Subgroups — #72234407

Order 8192.

1026 conjugacy classes of subgroups.

The subgroup lattice has 373 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72234392	C₂	1	2	yes	maximal
2	#72234377	—	2	2	no	—
3	#72234377	—	2	2	no	—
4	#72234371	C₄	1	4	yes	—
5	#72234367	—	2	4	no	—
6	C₈×M₁₂₈	C₈	1	8	yes	—
7	#72234363	—	2	4	no	—
8	#72234269	C₂	1	1	yes	maximal, characteristic
9	#72234265	—	2	2	no	—
10	#72234241	C₄	1	1	yes	characteristic
11	#72234241	C₂²	1	1	yes	characteristic
12	#72234237	—	2	1	no	—
13	#72234237	—	2	1	no	—
14	#72234237	D₄	1	2	yes	—
15	#72234237	D₄	1	1	yes	characteristic
16	#72234237	Q₈	1	1	yes	characteristic
17	C₈×#1397	C₈	1	2	yes	—
18	C₈×#1397	C₂×C₄	1	2	yes	—
19	#72234231	C₄	1	1	yes	characteristic
20	#72234227	—	2	1	no	—
21	#72234226	C₈	1	2	yes	—
22	#72234224	C₂×C₄	1	1	yes	characteristic
23	#72210645	—	2	1	no	—
24	C₄×M₁₂₈	—	2	8	no	—
25	C₄×#1397	—	2	2	no	—
26	C₄×#1397	C₂²⋊C₄	1	2	yes	—
27	C₄×#1397	C₄⋊C₄	1	2	yes	—
28	C₂×#7329	—	2	2	no	—
29	C₂×#7329	M₁₆	1	4	yes	—
30	C₂×C₈×C₃₂	C₁₆	1	4	yes	—
31	C₂×C₈×C₃₂	C₂×C₈	1	4	yes	—
32	#126530	—	2	2	no	—
33	C₄×#1311	C₂²⋊C₄	1	1	yes	characteristic
34	C₄×#1311	C₄⋊C₄	1	1	yes	characteristic
35	C₄×#1311	M₁₆	1	1	yes	characteristic
36	C₄×C₈×C₁₆	C₄²	1	1	yes	characteristic
37	C₄×C₈×C₁₆	C₂×C₈	1	1	yes	characteristic
38	C₄×C₈×C₁₆	C₂×C₈	1	1	yes	characteristic
39	C₂×#7052	—	2	2	no	—
40	C₂×C₁₆²	C₁₆	1	4	yes	—
41	#94852	M₁₆	1	1	yes	characteristic
42	#94756	—	4	4	no	—
43	#94756	—	4	4	no	—
44	#94508	—	4	8	no	—
45	#94504	—	2	8	no	—
46	C₈×C₆₄	C₁₆	1	16	yes	complemented
47	#93990	—	4	2	no	—
48	#93990	—	4	1	no	—
49	#93990	—	4	1	no	—
50	#93990	—	4	2	no	—
51	#93990	—	4	2	no	—
52	#93249	—	4	2	no	—
53	#92896	—	4	2	no	—
54	C₂×M₁₂₈	—	4	8	no	—
55	C₂×#1397	—	2	2	no	—
56	C₂×#1397	—	2	2	no	—
57	C₂×#1397	—	2	2	no	—
58	C₂×#1397	—	2	2	no	—
59	C₂×#1397	—	4	2	no	—
60	C₂×#1397	—	4	2	no	—
61	C₂×C₄×C₃₂	—	2	4	no	—
62	C₂×C₄×C₃₂	C₂²⋊C₈	1	4	yes	—
63	C₂×C₄×C₃₂	C₄⋊C₈	1	4	yes	—
64	#12590	—	4	2	no	—
65	C₄²×C₁₆	#95	1	1	yes	characteristic
66	C₄²×C₁₆	C₄×C₈	1	1	yes	characteristic
67	C₄²×C₁₆	C₂²⋊C₈	1	1	yes	characteristic
68	C₄²×C₁₆	C₄⋊C₈	1	1	yes	characteristic
69	C₂×#1312	—	4	2	no	—
