Subgroups — C₂×C₄×#4072 #12257485

Order 1280.

4462 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄²×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×#4072	C₂	1	2	yes	maximal
3	C₂×C₄×C₂²⋊Dic₅	C₂	1	4	yes	maximal
4	C₄×#4072	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₄×C₄⋊Dic₅	C₂	1	4	yes	maximal, complemented
6	C₂×#65221	C₂	1	2	yes	maximal, complemented
7	C₂×C₄²×Dic₅	C₂	1	2	yes	maximal, complemented
8	C₂⁴×C₄×C₅	C₂²	1	2	yes	—
9	C₂⁴×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₅	C₄	1	16	yes	complemented
11	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₅	C₂²	1	2	yes	—
13	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₅	C₂²	1	8	yes	—
15	C₂²×C₂²⋊Dic₅	C₂²	1	4	yes	—
16	C₂×#4072	C₄	1	32	yes	direct product factor
17	C₂×#4072	C₂²	1	16	yes	—
18	C₂²×C₄⋊Dic₅	C₂²	1	4	yes	—
19	C₂²×C₄×Dic₅	C₂²	1	8	yes	—
20	C₂²×C₄×Dic₅	C₂²	1	4	yes	—
21	C₄×C₂²⋊Dic₅	C₂²	1	32	yes	—
22	C₂×#3963	C₂²	1	8	yes	—
23	C₄×C₄⋊Dic₅	C₂²	1	32	yes	complemented
24	#65221	C₂²	1	16	yes	complemented
25	C₄²×Dic₅	C₂²	1	16	yes	complemented
26	C₂×C₄×#117	—	5	1	no	maximal, Sylow
27	C₂⁵×C₅	C₂³	1	1	yes	characteristic
28	C₂³×C₄×C₅	—	2	16	no	—
29	C₂³×C₄×C₅	C₂×C₄	1	16	yes	—
30	C₂³×C₄×C₅	C₂×C₄	1	16	yes	—
31	C₂³×C₄×C₅	C₂×C₄	1	8	yes	—
32	C₂³×C₄×C₅	C₂³	1	2	yes	—
33	C₂³×C₄×C₅	C₂³	1	8	yes	—
34	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂³×C₄×C₅	C₂³	1	4	yes	—
37	C₂³×C₄×C₅	C₂³	1	2	yes	—
38	C₂³×Dic₅	C₂³	1	8	yes	—
39	C₂×C₄²×C₅	C₂×C₄	1	8	yes	—
40	C₂×C₄²×C₅	C₂×C₄	1	16	yes	—
41	C₂×C₄²×C₅	C₂×C₄	1	8	yes	—
42	C₂×C₄²×C₅	C₂×C₄	1	64	yes	complemented
43	C₂×C₄²×C₅	C₂³	1	4	yes	—
44	C₂×C₄²×C₅	C₂³	1	4	yes	—
45	C₂×C₄²×C₅	C₂³	1	8	yes	—
46	C₂×C₂²⋊Dic₅	C₂×C₄	1	32	yes	—
47	C₂×C₂²⋊Dic₅	C₂³	1	16	yes	—
48	#4072	C₂×C₄	1	128	yes	direct product factor
49	C₂×C₄⋊Dic₅	C₂×C₄	1	32	yes	complemented
50	C₂×C₄⋊Dic₅	C₂³	1	16	yes	—
51	C₂×C₄×Dic₅	—	2	32	no	—
52	C₂×C₄×Dic₅	C₂×C₄	1	32	yes	complemented
53	C₂×C₄×Dic₅	C₂³	1	32	yes	—
54	C₂×C₄×Dic₅	C₂³	1	16	yes	—
55	#3963	C₂³	1	32	yes	—
56	C₂²×#117	—	5	2	no	—
