Subgroups — C₂²×#10623248 #15660683

Order 2816.

20038 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄×D₁₁	C₂	1	4	yes	maximal, complemented
2	C₂×#10623248	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#66991	C₂	1	4	yes	maximal, complemented
4	C₂²×#66990	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄×D₄₄	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₁×#127	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₁₁	C₂²	1	4	yes	complemented
8	C₂³×D₄×C₁₁	C₂²	1	4	yes	complemented
9	C₂³×D₄×C₁₁	C₂²	1	2	yes	complemented
10	C₂³×C₂²⋊D₁₁	C₂²	1	8	yes	complemented
11	C₂²×D₄×D₁₁	—	2	64	no	—
12	C₂²×D₄×D₁₁	C₂²	1	96	yes	complemented
13	C₂³×D₄₄	C₂²	1	2	yes	—
14	C₂³×C₄×D₁₁	C₂²	1	4	yes	complemented
15	C₂×C₁₁×#127	C₂²	1	24	yes	complemented
16	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic, complemented
17	C₂²×C₂²⋊Dic₁₁	C₂²	1	4	yes	complemented
18	C₂×#66991	C₂²	1	96	yes	complemented
19	C₂×#66990	C₂²	1	96	yes	complemented
20	C₂²×(C₂×D₁₁)⋊C₄	C₂²	1	4	yes	complemented
21	C₂²×C₄⋊Dic₁₁	C₂²	1	2	yes	complemented
22	#10623248	C₂²	1	256	yes	direct product factor
23	C₂×C₄×D₄₄	C₂²	1	48	yes	complemented
24	C₂⁵×C₁₁	C₂³	1	4	yes	—
25	C₂⁴×D₁₁	—	2	64	no	—
26	C₂⁴×D₁₁	—	4	16	no	—
27	C₂⁴×D₁₁	C₂³	1	4	yes	—
28	C₂⁴×D₁₁	C₂³	1	48	yes	complemented
29	C₂²×D₄×C₁₁	—	2	32	no	—
30	C₂²×D₄×C₁₁	—	4	8	no	—
31	C₂²×D₄×C₁₁	C₂³	1	48	yes	complemented
32	C₂²×D₄×C₁₁	C₂³	1	24	yes	complemented
33	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₁₁	C₂³	1	2	yes	—
35	C₂²×C₂²⋊D₁₁	—	2	64	no	—
36	C₂²×C₂²⋊D₁₁	C₂³	1	96	yes	complemented
37	C₂³×Dic₁₁	C₂³	1	4	yes	—
38	C₂×D₄×D₁₁	—	2	768	no	—
39	C₂×D₄×D₁₁	C₂³	1	256	yes	complemented
40	C₂²×D₄₄	D₄	1	32	yes	complemented
41	C₂²×D₄₄	C₂³	1	24	yes	—
42	C₂²×C₄×D₁₁	—	2	32	no	—
43	C₂²×C₄×D₁₁	C₂³	1	48	yes	complemented
44	C₁₁×#127	C₂³	1	64	yes	complemented
45	C₂×C₄²×C₁₁	C₂³	1	12	yes	complemented
46	C₂×C₂²⋊Dic₁₁	C₂³	1	48	yes	complemented
47	#66991	C₂³	1	256	yes	complemented
48	#66990	C₂³	1	256	yes	complemented
49	C₂×(C₂×D₁₁)⋊C₄	C₂³	1	48	yes	complemented
50	C₂×C₄⋊Dic₁₁	C₂³	1	24	yes	complemented
51	C₄×D₄₄	C₂³	1	128	yes	complemented
52	C₂²×D₄²	—	11	1	no	maximal, Sylow
53	C₂⁴×C₁₁	—	2	32	no	—
