Subgroups — C₂×C₁₁×#1688 #14596337

Order 2816.

850 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₁×#460	C₂	1	2	yes	maximal, complemented
3	C₂×C₁₁×#446	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₁×#1688	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₁×#334	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₁×#331	C₂	1	2	yes	maximal, complemented
7	C₂²×C₁₁×SD₁₆	—	2	2	no	—
8	C₂²×C₁₁×D₈	—	2	1	no	—
9	C₂²×C₁₁×D₈	—	2	1	no	—
10	C₂×C₁₁×#127	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₁₁	C₂²	1	1	yes	characteristic
12	C₁₁×#493	—	2	2	no	—
13	C₁₁×#493	C₂²	1	4	yes	—
14	C₁₁×#460	—	2	4	no	—
15	C₁₁×#460	C₂²	1	8	yes	complemented
16	C₁₁×#446	—	2	2	no	—
17	C₁₁×#446	C₂²	1	4	yes	complemented
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×#102	—	2	2	no	—
22	C₂×C₁₁×#97	—	2	1	no	—
23	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
24	C₁₁×#334	C₂²	1	4	yes	complemented
25	C₁₁×#331	C₂²	1	8	yes	complemented
26	C₂²×Q₈×C₁₁	D₄	1	1	yes	characteristic
27	C₂²×D₄×C₁₁	—	2	2	no	—
28	C₂²×D₄×C₁₁	—	2	2	no	—
29	C₂²×D₄×C₁₁	—	2	2	no	—
30	C₂×C₁₁×SD₁₆	—	2	8	no	—
31	C₂×C₁₁×SD₁₆	—	4	8	no	—
32	C₂×C₁₁×D₈	—	2	4	no	—
33	C₂×C₁₁×D₈	—	2	4	no	—
34	C₂×C₁₁×D₈	—	4	4	no	—
35	C₂×C₁₁×D₈	—	4	4	no	—
36	C₂²×C₈×C₁₁	—	2	1	no	—
37	C₂²×C₈×C₁₁	—	2	2	no	—
38	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
40	C₁₁×#127	—	2	2	no	—
41	C₁₁×#127	—	2	2	no	—
42	C₁₁×#127	—	2	2	no	—
43	C₁₁×#127	C₂³	1	4	yes	—
44	C₄×Q₈×C₁₁	—	2	1	no	—
45	C₄×Q₈×C₁₁	—	2	1	no	—
46	C₄×Q₈×C₁₁	D₄	1	2	yes	—
47	C₄×Q₈×C₁₁	C₂³	1	2	yes	—
48	C₂×C₁₁×C₄⋊C₄	—	2	1	no	—
49	C₂×C₁₁×C₄⋊C₄	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₁₁	—	2	1	no	—
51	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
52	C₁₁×C₄⋊C₈	—	2	4	no	—
53	C₁₁×C₄⋊C₈	—	2	1	no	—
54	C₁₁×C₄⋊C₈	—	2	2	no	—
55	C₁₁×C₄⋊C₈	C₂³	1	2	yes	—
56	C₁₁×C₄⋊C₈	C₂³	1	4	yes	—
57	C₁₁×#102	—	2	8	no	—
58	C₁₁×#97	—	2	4	no	—
59	C₄×C₈×C₁₁	D₄	1	2	yes	—
60	C₄×C₈×C₁₁	C₂³	1	2	yes	—
61	C₂×#1688	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂⁴×C₁₁	—	4	2	no	—
63	C₂×Q₈×C₁₁	—	2	1	no	—
64	C₂×Q₈×C₁₁	—	2	4	no	—
65	C₂×Q₈×C₁₁	C₂×D₄	1	2	yes	—
66	C₂×D₄×C₁₁	—	2	4	no	—
67	C₂×D₄×C₁₁	—	2	4	no	—
68	C₂×D₄×C₁₁	—	2	4	no	—
69	C₂×D₄×C₁₁	—	4	2	no	—
70	C₂×D₄×C₁₁	—	4	2	no	—
71	C₂×D₄×C₁₁	—	4	2	no	—
72	C₂×D₄×C₁₁	—	4	4	no	—
73	C₂×D₄×C₁₁	—	4	4	no	—
74	C₂×D₄×C₁₁	—	4	4	no	—
75	C₂×D₄×C₁₁	—	4	4	no	—
76	C₂×D₄×C₁₁	—	8	4	no	—
77	C₂²×C₄×C₁₁	—	2	1	no	—
78	C₂²×C₄×C₁₁	—	2	1	no	—
79	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
82	C₁₁×SD₁₆	—	4	16	no	—
83	C₁₁×D₈	—	4	8	no	—
84	C₁₁×D₈	—	4	8	no	—
85	C₂×C₈×C₁₁	—	2	2	no	—
