Subgroups — C₂²×M₁₆×Dic₁₁ #15617159

Order 2816.

3950 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#4318	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄×Dic₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂×M₁₆×Dic₁₁	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#66879	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×Dic₁₁	C₂	1	2	yes	maximal, complemented
6	C₂³×C₁₁×M₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×C₄×C₁₁	C₂²	1	1	yes	characteristic
8	C₂⁴×Dic₁₁	C₄	1	2	yes	—
9	C₂²×C₁₁×M₁₆	C₄	1	16	yes	complemented
10	C₂²×C₁₁×M₁₆	C₂²	1	12	yes	—
11	C₂³×C₈×C₁₁	C₂²	1	2	yes	—
12	C₂²×C₄×Dic₁₁	C₄	1	12	yes	—
13	C₂²×C₄×Dic₁₁	C₄	1	2	yes	—
14	C₂²×C₄×Dic₁₁	C₂²	1	1	yes	characteristic
15	C₂²×C₄×Dic₁₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄×Dic₁₁	C₂²	1	12	yes	—
17	C₂²×#4318	C₄	1	16	yes	complemented
18	C₂²×#4318	C₂²	1	12	yes	—
19	C₂³×C₁₁⋊C₈	C₂²	1	2	yes	—
20	M₁₆×Dic₁₁	C₂²	1	64	yes	direct product factor
21	C₂×#66879	C₂²	1	24	yes	complemented
22	C₂×C₈×Dic₁₁	C₂²	1	24	yes	complemented
23	C₂⁵×C₁₁	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₁	C₂×C₄	1	6	yes	—
25	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₁	C₂×C₄	1	8	yes	—
27	C₂³×C₄×C₁₁	C₂×C₄	1	8	yes	—
28	C₂³×C₄×C₁₁	C₂³	1	6	yes	—
29	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
30	C₂³×Dic₁₁	—	2	16	no	—
31	C₂³×Dic₁₁	C₂×C₄	1	24	yes	—
32	C₂³×Dic₁₁	C₂×C₄	1	2	yes	—
33	C₂³×Dic₁₁	C₂×C₄	1	2	yes	—
34	C₂×C₁₁×M₁₆	C₂×C₄	1	96	yes	complemented
35	C₂×C₁₁×M₁₆	C₂³	1	16	yes	—
36	C₂²×C₈×C₁₁	C₂×C₄	1	16	yes	complemented
37	C₂²×C₈×C₁₁	C₂³	1	12	yes	—
38	C₂×C₄×Dic₁₁	C₂×C₄	1	6	yes	—
39	C₂×C₄×Dic₁₁	C₂×C₄	1	48	yes	—
40	C₂×C₄×Dic₁₁	C₂×C₄	1	24	yes	—
41	C₂×C₄×Dic₁₁	C₂×C₄	1	6	yes	—
42	C₂×C₄×Dic₁₁	C₂³	1	16	yes	—
43	C₂×C₄×Dic₁₁	C₂³	1	6	yes	—
44	C₂×C₄×Dic₁₁	C₂³	1	6	yes	—
45	C₂×#4318	C₂×C₄	1	96	yes	complemented
46	C₂×#4318	C₂³	1	16	yes	—
47	C₂²×C₁₁⋊C₈	C₂×C₄	1	16	yes	complemented
48	C₂²×C₁₁⋊C₈	C₂³	1	12	yes	—
