Subgroups — C₂²×#66051 #13006210

Order 1280.

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×D₅	C₂	1	4	yes	maximal, complemented
2	C₂²×C₅×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#4084	C₂	1	4	yes	maximal, complemented
4	C₂²×#4083	C₂	1	4	yes	maximal, complemented
5	C₂×#66051	C₂	1	48	yes	maximal, direct product factor
6	C₂²×C₄×D₂₀	C₂	1	2	yes	maximal, complemented
7	C₂⁵×D₅	C₂²	1	4	yes	complemented
8	C₂³×C₅×D₄	C₂²	1	4	yes	complemented
9	C₂³×C₅×D₄	C₂²	1	2	yes	complemented
10	C₂³×C₂²⋊D₅	C₂²	1	8	yes	complemented
11	C₂²×D₄×D₅	—	2	64	no	—
12	C₂²×D₄×D₅	C₂²	1	96	yes	complemented
13	C₂³×D₂₀	C₂²	1	2	yes	—
14	C₂³×C₄×D₅	C₂²	1	4	yes	complemented
15	C₂×C₅×#127	C₂²	1	24	yes	complemented
16	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic, complemented
17	C₂²×C₂²⋊Dic₅	C₂²	1	4	yes	complemented
18	C₂×#4084	C₂²	1	96	yes	complemented
19	C₂×#4083	C₂²	1	96	yes	complemented
20	C₂²×#685	C₂²	1	4	yes	complemented
21	C₂²×C₄⋊Dic₅	C₂²	1	2	yes	complemented
22	#66051	C₂²	1	256	yes	direct product factor
23	C₂×C₄×D₂₀	C₂²	1	48	yes	complemented
24	C₂²×D₄²	—	5	1	no	maximal, Sylow
25	C₂⁵×C₅	C₂³	1	4	yes	—
26	C₂⁴×D₅	—	2	64	no	—
27	C₂⁴×D₅	—	4	16	no	—
28	C₂⁴×D₅	C₂³	1	4	yes	—
29	C₂⁴×D₅	C₂³	1	48	yes	complemented
30	C₂²×C₅×D₄	—	2	32	no	—
31	C₂²×C₅×D₄	—	4	8	no	—
32	C₂²×C₅×D₄	C₂³	1	48	yes	complemented
33	C₂²×C₅×D₄	C₂³	1	24	yes	complemented
34	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂³×C₄×C₅	C₂³	1	2	yes	—
36	C₂²×C₂²⋊D₅	—	2	64	no	—
37	C₂²×C₂²⋊D₅	C₂³	1	96	yes	complemented
38	C₂³×Dic₅	C₂³	1	4	yes	—
39	C₂×D₄×D₅	—	2	768	no	—
40	C₂×D₄×D₅	C₂³	1	256	yes	complemented
41	C₂²×D₂₀	D₄	1	32	yes	complemented
42	C₂²×D₂₀	C₂³	1	24	yes	—
43	C₂²×C₄×D₅	—	2	32	no	—
44	C₂²×C₄×D₅	C₂³	1	48	yes	complemented
45	C₅×#127	C₂³	1	64	yes	complemented
46	C₂×C₄²×C₅	C₂³	1	12	yes	complemented
47	C₂×C₂²⋊Dic₅	C₂³	1	48	yes	complemented
48	#4084	C₂³	1	256	yes	complemented
49	#4083	C₂³	1	256	yes	complemented
50	C₂×#685	C₂³	1	48	yes	complemented
51	C₂×C₄⋊Dic₅	C₂³	1	24	yes	complemented
52	C₄×D₂₀	C₂³	1	128	yes	complemented
53	C₂⁴×D₄	—	5	4	no	—
54	C₂×D₄²	—	5	48	no	—
