Subgroups — C₂×C₁₁×#2078 #14600837

Order 2816.

950 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₁×#107	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₁×#417	C₂	1	2	yes	maximal
3	C₂×C₁₁×#408	C₂	1	1	yes	maximal, characteristic
4	C₁₁×#2078	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₁×#343	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₁×#341	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₁₁×#41	C₂²	1	1	yes	characteristic
8	C₂²×C₁₁×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
11	C₂²×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₁₁×#107	—	2	2	no	—
13	C₂×C₁₁×#107	—	2	2	no	—
14	C₂×C₁₁×#107	C₂²	1	4	yes	complemented
15	C₁₁×#417	—	2	4	no	—
16	C₁₁×#417	C₂²	1	8	yes	—
17	C₂×C₁₁×#103	—	2	2	no	—
18	C₂×C₁₁×#102	—	2	2	no	—
19	C₁₁×#408	—	2	2	no	—
20	C₁₁×#408	C₂²	1	4	yes	—
21	C₂×C₁₁×C₂²⋊C₈	—	2	2	no	—
22	C₁₁×#343	C₂²	1	8	yes	complemented
23	C₁₁×#341	C₂²	1	4	yes	complemented
24	C₂×C₁₁×#41	—	2	1	no	—
25	C₂×C₁₁×#41	—	2	1	no	—
26	C₂×C₁₁×#41	—	2	4	no	—
27	C₂×C₁₁×#41	—	2	4	no	—
28	C₂×C₁₁×#41	D₄	1	2	yes	—
29	C₂×C₁₁×#41	C₂³	1	2	yes	—
30	C₂²×Q₈×C₁₁	D₄	1	1	yes	characteristic
31	C₂²×D₄×C₁₁	—	2	1	no	—
32	C₂²×D₄×C₁₁	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₁₁	—	2	1	no	—
34	C₂³×C₄×C₁₁	—	2	2	no	—
35	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
36	C₂×C₁₁×M₁₆	—	2	1	no	—
37	C₂×C₁₁×M₁₆	—	2	2	no	—
38	C₂×C₁₁×M₁₆	—	2	4	no	—
39	C₂×C₁₁×M₁₆	—	2	4	no	—
40	C₂×C₁₁×M₁₆	—	2	4	no	—
41	C₂×C₁₁×M₁₆	C₂³	1	4	yes	—
42	C₂×C₁₁×M₁₆	C₂³	1	2	yes	—
43	C₂²×C₈×C₁₁	—	2	4	no	—
44	C₂²×C₈×C₁₁	—	2	1	no	—
45	C₂²×C₈×C₁₁	—	2	2	no	—
46	C₂²×C₈×C₁₁	D₄	1	2	yes	—
47	C₂²×C₈×C₁₁	D₄	1	2	yes	—
48	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
49	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
50	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
51	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
52	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
53	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
54	C₂×C₁₁×C₄⋊C₄	C₂³	1	4	yes	—
55	C₁₁×#107	—	2	8	no	—
56	C₁₁×#103	—	2	8	no	—
57	C₁₁×#102	—	2	8	no	—
58	C₁₁×C₂²⋊C₈	—	2	8	no	—
59	C₂×#2078	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂⁴×C₁₁	—	2	1	no	—
61	C₂⁴×C₁₁	#41	1	1	yes	characteristic
