Subgroups — C₂²×C₁₁×#442 #14637347

Order 2816.

3654 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₁×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×D₄×C₁₁	C₂	1	2	yes	maximal, complemented
3	C₂×C₁₁×#442	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₁₁×#106	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₁₁×#102	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₁×#97	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄×C₁₁	C₂²	1	2	yes	—
8	C₂⁴×C₄×C₁₁	—	2	2	no	—
9	C₂²×C₁₁×D₈	C₄	1	16	yes	complemented
10	C₂²×C₁₁×D₈	C₂²	1	12	yes	—
11	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₁₁	—	2	12	no	—
14	C₂×C₄×D₄×C₁₁	C₂²	1	24	yes	complemented
15	C₂²×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂²×C₁₁×C₂²⋊C₄	—	2	2	no	—
17	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
18	C₁₁×#442	C₂²	1	64	yes	direct product factor
19	C₂×C₁₁×#106	C₂²	1	12	yes	complemented
20	C₂×C₁₁×#102	C₂²	1	24	yes	complemented
21	C₂×C₁₁×#97	C₂²	1	12	yes	complemented
22	C₂⁵×C₁₁	—	2	2	no	—
23	C₂²×D₄×C₁₁	—	2	6	no	—
24	C₂²×D₄×C₁₁	—	2	8	no	—
25	C₂²×D₄×C₁₁	C₂×C₄	1	16	yes	complemented
26	C₂²×D₄×C₁₁	C₂³	1	12	yes	—
27	C₂³×C₄×C₁₁	—	2	2	no	—
28	C₂³×C₄×C₁₁	—	2	24	no	—
29	C₂³×C₄×C₁₁	—	4	16	no	—
30	C₂³×C₄×C₁₁	D₄	1	2	yes	—
31	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
32	C₂×C₁₁×D₈	C₂×C₄	1	96	yes	complemented
33	C₂×C₁₁×D₈	C₂³	1	16	yes	—
34	C₂²×C₈×C₁₁	—	2	4	no	—
35	C₂²×C₈×C₁₁	C₂×C₄	1	8	yes	complemented
36	C₂²×C₈×C₁₁	C₂³	1	6	yes	—
37	C₂²×C₈×C₁₁	C₂³	1	6	yes	—
38	C₄×D₄×C₁₁	—	2	48	no	—
39	C₄×D₄×C₁₁	C₂³	1	32	yes	complemented
40	C₂×C₁₁×C₄⋊C₄	—	2	6	no	—
41	C₂×C₁₁×C₄⋊C₄	C₂³	1	12	yes	—
42	C₂×C₁₁×C₂²⋊C₄	—	2	24	no	—
43	C₂×C₄²×C₁₁	D₄	1	6	yes	—
44	C₂×C₄²×C₁₁	C₂³	1	6	yes	—
45	C₁₁×#106	C₂³	1	16	yes	complemented
46	C₁₁×#102	C₂³	1	32	yes	complemented
47	C₁₁×#97	C₂³	1	16	yes	complemented
48	C₂²×#442	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂⁴×C₁₁	—	2	16	no	—
50	C₂⁴×C₁₁	—	2	12	no	—
51	C₂⁴×C₁₁	—	4	8	no	—
