Subgroups — C₁₁×#23157 #14600350

Order 2816.

806 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×#500	C₂	1	1	yes	maximal, characteristic
2	C₁₁×#2365	C₂	1	2	yes	maximal
3	C₁₁×#1988	C₂	1	2	yes	maximal
4	C₁₁×#1942	C₂	1	2	yes	maximal
5	C₁₁×#1918	C₂	1	4	yes	maximal
6	C₄×C₁₁×#107	C₂	1	1	yes	maximal, characteristic
7	C₄×C₁₁×#107	C₂	1	2	yes	maximal
8	C₈×C₁₁×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₁×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₁₁	—	2	2	no	—
11	C₁₁×#500	C₄	1	8	yes	complemented
12	C₂×C₁₁×#107	C₂²	1	1	yes	characteristic
13	C₂×C₁₁×#107	C₂²	1	2	yes	—
14	C₂×C₁₁×#107	C₂²	1	2	yes	—
15	C₂×C₁₁×#107	C₂²	1	1	yes	characteristic
16	C₂×C₁₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₁₁	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
19	C₁₁×#391	—	2	2	no	—
20	C₁₁×#389	C₂²	1	4	yes	—
21	C₁₁×#387	—	2	4	no	—
22	C₄×C₁₁×C₄⋊C₄	C₂²	1	4	yes	—
23	C₄×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
24	C₄×C₁₁×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₁₁×#340	C₂²	1	2	yes	—
26	C₁₁×#340	C₂²	1	4	yes	—
27	C₁₁×#340	C₂²	1	4	yes	—
28	C₁₁×#336	C₂²	1	2	yes	—
29	C₂²×Q₈×C₁₁	—	2	2	no	—
30	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
33	C₁₁×C₄⋊Q₈	—	2	4	no	—
34	C₁₁×C₄⋊Q₈	C₂×C₄	1	8	yes	—
35	C₄×Q₈×C₁₁	—	4	8	no	—
36	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
37	C₂×C₁₁×C₄⋊C₄	—	2	4	no	—
38	C₂×C₁₁×C₄⋊C₄	D₄	1	1	yes	characteristic
39	C₂×C₁₁×C₄⋊C₄	D₄	1	1	yes	characteristic
40	C₂×C₁₁×C₄⋊C₄	C₂³	1	2	yes	—
41	C₂×C₁₁×C₄⋊C₄	C₂³	1	4	yes	—
42	C₂×C₁₁×C₄⋊C₄	C₂³	1	2	yes	—
43	C₂×C₄²×C₁₁	—	2	4	no	—
44	C₂×C₄²×C₁₁	—	2	2	no	—
45	C₂×C₄²×C₁₁	C₂³	1	2	yes	—
46	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
47	C₁₁×#107	—	2	2	no	—
48	C₁₁×#107	C₂×C₄	1	8	yes	—
49	C₁₁×#107	C₂×C₄	1	8	yes	—
50	C₁₁×#107	Q₈	1	4	yes	complemented
51	C₁₁×C₄⋊C₈	—	2	2	no	—
52	C₄×C₈×C₁₁	—	2	4	no	—
53	C₄×C₈×C₁₁	C₂×C₄	1	4	yes	—
54	C₁₁×#95	—	2	2	no	—
55	C₁₁×#95	—	2	4	no	—
56	C₁₁×#95	—	2	4	no	—
57	C₁₁×#95	D₄	1	2	yes	—
58	#23157	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂×Q₈×C₁₁	—	2	8	no	—
60	C₂×Q₈×C₁₁	—	4	4	no	—
61	C₂×Q₈×C₁₁	—	4	4	no	—
62	C₂²×C₄×C₁₁	—	2	2	no	—
63	C₂²×C₄×C₁₁	—	2	4	no	—
64	C₂²×C₄×C₁₁	—	2	2	no	—
65	C₂²×C₄×C₁₁	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₁	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
70	C₂×C₈×C₁₁	—	2	2	no	—
71	C₂×C₈×C₁₁	—	2	2	no	—
