Subgroups — C₂×C₁₁×#1670 #14596306

Order 2816.

702 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₁×#475	C₂	1	2	yes	maximal
3	C₂×C₈×Q₈×C₁₁	C₂	1	1	yes	maximal, characteristic
4	C₁₁×#1670	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₁×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₁×#332	C₂	1	2	yes	maximal
7	C₂²×C₁₁×Q₁₆	—	2	1	no	—
8	C₂²×C₁₁×Q₁₆	—	2	1	no	—
9	C₂×C₁₁×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₁₁	C₂²	1	1	yes	characteristic
11	C₁₁×#494	—	2	2	no	—
12	C₁₁×#494	C₂²	1	4	yes	—
13	C₁₁×#475	—	2	4	no	—
14	C₁₁×#475	C₂²	1	8	yes	—
15	C₈×Q₈×C₁₁	—	2	2	no	—
16	C₈×Q₈×C₁₁	C₂²	1	4	yes	—
17	C₂×C₁₁×#106	—	2	2	no	—
18	C₂×C₁₁×C₄⋊C₈	—	2	1	no	—
19	C₂×C₁₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₁₁×#103	—	2	2	no	—
22	C₂×C₄×C₈×C₁₁	—	2	1	no	—
23	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
24	C₁₁×#335	C₂²	1	4	yes	—
25	C₁₁×#332	C₂²	1	8	yes	—
26	C₂²×Q₈×C₁₁	—	2	2	no	—
27	C₂²×Q₈×C₁₁	D₄	1	1	yes	characteristic
28	C₂×C₁₁×Q₁₆	—	2	4	no	—
29	C₂×C₁₁×Q₁₆	—	2	4	no	—
30	C₂×C₁₁×Q₁₆	—	4	4	no	—
31	C₂×C₁₁×Q₁₆	—	4	4	no	—
32	C₂²×C₈×C₁₁	—	2	1	no	—
33	C₂²×C₈×C₁₁	—	2	2	no	—
34	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
36	C₁₁×C₄⋊Q₈	—	2	2	no	—
37	C₁₁×C₄⋊Q₈	—	2	2	no	—
38	C₁₁×C₄⋊Q₈	—	2	2	no	—
39	C₁₁×C₄⋊Q₈	C₂³	1	4	yes	—
40	C₄×Q₈×C₁₁	—	2	1	no	—
41	C₄×Q₈×C₁₁	—	2	1	no	—
42	C₄×Q₈×C₁₁	D₄	1	2	yes	—
43	C₄×Q₈×C₁₁	C₂³	1	2	yes	—
44	C₂×C₁₁×C₄⋊C₄	—	2	1	no	—
45	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
46	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
47	C₂×C₁₁×C₄⋊C₄	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₁₁	—	2	1	no	—
49	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
50	C₁₁×#106	—	2	8	no	—
51	C₁₁×C₄⋊C₈	—	2	4	no	—
52	C₁₁×C₄⋊C₈	—	2	1	no	—
53	C₁₁×C₄⋊C₈	—	2	2	no	—
54	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
55	C₁₁×C₄⋊C₈	C₂³	1	4	yes	—
56	C₁₁×#103	—	2	8	no	—
57	C₄×C₈×C₁₁	—	2	4	no	—
58	C₄×C₈×C₁₁	D₄	1	2	yes	—
59	C₄×C₈×C₁₁	C₂³	1	2	yes	—
60	C₂×#1670	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂×Q₈×C₁₁	—	2	1	no	—
62	C₂×Q₈×C₁₁	—	2	2	no	—
63	C₂×Q₈×C₁₁	—	2	4	no	—
64	C₂×Q₈×C₁₁	—	4	2	no	—
65	C₂×Q₈×C₁₁	—	4	4	no	—
66	C₂×Q₈×C₁₁	—	4	4	no	—
67	C₂×Q₈×C₁₁	Q₁₆	1	4	yes	complemented
68	C₂×Q₈×C₁₁	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₁₁	—	2	1	no	—
70	C₂²×C₄×C₁₁	—	2	1	no	—
