Subgroups — C₂×C₁₁×#1721 #14596807

Order 2816.

1146 conjugacy classes of subgroups.

The subgroup lattice has 284 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, complemented
2	C₂×C₁₁×#493	C₂	1	2	yes	maximal, complemented
3	C₂×C₁₁×#492	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₁×#486	C₂	1	2	yes	maximal, complemented
5	C₁₁×#1721	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₁₁×#322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₁₁×SD₁₆	—	2	2	no	—
8	C₂²×C₁₁×D₈	—	2	4	no	—
9	C₂²×C₁₁×D₈	—	2	2	no	—
10	C₂×C₁₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₁×#127	C₂²	1	2	yes	—
12	C₂×C₁₁×#127	C₂²	1	1	yes	characteristic
13	C₁₁×#493	—	2	2	no	—
14	C₁₁×#493	—	2	4	no	—
15	C₁₁×#493	C₂²	1	4	yes	complemented
16	C₁₁×#493	C₂²	1	8	yes	complemented
17	C₁₁×#492	—	2	2	no	—
18	C₁₁×#492	C₂²	1	4	yes	complemented
19	C₁₁×#486	—	2	4	no	—
20	C₁₁×#486	C₂²	1	8	yes	complemented
21	C₂×C₁₁×#102	—	2	4	no	—
22	C₂×C₄×C₈×C₁₁	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
24	C₁₁×#322	C₂²	1	4	yes	complemented
25	C₂²×Q₈×C₁₁	—	2	1	no	—
26	C₂²×D₄×C₁₁	—	2	2	no	—
27	C₂²×D₄×C₁₁	—	2	2	no	—
28	C₂²×D₄×C₁₁	—	2	2	no	—
29	C₂²×D₄×C₁₁	—	2	2	no	—
30	C₂²×D₄×C₁₁	—	2	1	no	—
31	C₂×C₁₁×SD₁₆	—	2	8	no	—
32	C₂×C₁₁×SD₁₆	—	4	8	no	—
33	C₂×C₁₁×D₈	—	2	16	no	—
34	C₂×C₁₁×D₈	—	2	8	no	—
35	C₂×C₁₁×D₈	—	4	16	no	—
36	C₂×C₁₁×D₈	—	4	8	no	—
37	C₂²×C₈×C₁₁	D₄	1	4	yes	—
38	C₂²×C₈×C₁₁	D₄	1	2	yes	—
39	C₁₁×C₄⋊Q₈	—	2	2	no	—
40	C₁₁×C₄⋊Q₈	—	2	1	no	—
41	C₁₁×C₄⋊Q₈	C₂³	1	2	yes	—
42	C₁₁×#127	—	2	1	no	—
43	C₁₁×#127	—	2	2	no	—
44	C₁₁×#127	—	2	2	no	—
45	C₁₁×#127	—	2	2	no	—
46	C₁₁×#127	—	2	2	no	—
47	C₁₁×#127	C₂³	1	2	yes	—
48	C₁₁×#127	C₂³	1	4	yes	—
49	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
50	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
51	C₁₁×#102	—	2	16	no	—
52	C₄×C₈×C₁₁	D₄	1	2	yes	—
53	C₄×C₈×C₁₁	D₄	1	4	yes	—
54	C₄×C₈×C₁₁	C₂³	1	2	yes	—
55	C₄×C₈×C₁₁	C₂³	1	4	yes	—
56	C₂×#1721	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂⁴×C₁₁	—	4	2	no	—
58	C₂⁴×C₁₁	—	4	1	no	—
59	C₂×Q₈×C₁₁	—	2	2	no	—
60	C₂×Q₈×C₁₁	—	4	1	no	—
61	C₂×Q₈×C₁₁	—	4	4	no	—
62	C₂×D₄×C₁₁	—	2	4	no	—
63	C₂×D₄×C₁₁	—	2	4	no	—
64	C₂×D₄×C₁₁	—	2	4	no	—
65	C₂×D₄×C₁₁	—	2	4	no	—
66	C₂×D₄×C₁₁	—	2	2	no	—
67	C₂×D₄×C₁₁	—	4	1	no	—
68	C₂×D₄×C₁₁	—	4	2	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	8	no	—
73	C₂×D₄×C₁₁	—	4	4	no	—
