Subgroups — C₁₁×#13214 #14590407

Order 2816.

490 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×#495	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₁×#362	C₂	1	2	yes	maximal, complemented
3	C₂×C₁₁×#361	C₂	1	2	yes	maximal
4	C₄×C₁₁×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₁×#1848	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₁₁×Q₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₁₁×SD₁₆	—	2	1	no	—
8	C₂²×C₁₁×D₈	C₂²	1	1	yes	characteristic
9	C₂×C₁₁×#124	—	2	1	no	—
10	C₂×C₁₁×M₃₂	C₂²	1	2	yes	—
11	C₁₁×#495	—	2	1	no	—
12	C₁₁×#495	—	2	1	no	—
13	C₁₁×#495	C₄	1	2	yes	—
14	C₁₁×#495	C₄	1	2	yes	—
15	C₂×C₁₁×#108	—	2	1	no	—
16	C₂×C₁₁×#108	C₂²	1	2	yes	—
17	C₁₁×#423	—	2	1	no	—
18	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
19	C₁₁×#362	—	2	4	no	—
20	C₁₁×#361	—	2	4	no	—
21	C₁₁×#346	—	2	1	no	—
22	C₄×C₁₁×C₁₆	—	2	1	no	—
23	C₂²×Q₈×C₁₁	—	2	1	no	—
24	C₂²×D₄×C₁₁	—	2	1	no	—
25	C₂×C₁₁×Q₁₆	—	2	2	no	—
26	C₂×C₁₁×Q₁₆	—	2	1	no	—
27	C₂×C₁₁×Q₁₆	—	2	2	no	—
28	C₂×C₁₁×Q₁₆	C₂×C₄	1	2	yes	—
29	C₂×C₁₁×SD₁₆	—	2	2	no	—
30	C₂×C₁₁×SD₁₆	—	2	2	no	—
31	C₂×C₁₁×SD₁₆	—	4	2	no	—
32	C₂×C₁₁×SD₁₆	—	4	2	no	—
33	C₂×C₁₁×D₈	—	2	2	no	—
34	C₂×C₁₁×D₈	—	2	1	no	—
35	C₂×C₁₁×D₈	—	2	2	no	—
36	C₂×C₁₁×D₈	C₂×C₄	1	2	yes	—
37	C₂×C₁₁×M₁₆	—	2	2	no	—
38	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
40	C₁₁×#124	—	2	2	no	—
41	C₁₁×#124	—	2	2	no	—
42	C₁₁×#124	—	4	2	no	—
43	C₂×C₁₁×C₂²⋊C₄	—	4	2	no	—
44	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
45	C₁₁×M₃₂	—	2	2	no	—
46	C₁₁×M₃₂	D₄	1	4	yes	—
47	C₂×C₁₁×C₁₆	—	2	2	no	—
48	C₁₁×#108	—	2	2	no	—
49	C₁₁×#108	—	2	2	no	—
50	C₁₁×#108	—	4	2	no	—
51	C₁₁×C₄⋊C₈	—	4	2	no	—
52	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
55	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
56	#13214	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂⁴×C₁₁	—	4	1	no	—
58	C₂×Q₈×C₁₁	—	2	2	no	—
59	C₂×Q₈×C₁₁	—	4	1	no	—
60	C₂×Q₈×C₁₁	—	4	1	no	—
61	C₂×Q₈×C₁₁	—	4	1	no	—
62	C₂×Q₈×C₁₁	—	4	1	no	—
63	C₂×Q₈×C₁₁	—	4	1	no	—
64	C₂×D₄×C₁₁	—	2	2	no	—
65	C₂×D₄×C₁₁	—	4	1	no	—
66	C₂×D₄×C₁₁	—	4	1	no	—
67	C₂×D₄×C₁₁	—	4	1	no	—
68	C₂×D₄×C₁₁	—	4	1	no	—
69	C₂×D₄×C₁₁	—	4	1	no	—
70	C₂²×C₄×C₁₁	—	4	1	no	—
71	C₂²×C₄×C₁₁	SD₁₆	1	2	yes	—
72	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