54	C₂⁴×C₁₁	—	2	16	no	—
55	C₂⁴×C₁₁	C₂⁴	1	24	yes	—
56	C₂⁴×C₁₁	C₂⁴	1	1	yes	characteristic
57	C₂³×D₁₁	—	2	16	no	—
58	C₂³×D₁₁	—	2	16	no	—
59	C₂³×D₁₁	—	2	384	no	—
60	C₂³×D₁₁	—	4	96	no	—
61	C₂³×D₁₁	—	4	256	no	—
62	C₂³×D₁₁	C₂×D₄	1	32	yes	complemented
63	C₂³×D₁₁	C₂⁴	1	24	yes	—
64	C₂³×D₁₁	C₂⁴	1	64	yes	complemented
65	C₂×D₄×C₁₁	—	2	192	no	—
66	C₂×D₄×C₁₁	—	4	48	no	—
67	C₂×D₄×C₁₁	C₂⁴	1	64	yes	complemented
68	C₂×D₄×C₁₁	C₂⁴	1	32	yes	complemented
69	C₂²×C₄×C₁₁	—	2	4	no	—
70	C₂²×C₄×C₁₁	C₂×D₄	1	16	yes	complemented
71	C₂²×C₄×C₁₁	C₂⁴	1	6	yes	—
72	C₂²×C₄×C₁₁	C₂⁴	1	12	yes	—
73	C₂×C₂²⋊D₁₁	—	2	384	no	—
74	C₂×C₂²⋊D₁₁	C₂⁴	1	128	yes	complemented
75	C₂²×Dic₁₁	—	2	16	no	—
76	C₂²×Dic₁₁	C₂⁴	1	24	yes	—
77	D₄×D₁₁	—	2	1024	no	—
78	C₂×D₄₄	C₂×D₄	1	192	yes	complemented
79	C₂×D₄₄	C₂⁴	1	32	yes	—
80	C₂×C₄×D₁₁	—	2	192	no	—
81	C₂×C₄×D₁₁	C₂⁴	1	64	yes	complemented
82	C₄²×C₁₁	C₂⁴	1	16	yes	complemented
83	C₂²⋊Dic₁₁	C₂⁴	1	64	yes	complemented
84	(C₂×D₁₁)⋊C₄	C₂⁴	1	64	yes	complemented
85	C₄⋊Dic₁₁	C₂⁴	1	32	yes	complemented
86	C₂⁴×D₄	—	11	4	no	—
87	C₂×D₄²	—	11	48	no	—
88	C₂²×#127	D₁₁	1	1	yes	characteristic, complemented
89	C₂²×#121	—	11	4	no	—
90	C₂²×#120	—	11	4	no	—
91	C₂²×C₄×D₄	—	11	2	no	—
92	C₂³×C₁₁	—	2	96	no	—
93	C₂³×C₁₁	—	2	48	no	—
94	C₂³×C₁₁	—	4	64	no	—
95	C₂³×C₁₁	C₂²×D₄	1	8	yes	—
96	C₂³×C₁₁	H(5,𝔽₂)	1	4	yes	—
97	C₂³×C₁₁	C₂⁵	1	16	yes	—
98	C₂³×C₁₁	C₂⁵	1	3	yes	—
99	C₂²×D₁₁	—	2	128	no	—
100	C₂²×D₁₁	—	2	48	no	—
101	C₂²×D₁₁	—	2	48	no	—
102	C₂²×D₁₁	—	2	256	no	—
103	C₂²×D₁₁	—	4	64	no	—
104	C₂²×D₁₁	—	4	768	no	—
105	C₂²×D₁₁	C₂²×D₄	1	96	yes	complemented
106	C₂²×D₁₁	C₂⁵	1	16	yes	—
107	D₄×C₁₁	—	2	128	no	—
108	D₄×C₁₁	—	4	32	no	—
109	C₂×C₄×C₁₁	—	2	12	no	—
110	C₂×C₄×C₁₁	C₂²×D₄	1	48	yes	complemented
111	C₂×C₄×C₁₁	C₂⁵	1	4	yes	—
112	C₂×C₄×C₁₁	C₂⁵	1	8	yes	—
113	C₂²⋊D₁₁	—	2	256	no	—
114	C₂×Dic₁₁	—	2	48	no	—
115	C₂×Dic₁₁	C₂⁵	1	16	yes	—
116	D₄₄	C₂²×D₄	1	128	yes	complemented
117	C₄×D₁₁	—	2	128	no	—
118	C₂⁶	—	11	4	no	—
119	C₂³×D₄	—	11	8	no	—
120	C₂³×D₄	—	11	96	no	—
121	C₂³×D₄	—	11	2	no	—
122	C₂³×D₄	—	22	64	no	—
123	C₂³×D₄	C₂×D₁₁	1	4	yes	complemented
124	C₂³×D₄	C₂×D₁₁	1	2	yes	complemented
125	C₂⁴×C₄	—	11	4	no	—
126	D₄²	—	11	256	no	—
127	C₂×#127	C₂×D₁₁	1	24	yes	complemented