71	C₂²×C₄×C₁₁	—	4	2	no	—
72	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
75	C₁₁×Q₁₆	—	4	8	no	—
76	C₁₁×Q₁₆	—	4	8	no	—
77	C₂×C₈×C₁₁	—	2	2	no	—
78	C₂×C₈×C₁₁	—	2	2	no	—
79	C₂×C₈×C₁₁	—	2	4	no	—
80	C₂×C₈×C₁₁	—	4	4	no	—
81	C₂×C₈×C₁₁	—	4	2	no	—
82	C₂×C₈×C₁₁	SD₁₆	1	4	yes	—
83	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
85	C₁₁×C₄⋊C₄	—	2	2	no	—
86	C₁₁×C₄⋊C₄	—	2	1	no	—
87	C₁₁×C₄⋊C₄	—	2	2	no	—
88	C₁₁×C₄⋊C₄	—	2	4	no	—
89	C₁₁×C₄⋊C₄	—	2	4	no	—
90	C₁₁×C₄⋊C₄	—	4	2	no	—
91	C₁₁×C₄⋊C₄	—	4	2	no	—
92	C₁₁×C₄⋊C₄	C₂×D₄	1	2	yes	—
93	C₄²×C₁₁	—	2	2	no	—
94	C₄²×C₁₁	—	2	2	no	—
95	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₁₁	#41	1	1	yes	characteristic
98	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
99	C₂×#494	C₂×C₁₁	1	1	yes	characteristic
100	C₂×#475	C₂×C₁₁	1	2	yes	—
101	C₂×C₈×Q₈	C₂×C₁₁	1	1	yes	characteristic
102	#1670	C₂×C₁₁	1	8	yes	direct product factor
103	C₂×#335	C₂×C₁₁	1	1	yes	characteristic
104	C₂×#332	C₂×C₁₁	1	2	yes	—
105	C₂³×C₁₁	#121	1	1	yes	characteristic
106	Q₈×C₁₁	—	2	2	no	—
107	Q₈×C₁₁	—	2	4	no	—
108	Q₈×C₁₁	—	4	4	no	—
109	Q₈×C₁₁	—	4	4	no	—
110	Q₈×C₁₁	—	8	4	no	—
111	Q₈×C₁₁	C₂×Q₁₆	1	4	yes	complemented
112	C₂×C₄×C₁₁	—	2	2	no	—
113	C₂×C₄×C₁₁	—	2	4	no	—
114	C₂×C₄×C₁₁	—	2	2	no	—
115	C₂×C₄×C₁₁	—	2	2	no	—
116	C₂×C₄×C₁₁	—	4	2	no	—
117	C₂×C₄×C₁₁	—	4	4	no	—
118	C₂×C₄×C₁₁	—	8	4	no	—
119	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
120	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
121	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
122	C₂×C₄×C₁₁	C₂×SD₁₆	1	2	yes	—
123	C₂×C₄×C₁₁	C₂×Q₁₆	1	2	yes	—
124	C₂×C₄×C₁₁	#135	1	2	yes	—
125	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	2	yes	—
126	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
129	C₈×C₁₁	—	2	2	no	—
130	C₈×C₁₁	—	4	4	no	—
131	C₈×C₁₁	—	4	2	no	—
132	C₈×C₁₁	C₂×SD₁₆	1	4	yes	—
133	C₂²×Q₁₆	—	2	1	no	—
134	C₂²×Q₁₆	—	2	1	no	—
135	C₂×C₄⋊Q₈	C₂²×C₁₁	1	2	yes	—
136	C₂×C₄×Q₈	C₂²×C₁₁	1	1	yes	characteristic
137	#494	—	2	2	no	—
138	#494	C₂²×C₁₁	1	4	yes	—
139	#475	—	2	4	no	—
140	#475	C₂²×C₁₁	1	8	yes	—
141	C₈×Q₈	—	2	2	no	—
142	C₈×Q₈	C₂²×C₁₁	1	4	yes	—
143	C₂×#106	—	2	2	no	—
144	C₂×C₄⋊C₈	—	2	1	no	—
145	C₂×C₄⋊C₈	C₂²×C₁₁	1	2	yes	—
146	C₂×C₄⋊C₈	C₂²×C₁₁	1	1	yes	characteristic
147	C₂×#103	—	2	2	no	—
148	C₂×C₄×C₈	—	2	1	no	—
149	C₂×C₄×C₈	C₂²×C₁₁	1	1	yes	characteristic
