Subgroups — #14746114

Order 2816.

590 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13081667	C₂	1	2	yes	maximal, complemented
2	#13081629	C₂	1	1	yes	maximal, characteristic
3	#13081508	C₂	1	1	yes	maximal, characteristic, complemented
4	#13068208	C₂	1	1	yes	maximal, characteristic
5	#13068089	C₂	1	2	yes	maximal, complemented
6	#13068010	C₂	1	1	yes	maximal, characteristic, complemented
7	#13067976	C₂	1	1	yes	maximal, characteristic
8	#13067804	C₂	1	2	yes	maximal, complemented
9	#13067714	C₂	1	2	yes	maximal, complemented
10	#13067514	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×C₁₁×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#10623116	C₂²	1	2	yes	—
13	#10623098	C₂²	1	2	yes	—
14	C₈×Q₈×C₁₁	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₁₁	C₂²	1	1	yes	characteristic
16	C₄×C₁₁×Q₁₆	C₂²	1	1	yes	characteristic
17	#10622803	—	2	2	no	—
18	#10622644	C₂²	1	2	yes	—
19	#10622640	C₂²	1	2	yes	—
20	#10622502	—	2	2	no	—
21	#10622464	C₂²	1	2	yes	—
22	#10622461	C₂²	1	2	yes	—
23	#10622457	—	2	2	no	—
24	#10622442	—	2	2	no	—
25	#10622431	—	2	2	no	—
26	#10622411	—	2	2	no	—
27	C₄×Dic₄₄	—	2	1	no	—
28	C₄×Dic₄₄	C₂²	1	1	yes	characteristic
29	#10622309	—	2	1	no	—
30	#10622308	C₂²	1	1	yes	characteristic
31	#10622307	C₂²	1	2	yes	—
32	#10622306	C₂²	1	2	yes	—
33	C₄×D₈₈	—	2	1	no	—
34	#10622296	—	2	1	no	—
35	#10622295	C₂²	1	1	yes	characteristic
36	#10622292	C₂²	1	2	yes	—
37	C₁₁×#335	C₂²	1	1	yes	characteristic
38	C₁₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
39	C₁₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
40	C₈²×C₁₁	C₂²	1	1	yes	characteristic
41	#10622124	C₂²	1	2	yes	—
42	#10622123	C₂²	1	2	yes	—
43	#10622096	C₂²	1	1	yes	characteristic
44	#10622094	C₂²	1	2	yes	—
45	C₂×C₁₁×Q₁₆	D₄	1	1	yes	characteristic
46	C₄×Q₈×C₁₁	C₂³	1	1	yes	characteristic
47	C₄×Q₈×C₁₁	C₂³	1	1	yes	characteristic
48	#67000	—	2	2	no	—
49	#66997	—	2	2	no	—
50	C₂×#4326	—	2	2	no	—
51	C₂×#4325	—	2	2	no	—
52	C₂×Dic₄₄	D₄	1	1	yes	characteristic
53	C₂×D₈₈	—	2	1	no	—
54	C₂×D₈₈	—	2	1	no	—
55	C₂×#4314	—	2	2	no	—
56	C₂×#4314	—	2	2	no	—
57	C₂×#4314	—	2	2	no	—
58	#66947	—	4	2	no	—
59	#66946	—	2	2	no	—
60	#66942	—	4	2	no	—
61	#66940	—	2	2	no	—
62	#66928	—	2	2	no	—
63	#66927	C₂³	1	2	yes	—
64	C₄×D₄₄	—	2	2	no	—
