Subgroups — C₂²×C₁₁×#489 #14637799

Order 2816.

2486 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₁×#124	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₁×#489	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₁₁×#103	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₁×#102	C₂	1	2	yes	maximal
6	C₂²×C₁₁×#97	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×Q₈×C₁₁	—	2	1	no	—
8	C₂³×Q₈×C₁₁	C₂²	1	1	yes	characteristic
9	C₂³×D₄×C₁₁	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₁₁	C₂²	1	2	yes	—
11	C₂×C₁₁×C₄⋊Q₈	—	2	6	no	—
12	C₂×C₁₁×C₄⋊Q₈	C₂²	1	12	yes	complemented
13	C₂×C₁₁×#124	—	2	6	no	—
14	C₂×C₁₁×#124	C₂²	1	12	yes	—
15	C₂²×C₁₁×C₄⋊C₄	—	2	1	no	—
16	C₂²×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂²×C₁₁×C₂²⋊C₄	—	2	1	no	—
18	C₂²×C₁₁×C₂²⋊C₄	—	2	1	no	—
19	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
20	C₁₁×#489	C₂²	1	64	yes	direct product factor
21	C₂×C₁₁×#103	C₂²	1	24	yes	complemented
22	C₂×C₁₁×#102	C₂²	1	24	yes	—
23	C₂×C₁₁×#97	C₂²	1	12	yes	complemented
24	C₂⁵×C₁₁	—	2	1	no	—
25	C₂²×Q₈×C₁₁	—	2	3	no	—
26	C₂²×Q₈×C₁₁	—	2	12	no	—
27	C₂²×Q₈×C₁₁	—	2	8	no	—
28	C₂²×Q₈×C₁₁	—	4	8	no	—
29	C₂²×Q₈×C₁₁	C₂³	1	6	yes	—
30	C₂²×D₄×C₁₁	—	2	3	no	—
31	C₂²×D₄×C₁₁	—	2	8	no	—
32	C₂²×D₄×C₁₁	C₂³	1	6	yes	—
33	C₂³×C₄×C₁₁	—	2	1	no	—
34	C₂³×C₄×C₁₁	—	2	2	no	—
35	C₂³×C₄×C₁₁	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₁₁	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₁	—	2	8	no	—
39	C₂²×C₈×C₁₁	C₂³	1	12	yes	—
40	C₁₁×C₄⋊Q₈	—	2	24	no	—
41	C₁₁×C₄⋊Q₈	C₂³	1	16	yes	complemented
42	C₁₁×#124	—	2	24	no	—
43	C₁₁×#124	C₂³	1	16	yes	—
44	C₂×C₁₁×C₄⋊C₄	—	2	12	no	—
45	C₂×C₁₁×C₄⋊C₄	—	2	6	no	—
46	C₂×C₁₁×C₄⋊C₄	C₂³	1	12	yes	—
47	C₂×C₁₁×C₂²⋊C₄	—	2	12	no	—
48	C₂×C₁₁×C₂²⋊C₄	—	2	12	no	—
49	C₂×C₄²×C₁₁	D₄	1	6	yes	—
50	C₂×C₄²×C₁₁	C₂³	1	6	yes	—
51	C₁₁×#103	C₂³	1	32	yes	complemented
52	C₁₁×#102	C₂³	1	32	yes	—
53	C₁₁×#97	C₂³	1	16	yes	complemented