128	C₂×#121	—	11	96	no	—
129	C₂×#120	—	11	96	no	—
130	C₂×C₄×D₄	—	11	48	no	—
131	C₂²×C₄⋊C₄	—	11	2	no	—
132	C₂²×C₂²⋊C₄	—	11	4	no	—
133	C₂²×C₂²⋊C₄	—	11	4	no	—
134	C₂²×C₄²	C₂×D₁₁	1	1	yes	characteristic, complemented
135	C₂²×C₁₁	—	2	32	no	—
136	C₂²×C₁₁	—	2	16	no	—
137	C₂²×C₁₁	—	4	96	no	—
138	C₂²×C₁₁	D₄²	1	16	yes	complemented
139	C₂²×C₁₁	C₂³×D₄	1	12	yes	—
140	C₂²×C₁₁	C₂×H(5,𝔽₂)	1	6	yes	—
141	C₂²×C₁₁	C₂⁶	1	1	yes	characteristic
142	C₂×D₁₁	—	2	16	no	—
143	C₂×D₁₁	—	2	16	no	—
144	C₂×D₁₁	—	2	192	no	—
145	C₂×D₁₁	—	4	256	no	—
146	C₂×D₁₁	C₂³×D₄	1	32	yes	complemented
147	C₄×C₁₁	—	2	4	no	—
148	C₄×C₁₁	C₂³×D₄	1	16	yes	complemented
149	Dic₁₁	—	2	16	no	—
150	C₂⁵	—	11	4	no	—
151	C₂⁵	—	11	48	no	—
152	C₂⁵	—	22	64	no	—
153	C₂⁵	—	44	16	no	—
154	C₂⁵	C₂²×D₁₁	1	4	yes	—
155	C₂²×D₄	—	2	32	no	—
156	C₂²×D₄	—	4	8	no	—
157	C₂²×D₄	—	11	96	no	—
158	C₂²×D₄	—	11	256	no	—
159	C₂²×D₄	—	11	32	no	—
160	C₂²×D₄	—	11	24	no	—
161	C₂²×D₄	—	22	768	no	—
162	C₂²×D₄	—	22	64	no	—
163	C₂²×D₄	C₂²×D₁₁	1	48	yes	complemented
164	C₂²×D₄	C₂²×D₁₁	1	24	yes	complemented
165	C₂³×C₄	—	11	4	no	—
166	C₂³×C₄	—	11	48	no	—
167	C₂³×C₄	—	22	32	no	—
168	C₂³×C₄	C₂²×D₁₁	1	1	yes	characteristic
169	C₂³×C₄	C₂²×D₁₁	1	2	yes	—
170	#127	C₂²×D₁₁	1	64	yes	complemented
171	#121	—	11	256	no	—
172	#120	—	11	256	no	—
173	C₄×D₄	—	11	128	no	—
174	C₂×C₄⋊C₄	—	11	24	no	—
175	C₂×C₂²⋊C₄	—	11	48	no	—
176	C₂×C₂²⋊C₄	—	11	48	no	—
177	C₂×C₄²	C₂²×D₁₁	1	12	yes	complemented
178	C₂×C₁₁	—	4	16	no	—
179	C₂×C₁₁	C₂×D₄²	1	12	yes	complemented
180	C₂×C₁₁	C₂⁴×D₄	1	2	yes	—
181	C₂×C₁₁	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
182	D₁₁	—	2	32	no	—
183	C₂⁴	—	2	32	no	—
184	C₂⁴	—	2	16	no	—
185	C₂⁴	—	11	32	no	—
186	C₂⁴	—	11	24	no	—
187	C₂⁴	—	11	64	no	—
188	C₂⁴	—	22	384	no	—
189	C₂⁴	—	22	16	no	—
190	C₂⁴	—	22	16	no	—
191	C₂⁴	—	44	96	no	—
192	C₂⁴	—	44	256	no	—
193	C₂⁴	C₂³×D₁₁	1	24	yes	—
194	C₂⁴	C₂³×D₁₁	1	1	yes	characteristic
195	C₂×D₄	—	2	192	no	—
196	C₂×D₄	—	4	48	no	—
197	C₂×D₄	—	11	128	no	—
198	C₂×D₄	—	11	192	no	—
199	C₂×D₄	—	11	32	no	—
200	C₂×D₄	—	22	1024	no	—
201	C₂×D₄	—	22	384	no	—
202	C₂×D₄	C₂³×D₁₁	1	64	yes	complemented
203	C₂×D₄	C₂³×D₁₁	1	32	yes	complemented
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	11	24	no	—
