Subgroups — C₅×#20538 #11943258

Order 1280.

922 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#431	C₂	1	2	yes	maximal
2	C₅×#2285	C₂	1	1	yes	maximal, characteristic
3	C₅×#1770	C₂	1	4	yes	maximal, complemented
4	C₅×C₈×C₄⋊C₄	C₂	1	4	yes	maximal, complemented
5	C₅×#1750	C₂	1	4	yes	maximal
6	C₂×C₅×#117	C₂²	1	2	yes	—
7	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₅×#431	C₄	1	16	yes	complemented
9	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₅×C₂²⋊C₈	C₂²	1	4	yes	—
12	C₂×C₅×#97	C₂²	1	4	yes	—
13	C₂×C₄×C₅×C₈	C₂²	1	4	yes	—
14	C₂×C₄×C₅×C₈	C₂²	1	4	yes	—
15	C₅×#384	C₄	1	2	yes	—
16	C₅×#382	C₄	1	2	yes	—
17	C₅×#380	C₄	1	4	yes	—
18	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
19	C₅×#336	C₂²	1	8	yes	—
20	#20538	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂²×C₅×C₈	C₂³	1	4	yes	—
24	C₂²×C₅×C₈	C₂³	1	4	yes	—
25	C₅×#117	—	4	2	no	—
26	C₅×#117	—	4	2	no	—
27	C₂×C₅×C₄⋊C₄	C₂×C₄	1	8	yes	—
28	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
29	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅	C₂×C₄	1	8	yes	—
33	C₂×C₄²×C₅	C₂³	1	4	yes	—
34	C₂×C₄²×C₅	C₂³	1	2	yes	—
35	C₅×C₄⋊C₈	C₂×C₄	1	8	yes	—
36	C₅×C₄⋊C₈	C₂×C₄	1	8	yes	—
37	C₅×C₂²⋊C₈	C₂×C₄	1	16	yes	—
38	C₅×#97	—	4	4	no	—
39	C₄×C₅×C₈	—	4	4	no	—
40	C₄×C₅×C₈	C₂×C₄	1	16	yes	—
41	C₅×#95	C₂×C₄	1	8	yes	—
42	C₂×#431	C₂×C₅	1	2	yes	—
43	#2285	C₂×C₅	1	1	yes	characteristic
44	#1770	C₂×C₅	1	4	yes	complemented
45	C₈×C₄⋊C₄	C₂×C₅	1	4	yes	complemented
46	#1750	C₂×C₅	1	4	yes	—
47	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	—	2	8	no	—
49	C₂²×C₄×C₅	—	2	1	no	—
50	C₂²×C₄×C₅	—	2	1	no	—
51	C₂²×C₄×C₅	—	2	2	no	—
52	C₂²×C₄×C₅	C₄²	1	8	yes	—
53	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
54	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
59	C₂×C₅×C₈	—	2	4	no	—
60	C₂×C₅×C₈	—	2	4	no	—
61	C₂×C₅×C₈	—	2	4	no	—
62	C₂×C₅×C₈	—	2	4	no	—
63	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
64	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
65	C₅×C₄⋊C₄	—	2	16	no	—
66	C₅×C₄⋊C₄	—	4	4	no	—
67	C₅×C₂²⋊C₄	—	4	4	no	—
68	C₄²×C₅	—	2	4	no	—
69	C₄²×C₅	—	2	4	no	—
70	C₄²×C₅	C₄²	1	4	yes	—
71	C₄²×C₅	C₄²	1	4	yes	—
72	C₄²×C₅	C₂²×C₄	1	4	yes	—
73	C₄²×C₅	C₂²×C₄	1	4	yes	—
74	C₂×#117	C₂²×C₅	1	2	yes	—
75	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
76	#431	C₄×C₅	1	16	yes	complemented
77	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
78	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
79	C₂×C₂²⋊C₈	C₂²×C₅	1	4	yes	—
80	C₂×#97	C₂²×C₅	1	4	yes	—
81	C₂×C₄×C₈	C₂²×C₅	1	4	yes	—
82	C₂×C₄×C₈	C₂²×C₅	1	4	yes	—
83	#384	C₄×C₅	1	2	yes	—
84	#382	C₄×C₅	1	2	yes	—
85	#380	C₄×C₅	1	4	yes	—
86	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
87	#336	C₂²×C₅	1	8	yes	—
88	C₂³×C₅	—	2	1	no	—
89	C₂³×C₅	—	2	1	no	—
90	C₂³×C₅	—	2	4	no	—
91	C₂³×C₅	C₂×C₄²	1	2	yes	—
92	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₅	—	2	1	no	—
94	C₂×C₄×C₅	—	2	1	no	—
95	C₂×C₄×C₅	—	2	4	no	—
96	C₂×C₄×C₅	—	2	4	no	—
97	C₂×C₄×C₅	—	2	4	no	—
98	C₂×C₄×C₅	—	2	16	no	—
99	C₂×C₄×C₅	—	2	8	no	—
100	C₂×C₄×C₅	—	2	16	no	—
101	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
103	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
104	C₂×C₄×C₅	#131	1	8	yes	—
105	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
106	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
107	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
108	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
109	C₅×C₈	—	2	8	no	—
110	C₅×C₈	—	2	8	no	—
111	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
112	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
113	C₂²×C₈	C₂³×C₅	1	4	yes	—
114	C₂²×C₈	C₂³×C₅	1	4	yes	—
115	#117	—	4	2	no	—
116	#117	—	4	2	no	—
