Subgroups — C₁₁×#2946 #13065784

Order 1408.

506 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
2	C₁₁×#520	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₁×#518	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₁×#442	C₂	1	2	yes	maximal, complemented
5	C₁₁×#440	C₂	1	2	yes	maximal, complemented
6	C₄×C₁₁×SD₁₆	C₂	1	2	yes	maximal, complemented
7	C₄×C₁₁×D₈	C₂	1	2	yes	maximal, complemented
8	C₁₁×#428	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₁×#416	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₁×#102	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₁×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₁×#41	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₁₁	C₂²	1	1	yes	characteristic
14	C₁₁×AGL(1,ℤ/8)	C₄	1	8	yes	complemented
15	C₂×C₁₁×SD₁₆	C₂²	1	2	yes	—
16	C₂×C₁₁×D₈	C₂²	1	2	yes	—
17	C₂×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₁₁	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₁₁	C₂²	1	2	yes	complemented
20	C₄×D₄×C₁₁	—	2	1	no	—
21	C₄×D₄×C₁₁	—	2	1	no	—
22	C₄×D₄×C₁₁	—	2	2	no	—
23	C₄×D₄×C₁₁	C₂²	1	2	yes	—
24	C₄×D₄×C₁₁	C₂²	1	2	yes	—
25	C₄×D₄×C₁₁	C₂²	1	2	yes	complemented
26	C₁₁×#117	—	2	1	no	—
27	C₁₁×#117	C₂²	1	1	yes	characteristic
28	C₁₁×#117	C₂²	1	1	yes	characteristic
29	C₂×C₁₁×C₄⋊C₄	—	2	1	no	—
30	C₂×C₁₁×C₄⋊C₄	C₂²	1	1	yes	characteristic
31	C₂×C₁₁×C₂²⋊C₄	—	2	2	no	—
32	C₁₁×#107	C₂²	1	2	yes	complemented
33	C₁₁×#106	C₂²	1	2	yes	complemented
34	C₁₁×#103	C₂²	1	1	yes	characteristic, complemented
35	C₁₁×#103	C₂²	1	1	yes	characteristic, complemented
36	C₁₁×#102	C₂²	1	2	yes	—
37	C₁₁×#102	C₂²	1	2	yes	—
38	C₁₁×#102	C₂²	1	1	yes	characteristic, complemented
39	C₁₁×#102	C₂²	1	1	yes	characteristic, complemented
40	C₁₁×#97	C₂²	1	2	yes	complemented
41	C₄×C₈×C₁₁	C₂²	1	2	yes	complemented
42	C₂⁴×C₁₁	—	2	1	no	—
43	C₁₁×#41	—	2	2	no	—
44	C₁₁×#41	C₂×C₄	1	4	yes	complemented
45	C₂×Q₈×C₁₁	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₁₁	—	2	1	no	—
47	C₂×D₄×C₁₁	—	2	1	no	—
48	C₂×D₄×C₁₁	—	2	2	no	—
49	C₂×D₄×C₁₁	C₂×C₄	1	4	yes	complemented
50	C₂×D₄×C₁₁	C₂³	1	2	yes	—
51	C₂×D₄×C₁₁	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₁₁	—	2	1	no	—
53	C₂²×C₄×C₁₁	—	2	1	no	—
54	C₂²×C₄×C₁₁	—	2	2	no	—
55	C₂²×C₄×C₁₁	—	2	2	no	—
56	C₂²×C₄×C₁₁	—	2	2	no	—
57	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
58	C₁₁×SD₁₆	C₂×C₄	1	8	yes	complemented
59	C₁₁×D₈	C₂×C₄	1	8	yes	complemented
60	C₁₁×M₁₆	C₂×C₄	1	4	yes	complemented
61	C₂×C₈×C₁₁	—	2	2	no	—
62	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₁₁	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₁₁	C₂³	1	2	yes	—
65	C₁₁×C₄⋊C₄	—	2	1	no	—
66	C₁₁×C₄⋊C₄	—	2	2	no	—
67	C₁₁×C₄⋊C₄	D₄	1	1	yes	characteristic
68	C₁₁×C₄⋊C₄	D₄	1	1	yes	characteristic
