Subgroups — M₁₆×Dic₁₇₂ #25068059

Order 11008.

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×Dic₈₆	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic
3	#19133493	C₂	1	2	yes	maximal, complemented
4	#19133447	C₂	1	4	yes	maximal, complemented
5	C₈×Dic₁₇₂	C₂	1	2	yes	maximal, complemented
6	#19133354	C₂	1	1	yes	maximal, characteristic
7	#19133025	C₂	1	2	yes	maximal
8	C₈×M₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₇₂	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₈₆	C₂²	1	2	yes	—
11	#14583029	C₂²	1	4	yes	—
12	#14583027	—	2	4	no	—
13	C₈×Dic₈₆	C₂²	1	4	yes	—
14	C₄×Dic₁₇₂	C₄	1	2	yes	—
15	C₄×Dic₁₇₂	C₄	1	1	yes	characteristic
16	C₄×Dic₁₇₂	C₄	1	1	yes	characteristic
17	C₄×Dic₁₇₂	C₂²	1	2	yes	—
18	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
19	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
20	C₂×#13063259	C₄	1	1	yes	characteristic
21	#14582923	C₂²	1	2	yes	—
22	#14582922	—	2	2	no	—
23	C₂×#13063250	C₄	1	2	yes	—
24	M₁₆×Dic₄₃	—	2	2	no	—
25	#14582652	C₂²	1	2	yes	—
26	#14582652	C₂²	1	2	yes	—
27	#14582631	C₂²	1	4	yes	—
28	#14582631	C₂²	1	4	yes	—
29	C₄₃×#431	C₂²	1	1	yes	characteristic
30	C₄×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
32	C₄₃×#322	C₂²	1	2	yes	—
33	C₈²×C₄₃	C₂²	1	2	yes	—
34	C₂²×Dic₈₆	C₂×C₄	1	2	yes	—
35	C₂×Dic₁₇₂	—	2	4	no	—
36	C₂×Dic₁₇₂	C₂×C₄	1	2	yes	—
37	C₂×Dic₁₇₂	C₂×C₄	1	1	yes	characteristic
38	C₂×Dic₁₇₂	C₂×C₄	1	1	yes	characteristic
39	C₂×#10621877	—	2	2	no	—
40	C₄×Dic₈₆	—	2	2	no	—
41	C₄×Dic₈₆	—	2	2	no	—
42	C₄×Dic₈₆	C₂×C₄	1	2	yes	—
43	C₄×Dic₈₆	C₂×C₄	1	2	yes	—
44	C₄×Dic₈₆	C₂³	1	2	yes	—
45	C₄×Dic₈₆	C₂³	1	2	yes	—
46	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
47	C₂×Dic₄₃⋊C₄	—	2	2	no	—
48	C₂×C₄×Dic₄₃	—	2	2	no	—
49	#13063259	C₂×C₄	1	2	yes	—
50	#13063259	C₂×C₄	1	1	yes	characteristic
51	#13063259	C₂×C₄	1	1	yes	characteristic
52	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
53	C₄⋊(C₄₃⋊C₈)	C₂³	1	4	yes	—
54	Dic₄₃⋊C₈	—	2	4	no	—
55	Dic₄₃⋊C₈	—	2	4	no	—
56	#13063250	C₂×C₄	1	2	yes	—
57	#13063250	C₂×C₄	1	2	yes	—
58	#13063250	C₂×C₄	1	2	yes	—
59	#13063250	C₂×C₄	1	2	yes	—
60	#13063235	—	2	4	no	—
61	C₈×Dic₄₃	—	2	4	no	—
62	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
63	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₄₃	—	2	1	no	—
68	C₂²⋊C₈×C₄₃	—	2	1	no	—
69	#97×C₄₃	D₄	1	2	yes	—
70	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₄₃	C₂³	1	2	yes	—
73	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₄₃	C₂³	1	2	yes	—
76	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
79	C₂²×Dic₄₃	—	2	2	no	—
80	C₂²×Dic₄₃	—	2	2	no	—
