Subgroups — #72234409

Order 16384.

2159 conjugacy classes of subgroups.

The subgroup lattice has 381 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72234407	C₂	1	4	yes	maximal
2	#72234405	C₂	1	2	yes	maximal
3	#72234402	C₄	1	4	yes	—
4	#72234401	—	2	8	no	—
5	#72234400	C₄	1	4	yes	—
6	#72234397	C₈	1	8	yes	—
7	#72234395	—	2	16	no	—
8	C₁₆×M₁₂₈	C₈	1	8	yes	—
9	#72234393	C₁₆	1	16	yes	complemented
10	#72234392	C₂²	1	4	yes	—
11	#72234377	—	2	8	no	—
12	#72234371	C₂×C₄	1	8	yes	—
13	#72234367	—	2	8	no	—
14	C₈×M₁₂₈	C₂×C₈	1	16	yes	—
15	#72234363	—	2	8	no	—
16	#72234276	C₂	1	1	yes	maximal, characteristic
17	#72234269	C₄	1	1	yes	characteristic
18	#72234269	C₂²	1	2	yes	—
19	#72234265	—	2	4	no	—
20	#72234265	—	2	2	no	—
21	#72234262	C₄	1	1	yes	characteristic
22	#72234256	C₄	1	2	yes	—
23	#72234256	C₂²	1	1	yes	characteristic
24	#72234250	C₈	1	4	yes	—
25	#72234250	C₂×C₄	1	2	yes	—
26	C₂×#93666	—	2	8	no	—
27	C₂×#93666	M₁₆	1	8	yes	—
28	#72234246	C₈	1	4	yes	—
29	#72234246	C₂×C₄	1	2	yes	—
30	C₂×#93664	C₁₆	1	8	yes	—
31	C₂×#93664	C₂×C₈	1	4	yes	—
32	#72234241	C₂×C₄	1	2	yes	—
33	#72234241	C₂³	1	1	yes	characteristic
34	#72234237	—	2	2	no	—
35	#72234237	—	2	2	no	—
36	#72234237	#41	1	4	yes	—
37	C₈×#1397	C₂×C₈	1	4	yes	—
38	C₈×#1397	C₂²×C₄	1	2	yes	—
39	#72234235	C₂×C₄	1	1	yes	characteristic
40	C₄×#7329	C₄²	1	1	yes	characteristic
41	C₄×#7329	C₂×C₈	1	2	yes	—
42	#72234231	C₂×C₄	1	2	yes	—
43	#72234231	C₂×C₄	1	1	yes	characteristic
44	#72234227	—	2	2	no	—
45	#72234227	M₁₆	1	2	yes	—
46	#72234226	C₄²	1	2	yes	—
47	#72234226	C₂×C₈	1	4	yes	—
48	#72234224	C₂²×C₄	1	1	yes	characteristic
49	#72210719	—	2	2	no	—
50	#72210645	—	2	2	no	—
51	#72210645	M₁₆	1	2	yes	—
52	C₄×C₈×C₃₂	C₄²	1	1	yes	characteristic
53	C₄×C₈×C₃₂	C₂×C₈	1	2	yes	—
54	C₂×C₁₆×C₃₂	C₁₆	1	8	yes	—
55	C₂×C₁₆×C₃₂	C₂×C₈	1	4	yes	—
56	C₁₆×C₆₄	C₁₆	1	16	yes	complemented
57	C₄×M₁₂₈	—	2	16	no	—
