Subgroups — C₁₁×#23187 #14600380

Order 2816.

946 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₁×#441	C₂	1	2	yes	maximal
3	C₁₁×#2365	C₂	1	2	yes	maximal
4	C₁₁×#1981	C₂	1	2	yes	maximal
5	C₁₁×#1971	C₂	1	2	yes	maximal
6	C₁₁×#1878	C₂	1	4	yes	maximal
7	C₄×C₁₁×#106	C₂	1	1	yes	maximal, characteristic
8	C₁₁×#1770	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₁×Q₁₆	C₂²	1	2	yes	—
10	C₂²×C₁₁×Q₁₆	C₂²	1	2	yes	—
11	C₂×C₁₁×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₂×C₄×Q₈×C₁₁	C₂²	1	4	yes	—
13	C₁₁×#494	C₄	1	8	yes	complemented
14	C₁₁×#441	—	2	8	no	—
15	C₂×C₁₁×#106	C₂²	1	1	yes	characteristic
16	C₂×C₁₁×#106	C₂²	1	1	yes	characteristic
17	C₂×C₁₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₁₁×#103	C₂²	1	4	yes	—
19	C₂×C₁₁×#103	C₂²	1	4	yes	—
20	C₂×C₁₁×#97	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
22	C₁₁×#391	C₂²	1	4	yes	—
23	C₁₁×#389	—	2	2	no	—
24	C₁₁×#387	—	2	4	no	—
25	C₄×C₁₁×C₄⋊C₄	—	2	2	no	—
26	C₄×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
27	C₄×C₁₁×C₄⋊C₄	C₂²	1	1	yes	characteristic
28	C₁₁×#340	C₂²	1	2	yes	—
29	C₁₁×#336	C₂²	1	2	yes	—
30	C₂²×Q₈×C₁₁	C₂³	1	4	yes	—
31	C₂×C₁₁×Q₁₆	—	2	4	no	—
32	C₂×C₁₁×Q₁₆	—	2	4	no	—
33	C₂×C₁₁×Q₁₆	—	4	2	no	—
34	C₂×C₁₁×Q₁₆	—	4	2	no	—
35	C₂×C₁₁×Q₁₆	C₂×C₄	1	8	yes	—
36	C₂×C₁₁×Q₁₆	C₂×C₄	1	8	yes	—
37	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
40	C₁₁×C₄⋊Q₈	—	2	4	no	—
41	C₁₁×C₄⋊Q₈	C₂×C₄	1	8	yes	—
42	C₄×Q₈×C₁₁	—	2	8	no	—
43	C₄×Q₈×C₁₁	—	4	4	no	—
44	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
45	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
46	C₂×C₁₁×C₄⋊C₄	—	2	4	no	—
47	C₂×C₁₁×C₄⋊C₄	D₄	1	2	yes	—
48	C₂×C₁₁×C₄⋊C₄	C₂³	1	2	yes	—
49	C₂×C₁₁×C₄⋊C₄	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₁	—	2	2	no	—
51	C₂×C₄²×C₁₁	—	2	4	no	—
52	C₂×C₄²×C₁₁	C₂³	1	2	yes	—
53	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
54	C₁₁×#106	—	2	2	no	—
55	C₁₁×#106	D₄	1	4	yes	—
56	C₁₁×C₄⋊C₈	—	2	2	no	—
57	C₁₁×#103	—	2	8	no	—
58	C₁₁×#103	—	4	4	no	—
59	C₁₁×#97	—	2	4	no	—
60	C₄×C₈×C₁₁	C₂×C₄	1	4	yes	—
61	C₁₁×#95	—	2	4	no	—
62	C₁₁×#95	—	2	2	no	—
63	C₁₁×#95	—	2	4	no	—
64	C₁₁×#95	D₄	1	2	yes	—
65	#23187	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂×Q₈×C₁₁	—	2	4	no	—
67	C₂×Q₈×C₁₁	—	2	4	no	—
68	C₂×Q₈×C₁₁	—	2	4	no	—
69	C₂×Q₈×C₁₁	—	2	4	no	—
70	C₂×Q₈×C₁₁	—	2	4	no	—
