Subgroups — C₄×C₅×#499 #11944441

Order 1280.

774 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	C₂×C₅×#499	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×#125	C₂	1	2	yes	maximal
3	C₅×#1991	C₂	1	2	yes	maximal
4	C₅×#1942	C₂	1	2	yes	maximal
5	C₅×#1941	C₂	1	2	yes	maximal
6	C₅×#1846	C₂	1	1	yes	maximal, characteristic
7	C₄×C₅×#107	C₂	1	2	yes	maximal
8	C₄×C₅×#106	C₂	1	2	yes	maximal
9	C₄²×C₅×C₈	C₂	1	1	yes	maximal, characteristic
10	C₂×C₅×#125	C₂²	1	2	yes	—
11	C₅×#499	C₄	1	8	yes	direct product factor
12	C₂×C₅×#107	C₂²	1	2	yes	—
13	C₂×C₅×#107	C₂²	1	2	yes	—
14	C₂×C₅×#106	C₂²	1	2	yes	—
15	C₂×C₅×#106	C₂²	1	2	yes	—
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₅×#389	C₂²	1	4	yes	—
23	C₅×#387	—	2	4	no	—
24	C₅×#382	C₂²	1	2	yes	—
25	C₄×C₅×C₄⋊C₄	—	2	4	no	—
26	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
27	C₄³×C₅	C₂²	1	1	yes	characteristic
28	C₅×#340	C₂²	1	4	yes	—
29	C₅×#340	C₂²	1	4	yes	—
30	C₄×#499	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
32	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₅×#125	—	2	4	no	—
36	C₅×#125	C₂×C₄	1	8	yes	—
37	C₂×C₅×C₄⋊C₄	—	2	4	no	—
38	C₂×C₅×C₄⋊C₄	—	2	4	no	—
39	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
40	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
41	C₂×C₄²×C₅	—	2	4	no	—
42	C₂×C₄²×C₅	D₄	1	2	yes	—
43	C₂×C₄²×C₅	D₄	1	2	yes	—
44	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₅×#107	—	2	4	no	—
48	C₅×#107	C₂×C₄	1	8	yes	—
49	C₅×#106	—	2	4	no	—
50	C₅×#106	C₂×C₄	1	8	yes	—
51	C₄×C₅×C₈	—	2	2	no	—
52	C₄×C₅×C₈	—	2	2	no	—
53	C₄×C₅×C₈	—	2	2	no	—
54	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
55	C₄×C₅×C₈	Q₈	1	4	yes	—
56	C₄×C₅×C₈	Q₈	1	4	yes	—
57	C₅×#95	—	2	4	no	—
58	C₅×#95	—	2	4	no	—
59	C₅×#95	—	2	4	no	—
60	C₂×#499	C₂×C₅	1	1	yes	characteristic
61	C₄×#125	C₂×C₅	1	2	yes	—
62	#1991	C₂×C₅	1	2	yes	—
63	#1942	C₂×C₅	1	2	yes	—
64	#1941	C₂×C₅	1	2	yes	—
65	#1846	C₂×C₅	1	1	yes	characteristic
66	C₄×#107	C₂×C₅	1	2	yes	—
67	C₄×#106	C₂×C₅	1	2	yes	—
68	C₄²×C₈	C₂×C₅	1	1	yes	characteristic
69	C₂²×C₄×C₅	—	2	4	no	—
70	C₂²×C₄×C₅	—	2	4	no	—
71	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
76	C₂×C₅×C₈	—	2	1	no	—
77	C₂×C₅×C₈	—	2	1	no	—
78	C₂×C₅×C₈	—	2	1	no	—
79	C₂×C₅×C₈	—	2	1	no	—
80	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
81	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
82	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
83	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
84	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
85	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
86	C₂×C₅×C₈	#41	1	2	yes	—
87	C₂×C₅×C₈	#41	1	2	yes	—
88	C₅×C₄⋊C₄	—	2	16	no	—
89	C₅×C₄⋊C₄	—	2	4	no	—
90	C₅×C₄⋊C₄	—	2	4	no	—
91	C₅×C₄⋊C₄	—	2	4	no	—
92	C₅×C₄⋊C₄	—	4	8	no	—
93	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
94	C₄²×C₅	—	2	4	no	—
95	C₄²×C₅	—	2	4	no	—
96	C₄²×C₅	—	2	1	no	—
97	C₄²×C₅	—	2	1	no	—
98	C₄²×C₅	—	4	8	no	—
99	C₄²×C₅	C₂²×C₄	1	2	yes	—
100	C₄²×C₅	C₂×Q₈	1	2	yes	—
101	C₄²×C₅	C₂×Q₈	1	2	yes	—
102	C₄²×C₅	#41	1	2	yes	—
103	C₂×#125	C₂²×C₅	1	2	yes	—
104	#499	C₄×C₅	1	8	yes	direct product factor
105	C₂×#107	C₂²×C₅	1	2	yes	—
106	C₂×#107	C₂²×C₅	1	2	yes	—
107	C₂×#106	C₂²×C₅	1	2	yes	—
108	C₂×#106	C₂²×C₅	1	2	yes	—
109	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
110	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
111	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
112	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
113	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
114	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
115	#389	C₂²×C₅	1	4	yes	—
116	#387	—	2	4	no	—
117	#382	C₂²×C₅	1	2	yes	—
118	C₄×C₄⋊C₄	—	2	4	no	—
119	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
120	C₄³	C₂²×C₅	1	1	yes	characteristic
121	#340	C₂²×C₅	1	4	yes	—
122	#340	C₂²×C₅	1	4	yes	—
123	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	2	2	no	—
129	C₂×C₄×C₅	—	2	8	no	—
130	C₂×C₄×C₅	—	4	4	no	—
131	C₂×C₄×C₅	—	4	4	no	—
132	C₂×C₄×C₅	—	4	4	no	—
133	C₂×C₄×C₅	—	4	4	no	—
134	C₂×C₄×C₅	—	4	4	no	—
135	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
136	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
137	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
138	C₂×C₄×C₅	#135	1	4	yes	—
139	C₂×C₄×C₅	#135	1	4	yes	—
140	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
146	C₅×C₈	—	2	2	no	—
147	C₅×C₈	—	2	2	no	—
148	C₅×C₈	C₄×Q₈	1	4	yes	—
149	C₅×C₈	C₄×Q₈	1	4	yes	—
150	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
151	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
152	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
153	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
154	#125	—	2	4	no	—