57	C₂³×C₄²	D₅	1	1	yes	characteristic
58	C₄×#117	—	5	16	no	—
59	C₂×C₄×C₄⋊C₄	—	5	4	no	—
60	C₂×C₄×C₂²⋊C₄	—	5	4	no	—
61	C₂×#376	—	5	2	no	—
62	C₂×C₄³	—	5	2	no	—
63	C₂⁴×C₅	—	2	8	no	—
64	C₂⁴×C₅	C₂²×C₄	1	8	yes	—
65	C₂⁴×C₅	C₂²×C₄	1	4	yes	—
66	C₂⁴×C₅	C₂⁴	1	2	yes	—
67	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
68	C₂²×C₄×C₅	—	2	8	no	—
69	C₂²×C₄×C₅	—	2	64	no	—
70	C₂²×C₄×C₅	—	2	32	no	—
71	C₂²×C₄×C₅	C₄²	1	64	yes	complemented
72	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
73	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
74	C₂²×C₄×C₅	C₂²×C₄	1	32	yes	—
75	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
76	C₂²×C₄×C₅	C₂²×C₄	1	32	yes	—
77	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
78	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₅	C₂²×C₄	1	16	yes	—
80	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
81	C₂²×C₄×C₅	#41	1	16	yes	—
82	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
83	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
84	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
85	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
86	C₂²×Dic₅	—	2	32	no	—
87	C₂²×Dic₅	C₂²×C₄	1	32	yes	—
88	C₂²×Dic₅	C₂⁴	1	16	yes	—
89	C₄²×C₅	C₂²×C₄	1	16	yes	—
90	C₄²×C₅	C₂²×C₄	1	32	yes	—
91	C₄²×C₅	C₂²×C₄	1	16	yes	—
92	C₂²⋊Dic₅	C₂²×C₄	1	64	yes	—
93	C₄⋊Dic₅	C₂²×C₄	1	64	yes	complemented
94	C₄×Dic₅	—	2	64	no	—
95	C₄×Dic₅	C₂²×C₄	1	64	yes	complemented
96	C₂⁴×C₄	C₂×D₅	1	2	yes	—
97	C₂⁴×C₄	C₂×D₅	1	1	yes	characteristic
98	C₂×#117	—	5	32	no	—
99	C₂×#117	—	5	16	no	—
100	C₂²×C₄⋊C₄	—	5	4	no	—
101	C₂²×C₂²⋊C₄	—	5	4	no	—
102	C₂²×C₄²	—	5	4	no	—
103	C₂²×C₄²	—	5	8	no	—
104	C₂²×C₄²	Dic₅	1	16	yes	complemented
105	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
106	C₂²×C₄²	C₂×D₅	1	2	yes	—
107	C₂²×C₄²	C₂×D₅	1	8	yes	—
108	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
109	C₄×C₄⋊C₄	—	5	32	no	—
110	C₄×C₂²⋊C₄	—	5	32	no	—
111	#376	—	5	16	no	—
112	C₂×#95	—	5	8	no	—
113	C₄³	—	5	16	no	—
114	C₂³×C₅	—	2	16	no	—
115	C₂³×C₅	—	2	32	no	—
