Subgroups — C₁₁×#21974 #14599167

Order 2816.

898 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₁×#423	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₁×#117	C₂	1	1	yes	maximal, characteristic
4	C₁₁×#1939	C₂	1	4	yes	maximal, complemented
5	C₁₁×#1851	C₂	1	4	yes	maximal
6	C₁₁×#1842	C₂	1	2	yes	maximal, complemented
7	C₁₁×#1831	C₂	1	2	yes	maximal, complemented
8	C₂²×C₁₁×M₁₆	C₂²	1	2	yes	—
9	C₂×C₁₁×#117	C₄	1	2	yes	—
10	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
11	C₁₁×#431	—	2	4	no	—
12	C₁₁×#423	—	2	4	no	—
13	C₂×C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
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	8	yes	—
18	C₂×C₁₁×C₂²⋊C₈	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₁	C₂²	1	2	yes	—
20	C₄×C₁₁×C₄⋊C₄	C₄	1	1	yes	characteristic
21	C₄×C₁₁×C₄⋊C₄	C₄	1	1	yes	characteristic
22	C₄×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
23	C₄×C₁₁×C₂²⋊C₄	C₄	1	2	yes	—
24	C₄×C₁₁×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₁₁×#376	C₄	1	2	yes	—
26	C₄³×C₁₁	C₂²	1	2	yes	—
27	C₁₁×#336	C₂²	1	8	yes	—
28	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
31	C₂×C₁₁×M₁₆	—	2	4	no	—
32	C₂×C₁₁×M₁₆	—	4	4	no	—
33	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₁	C₂³	1	4	yes	—
35	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
36	C₁₁×#117	—	2	8	no	—
37	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	2	yes	—
39	C₂×C₁₁×C₂²⋊C₄	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₁₁	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₁₁	D₄	1	4	yes	—
46	C₂×C₄²×C₁₁	Q₈	1	4	yes	—
47	C₂×C₄²×C₁₁	C₂³	1	4	yes	—
48	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
50	C₁₁×C₄⋊C₈	—	2	4	no	—
51	C₁₁×C₄⋊C₈	—	2	4	no	—
52	C₁₁×C₄⋊C₈	—	2	2	no	—
53	C₁₁×C₄⋊C₈	—	2	2	no	—
54	C₁₁×C₄⋊C₈	—	4	8	no	—
55	C₁₁×C₂²⋊C₈	—	2	4	no	—
56	C₄×C₈×C₁₁	—	2	4	no	—
57	C₁₁×#95	C₂×C₄	1	4	yes	—
58	C₁₁×#95	C₂×C₄	1	2	yes	—
59	C₁₁×#95	C₂×C₄	1	2	yes	—
60	#21974	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂⁴×C₁₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₁	—	2	4	no	—
63	C₂²×C₄×C₁₁	—	2	2	no	—
64	C₂²×C₄×C₁₁	—	2	2	no	—
65	C₂²×C₄×C₁₁	—	2	2	no	—
66	C₂²×C₄×C₁₁	—	2	2	no	—
67	C₂²×C₄×C₁₁	C₄⋊C₄	1	4	yes	—
68	C₂²×C₄×C₁₁	C₄⋊C₄	1	4	yes	—
69	C₂²×C₄×C₁₁	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₁	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₁	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₁₁	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
