Subgroups — C₁₁×#3154 #13065992

Order 1408.

564 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×#137	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₁×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₁×#135	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₁₁×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₁×#536	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₁×#535	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₁×#497	C₂	1	2	yes	maximal, complemented
8	C₁₁×#496	C₂	1	2	yes	maximal, complemented
9	C₁₁×#495	C₂	1	2	yes	maximal, complemented
10	C₁₁×#492	C₂	1	2	yes	maximal, complemented
11	C₁₁×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₁×#41	C₂²	1	1	yes	characteristic
13	C₂×C₁₁×#41	C₂²	1	1	yes	characteristic
14	C₂²×Q₈×C₁₁	C₂²	1	1	yes	characteristic
15	C₂²×D₄×C₁₁	C₂²	1	1	yes	characteristic
16	C₁₁×#137	—	2	4	no	—
17	C₁₁×AGL(1,ℤ/8)	—	2	4	no	—
18	C₁₁×#135	—	2	4	no	—
19	C₂×C₁₁×Q₁₆	C₂²	1	2	yes	complemented
20	C₂×C₁₁×Q₁₆	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₁₁×SD₁₆	—	2	4	no	—
22	C₂×C₁₁×SD₁₆	C₂²	1	2	yes	—
23	C₂×C₁₁×SD₁₆	C₂²	1	2	yes	—
24	C₂×C₁₁×SD₁₆	C₂²	1	2	yes	—
25	C₂×C₁₁×SD₁₆	C₂²	1	1	yes	characteristic, complemented
26	C₂×C₁₁×SD₁₆	C₂²	1	2	yes	complemented
27	C₂×C₁₁×SD₁₆	C₂²	1	1	yes	characteristic, complemented
28	C₂×C₁₁×D₈	C₂²	1	2	yes	—
29	C₂×C₁₁×D₈	C₂²	1	1	yes	characteristic, complemented
30	C₂×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
31	C₂²×C₈×C₁₁	C₂²	1	1	yes	characteristic
32	C₁₁×C₄⋊Q₈	C₂²	1	2	yes	complemented
33	C₁₁×#127	C₂²	1	2	yes	—
34	C₁₁×#124	C₂²	1	2	yes	—
35	C₁₁×#124	C₂²	1	2	yes	complemented
36	C₁₁×#123	—	2	2	no	—
37	C₁₁×C₂²⋊Q₈	—	2	2	no	—
38	C₁₁×#121	—	2	2	no	—
39	C₁₁×#120	—	2	2	no	—
40	C₁₁×#117	C₂²	1	1	yes	characteristic
41	C₁₁×#97	C₂²	1	2	yes	complemented
42	C₄×C₈×C₁₁	C₂²	1	2	yes	complemented
43	C₂⁴×C₁₁	—	2	1	no	—
44	C₁₁×#41	—	2	2	no	—
45	C₁₁×#41	—	2	2	no	—
46	C₁₁×#41	—	2	2	no	—
47	C₁₁×#41	—	2	2	no	—
48	C₂×Q₈×C₁₁	—	2	1	no	—
49	C₂×Q₈×C₁₁	—	2	2	no	—
50	C₂×Q₈×C₁₁	—	2	2	no	—
51	C₂×Q₈×C₁₁	C₂³	1	1	yes	characteristic
52	C₂×Q₈×C₁₁	C₂³	1	2	yes	—
53	C₂×Q₈×C₁₁	C₂³	1	1	yes	characteristic
54	C₂×D₄×C₁₁	—	2	1	no	—
55	C₂×D₄×C₁₁	—	2	1	no	—