55	C₂²×#127	D₅	1	1	yes	characteristic, complemented
56	C₂²×#121	—	5	4	no	—
57	C₂²×#120	—	5	4	no	—
58	C₂²×C₄×D₄	—	5	2	no	—
59	C₂⁴×C₅	—	2	32	no	—
60	C₂⁴×C₅	—	2	16	no	—
61	C₂⁴×C₅	C₂⁴	1	24	yes	—
62	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
63	C₂³×D₅	—	2	16	no	—
64	C₂³×D₅	—	2	16	no	—
65	C₂³×D₅	—	2	384	no	—
66	C₂³×D₅	—	4	96	no	—
67	C₂³×D₅	—	4	256	no	—
68	C₂³×D₅	C₂×D₄	1	32	yes	complemented
69	C₂³×D₅	C₂⁴	1	24	yes	—
70	C₂³×D₅	C₂⁴	1	64	yes	complemented
71	C₂×C₅×D₄	—	2	192	no	—
72	C₂×C₅×D₄	—	4	48	no	—
73	C₂×C₅×D₄	C₂⁴	1	64	yes	complemented
74	C₂×C₅×D₄	C₂⁴	1	32	yes	complemented
75	C₂²×C₄×C₅	—	2	4	no	—
76	C₂²×C₄×C₅	C₂×D₄	1	16	yes	complemented
77	C₂²×C₄×C₅	C₂⁴	1	6	yes	—
78	C₂²×C₄×C₅	C₂⁴	1	12	yes	—
79	C₂×C₂²⋊D₅	—	2	384	no	—
80	C₂×C₂²⋊D₅	C₂⁴	1	128	yes	complemented
81	C₂²×Dic₅	—	2	16	no	—
82	C₂²×Dic₅	C₂⁴	1	24	yes	—
83	D₄×D₅	—	2	1024	no	—
84	C₂×D₂₀	C₂×D₄	1	192	yes	complemented
85	C₂×D₂₀	C₂⁴	1	32	yes	—
86	C₂×C₄×D₅	—	2	192	no	—
87	C₂×C₄×D₅	C₂⁴	1	64	yes	complemented
88	C₄²×C₅	C₂⁴	1	16	yes	complemented
89	C₂²⋊Dic₅	C₂⁴	1	64	yes	complemented
90	#685	C₂⁴	1	64	yes	complemented
91	C₄⋊Dic₅	C₂⁴	1	32	yes	complemented
92	C₂⁶	—	5	4	no	—
93	C₂³×D₄	—	5	8	no	—
94	C₂³×D₄	—	5	96	no	—
95	C₂³×D₄	—	5	2	no	—
96	C₂³×D₄	—	10	64	no	—
97	C₂³×D₄	C₂×D₅	1	4	yes	complemented
98	C₂³×D₄	C₂×D₅	1	2	yes	complemented
99	C₂⁴×C₄	—	5	4	no	—
100	D₄²	—	5	256	no	—
101	C₂×#127	C₂×D₅	1	24	yes	complemented
102	C₂×#121	—	5	96	no	—
103	C₂×#120	—	5	96	no	—
104	C₂×C₄×D₄	—	5	48	no	—
105	C₂²×C₄⋊C₄	—	5	2	no	—
106	C₂²×C₂²⋊C₄	—	5	4	no	—
107	C₂²×C₂²⋊C₄	—	5	4	no	—
108	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic, complemented
109	C₂³×C₅	—	2	96	no	—
110	C₂³×C₅	—	2	48	no	—
111	C₂³×C₅	—	4	64	no	—
112	C₂³×C₅	C₂²×D₄	1	8	yes	—
113	C₂³×C₅	H(5,𝔽₂)	1	4	yes	—
114	C₂³×C₅	C₂⁵	1	16	yes	—
115	C₂³×C₅	C₂⁵	1	3	yes	—
116	C₂²×D₅	—	2	128	no	—
117	C₂²×D₅	—	2	48	no	—
118	C₂²×D₅	—	2	48	no	—
119	C₂²×D₅	—	2	256	no	—
120	C₂²×D₅	—	4	64	no	—
121	C₂²×D₅	—	4	768	no	—
122	C₂²×D₅	C₂²×D₄	1	96	yes	complemented
123	C₂²×D₅	C₂⁵	1	16	yes	—
124	C₅×D₄	—	2	128	no	—
125	C₅×D₄	—	4	32	no	—
126	C₂×C₄×C₅	—	2	12	no	—