52	C₂⁴×C₁₁	—	4	8	no	—
53	C₂⁴×C₁₁	C₂×D₄	1	1	yes	characteristic
54	C₂×D₄×C₁₁	—	2	48	no	—
55	C₂×D₄×C₁₁	—	2	12	no	—
56	C₂×D₄×C₁₁	—	2	24	no	—
57	C₂×D₄×C₁₁	C₂²×C₄	1	48	yes	complemented
58	C₂×D₄×C₁₁	C₂⁴	1	8	yes	—
59	C₂²×C₄×C₁₁	—	2	8	no	—
60	C₂²×C₄×C₁₁	—	2	12	no	—
61	C₂²×C₄×C₁₁	—	2	32	no	—
62	C₂²×C₄×C₁₁	—	4	96	no	—
63	C₂²×C₄×C₁₁	—	4	8	no	—
64	C₂²×C₄×C₁₁	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₁₁	C₂×D₄	1	12	yes	—
66	C₂²×C₄×C₁₁	C₂×D₄	1	3	yes	—
67	C₂²×C₄×C₁₁	#41	1	4	yes	—
68	C₂²×C₄×C₁₁	C₂⁴	1	3	yes	—
69	C₁₁×D₈	C₂²×C₄	1	64	yes	complemented
70	C₂×C₈×C₁₁	—	2	12	no	—
71	C₂×C₈×C₁₁	C₂²×C₄	1	24	yes	complemented
72	C₂×C₈×C₁₁	C₂⁴	1	4	yes	—
73	C₂×C₈×C₁₁	C₂⁴	1	4	yes	—
74	C₁₁×C₄⋊C₄	—	2	12	no	—
75	C₁₁×C₄⋊C₄	C₂⁴	1	8	yes	—
76	C₁₁×C₂²⋊C₄	—	2	32	no	—
77	C₄²×C₁₁	C₂×D₄	1	12	yes	—
78	C₄²×C₁₁	C₂⁴	1	4	yes	—
79	C₂³×D₈	C₂×C₁₁	1	1	yes	characteristic
80	C₂²×C₄×D₄	C₂×C₁₁	1	2	yes	complemented
81	C₂×#442	C₂×C₁₁	1	24	yes	direct product factor
82	C₂²×#106	C₂×C₁₁	1	1	yes	characteristic, complemented
83	C₂²×#102	C₂×C₁₁	1	2	yes	complemented
84	C₂²×#97	C₂×C₁₁	1	1	yes	characteristic, complemented
85	C₂³×C₁₁	—	2	48	no	—
86	C₂³×C₁₁	—	2	8	no	—
87	C₂³×C₁₁	—	4	24	no	—
88	C₂³×C₁₁	—	4	24	no	—
89	C₂³×C₁₁	—	4	64	no	—
90	C₂³×C₁₁	C₄×D₄	1	4	yes	—
91	C₂³×C₁₁	AGL(1,ℤ/8)	1	8	yes	—
92	C₂³×C₁₁	C₂²×D₄	1	3	yes	—
93	D₄×C₁₁	—	2	48	no	—
94	D₄×C₁₁	—	2	8	no	—
95	D₄×C₁₁	C₂³×C₄	1	16	yes	complemented
96	C₂×C₄×C₁₁	—	2	24	no	—
97	C₂×C₄×C₁₁	—	2	8	no	—
98	C₂×C₄×C₁₁	—	4	64	no	—
99	C₂×C₄×C₁₁	—	4	24	no	—
100	C₂×C₄×C₁₁	C₄×D₄	1	12	yes	—
101	C₂×C₄×C₁₁	C₂³×C₄	1	6	yes	—
102	C₂×C₄×C₁₁	C₂²×D₄	1	8	yes	—
103	C₂×C₄×C₁₁	C₂²×D₄	1	3	yes	—
104	C₂×C₄×C₁₁	C₂×#41	1	6	yes	—
105	C₂×C₄×C₁₁	C₂⁵	1	1	yes	characteristic
106	C₈×C₁₁	—	2	4	no	—
107	C₈×C₁₁	C₂³×C₄	1	8	yes	complemented
108	C₂³×D₄	C₂²×C₁₁	1	2	yes	—
109	C₂⁴×C₄	—	2	2	no	—
110	C₂²×D₈	C₄×C₁₁	1	16	yes	complemented
111	C₂²×D₈	C₂²×C₁₁	1	12	yes	—
112	C₂³×C₈	C₂²×C₁₁	1	1	yes	characteristic
113	C₂³×C₈	C₂²×C₁₁	1	1	yes	characteristic
