Subgroups — M₁₆×Dic₃₅₆ #36525890

Order 22784.

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×Dic₁₇₈	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic
3	#26019150	C₂	1	2	yes	maximal, complemented
4	#26019104	C₂	1	4	yes	maximal, complemented
5	C₈×Dic₃₅₆	C₂	1	2	yes	maximal, complemented
6	#26019011	C₂	1	1	yes	maximal, characteristic
7	#26018682	C₂	1	2	yes	maximal
8	C₈×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₃₅₆	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₁₇₈	C₂²	1	2	yes	—
11	#19151728	C₂²	1	4	yes	—
12	#19151726	—	2	4	no	—
13	C₈×Dic₁₇₈	C₂²	1	4	yes	—
14	C₄×Dic₃₅₆	C₄	1	1	yes	characteristic
15	C₄×Dic₃₅₆	C₄	1	2	yes	—
16	C₄×Dic₃₅₆	C₄	1	1	yes	characteristic
17	C₄×Dic₃₅₆	C₂²	1	2	yes	—
18	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
19	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
20	C₂×#15667946	C₄	1	1	yes	characteristic
21	#19151622	C₂²	1	2	yes	—
22	#19151621	—	2	2	no	—
23	C₂×#15667937	C₄	1	2	yes	—
24	M₁₆×Dic₈₉	—	2	2	no	—
25	#19151351	C₂²	1	2	yes	—
26	#19151351	C₂²	1	2	yes	—
27	#19151330	C₂²	1	4	yes	—
28	#19151330	C₂²	1	4	yes	—
29	#431×C₈₉	C₂²	1	1	yes	characteristic
30	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
32	#322×C₈₉	C₂²	1	2	yes	—
33	C₈²×C₈₉	C₂²	1	2	yes	—
34	C₂²×Dic₁₇₈	C₂×C₄	1	2	yes	—
35	C₂×Dic₃₅₆	—	2	4	no	—
36	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
37	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
38	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
39	C₂×#13083558	—	2	2	no	—
40	C₄×Dic₁₇₈	—	2	2	no	—
41	C₄×Dic₁₇₈	—	2	2	no	—
42	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
43	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
44	C₄×Dic₁₇₈	C₂³	1	2	yes	—
45	C₄×Dic₁₇₈	C₂³	1	2	yes	—
46	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
47	C₂×#13083553	—	2	2	no	—
48	C₂×C₄×Dic₈₉	—	2	2	no	—
49	#15667946	C₂×C₄	1	2	yes	—
50	#15667946	C₂×C₄	1	1	yes	characteristic
51	#15667946	C₂×C₄	1	1	yes	characteristic
52	C₄⋊#10623494	—	2	2	no	—
53	C₄⋊#10623494	C₂³	1	4	yes	—
54	#15667941	—	2	4	no	—
55	#15667941	—	2	4	no	—
56	#15667937	C₂×C₄	1	2	yes	—
57	#15667937	C₂×C₄	1	2	yes	—
58	#15667937	C₂×C₄	1	2	yes	—
59	#15667937	C₂×C₄	1	2	yes	—
60	#15667922	—	2	4	no	—
61	C₈×Dic₈₉	—	2	4	no	—
62	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
63	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₈₉	—	2	1	no	—
68	C₂²⋊C₈×C₈₉	—	2	1	no	—
69	#97×C₈₉	D₄	1	2	yes	—
70	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₈₉	C₂³	1	2	yes	—
73	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₈₉	C₂³	1	2	yes	—
76	C₂×Dic₁₇₈	—	2	4	no	—
77	C₂×Dic₁₇₈	—	2	2	no	—
78	C₂×Dic₁₇₈	—	4	2	no	—
79	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
80	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
81	C₂²×Dic₈₉	—	2	2	no	—
82	C₂²×Dic₈₉	—	2	2	no	—
83	Dic₃₅₆	—	2	4	no	—
84	Dic₃₅₆	—	2	2	no	—
85	Dic₃₅₆	M₁₆	1	4	yes	direct product factor
86	#13083558	—	4	4	no	—
87	C₂×#10623494	—	2	4	no	—
88	C₄⋊Dic₈₉	—	2	2	no	—
89	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
90	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
91	#13083553	—	2	2	no	—
92	#13083553	—	2	2	no	—
93	#13083553	—	2	2	no	—
94	#13083553	—	2	2	no	—
95	C₄×Dic₈₉	—	2	2	no	—
96	C₄×Dic₈₉	—	2	2	no	—
97	C₄×Dic₈₉	—	2	2	no	—
98	C₄×Dic₈₉	—	2	2	no	—
99	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
102	M₁₆×C₈₉	—	2	2	no	—
103	M₁₆×C₈₉	Q₁₆	1	4	yes	complemented
104	C₂×C₈×C₈₉	—	2	2	no	—
105	C₂×C₈×C₈₉	—	2	2	no	—
106	C₂×C₈×C₈₉	—	4	1	no	—
107	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