74	C₂×D₄×C₁₁	—	4	8	no	—
75	C₂×D₄×C₁₁	—	4	8	no	—
76	C₂×D₄×C₁₁	—	4	8	no	—
77	C₂²×C₄×C₁₁	—	4	1	no	—
78	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₁	C₂×D₄	1	2	yes	—
80	C₁₁×SD₁₆	—	4	16	no	—
81	C₁₁×D₈	—	4	32	no	—
82	C₁₁×D₈	—	4	16	no	—
83	C₂×C₈×C₁₁	—	2	8	no	—
84	C₂×C₈×C₁₁	D₈	1	8	yes	complemented
85	C₂×C₈×C₁₁	C₂×D₄	1	8	yes	—
86	C₂×C₈×C₁₁	C₂×D₄	1	4	yes	—
87	C₁₁×C₄⋊C₄	—	2	4	no	—
88	C₁₁×C₄⋊C₄	—	4	2	no	—
89	C₄²×C₁₁	C₂×D₄	1	2	yes	—
90	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
92	C₂×#493	C₂×C₁₁	1	1	yes	characteristic, complemented
93	C₂×#493	C₂×C₁₁	1	2	yes	complemented
94	C₂×#492	C₂×C₁₁	1	1	yes	characteristic, complemented
95	C₂×#486	C₂×C₁₁	1	2	yes	complemented
96	#1721	C₂×C₁₁	1	8	yes	direct product factor
97	C₂×#322	C₂×C₁₁	1	1	yes	characteristic, complemented
98	C₂³×C₁₁	—	4	4	no	—
99	C₂³×C₁₁	—	4	2	no	—
100	C₂³×C₁₁	—	8	4	no	—
101	C₂³×C₁₁	—	8	2	no	—
102	C₂³×C₁₁	—	8	4	no	—
103	C₂³×C₁₁	—	8	4	no	—
104	C₂³×C₁₁	—	8	4	no	—
105	C₂³×C₁₁	#127	1	1	yes	characteristic
106	Q₈×C₁₁	—	4	4	no	—
107	Q₈×C₁₁	—	8	2	no	—
108	D₄×C₁₁	—	4	8	no	—
109	D₄×C₁₁	—	4	4	no	—
110	D₄×C₁₁	—	4	8	no	—
111	D₄×C₁₁	—	4	8	no	—
112	D₄×C₁₁	—	4	8	no	—
113	D₄×C₁₁	—	8	4	no	—
114	D₄×C₁₁	—	8	2	no	—
115	D₄×C₁₁	—	8	4	no	—
116	D₄×C₁₁	—	8	4	no	—
117	D₄×C₁₁	—	8	4	no	—
118	C₂×C₄×C₁₁	—	4	2	no	—
119	C₂×C₄×C₁₁	—	8	2	no	—
120	C₂×C₄×C₁₁	#127	1	1	yes	characteristic
121	C₂×C₄×C₁₁	#127	1	2	yes	—
122	C₂×C₄×C₁₁	C₂×D₈	1	4	yes	—
123	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	8	yes	—
124	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₁	C₂²×D₄	1	2	yes	—
126	C₈×C₁₁	—	2	8	no	—
127	C₈×C₁₁	C₂×D₈	1	8	yes	complemented
128	C₂²×SD₁₆	—	2	2	no	—
129	C₂²×D₈	—	2	4	no	—
130	C₂²×D₈	—	2	2	no	—
131	C₂×C₄⋊Q₈	C₂²×C₁₁	1	1	yes	characteristic
132	C₂×#127	C₂²×C₁₁	1	2	yes	—
133	C₂×#127	C₂²×C₁₁	1	1	yes	characteristic
134	#493	—	2	4	no	—
135	#493	—	2	2	no	—
136	#493	C₂²×C₁₁	1	8	yes	complemented
137	#493	C₂²×C₁₁	1	4	yes	complemented
138	#492	—	2	2	no	—
139	#492	C₂²×C₁₁	1	4	yes	complemented
140	#486	—	2	4	no	—
141	#486	C₂²×C₁₁	1	8	yes	complemented
142	C₂×#102	—	2	4	no	—
143	C₂×C₄×C₈	C₂²×C₁₁	1	2	yes	—
144	C₂×C₄×C₈	C₂²×C₁₁	1	1	yes	characteristic
145	#322	C₂²×C₁₁	1	4	yes	complemented
146	C₂²×C₁₁	—	8	4	no	—
147	C₂²×C₁₁	—	8	2	no	—
148	C₂²×C₁₁	—	8	4	no	—
149	C₂²×C₁₁	—	8	4	no	—
150	C₂²×C₁₁	—	8	4	no	—
