Subgroups — C₂³×C₁₁×#121 #14639849

Order 2816.

19998 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₄×C₁₁	C₂	1	2	yes	maximal, complemented
2	C₂⁴×D₄×C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂⁵×C₄×C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₁₁×#121	C₂	1	56	yes	maximal, direct product factor
5	C₂³×C₁₁×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₁₁×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
7	C₂⁶×C₁₁	C₂²	1	2	yes	—
8	C₂⁶×C₁₁	C₂²	1	1	yes	characteristic
9	C₂³×D₄×C₁₁	—	2	32	no	—
10	C₂³×D₄×C₁₁	—	2	16	no	—
11	C₂³×D₄×C₁₁	C₂²	1	56	yes	complemented
12	C₂³×D₄×C₁₁	C₂²	1	28	yes	complemented
13	C₂⁴×C₄×C₁₁	—	2	16	no	—
14	C₂⁴×C₄×C₁₁	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₁₁	C₂²	1	2	yes	—
16	C₂⁴×C₄×C₁₁	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₁₁	C₂²	1	28	yes	complemented
18	C₂×C₁₁×#121	C₂²	1	448	yes	direct product factor
19	C₂²×C₁₁×C₄⋊C₄	C₂²	1	28	yes	complemented
20	C₂²×C₁₁×C₂²⋊C₄	C₂²	1	56	yes	complemented
21	C₂⁵×C₁₁	—	2	8	no	—
22	C₂⁵×C₁₁	—	2	8	no	—
23	C₂⁵×C₁₁	—	2	16	no	—
24	C₂⁵×C₁₁	—	2	16	no	—
25	C₂⁵×C₁₁	—	2	16	no	—
26	C₂⁵×C₁₁	D₄	1	16	yes	complemented
27	C₂⁵×C₁₁	C₂³	1	28	yes	—
28	C₂⁵×C₁₁	C₂³	1	14	yes	—
29	C₂⁵×C₁₁	C₂³	1	1	yes	characteristic
30	C₂²×D₄×C₁₁	—	2	448	no	—
31	C₂²×D₄×C₁₁	—	2	224	no	—
32	C₂²×D₄×C₁₁	C₂³	1	224	yes	complemented
33	C₂²×D₄×C₁₁	C₂³	1	112	yes	complemented
34	C₂³×C₄×C₁₁	—	2	8	no	—
35	C₂³×C₄×C₁₁	—	2	16	no	—
36	C₂³×C₄×C₁₁	—	2	224	no	—
37	C₂³×C₄×C₁₁	D₄	1	16	yes	complemented
38	C₂³×C₄×C₁₁	C₂³	1	14	yes	—
39	C₂³×C₄×C₁₁	C₂³	1	28	yes	—
40	C₂³×C₄×C₁₁	C₂³	1	14	yes	—
41	C₂³×C₄×C₁₁	C₂³	1	112	yes	complemented
42	C₁₁×#121	C₂³	1	512	yes	direct product factor
43	C₂×C₁₁×C₄⋊C₄	C₂³	1	112	yes	complemented
44	C₂×C₁₁×C₂²⋊C₄	C₂³	1	224	yes	complemented
45	C₂³×#121	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂⁴×C₁₁	—	2	128	no	—
47	C₂⁴×C₁₁	—	2	56	no	—
48	C₂⁴×C₁₁	—	2	56	no	—
49	C₂⁴×C₁₁	—	2	112	no	—
50	C₂⁴×C₁₁	—	2	112	no	—
51	C₂⁴×C₁₁	—	2	112	no	—
52	C₂⁴×C₁₁	—	4	128	no	—
53	C₂⁴×C₁₁	C₂×D₄	1	8	yes	—
54	C₂⁴×C₁₁	C₂×D₄	1	8	yes	—
55	C₂⁴×C₁₁	C₂×D₄	1	112	yes	complemented
