Subgroups — C₂²×C₁₁×#408 #14637254

Order 2816.

5326 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₁×#41	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₁₁×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂⁴×C₈×C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₁×#408	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₁₁×C₂²⋊C₈	C₂	1	4	yes	maximal, complemented
6	C₂²×C₁₁×#41	—	2	8	no	—
7	C₂²×C₁₁×#41	—	2	8	no	—
8	C₂²×C₁₁×#41	C₄	1	12	yes	—
9	C₂²×C₁₁×#41	C₂²	1	12	yes	—
10	C₂³×Q₈×C₁₁	C₄	1	1	yes	characteristic
11	C₂³×D₄×C₁₁	C₄	1	2	yes	—
12	C₂³×D₄×C₁₁	C₄	1	1	yes	characteristic
13	C₂⁴×C₄×C₁₁	C₂²	1	2	yes	—
14	C₂⁴×C₄×C₁₁	C₂²	1	1	yes	characteristic
15	C₂²×C₁₁×M₁₆	—	2	8	no	—
16	C₂²×C₁₁×M₁₆	C₂²	1	12	yes	complemented
17	C₂³×C₈×C₁₁	—	2	8	no	—
18	C₂³×C₈×C₁₁	C₂²	1	2	yes	—
19	C₂³×C₈×C₁₁	C₂²	1	2	yes	—
20	C₂³×C₈×C₁₁	C₂²	1	12	yes	complemented
21	C₁₁×#408	C₂²	1	64	yes	direct product factor
22	C₂×C₁₁×C₂²⋊C₈	C₂²	1	48	yes	complemented
23	C₂⁵×C₁₁	C₂×C₄	1	2	yes	—
24	C₂⁵×C₁₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₁×#41	—	2	96	no	—
26	C₂×C₁₁×#41	—	2	48	no	—
27	C₂×C₁₁×#41	—	2	48	no	—
28	C₂×C₁₁×#41	C₂×C₄	1	48	yes	—
29	C₂×C₁₁×#41	C₂³	1	16	yes	—
30	C₂²×Q₈×C₁₁	—	2	8	no	—
31	C₂²×Q₈×C₁₁	C₂×C₄	1	12	yes	—
32	C₂²×D₄×C₁₁	—	2	16	no	—
33	C₂²×D₄×C₁₁	—	2	8	no	—
34	C₂²×D₄×C₁₁	C₂×C₄	1	24	yes	—
35	C₂²×D₄×C₁₁	C₂×C₄	1	12	yes	—
36	C₂³×C₄×C₁₁	—	2	8	no	—
37	C₂³×C₄×C₁₁	—	2	8	no	—
38	C₂³×C₄×C₁₁	C₂×C₄	1	2	yes	—
39	C₂³×C₄×C₁₁	C₂×C₄	1	12	yes	—
40	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₁₁	C₂×C₄	1	6	yes	—
42	C₂³×C₄×C₁₁	D₄	1	8	yes	—
43	C₂³×C₄×C₁₁	D₄	1	8	yes	—
44	C₂³×C₄×C₁₁	C₂³	1	12	yes	—
45	C₂³×C₄×C₁₁	C₂³	1	6	yes	—
46	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
47	C₂×C₁₁×M₁₆	—	2	48	no	—
48	C₂×C₁₁×M₁₆	C₂³	1	16	yes	complemented
