Subgroups — C₁₁×#22215 #14599408

Order 2816.

782 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×#446	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₁×#436	C₂	1	1	yes	maximal, characteristic
3	C₁₁×#2398	C₂	1	1	yes	maximal, characteristic
4	C₁₁×#1985	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₁×#1955	C₂	1	1	yes	maximal, characteristic
6	C₁₁×#1924	C₂	1	4	yes	maximal, complemented
7	C₁₁×#1851	C₂	1	4	yes	maximal, complemented
8	C₁₁×#1843	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₁×#1746	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₁₁×M₁₆	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₁₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₁₁	C₂²	1	1	yes	characteristic
13	C₁₁×#446	—	2	4	no	—
14	C₁₁×#436	—	2	4	no	—
15	C₂×C₁₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₁₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₁₁×C₄⋊C₈	C₂²	1	4	yes	—
19	C₂×C₁₁×C₂²⋊C₈	C₂²	1	2	yes	—
20	C₂×C₁₁×C₂²⋊C₈	C₂²	1	4	yes	—
21	C₂×C₁₁×#97	C₂²	1	2	yes	—
22	C₂×C₁₁×#97	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
24	C₁₁×#400	C₄	1	2	yes	—
25	C₁₁×#397	C₄	1	2	yes	—
26	C₁₁×#394	C₄	1	2	yes	—
27	C₁₁×#393	C₄	1	2	yes	—
28	C₄×C₁₁×C₂²⋊C₄	C₂²	1	4	yes	—
29	C₁₁×#376	C₂²	1	1	yes	characteristic
30	C₁₁×#336	C₂²	1	2	yes	—
31	C₁₁×#336	C₂²	1	4	yes	—
32	C₁₁×#336	C₂²	1	2	yes	—
33	C₂²×Q₈×C₁₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×D₄×C₁₁	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₁₁	C₂³	1	2	yes	—
36	C₂×C₁₁×M₁₆	—	2	4	no	—
37	C₂×C₁₁×M₁₆	—	4	4	no	—
38	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
39	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
43	C₄×Q₈×C₁₁	—	2	2	no	—
44	C₄×Q₈×C₁₁	—	2	2	no	—
45	C₄×D₄×C₁₁	—	2	2	no	—
46	C₄×D₄×C₁₁	—	2	2	no	—
47	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
48	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	2	yes	—
49	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
50	C₂×C₁₁×C₂²⋊C₄	C₂×C₄	1	2	yes	—
51	C₂×C₁₁×C₂²⋊C₄	C₂×C₄	1	4	yes	—
52	C₂×C₁₁×C₂²⋊C₄	C₂×C₄	1	4	yes	—
53	C₂×C₄²×C₁₁	C₂³	1	2	yes	—
54	C₂×C₄²×C₁₁	C₂³	1	2	yes	—
55	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
56	C₁₁×C₄⋊C₈	—	2	4	no	—
57	C₁₁×C₄⋊C₈	—	2	2	no	—
58	C₁₁×C₄⋊C₈	—	2	2	no	—
59	C₁₁×C₄⋊C₈	—	4	4	no	—
60	C₁₁×C₂²⋊C₈	—	2	4	no	—
61	C₁₁×C₂²⋊C₈	—	4	4	no	—
62	C₁₁×#97	—	2	4	no	—
63	C₁₁×#97	—	2	2	no	—
64	C₄×C₈×C₁₁	D₄	1	4	yes	—
65	C₁₁×#95	C₂×C₄	1	2	yes	—
66	C₁₁×#95	C₂×C₄	1	2	yes	—
67	C₁₁×#95	C₂×C₄	1	4	yes	—
68	C₁₁×#95	C₂×C₄	1	4	yes	—