62	C₁₁×#41	—	2	8	no	—
63	C₁₁×#41	—	2	4	no	—
64	C₁₁×#41	—	2	4	no	—
65	C₁₁×#41	—	4	2	no	—
66	C₁₁×#41	—	4	2	no	—
67	C₁₁×#41	—	4	4	no	—
68	C₂×Q₈×C₁₁	—	2	1	no	—
69	C₂×Q₈×C₁₁	—	2	4	no	—
70	C₂×Q₈×C₁₁	C₂×D₄	1	2	yes	—
71	C₂×D₄×C₁₁	—	2	2	no	—
72	C₂×D₄×C₁₁	—	2	2	no	—
73	C₂×D₄×C₁₁	—	2	1	no	—
74	C₂×D₄×C₁₁	—	2	4	no	—
75	C₂×D₄×C₁₁	—	4	4	no	—
76	C₂×D₄×C₁₁	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₁₁	—	2	1	no	—
78	C₂²×C₄×C₁₁	—	2	2	no	—
79	C₂²×C₄×C₁₁	—	2	2	no	—
80	C₂²×C₄×C₁₁	—	2	4	no	—
81	C₂²×C₄×C₁₁	—	4	2	no	—
82	C₂²×C₄×C₁₁	—	4	2	no	—
83	C₂²×C₄×C₁₁	—	4	2	no	—
84	C₂²×C₄×C₁₁	—	4	2	no	—
85	C₂²×C₄×C₁₁	—	8	4	no	—
86	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
91	C₂²×C₄×C₁₁	#41	1	4	yes	—
92	C₂²×C₄×C₁₁	#41	1	4	yes	—
93	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
94	C₁₁×M₁₆	—	2	4	no	—
95	C₁₁×M₁₆	—	2	4	no	—
96	C₁₁×M₁₆	—	2	4	no	—
97	C₁₁×M₁₆	—	4	4	no	—
98	C₁₁×M₁₆	—	4	2	no	—
99	C₂×C₈×C₁₁	—	2	4	no	—
100	C₂×C₈×C₁₁	—	2	4	no	—
101	C₂×C₈×C₁₁	—	2	2	no	—
102	C₂×C₈×C₁₁	—	2	4	no	—
103	C₂×C₈×C₁₁	—	2	4	no	—
104	C₂×C₈×C₁₁	—	4	4	no	—
105	C₂×C₈×C₁₁	—	4	2	no	—
106	C₂×C₈×C₁₁	C₂×D₄	1	4	yes	—
107	C₁₁×C₄⋊C₄	—	2	4	no	—
108	C₁₁×C₄⋊C₄	—	2	4	no	—
109	C₁₁×C₄⋊C₄	—	4	2	no	—
110	C₁₁×C₄⋊C₄	—	4	2	no	—
111	C₂²×#107	C₂×C₁₁	1	1	yes	characteristic, complemented
112	C₂×#417	C₂×C₁₁	1	2	yes	—
113	C₂×#408	C₂×C₁₁	1	1	yes	characteristic
114	#2078	C₂×C₁₁	1	8	yes	direct product factor
115	C₂×#343	C₂×C₁₁	1	2	yes	complemented
116	C₂×#341	C₂×C₁₁	1	1	yes	characteristic, complemented
117	C₂³×C₁₁	—	2	2	no	—
118	C₂³×C₁₁	—	2	2	no	—
119	C₂³×C₁₁	—	2	2	no	—
120	C₂³×C₁₁	—	4	2	no	—
121	C₂³×C₁₁	—	4	2	no	—
122	C₂³×C₁₁	—	4	2	no	—
123	C₂³×C₁₁	#121	1	1	yes	characteristic
124	C₂³×C₁₁	#121	1	1	yes	characteristic
125	C₂³×C₁₁	#126	1	4	yes	—
126	C₂³×C₁₁	C₂×#41	1	1	yes	characteristic
127	Q₈×C₁₁	—	2	4	no	—
128	Q₈×C₁₁	—	4	2	no	—
129	D₄×C₁₁	—	2	4	no	—
130	D₄×C₁₁	—	4	4	no	—
131	D₄×C₁₁	—	4	4	no	—
132	D₄×C₁₁	—	4	2	no	—
133	C₂×C₄×C₁₁	—	2	2	no	—
134	C₂×C₄×C₁₁	—	4	2	no	—
135	C₂×C₄×C₁₁	—	4	2	no	—
136	C₂×C₄×C₁₁	—	4	2	no	—
137	C₂×C₄×C₁₁	—	4	4	no	—
138	C₂×C₄×C₁₁	—	4	4	no	—
139	C₂×C₄×C₁₁	—	8	4	no	—
140	C₂×C₄×C₁₁	—	8	4	no	—
141	C₂×C₄×C₁₁	—	8	4	no	—
142	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
143	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
