Subgroups — C₁₁×#11365 #14588558

Order 2816.

634 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₁×#107	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×#107	—	2	2	no	—
12	C₂×C₁₁×#107	—	2	2	no	—
13	C₂×C₁₁×#107	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×#107	—	2	8	no	—
49	#11365	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×#107	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	2	no	—
97	C₂³×C₁₁	—	2	2	no	—
98	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
99	C₂³×C₁₁	#121	1	1	yes	characteristic
100	C₂³×C₁₁	#124	1	1	yes	characteristic
101	C₂³×C₁₁	#125	1	2	yes	—
102	C₂³×C₁₁	#126	1	2	yes	—
103	C₂×C₄×C₁₁	—	4	4	no	—
104	C₂×C₄×C₁₁	—	4	2	no	—
105	C₂×C₄×C₁₁	—	4	2	no	—
106	C₂×C₄×C₁₁	—	4	2	no	—
107	C₂×C₄×C₁₁	—	8	4	no	—
108	C₂×C₄×C₁₁	—	8	4	no	—
109	C₂×C₄×C₁₁	—	8	4	no	—
110	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₁	#117	1	2	yes	—
112	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₁	C₄×Q₈	1	2	yes	—
114	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
115	C₂×C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₁₁	#123	1	1	yes	characteristic
119	C₂×C₄×C₁₁	#123	1	1	yes	characteristic
120	C₂×C₄×C₁₁	#123	1	2	yes	—
121	C₂×C₄×C₁₁	#123	1	2	yes	—
122	C₂×C₄×C₁₁	#123	1	2	yes	—
123	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
124	C₂×C₄×C₁₁	#125	1	2	yes	—
125	C₂×C₄×C₁₁	#126	1	2	yes	—
126	C₂×C₄×C₁₁	#127	1	1	yes	characteristic
127	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
128	C₈×C₁₁	—	2	4	no	—
129	C₈×C₁₁	—	4	2	no	—
130	C₈×C₁₁	—	4	4	no	—
131	C₂²×M₁₆	C₂²×C₁₁	1	1	yes	characteristic
132	C₂²×M₁₆	C₂²×C₁₁	1	2	yes	—
133	C₂³×C₈	C₂²×C₁₁	1	1	yes	characteristic
134	C₂×#117	C₂²×C₁₁	1	1	yes	characteristic
135	C₂²×C₄⋊C₄	C₂²×C₁₁	1	2	yes	—
136	C₂×#107	—	2	2	no	—
137	C₂×#107	—	2	2	no	—
138	C₂×#107	C₄×C₁₁	1	4	yes	complemented
139	#343	—	2	4	no	—
140	#341	—	2	2	no	—
141	#340	—	2	4	no	—
142	#336	—	2	2	no	—
143	C₂²×C₁₁	—	2	4	no	—
144	C₂²×C₁₁	—	2	4	no	—
145	C₂²×C₁₁	—	2	1	no	—
146	C₂²×C₁₁	—	2	1	no	—
147	C₂²×C₁₁	—	2	1	no	—
148	C₂²×C₁₁	—	2	1	no	—
149	C₂²×C₁₁	#387	1	2	yes	—
150	C₂²×C₁₁	#390	1	1	yes	characteristic
151	C₂²×C₁₁	#397	1	1	yes	characteristic
152	C₂²×C₁₁	#398	1	1	yes	characteristic
153	C₂²×C₁₁	#400	1	2	yes	—
154	C₂²×C₁₁	#472	1	2	yes	—
155	C₂²×C₁₁	#473	1	2	yes	—
156	C₄×C₁₁	—	8	2	no	—
157	C₄×C₁₁	—	8	4	no	—
158	C₄×C₁₁	#381	1	1	yes	characteristic
159	C₄×C₁₁	#387	1	2	yes	—
160	C₄×C₁₁	#391	1	1	yes	characteristic
161	C₄×C₁₁	#397	1	1	yes	characteristic
162	C₄×C₁₁	#398	1	1	yes	characteristic
163	C₄×C₁₁	#399	1	2	yes	—
164	C₂³×C₄	—	2	2	no	—
165	C₂³×C₄	C₂³×C₁₁	1	1	yes	characteristic
166	C₂×M₁₆	—	2	1	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	D₄×C₁₁	1	2	yes	—
175	C₂×M₁₆	Q₈×C₁₁	1	4	yes	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	C₂×C₄×C₁₁	1	2	yes	—
181	C₂²×C₈	D₄×C₁₁	1	2	yes	—
182	C₂²×C₈	D₄×C₁₁	1	2	yes	—
183	#117	—	2	2	no	—
184	#117	—	2	2	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	C₂×C₄×C₁₁	1	4	yes	—
191	C₂×C₄⋊C₄	D₄×C₁₁	1	2	yes	—
192	C₂×C₂²⋊C₄	—	2	1	no	—
193	C₂×C₄²	—	2	1	no	—
194	#107	—	2	8	no	—
195	C₂×C₁₁	—	2	2	no	—
