Subgroups — C₁₁×#19192 #14596385

Order 2816.

674 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁×#3206	C₂	1	2	yes	maximal
2	C₂×C₁₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₁×#2931	C₂	1	1	yes	maximal, characteristic
4	C₁₁×#1693	C₂	1	4	yes	maximal, complemented
5	C₁₁×#1673	C₂	1	4	yes	maximal, complemented
6	C₁₁×#1564	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₁×#1549	C₂	1	2	yes	maximal, complemented
8	C₂×C₁₁×#131	—	2	1	no	—
9	C₁₁×#532	C₂²	1	1	yes	characteristic
10	C₂×C₁₁×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₁₁×#500	—	2	1	no	—
12	C₁₁×#500	—	2	1	no	—
13	C₁₁×#500	C₂²	1	2	yes	—
14	C₁₁×#500	C₂²	1	2	yes	—
15	C₁₁×#492	—	2	2	no	—
16	C₁₁×#487	—	2	2	no	—
17	C₁₁×#461	C₂²	1	8	yes	—
18	C₁₁×#448	—	2	4	no	—
19	C₁₁×#436	—	2	1	no	—
20	C₁₁×#436	C₂²	1	2	yes	—
21	C₈×D₄×C₁₁	—	2	1	no	—
22	C₈×D₄×C₁₁	C₂²	1	2	yes	—
23	C₁₁×#431	C₂²	1	2	yes	—
24	C₂×C₁₁×#107	—	2	1	no	—
25	C₂×C₁₁×#107	—	2	1	no	—
26	C₁₁×#423	C₂²	1	1	yes	characteristic
27	C₁₁×#408	—	2	1	no	—
28	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
29	C₁₁×#335	C₂²	1	2	yes	—
30	C₁₁×#334	C₂²	1	2	yes	—
31	C₁₁×#332	C₂²	1	8	yes	—
32	C₂×C₁₁×#41	—	2	1	no	—
33	C₂²×Q₈×C₁₁	—	2	2	no	—
34	C₂×C₁₁×SD₁₆	—	2	8	no	—
35	C₁₁×#131	—	4	4	no	—
36	C₂×C₁₁×M₁₆	—	2	1	no	—
37	C₂×C₁₁×M₁₆	—	2	2	no	—
38	C₂²×C₈×C₁₁	—	2	2	no	—
39	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
41	C₁₁×C₄⋊Q₈	—	2	2	no	—
42	C₁₁×C₄⋊Q₈	—	2	2	no	—
43	C₁₁×C₄⋊Q₈	—	2	2	no	—
44	C₁₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
45	C₁₁×C₄⋊Q₈	C₂³	1	4	yes	—
46	C₁₁×#127	D₄	1	1	yes	characteristic
47	C₁₁×#124	—	2	1	no	—
48	C₁₁×#121	—	2	1	no	—
49	C₁₁×#121	D₄	1	2	yes	—
50	C₄×D₄×C₁₁	—	2	1	no	—
51	C₄×D₄×C₁₁	C₂³	1	2	yes	—
52	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
53	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
54	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
55	C₁₁×#107	—	2	2	no	—
56	C₁₁×#107	—	2	2	no	—
57	C₁₁×#107	—	2	2	no	—
58	C₁₁×#107	—	2	2	no	—
59	C₁₁×C₄⋊C₈	—	2	1	no	—
60	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
61	C₁₁×C₄⋊C₈	C₂³	1	4	yes	—
62	C₁₁×#103	—	2	8	no	—
63	C₁₁×C₂²⋊C₈	—	2	2	no	—
64	C₁₁×C₂²⋊C₈	—	2	2	no	—
65	C₁₁×C₂²⋊C₈	—	2	2	no	—
66	C₄×C₈×C₁₁	—	2	2	no	—
67	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
68	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
69	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
71	#19192	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₁₁×#41	—	4	2	no	—
73	C₁₁×#41	—	4	2	no	—
74	C₂×Q₈×C₁₁	—	2	1	no	—
