Subgroups — C₂²×C₁₁×#559 #14639903

Order 2816.

8710 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×#559	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₁₁×#127	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₁×#124	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₁×#123	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄×Q₈×C₁₁	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×D₄×C₁₁	C₂	1	4	yes	maximal, complemented
7	C₂³×Q₈×C₁₁	C₂²	1	2	yes	—
8	C₂³×D₄×C₁₁	C₂²	1	4	yes	—
9	C₂³×D₄×C₁₁	C₂²	1	2	yes	—
10	C₂⁴×C₄×C₁₁	C₂²	1	4	yes	—
11	C₁₁×#559	C₂²	1	256	yes	direct product factor
12	C₂×C₁₁×#127	C₂²	1	24	yes	—
13	C₂×C₁₁×#124	C₂²	1	96	yes	complemented
14	C₂×C₁₁×#123	C₂²	1	96	yes	complemented
15	C₂×C₄×Q₈×C₁₁	C₂²	1	48	yes	complemented
16	C₂×C₄×D₄×C₁₁	C₂²	1	96	yes	complemented
17	C₂²×C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂²×C₁₁×C₄⋊C₄	C₂²	1	4	yes	—
19	C₂²×C₁₁×C₂²⋊C₄	C₂²	1	8	yes	—
20	C₂²×C₁₁×C₂²⋊C₄	C₂²	1	4	yes	—
21	C₂²×C₄²×C₁₁	C₂²	1	4	yes	—
22	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
23	C₂⁵×C₁₁	C₂³	1	4	yes	—
24	C₂²×Q₈×C₁₁	—	2	16	no	—
25	C₂²×Q₈×C₁₁	C₂³	1	24	yes	—
26	C₂²×D₄×C₁₁	—	2	32	no	—
27	C₂²×D₄×C₁₁	—	4	8	no	—
28	C₂²×D₄×C₁₁	C₂³	1	48	yes	—
29	C₂²×D₄×C₁₁	C₂³	1	24	yes	—
30	C₂³×C₄×C₁₁	—	4	16	no	—
31	C₂³×C₄×C₁₁	C₂³	1	4	yes	—
32	C₂³×C₄×C₁₁	C₂³	1	4	yes	—
33	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₁₁	C₂³	1	48	yes	—
35	C₂³×C₄×C₁₁	C₂³	1	2	yes	—
36	C₁₁×#127	C₂³	1	64	yes	—
37	C₁₁×#124	C₂³	1	256	yes	complemented
38	C₁₁×#123	C₂³	1	256	yes	complemented
39	C₄×Q₈×C₁₁	C₂³	1	128	yes	complemented
40	C₄×D₄×C₁₁	C₂³	1	256	yes	complemented
41	C₂×C₁₁×C₄⋊C₄	C₂³	1	24	yes	—
42	C₂×C₁₁×C₄⋊C₄	C₂³	1	48	yes	—
43	C₂×C₁₁×C₂²⋊C₄	C₂³	1	96	yes	—
44	C₂×C₁₁×C₂²⋊C₄	C₂³	1	48	yes	—
45	C₂×C₄²×C₁₁	C₂³	1	48	yes	—
46	C₂×C₄²×C₁₁	C₂³	1	12	yes	—
47	C₂²×#559	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂⁴×C₁₁	—	2	32	no	—
49	C₂⁴×C₁₁	—	2	16	no	—
50	C₂⁴×C₁₁	C₂⁴	1	24	yes	—
51	C₂⁴×C₁₁	C₂⁴	1	1	yes	characteristic
52	C₂×Q₈×C₁₁	—	2	96	no	—
53	C₂×Q₈×C₁₁	C₂⁴	1	32	yes	—
54	C₂×D₄×C₁₁	—	2	192	no	—
55	C₂×D₄×C₁₁	—	4	48	no	—