49	C₂²×C₈×C₁₁	—	2	8	no	—
50	C₂²×C₈×C₁₁	—	2	48	no	—
51	C₂²×C₈×C₁₁	—	2	8	no	—
52	C₂²×C₈×C₁₁	C₂³	1	12	yes	—
53	C₂²×C₈×C₁₁	C₂³	1	12	yes	—
54	C₂²×C₈×C₁₁	C₂³	1	16	yes	complemented
55	C₁₁×C₂²⋊C₈	C₂³	1	64	yes	complemented
56	C₂²×#408	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂⁴×C₁₁	—	2	8	no	—
58	C₂⁴×C₁₁	—	2	8	no	—
59	C₂⁴×C₁₁	—	2	16	no	—
60	C₂⁴×C₁₁	C₂²⋊C₄	1	8	yes	—
61	C₂⁴×C₁₁	C₂²×C₄	1	12	yes	—
62	C₂⁴×C₁₁	C₂²×C₄	1	6	yes	—
63	C₂⁴×C₁₁	C₂²×C₄	1	1	yes	characteristic
64	C₁₁×#41	—	2	192	no	—
65	C₁₁×#41	—	2	32	no	—
66	C₁₁×#41	—	2	32	no	—
67	C₂×Q₈×C₁₁	—	2	48	no	—
68	C₂×Q₈×C₁₁	C₂²×C₄	1	16	yes	—
69	C₂×D₄×C₁₁	—	2	96	no	—
70	C₂×D₄×C₁₁	—	2	48	no	—
71	C₂×D₄×C₁₁	C₂²×C₄	1	32	yes	—
72	C₂×D₄×C₁₁	C₂²×C₄	1	16	yes	—
73	C₂²×C₄×C₁₁	—	2	8	no	—
74	C₂²×C₄×C₁₁	—	2	48	no	—
75	C₂²×C₄×C₁₁	—	2	24	no	—
76	C₂²×C₄×C₁₁	—	2	24	no	—
77	C₂²×C₄×C₁₁	C₂²⋊C₄	1	8	yes	—
78	C₂²×C₄×C₁₁	C₂²⋊C₄	1	48	yes	—
79	C₂²×C₄×C₁₁	C₂²×C₄	1	12	yes	—
80	C₂²×C₄×C₁₁	C₂²×C₄	1	24	yes	—
81	C₂²×C₄×C₁₁	C₂²×C₄	1	3	yes	—
82	C₂²×C₄×C₁₁	C₂²×C₄	1	6	yes	—
83	C₂²×C₄×C₁₁	C₂²×C₄	1	12	yes	—
84	C₂²×C₄×C₁₁	C₂×D₄	1	4	yes	—
85	C₂²×C₄×C₁₁	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₁₁	C₂×D₄	1	24	yes	—
87	C₂²×C₄×C₁₁	C₂×D₄	1	24	yes	—
88	C₂²×C₄×C₁₁	C₂⁴	1	8	yes	—
89	C₂²×C₄×C₁₁	C₂⁴	1	3	yes	—
90	C₂²×C₄×C₁₁	C₂⁴	1	4	yes	—
91	C₁₁×M₁₆	—	2	32	no	—
92	C₂×C₈×C₁₁	—	2	24	no	—
93	C₂×C₈×C₁₁	—	2	32	no	—
94	C₂×C₈×C₁₁	—	2	24	no	—
95	C₂×C₈×C₁₁	C₂⁴	1	8	yes	—
96	C₂×C₈×C₁₁	C₂⁴	1	8	yes	—
97	C₂³×#41	C₂×C₁₁	1	1	yes	characteristic
98	C₂³×M₁₆	C₂×C₁₁	1	1	yes	characteristic, complemented
99	C₂⁴×C₈	C₂×C₁₁	1	1	yes	characteristic, complemented
100	C₂×#408	C₂×C₁₁	1	24	yes	direct product factor
101	C₂²×C₂²⋊C₈	C₂×C₁₁	1	4	yes	complemented
102	C₂³×C₁₁	—	2	24	no	—
103	C₂³×C₁₁	—	2	32	no	—
104	C₂³×C₁₁	—	2	24	no	—
