Subgroups — C₂²×#65501 #12257560

Order 1280.

8670 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₂²⋊Dic₅	C₂	1	2	yes	maximal
3	C₂³×C₄⋊Dic₅	C₂	1	2	yes	maximal, complemented
4	C₂×#65501	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#3963	C₂	1	2	yes	maximal, complemented
6	C₂⁶×C₅	C₂²	1	1	yes	characteristic
7	C₂⁴×C₄×C₅	—	2	16	no	—
8	C₂⁴×C₄×C₅	C₄	1	16	yes	complemented
9	C₂⁴×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂⁴×C₄×C₅	C₂²	1	12	yes	—
11	C₂⁴×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₂⁴×Dic₅	C₂²	1	4	yes	—
13	C₂²×C₂²⋊Dic₅	—	2	16	no	—
14	C₂²×C₂²⋊Dic₅	C₂²	1	24	yes	—
15	C₂²×C₄⋊Dic₅	—	2	16	no	—
16	C₂²×C₄⋊Dic₅	C₂²	1	24	yes	complemented
17	#65501	C₂²	1	64	yes	direct product factor
18	C₂×#3963	C₂²	1	24	yes	complemented
19	C₂²×#380	—	5	1	no	maximal, Sylow
20	C₂⁵×C₅	—	2	8	no	—
21	C₂⁵×C₅	—	2	8	no	—
22	C₂⁵×C₅	C₂×C₄	1	8	yes	—
23	C₂⁵×C₅	D₄	1	8	yes	—
24	C₂⁵×C₅	Q₈	1	8	yes	—
25	C₂⁵×C₅	C₂³	1	6	yes	—
26	C₂⁵×C₅	C₂³	1	1	yes	characteristic
27	C₂³×C₄×C₅	—	2	8	no	—
28	C₂³×C₄×C₅	—	2	128	no	—
29	C₂³×C₄×C₅	—	2	96	no	—
30	C₂³×C₄×C₅	C₂×C₄	1	8	yes	—
31	C₂³×C₄×C₅	C₂×C₄	1	8	yes	—
32	C₂³×C₄×C₅	C₂×C₄	1	96	yes	complemented
33	C₂³×C₄×C₅	D₄	1	16	yes	—
34	C₂³×C₄×C₅	C₂³	1	6	yes	—
35	C₂³×C₄×C₅	C₂³	1	16	yes	—
36	C₂³×C₄×C₅	C₂³	1	6	yes	—
37	C₂³×Dic₅	—	2	16	no	—
38	C₂³×Dic₅	—	2	16	no	—
39	C₂³×Dic₅	—	2	16	no	—
40	C₂³×Dic₅	C₂³	1	24	yes	—
41	C₂×C₂²⋊Dic₅	—	2	96	no	—
42	C₂×C₂²⋊Dic₅	C₂³	1	32	yes	—
43	C₂×C₄⋊Dic₅	—	2	96	no	—
44	C₂×C₄⋊Dic₅	C₂³	1	32	yes	complemented
45	#3963	C₂³	1	32	yes	complemented
46	C₂⁵×C₄	D₅	1	1	yes	characteristic
47	C₂³×C₄⋊C₄	—	5	2	no	—
48	C₂³×C₂²⋊C₄	—	5	2	no	—
49	C₂×#380	—	5	24	no	—
50	C₂²×#95	—	5	2	no	—
51	C₂⁴×C₅	—	2	32	no	—
52	C₂⁴×C₅	—	2	32	no	—
53	C₂⁴×C₅	—	2	64	no	—
54	C₂⁴×C₅	—	2	64	no	—
55	C₂⁴×C₅	—	2	24	no	—
56	C₂⁴×C₅	—	2	24	no	—
57	C₂⁴×C₅	C₄⋊C₄	1	64	yes	complemented
58	C₂⁴×C₅	C₂²×C₄	1	24	yes	—
59	C₂⁴×C₅	C₂²×C₄	1	4	yes	—
60	C₂⁴×C₅	C₂×D₄	1	24	yes	—
61	C₂⁴×C₅	C₂×D₄	1	4	yes	—
62	C₂⁴×C₅	C₂×D₄	1	8	yes	—