69	#22215	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂⁴×C₁₁	C₂²×C₄	1	2	yes	—
71	C₂×Q₈×C₁₁	—	2	2	no	—
72	C₂×Q₈×C₁₁	—	4	2	no	—
73	C₂×D₄×C₁₁	—	2	2	no	—
74	C₂×D₄×C₁₁	—	4	2	no	—
75	C₂²×C₄×C₁₁	—	2	2	no	—
76	C₂²×C₄×C₁₁	—	2	2	no	—
77	C₂²×C₄×C₁₁	—	2	4	no	—
78	C₂²×C₄×C₁₁	—	2	4	no	—
79	C₂²×C₄×C₁₁	C₂²×C₄	1	4	yes	—
80	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
86	C₁₁×M₁₆	—	4	8	no	—
87	C₂×C₈×C₁₁	—	2	2	no	—
88	C₂×C₈×C₁₁	—	2	2	no	—
89	C₂×C₈×C₁₁	—	2	2	no	—
90	C₂×C₈×C₁₁	—	2	2	no	—
91	C₂×C₈×C₁₁	—	2	2	no	—
92	C₂×C₈×C₁₁	—	2	4	no	—
93	C₂×C₈×C₁₁	—	2	4	no	—
94	C₂×C₈×C₁₁	—	2	2	no	—
95	C₂×C₈×C₁₁	—	2	2	no	—
96	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
98	C₁₁×C₄⋊C₄	—	2	2	no	—
99	C₁₁×C₄⋊C₄	—	2	2	no	—
100	C₁₁×C₄⋊C₄	—	2	4	no	—
101	C₁₁×C₂²⋊C₄	—	2	4	no	—
102	C₁₁×C₂²⋊C₄	—	4	4	no	—
103	C₁₁×C₂²⋊C₄	—	4	4	no	—
104	C₄²×C₁₁	—	2	2	no	—
105	C₄²×C₁₁	—	2	2	no	—
106	C₄²×C₁₁	—	2	2	no	—
107	C₄²×C₁₁	—	2	2	no	—
108	C₄²×C₁₁	C₂×D₄	1	2	yes	—
109	C₄²×C₁₁	#41	1	2	yes	—
110	C₂×#446	C₂×C₁₁	1	1	yes	characteristic, complemented
111	C₂×#436	C₂×C₁₁	1	1	yes	characteristic
112	#2398	C₂×C₁₁	1	1	yes	characteristic
113	#1985	C₂×C₁₁	1	1	yes	characteristic, complemented
114	#1955	C₂×C₁₁	1	1	yes	characteristic
115	#1924	C₂×C₁₁	1	4	yes	complemented
116	#1851	C₂×C₁₁	1	4	yes	complemented
117	#1843	C₂×C₁₁	1	1	yes	characteristic, complemented
118	#1746	C₂×C₁₁	1	1	yes	characteristic, complemented
119	C₂³×C₁₁	—	2	2	no	—
120	C₂³×C₁₁	—	2	4	no	—
121	C₂³×C₁₁	—	2	2	no	—
122	C₂³×C₁₁	—	2	4	no	—
123	C₂³×C₁₁	—	2	2	no	—
124	C₂³×C₁₁	C₂³×C₄	1	1	yes	characteristic
125	Q₈×C₁₁	—	4	4	no	—
126	D₄×C₁₁	—	4	4	no	—
127	C₂×C₄×C₁₁	—	2	2	no	—
128	C₂×C₄×C₁₁	—	2	2	no	—
129	C₂×C₄×C₁₁	—	2	2	no	—
130	C₂×C₄×C₁₁	—	2	2	no	—
131	C₂×C₄×C₁₁	—	2	2	no	—
132	C₂×C₄×C₁₁	—	2	4	no	—
133	C₂×C₄×C₁₁	—	2	4	no	—
134	C₂×C₄×C₁₁	—	2	2	no	—
135	C₂×C₄×C₁₁	—	2	4	no	—
136	C₂×C₄×C₁₁	—	2	4	no	—
137	C₂×C₄×C₁₁	—	2	4	no	—
138	C₂×C₄×C₁₁	—	2	2	no	—
139	C₂×C₄×C₁₁	—	4	2	no	—
140	C₂×C₄×C₁₁	—	4	2	no	—
141	C₂×C₄×C₁₁	—	4	4	no	—
142	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
143	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
144	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₁₁	H(5,𝔽₂)	1	2	yes	—
147	C₂×C₄×C₁₁	2₋¹⁺⁴	1	2	yes	—
148	C₈×C₁₁	—	2	2	no	—
149	C₈×C₁₁	—	2	2	no	—
150	C₈×C₁₁	—	4	2	no	—
151	C₈×C₁₁	—	4	2	no	—
152	C₈×C₁₁	—	4	2	no	—
153	C₂²×M₁₆	C₂²×C₁₁	1	2	yes	—
154	C₂×C₄×Q₈	C₂²×C₁₁	1	1	yes	characteristic
155	C₂×C₄×D₄	C₂²×C₁₁	1	1	yes	characteristic
156	#446	—	2	4	no	—
157	#436	—	2	4	no	—
158	C₂×C₄⋊C₈	C₂²×C₁₁	1	1	yes	characteristic