116	C₂³×C₅	—	2	8	no	—
117	C₂³×C₅	C₂×C₄²	1	16	yes	—
118	C₂³×C₅	C₂³×C₄	1	8	yes	—
119	C₂³×C₅	C₂³×C₄	1	4	yes	—
120	C₂³×C₅	C₂³×C₄	1	4	yes	—
121	C₂³×C₅	C₂³×C₄	1	2	yes	—
122	C₂³×C₅	C₂×#41	1	8	yes	—
123	C₂³×C₅	C₂⁵	1	1	yes	characteristic
124	C₂×C₄×C₅	—	2	16	no	—
125	C₂×C₄×C₅	—	2	8	no	—
126	C₂×C₄×C₅	—	2	64	no	—
127	C₂×C₄×C₅	C₂×C₄²	1	16	yes	—
128	C₂×C₄×C₅	C₂×C₄²	1	16	yes	—
129	C₂×C₄×C₅	C₂×C₄²	1	64	yes	complemented
130	C₂×C₄×C₅	#117	1	32	yes	complemented
131	C₂×C₄×C₅	C₂³×C₄	1	8	yes	—
132	C₂×C₄×C₅	C₂³×C₄	1	8	yes	—
133	C₂×C₄×C₅	C₂³×C₄	1	8	yes	—
134	C₂×C₄×C₅	C₂³×C₄	1	8	yes	—
135	C₂×C₄×C₅	C₂³×C₄	1	4	yes	—
136	C₂×C₄×C₅	C₂³×C₄	1	4	yes	—
137	C₂×C₄×C₅	C₂×#41	1	16	yes	—
138	C₂×Dic₅	—	2	64	no	—
139	C₂×Dic₅	—	2	32	no	—
140	C₂×Dic₅	C₂³×C₄	1	32	yes	—
141	C₂⁵	C₂²×D₅	1	1	yes	characteristic
142	C₂³×C₄	—	2	16	no	—
143	C₂³×C₄	—	5	8	no	—
144	C₂³×C₄	C₄×D₅	1	16	yes	—
145	C₂³×C₄	C₂×Dic₅	1	16	yes	—
146	C₂³×C₄	C₂×Dic₅	1	8	yes	—
147	C₂³×C₄	C₂²×D₅	1	2	yes	—
148	C₂³×C₄	C₂²×D₅	1	8	yes	—
149	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
150	C₂³×C₄	C₂²×D₅	1	4	yes	—
151	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
152	C₂³×C₄	C₂²×D₅	1	2	yes	—
153	#117	—	5	128	no	—
154	C₂×C₄⋊C₄	—	5	32	no	—
155	C₂×C₄⋊C₄	—	5	16	no	—
156	C₂×C₂²⋊C₄	—	5	32	no	—
157	C₂×C₂²⋊C₄	—	5	16	no	—
158	C₂×C₄²	—	5	32	no	—
159	C₂×C₄²	—	5	16	no	—
160	C₂×C₄²	—	5	32	no	—
161	C₂×C₄²	—	10	32	no	—
162	C₂×C₄²	C₂×Dic₅	1	8	yes	—
163	C₂×C₄²	C₂×Dic₅	1	16	yes	—
164	C₂×C₄²	C₂×Dic₅	1	8	yes	—
165	C₂×C₄²	C₂×Dic₅	1	64	yes	complemented
166	C₂×C₄²	C₂²×D₅	1	4	yes	—
167	C₂×C₄²	C₂²×D₅	1	4	yes	—
168	C₂×C₄²	C₂²×D₅	1	8	yes	—
169	#95	—	5	32	no	—
170	C₂²×C₅	—	2	8	no	—
171	C₂²×C₅	—	2	16	no	—
172	C₂²×C₅	—	2	32	no	—
173	C₂²×C₅	C₂²×C₄²	1	8	yes	—
174	C₂²×C₅	C₂²×C₄²	1	4	yes	—
175	C₂²×C₅	C₂×#117	1	8	yes	—
176	C₂²×C₅	C₄×#41	1	16	yes	—
177	C₂²×C₅	C₂⁴×C₄	1	2	yes	—
178	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
179	C₂²×C₅	C₂²×#41	1	4	yes	—
180	C₄×C₅	—	2	8	no	—
181	C₄×C₅	C₂²×C₄²	1	8	yes	—
182	C₄×C₅	C₂²×C₄²	1	8	yes	—
183	C₄×C₅	C₂×#117	1	16	yes	complemented