65	#66923	—	2	2	no	—
66	#66922	C₂³	1	2	yes	—
67	C₄×Dic₂₂	—	2	2	no	—
68	C₄×Dic₂₂	C₂³	1	2	yes	—
69	C₁₁×#107	D₄	1	1	yes	characteristic
70	C₁₁×C₄⋊C₈	—	2	1	no	—
71	C₁₁×C₄⋊C₈	—	2	1	no	—
72	C₁₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
73	C₁₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
74	C₁₁×#103	D₄	1	1	yes	characteristic
75	C₁₁×#103	D₄	1	1	yes	characteristic
76	C₄×C₈×C₁₁	—	2	1	no	—
77	C₄×C₈×C₁₁	—	2	1	no	—
78	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
81	#66885	—	2	2	no	—
82	#66885	—	4	2	no	—
83	#66882	—	2	1	no	—
84	#66882	—	2	1	no	—
85	#66882	D₄	1	1	yes	characteristic
86	#66881	—	2	2	no	—
87	#66880	—	2	2	no	—
88	#66880	D₄	1	2	yes	—
89	#66879	—	2	2	no	—
90	Dic₁₁⋊C₈	—	2	2	no	—
91	#66874	—	2	2	no	—
92	#66873	—	2	2	no	—
93	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
94	#19966	—	11	1	no	maximal, Sylow
95	C₂×Q₈×C₁₁	C₂×D₄	1	1	yes	characteristic
96	C₂×Q₈×C₁₁	C₂×D₄	1	1	yes	characteristic
97	C₂×D₄₄	—	2	2	no	—
98	C₂×D₄₄	—	2	2	no	—
99	C₂×D₄₄	—	2	2	no	—
100	C₂×C₄×D₁₁	—	4	2	no	—
101	C₂×Dic₂₂	—	2	2	no	—
102	C₂×Dic₂₂	C₂×D₄	1	2	yes	—
103	C₁₁×Q₁₆	—	2	2	no	—
104	C₂×C₈×C₁₁	—	2	1	no	—
105	C₂×C₈×C₁₁	—	2	1	no	—
106	C₂×C₈×C₁₁	—	2	1	no	—
107	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
111	C₁₁×C₄⋊C₄	—	2	1	no	—
112	C₁₁×C₄⋊C₄	—	2	1	no	—
113	C₁₁×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
114	C₁₁×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₁₁	—	2	1	no	—
116	C₄²×C₁₁	—	2	1	no	—
117	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
118	#4326	—	8	2	no	—
119	#4325	—	8	2	no	—
120	(C₂×D₁₁)⋊C₄	—	4	2	no	—
121	(C₂×D₁₁)⋊C₄	—	4	2	no	—
122	(C₂×D₁₁)⋊C₄	—	4	2	no	—
123	C₄⋊Dic₁₁	—	2	2	no	—
124	C₄⋊Dic₁₁	—	2	2	no	—
125	C₄⋊Dic₁₁	C₂×D₄	1	2	yes	—
126	Dic₁₁⋊C₄	—	2	2	no	—
127	Dic₁₁⋊C₄	—	4	2	no	—
128	Dic₁₁⋊C₄	—	4	2	no	—
129	Dic₁₁⋊C₄	—	4	2	no	—
130	C₄×Dic₁₁	—	2	2	no	—
131	C₄×Dic₁₁	—	4	2	no	—
132	C₂×C₁₁⋊C₈	—	2	2	no	—
133	Dic₄₄	—	2	2	no	—
134	D₈₈	—	4	2	no	—
135	D₈₈	—	8	1	no	—
136	#4314	—	4	4	no	—
137	#4314	—	8	2	no	—
138	#4314	—	8	2	no	—
139	#3400	—	11	2	no	—
140	#3396	—	11	1	no	—
141	#3383	—	11	1	no	—
142	#1706	—	11	1	no	—
143	#1686	—	11	2	no	—
144	#1674	—	11	1	no	—
145	#1668	—	11	1	no	—
146	#1640	—	11	2	no	—
147	#1626	—	11	2	no	—
148	#1591	—	11	1	no	—
