Subgroups — C₂²×C₁₁×#488 #14637797

Order 2816.

2310 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₁×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₁×#124	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₁×#488	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×C₄×C₈×C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×Q₈×C₁₁	C₂²	1	1	yes	characteristic
8	C₂³×D₄×C₁₁	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₁₁	C₂²	1	2	yes	—
10	C₂×C₁₁×#125	—	2	6	no	—
11	C₂×C₁₁×#125	C₂²	1	12	yes	complemented
12	C₂×C₁₁×#124	—	2	6	no	—
13	C₂×C₁₁×#124	C₂²	1	12	yes	—
14	C₂²×C₁₁×C₄⋊C₄	—	2	2	no	—
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×#488	C₂²	1	64	yes	direct product factor
19	C₂×C₁₁×#103	C₂²	1	24	yes	complemented
20	C₂×C₁₁×#102	C₂²	1	24	yes	—
21	C₂×C₄×C₈×C₁₁	C₂²	1	12	yes	complemented
22	C₂⁵×C₁₁	—	2	1	no	—
23	C₂²×Q₈×C₁₁	—	2	3	no	—
24	C₂²×Q₈×C₁₁	—	2	8	no	—
25	C₂²×Q₈×C₁₁	C₂³	1	6	yes	—
26	C₂²×D₄×C₁₁	—	2	3	no	—
27	C₂²×D₄×C₁₁	—	2	8	no	—
28	C₂²×D₄×C₁₁	C₂³	1	6	yes	—
29	C₂³×C₄×C₁₁	—	2	1	no	—
30	C₂³×C₄×C₁₁	—	2	2	no	—
31	C₂³×C₄×C₁₁	D₄	1	2	yes	—
32	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₁	—	2	8	no	—
34	C₂²×C₈×C₁₁	C₂³	1	12	yes	—
35	C₁₁×#125	—	2	24	no	—
36	C₁₁×#125	C₂³	1	16	yes	complemented
37	C₁₁×#124	—	2	24	no	—
38	C₁₁×#124	C₂³	1	16	yes	—
39	C₂×C₁₁×C₄⋊C₄	—	2	24	no	—
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×#103	C₂³	1	32	yes	complemented
46	C₁₁×#102	C₂³	1	32	yes	—
47	C₄×C₈×C₁₁	C₂³	1	16	yes	complemented
48	C₂²×#488	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂⁴×C₁₁	—	2	6	no	—
50	C₂⁴×C₁₁	—	4	4	no	—
51	C₂⁴×C₁₁	—	4	8	no	—
52	C₂⁴×C₁₁	C₂×D₄	1	1	yes	characteristic
53	C₂×Q₈×C₁₁	—	2	6	no	—
54	C₂×Q₈×C₁₁	—	2	24	no	—
55	C₂×Q₈×C₁₁	—	4	12	no	—
56	C₂×Q₈×C₁₁	C₂⁴	1	4	yes	—
57	C₂×D₄×C₁₁	—	2	6	no	—
58	C₂×D₄×C₁₁	—	2	24	no	—
59	C₂×D₄×C₁₁	—	4	12	no	—
60	C₂×D₄×C₁₁	C₂⁴	1	4	yes	—
61	C₂²×C₄×C₁₁	—	2	8	no	—
62	C₂²×C₄×C₁₁	—	2	6	no	—
63	C₂²×C₄×C₁₁	—	2	12	no	—
64	C₂²×C₄×C₁₁	—	4	4	no	—
65	C₂²×C₄×C₁₁	—	4	8	no	—