56	C₂⁴×C₁₁	#41	1	8	yes	—
57	C₂⁴×C₁₁	C₂⁴	1	56	yes	—
58	C₂⁴×C₁₁	C₂⁴	1	28	yes	—
59	C₂⁴×C₁₁	C₂⁴	1	7	yes	—
60	C₂×D₄×C₁₁	—	2	896	no	—
61	C₂×D₄×C₁₁	—	2	448	no	—
62	C₂×D₄×C₁₁	C₂⁴	1	128	yes	complemented
63	C₂×D₄×C₁₁	C₂⁴	1	64	yes	complemented
64	C₂²×C₄×C₁₁	—	2	56	no	—
65	C₂²×C₄×C₁₁	—	2	112	no	—
66	C₂²×C₄×C₁₁	—	2	448	no	—
67	C₂²×C₄×C₁₁	C₂×D₄	1	112	yes	complemented
68	C₂²×C₄×C₁₁	C₂⁴	1	28	yes	—
69	C₂²×C₄×C₁₁	C₂⁴	1	56	yes	—
70	C₂²×C₄×C₁₁	C₂⁴	1	28	yes	—
71	C₂²×C₄×C₁₁	C₂⁴	1	64	yes	complemented
72	C₁₁×C₄⋊C₄	C₂⁴	1	64	yes	complemented
73	C₁₁×C₂²⋊C₄	C₂⁴	1	128	yes	complemented
74	C₂⁴×D₄	C₂×C₁₁	1	2	yes	complemented
75	C₂⁴×D₄	C₂×C₁₁	1	1	yes	characteristic, complemented
76	C₂⁵×C₄	C₂×C₁₁	1	1	yes	characteristic, complemented
77	C₂²×#121	C₂×C₁₁	1	56	yes	direct product factor
78	C₂³×C₄⋊C₄	C₂×C₁₁	1	1	yes	characteristic, complemented
79	C₂³×C₂²⋊C₄	C₂×C₁₁	1	2	yes	complemented
80	C₂³×C₁₁	—	2	448	no	—
81	C₂³×C₁₁	—	2	56	no	—
82	C₂³×C₁₁	—	2	56	no	—
83	C₂³×C₁₁	—	2	112	no	—
84	C₂³×C₁₁	—	2	112	no	—
85	C₂³×C₁₁	—	2	112	no	—
86	C₂³×C₁₁	—	4	448	no	—
87	C₂³×C₁₁	#121	1	64	yes	direct product factor
88	C₂³×C₁₁	C₂²×D₄	1	28	yes	—
89	C₂³×C₁₁	C₂²×D₄	1	28	yes	—
90	C₂³×C₁₁	C₂²×D₄	1	112	yes	complemented
91	C₂³×C₁₁	C₂×#41	1	28	yes	—
92	C₂³×C₁₁	C₂⁵	1	16	yes	—
93	C₂³×C₁₁	C₂⁵	1	8	yes	—
94	C₂³×C₁₁	C₂⁵	1	7	yes	—
95	D₄×C₁₁	—	2	256	no	—
96	D₄×C₁₁	—	2	128	no	—
97	C₂×C₄×C₁₁	—	2	56	no	—
98	C₂×C₄×C₁₁	—	2	112	no	—
99	C₂×C₄×C₁₁	—	2	128	no	—
100	C₂×C₄×C₁₁	C₂²×D₄	1	112	yes	complemented
101	C₂×C₄×C₁₁	C₂⁵	1	8	yes	—
102	C₂×C₄×C₁₁	C₂⁵	1	16	yes	—
103	C₂×C₄×C₁₁	C₂⁵	1	8	yes	—
104	C₂⁶	C₂²×C₁₁	1	2	yes	—
105	C₂⁶	C₂²×C₁₁	1	1	yes	characteristic
106	C₂³×D₄	—	2	32	no	—
107	C₂³×D₄	—	2	16	no	—
108	C₂³×D₄	C₂²×C₁₁	1	56	yes	complemented
109	C₂³×D₄	C₂²×C₁₁	1	28	yes	complemented
110	C₂⁴×C₄	—	2	16	no	—
111	C₂⁴×C₄	C₂²×C₁₁	1	1	yes	characteristic
112	C₂⁴×C₄	C₂²×C₁₁	1	2	yes	—
113	C₂⁴×C₄	C₂²×C₁₁	1	1	yes	characteristic
114	C₂⁴×C₄	C₂²×C₁₁	1	28	yes	complemented
115	C₂×#121	C₂²×C₁₁	1	448	yes	direct product factor
116	C₂²×C₄⋊C₄	C₂²×C₁₁	1	28	yes	complemented
117	C₂²×C₂²⋊C₄	C₂²×C₁₁	1	56	yes	complemented
118	C₂²×C₁₁	—	2	8	no	—