58	C₄×#1397	—	2	4	no	—
59	C₄×#1397	#117	1	4	yes	—
60	C₄²×C₃₂	C₄×C₈	1	1	yes	characteristic
61	C₄²×C₃₂	C₄×C₈	1	1	yes	characteristic
62	C₄²×C₃₂	C₄×C₈	1	2	yes	—
63	C₂×#7329	—	2	4	no	—
64	C₂×#7329	#97	1	4	yes	—
65	C₂×#7329	M₃₂	1	8	yes	—
66	C₂×#7329	C₂×M₁₆	1	4	yes	—
67	C₂×C₈×C₃₂	C₄×C₈	1	2	yes	—
68	C₂×C₈×C₃₂	C₄×C₈	1	2	yes	—
69	C₂×C₈×C₃₂	C₂×C₁₆	1	8	yes	—
70	C₂×C₈×C₃₂	C₂×C₁₆	1	4	yes	—
71	C₂×C₈×C₃₂	C₂×C₁₆	1	4	yes	—
72	C₂×C₈×C₃₂	C₂²×C₈	1	4	yes	—
73	#126530	—	2	4	no	—
74	#126530	#97	1	4	yes	—
75	C₄×#1311	#117	1	2	yes	—
76	C₄×#1311	C₂×M₁₆	1	1	yes	characteristic
77	C₄×C₈×C₁₆	C₂×C₄²	1	1	yes	characteristic
78	C₄×C₈×C₁₆	C₂²×C₈	1	2	yes	—
79	C₂×#7052	—	2	4	no	—
80	C₂×#7052	#97	1	4	yes	—
81	C₂×C₁₆²	C₄×C₈	1	4	yes	—
82	C₂×C₁₆²	C₂×C₁₆	1	8	yes	—
83	#94852	C₂×M₁₆	1	1	yes	characteristic
84	#94756	—	4	16	no	—
85	#94508	—	4	16	no	—
86	#94504	—	2	16	no	—
87	C₈×C₆₄	C₂×C₁₆	1	32	yes	—
88	#93990	—	4	4	no	—
89	#93990	—	4	4	no	—
90	#93990	—	4	4	no	—
91	#93666	—	4	8	no	—
92	#93666	—	4	16	no	—
93	#93664	—	2	16	no	—
94	#93664	M₃₂	1	16	yes	—
95	C₁₆×C₃₂	C₃₂	1	32	yes	—
96	C₁₆×C₃₂	C₂×C₁₆	1	8	yes	—
97	C₁₆×C₃₂	C₂×C₁₆	1	8	yes	—
98	#93249	—	4	2	no	—
99	#93249	—	4	4	no	—
100	#92896	—	4	2	no	—
101	#92896	—	4	4	no	—
102	C₂×M₁₂₈	—	4	16	no	—
103	C₂×#1397	—	2	8	no	—
104	C₂×#1397	—	4	4	no	—
105	C₂×#1397	—	4	4	no	—
106	C₂×C₄×C₃₂	—	2	8	no	—
107	C₂×C₄×C₃₂	C₈²	1	1	yes	characteristic
108	C₂×C₄×C₃₂	C₈²	1	2	yes	—
109	C₂×C₄×C₃₂	C₈²	1	1	yes	characteristic
110	C₂×C₄×C₃₂	#322	1	4	yes	—
111	C₂×C₄×C₃₂	C₄×C₁₆	1	2	yes	—
112	C₂×C₄×C₃₂	C₄×C₁₆	1	4	yes	—
113	C₂×C₄×C₃₂	C₄×C₁₆	1	2	yes	—
114	C₂×C₄×C₃₂	#346	1	4	yes	—
115	C₂×C₄×C₃₂	#346	1	4	yes	—
116	C₂×C₄×C₃₂	#431	1	8	yes	—
117	#12590	—	2	4	no	—
118	#12590	—	4	4	no	—
119	C₄²×C₁₆	#376	1	1	yes	characteristic
120	C₄²×C₁₆	C₂×C₄×C₈	1	1	yes	characteristic
121	C₄²×C₁₆	#431	1	2	yes	—
122	C₂×#1312	—	2	4	no	—
123	C₂×#1312	—	4	4	no	—
