Subgroups — C₂²×C₁₁×#350 #14610531

Order 2816.

1174 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₁×#131	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₁×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₁₁×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₁×#350	C₂	1	12	yes	maximal, direct product factor
5	C₂²×C₁₁×#41	C₄	1	1	yes	characteristic
6	C₂×C₁₁×#131	—	2	3	no	—
7	C₂×C₁₁×#131	—	2	3	no	—
8	C₂×C₁₁×#131	C₄	1	6	yes	—
9	C₂×C₁₁×#131	C₂²	1	6	yes	—
10	C₂²×C₁₁×M₁₆	—	2	1	no	—
11	C₂²×C₁₁×M₁₆	C₄	1	1	yes	characteristic
12	C₂³×C₈×C₁₁	—	2	1	no	—
13	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
14	C₂×C₁₁×M₃₂	—	2	3	no	—
15	C₂×C₁₁×M₃₂	C₂²	1	6	yes	complemented
16	C₂²×C₁₁×C₁₆	—	2	1	no	—
17	C₂²×C₁₁×C₁₆	—	2	3	no	—
18	C₂²×C₁₁×C₁₆	—	2	1	no	—
19	C₂²×C₁₁×C₁₆	C₂²	1	6	yes	complemented
20	C₁₁×#350	C₂²	1	16	yes	direct product factor
21	C₂×C₁₁×#41	—	2	3	no	—
22	C₂×C₁₁×#41	—	2	3	no	—
23	C₂×C₁₁×#41	C₈	1	6	yes	—
24	C₂×C₁₁×#41	C₂×C₄	1	6	yes	—
25	C₂²×Q₈×C₁₁	C₈	1	1	yes	characteristic
26	C₂²×D₄×C₁₁	—	2	1	no	—
27	C₂²×D₄×C₁₁	C₈	1	1	yes	characteristic
28	C₂³×C₄×C₁₁	—	2	1	no	—
29	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
30	C₁₁×#131	—	2	12	no	—
31	C₁₁×#131	—	2	6	no	—
32	C₁₁×#131	—	2	6	no	—
33	C₁₁×#131	C₂×C₄	1	12	yes	—
34	C₁₁×#131	C₂³	1	4	yes	—
35	C₂×C₁₁×M₁₆	—	2	3	no	—
36	C₂×C₁₁×M₁₆	—	2	6	no	—
37	C₂×C₁₁×M₁₆	—	2	6	no	—
38	C₂×C₁₁×M₁₆	C₂×C₄	1	6	yes	—
39	C₂²×C₈×C₁₁	—	2	1	no	—
40	C₂²×C₈×C₁₁	—	2	6	no	—
41	C₂²×C₈×C₁₁	—	2	6	no	—
42	C₂²×C₈×C₁₁	C₂×C₄	1	3	yes	—
43	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₁₁	D₄	1	3	yes	—
45	C₂²×C₈×C₁₁	D₄	1	3	yes	—
46	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₁₁	C₂³	1	3	yes	—
48	C₁₁×M₃₂	—	2	6	no	—
49	C₁₁×M₃₂	C₂³	1	4	yes	complemented
50	C₂×C₁₁×C₁₆	—	2	6	no	—
51	C₂×C₁₁×C₁₆	—	2	6	no	—
52	C₂×C₁₁×C₁₆	—	2	6	no	—
53	C₂×C₁₁×C₁₆	C₂³	1	4	yes	complemented
54	C₂²×#350	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂⁴×C₁₁	—	2	1	no	—
56	C₂⁴×C₁₁	M₁₆	1	1	yes	characteristic
57	C₁₁×#41	—	2	12	no	—
58	C₁₁×#41	—	2	6	no	—