105	C₂³×C₁₁	—	2	48	no	—
106	C₂³×C₁₁	—	4	32	no	—
107	C₂³×C₁₁	C₂×C₂²⋊C₄	1	24	yes	—
108	C₂³×C₁₁	C₂×C₂²⋊C₄	1	4	yes	—
109	C₂³×C₁₁	#131	1	8	yes	—
110	C₂³×C₁₁	C₂³×C₄	1	8	yes	—
111	C₂³×C₁₁	C₂³×C₄	1	4	yes	—
112	C₂³×C₁₁	C₂³×C₄	1	3	yes	—
113	Q₈×C₁₁	—	2	32	no	—
114	D₄×C₁₁	—	2	64	no	—
115	D₄×C₁₁	—	2	32	no	—
116	C₂×C₄×C₁₁	—	2	24	no	—
117	C₂×C₄×C₁₁	—	2	48	no	—
118	C₂×C₄×C₁₁	—	2	8	no	—
119	C₂×C₄×C₁₁	—	2	8	no	—
120	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	12	yes	—
121	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	24	yes	—
122	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	48	yes	—
123	C₂×C₄×C₁₁	C₂³×C₄	1	8	yes	—
124	C₂×C₄×C₁₁	C₂³×C₄	1	3	yes	—
125	C₂×C₄×C₁₁	C₂³×C₄	1	4	yes	—
126	C₂×C₄×C₁₁	C₂²×D₄	1	6	yes	—
127	C₂×C₄×C₁₁	C₂²×D₄	1	6	yes	—
128	C₂×C₄×C₁₁	C₂²×D₄	1	8	yes	—
129	C₂×C₄×C₁₁	C₂²×D₄	1	8	yes	—
130	C₂×C₄×C₁₁	C₂⁵	1	1	yes	characteristic
131	C₈×C₁₁	—	2	8	no	—
132	C₈×C₁₁	—	2	8	no	—
133	C₂²×#41	—	2	8	no	—
134	C₂²×#41	—	2	8	no	—
135	C₂²×#41	C₄×C₁₁	1	12	yes	—
136	C₂²×#41	C₂²×C₁₁	1	12	yes	—
137	C₂³×Q₈	C₄×C₁₁	1	1	yes	characteristic
138	C₂³×D₄	C₄×C₁₁	1	2	yes	—
139	C₂³×D₄	C₄×C₁₁	1	1	yes	characteristic
140	C₂⁴×C₄	C₂²×C₁₁	1	2	yes	—
141	C₂⁴×C₄	C₂²×C₁₁	1	1	yes	characteristic
142	C₂²×M₁₆	—	2	8	no	—
143	C₂²×M₁₆	C₂²×C₁₁	1	12	yes	complemented
144	C₂³×C₈	—	2	8	no	—
145	C₂³×C₈	C₂²×C₁₁	1	2	yes	—
146	C₂³×C₈	C₂²×C₁₁	1	2	yes	—
147	C₂³×C₈	C₂²×C₁₁	1	12	yes	complemented
148	#408	C₂²×C₁₁	1	64	yes	direct product factor
149	C₂×C₂²⋊C₈	C₂²×C₁₁	1	48	yes	complemented
150	C₂²×C₁₁	—	2	48	no	—
151	C₂²×C₁₁	—	2	8	no	—
152	C₂²×C₁₁	—	2	8	no	—
153	C₂²×C₁₁	—	2	16	no	—
154	C₂²×C₁₁	—	4	48	no	—
155	C₂²×C₁₁	#408	1	16	yes	direct product factor
156	C₂²×C₁₁	C₂²×C₂²⋊C₄	1	8	yes	—
157	C₂²×C₁₁	C₂²×C₂²⋊C₄	1	6	yes	—
158	C₂²×C₁₁	C₂×#131	1	12	yes	—
