₁₆×D₅₃	C₈	semidirect
93. D₅₃⋊C₈	Q₁₆	direct (N × H)
94. Q₁₆×C₅₃	C₂×C₈	semidirect
101. #11719925	C₂×C₈	semidirect
115. Dic₂₁₂	C₂×C₈	semidirect
144. #88324	C₂×Q₁₆	semidirect
177. D₅₃	C₈×Q₁₆	semidirect
203. C₅₃	C₂×C₈×Q₁₆	semidirect
258. Q₁₆	D₅₃⋊C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×D₅₃⋊C₈	2
#20290303	2
#20290302	2
Q₁₆×#88324	2
D₅₃⋊#335	2
D₅₃⋊(Q₈⋊C₈)	2
C₈×D₅₃⋊C₈	2
C₄×Q₁₆×D₅₃	2
C₂×C₈×Q₁₆	53

Subgroups — Q16×D53⋊C8 #28316355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₁₆×D₅₃⋊C₈ #28316355