70	C₂×#1311	—	2	2	no	—
71	C₂×#1311	—	2	2	no	—
72	C₂×#1311	—	2	1	no	—
73	C₂×#1311	—	2	1	no	—
74	C₂×C₈×C₁₆	—	2	4	no	—
75	C₂×C₈×C₁₆	C₄×C₈	1	2	yes	—
76	C₂×C₈×C₁₆	#97	1	2	yes	—
77	C₂×C₈×C₁₆	C₂×C₁₆	1	2	yes	—
78	C₂×C₈×C₁₆	C₂×C₁₆	1	2	yes	—
79	C₂×C₈×C₁₆	M₃₂	1	2	yes	—
80	C₂×C₈×C₁₆	M₃₂	1	2	yes	—
81	#8131	—	2	2	no	—
82	C₄×C₈²	#97	1	1	yes	characteristic
83	C₄×C₈²	C₂²⋊C₈	1	1	yes	characteristic
84	C₄×C₈²	C₄⋊C₈	1	1	yes	characteristic
85	#7510	—	4	8	no	—
86	C₄×C₆₄	—	2	16	no	—
87	#7329	—	2	8	no	—
88	#7329	—	4	4	no	—
89	C₈×C₃₂	—	2	4	no	—
90	C₈×C₃₂	C₃₂	1	8	yes	—
91	C₈×C₃₂	C₂×C₁₆	1	8	yes	—
92	C₈×C₃₂	M₃₂	1	8	yes	—
93	#7052	—	4	4	no	—
94	C₁₆²	—	2	4	no	—
95	C₁₆²	C₃₂	1	8	yes	—
96	C₂²×C₃₂	—	2	4	no	—
97	C₂²×C₃₂	—	2	4	no	—
98	C₂²×C₃₂	—	4	4	no	—
99	C₂×C₄×C₁₆	—	2	1	no	—
100	C₂×C₄×C₁₆	—	2	1	no	—
101	C₂×C₄×C₁₆	—	2	1	no	—
102	C₂×C₄×C₁₆	—	2	1	no	—
103	C₂×C₄×C₁₆	—	4	4	no	—
104	C₂×C₄×C₁₆	C₈²	1	1	yes	characteristic
105	C₂×C₄×C₁₆	#322	1	1	yes	characteristic
106	C₂×C₄×C₁₆	#336	1	2	yes	—
107	C₂×C₄×C₁₆	H(3,ℤ/4)	1	2	yes	—
108	C₂×C₄×C₁₆	H(3,ℤ/4)	1	1	yes	characteristic
109	C₂×C₄×C₁₆	#338	1	1	yes	characteristic
110	C₂×C₄×C₁₆	C₄×C₁₆	1	1	yes	characteristic
111	C₂×C₄×C₁₆	C₄×C₁₆	1	1	yes	characteristic
112	C₂×C₄×C₁₆	#346	1	1	yes	characteristic
113	C₂×C₄×C₁₆	#346	1	1	yes	characteristic
114	C₂×C₄×C₁₆	C₂²⋊C₁₆	1	2	yes	—
115	C₂×C₄×C₁₆	C₄⋊C₁₆	1	2	yes	—
116	C₄²×C₈	—	2	1	no	—
117	C₄²×C₈	—	2	1	no	—
118	C₄²×C₈	#336	1	1	yes	characteristic
119	C₄²×C₈	#347	1	2	yes	—
120	C₂×C₈²	—	2	1	no	—
121	C₂×C₈²	—	2	1	no	—
122	C₂×C₈²	#322	1	2	yes	—
123	C₂×C₈²	#347	1	2	yes	—
124	C₂×C₈²	C₂²⋊C₁₆	1	2	yes	—
125	C₂×C₈²	C₄⋊C₁₆	1	2	yes	—
126	M₁₂₈	—	8	8	no	—
127	M₁₂₈	—	8	8	no	—
128	C₂×C₆₄	—	4	16	no	—
129	#1397	—	8	2	no	—
130	#1397	—	8	2	no	—
131	#1397	—	8	2	no	—
132	#1397	—	8	2	no	—
133	#1397	—	8	4	no	—
134	#1397	—	8	4	no	—
135	C₄×C₃₂	—	2	4	no	—
136	C₄×C₃₂	—	2	4	no	—
137	C₄×C₃₂	—	2	8	no	—
138	C₄×C₃₂	—	4	4	no	—
139	C₄×C₃₂	C₂²⋊C₁₆	1	8	yes	—
140	C₄×C₃₂	C₄⋊C₁₆	1	8	yes	—
141	#1374	—	8	4	no	—
142	#1312	—	8	4	no	—
143	#1311	—	8	2	no	—
144	#1311	—	8	2	no	—
145	#1311	—	8	1	no	—
146	#1311	—	8	1	no	—
147	C₈×C₁₆	—	2	4	no	—
148	C₈×C₁₆	—	2	4	no	—
149	C₈×C₁₆	—	2	8	no	—
150	C₈×C₁₆	—	4	4	no	—