79	C₁₁×M₁₆	—	4	8	no	—
80	C₂×C₈×C₁₁	—	2	2	no	—
81	C₂×C₈×C₁₁	—	2	2	no	—
82	C₂×C₈×C₁₁	—	2	4	no	—
83	C₂×C₈×C₁₁	—	2	4	no	—
84	C₂×C₈×C₁₁	—	2	8	no	—
85	C₂×C₈×C₁₁	—	2	4	no	—
86	C₁₁×C₄⋊C₄	—	2	4	no	—
87	C₁₁×C₄⋊C₄	M₁₆	1	8	yes	complemented
88	C₁₁×C₂²⋊C₄	—	2	8	no	—
89	C₄²×C₁₁	—	2	8	no	—
90	C₄²×C₁₁	—	2	4	no	—
91	C₄²×C₁₁	—	2	4	no	—
92	C₄²×C₁₁	—	2	4	no	—
93	C₄²×C₁₁	—	2	4	no	—
94	C₄²×C₁₁	C₄⋊C₄	1	4	yes	—
95	C₄²×C₁₁	C₄⋊C₄	1	4	yes	—
96	C₄²×C₁₁	C₂×D₄	1	2	yes	—
97	C₄²×C₁₁	C₂×D₄	1	2	yes	—
98	C₄²×C₁₁	C₂×Q₈	1	2	yes	—
99	C₄²×C₁₁	C₂×Q₈	1	2	yes	—
100	C₂×#431	C₂×C₁₁	1	1	yes	characteristic
101	C₂×#423	C₂×C₁₁	1	1	yes	characteristic
102	C₄×#117	C₂×C₁₁	1	1	yes	characteristic
103	#1939	C₂×C₁₁	1	4	yes	complemented
104	#1851	C₂×C₁₁	1	4	yes	—
105	#1842	C₂×C₁₁	1	2	yes	complemented
106	#1831	C₂×C₁₁	1	2	yes	complemented
107	C₂³×C₁₁	—	2	2	no	—
108	C₂³×C₁₁	—	2	2	no	—
109	C₂³×C₁₁	—	2	1	no	—
110	C₂³×C₁₁	—	2	1	no	—
111	C₂³×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
112	C₂³×C₁₁	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₁	—	2	4	no	—
114	C₂×C₄×C₁₁	—	2	2	no	—
115	C₂×C₄×C₁₁	—	2	2	no	—
116	C₂×C₄×C₁₁	—	2	4	no	—
117	C₂×C₄×C₁₁	—	2	1	no	—
118	C₂×C₄×C₁₁	—	2	1	no	—
119	C₂×C₄×C₁₁	—	2	2	no	—
120	C₂×C₄×C₁₁	—	2	2	no	—
121	C₂×C₄×C₁₁	—	2	4	no	—
122	C₂×C₄×C₁₁	—	2	8	no	—
123	C₂×C₄×C₁₁	—	2	8	no	—
124	C₂×C₄×C₁₁	—	2	4	no	—
125	C₂×C₄×C₁₁	—	2	4	no	—
126	C₂×C₄×C₁₁	—	2	4	no	—
127	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×C₄×C₁₁	C₂×M₁₆	1	8	yes	—
133	C₂×C₄×C₁₁	C₂×M₁₆	1	4	yes	—
134	C₂×C₄×C₁₁	#131	1	2	yes	—
135	C₂×C₄×C₁₁	#131	1	2	yes	—
136	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
138	C₂×C₄×C₁₁	H(5,𝔽₂)	1	2	yes	—
139	C₂×C₄×C₁₁	2₋¹⁺⁴	1	2	yes	—
140	C₈×C₁₁	—	4	2	no	—
141	C₈×C₁₁	—	4	4	no	—
142	C₈×C₁₁	—	4	2	no	—
143	C₂²×M₁₆	C₂²×C₁₁	1	2	yes	—
144	C₂×#117	C₄×C₁₁	1	2	yes	—
145	C₂²×C₄²	C₂²×C₁₁	1	1	yes	characteristic
146	#431	—	2	4	no	—
147	#423	—	2	4	no	—
148	C₂×C₄⋊C₈	C₂²×C₁₁	1	1	yes	characteristic
149	C₂×C₄⋊C₈	C₂²×C₁₁	1	2	yes	—
150	C₂×C₄⋊C₈	C₂²×C₁₁	1	1	yes	characteristic
151	C₂×C₄⋊C₈	C₂²×C₁₁	1	2	yes	—
152	C₂×C₄⋊C₈	C₂²×C₁₁	1	8	yes	—
153	C₂×C₂²⋊C₈	C₂²×C₁₁	1	2	yes	—
154	C₂×C₄×C₈	C₂²×C₁₁	1	2	yes	—
155	C₄×C₄⋊C₄	C₄×C₁₁	1	1	yes	characteristic
156	C₄×C₄⋊C₄	C₄×C₁₁	1	1	yes	characteristic
157	C₄×C₄⋊C₄	C₂²×C₁₁	1	2	yes	—
158	C₄×C₂²⋊C₄	C₄×C₁₁	1	2	yes	—
159	C₄×C₂²⋊C₄	C₂²×C₁₁	1	2	yes	—
160	#376	C₄×C₁₁	1	2	yes	—
161	C₄³	C₂²×C₁₁	1	2	yes	—
162	#336	C₂²×C₁₁	1	8	yes	—
163	C₂²×C₁₁	—	2	2	no	—
164	C₂²×C₁₁	—	2	2	no	—
165	C₂²×C₁₁	—	2	2	no	—
