Subgroups — #14725545

Order 2816.

586 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13079199	C₂	1	1	yes	maximal, characteristic
2	#13077558	C₂	1	2	yes	maximal
3	#13067514	C₂	1	1	yes	maximal, characteristic, complemented
4	#13067513	C₂	1	1	yes	maximal, characteristic, complemented
5	#13067469	C₂	1	2	yes	maximal, complemented
6	#13067466	C₂	1	2	yes	maximal, complemented
7	#13067422	C₂	1	1	yes	maximal, characteristic, complemented
8	D₁₁×#335	C₂	1	1	yes	maximal, characteristic
9	#13067017	C₂	1	2	yes	maximal
10	C₈×#4313	C₂	1	1	yes	maximal, characteristic
11	C₈×C₁₁×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#10623109	C₂²	1	2	yes	—
13	C₈×Q₈×C₁₁	C₂²	1	2	yes	—
14	C₄×C₁₁×Q₁₆	C₂²	1	1	yes	characteristic
15	#10622801	C₄	1	1	yes	characteristic
16	#10622794	C₄	1	1	yes	characteristic
17	C₄×#4326	C₂²	1	2	yes	—
18	#10622637	C₂²	1	2	yes	—
19	#10622500	C₄	1	1	yes	characteristic
20	#10622487	C₄	1	1	yes	characteristic
21	#10622452	—	2	2	no	—
22	D₁₁×C₄⋊C₈	C₂²	1	2	yes	—
23	#10622443	C₂²	1	2	yes	—
24	#10622441	—	2	2	no	—
25	#10622426	C₄	1	2	yes	—
26	#10622405	C₄	1	2	yes	—
27	C₄×Dic₄₄	C₂²	1	1	yes	characteristic
28	#10622302	—	2	2	no	—
29	C₄×#4313	C₂²	1	1	yes	characteristic
30	#10622298	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₁₁	C₂²	1	1	yes	characteristic
32	#10622292	—	2	2	no	—
33	#10622292	C₂²	1	2	yes	—
34	C₁₁×#335	C₂²	1	1	yes	characteristic
35	C₁₁×Q₈⋊C₈	C₂²	1	2	yes	—
36	C₈²×C₁₁	C₂²	1	1	yes	characteristic
37	Q₈⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
38	#10622122	C₂²	1	2	yes	—
39	#10622117	C₂²	1	2	yes	—
40	#10622096	C₂²	1	1	yes	characteristic
41	#10622094	C₂²	1	2	yes	—
42	#10622093	C₂²	1	1	yes	characteristic
43	#10622092	C₂²	1	1	yes	characteristic
44	C₈×C₁₁⋊C₈	C₂²	1	1	yes	characteristic
45	C₂×C₁₁×Q₁₆	C₂×C₄	1	1	yes	characteristic
46	C₄×Q₈×C₁₁	C₂³	1	2	yes	—
47	#66999	C₂×C₄	1	2	yes	—
48	#66997	C₂×C₄	1	2	yes	—
49	C₂×#4326	C₂×C₄	1	2	yes	—
50	C₂×Dic₄₄	C₂×C₄	1	1	yes	characteristic
51	C₂×#4313	D₄	1	1	yes	characteristic
52	C₂×#4313	D₄	1	1	yes	characteristic
53	C₂×C₈×D₁₁	—	2	2	no	—
54	C₂×C₈×D₁₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₈×D₁₁	C₂×C₄	1	1	yes	characteristic
56	#66950	C₂×C₄	1	2	yes	—
57	#66949	—	2	2	no	—
58	#66942	—	2	2	no	—
59	#66941	C₂×C₄	1	2	yes	—
60	#66925	—	2	2	no	—
61	C₄²×D₁₁	C₂³	1	1	yes	characteristic
62	C₄×Dic₂₂	—	2	2	no	—
63	C₄×Dic₂₂	C₂³	1	2	yes	—
