Subgroups — C₁₁×#21154 #14598347

Order 2816.

998 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×#427	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₁×#427	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₁×#97	C₂	1	1	yes	maximal, characteristic
4	C₁₁×#2296	C₂	1	2	yes	maximal
5	C₁₁×#1944	C₂	1	4	yes	maximal, complemented
6	C₁₁×#1943	C₂	1	4	yes	maximal, complemented
7	C₁₁×#1735	C₂	1	2	yes	maximal
8	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
10	C₂×C₁₁×#117	C₂²	1	2	yes	—
11	C₂×C₁₁×#117	C₂²	1	2	yes	—
12	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
13	C₁₁×#427	—	2	4	no	—
14	C₁₁×#427	C₄	1	8	yes	complemented
15	C₂×C₁₁×#107	C₂²	1	2	yes	—
16	C₂×C₁₁×#107	C₂²	1	2	yes	—
17	C₂×C₁₁×#106	C₂²	1	2	yes	—
18	C₂×C₁₁×#106	C₂²	1	2	yes	—
19	C₂×C₁₁×#97	—	2	4	no	—
20	C₂×C₁₁×#97	C₂²	1	2	yes	—
21	C₂×C₁₁×#97	C₂²	1	2	yes	—
22	C₁₁×#389	C₂²	1	8	yes	—
23	C₁₁×#385	—	2	4	no	—
24	C₄×C₁₁×C₄⋊C₄	—	2	4	no	—
25	C₁₁×#340	C₂²	1	8	yes	—
26	C₂³×C₄×C₁₁	D₄	1	2	yes	—
27	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₁₁	—	2	2	no	—
29	C₂²×C₈×C₁₁	—	2	2	no	—
30	C₂²×C₈×C₁₁	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₁₁	D₄	1	4	yes	—
32	C₂²×C₈×C₁₁	Q₈	1	4	yes	—
33	C₂²×C₈×C₁₁	Q₈	1	4	yes	—
34	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
37	C₁₁×#117	—	2	4	no	—
38	C₁₁×#117	—	2	4	no	—
39	C₁₁×#117	—	2	4	no	—
40	C₁₁×#117	C₂×C₄	1	8	yes	—
41	C₂×C₁₁×C₄⋊C₄	—	2	8	no	—
42	C₂×C₁₁×C₄⋊C₄	C₂³	1	4	yes	—
43	C₂×C₁₁×C₄⋊C₄	C₂³	1	4	yes	—
44	C₂×C₁₁×C₂²⋊C₄	—	2	2	no	—
45	C₂×C₁₁×C₂²⋊C₄	—	2	2	no	—
46	C₂×C₄²×C₁₁	—	2	2	no	—
47	C₂×C₄²×C₁₁	—	2	2	no	—
48	C₂×C₄²×C₁₁	—	2	2	no	—
49	C₂×C₄²×C₁₁	—	2	2	no	—
50	C₂×C₄²×C₁₁	D₄	1	2	yes	—
51	C₂×C₄²×C₁₁	C₂³	1	2	yes	—
52	C₁₁×#107	—	2	4	no	—
53	C₁₁×#107	C₂×C₄	1	8	yes	complemented
54	C₁₁×#106	—	2	4	no	—
55	C₁₁×#106	C₂×C₄	1	8	yes	complemented
56	C₁₁×#97	—	2	4	no	—
57	C₁₁×#97	—	2	4	no	—
58	C₁₁×#95	—	2	8	no	—
59	#21154	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂⁴×C₁₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₁	—	2	4	no	—
62	C₂²×C₄×C₁₁	—	2	4	no	—
63	C₂²×C₄×C₁₁	—	2	4	no	—
64	C₂²×C₄×C₁₁	—	4	4	no	—
65	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₁₁	C₂×D₄	1	4	yes	—
67	C₂²×C₄×C₁₁	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₁	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₁₁	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₁₁	#41	1	2	yes	—
