Subgroups — C₂×C₁₁×#2856 #14613815

Order 2816.

5694 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₁×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₁×#121	C₂	1	6	yes	maximal, complemented
3	C₂³×C₄²×C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₁×#2856	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₁₁×#390	C₂	1	6	yes	maximal, complemented
6	C₂×C₁₁×#384	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄×C₁₁	C₂²	1	4	yes	—
8	C₂³×D₄×C₁₁	C₂²	1	6	yes	—
9	C₂⁴×C₄×C₁₁	C₂²	1	3	yes	—
10	C₂×C₁₁×#127	—	2	8	no	—
11	C₂×C₁₁×#127	C₂²	1	8	yes	complemented
12	C₂×C₁₁×#121	—	2	48	no	—
13	C₂×C₁₁×#121	C₂²	1	48	yes	complemented
14	C₂²×C₁₁×C₄⋊C₄	C₂²	1	6	yes	—
15	C₂²×C₁₁×C₂²⋊C₄	C₂²	1	12	yes	—
16	C₂²×C₄²×C₁₁	—	2	8	no	—
17	C₂²×C₄²×C₁₁	C₂²	1	3	yes	—
18	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
19	C₂²×C₄²×C₁₁	C₂²	1	8	yes	complemented
20	C₁₁×#390	C₂²	1	48	yes	complemented
21	C₁₁×#384	C₂²	1	8	yes	complemented
22	C₂⁵×C₁₁	C₂³	1	4	yes	—
23	C₂⁵×C₁₁	C₂³	1	1	yes	characteristic
24	C₂²×D₄×C₁₁	—	2	48	no	—
25	C₂²×D₄×C₁₁	—	2	24	no	—
26	C₂²×D₄×C₁₁	—	2	24	no	—
27	C₂²×D₄×C₁₁	—	2	24	no	—
28	C₂²×D₄×C₁₁	C₂³	1	16	yes	—
29	C₂²×D₄×C₁₁	C₂³	1	24	yes	—
30	C₂³×C₄×C₁₁	—	2	12	no	—
31	C₂³×C₄×C₁₁	—	2	12	no	—
32	C₂³×C₄×C₁₁	D₄	1	24	yes	complemented
33	C₂³×C₄×C₁₁	C₂³	1	4	yes	—
34	C₂³×C₄×C₁₁	C₂³	1	3	yes	—
35	C₂³×C₄×C₁₁	C₂³	1	12	yes	—
36	C₂³×C₄×C₁₁	C₂³	1	3	yes	—
37	C₁₁×#127	—	2	32	no	—
38	C₁₁×#121	—	2	192	no	—
39	C₂×C₁₁×C₄⋊C₄	—	2	24	no	—
40	C₂×C₁₁×C₄⋊C₄	C₂³	1	24	yes	—
41	C₂×C₁₁×C₂²⋊C₄	—	2	48	no	—
42	C₂×C₁₁×C₂²⋊C₄	C₂³	1	48	yes	—
43	C₂×C₄²×C₁₁	—	2	12	no	—
44	C₂×C₄²×C₁₁	—	2	32	no	—
45	C₂×C₄²×C₁₁	D₄	1	8	yes	complemented
46	C₂×C₄²×C₁₁	C₂³	1	12	yes	—
47	C₂×C₄²×C₁₁	C₂³	1	4	yes	—
48	C₂×#2856	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂⁴×C₁₁	—	2	2	no	—
50	C₂⁴×C₁₁	—	2	6	no	—
51	C₂⁴×C₁₁	—	2	24	no	—
52	C₂⁴×C₁₁	—	2	8	no	—
53	C₂⁴×C₁₁	—	2	24	no	—
54	C₂⁴×C₁₁	C₂×D₄	1	12	yes	—
55	C₂⁴×C₁₁	C₂⁴	1	8	yes	—