73	C₁₁×Q₁₆	—	2	4	no	—
74	C₁₁×Q₁₆	—	4	2	no	—
75	C₁₁×SD₁₆	—	4	4	no	—
76	C₁₁×SD₁₆	—	4	4	no	—
77	C₁₁×D₈	—	2	4	no	—
78	C₁₁×D₈	—	4	2	no	—
79	C₁₁×M₁₆	—	4	2	no	—
80	C₁₁×M₁₆	—	4	2	no	—
81	C₂×C₈×C₁₁	—	4	2	no	—
82	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₁₁	D₈	1	2	yes	—
85	C₂×C₈×C₁₁	SD₁₆	1	2	yes	—
86	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₁₁	#41	1	2	yes	—
90	C₁₁×C₂²⋊C₄	—	4	4	no	—
91	C₁₁×C₂²⋊C₄	—	4	4	no	—
92	C₄²×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
93	C₄²×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
94	C₄²×C₁₁	D₈	1	2	yes	—
95	C₁₁×C₁₆	—	2	4	no	—
96	C₂×#495	C₂×C₁₁	1	1	yes	characteristic
97	C₂×#362	C₂×C₁₁	1	2	yes	complemented
98	C₂×#361	C₂×C₁₁	1	2	yes	—
99	C₄×M₃₂	C₂×C₁₁	1	1	yes	characteristic, complemented
100	#1848	C₂×C₁₁	1	1	yes	characteristic, complemented
101	C₂³×C₁₁	—	4	2	no	—
102	C₂³×C₁₁	—	8	4	no	—
103	C₂³×C₁₁	—	8	1	no	—
104	C₂³×C₁₁	—	8	1	no	—
105	C₂³×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
106	Q₈×C₁₁	—	4	2	no	—
107	Q₈×C₁₁	—	4	2	no	—
108	Q₈×C₁₁	—	8	1	no	—
109	Q₈×C₁₁	—	8	1	no	—
110	D₄×C₁₁	—	4	2	no	—
111	D₄×C₁₁	—	4	2	no	—
112	D₄×C₁₁	—	8	1	no	—
113	D₄×C₁₁	—	8	1	no	—
114	C₂×C₄×C₁₁	—	4	2	no	—
115	C₂×C₄×C₁₁	—	4	2	no	—
116	C₂×C₄×C₁₁	—	8	1	no	—
117	C₂×C₄×C₁₁	—	8	1	no	—
118	C₂×C₄×C₁₁	#102	1	2	yes	—
119	C₂×C₄×C₁₁	#102	1	2	yes	—
120	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
122	C₂×C₄×C₁₁	#127	1	1	yes	characteristic
123	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
126	C₈×C₁₁	—	8	2	no	—
127	C₈×C₁₁	#102	1	2	yes	—
128	C₈×C₁₁	#102	1	2	yes	—
129	C₈×C₁₁	C₄×D₄	1	2	yes	—
130	C₈×C₁₁	#121	1	2	yes	—
131	C₂²×Q₁₆	C₂²×C₁₁	1	1	yes	characteristic
132	C₂²×SD₁₆	—	2	1	no	—
133	C₂²×D₈	C₂²×C₁₁	1	1	yes	characteristic
134	C₂×#124	—	2	1	no	—
135	C₂×M₃₂	C₂²×C₁₁	1	2	yes	—
136	#495	—	2	1	no	—
137	#495	—	2	1	no	—
138	#495	C₄×C₁₁	1	2	yes	—
139	#495	C₄×C₁₁	1	2	yes	—
140	C₂×#108	—	2	1	no	—
141	C₂×#108	C₂²×C₁₁	1	2	yes	—
142	#423	—	2	1	no	—
143	C₂×C₄×C₈	C₂²×C₁₁	1	1	yes	characteristic
144	#362	—	2	4	no	—
145	#361	—	2	4	no	—
146	#346	—	2	1	no	—
147	C₄×C₁₆	—	2	1	no	—
148	C₂²×C₁₁	—	2	2	no	—
149	C₂²×C₁₁	—	8	4	no	—
150	C₂²×C₁₁	—	8	2	no	—
151	C₂²×C₁₁	—	16	2	no	—
152	C₂²×C₁₁	—	16	2	no	—
153	C₂²×C₁₁	C₂×#102	1	1	yes	characteristic