155	#125	C₂×C₄×C₅	1	8	yes	—
156	C₂×C₄⋊C₄	—	2	4	no	—
157	C₂×C₄⋊C₄	—	2	4	no	—
158	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
159	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
160	C₂×C₄²	—	2	4	no	—
161	C₂×C₄²	C₅×D₄	1	2	yes	—
162	C₂×C₄²	C₅×D₄	1	2	yes	—
163	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
164	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
165	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
166	#107	—	2	4	no	—
167	#107	C₂×C₄×C₅	1	8	yes	—
168	#106	—	2	4	no	—
169	#106	C₂×C₄×C₅	1	8	yes	—
170	C₄×C₈	—	2	2	no	—
171	C₄×C₈	—	2	2	no	—
172	C₄×C₈	—	2	2	no	—
173	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
174	C₄×C₈	C₅×Q₈	1	4	yes	—
175	C₄×C₈	C₅×Q₈	1	4	yes	—
176	#95	—	2	4	no	—
177	#95	—	2	4	no	—
178	#95	—	2	4	no	—
179	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
180	C₂²×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
181	C₂²×C₅	#532	1	1	yes	characteristic
182	C₂²×C₅	C₂×#135	1	2	yes	—
183	C₂²×C₅	C₂×#135	1	2	yes	—
184	C₄×C₅	—	2	2	no	—
185	C₄×C₅	—	2	4	no	—
186	C₄×C₅	—	2	4	no	—
187	C₄×C₅	—	4	8	no	—
188	C₄×C₅	—	8	4	no	—
189	C₄×C₅	#499	1	4	yes	direct product factor
190	C₄×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
191	C₄×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
192	C₄×C₅	#533	1	1	yes	characteristic
193	C₄×C₅	#533	1	1	yes	characteristic
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
197	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
199	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
200	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
206	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
207	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
208	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
209	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
210	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
211	C₂×C₈	C₅×#41	1	2	yes	—
212	C₂×C₈	C₅×#41	1	2	yes	—
213	C₄⋊C₄	—	2	16	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	4	8	no	—
218	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
219	C₄²	—	2	4	no	—
220	C₄²	—	2	4	no	—
221	C₄²	—	2	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	4	8	no	—
224	C₄²	C₂²×C₄×C₅	1	2	yes	—
225	C₄²	C₂×C₅×Q₈	1	2	yes	—
226	C₄²	C₂×C₅×Q₈	1	2	yes	—
227	C₄²	C₅×#41	1	2	yes	—
228	C₂×C₅	C₄×C₄⋊Q₈	1	1	yes	characteristic
229	C₂×C₅	C₄×#135	1	2	yes	—
230	C₂×C₅	C₂×#499	1	1	yes	characteristic
231	C₂×C₅	#3160	1	1	yes	characteristic
232	C₂×C₅	#3168	1	2	yes	—
233	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	8	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	—	4	4	no	—
245	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
246	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
247	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
248	C₂×C₄	C₅×#135	1	4	yes	—
249	C₂×C₄	C₅×#135	1	4	yes	—
250	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
251	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
252	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
253	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
254	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
255	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
256	C₈	—	2	2	no	—
257	C₈	—	2	2	no	—
258	C₈	C₄×C₅×Q₈	1	4	yes	—
259	C₈	C₄×C₅×Q₈	1	4	yes	—
260	C₅	C₄×#499	1	1	yes	characteristic, direct product factor, Sylow
261	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
262	C₂²	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
263	C₂²	C₅×#532	1	1	yes	characteristic
264	C₂²	C₂×C₅×#135	1	2	yes	—
265	C₂²	C₂×C₅×#135	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	2	4	no	—
268	C₄	—	2	4	no	—
269	C₄	—	4	8	no	—
270	C₄	—	8	4	no	—
271	C₄	C₅×#499	1	4	yes	direct product factor
272	C₄	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
273	C₄	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
274	C₄	C₅×#533	1	1	yes	characteristic
275	C₄	C₅×#533	1	1	yes	characteristic
276	C₂	C₄×C₅×C₄⋊Q₈	1	1	yes	characteristic
277	C₂	C₄×C₅×#135	1	2	yes	—
278	C₂	C₂×C₅×#499	1	1	yes	characteristic
279	C₂	C₅×#3160	1	1	yes	characteristic
280	C₂	C₅×#3168	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, 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
11. C₅×#499	C₄	direct (N × H)
30. C₄×#499	C₅	direct (N × H)
104. #499	C₄×C₅	direct (N × H)
189. C₄×C₅	#499	direct (N × H)
260. C₅	C₄×#499	direct (N × H)
271. C₄	C₅×#499	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×#499	2
C₄×C₅×#125	2
C₅×#1991	2
C₅×#1942	2
C₅×#1941	2
C₅×#1846	2
C₄×C₅×#107	2
C₄×C₅×#106	2
C₄²×C₅×C₈	2
C₄×#499	5

Subgroups — C4×C5×#499 #11944441

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×#499 #11944441