56	C₂⁴×C₁₁	C₂⁴	1	2	yes	—
57	C₂⁴×C₁₁	C₂⁴	1	1	yes	characteristic
58	C₂×D₄×C₁₁	—	2	96	no	—
59	C₂×D₄×C₁₁	—	2	48	no	—
60	C₂×D₄×C₁₁	—	2	48	no	—
61	C₂×D₄×C₁₁	—	2	48	no	—
62	C₂×D₄×C₁₁	—	4	64	no	—
63	C₂×D₄×C₁₁	—	4	96	no	—
64	C₂²×C₄×C₁₁	—	2	6	no	—
65	C₂²×C₄×C₁₁	—	2	6	no	—
66	C₂²×C₄×C₁₁	—	2	96	no	—
67	C₂²×C₄×C₁₁	—	2	24	no	—
68	C₂²×C₄×C₁₁	—	2	24	no	—
69	C₂²×C₄×C₁₁	—	2	24	no	—
70	C₂²×C₄×C₁₁	C₂×D₄	1	12	yes	—
71	C₂²×C₄×C₁₁	C₂×D₄	1	12	yes	—
72	C₂²×C₄×C₁₁	C₂×D₄	1	12	yes	—
73	C₂²×C₄×C₁₁	C₂×D₄	1	48	yes	complemented
74	C₂²×C₄×C₁₁	#41	1	12	yes	—
75	C₂²×C₄×C₁₁	C₂⁴	1	8	yes	—
76	C₂²×C₄×C₁₁	C₂⁴	1	6	yes	—
77	C₂²×C₄×C₁₁	C₂⁴	1	6	yes	—
78	C₁₁×C₄⋊C₄	—	2	48	no	—
79	C₁₁×C₂²⋊C₄	—	2	96	no	—
80	C₄²×C₁₁	—	2	24	no	—
81	C₄²×C₁₁	C₂×D₄	1	16	yes	complemented
82	C₂²×#127	C₂×C₁₁	1	1	yes	characteristic, complemented
83	C₂²×#121	C₂×C₁₁	1	6	yes	complemented
84	C₂³×C₄²	C₂×C₁₁	1	1	yes	characteristic, complemented
85	#2856	C₂×C₁₁	1	16	yes	direct product factor
86	C₂×#390	C₂×C₁₁	1	6	yes	complemented
87	C₂×#384	C₂×C₁₁	1	1	yes	characteristic, complemented
88	C₂³×C₁₁	—	2	24	no	—
89	C₂³×C₁₁	—	2	24	no	—
90	C₂³×C₁₁	—	2	2	no	—
91	C₂³×C₁₁	—	2	6	no	—
92	C₂³×C₁₁	—	2	24	no	—
93	C₂³×C₁₁	—	2	8	no	—
94	C₂³×C₁₁	—	2	24	no	—
95	C₂³×C₁₁	—	4	48	no	—
96	C₂³×C₁₁	—	4	16	no	—
97	C₂³×C₁₁	—	4	48	no	—
98	C₂³×C₁₁	#127	1	16	yes	complemented
99	C₂³×C₁₁	C₂²×D₄	1	12	yes	—
100	C₂³×C₁₁	C₂²×D₄	1	2	yes	—
101	C₂³×C₁₁	C₂²×D₄	1	6	yes	—
102	C₂³×C₁₁	C₂×#41	1	6	yes	—
103	C₂³×C₁₁	C₂⁵	1	1	yes	characteristic
104	D₄×C₁₁	—	4	64	no	—
105	D₄×C₁₁	—	4	96	no	—
106	C₂×C₄×C₁₁	—	2	24	no	—
107	C₂×C₄×C₁₁	—	2	6	no	—
108	C₂×C₄×C₁₁	—	2	6	no	—
109	C₂×C₄×C₁₁	—	2	96	no	—
110	C₂×C₄×C₁₁	—	2	24	no	—
111	C₂×C₄×C₁₁	—	4	16	no	—
112	C₂×C₄×C₁₁	#121	1	48	yes	complemented
113	C₂×C₄×C₁₁	C₂²×D₄	1	12	yes	—
114	C₂×C₄×C₁₁	C₂²×D₄	1	12	yes	—
115	C₂×C₄×C₁₁	C₂²×D₄	1	12	yes	—
116	C₂×C₄×C₁₁	C₂×#41	1	12	yes	—
117	C₂³×D₄	C₂²×C₁₁	1	4	yes	—
118	C₂³×D₄	C₂²×C₁₁	1	6	yes	—
119	C₂⁴×C₄	C₂²×C₁₁	1	3	yes	—
