Subgroups — M₁₆×Dic₂₁₂ #27355554

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	M₁₆×Dic₁₀₆	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
3	#20273613	C₂	1	2	yes	maximal, complemented
4	#20273567	C₂	1	4	yes	maximal, complemented
5	C₈×Dic₂₁₂	C₂	1	2	yes	maximal, complemented
6	#20273474	C₂	1	1	yes	maximal, characteristic
7	#20273145	C₂	1	2	yes	maximal
8	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₁₂	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₁₀₆	C₂²	1	2	yes	—
11	#16810753	C₂²	1	4	yes	—
12	#16810751	—	2	4	no	—
13	C₈×Dic₁₀₆	C₂²	1	4	yes	—
14	C₄×Dic₂₁₂	C₄	1	2	yes	—
15	C₄×Dic₂₁₂	C₄	1	1	yes	characteristic
16	C₄×Dic₂₁₂	C₄	1	1	yes	characteristic
17	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
18	C₄×Dic₂₁₂	C₂²	1	2	yes	—
19	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
20	C₂×#13133151	C₄	1	1	yes	characteristic
21	#16810647	C₂²	1	2	yes	—
22	#16810646	—	2	2	no	—
23	C₂×#13133142	C₄	1	2	yes	—
24	M₁₆×Dic₅₃	—	2	2	no	—
25	#16810376	C₂²	1	2	yes	—
26	#16810376	C₂²	1	2	yes	—
27	#16810355	C₂²	1	4	yes	—
28	#16810355	C₂²	1	4	yes	—
29	C₅₃×#431	C₂²	1	1	yes	characteristic
30	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#322	C₂²	1	2	yes	—
33	C₈²×C₅₃	C₂²	1	2	yes	—
34	C₂²×Dic₁₀₆	C₂×C₄	1	2	yes	—
35	C₂×Dic₂₁₂	—	2	4	no	—
36	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
37	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
38	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
39	C₂×#11719917	—	2	2	no	—
40	C₄×Dic₁₀₆	—	2	2	no	—
41	C₄×Dic₁₀₆	—	2	2	no	—
42	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
43	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
44	C₄×Dic₁₀₆	C₂³	1	2	yes	—
45	C₄×Dic₁₀₆	C₂³	1	2	yes	—
46	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
47	C₂×#11719919	—	2	2	no	—
48	C₂×C₄×Dic₅₃	—	2	2	no	—
49	#13133151	C₂×C₄	1	2	yes	—
50	#13133151	C₂×C₄	1	1	yes	characteristic
51	#13133151	C₂×C₄	1	1	yes	characteristic
52	C₄⋊#88322	—	2	2	no	—
53	C₄⋊#88322	C₂³	1	4	yes	—
54	#13133146	—	2	4	no	—
55	#13133146	—	2	4	no	—
56	#13133142	C₂×C₄	1	2	yes	—
57	#13133142	C₂×C₄	1	2	yes	—
58	#13133142	C₂×C₄	1	2	yes	—
59	#13133142	C₂×C₄	1	2	yes	—
60	#13133127	—	2	4	no	—
61	C₈×Dic₅₃	—	2	4	no	—
62	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
63	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₅₃	—	2	1	no	—
68	C₂²⋊C₈×C₅₃	—	2	1	no	—
69	#97×C₅₃	D₄	1	2	yes	—
70	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₅₃	C₂³	1	2	yes	—
73	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₅₃	C₂³	1	2	yes	—
76	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
79	C₂²×Dic₅₃	—	2	2	no	—
80	C₂²×Dic₅₃	—	2	2	no	—
81	C₂×Dic₁₀₆	—	2	4	no	—
82	C₂×Dic₁₀₆	—	2	2	no	—
83	C₂×Dic₁₀₆	—	4	2	no	—
84	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
85	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
86	M₁₆×C₅₃	—	2	2	no	—
87	M₁₆×C₅₃	Q₁₆	1	4	yes	complemented
88	C₂×C₈×C₅₃	—	2	2	no	—
89	C₂×C₈×C₅₃	—	2	2	no	—
90	C₂×C₈×C₅₃	—	4	1	no	—
91	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
97	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₅₃	#41	1	1	yes	characteristic
100	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
101	C₄⋊Dic₅₃	—	2	2	no	—
102	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
103	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
104	#11719919	—	2	2	no	—