54	C₂²×#489	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂⁴×C₁₁	—	2	6	no	—
56	C₂⁴×C₁₁	—	4	4	no	—
57	C₂⁴×C₁₁	—	4	4	no	—
58	C₂⁴×C₁₁	—	4	4	no	—
59	C₂⁴×C₁₁	C₂×D₄	1	1	yes	characteristic
60	C₂×Q₈×C₁₁	—	2	6	no	—
61	C₂×Q₈×C₁₁	—	2	16	no	—
62	C₂×Q₈×C₁₁	—	2	24	no	—
63	C₂×Q₈×C₁₁	—	4	48	no	—
64	C₂×Q₈×C₁₁	—	4	12	no	—
65	C₂×Q₈×C₁₁	C₂⁴	1	4	yes	—
66	C₂×D₄×C₁₁	—	2	6	no	—
67	C₂×D₄×C₁₁	—	2	24	no	—
68	C₂×D₄×C₁₁	—	4	12	no	—
69	C₂×D₄×C₁₁	C₂⁴	1	4	yes	—
70	C₂²×C₄×C₁₁	—	2	4	no	—
71	C₂²×C₄×C₁₁	—	2	4	no	—
72	C₂²×C₄×C₁₁	—	2	6	no	—
73	C₂²×C₄×C₁₁	—	2	12	no	—
74	C₂²×C₄×C₁₁	—	4	4	no	—
75	C₂²×C₄×C₁₁	—	4	8	no	—
76	C₂²×C₄×C₁₁	C₂×D₄	1	6	yes	—
77	C₂²×C₄×C₁₁	C₂×D₄	1	6	yes	—
78	C₂²×C₄×C₁₁	C₂×D₄	1	3	yes	—
79	C₂²×C₄×C₁₁	#41	1	8	yes	—
80	C₂²×C₄×C₁₁	C₂⁴	1	3	yes	—
81	C₂×C₈×C₁₁	—	2	24	no	—
82	C₂×C₈×C₁₁	C₂⁴	1	8	yes	—
83	C₁₁×C₄⋊C₄	—	2	16	no	—
84	C₁₁×C₄⋊C₄	—	2	12	no	—
85	C₁₁×C₄⋊C₄	C₂⁴	1	8	yes	—
86	C₁₁×C₂²⋊C₄	—	2	16	no	—
87	C₁₁×C₂²⋊C₄	—	2	16	no	—
88	C₄²×C₁₁	C₂×D₄	1	12	yes	—
89	C₄²×C₁₁	C₂⁴	1	4	yes	—
90	C₂²×C₄⋊Q₈	C₂×C₁₁	1	1	yes	characteristic, complemented
91	C₂²×#124	C₂×C₁₁	1	1	yes	characteristic
92	C₂×#489	C₂×C₁₁	1	24	yes	direct product factor
93	C₂²×#103	C₂×C₁₁	1	2	yes	complemented
94	C₂²×#102	C₂×C₁₁	1	2	yes	—
95	C₂²×#97	C₂×C₁₁	1	1	yes	characteristic, complemented
96	C₂³×C₁₁	—	2	4	no	—
97	C₂³×C₁₁	—	4	12	no	—
98	C₂³×C₁₁	—	4	12	no	—
99	C₂³×C₁₁	—	4	12	no	—
100	C₂³×C₁₁	—	8	16	no	—
101	C₂³×C₁₁	#124	1	4	yes	—
102	C₂³×C₁₁	AGL(1,ℤ/8)	1	4	yes	—
103	C₂³×C₁₁	#137	1	4	yes	—
104	C₂³×C₁₁	C₂²×D₄	1	3	yes	—
105	Q₈×C₁₁	—	2	8	no	—
106	Q₈×C₁₁	—	4	32	no	—
107	Q₈×C₁₁	—	4	12	no	—
108	D₄×C₁₁	—	2	8	no	—
109	D₄×C₁₁	—	4	12	no	—
110	C₂×C₄×C₁₁	—	2	12	no	—
111	C₂×C₄×C₁₁	—	2	12	no	—
112	C₂×C₄×C₁₁	—	2	4	no	—
113	C₂×C₄×C₁₁	—	2	8	no	—
114	C₂×C₄×C₁₁	—	4	12	no	—
115	C₂×C₄×C₁₁	—	4	24	no	—
116	C₂×C₄×C₁₁	#124	1	12	yes	—
117	C₂×C₄×C₁₁	C₂²×D₄	1	4	yes	—
118	C₂×C₄×C₁₁	C₂²×D₄	1	4	yes	—
119	C₂×C₄×C₁₁	C₂²×D₄	1	3	yes	—
120	C₂×C₄×C₁₁	C₂×#41	1	12	yes	—