120	C₂×#127	—	2	8	no	—
121	C₂×#127	C₂²×C₁₁	1	8	yes	complemented
122	C₂×#121	—	2	48	no	—
123	C₂×#121	C₂²×C₁₁	1	48	yes	complemented
124	C₂²×C₄⋊C₄	C₂²×C₁₁	1	6	yes	—
125	C₂²×C₂²⋊C₄	C₂²×C₁₁	1	12	yes	—
126	C₂²×C₄²	—	2	8	no	—
127	C₂²×C₄²	C₂²×C₁₁	1	3	yes	—
128	C₂²×C₄²	C₂²×C₁₁	1	1	yes	characteristic
129	C₂²×C₄²	C₂²×C₁₁	1	8	yes	complemented
130	#390	C₂²×C₁₁	1	48	yes	complemented
131	#384	C₂²×C₁₁	1	8	yes	complemented
132	C₂²×C₁₁	—	2	32	no	—
133	C₂²×C₁₁	—	2	12	no	—
134	C₂²×C₁₁	—	2	12	no	—
135	C₂²×C₁₁	—	4	24	no	—
136	C₂²×C₁₁	—	4	8	no	—
137	C₂²×C₁₁	—	4	24	no	—
138	C₂²×C₁₁	—	8	32	no	—
139	C₂²×C₁₁	C₂×#121	1	12	yes	—
140	C₂²×C₁₁	C₂×#127	1	4	yes	—
141	C₂²×C₁₁	C₂×#127	1	8	yes	complemented
142	C₂²×C₁₁	#532	1	12	yes	—
143	C₂²×C₁₁	C₂³×D₄	1	1	yes	characteristic
144	C₂²×C₁₁	C₂³×D₄	1	3	yes	—
145	C₂²×C₁₁	C₂²×#41	1	3	yes	—
146	C₄×C₁₁	—	2	12	no	—
147	C₄×C₁₁	—	4	8	no	—
148	C₄×C₁₁	C₂×#121	1	24	yes	complemented
149	C₂⁵	C₂³×C₁₁	1	4	yes	—
150	C₂⁵	C₂³×C₁₁	1	1	yes	characteristic
151	C₂²×D₄	—	2	48	no	—
152	C₂²×D₄	—	2	24	no	—
153	C₂²×D₄	—	2	24	no	—
154	C₂²×D₄	—	2	24	no	—
155	C₂²×D₄	C₂³×C₁₁	1	16	yes	—
156	C₂²×D₄	C₂³×C₁₁	1	24	yes	—
157	C₂³×C₄	—	2	12	no	—
158	C₂³×C₄	—	2	12	no	—
159	C₂³×C₄	D₄×C₁₁	1	24	yes	complemented
160	C₂³×C₄	C₂³×C₁₁	1	4	yes	—
161	C₂³×C₄	C₂³×C₁₁	1	3	yes	—
162	C₂³×C₄	C₂³×C₁₁	1	12	yes	—
163	C₂³×C₄	C₂³×C₁₁	1	3	yes	—
164	#127	—	2	32	no	—
165	#121	—	2	192	no	—
166	C₂×C₄⋊C₄	—	2	24	no	—
167	C₂×C₄⋊C₄	C₂³×C₁₁	1	24	yes	—
168	C₂×C₂²⋊C₄	—	2	48	no	—
169	C₂×C₂²⋊C₄	C₂³×C₁₁	1	48	yes	—
170	C₂×C₄²	—	2	12	no	—
171	C₂×C₄²	—	2	32	no	—
172	C₂×C₄²	D₄×C₁₁	1	8	yes	complemented
173	C₂×C₄²	C₂³×C₁₁	1	12	yes	—
174	C₂×C₄²	C₂³×C₁₁	1	4	yes	—
175	C₂×C₁₁	—	2	8	no	—
176	C₂×C₁₁	—	8	8	no	—
177	C₂×C₁₁	#2856	1	8	yes	direct product factor
178	C₂×C₁₁	C₂²×#121	1	3	yes	—
179	C₂×C₁₁	C₂²×#127	1	1	yes	characteristic
180	C₂×C₁₁	C₂×#532	1	3	yes	—
181	C₂⁴	—	2	2	no	—
182	C₂⁴	—	2	6	no	—
183	C₂⁴	—	2	24	no	—
184	C₂⁴	—	2	8	no	—
185	C₂⁴	—	2	24	no	—