127	C₂×C₄×C₅	C₂²×D₄	1	48	yes	complemented
128	C₂×C₄×C₅	C₂⁵	1	4	yes	—
129	C₂×C₄×C₅	C₂⁵	1	8	yes	—
130	C₂²⋊D₅	—	2	256	no	—
131	C₂×Dic₅	—	2	48	no	—
132	C₂×Dic₅	C₂⁵	1	16	yes	—
133	D₂₀	C₂²×D₄	1	128	yes	complemented
134	C₄×D₅	—	2	128	no	—
135	C₂⁵	—	5	4	no	—
136	C₂⁵	—	5	48	no	—
137	C₂⁵	—	10	64	no	—
138	C₂⁵	—	20	16	no	—
139	C₂⁵	C₂²×D₅	1	4	yes	—
140	C₂²×D₄	—	2	32	no	—
141	C₂²×D₄	—	4	8	no	—
142	C₂²×D₄	—	5	96	no	—
143	C₂²×D₄	—	5	256	no	—
144	C₂²×D₄	—	5	32	no	—
145	C₂²×D₄	—	5	24	no	—
146	C₂²×D₄	—	10	768	no	—
147	C₂²×D₄	—	10	64	no	—
148	C₂²×D₄	C₂²×D₅	1	48	yes	complemented
149	C₂²×D₄	C₂²×D₅	1	24	yes	complemented
150	C₂³×C₄	—	5	4	no	—
151	C₂³×C₄	—	5	48	no	—
152	C₂³×C₄	—	10	32	no	—
153	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
154	C₂³×C₄	C₂²×D₅	1	2	yes	—
155	#127	C₂²×D₅	1	64	yes	complemented
156	#121	—	5	256	no	—
157	#120	—	5	256	no	—
158	C₄×D₄	—	5	128	no	—
159	C₂×C₄⋊C₄	—	5	24	no	—
160	C₂×C₂²⋊C₄	—	5	48	no	—
161	C₂×C₂²⋊C₄	—	5	48	no	—
162	C₂×C₄²	C₂²×D₅	1	12	yes	complemented
163	C₂²×C₅	—	2	32	no	—
164	C₂²×C₅	—	2	16	no	—
165	C₂²×C₅	—	4	96	no	—
166	C₂²×C₅	D₄²	1	16	yes	complemented
167	C₂²×C₅	C₂³×D₄	1	12	yes	—
168	C₂²×C₅	C₂×H(5,𝔽₂)	1	6	yes	—
169	C₂²×C₅	C₂⁶	1	1	yes	characteristic
170	C₂×D₅	—	2	16	no	—
171	C₂×D₅	—	2	16	no	—
172	C₂×D₅	—	2	192	no	—
173	C₂×D₅	—	4	256	no	—
174	C₂×D₅	C₂³×D₄	1	32	yes	complemented
175	C₄×C₅	—	2	4	no	—
176	C₄×C₅	C₂³×D₄	1	16	yes	complemented
177	Dic₅	—	2	16	no	—
178	C₂⁴	—	2	32	no	—
179	C₂⁴	—	2	16	no	—
180	C₂⁴	—	5	32	no	—
181	C₂⁴	—	5	24	no	—
182	C₂⁴	—	5	64	no	—
183	C₂⁴	—	10	384	no	—
184	C₂⁴	—	10	16	no	—
185	C₂⁴	—	10	16	no	—
186	C₂⁴	—	20	96	no	—
187	C₂⁴	—	20	256	no	—
188	C₂⁴	C₂³×D₅	1	24	yes	—
189	C₂⁴	C₂³×D₅	1	1	yes	characteristic
190	C₂×D₄	—	2	192	no	—
191	C₂×D₄	—	4	48	no	—
192	C₂×D₄	—	5	128	no	—
193	C₂×D₄	—	5	192	no	—
194	C₂×D₄	—	5	32	no	—
195	C₂×D₄	—	10	1024	no	—
196	C₂×D₄	—	10	384	no	—
197	C₂×D₄	C₂³×D₅	1	64	yes	complemented
198	C₂×D₄	C₂³×D₅	1	32	yes	complemented
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	5	24	no	—
201	C₂²×C₄	—	5	64	no	—
202	C₂²×C₄	—	10	16	no	—
203	C₂²×C₄	—	10	192	no	—
204	C₂²×C₄	D₄×D₅	1	16	yes	complemented