49	#66879	C₂³	1	32	yes	complemented
50	C₈×Dic₁₁	C₂³	1	32	yes	complemented
51	C₂²×C₄×M₁₆	—	11	1	no	maximal, Sylow
52	C₂⁴×C₁₁	—	2	8	no	—
53	C₂⁴×C₁₁	C₄²	1	8	yes	—
54	C₂⁴×C₁₁	C₂²×C₄	1	6	yes	—
55	C₂⁴×C₁₁	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₁	C₄²	1	48	yes	—
57	C₂²×C₄×C₁₁	C₄²	1	8	yes	—
58	C₂²×C₄×C₁₁	C₂²×C₄	1	12	yes	—
59	C₂²×C₄×C₁₁	C₂²×C₄	1	6	yes	—
60	C₂²×C₄×C₁₁	C₂²×C₄	1	3	yes	—
61	C₂²×C₄×C₁₁	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₁₁	C₂²×C₄	1	24	yes	—
63	C₂²×C₄×C₁₁	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₁₁	C₂²×C₄	1	24	yes	—
65	C₂²×C₄×C₁₁	C₂⁴	1	3	yes	—
66	C₂²×C₄×C₁₁	C₂⁴	1	4	yes	—
67	C₂²×Dic₁₁	—	2	96	no	—
68	C₂²×Dic₁₁	—	2	8	no	—
69	C₂²×Dic₁₁	M₁₆	1	16	yes	direct product factor
70	C₂²×Dic₁₁	C₂²×C₄	1	32	yes	—
71	C₂²×Dic₁₁	C₂²×C₄	1	12	yes	—
72	C₂²×Dic₁₁	C₂²×C₄	1	12	yes	—
73	C₁₁×M₁₆	C₂²×C₄	1	64	yes	complemented
74	C₂×C₈×C₁₁	C₂²×C₄	1	48	yes	complemented
75	C₂×C₈×C₁₁	C₂⁴	1	8	yes	—
76	C₄×Dic₁₁	C₂²×C₄	1	12	yes	—
77	C₄×Dic₁₁	C₂²×C₄	1	32	yes	—
78	C₄×Dic₁₁	C₂²×C₄	1	12	yes	—
79	C₄×Dic₁₁	C₂⁴	1	4	yes	—
80	C₄×Dic₁₁	C₂⁴	1	4	yes	—
81	#4318	C₂²×C₄	1	64	yes	complemented
82	C₂×C₁₁⋊C₈	C₂²×C₄	1	48	yes	complemented
83	C₂×C₁₁⋊C₈	C₂⁴	1	8	yes	—
84	C₂³×M₁₆	—	11	1	no	—
85	C₂³×M₁₆	D₁₁	1	1	yes	characteristic
86	C₂³×C₄²	—	11	1	no	—
87	C₂×C₄×M₁₆	—	11	24	no	—
88	C₂²×#97	—	11	2	no	—
89	C₂²×C₄×C₈	—	11	2	no	—
90	C₂³×C₁₁	—	2	24	no	—
91	C₂³×C₁₁	—	2	32	no	—
92	C₂³×C₁₁	C₂×C₄²	1	24	yes	—
93	C₂³×C₁₁	C₂×C₄²	1	4	yes	—
94	C₂³×C₁₁	C₂×M₁₆	1	8	yes	—
95	C₂³×C₁₁	C₂³×C₄	1	4	yes	—
96	C₂³×C₁₁	C₂³×C₄	1	3	yes	—
97	C₂×C₄×C₁₁	C₂×C₄²	1	48	yes	—
98	C₂×C₄×C₁₁	C₂×C₄²	1	12	yes	—
99	C₂×C₄×C₁₁	C₂×C₄²	1	24	yes	—
100	C₂×C₄×C₁₁	C₂³×C₄	1	6	yes	—
101	C₂×C₄×C₁₁	C₂³×C₄	1	8	yes	—
102	C₂×C₄×C₁₁	C₂³×C₄	1	3	yes	—
103	C₂×C₄×C₁₁	C₂³×C₄	1	4	yes	—
104	C₂×C₄×C₁₁	C₂³×C₄	1	6	yes	—
105	C₂×C₄×C₁₁	C₂³×C₄	1	8	yes	—
106	C₂×C₄×C₁₁	C₂⁵	1	1	yes	characteristic
107	C₂×Dic₁₁	—	2	24	no	—
108	C₂×Dic₁₁	—	2	64	no	—
109	C₂×Dic₁₁	C₂×M₁₆	1	48	yes	direct product factor
