Subgroups — C₄×C₅×#493 #11944407

Order 1280.

1422 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	C₂×C₅×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×D₈	C₂	1	4	yes	maximal, complemented
3	C₄×C₅×#127	C₂	1	2	yes	maximal, complemented
4	C₅×#1970	C₂	1	4	yes	maximal, complemented
5	C₅×#1957	C₂	1	2	yes	maximal, complemented
6	C₅×#1845	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄²×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₅×D₈	C₂²	1	4	yes	—
9	C₂×C₅×#127	C₂²	1	2	yes	—
10	C₂×C₄×C₅×D₄	—	2	4	no	—
11	C₂×C₄×C₅×D₄	C₂²	1	4	yes	—
12	C₅×#493	C₄	1	8	yes	direct product factor
13	C₄×C₅×D₈	—	2	16	no	—
14	C₂×C₅×#107	C₂²	1	4	yes	—
15	C₂×C₅×#102	C₂²	1	8	yes	—
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
20	C₅×#390	—	2	4	no	—
21	C₅×#390	C₂²	1	4	yes	—
22	C₅×#384	C₂²	1	2	yes	—
23	C₄³×C₅	C₂²	1	1	yes	characteristic
24	C₄×#493	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂²×C₅×D₄	—	2	4	no	—
26	C₂²×C₅×D₄	C₂³	1	4	yes	—
27	C₂³×C₄×C₅	—	2	4	no	—
28	C₂×C₅×D₈	—	2	8	no	—
29	C₂×C₅×D₈	—	2	8	no	—
30	C₂×C₅×D₈	C₂×C₄	1	16	yes	complemented
31	C₂²×C₅×C₈	C₂³	1	2	yes	—
32	C₂²×C₅×C₈	C₂³	1	2	yes	—
33	C₅×#127	—	2	4	no	—
34	C₅×#127	C₂×C₄	1	8	yes	complemented
35	C₄×C₅×D₄	—	2	16	no	—
36	C₄×C₅×D₄	—	4	16	no	—
37	C₂×C₅×C₄⋊C₄	—	2	4	no	—
38	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
39	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
40	C₂×C₅×C₂²⋊C₄	—	2	8	no	—
41	C₂×C₄²×C₅	D₄	1	2	yes	—
42	C₂×C₄²×C₅	D₄	1	2	yes	—
43	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₅×#107	—	2	8	no	—
47	C₅×#102	—	2	16	no	—
48	C₄×C₅×C₈	—	2	4	no	—
49	C₄×C₅×C₈	—	2	2	no	—
50	C₄×C₅×C₈	C₂×C₄	1	4	yes	complemented
51	C₄×C₅×C₈	D₄	1	8	yes	complemented
52	C₂×#493	C₂×C₅	1	1	yes	characteristic
53	C₂×C₄×D₈	C₂×C₅	1	4	yes	complemented
54	C₄×#127	C₂×C₅	1	2	yes	complemented
55	#1970	C₂×C₅	1	4	yes	complemented
56	#1957	C₂×C₅	1	2	yes	complemented
57	#1845	C₂×C₅	1	1	yes	characteristic, complemented
58	C₄²×C₈	C₂×C₅	1	1	yes	characteristic, complemented
59	C₂⁴×C₅	—	2	4	no	—
60	C₂×C₅×D₄	—	2	16	no	—
61	C₂×C₅×D₄	—	2	8	no	—
62	C₂×C₅×D₄	—	2	8	no	—
63	C₂×C₅×D₄	—	2	4	no	—
64	C₂×C₅×D₄	—	4	8	no	—
65	C₂×C₅×D₄	—	4	8	no	—
66	C₂×C₅×D₄	C₂²×C₄	1	8	yes	—
67	C₂²×C₄×C₅	—	2	4	no	—
68	C₂²×C₄×C₅	—	4	8	no	—
69	C₂²×C₄×C₅	—	4	16	no	—
70	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
75	C₅×D₈	—	2	32	no	—
76	C₂×C₅×C₈	—	2	2	no	—
77	C₂×C₅×C₈	—	2	2	no	—