151	C₂²×C₁₁	—	16	8	no	—
152	C₂²×C₁₁	—	16	4	no	—
153	C₂²×C₁₁	#493	1	2	yes	—
154	C₂²×C₁₁	#496	1	4	yes	—
155	C₂²×C₁₁	C₂×#127	1	1	yes	characteristic
156	C₄×C₁₁	—	8	2	no	—
157	C₄×C₁₁	#493	1	2	yes	—
158	C₄×C₁₁	#496	1	4	yes	—
159	C₄×C₁₁	C₂²×D₈	1	2	yes	—
160	C₄×C₁₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
161	C₂²×Q₈	—	2	1	no	—
162	C₂²×D₄	—	2	2	no	—
163	C₂²×D₄	—	2	2	no	—
164	C₂²×D₄	—	2	2	no	—
165	C₂²×D₄	—	2	1	no	—
166	C₂²×D₄	—	2	2	no	—
167	C₂×SD₁₆	—	2	8	no	—
168	C₂×SD₁₆	—	4	8	no	—
169	C₂×D₈	—	2	16	no	—
170	C₂×D₈	—	2	8	no	—
171	C₂×D₈	—	4	16	no	—
172	C₂×D₈	—	4	8	no	—
173	C₂²×C₈	D₄×C₁₁	1	4	yes	—
174	C₂²×C₈	D₄×C₁₁	1	2	yes	—
175	C₄⋊Q₈	—	2	2	no	—
176	C₄⋊Q₈	—	2	1	no	—
177	C₄⋊Q₈	C₂³×C₁₁	1	2	yes	—
178	#127	—	2	2	no	—
179	#127	—	2	1	no	—
180	#127	—	2	2	no	—
181	#127	—	2	2	no	—
182	#127	—	2	2	no	—
183	#127	C₂³×C₁₁	1	4	yes	—
184	#127	C₂³×C₁₁	1	2	yes	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄²	C₂³×C₁₁	1	1	yes	characteristic
187	#102	—	2	16	no	—
188	C₄×C₈	D₄×C₁₁	1	4	yes	—
189	C₄×C₈	D₄×C₁₁	1	2	yes	—
190	C₄×C₈	C₂³×C₁₁	1	4	yes	—
191	C₄×C₈	C₂³×C₁₁	1	2	yes	—
192	C₂×C₁₁	—	16	4	no	—
193	C₂×C₁₁	—	16	2	no	—
194	C₂×C₁₁	#1721	1	4	yes	direct product factor
195	C₂×C₁₁	C₂×#493	1	1	yes	characteristic
196	C₂×C₁₁	C₂×#496	1	2	yes	—
197	C₂⁴	—	4	1	no	—
198	C₂⁴	—	4	2	no	—
199	C₂×Q₈	—	2	2	no	—
200	C₂×Q₈	—	4	1	no	—
201	C₂×Q₈	—	4	4	no	—
202	C₂×D₄	—	2	4	no	—
203	C₂×D₄	—	2	4	no	—
204	C₂×D₄	—	2	4	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	4	2	no	—
208	C₂×D₄	—	4	1	no	—
209	C₂×D₄	—	4	2	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	—	4	8	no	—
213	C₂×D₄	—	4	4	no	—
214	C₂×D₄	—	4	8	no	—
215	C₂×D₄	—	4	8	no	—
216	C₂×D₄	—	4	8	no	—
217	C₂²×C₄	—	4	1	no	—
218	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₄×C₁₁	1	2	yes	—
220	SD₁₆	—	4	16	no	—
221	D₈	—	4	32	no	—
222	D₈	—	4	16	no	—
223	C₂×C₈	—	2	8	no	—
224	C₂×C₈	C₁₁×D₈	1	8	yes	complemented
225	C₂×C₈	C₂×D₄×C₁₁	1	8	yes	—
226	C₂×C₈	C₂×D₄×C₁₁	1	4	yes	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₄²	C₂×D₄×C₁₁	1	2	yes	—
230	C₄²	C₂×D₄×C₁₁	1	1	yes	characteristic
231	C₄²	C₂⁴×C₁₁	1	1	yes	characteristic
232	C₁₁	C₂×#1721	1	1	yes	characteristic, direct product factor, Sylow
233	C₂³	—	4	4	no	—
234	C₂³	—	4	2	no	—
235	C₂³	—	8	4	no	—
236	C₂³	—	8	2	no	—
237	C₂³	—	8	4	no	—
238	C₂³	—	8	4	no	—
239	C₂³	—	8	4	no	—