124	C₂×#1311	—	2	2	no	—
125	C₂×#1311	—	2	4	no	—
126	C₂×C₈×C₁₆	—	2	8	no	—
127	C₂×C₈×C₁₆	#322	1	8	yes	—
128	C₂×C₈×C₁₆	C₂×C₄×C₈	1	2	yes	—
129	C₂×C₈×C₁₆	C₂×#97	1	2	yes	—
130	C₂×C₈×C₁₆	C₂²×C₁₆	1	4	yes	—
131	C₂×C₈×C₁₆	C₂×M₃₂	1	4	yes	—
132	#8131	—	2	2	no	—
133	C₄×C₈²	C₂×#97	1	1	yes	characteristic
134	C₄×C₈²	#431	1	2	yes	—
135	#7510	—	4	16	no	—
136	C₄×C₆₄	—	2	32	no	—
137	#7329	—	2	16	no	—
138	#7329	—	2	8	no	—
139	#7329	—	2	8	no	—
140	#7329	—	4	8	no	—
141	C₈×C₃₂	—	2	8	no	—
142	C₈×C₃₂	C₄×C₁₆	1	4	yes	—
143	C₈×C₃₂	C₄×C₁₆	1	4	yes	—
144	C₈×C₃₂	#346	1	8	yes	—
145	C₈×C₃₂	C₂×C₃₂	1	16	yes	—
146	C₈×C₃₂	C₂×C₃₂	1	16	yes	—
147	C₈×C₃₂	M₆₄	1	16	yes	—
148	C₈×C₃₂	C₂²×C₁₆	1	8	yes	—
149	C₈×C₃₂	C₂×M₃₂	1	8	yes	—
150	#7052	—	2	8	no	—
151	#7052	—	4	8	no	—
152	C₁₆²	—	2	8	no	—
153	C₁₆²	C₄×C₁₆	1	8	yes	—
154	C₁₆²	#346	1	8	yes	—
155	C₁₆²	C₂×C₃₂	1	16	yes	—
156	C₂²×C₃₂	—	2	8	no	—
157	C₂²×C₃₂	—	4	8	no	—
158	C₂²×C₃₂	C₈×C₁₆	1	1	yes	characteristic
159	C₂²×C₃₂	C₈×C₁₆	1	1	yes	characteristic
160	C₂²×C₃₂	C₈×C₁₆	1	2	yes	—
161	C₂²×C₃₂	#1311	1	4	yes	—
162	C₂²×C₃₂	#1312	1	8	yes	—
163	C₂×C₄×C₁₆	—	2	8	no	—
164	C₂×C₄×C₁₆	—	2	2	no	—
165	C₂×C₄×C₁₆	—	2	2	no	—
166	C₂×C₄×C₁₆	—	4	8	no	—
167	C₂×C₄×C₁₆	C₂×C₈²	1	1	yes	characteristic
168	C₂×C₄×C₁₆	C₂×#322	1	1	yes	characteristic
169	C₂×C₄×C₁₆	#1744	1	2	yes	—
170	C₂×C₄×C₁₆	#1747	1	4	yes	—
171	C₂×C₄×C₁₆	C₂×C₄×C₁₆	1	2	yes	—
172	C₂×C₄×C₁₆	C₂×#346	1	2	yes	—
173	C₂×C₄×C₁₆	#2162	1	4	yes	—
174	C₄²×C₈	—	2	2	no	—
175	C₄²×C₈	#1744	1	1	yes	characteristic
176	C₄²×C₈	#2110	1	2	yes	—
177	C₂×C₈²	—	2	2	no	—
178	C₂×C₈²	C₂×#322	1	2	yes	—
179	C₂×C₈²	#2110	1	2	yes	—
180	C₂×C₈²	#2162	1	4	yes	—
181	M₁₂₈	—	8	32	no	—
182	C₂×C₆₄	—	4	32	no	—
183	#1397	—	8	8	no	—
184	#1397	—	8	8	no	—
185	#1397	—	8	8	no	—
186	C₄×C₃₂	—	2	8	no	—
187	C₄×C₃₂	—	2	16	no	—
188	C₄×C₃₂	—	2	8	no	—
189	C₄×C₃₂	—	2	8	no	—
190	C₄×C₃₂	—	4	8	no	—
191	C₄×C₃₂	C₈×C₁₆	1	2	yes	—