69	C₁₁×C₄⋊C₄	C₂³	1	2	yes	—
70	C₁₁×C₄⋊C₄	C₂³	1	1	yes	characteristic
71	C₁₁×C₄⋊C₄	C₂³	1	1	yes	characteristic
72	C₁₁×C₂²⋊C₄	—	2	1	no	—
73	C₁₁×C₂²⋊C₄	—	2	2	no	—
74	C₁₁×C₂²⋊C₄	—	2	2	no	—
75	C₁₁×C₂²⋊C₄	—	2	2	no	—
76	C₁₁×C₂²⋊C₄	D₄	1	2	yes	—
77	C₄²×C₁₁	—	2	2	no	—
78	C₄²×C₁₁	—	2	1	no	—
79	C₄²×C₁₁	C₂³	1	2	yes	—
80	#2946	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₁₁	—	2	1	no	—
82	C₂³×C₁₁	—	2	2	no	—
83	C₂³×C₁₁	—	2	4	no	—
84	C₂³×C₁₁	—	8	1	no	—
85	C₂³×C₁₁	—	8	1	no	—
86	C₂³×C₁₁	C₂×D₄	1	1	yes	characteristic
87	Q₈×C₁₁	—	2	1	no	—
88	Q₈×C₁₁	C₂²×C₄	1	2	yes	—
89	D₄×C₁₁	—	2	4	no	—
90	D₄×C₁₁	—	2	4	no	—
91	D₄×C₁₁	—	2	2	no	—
92	D₄×C₁₁	—	2	1	no	—
93	D₄×C₁₁	C₂²×C₄	1	2	yes	—
94	D₄×C₁₁	C₂²×C₄	1	4	yes	—
95	C₂×C₄×C₁₁	—	2	1	no	—
96	C₂×C₄×C₁₁	—	2	4	no	—
97	C₂×C₄×C₁₁	—	2	1	no	—
98	C₂×C₄×C₁₁	—	2	1	no	—
99	C₂×C₄×C₁₁	—	2	2	no	—
100	C₂×C₄×C₁₁	—	8	2	no	—
101	C₂×C₄×C₁₁	—	8	2	no	—
102	C₂×C₄×C₁₁	—	8	2	no	—
103	C₂×C₄×C₁₁	—	8	1	no	—
104	C₂×C₄×C₁₁	C₂²×C₄	1	2	yes	—
105	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₁	#41	1	2	yes	—
109	C₂×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
110	C₈×C₁₁	—	2	1	no	—
111	C₈×C₁₁	—	2	1	no	—
112	C₈×C₁₁	C₂²×C₄	1	4	yes	—
113	C₂×AGL(1,ℤ/8)	C₂×C₁₁	1	1	yes	characteristic
114	#520	C₂×C₁₁	1	1	yes	characteristic, complemented
115	#518	C₂×C₁₁	1	1	yes	characteristic, complemented
116	#442	C₂×C₁₁	1	2	yes	complemented
117	#440	C₂×C₁₁	1	2	yes	complemented
118	C₄×SD₁₆	C₂×C₁₁	1	2	yes	complemented
119	C₄×D₈	C₂×C₁₁	1	2	yes	complemented
120	#428	C₂×C₁₁	1	1	yes	characteristic, complemented
121	#416	C₂×C₁₁	1	1	yes	characteristic, complemented
122	C₂×#102	C₂×C₁₁	1	1	yes	characteristic, complemented
123	#405	C₂×C₁₁	1	1	yes	characteristic, complemented
124	C₂²×C₁₁	—	2	1	no	—
125	C₂²×C₁₁	—	2	1	no	—
126	C₂²×C₁₁	—	2	2	no	—
127	C₂²×C₁₁	—	2	4	no	—
128	C₂²×C₁₁	—	4	1	no	—
129	C₂²×C₁₁	—	4	2	no	—
130	C₂²×C₁₁	—	4	2	no	—
131	C₂²×C₁₁	—	4	1	no	—
132	C₂²×C₁₁	—	8	4	no	—
133	C₂²×C₁₁	C₄×D₄	1	2	yes	—
134	C₂²×C₁₁	C₂²×D₄	1	1	yes	characteristic
135	C₄×C₁₁	—	2	2	no	—
136	C₄×C₁₁	—	2	2	no	—
137	C₄×C₁₁	—	2	2	no	—
138	C₄×C₁₁	—	4	1	no	—
139	C₄×C₁₁	—	4	2	no	—
140	C₄×C₁₁	—	4	1	no	—
141	C₄×C₁₁	C₄×D₄	1	2	yes	—
142	C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
143	C₄×C₁₁	C₂×#41	1	1	yes	characteristic
144	C₂×#41	C₂²×C₁₁	1	1	yes	characteristic
145	C₂²×D₄	C₂²×C₁₁	1	1	yes	characteristic
146	AGL(1,ℤ/8)	C₄×C₁₁	1	8	yes	complemented
147	C₂×SD₁₆	C₂²×C₁₁	1	2	yes	—
148	C₂×D₈	C₂²×C₁₁	1	2	yes	—
149	C₂×M₁₆	C₂²×C₁₁	1	1	yes	characteristic
150	C₂²×C₈	C₂²×C₁₁	1	1	yes	characteristic
151	C₄×Q₈	C₂²×C₁₁	1	2	yes	complemented
152	C₄×D₄	—	2	1	no	—
153	C₄×D₄	—	2	1	no	—
154	C₄×D₄	—	2	2	no	—
155	C₄×D₄	C₂²×C₁₁	1	2	yes	—
156	C₄×D₄	C₂²×C₁₁	1	2	yes	—
