Subgroups — C₁₁×#21155 #14598348

Order 2816.

982 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×#427	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₁×#106	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₁×#405	C₂	1	1	yes	maximal, characteristic
4	C₁₁×#2317	C₂	1	2	yes	maximal
5	C₁₁×#1943	C₂	1	4	yes	maximal, complemented
6	C₁₁×#1941	C₂	1	4	yes	maximal, complemented
7	C₁₁×#1736	C₂	1	2	yes	maximal
8	C₂²×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
10	C₂×C₁₁×#117	C₂²	1	2	yes	—
11	C₂×C₁₁×#117	C₂²	1	1	yes	characteristic
12	C₂²×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
13	C₁₁×#427	C₄	1	8	yes	complemented
14	C₂×C₁₁×#107	C₂²	1	2	yes	—
15	C₂×C₁₁×#106	—	2	4	no	—
16	C₂×C₁₁×#106	C₂²	1	2	yes	—
17	C₂×C₁₁×#106	C₂²	1	2	yes	—
18	C₂×C₁₁×#106	C₂²	1	2	yes	—
19	C₁₁×#405	—	2	4	no	—
20	C₂×C₁₁×#97	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₁	C₂²	1	2	yes	—
22	C₁₁×#389	C₂²	1	8	yes	—
23	C₁₁×#385	—	2	4	no	—
24	C₁₁×#384	—	2	2	no	—
25	C₁₁×#382	—	2	2	no	—
26	C₁₁×#340	C₂²	1	8	yes	—
27	C₂³×C₄×C₁₁	—	2	2	no	—
28	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
29	C₂×C₁₁×M₁₆	—	2	2	no	—
30	C₂×C₁₁×M₁₆	—	2	2	no	—
31	C₂×C₁₁×M₁₆	Q₈	1	4	yes	—
32	C₂²×C₈×C₁₁	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₁₁	D₄	1	4	yes	—
34	C₂²×C₈×C₁₁	Q₈	1	4	yes	—
35	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
38	C₁₁×#117	—	2	2	no	—
39	C₁₁×#117	—	2	4	no	—
40	C₁₁×#117	—	2	2	no	—
41	C₁₁×#117	C₂×C₄	1	8	yes	—
42	C₂×C₁₁×C₄⋊C₄	—	2	4	no	—
43	C₂×C₁₁×C₄⋊C₄	—	2	4	no	—
44	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
45	C₂×C₁₁×C₄⋊C₄	—	2	8	no	—
46	C₂×C₁₁×C₄⋊C₄	D₄	1	2	yes	—
47	C₂×C₁₁×C₄⋊C₄	C₂³	1	4	yes	—
48	C₂×C₁₁×C₄⋊C₄	C₂³	1	4	yes	—
49	C₂×C₁₁×C₂²⋊C₄	—	2	2	no	—
50	C₂×C₁₁×C₂²⋊C₄	D₄	1	2	yes	—
51	C₂×C₄²×C₁₁	—	2	2	no	—
52	C₂×C₄²×C₁₁	C₂³	1	2	yes	—
53	C₁₁×#107	C₂×C₄	1	8	yes	complemented
54	C₁₁×#106	—	2	4	no	—
55	C₁₁×#106	—	2	4	no	—
56	C₁₁×#106	C₂×C₄	1	8	yes	complemented
57	C₁₁×#97	—	2	4	no	—
58	C₄×C₈×C₁₁	—	2	4	no	—
59	C₁₁×#95	—	2	8	no	—
60	#21155	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂⁴×C₁₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₁	—	2	4	no	—
63	C₂²×C₄×C₁₁	—	2	4	no	—
64	C₂²×C₄×C₁₁	—	4	4	no	—
65	C₂²×C₄×C₁₁	—	8	4	no	—
66	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₁	C₂×D₄	1	4	yes	—
68	C₂²×C₄×C₁₁	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₁	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₁₁	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₁₁	#41	1	2	yes	—
