Subgroups — C₁₁×#23184 #14600377

Order 2816.

958 conjugacy classes of subgroups.

The subgroup lattice has 334 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×C₄×C₁₁×Q₁₆	C₂	1	2	yes	maximal
3	C₁₁×#2365	C₂	1	2	yes	maximal
4	C₁₁×#1981	C₂	1	2	yes	maximal
5	C₁₁×#1969	C₂	1	2	yes	maximal
6	C₁₁×#1878	C₂	1	4	yes	maximal
7	C₄×C₁₁×#107	C₂	1	1	yes	maximal, characteristic
8	C₈×C₁₁×C₄⋊C₄	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	C₄×C₁₁×Q₁₆	—	2	8	no	—
15	C₂×C₁₁×#107	C₂²	1	1	yes	characteristic
16	C₂×C₁₁×#107	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×C₄×C₈×C₁₁	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	1	yes	characteristic
48	C₂×C₁₁×C₄⋊C₄	D₄	1	1	yes	characteristic
49	C₂×C₁₁×C₄⋊C₄	C₂³	1	2	yes	—
50	C₂×C₁₁×C₄⋊C₄	C₂³	1	2	yes	—
51	C₂×C₄²×C₁₁	—	2	2	no	—
52	C₂×C₄²×C₁₁	—	2	4	no	—
53	C₂×C₄²×C₁₁	C₂³	1	2	yes	—
54	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
55	C₁₁×#107	—	2	2	no	—
56	C₁₁×#107	D₄	1	4	yes	—
57	C₁₁×C₄⋊C₈	—	2	2	no	—
58	C₁₁×#103	—	2	8	no	—
59	C₁₁×#103	—	4	4	no	—
60	C₄×C₈×C₁₁	—	2	4	no	—
61	C₄×C₈×C₁₁	C₂×C₄	1	4	yes	—
62	C₁₁×#95	—	2	4	no	—
63	C₁₁×#95	—	2	2	no	—
64	C₁₁×#95	—	2	4	no	—
65	C₁₁×#95	D₄	1	2	yes	—
66	#23184	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
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	—	2	4	no	—
72	C₂×Q₈×C₁₁	C₂²×C₄	1	8	yes	—
73	C₂²×C₄×C₁₁	—	2	4	no	—
74	C₂²×C₄×C₁₁	—	2	2	no	—
75	C₂²×C₄×C₁₁	—	2	2	no	—
76	C₂²×C₄×C₁₁	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₁	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
81	C₁₁×Q₁₆	—	2	16	no	—
82	C₁₁×Q₁₆	—	4	8	no	—
83	C₂×C₈×C₁₁	—	2	2	no	—
84	C₂×C₈×C₁₁	—	2	2	no	—
85	C₂×C₈×C₁₁	—	2	1	no	—
86	C₂×C₈×C₁₁	—	2	1	no	—
87	C₂×C₈×C₁₁	—	2	2	no	—
88	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₁₁	#41	1	2	yes	—
92	C₁₁×C₄⋊C₄	—	2	8	no	—
93	C₁₁×C₄⋊C₄	—	2	8	no	—
94	C₁₁×C₄⋊C₄	—	2	2	no	—
95	C₁₁×C₄⋊C₄	—	2	2	no	—
96	C₁₁×C₄⋊C₄	—	2	2	no	—
97	C₁₁×C₄⋊C₄	—	2	2	no	—
98	C₁₁×C₄⋊C₄	—	4	8	no	—
99	C₁₁×C₄⋊C₄	Q₁₆	1	4	yes	complemented
100	C₁₁×C₄⋊C₄	C₂×D₄	1	4	yes	—
101	C₄²×C₁₁	—	2	2	no	—
102	C₄²×C₁₁	—	2	2	no	—
103	C₄²×C₁₁	—	4	8	no	—
104	C₄²×C₁₁	—	4	4	no	—
105	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
106	C₄²×C₁₁	#41	1	2	yes	—
107	C₂×#494	C₂×C₁₁	1	1	yes	characteristic
108	C₂×C₄×Q₁₆	C₂×C₁₁	1	2	yes	—
109	#2365	C₂×C₁₁	1	2	yes	—