192	C₄×C₃₂	C₈×C₁₆	1	2	yes	—
193	C₄×C₃₂	C₈×C₁₆	1	4	yes	—
194	C₄×C₃₂	#1311	1	4	yes	—
195	C₄×C₃₂	#1311	1	4	yes	—
196	C₄×C₃₂	C₄×C₃₂	1	8	yes	—
197	C₄×C₃₂	C₄×C₃₂	1	8	yes	—
198	C₄×C₃₂	#1397	1	8	yes	—
199	C₄×C₃₂	#1397	1	8	yes	—
200	C₄×C₃₂	#1398	1	16	yes	—
201	C₄×C₃₂	#2162	1	16	yes	—
202	#1374	—	8	4	no	—
203	#1374	—	8	8	no	—
204	#1312	—	8	4	no	—
205	#1312	—	8	8	no	—
206	#1311	—	8	2	no	—
207	#1311	—	8	4	no	—
208	C₈×C₁₆	—	2	8	no	—
209	C₈×C₁₆	—	2	16	no	—
210	C₈×C₁₆	—	2	8	no	—
211	C₈×C₁₆	—	4	8	no	—
212	C₈×C₁₆	#1312	1	16	yes	—
213	C₈×C₁₆	C₂×C₄×C₁₆	1	4	yes	—
214	C₈×C₁₆	C₂×#346	1	4	yes	—
215	C₈×C₁₆	#2110	1	8	yes	—
216	C₈×C₁₆	C₂²×C₃₂	1	8	yes	—
217	C₈×C₁₆	C₂×M₆₄	1	8	yes	—
218	#1273	—	8	2	no	—
219	C₂²×C₁₆	—	2	4	no	—
220	C₂²×C₁₆	—	4	8	no	—
221	C₂²×C₁₆	—	8	8	no	—
222	C₂²×C₁₆	#8137	1	4	yes	—
223	C₂²×C₁₆	C₂×C₈×C₁₆	1	1	yes	characteristic
224	C₂²×C₁₆	C₂×#1311	1	1	yes	characteristic
225	C₂²×C₁₆	C₂×#1312	1	2	yes	—
226	C₂×C₄×C₈	—	2	4	no	—
227	C₂×C₄×C₈	—	4	2	no	—
228	C₂×C₄×C₈	—	4	4	no	—
229	C₂×C₄×C₈	#8131	1	1	yes	characteristic
230	C₂×C₄×C₈	#8137	1	2	yes	—
231	C₂×C₄×C₈	#11405	1	4	yes	—
232	C₂×C₄×C₈	#12590	1	2	yes	—
233	C₂×C₄×C₈	#12593	1	1	yes	characteristic
234	C₄³	#12593	1	1	yes	characteristic
235	C₂×C₃₂	—	2	8	no	—
236	C₂×C₃₂	—	2	8	no	—
237	C₂×C₃₂	—	2	16	no	—
238	C₂×C₃₂	—	4	16	no	—
239	C₂×C₃₂	—	4	16	no	—
240	C₂×C₃₂	—	8	8	no	—
241	C₂×C₃₂	—	8	8	no	—
242	C₂×C₃₂	C₁₆²	1	1	yes	characteristic
243	C₂×C₃₂	C₁₆²	1	2	yes	—
244	C₂×C₃₂	C₁₆²	1	1	yes	characteristic
245	C₂×C₃₂	#7052	1	4	yes	—
246	C₂×C₃₂	#7052	1	8	yes	—
247	C₂×C₃₂	C₈×C₃₂	1	4	yes	—
248	C₂×C₃₂	C₈×C₃₂	1	4	yes	—
249	C₂×C₃₂	#7329	1	4	yes	—
250	C₂×C₃₂	#7329	1	4	yes	—
251	C₂×C₃₂	#7330	1	16	yes	—
252	C₂×C₃₂	#7331	1	16	yes	—
253	C₄×C₁₆	—	2	4	no	—
254	C₄×C₁₆	—	2	16	no	—
255	C₄×C₁₆	—	4	4	no	—
256	C₄×C₁₆	—	4	4	no	—
257	C₄×C₁₆	—	4	4	no	—
258	C₄×C₁₆	—	4	8	no	—
259	C₄×C₁₆	—	4	16	no	—
260	C₄×C₁₆	—	8	8	no	—