73	C₂²×C₄×C₁₁	#41	1	2	yes	—
74	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
75	C₁₁×M₁₆	—	2	8	no	—
76	C₂×C₈×C₁₁	—	2	2	no	—
77	C₂×C₈×C₁₁	—	2	2	no	—
78	C₂×C₈×C₁₁	C₄⋊C₄	1	16	yes	complemented
79	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₁₁	C₂×Q₈	1	4	yes	—
84	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
85	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
86	C₁₁×C₄⋊C₄	—	2	4	no	—
87	C₁₁×C₄⋊C₄	—	2	4	no	—
88	C₁₁×C₄⋊C₄	—	2	4	no	—
89	C₁₁×C₄⋊C₄	—	2	4	no	—
90	C₁₁×C₄⋊C₄	—	4	4	no	—
91	C₁₁×C₄⋊C₄	—	8	8	no	—
92	C₁₁×C₄⋊C₄	C₂²×C₄	1	8	yes	—
93	C₁₁×C₂²⋊C₄	—	2	8	no	—
94	C₁₁×C₂²⋊C₄	—	2	4	no	—
95	C₄²×C₁₁	—	2	8	no	—
96	C₄²×C₁₁	—	2	2	no	—
97	C₄²×C₁₁	—	2	2	no	—
98	C₂×#427	C₂×C₁₁	1	1	yes	characteristic
99	C₂²×#106	C₂×C₁₁	1	1	yes	characteristic
100	C₂×#405	C₂×C₁₁	1	1	yes	characteristic
101	#2317	C₂×C₁₁	1	2	yes	—
102	#1943	C₂×C₁₁	1	4	yes	complemented
103	#1941	C₂×C₁₁	1	4	yes	complemented
104	#1736	C₂×C₁₁	1	2	yes	—
105	C₂³×C₁₁	—	2	4	no	—
106	C₂³×C₁₁	C₄×D₄	1	2	yes	—
107	C₂³×C₁₁	#121	1	2	yes	—
108	C₂³×C₁₁	C₄⋊Q₈	1	2	yes	—
109	C₂³×C₁₁	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₁	—	2	8	no	—
111	C₂×C₄×C₁₁	—	2	4	no	—
112	C₂×C₄×C₁₁	—	2	8	no	—
113	C₂×C₄×C₁₁	—	4	4	no	—
114	C₂×C₄×C₁₁	—	4	4	no	—
115	C₂×C₄×C₁₁	—	4	4	no	—
116	C₂×C₄×C₁₁	—	4	8	no	—
117	C₂×C₄×C₁₁	—	8	8	no	—
118	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	4	yes	—
119	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₁₁	C₄×Q₈	1	4	yes	—
121	C₂×C₄×C₁₁	#121	1	2	yes	—
122	C₂×C₄×C₁₁	C₂²⋊Q₈	1	4	yes	—
123	C₂×C₄×C₁₁	#125	1	4	yes	—
124	C₂×C₄×C₁₁	C₄⋊Q₈	1	2	yes	—
125	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
131	C₈×C₁₁	—	2	4	no	—
132	C₈×C₁₁	C₂×C₄⋊C₄	1	4	yes	—
133	C₈×C₁₁	C₂×C₄⋊C₄	1	4	yes	—
134	C₂²×M₁₆	C₂²×C₁₁	1	1	yes	characteristic
135	C₂³×C₈	C₂²×C₁₁	1	1	yes	characteristic
136	C₂×#117	C₂²×C₁₁	1	2	yes	—
137	C₂×#117	C₂²×C₁₁	1	1	yes	characteristic
138	C₂²×C₄⋊C₄	C₂²×C₁₁	1	2	yes	—
139	#427	C₄×C₁₁	1	8	yes	complemented
140	C₂×#107	C₂²×C₁₁	1	2	yes	—
141	C₂×#106	—	2	4	no	—
142	C₂×#106	C₂²×C₁₁	1	2	yes	—
143	C₂×#106	C₂²×C₁₁	1	2	yes	—
144	C₂×#106	C₂²×C₁₁	1	2	yes	—
145	#405	—	2	4	no	—
146	C₂×#97	C₂²×C₁₁	1	2	yes	—
147	C₂×C₄×C₈	C₂²×C₁₁	1	2	yes	—
148	#389	C₂²×C₁₁	1	8	yes	—
149	#385	—	2	4	no	—
150	#384	—	2	2	no	—
151	#382	—	2	2	no	—
152	#340	C₂²×C₁₁	1	8	yes	—
153	C₂²×C₁₁	—	2	4	no	—
154	C₂²×C₁₁	—	2	4	no	—
155	C₂²×C₁₁	—	2	4	no	—
156	C₂²×C₁₁	#389	1	4	yes	—
157	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
