Subgroups — Q₁₆×#11719936 #28317517

Order 13568.

638 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11719936	C₂	1	2	yes	maximal
2	#20290354	C₂	1	2	yes	maximal, complemented
3	#20290326	C₂	1	4	yes	maximal, complemented
4	Q₁₆×#88324	C₂	1	2	yes	maximal, complemented
5	#20290273	C₂	1	1	yes	maximal, characteristic
6	#20290044	C₂	1	2	yes	maximal
7	C₈×#11719936	C₂	1	1	yes	maximal, characteristic
8	C₄×Q₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
9	#16811673	—	2	4	no	—
10	#16811673	C₂²	1	4	yes	—
11	Q₈×#88324	C₂²	1	4	yes	—
12	#16811642	C₂²	1	1	yes	characteristic
13	#16811627	—	2	2	no	—
14	#16811627	C₂²	1	2	yes	—
15	C₄×#11719936	—	2	2	no	—
16	C₄×#11719936	C₂²	1	1	yes	characteristic
17	#16811486	C₂²	1	2	yes	—
18	#16811486	C₂²	1	2	yes	—
19	#16811473	C₂²	1	4	yes	—
20	Q₈⋊#88324	C₂²	1	4	yes	—
21	#88324⋊C₈	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×#11719936	—	2	2	no	—
35	C₂×#11719936	D₄	1	1	yes	characteristic
36	C₂×#11719936	D₄	1	1	yes	characteristic
37	C₄⋊#88324	—	2	2	no	—
38	C₄⋊#88324	—	2	1	no	—
39	C₄⋊#88324	—	2	4	no	—
40	C₄⋊#88324	—	2	4	no	—
41	C₄⋊#88324	C₂³	1	4	yes	—
42	#13133274	—	2	1	no	—
43	#13133272	—	2	4	no	—
44	#13133272	D₄	1	2	yes	—
45	C₄×#88324	—	2	4	no	—
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	—	2	4	no	—
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	—	4	2	no	—
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	4	no	—
87	C₂×#88324	—	2	2	no	—
88	C₂×#88324	C₂×D₄	1	2	yes	—
89	C₂×#88324	C₂×D₄	1	2	yes	—
90	#11719936	—	2	2	no	—
91	#11719936	—	4	4	no	—
92	#11719936	Q₁₆	1	4	yes	direct product factor
93	Q₁₆×C₅₃	M₁₆	1	4	yes	complemented
94	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₄⋊C₄×C₅₃	—	2	2	no	—
97	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
98	C₄²×C₅₃	—	2	2	no	—
99	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
100	#11719925	—	2	4	no	—
101	C₄⋊Dic₅₃	—	2	2	no	—
102	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
103	#11719919	—	2	2	no	—
104	#11719919	—	2	2	no	—
105	#11719919	—	2	2	no	—
106	C₄×Dic₅₃	—	2	2	no	—
107	C₄×Dic₅₃	—	2	2	no	—
108	C₄×Dic₅₃	—	2	2	no	—
109	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
110	C₄×Dic₅₃	#41	1	1	yes	characteristic
111	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
112	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
113	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
114	Dic₂₁₂	—	2	2	no	—
115	C₈×D₅₃	—	2	2	no	—
116	C₈×D₅₃	—	4	1	no	—
117	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
118	Q₈×C₅₃	—	2	2	no	—
119	Q₈×C₅₃	C₂×M₁₆	1	4	yes	—
120	C₂×C₄×C₅₃	—	2	2	no	—
121	C₂×C₄×C₅₃	—	2	2	no	—
122	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×Dic₅₃	—	2	2	no	—
126	C₂×Dic₅₃	—	2	2	no	—
127	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
128	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×Dic₅₃	C₂×#41	1	1	yes	characteristic
130	C₄×D₅₃	—	2	1	no	—
131	C₄×D₅₃	—	2	1	no	—
132	C₄×D₅₃	—	4	1	no	—
133	C₄×D₅₃	—	4	4	no	—
134	C₄×D₅₃	—	8	2	no	—