105	#11719919	—	2	2	no	—
106	#11719919	—	2	2	no	—
107	#11719919	—	2	2	no	—
108	C₄×Dic₅₃	—	2	2	no	—
109	C₄×Dic₅₃	—	2	2	no	—
110	C₄×Dic₅₃	—	2	2	no	—
111	C₄×Dic₅₃	—	2	2	no	—
112	#11719917	—	4	4	no	—
113	C₂×#88322	—	2	4	no	—
114	Dic₂₁₂	—	2	4	no	—
115	Dic₂₁₂	—	2	2	no	—
116	Dic₂₁₂	M₁₆	1	4	yes	direct product factor
117	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	—	2	1	no	—
119	C₂×C₄×C₅₃	—	2	1	no	—
120	C₂×C₄×C₅₃	—	4	1	no	—
121	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
125	C₂×C₄×C₅₃	#135	1	2	yes	—
126	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
129	C₂×Dic₅₃	—	2	2	no	—
130	C₂×Dic₅₃	—	2	2	no	—
131	C₂×Dic₅₃	—	2	2	no	—
132	C₂×Dic₅₃	—	2	2	no	—
133	C₂×Dic₅₃	—	4	4	no	—
134	C₂×Dic₅₃	—	8	2	no	—
135	Dic₁₀₆	—	2	2	no	—
136	Dic₁₀₆	—	2	2	no	—
137	Dic₁₀₆	—	2	2	no	—
138	Dic₁₀₆	—	4	4	no	—
139	Dic₁₀₆	C₂×M₁₆	1	4	yes	—
140	C₈×C₅₃	—	2	1	no	—
141	C₈×C₅₃	—	2	1	no	—
142	C₈×C₅₃	—	2	1	no	—
143	C₈×C₅₃	—	2	2	no	—
144	C₈×C₅₃	—	4	2	no	—
145	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
146	C₈×C₅₃	C₂×Q₁₆	1	4	yes	complemented
147	#88322	—	4	4	no	—
148	M₁₆×Q₁₆	—	53	1	no	maximal, Sylow
149	C₂²×C₅₃	—	2	1	no	—
150	C₂²×C₅₃	—	2	1	no	—
151	C₂²×C₅₃	C₄×Q₁₆	1	2	yes	—
152	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
153	C₄×C₅₃	—	2	1	no	—
154	C₄×C₅₃	—	2	1	no	—
155	C₄×C₅₃	—	2	1	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	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
160	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
161	Dic₅₃	—	2	4	no	—
162	Dic₅₃	—	4	2	no	—
163	Dic₅₃	—	4	2	no	—
164	Dic₅₃	—	4	2	no	—
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	D₅₃	1	1	yes	characteristic
173	C₂×C₅₃	—	2	2	no	—
174	C₂×C₅₃	D₄×M₁₆	1	1	yes	characteristic
175	C₂×C₅₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
176	C₂×C₅₃	#2957	1	1	yes	characteristic
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	1	no	—
186	C₄×Q₁₆	—	53	2	no	—
187	C₄×Q₁₆	—	53	1	no	—
188	#431	C₂×D₅₃	1	1	yes	characteristic
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	—	106	2	no	—
194	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
195	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
196	#335	—	53	2	no	—
197	#335	—	53	2	no	—
198	Q₈⋊C₈	—	53	4	no	—
199	Q₈⋊C₈	—	53	4	no	—
200	#322	C₂×D₅₃	1	2	yes	—
201	C₈²	C₂×D₅₃	1	2	yes	—
202	C₅₃	M₁₆×Q₁₆	1	1	yes	characteristic, complemented, Sylow
203	C₂²×Q₈	—	53	2	no	—
204	C₂×Q₁₆	—	53	1	no	—
205	C₂×Q₁₆	—	53	2	no	—
206	C₂×Q₁₆	—	53	1	no	—
207	C₂×Q₁₆	—	106	4	no	—
208	C₂×M₁₆	—	106	2	no	—
209	C₂×M₁₆	D₂₁₂	1	1	yes	characteristic
210	C₂×M₁₆	D₂₁₂	1	1	yes	characteristic
211	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
212	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
213	C₄×Q₈	—	53	2	no	—
214	C₄×Q₈	—	53	2	no	—
215	C₄×Q₈	—	53	2	no	—
216	C₄×Q₈	—	53	2	no	—
217	C₄×Q₈	—	106	2	no	—
218	C₄×Q₈	—	106	2	no	—
219	C₂×C₄⋊C₄	—	53	2	no	—
220	C₂×C₄⋊C₄	—	106	2	no	—
221	C₂×C₄²	—	106	2	no	—
222	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
223	#107	—	53	2	no	—
224	#107	—	53	1	no	—
225	#107	—	53	1	no	—
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	53	4	no	—
228	C₄⋊C₈	—	106	4	no	—
229	C₄⋊C₈	—	106	4	no	—
230	C₄⋊C₈	—	106	2	no	—
231	#103	—	53	2	no	—
232	#103	—	53	2	no	—
233	#103	—	53	2	no	—
234	#103	—	53	2	no	—
235	C₂²⋊C₈	—	2	1	no	—
236	#97	—	106	4	no	—
237	#97	D₂₁₂	1	2	yes	—
238	C₄×C₈	—	106	4	no	—