144	C₂×C₄×C₁₁	#123	1	2	yes	—
145	C₂×C₄×C₁₁	#123	1	2	yes	—
146	C₂×C₄×C₁₁	#123	1	4	yes	—
147	C₂×C₄×C₁₁	#123	1	4	yes	—
148	C₂×C₄×C₁₁	#124	1	2	yes	—
149	C₂×C₄×C₁₁	#124	1	2	yes	—
150	C₂×C₄×C₁₁	#126	1	4	yes	—
151	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₁₁	C₂×#41	1	2	yes	—
154	C₂×C₄×C₁₁	C₂×#41	1	2	yes	—
155	C₈×C₁₁	—	2	4	no	—
156	C₈×C₁₁	—	4	4	no	—
157	C₈×C₁₁	—	4	2	no	—
158	C₂²×#41	C₂²×C₁₁	1	1	yes	characteristic
159	C₂²×M₁₆	C₂²×C₁₁	1	1	yes	characteristic
160	C₂²×M₁₆	C₂²×C₁₁	1	2	yes	—
161	C₂³×C₈	C₂²×C₁₁	1	1	yes	characteristic
162	C₂²×C₄⋊C₄	C₂²×C₁₁	1	2	yes	—
163	C₂×#107	—	2	2	no	—
164	C₂×#107	—	2	2	no	—
165	C₂×#107	C₂²×C₁₁	1	4	yes	complemented
166	#417	—	2	4	no	—
167	#417	C₂²×C₁₁	1	8	yes	—
168	C₂×#103	—	2	2	no	—
169	C₂×#102	—	2	2	no	—
170	#408	—	2	2	no	—
171	#408	C₂²×C₁₁	1	4	yes	—
172	C₂×C₂²⋊C₈	—	2	2	no	—
173	#343	C₂²×C₁₁	1	8	yes	complemented
174	#341	C₂²×C₁₁	1	4	yes	complemented
175	C₂²×C₁₁	—	2	8	no	—
176	C₂²×C₁₁	—	2	1	no	—
177	C₂²×C₁₁	—	2	1	no	—
178	C₂²×C₁₁	—	2	1	no	—
179	C₂²×C₁₁	—	2	1	no	—
180	C₂²×C₁₁	—	4	2	no	—
181	C₂²×C₁₁	—	4	2	no	—
182	C₂²×C₁₁	—	4	2	no	—
183	C₂²×C₁₁	—	8	4	no	—
184	C₂²×C₁₁	#397	1	2	yes	—
185	C₂²×C₁₁	#397	1	2	yes	—
186	C₂²×C₁₁	#472	1	2	yes	—
187	C₂²×C₁₁	#473	1	2	yes	—
188	C₂²×C₁₁	C₂×#121	1	1	yes	characteristic
189	C₂²×C₁₁	C₂×#126	1	2	yes	—
190	C₄×C₁₁	—	4	2	no	—
191	C₄×C₁₁	—	8	4	no	—
192	C₄×C₁₁	#397	1	2	yes	—
193	C₄×C₁₁	#397	1	2	yes	—
194	C₄×C₁₁	C₂×#123	1	1	yes	characteristic
195	C₄×C₁₁	C₂×#123	1	2	yes	—
196	C₄×C₁₁	C₂×#124	1	1	yes	characteristic
197	C₂×#41	—	2	1	no	—
198	C₂×#41	—	2	1	no	—
199	C₂×#41	—	2	4	no	—
200	C₂×#41	—	2	4	no	—
201	C₂×#41	D₄×C₁₁	1	2	yes	—
202	C₂×#41	C₂³×C₁₁	1	2	yes	—
203	C₂²×Q₈	D₄×C₁₁	1	1	yes	characteristic
204	C₂²×D₄	—	2	1	no	—
205	C₂²×D₄	D₄×C₁₁	1	1	yes	characteristic
206	C₂³×C₄	—	2	1	no	—
207	C₂³×C₄	—	2	2	no	—
208	C₂³×C₄	C₂³×C₁₁	1	1	yes	characteristic
209	C₂×M₁₆	—	2	1	no	—
210	C₂×M₁₆	—	2	2	no	—
211	C₂×M₁₆	—	2	4	no	—
212	C₂×M₁₆	—	2	4	no	—
213	C₂×M₁₆	—	2	4	no	—
214	C₂×M₁₆	C₂³×C₁₁	1	4	yes	—
215	C₂×M₁₆	C₂³×C₁₁	1	2	yes	—
216	C₂²×C₈	—	2	4	no	—
217	C₂²×C₈	—	2	1	no	—
218	C₂²×C₈	—	2	2	no	—
219	C₂²×C₈	D₄×C₁₁	1	2	yes	—
220	C₂²×C₈	D₄×C₁₁	1	2	yes	—
221	C₂²×C₈	C₂³×C₁₁	1	2	yes	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	2	2	no	—
