Subgroups — C₁₁×#11366 #14588559

Order 2816.

634 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₁×#106	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₁×#343	C₂	1	2	yes	maximal
3	C₂×C₁₁×#341	C₂	1	1	yes	maximal, characteristic
4	C₁₁×#1736	C₂	1	2	yes	maximal
5	C₁₁×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₁₁×M₁₆	C₂²	1	2	yes	—
8	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
9	C₂×C₁₁×#117	C₂²	1	1	yes	characteristic
10	C₂²×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₁₁×#106	—	2	2	no	—
12	C₂×C₁₁×#106	—	2	2	no	—
13	C₂×C₁₁×#106	C₄	1	4	yes	complemented
14	C₁₁×#343	—	2	4	no	—
15	C₁₁×#341	—	2	2	no	—
16	C₁₁×#340	—	2	4	no	—
17	C₁₁×#336	—	2	2	no	—
18	C₂³×C₄×C₁₁	—	2	2	no	—
19	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
20	C₂×C₁₁×M₁₆	—	2	1	no	—
21	C₂×C₁₁×M₁₆	—	2	2	no	—
22	C₂×C₁₁×M₁₆	—	2	2	no	—
23	C₂×C₁₁×M₁₆	—	2	2	no	—
24	C₂×C₁₁×M₁₆	—	2	2	no	—
25	C₂×C₁₁×M₁₆	—	2	2	no	—
26	C₂×C₁₁×M₁₆	—	2	2	no	—
27	C₂×C₁₁×M₁₆	—	2	2	no	—
28	C₂×C₁₁×M₁₆	D₄	1	2	yes	—
29	C₂×C₁₁×M₁₆	Q₈	1	4	yes	—
30	C₂²×C₈×C₁₁	—	2	2	no	—
31	C₂²×C₈×C₁₁	—	2	2	no	—
32	C₂²×C₈×C₁₁	—	2	1	no	—
33	C₂²×C₈×C₁₁	—	2	2	no	—
34	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₁₁	D₄	1	2	yes	—
36	C₂²×C₈×C₁₁	D₄	1	2	yes	—
37	C₁₁×#117	—	2	2	no	—
38	C₁₁×#117	—	2	2	no	—
39	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
40	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
41	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
42	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
43	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
44	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	4	yes	—
45	C₂×C₁₁×C₄⋊C₄	D₄	1	2	yes	—
46	C₂×C₁₁×C₂²⋊C₄	—	2	1	no	—
47	C₂×C₄²×C₁₁	—	2	1	no	—
48	C₁₁×#106	—	2	8	no	—
49	#11366	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂⁴×C₁₁	#41	1	1	yes	characteristic
51	C₂²×C₄×C₁₁	—	2	2	no	—
52	C₂²×C₄×C₁₁	—	2	4	no	—
53	C₂²×C₄×C₁₁	—	4	2	no	—
54	C₂²×C₄×C₁₁	—	4	2	no	—
55	C₂²×C₄×C₁₁	—	8	4	no	—
56	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₁	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₁₁	#41	1	2	yes	—
63	C₂²×C₄×C₁₁	#41	1	2	yes	—
64	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
65	C₂²×C₄×C₁₁	#41	1	2	yes	—
66	C₁₁×M₁₆	—	2	4	no	—
67	C₁₁×M₁₆	—	2	4	no	—
68	C₁₁×M₁₆	—	2	4	no	—
69	C₁₁×M₁₆	—	4	2	no	—
70	C₁₁×M₁₆	—	4	2	no	—
71	C₁₁×M₁₆	—	4	2	no	—
72	C₂×C₈×C₁₁	—	2	4	no	—
73	C₂×C₈×C₁₁	—	2	2	no	—
74	C₂×C₈×C₁₁	—	2	4	no	—
75	C₂×C₈×C₁₁	—	2	2	no	—
76	C₂×C₈×C₁₁	—	2	2	no	—
77	C₂×C₈×C₁₁	—	2	4	no	—
78	C₂×C₈×C₁₁	—	4	1	no	—
79	C₂×C₈×C₁₁	—	4	2	no	—
80	C₂×C₈×C₁₁	—	4	1	no	—
81	C₂×C₈×C₁₁	—	4	2	no	—
82	C₂×C₈×C₁₁	C₂²⋊C₄	1	4	yes	—
83	C₁₁×C₄⋊C₄	—	2	4	no	—
84	C₁₁×C₄⋊C₄	—	2	4	no	—
85	C₁₁×C₄⋊C₄	—	2	4	no	—
86	C₁₁×C₄⋊C₄	—	4	2	no	—
87	C₁₁×C₄⋊C₄	—	4	2	no	—
88	C₁₁×C₂²⋊C₄	—	4	2	no	—
89	C₄²×C₁₁	—	4	1	no	—
90	C₄²×C₁₁	—	4	1	no	—
91	C₂²×#106	C₂×C₁₁	1	1	yes	characteristic
92	C₂×#343	C₂×C₁₁	1	2	yes	—