150	#335	C₂²×C₁₁	1	4	yes	—
151	#332	C₂²×C₁₁	1	8	yes	—
152	C₂²×C₁₁	#460	1	2	yes	—
153	C₂²×C₁₁	#467	1	2	yes	—
154	C₂²×C₁₁	#473	1	2	yes	—
155	C₂²×C₁₁	C₂×#121	1	1	yes	characteristic
156	C₄×C₁₁	—	2	4	no	—
157	C₄×C₁₁	—	2	1	no	—
158	C₄×C₁₁	—	2	1	no	—
159	C₄×C₁₁	—	4	2	no	—
160	C₄×C₁₁	—	8	4	no	—
161	C₄×C₁₁	#460	1	2	yes	—
162	C₄×C₁₁	#467	1	2	yes	—
163	C₄×C₁₁	C₂²×SD₁₆	1	1	yes	characteristic
164	C₄×C₁₁	C₂²×Q₁₆	1	1	yes	characteristic
165	C₄×C₁₁	C₂×#135	1	1	yes	characteristic
166	C₄×C₁₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₂²×Q₈	—	2	2	no	—
168	C₂²×Q₈	D₄×C₁₁	1	1	yes	characteristic
169	C₂×Q₁₆	—	2	4	no	—
170	C₂×Q₁₆	—	2	4	no	—
171	C₂×Q₁₆	—	4	4	no	—
172	C₂×Q₁₆	—	4	4	no	—
173	C₂²×C₈	—	2	1	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	D₄×C₁₁	1	1	yes	characteristic
176	C₂²×C₈	D₄×C₁₁	1	1	yes	characteristic
177	C₄⋊Q₈	—	2	2	no	—
178	C₄⋊Q₈	—	2	2	no	—
179	C₄⋊Q₈	—	2	2	no	—
180	C₄⋊Q₈	C₂³×C₁₁	1	4	yes	—
181	C₄×Q₈	—	2	1	no	—
182	C₄×Q₈	—	2	1	no	—
183	C₄×Q₈	D₄×C₁₁	1	2	yes	—
184	C₄×Q₈	C₂³×C₁₁	1	2	yes	—
185	C₂×C₄⋊C₄	—	2	1	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	D₄×C₁₁	1	1	yes	characteristic
189	C₂×C₄²	—	2	1	no	—
190	C₂×C₄²	C₂³×C₁₁	1	1	yes	characteristic
191	#106	—	2	8	no	—
192	C₄⋊C₈	—	2	4	no	—
193	C₄⋊C₈	—	2	1	no	—
194	C₄⋊C₈	—	2	2	no	—
195	C₄⋊C₈	C₂³×C₁₁	1	2	yes	—
196	C₄⋊C₈	C₂³×C₁₁	1	4	yes	—
197	#103	—	2	8	no	—
198	C₄×C₈	—	2	4	no	—
199	C₄×C₈	D₄×C₁₁	1	2	yes	—
200	C₄×C₈	C₂³×C₁₁	1	2	yes	—
201	C₂×C₁₁	#1670	1	4	yes	direct product factor
202	C₂×C₁₁	C₂×#460	1	1	yes	characteristic
203	C₂×C₁₁	C₂×#467	1	1	yes	characteristic
204	C₂×C₁₁	C₂×#473	1	1	yes	characteristic
205	C₂×Q₈	—	2	1	no	—
206	C₂×Q₈	—	2	2	no	—
207	C₂×Q₈	—	2	4	no	—
208	C₂×Q₈	—	4	2	no	—
209	C₂×Q₈	—	4	4	no	—
210	C₂×Q₈	—	4	4	no	—
211	C₂×Q₈	C₁₁×Q₁₆	1	4	yes	complemented
212	C₂×Q₈	C₂×D₄×C₁₁	1	2	yes	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	4	2	no	—
216	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
218	C₂²×C₄	C₁₁×#41	1	1	yes	characteristic
219	Q₁₆	—	4	8	no	—
220	Q₁₆	—	4	8	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	4	4	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	C₁₁×SD₁₆	1	4	yes	—
227	C₂×C₈	C₂×D₄×C₁₁	1	2	yes	—
228	C₂×C₈	C₂×D₄×C₁₁	1	2	yes	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	C₂×D₄×C₁₁	1	2	yes	—
237	C₄²	—	2	2	no	—