119	C₂²×C₁₁	—	2	8	no	—
120	C₂²×C₁₁	—	2	224	no	—
121	C₂²×C₁₁	—	2	16	no	—
122	C₂²×C₁₁	—	2	16	no	—
123	C₂²×C₁₁	—	2	16	no	—
124	C₂²×C₁₁	—	4	224	no	—
125	C₂²×C₁₁	C₂×#121	1	112	yes	direct product factor
126	C₂²×C₁₁	C₂³×D₄	1	14	yes	—
127	C₂²×C₁₁	C₂³×D₄	1	14	yes	—
128	C₂²×C₁₁	C₂³×D₄	1	16	yes	complemented
129	C₂²×C₁₁	C₂²×#41	1	14	yes	—
130	C₂²×C₁₁	C₂⁶	1	1	yes	characteristic
131	C₄×C₁₁	—	2	8	no	—
132	C₄×C₁₁	—	2	16	no	—
133	C₄×C₁₁	C₂³×D₄	1	16	yes	complemented
134	C₂⁵	—	2	8	no	—
135	C₂⁵	—	2	8	no	—
136	C₂⁵	—	2	16	no	—
137	C₂⁵	—	2	16	no	—
138	C₂⁵	—	2	16	no	—
139	C₂⁵	D₄×C₁₁	1	16	yes	complemented
140	C₂⁵	C₂³×C₁₁	1	28	yes	—
141	C₂⁵	C₂³×C₁₁	1	14	yes	—
142	C₂⁵	C₂³×C₁₁	1	1	yes	characteristic
143	C₂²×D₄	—	2	448	no	—
144	C₂²×D₄	—	2	224	no	—
145	C₂²×D₄	C₂³×C₁₁	1	224	yes	complemented
146	C₂²×D₄	C₂³×C₁₁	1	112	yes	complemented
147	C₂³×C₄	—	2	8	no	—
148	C₂³×C₄	—	2	16	no	—
149	C₂³×C₄	—	2	224	no	—
150	C₂³×C₄	D₄×C₁₁	1	16	yes	complemented
151	C₂³×C₄	C₂³×C₁₁	1	14	yes	—
152	C₂³×C₄	C₂³×C₁₁	1	28	yes	—
153	C₂³×C₄	C₂³×C₁₁	1	14	yes	—
154	C₂³×C₄	C₂³×C₁₁	1	112	yes	complemented
155	#121	C₂³×C₁₁	1	512	yes	direct product factor
156	C₂×C₄⋊C₄	C₂³×C₁₁	1	112	yes	complemented
157	C₂×C₂²⋊C₄	C₂³×C₁₁	1	224	yes	complemented
158	C₂×C₁₁	—	2	16	no	—
159	C₂×C₁₁	—	4	16	no	—
160	C₂×C₁₁	C₂²×#121	1	28	yes	direct product factor
161	C₂×C₁₁	C₂⁴×D₄	1	1	yes	characteristic
162	C₂×C₁₁	C₂⁴×D₄	1	1	yes	characteristic
163	C₂×C₁₁	C₂³×#41	1	1	yes	characteristic
164	C₂⁴	—	2	128	no	—
165	C₂⁴	—	2	56	no	—
166	C₂⁴	—	2	56	no	—
167	C₂⁴	—	2	112	no	—
168	C₂⁴	—	2	112	no	—
169	C₂⁴	—	2	112	no	—
170	C₂⁴	—	4	128	no	—
171	C₂⁴	C₂×D₄×C₁₁	1	8	yes	—
172	C₂⁴	C₂×D₄×C₁₁	1	8	yes	—
173	C₂⁴	C₂×D₄×C₁₁	1	112	yes	complemented
174	C₂⁴	C₁₁×#41	1	8	yes	—
175	C₂⁴	C₂⁴×C₁₁	1	56	yes	—
176	C₂⁴	C₂⁴×C₁₁	1	28	yes	—
177	C₂⁴	C₂⁴×C₁₁	1	7	yes	—
178	C₂×D₄	—	2	896	no	—
179	C₂×D₄	—	2	448	no	—
180	C₂×D₄	C₂⁴×C₁₁	1	128	yes	complemented
181	C₂×D₄	C₂⁴×C₁₁	1	64	yes	complemented
182	C₂²×C₄	—	2	56	no	—
183	C₂²×C₄	—	2	112	no	—
184	C₂²×C₄	—	2	448	no	—
185	C₂²×C₄	C₂×D₄×C₁₁	1	112	yes	complemented
186	C₂²×C₄	C₂⁴×C₁₁	1	28	yes	—