71	C₂×Q₈×C₁₁	C₂²×C₄	1	8	yes	—
72	C₂²×C₄×C₁₁	—	2	4	no	—
73	C₂²×C₄×C₁₁	—	2	2	no	—
74	C₂²×C₄×C₁₁	—	2	2	no	—
75	C₂²×C₄×C₁₁	C₂×D₄	1	4	yes	—
76	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
79	C₁₁×Q₁₆	—	2	16	no	—
80	C₁₁×Q₁₆	—	4	8	no	—
81	C₂×C₈×C₁₁	—	2	2	no	—
82	C₂×C₈×C₁₁	—	2	2	no	—
83	C₂×C₈×C₁₁	—	2	1	no	—
84	C₂×C₈×C₁₁	—	2	1	no	—
85	C₂×C₈×C₁₁	—	2	2	no	—
86	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₁₁	#41	1	2	yes	—
90	C₁₁×C₄⋊C₄	—	2	8	no	—
91	C₁₁×C₄⋊C₄	—	2	8	no	—
92	C₁₁×C₄⋊C₄	—	2	4	no	—
93	C₁₁×C₄⋊C₄	—	2	2	no	—
94	C₁₁×C₄⋊C₄	—	2	2	no	—
95	C₁₁×C₄⋊C₄	—	2	2	no	—
96	C₁₁×C₄⋊C₄	—	4	8	no	—
97	C₁₁×C₄⋊C₄	C₂×D₄	1	4	yes	—
98	C₄²×C₁₁	—	2	2	no	—
99	C₄²×C₁₁	—	2	2	no	—
100	C₄²×C₁₁	—	4	8	no	—
101	C₄²×C₁₁	—	4	4	no	—
102	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
103	C₄²×C₁₁	#41	1	2	yes	—
104	C₂×#494	C₂×C₁₁	1	1	yes	characteristic
105	C₂×#441	C₂×C₁₁	1	2	yes	—
106	#2365	C₂×C₁₁	1	2	yes	—
107	#1981	C₂×C₁₁	1	2	yes	—
108	#1971	C₂×C₁₁	1	2	yes	—
109	#1878	C₂×C₁₁	1	4	yes	—
110	C₄×#106	C₂×C₁₁	1	1	yes	characteristic
111	#1770	C₂×C₁₁	1	1	yes	characteristic
112	C₂³×C₁₁	C₂²×D₄	1	1	yes	characteristic
113	Q₈×C₁₁	—	2	8	no	—
114	Q₈×C₁₁	—	2	8	no	—
115	Q₈×C₁₁	—	4	4	no	—
116	Q₈×C₁₁	—	4	4	no	—
117	C₂×C₄×C₁₁	—	2	2	no	—
118	C₂×C₄×C₁₁	—	2	8	no	—
119	C₂×C₄×C₁₁	—	2	4	no	—
120	C₂×C₄×C₁₁	—	2	4	no	—
121	C₂×C₄×C₁₁	—	2	4	no	—
122	C₂×C₄×C₁₁	—	2	4	no	—
123	C₂×C₄×C₁₁	—	4	4	no	—
124	C₂×C₄×C₁₁	—	4	4	no	—
125	C₂×C₄×C₁₁	—	4	2	no	—
126	C₂×C₄×C₁₁	—	4	2	no	—
127	C₂×C₄×C₁₁	—	4	2	no	—
128	C₂×C₄×C₁₁	—	4	2	no	—
129	C₂×C₄×C₁₁	—	4	2	no	—
130	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
131	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₁₁	#137	1	4	yes	—
133	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
136	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
138	C₂×C₄×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
139	C₈×C₁₁	—	2	2	no	—
140	C₈×C₁₁	—	4	2	no	—
141	C₈×C₁₁	C₄×D₄	1	4	yes	—
142	C₂²×Q₁₆	C₂²×C₁₁	1	2	yes	—
143	C₂²×Q₁₆	C₂²×C₁₁	1	2	yes	—
144	C₂×C₄⋊Q₈	C₂²×C₁₁	1	2	yes	—
145	C₂×C₄×Q₈	C₂²×C₁₁	1	4	yes	—
146	#494	C₄×C₁₁	1	8	yes	complemented
147	#441	—	2	8	no	—
148	C₂×#106	C₂²×C₁₁	1	1	yes	characteristic
149	C₂×#106	C₂²×C₁₁	1	1	yes	characteristic
150	C₂×C₄⋊C₈	C₂²×C₁₁	1	1	yes	characteristic
151	C₂×#103	C₂²×C₁₁	1	4	yes	—