78	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
79	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
80	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
81	C₂×C₅×C₈	#41	1	4	yes	—
82	C₅×C₄⋊C₄	—	2	8	no	—
83	C₅×C₄⋊C₄	—	4	8	no	—
84	C₅×C₂²⋊C₄	—	4	8	no	—
85	C₅×C₂²⋊C₄	—	4	16	no	—
86	C₄²×C₅	—	2	4	no	—
87	C₄²×C₅	—	2	2	no	—
88	C₄²×C₅	D₈	1	8	yes	complemented
89	C₄²×C₅	C₂²×C₄	1	2	yes	—
90	C₄²×C₅	C₂×D₄	1	4	yes	—
91	C₄²×C₅	#41	1	2	yes	—
92	C₂²×D₈	C₂²×C₅	1	4	yes	—
93	C₂×#127	C₂²×C₅	1	2	yes	—
94	C₂×C₄×D₄	—	2	4	no	—
95	C₂×C₄×D₄	C₂²×C₅	1	4	yes	—
96	#493	C₄×C₅	1	8	yes	direct product factor
97	C₄×D₈	—	2	16	no	—
98	C₂×#107	C₂²×C₅	1	4	yes	—
99	C₂×#102	C₂²×C₅	1	8	yes	—
100	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
101	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
102	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
103	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
104	#390	—	2	4	no	—
105	#390	C₂²×C₅	1	4	yes	—
106	#384	C₂²×C₅	1	2	yes	—
107	C₄³	C₂²×C₅	1	1	yes	characteristic
108	C₂³×C₅	—	2	8	no	—
109	C₂³×C₅	—	4	4	no	—
110	C₂³×C₅	—	4	4	no	—
111	C₂³×C₅	—	4	8	no	—
112	C₂³×C₅	—	4	8	no	—
113	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
114	C₅×D₄	—	2	16	no	—
115	C₅×D₄	—	4	8	no	—
116	C₅×D₄	—	4	32	no	—
117	C₂×C₄×C₅	—	2	2	no	—
118	C₂×C₄×C₅	—	2	2	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	4	4	no	—
121	C₂×C₄×C₅	—	4	8	no	—
122	C₂×C₄×C₅	—	8	16	no	—
123	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
124	C₂×C₄×C₅	#127	1	2	yes	—
125	C₂×C₄×C₅	#127	1	2	yes	—
126	C₂×C₄×C₅	C₂×D₈	1	4	yes	—
127	C₂×C₄×C₅	C₂×D₈	1	4	yes	—
128	C₂×C₄×C₅	C₂×D₈	1	4	yes	—
129	C₂×C₄×C₅	#135	1	4	yes	—
130	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
132	C₂×C₄×C₅	C₂×#41	1	2	yes	—
133	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
134	C₅×C₈	—	2	4	no	—
135	C₅×C₈	C₄×D₄	1	8	yes	complemented
136	C₂²×D₄	—	2	4	no	—
137	C₂²×D₄	C₂³×C₅	1	4	yes	—
138	C₂³×C₄	—	2	4	no	—
139	C₂×D₈	—	2	8	no	—
140	C₂×D₈	—	2	8	no	—
141	C₂×D₈	C₂×C₄×C₅	1	16	yes	complemented
142	C₂²×C₈	C₂³×C₅	1	2	yes	—
143	C₂²×C₈	C₂³×C₅	1	2	yes	—
144	#127	—	2	4	no	—
145	#127	C₂×C₄×C₅	1	8	yes	complemented
146	C₄×D₄	—	2	16	no	—
147	C₄×D₄	—	4	16	no	—
148	C₂×C₄⋊C₄	—	2	4	no	—
149	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
150	C₂×C₂²⋊C₄	—	2	8	no	—
151	C₂×C₂²⋊C₄	—	2	4	no	—
152	C₂×C₄²	C₅×D₄	1	2	yes	—
153	C₂×C₄²	C₅×D₄	1	2	yes	—
154	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