158	C₂²×C₁₁	C₂×#121	1	1	yes	characteristic
159	C₂²×C₁₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
160	C₂²×C₁₁	#577	1	4	yes	—
161	C₄×C₁₁	—	4	8	no	—
162	C₄×C₁₁	—	4	4	no	—
163	C₄×C₁₁	—	8	4	no	—
164	C₄×C₁₁	#389	1	4	yes	—
165	C₄×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
166	C₄×C₁₁	C₂×C₄×Q₈	1	1	yes	characteristic
167	C₄×C₁₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
168	C₄×C₁₁	C₂×#125	1	1	yes	characteristic
169	C₂³×C₄	—	2	2	no	—
170	C₂³×C₄	C₂³×C₁₁	1	1	yes	characteristic
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	Q₈×C₁₁	1	4	yes	—
174	C₂²×C₈	C₂×C₄×C₁₁	1	4	yes	—
175	C₂²×C₈	D₄×C₁₁	1	4	yes	—
176	C₂²×C₈	Q₈×C₁₁	1	4	yes	—
177	C₂²×C₈	C₂³×C₁₁	1	2	yes	—
178	C₂²×C₈	C₂³×C₁₁	1	1	yes	characteristic
179	C₂²×C₈	C₂³×C₁₁	1	1	yes	characteristic
180	#117	—	2	2	no	—
181	#117	—	2	4	no	—
182	#117	—	2	2	no	—
183	#117	C₂×C₄×C₁₁	1	8	yes	—
184	C₂×C₄⋊C₄	—	2	4	no	—
185	C₂×C₄⋊C₄	—	2	4	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	8	no	—
188	C₂×C₄⋊C₄	D₄×C₁₁	1	2	yes	—
189	C₂×C₄⋊C₄	C₂³×C₁₁	1	4	yes	—
190	C₂×C₄⋊C₄	C₂³×C₁₁	1	4	yes	—
191	C₂×C₂²⋊C₄	—	2	2	no	—
192	C₂×C₂²⋊C₄	D₄×C₁₁	1	2	yes	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	C₂³×C₁₁	1	2	yes	—
195	#107	C₂×C₄×C₁₁	1	8	yes	complemented
196	#106	—	2	4	no	—
197	#106	—	2	4	no	—
198	#106	C₂×C₄×C₁₁	1	8	yes	complemented
199	#97	—	2	4	no	—
200	C₄×C₈	—	2	4	no	—
201	#95	—	2	8	no	—
202	C₂×C₁₁	—	2	4	no	—
203	C₂×C₁₁	C₂×#389	1	1	yes	characteristic
204	C₂×C₁₁	#2952	1	2	yes	—
205	C₂×C₁₁	#3057	1	2	yes	—
206	C₂×C₁₁	#3163	1	2	yes	—
207	C₂⁴	C₂×D₄×C₁₁	1	1	yes	characteristic
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	4	4	no	—
211	C₂²×C₄	—	8	4	no	—
212	C₂²×C₄	C₂²×C₄×C₁₁	1	2	yes	—
213	C₂²×C₄	C₂×D₄×C₁₁	1	4	yes	—
214	C₂²×C₄	C₂×D₄×C₁₁	1	2	yes	—
215	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
216	C₂²×C₄	C₂×Q₈×C₁₁	1	2	yes	—
217	C₂²×C₄	C₂×Q₈×C₁₁	1	2	yes	—
218	C₂²×C₄	C₁₁×#41	1	2	yes	—
219	C₂²×C₄	C₁₁×#41	1	2	yes	—
220	C₂²×C₄	C₂⁴×C₁₁	1	1	yes	characteristic
221	M₁₆	—	2	8	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	C₁₁×C₄⋊C₄	1	16	yes	complemented
225	C₂×C₈	C₂²×C₄×C₁₁	1	2	yes	—
226	C₂×C₈	C₂²×C₄×C₁₁	1	2	yes	—
227	C₂×C₈	C₂×D₄×C₁₁	1	2	yes	—
228	C₂×C₈	C₂×D₄×C₁₁	1	2	yes	—
229	C₂×C₈	C₂×Q₈×C₁₁	1	4	yes	—
230	C₂×C₈	C₂×Q₈×C₁₁	1	2	yes	—
231	C₂×C₈	C₂×Q₈×C₁₁	1	2	yes	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	4	4	no	—
237	C₄⋊C₄	—	8	8	no	—
238	C₄⋊C₄	C₂²×C₄×C₁₁	1	8	yes	—
239	C₂²⋊C₄	—	2	8	no	—
240	C₂²⋊C₄	—	2	4	no	—
