Subgroups — C₂²×C₁₁×#510 #14639823

Order 2816.

2078 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₁×#135	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₁₁×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₁×#510	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₁₁×Q₃₂	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₁×SD₃₂	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₁×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₁₁×#41	—	2	1	no	—
8	C₂³×Q₈×C₁₁	—	2	1	no	—
9	C₂×C₁₁×#135	—	2	6	no	—
10	C₂×C₁₁×#135	C₂²	1	12	yes	—
11	C₂²×C₁₁×Q₁₆	—	2	1	no	—
12	C₂²×C₁₁×Q₁₆	—	2	6	no	—
13	C₂²×C₁₁×Q₁₆	C₂²	1	2	yes	—
14	C₂²×C₁₁×Q₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₁₁×Q₁₆	C₂²	1	12	yes	complemented
16	C₂²×C₁₁×SD₁₆	—	2	1	no	—
17	C₂²×C₁₁×D₈	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
19	C₁₁×#510	C₂²	1	64	yes	direct product factor
20	C₂×C₁₁×Q₃₂	C₂²	1	24	yes	complemented
21	C₂×C₁₁×SD₃₂	C₂²	1	24	yes	complemented
22	C₂×C₁₁×M₃₂	C₂²	1	12	yes	complemented
23	C₂²×C₁₁×C₁₆	C₂²	1	2	yes	—
24	C₂×C₁₁×#41	—	2	12	no	—
25	C₂×C₁₁×#41	—	4	6	no	—
26	C₂²×Q₈×C₁₁	—	2	1	no	—
27	C₂²×Q₈×C₁₁	—	2	2	no	—
28	C₂²×Q₈×C₁₁	—	2	12	no	—
29	C₂²×Q₈×C₁₁	—	4	6	no	—
30	C₂²×Q₈×C₁₁	—	4	1	no	—
31	C₂²×D₄×C₁₁	—	2	1	no	—
32	C₂²×D₄×C₁₁	—	4	1	no	—
33	C₂³×C₄×C₁₁	—	4	1	no	—
34	C₂³×C₄×C₁₁	—	4	1	no	—
35	C₂³×C₄×C₁₁	D₄	1	1	yes	characteristic
36	C₁₁×#135	—	2	24	no	—
37	C₁₁×#135	C₂³	1	16	yes	—
38	C₂×C₁₁×Q₁₆	—	2	6	no	—
39	C₂×C₁₁×Q₁₆	—	2	12	no	—
40	C₂×C₁₁×Q₁₆	—	2	24	no	—
41	C₂×C₁₁×Q₁₆	—	2	3	no	—
42	C₂×C₁₁×Q₁₆	C₂³	1	12	yes	—
43	C₂×C₁₁×Q₁₆	C₂³	1	6	yes	—
44	C₂×C₁₁×Q₁₆	C₂³	1	16	yes	complemented
45	C₂×C₁₁×SD₁₆	—	2	12	no	—
46	C₂×C₁₁×D₈	—	2	3	no	—
47	C₂×C₁₁×D₈	C₂³	1	6	yes	—
48	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₁₁	D₄	1	6	yes	—
50	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₁₁	C₂³	1	6	yes	—
52	C₁₁×Q₃₂	C₂³	1	32	yes	complemented