72	C₂×C₈×C₁₁	—	2	1	no	—
73	C₂×C₈×C₁₁	—	2	1	no	—
74	C₂×C₈×C₁₁	—	2	2	no	—
75	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
78	C₂×C₈×C₁₁	#41	1	2	yes	—
79	C₁₁×C₄⋊C₄	—	2	8	no	—
80	C₁₁×C₄⋊C₄	—	2	2	no	—
81	C₁₁×C₄⋊C₄	—	2	4	no	—
82	C₁₁×C₄⋊C₄	—	2	2	no	—
83	C₁₁×C₄⋊C₄	—	2	2	no	—
84	C₁₁×C₄⋊C₄	—	2	2	no	—
85	C₁₁×C₄⋊C₄	—	4	8	no	—
86	C₁₁×C₄⋊C₄	D₈	1	4	yes	—
87	C₁₁×C₄⋊C₄	C₂²×C₄	1	8	yes	—
88	C₁₁×C₄⋊C₄	C₂×Q₈	1	4	yes	—
89	C₄²×C₁₁	—	2	2	no	—
90	C₄²×C₁₁	—	2	2	no	—
91	C₄²×C₁₁	—	4	8	no	—
92	C₄²×C₁₁	—	4	4	no	—
93	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
94	C₄²×C₁₁	#41	1	2	yes	—
95	C₂×#500	C₂×C₁₁	1	1	yes	characteristic
96	#2365	C₂×C₁₁	1	2	yes	—
97	#1988	C₂×C₁₁	1	2	yes	—
98	#1942	C₂×C₁₁	1	2	yes	—
99	#1918	C₂×C₁₁	1	4	yes	—
100	C₄×#107	C₂×C₁₁	1	1	yes	characteristic
101	C₄×#107	C₂×C₁₁	1	2	yes	—
102	C₈×C₄⋊C₄	C₂×C₁₁	1	1	yes	characteristic
103	C₂³×C₁₁	C₂²×D₄	1	1	yes	characteristic
104	Q₈×C₁₁	—	4	16	no	—
105	C₂×C₄×C₁₁	—	2	2	no	—
106	C₂×C₄×C₁₁	—	2	2	no	—
107	C₂×C₄×C₁₁	—	2	1	no	—
108	C₂×C₄×C₁₁	—	2	1	no	—
109	C₂×C₄×C₁₁	—	2	8	no	—
110	C₂×C₄×C₁₁	—	2	2	no	—
111	C₂×C₄×C₁₁	—	2	4	no	—
112	C₂×C₄×C₁₁	—	2	2	no	—
113	C₂×C₄×C₁₁	—	2	2	no	—
114	C₂×C₄×C₁₁	—	4	2	no	—
115	C₂×C₄×C₁₁	—	4	4	no	—
116	C₂×C₄×C₁₁	—	4	4	no	—
117	C₂×C₄×C₁₁	—	4	2	no	—
118	C₂×C₄×C₁₁	—	4	2	no	—
119	C₂×C₄×C₁₁	—	4	2	no	—
120	C₂×C₄×C₁₁	—	4	2	no	—
121	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₁	C₂×D₈	1	4	yes	—
124	C₂×C₄×C₁₁	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₁₁	#135	1	2	yes	—
126	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
129	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
131	C₂×C₄×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
132	C₈×C₁₁	—	2	2	no	—
133	C₈×C₁₁	—	4	2	no	—
134	C₈×C₁₁	C₄×Q₈	1	4	yes	complemented
135	C₂×C₄⋊Q₈	C₂²×C₁₁	1	2	yes	—
136	C₂×C₄×Q₈	—	2	2	no	—
137	#500	C₄×C₁₁	1	8	yes	complemented
138	C₂×#107	C₂²×C₁₁	1	1	yes	characteristic
139	C₂×#107	C₂²×C₁₁	1	2	yes	—
140	C₂×#107	C₂²×C₁₁	1	2	yes	—
141	C₂×#107	C₂²×C₁₁	1	1	yes	characteristic
142	C₂×C₄⋊C₈	C₂²×C₁₁	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂²×C₁₁	1	2	yes	—
144	C₂×C₄×C₈	C₂²×C₁₁	1	1	yes	characteristic
145	#391	—	2	2	no	—
146	#389	C₂²×C₁₁	1	4	yes	—
147	#387	—	2	4	no	—
148	C₄×C₄⋊C₄	C₂²×C₁₁	1	4	yes	—
149	C₄×C₄⋊C₄	C₂²×C₁₁	1	2	yes	—
150	C₄×C₄⋊C₄	C₂²×C₁₁	1	1	yes	characteristic
151	#340	C₂²×C₁₁	1	2	yes	—
152	#340	C₂²×C₁₁	1	4	yes	—