135	C₄×D₅₃	C₂×Q₁₆	1	2	yes	—
136	C₄×D₅₃	#135	1	2	yes	—
137	Dic₁₀₆	—	2	2	no	—
138	Dic₁₀₆	—	2	2	no	—
139	Dic₁₀₆	—	4	4	no	—
140	#88324	—	2	2	no	—
141	#88324	—	4	2	no	—
142	#88324	—	4	4	no	—
143	#88324	C₂×Q₁₆	1	4	yes	complemented
144	C₈×C₅₃	—	2	1	no	—
145	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
146	#88322	—	2	1	no	—
147	#88322	—	2	1	no	—
148	M₁₆×Q₁₆	—	53	1	no	maximal, Sylow
149	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
150	C₂×D₅₃	—	2	1	no	—
151	C₂×D₅₃	—	2	1	no	—
152	C₂×D₅₃	C₄×Q₁₆	1	2	yes	—
153	C₄×C₅₃	—	2	4	no	—
154	C₄×C₅₃	—	2	1	no	—
155	C₄×C₅₃	—	4	2	no	—
156	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
157	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
158	C₄×C₅₃	#568	1	1	yes	characteristic
159	Dic₅₃	—	2	1	no	—
160	Dic₅₃	—	2	1	no	—
161	Dic₅₃	—	4	2	no	—
162	Dic₅₃	—	4	2	no	—
163	Dic₅₃	C₂²×Q₁₆	1	1	yes	characteristic
164	Dic₅₃	C₂×#135	1	1	yes	characteristic
165	Q₈×M₁₆	—	53	2	no	—
166	C₂×C₄×Q₁₆	—	53	1	no	—
167	#1591	—	53	2	no	—
168	#1584	—	53	4	no	—
169	C₈×Q₁₆	—	53	2	no	—
170	#1567	—	53	1	no	—
171	#1487	—	53	2	no	—
172	C₈×M₁₆	—	53	1	no	—
173	C₂×C₅₃	D₄×M₁₆	1	1	yes	characteristic
174	C₂×C₅₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
175	C₂×C₅₃	#2957	1	1	yes	characteristic
176	D₅₃	—	2	2	no	—
177	C₂²×Q₁₆	—	53	1	no	—
178	C₂×C₄×Q₈	—	53	2	no	—
179	#446	—	53	4	no	—
180	#446	—	106	4	no	—
181	C₈×Q₈	—	53	4	no	—
182	C₄×Q₁₆	—	53	1	no	—
183	C₄×Q₁₆	—	53	2	no	—
184	C₄×Q₁₆	—	53	1	no	—
185	C₄×Q₁₆	—	53	2	no	—
186	C₄×Q₁₆	—	53	1	no	—
187	C₄×Q₁₆	#6157	1	1	yes	characteristic
188	#431	—	53	1	no	—
189	C₂×#107	—	53	1	no	—
190	#423	—	53	2	no	—
191	#423	—	106	2	no	—
192	C₂×#103	—	53	2	no	—
193	C₄×M₁₆	—	53	1	no	—
194	C₄×M₁₆	—	106	2	no	—
195	C₂×C₄×C₈	—	53	1	no	—
196	#335	—	53	2	no	—
197	#335	—	53	2	no	—
198	Q₈⋊C₈	—	53	4	no	—
199	Q₈⋊C₈	—	53	4	no	—
200	#322	—	53	2	no	—
201	C₈²	—	53	2	no	—
202	C₅₃	M₁₆×Q₁₆	1	1	yes	characteristic, complemented, Sylow
203	C₂²×Q₈	—	53	2	no	—
204	C₂×Q₁₆	—	53	2	no	—
205	C₂×Q₁₆	—	53	1	no	—
206	C₂×Q₁₆	—	106	4	no	—
207	C₂×Q₁₆	C₂×#6157	1	1	yes	characteristic
208	C₂×M₁₆	—	53	1	no	—
209	C₂×M₁₆	—	53	1	no	—
210	C₂×M₁₆	—	106	2	no	—
211	C₂²×C₈	—	53	1	no	—
212	C₂²×C₈	—	53	1	no	—
213	C₄×Q₈	—	53	2	no	—
214	C₄×Q₈	—	53	2	no	—
215	C₄×Q₈	—	53	2	no	—
216	C₄×Q₈	—	106	2	no	—
217	C₄×Q₈	—	106	2	no	—
218	C₄×Q₈	C₂×#6157	1	2	yes	—
219	C₂×C₄⋊C₄	—	53	2	no	—
220	C₂×C₄⋊C₄	—	106	2	no	—
221	C₂×C₄²	—	53	1	no	—
222	C₂×C₄²	—	106	2	no	—
223	#107	—	53	1	no	—
224	#107	—	53	2	no	—
225	#107	C₂×#6157	1	1	yes	characteristic
226	C₄⋊C₈	—	53	4	no	—
227	C₄⋊C₈	—	106	2	no	—
228	C₄⋊C₈	—	106	1	no	—
229	C₄⋊C₈	—	106	4	no	—
230	C₄⋊C₈	—	106	4	no	—
231	#103	—	53	2	no	—
232	#103	—	53	2	no	—
233	#103	—	53	2	no	—
234	#103	C₂×#6157	1	2	yes	—
235	C₂²⋊C₈	—	106	1	no	—
236	#97	—	53	2	no	—
237	#97	—	106	4	no	—
238	C₄×C₈	—	53	2	no	—
239	C₄×C₈	—	53	1	no	—
240	C₄×C₈	—	53	1	no	—
241	C₄×C₈	—	53	2	no	—
242	C₄×C₈	—	53	1	no	—
243	C₄×C₈	—	106	4	no	—
244	C₄×C₈	C₂×#6157	1	1	yes	characteristic