56	C₂×D₄×C₁₁	—	2	2	no	—
57	C₂×D₄×C₁₁	—	2	2	no	—
58	C₂×D₄×C₁₁	—	2	1	no	—
59	C₂×D₄×C₁₁	—	2	4	no	—
60	C₂×D₄×C₁₁	C₂³	1	2	yes	—
61	C₂×D₄×C₁₁	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₁₁	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₁	—	2	1	no	—
64	C₂²×C₄×C₁₁	—	2	1	no	—
65	C₂²×C₄×C₁₁	—	2	1	no	—
66	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
67	C₁₁×Q₁₆	—	2	2	no	—
68	C₁₁×Q₁₆	—	2	4	no	—
69	C₁₁×SD₁₆	—	2	4	no	—
70	C₁₁×SD₁₆	—	2	4	no	—
71	C₁₁×SD₁₆	—	2	4	no	—
72	C₁₁×SD₁₆	—	2	2	no	—
73	C₁₁×SD₁₆	—	2	2	no	—
74	C₁₁×SD₁₆	—	2	4	no	—
75	C₁₁×D₈	—	2	4	no	—
76	C₁₁×D₈	—	2	2	no	—
77	C₁₁×M₁₆	D₄	1	4	yes	complemented
78	C₂×C₈×C₁₁	D₄	1	4	yes	complemented
79	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₁₁	C₂³	1	2	yes	—
81	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
82	C₁₁×C₄⋊C₄	—	2	4	no	—
83	C₁₁×C₄⋊C₄	D₄	1	2	yes	—
84	C₁₁×C₂²⋊C₄	—	2	4	no	—
85	C₁₁×C₂²⋊C₄	—	2	4	no	—
86	C₁₁×C₂²⋊C₄	D₄	1	2	yes	—
87	C₄²×C₁₁	C₂³	1	2	yes	—
88	#3154	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₁₁	—	2	1	no	—
90	C₂³×C₁₁	—	2	2	no	—
91	C₂³×C₁₁	—	2	1	no	—
92	C₂³×C₁₁	—	4	2	no	—
93	C₂³×C₁₁	—	8	2	no	—
94	C₂³×C₁₁	C₂×D₄	1	1	yes	characteristic
95	Q₈×C₁₁	—	2	2	no	—
96	Q₈×C₁₁	—	2	1	no	—
97	Q₈×C₁₁	—	2	2	no	—
98	Q₈×C₁₁	—	2	1	no	—
99	Q₈×C₁₁	—	2	1	no	—
100	Q₈×C₁₁	—	2	1	no	—
101	Q₈×C₁₁	—	4	2	no	—
102	D₄×C₁₁	—	2	1	no	—
103	D₄×C₁₁	—	2	2	no	—
104	D₄×C₁₁	—	2	1	no	—
105	D₄×C₁₁	—	2	1	no	—
106	D₄×C₁₁	—	2	2	no	—
107	D₄×C₁₁	—	2	1	no	—
108	D₄×C₁₁	—	4	2	no	—
109	D₄×C₁₁	—	4	2	no	—
110	D₄×C₁₁	—	4	2	no	—
111	D₄×C₁₁	—	8	4	no	—
112	C₂×C₄×C₁₁	—	2	1	no	—
113	C₂×C₄×C₁₁	—	2	2	no	—
114	C₂×C₄×C₁₁	—	2	1	no	—
115	C₂×C₄×C₁₁	—	4	2	no	—
116	C₂×C₄×C₁₁	—	4	2	no	—
117	C₂×C₄×C₁₁	—	4	2	no	—
118	C₂×C₄×C₁₁	C₂×D₄	1	4	yes	—
119	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
121	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
122	C₂×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
123	C₈×C₁₁	C₂×D₄	1	2	yes	—
124	C₈×C₁₁	C₂×D₄	1	2	yes	—
125	C₈×C₁₁	C₂×D₄	1	4	yes	—
126	C₂×#137	C₂×C₁₁	1	1	yes	characteristic, complemented
127	C₂×AGL(1,ℤ/8)	C₂×C₁₁	1	1	yes	characteristic, complemented
128	C₂×#135	C₂×C₁₁	1	1	yes	characteristic, complemented