159	C₂²×C₁₁	C₂⁴×C₄	1	1	yes	characteristic
160	C₄×C₁₁	—	2	8	no	—
161	C₄×C₁₁	C₂²×C₂²⋊C₄	1	6	yes	—
162	C₄×C₁₁	C₂²×C₂²⋊C₄	1	8	yes	—
163	C₄×C₁₁	C₂³×D₄	1	1	yes	characteristic
164	C₄×C₁₁	C₂³×D₄	1	1	yes	characteristic
165	C₂⁵	C₂×C₄×C₁₁	1	2	yes	—
166	C₂⁵	C₂×C₄×C₁₁	1	1	yes	characteristic
167	C₂×#41	—	2	96	no	—
168	C₂×#41	—	2	48	no	—
169	C₂×#41	—	2	48	no	—
170	C₂×#41	C₂×C₄×C₁₁	1	48	yes	—
171	C₂×#41	C₂³×C₁₁	1	16	yes	—
172	C₂²×Q₈	—	2	8	no	—
173	C₂²×Q₈	C₂×C₄×C₁₁	1	12	yes	—
174	C₂²×D₄	—	2	16	no	—
175	C₂²×D₄	—	2	8	no	—
176	C₂²×D₄	C₂×C₄×C₁₁	1	24	yes	—
177	C₂²×D₄	C₂×C₄×C₁₁	1	12	yes	—
178	C₂³×C₄	—	2	8	no	—
179	C₂³×C₄	—	2	8	no	—
180	C₂³×C₄	C₂×C₄×C₁₁	1	2	yes	—
181	C₂³×C₄	C₂×C₄×C₁₁	1	12	yes	—
182	C₂³×C₄	C₂×C₄×C₁₁	1	1	yes	characteristic
183	C₂³×C₄	C₂×C₄×C₁₁	1	6	yes	—
184	C₂³×C₄	D₄×C₁₁	1	8	yes	—
185	C₂³×C₄	D₄×C₁₁	1	8	yes	—
186	C₂³×C₄	C₂³×C₁₁	1	12	yes	—
187	C₂³×C₄	C₂³×C₁₁	1	1	yes	characteristic
188	C₂³×C₄	C₂³×C₁₁	1	6	yes	—
189	C₂×M₁₆	—	2	48	no	—
190	C₂×M₁₆	C₂³×C₁₁	1	16	yes	complemented
191	C₂²×C₈	—	2	8	no	—
192	C₂²×C₈	—	2	48	no	—
193	C₂²×C₈	—	2	8	no	—
194	C₂²×C₈	C₂³×C₁₁	1	12	yes	—
195	C₂²×C₈	C₂³×C₁₁	1	12	yes	—
196	C₂²×C₈	C₂³×C₁₁	1	16	yes	complemented
197	C₂²⋊C₈	C₂³×C₁₁	1	64	yes	complemented
198	C₂×C₁₁	—	2	8	no	—
199	C₂×C₁₁	—	4	8	no	—
200	C₂×C₁₁	C₂×#408	1	12	yes	direct product factor
201	C₂×C₁₁	C₂³×C₂²⋊C₄	1	1	yes	characteristic
202	C₂×C₁₁	C₂²×#131	1	2	yes	—
203	C₂⁴	—	2	8	no	—
204	C₂⁴	—	2	8	no	—
205	C₂⁴	—	2	16	no	—
206	C₂⁴	C₁₁×C₂²⋊C₄	1	8	yes	—
207	C₂⁴	C₂²×C₄×C₁₁	1	12	yes	—
208	C₂⁴	C₂²×C₄×C₁₁	1	6	yes	—
209	C₂⁴	C₂²×C₄×C₁₁	1	1	yes	characteristic
210	#41	—	2	192	no	—
211	#41	—	2	32	no	—
212	#41	—	2	32	no	—
213	C₂×Q₈	—	2	48	no	—
214	C₂×Q₈	C₂²×C₄×C₁₁	1	16	yes	—
215	C₂×D₄	—	2	96	no	—
216	C₂×D₄	—	2	48	no	—
217	C₂×D₄	C₂²×C₄×C₁₁	1	32	yes	—
218	C₂×D₄	C₂²×C₄×C₁₁	1	16	yes	—