86	C₂×C₈×C₁₁	—	2	2	no	—
87	C₂×C₈×C₁₁	—	2	4	no	—
88	C₂×C₈×C₁₁	—	4	2	no	—
89	C₂×C₈×C₁₁	—	4	4	no	—
90	C₂×C₈×C₁₁	SD₁₆	1	4	yes	complemented
91	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
93	C₁₁×C₄⋊C₄	—	2	2	no	—
94	C₁₁×C₄⋊C₄	—	2	1	no	—
95	C₁₁×C₄⋊C₄	—	2	2	no	—
96	C₁₁×C₄⋊C₄	C₂×D₄	1	2	yes	—
97	C₄²×C₁₁	—	2	2	no	—
98	C₄²×C₁₁	—	2	2	no	—
99	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₁₁	#41	1	1	yes	characteristic
102	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
103	C₂×#493	C₂×C₁₁	1	1	yes	characteristic
104	C₂×#460	C₂×C₁₁	1	2	yes	complemented
105	C₂×#446	C₂×C₁₁	1	1	yes	characteristic, complemented
106	#1688	C₂×C₁₁	1	8	yes	direct product factor
107	C₂×#334	C₂×C₁₁	1	1	yes	characteristic, complemented
108	C₂×#331	C₂×C₁₁	1	2	yes	complemented
109	C₂³×C₁₁	—	4	4	no	—
110	C₂³×C₁₁	—	8	4	no	—
111	C₂³×C₁₁	—	8	4	no	—
112	C₂³×C₁₁	—	8	4	no	—
113	C₂³×C₁₁	#121	1	1	yes	characteristic
114	Q₈×C₁₁	—	2	4	no	—
115	Q₈×C₁₁	—	4	2	no	—
116	D₄×C₁₁	—	4	4	no	—
117	D₄×C₁₁	—	4	4	no	—
118	D₄×C₁₁	—	4	4	no	—
119	D₄×C₁₁	—	4	4	no	—
120	D₄×C₁₁	—	8	4	no	—
121	D₄×C₁₁	—	8	4	no	—
122	D₄×C₁₁	—	8	4	no	—
123	D₄×C₁₁	—	8	4	no	—
124	C₂×C₄×C₁₁	—	2	2	no	—
125	C₂×C₄×C₁₁	—	2	2	no	—
126	C₂×C₄×C₁₁	—	2	2	no	—
127	C₂×C₄×C₁₁	—	4	2	no	—
128	C₂×C₄×C₁₁	—	4	2	no	—
129	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
130	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
131	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
132	C₂×C₄×C₁₁	C₂×SD₁₆	1	2	yes	—
133	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	2	yes	—
134	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	4	yes	—
135	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
138	C₈×C₁₁	—	2	2	no	—
139	C₈×C₁₁	—	4	2	no	—
140	C₈×C₁₁	—	4	4	no	—
141	C₈×C₁₁	C₂×SD₁₆	1	4	yes	complemented
142	C₂²×SD₁₆	—	2	2	no	—
143	C₂²×D₈	—	2	1	no	—
144	C₂²×D₈	—	2	1	no	—
145	C₂×#127	C₂²×C₁₁	1	2	yes	—
146	C₂×C₄×Q₈	C₂²×C₁₁	1	1	yes	characteristic
147	#493	—	2	2	no	—
148	#493	C₂²×C₁₁	1	4	yes	—
149	#460	—	2	4	no	—
150	#460	C₂²×C₁₁	1	8	yes	complemented
151	#446	—	2	2	no	—
152	#446	C₂²×C₁₁	1	4	yes	complemented
153	C₂×C₄⋊C₈	—	2	1	no	—
154	C₂×C₄⋊C₈	C₂²×C₁₁	1	1	yes	characteristic
155	C₂×C₄⋊C₈	C₂²×C₁₁	1	2	yes	—
156	C₂×#102	—	2	2	no	—
157	C₂×#97	—	2	1	no	—
158	C₂×C₄×C₈	C₂²×C₁₁	1	1	yes	characteristic
159	#334	C₂²×C₁₁	1	4	yes	complemented
160	#331	C₂²×C₁₁	1	8	yes	complemented
161	C₂²×C₁₁	—	8	4	no	—
162	C₂²×C₁₁	—	8	4	no	—
163	C₂²×C₁₁	—	8	4	no	—
164	C₂²×C₁₁	—	16	8	no	—
165	C₂²×C₁₁	#460	1	2	yes	—
166	C₂²×C₁₁	#469	1	2	yes	—
167	C₂²×C₁₁	#472	1	2	yes	—
168	C₂²×C₁₁	C₂×#121	1	1	yes	characteristic