238	C₄²	—	2	2	no	—
239	C₄²	C₂×D₄×C₁₁	1	1	yes	characteristic
240	C₄²	C₂×D₄×C₁₁	1	1	yes	characteristic
241	C₄²	C₁₁×#41	1	1	yes	characteristic
242	C₄²	C₂⁴×C₁₁	1	1	yes	characteristic
243	C₁₁	C₂×#1670	1	1	yes	characteristic, direct product factor, Sylow
244	C₂³	C₁₁×#121	1	1	yes	characteristic
245	Q₈	—	2	2	no	—
246	Q₈	—	2	4	no	—
247	Q₈	—	4	4	no	—
248	Q₈	—	4	4	no	—
249	Q₈	—	8	4	no	—
250	Q₈	C₂×C₁₁×Q₁₆	1	4	yes	complemented
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	8	4	no	—
258	C₂×C₄	C₁₁×#121	1	1	yes	characteristic
259	C₂×C₄	C₁₁×#121	1	1	yes	characteristic
260	C₂×C₄	C₁₁×#121	1	1	yes	characteristic
261	C₂×C₄	C₂×C₁₁×SD₁₆	1	2	yes	—
262	C₂×C₄	C₂×C₁₁×Q₁₆	1	2	yes	—
263	C₂×C₄	C₁₁×#135	1	2	yes	—
264	C₂×C₄	C₁₁×AGL(1,ℤ/8)	1	2	yes	—
265	C₂×C₄	C₂²×D₄×C₁₁	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₄×C₁₁	1	1	yes	characteristic
267	C₂×C₄	C₂×C₁₁×#41	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	4	4	no	—
270	C₈	—	4	2	no	—
271	C₈	C₂×C₁₁×SD₁₆	1	4	yes	—
272	C₂²	C₁₁×#460	1	2	yes	—
273	C₂²	C₁₁×#467	1	2	yes	—
274	C₂²	C₁₁×#473	1	2	yes	—
275	C₂²	C₂×C₁₁×#121	1	1	yes	characteristic
276	C₄	—	2	4	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	4	2	no	—
280	C₄	—	8	4	no	—
281	C₄	C₁₁×#460	1	2	yes	—
282	C₄	C₁₁×#467	1	2	yes	—
283	C₄	C₂²×C₁₁×SD₁₆	1	1	yes	characteristic
284	C₄	C₂²×C₁₁×Q₁₆	1	1	yes	characteristic
285	C₄	C₂×C₁₁×#135	1	1	yes	characteristic
286	C₄	C₂×C₁₁×AGL(1,ℤ/8)	1	1	yes	characteristic
287	C₂	C₁₁×#1670	1	4	yes	direct product factor
288	C₂	C₂×C₁₁×#460	1	1	yes	characteristic
289	C₂	C₂×C₁₁×#467	1	1	yes	characteristic
290	C₂	C₂×C₁₁×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₁	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₁ ◁ C₂×C₄²×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₁×#1670	C₂	direct (N × H)
60. C₂×#1670	C₁₁	direct (N × H)
67. C₂×Q₈×C₁₁	Q₁₆	semidirect
102. #1670	C₂×C₁₁	direct (N × H)
111. Q₈×C₁₁	C₂×Q₁₆	semidirect
201. C₂×C₁₁	#1670	direct (N × H)
211. C₂×Q₈	C₁₁×Q₁₆	semidirect
243. C₁₁	C₂×#1670	direct (N × H)
250. Q₈	C₂×C₁₁×Q₁₆	semidirect
287. C₂	C₁₁×#1670	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₁×#494	2
C₂×C₁₁×#475	2
C₂×C₈×Q₈×C₁₁	2
C₁₁×#1670	2
C₂×C₁₁×#335	2
C₂×C₁₁×#332	2
C₂×#1670	11

Subgroups — C2×C11×#1670 #14596306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₁×#1670 #14596306