64	C₁₁×#107	C₂×C₄	1	1	yes	characteristic
65	C₁₁×C₄⋊C₈	—	2	2	no	—
66	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
67	C₁₁×#103	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₁₁	—	2	2	no	—
69	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
72	#66899	C₂×C₄	1	2	yes	—
73	(C₂×D₁₁)⋊C₈	—	2	1	no	—
74	#66882	C₂×C₄	1	1	yes	characteristic
75	#66880	C₂×C₄	1	2	yes	—
76	#66879	—	2	2	no	—
77	#66879	D₄	1	1	yes	characteristic
78	#66879	D₄	1	1	yes	characteristic
79	Dic₁₁⋊C₈	—	2	2	no	—
80	Dic₁₁⋊C₈	—	2	2	no	—
81	Dic₁₁⋊C₈	—	2	1	no	—
82	Dic₁₁⋊C₈	—	2	2	no	—
83	C₈×Dic₁₁	—	2	2	no	—
84	C₈×Dic₁₁	C₂×C₄	1	1	yes	characteristic
85	C₈×Dic₁₁	C₂×C₄	1	1	yes	characteristic
86	#66874	C₂×C₄	1	2	yes	—
87	#66871	C₂×C₄	1	2	yes	—
88	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
89	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
90	#66867	—	2	2	no	—
91	C₄×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
92	C₄×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
93	C₄×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
94	#18068	—	11	1	no	maximal, Sylow
95	C₂×Q₈×C₁₁	C₂²×C₄	1	2	yes	—
96	C₂×C₄×D₁₁	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×D₁₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×D₁₁	#41	1	1	yes	characteristic
99	C₂×Dic₂₂	—	2	2	no	—
100	C₂×Dic₂₂	C₂²×C₄	1	2	yes	—
101	C₁₁×Q₁₆	—	2	2	no	—
102	C₂×C₈×C₁₁	—	2	2	no	—
103	C₂×C₈×C₁₁	—	2	1	no	—
104	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
108	C₁₁×C₄⋊C₄	—	2	2	no	—
109	C₁₁×C₄⋊C₄	C₂²×C₄	1	2	yes	—
110	C₄²×C₁₁	—	2	2	no	—
111	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
112	#4326	—	4	2	no	—
113	(C₂×D₁₁)⋊C₄	—	2	2	no	—
114	(C₂×D₁₁)⋊C₄	—	2	2	no	—
115	C₄⋊Dic₁₁	—	2	2	no	—
116	C₄⋊Dic₁₁	C₂²×C₄	1	2	yes	—
117	Dic₁₁⋊C₄	—	2	2	no	—
118	Dic₁₁⋊C₄	—	2	2	no	—
119	Dic₁₁⋊C₄	—	2	2	no	—
120	Dic₁₁⋊C₄	—	2	2	no	—
121	Dic₁₁⋊C₄	—	2	2	no	—
122	C₄×Dic₁₁	—	2	2	no	—
123	C₄×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
124	C₄×Dic₁₁	C₂×D₄	1	1	yes	characteristic
125	C₄×Dic₁₁	#41	1	1	yes	characteristic
126	C₂×C₁₁⋊C₈	—	2	2	no	—
127	C₂×C₁₁⋊C₈	—	2	1	no	—
128	C₂×C₁₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₁₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₁₁⋊C₈	C₂×D₄	1	1	yes	characteristic
131	C₂×C₁₁⋊C₈	C₂×D₄	1	1	yes	characteristic
132	Dic₄₄	—	2	2	no	—
133	#4313	—	2	2	no	—
134	#4313	Q₁₆	1	4	yes	complemented