240	C₂³	C₁₁×#127	1	1	yes	characteristic
241	Q₈	—	4	4	no	—
242	Q₈	—	8	2	no	—
243	D₄	—	4	8	no	—
244	D₄	—	4	4	no	—
245	D₄	—	4	8	no	—
246	D₄	—	4	8	no	—
247	D₄	—	4	8	no	—
248	D₄	—	8	4	no	—
249	D₄	—	8	2	no	—
250	D₄	—	8	4	no	—
251	D₄	—	8	4	no	—
252	D₄	—	8	4	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	8	2	no	—
255	C₂×C₄	C₁₁×#127	1	1	yes	characteristic
256	C₂×C₄	C₁₁×#127	1	2	yes	—
257	C₂×C₄	C₂×C₁₁×D₈	1	4	yes	—
258	C₂×C₄	C₁₁×AGL(1,ℤ/8)	1	8	yes	—
259	C₂×C₄	C₂²×D₄×C₁₁	1	1	yes	characteristic
260	C₂×C₄	C₂²×D₄×C₁₁	1	2	yes	—
261	C₈	—	2	8	no	—
262	C₈	C₂×C₁₁×D₈	1	8	yes	complemented
263	C₂²	—	8	4	no	—
264	C₂²	—	8	4	no	—
265	C₂²	—	8	4	no	—
266	C₂²	—	8	4	no	—
267	C₂²	—	8	2	no	—
268	C₂²	—	16	8	no	—
269	C₂²	—	16	4	no	—
270	C₂²	C₁₁×#493	1	2	yes	—
271	C₂²	C₁₁×#496	1	4	yes	—
272	C₂²	C₂×C₁₁×#127	1	1	yes	characteristic
273	C₄	—	8	2	no	—
274	C₄	C₁₁×#493	1	2	yes	—
275	C₄	C₁₁×#496	1	4	yes	—
276	C₄	C₂²×C₁₁×D₈	1	2	yes	—
277	C₄	C₂×C₁₁×AGL(1,ℤ/8)	1	4	yes	—
278	C₂	—	16	2	no	—
279	C₂	—	16	4	no	—
280	C₂	C₁₁×#1721	1	4	yes	direct product factor
281	C₂	C₂×C₁₁×#493	1	1	yes	characteristic
282	C₂	C₂×C₁₁×#496	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₁	#127
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
1. C₂×C₁₁×#493	C₂	semidirect
2. C₂×C₁₁×#493	C₂	semidirect
3. C₂×C₁₁×#492	C₂	semidirect
4. C₂×C₁₁×#486	C₂	semidirect
5. C₁₁×#1721	C₂	direct (N × H)
6. C₂×C₁₁×#322	C₂	semidirect
15. C₁₁×#493	C₂²	semidirect
16. C₁₁×#493	C₂²	semidirect
18. C₁₁×#492	C₂²	semidirect
20. C₁₁×#486	C₂²	semidirect
24. C₁₁×#322	C₂²	semidirect
56. C₂×#1721	C₁₁	direct (N × H)
84. C₂×C₈×C₁₁	D₈	semidirect
92. C₂×#493	C₂×C₁₁	semidirect
93. C₂×#493	C₂×C₁₁	semidirect
94. C₂×#492	C₂×C₁₁	semidirect
95. C₂×#486	C₂×C₁₁	semidirect
96. #1721	C₂×C₁₁	direct (N × H)
97. C₂×#322	C₂×C₁₁	semidirect
127. C₈×C₁₁	C₂×D₈	semidirect
136. #493	C₂²×C₁₁	semidirect
137. #493	C₂²×C₁₁	semidirect
139. #492	C₂²×C₁₁	semidirect
141. #486	C₂²×C₁₁	semidirect
145. #322	C₂²×C₁₁	semidirect
194. C₂×C₁₁	#1721	direct (N × H)
224. C₂×C₈	C₁₁×D₈	semidirect
232. C₁₁	C₂×#1721	direct (N × H)
262. C₈	C₂×C₁₁×D₈	semidirect
280. C₂	C₁₁×#1721	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×#492	2
C₂×C₁₁×#486	2
C₁₁×#1721	2
C₂×C₁₁×#322	2
C₂×#1721	11

Subgroups — C2×C11×#1721 #14596807

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₁×#1721 #14596807