63	C₂⁴×C₅	C₂×Q₈	1	24	yes	—
64	C₂⁴×C₅	C₂×Q₈	1	4	yes	—
65	C₂⁴×C₅	#41	1	8	yes	—
66	C₂⁴×C₅	C₂⁴	1	4	yes	—
67	C₂⁴×C₅	C₂⁴	1	3	yes	—
68	C₂²×C₄×C₅	—	2	32	no	—
69	C₂²×C₄×C₅	—	2	24	no	—
70	C₂²×C₄×C₅	—	2	384	no	—
71	C₂²×C₄×C₅	—	2	64	no	—
72	C₂²×C₄×C₅	C₂²⋊C₄	1	64	yes	complemented
73	C₂²×C₄×C₅	C₂²×C₄	1	24	yes	—
74	C₂²×C₄×C₅	C₂²×C₄	1	24	yes	—
75	C₂²×C₄×C₅	C₂²×C₄	1	64	yes	complemented
76	C₂²×C₄×C₅	C₂×D₄	1	48	yes	—
77	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
78	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
79	C₂²×Dic₅	—	2	48	no	—
80	C₂²×Dic₅	—	2	48	no	—
81	C₂²×Dic₅	—	2	48	no	—
82	C₂²×Dic₅	—	4	64	no	—
83	C₂²×Dic₅	C₂⁴	1	16	yes	—
84	C₂²⋊Dic₅	—	2	64	no	—
85	C₄⋊Dic₅	—	2	64	no	—
86	C₂⁶	C₂×D₅	1	1	yes	characteristic
87	C₂⁴×C₄	—	2	16	no	—
88	C₂⁴×C₄	—	5	4	no	—
89	C₂⁴×C₄	Dic₅	1	16	yes	complemented
90	C₂⁴×C₄	C₂×D₅	1	1	yes	characteristic
91	C₂⁴×C₄	C₂×D₅	1	12	yes	—
92	C₂⁴×C₄	C₂×D₅	1	1	yes	characteristic
93	C₂²×C₄⋊C₄	—	5	24	no	—
94	C₂²×C₄⋊C₄	—	10	16	no	—
95	C₂²×C₂²⋊C₄	—	5	24	no	—
96	C₂²×C₂²⋊C₄	—	10	16	no	—
97	#380	—	5	64	no	—
98	C₂×#95	—	5	24	no	—
99	C₂³×C₅	—	2	48	no	—
100	C₂³×C₅	—	2	48	no	—
101	C₂³×C₅	—	2	96	no	—
102	C₂³×C₅	—	2	96	no	—
103	C₂³×C₅	—	2	8	no	—
104	C₂³×C₅	—	2	8	no	—
105	C₂³×C₅	—	2	256	no	—
106	C₂³×C₅	C₂×C₂²⋊C₄	1	16	yes	—
107	C₂³×C₅	C₂×C₄⋊C₄	1	16	yes	—
108	C₂³×C₅	C₂×C₄⋊C₄	1	96	yes	complemented
109	C₂³×C₅	#117	1	16	yes	—
110	C₂³×C₅	#121	1	32	yes	—
111	C₂³×C₅	C₂²⋊Q₈	1	32	yes	—
112	C₂³×C₅	C₂³×C₄	1	8	yes	—
113	C₂³×C₅	C₂³×C₄	1	6	yes	—
114	C₂³×C₅	C₂²×D₄	1	8	yes	—
115	C₂³×C₅	C₂²×D₄	1	6	yes	—
116	C₂³×C₅	C₂²×D₄	1	12	yes	—
117	C₂³×C₅	C₂²×Q₈	1	8	yes	—
118	C₂³×C₅	C₂²×Q₈	1	6	yes	—
119	C₂³×C₅	C₂×#41	1	12	yes	—
120	C₂³×C₅	C₂⁵	1	1	yes	characteristic
121	C₂×C₄×C₅	—	2	48	no	—
122	C₂×C₄×C₅	—	2	8	no	—
123	C₂×C₄×C₅	—	2	128	no	—
124	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	96	yes	complemented
125	C₂×C₄×C₅	C₂³×C₄	1	8	yes	—
126	C₂×C₄×C₅	C₂³×C₄	1	8	yes	—
127	C₂×C₄×C₅	C₂²×D₄	1	16	yes	—
128	C₂×Dic₅	—	2	16	no	—