93	C₂×#341	C₂×C₁₁	1	1	yes	characteristic
94	#1736	C₂×C₁₁	1	2	yes	—
95	#1729	C₂×C₁₁	1	1	yes	characteristic
96	C₂³×C₁₁	—	2	4	no	—
97	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
98	C₂³×C₁₁	#121	1	1	yes	characteristic
99	C₂³×C₁₁	#124	1	1	yes	characteristic
100	C₂³×C₁₁	#125	1	2	yes	—
101	C₂³×C₁₁	#126	1	2	yes	—
102	C₂×C₄×C₁₁	—	4	4	no	—
103	C₂×C₄×C₁₁	—	4	2	no	—
104	C₂×C₄×C₁₁	—	4	4	no	—
105	C₂×C₄×C₁₁	—	8	4	no	—
106	C₂×C₄×C₁₁	—	8	4	no	—
107	C₂×C₄×C₁₁	—	8	4	no	—
108	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₁	#117	1	2	yes	—
110	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₁	C₄×Q₈	1	2	yes	—
112	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
113	C₂×C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₁₁	#123	1	1	yes	characteristic
117	C₂×C₄×C₁₁	#123	1	1	yes	characteristic
118	C₂×C₄×C₁₁	#123	1	2	yes	—
119	C₂×C₄×C₁₁	#123	1	2	yes	—
120	C₂×C₄×C₁₁	#123	1	2	yes	—
121	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
122	C₂×C₄×C₁₁	#125	1	2	yes	—
123	C₂×C₄×C₁₁	#126	1	2	yes	—
124	C₂×C₄×C₁₁	#127	1	1	yes	characteristic
125	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
126	C₈×C₁₁	—	2	4	no	—
127	C₈×C₁₁	—	4	2	no	—
128	C₈×C₁₁	—	4	4	no	—
129	C₂²×M₁₆	C₂²×C₁₁	1	1	yes	characteristic
130	C₂²×M₁₆	C₂²×C₁₁	1	2	yes	—
131	C₂³×C₈	C₂²×C₁₁	1	1	yes	characteristic
132	C₂×#117	C₂²×C₁₁	1	1	yes	characteristic
133	C₂²×C₄⋊C₄	C₂²×C₁₁	1	2	yes	—
134	C₂×#106	—	2	2	no	—
135	C₂×#106	—	2	2	no	—
136	C₂×#106	C₄×C₁₁	1	4	yes	complemented
137	#343	—	2	4	no	—
138	#341	—	2	2	no	—
139	#340	—	2	4	no	—
140	#336	—	2	2	no	—
141	C₂²×C₁₁	—	2	4	no	—
142	C₂²×C₁₁	—	2	4	no	—
143	C₂²×C₁₁	—	2	2	no	—
144	C₂²×C₁₁	—	2	2	no	—
145	C₂²×C₁₁	#387	1	2	yes	—
146	C₂²×C₁₁	#390	1	1	yes	characteristic
147	C₂²×C₁₁	#397	1	1	yes	characteristic
148	C₂²×C₁₁	#398	1	1	yes	characteristic
149	C₂²×C₁₁	#400	1	2	yes	—
150	C₂²×C₁₁	#471	1	4	yes	—
151	C₄×C₁₁	—	8	2	no	—
152	C₄×C₁₁	—	8	4	no	—
153	C₄×C₁₁	#381	1	1	yes	characteristic
154	C₄×C₁₁	#387	1	2	yes	—
155	C₄×C₁₁	#391	1	1	yes	characteristic
156	C₄×C₁₁	#397	1	1	yes	characteristic
157	C₄×C₁₁	#398	1	1	yes	characteristic
158	C₄×C₁₁	#399	1	2	yes	—
159	C₂³×C₄	—	2	2	no	—
160	C₂³×C₄	C₂³×C₁₁	1	1	yes	characteristic
161	C₂×M₁₆	—	2	1	no	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	D₄×C₁₁	1	2	yes	—
170	C₂×M₁₆	Q₈×C₁₁	1	4	yes	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	2	1	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	C₂×C₄×C₁₁	1	2	yes	—
176	C₂²×C₈	D₄×C₁₁	1	2	yes	—
177	C₂²×C₈	D₄×C₁₁	1	2	yes	—
178	#117	—	2	2	no	—
179	#117	—	2	2	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	C₂×C₄×C₁₁	1	4	yes	—
186	C₂×C₄⋊C₄	D₄×C₁₁	1	2	yes	—
187	C₂×C₂²⋊C₄	—	2	1	no	—
188	C₂×C₄²	—	2	1	no	—
189	#106	—	2	8	no	—
190	C₂×C₁₁	—	2	2	no	—
191	C₂×C₁₁	—	2	2	no	—
192	C₂×C₁₁	#1445	1	1	yes	characteristic
193	C₂×C₁₁	#1977	1	2	yes	—
194	C₂×C₁₁	#2079	1	2	yes	—
195	C₂×C₁₁	#2090	1	2	yes	—
196	C₂⁴	C₁₁×#41	1	1	yes	characteristic