184	Dic₅	—	2	32	no	—
185	C₂⁴	—	2	8	no	—
186	C₂⁴	C₂×C₄×D₅	1	8	yes	—
187	C₂⁴	C₂²×Dic₅	1	4	yes	—
188	C₂⁴	C₂³×D₅	1	2	yes	—
189	C₂⁴	C₂³×D₅	1	1	yes	characteristic
190	C₂²×C₄	—	2	8	no	—
191	C₂²×C₄	—	2	32	no	—
192	C₂²×C₄	—	2	64	no	—
193	C₂²×C₄	—	5	32	no	—
194	C₂²×C₄	—	5	16	no	—
195	C₂²×C₄	—	10	32	no	—
196	C₂²×C₄	C₄×Dic₅	1	64	yes	complemented
197	C₂²×C₄	C₂×C₄×D₅	1	8	yes	—
198	C₂²×C₄	C₂×C₄×D₅	1	8	yes	—
199	C₂²×C₄	C₂×C₄×D₅	1	32	yes	—
200	C₂²×C₄	#709	1	16	yes	—
201	C₂²×C₄	C₂²×Dic₅	1	8	yes	—
202	C₂²×C₄	C₂²×Dic₅	1	32	yes	—
203	C₂²×C₄	C₂²×Dic₅	1	4	yes	—
204	C₂²×C₄	C₂²×Dic₅	1	4	yes	—
205	C₂²×C₄	C₂²×Dic₅	1	16	yes	—
206	C₂²×C₄	C₂²×Dic₅	1	8	yes	—
207	C₂²×C₄	C₂³×D₅	1	4	yes	—
208	C₂²×C₄	C₂³×D₅	1	2	yes	—
209	C₂²×C₄	C₂³×D₅	1	2	yes	—
210	C₂²×C₄	C₂³×D₅	1	4	yes	—
211	C₄⋊C₄	—	5	64	no	—
212	C₂²⋊C₄	—	5	64	no	—
213	C₄²	—	5	64	no	—
214	C₄²	—	10	64	no	—
215	C₄²	C₂²×Dic₅	1	16	yes	—
216	C₄²	C₂²×Dic₅	1	32	yes	—
217	C₄²	C₂²×Dic₅	1	16	yes	—
218	C₂×C₅	—	2	8	no	—
219	C₂×C₅	C₄×#117	1	8	yes	complemented
220	C₂×C₅	C₂³×C₄²	1	1	yes	characteristic
221	C₂×C₅	C₂²×#117	1	2	yes	—
222	C₂×C₅	C₂×C₄×#41	1	4	yes	—
223	C₂³	—	2	32	no	—
224	C₂³	—	2	16	no	—
225	C₂³	—	2	8	no	—
226	C₂³	C₂×C₄×Dic₅	1	16	yes	—
227	C₂³	C₂²×C₄×D₅	1	8	yes	—
228	C₂³	C₂²×C₄×D₅	1	4	yes	—
229	C₂³	C₂×#709	1	8	yes	—
230	C₂³	C₂³×Dic₅	1	4	yes	—
231	C₂³	C₂³×Dic₅	1	2	yes	—
232	C₂³	C₂⁴×D₅	1	1	yes	characteristic
233	C₂×C₄	—	2	16	no	—
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	2	64	no	—
236	C₂×C₄	—	5	32	no	—
237	C₂×C₄	—	10	64	no	—
238	C₂×C₄	—	10	32	no	—
239	C₂×C₄	C₂×C₄×Dic₅	1	16	yes	—
240	C₂×C₄	C₂×C₄×Dic₅	1	16	yes	—
241	C₂×C₄	C₂×C₄×Dic₅	1	64	yes	complemented
242	C₂×C₄	#4072	1	32	yes	direct product factor
243	C₂×C₄	C₂²×C₄×D₅	1	8	yes	—
244	C₂×C₄	C₂²×C₄×D₅	1	8	yes	—
245	C₂×C₄	C₂×#709	1	16	yes	—
246	C₂×C₄	C₂³×Dic₅	1	8	yes	—
247	C₂×C₄	C₂³×Dic₅	1	4	yes	—
248	C₂×C₄	C₂³×Dic₅	1	4	yes	—
249	C₂×C₄	C₂³×Dic₅	1	8	yes	—
250	C₅	C₂×C₄×#117	1	1	yes	characteristic, complemented, Sylow
251	C₂²	—	2	32	no	—
252	C₂²	—	2	16	no	—