129	C₂×Dic₅	—	2	16	no	—
130	C₂×Dic₅	—	2	16	no	—
131	C₂×Dic₅	—	4	96	no	—
132	C₂⁵	—	2	8	no	—
133	C₂⁵	—	2	8	no	—
134	C₂⁵	Dic₁₀	1	8	yes	—
135	C₂⁵	D₂₀	1	8	yes	—
136	C₂⁵	C₂×Dic₅	1	8	yes	—
137	C₂⁵	C₂²×D₅	1	6	yes	—
138	C₂⁵	C₂²×D₅	1	1	yes	characteristic
139	C₂³×C₄	—	2	8	no	—
140	C₂³×C₄	—	2	128	no	—
141	C₂³×C₄	—	2	96	no	—
142	C₂³×C₄	—	5	24	no	—
143	C₂³×C₄	—	10	16	no	—
144	C₂³×C₄	—	10	16	no	—
145	C₂³×C₄	—	10	16	no	—
146	C₂³×C₄	C₂×Dic₅	1	8	yes	—
147	C₂³×C₄	C₂×Dic₅	1	8	yes	—
148	C₂³×C₄	C₂×Dic₅	1	96	yes	complemented
149	C₂³×C₄	C₂²⋊D₅	1	16	yes	—
150	C₂³×C₄	C₂²×D₅	1	6	yes	—
151	C₂³×C₄	C₂²×D₅	1	16	yes	—
152	C₂³×C₄	C₂²×D₅	1	6	yes	—
153	C₂×C₄⋊C₄	—	5	32	no	—
154	C₂×C₄⋊C₄	—	10	96	no	—
155	C₂×C₂²⋊C₄	—	5	32	no	—
156	C₂×C₂²⋊C₄	—	10	96	no	—
157	#95	—	5	32	no	—
158	C₂²×C₅	—	2	8	no	—
159	C₂²×C₅	—	2	8	no	—
160	C₂²×C₅	—	2	16	no	—
161	C₂²×C₅	—	2	16	no	—
162	C₂²×C₅	—	2	192	no	—
163	C₂²×C₅	#380	1	64	yes	complemented
164	C₂²×C₅	C₂²×C₂²⋊C₄	1	12	yes	—
165	C₂²×C₅	C₂²×C₄⋊C₄	1	12	yes	—
166	C₂²×C₅	C₂²×C₄⋊C₄	1	16	yes	complemented
167	C₂²×C₅	C₂×#117	1	12	yes	—
168	C₂²×C₅	C₂×#121	1	24	yes	—
169	C₂²×C₅	C₂×C₂²⋊Q₈	1	24	yes	—
170	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
171	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
172	C₂²×C₅	C₂³×D₄	1	2	yes	—
173	C₂²×C₅	C₂³×Q₈	1	1	yes	characteristic
174	C₂²×C₅	C₂²×#41	1	2	yes	—
175	C₄×C₅	—	2	8	no	—
176	C₄×C₅	C₂²×C₂²⋊C₄	1	16	yes	complemented
177	Dic₅	—	4	16	no	—
178	C₂⁴	—	2	32	no	—
179	C₂⁴	—	2	24	no	—
180	C₂⁴	—	2	24	no	—
181	C₂⁴	—	2	64	no	—
182	C₂⁴	—	2	32	no	—
183	C₂⁴	—	2	64	no	—
184	C₂⁴	C₄⋊Dic₅	1	64	yes	complemented
185	C₂⁴	C₂×Dic₁₀	1	24	yes	—
186	C₂⁴	C₂×Dic₁₀	1	4	yes	—
187	C₂⁴	C₂×D₂₀	1	24	yes	—
188	C₂⁴	C₂×D₂₀	1	4	yes	—
189	C₂⁴	#709	1	8	yes	—
190	C₂⁴	C₂²×Dic₅	1	24	yes	—
191	C₂⁴	C₂²×Dic₅	1	4	yes	—
192	C₂⁴	C₂×C₂²⋊D₅	1	8	yes	—
193	C₂⁴	C₂³×D₅	1	4	yes	—
194	C₂⁴	C₂³×D₅	1	3	yes	—
195	C₂²×C₄	—	2	32	no	—
196	C₂²×C₄	—	2	24	no	—
197	C₂²×C₄	—	2	384	no	—
198	C₂²×C₄	—	2	64	no	—
199	C₂²×C₄	—	5	16	no	—
200	C₂²×C₄	—	10	48	no	—