121	C₂×C₄×C₁₁	C₂⁵	1	1	yes	characteristic
122	C₈×C₁₁	—	2	8	no	—
123	C₂³×Q₈	—	2	1	no	—
124	C₂³×Q₈	C₂²×C₁₁	1	1	yes	characteristic
125	C₂³×D₄	C₂²×C₁₁	1	1	yes	characteristic
126	C₂³×C₈	C₂²×C₁₁	1	2	yes	—
127	C₂×C₄⋊Q₈	—	2	6	no	—
128	C₂×C₄⋊Q₈	C₂²×C₁₁	1	12	yes	complemented
129	C₂×#124	—	2	6	no	—
130	C₂×#124	C₂²×C₁₁	1	12	yes	—
131	C₂²×C₄⋊C₄	—	2	1	no	—
132	C₂²×C₄⋊C₄	C₂²×C₁₁	1	2	yes	—
133	C₂²×C₂²⋊C₄	—	2	1	no	—
134	C₂²×C₂²⋊C₄	—	2	1	no	—
135	C₂²×C₄²	C₂²×C₁₁	1	1	yes	characteristic
136	#489	C₂²×C₁₁	1	64	yes	direct product factor
137	C₂×#103	C₂²×C₁₁	1	24	yes	complemented
138	C₂×#102	C₂²×C₁₁	1	24	yes	—
139	C₂×#97	C₂²×C₁₁	1	12	yes	complemented
140	C₂²×C₁₁	—	4	4	no	—
141	C₂²×C₁₁	—	4	4	no	—
142	C₂²×C₁₁	—	4	4	no	—
143	C₂²×C₁₁	—	8	24	no	—
144	C₂²×C₁₁	#489	1	16	yes	direct product factor
145	C₂²×C₁₁	C₂×#124	1	6	yes	—
146	C₂²×C₁₁	C₂×AGL(1,ℤ/8)	1	6	yes	—
147	C₂²×C₁₁	C₂×#137	1	6	yes	—
148	C₂²×C₁₁	C₂³×D₄	1	1	yes	characteristic
149	C₄×C₁₁	—	2	4	no	—
150	C₄×C₁₁	—	2	4	no	—
151	C₄×C₁₁	—	4	4	no	—
152	C₄×C₁₁	—	4	8	no	—
153	C₄×C₁₁	C₂×#124	1	6	yes	—
154	C₄×C₁₁	C₂²×#41	1	2	yes	—
155	C₂⁵	—	2	1	no	—
156	C₂²×Q₈	—	2	3	no	—
157	C₂²×Q₈	—	2	12	no	—
158	C₂²×Q₈	—	2	8	no	—
159	C₂²×Q₈	—	4	8	no	—
160	C₂²×Q₈	C₂³×C₁₁	1	6	yes	—
161	C₂²×D₄	—	2	3	no	—
162	C₂²×D₄	—	2	8	no	—
163	C₂²×D₄	C₂³×C₁₁	1	6	yes	—
164	C₂³×C₄	—	2	1	no	—
165	C₂³×C₄	—	2	2	no	—
166	C₂³×C₄	D₄×C₁₁	1	1	yes	characteristic
167	C₂³×C₄	D₄×C₁₁	1	1	yes	characteristic
168	C₂³×C₄	C₂³×C₁₁	1	1	yes	characteristic
169	C₂²×C₈	—	2	8	no	—
170	C₂²×C₈	C₂³×C₁₁	1	12	yes	—
171	C₄⋊Q₈	—	2	24	no	—
172	C₄⋊Q₈	C₂³×C₁₁	1	16	yes	complemented
173	#124	—	2	24	no	—
174	#124	C₂³×C₁₁	1	16	yes	—
175	C₂×C₄⋊C₄	—	2	12	no	—
176	C₂×C₄⋊C₄	—	2	6	no	—
177	C₂×C₄⋊C₄	C₂³×C₁₁	1	12	yes	—
178	C₂×C₂²⋊C₄	—	2	12	no	—
179	C₂×C₂²⋊C₄	—	2	12	no	—
180	C₂×C₄²	D₄×C₁₁	1	6	yes	—
181	C₂×C₄²	C₂³×C₁₁	1	6	yes	—
182	#103	C₂³×C₁₁	1	32	yes	complemented
183	#102	C₂³×C₁₁	1	32	yes	—
184	#97	C₂³×C₁₁	1	16	yes	complemented