205	C₂²×C₄	C₂³×D₅	1	6	yes	—
206	C₂²×C₄	C₂³×D₅	1	12	yes	—
207	C₄⋊C₄	—	5	32	no	—
208	C₂²⋊C₄	—	5	64	no	—
209	C₂²⋊C₄	—	5	64	no	—
210	C₄²	C₂³×D₅	1	16	yes	complemented
211	C₂×C₅	—	4	16	no	—
212	C₂×C₅	C₂×D₄²	1	12	yes	complemented
213	C₂×C₅	C₂⁴×D₄	1	2	yes	—
214	C₂×C₅	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
215	D₅	—	2	32	no	—
216	C₂³	—	2	96	no	—
217	C₂³	—	2	48	no	—
218	C₂³	—	4	64	no	—
219	C₂³	—	5	96	no	—
220	C₂³	—	5	16	no	—
221	C₂³	—	10	256	no	—
222	C₂³	—	10	128	no	—
223	C₂³	—	10	48	no	—
224	C₂³	—	10	48	no	—
225	C₂³	—	20	64	no	—
226	C₂³	—	20	768	no	—
227	C₂³	C₂×D₄×D₅	1	8	yes	—
228	C₂³	#4144	1	4	yes	—
229	C₂³	C₂⁴×D₅	1	16	yes	—
230	C₂³	C₂⁴×D₅	1	3	yes	—
231	D₄	—	2	128	no	—
232	D₄	—	4	32	no	—
233	D₄	—	5	128	no	—
234	D₄	—	10	256	no	—
235	C₂×C₄	—	2	12	no	—
236	C₂×C₄	—	5	16	no	—
237	C₂×C₄	—	10	48	no	—
238	C₂×C₄	—	10	128	no	—
239	C₂×C₄	C₂×D₄×D₅	1	48	yes	complemented
240	C₂×C₄	C₂⁴×D₅	1	4	yes	—
241	C₂×C₄	C₂⁴×D₅	1	8	yes	—
242	C₅	C₂²×D₄²	1	1	yes	characteristic, complemented, Sylow
243	C₂²	—	2	32	no	—
244	C₂²	—	2	16	no	—
245	C₂²	—	4	96	no	—
246	C₂²	—	5	32	no	—
247	C₂²	—	10	16	no	—
248	C₂²	—	10	16	no	—
249	C₂²	—	10	192	no	—
250	C₂²	—	20	256	no	—
251	C₂²	#66051	1	16	yes	direct product factor
252	C₂²	C₂²×D₄×D₅	1	12	yes	—
253	C₂²	C₂×#4144	1	6	yes	—
254	C₂²	C₂⁵×D₅	1	1	yes	characteristic
255	C₄	—	2	4	no	—
256	C₄	—	10	16	no	—
257	C₄	C₂²×D₄×D₅	1	16	yes	complemented
258	C₂	—	4	16	no	—
259	C₂	—	10	32	no	—
260	C₂	C₂×#66051	1	12	yes	direct product factor
261	C₂	C₂³×D₄×D₅	1	2	yes	—
262	C₂	C₂²×#4144	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁶
Frattini	C₂²	C₂⁵×D₅
Fitting	C₂²×C₅×#127	C₂
socle	C₂⁴×C₅	C₂⁴
hypercenter	C₂²×#127	D₅
hypocenter	C₅	C₂²×D₄²
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#127

Complements

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

N	H ≅ G/N	type
1. C₂³×D₄×D₅	C₂	semidirect
2. C₂²×C₅×#127	C₂	semidirect
3. C₂²×#4084	C₂	semidirect
4. C₂²×#4083	C₂	semidirect
5. C₂×#66051	C₂	direct (N × H)
6. C₂²×C₄×D₂₀	C₂	semidirect
7. C₂⁵×D₅	C₂²	semidirect
8. C₂³×C₅×D₄	C₂²	semidirect
9. C₂³×C₅×D₄	C₂²	semidirect