261	C₄×C₁₆	C₂×C₈×C₁₆	1	2	yes	—
262	C₄×C₁₆	C₂×#1311	1	2	yes	—
263	C₄×C₁₆	#11405	1	4	yes	—
264	C₄×C₁₆	#12590	1	4	yes	—
265	C₄×C₁₆	#12593	1	4	yes	—
266	C₄×C₁₆	C₂×C₄×C₃₂	1	4	yes	—
267	C₄×C₁₆	C₂×#1397	1	4	yes	—
268	C₄×C₁₆	C₂×#1398	1	8	yes	—
269	C₄×C₁₆	#13639	1	8	yes	—
270	C₈²	—	2	4	no	—
271	C₈²	—	4	2	no	—
272	C₈²	—	4	4	no	—
273	C₈²	C₂×#1312	1	4	yes	—
274	C₈²	#11405	1	4	yes	—
275	C₈²	#13639	1	8	yes	—
276	C₆₄	—	8	32	no	—
277	C₂²×C₈	—	2	4	no	—
278	C₂²×C₈	—	4	4	no	—
279	C₂²×C₈	#96658	1	1	yes	characteristic
280	C₂²×C₈	#128203	1	1	yes	characteristic
281	C₂²×C₈	#128225	1	2	yes	—
282	C₂×C₄²	—	2	2	no	—
283	C₂×C₄²	#128206	1	1	yes	characteristic
284	C₂×C₄²	#128225	1	2	yes	—
285	C₂×C₁₆	—	2	8	no	—
286	C₂×C₁₆	—	2	8	no	—
287	C₂×C₁₆	—	4	4	no	—
288	C₂×C₁₆	—	4	4	no	—
289	C₂×C₁₆	—	4	16	no	—
290	C₂×C₁₆	—	8	16	no	—
291	C₂×C₁₆	—	16	8	no	—
292	C₂×C₁₆	—	16	16	no	—
293	C₂×C₁₆	C₂×C₁₆²	1	1	yes	characteristic
294	C₂×C₁₆	C₂×#7052	1	2	yes	—
295	C₂×C₁₆	C₂×#7052	1	1	yes	characteristic
296	C₂×C₁₆	#122214	1	4	yes	—
297	C₂×C₁₆	#128225	1	8	yes	—
298	C₂×C₁₆	C₂×C₈×C₃₂	1	2	yes	—
299	C₂×C₁₆	C₂×#7329	1	2	yes	—
300	C₂×C₁₆	C₂×#7330	1	4	yes	—
301	C₂×C₁₆	C₂×#7331	1	4	yes	—
302	C₂×C₁₆	#149502	1	4	yes	—
303	C₄×C₈	—	2	4	no	—
304	C₄×C₈	—	2	8	no	—
305	C₄×C₈	—	2	2	no	—
306	C₄×C₈	—	2	4	no	—
307	C₄×C₈	—	4	4	no	—
308	C₄×C₈	—	16	2	no	—
309	C₄×C₈	—	16	4	no	—
310	C₄×C₈	#122214	1	8	yes	—
311	C₄×C₈	#128203	1	2	yes	—
312	C₄×C₈	#128206	1	2	yes	—
313	C₄×C₈	#151876	1	4	yes	—
314	C₄×C₈	#151879	1	4	yes	—
315	C₃₂	—	2	16	no	—
316	C₃₂	—	4	16	no	—
317	C₃₂	—	8	16	no	—
318	C₃₂	C₁₆×C₃₂	1	2	yes	—
319	C₃₂	C₁₆×C₃₂	1	2	yes	—
320	C₃₂	#93664	1	4	yes	—
321	C₃₂	#93665	1	8	yes	—
322	C₃₂	#93666	1	16	yes	—
323	C₂²×C₄	—	2	2	no	—
324	C₂²×C₄	#72210645	1	1	yes	characteristic
325	C₂²×C₄	#72210687	1	2	yes	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	8	no	—
329	C₂×C₈	—	4	8	no	—
