Subgroups — C₄×C₅×#500 #11944440

Order 1280.

878 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#500	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×C₄⋊Q₈	C₂	1	2	yes	maximal
3	C₅×#1990	C₂	1	2	yes	maximal
4	C₅×#1942	C₂	1	4	yes	maximal
5	C₅×#1845	C₂	1	1	yes	maximal, characteristic
6	C₄×C₅×#107	C₂	1	4	yes	maximal
7	C₄²×C₅×C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₄×C₅×Q₈	—	2	2	no	—
10	C₅×#500	C₄	1	8	yes	direct product factor
11	C₂×C₅×#107	C₂²	1	4	yes	—
12	C₂×C₅×#107	C₂²	1	4	yes	—
13	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
17	C₅×#391	—	2	2	no	—
18	C₅×#389	—	2	2	no	—
19	C₅×#389	C₂²	1	4	yes	—
20	C₅×#384	C₂²	1	2	yes	—
21	C₄×C₅×C₄⋊C₄	—	2	2	no	—
22	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
23	C₄³×C₅	C₂²	1	1	yes	characteristic
24	C₅×#340	C₂²	1	8	yes	—
25	C₄×#500	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂²×C₅×Q₈	—	2	2	no	—
27	C₂²×C₅×C₈	C₂³	1	2	yes	—
28	C₂²×C₅×C₈	C₂³	1	2	yes	—
29	C₅×C₄⋊Q₈	—	2	4	no	—
30	C₅×C₄⋊Q₈	C₂×C₄	1	8	yes	—
31	C₄×C₅×Q₈	—	4	8	no	—
32	C₂×C₅×C₄⋊C₄	—	2	2	no	—
33	C₂×C₅×C₄⋊C₄	—	2	2	no	—
34	C₂×C₅×C₄⋊C₄	—	2	2	no	—
35	C₂×C₅×C₄⋊C₄	—	2	4	no	—
36	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
37	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
38	C₂×C₄²×C₅	—	2	4	no	—
39	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
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×#107	C₂×C₄	1	16	yes	—
48	C₄×C₅×C₈	—	2	4	no	—
49	C₄×C₅×C₈	—	2	2	no	—
50	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
51	C₄×C₅×C₈	Q₈	1	8	yes	complemented
52	C₅×#95	—	2	4	no	—
53	C₅×#95	—	2	4	no	—
54	C₂×#500	C₂×C₅	1	1	yes	characteristic
55	C₄×C₄⋊Q₈	C₂×C₅	1	2	yes	—
56	#1990	C₂×C₅	1	2	yes	—
57	#1942	C₂×C₅	1	4	yes	—
58	#1845	C₂×C₅	1	1	yes	characteristic
59	C₄×#107	C₂×C₅	1	4	yes	—
60	C₄²×C₈	C₂×C₅	1	1	yes	characteristic
61	C₂×C₅×Q₈	—	2	8	no	—
62	C₂×C₅×Q₈	—	4	4	no	—
63	C₂×C₅×Q₈	—	4	4	no	—
64	C₂²×C₄×C₅	—	2	4	no	—
65	C₂²×C₄×C₅	—	2	4	no	—
66	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
73	C₂×C₅×C₈	—	2	2	no	—
74	C₂×C₅×C₈	—	2	2	no	—
75	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
76	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
77	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
78	C₂×C₅×C₈	#41	1	4	yes	—
79	C₅×C₄⋊C₄	—	2	8	no	—
80	C₅×C₄⋊C₄	—	2	4	no	—
81	C₅×C₄⋊C₄	—	2	8	no	—
82	C₅×C₄⋊C₄	—	4	8	no	—
83	C₅×C₄⋊C₄	—	4	4	no	—
84	C₅×C₄⋊C₄	—	4	4	no	—
85	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
86	C₄²×C₅	—	2	2	no	—
87	C₄²×C₅	—	2	2	no	—
88	C₄²×C₅	—	2	2	no	—
89	C₄²×C₅	—	4	8	no	—
90	C₄²×C₅	D₈	1	4	yes	—
91	C₄²×C₅	Q₁₆	1	4	yes	—
92	C₄²×C₅	C₂²×C₄	1	2	yes	—
93	C₄²×C₅	C₂×Q₈	1	4	yes	—
94	C₄²×C₅	#41	1	2	yes	—
95	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
96	C₂×C₄×Q₈	—	2	2	no	—
97	#500	C₄×C₅	1	8	yes	direct product factor
98	C₂×#107	C₂²×C₅	1	4	yes	—
99	C₂×#107	C₂²×C₅	1	4	yes	—
100	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
101	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
102	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
103	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
104	#391	—	2	2	no	—