129	C₂²×SD₁₆	C₂×C₁₁	1	1	yes	characteristic, complemented
130	#536	C₂×C₁₁	1	1	yes	characteristic, complemented
131	#535	C₂×C₁₁	1	1	yes	characteristic, complemented
132	#497	C₂×C₁₁	1	2	yes	complemented
133	#496	C₂×C₁₁	1	2	yes	complemented
134	#495	C₂×C₁₁	1	2	yes	complemented
135	#492	C₂×C₁₁	1	2	yes	complemented
136	#405	C₂×C₁₁	1	1	yes	characteristic, complemented
137	C₂²×C₁₁	—	2	2	no	—
138	C₂²×C₁₁	—	4	1	no	—
139	C₂²×C₁₁	—	4	1	no	—
140	C₂²×C₁₁	—	4	2	no	—
141	C₂²×C₁₁	—	4	2	no	—
142	C₂²×C₁₁	—	4	4	no	—
143	C₂²×C₁₁	—	4	2	no	—
144	C₂²×C₁₁	—	8	4	no	—
145	C₂²×C₁₁	#127	1	2	yes	—
146	C₂²×C₁₁	C₂²×D₄	1	1	yes	characteristic
147	C₄×C₁₁	—	2	4	no	—
148	C₄×C₁₁	—	4	1	no	—
149	C₄×C₁₁	—	4	1	no	—
150	C₄×C₁₁	—	4	2	no	—
151	C₄×C₁₁	#127	1	2	yes	—
152	C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
153	C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
154	C₂×#41	C₂²×C₁₁	1	1	yes	characteristic
155	C₂×#41	C₂²×C₁₁	1	1	yes	characteristic
156	C₂²×Q₈	C₂²×C₁₁	1	1	yes	characteristic
157	C₂²×D₄	C₂²×C₁₁	1	1	yes	characteristic
158	#137	—	2	4	no	—
159	AGL(1,ℤ/8)	—	2	4	no	—
160	#135	—	2	4	no	—
161	C₂×Q₁₆	C₂²×C₁₁	1	1	yes	characteristic, complemented
162	C₂×Q₁₆	C₂²×C₁₁	1	2	yes	complemented
163	C₂×SD₁₆	—	2	4	no	—
164	C₂×SD₁₆	C₂²×C₁₁	1	2	yes	—
165	C₂×SD₁₆	C₂²×C₁₁	1	2	yes	—
166	C₂×SD₁₆	C₂²×C₁₁	1	2	yes	—
167	C₂×SD₁₆	C₂²×C₁₁	1	1	yes	characteristic, complemented
168	C₂×SD₁₆	C₂²×C₁₁	1	1	yes	characteristic, complemented
169	C₂×SD₁₆	C₂²×C₁₁	1	2	yes	complemented
170	C₂×D₈	C₂²×C₁₁	1	2	yes	—
171	C₂×D₈	C₂²×C₁₁	1	1	yes	characteristic, complemented
172	C₂×M₁₆	C₂²×C₁₁	1	1	yes	characteristic
173	C₂²×C₈	C₂²×C₁₁	1	1	yes	characteristic
174	C₄⋊Q₈	C₂²×C₁₁	1	2	yes	complemented
175	#127	C₂²×C₁₁	1	2	yes	—
176	#124	C₂²×C₁₁	1	2	yes	—
177	#124	C₂²×C₁₁	1	2	yes	complemented
178	#123	—	2	2	no	—
179	C₂²⋊Q₈	—	2	2	no	—
180	#121	—	2	2	no	—
181	#120	—	2	2	no	—
182	#117	C₂²×C₁₁	1	1	yes	characteristic
183	#97	C₂²×C₁₁	1	2	yes	complemented
184	C₄×C₈	C₂²×C₁₁	1	2	yes	complemented
185	C₂×C₁₁	—	2	2	no	—
186	C₂×C₁₁	—	8	1	no	—
187	C₂×C₁₁	—	8	1	no	—
188	C₂×C₁₁	—	8	2	no	—
189	C₂×C₁₁	C₂×#127	1	1	yes	characteristic
190	C₂×C₁₁	#577	1	2	yes	—
191	C₂⁴	—	2	1	no	—
192	#41	—	2	2	no	—
193	#41	—	2	2	no	—