66	C₂²×C₄×C₁₁	C₂×D₄	1	12	yes	—
67	C₂²×C₄×C₁₁	C₂×D₄	1	3	yes	—
68	C₂²×C₄×C₁₁	#41	1	8	yes	—
69	C₂²×C₄×C₁₁	C₂⁴	1	3	yes	—
70	C₂×C₈×C₁₁	—	2	24	no	—
71	C₂×C₈×C₁₁	C₂⁴	1	8	yes	—
72	C₁₁×C₄⋊C₄	—	2	32	no	—
73	C₁₁×C₄⋊C₄	—	2	12	no	—
74	C₁₁×C₄⋊C₄	C₂⁴	1	8	yes	—
75	C₁₁×C₂²⋊C₄	—	2	32	no	—
76	C₄²×C₁₁	C₂×D₄	1	12	yes	—
77	C₄²×C₁₁	C₂⁴	1	4	yes	—
78	C₂²×#125	C₂×C₁₁	1	1	yes	characteristic, complemented
79	C₂²×#124	C₂×C₁₁	1	1	yes	characteristic
80	C₂×#488	C₂×C₁₁	1	24	yes	direct product factor
81	C₂²×#103	C₂×C₁₁	1	2	yes	complemented
82	C₂²×#102	C₂×C₁₁	1	2	yes	—
83	C₂²×C₄×C₈	C₂×C₁₁	1	1	yes	characteristic, complemented
84	C₂³×C₁₁	—	2	4	no	—
85	C₂³×C₁₁	—	4	12	no	—
86	C₂³×C₁₁	—	4	24	no	—
87	C₂³×C₁₁	—	8	16	no	—
88	C₂³×C₁₁	#124	1	4	yes	—
89	C₂³×C₁₁	#135	1	8	yes	—
90	C₂³×C₁₁	C₂²×D₄	1	3	yes	—
91	Q₈×C₁₁	—	2	8	no	—
92	Q₈×C₁₁	—	4	12	no	—
93	D₄×C₁₁	—	2	8	no	—
94	D₄×C₁₁	—	4	12	no	—
95	C₂×C₄×C₁₁	—	2	24	no	—
96	C₂×C₄×C₁₁	—	2	4	no	—
97	C₂×C₄×C₁₁	—	2	8	no	—
98	C₂×C₄×C₁₁	—	4	12	no	—
99	C₂×C₄×C₁₁	—	4	24	no	—
100	C₂×C₄×C₁₁	#124	1	12	yes	—
101	C₂×C₄×C₁₁	C₂²×D₄	1	8	yes	—
102	C₂×C₄×C₁₁	C₂²×D₄	1	3	yes	—
103	C₂×C₄×C₁₁	C₂×#41	1	12	yes	—
104	C₂×C₄×C₁₁	C₂⁵	1	1	yes	characteristic
105	C₈×C₁₁	—	2	8	no	—
106	C₂³×Q₈	C₂²×C₁₁	1	1	yes	characteristic
107	C₂³×D₄	C₂²×C₁₁	1	1	yes	characteristic
108	C₂³×C₈	C₂²×C₁₁	1	2	yes	—
109	C₂×#125	—	2	6	no	—
110	C₂×#125	C₂²×C₁₁	1	12	yes	complemented
111	C₂×#124	—	2	6	no	—
112	C₂×#124	C₂²×C₁₁	1	12	yes	—
113	C₂²×C₄⋊C₄	—	2	2	no	—
114	C₂²×C₄⋊C₄	C₂²×C₁₁	1	2	yes	—
115	C₂²×C₂²⋊C₄	—	2	2	no	—
116	C₂²×C₄²	C₂²×C₁₁	1	1	yes	characteristic
117	#488	C₂²×C₁₁	1	64	yes	direct product factor
118	C₂×#103	C₂²×C₁₁	1	24	yes	complemented
119	C₂×#102	C₂²×C₁₁	1	24	yes	—
120	C₂×C₄×C₈	C₂²×C₁₁	1	12	yes	complemented
121	C₂²×C₁₁	—	4	4	no	—
122	C₂²×C₁₁	—	4	8	no	—
123	C₂²×C₁₁	—	8	24	no	—
124	C₂²×C₁₁	#488	1	16	yes	direct product factor
125	C₂²×C₁₁	C₂×#124	1	6	yes	—
126	C₂²×C₁₁	C₂×#135	1	12	yes	—
127	C₂²×C₁₁	C₂³×D₄	1	1	yes	characteristic
128	C₄×C₁₁	—	2	8	no	—
129	C₄×C₁₁	—	4	4	no	—