166	C₂²×C₁₁	—	4	2	no	—
167	C₂²×C₁₁	—	4	2	no	—
168	C₂²×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
169	C₂²×C₁₁	#520	1	2	yes	—
170	C₂²×C₁₁	C₂²×M₁₆	1	2	yes	—
171	C₂²×C₁₁	C₂×#131	1	1	yes	characteristic
172	C₂²×C₁₁	C₂×#131	1	1	yes	characteristic
173	C₄×C₁₁	—	2	2	no	—
174	C₄×C₁₁	—	2	2	no	—
175	C₄×C₁₁	—	2	2	no	—
176	C₄×C₁₁	—	2	8	no	—
177	C₄×C₁₁	—	4	4	no	—
178	C₄×C₁₁	#423	1	4	yes	—
179	C₄×C₁₁	#424	1	4	yes	—
180	C₄×C₁₁	#535	1	1	yes	characteristic
181	C₄×C₁₁	#536	1	1	yes	characteristic
182	C₄×C₁₁	#544	1	2	yes	—
183	C₂³×C₄	C₂×C₄×C₁₁	1	1	yes	characteristic
184	C₂³×C₄	C₂×C₄×C₁₁	1	1	yes	characteristic
185	C₂³×C₄	C₂³×C₁₁	1	1	yes	characteristic
186	C₂×M₁₆	—	2	4	no	—
187	C₂×M₁₆	—	4	4	no	—
188	C₂²×C₈	C₂³×C₁₁	1	4	yes	—
189	C₂²×C₈	C₂³×C₁₁	1	2	yes	—
190	C₂²×C₈	C₂³×C₁₁	1	2	yes	—
191	#117	—	2	8	no	—
192	C₂×C₄⋊C₄	C₂×C₄×C₁₁	1	2	yes	—
193	C₂×C₄⋊C₄	C₂×C₄×C₁₁	1	2	yes	—
194	C₂×C₂²⋊C₄	C₂×C₄×C₁₁	1	4	yes	—
195	C₂×C₄²	C₂×C₄×C₁₁	1	4	yes	—
196	C₂×C₄²	C₂×C₄×C₁₁	1	2	yes	—
197	C₂×C₄²	C₂×C₄×C₁₁	1	2	yes	—
198	C₂×C₄²	C₂×C₄×C₁₁	1	1	yes	characteristic
199	C₂×C₄²	C₂×C₄×C₁₁	1	1	yes	characteristic
200	C₂×C₄²	D₄×C₁₁	1	4	yes	—
201	C₂×C₄²	Q₈×C₁₁	1	4	yes	—
202	C₂×C₄²	C₂³×C₁₁	1	4	yes	—
203	C₂×C₄²	C₂³×C₁₁	1	1	yes	characteristic
204	C₂×C₄²	C₂³×C₁₁	1	1	yes	characteristic
205	C₄⋊C₈	—	2	4	no	—
206	C₄⋊C₈	—	2	4	no	—
207	C₄⋊C₈	—	2	2	no	—
208	C₄⋊C₈	—	2	2	no	—
209	C₄⋊C₈	—	4	8	no	—
210	C₂²⋊C₈	—	2	4	no	—
211	C₄×C₈	—	2	4	no	—
212	#95	C₂×C₄×C₁₁	1	4	yes	—
213	#95	C₂×C₄×C₁₁	1	2	yes	—
214	#95	C₂×C₄×C₁₁	1	2	yes	—
215	C₂×C₁₁	—	4	2	no	—
216	C₂×C₁₁	#2317	1	1	yes	characteristic
217	C₂×C₁₁	C₂×#423	1	1	yes	characteristic
218	C₂×C₁₁	C₂×#424	1	1	yes	characteristic
219	C₂×C₁₁	#2990	1	2	yes	—
220	C₂×C₁₁	#2991	1	2	yes	—
221	C₂⁴	C₂²×C₄×C₁₁	1	1	yes	characteristic
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	C₁₁×C₄⋊C₄	1	4	yes	—
228	C₂²×C₄	C₁₁×C₄⋊C₄	1	4	yes	—
229	C₂²×C₄	C₂²×C₄×C₁₁	1	4	yes	—
230	C₂²×C₄	C₂²×C₄×C₁₁	1	1	yes	characteristic
231	C₂²×C₄	C₂²×C₄×C₁₁	1	1	yes	characteristic
232	C₂²×C₄	C₂²×C₄×C₁₁	1	1	yes	characteristic
233	C₂²×C₄	C₂²×C₄×C₁₁	1	4	yes	—
234	C₂²×C₄	C₂²×C₄×C₁₁	1	1	yes	characteristic
235	C₂²×C₄	C₂²×C₄×C₁₁	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₁₁	1	2	yes	—
237	C₂²×C₄	C₂×Q₈×C₁₁	1	2	yes	—
238	C₂²×C₄	C₂⁴×C₁₁	1	1	yes	characteristic
239	M₁₆	—	4	8	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	8	no	—
245	C₂×C₈	—	2	4	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	C₁₁×M₁₆	1	8	yes	complemented
