Subgroups — C₂×C₁₁×#2249 #14610905

Order 2816.

854 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₁×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₁×SD₃₂	C₂	1	2	yes	maximal, complemented
3	C₂²×C₁₁×D₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₁×#495	C₂	1	2	yes	maximal, complemented
5	C₁₁×#2249	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₄×C₁₁×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₁₁×Q₁₆	C₂²	1	2	yes	—
8	C₂²×C₁₁×SD₁₆	—	2	2	no	—
9	C₂²×C₁₁×D₈	C₂²	1	2	yes	—
10	C₂×C₁₁×#124	—	2	2	no	—
11	C₂×C₁₁×Q₃₂	—	2	4	no	—
12	C₂×C₁₁×Q₃₂	C₂²	1	4	yes	complemented
13	C₂×C₁₁×SD₃₂	—	2	8	no	—
14	C₂×C₁₁×SD₃₂	C₂²	1	8	yes	complemented
15	C₂×C₁₁×D₁₆	—	2	4	no	—
16	C₂×C₁₁×D₁₆	C₂²	1	4	yes	—
17	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
19	C₁₁×#495	—	2	4	no	—
20	C₁₁×#495	C₂²	1	8	yes	complemented
21	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
22	C₄×C₁₁×C₁₆	C₂²	1	4	yes	complemented
23	C₂²×Q₈×C₁₁	—	2	2	no	—
24	C₂²×D₄×C₁₁	—	2	2	no	—
25	C₂×C₁₁×Q₁₆	—	2	4	no	—
26	C₂×C₁₁×Q₁₆	—	2	4	no	—
27	C₂×C₁₁×Q₁₆	—	2	2	no	—
28	C₂×C₁₁×Q₁₆	C₂³	1	4	yes	—
29	C₂×C₁₁×SD₁₆	—	2	8	no	—
30	C₂×C₁₁×SD₁₆	—	4	8	no	—
31	C₂×C₁₁×D₈	—	2	4	no	—
32	C₂×C₁₁×D₈	—	2	4	no	—
33	C₂×C₁₁×D₈	—	2	2	no	—
34	C₂×C₁₁×D₈	C₂³	1	4	yes	—
35	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
37	C₁₁×#124	—	2	8	no	—
38	C₁₁×#124	—	4	4	no	—
39	C₂×C₁₁×C₂²⋊C₄	—	4	4	no	—
40	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
41	C₁₁×Q₃₂	—	2	8	no	—
42	C₁₁×SD₃₂	—	2	16	no	—
43	C₁₁×D₁₆	—	2	8	no	—
44	C₂×C₁₁×C₁₆	D₄	1	4	yes	—
45	C₂×C₁₁×C₁₆	D₄	1	4	yes	—
46	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
47	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
48	C₄×C₈×C₁₁	D₄	1	2	yes	—
49	C₄×C₈×C₁₁	C₂³	1	2	yes	—
50	C₂×#2249	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂⁴×C₁₁	—	4	2	no	—
52	C₂×Q₈×C₁₁	—	2	4	no	—
53	C₂×Q₈×C₁₁	—	4	2	no	—
54	C₂×Q₈×C₁₁	—	4	4	no	—
55	C₂×Q₈×C₁₁	—	4	4	no	—
56	C₂×D₄×C₁₁	—	2	4	no	—
57	C₂×D₄×C₁₁	—	4	4	no	—
58	C₂×D₄×C₁₁	—	4	4	no	—
59	C₂×D₄×C₁₁	—	4	2	no	—