112	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
113	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₈₉	#41	1	1	yes	characteristic
116	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
117	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	4	1	no	—
121	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
125	C₂×C₄×C₈₉	#135	1	2	yes	—
126	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
129	C₂×Dic₈₉	—	2	2	no	—
130	C₂×Dic₈₉	—	2	2	no	—
131	C₂×Dic₈₉	—	2	2	no	—
132	C₂×Dic₈₉	—	2	2	no	—
133	C₂×Dic₈₉	—	4	4	no	—
134	C₂×Dic₈₉	—	8	2	no	—
135	Dic₁₇₈	—	2	2	no	—
136	Dic₁₇₈	—	2	2	no	—
137	Dic₁₇₈	—	2	2	no	—
138	Dic₁₇₈	—	4	4	no	—
139	Dic₁₇₈	C₂×M₁₆	1	4	yes	—
140	C₈×C₈₉	—	2	1	no	—
141	C₈×C₈₉	—	2	1	no	—
142	C₈×C₈₉	—	2	1	no	—
143	C₈×C₈₉	—	2	2	no	—
144	C₈×C₈₉	—	4	2	no	—
145	C₈×C₈₉	C₂×M₁₆	1	2	yes	—
146	C₈×C₈₉	C₂×Q₁₆	1	4	yes	complemented
147	#10623494	—	4	4	no	—
148	C₂²×C₈₉	—	2	1	no	—
149	C₂²×C₈₉	—	2	1	no	—
150	C₂²×C₈₉	C₄×Q₁₆	1	2	yes	—
151	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
152	C₄×C₈₉	—	2	1	no	—
153	C₄×C₈₉	—	2	1	no	—
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	C₄×Q₁₆	1	2	yes	—
156	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
157	C₄×C₈₉	#568	1	1	yes	characteristic
158	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
159	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
160	Dic₈₉	—	2	4	no	—
161	Dic₈₉	—	4	2	no	—
162	Dic₈₉	—	4	2	no	—
163	Dic₈₉	—	4	2	no	—
164	M₁₆×Q₁₆	—	89	1	no	maximal, Sylow
165	C₂×C₈₉	—	2	2	no	—
166	C₂×C₈₉	D₄×M₁₆	1	1	yes	characteristic
167	C₂×C₈₉	C₂×C₄×Q₁₆	1	1	yes	characteristic
168	C₂×C₈₉	#2957	1	1	yes	characteristic
169	Q₈×M₁₆	—	89	2	no	—
170	C₂×C₄×Q₁₆	—	89	1	no	—
171	#1591	—	89	2	no	—
172	#1584	—	89	4	no	—
173	C₈×Q₁₆	—	89	2	no	—
174	#1567	—	89	1	no	—
175	#1487	—	89	2	no	—
176	C₈×M₁₆	D₈₉	1	1	yes	characteristic
177	C₈₉	M₁₆×Q₁₆	1	1	yes	characteristic, complemented, Sylow
178	C₂²×Q₁₆	—	89	1	no	—
179	C₂×C₄×Q₈	—	89	2	no	—
180	#446	—	89	4	no	—
181	#446	—	178	4	no	—
182	C₈×Q₈	—	89	4	no	—
183	C₄×Q₁₆	—	89	1	no	—
184	C₄×Q₁₆	—	89	2	no	—
185	C₄×Q₁₆	—	89	1	no	—
186	C₄×Q₁₆	—	89	1	no	—
187	C₄×Q₁₆	—	89	2	no	—
188	C₄×Q₁₆	—	89	1	no	—
189	#431	C₂×D₈₉	1	1	yes	characteristic
190	C₂×#107	—	89	1	no	—
191	#423	—	89	2	no	—
192	#423	—	178	2	no	—
193	C₂×#103	—	89	2	no	—
194	C₄×M₁₆	—	178	2	no	—
195	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
196	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
197	#335	—	89	2	no	—
198	#335	—	89	2	no	—
199	Q₈⋊C₈	—	89	4	no	—
200	Q₈⋊C₈	—	89	4	no	—
201	#322	C₂×D₈₉	1	2	yes	—
202	C₈²	C₂×D₈₉	1	2	yes	—
203	C₂²×Q₈	—	89	2	no	—
204	C₂×Q₁₆	—	89	1	no	—
205	C₂×Q₁₆	—	89	2	no	—
206	C₂×Q₁₆	—	89	1	no	—
207	C₂×Q₁₆	—	178	4	no	—
208	C₂×M₁₆	—	178	2	no	—
209	C₂×M₁₆	D₃₅₆	1	1	yes	characteristic
210	C₂×M₁₆	D₃₅₆	1	1	yes	characteristic
211	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
212	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
213	C₄×Q₈	—	89	2	no	—
214	C₄×Q₈	—	89	2	no	—
215	C₄×Q₈	—	89	2	no	—
216	C₄×Q₈	—	89	2	no	—
217	C₄×Q₈	—	178	2	no	—
218	C₄×Q₈	—	178	2	no	—
219	C₂×C₄⋊C₄	—	89	2	no	—
220	C₂×C₄⋊C₄	—	178	2	no	—
221	C₂×C₄²	—	178	2	no	—
222	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
223	#107	—	89	2	no	—
224	#107	—	89	1	no	—
225	#107	—	89	1	no	—
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	89	4	no	—
228	C₄⋊C₈	—	178	4	no	—
229	C₄⋊C₈	—	178	4	no	—
230	C₄⋊C₈	—	178	2	no	—
231	#103	—	89	2	no	—
232	#103	—	89	2	no	—
233	#103	—	89	2	no	—
234	#103	—	89	2	no	—