56	C₂×D₄×C₁₁	C₂⁴	1	64	yes	—
57	C₂×D₄×C₁₁	C₂⁴	1	32	yes	—
58	C₂²×C₄×C₁₁	—	2	16	no	—
59	C₂²×C₄×C₁₁	—	2	16	no	—
60	C₂²×C₄×C₁₁	—	2	4	no	—
61	C₂²×C₄×C₁₁	—	4	96	no	—
62	C₂²×C₄×C₁₁	#41	1	16	yes	—
63	C₂²×C₄×C₁₁	C₂⁴	1	24	yes	—
64	C₂²×C₄×C₁₁	C₂⁴	1	24	yes	—
65	C₂²×C₄×C₁₁	C₂⁴	1	6	yes	—
66	C₂²×C₄×C₁₁	C₂⁴	1	64	yes	—
67	C₂²×C₄×C₁₁	C₂⁴	1	12	yes	—
68	C₁₁×C₄⋊C₄	C₂⁴	1	32	yes	—
69	C₁₁×C₄⋊C₄	C₂⁴	1	64	yes	—
70	C₁₁×C₂²⋊C₄	C₂⁴	1	128	yes	—
71	C₁₁×C₂²⋊C₄	C₂⁴	1	64	yes	—
72	C₄²×C₁₁	C₂⁴	1	64	yes	—
73	C₄²×C₁₁	C₂⁴	1	16	yes	—
74	C₂×#559	C₂×C₁₁	1	48	yes	direct product factor
75	C₂²×#127	C₂×C₁₁	1	1	yes	characteristic
76	C₂²×#124	C₂×C₁₁	1	4	yes	complemented
77	C₂²×#123	C₂×C₁₁	1	4	yes	complemented
78	C₂²×C₄×Q₈	C₂×C₁₁	1	2	yes	complemented
79	C₂²×C₄×D₄	C₂×C₁₁	1	4	yes	complemented
80	C₂³×C₁₁	—	2	96	no	—
81	C₂³×C₁₁	—	2	48	no	—
82	C₂³×C₁₁	—	4	64	no	—
83	C₂³×C₁₁	C₂×#41	1	8	yes	—
84	C₂³×C₁₁	H(5,𝔽₂)	1	4	yes	—
85	C₂³×C₁₁	C₂⁵	1	16	yes	—
86	C₂³×C₁₁	C₂⁵	1	3	yes	—
87	Q₈×C₁₁	—	2	64	no	—
88	D₄×C₁₁	—	2	128	no	—
89	D₄×C₁₁	—	4	32	no	—
90	C₂×C₄×C₁₁	—	2	48	no	—
91	C₂×C₄×C₁₁	—	2	48	no	—
92	C₂×C₄×C₁₁	—	2	12	no	—
93	C₂×C₄×C₁₁	—	4	64	no	—
94	C₂×C₄×C₁₁	C₂×#41	1	48	yes	—
95	C₂×C₄×C₁₁	C₂⁵	1	16	yes	—
96	C₂×C₄×C₁₁	C₂⁵	1	16	yes	—
97	C₂×C₄×C₁₁	C₂⁵	1	4	yes	—
98	C₂×C₄×C₁₁	C₂⁵	1	8	yes	—
99	C₂³×Q₈	C₂²×C₁₁	1	2	yes	—
100	C₂³×D₄	C₂²×C₁₁	1	4	yes	—
101	C₂³×D₄	C₂²×C₁₁	1	2	yes	—
102	C₂⁴×C₄	C₂²×C₁₁	1	4	yes	—
103	#559	C₂²×C₁₁	1	256	yes	direct product factor
104	C₂×#127	C₂²×C₁₁	1	24	yes	—
105	C₂×#124	C₂²×C₁₁	1	96	yes	complemented
106	C₂×#123	C₂²×C₁₁	1	96	yes	complemented
107	C₂×C₄×Q₈	C₂²×C₁₁	1	48	yes	complemented
108	C₂×C₄×D₄	C₂²×C₁₁	1	96	yes	complemented
109	C₂²×C₄⋊C₄	C₂²×C₁₁	1	2	yes	—
110	C₂²×C₄⋊C₄	C₂²×C₁₁	1	4	yes	—
111	C₂²×C₂²⋊C₄	C₂²×C₁₁	1	8	yes	—
112	C₂²×C₂²⋊C₄	C₂²×C₁₁	1	4	yes	—
113	C₂²×C₄²	C₂²×C₁₁	1	4	yes	—
114	C₂²×C₄²	C₂²×C₁₁	1	1	yes	characteristic
115	C₂²×C₁₁	—	2	32	no	—
116	C₂²×C₁₁	—	2	16	no	—
117	C₂²×C₁₁	—	4	96	no	—
118	C₂²×C₁₁	#559	1	16	yes	direct product factor
119	C₂²×C₁₁	C₂²×#41	1	12	yes	—
120	C₂²×C₁₁	C₂×H(5,𝔽₂)	1	6	yes	—
121	C₂²×C₁₁	C₂⁶	1	1	yes	characteristic
122	C₄×C₁₁	—	2	16	no	—
123	C₄×C₁₁	—	2	16	no	—