201	C₂²×C₄	—	10	48	no	—
202	C₂²×C₄	—	10	48	no	—
203	C₂²×C₄	—	20	64	no	—
204	C₂²×C₄	C₂²⋊Dic₅	1	64	yes	complemented
205	C₂²×C₄	C₂²×Dic₅	1	24	yes	—
206	C₂²×C₄	C₂²×Dic₅	1	24	yes	—
207	C₂²×C₄	C₂²×Dic₅	1	64	yes	complemented
208	C₂²×C₄	C₂×C₂²⋊D₅	1	48	yes	—
209	C₂²×C₄	C₂³×D₅	1	4	yes	—
210	C₂²×C₄	C₂³×D₅	1	4	yes	—
211	C₄⋊C₄	—	10	64	no	—
212	C₂²⋊C₄	—	10	64	no	—
213	C₂×C₅	—	2	16	no	—
214	C₂×C₅	C₂×#380	1	24	yes	complemented
215	C₂×C₅	C₂³×C₂²⋊C₄	1	1	yes	characteristic
216	C₂×C₅	C₂³×C₄⋊C₄	1	1	yes	characteristic
217	C₂×C₅	C₂²×#117	1	1	yes	characteristic
218	C₂×C₅	C₂²×#121	1	2	yes	—
219	C₂×C₅	C₂²×C₂²⋊Q₈	1	2	yes	—
220	C₂³	—	2	96	no	—
221	C₂³	—	2	48	no	—
222	C₂³	—	2	96	no	—
223	C₂³	—	2	48	no	—
224	C₂³	—	2	256	no	—
225	C₂³	—	2	8	no	—
226	C₂³	—	2	8	no	—
227	C₂³	#4070	1	32	yes	—
228	C₂³	C₂×C₄⋊Dic₅	1	16	yes	—
229	C₂³	C₂×C₄⋊Dic₅	1	96	yes	complemented
230	C₂³	#4072	1	16	yes	—
231	C₂³	#4076	1	32	yes	—
232	C₂³	C₂×C₂²⋊Dic₅	1	16	yes	—
233	C₂³	C₂²×Dic₁₀	1	8	yes	—
234	C₂³	C₂²×Dic₁₀	1	6	yes	—
235	C₂³	C₂²×D₂₀	1	8	yes	—
236	C₂³	C₂²×D₂₀	1	6	yes	—
237	C₂³	C₂×#709	1	12	yes	—
238	C₂³	C₂³×Dic₅	1	8	yes	—
239	C₂³	C₂³×Dic₅	1	6	yes	—
240	C₂³	C₂²×C₂²⋊D₅	1	12	yes	—
241	C₂³	C₂⁴×D₅	1	1	yes	characteristic
242	C₂×C₄	—	2	48	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	128	no	—
245	C₂×C₄	—	10	16	no	—
246	C₂×C₄	—	10	16	no	—
247	C₂×C₄	—	10	16	no	—
248	C₂×C₄	—	20	96	no	—
249	C₂×C₄	C₂×C₂²⋊Dic₅	1	96	yes	complemented
250	C₂×C₄	C₂³×Dic₅	1	8	yes	—
251	C₂×C₄	C₂³×Dic₅	1	8	yes	—
252	C₂×C₄	C₂²×C₂²⋊D₅	1	16	yes	—
253	C₅	C₂²×#380	1	1	yes	characteristic, complemented, Sylow
254	C₂²	—	2	8	no	—
255	C₂²	—	2	192	no	—
256	C₂²	—	2	8	no	—
257	C₂²	—	2	16	no	—
258	C₂²	—	2	16	no	—
259	C₂²	#65501	1	64	yes	direct product factor
260	C₂²	C₂×#4070	1	24	yes	—
261	C₂²	C₂²×C₄⋊Dic₅	1	12	yes	—
262	C₂²	C₂²×C₄⋊Dic₅	1	16	yes	complemented
263	C₂²	C₂×#4072	1	12	yes	—
264	C₂²	C₂×#4076	1	24	yes	—
265	C₂²	C₂²×C₂²⋊Dic₅	1	12	yes	—
266	C₂²	C₂³×Dic₁₀	1	1	yes	characteristic