206	C₂²×C₄	—	11	64	no	—
207	C₂²×C₄	—	22	16	no	—
208	C₂²×C₄	—	22	192	no	—
209	C₂²×C₄	D₄×D₁₁	1	16	yes	complemented
210	C₂²×C₄	C₂³×D₁₁	1	6	yes	—
211	C₂²×C₄	C₂³×D₁₁	1	12	yes	—
212	C₄⋊C₄	—	11	32	no	—
213	C₂²⋊C₄	—	11	64	no	—
214	C₂²⋊C₄	—	11	64	no	—
215	C₄²	C₂³×D₁₁	1	16	yes	complemented
216	C₁₁	C₂²×D₄²	1	1	yes	characteristic, complemented, Sylow
217	C₂³	—	2	96	no	—
218	C₂³	—	2	48	no	—
219	C₂³	—	4	64	no	—
220	C₂³	—	11	96	no	—
221	C₂³	—	11	16	no	—
222	C₂³	—	22	256	no	—
223	C₂³	—	22	128	no	—
224	C₂³	—	22	48	no	—
225	C₂³	—	22	48	no	—
226	C₂³	—	44	64	no	—
227	C₂³	—	44	768	no	—
228	C₂³	C₂×D₄×D₁₁	1	8	yes	—
229	C₂³	#67037	1	4	yes	—
230	C₂³	C₂⁴×D₁₁	1	16	yes	—
231	C₂³	C₂⁴×D₁₁	1	3	yes	—
232	D₄	—	2	128	no	—
233	D₄	—	4	32	no	—
234	D₄	—	11	128	no	—
235	D₄	—	22	256	no	—
236	C₂×C₄	—	2	12	no	—
237	C₂×C₄	—	11	16	no	—
238	C₂×C₄	—	22	48	no	—
239	C₂×C₄	—	22	128	no	—
240	C₂×C₄	C₂×D₄×D₁₁	1	48	yes	complemented
241	C₂×C₄	C₂⁴×D₁₁	1	4	yes	—
242	C₂×C₄	C₂⁴×D₁₁	1	8	yes	—
243	C₂²	—	2	32	no	—
244	C₂²	—	2	16	no	—
245	C₂²	—	4	96	no	—
246	C₂²	—	11	32	no	—
247	C₂²	—	22	16	no	—
248	C₂²	—	22	16	no	—
249	C₂²	—	22	192	no	—
250	C₂²	—	44	256	no	—
251	C₂²	#10623248	1	16	yes	direct product factor
252	C₂²	C₂²×D₄×D₁₁	1	12	yes	—
253	C₂²	C₂×#67037	1	6	yes	—
254	C₂²	C₂⁵×D₁₁	1	1	yes	characteristic
255	C₄	—	2	4	no	—
256	C₄	—	22	16	no	—
257	C₄	C₂²×D₄×D₁₁	1	16	yes	complemented
258	C₂	—	4	16	no	—
259	C₂	—	22	32	no	—
260	C₂	C₂×#10623248	1	12	yes	direct product factor
261	C₂	C₂³×D₄×D₁₁	1	2	yes	—
262	C₂	C₂²×#67037	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₁₁
commutator, cosocle	C₂²×C₁₁	C₂⁶
Frattini	C₂²	C₂⁵×D₁₁
Fitting	C₂²×C₁₁×#127	C₂
socle	C₂⁴×C₁₁	C₂⁴
hypercenter	C₂²×#127	D₁₁
hypocenter	C₁₁	C₂²×D₄²
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#127

Complements

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

N	H ≅ G/N	type
1. C₂³×D₄×D₁₁	C₂	semidirect
2. C₂×#10623248	C₂	direct (N × H)
3. C₂²×#66991	C₂	semidirect
4. C₂²×#66990	C₂	semidirect
5. C₂²×C₄×D₄₄	C₂	semidirect
6. C₂²×C₁₁×#127	C₂	semidirect
7. C₂⁵×D₁₁	C₂²	semidirect
8. C₂³×D₄×C₁₁	C₂²	semidirect
9. C₂³×D₄×C₁₁	C₂²	semidirect
10. C₂³×C₂²⋊D₁₁	C₂²	semidirect