330	C₂×C₈	—	8	4	no	—
331	C₂×C₈	—	8	8	no	—
332	C₂×C₈	#72210645	1	2	yes	—
333	C₂×C₈	#72210650	1	4	yes	—
334	C₂×C₈	#72234227	1	1	yes	characteristic
335	C₂×C₈	#72234237	1	2	yes	—
336	C₂×C₈	#72234260	1	1	yes	characteristic
337	C₂×C₈	#72234277	1	4	yes	—
338	C₂×C₈	#72234287	1	2	yes	—
339	C₄²	—	2	2	no	—
340	C₄²	—	8	2	no	—
341	C₄²	#72210687	1	2	yes	—
342	C₄²	#72234227	1	2	yes	—
343	C₄²	#72234277	1	4	yes	—
344	C₁₆	—	2	8	no	—
345	C₁₆	—	4	8	no	—
346	C₁₆	—	8	16	no	—
347	C₁₆	—	16	16	no	—
348	C₁₆	C₂×C₁₆×C₃₂	1	1	yes	characteristic
349	C₁₆	C₂×#93664	1	1	yes	characteristic
350	C₁₆	C₂×#93666	1	4	yes	—
351	C₁₆	#72234279	1	8	yes	—
352	C₁₆	C₂×#93665	1	2	yes	—
353	C₂³	#72210694	1	1	yes	characteristic
354	C₂×C₄	—	2	4	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	4	4	no	—
357	C₂×C₄	#72210664	1	2	yes	—
358	C₂×C₄	#72210694	1	1	yes	characteristic
359	C₂×C₄	#72234261	1	1	yes	characteristic
360	C₂×C₄	#72234267	1	4	yes	—
361	C₂×C₄	#72234289	1	2	yes	—
362	C₈	—	2	4	no	—
363	C₈	—	4	8	no	—
364	C₈	—	8	8	no	—
365	C₈	#72210664	1	4	yes	—
366	C₈	#72234303	1	2	yes	—
367	C₈	#72234325	1	1	yes	characteristic
368	C₈	#72234335	1	1	yes	characteristic
369	C₂²	—	2	2	no	—
370	C₂²	#72234269	1	1	yes	characteristic
371	C₂²	#72234291	1	2	yes	—
372	C₄	—	2	4	no	—
373	C₄	—	4	4	no	—
374	C₄	#72234262	1	1	yes	characteristic
375	C₄	#72234291	1	2	yes	—
376	C₄	#72234340	1	1	yes	characteristic
377	C₂	—	2	2	no	—
378	C₂	#72234350	1	1	yes	characteristic
379	C₂	#72234408	1	2	yes	—

	subgroup	G/H
center	C₂×C₃₂	C₁₆²
commutator	C₁₆	C₂×C₁₆×C₃₂
Frattini, cosocle	#72234241	C₂³
socle	C₂²	#72234269
prime radical	C₂³	#72210694
LCS	G ▷ C₁₆ ▷ C₁
UCS	C₁ ◁ C₂×C₃₂ ◁ G
LECS	G ▷ #72234241 ▷ C₄²×C₁₆ ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
9. #72234393	C₁₆	semidirect
56. C₁₆×C₆₄	C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#72234407	2
#72234405	2
#72234276	2