53	C₁₁×SD₃₂	C₂³	1	32	yes	complemented
54	C₁₁×M₃₂	C₂³	1	16	yes	complemented
55	C₂×C₁₁×C₁₆	C₂³	1	12	yes	—
56	C₂²×#510	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂⁴×C₁₁	—	4	1	no	—
58	C₂⁴×C₁₁	D₈	1	1	yes	characteristic
59	C₁₁×#41	—	2	16	no	—
60	C₁₁×#41	—	4	24	no	—
61	C₂×Q₈×C₁₁	—	2	6	no	—
62	C₂×Q₈×C₁₁	—	2	12	no	—
63	C₂×Q₈×C₁₁	—	2	16	no	—
64	C₂×Q₈×C₁₁	—	4	12	no	—
65	C₂×Q₈×C₁₁	—	4	6	no	—
66	C₂×Q₈×C₁₁	—	4	24	no	—
67	C₂×Q₈×C₁₁	—	4	3	no	—
68	C₂×D₄×C₁₁	—	2	6	no	—
69	C₂×D₄×C₁₁	—	4	12	no	—
70	C₂×D₄×C₁₁	—	4	3	no	—
71	C₂²×C₄×C₁₁	—	4	12	no	—
72	C₂²×C₄×C₁₁	—	4	1	no	—
73	C₂²×C₄×C₁₁	—	4	2	no	—
74	C₂²×C₄×C₁₁	—	4	12	no	—
75	C₂²×C₄×C₁₁	D₈	1	6	yes	—
76	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₁₁	C₂×D₄	1	6	yes	—
78	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
79	C₁₁×Q₁₆	—	2	12	no	—
80	C₁₁×Q₁₆	—	2	16	no	—
81	C₁₁×Q₁₆	—	2	6	no	—
82	C₁₁×Q₁₆	C₂⁴	1	8	yes	—
83	C₁₁×Q₁₆	C₂⁴	1	4	yes	—
84	C₁₁×SD₁₆	—	2	16	no	—
85	C₁₁×D₈	—	2	6	no	—
86	C₁₁×D₈	C₂⁴	1	4	yes	—
87	C₂×C₈×C₁₁	C₂×D₄	1	3	yes	—
88	C₂×C₈×C₁₁	C₂×D₄	1	6	yes	—
89	C₂×C₈×C₁₁	C₂×D₄	1	12	yes	—
90	C₂×C₈×C₁₁	C₂⁴	1	3	yes	—
91	C₂×C₈×C₁₁	C₂⁴	1	4	yes	—
92	C₁₁×C₁₆	C₂⁴	1	8	yes	—
93	C₂²×#135	C₂×C₁₁	1	1	yes	characteristic
94	C₂³×Q₁₆	C₂×C₁₁	1	1	yes	characteristic, complemented
95	C₂×#510	C₂×C₁₁	1	24	yes	direct product factor
96	C₂²×Q₃₂	C₂×C₁₁	1	2	yes	complemented
97	C₂²×SD₃₂	C₂×C₁₁	1	2	yes	complemented
98	C₂²×M₃₂	C₂×C₁₁	1	1	yes	characteristic, complemented
99	C₂³×C₁₁	—	2	4	no	—
100	C₂³×C₁₁	—	4	6	no	—
101	C₂³×C₁₁	—	8	4	no	—
102	C₂³×C₁₁	C₂×D₈	1	6	yes	—
103	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
104	Q₈×C₁₁	—	2	4	no	—
105	Q₈×C₁₁	—	2	8	no	—
106	Q₈×C₁₁	—	4	16	no	—
107	Q₈×C₁₁	—	4	12	no	—
108	Q₈×C₁₁	—	4	6	no	—
109	D₄×C₁₁	—	2	4	no	—
110	D₄×C₁₁	—	4	16	no	—
111	D₄×C₁₁	—	4	6	no	—
112	C₂×C₄×C₁₁	—	4	6	no	—
113	C₂×C₄×C₁₁	—	4	16	no	—
114	C₂×C₄×C₁₁	—	4	12	no	—
115	C₂×C₄×C₁₁	—	4	16	no	—
116	C₂×C₄×C₁₁	C₂×D₈	1	6	yes	—