185	C₂×C₁₁	—	8	4	no	—
186	C₂×C₁₁	C₂×#489	1	12	yes	direct product factor
187	C₂×C₁₁	C₂²×#124	1	1	yes	characteristic
188	C₂×C₁₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
189	C₂×C₁₁	C₂²×#137	1	1	yes	characteristic
190	C₂⁴	—	2	6	no	—
191	C₂⁴	—	4	4	no	—
192	C₂⁴	—	4	4	no	—
193	C₂⁴	—	4	4	no	—
194	C₂⁴	C₂×D₄×C₁₁	1	1	yes	characteristic
195	C₂×Q₈	—	2	6	no	—
196	C₂×Q₈	—	2	16	no	—
197	C₂×Q₈	—	2	24	no	—
198	C₂×Q₈	—	4	48	no	—
199	C₂×Q₈	—	4	12	no	—
200	C₂×Q₈	C₂⁴×C₁₁	1	4	yes	—
201	C₂×D₄	—	2	6	no	—
202	C₂×D₄	—	2	24	no	—
203	C₂×D₄	—	4	12	no	—
204	C₂×D₄	C₂⁴×C₁₁	1	4	yes	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	6	no	—
208	C₂²×C₄	—	2	12	no	—
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	—	4	8	no	—
211	C₂²×C₄	C₂×D₄×C₁₁	1	6	yes	—
212	C₂²×C₄	C₂×D₄×C₁₁	1	6	yes	—
213	C₂²×C₄	C₂×D₄×C₁₁	1	3	yes	—
214	C₂²×C₄	C₁₁×#41	1	8	yes	—
215	C₂²×C₄	C₂⁴×C₁₁	1	3	yes	—
216	C₂×C₈	—	2	24	no	—
217	C₂×C₈	C₂⁴×C₁₁	1	8	yes	—
218	C₄⋊C₄	—	2	16	no	—
219	C₄⋊C₄	—	2	12	no	—
220	C₄⋊C₄	C₂⁴×C₁₁	1	8	yes	—
221	C₂²⋊C₄	—	2	16	no	—
222	C₂²⋊C₄	—	2	16	no	—
223	C₄²	C₂×D₄×C₁₁	1	12	yes	—
224	C₄²	C₂⁴×C₁₁	1	4	yes	—
225	C₁₁	C₂²×#489	1	1	yes	characteristic, direct product factor, Sylow
226	C₂³	—	2	4	no	—
227	C₂³	—	4	12	no	—
228	C₂³	—	4	12	no	—
229	C₂³	—	4	12	no	—
230	C₂³	—	8	16	no	—
231	C₂³	C₁₁×#124	1	4	yes	—
232	C₂³	C₁₁×AGL(1,ℤ/8)	1	4	yes	—
233	C₂³	C₁₁×#137	1	4	yes	—
234	C₂³	C₂²×D₄×C₁₁	1	3	yes	—
235	Q₈	—	2	8	no	—
236	Q₈	—	4	32	no	—
237	Q₈	—	4	12	no	—
238	D₄	—	2	8	no	—
239	D₄	—	4	12	no	—
240	C₂×C₄	—	2	12	no	—
241	C₂×C₄	—	2	12	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	4	12	no	—
245	C₂×C₄	—	4	24	no	—
246	C₂×C₄	C₁₁×#124	1	12	yes	—
247	C₂×C₄	C₂²×D₄×C₁₁	1	4	yes	—
248	C₂×C₄	C₂²×D₄×C₁₁	1	4	yes	—
249	C₂×C₄	C₂²×D₄×C₁₁	1	3	yes	—
250	C₂×C₄	C₂×C₁₁×#41	1	12	yes	—
251	C₂×C₄	C₂⁵×C₁₁	1	1	yes	characteristic
252	C₈	—	2	8	no	—
253	C₂²	—	4	4	no	—
254	C₂²	—	4	4	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	8	24	no	—