59	C₁₁×#41	—	2	6	no	—
60	C₁₁×#41	C₂×C₈	1	12	yes	—
61	C₁₁×#41	C₂²×C₄	1	4	yes	—
62	C₂×Q₈×C₁₁	—	2	3	no	—
63	C₂×Q₈×C₁₁	C₂×C₈	1	6	yes	—
64	C₂×D₄×C₁₁	—	2	6	no	—
65	C₂×D₄×C₁₁	—	2	6	no	—
66	C₂×D₄×C₁₁	—	2	3	no	—
67	C₂×D₄×C₁₁	C₂×C₈	1	6	yes	—
68	C₂²×C₄×C₁₁	—	2	1	no	—
69	C₂²×C₄×C₁₁	—	2	6	no	—
70	C₂²×C₄×C₁₁	—	2	6	no	—
71	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₁	C₂²⋊C₄	1	3	yes	—
73	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄×C₁₁	C₂×C₈	1	3	yes	—
75	C₂²×C₄×C₁₁	M₁₆	1	3	yes	—
76	C₂²×C₄×C₁₁	C₂²×C₄	1	3	yes	—
77	C₁₁×M₁₆	—	2	12	no	—
78	C₁₁×M₁₆	—	2	4	no	—
79	C₁₁×M₁₆	—	2	6	no	—
80	C₁₁×M₁₆	C₂²×C₄	1	4	yes	—
81	C₂×C₈×C₁₁	—	2	12	no	—
82	C₂×C₈×C₁₁	—	2	4	no	—
83	C₂×C₈×C₁₁	—	2	6	no	—
84	C₂×C₈×C₁₁	C₂²⋊C₄	1	3	yes	—
85	C₂×C₈×C₁₁	C₂²⋊C₄	1	3	yes	—
86	C₂×C₈×C₁₁	C₂²⋊C₄	1	6	yes	—
87	C₂×C₈×C₁₁	C₂²×C₄	1	3	yes	—
88	C₂×C₈×C₁₁	C₂×D₄	1	3	yes	—
89	C₂×C₈×C₁₁	C₂×D₄	1	3	yes	—
90	C₂×C₈×C₁₁	C₂×D₄	1	3	yes	—
91	C₂×C₈×C₁₁	C₂×D₄	1	3	yes	—
92	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
93	C₁₁×C₁₆	—	2	4	no	—
94	C₁₁×C₁₆	—	2	4	no	—
95	C₂²×#131	C₂×C₁₁	1	1	yes	characteristic
96	C₂²×M₃₂	C₂×C₁₁	1	1	yes	characteristic, complemented
97	C₂³×C₁₆	C₂×C₁₁	1	1	yes	characteristic, complemented
98	C₂×#350	C₂×C₁₁	1	12	yes	direct product factor
99	C₂³×C₁₁	—	2	4	no	—
100	C₂³×C₁₁	—	2	6	no	—
101	C₂³×C₁₁	—	4	4	no	—
102	C₂³×C₁₁	C₂²⋊C₈	1	3	yes	—
103	C₂³×C₁₁	C₂²⋊C₈	1	1	yes	characteristic
104	C₂³×C₁₁	C₂×M₁₆	1	3	yes	—
105	Q₈×C₁₁	—	2	6	no	—
106	Q₈×C₁₁	C₂²×C₈	1	4	yes	—
107	D₄×C₁₁	—	2	4	no	—
108	D₄×C₁₁	—	2	12	no	—
109	D₄×C₁₁	—	2	6	no	—
110	D₄×C₁₁	C₂²×C₈	1	4	yes	—
111	C₂×C₄×C₁₁	—	2	6	no	—
112	C₂×C₄×C₁₁	—	2	4	no	—
113	C₂×C₄×C₁₁	—	2	12	no	—
114	C₂×C₄×C₁₁	C₂²⋊C₈	1	3	yes	—
115	C₂×C₄×C₁₁	C₂²⋊C₈	1	3	yes	—
116	C₂×C₄×C₁₁	C₂²⋊C₈	1	6	yes	—
117	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	3	yes	—
118	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	3	yes	—
119	C₂×C₄×C₁₁	C₂²×C₈	1	3	yes	—
120	C₂×C₄×C₁₁	C₂²×C₈	1	3	yes	—
121	C₂×C₄×C₁₁	C₂×M₁₆	1	3	yes	—
122	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
