Subgroups — M₁₆×Dic₅₂ #15834381

Order 3328.

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×Dic₂₆	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic
3	#13113734	C₂	1	2	yes	maximal, complemented
4	#13113688	C₂	1	4	yes	maximal, complemented
5	C₈×Dic₅₂	C₂	1	2	yes	maximal, complemented
6	#13113595	C₂	1	1	yes	maximal, characteristic
7	#13113266	C₂	1	2	yes	maximal
8	C₈×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₅₂	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₂₆	C₂²	1	2	yes	—
11	C₁₃×#431	C₂²	1	1	yes	characteristic
12	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	#11718779	—	2	2	no	—
15	M₁₆×Dic₁₃	—	2	2	no	—
16	C₂×#88058	C₄	1	1	yes	characteristic
17	C₂×#88056	C₄	1	2	yes	—
18	#11718585	C₂²	1	2	yes	—
19	#11718362	—	2	4	no	—
20	C₄×Dic₅₂	C₄	1	2	yes	—
21	C₄×Dic₅₂	C₄	1	1	yes	characteristic
22	C₄×Dic₅₂	C₄	1	1	yes	characteristic
23	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
24	C₄×Dic₅₂	C₂²	1	2	yes	—
25	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
26	#11718340	C₂²	1	4	yes	—
27	C₈×Dic₂₆	C₂²	1	4	yes	—
28	C₁₃×#322	C₂²	1	2	yes	—
29	C₈²×C₁₃	C₂²	1	2	yes	—
30	#11718050	C₂²	1	2	yes	—
31	#11718050	C₂²	1	2	yes	—
32	#11718048	C₂²	1	4	yes	—
33	#11718048	C₂²	1	4	yes	—
34	C₂²×Dic₂₆	C₂×C₄	1	2	yes	—
35	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
36	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
41	C₂×#6101	—	2	2	no	—
42	C₂×C₄×Dic₁₃	—	2	2	no	—
43	C₂×#6099	—	2	2	no	—
44	C₂×Dic₅₂	—	2	4	no	—
45	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
46	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
47	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
48	C₄×Dic₂₆	—	2	2	no	—
49	C₄×Dic₂₆	—	2	2	no	—
50	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
51	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
52	C₄×Dic₂₆	C₂³	1	2	yes	—
53	C₄×Dic₂₆	C₂³	1	2	yes	—
54	C₁₃×C₄⋊C₈	—	2	1	no	—
55	C₁₃×C₂²⋊C₈	—	2	1	no	—
56	C₁₃×#97	D₄	1	2	yes	—
57	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₁₃	C₂³	1	2	yes	—
60	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₁₃	C₂³	1	2	yes	—
63	#88058	C₂×C₄	1	2	yes	—
64	#88058	C₂×C₄	1	1	yes	characteristic
65	#88058	C₂×C₄	1	1	yes	characteristic
66	#88056	C₂×C₄	1	2	yes	—
67	#88056	C₂×C₄	1	2	yes	—
68	#88056	C₂×C₄	1	2	yes	—
69	#88056	C₂×C₄	1	2	yes	—
70	#88055	—	2	4	no	—
71	#88054	—	2	4	no	—
72	#88054	—	2	4	no	—
73	C₈×Dic₁₃	—	2	4	no	—
74	C₄⋊#1055	—	2	2	no	—
75	C₄⋊#1055	C₂³	1	4	yes	—
76	M₁₆×Q₁₆	—	13	1	no	maximal, Sylow
77	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
80	C₂²×Dic₁₃	—	2	2	no	—
81	C₂²×Dic₁₃	—	2	2	no	—
82	C₂×Dic₂₆	—	2	4	no	—
83	C₂×Dic₂₆	—	2	2	no	—
84	C₂×Dic₂₆	—	4	2	no	—
85	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
86	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
87	C₁₃×M₁₆	—	2	2	no	—
88	C₁₃×M₁₆	Q₁₆	1	4	yes	complemented
89	C₂×C₈×C₁₃	—	2	2	no	—
90	C₂×C₈×C₁₃	—	2	2	no	—
91	C₂×C₈×C₁₃	—	4	1	no	—
92	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
98	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₁₃	#41	1	1	yes	characteristic
101	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
102	C₄⋊Dic₁₃	—	2	2	no	—
103	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
104	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
105	#6101	—	2	2	no	—
106	#6101	—	2	2	no	—
107	#6101	—	2	2	no	—
108	#6101	—	2	2	no	—
109	C₄×Dic₁₃	—	2	2	no	—
110	C₄×Dic₁₃	—	2	2	no	—
111	C₄×Dic₁₃	—	2	2	no	—
112	C₄×Dic₁₃	—	2	2	no	—