151	C₈×C₁₆	C₄×C₁₆	1	4	yes	—
152	C₈×C₁₆	#346	1	4	yes	—
153	C₈×C₁₆	#347	1	8	yes	—
154	C₈×C₁₆	C₂×C₃₂	1	4	yes	—
155	C₈×C₁₆	C₂×C₃₂	1	4	yes	—
156	C₈×C₁₆	M₆₄	1	4	yes	—
157	C₈×C₁₆	M₆₄	1	4	yes	—
158	#1273	—	8	2	no	—
159	C₂²×C₁₆	—	2	2	no	—
160	C₂²×C₁₆	—	2	2	no	—
161	C₂²×C₁₆	—	8	4	no	—
162	C₂²×C₁₆	#1274	1	2	yes	—
163	C₂²×C₁₆	#1275	1	2	yes	—
164	C₂²×C₁₆	C₈×C₁₆	1	1	yes	characteristic
165	C₂²×C₁₆	#1311	1	1	yes	characteristic
166	C₂²×C₁₆	#1312	1	1	yes	characteristic
167	C₂²×C₁₆	#1312	1	1	yes	characteristic
168	C₂×C₄×C₈	—	2	2	no	—
169	C₂×C₄×C₈	—	2	2	no	—
170	C₂×C₄×C₈	—	4	2	no	—
171	C₂×C₄×C₈	—	4	2	no	—
172	C₂×C₄×C₈	—	4	2	no	—
173	C₂×C₄×C₈	#1273	1	1	yes	characteristic
174	C₂×C₄×C₈	#1274	1	1	yes	characteristic
175	C₂×C₄×C₈	#1275	1	1	yes	characteristic
176	C₂×C₄×C₈	#1313	1	4	yes	—
177	C₂×C₄×C₈	#1374	1	1	yes	characteristic
178	C₂×C₄×C₈	#1374	1	1	yes	characteristic
179	C₂×C₄×C₈	#1375	1	1	yes	characteristic
180	C₄³	#1375	1	1	yes	characteristic
181	C₂×C₃₂	—	2	8	no	—
182	C₂×C₃₂	—	2	8	no	—
183	C₂×C₃₂	—	4	4	no	—
184	C₂×C₃₂	—	4	4	no	—
185	C₂×C₃₂	—	4	4	no	—
186	C₂×C₃₂	—	4	4	no	—
187	C₂×C₃₂	—	4	8	no	—
188	C₂×C₃₂	—	8	4	no	—
189	C₂×C₃₂	—	8	4	no	—
190	C₄×C₁₆	—	2	2	no	—
191	C₄×C₁₆	—	2	2	no	—
192	C₄×C₁₆	—	4	2	no	—
193	C₄×C₁₆	—	4	1	no	—
194	C₄×C₁₆	—	4	1	no	—
195	C₄×C₁₆	—	4	1	no	—
196	C₄×C₁₆	—	4	1	no	—
197	C₄×C₁₆	—	4	2	no	—
198	C₄×C₁₆	—	4	2	no	—
199	C₄×C₁₆	—	4	2	no	—
200	C₄×C₁₆	—	4	8	no	—
201	C₄×C₁₆	—	8	4	no	—
202	C₄×C₁₆	C₈×C₁₆	1	2	yes	—
203	C₄×C₁₆	#1311	1	2	yes	—
204	C₄×C₁₆	#1313	1	4	yes	—
205	C₄×C₁₆	#1374	1	4	yes	—
206	C₄×C₁₆	#1375	1	4	yes	—
207	C₄×C₁₆	C₄×C₃₂	1	2	yes	—
208	C₄×C₁₆	C₄×C₃₂	1	2	yes	—
209	C₄×C₁₆	#1397	1	2	yes	—
210	C₄×C₁₆	#1397	1	2	yes	—
211	C₄×C₁₆	#1398	1	4	yes	—
212	C₄×C₁₆	#1398	1	4	yes	—
213	C₄×C₁₆	C₂²⋊C₃₂	1	4	yes	—
214	C₄×C₁₆	C₄⋊C₃₂	1	4	yes	—
215	C₈²	—	2	2	no	—
216	C₈²	—	2	2	no	—
217	C₈²	—	4	2	no	—
218	C₈²	—	4	2	no	—
219	C₈²	—	4	1	no	—
220	C₈²	—	4	1	no	—
221	C₈²	#1312	1	4	yes	—
222	C₈²	#1313	1	4	yes	—
223	C₈²	C₂²⋊C₃₂	1	4	yes	—
224	C₈²	C₄⋊C₃₂	1	4	yes	—
225	C₆₄	—	8	16	no	—
226	C₂²×C₈	—	2	4	no	—
227	C₂²×C₈	—	4	2	no	—
228	C₂²×C₈	—	4	2	no	—
229	C₂²×C₈	#7014	1	1	yes	characteristic