72	C₂²×C₄×C₁₁	#41	1	2	yes	—
73	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
74	C₂×C₈×C₁₁	—	2	8	no	—
75	C₂×C₈×C₁₁	—	2	2	no	—
76	C₂×C₈×C₁₁	—	2	2	no	—
77	C₂×C₈×C₁₁	C₄⋊C₄	1	16	yes	complemented
78	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₁₁	C₂×Q₈	1	4	yes	—
83	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
84	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
85	C₁₁×C₄⋊C₄	—	2	4	no	—
86	C₁₁×C₄⋊C₄	—	2	4	no	—
87	C₁₁×C₄⋊C₄	—	2	4	no	—
88	C₁₁×C₄⋊C₄	—	4	16	no	—
89	C₁₁×C₄⋊C₄	C₂²×C₄	1	8	yes	—
90	C₁₁×C₂²⋊C₄	—	2	8	no	—
91	C₁₁×C₂²⋊C₄	—	4	4	no	—
92	C₄²×C₁₁	—	2	8	no	—
93	C₄²×C₁₁	—	2	4	no	—
94	C₄²×C₁₁	—	2	2	no	—
95	C₄²×C₁₁	—	2	2	no	—
96	C₄²×C₁₁	—	4	4	no	—
97	C₂×#427	C₂×C₁₁	1	1	yes	characteristic
98	C₂×#427	C₂×C₁₁	1	1	yes	characteristic
99	C₂²×#97	C₂×C₁₁	1	1	yes	characteristic
100	#2296	C₂×C₁₁	1	2	yes	—
101	#1944	C₂×C₁₁	1	4	yes	complemented
102	#1943	C₂×C₁₁	1	4	yes	complemented
103	#1735	C₂×C₁₁	1	2	yes	—
104	C₂³×C₁₁	—	2	4	no	—
105	C₂³×C₁₁	C₄×D₄	1	2	yes	—
106	C₂³×C₁₁	#121	1	2	yes	—
107	C₂³×C₁₁	C₄⋊Q₈	1	2	yes	—
108	C₂³×C₁₁	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₁	—	2	8	no	—
110	C₂×C₄×C₁₁	—	2	4	no	—
111	C₂×C₄×C₁₁	—	2	8	no	—
112	C₂×C₄×C₁₁	—	4	8	no	—
113	C₂×C₄×C₁₁	—	4	8	no	—
114	C₂×C₄×C₁₁	—	4	4	no	—
115	C₂×C₄×C₁₁	—	4	8	no	—
116	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	4	yes	—
117	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₁₁	C₄×Q₈	1	4	yes	—
119	C₂×C₄×C₁₁	#121	1	2	yes	—
120	C₂×C₄×C₁₁	C₂²⋊Q₈	1	4	yes	—
121	C₂×C₄×C₁₁	#125	1	4	yes	—
122	C₂×C₄×C₁₁	C₄⋊Q₈	1	2	yes	—
123	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
129	C₈×C₁₁	—	2	4	no	—
130	C₈×C₁₁	C₂×C₄⋊C₄	1	4	yes	—
131	C₈×C₁₁	C₂×C₄⋊C₄	1	4	yes	—
132	C₂³×C₈	C₂²×C₁₁	1	1	yes	characteristic
133	C₂³×C₈	C₂²×C₁₁	1	1	yes	characteristic
134	C₂×#117	C₂²×C₁₁	1	2	yes	—
135	C₂×#117	C₂²×C₁₁	1	2	yes	—
136	C₂²×C₄²	C₂²×C₁₁	1	1	yes	characteristic
137	#427	—	2	4	no	—
138	#427	C₄×C₁₁	1	8	yes	complemented
139	C₂×#107	C₂²×C₁₁	1	2	yes	—
140	C₂×#107	C₂²×C₁₁	1	2	yes	—
141	C₂×#106	C₂²×C₁₁	1	2	yes	—
142	C₂×#106	C₂²×C₁₁	1	2	yes	—
143	C₂×#97	—	2	4	no	—
144	C₂×#97	C₂²×C₁₁	1	2	yes	—
145	C₂×#97	C₂²×C₁₁	1	2	yes	—
146	#389	C₂²×C₁₁	1	8	yes	—
147	#385	—	2	4	no	—
148	C₄×C₄⋊C₄	—	2	4	no	—
149	#340	C₂²×C₁₁	1	8	yes	—
150	C₂²×C₁₁	—	2	4	no	—
151	C₂²×C₁₁	—	2	4	no	—
152	C₂²×C₁₁	—	2	4	no	—
153	C₂²×C₁₁	#389	1	4	yes	—
