Subgroups — C₄×#64771 #12004481

Order 1280.

1182 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄⋊Dic₅	C₂	1	2	yes	maximal
3	C₂×#64771	C₂	1	4	yes	maximal
4	C₂³×C₅×C₈	C₂²	1	2	yes	—
5	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
6	C₂²×C₄⋊Dic₅	C₂²	1	4	yes	—
7	C₂²×C₄×Dic₅	—	2	2	no	—
8	C₂×C₄×C₅×C₈	—	2	4	no	—
9	C₂×C₄×C₅×C₈	C₄	1	4	yes	complemented
10	C₂×#3963	—	2	2	no	—
11	C₄×C₄⋊Dic₅	—	2	4	no	—
12	C₄×C₄⋊Dic₅	C₄	1	8	yes	complemented
13	#64771	C₄	1	16	yes	direct product factor
14	C₄×#340	—	5	1	no	maximal, Sylow
15	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
16	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
17	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
18	C₂³×Dic₅	—	2	4	no	—
19	C₂²×C₅×C₈	—	2	4	no	—
20	C₂²×C₅×C₈	—	2	4	no	—
21	C₂²×C₅×C₈	C₂×C₄	1	8	yes	—
22	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₅	D₄	1	4	yes	—
26	C₂×C₄²×C₅	Q₈	1	2	yes	—
27	C₂×C₄⋊Dic₅	—	2	4	no	—
28	C₂×C₄⋊Dic₅	—	2	8	no	—
29	C₂×C₄⋊Dic₅	C₂×C₄	1	16	yes	—
30	C₂×C₄⋊Dic₅	C₂×C₄	1	8	yes	—
31	C₂×C₄×Dic₅	—	2	8	no	—
32	C₂×C₄×Dic₅	—	4	4	no	—
33	C₂×C₄×Dic₅	—	4	4	no	—
34	C₄×C₅×C₈	—	2	8	no	—
35	#3963	—	2	8	no	—
36	C₂²×C₄×C₈	D₅	1	1	yes	characteristic
37	C₂×C₄×C₄⋊C₄	—	5	2	no	—
38	C₂×#340	—	5	4	no	—
39	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅	—	2	4	no	—
41	C₂²×C₄×C₅	C₂²⋊C₄	1	4	yes	—
42	C₂²×C₄×C₅	C₂²⋊C₄	1	4	yes	—
43	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₅	D₈	1	2	yes	—
46	C₂²×C₄×C₅	SD₁₆	1	4	yes	—
47	C₂²×C₄×C₅	Q₁₆	1	2	yes	—
48	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
50	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₅	#41	1	4	yes	—
54	C₂²×Dic₅	—	2	8	no	—
55	C₂²×Dic₅	—	2	16	no	—
56	C₂²×Dic₅	—	4	4	no	—
57	C₂²×Dic₅	—	4	4	no	—
58	C₂²×Dic₅	—	4	8	no	—
59	C₂×C₅×C₈	—	2	4	no	—
60	C₂×C₅×C₈	—	2	4	no	—
61	C₂×C₅×C₈	—	2	16	no	—
62	C₂×C₅×C₈	C₄²	1	8	yes	complemented
63	C₄²×C₅	C₄²	1	4	yes	—
64	C₄²×C₅	C₂²⋊C₄	1	4	yes	—
65	C₄²×C₅	C₄⋊C₄	1	8	yes	—
66	C₄⋊Dic₅	—	2	8	no	—
67	C₄⋊Dic₅	—	2	16	no	—
68	C₄⋊Dic₅	C₄²	1	16	yes	complemented
69	C₄×Dic₅	—	4	16	no	—
70	C₂³×C₈	C₂×D₅	1	2	yes	—
71	C₂²×C₄⋊C₄	—	5	4	no	—
72	C₂²×C₄²	—	10	2	no	—
73	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
74	C₂×C₄×C₈	—	2	4	no	—
75	C₂×C₄×C₈	Dic₅	1	4	yes	complemented
76	C₄×C₄⋊C₄	—	5	8	no	—
77	C₄×C₄⋊C₄	—	10	4	no	—
78	C₂×#95	—	10	2	no	—
79	#340	—	5	16	no	—
80	C₂³×C₅	C₂×C₂²⋊C₄	1	2	yes	—
81	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂³×C₅	C₄×D₄	1	4	yes	—
83	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
84	C₂³×C₅	C₂×SD₁₆	1	2	yes	—
85	C₂³×C₅	C₂×Q₁₆	1	1	yes	characteristic
86	C₂³×C₅	#135	1	4	yes	—
87	C₂×C₄×C₅	—	2	8	no	—
88	C₂×C₄×C₅	—	2	4	no	—
89	C₂×C₄×C₅	#95	1	4	yes	—
90	C₂×C₄×C₅	#95	1	4	yes	—
91	C₂×C₄×C₅	#102	1	8	yes	—