155	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
156	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
157	#107	—	2	8	no	—
158	#102	—	2	16	no	—
159	C₄×C₈	—	2	4	no	—
160	C₄×C₈	—	2	2	no	—
161	C₄×C₈	C₂×C₄×C₅	1	4	yes	complemented
162	C₄×C₈	C₅×D₄	1	8	yes	complemented
163	C₂²×C₅	—	4	8	no	—
164	C₂²×C₅	—	4	16	no	—
165	C₂²×C₅	—	8	4	no	—
166	C₂²×C₅	—	8	4	no	—
167	C₂²×C₅	—	8	8	no	—
168	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
169	C₂²×C₅	C₂×#127	1	1	yes	characteristic
170	C₂²×C₅	#532	1	1	yes	characteristic
171	C₂²×C₅	C₂²×D₈	1	2	yes	—
172	C₂²×C₅	C₂×#135	1	2	yes	—
173	C₄×C₅	—	2	2	no	—
174	C₄×C₅	—	2	4	no	—
175	C₄×C₅	—	8	4	no	—
176	C₄×C₅	C₄×D₈	1	8	yes	complemented
177	C₄×C₅	#493	1	4	yes	direct product factor
178	C₄×C₅	C₂×C₄×D₄	1	2	yes	—
179	C₄×C₅	#532	1	2	yes	—
180	C₂⁴	—	2	4	no	—
181	C₂×D₄	—	2	16	no	—
182	C₂×D₄	—	2	8	no	—
183	C₂×D₄	—	2	8	no	—
184	C₂×D₄	—	2	4	no	—
185	C₂×D₄	—	4	8	no	—
186	C₂×D₄	—	4	8	no	—
187	C₂×D₄	C₂²×C₄×C₅	1	8	yes	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	4	8	no	—
190	C₂²×C₄	—	4	16	no	—
191	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
193	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
194	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
195	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
196	D₈	—	2	32	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
200	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
201	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
202	C₂×C₈	C₅×#41	1	4	yes	—
203	C₄⋊C₄	—	2	8	no	—
204	C₄⋊C₄	—	4	8	no	—
205	C₂²⋊C₄	—	4	8	no	—
206	C₂²⋊C₄	—	4	16	no	—
207	C₄²	—	2	4	no	—
208	C₄²	—	2	2	no	—
209	C₄²	C₅×D₈	1	8	yes	complemented
210	C₄²	C₂²×C₄×C₅	1	2	yes	—
211	C₄²	C₂×C₅×D₄	1	4	yes	—
212	C₄²	C₅×#41	1	2	yes	—
213	C₂×C₅	—	8	8	no	—
214	C₂×C₅	C₄×#127	1	1	yes	characteristic
215	C₂×C₅	C₂×C₄×D₈	1	2	yes	—
216	C₂×C₅	C₂×#493	1	1	yes	characteristic
217	C₂×C₅	#3147	1	1	yes	characteristic
218	C₂×C₅	#3169	1	2	yes	—
219	C₂³	—	2	8	no	—
220	C₂³	—	4	4	no	—
221	C₂³	—	4	4	no	—
222	C₂³	—	4	8	no	—
223	C₂³	—	4	8	no	—
224	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
225	D₄	—	2	16	no	—
226	D₄	—	4	32	no	—
227	D₄	—	4	8	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	8	no	—
233	C₂×C₄	—	8	16	no	—
234	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
235	C₂×C₄	C₅×#127	1	2	yes	—
236	C₂×C₄	C₅×#127	1	2	yes	—
237	C₂×C₄	C₂×C₅×D₈	1	4	yes	—