117	C₂×C₄×C₁₁	C₂×D₈	1	3	yes	—
118	C₂×C₄×C₁₁	C₂×D₈	1	12	yes	—
119	C₂×C₄×C₁₁	C₂²×D₄	1	3	yes	—
120	C₂×C₄×C₁₁	C₂²×D₄	1	4	yes	—
121	C₈×C₁₁	C₂²×D₄	1	3	yes	—
122	C₈×C₁₁	C₂²×D₄	1	4	yes	—
123	C₈×C₁₁	C₂⁵	1	1	yes	characteristic
124	C₂²×#41	—	2	1	no	—
125	C₂³×Q₈	—	2	1	no	—
126	C₂×#135	—	2	6	no	—
127	C₂×#135	C₂²×C₁₁	1	12	yes	—
128	C₂²×Q₁₆	—	2	1	no	—
129	C₂²×Q₁₆	—	2	6	no	—
130	C₂²×Q₁₆	C₂²×C₁₁	1	2	yes	—
131	C₂²×Q₁₆	C₂²×C₁₁	1	1	yes	characteristic
132	C₂²×Q₁₆	C₂²×C₁₁	1	12	yes	complemented
133	C₂²×SD₁₆	—	2	1	no	—
134	C₂²×D₈	C₂²×C₁₁	1	1	yes	characteristic
135	C₂³×C₈	C₂²×C₁₁	1	1	yes	characteristic
136	#510	C₂²×C₁₁	1	64	yes	direct product factor
137	C₂×Q₃₂	C₂²×C₁₁	1	24	yes	complemented
138	C₂×SD₃₂	C₂²×C₁₁	1	24	yes	complemented
139	C₂×M₃₂	C₂²×C₁₁	1	12	yes	complemented
140	C₂²×C₁₆	C₂²×C₁₁	1	2	yes	—
141	C₂²×C₁₁	—	2	12	no	—
142	C₂²×C₁₁	—	4	4	no	—
143	C₂²×C₁₁	—	8	12	no	—
144	C₂²×C₁₁	#510	1	4	yes	direct product factor
145	C₂²×C₁₁	C₂²×D₈	1	4	yes	—
146	C₂²×C₁₁	C₂²×D₈	1	3	yes	—
147	C₄×C₁₁	—	4	4	no	—
148	C₄×C₁₁	—	4	8	no	—
149	C₄×C₁₁	C₂²×D₈	1	4	yes	—
150	C₄×C₁₁	C₂²×D₈	1	3	yes	—
151	C₄×C₁₁	C₂³×D₄	1	1	yes	characteristic
152	C₂×#41	—	2	12	no	—
153	C₂×#41	—	4	6	no	—
154	C₂²×Q₈	—	2	1	no	—
155	C₂²×Q₈	—	2	2	no	—
156	C₂²×Q₈	—	2	12	no	—
157	C₂²×Q₈	—	4	1	no	—
158	C₂²×Q₈	—	4	6	no	—
159	C₂²×D₄	—	2	1	no	—
160	C₂²×D₄	—	4	1	no	—
161	C₂³×C₄	—	4	1	no	—
162	C₂³×C₄	—	4	1	no	—
163	C₂³×C₄	D₄×C₁₁	1	1	yes	characteristic
164	#135	—	2	24	no	—
165	#135	C₂³×C₁₁	1	16	yes	—
166	C₂×Q₁₆	—	2	6	no	—
167	C₂×Q₁₆	—	2	12	no	—
168	C₂×Q₁₆	—	2	24	no	—
169	C₂×Q₁₆	—	2	3	no	—
170	C₂×Q₁₆	C₂³×C₁₁	1	12	yes	—
171	C₂×Q₁₆	C₂³×C₁₁	1	6	yes	—
172	C₂×Q₁₆	C₂³×C₁₁	1	16	yes	complemented
173	C₂×SD₁₆	—	2	12	no	—
174	C₂×D₈	—	2	3	no	—
175	C₂×D₈	C₂³×C₁₁	1	6	yes	—
176	C₂²×C₈	D₄×C₁₁	1	1	yes	characteristic
177	C₂²×C₈	D₄×C₁₁	1	6	yes	—
178	C₂²×C₈	C₂³×C₁₁	1	1	yes	characteristic
179	C₂²×C₈	C₂³×C₁₁	1	6	yes	—
180	Q₃₂	C₂³×C₁₁	1	32	yes	complemented