123	C₈×C₁₁	—	2	4	no	—
124	C₈×C₁₁	C₂×C₂²⋊C₄	1	3	yes	—
125	C₈×C₁₁	C₂×C₂²⋊C₄	1	3	yes	—
126	C₈×C₁₁	C₂²×D₄	1	1	yes	characteristic
127	C₈×C₁₁	C₂²×D₄	1	1	yes	characteristic
128	C₂²×#41	C₄×C₁₁	1	1	yes	characteristic
129	C₂×#131	—	2	3	no	—
130	C₂×#131	—	2	3	no	—
131	C₂×#131	C₄×C₁₁	1	6	yes	—
132	C₂×#131	C₂²×C₁₁	1	6	yes	—
133	C₂²×M₁₆	—	2	1	no	—
134	C₂²×M₁₆	C₄×C₁₁	1	1	yes	characteristic
135	C₂³×C₈	—	2	1	no	—
136	C₂³×C₈	C₂²×C₁₁	1	1	yes	characteristic
137	C₂×M₃₂	—	2	3	no	—
138	C₂×M₃₂	C₂²×C₁₁	1	6	yes	complemented
139	C₂²×C₁₆	—	2	1	no	—
140	C₂²×C₁₆	—	2	3	no	—
141	C₂²×C₁₆	—	2	1	no	—
142	C₂²×C₁₆	C₂²×C₁₁	1	6	yes	complemented
143	#350	C₂²×C₁₁	1	16	yes	direct product factor
144	C₂²×C₁₁	—	2	6	no	—
145	C₂²×C₁₁	—	2	6	no	—
146	C₂²×C₁₁	—	2	4	no	—
147	C₂²×C₁₁	—	4	12	no	—
148	C₂²×C₁₁	#350	1	4	yes	direct product factor
149	C₂²×C₁₁	C₂×C₂²⋊C₈	1	3	yes	—
150	C₂²×C₁₁	C₂×C₂²⋊C₈	1	3	yes	—
151	C₂²×C₁₁	C₂²×M₁₆	1	1	yes	characteristic
152	C₄×C₁₁	—	2	4	no	—
153	C₄×C₁₁	C₂×C₂²⋊C₈	1	3	yes	—
154	C₄×C₁₁	C₂×C₂²⋊C₈	1	3	yes	—
155	C₄×C₁₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
156	C₄×C₁₁	C₂³×C₈	1	1	yes	characteristic
157	C₂×#41	—	2	3	no	—
158	C₂×#41	—	2	3	no	—
159	C₂×#41	C₈×C₁₁	1	6	yes	—
160	C₂×#41	C₂×C₄×C₁₁	1	6	yes	—
161	C₂²×Q₈	C₈×C₁₁	1	1	yes	characteristic
162	C₂²×D₄	—	2	1	no	—
163	C₂²×D₄	C₈×C₁₁	1	1	yes	characteristic
164	C₂³×C₄	—	2	1	no	—
165	C₂³×C₄	C₂×C₄×C₁₁	1	1	yes	characteristic
166	#131	—	2	12	no	—
167	#131	—	2	6	no	—
168	#131	—	2	6	no	—
169	#131	C₂×C₄×C₁₁	1	12	yes	—
170	#131	C₂³×C₁₁	1	4	yes	—
171	C₂×M₁₆	—	2	3	no	—
172	C₂×M₁₆	—	2	6	no	—
173	C₂×M₁₆	—	2	6	no	—
174	C₂×M₁₆	C₂×C₄×C₁₁	1	6	yes	—
175	C₂²×C₈	—	2	1	no	—
176	C₂²×C₈	—	2	6	no	—
177	C₂²×C₈	—	2	6	no	—
178	C₂²×C₈	C₂×C₄×C₁₁	1	3	yes	—
179	C₂²×C₈	D₄×C₁₁	1	1	yes	characteristic
180	C₂²×C₈	D₄×C₁₁	1	1	yes	characteristic
181	C₂²×C₈	D₄×C₁₁	1	3	yes	—
182	C₂²×C₈	D₄×C₁₁	1	3	yes	—
183	C₂²×C₈	C₂³×C₁₁	1	3	yes	—
184	M₃₂	—	2	6	no	—
185	M₃₂	C₂³×C₁₁	1	4	yes	complemented
186	C₂×C₁₆	—	2	6	no	—
187	C₂×C₁₆	—	2	6	no	—
188	C₂×C₁₆	—	2	6	no	—