159	C₂×C₄⋊C₈	C₂²×C₁₁	1	1	yes	characteristic
160	C₂×C₄⋊C₈	C₂²×C₁₁	1	2	yes	—
161	C₂×C₄⋊C₈	C₂²×C₁₁	1	4	yes	—
162	C₂×C₂²⋊C₈	C₂²×C₁₁	1	2	yes	—
163	C₂×C₂²⋊C₈	C₂²×C₁₁	1	4	yes	—
164	C₂×#97	C₂²×C₁₁	1	2	yes	—
165	C₂×#97	C₂²×C₁₁	1	1	yes	characteristic
166	C₂×C₄×C₈	C₂²×C₁₁	1	1	yes	characteristic
167	#400	C₄×C₁₁	1	2	yes	—
168	#397	C₄×C₁₁	1	2	yes	—
169	#394	C₄×C₁₁	1	2	yes	—
170	#393	C₄×C₁₁	1	2	yes	—
171	C₄×C₂²⋊C₄	C₂²×C₁₁	1	4	yes	—
172	#376	C₂²×C₁₁	1	1	yes	characteristic
173	#336	C₂²×C₁₁	1	2	yes	—
174	#336	C₂²×C₁₁	1	4	yes	—
175	#336	C₂²×C₁₁	1	2	yes	—
176	C₂²×C₁₁	—	4	2	no	—
177	C₂²×C₁₁	—	4	2	no	—
178	C₂²×C₁₁	—	4	2	no	—
179	C₂²×C₁₁	—	4	4	no	—
180	C₂²×C₁₁	—	4	4	no	—
181	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
182	C₂²×C₁₁	#520	1	2	yes	—
183	C₂²×C₁₁	#568	1	2	yes	—
184	C₂²×C₁₁	#568	1	2	yes	—
185	C₄×C₁₁	—	2	2	no	—
186	C₄×C₁₁	—	2	2	no	—
187	C₄×C₁₁	—	4	2	no	—
188	C₄×C₁₁	—	4	2	no	—
189	C₄×C₁₁	—	4	2	no	—
190	C₄×C₁₁	—	4	4	no	—
191	C₄×C₁₁	#535	1	1	yes	characteristic
192	C₄×C₁₁	#536	1	1	yes	characteristic
193	C₄×C₁₁	#542	1	2	yes	—
194	C₂²×Q₈	C₂×C₄×C₁₁	1	1	yes	characteristic
195	C₂²×D₄	C₂×C₄×C₁₁	1	1	yes	characteristic
196	C₂³×C₄	C₂³×C₁₁	1	2	yes	—
197	C₂×M₁₆	—	2	4	no	—
198	C₂×M₁₆	—	4	4	no	—
199	C₂²×C₈	C₂³×C₁₁	1	2	yes	—
200	C₂²×C₈	C₂³×C₁₁	1	2	yes	—
201	C₂²×C₈	C₂³×C₁₁	1	2	yes	—
202	C₂²×C₈	C₂³×C₁₁	1	1	yes	characteristic
203	C₂²×C₈	C₂³×C₁₁	1	1	yes	characteristic
204	C₄×Q₈	—	2	2	no	—
205	C₄×Q₈	—	2	2	no	—
206	C₄×D₄	—	2	2	no	—
207	C₄×D₄	—	2	2	no	—
208	C₂×C₄⋊C₄	C₂×C₄×C₁₁	1	1	yes	characteristic
209	C₂×C₄⋊C₄	C₂×C₄×C₁₁	1	2	yes	—
210	C₂×C₄⋊C₄	C₂×C₄×C₁₁	1	1	yes	characteristic
211	C₂×C₂²⋊C₄	C₂×C₄×C₁₁	1	2	yes	—
212	C₂×C₂²⋊C₄	C₂×C₄×C₁₁	1	4	yes	—
213	C₂×C₂²⋊C₄	C₂×C₄×C₁₁	1	4	yes	—
214	C₂×C₄²	C₂³×C₁₁	1	2	yes	—
215	C₂×C₄²	C₂³×C₁₁	1	2	yes	—
216	C₂×C₄²	C₂³×C₁₁	1	1	yes	characteristic
217	C₄⋊C₈	—	2	4	no	—
218	C₄⋊C₈	—	2	2	no	—
219	C₄⋊C₈	—	2	2	no	—
220	C₄⋊C₈	—	4	4	no	—
221	C₂²⋊C₈	—	2	4	no	—
222	C₂²⋊C₈	—	4	4	no	—
223	#97	—	2	4	no	—
224	#97	—	2	2	no	—
225	C₄×C₈	D₄×C₁₁	1	4	yes	—
226	#95	C₂×C₄×C₁₁	1	2	yes	—
227	#95	C₂×C₄×C₁₁	1	2	yes	—
228	#95	C₂×C₄×C₁₁	1	4	yes	—
229	#95	C₂×C₄×C₁₁	1	4	yes	—
230	C₂×C₁₁	—	8	2	no	—
231	C₂×C₁₁	#2326	1	1	yes	characteristic
232	C₂×C₁₁	#2930	1	1	yes	characteristic
233	C₂×C₁₁	#2930	1	1	yes	characteristic
234	C₂×C₁₁	#2995	1	4	yes	—
235	C₂⁴	C₂²×C₄×C₁₁	1	2	yes	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	4	2	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	4	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	C₂²×C₄×C₁₁	1	4	yes	—