152	C₂×#103	C₂²×C₁₁	1	4	yes	—
153	C₂×#97	C₂²×C₁₁	1	2	yes	—
154	C₂×C₄×C₈	C₂²×C₁₁	1	1	yes	characteristic
155	#391	C₂²×C₁₁	1	4	yes	—
156	#389	—	2	2	no	—
157	#387	—	2	4	no	—
158	C₄×C₄⋊C₄	—	2	2	no	—
159	C₄×C₄⋊C₄	C₂²×C₁₁	1	2	yes	—
160	C₄×C₄⋊C₄	C₂²×C₁₁	1	1	yes	characteristic
161	#340	C₂²×C₁₁	1	2	yes	—
162	#336	C₂²×C₁₁	1	2	yes	—
163	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
164	C₂²×C₁₁	D₄²	1	1	yes	characteristic
165	C₂²×C₁₁	#550	1	1	yes	characteristic
166	C₂²×C₁₁	C₂×#137	1	2	yes	—
167	C₂²×C₁₁	#578	1	2	yes	—
168	C₄×C₁₁	—	2	2	no	—
169	C₄×C₁₁	—	4	8	no	—
170	C₄×C₁₁	—	4	4	no	—
171	C₄×C₁₁	—	4	4	no	—
172	C₄×C₁₁	—	8	2	no	—
173	C₄×C₁₁	#441	1	4	yes	complemented
174	C₄×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
175	C₄×C₁₁	#518	1	1	yes	characteristic
176	C₄×C₁₁	#550	1	1	yes	characteristic
177	C₄×C₁₁	#559	1	1	yes	characteristic
178	C₂²×Q₈	C₂³×C₁₁	1	4	yes	—
179	C₂×Q₁₆	—	2	4	no	—
180	C₂×Q₁₆	—	2	4	no	—
181	C₂×Q₁₆	—	4	2	no	—
182	C₂×Q₁₆	—	4	2	no	—
183	C₂×Q₁₆	C₂×C₄×C₁₁	1	8	yes	—
184	C₂×Q₁₆	C₂×C₄×C₁₁	1	8	yes	—
185	C₂²×C₈	C₂³×C₁₁	1	2	yes	—
186	C₂²×C₈	C₂³×C₁₁	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₁₁	1	1	yes	characteristic
188	C₄⋊Q₈	—	2	4	no	—
189	C₄⋊Q₈	C₂×C₄×C₁₁	1	8	yes	—
190	C₄×Q₈	—	2	8	no	—
191	C₄×Q₈	—	4	4	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	4	no	—
195	C₂×C₄⋊C₄	D₄×C₁₁	1	2	yes	—
196	C₂×C₄⋊C₄	C₂³×C₁₁	1	2	yes	—
197	C₂×C₄⋊C₄	C₂³×C₁₁	1	2	yes	—
198	C₂×C₄²	—	2	4	no	—
199	C₂×C₄²	—	2	2	no	—
200	C₂×C₄²	C₂³×C₁₁	1	2	yes	—
201	C₂×C₄²	C₂³×C₁₁	1	1	yes	characteristic
202	#106	—	2	2	no	—
203	#106	D₄×C₁₁	1	4	yes	—
204	C₄⋊C₈	—	2	2	no	—
205	#103	—	2	8	no	—
206	#103	—	4	4	no	—
207	#97	—	2	4	no	—
208	C₄×C₈	C₂×C₄×C₁₁	1	4	yes	—
209	#95	—	2	4	no	—
210	#95	—	2	2	no	—
211	#95	—	2	4	no	—
212	#95	D₄×C₁₁	1	2	yes	—
213	C₂×C₁₁	#2366	1	1	yes	characteristic
214	C₂×C₁₁	C₂×#441	1	1	yes	characteristic
215	C₂×C₁₁	#2950	1	1	yes	characteristic
216	C₂×C₁₁	#3276	1	2	yes	—
217	C₂×C₁₁	#3398	1	2	yes	—
218	C₂×Q₈	—	2	4	no	—
219	C₂×Q₈	—	2	4	no	—
220	C₂×Q₈	—	2	4	no	—
221	C₂×Q₈	—	2	4	no	—
222	C₂×Q₈	—	2	4	no	—
223	C₂×Q₈	C₂²×C₄×C₁₁	1	8	yes	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	C₂×D₄×C₁₁	1	4	yes	—
228	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
230	C₂²×C₄	C₂⁴×C₁₁	1	1	yes	characteristic
231	Q₁₆	—	2	16	no	—
