Subgroups — C₁₁×#3008 #13065846

Order 1408.

784 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₁×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₁₁×D₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₁×#535	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₁×#449	C₂	1	4	yes	maximal, complemented
6	C₁₁×#447	C₂	1	4	yes	maximal, complemented
7	C₁₁×#418	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₁×#416	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₁×#408	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₁×H(5,𝔽₂)	—	2	4	no	—
11	C₁₁×H(5,𝔽₂)	—	2	4	no	—
12	C₂×C₁₁×#41	—	2	1	no	—
13	C₂×C₁₁×#41	C₂²	1	1	yes	characteristic
14	C₂×C₁₁×#41	C₂²	1	1	yes	characteristic
15	C₂×C₁₁×#41	C₂²	1	2	yes	complemented
16	C₂²×D₄×C₁₁	—	2	1	no	—
17	C₂²×D₄×C₁₁	—	2	2	no	—
18	C₂²×D₄×C₁₁	C₂²	1	1	yes	characteristic
19	C₂²×D₄×C₁₁	C₂²	1	1	yes	characteristic, complemented
20	C₂²×D₄×C₁₁	C₂²	1	2	yes	complemented
21	C₁₁×AGL(1,ℤ/8)	—	2	4	no	—
22	C₂×C₁₁×SD₁₆	C₂²	1	2	yes	complemented
23	C₂×C₁₁×D₈	—	2	4	no	—
24	C₂×C₁₁×D₈	C₂²	1	4	yes	complemented
25	C₂×C₁₁×D₈	C₂²	1	2	yes	complemented
26	C₂×C₁₁×M₁₆	C₂²	1	1	yes	characteristic, complemented
27	C₂²×C₈×C₁₁	C₂²	1	1	yes	characteristic, complemented
28	C₁₁×#127	—	2	1	no	—
29	C₁₁×#124	—	2	1	no	—
30	C₁₁×#121	C₂²	1	4	yes	complemented
31	C₁₁×#120	—	2	2	no	—
32	C₁₁×#117	C₂²	1	1	yes	characteristic, complemented
33	C₁₁×#103	C₂²	1	2	yes	complemented
34	C₁₁×#102	C₂²	1	2	yes	complemented
35	C₁₁×#102	C₂²	1	4	yes	complemented
36	C₁₁×C₂²⋊C₈	C₂²	1	4	yes	complemented
37	C₂⁴×C₁₁	—	2	1	no	—
38	C₂⁴×C₁₁	—	2	2	no	—
39	C₂⁴×C₁₁	—	2	1	no	—
40	C₁₁×#41	—	2	8	no	—
41	C₁₁×#41	—	2	4	no	—
42	C₁₁×#41	—	2	4	no	—
43	C₁₁×#41	—	2	2	no	—
44	C₁₁×#41	—	2	2	no	—
45	C₁₁×#41	—	2	2	no	—
46	C₁₁×#41	D₄	1	4	yes	complemented
47	C₂×Q₈×C₁₁	D₄	1	1	yes	characteristic
48	C₂×Q₈×C₁₁	C₂³	1	1	yes	characteristic
49	C₂×D₄×C₁₁	—	2	16	no	—
50	C₂×D₄×C₁₁	—	2	8	no	—
51	C₂×D₄×C₁₁	—	2	1	no	—
52	C₂×D₄×C₁₁	—	2	2	no	—
53	C₂×D₄×C₁₁	—	2	4	no	—
54	C₂×D₄×C₁₁	—	2	4	no	—
55	C₂×D₄×C₁₁	—	2	4	no	—
56	C₂×D₄×C₁₁	—	2	1	no	—
57	C₂×D₄×C₁₁	—	2	2	no	—
58	C₂×D₄×C₁₁	—	2	2	no	—