189	C₂×C₁₆	C₂³×C₁₁	1	4	yes	complemented
190	C₂×C₁₁	—	2	1	no	—
191	C₂×C₁₁	—	2	3	no	—
192	C₂×C₁₁	—	4	4	no	—
193	C₂×C₁₁	C₂×#350	1	6	yes	direct product factor
194	C₂×C₁₁	C₂²×C₂²⋊C₈	1	1	yes	characteristic
195	C₂⁴	—	2	1	no	—
196	C₂⁴	C₁₁×M₁₆	1	1	yes	characteristic
197	#41	—	2	12	no	—
198	#41	—	2	6	no	—
199	#41	—	2	6	no	—
200	#41	C₂×C₈×C₁₁	1	12	yes	—
201	#41	C₂²×C₄×C₁₁	1	4	yes	—
202	C₂×Q₈	—	2	3	no	—
203	C₂×Q₈	C₂×C₈×C₁₁	1	6	yes	—
204	C₂×D₄	—	2	6	no	—
205	C₂×D₄	—	2	6	no	—
206	C₂×D₄	—	2	3	no	—
207	C₂×D₄	C₂×C₈×C₁₁	1	6	yes	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	6	no	—
210	C₂²×C₄	—	2	6	no	—
211	C₂²×C₄	C₁₁×C₂²⋊C₄	1	1	yes	characteristic
212	C₂²×C₄	C₁₁×C₂²⋊C₄	1	3	yes	—
213	C₂²×C₄	C₂×C₈×C₁₁	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₈×C₁₁	1	3	yes	—
215	C₂²×C₄	C₁₁×M₁₆	1	3	yes	—
216	C₂²×C₄	C₂²×C₄×C₁₁	1	3	yes	—
217	M₁₆	—	2	6	no	—
218	M₁₆	—	2	12	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	C₂²×C₄×C₁₁	1	4	yes	—
221	C₂×C₈	—	2	6	no	—
222	C₂×C₈	—	2	12	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	C₁₁×C₂²⋊C₄	1	3	yes	—
225	C₂×C₈	C₁₁×C₂²⋊C₄	1	3	yes	—
226	C₂×C₈	C₁₁×C₂²⋊C₄	1	6	yes	—
227	C₂×C₈	C₂²×C₄×C₁₁	1	3	yes	—
228	C₂×C₈	C₂×D₄×C₁₁	1	3	yes	—
229	C₂×C₈	C₂×D₄×C₁₁	1	3	yes	—
230	C₂×C₈	C₂×D₄×C₁₁	1	3	yes	—
231	C₂×C₈	C₂×D₄×C₁₁	1	3	yes	—
232	C₂×C₈	C₂⁴×C₁₁	1	1	yes	characteristic
233	C₁₆	—	2	4	no	—
234	C₁₆	—	2	4	no	—
235	C₁₁	C₂²×#350	1	1	yes	characteristic, direct product factor, Sylow
236	C₂³	—	2	4	no	—
237	C₂³	—	2	6	no	—
238	C₂³	—	4	4	no	—
239	C₂³	C₁₁×C₂²⋊C₈	1	3	yes	—
240	C₂³	C₁₁×C₂²⋊C₈	1	1	yes	characteristic
241	C₂³	C₂×C₁₁×M₁₆	1	3	yes	—
242	Q₈	—	2	6	no	—
243	Q₈	C₂²×C₈×C₁₁	1	4	yes	—
244	D₄	—	2	12	no	—
245	D₄	—	2	4	no	—
246	D₄	—	2	6	no	—
247	D₄	C₂²×C₈×C₁₁	1	4	yes	—
248	C₂×C₄	—	2	6	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	12	no	—
251	C₂×C₄	C₁₁×C₂²⋊C₈	1	3	yes	—
252	C₂×C₄	C₁₁×C₂²⋊C₈	1	3	yes	—
253	C₂×C₄	C₁₁×C₂²⋊C₈	1	6	yes	—
254	C₂×C₄	C₂×C₁₁×C₂²⋊C₄	1	3	yes	—
255	C₂×C₄	C₂×C₁₁×C₂²⋊C₄	1	3	yes	—
256	C₂×C₄	C₂²×C₈×C₁₁	1	3	yes	—
257	C₂×C₄	C₂²×C₈×C₁₁	1	3	yes	—