169	C₄×C₁₁	—	2	1	no	—
170	C₄×C₁₁	—	2	1	no	—
171	C₄×C₁₁	—	4	2	no	—
172	C₄×C₁₁	—	4	2	no	—
173	C₄×C₁₁	#460	1	2	yes	—
174	C₄×C₁₁	#469	1	2	yes	—
175	C₄×C₁₁	C₂²×SD₁₆	1	1	yes	characteristic
176	C₄×C₁₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
177	C₄×C₁₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
178	C₂²×Q₈	D₄×C₁₁	1	1	yes	characteristic
179	C₂²×D₄	—	2	2	no	—
180	C₂²×D₄	—	2	2	no	—
181	C₂²×D₄	—	2	2	no	—
182	C₂×SD₁₆	—	2	8	no	—
183	C₂×SD₁₆	—	4	8	no	—
184	C₂×D₈	—	2	4	no	—
185	C₂×D₈	—	2	4	no	—
186	C₂×D₈	—	4	4	no	—
187	C₂×D₈	—	4	4	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	D₄×C₁₁	1	1	yes	characteristic
191	C₂²×C₈	D₄×C₁₁	1	1	yes	characteristic
192	#127	—	2	2	no	—
193	#127	—	2	2	no	—
194	#127	—	2	2	no	—
195	#127	C₂³×C₁₁	1	4	yes	—
196	C₄×Q₈	—	2	1	no	—
197	C₄×Q₈	—	2	1	no	—
198	C₄×Q₈	D₄×C₁₁	1	2	yes	—
199	C₄×Q₈	C₂³×C₁₁	1	2	yes	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	D₄×C₁₁	1	1	yes	characteristic
202	C₂×C₄²	—	2	1	no	—
203	C₂×C₄²	C₂³×C₁₁	1	1	yes	characteristic
204	C₄⋊C₈	—	2	4	no	—
205	C₄⋊C₈	—	2	1	no	—
206	C₄⋊C₈	—	2	2	no	—
207	C₄⋊C₈	C₂³×C₁₁	1	2	yes	—
208	C₄⋊C₈	C₂³×C₁₁	1	4	yes	—
209	#102	—	2	8	no	—
210	#97	—	2	4	no	—
211	C₄×C₈	D₄×C₁₁	1	2	yes	—
212	C₄×C₈	C₂³×C₁₁	1	2	yes	—
213	C₂×C₁₁	—	16	4	no	—
214	C₂×C₁₁	#1688	1	4	yes	direct product factor
215	C₂×C₁₁	C₂×#460	1	1	yes	characteristic
216	C₂×C₁₁	C₂×#469	1	1	yes	characteristic
217	C₂×C₁₁	C₂×#472	1	1	yes	characteristic
218	C₂⁴	—	4	2	no	—
219	C₂×Q₈	—	2	1	no	—
220	C₂×Q₈	—	2	4	no	—
221	C₂×Q₈	C₂×D₄×C₁₁	1	2	yes	—
222	C₂×D₄	—	2	4	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	2	4	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	4	4	no	—
229	C₂×D₄	—	4	4	no	—
230	C₂×D₄	—	4	4	no	—
231	C₂×D₄	—	4	4	no	—
232	C₂×D₄	—	8	4	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
237	C₂²×C₄	C₁₁×#41	1	1	yes	characteristic
238	SD₁₆	—	4	16	no	—
239	D₈	—	4	8	no	—
240	D₈	—	4	8	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	4	4	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	C₁₁×SD₁₆	1	4	yes	complemented
247	C₂×C₈	C₂×D₄×C₁₁	1	2	yes	—
248	C₂×C₈	C₂×D₄×C₁₁	1	2	yes	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	C₂×D₄×C₁₁	1	2	yes	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	C₂×D₄×C₁₁	1	1	yes	characteristic
256	C₄²	C₂×D₄×C₁₁	1	1	yes	characteristic
257	C₄²	C₁₁×#41	1	1	yes	characteristic
258	C₄²	C₂⁴×C₁₁	1	1	yes	characteristic
259	C₁₁	C₂×#1688	1	1	yes	characteristic, direct product factor, Sylow
260	C₂³	—	4	4	no	—
261	C₂³	—	8	4	no	—
262	C₂³	—	8	4	no	—
263	C₂³	—	8	4	no	—
264	C₂³	C₁₁×#121	1	1	yes	characteristic