154	C₂²×C₁₁	#486	1	1	yes	characteristic
155	C₂²×C₁₁	#492	1	1	yes	characteristic
156	C₄×C₁₁	—	2	2	no	—
157	C₄×C₁₁	—	2	2	no	—
158	C₄×C₁₁	—	8	2	no	—
159	C₄×C₁₁	#390	1	1	yes	characteristic
160	C₄×C₁₁	C₂×#102	1	1	yes	characteristic
161	C₄×C₁₁	#486	1	1	yes	characteristic
162	C₄×C₁₁	#493	1	1	yes	characteristic
163	C₂²×Q₈	—	2	1	no	—
164	C₂²×D₄	—	2	1	no	—
165	C₂×Q₁₆	—	2	2	no	—
166	C₂×Q₁₆	—	2	1	no	—
167	C₂×Q₁₆	—	2	2	no	—
168	C₂×Q₁₆	C₂×C₄×C₁₁	1	2	yes	—
169	C₂×SD₁₆	—	2	2	no	—
170	C₂×SD₁₆	—	2	2	no	—
171	C₂×SD₁₆	—	4	2	no	—
172	C₂×SD₁₆	—	4	2	no	—
173	C₂×D₈	—	2	2	no	—
174	C₂×D₈	—	2	1	no	—
175	C₂×D₈	—	2	2	no	—
176	C₂×D₈	C₂×C₄×C₁₁	1	2	yes	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂²×C₈	D₄×C₁₁	1	1	yes	characteristic
179	C₂²×C₈	C₂³×C₁₁	1	1	yes	characteristic
180	#124	—	2	2	no	—
181	#124	—	2	2	no	—
182	#124	—	4	2	no	—
183	C₂×C₂²⋊C₄	—	4	2	no	—
184	C₂×C₄²	D₄×C₁₁	1	1	yes	characteristic
185	M₃₂	—	2	2	no	—
186	M₃₂	D₄×C₁₁	1	4	yes	—
187	C₂×C₁₆	—	2	2	no	—
188	#108	—	2	2	no	—
189	#108	—	2	2	no	—
190	#108	—	4	2	no	—
191	C₄⋊C₈	—	4	2	no	—
192	C₄×C₈	C₂×C₄×C₁₁	1	1	yes	characteristic
193	C₄×C₈	C₂×C₄×C₁₁	1	1	yes	characteristic
194	C₄×C₈	D₄×C₁₁	1	1	yes	characteristic
195	C₄×C₈	D₄×C₁₁	1	1	yes	characteristic
196	C₂×C₁₁	—	2	1	no	—
197	C₂×C₁₁	—	2	1	no	—
198	C₂×C₁₁	—	16	2	no	—
199	C₂×C₁₁	#1957	1	1	yes	characteristic
200	C₂×C₁₁	#2148	1	2	yes	—
201	C₂⁴	—	4	1	no	—
202	C₂×Q₈	—	2	2	no	—
203	C₂×Q₈	—	4	1	no	—
204	C₂×Q₈	—	4	1	no	—
205	C₂×Q₈	—	4	1	no	—
206	C₂×Q₈	—	4	1	no	—
207	C₂×Q₈	—	4	1	no	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	—	4	1	no	—
210	C₂×D₄	—	4	1	no	—
211	C₂×D₄	—	4	1	no	—
212	C₂×D₄	—	4	1	no	—
213	C₂×D₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	C₁₁×SD₁₆	1	2	yes	—
216	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
217	Q₁₆	—	2	4	no	—
218	Q₁₆	—	4	2	no	—
219	SD₁₆	—	4	4	no	—
220	SD₁₆	—	4	4	no	—
221	D₈	—	2	4	no	—
222	D₈	—	4	2	no	—
223	M₁₆	—	4	2	no	—
224	M₁₆	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	C₁₁×C₂²⋊C₄	1	1	yes	characteristic
227	C₂×C₈	C₁₁×C₂²⋊C₄	1	1	yes	characteristic
228	C₂×C₈	C₁₁×D₈	1	2	yes	—
229	C₂×C₈	C₁₁×SD₁₆	1	2	yes	—
230	C₂×C₈	C₂²×C₄×C₁₁	1	1	yes	characteristic
231	C₂×C₈	C₂×D₄×C₁₁	1	2	yes	—
232	C₂×C₈	C₂×D₄×C₁₁	1	1	yes	characteristic
233	C₂×C₈	C₁₁×#41	1	2	yes	—