124	C₄×C₁₁	—	2	4	no	—
125	C₄×C₁₁	C₂²×#41	1	16	yes	—
126	C₂⁵	C₂³×C₁₁	1	4	yes	—
127	C₂²×Q₈	—	2	16	no	—
128	C₂²×Q₈	C₂³×C₁₁	1	24	yes	—
129	C₂²×D₄	—	2	32	no	—
130	C₂²×D₄	—	4	8	no	—
131	C₂²×D₄	C₂³×C₁₁	1	48	yes	—
132	C₂²×D₄	C₂³×C₁₁	1	24	yes	—
133	C₂³×C₄	—	4	16	no	—
134	C₂³×C₄	C₂³×C₁₁	1	4	yes	—
135	C₂³×C₄	C₂³×C₁₁	1	4	yes	—
136	C₂³×C₄	C₂³×C₁₁	1	1	yes	characteristic
137	C₂³×C₄	C₂³×C₁₁	1	48	yes	—
138	C₂³×C₄	C₂³×C₁₁	1	2	yes	—
139	#127	C₂³×C₁₁	1	64	yes	—
140	#124	C₂³×C₁₁	1	256	yes	complemented
141	#123	C₂³×C₁₁	1	256	yes	complemented
142	C₄×Q₈	C₂³×C₁₁	1	128	yes	complemented
143	C₄×D₄	C₂³×C₁₁	1	256	yes	complemented
144	C₂×C₄⋊C₄	C₂³×C₁₁	1	24	yes	—
145	C₂×C₄⋊C₄	C₂³×C₁₁	1	48	yes	—
146	C₂×C₂²⋊C₄	C₂³×C₁₁	1	96	yes	—
147	C₂×C₂²⋊C₄	C₂³×C₁₁	1	48	yes	—
148	C₂×C₄²	C₂³×C₁₁	1	48	yes	—
149	C₂×C₄²	C₂³×C₁₁	1	12	yes	—
150	C₂×C₁₁	—	4	16	no	—
151	C₂×C₁₁	C₂×#559	1	12	yes	direct product factor
152	C₂×C₁₁	C₂³×#41	1	2	yes	—
153	C₂×C₁₁	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
154	C₂⁴	—	2	32	no	—
155	C₂⁴	—	2	16	no	—
156	C₂⁴	C₂⁴×C₁₁	1	24	yes	—
157	C₂⁴	C₂⁴×C₁₁	1	1	yes	characteristic
158	C₂×Q₈	—	2	96	no	—
159	C₂×Q₈	C₂⁴×C₁₁	1	32	yes	—
160	C₂×D₄	—	2	192	no	—
161	C₂×D₄	—	4	48	no	—
162	C₂×D₄	C₂⁴×C₁₁	1	64	yes	—
163	C₂×D₄	C₂⁴×C₁₁	1	32	yes	—
164	C₂²×C₄	—	2	16	no	—
165	C₂²×C₄	—	2	16	no	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	—	4	96	no	—
168	C₂²×C₄	C₁₁×#41	1	16	yes	—
169	C₂²×C₄	C₂⁴×C₁₁	1	24	yes	—
170	C₂²×C₄	C₂⁴×C₁₁	1	24	yes	—
171	C₂²×C₄	C₂⁴×C₁₁	1	6	yes	—
172	C₂²×C₄	C₂⁴×C₁₁	1	64	yes	—
173	C₂²×C₄	C₂⁴×C₁₁	1	12	yes	—
174	C₄⋊C₄	C₂⁴×C₁₁	1	32	yes	—
175	C₄⋊C₄	C₂⁴×C₁₁	1	64	yes	—
176	C₂²⋊C₄	C₂⁴×C₁₁	1	128	yes	—
177	C₂²⋊C₄	C₂⁴×C₁₁	1	64	yes	—
178	C₄²	C₂⁴×C₁₁	1	64	yes	—
179	C₄²	C₂⁴×C₁₁	1	16	yes	—
180	C₁₁	C₂²×#559	1	1	yes	characteristic, direct product factor, Sylow
181	C₂³	—	2	96	no	—
182	C₂³	—	2	48	no	—
183	C₂³	—	4	64	no	—
184	C₂³	C₂×C₁₁×#41	1	8	yes	—
185	C₂³	C₁₁×H(5,𝔽₂)	1	4	yes	—
186	C₂³	C₂⁵×C₁₁	1	16	yes	—
187	C₂³	C₂⁵×C₁₁	1	3	yes	—
188	Q₈	—	2	64	no	—
189	D₄	—	2	128	no	—
190	D₄	—	4	32	no	—
191	C₂×C₄	—	2	48	no	—
192	C₂×C₄	—	2	48	no	—
193	C₂×C₄	—	2	12	no	—