253	C₂²	—	2	8	no	—
254	C₂²	C₄×#709	1	16	yes	—
255	C₂²	C₂²×C₄×Dic₅	1	8	yes	—
256	C₂²	C₂²×C₄×Dic₅	1	4	yes	—
257	C₂²	C₂×#4072	1	8	yes	—
258	C₂²	C₂³×C₄×D₅	1	2	yes	—
259	C₂²	C₂²×#709	1	4	yes	—
260	C₂²	C₂⁴×Dic₅	1	1	yes	characteristic
261	C₄	—	2	8	no	—
262	C₄	—	10	32	no	—
263	C₄	C₂²×C₄×Dic₅	1	8	yes	—
264	C₄	C₂²×C₄×Dic₅	1	8	yes	—
265	C₄	C₂×#4072	1	16	yes	direct product factor
266	C₂	—	2	8	no	—
267	C₂	C₄×#4072	1	8	yes	direct product factor
268	C₂	C₂×C₄×#709	1	4	yes	—
269	C₂	C₂³×C₄×Dic₅	1	1	yes	characteristic
270	C₂	C₂²×#4072	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄²	C₂×D₅
commutator	C₂×C₅	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₅
Fitting	C₂³×C₄²×C₅	C₂
socle	C₂⁴×C₅	C₂⁴
prime radical	C₂⁵×C₅	C₂³
cosocle	C₂³×C₅	C₂⁵
hypercenter	C₂³×C₄²	D₅
hypocenter	C₅	C₂×C₄×#117
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄² ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
4. C₄×#4072	C₂	direct (N × H)
5. C₂×C₄×C₄⋊Dic₅	C₂	semidirect
6. C₂×#65221	C₂	semidirect
7. C₂×C₄²×Dic₅	C₂	semidirect
10. C₂²×C₄²×C₅	C₄	semidirect
16. C₂×#4072	C₄	direct (N × H)
23. C₄×C₄⋊Dic₅	C₂²	semidirect
24. #65221	C₂²	semidirect
25. C₄²×Dic₅	C₂²	semidirect
42. C₂×C₄²×C₅	C₂×C₄	semidirect
48. #4072	C₂×C₄	direct (N × H)
49. C₂×C₄⋊Dic₅	C₂×C₄	semidirect
52. C₂×C₄×Dic₅	C₂×C₄	semidirect
71. C₂²×C₄×C₅	C₄²	semidirect
93. C₄⋊Dic₅	C₂²×C₄	semidirect
95. C₄×Dic₅	C₂²×C₄	semidirect
104. C₂²×C₄²	Dic₅	semidirect
129. C₂×C₄×C₅	C₂×C₄²	semidirect
130. C₂×C₄×C₅	#117	semidirect
165. C₂×C₄²	C₂×Dic₅	semidirect
183. C₄×C₅	C₂×#117	semidirect
196. C₂²×C₄	C₄×Dic₅	semidirect
219. C₂×C₅	C₄×#117	semidirect
241. C₂×C₄	C₂×C₄×Dic₅	semidirect
242. C₂×C₄	#4072	direct (N × H)
250. C₅	C₂×C₄×#117	semidirect
265. C₄	C₂×#4072	direct (N × H)
267. C₂	C₄×#4072	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×C₄²×C₅	2
C₂²×#4072	2
C₂×C₄×C₂²⋊Dic₅	2
C₄×#4072	2
C₂×C₄×C₄⋊Dic₅	2
C₂×#65221	2
C₂×C₄²×Dic₅	2
C₂×C₄×#117	5

Subgroups — C2×C4×#4072 #12257485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#4072 #12257485