59	C₂×D₄×C₁₁	—	2	2	no	—
60	C₂×D₄×C₁₁	—	2	2	no	—
61	C₂×D₄×C₁₁	—	2	1	no	—
62	C₂×D₄×C₁₁	D₄	1	2	yes	—
63	C₂×D₄×C₁₁	D₄	1	1	yes	characteristic
64	C₂×D₄×C₁₁	D₄	1	4	yes	complemented
65	C₂×D₄×C₁₁	C₂³	1	2	yes	—
66	C₂×D₄×C₁₁	C₂³	1	2	yes	—
67	C₂×D₄×C₁₁	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₁₁	—	2	1	no	—
69	C₂²×C₄×C₁₁	—	2	2	no	—
70	C₂²×C₄×C₁₁	C₂³	1	2	yes	—
71	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
72	C₁₁×SD₁₆	—	2	4	no	—
73	C₁₁×D₈	—	2	8	no	—
74	C₁₁×D₈	—	2	4	no	—
75	C₁₁×M₁₆	—	2	2	no	—
76	C₂×C₈×C₁₁	—	2	2	no	—
77	C₂×C₈×C₁₁	C₂³	1	2	yes	—
78	C₂×C₈×C₁₁	C₂³	1	2	yes	—
79	C₁₁×C₄⋊C₄	C₂³	1	2	yes	—
80	C₁₁×C₂²⋊C₄	—	2	1	no	—
81	C₁₁×C₂²⋊C₄	—	2	4	no	—
82	C₄²×C₁₁	—	2	1	no	—
83	#3008	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂³×C₁₁	—	2	8	no	—
85	C₂³×C₁₁	—	2	1	no	—
86	C₂³×C₁₁	—	2	2	no	—
87	C₂³×C₁₁	—	2	4	no	—
88	C₂³×C₁₁	—	2	2	no	—
89	C₂³×C₁₁	—	4	2	no	—
90	C₂³×C₁₁	—	8	2	no	—
91	C₂³×C₁₁	—	8	2	no	—
92	C₂³×C₁₁	—	8	4	no	—
93	C₂³×C₁₁	C₂×D₄	1	2	yes	—
94	C₂³×C₁₁	C₂×D₄	1	1	yes	characteristic
95	Q₈×C₁₁	—	2	2	no	—
96	Q₈×C₁₁	—	2	1	no	—
97	Q₈×C₁₁	C₂×D₄	1	2	yes	—
98	D₄×C₁₁	—	2	8	no	—
99	D₄×C₁₁	—	2	8	no	—
100	D₄×C₁₁	—	2	4	no	—
101	D₄×C₁₁	—	2	4	no	—
102	D₄×C₁₁	—	2	2	no	—
103	D₄×C₁₁	—	2	4	no	—
104	D₄×C₁₁	—	2	2	no	—
105	D₄×C₁₁	—	2	1	no	—
106	D₄×C₁₁	—	2	2	no	—
107	D₄×C₁₁	—	2	2	no	—
108	D₄×C₁₁	—	4	2	no	—
109	D₄×C₁₁	—	8	4	no	—
110	D₄×C₁₁	C₂×D₄	1	2	yes	—
111	D₄×C₁₁	C₂×D₄	1	4	yes	—
112	C₂×C₄×C₁₁	—	2	8	no	—
113	C₂×C₄×C₁₁	—	2	1	no	—
114	C₂×C₄×C₁₁	—	2	4	no	—
115	C₂×C₄×C₁₁	—	2	2	no	—
116	C₂×C₄×C₁₁	—	2	2	no	—
117	C₂×C₄×C₁₁	—	2	2	no	—
118	C₂×C₄×C₁₁	C₂×D₄	1	2	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	—	2	2	no	—
124	C₈×C₁₁	—	2	2	no	—
125	C₂×H(5,𝔽₂)	C₂×C₁₁	1	1	yes	characteristic, complemented
126	C₂×AGL(1,ℤ/8)	C₂×C₁₁	1	1	yes	characteristic, complemented
127	C₂²×D₈	C₂×C₁₁	1	1	yes	characteristic, complemented
128	#535	C₂×C₁₁	1	1	yes	characteristic, complemented
129	#449	C₂×C₁₁	1	4	yes	complemented
130	#447	C₂×C₁₁	1	4	yes	complemented
131	#418	C₂×C₁₁	1	1	yes	characteristic, complemented
132	#416	C₂×C₁₁	1	1	yes	characteristic, complemented