257	C₂²	C₁₁×#489	1	16	yes	direct product factor
258	C₂²	C₂×C₁₁×#124	1	6	yes	—
259	C₂²	C₂×C₁₁×AGL(1,ℤ/8)	1	6	yes	—
260	C₂²	C₂×C₁₁×#137	1	6	yes	—
261	C₂²	C₂³×D₄×C₁₁	1	1	yes	characteristic
262	C₄	—	2	4	no	—
263	C₄	—	2	4	no	—
264	C₄	—	4	4	no	—
265	C₄	—	4	8	no	—
266	C₄	C₂×C₁₁×#124	1	6	yes	—
267	C₄	C₂²×C₁₁×#41	1	2	yes	—
268	C₂	—	8	4	no	—
269	C₂	C₂×C₁₁×#489	1	12	yes	direct product factor
270	C₂	C₂²×C₁₁×#124	1	1	yes	characteristic
271	C₂	C₂²×C₁₁×AGL(1,ℤ/8)	1	1	yes	characteristic
272	C₂	C₂²×C₁₁×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₁₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₁
prime radical	C₂³×D₄×C₁₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₁ ◁ C₂²×C₄²×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₁×C₄⋊Q₈	C₂	semidirect
3. C₂×C₁₁×#489	C₂	direct (N × H)
4. C₂²×C₁₁×#103	C₂	semidirect
6. C₂²×C₁₁×#97	C₂	semidirect
12. C₂×C₁₁×C₄⋊Q₈	C₂²	semidirect
20. C₁₁×#489	C₂²	direct (N × H)
21. C₂×C₁₁×#103	C₂²	semidirect
23. C₂×C₁₁×#97	C₂²	semidirect
41. C₁₁×C₄⋊Q₈	C₂³	semidirect
51. C₁₁×#103	C₂³	semidirect
53. C₁₁×#97	C₂³	semidirect
54. C₂²×#489	C₁₁	direct (N × H)
90. C₂²×C₄⋊Q₈	C₂×C₁₁	semidirect
92. C₂×#489	C₂×C₁₁	direct (N × H)
93. C₂²×#103	C₂×C₁₁	semidirect
95. C₂²×#97	C₂×C₁₁	semidirect
128. C₂×C₄⋊Q₈	C₂²×C₁₁	semidirect
136. #489	C₂²×C₁₁	direct (N × H)
137. C₂×#103	C₂²×C₁₁	semidirect
139. C₂×#97	C₂²×C₁₁	semidirect
144. C₂²×C₁₁	#489	direct (N × H)
172. C₄⋊Q₈	C₂³×C₁₁	semidirect
182. #103	C₂³×C₁₁	semidirect
184. #97	C₂³×C₁₁	semidirect
186. C₂×C₁₁	C₂×#489	direct (N × H)
225. C₁₁	C₂²×#489	direct (N × H)
257. C₂²	C₁₁×#489	direct (N × H)
269. C₂	C₂×C₁₁×#489	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×C₄⋊Q₈	2
C₂²×C₁₁×#124	2
C₂×C₁₁×#489	2
C₂²×C₁₁×#103	2
C₂²×C₁₁×#102	2
C₂²×C₁₁×#97	2
C₂²×#489	11

Subgroups — C22×C11×#489 #14637799

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₁×#489 #14637799