130	C₄×C₁₁	—	4	8	no	—
131	C₄×C₁₁	C₂×#124	1	6	yes	—
132	C₄×C₁₁	C₂²×#41	1	2	yes	—
133	C₂⁵	—	2	1	no	—
134	C₂²×Q₈	—	2	3	no	—
135	C₂²×Q₈	—	2	8	no	—
136	C₂²×Q₈	C₂³×C₁₁	1	6	yes	—
137	C₂²×D₄	—	2	3	no	—
138	C₂²×D₄	—	2	8	no	—
139	C₂²×D₄	C₂³×C₁₁	1	6	yes	—
140	C₂³×C₄	—	2	1	no	—
141	C₂³×C₄	—	2	2	no	—
142	C₂³×C₄	D₄×C₁₁	1	2	yes	—
143	C₂³×C₄	C₂³×C₁₁	1	1	yes	characteristic
144	C₂²×C₈	—	2	8	no	—
145	C₂²×C₈	C₂³×C₁₁	1	12	yes	—
146	#125	—	2	24	no	—
147	#125	C₂³×C₁₁	1	16	yes	complemented
148	#124	—	2	24	no	—
149	#124	C₂³×C₁₁	1	16	yes	—
150	C₂×C₄⋊C₄	—	2	24	no	—
151	C₂×C₄⋊C₄	—	2	6	no	—
152	C₂×C₄⋊C₄	C₂³×C₁₁	1	12	yes	—
153	C₂×C₂²⋊C₄	—	2	24	no	—
154	C₂×C₄²	D₄×C₁₁	1	6	yes	—
155	C₂×C₄²	C₂³×C₁₁	1	6	yes	—
156	#103	C₂³×C₁₁	1	32	yes	complemented
157	#102	C₂³×C₁₁	1	32	yes	—
158	C₄×C₈	C₂³×C₁₁	1	16	yes	complemented
159	C₂×C₁₁	—	8	4	no	—
160	C₂×C₁₁	C₂×#488	1	12	yes	direct product factor
161	C₂×C₁₁	C₂²×#124	1	1	yes	characteristic
162	C₂×C₁₁	C₂²×#135	1	2	yes	—
163	C₂⁴	—	2	6	no	—
164	C₂⁴	—	4	4	no	—
165	C₂⁴	—	4	8	no	—
166	C₂⁴	C₂×D₄×C₁₁	1	1	yes	characteristic
167	C₂×Q₈	—	2	6	no	—
168	C₂×Q₈	—	2	24	no	—
169	C₂×Q₈	—	4	12	no	—
170	C₂×Q₈	C₂⁴×C₁₁	1	4	yes	—
171	C₂×D₄	—	2	6	no	—
172	C₂×D₄	—	2	24	no	—
173	C₂×D₄	—	4	12	no	—
174	C₂×D₄	C₂⁴×C₁₁	1	4	yes	—
175	C₂²×C₄	—	2	8	no	—
176	C₂²×C₄	—	2	6	no	—
177	C₂²×C₄	—	2	12	no	—
178	C₂²×C₄	—	4	4	no	—
179	C₂²×C₄	—	4	8	no	—
180	C₂²×C₄	C₂×D₄×C₁₁	1	12	yes	—
181	C₂²×C₄	C₂×D₄×C₁₁	1	3	yes	—
182	C₂²×C₄	C₁₁×#41	1	8	yes	—
183	C₂²×C₄	C₂⁴×C₁₁	1	3	yes	—
184	C₂×C₈	—	2	24	no	—
185	C₂×C₈	C₂⁴×C₁₁	1	8	yes	—
186	C₄⋊C₄	—	2	32	no	—
187	C₄⋊C₄	—	2	12	no	—
188	C₄⋊C₄	C₂⁴×C₁₁	1	8	yes	—
189	C₂²⋊C₄	—	2	32	no	—
190	C₄²	C₂×D₄×C₁₁	1	12	yes	—
191	C₄²	C₂⁴×C₁₁	1	4	yes	—
192	C₁₁	C₂²×#488	1	1	yes	characteristic, direct product factor, Sylow
193	C₂³	—	2	4	no	—
194	C₂³	—	4	12	no	—
195	C₂³	—	4	24	no	—
196	C₂³	—	8	16	no	—
197	C₂³	C₁₁×#124	1	4	yes	—
198	C₂³	C₁₁×#135	1	8	yes	—
199	C₂³	C₂²×D₄×C₁₁	1	3	yes	—
200	Q₈	—	2	8	no	—
201	Q₈	—	4	12	no	—
202	D₄	—	2	8	no	—