75	C₂×Q₈×C₁₁	—	2	4	no	—
76	C₂×Q₈×C₁₁	—	4	2	no	—
77	C₂×Q₈×C₁₁	—	8	4	no	—
78	C₂×D₄×C₁₁	—	2	2	no	—
79	C₂×D₄×C₁₁	—	2	1	no	—
80	C₂×D₄×C₁₁	—	2	1	no	—
81	C₂×D₄×C₁₁	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₁₁	—	2	2	no	—
83	C₂²×C₄×C₁₁	—	4	2	no	—
84	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
87	C₁₁×SD₁₆	—	8	8	no	—
88	C₁₁×M₁₆	—	4	4	no	—
89	C₁₁×M₁₆	—	4	2	no	—
90	C₂×C₈×C₁₁	—	2	2	no	—
91	C₂×C₈×C₁₁	—	2	4	no	—
92	C₂×C₈×C₁₁	—	2	2	no	—
93	C₂×C₈×C₁₁	—	4	4	no	—
94	C₂×C₈×C₁₁	SD₁₆	1	4	yes	complemented
95	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
99	C₁₁×C₄⋊C₄	—	2	1	no	—
100	C₁₁×C₄⋊C₄	—	2	4	no	—
101	C₁₁×C₄⋊C₄	—	2	4	no	—
102	C₁₁×C₄⋊C₄	—	4	2	no	—
103	C₁₁×C₄⋊C₄	—	4	2	no	—
104	C₁₁×C₄⋊C₄	C₂×D₄	1	2	yes	—
105	C₁₁×C₂²⋊C₄	—	2	2	no	—
106	C₁₁×C₂²⋊C₄	—	2	2	no	—
107	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₁₁	#41	1	1	yes	characteristic
110	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
111	#3206	C₂×C₁₁	1	2	yes	—
112	C₂×#500	C₂×C₁₁	1	1	yes	characteristic, complemented
113	#2931	C₂×C₁₁	1	1	yes	characteristic
114	#1693	C₂×C₁₁	1	4	yes	complemented
115	#1673	C₂×C₁₁	1	4	yes	complemented
116	#1564	C₂×C₁₁	1	1	yes	characteristic, complemented
117	#1549	C₂×C₁₁	1	2	yes	complemented
118	C₂³×C₁₁	—	2	2	no	—
119	C₂³×C₁₁	#121	1	1	yes	characteristic
120	Q₈×C₁₁	—	4	2	no	—
121	Q₈×C₁₁	—	4	8	no	—
122	Q₈×C₁₁	—	8	4	no	—
123	D₄×C₁₁	—	2	4	no	—
124	D₄×C₁₁	—	4	4	no	—
125	D₄×C₁₁	—	4	2	no	—
126	D₄×C₁₁	—	4	2	no	—
127	C₂×C₄×C₁₁	—	2	2	no	—
128	C₂×C₄×C₁₁	—	2	2	no	—
129	C₂×C₄×C₁₁	—	2	2	no	—
130	C₂×C₄×C₁₁	—	4	2	no	—
131	C₂×C₄×C₁₁	—	4	2	no	—
132	C₂×C₄×C₁₁	—	4	4	no	—
133	C₂×C₄×C₁₁	—	8	4	no	—
134	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
135	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
136	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
137	C₂×C₄×C₁₁	C₂×SD₁₆	1	2	yes	—
138	C₂×C₄×C₁₁	#137	1	2	yes	—
139	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
142	C₈×C₁₁	—	2	1	no	—
143	C₈×C₁₁	—	2	1	no	—
144	C₈×C₁₁	—	4	2	no	—
145	C₈×C₁₁	—	4	4	no	—
146	C₈×C₁₁	C₂×SD₁₆	1	2	yes	—
147	C₈×C₁₁	C₂×SD₁₆	1	2	yes	—
148	C₂×#131	—	2	1	no	—
149	#532	C₂²×C₁₁	1	1	yes	characteristic
150	C₂×C₄⋊Q₈	C₂²×C₁₁	1	2	yes	—
151	#500	—	2	1	no	—
152	#500	—	2	1	no	—
153	#500	C₂²×C₁₁	1	2	yes	—
154	#500	C₂²×C₁₁	1	2	yes	—
155	#492	—	2	2	no	—
156	#487	—	2	2	no	—
157	#461	C₂²×C₁₁	1	8	yes	—
158	#448	—	2	4	no	—
159	#436	—	2	1	no	—
160	#436	C₂²×C₁₁	1	2	yes	—
161	C₈×D₄	—	2	1	no	—
162	C₈×D₄	C₂²×C₁₁	1	2	yes	—
163	#431	C₂²×C₁₁	1	2	yes	—
164	C₂×#107	—	2	1	no	—