117	C₂×C₄⋊C₄	C₂×C₄×C₅	1	8	yes	—
118	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
119	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	4	yes	—
120	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
121	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
122	C₂×C₄²	C₂×C₄×C₅	1	8	yes	—
123	C₂×C₄²	C₂³×C₅	1	4	yes	—
124	C₂×C₄²	C₂³×C₅	1	2	yes	—
125	C₄⋊C₈	C₂×C₄×C₅	1	8	yes	—
126	C₄⋊C₈	C₂×C₄×C₅	1	8	yes	—
127	C₂²⋊C₈	C₂×C₄×C₅	1	16	yes	—
128	#97	—	4	4	no	—
129	C₄×C₈	—	4	4	no	—
130	C₄×C₈	C₂×C₄×C₅	1	16	yes	—
131	#95	C₂×C₄×C₅	1	8	yes	—
132	C₂²×C₅	—	2	4	no	—
133	C₂²×C₅	—	2	2	no	—
134	C₂²×C₅	—	4	2	no	—
135	C₂²×C₅	—	4	2	no	—
136	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
137	C₂²×C₅	#518	1	1	yes	characteristic
138	C₂²×C₅	#519	1	1	yes	characteristic
139	C₂²×C₅	C₂×#131	1	4	yes	—
140	C₄×C₅	—	2	4	no	—
141	C₄×C₅	—	2	2	no	—
142	C₄×C₅	—	4	8	no	—
143	C₄×C₅	#405	1	8	yes	—
144	C₄×C₅	#520	1	2	yes	—
145	C₄×C₅	#520	1	2	yes	—
146	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
147	C₂²×C₄	—	2	8	no	—
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	C₄²×C₅	1	8	yes	—
152	C₂²×C₄	C₂²×C₄×C₅	1	8	yes	—
153	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
154	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
155	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
156	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
157	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	C₂²×C₄×C₅	1	8	yes	—
163	C₂×C₈	C₂²×C₄×C₅	1	8	yes	—
164	C₄⋊C₄	—	2	16	no	—
165	C₄⋊C₄	—	4	4	no	—
166	C₂²⋊C₄	—	4	4	no	—
167	C₄²	—	2	4	no	—
168	C₄²	—	2	4	no	—
169	C₄²	C₄²×C₅	1	4	yes	—
170	C₄²	C₄²×C₅	1	4	yes	—
171	C₄²	C₂²×C₄×C₅	1	4	yes	—
172	C₄²	C₂²×C₄×C₅	1	4	yes	—
173	C₂×C₅	—	4	2	no	—
174	C₂×C₅	#2274	1	1	yes	characteristic
175	C₂×C₅	C₂×#405	1	2	yes	—
176	C₂×C₅	#2979	1	2	yes	—
177	C₂×C₅	#2986	1	2	yes	—
178	C₂³	—	2	1	no	—
179	C₂³	—	2	1	no	—
180	C₂³	—	2	4	no	—
181	C₂³	C₂×C₄²×C₅	1	2	yes	—
182	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	16	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	8	no	—
190	C₂×C₄	—	2	16	no	—
191	C₂×C₄	C₂×C₄²×C₅	1	2	yes	—
192	C₂×C₄	C₂×C₄²×C₅	1	4	yes	—
193	C₂×C₄	C₂×C₄²×C₅	1	4	yes	—
194	C₂×C₄	C₅×#131	1	8	yes	—
195	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
196	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
197	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
198	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
199	C₈	—	2	8	no	—
200	C₈	—	2	8	no	—
201	C₅	#20538	1	1	yes	characteristic, direct product factor, Sylow
202	C₂²	—	2	2	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	C₂²×C₄²×C₅	1	1	yes	characteristic
207	C₂²	C₅×#518	1	1	yes	characteristic
208	C₂²	C₅×#519	1	1	yes	characteristic
209	C₂²	C₂×C₅×#131	1	4	yes	—
210	C₄	—	2	2	no	—
211	C₄	—	2	4	no	—
212	C₄	—	4	8	no	—
213	C₄	C₅×#405	1	8	yes	—
214	C₄	C₅×#520	1	2	yes	—
215	C₄	C₅×#520	1	2	yes	—
216	C₂	—	4	2	no	—
217	C₂	C₅×#2274	1	1	yes	characteristic
218	C₂	C₂×C₅×#405	1	2	yes	—
219	C₂	C₅×#2979	1	2	yes	—
220	C₂	C₅×#2986	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂⁴
commutator	C₂²	C₂²×C₄²×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂³×C₅	C₂³×C₄
prime radical	C₂⁴×C₅	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1770	C₂	semidirect
4. C₅×C₈×C₄⋊C₄	C₂	semidirect
8. C₅×#431	C₄	semidirect
20. #20538	C₅	direct (N × H)
44. #1770	C₂×C₅	semidirect
45. C₈×C₄⋊C₄	C₂×C₅	semidirect
76. #431	C₄×C₅	semidirect
201. C₅	#20538	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×#431	2
C₅×#2285	2
C₅×#1770	2
C₅×C₈×C₄⋊C₄	2
C₅×#1750	2
#20538	5

Subgroups — C5×#20538 #11943258

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#20538 #11943258