196	C₂×C₁₁	—	2	2	no	—
197	C₂×C₁₁	#1445	1	1	yes	characteristic
198	C₂×C₁₁	#1978	1	1	yes	characteristic
199	C₂×C₁₁	#1979	1	1	yes	characteristic
200	C₂×C₁₁	#2078	1	2	yes	—
201	C₂×C₁₁	#2089	1	2	yes	—
202	C₂⁴	C₁₁×#41	1	1	yes	characteristic
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	4	2	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	—	8	4	no	—
208	C₂²×C₄	C₂²×C₄×C₁₁	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
213	C₂²×C₄	C₂×Q₈×C₁₁	1	2	yes	—
214	C₂²×C₄	C₁₁×#41	1	2	yes	—
215	C₂²×C₄	C₁₁×#41	1	2	yes	—
216	C₂²×C₄	C₁₁×#41	1	1	yes	characteristic
217	C₂²×C₄	C₁₁×#41	1	2	yes	—
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	4	2	no	—
222	M₁₆	—	4	2	no	—
223	M₁₆	—	4	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	C₁₁×C₂²⋊C₄	1	4	yes	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₄²	—	4	1	no	—
242	C₄²	—	4	1	no	—
243	C₁₁	#11365	1	1	yes	characteristic, direct product factor, Sylow
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	C₄×D₄×C₁₁	1	1	yes	characteristic
247	C₂³	C₁₁×#121	1	1	yes	characteristic
248	C₂³	C₁₁×#124	1	1	yes	characteristic
249	C₂³	C₁₁×#125	1	2	yes	—
250	C₂³	C₁₁×#126	1	2	yes	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	8	4	no	—
256	C₂×C₄	—	8	4	no	—
257	C₂×C₄	—	8	4	no	—
258	C₂×C₄	C₂×C₁₁×C₂²⋊C₄	1	1	yes	characteristic
259	C₂×C₄	C₁₁×#117	1	2	yes	—
260	C₂×C₄	C₄×D₄×C₁₁	1	1	yes	characteristic
261	C₂×C₄	C₄×Q₈×C₁₁	1	2	yes	—
262	C₂×C₄	C₁₁×#121	1	1	yes	characteristic
263	C₂×C₄	C₁₁×C₂²⋊Q₈	1	2	yes	—
264	C₂×C₄	C₁₁×C₂²⋊Q₈	1	2	yes	—
265	C₂×C₄	C₁₁×C₂²⋊Q₈	1	2	yes	—
266	C₂×C₄	C₁₁×#123	1	1	yes	characteristic
267	C₂×C₄	C₁₁×#123	1	1	yes	characteristic
268	C₂×C₄	C₁₁×#123	1	2	yes	—
269	C₂×C₄	C₁₁×#123	1	2	yes	—
270	C₂×C₄	C₁₁×#123	1	2	yes	—
271	C₂×C₄	C₁₁×#124	1	1	yes	characteristic
272	C₂×C₄	C₁₁×#125	1	2	yes	—
273	C₂×C₄	C₁₁×#126	1	2	yes	—
274	C₂×C₄	C₁₁×#127	1	1	yes	characteristic
275	C₂×C₄	C₁₁×C₄⋊Q₈	1	1	yes	characteristic
276	C₈	—	2	4	no	—
277	C₈	—	4	2	no	—
278	C₈	—	4	4	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	2	4	no	—
285	C₂²	C₁₁×#387	1	2	yes	—
286	C₂²	C₁₁×#390	1	1	yes	characteristic
287	C₂²	C₁₁×#397	1	1	yes	characteristic
288	C₂²	C₁₁×#398	1	1	yes	characteristic
289	C₂²	C₁₁×#400	1	2	yes	—
290	C₂²	C₁₁×#472	1	2	yes	—
291	C₂²	C₁₁×#473	1	2	yes	—
292	C₄	—	8	4	no	—
293	C₄	—	8	2	no	—
294	C₄	C₁₁×#381	1	1	yes	characteristic
295	C₄	C₁₁×#387	1	2	yes	—
296	C₄	C₁₁×#391	1	1	yes	characteristic
297	C₄	C₁₁×#397	1	1	yes	characteristic
298	C₄	C₁₁×#398	1	1	yes	characteristic
299	C₄	C₁₁×#399	1	2	yes	—
300	C₂	—	2	2	no	—
301	C₂	—	2	2	no	—
302	C₂	C₁₁×#1445	1	1	yes	characteristic
303	C₂	C₁₁×#1978	1	1	yes	characteristic
304	C₂	C₁₁×#1979	1	1	yes	characteristic
305	C₂	C₁₁×#2078	1	2	yes	—
306	C₂	C₁₁×#2089	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×#107	C₄	semidirect
49. #11365	C₁₁	direct (N × H)
138. C₂×#107	C₄×C₁₁	semidirect
243. C₁₁	#11365	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×#107	2
C₂×C₁₁×#343	2
C₂×C₁₁×#341	2
C₁₁×#1736	2
C₁₁×#1729	2
#11365	11

Subgroups — C11×#11365 #14588558

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#11365 #14588558