Subgroups — C₁₁×#18068 #14595261

Order 2816.

586 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁×#3170	C₂	1	1	yes	maximal, characteristic
2	C₁₁×#2962	C₂	1	2	yes	maximal
3	C₁₁×#1591	C₂	1	2	yes	maximal, complemented
4	C₁₁×#1584	C₂	1	4	yes	maximal, complemented
5	C₈×C₁₁×Q₁₆	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₁×#335	C₂	1	1	yes	maximal, characteristic
7	C₁₁×#1481	C₂	1	2	yes	maximal
8	C₈×C₁₁×M₁₆	C₂	1	1	yes	maximal, characteristic
9	C₁₁×#533	C₂²	1	2	yes	—
10	C₁₁×#500	C₄	1	1	yes	characteristic
11	C₁₁×#494	C₄	1	1	yes	characteristic
12	C₁₁×#487	C₄	1	2	yes	—
13	C₁₁×#467	C₄	1	2	yes	—
14	C₁₁×#467	C₄	1	2	yes	—
15	C₁₁×#446	—	2	4	no	—
16	C₁₁×#446	C₂²	1	4	yes	—
17	C₈×Q₈×C₁₁	C₂²	1	4	yes	—
18	C₄×C₁₁×Q₁₆	C₂²	1	2	yes	—
19	C₄×C₁₁×Q₁₆	C₂²	1	2	yes	—
20	C₁₁×#431	C₂²	1	1	yes	characteristic
21	C₁₁×#424	—	2	2	no	—
22	C₂×C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
23	C₁₁×#405	—	2	2	no	—
24	C₄×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
26	C₁₁×#335	C₂²	1	1	yes	characteristic
27	C₁₁×#335	C₂²	1	2	yes	—
28	C₁₁×#335	C₂²	1	1	yes	characteristic
29	C₁₁×Q₈⋊C₈	C₂²	1	4	yes	—
30	C₁₁×Q₈⋊C₈	C₂²	1	4	yes	—
31	C₁₁×#322	C₂²	1	2	yes	—
32	C₈²×C₁₁	C₂²	1	2	yes	—
33	C₂×C₁₁×Q₁₆	C₂×C₄	1	2	yes	—
34	C₂×C₁₁×Q₁₆	C₂×C₄	1	2	yes	—
35	C₂×C₁₁×M₁₆	D₄	1	1	yes	characteristic
36	C₂×C₁₁×M₁₆	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₁	—	2	2	no	—
38	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
40	C₁₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
41	C₁₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
42	C₁₁×#125	—	2	2	no	—
43	C₁₁×C₂²⋊Q₈	—	2	2	no	—
44	C₁₁×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
45	C₄×Q₈×C₁₁	—	2	2	no	—
46	C₄×Q₈×C₁₁	C₂³	1	4	yes	—
47	C₁₁×#117	—	2	2	no	—
48	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
49	C₁₁×#107	C₂×C₄	1	2	yes	—
50	C₁₁×#107	C₂×C₄	1	2	yes	—
51	C₁₁×C₄⋊C₈	—	2	4	no	—
52	C₁₁×C₄⋊C₈	—	2	4	no	—
53	C₁₁×C₄⋊C₈	—	2	2	no	—
54	C₁₁×C₄⋊C₈	—	2	1	no	—
55	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
56	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
57	C₁₁×#103	C₂×C₄	1	4	yes	—
58	C₁₁×#103	C₂×C₄	1	4	yes	—
59	C₁₁×C₂²⋊C₈	—	2	1	no	—
60	C₁₁×#97	—	2	4	no	—
61	C₁₁×#97	D₄	1	2	yes	—
62	C₄×C₈×C₁₁	—	2	4	no	—
63	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₁₁	C₂³	1	2	yes	—
66	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₁₁	C₂³	1	2	yes	—