186	C₂⁴	C₂×D₄×C₁₁	1	12	yes	—
187	C₂⁴	C₂⁴×C₁₁	1	8	yes	—
188	C₂⁴	C₂⁴×C₁₁	1	2	yes	—
189	C₂⁴	C₂⁴×C₁₁	1	1	yes	characteristic
190	C₂×D₄	—	2	96	no	—
191	C₂×D₄	—	2	48	no	—
192	C₂×D₄	—	2	48	no	—
193	C₂×D₄	—	2	48	no	—
194	C₂×D₄	—	4	64	no	—
195	C₂×D₄	—	4	96	no	—
196	C₂²×C₄	—	2	6	no	—
197	C₂²×C₄	—	2	6	no	—
198	C₂²×C₄	—	2	96	no	—
199	C₂²×C₄	—	2	24	no	—
200	C₂²×C₄	—	2	24	no	—
201	C₂²×C₄	—	2	24	no	—
202	C₂²×C₄	C₂×D₄×C₁₁	1	12	yes	—
203	C₂²×C₄	C₂×D₄×C₁₁	1	12	yes	—
204	C₂²×C₄	C₂×D₄×C₁₁	1	12	yes	—
205	C₂²×C₄	C₂×D₄×C₁₁	1	48	yes	complemented
206	C₂²×C₄	C₁₁×#41	1	12	yes	—
207	C₂²×C₄	C₂⁴×C₁₁	1	8	yes	—
208	C₂²×C₄	C₂⁴×C₁₁	1	6	yes	—
209	C₂²×C₄	C₂⁴×C₁₁	1	6	yes	—
210	C₄⋊C₄	—	2	48	no	—
211	C₂²⋊C₄	—	2	96	no	—
212	C₄²	—	2	24	no	—
213	C₄²	C₂×D₄×C₁₁	1	16	yes	complemented
214	C₁₁	C₂×#2856	1	1	yes	characteristic, direct product factor, Sylow
215	C₂³	—	2	24	no	—
216	C₂³	—	2	24	no	—
217	C₂³	—	2	2	no	—
218	C₂³	—	2	6	no	—
219	C₂³	—	2	24	no	—
220	C₂³	—	2	8	no	—
221	C₂³	—	2	24	no	—
222	C₂³	—	4	48	no	—
223	C₂³	—	4	16	no	—
224	C₂³	—	4	48	no	—
225	C₂³	C₁₁×#127	1	16	yes	complemented
226	C₂³	C₂²×D₄×C₁₁	1	12	yes	—
227	C₂³	C₂²×D₄×C₁₁	1	2	yes	—
228	C₂³	C₂²×D₄×C₁₁	1	6	yes	—
229	C₂³	C₂×C₁₁×#41	1	6	yes	—
230	C₂³	C₂⁵×C₁₁	1	1	yes	characteristic
231	D₄	—	4	64	no	—
232	D₄	—	4	96	no	—
233	C₂×C₄	—	2	24	no	—
234	C₂×C₄	—	2	6	no	—
235	C₂×C₄	—	2	6	no	—
236	C₂×C₄	—	2	96	no	—
237	C₂×C₄	—	2	24	no	—
238	C₂×C₄	—	4	16	no	—
239	C₂×C₄	C₁₁×#121	1	48	yes	complemented
240	C₂×C₄	C₂²×D₄×C₁₁	1	12	yes	—
241	C₂×C₄	C₂²×D₄×C₁₁	1	12	yes	—
242	C₂×C₄	C₂²×D₄×C₁₁	1	12	yes	—
243	C₂×C₄	C₂×C₁₁×#41	1	12	yes	—
244	C₂²	—	2	12	no	—
245	C₂²	—	2	12	no	—
246	C₂²	—	2	32	no	—
247	C₂²	—	4	24	no	—
248	C₂²	—	4	8	no	—
249	C₂²	—	4	24	no	—
250	C₂²	—	8	32	no	—
251	C₂²	C₂×C₁₁×#121	1	12	yes	—
252	C₂²	C₂×C₁₁×#127	1	4	yes	—
253	C₂²	C₂×C₁₁×#127	1	8	yes	complemented
254	C₂²	C₁₁×#532	1	12	yes	—
255	C₂²	C₂³×D₄×C₁₁	1	1	yes	characteristic
256	C₂²	C₂³×D₄×C₁₁	1	3	yes	—