187	C₂²×C₄	C₂⁴×C₁₁	1	56	yes	—
188	C₂²×C₄	C₂⁴×C₁₁	1	28	yes	—
189	C₂²×C₄	C₂⁴×C₁₁	1	64	yes	complemented
190	C₄⋊C₄	C₂⁴×C₁₁	1	64	yes	complemented
191	C₂²⋊C₄	C₂⁴×C₁₁	1	128	yes	complemented
192	C₁₁	C₂³×#121	1	1	yes	characteristic, direct product factor, Sylow
193	C₂³	—	2	448	no	—
194	C₂³	—	2	56	no	—
195	C₂³	—	2	56	no	—
196	C₂³	—	2	112	no	—
197	C₂³	—	2	112	no	—
198	C₂³	—	2	112	no	—
199	C₂³	—	4	448	no	—
200	C₂³	C₁₁×#121	1	64	yes	direct product factor
201	C₂³	C₂²×D₄×C₁₁	1	28	yes	—
202	C₂³	C₂²×D₄×C₁₁	1	28	yes	—
203	C₂³	C₂²×D₄×C₁₁	1	112	yes	complemented
204	C₂³	C₂×C₁₁×#41	1	28	yes	—
205	C₂³	C₂⁵×C₁₁	1	16	yes	—
206	C₂³	C₂⁵×C₁₁	1	8	yes	—
207	C₂³	C₂⁵×C₁₁	1	7	yes	—
208	D₄	—	2	256	no	—
209	D₄	—	2	128	no	—
210	C₂×C₄	—	2	56	no	—
211	C₂×C₄	—	2	112	no	—
212	C₂×C₄	—	2	128	no	—
213	C₂×C₄	C₂²×D₄×C₁₁	1	112	yes	complemented
214	C₂×C₄	C₂⁵×C₁₁	1	8	yes	—
215	C₂×C₄	C₂⁵×C₁₁	1	16	yes	—
216	C₂×C₄	C₂⁵×C₁₁	1	8	yes	—
217	C₂²	—	2	224	no	—
218	C₂²	—	2	8	no	—
219	C₂²	—	2	8	no	—
220	C₂²	—	2	16	no	—
221	C₂²	—	2	16	no	—
222	C₂²	—	2	16	no	—
223	C₂²	—	4	224	no	—
224	C₂²	C₂×C₁₁×#121	1	112	yes	direct product factor
225	C₂²	C₂³×D₄×C₁₁	1	14	yes	—
226	C₂²	C₂³×D₄×C₁₁	1	14	yes	—
227	C₂²	C₂³×D₄×C₁₁	1	16	yes	complemented
228	C₂²	C₂²×C₁₁×#41	1	14	yes	—
229	C₂²	C₂⁶×C₁₁	1	1	yes	characteristic
230	C₄	—	2	8	no	—
231	C₄	—	2	16	no	—
232	C₄	C₂³×D₄×C₁₁	1	16	yes	complemented
233	C₂	—	2	16	no	—
234	C₂	—	4	16	no	—
235	C₂	C₂²×C₁₁×#121	1	28	yes	direct product factor
236	C₂	C₂⁴×D₄×C₁₁	1	1	yes	characteristic
237	C₂	C₂⁴×D₄×C₁₁	1	1	yes	characteristic
238	C₂	C₂³×C₁₁×#41	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁵×C₁₁	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁶×C₁₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×D₄×C₁₁	C₂	semidirect
2. C₂⁴×D₄×C₁₁	C₂	semidirect
3. C₂⁵×C₄×C₁₁	C₂	semidirect
4. C₂²×C₁₁×#121	C₂	direct (N × H)
5. C₂³×C₁₁×C₄⋊C₄	C₂	semidirect
6. C₂³×C₁₁×C₂²⋊C₄	C₂	semidirect
11. C₂³×D₄×C₁₁	C₂²	semidirect
12. C₂³×D₄×C₁₁	C₂²	semidirect
17. C₂⁴×C₄×C₁₁	C₂²	semidirect
18. C₂×C₁₁×#121	C₂²	direct (N × H)
19. C₂²×C₁₁×C₄⋊C₄	C₂²	semidirect
20. C₂²×C₁₁×C₂²⋊C₄	C₂²	semidirect