105	#389	—	2	2	no	—
106	#389	C₂²×C₅	1	4	yes	—
107	#384	C₂²×C₅	1	2	yes	—
108	C₄×C₄⋊C₄	—	2	2	no	—
109	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
110	C₄³	C₂²×C₅	1	1	yes	characteristic
111	#340	C₂²×C₅	1	8	yes	—
112	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
113	C₅×Q₈	—	4	16	no	—
114	C₂×C₄×C₅	—	2	1	no	—
115	C₂×C₄×C₅	—	2	1	no	—
116	C₂×C₄×C₅	—	2	1	no	—
117	C₂×C₄×C₅	—	2	1	no	—
118	C₂×C₄×C₅	—	2	1	no	—
119	C₂×C₄×C₅	—	2	1	no	—
120	C₂×C₄×C₅	—	2	8	no	—
121	C₂×C₄×C₅	—	4	4	no	—
122	C₂×C₄×C₅	—	4	4	no	—
123	C₂×C₄×C₅	—	4	4	no	—
124	C₂×C₄×C₅	—	4	4	no	—
125	C₂×C₄×C₅	—	4	4	no	—
126	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
129	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
130	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
134	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
135	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
136	C₂×C₄×C₅	#135	1	2	yes	—
137	C₂×C₄×C₅	#135	1	2	yes	—
138	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
140	C₂×C₄×C₅	C₂×#41	1	2	yes	—
141	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
142	C₅×C₈	—	2	4	no	—
143	C₅×C₈	C₄×Q₈	1	8	yes	complemented
144	C₂²×Q₈	—	2	2	no	—
145	C₂²×C₈	C₂³×C₅	1	2	yes	—
146	C₂²×C₈	C₂³×C₅	1	2	yes	—
147	C₄⋊Q₈	—	2	4	no	—
148	C₄⋊Q₈	C₂×C₄×C₅	1	8	yes	—
149	C₄×Q₈	—	4	8	no	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	—	2	4	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
155	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
156	C₂×C₄²	—	2	4	no	—
157	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
158	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
159	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
160	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
161	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
162	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
163	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
164	#107	—	2	8	no	—
165	#107	C₂×C₄×C₅	1	16	yes	—
166	C₄×C₈	—	2	4	no	—
167	C₄×C₈	—	2	2	no	—
168	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
169	C₄×C₈	C₅×Q₈	1	8	yes	complemented
170	#95	—	2	4	no	—
171	#95	—	2	4	no	—
172	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
173	C₂²×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
174	C₂²×C₅	#532	1	1	yes	characteristic
175	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
176	C₂²×C₅	C₂²×Q₁₆	1	1	yes	characteristic
177	C₂²×C₅	C₂×#135	1	1	yes	characteristic
178	C₂²×C₅	C₂×#135	1	1	yes	characteristic
179	C₄×C₅	—	2	2	no	—
180	C₄×C₅	—	2	2	no	—
181	C₄×C₅	—	2	2	no	—
182	C₄×C₅	—	4	8	no	—
183	C₄×C₅	—	8	4	no	—
184	C₄×C₅	C₄×D₈	1	4	yes	—
185	C₄×C₅	C₄×Q₁₆	1	4	yes	—
186	C₄×C₅	#500	1	4	yes	direct product factor
187	C₄×C₅	C₂×C₄×Q₈	1	2	yes	—
188	C₄×C₅	#533	1	2	yes	—
189	C₂×Q₈	—	2	8	no	—
190	C₂×Q₈	—	4	4	no	—
191	C₂×Q₈	—	4	4	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
200	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
204	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
205	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
206	C₂×C₈	C₅×#41	1	4	yes	—