265	Q₈	—	2	4	no	—
266	Q₈	—	4	2	no	—
267	D₄	—	4	4	no	—
268	D₄	—	4	4	no	—
269	D₄	—	4	4	no	—
270	D₄	—	4	4	no	—
271	D₄	—	8	4	no	—
272	D₄	—	8	4	no	—
273	D₄	—	8	4	no	—
274	D₄	—	8	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	C₁₁×#121	1	1	yes	characteristic
281	C₂×C₄	C₁₁×#121	1	1	yes	characteristic
282	C₂×C₄	C₁₁×#121	1	1	yes	characteristic
283	C₂×C₄	C₂×C₁₁×SD₁₆	1	2	yes	—
284	C₂×C₄	C₁₁×AGL(1,ℤ/8)	1	2	yes	—
285	C₂×C₄	C₁₁×AGL(1,ℤ/8)	1	4	yes	—
286	C₂×C₄	C₂²×D₄×C₁₁	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₄×C₁₁	1	1	yes	characteristic
288	C₂×C₄	C₂×C₁₁×#41	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	4	4	no	—
291	C₈	—	4	2	no	—
292	C₈	C₂×C₁₁×SD₁₆	1	4	yes	complemented
293	C₂²	—	8	4	no	—
294	C₂²	—	8	4	no	—
295	C₂²	—	8	4	no	—
296	C₂²	—	16	8	no	—
297	C₂²	C₁₁×#460	1	2	yes	—
298	C₂²	C₁₁×#469	1	2	yes	—
299	C₂²	C₁₁×#472	1	2	yes	—
300	C₂²	C₂×C₁₁×#121	1	1	yes	characteristic
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	4	2	no	—
304	C₄	—	4	2	no	—
305	C₄	C₁₁×#460	1	2	yes	—
306	C₄	C₁₁×#469	1	2	yes	—
307	C₄	C₂²×C₁₁×SD₁₆	1	1	yes	characteristic
308	C₄	C₂×C₁₁×AGL(1,ℤ/8)	1	1	yes	characteristic
309	C₄	C₂×C₁₁×AGL(1,ℤ/8)	1	2	yes	—
310	C₂	—	16	4	no	—
311	C₂	C₁₁×#1688	1	4	yes	direct product factor
312	C₂	C₂×C₁₁×#460	1	1	yes	characteristic
313	C₂	C₂×C₁₁×#469	1	1	yes	characteristic
314	C₂	C₂×C₁₁×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₁	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₁
prime radical	C₂×C₁₁×#493	C₂
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
2. C₂×C₁₁×#460	C₂	semidirect
3. C₂×C₁₁×#446	C₂	semidirect
4. C₁₁×#1688	C₂	direct (N × H)
5. C₂×C₁₁×#334	C₂	semidirect
6. C₂×C₁₁×#331	C₂	semidirect
15. C₁₁×#460	C₂²	semidirect
17. C₁₁×#446	C₂²	semidirect
24. C₁₁×#334	C₂²	semidirect
25. C₁₁×#331	C₂²	semidirect
61. C₂×#1688	C₁₁	direct (N × H)
90. C₂×C₈×C₁₁	SD₁₆	semidirect
104. C₂×#460	C₂×C₁₁	semidirect
105. C₂×#446	C₂×C₁₁	semidirect
106. #1688	C₂×C₁₁	direct (N × H)
107. C₂×#334	C₂×C₁₁	semidirect
108. C₂×#331	C₂×C₁₁	semidirect
141. C₈×C₁₁	C₂×SD₁₆	semidirect
150. #460	C₂²×C₁₁	semidirect
152. #446	C₂²×C₁₁	semidirect
159. #334	C₂²×C₁₁	semidirect
160. #331	C₂²×C₁₁	semidirect
214. C₂×C₁₁	#1688	direct (N × H)
246. C₂×C₈	C₁₁×SD₁₆	semidirect
259. C₁₁	C₂×#1688	direct (N × H)
292. C₈	C₂×C₁₁×SD₁₆	semidirect
311. C₂	C₁₁×#1688	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×#493	2
C₂×C₁₁×#460	2
C₂×C₁₁×#446	2
C₁₁×#1688	2
C₂×C₁₁×#334	2
C₂×C₁₁×#331	2
C₂×#1688	11

Subgroups — C2×C11×#1688 #14596337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₁×#1688 #14596337