248	C₂²⋊C₄	—	2	8	no	—
249	C₄²	—	2	8	no	—
250	C₄²	—	2	4	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	2	4	no	—
253	C₄²	—	2	4	no	—
254	C₄²	C₁₁×C₄⋊C₄	1	4	yes	—
255	C₄²	C₁₁×C₄⋊C₄	1	4	yes	—
256	C₄²	C₂×D₄×C₁₁	1	2	yes	—
257	C₄²	C₂×D₄×C₁₁	1	2	yes	—
258	C₄²	C₂×Q₈×C₁₁	1	2	yes	—
259	C₄²	C₂×Q₈×C₁₁	1	2	yes	—
260	C₁₁	#21974	1	1	yes	characteristic, direct product factor, Sylow
261	C₂³	—	2	2	no	—
262	C₂³	—	2	2	no	—
263	C₂³	—	2	1	no	—
264	C₂³	—	2	1	no	—
265	C₂³	C₂×C₁₁×C₄⋊C₄	1	2	yes	—
266	C₂³	C₂³×C₄×C₁₁	1	1	yes	characteristic
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	C₂×C₁₁×C₄⋊C₄	1	2	yes	—
282	C₂×C₄	C₂×C₁₁×C₄⋊C₄	1	2	yes	—
283	C₂×C₄	C₂×C₁₁×C₄⋊C₄	1	2	yes	—
284	C₂×C₄	C₂×C₁₁×C₄⋊C₄	1	2	yes	—
285	C₂×C₄	C₂×C₁₁×C₄⋊C₄	1	2	yes	—
286	C₂×C₄	C₂×C₁₁×M₁₆	1	8	yes	—
287	C₂×C₄	C₂×C₁₁×M₁₆	1	4	yes	—
288	C₂×C₄	C₁₁×#131	1	2	yes	—
289	C₂×C₄	C₁₁×#131	1	2	yes	—
290	C₂×C₄	C₂²×D₄×C₁₁	1	1	yes	characteristic
291	C₂×C₄	C₂²×Q₈×C₁₁	1	1	yes	characteristic
292	C₂×C₄	C₁₁×H(5,𝔽₂)	1	2	yes	—
293	C₂×C₄	C₁₁×2₋¹⁺⁴	1	2	yes	—
294	C₈	—	4	4	no	—
295	C₈	—	4	2	no	—
296	C₈	—	4	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	4	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	C₂²×C₁₁×C₄⋊C₄	1	1	yes	characteristic
303	C₂²	C₁₁×#520	1	2	yes	—
304	C₂²	C₂²×C₁₁×M₁₆	1	2	yes	—
305	C₂²	C₂×C₁₁×#131	1	1	yes	characteristic
306	C₂²	C₂×C₁₁×#131	1	1	yes	characteristic
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	8	no	—
310	C₄	—	2	2	no	—
311	C₄	—	4	4	no	—
312	C₄	C₁₁×#423	1	4	yes	—
313	C₄	C₁₁×#424	1	4	yes	—
314	C₄	C₁₁×#535	1	1	yes	characteristic
315	C₄	C₁₁×#536	1	1	yes	characteristic
316	C₄	C₁₁×#544	1	2	yes	—
317	C₂	—	4	2	no	—
318	C₂	C₁₁×#2317	1	1	yes	characteristic
319	C₂	C₂×C₁₁×#423	1	1	yes	characteristic
320	C₂	C₂×C₁₁×#424	1	1	yes	characteristic
321	C₂	C₁₁×#2990	1	2	yes	—
322	C₂	C₁₁×#2991	1	2	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	C₂⁴×C₁₁	C₂²×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
4. C₁₁×#1939	C₂	semidirect
6. C₁₁×#1842	C₂	semidirect
7. C₁₁×#1831	C₂	semidirect
60. #21974	C₁₁	direct (N × H)
87. C₁₁×C₄⋊C₄	M₁₆	semidirect
103. #1939	C₂×C₁₁	semidirect
105. #1842	C₂×C₁₁	semidirect
106. #1831	C₂×C₁₁	semidirect
247. C₄⋊C₄	C₁₁×M₁₆	semidirect
260. C₁₁	#21974	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×#431	2
C₂×C₁₁×#423	2
C₄×C₁₁×#117	2
C₁₁×#1939	2
C₁₁×#1851	2
C₁₁×#1842	2
C₁₁×#1831	2
#21974	11

Subgroups — C11×#21974 #14599167

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#21974 #14599167