60	C₂²×C₄×C₁₁	—	4	2	no	—
61	C₂²×C₄×C₁₁	D₈	1	2	yes	—
62	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
63	C₁₁×Q₁₆	—	2	4	no	—
64	C₁₁×Q₁₆	—	2	4	no	—
65	C₁₁×Q₁₆	—	4	4	no	—
66	C₁₁×SD₁₆	—	4	16	no	—
67	C₁₁×D₈	—	2	4	no	—
68	C₁₁×D₈	—	2	4	no	—
69	C₁₁×D₈	—	4	4	no	—
70	C₂×C₈×C₁₁	D₈	1	4	yes	—
71	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
76	C₁₁×C₂²⋊C₄	—	4	16	no	—
77	C₄²×C₁₁	D₈	1	2	yes	—
78	C₄²×C₁₁	C₂×D₄	1	2	yes	—
79	C₁₁×C₁₆	C₂×D₄	1	4	yes	—
80	C₁₁×C₁₆	C₂×D₄	1	4	yes	—
81	C₂²×Q₃₂	C₂×C₁₁	1	1	yes	characteristic, complemented
82	C₂²×SD₃₂	C₂×C₁₁	1	2	yes	complemented
83	C₂²×D₁₆	C₂×C₁₁	1	1	yes	characteristic
84	C₂×#495	C₂×C₁₁	1	2	yes	complemented
85	#2249	C₂×C₁₁	1	8	yes	direct product factor
86	C₂×C₄×C₁₆	C₂×C₁₁	1	1	yes	characteristic, complemented
87	C₂³×C₁₁	—	4	4	no	—
88	C₂³×C₁₁	—	8	4	no	—
89	C₂³×C₁₁	—	8	8	no	—
90	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
91	Q₈×C₁₁	—	4	4	no	—
92	Q₈×C₁₁	—	4	4	no	—
93	Q₈×C₁₁	—	8	4	no	—
94	D₄×C₁₁	—	4	4	no	—
95	D₄×C₁₁	—	4	4	no	—
96	D₄×C₁₁	—	8	4	no	—
97	C₂×C₄×C₁₁	—	2	4	no	—
98	C₂×C₄×C₁₁	—	4	4	no	—
99	C₂×C₄×C₁₁	—	8	4	no	—
100	C₂×C₄×C₁₁	#127	1	2	yes	—
101	C₂×C₄×C₁₁	C₂×D₈	1	4	yes	—
102	C₂×C₄×C₁₁	C₂×D₈	1	2	yes	—
103	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
105	C₈×C₁₁	#127	1	2	yes	—
106	C₈×C₁₁	C₂×D₈	1	4	yes	—
107	C₈×C₁₁	C₂²×D₄	1	1	yes	characteristic
108	C₈×C₁₁	C₂²×D₄	1	1	yes	characteristic
109	C₂²×Q₁₆	C₂²×C₁₁	1	2	yes	—
110	C₂²×SD₁₆	—	2	2	no	—
111	C₂²×D₈	C₂²×C₁₁	1	2	yes	—
112	C₂×#124	—	2	2	no	—
113	C₂×Q₃₂	—	2	4	no	—
114	C₂×Q₃₂	C₂²×C₁₁	1	4	yes	complemented
115	C₂×SD₃₂	—	2	8	no	—
116	C₂×SD₃₂	C₂²×C₁₁	1	8	yes	complemented
117	C₂×D₁₆	—	2	4	no	—
118	C₂×D₁₆	C₂²×C₁₁	1	4	yes	—
119	C₂²×C₁₆	C₂²×C₁₁	1	1	yes	characteristic
120	C₂²×C₁₆	C₂²×C₁₁	1	1	yes	characteristic
121	#495	—	2	4	no	—
122	#495	C₂²×C₁₁	1	8	yes	complemented
123	C₂×C₄×C₈	C₂²×C₁₁	1	1	yes	characteristic
124	C₄×C₁₆	C₂²×C₁₁	1	4	yes	complemented
125	C₂²×C₁₁	—	8	4	no	—
126	C₂²×C₁₁	—	8	8	no	—
127	C₂²×C₁₁	—	16	8	no	—
128	C₂²×C₁₁	#493	1	2	yes	—
129	C₂²×C₁₁	#508	1	4	yes	—