110	#1981	C₂×C₁₁	1	2	yes	—
111	#1969	C₂×C₁₁	1	2	yes	—
112	#1878	C₂×C₁₁	1	4	yes	—
113	C₄×#107	C₂×C₁₁	1	1	yes	characteristic
114	C₈×C₄⋊C₄	C₂×C₁₁	1	1	yes	characteristic
115	C₂³×C₁₁	C₂²×D₄	1	1	yes	characteristic
116	Q₈×C₁₁	—	2	8	no	—
117	Q₈×C₁₁	—	2	8	no	—
118	Q₈×C₁₁	—	4	4	no	—
119	Q₈×C₁₁	—	4	4	no	—
120	C₂×C₄×C₁₁	—	2	2	no	—
121	C₂×C₄×C₁₁	—	2	1	no	—
122	C₂×C₄×C₁₁	—	2	1	no	—
123	C₂×C₄×C₁₁	—	2	2	no	—
124	C₂×C₄×C₁₁	—	2	8	no	—
125	C₂×C₄×C₁₁	—	2	2	no	—
126	C₂×C₄×C₁₁	—	2	4	no	—
127	C₂×C₄×C₁₁	—	2	2	no	—
128	C₂×C₄×C₁₁	—	2	2	no	—
129	C₂×C₄×C₁₁	—	4	4	no	—
130	C₂×C₄×C₁₁	—	4	4	no	—
131	C₂×C₄×C₁₁	—	4	2	no	—
132	C₂×C₄×C₁₁	—	4	2	no	—
133	C₂×C₄×C₁₁	—	4	2	no	—
134	C₂×C₄×C₁₁	—	4	2	no	—
135	C₂×C₄×C₁₁	—	4	2	no	—
136	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
137	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
138	C₂×C₄×C₁₁	C₂×Q₁₆	1	4	yes	—
139	C₂×C₄×C₁₁	C₂×Q₁₆	1	2	yes	—
140	C₂×C₄×C₁₁	#135	1	2	yes	—
141	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
144	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
145	C₂×C₄×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂×C₄×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
147	C₈×C₁₁	—	2	2	no	—
148	C₈×C₁₁	—	4	2	no	—
149	C₈×C₁₁	C₄×D₄	1	4	yes	—
150	C₂²×Q₁₆	C₂²×C₁₁	1	2	yes	—
151	C₂²×Q₁₆	C₂²×C₁₁	1	2	yes	—
152	C₂×C₄⋊Q₈	C₂²×C₁₁	1	2	yes	—
153	C₂×C₄×Q₈	C₂²×C₁₁	1	4	yes	—
154	#494	C₄×C₁₁	1	8	yes	complemented
155	C₄×Q₁₆	—	2	8	no	—
156	C₂×#107	C₂²×C₁₁	1	1	yes	characteristic
157	C₂×#107	C₂²×C₁₁	1	1	yes	characteristic
158	C₂×C₄⋊C₈	C₂²×C₁₁	1	1	yes	characteristic
159	C₂×#103	C₂²×C₁₁	1	4	yes	—
160	C₂×#103	C₂²×C₁₁	1	4	yes	—
161	C₂×C₄×C₈	C₂²×C₁₁	1	2	yes	—
162	C₂×C₄×C₈	C₂²×C₁₁	1	1	yes	characteristic
163	#391	C₂²×C₁₁	1	4	yes	—
164	#389	—	2	2	no	—
165	#387	—	2	4	no	—
166	C₄×C₄⋊C₄	—	2	2	no	—
167	C₄×C₄⋊C₄	C₂²×C₁₁	1	2	yes	—
168	C₄×C₄⋊C₄	C₂²×C₁₁	1	1	yes	characteristic
169	#340	C₂²×C₁₁	1	2	yes	—
170	#336	C₂²×C₁₁	1	2	yes	—
171	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂²×C₁₁	D₄²	1	1	yes	characteristic
173	C₂²×C₁₁	#550	1	1	yes	characteristic
174	C₂²×C₁₁	C₂²×Q₁₆	1	1	yes	characteristic
175	C₂²×C₁₁	C₂×#135	1	1	yes	characteristic
176	C₂²×C₁₁	#578	1	1	yes	characteristic
177	C₂²×C₁₁	#578	1	1	yes	characteristic
178	C₄×C₁₁	—	2	2	no	—
179	C₄×C₁₁	—	2	4	no	—
180	C₄×C₁₁	—	4	8	no	—
181	C₄×C₁₁	—	4	2	no	—