69	#18068	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂×Q₈×C₁₁	—	2	2	no	—
71	C₂×Q₈×C₁₁	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
75	C₁₁×Q₁₆	—	2	4	no	—
76	C₁₁×Q₁₆	—	4	2	no	—
77	C₁₁×M₁₆	—	2	2	no	—
78	C₁₁×M₁₆	Q₁₆	1	4	yes	complemented
79	C₂×C₈×C₁₁	—	2	2	no	—
80	C₂×C₈×C₁₁	—	2	2	no	—
81	C₂×C₈×C₁₁	—	2	2	no	—
82	C₂×C₈×C₁₁	—	2	2	no	—
83	C₂×C₈×C₁₁	—	4	1	no	—
84	C₂×C₈×C₁₁	—	4	4	no	—
85	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
91	C₁₁×C₄⋊C₄	—	2	2	no	—
92	C₁₁×C₄⋊C₄	—	2	2	no	—
93	C₁₁×C₄⋊C₄	—	2	2	no	—
94	C₁₁×C₄⋊C₄	—	2	4	no	—
95	C₁₁×C₄⋊C₄	—	2	4	no	—
96	C₁₁×C₄⋊C₄	C₂²×C₄	1	4	yes	—
97	C₁₁×C₂²⋊C₄	—	2	4	no	—
98	C₄²×C₁₁	—	2	4	no	—
99	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₁₁	#41	1	1	yes	characteristic
102	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
103	#3170	C₂×C₁₁	1	1	yes	characteristic
104	#2962	C₂×C₁₁	1	2	yes	—
105	#1591	C₂×C₁₁	1	2	yes	complemented
106	#1584	C₂×C₁₁	1	4	yes	complemented
107	C₈×Q₁₆	C₂×C₁₁	1	2	yes	complemented
108	C₂×#335	C₂×C₁₁	1	1	yes	characteristic
109	#1481	C₂×C₁₁	1	2	yes	—
110	C₈×M₁₆	C₂×C₁₁	1	1	yes	characteristic
111	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
112	Q₈×C₁₁	—	2	4	no	—
113	Q₈×C₁₁	—	2	4	no	—
114	Q₈×C₁₁	—	8	2	no	—
115	C₂×C₄×C₁₁	—	2	1	no	—
116	C₂×C₄×C₁₁	—	2	1	no	—
117	C₂×C₄×C₁₁	—	2	4	no	—
118	C₂×C₄×C₁₁	—	2	4	no	—
119	C₂×C₄×C₁₁	—	4	1	no	—
120	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₁	C₂×Q₁₆	1	2	yes	—
124	C₂×C₄×C₁₁	#135	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	C₂×#41	1	1	yes	characteristic
128	C₈×C₁₁	—	2	1	no	—
129	C₈×C₁₁	—	2	1	no	—
130	C₈×C₁₁	—	2	1	no	—
131	C₈×C₁₁	—	2	2	no	—
132	C₈×C₁₁	—	4	2	no	—
133	C₈×C₁₁	—	4	2	no	—
134	C₈×C₁₁	—	4	2	no	—
135	C₈×C₁₁	#131	1	2	yes	—
136	C₈×C₁₁	C₂×Q₁₆	1	4	yes	—
137	#533	C₂²×C₁₁	1	2	yes	—
138	#500	C₄×C₁₁	1	1	yes	characteristic
139	#494	C₄×C₁₁	1	1	yes	characteristic
140	#487	C₄×C₁₁	1	2	yes	—
141	#467	C₄×C₁₁	1	2	yes	—
142	#467	C₄×C₁₁	1	2	yes	—
143	#446	—	2	4	no	—
144	#446	C₂²×C₁₁	1	4	yes	—
145	C₈×Q₈	C₂²×C₁₁	1	4	yes	—
146	C₄×Q₁₆	C₂²×C₁₁	1	2	yes	—
147	C₄×Q₁₆	C₂²×C₁₁	1	2	yes	—
148	#431	C₂²×C₁₁	1	1	yes	characteristic
149	#424	—	2	2	no	—
150	C₂×C₄⋊C₈	C₂²×C₁₁	1	2	yes	—
151	#405	—	2	2	no	—
152	C₄×M₁₆	C₂²×C₁₁	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂²×C₁₁	1	1	yes	characteristic
154	#335	C₂²×C₁₁	1	1	yes	characteristic