154	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
155	C₂²×C₁₁	C₂×#121	1	1	yes	characteristic
156	C₂²×C₁₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
157	C₂²×C₁₁	#575	1	4	yes	—
158	C₄×C₁₁	—	4	8	no	—
159	C₄×C₁₁	—	4	4	no	—
160	C₄×C₁₁	—	8	4	no	—
161	C₄×C₁₁	#389	1	4	yes	—
162	C₄×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
163	C₄×C₁₁	C₂×C₄×Q₈	1	1	yes	characteristic
164	C₄×C₁₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
165	C₄×C₁₁	C₂×#125	1	1	yes	characteristic
166	C₂³×C₄	D₄×C₁₁	1	2	yes	—
167	C₂³×C₄	C₂³×C₁₁	1	1	yes	characteristic
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	C₂×C₄×C₁₁	1	4	yes	—
171	C₂²×C₈	D₄×C₁₁	1	4	yes	—
172	C₂²×C₈	Q₈×C₁₁	1	4	yes	—
173	C₂²×C₈	Q₈×C₁₁	1	4	yes	—
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	#117	—	2	4	no	—
178	#117	—	2	4	no	—
179	#117	—	2	4	no	—
180	#117	C₂×C₄×C₁₁	1	8	yes	—
181	C₂×C₄⋊C₄	—	2	8	no	—
182	C₂×C₄⋊C₄	C₂³×C₁₁	1	4	yes	—
183	C₂×C₄⋊C₄	C₂³×C₁₁	1	4	yes	—
184	C₂×C₂²⋊C₄	—	2	2	no	—
185	C₂×C₂²⋊C₄	—	2	2	no	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	D₄×C₁₁	1	2	yes	—
191	C₂×C₄²	C₂³×C₁₁	1	2	yes	—
192	#107	—	2	4	no	—
193	#107	C₂×C₄×C₁₁	1	8	yes	complemented
194	#106	—	2	4	no	—
195	#106	C₂×C₄×C₁₁	1	8	yes	complemented
196	#97	—	2	4	no	—
197	#97	—	2	4	no	—
198	#95	—	2	8	no	—
199	C₂×C₁₁	—	2	4	no	—
200	C₂×C₁₁	C₂×#389	1	1	yes	characteristic
201	C₂×C₁₁	#2943	1	2	yes	—
202	C₂×C₁₁	#3054	1	2	yes	—
203	C₂×C₁₁	#3162	1	2	yes	—
204	C₂⁴	C₂×D₄×C₁₁	1	1	yes	characteristic
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	4	4	no	—
209	C₂²×C₄	C₂²×C₄×C₁₁	1	2	yes	—
210	C₂²×C₄	C₂×D₄×C₁₁	1	4	yes	—
211	C₂²×C₄	C₂×D₄×C₁₁	1	2	yes	—
212	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
213	C₂²×C₄	C₂×Q₈×C₁₁	1	2	yes	—
214	C₂²×C₄	C₂×Q₈×C₁₁	1	2	yes	—
215	C₂²×C₄	C₁₁×#41	1	2	yes	—
216	C₂²×C₄	C₁₁×#41	1	2	yes	—
217	C₂²×C₄	C₂⁴×C₁₁	1	1	yes	characteristic
218	C₂×C₈	—	2	8	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	C₁₁×C₄⋊C₄	1	16	yes	complemented
222	C₂×C₈	C₂²×C₄×C₁₁	1	2	yes	—
223	C₂×C₈	C₂²×C₄×C₁₁	1	2	yes	—
224	C₂×C₈	C₂×D₄×C₁₁	1	2	yes	—
225	C₂×C₈	C₂×D₄×C₁₁	1	2	yes	—
226	C₂×C₈	C₂×Q₈×C₁₁	1	4	yes	—
227	C₂×C₈	C₂×Q₈×C₁₁	1	2	yes	—
228	C₂×C₈	C₂×Q₈×C₁₁	1	2	yes	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	4	16	no	—
233	C₄⋊C₄	C₂²×C₄×C₁₁	1	8	yes	—
234	C₂²⋊C₄	—	2	8	no	—
235	C₂²⋊C₄	—	4	4	no	—