157	C₄×D₄	C₂²×C₁₁	1	2	yes	complemented
158	#117	—	2	1	no	—
159	#117	C₂²×C₁₁	1	1	yes	characteristic
160	#117	C₂²×C₁₁	1	1	yes	characteristic
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	C₂²×C₁₁	1	1	yes	characteristic
163	C₂×C₂²⋊C₄	—	2	2	no	—
164	#107	C₂²×C₁₁	1	2	yes	complemented
165	#106	C₂²×C₁₁	1	2	yes	complemented
166	#103	C₂²×C₁₁	1	1	yes	characteristic, complemented
167	#103	C₂²×C₁₁	1	1	yes	characteristic, complemented
168	#102	C₂²×C₁₁	1	2	yes	—
169	#102	C₂²×C₁₁	1	2	yes	—
170	#102	C₂²×C₁₁	1	1	yes	characteristic, complemented
171	#102	C₂²×C₁₁	1	1	yes	characteristic, complemented
172	#97	C₂²×C₁₁	1	2	yes	complemented
173	C₄×C₈	C₂²×C₁₁	1	2	yes	complemented
174	C₂×C₁₁	—	2	2	no	—
175	C₂×C₁₁	—	4	2	no	—
176	C₂×C₁₁	—	4	4	no	—
177	C₂×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
178	C₂×C₁₁	#576	1	1	yes	characteristic
179	C₂×C₁₁	#577	1	1	yes	characteristic
180	C₂⁴	—	2	1	no	—
181	#41	—	2	2	no	—
182	#41	C₂×C₄×C₁₁	1	4	yes	complemented
183	C₂×Q₈	C₂³×C₁₁	1	1	yes	characteristic
184	C₂×D₄	—	2	1	no	—
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	C₂×C₄×C₁₁	1	4	yes	complemented
188	C₂×D₄	C₂³×C₁₁	1	2	yes	—
189	C₂×D₄	C₂³×C₁₁	1	1	yes	characteristic
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	C₂³×C₁₁	1	1	yes	characteristic
196	SD₁₆	C₂×C₄×C₁₁	1	8	yes	complemented
197	D₈	C₂×C₄×C₁₁	1	8	yes	complemented
198	M₁₆	C₂×C₄×C₁₁	1	4	yes	complemented
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	C₂³×C₁₁	1	1	yes	characteristic
201	C₂×C₈	C₂³×C₁₁	1	1	yes	characteristic
202	C₂×C₈	C₂³×C₁₁	1	2	yes	—
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	D₄×C₁₁	1	1	yes	characteristic
206	C₄⋊C₄	D₄×C₁₁	1	1	yes	characteristic
207	C₄⋊C₄	C₂³×C₁₁	1	2	yes	—
208	C₄⋊C₄	C₂³×C₁₁	1	1	yes	characteristic
209	C₄⋊C₄	C₂³×C₁₁	1	1	yes	characteristic
210	C₂²⋊C₄	—	2	1	no	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	D₄×C₁₁	1	2	yes	—
215	C₄²	—	2	2	no	—
216	C₄²	—	2	1	no	—
217	C₄²	C₂³×C₁₁	1	2	yes	—
218	C₁₁	#2946	1	1	yes	characteristic, direct product factor, Sylow
219	C₂³	—	2	1	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	4	no	—
222	C₂³	—	8	1	no	—
223	C₂³	—	8	1	no	—
224	C₂³	C₂×D₄×C₁₁	1	1	yes	characteristic
225	Q₈	—	2	1	no	—
226	Q₈	C₂²×C₄×C₁₁	1	2	yes	—
227	D₄	—	2	4	no	—
228	D₄	—	2	4	no	—
229	D₄	—	2	2	no	—
230	D₄	—	2	1	no	—
231	D₄	C₂²×C₄×C₁₁	1	2	yes	—
232	D₄	C₂²×C₄×C₁₁	1	4	yes	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	—	8	2	no	—
241	C₂×C₄	—	8	1	no	—
242	C₂×C₄	C₂²×C₄×C₁₁	1	2	yes	—
243	C₂×C₄	C₂×D₄×C₁₁	1	2	yes	—
244	C₂×C₄	C₂×D₄×C₁₁	1	2	yes	—
245	C₂×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
246	C₂×C₄	C₁₁×#41	1	2	yes	—
247	C₂×C₄	C₂⁴×C₁₁	1	1	yes	characteristic
248	C₈	—	2	1	no	—