26. C₂⁵×C₁₁	D₄	semidirect
32. C₂²×D₄×C₁₁	C₂³	semidirect
33. C₂²×D₄×C₁₁	C₂³	semidirect
37. C₂³×C₄×C₁₁	D₄	semidirect
41. C₂³×C₄×C₁₁	C₂³	semidirect
42. C₁₁×#121	C₂³	direct (N × H)
43. C₂×C₁₁×C₄⋊C₄	C₂³	semidirect
44. C₂×C₁₁×C₂²⋊C₄	C₂³	semidirect
45. C₂³×#121	C₁₁	direct (N × H)
55. C₂⁴×C₁₁	C₂×D₄	semidirect
62. C₂×D₄×C₁₁	C₂⁴	semidirect
63. C₂×D₄×C₁₁	C₂⁴	semidirect
67. C₂²×C₄×C₁₁	C₂×D₄	semidirect
71. C₂²×C₄×C₁₁	C₂⁴	semidirect
72. C₁₁×C₄⋊C₄	C₂⁴	semidirect
73. C₁₁×C₂²⋊C₄	C₂⁴	semidirect
74. C₂⁴×D₄	C₂×C₁₁	semidirect
75. C₂⁴×D₄	C₂×C₁₁	semidirect
76. C₂⁵×C₄	C₂×C₁₁	semidirect
77. C₂²×#121	C₂×C₁₁	direct (N × H)
78. C₂³×C₄⋊C₄	C₂×C₁₁	semidirect
79. C₂³×C₂²⋊C₄	C₂×C₁₁	semidirect
87. C₂³×C₁₁	#121	direct (N × H)
90. C₂³×C₁₁	C₂²×D₄	semidirect
100. C₂×C₄×C₁₁	C₂²×D₄	semidirect
108. C₂³×D₄	C₂²×C₁₁	semidirect
109. C₂³×D₄	C₂²×C₁₁	semidirect
114. C₂⁴×C₄	C₂²×C₁₁	semidirect
115. C₂×#121	C₂²×C₁₁	direct (N × H)
116. C₂²×C₄⋊C₄	C₂²×C₁₁	semidirect
117. C₂²×C₂²⋊C₄	C₂²×C₁₁	semidirect
125. C₂²×C₁₁	C₂×#121	direct (N × H)
128. C₂²×C₁₁	C₂³×D₄	semidirect
133. C₄×C₁₁	C₂³×D₄	semidirect
139. C₂⁵	D₄×C₁₁	semidirect
145. C₂²×D₄	C₂³×C₁₁	semidirect
146. C₂²×D₄	C₂³×C₁₁	semidirect
150. C₂³×C₄	D₄×C₁₁	semidirect
154. C₂³×C₄	C₂³×C₁₁	semidirect
155. #121	C₂³×C₁₁	direct (N × H)
156. C₂×C₄⋊C₄	C₂³×C₁₁	semidirect
157. C₂×C₂²⋊C₄	C₂³×C₁₁	semidirect
160. C₂×C₁₁	C₂²×#121	direct (N × H)
173. C₂⁴	C₂×D₄×C₁₁	semidirect
180. C₂×D₄	C₂⁴×C₁₁	semidirect
181. C₂×D₄	C₂⁴×C₁₁	semidirect
185. C₂²×C₄	C₂×D₄×C₁₁	semidirect
189. C₂²×C₄	C₂⁴×C₁₁	semidirect
190. C₄⋊C₄	C₂⁴×C₁₁	semidirect
191. C₂²⋊C₄	C₂⁴×C₁₁	semidirect
192. C₁₁	C₂³×#121	direct (N × H)
200. C₂³	C₁₁×#121	direct (N × H)
203. C₂³	C₂²×D₄×C₁₁	semidirect
213. C₂×C₄	C₂²×D₄×C₁₁	semidirect
224. C₂²	C₂×C₁₁×#121	direct (N × H)
227. C₂²	C₂³×D₄×C₁₁	semidirect
232. C₄	C₂³×D₄×C₁₁	semidirect
235. C₂	C₂²×C₁₁×#121	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₄×C₁₁	2
C₂⁵×C₄×C₁₁	2
C₂²×C₁₁×#121	2
C₂³×C₁₁×C₄⋊C₄	2
C₂³×C₁₁×C₂²⋊C₄	2
C₂³×#121	11

Subgroups — C23×C11×#121 #14639849

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₁₁×#121 #14639849