203	D₄	—	4	12	no	—
204	C₂×C₄	—	2	24	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	8	no	—
207	C₂×C₄	—	4	12	no	—
208	C₂×C₄	—	4	24	no	—
209	C₂×C₄	C₁₁×#124	1	12	yes	—
210	C₂×C₄	C₂²×D₄×C₁₁	1	8	yes	—
211	C₂×C₄	C₂²×D₄×C₁₁	1	3	yes	—
212	C₂×C₄	C₂×C₁₁×#41	1	12	yes	—
213	C₂×C₄	C₂⁵×C₁₁	1	1	yes	characteristic
214	C₈	—	2	8	no	—
215	C₂²	—	4	4	no	—
216	C₂²	—	4	8	no	—
217	C₂²	—	8	24	no	—
218	C₂²	C₁₁×#488	1	16	yes	direct product factor
219	C₂²	C₂×C₁₁×#124	1	6	yes	—
220	C₂²	C₂×C₁₁×#135	1	12	yes	—
221	C₂²	C₂³×D₄×C₁₁	1	1	yes	characteristic
222	C₄	—	2	8	no	—
223	C₄	—	4	4	no	—
224	C₄	—	4	8	no	—
225	C₄	C₂×C₁₁×#124	1	6	yes	—
226	C₄	C₂²×C₁₁×#41	1	2	yes	—
227	C₂	—	8	4	no	—
228	C₂	C₂×C₁₁×#488	1	12	yes	direct product factor
229	C₂	C₂²×C₁₁×#124	1	1	yes	characteristic
230	C₂	C₂²×C₁₁×#135	1	2	yes	—

	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×#125	C₂	semidirect
3. C₂×C₁₁×#488	C₂	direct (N × H)
4. C₂²×C₁₁×#103	C₂	semidirect
6. C₂²×C₄×C₈×C₁₁	C₂	semidirect
11. C₂×C₁₁×#125	C₂²	semidirect
18. C₁₁×#488	C₂²	direct (N × H)
19. C₂×C₁₁×#103	C₂²	semidirect
21. C₂×C₄×C₈×C₁₁	C₂²	semidirect
36. C₁₁×#125	C₂³	semidirect
45. C₁₁×#103	C₂³	semidirect
47. C₄×C₈×C₁₁	C₂³	semidirect
48. C₂²×#488	C₁₁	direct (N × H)
78. C₂²×#125	C₂×C₁₁	semidirect
80. C₂×#488	C₂×C₁₁	direct (N × H)
81. C₂²×#103	C₂×C₁₁	semidirect
83. C₂²×C₄×C₈	C₂×C₁₁	semidirect
110. C₂×#125	C₂²×C₁₁	semidirect
117. #488	C₂²×C₁₁	direct (N × H)
118. C₂×#103	C₂²×C₁₁	semidirect
120. C₂×C₄×C₈	C₂²×C₁₁	semidirect
124. C₂²×C₁₁	#488	direct (N × H)
147. #125	C₂³×C₁₁	semidirect
156. #103	C₂³×C₁₁	semidirect
158. C₄×C₈	C₂³×C₁₁	semidirect
160. C₂×C₁₁	C₂×#488	direct (N × H)
192. C₁₁	C₂²×#488	direct (N × H)
218. C₂²	C₁₁×#488	direct (N × H)
228. C₂	C₂×C₁₁×#488	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×#125	2
C₂²×C₁₁×#124	2
C₂×C₁₁×#488	2
C₂²×C₁₁×#103	2
C₂²×C₁₁×#102	2
C₂²×C₄×C₈×C₁₁	2
C₂²×#488	11

Subgroups — C22×C11×#488 #14637797

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₁×#488 #14637797