133	#408	C₂×C₁₁	1	1	yes	characteristic, complemented
134	C₂²×C₁₁	—	2	2	no	—
135	C₂²×C₁₁	—	2	2	no	—
136	C₂²×C₁₁	—	2	4	no	—
137	C₂²×C₁₁	—	2	2	no	—
138	C₂²×C₁₁	—	2	4	no	—
139	C₂²×C₁₁	—	4	4	no	—
140	C₂²×C₁₁	—	4	2	no	—
141	C₂²×C₁₁	—	4	2	no	—
142	C₂²×C₁₁	—	4	4	no	—
143	C₂²×C₁₁	—	8	4	no	—
144	C₂²×C₁₁	—	8	4	no	—
145	C₂²×C₁₁	—	8	8	no	—
146	C₂²×C₁₁	#120	1	2	yes	—
147	C₂²×C₁₁	C₂²×D₄	1	1	yes	characteristic
148	C₄×C₁₁	—	2	2	no	—
149	C₄×C₁₁	—	2	2	no	—
150	C₄×C₁₁	—	4	2	no	—
151	C₄×C₁₁	#120	1	2	yes	—
152	C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
153	C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
154	H(5,𝔽₂)	—	2	4	no	—
155	H(5,𝔽₂)	—	2	4	no	—
156	C₂×#41	—	2	1	no	—
157	C₂×#41	C₂²×C₁₁	1	1	yes	characteristic
158	C₂×#41	C₂²×C₁₁	1	1	yes	characteristic
159	C₂×#41	C₂²×C₁₁	1	2	yes	complemented
160	C₂²×D₄	—	2	1	no	—
161	C₂²×D₄	—	2	2	no	—
162	C₂²×D₄	C₂²×C₁₁	1	1	yes	characteristic
163	C₂²×D₄	C₂²×C₁₁	1	1	yes	characteristic, complemented
164	C₂²×D₄	C₂²×C₁₁	1	2	yes	complemented
165	AGL(1,ℤ/8)	—	2	4	no	—
166	C₂×SD₁₆	C₂²×C₁₁	1	2	yes	complemented
167	C₂×D₈	—	2	4	no	—
168	C₂×D₈	C₂²×C₁₁	1	4	yes	complemented
169	C₂×D₈	C₂²×C₁₁	1	2	yes	complemented
170	C₂×M₁₆	C₂²×C₁₁	1	1	yes	characteristic, complemented
171	C₂²×C₈	C₂²×C₁₁	1	1	yes	characteristic, complemented
172	#127	—	2	1	no	—
173	#124	—	2	1	no	—
174	#121	C₂²×C₁₁	1	4	yes	complemented
175	#120	—	2	2	no	—
176	#117	C₂²×C₁₁	1	1	yes	characteristic, complemented
177	#103	C₂²×C₁₁	1	2	yes	complemented
178	#102	C₂²×C₁₁	1	2	yes	complemented
179	#102	C₂²×C₁₁	1	4	yes	complemented
180	C₂²⋊C₈	C₂²×C₁₁	1	4	yes	complemented
181	C₂×C₁₁	—	2	2	no	—
182	C₂×C₁₁	—	4	2	no	—
183	C₂×C₁₁	—	4	4	no	—
184	C₂×C₁₁	—	4	2	no	—
185	C₂×C₁₁	—	8	2	no	—
186	C₂×C₁₁	C₂×#120	1	1	yes	characteristic
187	C₂×C₁₁	#576	1	2	yes	—
188	C₂⁴	—	2	1	no	—
189	C₂⁴	—	2	2	no	—
190	C₂⁴	—	2	1	no	—
191	#41	—	2	8	no	—
192	#41	—	2	4	no	—
193	#41	—	2	4	no	—
194	#41	—	2	2	no	—
195	#41	—	2	2	no	—
196	#41	—	2	2	no	—
197	#41	D₄×C₁₁	1	4	yes	complemented
198	C₂×Q₈	D₄×C₁₁	1	1	yes	characteristic
199	C₂×Q₈	C₂³×C₁₁	1	1	yes	characteristic
200	C₂×D₄	—	2	16	no	—
201	C₂×D₄	—	2	8	no	—