219	C₂²×C₄	—	2	8	no	—
220	C₂²×C₄	—	2	48	no	—
221	C₂²×C₄	—	2	24	no	—
222	C₂²×C₄	—	2	24	no	—
223	C₂²×C₄	C₁₁×C₂²⋊C₄	1	8	yes	—
224	C₂²×C₄	C₁₁×C₂²⋊C₄	1	48	yes	—
225	C₂²×C₄	C₂²×C₄×C₁₁	1	12	yes	—
226	C₂²×C₄	C₂²×C₄×C₁₁	1	24	yes	—
227	C₂²×C₄	C₂²×C₄×C₁₁	1	3	yes	—
228	C₂²×C₄	C₂²×C₄×C₁₁	1	6	yes	—
229	C₂²×C₄	C₂²×C₄×C₁₁	1	12	yes	—
230	C₂²×C₄	C₂×D₄×C₁₁	1	4	yes	—
231	C₂²×C₄	C₂×D₄×C₁₁	1	4	yes	—
232	C₂²×C₄	C₂×D₄×C₁₁	1	24	yes	—
233	C₂²×C₄	C₂×D₄×C₁₁	1	24	yes	—
234	C₂²×C₄	C₂⁴×C₁₁	1	8	yes	—
235	C₂²×C₄	C₂⁴×C₁₁	1	3	yes	—
236	C₂²×C₄	C₂⁴×C₁₁	1	4	yes	—
237	M₁₆	—	2	32	no	—
238	C₂×C₈	—	2	24	no	—
239	C₂×C₈	—	2	32	no	—
240	C₂×C₈	—	2	24	no	—
241	C₂×C₈	C₂⁴×C₁₁	1	8	yes	—
242	C₂×C₈	C₂⁴×C₁₁	1	8	yes	—
243	C₁₁	C₂²×#408	1	1	yes	characteristic, direct product factor, Sylow
244	C₂³	—	2	32	no	—
245	C₂³	—	2	24	no	—
246	C₂³	—	2	24	no	—
247	C₂³	—	2	48	no	—
248	C₂³	—	4	32	no	—
249	C₂³	C₂×C₁₁×C₂²⋊C₄	1	24	yes	—
250	C₂³	C₂×C₁₁×C₂²⋊C₄	1	4	yes	—
251	C₂³	C₁₁×#131	1	8	yes	—
252	C₂³	C₂³×C₄×C₁₁	1	8	yes	—
253	C₂³	C₂³×C₄×C₁₁	1	4	yes	—
254	C₂³	C₂³×C₄×C₁₁	1	3	yes	—
255	Q₈	—	2	32	no	—
256	D₄	—	2	64	no	—
257	D₄	—	2	32	no	—
258	C₂×C₄	—	2	24	no	—
259	C₂×C₄	—	2	48	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	C₂×C₁₁×C₂²⋊C₄	1	12	yes	—
263	C₂×C₄	C₂×C₁₁×C₂²⋊C₄	1	24	yes	—
264	C₂×C₄	C₂×C₁₁×C₂²⋊C₄	1	48	yes	—
265	C₂×C₄	C₂³×C₄×C₁₁	1	8	yes	—
266	C₂×C₄	C₂³×C₄×C₁₁	1	3	yes	—
267	C₂×C₄	C₂³×C₄×C₁₁	1	4	yes	—
268	C₂×C₄	C₂²×D₄×C₁₁	1	6	yes	—
269	C₂×C₄	C₂²×D₄×C₁₁	1	6	yes	—
270	C₂×C₄	C₂²×D₄×C₁₁	1	8	yes	—
271	C₂×C₄	C₂²×D₄×C₁₁	1	8	yes	—
272	C₂×C₄	C₂⁵×C₁₁	1	1	yes	characteristic
273	C₈	—	2	8	no	—
274	C₈	—	2	8	no	—
275	C₂²	—	2	8	no	—
276	C₂²	—	2	48	no	—
277	C₂²	—	2	8	no	—
278	C₂²	—	2	16	no	—
279	C₂²	—	4	48	no	—
280	C₂²	C₁₁×#408	1	16	yes	direct product factor
281	C₂²	C₂²×C₁₁×C₂²⋊C₄	1	8	yes	—