241	C₄²	—	2	8	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₁₁	#21155	1	1	yes	characteristic, direct product factor, Sylow
245	C₂³	—	2	4	no	—
246	C₂³	C₄×D₄×C₁₁	1	2	yes	—
247	C₂³	C₁₁×#121	1	2	yes	—
248	C₂³	C₁₁×C₄⋊Q₈	1	2	yes	—
249	C₂³	C₂²×D₄×C₁₁	1	1	yes	characteristic
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	8	no	—
257	C₂×C₄	—	8	8	no	—
258	C₂×C₄	C₂×C₁₁×C₄⋊C₄	1	4	yes	—
259	C₂×C₄	C₄×D₄×C₁₁	1	2	yes	—
260	C₂×C₄	C₄×Q₈×C₁₁	1	4	yes	—
261	C₂×C₄	C₁₁×#121	1	2	yes	—
262	C₂×C₄	C₁₁×C₂²⋊Q₈	1	4	yes	—
263	C₂×C₄	C₁₁×#125	1	4	yes	—
264	C₂×C₄	C₁₁×C₄⋊Q₈	1	2	yes	—
265	C₂×C₄	C₂³×C₄×C₁₁	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₄×C₁₁	1	1	yes	characteristic
267	C₂×C₄	C₂²×Q₈×C₁₁	1	1	yes	characteristic
268	C₂×C₄	C₂²×Q₈×C₁₁	1	1	yes	characteristic
269	C₂×C₄	C₂×C₁₁×#41	1	1	yes	characteristic
270	C₂×C₄	C₂×C₁₁×#41	1	1	yes	characteristic
271	C₈	—	2	4	no	—
272	C₈	C₂×C₁₁×C₄⋊C₄	1	4	yes	—
273	C₈	C₂×C₁₁×C₄⋊C₄	1	4	yes	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	4	no	—
276	C₂²	—	2	4	no	—
277	C₂²	C₁₁×#389	1	4	yes	—
278	C₂²	C₂×C₄×D₄×C₁₁	1	1	yes	characteristic
279	C₂²	C₂×C₁₁×#121	1	1	yes	characteristic
280	C₂²	C₂×C₁₁×C₄⋊Q₈	1	1	yes	characteristic
281	C₂²	C₁₁×#577	1	4	yes	—
282	C₄	—	4	8	no	—
283	C₄	—	4	4	no	—
284	C₄	—	8	4	no	—
285	C₄	C₁₁×#389	1	4	yes	—
286	C₄	C₂²×C₁₁×C₄⋊C₄	1	1	yes	characteristic
287	C₄	C₂×C₄×Q₈×C₁₁	1	1	yes	characteristic
288	C₄	C₂×C₁₁×C₂²⋊Q₈	1	1	yes	characteristic
289	C₄	C₂×C₁₁×#125	1	1	yes	characteristic
290	C₂	—	2	4	no	—
291	C₂	C₂×C₁₁×#389	1	1	yes	characteristic
292	C₂	C₁₁×#2952	1	2	yes	—
293	C₂	C₁₁×#3057	1	2	yes	—
294	C₂	C₁₁×#3163	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₁
prime radical	C₂⁴×C₁₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₁ ◁ C₂×C₁₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
5. C₁₁×#1943	C₂	semidirect
6. C₁₁×#1941	C₂	semidirect
13. C₁₁×#427	C₄	semidirect
53. C₁₁×#107	C₂×C₄	semidirect
56. C₁₁×#106	C₂×C₄	semidirect
60. #21155	C₁₁	direct (N × H)
78. C₂×C₈×C₁₁	C₄⋊C₄	semidirect
102. #1943	C₂×C₁₁	semidirect
103. #1941	C₂×C₁₁	semidirect
139. #427	C₄×C₁₁	semidirect
195. #107	C₂×C₄×C₁₁	semidirect
198. #106	C₂×C₄×C₁₁	semidirect
224. C₂×C₈	C₁₁×C₄⋊C₄	semidirect
244. C₁₁	#21155	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₁×#427	2
C₂²×C₁₁×#106	2
C₂×C₁₁×#405	2
C₁₁×#2317	2
C₁₁×#1943	2
C₁₁×#1941	2
C₁₁×#1736	2
#21155	11

Subgroups — C11×#21155 #14598348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#21155 #14598348