155	#335	C₂²×C₁₁	1	2	yes	—
156	#335	C₂²×C₁₁	1	1	yes	characteristic
157	Q₈⋊C₈	C₂²×C₁₁	1	4	yes	—
158	Q₈⋊C₈	C₂²×C₁₁	1	4	yes	—
159	#322	C₂²×C₁₁	1	2	yes	—
160	C₈²	C₂²×C₁₁	1	2	yes	—
161	C₂²×C₁₁	—	2	1	no	—
162	C₂²×C₁₁	—	2	1	no	—
163	C₂²×C₁₁	C₄×Q₁₆	1	2	yes	—
164	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
165	C₄×C₁₁	—	2	1	no	—
166	C₄×C₁₁	—	2	1	no	—
167	C₄×C₁₁	—	2	1	no	—
168	C₄×C₁₁	—	4	4	no	—
169	C₄×C₁₁	—	4	4	no	—
170	C₄×C₁₁	C₄×Q₁₆	1	2	yes	—
171	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
172	C₄×C₁₁	#568	1	1	yes	characteristic
173	C₄×C₁₁	C₂²×Q₁₆	1	1	yes	characteristic
174	C₄×C₁₁	C₂×#135	1	1	yes	characteristic
175	C₂×Q₁₆	C₂×C₄×C₁₁	1	2	yes	—
176	C₂×Q₁₆	C₂×C₄×C₁₁	1	2	yes	—
177	C₂×M₁₆	D₄×C₁₁	1	1	yes	characteristic
178	C₂×M₁₆	D₄×C₁₁	1	1	yes	characteristic
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	C₂×C₄×C₁₁	1	1	yes	characteristic
181	C₂²×C₈	C₂×C₄×C₁₁	1	1	yes	characteristic
182	C₄⋊Q₈	C₂×C₄×C₁₁	1	2	yes	—
183	C₄⋊Q₈	C₂×C₄×C₁₁	1	2	yes	—
184	#125	—	2	2	no	—
185	C₂²⋊Q₈	—	2	2	no	—
186	C₂²⋊Q₈	C₂×C₄×C₁₁	1	4	yes	—
187	C₄×Q₈	—	2	2	no	—
188	C₄×Q₈	C₂³×C₁₁	1	4	yes	—
189	#117	—	2	2	no	—
190	C₂×C₄²	C₂³×C₁₁	1	1	yes	characteristic
191	#107	C₂×C₄×C₁₁	1	2	yes	—
192	#107	C₂×C₄×C₁₁	1	2	yes	—
193	C₄⋊C₈	—	2	4	no	—
194	C₄⋊C₈	—	2	4	no	—
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	—	2	1	no	—
197	C₄⋊C₈	C₂³×C₁₁	1	2	yes	—
198	C₄⋊C₈	C₂³×C₁₁	1	2	yes	—
199	#103	C₂×C₄×C₁₁	1	4	yes	—
200	#103	C₂×C₄×C₁₁	1	4	yes	—
201	C₂²⋊C₈	—	2	1	no	—
202	#97	—	2	4	no	—
203	#97	D₄×C₁₁	1	2	yes	—
204	C₄×C₈	—	2	4	no	—
205	C₄×C₈	C₂×C₄×C₁₁	1	1	yes	characteristic
206	C₄×C₈	C₂×C₄×C₁₁	1	1	yes	characteristic
207	C₄×C₈	C₂³×C₁₁	1	2	yes	—
208	C₄×C₈	C₂³×C₁₁	1	1	yes	characteristic
209	C₄×C₈	C₂³×C₁₁	1	1	yes	characteristic
210	C₄×C₈	C₂³×C₁₁	1	2	yes	—
211	C₂×C₁₁	—	2	2	no	—
212	C₂×C₁₁	#2935	1	1	yes	characteristic
213	C₂×C₁₁	C₂×C₄×Q₁₆	1	1	yes	characteristic
214	C₂×C₁₁	#2957	1	1	yes	characteristic
215	C₂×Q₈	—	2	2	no	—
216	C₂×Q₈	C₂²×C₄×C₁₁	1	4	yes	—
217	C₂²×C₄	C₂²×C₄×C₁₁	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
219	C₂²×C₄	C₁₁×#41	1	1	yes	characteristic
220	Q₁₆	—	2	4	no	—
221	Q₁₆	—	4	2	no	—
222	M₁₆	—	2	2	no	—
223	M₁₆	C₁₁×Q₁₆	1	4	yes	complemented
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	4	no	—
230	C₂×C₈	C₂²×C₄×C₁₁	1	1	yes	characteristic
231	C₂×C₈	C₂²×C₄×C₁₁	1	1	yes	characteristic
232	C₂×C₈	C₂²×C₄×C₁₁	1	1	yes	characteristic