236	C₄²	—	2	8	no	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	4	4	no	—
241	C₁₁	#21154	1	1	yes	characteristic, direct product factor, Sylow
242	C₂³	—	2	4	no	—
243	C₂³	C₄×D₄×C₁₁	1	2	yes	—
244	C₂³	C₁₁×#121	1	2	yes	—
245	C₂³	C₁₁×C₄⋊Q₈	1	2	yes	—
246	C₂³	C₂²×D₄×C₁₁	1	1	yes	characteristic
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	8	no	—
250	C₂×C₄	—	4	8	no	—
251	C₂×C₄	—	4	8	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	8	no	—
254	C₂×C₄	C₂×C₁₁×C₄⋊C₄	1	4	yes	—
255	C₂×C₄	C₄×D₄×C₁₁	1	2	yes	—
256	C₂×C₄	C₄×Q₈×C₁₁	1	4	yes	—
257	C₂×C₄	C₁₁×#121	1	2	yes	—
258	C₂×C₄	C₁₁×C₂²⋊Q₈	1	4	yes	—
259	C₂×C₄	C₁₁×#125	1	4	yes	—
260	C₂×C₄	C₁₁×C₄⋊Q₈	1	2	yes	—
261	C₂×C₄	C₂³×C₄×C₁₁	1	1	yes	characteristic
262	C₂×C₄	C₂²×D₄×C₁₁	1	1	yes	characteristic
263	C₂×C₄	C₂²×Q₈×C₁₁	1	1	yes	characteristic
264	C₂×C₄	C₂²×Q₈×C₁₁	1	1	yes	characteristic
265	C₂×C₄	C₂×C₁₁×#41	1	1	yes	characteristic
266	C₂×C₄	C₂×C₁₁×#41	1	1	yes	characteristic
267	C₈	—	2	4	no	—
268	C₈	C₂×C₁₁×C₄⋊C₄	1	4	yes	—
269	C₈	C₂×C₁₁×C₄⋊C₄	1	4	yes	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	2	4	no	—
273	C₂²	C₁₁×#389	1	4	yes	—
274	C₂²	C₂×C₄×D₄×C₁₁	1	1	yes	characteristic
275	C₂²	C₂×C₁₁×#121	1	1	yes	characteristic
276	C₂²	C₂×C₁₁×C₄⋊Q₈	1	1	yes	characteristic
277	C₂²	C₁₁×#575	1	4	yes	—
278	C₄	—	4	8	no	—
279	C₄	—	4	4	no	—
280	C₄	—	8	4	no	—
281	C₄	C₁₁×#389	1	4	yes	—
282	C₄	C₂²×C₁₁×C₄⋊C₄	1	1	yes	characteristic
283	C₄	C₂×C₄×Q₈×C₁₁	1	1	yes	characteristic
284	C₄	C₂×C₁₁×C₂²⋊Q₈	1	1	yes	characteristic
285	C₄	C₂×C₁₁×#125	1	1	yes	characteristic
286	C₂	—	2	4	no	—
287	C₂	C₂×C₁₁×#389	1	1	yes	characteristic
288	C₂	C₁₁×#2943	1	2	yes	—
289	C₂	C₁₁×#3054	1	2	yes	—
290	C₂	C₁₁×#3162	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
5. C₁₁×#1944	C₂	semidirect
6. C₁₁×#1943	C₂	semidirect
14. C₁₁×#427	C₄	semidirect
53. C₁₁×#107	C₂×C₄	semidirect
55. C₁₁×#106	C₂×C₄	semidirect
59. #21154	C₁₁	direct (N × H)
77. C₂×C₈×C₁₁	C₄⋊C₄	semidirect
101. #1944	C₂×C₁₁	semidirect
102. #1943	C₂×C₁₁	semidirect
138. #427	C₄×C₁₁	semidirect
193. #107	C₂×C₄×C₁₁	semidirect
195. #106	C₂×C₄×C₁₁	semidirect
221. C₂×C₈	C₁₁×C₄⋊C₄	semidirect
241. C₁₁	#21154	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×#427	2
C₂²×C₁₁×#97	2
C₁₁×#2296	2
C₁₁×#1944	2
C₁₁×#1943	2
C₁₁×#1735	2
#21154	11

Subgroups — C11×#21154 #14598347

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#21154 #14598347