81	C₂×Dic₈₆	—	2	4	no	—
82	C₂×Dic₈₆	—	2	2	no	—
83	C₂×Dic₈₆	—	4	2	no	—
84	C₂×Dic₈₆	C₂²×C₄	1	2	yes	—
85	C₂×Dic₈₆	C₂²×C₄	1	2	yes	—
86	M₁₆×C₄₃	—	2	2	no	—
87	M₁₆×C₄₃	Q₁₆	1	4	yes	complemented
88	C₂×C₈×C₄₃	—	2	2	no	—
89	C₂×C₈×C₄₃	—	2	2	no	—
90	C₂×C₈×C₄₃	—	4	1	no	—
91	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
97	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₄₃	#41	1	1	yes	characteristic
100	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
101	C₄⋊Dic₄₃	—	2	2	no	—
102	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
103	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
104	Dic₄₃⋊C₄	—	2	2	no	—
105	Dic₄₃⋊C₄	—	2	2	no	—
106	Dic₄₃⋊C₄	—	2	2	no	—
107	Dic₄₃⋊C₄	—	2	2	no	—
108	C₄×Dic₄₃	—	2	2	no	—
109	C₄×Dic₄₃	—	2	2	no	—
110	C₄×Dic₄₃	—	2	2	no	—
111	C₄×Dic₄₃	—	2	2	no	—
112	#10621877	—	4	4	no	—
113	C₂×C₄₃⋊C₈	—	2	4	no	—
114	Dic₁₇₂	—	2	4	no	—
115	Dic₁₇₂	—	2	2	no	—
116	Dic₁₇₂	M₁₆	1	4	yes	direct product factor
117	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₃	—	2	1	no	—
119	C₂×C₄×C₄₃	—	2	1	no	—
120	C₂×C₄×C₄₃	—	4	1	no	—
121	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
125	C₂×C₄×C₄₃	#135	1	2	yes	—
126	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
129	C₂×Dic₄₃	—	2	2	no	—
130	C₂×Dic₄₃	—	2	2	no	—
131	C₂×Dic₄₃	—	2	2	no	—
132	C₂×Dic₄₃	—	2	2	no	—
133	C₂×Dic₄₃	—	4	4	no	—
134	C₂×Dic₄₃	—	8	2	no	—
135	Dic₈₆	—	2	2	no	—
136	Dic₈₆	—	2	2	no	—
137	Dic₈₆	—	2	2	no	—
138	Dic₈₆	—	4	4	no	—
139	Dic₈₆	C₂×M₁₆	1	4	yes	—
140	C₈×C₄₃	—	2	1	no	—
141	C₈×C₄₃	—	2	1	no	—
142	C₈×C₄₃	—	2	1	no	—
143	C₈×C₄₃	—	2	2	no	—
144	C₈×C₄₃	—	4	2	no	—
145	C₈×C₄₃	C₂×M₁₆	1	2	yes	—
146	C₈×C₄₃	C₂×Q₁₆	1	4	yes	complemented
147	C₄₃⋊C₈	—	4	4	no	—
148	M₁₆×Q₁₆	—	43	1	no	maximal, Sylow
149	C₂²×C₄₃	—	2	1	no	—
150	C₂²×C₄₃	—	2	1	no	—
151	C₂²×C₄₃	C₄×Q₁₆	1	2	yes	—
152	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
153	C₄×C₄₃	—	2	1	no	—
154	C₄×C₄₃	—	2	1	no	—
155	C₄×C₄₃	—	2	1	no	—
156	C₄×C₄₃	C₄×Q₁₆	1	2	yes	—
157	C₄×C₄₃	C₂²×M₁₆	1	1	yes	characteristic
158	C₄×C₄₃	#568	1	1	yes	characteristic
159	C₄×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
160	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
161	Dic₄₃	—	2	4	no	—
162	Dic₄₃	—	4	2	no	—
163	Dic₄₃	—	4	2	no	—
164	Dic₄₃	—	4	2	no	—
165	Q₈×M₁₆	—	43	2	no	—
166	C₂×C₄×Q₁₆	—	43	1	no	—
167	#1591	—	43	2	no	—
168	#1584	—	43	4	no	—
169	C₈×Q₁₆	—	43	2	no	—
170	#1567	—	43	1	no	—
171	#1487	—	43	2	no	—
172	C₈×M₁₆	D₄₃	1	1	yes	characteristic
173	C₂×C₄₃	—	2	2	no	—
174	C₂×C₄₃	D₄×M₁₆	1	1	yes	characteristic
175	C₂×C₄₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