258	C₂×C₄	C₂×C₁₁×M₁₆	1	3	yes	—
259	C₂×C₄	C₂³×C₄×C₁₁	1	1	yes	characteristic
260	C₈	—	2	4	no	—
261	C₈	C₂×C₁₁×C₂²⋊C₄	1	3	yes	—
262	C₈	C₂×C₁₁×C₂²⋊C₄	1	3	yes	—
263	C₈	C₂²×D₄×C₁₁	1	1	yes	characteristic
264	C₈	C₂²×D₄×C₁₁	1	1	yes	characteristic
265	C₂²	—	2	6	no	—
266	C₂²	—	2	6	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	4	12	no	—
269	C₂²	C₁₁×#350	1	4	yes	direct product factor
270	C₂²	C₂×C₁₁×C₂²⋊C₈	1	3	yes	—
271	C₂²	C₂×C₁₁×C₂²⋊C₈	1	3	yes	—
272	C₂²	C₂²×C₁₁×M₁₆	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	C₂×C₁₁×C₂²⋊C₈	1	3	yes	—
275	C₄	C₂×C₁₁×C₂²⋊C₈	1	3	yes	—
276	C₄	C₂²×C₁₁×C₂²⋊C₄	1	1	yes	characteristic
277	C₄	C₂³×C₈×C₁₁	1	1	yes	characteristic
278	C₂	—	2	3	no	—
279	C₂	—	2	1	no	—
280	C₂	—	4	4	no	—
281	C₂	C₂×C₁₁×#350	1	6	yes	direct product factor
282	C₂	C₂²×C₁₁×C₂²⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈×C₁₁	D₄
commutator	C₄	C₂³×C₈×C₁₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₁
socle	C₂³×C₁₁	C₂²⋊C₈
prime radical	C₂²×C₁₁×#41	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₁₁ ◁ C₂³×C₈×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₁₁×M₃₂	C₂	semidirect
3. C₂³×C₁₁×C₁₆	C₂	semidirect
4. C₂×C₁₁×#350	C₂	direct (N × H)
15. C₂×C₁₁×M₃₂	C₂²	semidirect
19. C₂²×C₁₁×C₁₆	C₂²	semidirect
20. C₁₁×#350	C₂²	direct (N × H)
49. C₁₁×M₃₂	C₂³	semidirect
53. C₂×C₁₁×C₁₆	C₂³	semidirect
54. C₂²×#350	C₁₁	direct (N × H)
96. C₂²×M₃₂	C₂×C₁₁	semidirect
97. C₂³×C₁₆	C₂×C₁₁	semidirect
98. C₂×#350	C₂×C₁₁	direct (N × H)
138. C₂×M₃₂	C₂²×C₁₁	semidirect
142. C₂²×C₁₆	C₂²×C₁₁	semidirect
143. #350	C₂²×C₁₁	direct (N × H)
148. C₂²×C₁₁	#350	direct (N × H)
185. M₃₂	C₂³×C₁₁	semidirect
189. C₂×C₁₆	C₂³×C₁₁	semidirect
193. C₂×C₁₁	C₂×#350	direct (N × H)
235. C₁₁	C₂²×#350	direct (N × H)
269. C₂²	C₁₁×#350	direct (N × H)
281. C₂	C₂×C₁₁×#350	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×#131	2
C₂²×C₁₁×M₃₂	2
C₂³×C₁₁×C₁₆	2
C₂×C₁₁×#350	2
C₂²×#350	11

Subgroups — C22×C11×#350 #14610531

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₁×#350 #14610531