194	#41	—	2	2	no	—
195	#41	—	2	2	no	—
196	C₂×Q₈	—	2	1	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×Q₈	—	2	2	no	—
199	C₂×Q₈	C₂³×C₁₁	1	1	yes	characteristic
200	C₂×Q₈	C₂³×C₁₁	1	2	yes	—
201	C₂×Q₈	C₂³×C₁₁	1	1	yes	characteristic
202	C₂×D₄	—	2	1	no	—
203	C₂×D₄	—	2	1	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	2	1	no	—
207	C₂×D₄	—	2	4	no	—
208	C₂×D₄	C₂³×C₁₁	1	2	yes	—
209	C₂×D₄	C₂³×C₁₁	1	1	yes	characteristic
210	C₂×D₄	C₂³×C₁₁	1	1	yes	characteristic
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	C₂³×C₁₁	1	1	yes	characteristic
215	Q₁₆	—	2	2	no	—
216	Q₁₆	—	2	4	no	—
217	SD₁₆	—	2	4	no	—
218	SD₁₆	—	2	4	no	—
219	SD₁₆	—	2	4	no	—
220	SD₁₆	—	2	2	no	—
221	SD₁₆	—	2	2	no	—
222	SD₁₆	—	2	4	no	—
223	D₈	—	2	4	no	—
224	D₈	—	2	2	no	—
225	M₁₆	D₄×C₁₁	1	4	yes	complemented
226	C₂×C₈	D₄×C₁₁	1	4	yes	complemented
227	C₂×C₈	C₂³×C₁₁	1	1	yes	characteristic
228	C₂×C₈	C₂³×C₁₁	1	2	yes	—
229	C₂×C₈	C₂³×C₁₁	1	1	yes	characteristic
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	D₄×C₁₁	1	2	yes	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₂²⋊C₄	D₄×C₁₁	1	2	yes	—
235	C₄²	C₂³×C₁₁	1	2	yes	—
236	C₁₁	#3154	1	1	yes	characteristic, direct product factor, Sylow
237	C₂³	—	2	1	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	2	1	no	—
240	C₂³	—	4	2	no	—
241	C₂³	—	8	2	no	—
242	C₂³	C₂×D₄×C₁₁	1	1	yes	characteristic
243	Q₈	—	2	2	no	—
244	Q₈	—	2	1	no	—
245	Q₈	—	2	2	no	—
246	Q₈	—	2	1	no	—
247	Q₈	—	2	1	no	—
248	Q₈	—	2	1	no	—
249	Q₈	—	4	2	no	—
250	D₄	—	2	1	no	—
251	D₄	—	2	2	no	—
252	D₄	—	2	1	no	—
253	D₄	—	2	1	no	—
254	D₄	—	2	2	no	—
255	D₄	—	2	1	no	—
256	D₄	—	4	2	no	—
257	D₄	—	4	2	no	—
258	D₄	—	4	2	no	—
259	D₄	—	8	4	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	C₂×D₄×C₁₁	1	4	yes	—
267	C₂×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
268	C₂×C₄	C₂×D₄×C₁₁	1	2	yes	—
269	C₂×C₄	C₂×D₄×C₁₁	1	2	yes	—
270	C₂×C₄	C₂⁴×C₁₁	1	1	yes	characteristic
271	C₈	C₂×D₄×C₁₁	1	2	yes	—
272	C₈	C₂×D₄×C₁₁	1	2	yes	—
273	C₈	C₂×D₄×C₁₁	1	4	yes	—
274	C₂²	—	2	2	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	4	1	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	4	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	8	4	no	—
282	C₂²	C₁₁×#127	1	2	yes	—