182	C₄×C₁₁	—	4	4	no	—
183	C₄×C₁₁	—	8	2	no	—
184	C₄×C₁₁	C₄×Q₁₆	1	4	yes	complemented
185	C₄×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
186	C₄×C₁₁	#518	1	1	yes	characteristic
187	C₄×C₁₁	#550	1	1	yes	characteristic
188	C₄×C₁₁	#559	1	1	yes	characteristic
189	C₂²×Q₈	C₂³×C₁₁	1	4	yes	—
190	C₂×Q₁₆	—	2	4	no	—
191	C₂×Q₁₆	—	2	4	no	—
192	C₂×Q₁₆	—	4	2	no	—
193	C₂×Q₁₆	—	4	2	no	—
194	C₂×Q₁₆	C₂×C₄×C₁₁	1	8	yes	—
195	C₂×Q₁₆	C₂×C₄×C₁₁	1	8	yes	—
196	C₂²×C₈	C₂³×C₁₁	1	2	yes	—
197	C₂²×C₈	C₂³×C₁₁	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₁₁	1	1	yes	characteristic
199	C₄⋊Q₈	—	2	4	no	—
200	C₄⋊Q₈	C₂×C₄×C₁₁	1	8	yes	—
201	C₄×Q₈	—	2	8	no	—
202	C₄×Q₈	—	4	4	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	D₄×C₁₁	1	1	yes	characteristic
207	C₂×C₄⋊C₄	D₄×C₁₁	1	1	yes	characteristic
208	C₂×C₄⋊C₄	C₂³×C₁₁	1	2	yes	—
209	C₂×C₄⋊C₄	C₂³×C₁₁	1	2	yes	—
210	C₂×C₄²	—	2	4	no	—
211	C₂×C₄²	—	2	2	no	—
212	C₂×C₄²	C₂³×C₁₁	1	2	yes	—
213	C₂×C₄²	C₂³×C₁₁	1	1	yes	characteristic
214	#107	—	2	2	no	—
215	#107	D₄×C₁₁	1	4	yes	—
216	C₄⋊C₈	—	2	2	no	—
217	#103	—	2	8	no	—
218	#103	—	4	4	no	—
219	C₄×C₈	—	2	4	no	—
220	C₄×C₈	C₂×C₄×C₁₁	1	4	yes	—
221	#95	—	2	4	no	—
222	#95	—	2	2	no	—
223	#95	—	2	4	no	—
224	#95	D₄×C₁₁	1	2	yes	—
225	C₂×C₁₁	#2366	1	1	yes	characteristic
226	C₂×C₁₁	C₂×C₄×Q₁₆	1	1	yes	characteristic
227	C₂×C₁₁	#2950	1	1	yes	characteristic
228	C₂×C₁₁	#3283	1	1	yes	characteristic
229	C₂×C₁₁	D₄×Q₁₆	1	1	yes	characteristic
230	C₂×C₁₁	#3393	1	1	yes	characteristic
231	C₂×C₁₁	#3395	1	1	yes	characteristic
232	C₂×Q₈	—	2	4	no	—
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	2	4	no	—
235	C₂×Q₈	—	2	4	no	—
236	C₂×Q₈	—	2	4	no	—
237	C₂×Q₈	C₂²×C₄×C₁₁	1	8	yes	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	C₂×D₄×C₁₁	1	2	yes	—
242	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₁₁	1	2	yes	—
244	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
245	C₂²×C₄	C₂⁴×C₁₁	1	1	yes	characteristic
246	Q₁₆	—	2	16	no	—
247	Q₁₆	—	4	8	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	C₂²×C₄×C₁₁	1	2	yes	—
254	C₂×C₈	C₂²×C₄×C₁₁	1	2	yes	—
255	C₂×C₈	C₂×D₄×C₁₁	1	2	yes	—
256	C₂×C₈	C₁₁×#41	1	2	yes	—
257	C₄⋊C₄	—	2	8	no	—
258	C₄⋊C₄	—	2	8	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	4	8	no	—
264	C₄⋊C₄	C₁₁×Q₁₆	1	4	yes	complemented
265	C₄⋊C₄	C₂×D₄×C₁₁	1	4	yes	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	4	8	no	—