110	C₂×Dic₁₁	C₂³×C₄	1	8	yes	—
111	C₂×Dic₁₁	C₂³×C₄	1	8	yes	—
112	C₈×C₁₁	C₂³×C₄	1	16	yes	complemented
113	C₁₁⋊C₈	C₂³×C₄	1	16	yes	complemented
114	C₂⁴×C₄	—	11	2	no	—
115	C₂⁴×C₄	C₂×D₁₁	1	1	yes	characteristic
116	C₂²×M₁₆	—	11	12	no	—
117	C₂²×M₁₆	—	11	16	no	—
118	C₂²×M₁₆	Dic₁₁	1	16	yes	direct product factor
119	C₂²×M₁₆	C₂×D₁₁	1	12	yes	—
120	C₂³×C₈	—	11	2	no	—
121	C₂³×C₈	C₂×D₁₁	1	2	yes	—
122	C₂²×C₄²	—	11	12	no	—
123	C₂²×C₄²	—	11	1	no	—
124	C₂²×C₄²	—	11	1	no	—
125	C₂²×C₄²	—	11	2	no	—
126	C₂²×C₄²	—	11	12	no	—
127	C₄×M₁₆	—	11	64	no	—
128	C₂×#97	—	11	24	no	—
129	C₂×C₄×C₈	—	11	24	no	—
130	C₂²×C₁₁	—	2	48	no	—
131	C₂²×C₁₁	—	2	8	no	—
132	C₂²×C₁₁	C₄×M₁₆	1	16	yes	complemented
133	C₂²×C₁₁	C₂²×C₄²	1	8	yes	—
134	C₂²×C₁₁	C₂²×C₄²	1	6	yes	—
135	C₂²×C₁₁	C₂²×M₁₆	1	12	yes	—
136	C₂²×C₁₁	C₂⁴×C₄	1	1	yes	characteristic
137	C₄×C₁₁	C₂²×C₄²	1	6	yes	—
138	C₄×C₁₁	C₂²×C₄²	1	8	yes	—
139	C₄×C₁₁	C₂⁴×C₄	1	1	yes	characteristic
140	C₄×C₁₁	C₂⁴×C₄	1	1	yes	characteristic
141	Dic₁₁	—	2	8	no	—
142	Dic₁₁	C₂²×M₁₆	1	16	yes	direct product factor
143	C₂⁵	C₄×D₁₁	1	1	yes	characteristic
144	C₂³×C₄	—	11	2	no	—
145	C₂³×C₄	—	11	24	no	—
146	C₂³×C₄	—	11	2	no	—
147	C₂³×C₄	—	22	16	no	—
148	C₂³×C₄	C₄×D₁₁	1	1	yes	characteristic
149	C₂³×C₄	C₄×D₁₁	1	6	yes	—
150	C₂³×C₄	C₄×D₁₁	1	8	yes	—
151	C₂³×C₄	C₂×Dic₁₁	1	8	yes	—
152	C₂³×C₄	C₂²×D₁₁	1	6	yes	—
153	C₂³×C₄	C₂²×D₁₁	1	1	yes	characteristic
154	C₂×M₁₆	—	11	96	no	—
155	C₂×M₁₆	—	11	16	no	—
156	C₂×M₁₆	C₂×Dic₁₁	1	96	yes	direct product factor
157	C₂×M₁₆	C₂²×D₁₁	1	16	yes	—
158	C₂²×C₈	—	11	16	no	—
159	C₂²×C₈	—	11	12	no	—
160	C₂²×C₈	C₂×Dic₁₁	1	16	yes	complemented
161	C₂²×C₈	C₂²×D₁₁	1	12	yes	—
162	C₂×C₄²	—	11	6	no	—
163	C₂×C₄²	—	11	24	no	—
164	C₂×C₄²	—	11	48	no	—
165	C₂×C₄²	—	11	6	no	—
166	C₂×C₄²	—	11	6	no	—
167	C₂×C₄²	—	11	6	no	—
168	C₂×C₄²	—	11	16	no	—
169	#97	—	11	32	no	—
170	C₄×C₈	—	11	32	no	—
171	C₂×C₁₁	—	2	8	no	—
172	C₂×C₁₁	C₂×C₄×M₁₆	1	12	yes	complemented
173	C₂×C₁₁	C₂³×C₄²	1	1	yes	characteristic