114	C₂×C₄×D₄	—	2	12	no	—
115	C₂×C₄×D₄	C₂²×C₁₁	1	24	yes	complemented
116	C₂²×C₄⋊C₄	C₂²×C₁₁	1	2	yes	—
117	C₂²×C₂²⋊C₄	—	2	2	no	—
118	C₂²×C₄²	C₂²×C₁₁	1	1	yes	characteristic
119	#442	C₂²×C₁₁	1	64	yes	direct product factor
120	C₂×#106	C₂²×C₁₁	1	12	yes	complemented
121	C₂×#102	C₂²×C₁₁	1	24	yes	complemented
122	C₂×#97	C₂²×C₁₁	1	12	yes	complemented
123	C₂²×C₁₁	—	2	16	no	—
124	C₂²×C₁₁	—	4	96	no	—
125	C₂²×C₁₁	—	4	8	no	—
126	C₂²×C₁₁	—	4	8	no	—
127	C₂²×C₁₁	#442	1	16	yes	direct product factor
128	C₂²×C₁₁	C₂×C₄×D₄	1	6	yes	—
129	C₂²×C₁₁	C₂×AGL(1,ℤ/8)	1	12	yes	—
130	C₂²×C₁₁	C₂³×D₄	1	1	yes	characteristic
131	C₄×C₁₁	—	2	8	no	—
132	C₄×C₁₁	—	4	8	no	—
133	C₄×C₁₁	C₂×C₄×D₄	1	6	yes	—
134	C₄×C₁₁	C₂⁴×C₄	1	1	yes	characteristic
135	C₄×C₁₁	C₂²×#41	1	1	yes	characteristic
136	C₂⁵	—	2	2	no	—
137	C₂²×D₄	—	2	6	no	—
138	C₂²×D₄	—	2	8	no	—
139	C₂²×D₄	C₂×C₄×C₁₁	1	16	yes	complemented
140	C₂²×D₄	C₂³×C₁₁	1	12	yes	—
141	C₂³×C₄	—	2	2	no	—
142	C₂³×C₄	—	2	24	no	—
143	C₂³×C₄	—	4	16	no	—
144	C₂³×C₄	D₄×C₁₁	1	2	yes	—
145	C₂³×C₄	C₂³×C₁₁	1	1	yes	characteristic
146	C₂×D₈	C₂×C₄×C₁₁	1	96	yes	complemented
147	C₂×D₈	C₂³×C₁₁	1	16	yes	—
148	C₂²×C₈	—	2	4	no	—
149	C₂²×C₈	C₂×C₄×C₁₁	1	8	yes	complemented
150	C₂²×C₈	C₂³×C₁₁	1	6	yes	—
151	C₂²×C₈	C₂³×C₁₁	1	6	yes	—
152	C₄×D₄	—	2	48	no	—
153	C₄×D₄	C₂³×C₁₁	1	32	yes	complemented
154	C₂×C₄⋊C₄	—	2	6	no	—
155	C₂×C₄⋊C₄	C₂³×C₁₁	1	12	yes	—
156	C₂×C₂²⋊C₄	—	2	24	no	—
157	C₂×C₄²	D₄×C₁₁	1	6	yes	—
158	C₂×C₄²	C₂³×C₁₁	1	6	yes	—
159	#106	C₂³×C₁₁	1	16	yes	complemented
160	#102	C₂³×C₁₁	1	32	yes	complemented
161	#97	C₂³×C₁₁	1	16	yes	complemented
162	C₂×C₁₁	—	4	16	no	—
163	C₂×C₁₁	C₂×#442	1	12	yes	direct product factor
164	C₂×C₁₁	C₂²×C₄×D₄	1	1	yes	characteristic
165	C₂×C₁₁	C₂²×AGL(1,ℤ/8)	1	2	yes	—
166	C₂⁴	—	2	16	no	—
167	C₂⁴	—	2	12	no	—
168	C₂⁴	—	4	8	no	—
169	C₂⁴	—	4	8	no	—
170	C₂⁴	C₂×D₄×C₁₁	1	1	yes	characteristic
171	C₂×D₄	—	2	48	no	—
172	C₂×D₄	—	2	12	no	—
173	C₂×D₄	—	2	24	no	—
174	C₂×D₄	C₂²×C₄×C₁₁	1	48	yes	complemented