92	C₂×C₄×C₅	#103	1	8	yes	—
93	C₂×C₄×C₅	#106	1	4	yes	—
94	C₂×C₄×C₅	#107	1	4	yes	—
95	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
96	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
98	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₅	#117	1	4	yes	—
101	C₂×C₄×C₅	#117	1	2	yes	—
102	C₂×C₄×C₅	C₄×D₄	1	8	yes	—
103	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
104	C₂×Dic₅	—	2	16	no	—
105	C₂×Dic₅	—	4	8	no	—
106	C₂×Dic₅	—	4	8	no	—
107	C₂×Dic₅	—	4	16	no	—
108	C₂×Dic₅	—	4	16	no	—
109	C₅×C₈	—	2	8	no	—
110	C₂³×C₄	—	10	4	no	—
111	C₂³×C₄	D₂₀	1	1	yes	characteristic
112	C₂³×C₄	D₂₀	1	1	yes	characteristic
113	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
114	C₂²×C₈	—	2	4	no	—
115	C₂²×C₈	—	2	4	no	—
116	C₂²×C₈	C₄×D₅	1	8	yes	—
117	C₂²×C₈	C₂×Dic₅	1	4	yes	—
118	C₂×C₄⋊C₄	—	5	16	no	—
119	C₂×C₄⋊C₄	—	5	8	no	—
120	C₂×C₄⋊C₄	—	10	8	no	—
121	C₂×C₄⋊C₄	—	10	4	no	—
122	C₂×C₄²	—	10	8	no	—
123	C₂×C₄²	—	20	4	no	—
124	C₂×C₄²	—	20	4	no	—
125	C₂×C₄²	Dic₁₀	1	2	yes	—
126	C₂×C₄²	C₄×D₅	1	4	yes	—
127	C₂×C₄²	C₂×Dic₅	1	2	yes	—
128	C₂×C₄²	C₂²⋊D₅	1	4	yes	—
129	C₄×C₈	—	2	8	no	—
130	#95	—	10	8	no	—
131	C₂²×C₅	C₂×#95	1	1	yes	characteristic
132	C₂²×C₅	C₄×C₂²⋊C₄	1	4	yes	—
133	C₂²×C₅	C₄×C₄⋊C₄	1	2	yes	—
134	C₂²×C₅	C₂×#102	1	2	yes	—
135	C₂²×C₅	C₂×#103	1	2	yes	—
136	C₂²×C₅	#416	1	4	yes	—
137	C₂²×C₅	C₂×#106	1	1	yes	characteristic
138	C₂²×C₅	C₂×#107	1	1	yes	characteristic
139	C₂²×C₅	#427	1	2	yes	—
140	C₂²×C₅	C₄×D₈	1	4	yes	—
141	C₂²×C₅	C₄×SD₁₆	1	8	yes	—
142	C₂²×C₅	C₄×Q₁₆	1	4	yes	—
143	C₄×C₅	—	2	4	no	—
144	C₄×C₅	#340	1	8	yes	complemented
145	C₄×C₅	C₄³	1	1	yes	characteristic
146	C₄×C₅	#376	1	1	yes	characteristic
147	C₄×C₅	C₄×C₂²⋊C₄	1	2	yes	—
148	C₄×C₅	C₄×C₄⋊C₄	1	4	yes	—
149	Dic₅	—	4	16	no	—
150	C₂⁴	C₂×D₂₀	1	1	yes	characteristic
151	C₂²×C₄	—	2	4	no	—
152	C₂²×C₄	—	10	16	no	—
153	C₂²×C₄	—	10	8	no	—
154	C₂²×C₄	—	20	4	no	—
155	C₂²×C₄	—	20	4	no	—
156	C₂²×C₄	—	20	8	no	—
157	C₂²×C₄	#677	1	4	yes	—
158	C₂²×C₄	D₄₀	1	2	yes	—
159	C₂²×C₄	Dic₂₀	1	2	yes	—
160	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
161	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
162	C₂²×C₄	#685	1	4	yes	—
163	C₂²×C₄	#685	1	4	yes	—
164	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₄×D₅	1	4	yes	—
166	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
167	C₂²×C₄	#709	1	4	yes	—
168	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
169	C₂²×C₄	C₂×C₂²⋊D₅	1	2	yes	—
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	2	16	no	—
173	C₂×C₈	C₄×Dic₅	1	8	yes	complemented
174	C₄⋊C₄	—	5	16	no	—
175	C₄⋊C₄	—	10	8	no	—
176	C₄⋊C₄	—	10	16	no	—
177	C₄²	—	20	16	no	—
178	C₄²	C₄×Dic₅	1	4	yes	—
179	C₄²	#683	1	8	yes	—
180	C₄²	C₂²⋊Dic₅	1	4	yes	—
181	C₂×C₅	C₄×#95	1	1	yes	characteristic