181	SD₃₂	C₂³×C₁₁	1	32	yes	complemented
182	M₃₂	C₂³×C₁₁	1	16	yes	complemented
183	C₂×C₁₆	C₂³×C₁₁	1	12	yes	—
184	C₂×C₁₁	—	2	4	no	—
185	C₂×C₁₁	—	8	4	no	—
186	C₂×C₁₁	C₂×#510	1	6	yes	direct product factor
187	C₂×C₁₁	C₂³×D₈	1	1	yes	characteristic
188	C₂⁴	—	4	1	no	—
189	C₂⁴	C₁₁×D₈	1	1	yes	characteristic
190	#41	—	2	16	no	—
191	#41	—	4	24	no	—
192	C₂×Q₈	—	2	6	no	—
193	C₂×Q₈	—	2	12	no	—
194	C₂×Q₈	—	2	16	no	—
195	C₂×Q₈	—	4	12	no	—
196	C₂×Q₈	—	4	6	no	—
197	C₂×Q₈	—	4	24	no	—
198	C₂×Q₈	—	4	3	no	—
199	C₂×D₄	—	2	6	no	—
200	C₂×D₄	—	4	12	no	—
201	C₂×D₄	—	4	3	no	—
202	C₂²×C₄	—	4	1	no	—
203	C₂²×C₄	—	4	12	no	—
204	C₂²×C₄	—	4	2	no	—
205	C₂²×C₄	—	4	12	no	—
206	C₂²×C₄	C₁₁×D₈	1	1	yes	characteristic
207	C₂²×C₄	C₁₁×D₈	1	6	yes	—
208	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₄×C₁₁	1	6	yes	—
210	Q₁₆	—	2	12	no	—
211	Q₁₆	—	2	16	no	—
212	Q₁₆	—	2	6	no	—
213	Q₁₆	C₂⁴×C₁₁	1	8	yes	—
214	Q₁₆	C₂⁴×C₁₁	1	4	yes	—
215	SD₁₆	—	2	16	no	—
216	D₈	—	2	6	no	—
217	D₈	C₂⁴×C₁₁	1	4	yes	—
218	C₂×C₈	C₂×D₄×C₁₁	1	3	yes	—
219	C₂×C₈	C₂×D₄×C₁₁	1	6	yes	—
220	C₂×C₈	C₂×D₄×C₁₁	1	12	yes	—
221	C₂×C₈	C₂⁴×C₁₁	1	3	yes	—
222	C₂×C₈	C₂⁴×C₁₁	1	4	yes	—
223	C₁₆	C₂⁴×C₁₁	1	8	yes	—
224	C₁₁	C₂²×#510	1	1	yes	characteristic, direct product factor, Sylow
225	C₂³	—	2	4	no	—
226	C₂³	—	4	6	no	—
227	C₂³	—	8	4	no	—
228	C₂³	C₂×C₁₁×D₈	1	6	yes	—
229	C₂³	C₂×C₁₁×D₈	1	1	yes	characteristic
230	Q₈	—	2	4	no	—
231	Q₈	—	2	8	no	—
232	Q₈	—	4	16	no	—
233	Q₈	—	4	12	no	—
234	Q₈	—	4	6	no	—
235	D₄	—	2	4	no	—
236	D₄	—	4	16	no	—
237	D₄	—	4	6	no	—
238	C₂×C₄	—	4	6	no	—
239	C₂×C₄	—	4	16	no	—
240	C₂×C₄	—	4	12	no	—
241	C₂×C₄	—	4	16	no	—
242	C₂×C₄	C₂×C₁₁×D₈	1	6	yes	—
243	C₂×C₄	C₂×C₁₁×D₈	1	3	yes	—
244	C₂×C₄	C₂×C₁₁×D₈	1	12	yes	—
245	C₂×C₄	C₂²×D₄×C₁₁	1	3	yes	—
246	C₂×C₄	C₂²×D₄×C₁₁	1	4	yes	—
247	C₈	C₂²×D₄×C₁₁	1	3	yes	—
248	C₈	C₂²×D₄×C₁₁	1	4	yes	—
249	C₈	C₂⁵×C₁₁	1	1	yes	characteristic
250	C₂²	—	2	12	no	—
251	C₂²	—	4	4	no	—
252	C₂²	—	8	12	no	—