233	C₂×C₈	C₂²×C₄×C₁₁	1	1	yes	characteristic
234	C₂×C₈	C₂×D₄×C₁₁	1	2	yes	—
235	C₂×C₈	C₂×D₄×C₁₁	1	2	yes	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	C₂²×C₄×C₁₁	1	4	yes	—
242	C₂²⋊C₄	—	2	4	no	—
243	C₄²	—	2	4	no	—
244	C₄²	C₂²×C₄×C₁₁	1	1	yes	characteristic
245	C₄²	C₂×D₄×C₁₁	1	1	yes	characteristic
246	C₄²	C₁₁×#41	1	1	yes	characteristic
247	C₄²	C₂⁴×C₁₁	1	1	yes	characteristic
248	C₁₁	#18068	1	1	yes	characteristic, direct product factor, Sylow
249	C₂³	C₄×D₄×C₁₁	1	1	yes	characteristic
250	Q₈	—	2	4	no	—
251	Q₈	—	2	4	no	—
252	Q₈	—	8	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	C₄×D₄×C₁₁	1	1	yes	characteristic
259	C₂×C₄	C₄×D₄×C₁₁	1	1	yes	characteristic
260	C₂×C₄	C₄×D₄×C₁₁	1	1	yes	characteristic
261	C₂×C₄	C₂×C₁₁×Q₁₆	1	2	yes	—
262	C₂×C₄	C₁₁×#135	1	2	yes	—
263	C₂×C₄	C₂³×C₄×C₁₁	1	1	yes	characteristic
264	C₂×C₄	C₂²×D₄×C₁₁	1	1	yes	characteristic
265	C₂×C₄	C₂×C₁₁×#41	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	2	2	no	—
270	C₈	—	4	2	no	—
271	C₈	—	4	2	no	—
272	C₈	—	4	2	no	—
273	C₈	C₁₁×#131	1	2	yes	—
274	C₈	C₂×C₁₁×Q₁₆	1	4	yes	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	C₄×C₁₁×Q₁₆	1	2	yes	—
278	C₂²	C₂×C₄×D₄×C₁₁	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	4	4	no	—
283	C₄	—	4	4	no	—
284	C₄	C₄×C₁₁×Q₁₆	1	2	yes	—
285	C₄	C₂×C₁₁×#131	1	1	yes	characteristic
286	C₄	C₁₁×#568	1	1	yes	characteristic
287	C₄	C₂²×C₁₁×Q₁₆	1	1	yes	characteristic
288	C₄	C₂×C₁₁×#135	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	C₁₁×#2935	1	1	yes	characteristic
291	C₂	C₂×C₄×C₁₁×Q₁₆	1	1	yes	characteristic
292	C₂	C₁₁×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₁
Frattini, cosocle	C₄²	C₂⁴×C₁₁
socle	C₂²×C₁₁	C₂×C₄×D₄
prime radical	C₂³×C₁₁	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₁ ◁ C₄×C₁₁×M₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₁×#1591	C₂	semidirect
4. C₁₁×#1584	C₂	semidirect
5. C₈×C₁₁×Q₁₆	C₂	semidirect
69. #18068	C₁₁	direct (N × H)
78. C₁₁×M₁₆	Q₁₆	semidirect
105. #1591	C₂×C₁₁	semidirect
106. #1584	C₂×C₁₁	semidirect
107. C₈×Q₁₆	C₂×C₁₁	semidirect
223. M₁₆	C₁₁×Q₁₆	semidirect
248. C₁₁	#18068	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×#3170	2
C₁₁×#2962	2
C₁₁×#1591	2
C₁₁×#1584	2
C₈×C₁₁×Q₁₆	2
C₂×C₁₁×#335	2
C₁₁×#1481	2
C₈×C₁₁×M₁₆	2
#18068	11

Subgroups — C11×#18068 #14595261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#18068 #14595261