Subgroups — C₁₁×#11367 #14588560

Order 2816.

622 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	C₂×C₁₁×#430	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₁×#344	C₂	1	2	yes	maximal
3	C₂×C₁₁×#339	C₂	1	2	yes	maximal
4	C₂×C₁₁×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
5	C₁₁×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₁×M₁₆	C₂²	1	2	yes	—
7	C₂²×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₁₁×#117	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
11	C₁₁×#430	—	2	2	no	—
12	C₁₁×#430	C₄	1	4	yes	complemented
13	C₂×C₁₁×#108	—	2	2	no	—
14	C₁₁×#344	—	2	4	no	—
15	C₁₁×#339	—	2	4	no	—
16	C₁₁×H(3,ℤ/4)	—	2	2	no	—
17	C₁₁×#336	—	2	2	no	—
18	C₂³×C₄×C₁₁	—	2	1	no	—
19	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
20	C₂×C₁₁×M₁₆	—	2	2	no	—
21	C₂×C₁₁×M₁₆	—	2	2	no	—
22	C₂×C₁₁×M₁₆	—	2	2	no	—
23	C₂×C₁₁×M₁₆	—	2	1	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	—	2	2	no	—
29	C₂×C₁₁×M₁₆	—	2	2	no	—
30	C₂×C₁₁×M₁₆	C₂×C₄	1	4	yes	—
31	C₂×C₁₁×M₁₆	C₂×C₄	1	2	yes	—
32	C₂×C₁₁×M₁₆	D₄	1	2	yes	—
33	C₂×C₁₁×M₁₆	Q₈	1	2	yes	—
34	C₂²×C₈×C₁₁	—	2	1	no	—
35	C₂²×C₈×C₁₁	—	2	2	no	—
36	C₂²×C₈×C₁₁	D₄	1	2	yes	—
37	C₁₁×#117	—	2	2	no	—
38	C₁₁×#117	—	2	2	no	—
39	C₁₁×#117	—	2	2	no	—
40	C₁₁×#117	D₄	1	4	yes	—
41	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
42	C₂×C₁₁×C₂²⋊C₄	—	2	2	no	—
43	C₂×C₄²×C₁₁	—	2	2	no	—
44	C₂×C₄²×C₁₁	—	2	2	no	—
45	C₂×C₄²×C₁₁	—	2	2	no	—
46	C₂×C₄²×C₁₁	—	2	1	no	—
47	C₂×C₄²×C₁₁	Q₈	1	2	yes	—
48	C₁₁×#108	—	2	8	no	—
49	#11367	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	—	4	4	no	—
53	C₂²×C₄×C₁₁	—	4	2	no	—
54	C₂²×C₄×C₁₁	—	4	2	no	—
55	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₁	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₁	#41	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	C₁₁×M₁₆	—	2	4	no	—
66	C₁₁×M₁₆	—	2	4	no	—
67	C₁₁×M₁₆	—	2	4	no	—
68	C₁₁×M₁₆	—	2	4	no	—
69	C₁₁×M₁₆	—	2	4	no	—
70	C₁₁×M₁₆	—	4	2	no	—
71	C₁₁×M₁₆	—	4	2	no	—
72	C₁₁×M₁₆	—	4	2	no	—
73	C₂×C₈×C₁₁	—	2	4	no	—
74	C₂×C₈×C₁₁	—	2	2	no	—
75	C₂×C₈×C₁₁	—	2	2	no	—
76	C₂×C₈×C₁₁	—	2	2	no	—
77	C₂×C₈×C₁₁	—	4	2	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	C₂²⋊C₄	1	4	yes	—
82	C₁₁×C₄⋊C₄	—	2	4	no	—
83	C₁₁×C₄⋊C₄	—	4	2	no	—
84	C₁₁×C₂²⋊C₄	—	2	4	no	—
85	C₁₁×C₂²⋊C₄	—	4	2	no	—
86	C₄²×C₁₁	—	2	4	no	—
87	C₄²×C₁₁	—	2	4	no	—
88	C₄²×C₁₁	—	4	1	no	—
89	C₄²×C₁₁	—	4	2	no	—
90	C₄²×C₁₁	—	4	1	no	—
91	C₂×#430	C₂×C₁₁	1	1	yes	characteristic
92	C₂×#344	C₂×C₁₁	1	2	yes	—
93	C₂×#339	C₂×C₁₁	1	2	yes	—
94	C₂×H(3,ℤ/4)	C₂×C₁₁	1	1	yes	characteristic
95	#1728	C₂×C₁₁	1	1	yes	characteristic
96	C₂³×C₁₁	—	4	2	no	—
97	C₂³×C₁₁	C₄×Q₈	1	1	yes	characteristic
98	C₂³×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂³×C₁₁	#123	1	2	yes	—
100	C₂³×C₁₁	#124	1	1	yes	characteristic