194	C₂×C₄	—	4	64	no	—
195	C₂×C₄	C₂×C₁₁×#41	1	48	yes	—
196	C₂×C₄	C₂⁵×C₁₁	1	16	yes	—
197	C₂×C₄	C₂⁵×C₁₁	1	16	yes	—
198	C₂×C₄	C₂⁵×C₁₁	1	4	yes	—
199	C₂×C₄	C₂⁵×C₁₁	1	8	yes	—
200	C₂²	—	2	32	no	—
201	C₂²	—	2	16	no	—
202	C₂²	—	4	96	no	—
203	C₂²	C₁₁×#559	1	16	yes	direct product factor
204	C₂²	C₂²×C₁₁×#41	1	12	yes	—
205	C₂²	C₂×C₁₁×H(5,𝔽₂)	1	6	yes	—
206	C₂²	C₂⁶×C₁₁	1	1	yes	characteristic
207	C₄	—	2	16	no	—
208	C₄	—	2	16	no	—
209	C₄	—	2	4	no	—
210	C₄	C₂²×C₁₁×#41	1	16	yes	—
211	C₂	—	4	16	no	—
212	C₂	C₂×C₁₁×#559	1	12	yes	direct product factor
213	C₂	C₂³×C₁₁×#41	1	2	yes	—
214	C₂	C₂²×C₁₁×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₁₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₁₁
prime radical	C₂²×C₁₁×#127	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₁×#559	C₂	direct (N × H)
3. C₂²×C₁₁×#124	C₂	semidirect
4. C₂²×C₁₁×#123	C₂	semidirect
5. C₂²×C₄×Q₈×C₁₁	C₂	semidirect
6. C₂²×C₄×D₄×C₁₁	C₂	semidirect
11. C₁₁×#559	C₂²	direct (N × H)
13. C₂×C₁₁×#124	C₂²	semidirect
14. C₂×C₁₁×#123	C₂²	semidirect
15. C₂×C₄×Q₈×C₁₁	C₂²	semidirect
16. C₂×C₄×D₄×C₁₁	C₂²	semidirect
37. C₁₁×#124	C₂³	semidirect
38. C₁₁×#123	C₂³	semidirect
39. C₄×Q₈×C₁₁	C₂³	semidirect
40. C₄×D₄×C₁₁	C₂³	semidirect
47. C₂²×#559	C₁₁	direct (N × H)
74. C₂×#559	C₂×C₁₁	direct (N × H)
76. C₂²×#124	C₂×C₁₁	semidirect
77. C₂²×#123	C₂×C₁₁	semidirect
78. C₂²×C₄×Q₈	C₂×C₁₁	semidirect
79. C₂²×C₄×D₄	C₂×C₁₁	semidirect
103. #559	C₂²×C₁₁	direct (N × H)
105. C₂×#124	C₂²×C₁₁	semidirect
106. C₂×#123	C₂²×C₁₁	semidirect
107. C₂×C₄×Q₈	C₂²×C₁₁	semidirect
108. C₂×C₄×D₄	C₂²×C₁₁	semidirect
118. C₂²×C₁₁	#559	direct (N × H)
140. #124	C₂³×C₁₁	semidirect
141. #123	C₂³×C₁₁	semidirect
142. C₄×Q₈	C₂³×C₁₁	semidirect
143. C₄×D₄	C₂³×C₁₁	semidirect
151. C₂×C₁₁	C₂×#559	direct (N × H)
180. C₁₁	C₂²×#559	direct (N × H)
203. C₂²	C₁₁×#559	direct (N × H)
212. C₂	C₂×C₁₁×#559	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₁×#559	2
C₂²×C₁₁×#127	2
C₂²×C₁₁×#124	2
C₂²×C₁₁×#123	2
C₂²×C₄×Q₈×C₁₁	2
C₂²×C₄×D₄×C₁₁	2
C₂²×#559	11

Subgroups — C22×C11×#559 #14639903

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₁×#559 #14639903