182	C₂×C₅	C₂×#340	1	1	yes	characteristic
183	C₂×C₅	#1735	1	1	yes	characteristic
184	C₂×C₅	C₄×#102	1	4	yes	—
185	C₂×C₅	C₄×#103	1	4	yes	—
186	C₂×C₅	C₄×#106	1	2	yes	—
187	C₂×C₅	C₄×#107	1	2	yes	—
188	C₂³	C₄×D₂₀	1	4	yes	—
189	C₂³	C₂×#677	1	2	yes	—
190	C₂³	C₂×D₄₀	1	1	yes	characteristic
191	C₂³	#4050	1	4	yes	—
192	C₂³	C₂×Dic₂₀	1	1	yes	characteristic
193	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
194	C₂³	C₂×#685	1	2	yes	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	8	no	—
197	C₂×C₄	—	10	16	no	—
198	C₂×C₄	—	20	8	no	—
199	C₂×C₄	—	20	8	no	—
200	C₂×C₄	—	20	16	no	—
201	C₂×C₄	—	20	16	no	—
202	C₂×C₄	#3948	1	8	yes	—
203	C₂×C₄	#3949	1	4	yes	—
204	C₂×C₄	#3950	1	4	yes	—
205	C₂×C₄	#3953	1	8	yes	—
206	C₂×C₄	#3963	1	4	yes	—
207	C₂×C₄	#3963	1	4	yes	—
208	C₂×C₄	C₄×Dic₁₀	1	4	yes	—
209	C₂×C₄	C₄²×D₅	1	4	yes	—
210	C₂×C₄	#4018	1	4	yes	—
211	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
213	C₂×C₄	C₂×#683	1	2	yes	—
214	C₂×C₄	#4072	1	2	yes	—
215	C₂×C₄	C₄×C₂²⋊D₅	1	8	yes	—
216	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
217	C₈	—	2	8	no	—
218	C₅	C₄×#340	1	1	yes	characteristic, complemented, Sylow
219	C₂²	C₄×#677	1	8	yes	—
220	C₂²	C₄×D₄₀	1	4	yes	—
221	C₂²	C₄×Dic₂₀	1	4	yes	—
222	C₂²	C₄×C₄⋊Dic₅	1	2	yes	—
223	C₂²	C₄×#685	1	4	yes	—
224	C₂²	C₂×#3948	1	2	yes	—
225	C₂²	C₂×#3949	1	1	yes	characteristic
226	C₂²	C₂×#3950	1	1	yes	characteristic
227	C₂²	#65395	1	2	yes	—
228	C₂²	C₂×#3953	1	2	yes	—
229	C₂²	#65402	1	4	yes	—
230	C₂²	C₂×#3963	1	1	yes	characteristic
231	C₄	—	2	4	no	—
232	C₄	—	20	16	no	—
233	C₄	#64771	1	8	yes	direct product factor
234	C₄	C₄²×Dic₅	1	1	yes	characteristic
235	C₄	C₄×#683	1	4	yes	—
236	C₄	#65221	1	1	yes	characteristic
237	C₄	C₄×C₂²⋊Dic₅	1	2	yes	—
238	C₂	C₄×#3953	1	4	yes	—
239	C₂	C₄×#3948	1	4	yes	—
240	C₂	C₄×#3949	1	2	yes	—
241	C₂	C₄×#3950	1	2	yes	—
242	C₂	C₄×#3963	1	1	yes	characteristic
243	C₂	C₂×#64771	1	1	yes	characteristic
244	C₂	#10602762	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	D₂₀
commutator	C₄×C₅	C₄³
Frattini	C₂³×C₄	C₂²×D₅
Fitting	C₂²×C₄×C₅×C₈	C₂
socle, prime radical	C₂⁴×C₅	C₂×D₄
cosocle	C₂³×C₄×C₅	C₂³
hypercenter	C₂²×C₄×C₈	D₅
hypocenter	C₅	C₄×#340
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
9. C₂×C₄×C₅×C₈	C₄	semidirect
12. C₄×C₄⋊Dic₅	C₄	semidirect
13. #64771	C₄	direct (N × H)
62. C₂×C₅×C₈	C₄²	semidirect
68. C₄⋊Dic₅	C₄²	semidirect
75. C₂×C₄×C₈	Dic₅	semidirect
144. C₄×C₅	#340	semidirect
173. C₂×C₈	C₄×Dic₅	semidirect
218. C₅	C₄×#340	semidirect
233. C₄	#64771	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₅×C₈	2
C₂×C₄×C₄⋊Dic₅	2
C₂×#64771	2
C₄×#340	5

Subgroups — C4×#64771 #12004481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#64771 #12004481