101	C₂³×C₁₁	#126	1	2	yes	—
102	C₂×C₄×C₁₁	—	2	4	no	—
103	C₂×C₄×C₁₁	—	4	2	no	—
104	C₂×C₄×C₁₁	—	4	8	no	—
105	C₂×C₄×C₁₁	—	4	2	no	—
106	C₂×C₄×C₁₁	—	4	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	2	yes	—
111	C₂×C₄×C₁₁	C₄×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₁₁	#121	1	2	yes	—
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	1	yes	characteristic
116	C₂×C₄×C₁₁	#123	1	2	yes	—
117	C₂×C₄×C₁₁	#123	1	1	yes	characteristic
118	C₂×C₄×C₁₁	#123	1	1	yes	characteristic
119	C₂×C₄×C₁₁	#123	1	2	yes	—
120	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
121	C₂×C₄×C₁₁	#125	1	2	yes	—
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	4	no	—
128	C₈×C₁₁	—	4	2	no	—
129	C₂²×M₁₆	C₂²×C₁₁	1	2	yes	—
130	C₂²×M₁₆	C₂²×C₁₁	1	1	yes	characteristic
131	C₂²×M₁₆	C₂²×C₁₁	1	1	yes	characteristic
132	C₂×#117	C₂²×C₁₁	1	2	yes	—
133	C₂²×C₄²	C₂²×C₁₁	1	1	yes	characteristic
134	#430	—	2	2	no	—
135	#430	C₄×C₁₁	1	4	yes	complemented
136	C₂×#108	—	2	2	no	—
137	#344	—	2	4	no	—
138	#339	—	2	4	no	—
139	H(3,ℤ/4)	—	2	2	no	—
140	#336	—	2	2	no	—
141	C₂²×C₁₁	—	2	2	no	—
142	C₂²×C₁₁	—	2	4	no	—
143	C₂²×C₁₁	—	4	2	no	—
144	C₂²×C₁₁	—	4	1	no	—
145	C₂²×C₁₁	—	4	1	no	—
146	C₂²×C₁₁	#387	1	2	yes	—
147	C₂²×C₁₁	#391	1	1	yes	characteristic
148	C₂²×C₁₁	#397	1	2	yes	—
149	C₂²×C₁₁	#399	1	1	yes	characteristic
150	C₂²×C₁₁	#400	1	1	yes	characteristic
151	C₄×C₁₁	—	4	4	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	#390	1	1	yes	characteristic
156	C₄×C₁₁	#398	1	2	yes	—
157	C₄×C₁₁	#399	1	1	yes	characteristic
158	C₄×C₁₁	#400	1	1	yes	characteristic
159	C₂³×C₄	—	2	1	no	—
160	C₂³×C₄	C₂³×C₁₁	1	1	yes	characteristic
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	1	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	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	C₂×C₄×C₁₁	1	4	yes	—
172	C₂×M₁₆	C₂×C₄×C₁₁	1	2	yes	—
173	C₂×M₁₆	D₄×C₁₁	1	2	yes	—
174	C₂×M₁₆	Q₈×C₁₁	1	2	yes	—
175	C₂²×C₈	—	2	1	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	D₄×C₁₁	1	2	yes	—
178	#117	—	2	2	no	—
179	#117	—	2	2	no	—
180	#117	—	2	2	no	—
181	#117	D₄×C₁₁	1	4	yes	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₂²⋊C₄	—	2	2	no	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	2	1	no	—
188	C₂×C₄²	Q₈×C₁₁	1	2	yes	—
189	#108	—	2	8	no	—
190	C₂×C₁₁	—	2	1	no	—
191	C₂×C₁₁	—	2	2	no	—
192	C₂×C₁₁	—	2	2	no	—
193	C₂×C₁₁	—	2	1	no	—
194	C₂×C₁₁	#1445	1	1	yes	characteristic
195	C₂×C₁₁	#2102	1	2	yes	—
196	C₂⁴	C₁₁×#41	1	1	yes	characteristic
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	4	4	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	C₂²×C₄×C₁₁	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₁₁	1	2	yes	—
205	C₂²×C₄	C₂×Q₈×C₁₁	1	1	yes	characteristic