153	#340	C₂²×C₁₁	1	4	yes	—
154	#336	C₂²×C₁₁	1	2	yes	—
155	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
156	C₂²×C₁₁	#547	1	1	yes	characteristic
157	C₂²×C₁₁	D₄×Q₈	1	1	yes	characteristic
158	C₂²×C₁₁	C₂²×D₈	1	1	yes	characteristic
159	C₂²×C₁₁	C₂×#135	1	1	yes	characteristic
160	C₂²×C₁₁	#578	1	1	yes	characteristic
161	C₂²×C₁₁	#578	1	1	yes	characteristic
162	C₄×C₁₁	—	2	4	no	—
163	C₄×C₁₁	—	2	2	no	—
164	C₄×C₁₁	—	4	2	no	—
165	C₄×C₁₁	—	4	4	no	—
166	C₄×C₁₁	—	4	8	no	—
167	C₄×C₁₁	—	8	2	no	—
168	C₄×C₁₁	C₄×D₈	1	4	yes	—
169	C₄×C₁₁	C₂×C₄×Q₈	1	1	yes	characteristic
170	C₄×C₁₁	#519	1	1	yes	characteristic
171	C₄×C₁₁	#556	1	1	yes	characteristic
172	C₄×C₁₁	#557	1	1	yes	characteristic
173	C₂²×Q₈	—	2	2	no	—
174	C₂²×C₈	C₂³×C₁₁	1	2	yes	—
175	C₂²×C₈	C₂³×C₁₁	1	1	yes	characteristic
176	C₂²×C₈	C₂³×C₁₁	1	1	yes	characteristic
177	C₄⋊Q₈	—	2	4	no	—
178	C₄⋊Q₈	C₂×C₄×C₁₁	1	8	yes	—
179	C₄×Q₈	—	4	8	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	4	no	—
182	C₂×C₄⋊C₄	D₄×C₁₁	1	1	yes	characteristic
183	C₂×C₄⋊C₄	D₄×C₁₁	1	1	yes	characteristic
184	C₂×C₄⋊C₄	C₂³×C₁₁	1	2	yes	—
185	C₂×C₄⋊C₄	C₂³×C₁₁	1	4	yes	—
186	C₂×C₄⋊C₄	C₂³×C₁₁	1	2	yes	—
187	C₂×C₄²	—	2	4	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	C₂³×C₁₁	1	2	yes	—
190	C₂×C₄²	C₂³×C₁₁	1	1	yes	characteristic
191	#107	—	2	2	no	—
192	#107	C₂×C₄×C₁₁	1	8	yes	—
193	#107	C₂×C₄×C₁₁	1	8	yes	—
194	#107	Q₈×C₁₁	1	4	yes	complemented
195	C₄⋊C₈	—	2	2	no	—
196	C₄×C₈	—	2	4	no	—
197	C₄×C₈	C₂×C₄×C₁₁	1	4	yes	—
198	#95	—	2	2	no	—
199	#95	—	2	4	no	—
200	#95	—	2	4	no	—
201	#95	D₄×C₁₁	1	2	yes	—
202	C₂×C₁₁	#2365	1	1	yes	characteristic
203	C₂×C₁₁	C₂×C₄×D₈	1	1	yes	characteristic
204	C₂×C₁₁	#2950	1	1	yes	characteristic
205	C₂×C₁₁	#3334	1	1	yes	characteristic
206	C₂×C₁₁	#3342	1	1	yes	characteristic
207	C₂×C₁₁	Q₈×D₈	1	1	yes	characteristic
208	C₂×C₁₁	#3383	1	1	yes	characteristic
209	C₂×Q₈	—	2	8	no	—
210	C₂×Q₈	—	4	4	no	—
211	C₂×Q₈	—	4	4	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	C₂×D₄×C₁₁	1	2	yes	—
216	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₁₁	1	2	yes	—
218	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
219	C₂²×C₄	C₂⁴×C₁₁	1	1	yes	characteristic
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	C₂²×C₄×C₁₁	1	2	yes	—
226	C₂×C₈	C₂²×C₄×C₁₁	1	2	yes	—
227	C₂×C₈	C₂×Q₈×C₁₁	1	2	yes	—
228	C₂×C₈	C₁₁×#41	1	2	yes	—
229	C₄⋊C₄	—	2	8	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	4	8	no	—