224	C₂×C₄⋊C₄	—	2	2	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	C₂³×C₁₁	1	4	yes	—
228	#107	—	2	8	no	—
229	#103	—	2	8	no	—
230	#102	—	2	8	no	—
231	C₂²⋊C₈	—	2	8	no	—
232	C₂×C₁₁	—	2	2	no	—
233	C₂×C₁₁	—	2	2	no	—
234	C₂×C₁₁	—	8	2	no	—
235	C₂×C₁₁	#2078	1	4	yes	direct product factor
236	C₂×C₁₁	C₂×#397	1	1	yes	characteristic
237	C₂×C₁₁	C₂×#472	1	1	yes	characteristic
238	C₂×C₁₁	C₂×#473	1	1	yes	characteristic
239	C₂⁴	—	2	1	no	—
240	C₂⁴	C₁₁×#41	1	1	yes	characteristic
241	#41	—	2	8	no	—
242	#41	—	2	4	no	—
243	#41	—	2	4	no	—
244	#41	—	4	2	no	—
245	#41	—	4	2	no	—
246	#41	—	4	4	no	—
247	C₂×Q₈	—	2	1	no	—
248	C₂×Q₈	—	2	4	no	—
249	C₂×Q₈	C₂×D₄×C₁₁	1	2	yes	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	2	1	no	—
253	C₂×D₄	—	2	4	no	—
254	C₂×D₄	—	4	4	no	—
255	C₂×D₄	C₂×D₄×C₁₁	1	2	yes	—
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	4	no	—
260	C₂²×C₄	—	4	2	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	—	4	2	no	—
263	C₂²×C₄	—	4	2	no	—
264	C₂²×C₄	—	8	4	no	—
265	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
266	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
269	C₂²×C₄	C₁₁×#41	1	1	yes	characteristic
270	C₂²×C₄	C₁₁×#41	1	4	yes	—
271	C₂²×C₄	C₁₁×#41	1	4	yes	—
272	C₂²×C₄	C₂⁴×C₁₁	1	1	yes	characteristic
273	M₁₆	—	2	4	no	—
274	M₁₆	—	2	4	no	—
275	M₁₆	—	2	4	no	—
276	M₁₆	—	4	4	no	—
277	M₁₆	—	4	2	no	—
278	C₂×C₈	—	2	4	no	—
279	C₂×C₈	—	2	4	no	—
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	4	no	—
283	C₂×C₈	—	4	4	no	—
284	C₂×C₈	—	4	2	no	—
285	C₂×C₈	C₂×D₄×C₁₁	1	4	yes	—
286	C₄⋊C₄	—	2	4	no	—
287	C₄⋊C₄	—	2	4	no	—
288	C₄⋊C₄	—	4	2	no	—
289	C₄⋊C₄	—	4	2	no	—
290	C₁₁	C₂×#2078	1	1	yes	characteristic, direct product factor, Sylow
291	C₂³	—	2	2	no	—
292	C₂³	—	2	2	no	—
293	C₂³	—	2	2	no	—
294	C₂³	—	4	2	no	—
295	C₂³	—	4	2	no	—
296	C₂³	—	4	2	no	—
297	C₂³	C₁₁×#121	1	1	yes	characteristic
298	C₂³	C₁₁×#121	1	1	yes	characteristic
299	C₂³	C₁₁×#126	1	4	yes	—
300	C₂³	C₂×C₁₁×#41	1	1	yes	characteristic
301	Q₈	—	2	4	no	—
302	Q₈	—	4	2	no	—
303	D₄	—	2	4	no	—
304	D₄	—	4	4	no	—
305	D₄	—	4	4	no	—
306	D₄	—	4	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	4	no	—
312	C₂×C₄	—	4	4	no	—
313	C₂×C₄	—	8	4	no	—
314	C₂×C₄	—	8	4	no	—
315	C₂×C₄	—	8	4	no	—
316	C₂×C₄	C₁₁×#121	1	1	yes	characteristic