149	C₈×Q₁₆	D₁₁	1	1	yes	characteristic, complemented
150	C₂²×D₁₁	—	4	2	no	—
151	Q₈×C₁₁	—	2	1	no	—
152	Q₈×C₁₁	—	2	1	no	—
153	Q₈×C₁₁	—	2	1	no	—
154	Q₈×C₁₁	—	2	1	no	—
155	C₂×C₄×C₁₁	—	2	1	no	—
156	C₂×C₄×C₁₁	—	2	1	no	—
157	C₂×C₄×C₁₁	—	2	1	no	—
158	C₂×C₄×C₁₁	—	2	1	no	—
159	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂×Dic₁₁	—	2	2	no	—
163	C₂×Dic₁₁	—	2	2	no	—
164	C₂×Dic₁₁	—	4	2	no	—
165	D₄₄	—	4	2	no	—
166	D₄₄	—	4	2	no	—
167	D₄₄	—	4	2	no	—
168	D₄₄	—	4	2	no	—
169	D₄₄	—	8	2	no	—
170	C₄×D₁₁	—	16	2	no	—
171	Dic₂₂	—	2	2	no	—
172	Dic₂₂	—	2	2	no	—
173	Dic₂₂	—	4	2	no	—
174	Dic₂₂	—	4	2	no	—
175	C₈×C₁₁	—	2	1	no	—
176	C₈×C₁₁	—	2	1	no	—
177	C₈×C₁₁	—	4	1	no	—
178	C₈×C₁₁	—	4	1	no	—
179	C₈×C₁₁	—	4	1	no	—
180	C₈×C₁₁	#135	1	2	yes	—
181	C₈×C₁₁	#137	1	2	yes	—
182	C₁₁⋊C₈	—	4	2	no	—
183	#559	—	11	2	no	—
184	#557	—	11	2	no	—
185	#500	—	11	1	no	—
186	#499	—	22	1	no	—
187	#497	—	22	2	no	—
188	#495	—	22	1	no	—
189	#493	—	11	1	no	—
190	#492	—	11	2	no	—
191	#486	—	11	2	no	—
192	#479	—	22	2	no	—
193	#475	—	11	2	no	—
194	#464	—	22	2	no	—
195	#463	—	22	2	no	—
196	#462	—	11	2	no	—
197	#461	—	22	2	no	—
198	#460	—	11	2	no	—
199	#460	—	11	2	no	—
200	#446	—	11	2	no	—
201	C₈×Q₈	C₂×D₁₁	1	1	yes	characteristic
202	C₈×Q₈	C₂×D₁₁	1	1	yes	characteristic
203	#440	—	22	2	no	—
204	C₄×Q₁₆	—	11	1	no	—
205	C₄×Q₁₆	—	22	1	no	—
206	C₄×Q₁₆	C₂×D₁₁	1	1	yes	characteristic
207	C₄×D₈	—	22	1	no	—
208	#335	—	11	1	no	—
209	#335	C₂×D₁₁	1	1	yes	characteristic
210	#332	—	11	2	no	—
211	#331	—	11	2	no	—
212	Q₈⋊C₈	—	11	2	no	—
213	Q₈⋊C₈	C₂×D₁₁	1	1	yes	characteristic
214	Q₈⋊C₈	C₂×D₁₁	1	1	yes	characteristic
215	C₈²	C₂×D₁₁	1	1	yes	characteristic
216	C₂²×C₁₁	D₄²	1	1	yes	characteristic
217	C₂×D₁₁	—	8	2	no	—
218	C₂×D₁₁	—	8	2	no	—
219	C₂×D₁₁	—	8	2	no	—
220	C₄×C₁₁	—	2	1	no	—
221	C₄×C₁₁	—	4	1	no	—
222	C₄×C₁₁	—	4	1	no	—
223	C₄×C₁₁	—	4	1	no	—
224	C₄×C₁₁	—	4	1	no	—
225	C₄×C₁₁	C₂×#135	1	1	yes	characteristic
226	C₄×C₁₁	C₂×#137	1	1	yes	characteristic
227	C₄×C₁₁	#576	1	1	yes	characteristic
228	C₄×C₁₁	#577	1	1	yes	characteristic
229	Dic₁₁	—	4	2	no	—
230	Dic₁₁	—	4	2	no	—
231	Dic₁₁	—	8	2	no	—
232	C₂×Q₁₆	—	11	1	no	—
233	C₂×Q₁₆	—	22	2	no	—
234	C₂×Q₁₆	D₄₄	1	1	yes	characteristic
235	C₂×SD₁₆	—	22	2	no	—
236	C₂×SD₁₆	—	22	2	no	—
237	C₂×SD₁₆	—	22	2	no	—
238	C₂×SD₁₆	—	22	2	no	—
239	C₂×D₈	—	22	1	no	—