113	#6099	—	4	4	no	—
114	C₂×#1055	—	2	4	no	—
115	Dic₅₂	—	2	4	no	—
116	Dic₅₂	—	2	2	no	—
117	Dic₅₂	M₁₆	1	4	yes	direct product factor
118	Q₈×M₁₆	—	13	2	no	—
119	C₂×C₄×Q₁₆	—	13	1	no	—
120	#1591	—	13	2	no	—
121	#1584	—	13	4	no	—
122	C₈×Q₁₆	—	13	2	no	—
123	#1567	—	13	1	no	—
124	#1487	—	13	2	no	—
125	C₈×M₁₆	D₁₃	1	1	yes	characteristic
126	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	—	2	1	no	—
128	C₂×C₄×C₁₃	—	2	1	no	—
129	C₂×C₄×C₁₃	—	4	1	no	—
130	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
134	C₂×C₄×C₁₃	#135	1	2	yes	—
135	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
138	C₂×Dic₁₃	—	2	2	no	—
139	C₂×Dic₁₃	—	2	2	no	—
140	C₂×Dic₁₃	—	2	2	no	—
141	C₂×Dic₁₃	—	2	2	no	—
142	C₂×Dic₁₃	—	4	4	no	—
143	C₂×Dic₁₃	—	8	2	no	—
144	Dic₂₆	—	2	2	no	—
145	Dic₂₆	—	2	2	no	—
146	Dic₂₆	—	2	2	no	—
147	Dic₂₆	—	4	4	no	—
148	Dic₂₆	C₂×M₁₆	1	4	yes	—
149	C₈×C₁₃	—	2	1	no	—
150	C₈×C₁₃	—	2	1	no	—
151	C₈×C₁₃	—	2	1	no	—
152	C₈×C₁₃	—	2	2	no	—
153	C₈×C₁₃	—	4	2	no	—
154	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
155	C₈×C₁₃	C₂×Q₁₆	1	4	yes	complemented
156	#1055	—	4	4	no	—
157	C₂²×Q₁₆	—	13	1	no	—
158	C₂×C₄×Q₈	—	13	2	no	—
159	#446	—	13	4	no	—
160	#446	—	26	4	no	—
161	C₈×Q₈	—	13	4	no	—
162	C₄×Q₁₆	—	13	1	no	—
163	C₄×Q₁₆	—	13	2	no	—
164	C₄×Q₁₆	—	13	1	no	—
165	C₄×Q₁₆	—	13	1	no	—
166	C₄×Q₁₆	—	13	2	no	—
167	C₄×Q₁₆	—	13	1	no	—
168	#431	C₂×D₁₃	1	1	yes	characteristic
169	C₂×#107	—	13	1	no	—
170	#423	—	13	2	no	—
171	#423	—	26	2	no	—
172	C₂×#103	—	13	2	no	—
173	C₄×M₁₆	—	26	2	no	—
174	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
175	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
176	#335	—	13	2	no	—
177	#335	—	13	2	no	—
178	Q₈⋊C₈	—	13	4	no	—
179	Q₈⋊C₈	—	13	4	no	—
180	#322	C₂×D₁₃	1	2	yes	—
181	C₈²	C₂×D₁₃	1	2	yes	—
182	C₂²×C₁₃	—	2	1	no	—
183	C₂²×C₁₃	—	2	1	no	—
184	C₂²×C₁₃	C₄×Q₁₆	1	2	yes	—
185	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	—	2	1	no	—
188	C₄×C₁₃	—	2	1	no	—
189	C₄×C₁₃	C₄×Q₁₆	1	2	yes	—
190	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
191	C₄×C₁₃	#568	1	1	yes	characteristic
192	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
193	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
194	Dic₁₃	—	2	4	no	—
195	Dic₁₃	—	4	2	no	—
196	Dic₁₃	—	4	2	no	—
197	Dic₁₃	—	4	2	no	—
198	C₂²×Q₈	—	13	2	no	—
199	C₂×Q₁₆	—	13	1	no	—
200	C₂×Q₁₆	—	13	2	no	—
201	C₂×Q₁₆	—	13	1	no	—
202	C₂×Q₁₆	—	26	4	no	—
203	C₂×M₁₆	—	26	2	no	—
204	C₂×M₁₆	D₅₂	1	1	yes	characteristic
205	C₂×M₁₆	D₅₂	1	1	yes	characteristic
206	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
207	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
208	C₄×Q₈	—	13	2	no	—
209	C₄×Q₈	—	13	2	no	—
210	C₄×Q₈	—	13	2	no	—
211	C₄×Q₈	—	13	2	no	—
212	C₄×Q₈	—	26	2	no	—
213	C₄×Q₈	—	26	2	no	—
214	C₂×C₄⋊C₄	—	13	2	no	—
215	C₂×C₄⋊C₄	—	26	2	no	—
216	C₂×C₄²	—	26	2	no	—
217	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
218	#107	—	13	2	no	—
219	#107	—	13	1	no	—
220	#107	—	13	1	no	—
221	C₄⋊C₈	—	2	1	no	—
222	C₄⋊C₈	—	13	4	no	—
223	C₄⋊C₈	—	26	4	no	—
224	C₄⋊C₈	—	26	4	no	—
225	C₄⋊C₈	—	26	2	no	—
226	#103	—	13	2	no	—
227	#103	—	13	2	no	—
228	#103	—	13	2	no	—
229	#103	—	13	2	no	—
230	C₂²⋊C₈	—	2	1	no	—
231	#97	—	26	4	no	—
232	#97	D₅₂	1	2	yes	—
233	C₄×C₈	—	26	4	no	—
234	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
235	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₁₃	1	2	yes	—
237	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
238	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₁₃	1	2	yes	—