130	C₂²×C₁₁	C₂²×D₈	1	1	yes	characteristic
131	C₄×C₁₁	—	2	4	no	—
132	C₄×C₁₁	—	8	4	no	—
133	C₄×C₁₁	#493	1	2	yes	—
134	C₄×C₁₁	C₂×#127	1	1	yes	characteristic
135	C₄×C₁₁	C₂²×D₈	1	1	yes	characteristic
136	C₂²×Q₈	—	2	2	no	—
137	C₂²×D₄	—	2	2	no	—
138	C₂×Q₁₆	—	2	4	no	—
139	C₂×Q₁₆	—	2	4	no	—
140	C₂×Q₁₆	—	2	2	no	—
141	C₂×Q₁₆	C₂³×C₁₁	1	4	yes	—
142	C₂×SD₁₆	—	2	8	no	—
143	C₂×SD₁₆	—	4	8	no	—
144	C₂×D₈	—	2	4	no	—
145	C₂×D₈	—	2	4	no	—
146	C₂×D₈	—	2	2	no	—
147	C₂×D₈	C₂³×C₁₁	1	4	yes	—
148	C₂²×C₈	D₄×C₁₁	1	1	yes	characteristic
149	C₂²×C₈	C₂³×C₁₁	1	1	yes	characteristic
150	#124	—	2	8	no	—
151	#124	—	4	4	no	—
152	C₂×C₂²⋊C₄	—	4	4	no	—
153	C₂×C₄²	D₄×C₁₁	1	1	yes	characteristic
154	Q₃₂	—	2	8	no	—
155	SD₃₂	—	2	16	no	—
156	D₁₆	—	2	8	no	—
157	C₂×C₁₆	D₄×C₁₁	1	4	yes	—
158	C₂×C₁₆	D₄×C₁₁	1	4	yes	—
159	C₂×C₁₆	C₂³×C₁₁	1	2	yes	—
160	C₂×C₁₆	C₂³×C₁₁	1	2	yes	—
161	C₄×C₈	D₄×C₁₁	1	2	yes	—
162	C₄×C₈	C₂³×C₁₁	1	2	yes	—
163	C₂×C₁₁	—	16	4	no	—
164	C₂×C₁₁	#2249	1	4	yes	direct product factor
165	C₂×C₁₁	C₂×#493	1	1	yes	characteristic
166	C₂×C₁₁	C₂×#508	1	2	yes	—
167	C₂⁴	—	4	2	no	—
168	C₂×Q₈	—	2	4	no	—
169	C₂×Q₈	—	4	2	no	—
170	C₂×Q₈	—	4	4	no	—
171	C₂×Q₈	—	4	4	no	—
172	C₂×D₄	—	2	4	no	—
173	C₂×D₄	—	4	4	no	—
174	C₂×D₄	—	4	4	no	—
175	C₂×D₄	—	4	2	no	—
176	C₂²×C₄	—	4	2	no	—
177	C₂²×C₄	C₁₁×D₈	1	2	yes	—
178	C₂²×C₄	C₂×D₄×C₁₁	1	1	yes	characteristic
179	Q₁₆	—	2	4	no	—
180	Q₁₆	—	2	4	no	—
181	Q₁₆	—	4	4	no	—
182	SD₁₆	—	4	16	no	—
183	D₈	—	2	4	no	—
184	D₈	—	2	4	no	—
185	D₈	—	4	4	no	—
186	C₂×C₈	C₁₁×D₈	1	4	yes	—
187	C₂×C₈	C₂×D₄×C₁₁	1	1	yes	characteristic
188	C₂×C₈	C₂×D₄×C₁₁	1	2	yes	—
189	C₂×C₈	C₂×D₄×C₁₁	1	2	yes	—
190	C₂×C₈	C₂×D₄×C₁₁	1	2	yes	—
191	C₂×C₈	C₂⁴×C₁₁	1	1	yes	characteristic
192	C₂²⋊C₄	—	4	16	no	—
193	C₄²	C₁₁×D₈	1	2	yes	—
194	C₄²	C₂×D₄×C₁₁	1	2	yes	—
195	C₁₆	C₂×D₄×C₁₁	1	4	yes	—
196	C₁₆	C₂×D₄×C₁₁	1	4	yes	—
197	C₁₁	C₂×#2249	1	1	yes	characteristic, direct product factor, Sylow
198	C₂³	—	4	4	no	—
199	C₂³	—	8	4	no	—
200	C₂³	—	8	8	no	—
201	C₂³	C₂×C₁₁×D₈	1	1	yes	characteristic
202	Q₈	—	4	4	no	—
203	Q₈	—	4	4	no	—