135	C₈×D₁₁	—	2	2	no	—
136	C₈×D₁₁	—	4	1	no	—
137	C₈×D₁₁	—	4	4	no	—
138	#3170	—	11	1	no	—
139	#2962	—	11	2	no	—
140	#1591	—	11	1	no	—
141	#1591	—	11	1	no	—
142	#1584	—	11	2	no	—
143	#1584	—	11	2	no	—
144	C₈×Q₁₆	—	11	1	no	—
145	C₈×Q₁₆	D₁₁	1	1	yes	characteristic, complemented
146	C₂×#335	—	11	1	no	—
147	#1481	—	11	2	no	—
148	C₈×M₁₆	—	11	1	no	—
149	C₂²×D₁₁	C₄×D₄	1	1	yes	characteristic
150	Q₈×C₁₁	—	2	2	no	—
151	Q₈×C₁₁	—	2	2	no	—
152	C₂×C₄×C₁₁	—	2	2	no	—
153	C₂×C₄×C₁₁	—	2	2	no	—
154	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
155	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
157	C₂×Dic₁₁	—	2	2	no	—
158	C₂×Dic₁₁	—	2	2	no	—
159	C₂×Dic₁₁	C₄×D₄	1	1	yes	characteristic
160	C₂×Dic₁₁	C₄×D₄	1	1	yes	characteristic
161	C₂×Dic₁₁	C₄×D₄	1	1	yes	characteristic
162	C₄×D₁₁	—	2	1	no	—
163	C₄×D₁₁	—	2	1	no	—
164	C₄×D₁₁	—	4	1	no	—
165	C₄×D₁₁	C₂×Q₁₆	1	2	yes	—
166	C₄×D₁₁	#135	1	2	yes	—
167	Dic₂₂	—	2	2	no	—
168	Dic₂₂	—	2	2	no	—
169	Dic₂₂	—	8	2	no	—
170	C₈×C₁₁	—	2	1	no	—
171	C₈×C₁₁	—	2	1	no	—
172	C₈×C₁₁	—	4	1	no	—
173	C₈×C₁₁	—	4	2	no	—
174	C₈×C₁₁	#131	1	2	yes	—
175	C₈×C₁₁	C₂×Q₁₆	1	2	yes	—
176	C₁₁⋊C₈	—	2	1	no	—
177	C₁₁⋊C₈	—	2	1	no	—
178	C₁₁⋊C₈	—	2	1	no	—
179	C₁₁⋊C₈	—	4	1	no	—
180	C₁₁⋊C₈	—	4	2	no	—
181	C₁₁⋊C₈	C₂×Q₁₆	1	2	yes	—
182	#533	—	11	2	no	—
183	#500	—	11	1	no	—
184	#494	—	11	1	no	—
185	#487	—	11	2	no	—
186	#467	—	11	1	no	—
187	#467	—	11	1	no	—
188	#467	—	11	2	no	—
189	#446	—	11	2	no	—
190	#446	—	11	2	no	—
191	#446	—	22	2	no	—
192	#446	—	22	2	no	—
193	C₈×Q₈	—	11	2	no	—
194	C₈×Q₈	C₂×D₁₁	1	2	yes	—
195	C₄×Q₁₆	—	11	2	no	—
196	C₄×Q₁₆	—	11	1	no	—
197	C₄×Q₁₆	C₂×D₁₁	1	1	yes	characteristic
198	#431	—	11	1	no	—
199	#424	—	22	2	no	—
200	C₂×C₄⋊C₈	—	11	2	no	—
201	#405	—	22	2	no	—
202	C₄×M₁₆	—	11	1	no	—
203	C₂×C₄×C₈	—	11	1	no	—
204	#335	—	11	2	no	—
205	#335	—	11	1	no	—
206	#335	C₂×D₁₁	1	1	yes	characteristic
207	Q₈⋊C₈	—	11	2	no	—
208	Q₈⋊C₈	—	11	2	no	—
209	Q₈⋊C₈	—	11	2	no	—
210	Q₈⋊C₈	C₂×D₁₁	1	2	yes	—
211	#322	—	11	1	no	—
212	#322	—	11	1	no	—
213	C₈²	—	11	1	no	—
214	C₈²	C₂×D₁₁	1	1	yes	characteristic
215	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
216	C₂×D₁₁	—	2	1	no	—
217	C₂×D₁₁	—	2	1	no	—
218	C₂×D₁₁	C₄×Q₁₆	1	2	yes	—
219	C₄×C₁₁	—	2	1	no	—
220	C₄×C₁₁	—	4	2	no	—
221	C₄×C₁₁	—	4	2	no	—
222	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
223	C₄×C₁₁	#568	1	1	yes	characteristic