176	C₂×C₄₃	#2957	1	1	yes	characteristic
177	C₂²×Q₁₆	—	43	1	no	—
178	C₂×C₄×Q₈	—	43	2	no	—
179	#446	—	43	4	no	—
180	#446	—	86	4	no	—
181	C₈×Q₈	—	43	4	no	—
182	C₄×Q₁₆	—	43	1	no	—
183	C₄×Q₁₆	—	43	2	no	—
184	C₄×Q₁₆	—	43	1	no	—
185	C₄×Q₁₆	—	43	1	no	—
186	C₄×Q₁₆	—	43	2	no	—
187	C₄×Q₁₆	—	43	1	no	—
188	#431	C₂×D₄₃	1	1	yes	characteristic
189	C₂×#107	—	43	1	no	—
190	#423	—	43	2	no	—
191	#423	—	86	2	no	—
192	C₂×#103	—	43	2	no	—
193	C₄×M₁₆	—	86	2	no	—
194	C₄×M₁₆	C₂×D₄₃	1	1	yes	characteristic
195	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
196	#335	—	43	2	no	—
197	#335	—	43	2	no	—
198	Q₈⋊C₈	—	43	4	no	—
199	Q₈⋊C₈	—	43	4	no	—
200	#322	C₂×D₄₃	1	2	yes	—
201	C₈²	C₂×D₄₃	1	2	yes	—
202	C₄₃	M₁₆×Q₁₆	1	1	yes	characteristic, complemented, Sylow
203	C₂²×Q₈	—	43	2	no	—
204	C₂×Q₁₆	—	43	1	no	—
205	C₂×Q₁₆	—	43	2	no	—
206	C₂×Q₁₆	—	43	1	no	—
207	C₂×Q₁₆	—	86	4	no	—
208	C₂×M₁₆	—	86	2	no	—
209	C₂×M₁₆	D₁₇₂	1	1	yes	characteristic
210	C₂×M₁₆	D₁₇₂	1	1	yes	characteristic
211	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
212	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
213	C₄×Q₈	—	43	2	no	—
214	C₄×Q₈	—	43	2	no	—
215	C₄×Q₈	—	43	2	no	—
216	C₄×Q₈	—	43	2	no	—
217	C₄×Q₈	—	86	2	no	—
218	C₄×Q₈	—	86	2	no	—
219	C₂×C₄⋊C₄	—	43	2	no	—
220	C₂×C₄⋊C₄	—	86	2	no	—
221	C₂×C₄²	—	86	2	no	—
222	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
223	#107	—	43	2	no	—
224	#107	—	43	1	no	—
225	#107	—	43	1	no	—
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	43	4	no	—
228	C₄⋊C₈	—	86	4	no	—
229	C₄⋊C₈	—	86	4	no	—
230	C₄⋊C₈	—	86	2	no	—
231	#103	—	43	2	no	—
232	#103	—	43	2	no	—
233	#103	—	43	2	no	—
234	#103	—	43	2	no	—
235	C₂²⋊C₈	—	2	1	no	—
236	#97	—	86	4	no	—
237	#97	D₁₇₂	1	2	yes	—
238	C₄×C₈	—	86	4	no	—
239	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
240	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₄₃	1	2	yes	—
242	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₄₃	1	2	yes	—
245	C₂×Q₈	—	43	2	no	—
246	C₂×Q₈	—	43	2	no	—
247	C₂×Q₈	—	86	4	no	—
248	C₂×Q₈	—	86	2	no	—
249	C₂×Q₈	—	172	2	no	—
250	C₂²×C₄	—	86	2	no	—
251	C₂²×C₄	—	86	2	no	—
252	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
254	C₂²×C₄	#10621898	1	1	yes	characteristic
255	Q₁₆	—	43	4	no	—
256	Q₁₆	—	86	4	no	—
257	Q₁₆	—	86	2	no	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	172	4	no	—
260	M₁₆	Dic₁₇₂	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	—	86	4	no	—
265	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
269	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
270	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
271	C₄⋊C₄	—	43	2	no	—