240	C₂×C₁₃	—	2	2	no	—
241	C₂×C₁₃	D₄×M₁₆	1	1	yes	characteristic
242	C₂×C₁₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
243	C₂×C₁₃	#2957	1	1	yes	characteristic
244	C₂×Q₈	—	13	2	no	—
245	C₂×Q₈	—	13	2	no	—
246	C₂×Q₈	—	26	4	no	—
247	C₂×Q₈	—	26	2	no	—
248	C₂×Q₈	—	52	2	no	—
249	C₂²×C₄	—	26	2	no	—
250	C₂²×C₄	—	26	2	no	—
251	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
253	C₂²×C₄	#6127	1	1	yes	characteristic
254	Q₁₆	—	13	4	no	—
255	Q₁₆	—	26	4	no	—
256	Q₁₆	—	26	2	no	—
257	M₁₆	—	2	2	no	—
258	M₁₆	—	52	4	no	—
259	M₁₆	Dic₅₂	1	4	yes	direct product factor
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	26	4	no	—
264	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
265	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
268	C₂×C₈	C₂×D₅₂	1	2	yes	—
269	C₂×C₈	C₂×D₅₂	1	2	yes	—
270	C₄⋊C₄	—	13	2	no	—
271	C₄⋊C₄	—	13	2	no	—
272	C₄⋊C₄	—	26	2	no	—
273	C₄⋊C₄	—	26	2	no	—
274	C₄⋊C₄	—	26	2	no	—
275	C₄⋊C₄	—	26	2	no	—
276	C₄⋊C₄	—	26	2	no	—
277	C₄²	—	26	2	no	—
278	C₄²	—	26	2	no	—
279	C₄²	—	26	2	no	—
280	C₄²	—	26	2	no	—
281	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
282	C₄²	C₂×D₅₂	1	1	yes	characteristic
283	C₄²	#6127	1	1	yes	characteristic
284	C₄²	C₂³×D₁₃	1	1	yes	characteristic
285	C₁₃	M₁₆×Q₁₆	1	1	yes	characteristic, complemented, Sylow
286	C₂³	C₄×D₅₂	1	1	yes	characteristic
287	Q₈	—	13	4	no	—
288	Q₈	—	26	2	no	—
289	Q₈	—	26	2	no	—
290	Q₈	—	26	2	no	—
291	Q₈	—	52	4	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	26	2	no	—
296	C₂×C₄	—	26	2	no	—
297	C₂×C₄	—	26	2	no	—
298	C₂×C₄	—	26	2	no	—
299	C₂×C₄	—	52	4	no	—
300	C₂×C₄	—	104	2	no	—
301	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
302	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
303	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
304	C₂×C₄	#88158	1	2	yes	—
305	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
306	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
308	C₂×C₄	C₂×#6127	1	1	yes	characteristic
309	C₈	—	2	2	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	4	2	no	—
314	C₈	—	52	4	no	—
315	C₈	C₂×Dic₅₂	1	4	yes	complemented
316	C₈	M₁₆×D₁₃	1	2	yes	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	C₄×Dic₅₂	1	2	yes	—
320	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	26	4	no	—
325	C₄	—	52	2	no	—
326	C₄	—	52	2	no	—
327	C₄	—	52	2	no	—
328	C₄	C₄×Dic₅₂	1	2	yes	—
329	C₄	C₂×#88158	1	1	yes	characteristic
330	C₄	C₂²×Dic₅₂	1	1	yes	characteristic
331	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
332	C₄	#11719450	1	1	yes	characteristic
333	C₂	—	2	2	no	—
334	C₂	M₁₆×D₅₂	1	1	yes	characteristic
335	C₂	C₂×C₄×Dic₅₂	1	1	yes	characteristic
336	C₂	#13123754	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₈×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₈×M₁₆	D₁₃
hypocenter	C₁₃	M₁₆×Q₁₆
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
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. #13113734	C₂	semidirect
4. #13113688	C₂	semidirect
5. C₈×Dic₅₂	C₂	semidirect
88. C₁₃×M₁₆	Q₁₆	semidirect
117. Dic₅₂	M₁₆	direct (N × H)
155. C₈×C₁₃	C₂×Q₁₆	semidirect
259. M₁₆	Dic₅₂	direct (N × H)
285. C₁₃	M₁₆×Q₁₆	semidirect
315. C₈	C₂×Dic₅₂	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×Dic₂₆	2
C₂×C₄×Dic₅₂	2
#13113734	2
#13113688	2
C₈×Dic₅₂	2
#13113595	2
#13113266	2
C₈×C₁₃×M₁₆	2
M₁₆×Q₁₆	13

Subgroups — M16×Dic52 #15834381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₆×Dic₅₂ #15834381