236	C₄⋊C₄	C₁₁×D₈	1	4	yes	—
237	C₄⋊C₄	C₂²×C₄×C₁₁	1	8	yes	—
238	C₄⋊C₄	C₂×Q₈×C₁₁	1	4	yes	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	4	8	no	—
242	C₄²	—	4	4	no	—
243	C₄²	C₂²×C₄×C₁₁	1	2	yes	—
244	C₄²	C₁₁×#41	1	2	yes	—
245	C₁₁	#23157	1	1	yes	characteristic, direct product factor, Sylow
246	C₂³	C₂²×D₄×C₁₁	1	1	yes	characteristic
247	Q₈	—	4	16	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	C₄×D₄×C₁₁	1	2	yes	—
265	C₂×C₄	C₄×D₄×C₁₁	1	2	yes	—
266	C₂×C₄	C₂×C₁₁×D₈	1	4	yes	—
267	C₂×C₄	C₂×C₁₁×D₈	1	2	yes	—
268	C₂×C₄	C₁₁×#135	1	2	yes	—
269	C₂×C₄	C₂³×C₄×C₁₁	1	1	yes	characteristic
270	C₂×C₄	C₂²×Q₈×C₁₁	1	1	yes	characteristic
271	C₂×C₄	C₂×C₁₁×#41	1	1	yes	characteristic
272	C₂×C₄	C₂×C₁₁×#41	1	1	yes	characteristic
273	C₂×C₄	C₁₁×2₋¹⁺⁴	1	1	yes	characteristic
274	C₂×C₄	C₁₁×2₋¹⁺⁴	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	—	4	2	no	—
277	C₈	C₄×Q₈×C₁₁	1	4	yes	complemented
278	C₂²	C₂×C₄×D₄×C₁₁	1	1	yes	characteristic
279	C₂²	C₁₁×#547	1	1	yes	characteristic
280	C₂²	D₄×Q₈×C₁₁	1	1	yes	characteristic
281	C₂²	C₂²×C₁₁×D₈	1	1	yes	characteristic
282	C₂²	C₂×C₁₁×#135	1	1	yes	characteristic
283	C₂²	C₁₁×#578	1	1	yes	characteristic
284	C₂²	C₁₁×#578	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	2	4	no	—
287	C₄	—	4	8	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	4	no	—
290	C₄	—	8	2	no	—
291	C₄	C₄×C₁₁×D₈	1	4	yes	—
292	C₄	C₂×C₄×Q₈×C₁₁	1	1	yes	characteristic
293	C₄	C₁₁×#519	1	1	yes	characteristic
294	C₄	C₁₁×#556	1	1	yes	characteristic
295	C₄	C₁₁×#557	1	1	yes	characteristic
296	C₂	C₁₁×#2365	1	1	yes	characteristic
297	C₂	C₂×C₄×C₁₁×D₈	1	1	yes	characteristic
298	C₂	C₁₁×#2950	1	1	yes	characteristic
299	C₂	C₁₁×#3334	1	1	yes	characteristic
300	C₂	C₁₁×#3342	1	1	yes	characteristic
301	C₂	Q₈×C₁₁×D₈	1	1	yes	characteristic
302	C₂	C₁₁×#3383	1	1	yes	characteristic

	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
11. C₁₁×#500	C₄	semidirect
50. C₁₁×#107	Q₈	semidirect
58. #23157	C₁₁	direct (N × H)
134. C₈×C₁₁	C₄×Q₈	semidirect
137. #500	C₄×C₁₁	semidirect
194. #107	Q₈×C₁₁	semidirect
245. C₁₁	#23157	direct (N × H)
277. C₈	C₄×Q₈×C₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₁×#500	2
C₁₁×#2365	2
C₁₁×#1988	2
C₁₁×#1942	2
C₁₁×#1918	2
C₄×C₁₁×#107	2
C₈×C₁₁×C₄⋊C₄	2
#23157	11

Subgroups — C11×#23157 #14600350

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#23157 #14600350