272	C₄⋊C₄	—	43	2	no	—
273	C₄⋊C₄	—	86	2	no	—
274	C₄⋊C₄	—	86	2	no	—
275	C₄⋊C₄	—	86	2	no	—
276	C₄⋊C₄	—	86	2	no	—
277	C₄⋊C₄	—	86	2	no	—
278	C₄²	—	86	2	no	—
279	C₄²	—	86	2	no	—
280	C₄²	—	86	2	no	—
281	C₄²	—	86	2	no	—
282	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
283	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
284	C₄²	#10621898	1	1	yes	characteristic
285	C₄²	C₂³×D₄₃	1	1	yes	characteristic
286	C₂³	C₄×D₁₇₂	1	1	yes	characteristic
287	Q₈	—	43	4	no	—
288	Q₈	—	86	2	no	—
289	Q₈	—	86	2	no	—
290	Q₈	—	86	2	no	—
291	Q₈	—	172	4	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	86	2	no	—
296	C₂×C₄	—	86	2	no	—
297	C₂×C₄	—	86	2	no	—
298	C₂×C₄	—	86	2	no	—
299	C₂×C₄	—	344	2	no	—
300	C₂×C₄	—	172	4	no	—
301	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
302	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
303	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
304	C₂×C₄	C₂×Dic₁₇₂	1	2	yes	—
305	C₂×C₄	#13063347	1	2	yes	—
306	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
308	C₂×C₄	C₂×#10621898	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	—	172	4	no	—
315	C₈	M₁₆×D₄₃	1	2	yes	—
316	C₈	C₂×Dic₁₇₂	1	4	yes	complemented
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	C₄×Dic₁₇₂	1	2	yes	—
320	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	86	4	no	—
325	C₄	—	172	2	no	—
326	C₄	—	172	2	no	—
327	C₄	—	172	2	no	—
328	C₄	C₄×Dic₁₇₂	1	2	yes	—
329	C₄	C₂×M₁₆×D₄₃	1	1	yes	characteristic
330	C₄	#14583665	1	1	yes	characteristic
331	C₄	C₂²×Dic₁₇₂	1	1	yes	characteristic
332	C₄	C₂×#13063347	1	1	yes	characteristic
333	C₂	—	2	2	no	—
334	C₂	M₁₆×D₁₇₂	1	1	yes	characteristic
335	C₂	C₂×C₄×Dic₁₇₂	1	1	yes	characteristic
336	C₂	#19143513	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₄²	C₂³×D₄₃
Fitting	C₈×M₁₆×C₄₃	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂³×C₄₃	C₄×D₄
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	C₈×M₁₆	D₄₃
hypocenter	C₄₃	M₁₆×Q₁₆
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
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. #19133493	C₂	semidirect
4. #19133447	C₂	semidirect
5. C₈×Dic₁₇₂	C₂	semidirect
87. M₁₆×C₄₃	Q₁₆	semidirect
116. Dic₁₇₂	M₁₆	direct (N × H)
146. C₈×C₄₃	C₂×Q₁₆	semidirect
202. C₄₃	M₁₆×Q₁₆	semidirect
260. M₁₆	Dic₁₇₂	direct (N × H)
316. C₈	C₂×Dic₁₇₂	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×Dic₈₆	2
C₂×C₄×Dic₁₇₂	2
#19133493	2
#19133447	2
C₈×Dic₁₇₂	2
#19133354	2
#19133025	2
C₈×M₁₆×C₄₃	2
M₁₆×Q₁₆	43

Subgroups — M16×Dic172 #25068059

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₆×Dic₁₇₂ #25068059