283	C₂²	C₂²×D₄×C₁₁	1	1	yes	characteristic
284	C₄	—	2	4	no	—
285	C₄	—	4	1	no	—
286	C₄	—	4	1	no	—
287	C₄	—	4	2	no	—
288	C₄	C₁₁×#127	1	2	yes	—
289	C₄	C₂²×D₄×C₁₁	1	1	yes	characteristic
290	C₄	C₂²×D₄×C₁₁	1	1	yes	characteristic
291	C₂	—	2	2	no	—
292	C₂	—	8	2	no	—
293	C₂	—	8	1	no	—
294	C₂	—	8	1	no	—
295	C₂	C₂×C₁₁×#127	1	1	yes	characteristic
296	C₂	C₁₁×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₁ ◁ C₁₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₁×#137	C₂	semidirect
2. C₂×C₁₁×AGL(1,ℤ/8)	C₂	semidirect
3. C₂×C₁₁×#135	C₂	semidirect
4. C₂²×C₁₁×SD₁₆	C₂	semidirect
5. C₁₁×#536	C₂	semidirect
6. C₁₁×#535	C₂	semidirect
7. C₁₁×#497	C₂	semidirect
8. C₁₁×#496	C₂	semidirect
9. C₁₁×#495	C₂	semidirect
10. C₁₁×#492	C₂	semidirect
11. C₁₁×#405	C₂	semidirect
19. C₂×C₁₁×Q₁₆	C₂²	semidirect
20. C₂×C₁₁×Q₁₆	C₂²	semidirect
25. C₂×C₁₁×SD₁₆	C₂²	semidirect
26. C₂×C₁₁×SD₁₆	C₂²	semidirect
27. C₂×C₁₁×SD₁₆	C₂²	semidirect
29. C₂×C₁₁×D₈	C₂²	semidirect
32. C₁₁×C₄⋊Q₈	C₂²	semidirect
35. C₁₁×#124	C₂²	semidirect
41. C₁₁×#97	C₂²	semidirect
42. C₄×C₈×C₁₁	C₂²	semidirect
77. C₁₁×M₁₆	D₄	semidirect
78. C₂×C₈×C₁₁	D₄	semidirect
88. #3154	C₁₁	direct (N × H)
126. C₂×#137	C₂×C₁₁	semidirect
127. C₂×AGL(1,ℤ/8)	C₂×C₁₁	semidirect
128. C₂×#135	C₂×C₁₁	semidirect
129. C₂²×SD₁₆	C₂×C₁₁	semidirect
130. #536	C₂×C₁₁	semidirect
131. #535	C₂×C₁₁	semidirect
132. #497	C₂×C₁₁	semidirect
133. #496	C₂×C₁₁	semidirect
134. #495	C₂×C₁₁	semidirect
135. #492	C₂×C₁₁	semidirect
136. #405	C₂×C₁₁	semidirect
161. C₂×Q₁₆	C₂²×C₁₁	semidirect
162. C₂×Q₁₆	C₂²×C₁₁	semidirect
167. C₂×SD₁₆	C₂²×C₁₁	semidirect
168. C₂×SD₁₆	C₂²×C₁₁	semidirect
169. C₂×SD₁₆	C₂²×C₁₁	semidirect
171. C₂×D₈	C₂²×C₁₁	semidirect
174. C₄⋊Q₈	C₂²×C₁₁	semidirect
177. #124	C₂²×C₁₁	semidirect
183. #97	C₂²×C₁₁	semidirect
184. C₄×C₈	C₂²×C₁₁	semidirect
225. M₁₆	D₄×C₁₁	semidirect
226. C₂×C₈	D₄×C₁₁	semidirect
236. C₁₁	#3154	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×#137	2
C₂×C₁₁×AGL(1,ℤ/8)	2
C₂×C₁₁×#135	2
C₂²×C₁₁×SD₁₆	2
C₁₁×#536	2
C₁₁×#535	2
C₁₁×#497	2
C₁₁×#496	2
C₁₁×#495	2
C₁₁×#492	2
C₁₁×#405	2
#3154	11

Subgroups — C11×#3154 #13065992

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#3154 #13065992