10. C₂³×C₂²⋊D₅	C₂²	semidirect
12. C₂²×D₄×D₅	C₂²	semidirect
14. C₂³×C₄×D₅	C₂²	semidirect
15. C₂×C₅×#127	C₂²	semidirect
16. C₂²×C₄²×C₅	C₂²	semidirect
17. C₂²×C₂²⋊Dic₅	C₂²	semidirect
18. C₂×#4084	C₂²	semidirect
19. C₂×#4083	C₂²	semidirect
20. C₂²×#685	C₂²	semidirect
21. C₂²×C₄⋊Dic₅	C₂²	semidirect
22. #66051	C₂²	direct (N × H)
23. C₂×C₄×D₂₀	C₂²	semidirect
29. C₂⁴×D₅	C₂³	semidirect
32. C₂²×C₅×D₄	C₂³	semidirect
33. C₂²×C₅×D₄	C₂³	semidirect
37. C₂²×C₂²⋊D₅	C₂³	semidirect
40. C₂×D₄×D₅	C₂³	semidirect
41. C₂²×D₂₀	D₄	semidirect
44. C₂²×C₄×D₅	C₂³	semidirect
45. C₅×#127	C₂³	semidirect
46. C₂×C₄²×C₅	C₂³	semidirect
47. C₂×C₂²⋊Dic₅	C₂³	semidirect
48. #4084	C₂³	semidirect
49. #4083	C₂³	semidirect
50. C₂×#685	C₂³	semidirect
51. C₂×C₄⋊Dic₅	C₂³	semidirect
52. C₄×D₂₀	C₂³	semidirect
55. C₂²×#127	D₅	semidirect
68. C₂³×D₅	C₂×D₄	semidirect
70. C₂³×D₅	C₂⁴	semidirect
73. C₂×C₅×D₄	C₂⁴	semidirect
74. C₂×C₅×D₄	C₂⁴	semidirect
76. C₂²×C₄×C₅	C₂×D₄	semidirect
80. C₂×C₂²⋊D₅	C₂⁴	semidirect
84. C₂×D₂₀	C₂×D₄	semidirect
87. C₂×C₄×D₅	C₂⁴	semidirect
88. C₄²×C₅	C₂⁴	semidirect
89. C₂²⋊Dic₅	C₂⁴	semidirect
90. #685	C₂⁴	semidirect
91. C₄⋊Dic₅	C₂⁴	semidirect
97. C₂³×D₄	C₂×D₅	semidirect
98. C₂³×D₄	C₂×D₅	semidirect
101. C₂×#127	C₂×D₅	semidirect
108. C₂²×C₄²	C₂×D₅	semidirect
122. C₂²×D₅	C₂²×D₄	semidirect
127. C₂×C₄×C₅	C₂²×D₄	semidirect
133. D₂₀	C₂²×D₄	semidirect
148. C₂²×D₄	C₂²×D₅	semidirect
149. C₂²×D₄	C₂²×D₅	semidirect
155. #127	C₂²×D₅	semidirect
162. C₂×C₄²	C₂²×D₅	semidirect
166. C₂²×C₅	D₄²	semidirect
174. C₂×D₅	C₂³×D₄	semidirect
176. C₄×C₅	C₂³×D₄	semidirect
197. C₂×D₄	C₂³×D₅	semidirect
198. C₂×D₄	C₂³×D₅	semidirect
204. C₂²×C₄	D₄×D₅	semidirect
210. C₄²	C₂³×D₅	semidirect
212. C₂×C₅	C₂×D₄²	semidirect
239. C₂×C₄	C₂×D₄×D₅	semidirect
242. C₅	C₂²×D₄²	semidirect
251. C₂²	#66051	direct (N × H)
257. C₄	C₂²×D₄×D₅	semidirect
260. C₂	C₂×#66051	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×D₅	2
C₂²×C₅×#127	2
C₂²×#4084	2
C₂²×#4083	2
C₂×#66051	2
C₂²×C₄×D₂₀	2
C₂²×D₄²	5

Subgroups — C22×#66051 #13006210

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#66051 #13006210