267	C₂²	C₂³×D₂₀	1	1	yes	characteristic
268	C₂²	C₂²×#709	1	2	yes	—
269	C₂²	C₂⁴×Dic₅	1	1	yes	characteristic
270	C₂²	C₂³×C₂²⋊D₅	1	2	yes	—
271	C₄	—	2	8	no	—
272	C₄	—	20	16	no	—
273	C₄	C₂²×C₂²⋊Dic₅	1	16	yes	complemented
274	C₂	—	2	16	no	—
275	C₂	C₂×#65501	1	24	yes	direct product factor
276	C₂	C₂³×C₄⋊Dic₅	1	1	yes	characteristic
277	C₂	C₂²×#4072	1	1	yes	characteristic
278	C₂	C₂²×#4070	1	2	yes	—
279	C₂	C₂²×#4076	1	2	yes	—
280	C₂	C₂³×C₂²⋊Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₅
commutator	C₂²×C₅	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₅
Fitting	C₂⁵×C₄×C₅	C₂
socle	C₂⁵×C₅	C₂³
prime radical	C₂⁶×C₅	C₂²
cosocle	C₂³×C₅	C₂⁵
hypercenter	C₂⁵×C₄	D₅
hypocenter	C₅	C₂²×#380
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁵ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
3. C₂³×C₄⋊Dic₅	C₂	semidirect
4. C₂×#65501	C₂	direct (N × H)
5. C₂²×#3963	C₂	semidirect
8. C₂⁴×C₄×C₅	C₄	semidirect
16. C₂²×C₄⋊Dic₅	C₂²	semidirect
17. #65501	C₂²	direct (N × H)
18. C₂×#3963	C₂²	semidirect
32. C₂³×C₄×C₅	C₂×C₄	semidirect
44. C₂×C₄⋊Dic₅	C₂³	semidirect
45. #3963	C₂³	semidirect
57. C₂⁴×C₅	C₄⋊C₄	semidirect
72. C₂²×C₄×C₅	C₂²⋊C₄	semidirect
75. C₂²×C₄×C₅	C₂²×C₄	semidirect
89. C₂⁴×C₄	Dic₅	semidirect
108. C₂³×C₅	C₂×C₄⋊C₄	semidirect
124. C₂×C₄×C₅	C₂×C₂²⋊C₄	semidirect
148. C₂³×C₄	C₂×Dic₅	semidirect
163. C₂²×C₅	#380	semidirect
166. C₂²×C₅	C₂²×C₄⋊C₄	semidirect
176. C₄×C₅	C₂²×C₂²⋊C₄	semidirect
184. C₂⁴	C₄⋊Dic₅	semidirect
204. C₂²×C₄	C₂²⋊Dic₅	semidirect
207. C₂²×C₄	C₂²×Dic₅	semidirect
214. C₂×C₅	C₂×#380	semidirect
229. C₂³	C₂×C₄⋊Dic₅	semidirect
249. C₂×C₄	C₂×C₂²⋊Dic₅	semidirect
253. C₅	C₂²×#380	semidirect
259. C₂²	#65501	direct (N × H)
262. C₂²	C₂²×C₄⋊Dic₅	semidirect
273. C₄	C₂²×C₂²⋊Dic₅	semidirect
275. C₂	C₂×#65501	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₄×C₅	2
C₂³×C₂²⋊Dic₅	2
C₂³×C₄⋊Dic₅	2
C₂×#65501	2
C₂²×#3963	2
C₂²×#380	5

Subgroups — C22×#65501 #12257560

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#65501 #12257560