206	C₂²×C₄	C₂×Q₈×C₁₁	1	1	yes	characteristic
207	C₂²×C₄	C₁₁×#41	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	M₁₆	—	2	4	no	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	2	4	no	—
216	M₁₆	—	4	2	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	4	2	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	—	4	2	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	C₁₁×C₂²⋊C₄	1	4	yes	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₂²⋊C₄	—	2	4	no	—
231	C₂²⋊C₄	—	4	2	no	—
232	C₄²	—	2	4	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	4	1	no	—
235	C₄²	—	4	2	no	—
236	C₄²	—	4	1	no	—
237	C₁₁	#11367	1	1	yes	characteristic, direct product factor, Sylow
238	C₂³	—	4	2	no	—
239	C₂³	C₄×Q₈×C₁₁	1	1	yes	characteristic
240	C₂³	C₁₁×C₂²⋊Q₈	1	1	yes	characteristic
241	C₂³	C₁₁×#123	1	2	yes	—
242	C₂³	C₁₁×#124	1	1	yes	characteristic
243	C₂³	C₁₁×#126	1	2	yes	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	8	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	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	2	yes	—
253	C₂×C₄	C₄×Q₈×C₁₁	1	1	yes	characteristic
254	C₂×C₄	C₁₁×#121	1	2	yes	—
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	1	yes	characteristic
258	C₂×C₄	C₁₁×#123	1	2	yes	—
259	C₂×C₄	C₁₁×#123	1	1	yes	characteristic
260	C₂×C₄	C₁₁×#123	1	1	yes	characteristic
261	C₂×C₄	C₁₁×#123	1	2	yes	—
262	C₂×C₄	C₁₁×#124	1	1	yes	characteristic
263	C₂×C₄	C₁₁×#125	1	2	yes	—
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	4	no	—
270	C₈	—	4	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	4	1	no	—
276	C₂²	C₁₁×#387	1	2	yes	—
277	C₂²	C₁₁×#391	1	1	yes	characteristic
278	C₂²	C₁₁×#397	1	2	yes	—
279	C₂²	C₁₁×#399	1	1	yes	characteristic
280	C₂²	C₁₁×#400	1	1	yes	characteristic
281	C₄	—	4	4	no	—
282	C₄	—	8	4	no	—
283	C₄	C₁₁×#381	1	1	yes	characteristic
284	C₄	C₁₁×#387	1	2	yes	—
285	C₄	C₁₁×#390	1	1	yes	characteristic
286	C₄	C₁₁×#398	1	2	yes	—
287	C₄	C₁₁×#399	1	1	yes	characteristic
288	C₄	C₁₁×#400	1	1	yes	characteristic
289	C₂	—	2	1	no	—
290	C₂	—	2	2	no	—
291	C₂	—	2	2	no	—
292	C₂	—	2	1	no	—
293	C₂	C₁₁×#1445	1	1	yes	characteristic
294	C₂	C₁₁×#2102	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₁	#124
commutator	C₂²×C₄	C₂²×C₄×C₁₁
Frattini, cosocle	C₂³×C₄	C₂³×C₁₁
socle	C₂²×C₁₁	#400
prime radical	C₂⁴×C₁₁	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₁ ◁ C₂³×C₄×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₁₁×#430	C₄	semidirect
49. #11367	C₁₁	direct (N × H)
135. #430	C₄×C₁₁	semidirect
237. C₁₁	#11367	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×#430	2
C₂×C₁₁×#344	2
C₂×C₁₁×#339	2
C₂×C₁₁×H(3,ℤ/4)	2
C₁₁×#1728	2
#11367	11

Subgroups — C11×#11367 #14588560

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#11367 #14588560