175	C₂×D₄	C₂⁴×C₁₁	1	8	yes	—
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	—	2	12	no	—
178	C₂²×C₄	—	2	32	no	—
179	C₂²×C₄	—	4	96	no	—
180	C₂²×C₄	—	4	8	no	—
181	C₂²×C₄	C₂²×C₄×C₁₁	1	4	yes	—
182	C₂²×C₄	C₂×D₄×C₁₁	1	12	yes	—
183	C₂²×C₄	C₂×D₄×C₁₁	1	3	yes	—
184	C₂²×C₄	C₁₁×#41	1	4	yes	—
185	C₂²×C₄	C₂⁴×C₁₁	1	3	yes	—
186	D₈	C₂²×C₄×C₁₁	1	64	yes	complemented
187	C₂×C₈	—	2	12	no	—
188	C₂×C₈	C₂²×C₄×C₁₁	1	24	yes	complemented
189	C₂×C₈	C₂⁴×C₁₁	1	4	yes	—
190	C₂×C₈	C₂⁴×C₁₁	1	4	yes	—
191	C₄⋊C₄	—	2	12	no	—
192	C₄⋊C₄	C₂⁴×C₁₁	1	8	yes	—
193	C₂²⋊C₄	—	2	32	no	—
194	C₄²	C₂×D₄×C₁₁	1	12	yes	—
195	C₄²	C₂⁴×C₁₁	1	4	yes	—
196	C₁₁	C₂²×#442	1	1	yes	characteristic, direct product factor, Sylow
197	C₂³	—	2	48	no	—
198	C₂³	—	2	8	no	—
199	C₂³	—	4	24	no	—
200	C₂³	—	4	24	no	—
201	C₂³	—	4	64	no	—
202	C₂³	C₄×D₄×C₁₁	1	4	yes	—
203	C₂³	C₁₁×AGL(1,ℤ/8)	1	8	yes	—
204	C₂³	C₂²×D₄×C₁₁	1	3	yes	—
205	D₄	—	2	48	no	—
206	D₄	—	2	8	no	—
207	D₄	C₂³×C₄×C₁₁	1	16	yes	complemented
208	C₂×C₄	—	2	24	no	—
209	C₂×C₄	—	2	8	no	—
210	C₂×C₄	—	4	64	no	—
211	C₂×C₄	—	4	24	no	—
212	C₂×C₄	C₄×D₄×C₁₁	1	12	yes	—
213	C₂×C₄	C₂³×C₄×C₁₁	1	6	yes	—
214	C₂×C₄	C₂²×D₄×C₁₁	1	8	yes	—
215	C₂×C₄	C₂²×D₄×C₁₁	1	3	yes	—
216	C₂×C₄	C₂×C₁₁×#41	1	6	yes	—
217	C₂×C₄	C₂⁵×C₁₁	1	1	yes	characteristic
218	C₈	—	2	4	no	—
219	C₈	C₂³×C₄×C₁₁	1	8	yes	complemented
220	C₂²	—	2	16	no	—
221	C₂²	—	4	96	no	—
222	C₂²	—	4	8	no	—
223	C₂²	—	4	8	no	—
224	C₂²	C₁₁×#442	1	16	yes	direct product factor
225	C₂²	C₂×C₄×D₄×C₁₁	1	6	yes	—
226	C₂²	C₂×C₁₁×AGL(1,ℤ/8)	1	12	yes	—
227	C₂²	C₂³×D₄×C₁₁	1	1	yes	characteristic
228	C₄	—	2	8	no	—
229	C₄	—	4	8	no	—
230	C₄	C₂×C₄×D₄×C₁₁	1	6	yes	—
231	C₄	C₂⁴×C₄×C₁₁	1	1	yes	characteristic
232	C₄	C₂²×C₁₁×#41	1	1	yes	characteristic
233	C₂	—	4	16	no	—
234	C₂	C₂×C₁₁×#442	1	12	yes	direct product factor
235	C₂	C₂²×C₄×D₄×C₁₁	1	1	yes	characteristic
236	C₂	C₂²×C₁₁×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₁	C₂×D₄
commutator	C₄	C₂⁴×C₄×C₁₁
Frattini, cosocle	C₂×C₄	C₂⁵×C₁₁
prime radical	C₂³×C₁₁×D₈	C₂