282	C₂²	C₂²×C₁₁×C₂²⋊C₄	1	6	yes	—
283	C₂²	C₂×C₁₁×#131	1	12	yes	—
284	C₂²	C₂⁴×C₄×C₁₁	1	1	yes	characteristic
285	C₄	—	2	8	no	—
286	C₄	C₂²×C₁₁×C₂²⋊C₄	1	6	yes	—
287	C₄	C₂²×C₁₁×C₂²⋊C₄	1	8	yes	—
288	C₄	C₂³×D₄×C₁₁	1	1	yes	characteristic
289	C₄	C₂³×D₄×C₁₁	1	1	yes	characteristic
290	C₂	—	2	8	no	—
291	C₂	—	4	8	no	—
292	C₂	C₂×C₁₁×#408	1	12	yes	direct product factor
293	C₂	C₂³×C₁₁×C₂²⋊C₄	1	1	yes	characteristic
294	C₂	C₂²×C₁₁×#131	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₁₁	C₂³
commutator	C₂²	C₂⁴×C₄×C₁₁
Frattini, cosocle	C₂×C₄	C₂⁵×C₁₁
socle	C₂⁴×C₁₁	C₂²×C₄
prime radical	C₂³×C₁₁×#41	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₁₁×M₁₆	C₂	semidirect
3. C₂⁴×C₈×C₁₁	C₂	semidirect
4. C₂×C₁₁×#408	C₂	direct (N × H)
5. C₂²×C₁₁×C₂²⋊C₈	C₂	semidirect
16. C₂²×C₁₁×M₁₆	C₂²	semidirect
20. C₂³×C₈×C₁₁	C₂²	semidirect
21. C₁₁×#408	C₂²	direct (N × H)
22. C₂×C₁₁×C₂²⋊C₈	C₂²	semidirect
48. C₂×C₁₁×M₁₆	C₂³	semidirect
54. C₂²×C₈×C₁₁	C₂³	semidirect
55. C₁₁×C₂²⋊C₈	C₂³	semidirect
56. C₂²×#408	C₁₁	direct (N × H)
98. C₂³×M₁₆	C₂×C₁₁	semidirect
99. C₂⁴×C₈	C₂×C₁₁	semidirect
100. C₂×#408	C₂×C₁₁	direct (N × H)
101. C₂²×C₂²⋊C₈	C₂×C₁₁	semidirect
143. C₂²×M₁₆	C₂²×C₁₁	semidirect
147. C₂³×C₈	C₂²×C₁₁	semidirect
148. #408	C₂²×C₁₁	direct (N × H)
149. C₂×C₂²⋊C₈	C₂²×C₁₁	semidirect
155. C₂²×C₁₁	#408	direct (N × H)
190. C₂×M₁₆	C₂³×C₁₁	semidirect
196. C₂²×C₈	C₂³×C₁₁	semidirect
197. C₂²⋊C₈	C₂³×C₁₁	semidirect
200. C₂×C₁₁	C₂×#408	direct (N × H)
243. C₁₁	C₂²×#408	direct (N × H)
280. C₂²	C₁₁×#408	direct (N × H)
292. C₂	C₂×C₁₁×#408	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×C₁₁×#41	2
C₂³×C₁₁×M₁₆	2
C₂⁴×C₈×C₁₁	2
C₂×C₁₁×#408	2
C₂²×C₁₁×C₂²⋊C₈	2
C₂²×#408	11

Subgroups — C22×C11×#408 #14637254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₁×#408 #14637254