238	C₂×C₄	C₂×C₅×D₈	1	4	yes	—
239	C₂×C₄	C₂×C₅×D₈	1	4	yes	—
240	C₂×C₄	C₅×#135	1	4	yes	—
241	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
242	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
243	C₂×C₄	C₂×C₅×#41	1	2	yes	—
244	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
245	C₈	—	2	4	no	—
246	C₈	C₄×C₅×D₄	1	8	yes	complemented
247	C₅	C₄×#493	1	1	yes	characteristic, direct product factor, Sylow
248	C₂²	—	4	8	no	—
249	C₂²	—	4	16	no	—
250	C₂²	—	8	4	no	—
251	C₂²	—	8	4	no	—
252	C₂²	—	8	8	no	—
253	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
254	C₂²	C₂×C₅×#127	1	1	yes	characteristic
255	C₂²	C₅×#532	1	1	yes	characteristic
256	C₂²	C₂²×C₅×D₈	1	2	yes	—
257	C₂²	C₂×C₅×#135	1	2	yes	—
258	C₄	—	2	2	no	—
259	C₄	—	2	4	no	—
260	C₄	—	8	4	no	—
261	C₄	C₄×C₅×D₈	1	8	yes	complemented
262	C₄	C₅×#493	1	4	yes	direct product factor
263	C₄	C₂×C₄×C₅×D₄	1	2	yes	—
264	C₄	C₅×#532	1	2	yes	—
265	C₂	—	8	8	no	—
266	C₂	C₄×C₅×#127	1	1	yes	characteristic
267	C₂	C₂×C₄×C₅×D₈	1	2	yes	—
268	C₂	C₂×C₅×#493	1	1	yes	characteristic
269	C₂	C₅×#3147	1	1	yes	characteristic
270	C₂	C₅×#3169	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂³×C₅	C₂²×D₄
prime radical	C₂×C₅×#493	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₅×D₈	C₂	semidirect
3. C₄×C₅×#127	C₂	semidirect
4. C₅×#1970	C₂	semidirect
5. C₅×#1957	C₂	semidirect
6. C₅×#1845	C₂	semidirect
7. C₄²×C₅×C₈	C₂	semidirect
12. C₅×#493	C₄	direct (N × H)
24. C₄×#493	C₅	direct (N × H)
30. C₂×C₅×D₈	C₂×C₄	semidirect
34. C₅×#127	C₂×C₄	semidirect
50. C₄×C₅×C₈	C₂×C₄	semidirect
51. C₄×C₅×C₈	D₄	semidirect
53. C₂×C₄×D₈	C₂×C₅	semidirect
54. C₄×#127	C₂×C₅	semidirect
55. #1970	C₂×C₅	semidirect
56. #1957	C₂×C₅	semidirect
57. #1845	C₂×C₅	semidirect
58. C₄²×C₈	C₂×C₅	semidirect
88. C₄²×C₅	D₈	semidirect
96. #493	C₄×C₅	direct (N × H)
135. C₅×C₈	C₄×D₄	semidirect
141. C₂×D₈	C₂×C₄×C₅	semidirect
145. #127	C₂×C₄×C₅	semidirect
161. C₄×C₈	C₂×C₄×C₅	semidirect
162. C₄×C₈	C₅×D₄	semidirect
176. C₄×C₅	C₄×D₈	semidirect
177. C₄×C₅	#493	direct (N × H)
209. C₄²	C₅×D₈	semidirect
246. C₈	C₄×C₅×D₄	semidirect
247. C₅	C₄×#493	direct (N × H)
261. C₄	C₄×C₅×D₈	semidirect
262. C₄	C₅×#493	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#493	2
C₂×C₄×C₅×D₈	2
C₄×C₅×#127	2
C₅×#1970	2
C₅×#1957	2
C₅×#1845	2
C₄²×C₅×C₈	2
C₄×#493	5

Subgroups — C4×C5×#493 #11944407

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×#493 #11944407