174	C₂×C₁₁	C₂³×M₁₆	1	2	yes	—
175	C₂⁴	—	2	8	no	—
176	C₂⁴	C₄×Dic₁₁	1	8	yes	—
177	C₂⁴	C₂×C₄×D₁₁	1	6	yes	—
178	C₂⁴	C₂×C₄×D₁₁	1	1	yes	characteristic
179	C₂²×C₄	—	11	16	no	—
180	C₂²×C₄	—	11	12	no	—
181	C₂²×C₄	—	11	32	no	—
182	C₂²×C₄	—	11	12	no	—
183	C₂²×C₄	—	22	96	no	—
184	C₂²×C₄	—	22	8	no	—
185	C₂²×C₄	C₄×Dic₁₁	1	48	yes	—
186	C₂²×C₄	C₄×Dic₁₁	1	8	yes	—
187	C₂²×C₄	C₂×C₄×D₁₁	1	12	yes	—
188	C₂²×C₄	C₂×C₄×D₁₁	1	6	yes	—
189	C₂²×C₄	C₂×C₄×D₁₁	1	3	yes	—
190	C₂²×C₄	C₂×C₄×D₁₁	1	4	yes	—
191	C₂²×C₄	C₂×C₄×D₁₁	1	24	yes	—
192	C₂²×C₄	C₂²×Dic₁₁	1	4	yes	—
193	C₂²×C₄	C₂²×Dic₁₁	1	24	yes	—
194	C₂²×C₄	C₂³×D₁₁	1	3	yes	—
195	C₂²×C₄	C₂³×D₁₁	1	4	yes	—
196	M₁₆	—	11	64	no	—
197	M₁₆	C₂²×Dic₁₁	1	64	yes	direct product factor
198	C₂×C₈	—	11	48	no	—
199	C₂×C₈	—	11	8	no	—
200	C₂×C₈	C₂²×Dic₁₁	1	48	yes	complemented
201	C₂×C₈	C₂³×D₁₁	1	8	yes	—
202	C₄²	—	11	12	no	—
203	C₄²	—	11	32	no	—
204	C₄²	—	11	12	no	—
205	C₄²	—	11	4	no	—
206	C₄²	—	11	4	no	—
207	C₁₁	C₂²×C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
208	C₂³	—	2	32	no	—
209	C₂³	—	2	24	no	—
210	C₂³	M₁₆×D₁₁	1	8	yes	—
211	C₂³	C₂×C₄×Dic₁₁	1	24	yes	—
212	C₂³	C₂×C₄×Dic₁₁	1	4	yes	—
213	C₂³	C₂²×C₄×D₁₁	1	4	yes	—
214	C₂³	C₂²×C₄×D₁₁	1	3	yes	—
215	C₂×C₄	—	11	48	no	—
216	C₂×C₄	—	11	8	no	—
217	C₂×C₄	—	11	8	no	—
218	C₂×C₄	—	22	64	no	—
219	C₂×C₄	—	22	24	no	—
220	C₂×C₄	C₂×C₄×Dic₁₁	1	48	yes	—
221	C₂×C₄	C₂×C₄×Dic₁₁	1	24	yes	—
222	C₂×C₄	C₂×C₄×Dic₁₁	1	12	yes	—
223	C₂×C₄	C₂²×C₄×D₁₁	1	4	yes	—
224	C₂×C₄	C₂²×C₄×D₁₁	1	3	yes	—
225	C₂×C₄	C₂²×C₄×D₁₁	1	6	yes	—
226	C₂×C₄	C₂²×C₄×D₁₁	1	8	yes	—
227	C₂×C₄	C₂³×Dic₁₁	1	6	yes	—
228	C₂×C₄	C₂³×Dic₁₁	1	8	yes	—
229	C₂×C₄	C₂⁴×D₁₁	1	1	yes	characteristic
230	C₈	—	11	16	no	—
231	C₈	C₂³×Dic₁₁	1	16	yes	complemented
232	C₂²	—	2	48	no	—
233	C₂²	—	2	8	no	—
234	C₂²	M₁₆×Dic₁₁	1	16	yes	direct product factor
235	C₂²	C₂×M₁₆×D₁₁	1	12	yes	—
236	C₂²	C₂²×C₄×Dic₁₁	1	8	yes	—
237	C₂²	C₂²×C₄×Dic₁₁	1	6	yes	—
238	C₂²	C₂³×C₄×D₁₁	1	1	yes	characteristic