207	C₄⋊C₄	—	2	8	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	2	8	no	—
210	C₄⋊C₄	—	4	8	no	—
211	C₄⋊C₄	—	4	4	no	—
212	C₄⋊C₄	—	4	4	no	—
213	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
214	C₄²	—	2	2	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	4	8	no	—
218	C₄²	C₅×D₈	1	4	yes	—
219	C₄²	C₅×Q₁₆	1	4	yes	—
220	C₄²	C₂²×C₄×C₅	1	2	yes	—
221	C₄²	C₂×C₅×Q₈	1	4	yes	—
222	C₄²	C₅×#41	1	2	yes	—
223	C₂×C₅	C₄×C₄⋊Q₈	1	1	yes	characteristic
224	C₂×C₅	C₂×C₄×D₈	1	1	yes	characteristic
225	C₂×C₅	C₂×C₄×Q₁₆	1	1	yes	characteristic
226	C₂×C₅	C₂×#500	1	1	yes	characteristic
227	C₂×C₅	#3160	1	1	yes	characteristic
228	C₂×C₅	#3169	1	1	yes	characteristic
229	C₂×C₅	#3170	1	1	yes	characteristic
230	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
231	Q₈	—	4	16	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
245	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
246	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
247	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
248	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
249	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
250	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
251	C₂×C₄	C₂×C₅×Q₁₆	1	2	yes	—
252	C₂×C₄	C₂×C₅×Q₁₆	1	2	yes	—
253	C₂×C₄	C₂×C₅×Q₁₆	1	2	yes	—
254	C₂×C₄	C₅×#135	1	2	yes	—
255	C₂×C₄	C₅×#135	1	2	yes	—
256	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
257	C₂×C₄	C₂²×C₅×Q₈	1	2	yes	—
258	C₂×C₄	C₂×C₅×#41	1	2	yes	—
259	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
260	C₈	—	2	4	no	—
261	C₈	C₄×C₅×Q₈	1	8	yes	complemented
262	C₅	C₄×#500	1	1	yes	characteristic, direct product factor, Sylow
263	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
264	C₂²	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
265	C₂²	C₅×#532	1	1	yes	characteristic
266	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
267	C₂²	C₂²×C₅×Q₁₆	1	1	yes	characteristic
268	C₂²	C₂×C₅×#135	1	1	yes	characteristic
269	C₂²	C₂×C₅×#135	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	4	8	no	—
274	C₄	—	8	4	no	—
275	C₄	C₄×C₅×D₈	1	4	yes	—
276	C₄	C₄×C₅×Q₁₆	1	4	yes	—
277	C₄	C₅×#500	1	4	yes	direct product factor
278	C₄	C₂×C₄×C₅×Q₈	1	2	yes	—
279	C₄	C₅×#533	1	2	yes	—
280	C₂	C₄×C₅×C₄⋊Q₈	1	1	yes	characteristic
281	C₂	C₂×C₄×C₅×D₈	1	1	yes	characteristic
282	C₂	C₂×C₄×C₅×Q₁₆	1	1	yes	characteristic
283	C₂	C₂×C₅×#500	1	1	yes	characteristic
284	C₂	C₅×#3160	1	1	yes	characteristic
285	C₂	C₅×#3169	1	1	yes	characteristic
286	C₂	C₅×#3170	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle, prime radical	C₂³×C₅	C₂²×D₄
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
10. C₅×#500	C₄	direct (N × H)
25. C₄×#500	C₅	direct (N × H)
51. C₄×C₅×C₈	Q₈	semidirect
97. #500	C₄×C₅	direct (N × H)
143. C₅×C₈	C₄×Q₈	semidirect
169. C₄×C₈	C₅×Q₈	semidirect
186. C₄×C₅	#500	direct (N × H)
261. C₈	C₄×C₅×Q₈	semidirect
262. C₅	C₄×#500	direct (N × H)
277. C₄	C₅×#500	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×#500	2
C₄×C₅×C₄⋊Q₈	2
C₅×#1990	2
C₅×#1942	2
C₅×#1845	2
C₄×C₅×#107	2
C₄²×C₅×C₈	2
C₄×#500	5

Subgroups — C4×C5×#500 #11944440

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×#500 #11944440