224	C₄×C₁₁	C₂²×Q₁₆	1	1	yes	characteristic
225	C₄×C₁₁	C₂×#135	1	1	yes	characteristic
226	Dic₁₁	—	2	1	no	—
227	Dic₁₁	—	2	1	no	—
228	Dic₁₁	—	4	2	no	—
229	Dic₁₁	—	4	2	no	—
230	Dic₁₁	C₄×Q₁₆	1	2	yes	—
231	C₂×Q₁₆	—	11	2	no	—
232	C₂×Q₁₆	—	11	1	no	—
233	C₂×Q₁₆	C₄×D₁₁	1	1	yes	characteristic
234	C₂×M₁₆	—	11	1	no	—
235	C₂×M₁₆	—	11	1	no	—
236	C₂²×C₈	—	11	1	no	—
237	C₂²×C₈	—	11	1	no	—
238	C₂²×C₈	—	22	2	no	—
239	C₄⋊Q₈	—	11	2	no	—
240	C₄⋊Q₈	—	11	2	no	—
241	#125	—	22	2	no	—
242	C₂²⋊Q₈	—	11	2	no	—
243	C₂²⋊Q₈	—	11	2	no	—
244	C₂²⋊Q₈	—	22	2	no	—
245	C₄×Q₈	—	11	2	no	—
246	C₄×Q₈	—	22	2	no	—
247	C₄×Q₈	C₂²×D₁₁	1	2	yes	—
248	#117	—	22	2	no	—
249	C₂×C₄²	—	11	1	no	—
250	#107	—	11	1	no	—
251	#107	—	11	2	no	—
252	#107	C₄×D₁₁	1	1	yes	characteristic
253	C₄⋊C₈	—	2	2	no	—
254	C₄⋊C₈	—	11	2	no	—
255	C₄⋊C₈	—	22	2	no	—
256	C₄⋊C₈	—	22	2	no	—
257	C₄⋊C₈	—	22	1	no	—
258	C₄⋊C₈	—	22	2	no	—
259	C₄⋊C₈	—	22	2	no	—
260	C₄⋊C₈	C₂²×D₁₁	1	2	yes	—
261	#103	—	11	2	no	—
262	#103	—	11	2	no	—
263	#103	—	11	2	no	—
264	#103	C₄×D₁₁	1	2	yes	—
265	C₂²⋊C₈	—	22	1	no	—
266	#97	—	11	1	no	—
267	#97	—	11	1	no	—
268	#97	—	22	2	no	—
269	#97	—	22	2	no	—
270	C₄×C₈	—	2	2	no	—
271	C₄×C₈	—	11	1	no	—
272	C₄×C₈	—	11	1	no	—
273	C₄×C₈	—	11	1	no	—
274	C₄×C₈	—	11	1	no	—
275	C₄×C₈	—	11	1	no	—
276	C₄×C₈	—	22	2	no	—
277	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
278	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
280	C₂×C₁₁	#2935	1	1	yes	characteristic
281	C₂×C₁₁	C₂×C₄×Q₁₆	1	1	yes	characteristic
282	C₂×C₁₁	#2957	1	1	yes	characteristic
283	D₁₁	—	2	2	no	—
284	C₂×Q₈	—	11	2	no	—
285	C₂×Q₈	—	22	2	no	—
286	C₂×Q₈	C₂×C₄×D₁₁	1	2	yes	—
287	C₂²×C₄	—	11	1	no	—
288	C₂²×C₄	—	11	1	no	—
289	C₂²×C₄	—	11	1	no	—
290	Q₁₆	—	2	2	no	—
291	Q₁₆	—	22	2	no	—
292	Q₁₆	—	44	2	no	—
293	M₁₆	—	11	4	no	—
294	M₁₆	—	22	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	1	no	—
297	C₂×C₈	—	11	1	no	—
298	C₂×C₈	—	11	1	no	—
299	C₂×C₈	—	11	1	no	—
300	C₂×C₈	—	11	1	no	—
301	C₂×C₈	—	22	2	no	—
302	C₂×C₈	—	22	2	no	—
303	C₂×C₈	—	22	1	no	—
304	C₂×C₈	—	44	1	no	—
305	C₂×C₈	—	44	4	no	—
306	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
307	C₂×C₈	C₂×C₄×D₁₁	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	2	no	—
311	C₄⋊C₄	—	11	2	no	—
312	C₄⋊C₄	—	22	2	no	—
313	C₄⋊C₄	—	22	2	no	—
314	C₄⋊C₄	—	22	2	no	—
315	C₄⋊C₄	—	22	2	no	—
316	C₄⋊C₄	—	22	2	no	—
317	C₄⋊C₄	—	22	2	no	—
318	C₄⋊C₄	C₂×C₄×D₁₁	1	2	yes	—