230	C₂²×C₈	#7136	1	1	yes	characteristic
231	C₂²×C₈	#7142	1	1	yes	characteristic
232	C₂²×C₈	#7145	1	1	yes	characteristic
233	C₂×C₄²	—	2	2	no	—
234	C₂×C₄²	#7137	1	1	yes	characteristic
235	C₂×C₄²	#7142	1	1	yes	characteristic
236	C₂×C₄²	#7145	1	1	yes	characteristic
237	C₂×C₁₆	—	2	2	no	—
238	C₂×C₁₆	—	2	2	no	—
239	C₂×C₁₆	—	2	2	no	—
240	C₂×C₁₆	—	2	2	no	—
241	C₂×C₁₆	—	2	4	no	—
242	C₂×C₁₆	—	2	4	no	—
243	C₂×C₁₆	—	4	2	no	—
244	C₂×C₁₆	—	4	2	no	—
245	C₂×C₁₆	—	4	2	no	—
246	C₂×C₁₆	—	4	2	no	—
247	C₂×C₁₆	—	8	8	no	—
248	C₂×C₁₆	—	16	8	no	—
249	C₂×C₁₆	C₁₆²	1	1	yes	characteristic
250	C₂×C₁₆	#7052	1	1	yes	characteristic
251	C₂×C₁₆	#7052	1	2	yes	—
252	C₂×C₁₆	#7053	1	2	yes	—
253	C₂×C₁₆	#7053	1	2	yes	—
254	C₂×C₁₆	#7142	1	4	yes	—
255	C₂×C₁₆	#7145	1	4	yes	—
256	C₂×C₁₆	C₈×C₃₂	1	1	yes	characteristic
257	C₂×C₁₆	C₈×C₃₂	1	1	yes	characteristic
258	C₂×C₁₆	#7329	1	1	yes	characteristic
259	C₂×C₁₆	#7329	1	1	yes	characteristic
260	C₂×C₁₆	#7330	1	2	yes	—
261	C₂×C₁₆	#7330	1	2	yes	—
262	C₂×C₁₆	#7331	1	2	yes	—
263	C₂×C₁₆	#7331	1	2	yes	—
264	C₂×C₁₆	#7332	1	2	yes	—
265	C₂×C₁₆	#7332	1	2	yes	—
266	C₄×C₈	—	2	2	no	—
267	C₄×C₈	—	2	1	no	—
268	C₄×C₈	—	2	1	no	—
269	C₄×C₈	—	2	2	no	—
270	C₄×C₈	—	2	4	no	—
271	C₄×C₈	—	2	4	no	—
272	C₄×C₈	—	2	4	no	—
273	C₄×C₈	—	4	2	no	—
274	C₄×C₈	—	4	2	no	—
275	C₄×C₈	—	16	2	no	—
276	C₄×C₈	—	16	2	no	—
277	C₄×C₈	—	16	2	no	—
278	C₄×C₈	#7053	1	8	yes	—
279	C₄×C₈	#7136	1	2	yes	—
280	C₄×C₈	#7137	1	2	yes	—
281	C₄×C₈	#7465	1	2	yes	—
282	C₄×C₈	#7465	1	2	yes	—
283	C₄×C₈	#7466	1	2	yes	—
284	C₄×C₈	#7466	1	2	yes	—
285	C₃₂	—	4	8	no	—
286	C₃₂	—	4	8	no	—
287	C₃₂	—	8	8	no	—
288	C₂²×C₄	—	2	1	no	—
289	C₂²×C₄	—	2	1	no	—
290	C₂²×C₄	#93249	1	1	yes	characteristic
291	C₂²×C₄	#93995	1	2	yes	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	2	8	no	—
297	C₂×C₈	—	4	4	no	—
298	C₂×C₈	—	4	4	no	—
299	C₂×C₈	—	8	4	no	—
300	C₂×C₈	—	8	4	no	—
301	C₂×C₈	—	8	4	no	—
302	C₂×C₈	#92895	1	1	yes	characteristic
303	C₂×C₈	#92896	1	1	yes	characteristic
304	C₂×C₈	#93249	1	2	yes	—
305	C₂×C₈	H(3,ℤ/8)	1	1	yes	characteristic
306	C₂×C₈	H(3,ℤ/8)	1	2	yes	—
307	C₂×C₈	#93255	1	1	yes	characteristic
308	C₂×C₈	#93989	1	1	yes	characteristic
309	C₂×C₈	#93989	1	1	yes	characteristic