240	C₂×D₈	—	22	1	no	—
241	C₄⋊Q₈	—	11	2	no	—
242	C₄⋊Q₈	—	22	2	no	—
243	#127	—	11	2	no	—
244	#125	—	22	2	no	—
245	#125	—	44	2	no	—
246	#124	—	22	2	no	—
247	#124	—	22	2	no	—
248	#123	—	44	2	no	—
249	C₄×Q₈	—	11	2	no	—
250	C₄×Q₈	—	22	2	no	—
251	C₄×Q₈	—	22	2	no	—
252	C₄×Q₈	C₂²×D₁₁	1	1	yes	characteristic
253	C₄×Q₈	C₂²×D₁₁	1	1	yes	characteristic
254	C₄×D₄	—	22	2	no	—
255	C₄×D₄	—	22	2	no	—
256	#107	—	11	1	no	—
257	#107	—	22	1	no	—
258	#107	—	22	1	no	—
259	#107	D₄₄	1	1	yes	characteristic
260	#106	—	22	2	no	—
261	C₄⋊C₈	—	2	1	no	—
262	C₄⋊C₈	—	2	1	no	—
263	C₄⋊C₈	—	11	2	no	—
264	C₄⋊C₈	—	22	2	no	—
265	C₄⋊C₈	C₂²×D₁₁	1	1	yes	characteristic
266	C₄⋊C₈	C₂²×D₁₁	1	1	yes	characteristic
267	#103	—	11	2	no	—
268	#103	—	22	2	no	—
269	#103	—	22	2	no	—
270	#103	D₄₄	1	1	yes	characteristic
271	#103	D₄₄	1	1	yes	characteristic
272	#102	—	22	2	no	—
273	#102	—	22	2	no	—
274	#102	—	44	2	no	—
275	#97	—	22	2	no	—
276	C₄×C₈	—	2	1	no	—
277	C₄×C₈	—	2	1	no	—
278	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
281	C₂×C₁₁	#3278	1	1	yes	characteristic
282	C₂×C₁₁	#3287	1	1	yes	characteristic
283	C₂×C₁₁	#3290	1	1	yes	characteristic
284	D₁₁	—	16	2	no	—
285	C₂×Q₈	—	11	2	no	—
286	C₂×Q₈	—	22	2	no	—
287	C₂×Q₈	C₂×D₄₄	1	1	yes	characteristic
288	C₂×Q₈	C₂×D₄₄	1	1	yes	characteristic
289	C₂×D₄	—	22	2	no	—
290	C₂×D₄	—	22	2	no	—
291	C₂×D₄	—	22	2	no	—
292	C₂²×C₄	—	44	2	no	—
293	Q₁₆	—	2	2	no	—
294	Q₁₆	—	22	2	no	—
295	Q₁₆	—	88	2	no	—
296	SD₁₆	—	44	4	no	—
297	SD₁₆	—	88	2	no	—
298	SD₁₆	—	88	2	no	—
299	SD₁₆	—	88	2	no	—
300	D₈	—	44	2	no	—
301	D₈	—	88	1	no	—
302	C₂×C₈	—	2	1	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	22	2	no	—
306	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
307	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
308	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
309	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	2	1	no	—
312	C₄⋊C₄	—	11	2	no	—
313	C₄⋊C₄	—	22	2	no	—
314	C₄⋊C₄	—	22	2	no	—
315	C₄⋊C₄	—	22	2	no	—
316	C₄⋊C₄	—	44	2	no	—
317	C₄⋊C₄	—	44	2	no	—
318	C₄⋊C₄	—	44	2	no	—
319	C₄⋊C₄	C₂×D₄₄	1	1	yes	characteristic
320	C₄⋊C₄	C₂×D₄₄	1	1	yes	characteristic
321	C₂²⋊C₄	—	44	2	no	—
322	C₂²⋊C₄	—	44	2	no	—
323	C₂²⋊C₄	—	44	2	no	—
324	C₄²	—	2	1	no	—
325	C₄²	—	2	1	no	—
326	C₄²	—	22	2	no	—