202	C₂×D₄	—	2	1	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	2	4	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	2	4	no	—
207	C₂×D₄	—	2	1	no	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	—	2	1	no	—
213	C₂×D₄	D₄×C₁₁	1	2	yes	—
214	C₂×D₄	D₄×C₁₁	1	1	yes	characteristic
215	C₂×D₄	D₄×C₁₁	1	4	yes	complemented
216	C₂×D₄	C₂³×C₁₁	1	2	yes	—
217	C₂×D₄	C₂³×C₁₁	1	2	yes	—
218	C₂×D₄	C₂³×C₁₁	1	1	yes	characteristic
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	C₂³×C₁₁	1	2	yes	—
222	C₂²×C₄	C₂³×C₁₁	1	1	yes	characteristic
223	SD₁₆	—	2	4	no	—
224	D₈	—	2	8	no	—
225	D₈	—	2	4	no	—
226	M₁₆	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	C₂³×C₁₁	1	2	yes	—
229	C₂×C₈	C₂³×C₁₁	1	2	yes	—
230	C₄⋊C₄	C₂³×C₁₁	1	2	yes	—
231	C₂²⋊C₄	—	2	1	no	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₄²	—	2	1	no	—
234	C₁₁	#3008	1	1	yes	characteristic, direct product factor, Sylow
235	C₂³	—	2	8	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	2	no	—
238	C₂³	—	2	4	no	—
239	C₂³	—	2	2	no	—
240	C₂³	—	4	2	no	—
241	C₂³	—	8	2	no	—
242	C₂³	—	8	2	no	—
243	C₂³	—	8	4	no	—
244	C₂³	C₂×D₄×C₁₁	1	2	yes	—
245	C₂³	C₂×D₄×C₁₁	1	1	yes	characteristic
246	Q₈	—	2	2	no	—
247	Q₈	—	2	1	no	—
248	Q₈	C₂×D₄×C₁₁	1	2	yes	—
249	D₄	—	2	8	no	—
250	D₄	—	2	8	no	—
251	D₄	—	2	4	no	—
252	D₄	—	2	4	no	—
253	D₄	—	2	2	no	—
254	D₄	—	2	4	no	—
255	D₄	—	2	2	no	—
256	D₄	—	2	1	no	—
257	D₄	—	2	2	no	—
258	D₄	—	2	2	no	—
259	D₄	—	4	2	no	—
260	D₄	—	8	4	no	—
261	D₄	C₂×D₄×C₁₁	1	2	yes	—
262	D₄	C₂×D₄×C₁₁	1	4	yes	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	C₂×D₄×C₁₁	1	2	yes	—
270	C₂×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
271	C₂×C₄	C₂×D₄×C₁₁	1	2	yes	—
272	C₂×C₄	C₂×D₄×C₁₁	1	2	yes	—
273	C₂×C₄	C₂⁴×C₁₁	1	1	yes	characteristic
274	C₈	—	2	2	no	—
275	C₈	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	4	4	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	4	no	—
285	C₂²	—	8	4	no	—
286	C₂²	—	8	4	no	—
287	C₂²	—	8	8	no	—
288	C₂²	C₁₁×#120	1	2	yes	—
289	C₂²	C₂²×D₄×C₁₁	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	4	2	no	—
293	C₄	C₁₁×#120	1	2	yes	—
294	C₄	C₂²×D₄×C₁₁	1	1	yes	characteristic