269	C₄²	—	4	4	no	—
270	C₄²	C₂²×C₄×C₁₁	1	2	yes	—
271	C₄²	C₁₁×#41	1	2	yes	—
272	C₁₁	#23184	1	1	yes	characteristic, direct product factor, Sylow
273	C₂³	C₂²×D₄×C₁₁	1	1	yes	characteristic
274	Q₈	—	2	8	no	—
275	Q₈	—	2	8	no	—
276	Q₈	—	4	4	no	—
277	Q₈	—	4	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	8	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	C₄×D₄×C₁₁	1	2	yes	—
295	C₂×C₄	C₄×D₄×C₁₁	1	2	yes	—
296	C₂×C₄	C₂×C₁₁×Q₁₆	1	4	yes	—
297	C₂×C₄	C₂×C₁₁×Q₁₆	1	2	yes	—
298	C₂×C₄	C₁₁×#135	1	2	yes	—
299	C₂×C₄	C₂³×C₄×C₁₁	1	1	yes	characteristic
300	C₂×C₄	C₂²×D₄×C₁₁	1	1	yes	characteristic
301	C₂×C₄	C₂×C₁₁×#41	1	1	yes	characteristic
302	C₂×C₄	C₂×C₁₁×#41	1	1	yes	characteristic
303	C₂×C₄	C₁₁×H(5,𝔽₂)	1	1	yes	characteristic
304	C₂×C₄	C₁₁×H(5,𝔽₂)	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	4	2	no	—
307	C₈	C₄×D₄×C₁₁	1	4	yes	—
308	C₂²	C₂×C₄×D₄×C₁₁	1	1	yes	characteristic
309	C₂²	D₄²×C₁₁	1	1	yes	characteristic
310	C₂²	C₁₁×#550	1	1	yes	characteristic
311	C₂²	C₂²×C₁₁×Q₁₆	1	1	yes	characteristic
312	C₂²	C₂×C₁₁×#135	1	1	yes	characteristic
313	C₂²	C₁₁×#578	1	1	yes	characteristic
314	C₂²	C₁₁×#578	1	1	yes	characteristic
315	C₄	—	2	2	no	—
316	C₄	—	2	4	no	—
317	C₄	—	4	2	no	—
318	C₄	—	4	4	no	—
319	C₄	—	4	8	no	—
320	C₄	—	8	2	no	—
321	C₄	C₄×C₁₁×Q₁₆	1	4	yes	complemented
322	C₄	C₂×C₄×D₄×C₁₁	1	1	yes	characteristic
323	C₄	C₁₁×#518	1	1	yes	characteristic
324	C₄	C₁₁×#550	1	1	yes	characteristic
325	C₄	C₁₁×#559	1	1	yes	characteristic
326	C₂	C₁₁×#2366	1	1	yes	characteristic
327	C₂	C₂×C₄×C₁₁×Q₁₆	1	1	yes	characteristic
328	C₂	C₁₁×#2950	1	1	yes	characteristic
329	C₂	C₁₁×#3283	1	1	yes	characteristic
330	C₂	D₄×C₁₁×Q₁₆	1	1	yes	characteristic
331	C₂	C₁₁×#3393	1	1	yes	characteristic
332	C₂	C₁₁×#3395	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
13. C₁₁×#494	C₄	semidirect
66. #23184	C₁₁	direct (N × H)
99. C₁₁×C₄⋊C₄	Q₁₆	semidirect
154. #494	C₄×C₁₁	semidirect
184. C₄×C₁₁	C₄×Q₁₆	semidirect
264. C₄⋊C₄	C₁₁×Q₁₆	semidirect
272. C₁₁	#23184	direct (N × H)
321. C₄	C₄×C₁₁×Q₁₆	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×C₄×C₁₁×Q₁₆	2
C₁₁×#2365	2
C₁₁×#1981	2
C₁₁×#1969	2
C₁₁×#1878	2
C₄×C₁₁×#107	2
C₈×C₁₁×C₄⋊C₄	2
#23184	11

Subgroups — C11×#23184 #14600377

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#23184 #14600377