310	C₂×C₈	#93990	1	1	yes	characteristic
311	C₂×C₈	#93990	1	1	yes	characteristic
312	C₂×C₈	#93998	1	2	yes	—
313	C₂×C₈	#94005	1	2	yes	—
314	C₄²	—	2	1	no	—
315	C₄²	—	2	1	no	—
316	C₄²	—	8	2	no	—
317	C₄²	#92896	1	2	yes	—
318	C₄²	#93995	1	2	yes	—
319	C₄²	#93998	1	2	yes	—
320	C₄²	#94005	1	2	yes	—
321	C₁₆	—	2	4	no	—
322	C₁₆	—	2	4	no	—
323	C₁₆	—	4	4	no	—
324	C₁₆	—	4	4	no	—
325	C₁₆	—	16	8	no	—
326	C₁₆	C₁₆×C₃₂	1	1	yes	characteristic
327	C₁₆	#93664	1	1	yes	characteristic
328	C₁₆	#93665	1	1	yes	characteristic
329	C₁₆	#93665	1	1	yes	characteristic
330	C₁₆	#93666	1	4	yes	—
331	C₁₆	#93667	1	4	yes	—
332	C₁₆	#93667	1	4	yes	—
333	C₂³	#72210682	1	1	yes	characteristic
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	#72210629	1	1	yes	characteristic
340	C₂×C₄	#72210635	1	1	yes	characteristic
341	C₂×C₄	#72210682	1	1	yes	characteristic
342	C₂×C₄	#72210714	1	1	yes	characteristic
343	C₂×C₄	#72234264	1	4	yes	—
344	C₂×C₄	#72234282	1	1	yes	characteristic
345	C₂×C₄	#72234282	1	1	yes	characteristic
346	C₈	—	2	2	no	—
347	C₈	—	2	2	no	—
348	C₈	—	4	8	no	—
349	C₈	—	8	4	no	—
350	C₈	—	8	4	no	—
351	C₈	#72210629	1	2	yes	—
352	C₈	#72210635	1	2	yes	—
353	C₈	#72234299	1	1	yes	characteristic
354	C₈	#72234299	1	1	yes	characteristic
355	C₈	#72234310	1	1	yes	characteristic
356	C₈	#72234311	1	1	yes	characteristic
357	C₂²	—	2	2	no	—
358	C₂²	#72234265	1	1	yes	characteristic
359	C₂²	#72234283	1	1	yes	characteristic
360	C₂²	#72234284	1	1	yes	characteristic
361	C₄	—	2	4	no	—
362	C₄	—	4	2	no	—
363	C₄	—	4	2	no	—
364	C₄	#72210719	1	1	yes	characteristic
365	C₄	#72234283	1	1	yes	characteristic
366	C₄	#72234284	1	1	yes	characteristic
367	C₄	#72234312	1	1	yes	characteristic
368	C₂	—	2	1	no	—
369	C₂	—	2	1	no	—
370	C₂	#72234313	1	1	yes	characteristic
371	C₂	#72234406	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₁₆²
commutator	C₁₆	C₁₆×C₃₂
Frattini, cosocle	#72234241	C₂²
socle	C₂²	#72234265
prime radical	C₂³	#72210682
LCS	G ▷ C₁₆ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ 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
46. C₈×C₆₄	C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#72234392	2
#72234269	2