317	C₂×C₄	C₁₁×#121	1	1	yes	characteristic
318	C₂×C₄	C₁₁×#123	1	2	yes	—
319	C₂×C₄	C₁₁×#123	1	2	yes	—
320	C₂×C₄	C₁₁×#123	1	4	yes	—
321	C₂×C₄	C₁₁×#123	1	4	yes	—
322	C₂×C₄	C₁₁×#124	1	2	yes	—
323	C₂×C₄	C₁₁×#124	1	2	yes	—
324	C₂×C₄	C₁₁×#126	1	4	yes	—
325	C₂×C₄	C₂²×D₄×C₁₁	1	1	yes	characteristic
326	C₂×C₄	C₂²×D₄×C₁₁	1	1	yes	characteristic
327	C₂×C₄	C₂×C₁₁×#41	1	2	yes	—
328	C₂×C₄	C₂×C₁₁×#41	1	2	yes	—
329	C₈	—	2	4	no	—
330	C₈	—	4	4	no	—
331	C₈	—	4	2	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	8	no	—
337	C₂²	—	4	2	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	4	2	no	—
340	C₂²	—	8	4	no	—
341	C₂²	C₁₁×#397	1	2	yes	—
342	C₂²	C₁₁×#397	1	2	yes	—
343	C₂²	C₁₁×#472	1	2	yes	—
344	C₂²	C₁₁×#473	1	2	yes	—
345	C₂²	C₂×C₁₁×#121	1	1	yes	characteristic
346	C₂²	C₂×C₁₁×#126	1	2	yes	—
347	C₄	—	4	2	no	—
348	C₄	—	8	4	no	—
349	C₄	C₁₁×#397	1	2	yes	—
350	C₄	C₁₁×#397	1	2	yes	—
351	C₄	C₂×C₁₁×#123	1	1	yes	characteristic
352	C₄	C₂×C₁₁×#123	1	2	yes	—
353	C₄	C₂×C₁₁×#124	1	1	yes	characteristic
354	C₂	—	2	2	no	—
355	C₂	—	2	2	no	—
356	C₂	—	8	2	no	—
357	C₂	C₁₁×#2078	1	4	yes	direct product factor
358	C₂	C₂×C₁₁×#397	1	1	yes	characteristic
359	C₂	C₂×C₁₁×#472	1	1	yes	characteristic
360	C₂	C₂×C₁₁×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₁	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₁
prime radical	C₂²×C₁₁×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₁ ◁ C₂³×C₄×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₁×#107	C₂	semidirect
4. C₁₁×#2078	C₂	direct (N × H)
5. C₂×C₁₁×#343	C₂	semidirect
6. C₂×C₁₁×#341	C₂	semidirect
14. C₂×C₁₁×#107	C₂²	semidirect
22. C₁₁×#343	C₂²	semidirect
23. C₁₁×#341	C₂²	semidirect
59. C₂×#2078	C₁₁	direct (N × H)
111. C₂²×#107	C₂×C₁₁	semidirect
114. #2078	C₂×C₁₁	direct (N × H)
115. C₂×#343	C₂×C₁₁	semidirect
116. C₂×#341	C₂×C₁₁	semidirect
165. C₂×#107	C₂²×C₁₁	semidirect
173. #343	C₂²×C₁₁	semidirect
174. #341	C₂²×C₁₁	semidirect
235. C₂×C₁₁	#2078	direct (N × H)
290. C₁₁	C₂×#2078	direct (N × H)
357. C₂	C₁₁×#2078	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×#107	2
C₂×C₁₁×#417	2
C₂×C₁₁×#408	2
C₁₁×#2078	2
C₂×C₁₁×#343	2
C₂×C₁₁×#341	2
C₂×#2078	11

Subgroups — C2×C11×#2078 #14600837

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₁×#2078 #14600837