253	C₂²	C₁₁×#510	1	4	yes	direct product factor
254	C₂²	C₂²×C₁₁×D₈	1	4	yes	—
255	C₂²	C₂²×C₁₁×D₈	1	3	yes	—
256	C₄	—	4	4	no	—
257	C₄	—	4	8	no	—
258	C₄	C₂²×C₁₁×D₈	1	4	yes	—
259	C₄	C₂²×C₁₁×D₈	1	3	yes	—
260	C₄	C₂³×D₄×C₁₁	1	1	yes	characteristic
261	C₂	—	2	4	no	—
262	C₂	—	8	4	no	—
263	C₂	C₂×C₁₁×#510	1	6	yes	direct product factor
264	C₂	C₂³×C₁₁×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₁	C₂×D₈
commutator, Frattini, cosocle	C₈	C₂⁵×C₁₁
prime radical	C₂²×C₁₁×#135	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₁ ◁ C₂³×C₄×C₁₁ ◁ C₂³×C₈×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₁₁×Q₁₆	C₂	semidirect
3. C₂×C₁₁×#510	C₂	direct (N × H)
4. C₂²×C₁₁×Q₃₂	C₂	semidirect
5. C₂²×C₁₁×SD₃₂	C₂	semidirect
6. C₂²×C₁₁×M₃₂	C₂	semidirect
15. C₂²×C₁₁×Q₁₆	C₂²	semidirect
19. C₁₁×#510	C₂²	direct (N × H)
20. C₂×C₁₁×Q₃₂	C₂²	semidirect
21. C₂×C₁₁×SD₃₂	C₂²	semidirect
22. C₂×C₁₁×M₃₂	C₂²	semidirect
44. C₂×C₁₁×Q₁₆	C₂³	semidirect
52. C₁₁×Q₃₂	C₂³	semidirect
53. C₁₁×SD₃₂	C₂³	semidirect
54. C₁₁×M₃₂	C₂³	semidirect
56. C₂²×#510	C₁₁	direct (N × H)
94. C₂³×Q₁₆	C₂×C₁₁	semidirect
95. C₂×#510	C₂×C₁₁	direct (N × H)
96. C₂²×Q₃₂	C₂×C₁₁	semidirect
97. C₂²×SD₃₂	C₂×C₁₁	semidirect
98. C₂²×M₃₂	C₂×C₁₁	semidirect
132. C₂²×Q₁₆	C₂²×C₁₁	semidirect
136. #510	C₂²×C₁₁	direct (N × H)
137. C₂×Q₃₂	C₂²×C₁₁	semidirect
138. C₂×SD₃₂	C₂²×C₁₁	semidirect
139. C₂×M₃₂	C₂²×C₁₁	semidirect
144. C₂²×C₁₁	#510	direct (N × H)
172. C₂×Q₁₆	C₂³×C₁₁	semidirect
180. Q₃₂	C₂³×C₁₁	semidirect
181. SD₃₂	C₂³×C₁₁	semidirect
182. M₃₂	C₂³×C₁₁	semidirect
186. C₂×C₁₁	C₂×#510	direct (N × H)
224. C₁₁	C₂²×#510	direct (N × H)
253. C₂²	C₁₁×#510	direct (N × H)
263. C₂	C₂×C₁₁×#510	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×#135	2
C₂³×C₁₁×Q₁₆	2
C₂×C₁₁×#510	2
C₂²×C₁₁×Q₃₂	2
C₂²×C₁₁×SD₃₂	2
C₂²×C₁₁×M₃₂	2
C₂²×#510	11

Subgroups — C22×C11×#510 #14639823

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₁×#510 #14639823