239	C₄	—	11	16	no	—
240	C₄	—	22	8	no	—
241	C₄	C₂²×C₄×Dic₁₁	1	6	yes	—
242	C₄	C₂²×C₄×Dic₁₁	1	8	yes	—
243	C₄	C₂³×C₄×D₁₁	1	1	yes	characteristic
244	C₄	C₂⁴×Dic₁₁	1	1	yes	characteristic
245	C₂	—	2	8	no	—
246	C₂	C₂×M₁₆×Dic₁₁	1	12	yes	direct product factor
247	C₂	C₂³×C₄×Dic₁₁	1	1	yes	characteristic
248	C₂	C₂²×M₁₆×D₁₁	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₁₁
commutator	C₂×C₁₁	C₂³×C₄²
Frattini	C₂×C₄	C₂⁴×D₁₁
Fitting	C₂³×C₁₁×M₁₆	C₂
socle	C₂⁴×C₁₁	C₂²×C₄
prime radical	C₂⁵×C₁₁	C₂×C₄
cosocle	C₂×C₄×C₁₁	C₂⁵
hypercenter	C₂³×M₁₆	D₁₁
hypocenter	C₁₁	C₂²×C₄×M₁₆
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×M₁₆×Dic₁₁	C₂	direct (N × H)
4. C₂²×#66879	C₂	semidirect
5. C₂²×C₈×Dic₁₁	C₂	semidirect
9. C₂²×C₁₁×M₁₆	C₄	semidirect
17. C₂²×#4318	C₄	semidirect
20. M₁₆×Dic₁₁	C₂²	direct (N × H)
21. C₂×#66879	C₂²	semidirect
22. C₂×C₈×Dic₁₁	C₂²	semidirect
34. C₂×C₁₁×M₁₆	C₂×C₄	semidirect
36. C₂²×C₈×C₁₁	C₂×C₄	semidirect
45. C₂×#4318	C₂×C₄	semidirect
47. C₂²×C₁₁⋊C₈	C₂×C₄	semidirect
49. #66879	C₂³	semidirect
50. C₈×Dic₁₁	C₂³	semidirect
69. C₂²×Dic₁₁	M₁₆	direct (N × H)
73. C₁₁×M₁₆	C₂²×C₄	semidirect
74. C₂×C₈×C₁₁	C₂²×C₄	semidirect
81. #4318	C₂²×C₄	semidirect
82. C₂×C₁₁⋊C₈	C₂²×C₄	semidirect
109. C₂×Dic₁₁	C₂×M₁₆	direct (N × H)
112. C₈×C₁₁	C₂³×C₄	semidirect
113. C₁₁⋊C₈	C₂³×C₄	semidirect
118. C₂²×M₁₆	Dic₁₁	direct (N × H)
132. C₂²×C₁₁	C₄×M₁₆	semidirect
142. Dic₁₁	C₂²×M₁₆	direct (N × H)
156. C₂×M₁₆	C₂×Dic₁₁	direct (N × H)
160. C₂²×C₈	C₂×Dic₁₁	semidirect
172. C₂×C₁₁	C₂×C₄×M₁₆	semidirect
197. M₁₆	C₂²×Dic₁₁	direct (N × H)
200. C₂×C₈	C₂²×Dic₁₁	semidirect
207. C₁₁	C₂²×C₄×M₁₆	semidirect
231. C₈	C₂³×Dic₁₁	semidirect
234. C₂²	M₁₆×Dic₁₁	direct (N × H)
246. C₂	C₂×M₁₆×Dic₁₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#4318	2
C₂³×C₄×Dic₁₁	2
C₂×M₁₆×Dic₁₁	2
C₂²×#66879	2
C₂²×C₈×Dic₁₁	2
C₂³×C₁₁×M₁₆	2
C₂²×C₄×M₁₆	11

Subgroups — C22×M16×Dic11 #15617159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×M₁₆×Dic₁₁ #15617159