232	Q₁₆	—	4	8	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	C₂²×C₄×C₁₁	1	2	yes	—
239	C₂×C₈	C₂²×C₄×C₁₁	1	2	yes	—
240	C₂×C₈	C₂×D₄×C₁₁	1	2	yes	—
241	C₂×C₈	C₁₁×#41	1	2	yes	—
242	C₄⋊C₄	—	2	8	no	—
243	C₄⋊C₄	—	2	8	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	4	8	no	—
249	C₄⋊C₄	C₂×D₄×C₁₁	1	4	yes	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	4	8	no	—
253	C₄²	—	4	4	no	—
254	C₄²	C₂²×C₄×C₁₁	1	2	yes	—
255	C₄²	C₁₁×#41	1	2	yes	—
256	C₁₁	#23187	1	1	yes	characteristic, direct product factor, Sylow
257	C₂³	C₂²×D₄×C₁₁	1	1	yes	characteristic
258	Q₈	—	2	8	no	—
259	Q₈	—	2	8	no	—
260	Q₈	—	4	4	no	—
261	Q₈	—	4	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	C₄×D₄×C₁₁	1	2	yes	—
276	C₂×C₄	C₄×D₄×C₁₁	1	2	yes	—
277	C₂×C₄	C₁₁×#137	1	4	yes	—
278	C₂×C₄	C₂³×C₄×C₁₁	1	1	yes	characteristic
279	C₂×C₄	C₂²×D₄×C₁₁	1	1	yes	characteristic
280	C₂×C₄	C₂×C₁₁×#41	1	1	yes	characteristic
281	C₂×C₄	C₂×C₁₁×#41	1	1	yes	characteristic
282	C₂×C₄	C₁₁×H(5,𝔽₂)	1	1	yes	characteristic
283	C₂×C₄	C₁₁×H(5,𝔽₂)	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	—	4	2	no	—
286	C₈	C₄×D₄×C₁₁	1	4	yes	—
287	C₂²	C₂×C₄×D₄×C₁₁	1	1	yes	characteristic
288	C₂²	D₄²×C₁₁	1	1	yes	characteristic
289	C₂²	C₁₁×#550	1	1	yes	characteristic
290	C₂²	C₂×C₁₁×#137	1	2	yes	—
291	C₂²	C₁₁×#578	1	2	yes	—
292	C₄	—	2	2	no	—
293	C₄	—	4	4	no	—
294	C₄	—	4	4	no	—
295	C₄	—	4	8	no	—
296	C₄	—	8	2	no	—
297	C₄	C₁₁×#441	1	4	yes	complemented
298	C₄	C₂×C₄×D₄×C₁₁	1	1	yes	characteristic
299	C₄	C₁₁×#518	1	1	yes	characteristic
300	C₄	C₁₁×#550	1	1	yes	characteristic
301	C₄	C₁₁×#559	1	1	yes	characteristic
302	C₂	C₁₁×#2366	1	1	yes	characteristic
303	C₂	C₂×C₁₁×#441	1	1	yes	characteristic
304	C₂	C₁₁×#2950	1	1	yes	characteristic
305	C₂	C₁₁×#3276	1	2	yes	—
306	C₂	C₁₁×#3398	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₁ ◁ C₄×C₁₁×C₄⋊C₄ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₁₁×#494	C₄	semidirect
65. #23187	C₁₁	direct (N × H)
146. #494	C₄×C₁₁	semidirect
173. C₄×C₁₁	#441	semidirect
256. C₁₁	#23187	direct (N × H)
297. C₄	C₁₁×#441	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₁×#494	2
C₂×C₁₁×#441	2
C₁₁×#2365	2
C₁₁×#1981	2
C₁₁×#1971	2
C₁₁×#1878	2
C₄×C₁₁×#106	2
C₁₁×#1770	2
#23187	11

Subgroups — C11×#23187 #14600380

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#23187 #14600380