165	C₂×#107	—	2	1	no	—
166	#423	C₂²×C₁₁	1	1	yes	characteristic
167	#408	—	2	1	no	—
168	C₂×C₄×C₈	C₂²×C₁₁	1	1	yes	characteristic
169	#335	C₂²×C₁₁	1	2	yes	—
170	#334	C₂²×C₁₁	1	2	yes	—
171	#332	C₂²×C₁₁	1	8	yes	—
172	C₂²×C₁₁	—	2	1	no	—
173	C₂²×C₁₁	—	2	1	no	—
174	C₂²×C₁₁	—	4	2	no	—
175	C₂²×C₁₁	—	4	2	no	—
176	C₂²×C₁₁	—	4	2	no	—
177	C₂²×C₁₁	#461	1	2	yes	—
178	C₂²×C₁₁	C₂×#121	1	1	yes	characteristic
179	C₄×C₁₁	—	2	1	no	—
180	C₄×C₁₁	—	2	1	no	—
181	C₄×C₁₁	—	4	2	no	—
182	C₄×C₁₁	—	8	4	no	—
183	C₄×C₁₁	#461	1	2	yes	—
184	C₄×C₁₁	#473	1	2	yes	—
185	C₄×C₁₁	C₂²×SD₁₆	1	1	yes	characteristic
186	C₄×C₁₁	C₂×#137	1	1	yes	characteristic
187	C₄×C₁₁	#576	1	2	yes	—
188	C₂×#41	—	2	1	no	—
189	C₂²×Q₈	—	2	2	no	—
190	C₂×SD₁₆	—	2	8	no	—
191	#131	—	4	4	no	—
192	C₂×M₁₆	—	2	1	no	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	D₄×C₁₁	1	1	yes	characteristic
196	C₂²×C₈	D₄×C₁₁	1	1	yes	characteristic
197	C₄⋊Q₈	—	2	2	no	—
198	C₄⋊Q₈	—	2	2	no	—
199	C₄⋊Q₈	—	2	2	no	—
200	C₄⋊Q₈	D₄×C₁₁	1	1	yes	characteristic
201	C₄⋊Q₈	C₂³×C₁₁	1	4	yes	—
202	#127	D₄×C₁₁	1	1	yes	characteristic
203	#124	—	2	1	no	—
204	#121	—	2	1	no	—
205	#121	D₄×C₁₁	1	2	yes	—
206	C₄×D₄	—	2	1	no	—
207	C₄×D₄	C₂³×C₁₁	1	2	yes	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄²	C₂³×C₁₁	1	1	yes	characteristic
211	#107	—	2	2	no	—
212	#107	—	2	2	no	—
213	#107	—	2	2	no	—
214	#107	—	2	2	no	—
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	C₂³×C₁₁	1	2	yes	—
217	C₄⋊C₈	C₂³×C₁₁	1	4	yes	—
218	#103	—	2	8	no	—
219	C₂²⋊C₈	—	2	2	no	—
220	C₂²⋊C₈	—	2	2	no	—
221	C₂²⋊C₈	—	2	2	no	—
222	C₄×C₈	—	2	2	no	—
223	C₄×C₈	D₄×C₁₁	1	1	yes	characteristic
224	C₄×C₈	D₄×C₁₁	1	1	yes	characteristic
225	C₄×C₈	C₂³×C₁₁	1	1	yes	characteristic
226	C₄×C₈	C₂³×C₁₁	1	1	yes	characteristic
227	C₂×C₁₁	—	2	2	no	—
228	C₂×C₁₁	—	8	2	no	—
229	C₂×C₁₁	C₂×#461	1	1	yes	characteristic
230	C₂×C₁₁	#3058	1	1	yes	characteristic
231	C₂×C₁₁	C₂×#473	1	1	yes	characteristic
232	#41	—	4	2	no	—
233	#41	—	4	2	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×Q₈	—	2	4	no	—
236	C₂×Q₈	—	4	2	no	—
237	C₂×Q₈	—	8	4	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	C₂×D₄×C₁₁	1	2	yes	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	4	2	no	—
244	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
246	C₂²×C₄	C₁₁×#41	1	1	yes	characteristic
247	SD₁₆	—	8	8	no	—
248	M₁₆	—	4	2	no	—
249	M₁₆	—	4	4	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	4	4	no	—
254	C₂×C₈	C₁₁×SD₁₆	1	4	yes	complemented
255	C₂×C₈	C₂×D₄×C₁₁	1	1	yes	characteristic