239	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
240	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₅₃	1	2	yes	—
242	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₅₃	1	2	yes	—
245	C₂×Q₈	—	53	2	no	—
246	C₂×Q₈	—	53	2	no	—
247	C₂×Q₈	—	106	4	no	—
248	C₂×Q₈	—	106	2	no	—
249	C₂×Q₈	—	212	2	no	—
250	C₂²×C₄	—	106	2	no	—
251	C₂²×C₄	—	106	2	no	—
252	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
254	C₂²×C₄	#11719945	1	1	yes	characteristic
255	Q₁₆	—	53	4	no	—
256	Q₁₆	—	106	4	no	—
257	Q₁₆	—	106	2	no	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	212	4	no	—
260	M₁₆	Dic₂₁₂	1	4	yes	direct product factor
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	106	4	no	—
265	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
269	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
270	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
271	C₄⋊C₄	—	53	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	—	106	2	no	—
278	C₄²	—	106	2	no	—
279	C₄²	—	106	2	no	—
280	C₄²	—	106	2	no	—
281	C₄²	—	106	2	no	—
282	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
283	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
284	C₄²	#11719945	1	1	yes	characteristic
285	C₄²	C₂³×D₅₃	1	1	yes	characteristic
286	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
287	Q₈	—	53	4	no	—
288	Q₈	—	106	2	no	—
289	Q₈	—	106	2	no	—
290	Q₈	—	106	2	no	—
291	Q₈	—	212	4	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	106	2	no	—
296	C₂×C₄	—	106	2	no	—
297	C₂×C₄	—	106	2	no	—
298	C₂×C₄	—	106	2	no	—
299	C₂×C₄	—	424	2	no	—
300	C₂×C₄	—	212	4	no	—
301	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
302	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
303	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
304	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
305	C₂×C₄	#13133239	1	2	yes	—
306	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
308	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	2	2	no	—
313	C₈	—	4	2	no	—
314	C₈	—	212	4	no	—
315	C₈	M₁₆×D₅₃	1	2	yes	—
316	C₈	C₂×Dic₂₁₂	1	4	yes	complemented
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	C₄×Dic₂₁₂	1	2	yes	—
320	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	106	4	no	—
325	C₄	—	212	2	no	—
326	C₄	—	212	2	no	—
327	C₄	—	212	2	no	—
328	C₄	C₄×Dic₂₁₂	1	2	yes	—
329	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
330	C₄	#16811389	1	1	yes	characteristic
331	C₄	C₂²×Dic₂₁₂	1	1	yes	characteristic
332	C₄	C₂×#13133239	1	1	yes	characteristic
333	C₂	—	2	2	no	—
334	C₂	M₁₆×D₂₁₂	1	1	yes	characteristic
335	C₂	C₂×C₄×Dic₂₁₂	1	1	yes	characteristic
336	C₂	#20283633	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₈×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₈×M₁₆	D₅₃
hypocenter	C₅₃	M₁₆×Q₁₆
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20273613	C₂	semidirect
4. #20273567	C₂	semidirect
5. C₈×Dic₂₁₂	C₂	semidirect
87. M₁₆×C₅₃	Q₁₆	semidirect
116. Dic₂₁₂	M₁₆	direct (N × H)
146. C₈×C₅₃	C₂×Q₁₆	semidirect
202. C₅₃	M₁₆×Q₁₆	semidirect
260. M₁₆	Dic₂₁₂	direct (N × H)
316. C₈	C₂×Dic₂₁₂	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×Dic₁₀₆	2
C₂×C₄×Dic₂₁₂	2
#20273613	2
#20273567	2
C₈×Dic₂₁₂	2
#20273474	2
#20273145	2
C₈×M₁₆×C₅₃	2
M₁₆×Q₁₆	53

Subgroups — M16×Dic212 #27355554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₆×Dic₂₁₂ #27355554