319	C₂²⋊C₄	—	22	2	no	—
320	C₂²⋊C₄	—	22	2	no	—
321	C₄²	—	2	2	no	—
322	C₄²	—	11	1	no	—
323	C₄²	—	11	1	no	—
324	C₄²	—	11	1	no	—
325	C₄²	—	22	2	no	—
326	C₄²	C₂³×D₁₁	1	1	yes	characteristic
327	C₁₁	#18068	1	1	yes	characteristic, complemented, Sylow
328	C₂³	—	11	1	no	—
329	Q₈	—	2	2	no	—
330	Q₈	—	2	2	no	—
331	Q₈	—	22	2	no	—
332	Q₈	—	22	2	no	—
333	Q₈	—	88	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	11	1	no	—
337	C₂×C₄	—	11	2	no	—
338	C₂×C₄	—	11	2	no	—
339	C₂×C₄	—	11	1	no	—
340	C₂×C₄	—	11	1	no	—
341	C₂×C₄	—	22	1	no	—
342	C₂×C₄	—	22	1	no	—
343	C₂×C₄	—	22	2	no	—
344	C₂×C₄	—	22	2	no	—
345	C₂×C₄	—	44	1	no	—
346	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
347	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
348	C₂×C₄	#41×D₁₁	1	1	yes	characteristic
349	C₈	—	2	1	no	—
350	C₈	—	2	1	no	—
351	C₈	—	4	1	no	—
352	C₈	—	4	2	no	—
353	C₈	—	11	2	no	—
354	C₈	—	22	1	no	—
355	C₈	—	22	1	no	—
356	C₈	—	22	1	no	—
357	C₈	—	44	1	no	—
358	C₈	—	44	2	no	—
359	C₈	#66954	1	2	yes	—
360	C₈	Q₁₆×D₁₁	1	2	yes	—
361	C₂²	—	11	2	no	—
362	C₂²	—	22	1	no	—
363	C₂²	—	22	1	no	—
364	C₂²	C₄×D₄×D₁₁	1	1	yes	characteristic
365	C₄	—	2	1	no	—
366	C₄	—	4	2	no	—
367	C₄	—	4	2	no	—
368	C₄	—	11	2	no	—
369	C₄	—	22	1	no	—
370	C₄	—	22	1	no	—
371	C₄	—	44	2	no	—
372	C₄	—	44	2	no	—
373	C₄	C₂×#66954	1	1	yes	characteristic
374	C₄	#10623281	1	1	yes	characteristic
375	C₄	C₂×Q₁₆×D₁₁	1	1	yes	characteristic
376	C₄	D₁₁×#135	1	1	yes	characteristic
377	C₂	—	22	2	no	—
378	C₂	#13077355	1	1	yes	characteristic
379	C₂	C₄×Q₁₆×D₁₁	1	1	yes	characteristic
380	C₂	#13077528	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₁
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₄²	C₂³×D₁₁
Fitting	C₈×C₁₁×Q₁₆	C₂
socle	C₂²×C₁₁	C₂×C₄×D₄
prime radical	C₂²×D₁₁	C₄×D₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	C₈×Q₁₆	D₁₁
hypocenter	C₁₁	#18068
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #13067514	C₂	semidirect
4. #13067513	C₂	semidirect
5. #13067469	C₂	semidirect
6. #13067466	C₂	semidirect
7. #13067422	C₂	semidirect
11. C₈×C₁₁×Q₁₆	C₂	semidirect
134. #4313	Q₁₆	semidirect
145. C₈×Q₁₆	D₁₁	semidirect
327. C₁₁	#18068	semidirect

Maximal subgroups

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

subgroup	index
#13079199	2
#13077558	2
#13067514	2
#13067513	2
#13067469	2
#13067466	2
#13067422	2
D₁₁×#335	2
#13067017	2
C₈×#4313	2
C₈×C₁₁×Q₁₆	2
#18068	11