256	C₂×C₈	C₂×D₄×C₁₁	1	1	yes	characteristic
257	C₂×C₈	C₂×D₄×C₁₁	1	1	yes	characteristic
258	C₂×C₈	C₂×D₄×C₁₁	1	1	yes	characteristic
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	C₂×D₄×C₁₁	1	2	yes	—
265	C₂²⋊C₄	—	2	2	no	—
266	C₂²⋊C₄	—	2	2	no	—
267	C₄²	C₂×D₄×C₁₁	1	1	yes	characteristic
268	C₄²	C₂×D₄×C₁₁	1	1	yes	characteristic
269	C₄²	C₁₁×#41	1	1	yes	characteristic
270	C₄²	C₂⁴×C₁₁	1	1	yes	characteristic
271	C₁₁	#19192	1	1	yes	characteristic, direct product factor, Sylow
272	C₂³	—	2	2	no	—
273	C₂³	C₁₁×#121	1	1	yes	characteristic
274	Q₈	—	4	2	no	—
275	Q₈	—	4	8	no	—
276	Q₈	—	8	4	no	—
277	D₄	—	2	4	no	—
278	D₄	—	4	4	no	—
279	D₄	—	4	2	no	—
280	D₄	—	4	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	8	4	no	—
288	C₂×C₄	C₁₁×#121	1	1	yes	characteristic
289	C₂×C₄	C₁₁×#121	1	1	yes	characteristic
290	C₂×C₄	C₁₁×#121	1	1	yes	characteristic
291	C₂×C₄	C₂×C₁₁×SD₁₆	1	2	yes	—
292	C₂×C₄	C₁₁×#137	1	2	yes	—
293	C₂×C₄	C₂²×D₄×C₁₁	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₄×C₁₁	1	1	yes	characteristic
295	C₂×C₄	C₂×C₁₁×#41	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	4	2	no	—
299	C₈	—	4	4	no	—
300	C₈	C₂×C₁₁×SD₁₆	1	2	yes	—
301	C₈	C₂×C₁₁×SD₁₆	1	2	yes	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	—	4	2	no	—
307	C₂²	C₁₁×#461	1	2	yes	—
308	C₂²	C₂×C₁₁×#121	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	4	2	no	—
312	C₄	—	8	4	no	—
313	C₄	C₁₁×#461	1	2	yes	—
314	C₄	C₁₁×#473	1	2	yes	—
315	C₄	C₂²×C₁₁×SD₁₆	1	1	yes	characteristic
316	C₄	C₂×C₁₁×#137	1	1	yes	characteristic
317	C₄	C₁₁×#576	1	2	yes	—
318	C₂	—	2	2	no	—
319	C₂	—	8	2	no	—
320	C₂	C₂×C₁₁×#461	1	1	yes	characteristic
321	C₂	C₁₁×#3058	1	1	yes	characteristic
322	C₂	C₂×C₁₁×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₁
prime radical	C₁₁×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₁ ◁ C₂×C₄²×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₁×#500	C₂	semidirect
4. C₁₁×#1693	C₂	semidirect
5. C₁₁×#1673	C₂	semidirect
6. C₁₁×#1564	C₂	semidirect
7. C₁₁×#1549	C₂	semidirect
71. #19192	C₁₁	direct (N × H)
94. C₂×C₈×C₁₁	SD₁₆	semidirect
112. C₂×#500	C₂×C₁₁	semidirect
114. #1693	C₂×C₁₁	semidirect
115. #1673	C₂×C₁₁	semidirect
116. #1564	C₂×C₁₁	semidirect
117. #1549	C₂×C₁₁	semidirect
254. C₂×C₈	C₁₁×SD₁₆	semidirect
271. C₁₁	#19192	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₁×#3206	2
C₂×C₁₁×#500	2
C₁₁×#2931	2
C₁₁×#1693	2
C₁₁×#1673	2
C₁₁×#1564	2
C₁₁×#1549	2
#19192	11

Subgroups — C11×#19192 #14596385

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#19192 #14596385