295	C₄	C₂²×D₄×C₁₁	1	1	yes	characteristic
296	C₂	—	2	2	no	—
297	C₂	—	4	2	no	—
298	C₂	—	4	4	no	—
299	C₂	—	4	2	no	—
300	C₂	—	8	2	no	—
301	C₂	C₂×C₁₁×#120	1	1	yes	characteristic
302	C₂	C₁₁×#576	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 ◁ C₂×C₁₁×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₁×H(5,𝔽₂)	C₂	semidirect
2. C₂×C₁₁×AGL(1,ℤ/8)	C₂	semidirect
3. C₂²×C₁₁×D₈	C₂	semidirect
4. C₁₁×#535	C₂	semidirect
5. C₁₁×#449	C₂	semidirect
6. C₁₁×#447	C₂	semidirect
7. C₁₁×#418	C₂	semidirect
8. C₁₁×#416	C₂	semidirect
9. C₁₁×#408	C₂	semidirect
15. C₂×C₁₁×#41	C₂²	semidirect
19. C₂²×D₄×C₁₁	C₂²	semidirect
20. C₂²×D₄×C₁₁	C₂²	semidirect
22. C₂×C₁₁×SD₁₆	C₂²	semidirect
24. C₂×C₁₁×D₈	C₂²	semidirect
25. C₂×C₁₁×D₈	C₂²	semidirect
26. C₂×C₁₁×M₁₆	C₂²	semidirect
27. C₂²×C₈×C₁₁	C₂²	semidirect
30. C₁₁×#121	C₂²	semidirect
32. C₁₁×#117	C₂²	semidirect
33. C₁₁×#103	C₂²	semidirect
34. C₁₁×#102	C₂²	semidirect
35. C₁₁×#102	C₂²	semidirect
36. C₁₁×C₂²⋊C₈	C₂²	semidirect
46. C₁₁×#41	D₄	semidirect
64. C₂×D₄×C₁₁	D₄	semidirect
83. #3008	C₁₁	direct (N × H)
125. C₂×H(5,𝔽₂)	C₂×C₁₁	semidirect
126. C₂×AGL(1,ℤ/8)	C₂×C₁₁	semidirect
127. C₂²×D₈	C₂×C₁₁	semidirect
128. #535	C₂×C₁₁	semidirect
129. #449	C₂×C₁₁	semidirect
130. #447	C₂×C₁₁	semidirect
131. #418	C₂×C₁₁	semidirect
132. #416	C₂×C₁₁	semidirect
133. #408	C₂×C₁₁	semidirect
159. C₂×#41	C₂²×C₁₁	semidirect
163. C₂²×D₄	C₂²×C₁₁	semidirect
164. C₂²×D₄	C₂²×C₁₁	semidirect
166. C₂×SD₁₆	C₂²×C₁₁	semidirect
168. C₂×D₈	C₂²×C₁₁	semidirect
169. C₂×D₈	C₂²×C₁₁	semidirect
170. C₂×M₁₆	C₂²×C₁₁	semidirect
171. C₂²×C₈	C₂²×C₁₁	semidirect
174. #121	C₂²×C₁₁	semidirect
176. #117	C₂²×C₁₁	semidirect
177. #103	C₂²×C₁₁	semidirect
178. #102	C₂²×C₁₁	semidirect
179. #102	C₂²×C₁₁	semidirect
180. C₂²⋊C₈	C₂²×C₁₁	semidirect
197. #41	D₄×C₁₁	semidirect
215. C₂×D₄	D₄×C₁₁	semidirect
234. C₁₁	#3008	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×H(5,𝔽₂)	2
C₂×C₁₁×AGL(1,ℤ/8)	2
C₂²×C₁₁×D₈	2
C₁₁×#535	2
C₁₁×#449	2
C₁₁×#447	2
C₁₁×#418	2
C₁₁×#416	2
C₁₁×#408	2
#3008	11

Subgroups — C11×#3008 #13065846

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#3008 #13065846