245	C₂×Q₈	—	53	2	no	—
246	C₂×Q₈	—	106	2	no	—
247	C₂×Q₈	—	106	4	no	—
248	C₂×Q₈	—	212	2	no	—
249	C₂×Q₈	C₂²×#6157	1	2	yes	—
250	C₂²×C₄	—	53	1	no	—
251	C₂²×C₄	—	53	1	no	—
252	C₂²×C₄	—	53	1	no	—
253	C₂²×C₄	—	106	2	no	—
254	C₂²×C₄	—	106	2	no	—
255	Q₁₆	—	106	4	no	—
256	Q₁₆	—	106	2	no	—
257	Q₁₆	#11719936	1	4	yes	direct product factor
258	M₁₆	—	53	4	no	—
259	M₁₆	—	106	2	no	—
260	M₁₆	—	212	4	no	—
261	C₂×C₈	—	53	2	no	—
262	C₂×C₈	—	53	2	no	—
263	C₂×C₈	—	53	1	no	—
264	C₂×C₈	—	53	1	no	—
265	C₂×C₈	—	106	2	no	—
266	C₂×C₈	—	106	2	no	—
267	C₂×C₈	—	106	4	no	—
268	C₂×C₈	—	212	1	no	—
269	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
270	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	53	2	no	—
273	C₄⋊C₄	—	106	2	no	—
274	C₄⋊C₄	—	106	2	no	—
275	C₄⋊C₄	—	106	2	no	—
276	C₄⋊C₄	—	106	2	no	—
277	C₄⋊C₄	C₂²×#6157	1	2	yes	—
278	C₄²	—	2	2	no	—
279	C₄²	—	53	1	no	—
280	C₄²	—	53	1	no	—
281	C₄²	—	53	1	no	—
282	C₄²	—	106	2	no	—
283	C₄²	—	106	2	no	—
284	C₄²	—	106	2	no	—
285	C₄²	C₂²×#6157	1	1	yes	characteristic
286	C₂³	—	53	1	no	—
287	Q₈	—	2	2	no	—
288	Q₈	—	106	2	no	—
289	Q₈	—	106	2	no	—
290	Q₈	—	212	4	no	—
291	Q₈	C₂×#11719936	1	4	yes	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	53	1	no	—
295	C₂×C₄	—	53	2	no	—
296	C₂×C₄	—	53	2	no	—
297	C₂×C₄	—	53	1	no	—
298	C₂×C₄	—	53	1	no	—
299	C₂×C₄	—	106	1	no	—
300	C₂×C₄	—	106	1	no	—
301	C₂×C₄	—	106	2	no	—
302	C₂×C₄	—	106	2	no	—
303	C₂×C₄	—	424	2	no	—
304	C₂×C₄	—	212	1	no	—
305	C₂×C₄	—	212	4	no	—
306	C₂×C₄	D₄×#6157	1	1	yes	characteristic
307	C₂×C₄	D₄×#6157	1	1	yes	characteristic
308	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	53	4	no	—
311	C₈	—	106	2	no	—
312	C₈	—	106	1	no	—
313	C₈	—	106	1	no	—
314	C₈	—	212	4	no	—
315	C₈	—	212	2	no	—
316	C₈	C₂×#11719936	1	2	yes	—
317	C₂²	—	53	2	no	—
318	C₂²	—	106	1	no	—
319	C₂²	—	106	1	no	—
320	C₂²	C₂×D₄×#6157	1	1	yes	characteristic
321	C₄	—	2	4	no	—
322	C₄	—	2	1	no	—
323	C₄	—	4	2	no	—
324	C₄	—	53	1	no	—
325	C₄	—	53	1	no	—
326	C₄	—	106	1	no	—
327	C₄	—	106	1	no	—
328	C₄	—	212	2	no	—
329	C₄	—	212	2	no	—
330	C₄	Q₁₆×#6157	1	2	yes	—
331	C₄	C₂²×#11719936	1	1	yes	characteristic
332	C₄	#16811702	1	1	yes	characteristic
333	C₂	—	106	2	no	—
334	C₂	D₄×#11719936	1	1	yes	characteristic
335	C₂	C₂×Q₁₆×#6157	1	1	yes	characteristic
336	C₂	#20291492	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#6157
commutator	C₂×C₄×C₅₃	C₂³×C₄
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	M₁₆×Q₁₆
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #20290354	C₂	semidirect
3. #20290326	C₂	semidirect
4. Q₁₆×#88324	C₂	semidirect
92. #11719936	Q₁₆	direct (N × H)
93. Q₁₆×C₅₃	M₁₆	semidirect
143. #88324	C₂×Q₁₆	semidirect
202. C₅₃	M₁₆×Q₁₆	semidirect
257. Q₁₆	#11719936	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×#11719936	2
#20290354	2
#20290326	2
Q₁₆×#88324	2
#20290273	2
#20290044	2
C₈×#11719936	2
C₄×Q₁₆×D₅₃	2
M₁₆×Q₁₆	53

Subgroups — Q16×#11719936 #28317517

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₁₆×#11719936 #28317517