234	C₂²⋊C₄	—	4	4	no	—
235	C₂²⋊C₄	—	4	4	no	—
236	C₄²	C₁₁×C₂²⋊C₄	1	1	yes	characteristic
237	C₄²	C₁₁×C₂²⋊C₄	1	1	yes	characteristic
238	C₄²	C₁₁×D₈	1	2	yes	—
239	C₁₆	—	2	4	no	—
240	C₁₁	#13214	1	1	yes	characteristic, direct product factor, Sylow
241	C₂³	—	4	2	no	—
242	C₂³	—	8	4	no	—
243	C₂³	—	8	1	no	—
244	C₂³	—	8	1	no	—
245	C₂³	C₂×C₁₁×SD₁₆	1	1	yes	characteristic
246	Q₈	—	4	2	no	—
247	Q₈	—	4	2	no	—
248	Q₈	—	8	1	no	—
249	Q₈	—	8	1	no	—
250	D₄	—	4	2	no	—
251	D₄	—	4	2	no	—
252	D₄	—	8	1	no	—
253	D₄	—	8	1	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	8	1	no	—
257	C₂×C₄	—	8	1	no	—
258	C₂×C₄	C₁₁×#102	1	2	yes	—
259	C₂×C₄	C₁₁×#102	1	2	yes	—
260	C₂×C₄	C₂×C₁₁×C₂²⋊C₄	1	1	yes	characteristic
261	C₂×C₄	C₁₁×#124	1	1	yes	characteristic
262	C₂×C₄	C₁₁×#127	1	1	yes	characteristic
263	C₂×C₄	C₂×C₁₁×D₈	1	1	yes	characteristic
264	C₂×C₄	C₂×C₁₁×D₈	1	1	yes	characteristic
265	C₂×C₄	C₂×C₁₁×SD₁₆	1	1	yes	characteristic
266	C₈	—	8	2	no	—
267	C₈	C₁₁×#102	1	2	yes	—
268	C₈	C₁₁×#102	1	2	yes	—
269	C₈	C₄×D₄×C₁₁	1	2	yes	—
270	C₈	C₁₁×#121	1	2	yes	—
271	C₂²	—	2	2	no	—
272	C₂²	—	8	4	no	—
273	C₂²	—	8	2	no	—
274	C₂²	—	16	2	no	—
275	C₂²	—	16	2	no	—
276	C₂²	C₂×C₁₁×#102	1	1	yes	characteristic
277	C₂²	C₁₁×#486	1	1	yes	characteristic
278	C₂²	C₁₁×#492	1	1	yes	characteristic
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	8	2	no	—
282	C₄	C₁₁×#390	1	1	yes	characteristic
283	C₄	C₂×C₁₁×#102	1	1	yes	characteristic
284	C₄	C₁₁×#486	1	1	yes	characteristic
285	C₄	C₁₁×#493	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	—	16	2	no	—
289	C₂	C₁₁×#1957	1	1	yes	characteristic
290	C₂	C₁₁×#2148	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₁
prime radical	C₂²×C₁₁×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₁ ◁ C₂×C₄²×C₁₁ ◁ C₂×C₄×C₈×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₁×#362	C₂	semidirect
4. C₄×C₁₁×M₃₂	C₂	semidirect
5. C₁₁×#1848	C₂	semidirect
56. #13214	C₁₁	direct (N × H)
97. C₂×#362	C₂×C₁₁	semidirect
99. C₄×M₃₂	C₂×C₁₁	semidirect
100. #1848	C₂×C₁₁	semidirect
240. C₁₁	#13214	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×#495	2
C₂×C₁₁×#362	2
C₂×C₁₁×#361	2
C₄×C₁₁×M₃₂	2
C₁₁×#1848	2
#13214	11

Subgroups — C11×#13214 #14590407

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#13214 #14590407