12. C₂²×D₄×D₁₁	C₂²	semidirect
14. C₂³×C₄×D₁₁	C₂²	semidirect
15. C₂×C₁₁×#127	C₂²	semidirect
16. C₂²×C₄²×C₁₁	C₂²	semidirect
17. C₂²×C₂²⋊Dic₁₁	C₂²	semidirect
18. C₂×#66991	C₂²	semidirect
19. C₂×#66990	C₂²	semidirect
20. C₂²×(C₂×D₁₁)⋊C₄	C₂²	semidirect
21. C₂²×C₄⋊Dic₁₁	C₂²	semidirect
22. #10623248	C₂²	direct (N × H)
23. C₂×C₄×D₄₄	C₂²	semidirect
28. C₂⁴×D₁₁	C₂³	semidirect
31. C₂²×D₄×C₁₁	C₂³	semidirect
32. C₂²×D₄×C₁₁	C₂³	semidirect
36. C₂²×C₂²⋊D₁₁	C₂³	semidirect
39. C₂×D₄×D₁₁	C₂³	semidirect
40. C₂²×D₄₄	D₄	semidirect
43. C₂²×C₄×D₁₁	C₂³	semidirect
44. C₁₁×#127	C₂³	semidirect
45. C₂×C₄²×C₁₁	C₂³	semidirect
46. C₂×C₂²⋊Dic₁₁	C₂³	semidirect
47. #66991	C₂³	semidirect
48. #66990	C₂³	semidirect
49. C₂×(C₂×D₁₁)⋊C₄	C₂³	semidirect
50. C₂×C₄⋊Dic₁₁	C₂³	semidirect
51. C₄×D₄₄	C₂³	semidirect
62. C₂³×D₁₁	C₂×D₄	semidirect
64. C₂³×D₁₁	C₂⁴	semidirect
67. C₂×D₄×C₁₁	C₂⁴	semidirect
68. C₂×D₄×C₁₁	C₂⁴	semidirect
70. C₂²×C₄×C₁₁	C₂×D₄	semidirect
74. C₂×C₂²⋊D₁₁	C₂⁴	semidirect
78. C₂×D₄₄	C₂×D₄	semidirect
81. C₂×C₄×D₁₁	C₂⁴	semidirect
82. C₄²×C₁₁	C₂⁴	semidirect
83. C₂²⋊Dic₁₁	C₂⁴	semidirect
84. (C₂×D₁₁)⋊C₄	C₂⁴	semidirect
85. C₄⋊Dic₁₁	C₂⁴	semidirect
88. C₂²×#127	D₁₁	semidirect
105. C₂²×D₁₁	C₂²×D₄	semidirect
110. C₂×C₄×C₁₁	C₂²×D₄	semidirect
116. D₄₄	C₂²×D₄	semidirect
123. C₂³×D₄	C₂×D₁₁	semidirect
124. C₂³×D₄	C₂×D₁₁	semidirect
127. C₂×#127	C₂×D₁₁	semidirect
134. C₂²×C₄²	C₂×D₁₁	semidirect
138. C₂²×C₁₁	D₄²	semidirect
146. C₂×D₁₁	C₂³×D₄	semidirect
148. C₄×C₁₁	C₂³×D₄	semidirect
163. C₂²×D₄	C₂²×D₁₁	semidirect
164. C₂²×D₄	C₂²×D₁₁	semidirect
170. #127	C₂²×D₁₁	semidirect
177. C₂×C₄²	C₂²×D₁₁	semidirect
179. C₂×C₁₁	C₂×D₄²	semidirect
202. C₂×D₄	C₂³×D₁₁	semidirect
203. C₂×D₄	C₂³×D₁₁	semidirect
209. C₂²×C₄	D₄×D₁₁	semidirect
215. C₄²	C₂³×D₁₁	semidirect
216. C₁₁	C₂²×D₄²	semidirect
240. C₂×C₄	C₂×D₄×D₁₁	semidirect
251. C₂²	#10623248	direct (N × H)
257. C₄	C₂²×D₄×D₁₁	semidirect
260. C₂	C₂×#10623248	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×D₄×D₁₁	2
C₂×#10623248	2
C₂²×#66991	2
C₂²×#66990	2
C₂²×C₄×D₄₄	2
C₂²×C₁₁×#127	2
C₂²×D₄²	11

Subgroups — C22×#10623248 #15660683

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#10623248 #15660683