LCS	G ▷ 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
2. C₂²×C₄×D₄×C₁₁	C₂	semidirect
3. C₂×C₁₁×#442	C₂	direct (N × H)
4. C₂²×C₁₁×#106	C₂	semidirect
5. C₂²×C₁₁×#102	C₂	semidirect
6. C₂²×C₁₁×#97	C₂	semidirect
9. C₂²×C₁₁×D₈	C₄	semidirect
14. C₂×C₄×D₄×C₁₁	C₂²	semidirect
18. C₁₁×#442	C₂²	direct (N × H)
19. C₂×C₁₁×#106	C₂²	semidirect
20. C₂×C₁₁×#102	C₂²	semidirect
21. C₂×C₁₁×#97	C₂²	semidirect
25. C₂²×D₄×C₁₁	C₂×C₄	semidirect
32. C₂×C₁₁×D₈	C₂×C₄	semidirect
35. C₂²×C₈×C₁₁	C₂×C₄	semidirect
39. C₄×D₄×C₁₁	C₂³	semidirect
45. C₁₁×#106	C₂³	semidirect
46. C₁₁×#102	C₂³	semidirect
47. C₁₁×#97	C₂³	semidirect
48. C₂²×#442	C₁₁	direct (N × H)
57. C₂×D₄×C₁₁	C₂²×C₄	semidirect
69. C₁₁×D₈	C₂²×C₄	semidirect
71. C₂×C₈×C₁₁	C₂²×C₄	semidirect
80. C₂²×C₄×D₄	C₂×C₁₁	semidirect
81. C₂×#442	C₂×C₁₁	direct (N × H)
82. C₂²×#106	C₂×C₁₁	semidirect
83. C₂²×#102	C₂×C₁₁	semidirect
84. C₂²×#97	C₂×C₁₁	semidirect
95. D₄×C₁₁	C₂³×C₄	semidirect
107. C₈×C₁₁	C₂³×C₄	semidirect
110. C₂²×D₈	C₄×C₁₁	semidirect
115. C₂×C₄×D₄	C₂²×C₁₁	semidirect
119. #442	C₂²×C₁₁	direct (N × H)
120. C₂×#106	C₂²×C₁₁	semidirect
121. C₂×#102	C₂²×C₁₁	semidirect
122. C₂×#97	C₂²×C₁₁	semidirect
127. C₂²×C₁₁	#442	direct (N × H)
139. C₂²×D₄	C₂×C₄×C₁₁	semidirect
146. C₂×D₈	C₂×C₄×C₁₁	semidirect
149. C₂²×C₈	C₂×C₄×C₁₁	semidirect
153. C₄×D₄	C₂³×C₁₁	semidirect
159. #106	C₂³×C₁₁	semidirect
160. #102	C₂³×C₁₁	semidirect
161. #97	C₂³×C₁₁	semidirect
163. C₂×C₁₁	C₂×#442	direct (N × H)
174. C₂×D₄	C₂²×C₄×C₁₁	semidirect
186. D₈	C₂²×C₄×C₁₁	semidirect
188. C₂×C₈	C₂²×C₄×C₁₁	semidirect
196. C₁₁	C₂²×#442	direct (N × H)
207. D₄	C₂³×C₄×C₁₁	semidirect
219. C₈	C₂³×C₄×C₁₁	semidirect
224. C₂²	C₁₁×#442	direct (N × H)
234. C₂	C₂×C₁₁×#442	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₁₁×D₈	2
C₂²×C₄×D₄×C₁₁	2
C₂×C₁₁×#442	2
C₂²×C₁₁×#106	2
C₂²×C₁₁×#102	2
C₂²×C₁₁×#97	2
C₂²×#442	11

Subgroups — C22×C11×#442 #14637347

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₁×#442 #14637347