327	C₄²	—	44	2	no	—
328	C₄²	C₂³×D₁₁	1	1	yes	characteristic
329	C₁₁	#19966	1	1	yes	characteristic, complemented, Sylow
330	C₂³	—	44	2	no	—
331	Q₈	—	2	1	no	—
332	Q₈	—	2	1	no	—
333	Q₈	—	2	1	no	—
334	Q₈	—	2	1	no	—
335	Q₈	—	22	2	no	—
336	Q₈	—	22	2	no	—
337	Q₈	—	44	2	no	—
338	Q₈	—	44	2	no	—
339	D₄	—	44	2	no	—
340	D₄	—	44	2	no	—
341	D₄	—	44	2	no	—
342	D₄	—	44	2	no	—
343	D₄	—	88	2	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	22	2	no	—
349	C₂×C₄	—	22	2	no	—
350	C₂×C₄	—	44	2	no	—
351	C₂×C₄	—	176	2	no	—
352	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
353	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
354	C₂×C₄	#67042	1	1	yes	characteristic
355	C₈	—	2	1	no	—
356	C₈	—	2	1	no	—
357	C₈	—	4	1	no	—
358	C₈	—	4	1	no	—
359	C₈	—	4	1	no	—
360	C₈	—	44	2	no	—
361	C₈	#66957	1	2	yes	—
362	C₈	#66971	1	2	yes	—
363	C₂²	—	88	2	no	—
364	C₂²	—	88	2	no	—
365	C₂²	—	88	2	no	—
366	C₂²	D₄×D₄₄	1	1	yes	characteristic
367	C₄	—	2	1	no	—
368	C₄	—	4	1	no	—
369	C₄	—	4	1	no	—
370	C₄	—	4	1	no	—
371	C₄	—	4	1	no	—
372	C₄	—	44	2	no	—
373	C₄	—	44	2	no	—
374	C₄	—	88	2	no	—
375	C₄	C₂×#66957	1	1	yes	characteristic
376	C₄	#10623283	1	1	yes	characteristic
377	C₄	C₂×#66971	1	1	yes	characteristic
378	C₄	#10623301	1	1	yes	characteristic
379	C₂	—	176	2	no	—
380	C₂	#13080283	1	1	yes	characteristic
381	C₂	#13080376	1	1	yes	characteristic
382	C₂	#13080403	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₄₄
commutator, cosocle	C₄²×C₁₁	C₂⁴
Frattini	C₄²	C₂³×D₁₁
Fitting	C₈×C₁₁×Q₁₆	C₂
socle	C₂²×C₁₁	D₄²
prime radical	#10622308	C₂²
hypercenter	C₈×Q₁₆	D₁₁
hypocenter	C₁₁	#19966
LCS	G ▷ C₄²×C₁₁ ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆

Complements

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

N	H ≅ G/N	type
1. #13081667	C₂	semidirect
3. #13081508	C₂	semidirect
5. #13068089	C₂	semidirect
6. #13068010	C₂	semidirect
8. #13067804	C₂	semidirect
9. #13067714	C₂	semidirect
10. #13067514	C₂	semidirect
11. C₈×C₁₁×Q₁₆	C₂	semidirect
149. C₈×Q₁₆	D₁₁	semidirect
329. C₁₁	#19966	semidirect

Maximal subgroups

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

subgroup	index
#13081667	2
#13081629	2
#13081508	2
#13068208	2
#13068089	2
#13068010	2
#13067976	2
#13067804	2
#13067714	2
#13067514	2
C₈×C₁₁×Q₁₆	2
#19966	11