204	Q₈	—	8	4	no	—
205	D₄	—	4	4	no	—
206	D₄	—	4	4	no	—
207	D₄	—	8	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	8	4	no	—
211	C₂×C₄	C₁₁×#127	1	2	yes	—
212	C₂×C₄	C₂×C₁₁×D₈	1	4	yes	—
213	C₂×C₄	C₂×C₁₁×D₈	1	1	yes	characteristic
214	C₂×C₄	C₂×C₁₁×D₈	1	2	yes	—
215	C₂×C₄	C₂²×D₄×C₁₁	1	1	yes	characteristic
216	C₈	C₁₁×#127	1	2	yes	—
217	C₈	C₂×C₁₁×D₈	1	4	yes	—
218	C₈	C₂²×D₄×C₁₁	1	1	yes	characteristic
219	C₈	C₂²×D₄×C₁₁	1	1	yes	characteristic
220	C₂²	—	8	8	no	—
221	C₂²	—	8	4	no	—
222	C₂²	—	16	8	no	—
223	C₂²	C₁₁×#493	1	2	yes	—
224	C₂²	C₁₁×#508	1	4	yes	—
225	C₂²	C₂²×C₁₁×D₈	1	1	yes	characteristic
226	C₄	—	2	4	no	—
227	C₄	—	8	4	no	—
228	C₄	C₁₁×#493	1	2	yes	—
229	C₄	C₂×C₁₁×#127	1	1	yes	characteristic
230	C₄	C₂²×C₁₁×D₈	1	1	yes	characteristic
231	C₂	—	16	4	no	—
232	C₂	C₁₁×#2249	1	4	yes	direct product factor
233	C₂	C₂×C₁₁×#493	1	1	yes	characteristic
234	C₂	C₂×C₁₁×#508	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₁
prime radical	C₂²×C₁₁×D₁₆	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₁ ◁ C₂×C₄²×C₁₁ ◁ C₂×C₄×C₈×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₁×Q₃₂	C₂	semidirect
2. C₂²×C₁₁×SD₃₂	C₂	semidirect
4. C₂×C₁₁×#495	C₂	semidirect
5. C₁₁×#2249	C₂	direct (N × H)
6. C₂×C₄×C₁₁×C₁₆	C₂	semidirect
12. C₂×C₁₁×Q₃₂	C₂²	semidirect
14. C₂×C₁₁×SD₃₂	C₂²	semidirect
20. C₁₁×#495	C₂²	semidirect
22. C₄×C₁₁×C₁₆	C₂²	semidirect
50. C₂×#2249	C₁₁	direct (N × H)
81. C₂²×Q₃₂	C₂×C₁₁	semidirect
82. C₂²×SD₃₂	C₂×C₁₁	semidirect
84. C₂×#495	C₂×C₁₁	semidirect
85. #2249	C₂×C₁₁	direct (N × H)
86. C₂×C₄×C₁₆	C₂×C₁₁	semidirect
114. C₂×Q₃₂	C₂²×C₁₁	semidirect
116. C₂×SD₃₂	C₂²×C₁₁	semidirect
122. #495	C₂²×C₁₁	semidirect
124. C₄×C₁₆	C₂²×C₁₁	semidirect
164. C₂×C₁₁	#2249	direct (N × H)
197. C₁₁	C₂×#2249	direct (N × H)
232. C₂	C₁₁×#2249	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×Q₃₂	2
C₂²×C₁₁×SD₃₂	2
C₂²×C₁₁×D₁₆	2
C₂×C₁₁×#495	2
C₁₁×#2249	2
C₂×C₄×C₁₁×C₁₆	2
C₂×#2249	11

Subgroups — C2×C11×#2249 #14610905

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₁×#2249 #14610905