257	C₂²	C₂²×C₁₁×#41	1	3	yes	—
258	C₄	—	2	12	no	—
259	C₄	—	4	8	no	—
260	C₄	C₂×C₁₁×#121	1	24	yes	complemented
261	C₂	—	2	8	no	—
262	C₂	—	8	8	no	—
263	C₂	C₁₁×#2856	1	8	yes	direct product factor
264	C₂	C₂²×C₁₁×#121	1	3	yes	—
265	C₂	C₂²×C₁₁×#127	1	1	yes	characteristic
266	C₂	C₂×C₁₁×#532	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₁
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₁×#127	C₂	semidirect
2. C₂²×C₁₁×#121	C₂	semidirect
3. C₂³×C₄²×C₁₁	C₂	semidirect
4. C₁₁×#2856	C₂	direct (N × H)
5. C₂×C₁₁×#390	C₂	semidirect
6. C₂×C₁₁×#384	C₂	semidirect
11. C₂×C₁₁×#127	C₂²	semidirect
13. C₂×C₁₁×#121	C₂²	semidirect
19. C₂²×C₄²×C₁₁	C₂²	semidirect
20. C₁₁×#390	C₂²	semidirect
21. C₁₁×#384	C₂²	semidirect
32. C₂³×C₄×C₁₁	D₄	semidirect
45. C₂×C₄²×C₁₁	D₄	semidirect
48. C₂×#2856	C₁₁	direct (N × H)
73. C₂²×C₄×C₁₁	C₂×D₄	semidirect
81. C₄²×C₁₁	C₂×D₄	semidirect
82. C₂²×#127	C₂×C₁₁	semidirect
83. C₂²×#121	C₂×C₁₁	semidirect
84. C₂³×C₄²	C₂×C₁₁	semidirect
85. #2856	C₂×C₁₁	direct (N × H)
86. C₂×#390	C₂×C₁₁	semidirect
87. C₂×#384	C₂×C₁₁	semidirect
98. C₂³×C₁₁	#127	semidirect
112. C₂×C₄×C₁₁	#121	semidirect
121. C₂×#127	C₂²×C₁₁	semidirect
123. C₂×#121	C₂²×C₁₁	semidirect
129. C₂²×C₄²	C₂²×C₁₁	semidirect
130. #390	C₂²×C₁₁	semidirect
131. #384	C₂²×C₁₁	semidirect
141. C₂²×C₁₁	C₂×#127	semidirect
148. C₄×C₁₁	C₂×#121	semidirect
159. C₂³×C₄	D₄×C₁₁	semidirect
172. C₂×C₄²	D₄×C₁₁	semidirect
177. C₂×C₁₁	#2856	direct (N × H)
205. C₂²×C₄	C₂×D₄×C₁₁	semidirect
213. C₄²	C₂×D₄×C₁₁	semidirect
214. C₁₁	C₂×#2856	direct (N × H)
225. C₂³	C₁₁×#127	semidirect
239. C₂×C₄	C₁₁×#121	semidirect
253. C₂²	C₂×C₁₁×#127	semidirect
260. C₄	C₂×C₁₁×#121	semidirect
263. C₂	C₁₁×#2856	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₁₁×#127	2
C₂²×C₁₁×#121	2
C₂³×C₄²×C₁₁	2
C₁₁×#2856	2
C₂×C₁₁×#390	2
C₂×C₁₁×#384	2
C₂×#2856	11

Subgroups — C2×C11×#2856 #14613815

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₁×#2856 #14613815