249	C₈	—	2	1	no	—
250	C₈	C₂²×C₄×C₁₁	1	4	yes	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	4	1	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	8	4	no	—
260	C₂²	C₄×D₄×C₁₁	1	2	yes	—
261	C₂²	C₂²×D₄×C₁₁	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	4	1	no	—
266	C₄	—	4	2	no	—
267	C₄	—	4	1	no	—
268	C₄	C₄×D₄×C₁₁	1	2	yes	—
269	C₄	C₂³×C₄×C₁₁	1	1	yes	characteristic
270	C₄	C₂×C₁₁×#41	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	4	4	no	—
273	C₂	—	4	2	no	—
274	C₂	C₂×C₄×D₄×C₁₁	1	1	yes	characteristic
275	C₂	C₁₁×#576	1	1	yes	characteristic
276	C₂	C₁₁×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₁	C₂²×D₄
commutator	C₄	C₂³×C₄×C₁₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₁
prime radical	C₂×C₁₁×AGL(1,ℤ/8)	C₂
LCS	G ▷ 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
2. C₁₁×#520	C₂	semidirect
3. C₁₁×#518	C₂	semidirect
4. C₁₁×#442	C₂	semidirect
5. C₁₁×#440	C₂	semidirect
6. C₄×C₁₁×SD₁₆	C₂	semidirect
7. C₄×C₁₁×D₈	C₂	semidirect
8. C₁₁×#428	C₂	semidirect
9. C₁₁×#416	C₂	semidirect
10. C₂×C₁₁×#102	C₂	semidirect
11. C₁₁×#405	C₂	semidirect
14. C₁₁×AGL(1,ℤ/8)	C₄	semidirect
19. C₄×Q₈×C₁₁	C₂²	semidirect
25. C₄×D₄×C₁₁	C₂²	semidirect
32. C₁₁×#107	C₂²	semidirect
33. C₁₁×#106	C₂²	semidirect
34. C₁₁×#103	C₂²	semidirect
35. C₁₁×#103	C₂²	semidirect
38. C₁₁×#102	C₂²	semidirect
39. C₁₁×#102	C₂²	semidirect
40. C₁₁×#97	C₂²	semidirect
41. C₄×C₈×C₁₁	C₂²	semidirect
44. C₁₁×#41	C₂×C₄	semidirect
49. C₂×D₄×C₁₁	C₂×C₄	semidirect
58. C₁₁×SD₁₆	C₂×C₄	semidirect
59. C₁₁×D₈	C₂×C₄	semidirect
60. C₁₁×M₁₆	C₂×C₄	semidirect
80. #2946	C₁₁	direct (N × H)
114. #520	C₂×C₁₁	semidirect
115. #518	C₂×C₁₁	semidirect
116. #442	C₂×C₁₁	semidirect
117. #440	C₂×C₁₁	semidirect
118. C₄×SD₁₆	C₂×C₁₁	semidirect
119. C₄×D₈	C₂×C₁₁	semidirect
120. #428	C₂×C₁₁	semidirect
121. #416	C₂×C₁₁	semidirect
122. C₂×#102	C₂×C₁₁	semidirect
123. #405	C₂×C₁₁	semidirect
146. AGL(1,ℤ/8)	C₄×C₁₁	semidirect
151. C₄×Q₈	C₂²×C₁₁	semidirect
157. C₄×D₄	C₂²×C₁₁	semidirect
164. #107	C₂²×C₁₁	semidirect
165. #106	C₂²×C₁₁	semidirect
166. #103	C₂²×C₁₁	semidirect
167. #103	C₂²×C₁₁	semidirect
170. #102	C₂²×C₁₁	semidirect
171. #102	C₂²×C₁₁	semidirect
172. #97	C₂²×C₁₁	semidirect
173. C₄×C₈	C₂²×C₁₁	semidirect
182. #41	C₂×C₄×C₁₁	semidirect
187. C₂×D₄	C₂×C₄×C₁₁	semidirect
196. SD₁₆	C₂×C₄×C₁₁	semidirect
197. D₈	C₂×C₄×C₁₁	semidirect
198. M₁₆	C₂×C₄×C₁₁	semidirect
218. C₁₁	#2946	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×AGL(1,ℤ/8)	2
C₁₁×#520	2
C₁₁×#518	2
C₁₁×#442	2
C₁₁×#440	2
C₄×C₁₁×SD₁₆	2
C₄×C₁₁×D₈	2
C₁₁×#428	2
C₁₁×#416	2
C₂×C₁₁×#102	2
C₁₁×#405	2
#2946	11

Subgroups — C11×#2946 #13065784

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#2946 #13065784