245	C₂²×C₄	C₂²×C₄×C₁₁	1	2	yes	—
246	C₂²×C₄	C₂²×C₄×C₁₁	1	2	yes	—
247	C₂²×C₄	C₂²×C₄×C₁₁	1	2	yes	—
248	C₂²×C₄	C₂²×C₄×C₁₁	1	1	yes	characteristic
249	C₂²×C₄	C₂²×C₄×C₁₁	1	1	yes	characteristic
250	C₂²×C₄	C₂⁴×C₁₁	1	1	yes	characteristic
251	M₁₆	—	4	8	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	C₂×D₄×C₁₁	1	2	yes	—
262	C₂×C₈	C₂×D₄×C₁₁	1	2	yes	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₂²⋊C₄	—	2	4	no	—
267	C₂²⋊C₄	—	4	4	no	—
268	C₂²⋊C₄	—	4	4	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	C₂×D₄×C₁₁	1	2	yes	—
274	C₄²	C₁₁×#41	1	2	yes	—
275	C₁₁	#22215	1	1	yes	characteristic, direct product factor, Sylow
276	C₂³	—	2	2	no	—
277	C₂³	—	2	4	no	—
278	C₂³	—	2	2	no	—
279	C₂³	—	2	4	no	—
280	C₂³	—	2	2	no	—
281	C₂³	C₂³×C₄×C₁₁	1	1	yes	characteristic
282	Q₈	—	4	4	no	—
283	D₄	—	4	4	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	C₄×D₄×C₁₁	1	2	yes	—
300	C₂×C₄	C₄×D₄×C₁₁	1	2	yes	—
301	C₂×C₄	C₂²×D₄×C₁₁	1	1	yes	characteristic
302	C₂×C₄	C₂×C₁₁×#41	1	1	yes	characteristic
303	C₂×C₄	C₁₁×H(5,𝔽₂)	1	2	yes	—
304	C₂×C₄	C₁₁×2₋¹⁺⁴	1	2	yes	—
305	C₈	—	2	2	no	—
306	C₈	—	2	2	no	—
307	C₈	—	4	2	no	—
308	C₈	—	4	2	no	—
309	C₈	—	4	2	no	—
310	C₂²	—	4	2	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	4	4	no	—
313	C₂²	—	4	4	no	—
314	C₂²	—	4	2	no	—
315	C₂²	C₂×C₄×D₄×C₁₁	1	1	yes	characteristic
316	C₂²	C₁₁×#520	1	2	yes	—
317	C₂²	C₁₁×#568	1	2	yes	—
318	C₂²	C₁₁×#568	1	2	yes	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	4	2	no	—
322	C₄	—	4	2	no	—
323	C₄	—	4	4	no	—
324	C₄	—	4	2	no	—
325	C₄	C₁₁×#535	1	1	yes	characteristic
326	C₄	C₁₁×#536	1	1	yes	characteristic
327	C₄	C₁₁×#542	1	2	yes	—
328	C₂	—	8	2	no	—
329	C₂	C₁₁×#2326	1	1	yes	characteristic
330	C₂	C₁₁×#2930	1	1	yes	characteristic
331	C₂	C₁₁×#2930	1	1	yes	characteristic
332	C₂	C₁₁×#2995	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₁×#446	C₂	semidirect
4. C₁₁×#1985	C₂	semidirect
6. C₁₁×#1924	C₂	semidirect
7. C₁₁×#1851	C₂	semidirect
8. C₁₁×#1843	C₂	semidirect
9. C₁₁×#1746	C₂	semidirect
69. #22215	C₁₁	direct (N × H)
110. C₂×#446	C₂×C₁₁	semidirect
113. #1985	C₂×C₁₁	semidirect
115. #1924	C₂×C₁₁	semidirect
116. #1851	C₂×C₁₁	semidirect
117. #1843	C₂×C₁₁	semidirect
118. #1746	C₂×C₁₁	semidirect
275. C₁₁	#22215	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×#446	2
C₂×C₁₁×#436	2
C₁₁×#2398	2
C₁₁×#1985	2
C₁₁×#1955	2
C₁₁×#1924	2
C₁₁×#1851	2
C₁₁×#1843	2
C₁₁×#1746	2
#22215	11

Subgroups — C11×#22215 #14599408

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#22215 #14599408