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	8	4	no	—
202	C₂²×C₄	C₂²×C₄×C₁₁	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
207	C₂²×C₄	C₂×Q₈×C₁₁	1	2	yes	—
208	C₂²×C₄	C₁₁×#41	1	2	yes	—
209	C₂²×C₄	C₁₁×#41	1	2	yes	—
210	C₂²×C₄	C₁₁×#41	1	1	yes	characteristic
211	C₂²×C₄	C₁₁×#41	1	2	yes	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	4	2	no	—
217	M₁₆	—	4	2	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	C₁₁×C₂²⋊C₄	1	4	yes	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₂²⋊C₄	—	4	2	no	—
235	C₄²	—	4	1	no	—
236	C₄²	—	4	1	no	—
237	C₁₁	#11366	1	1	yes	characteristic, direct product factor, Sylow
238	C₂³	—	2	4	no	—
239	C₂³	C₄×D₄×C₁₁	1	1	yes	characteristic
240	C₂³	C₁₁×#121	1	1	yes	characteristic
241	C₂³	C₁₁×#124	1	1	yes	characteristic
242	C₂³	C₁₁×#125	1	2	yes	—
243	C₂³	C₁₁×#126	1	2	yes	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	8	4	no	—
248	C₂×C₄	—	8	4	no	—
249	C₂×C₄	—	8	4	no	—
250	C₂×C₄	C₂×C₁₁×C₂²⋊C₄	1	1	yes	characteristic
251	C₂×C₄	C₁₁×#117	1	2	yes	—
252	C₂×C₄	C₄×D₄×C₁₁	1	1	yes	characteristic
253	C₂×C₄	C₄×Q₈×C₁₁	1	2	yes	—
254	C₂×C₄	C₁₁×#121	1	1	yes	characteristic
255	C₂×C₄	C₁₁×C₂²⋊Q₈	1	2	yes	—
256	C₂×C₄	C₁₁×C₂²⋊Q₈	1	2	yes	—
257	C₂×C₄	C₁₁×C₂²⋊Q₈	1	2	yes	—
258	C₂×C₄	C₁₁×#123	1	1	yes	characteristic
259	C₂×C₄	C₁₁×#123	1	1	yes	characteristic
260	C₂×C₄	C₁₁×#123	1	2	yes	—
261	C₂×C₄	C₁₁×#123	1	2	yes	—
262	C₂×C₄	C₁₁×#123	1	2	yes	—
263	C₂×C₄	C₁₁×#124	1	1	yes	characteristic
264	C₂×C₄	C₁₁×#125	1	2	yes	—
265	C₂×C₄	C₁₁×#126	1	2	yes	—
266	C₂×C₄	C₁₁×#127	1	1	yes	characteristic
267	C₂×C₄	C₁₁×C₄⋊Q₈	1	1	yes	characteristic
268	C₈	—	2	4	no	—
269	C₈	—	4	2	no	—
270	C₈	—	4	4	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	C₁₁×#387	1	2	yes	—
276	C₂²	C₁₁×#390	1	1	yes	characteristic
277	C₂²	C₁₁×#397	1	1	yes	characteristic
278	C₂²	C₁₁×#398	1	1	yes	characteristic
279	C₂²	C₁₁×#400	1	2	yes	—
280	C₂²	C₁₁×#471	1	4	yes	—
281	C₄	—	8	4	no	—
282	C₄	—	8	2	no	—
283	C₄	C₁₁×#381	1	1	yes	characteristic
284	C₄	C₁₁×#387	1	2	yes	—
285	C₄	C₁₁×#391	1	1	yes	characteristic
286	C₄	C₁₁×#397	1	1	yes	characteristic
287	C₄	C₁₁×#398	1	1	yes	characteristic
288	C₄	C₁₁×#399	1	2	yes	—
289	C₂	—	2	2	no	—
290	C₂	—	2	2	no	—
291	C₂	C₁₁×#1445	1	1	yes	characteristic
292	C₂	C₁₁×#1977	1	2	yes	—
293	C₂	C₁₁×#2079	1	2	yes	—
294	C₂	C₁₁×#2090	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₁	#121
commutator	C₂²×C₄	C₂²×C₄×C₁₁
Frattini, cosocle	C₂³×C₄	C₂³×C₁₁
prime radical	C₂⁴×C₁₁	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₁ ◁ C₂³×C₄×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₂×C₁₁×#106	C₄	semidirect
49. #11366	C₁₁	direct (N × H)
136. C₂×#106	C₄×C₁₁	semidirect
237. C₁₁	#11366	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₁×#106	2
C₂×C₁₁×#343	2
C₂×C₁₁×#341	2
C₁₁×#1736	2
C₁₁×#1729	2
#11366	11

Subgroups — C11×#11366 #14588559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#11366 #14588559