235	C₂²⋊C₈	—	2	1	no	—
236	#97	—	178	4	no	—
237	#97	D₃₅₆	1	2	yes	—
238	C₄×C₈	—	178	4	no	—
239	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
240	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₈₉	1	2	yes	—
242	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₈₉	1	2	yes	—
245	C₂×Q₈	—	89	2	no	—
246	C₂×Q₈	—	89	2	no	—
247	C₂×Q₈	—	356	2	no	—
248	C₂×Q₈	—	178	4	no	—
249	C₂×Q₈	—	178	2	no	—
250	C₂²×C₄	—	178	2	no	—
251	C₂²×C₄	—	178	2	no	—
252	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
254	C₂²×C₄	#13083574	1	1	yes	characteristic
255	Q₁₆	—	89	4	no	—
256	Q₁₆	—	178	4	no	—
257	Q₁₆	—	178	2	no	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	356	4	no	—
260	M₁₆	Dic₃₅₆	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	—	178	4	no	—
265	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
269	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
270	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
271	C₄⋊C₄	—	89	2	no	—
272	C₄⋊C₄	—	89	2	no	—
273	C₄⋊C₄	—	178	2	no	—
274	C₄⋊C₄	—	178	2	no	—
275	C₄⋊C₄	—	178	2	no	—
276	C₄⋊C₄	—	178	2	no	—
277	C₄⋊C₄	—	178	2	no	—
278	C₄²	—	178	2	no	—
279	C₄²	—	178	2	no	—
280	C₄²	—	178	2	no	—
281	C₄²	—	178	2	no	—
282	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
283	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
284	C₄²	#13083574	1	1	yes	characteristic
285	C₄²	C₂³×D₈₉	1	1	yes	characteristic
286	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
287	Q₈	—	89	4	no	—
288	Q₈	—	356	4	no	—
289	Q₈	—	178	2	no	—
290	Q₈	—	178	2	no	—
291	Q₈	—	178	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	356	4	no	—
296	C₂×C₄	—	178	2	no	—
297	C₂×C₄	—	178	2	no	—
298	C₂×C₄	—	178	2	no	—
299	C₂×C₄	—	178	2	no	—
300	C₂×C₄	—	712	2	no	—
301	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
302	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
303	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
304	C₂×C₄	C₂×Dic₃₅₆	1	2	yes	—
305	C₂×C₄	#15668034	1	2	yes	—
306	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
308	C₂×C₄	C₂×#13083574	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	—	356	4	no	—
315	C₈	M₁₆×D₈₉	1	2	yes	—
316	C₈	C₂×Dic₃₅₆	1	4	yes	complemented
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	C₄×Dic₃₅₆	1	2	yes	—
320	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	356	2	no	—
325	C₄	—	356	2	no	—
326	C₄	—	356	2	no	—
327	C₄	—	178	4	no	—
328	C₄	C₄×Dic₃₅₆	1	2	yes	—
329	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
330	C₄	#19152364	1	1	yes	characteristic
331	C₄	C₂²×Dic₃₅₆	1	1	yes	characteristic
332	C₄	C₂×#15668034	1	1	yes	characteristic
333	C₂	—	2	2	no	—
334	C₂	M₁₆×D₃₅₆	1	1	yes	characteristic
335	C₂	C₂×C₄×Dic₃₅₆	1	1	yes	characteristic
336	C₂	#26029170	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈×M₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₈×M₁₆	D₈₉
hypocenter	C₈₉	M₁₆×Q₁₆
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
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. #26019150	C₂	semidirect
4. #26019104	C₂	semidirect
5. C₈×Dic₃₅₆	C₂	semidirect
85. Dic₃₅₆	M₁₆	direct (N × H)
103. M₁₆×C₈₉	Q₁₆	semidirect
146. C₈×C₈₉	C₂×Q₁₆	semidirect
177. C₈₉	M₁₆×Q₁₆	semidirect
260. M₁₆	Dic₃₅₆	direct (N × H)
316. C₈	C₂×Dic₃₅₆	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×Dic₁₇₈	2
C₂×C₄×Dic₃₅₆	2
#26019150	2
#26019104	2
C₈×Dic₃₅₆	2
#26019011	2
#26018682	2
C₈×M₁₆×C₈₉	2
M₁₆×Q₁₆	89

Subgroups — M16×Dic356 #36525890

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₆×Dic₃₅₆ #36525890