Subgroups — C₅×#8287 #11931007

Order 1280.

398 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#107	C₂	1	1	yes	maximal, characteristic
2	C₅×#1770	C₂	1	2	yes	maximal
3	C₂×C₅×#334	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#322	C₂	1	1	yes	maximal, characteristic
5	C₅×#1279	C₂	1	2	yes	maximal
6	C₂×C₅×#107	C₄	1	2	yes	—
7	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₅×#97	—	2	2	no	—
9	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₅×#340	C₄	1	2	yes	—
14	C₅×#336	—	2	2	no	—
15	C₅×#334	C₄	1	4	yes	complemented
16	C₅×#322	C₄	1	4	yes	complemented
17	#8287	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂²×C₅×C₈	—	2	2	no	—
19	C₂²×C₅×C₈	—	2	1	no	—
20	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₅×C₈	D₄	1	2	yes	—
23	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₅	—	2	2	no	—
25	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
26	C₅×#107	—	2	4	no	—
27	C₅×C₄⋊C₈	—	2	2	no	—
28	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
29	C₅×#97	—	4	4	no	—
30	C₄×C₅×C₈	—	2	1	no	—
31	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
33	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
35	C₄×C₅×C₈	Q₈	1	2	yes	—
36	C₅×#95	—	2	2	no	—
37	C₄×#107	C₂×C₅	1	1	yes	characteristic
38	#1770	C₂×C₅	1	2	yes	—
39	C₂×#334	C₂×C₅	1	1	yes	characteristic
40	C₂×#322	C₂×C₅	1	1	yes	characteristic
41	#1279	C₂×C₅	1	2	yes	—
42	C₂²×C₄×C₅	—	2	4	no	—
43	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅	#41	1	1	yes	characteristic
46	C₂×C₅×C₈	—	2	2	no	—
47	C₂×C₅×C₈	—	2	2	no	—
48	C₂×C₅×C₈	—	2	2	no	—
49	C₂×C₅×C₈	—	2	2	no	—
50	C₂×C₅×C₈	—	2	4	no	—
51	C₂×C₅×C₈	—	4	2	no	—
52	C₂×C₅×C₈	—	4	2	no	—
53	C₂×C₅×C₈	—	4	2	no	—
54	C₂×C₅×C₈	C₄²	1	2	yes	—
55	C₂×C₅×C₈	C₄²	1	2	yes	—
56	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
57	C₂×C₅×C₈	M₁₆	1	4	yes	—
58	C₅×C₄⋊C₄	—	2	4	no	—
59	C₅×C₄⋊C₄	—	2	4	no	—
60	C₄²×C₅	—	4	2	no	—
61	C₄²×C₅	—	4	2	no	—
62	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
65	C₄²×C₅	#41	1	1	yes	characteristic
66	C₂×#107	C₄×C₅	1	2	yes	—
67	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
68	C₂×#97	—	2	2	no	—
69	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
70	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
72	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
73	#340	C₄×C₅	1	2	yes	—
74	#336	—	2	2	no	—
75	#334	C₄×C₅	1	4	yes	complemented
76	#322	C₄×C₅	1	4	yes	complemented
77	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
78	C₂×C₄×C₅	—	2	2	no	—
79	C₂×C₄×C₅	—	4	4	no	—
80	C₂×C₄×C₅	—	4	4	no	—
81	C₂×C₄×C₅	—	4	4	no	—
82	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
83	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅	#117	1	1	yes	characteristic
85	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
87	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
88	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
89	C₂×C₄×C₅	#131	1	2	yes	—
90	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
91	C₂×C₄×C₅	#137	1	2	yes	—
92	C₅×C₈	—	2	2	no	—
93	C₅×C₈	—	2	4	no	—
94	C₅×C₈	—	4	2	no	—
95	C₅×C₈	—	4	4	no	—
96	C₅×C₈	#97	1	4	yes	complemented
97	C₂²×C₈	—	2	2	no	—
98	C₂²×C₈	—	2	1	no	—
99	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
100	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
101	C₂²×C₈	C₅×D₄	1	2	yes	—
102	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
103	C₂×C₄²	—	2	2	no	—
104	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
105	#107	—	2	4	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	C₂×C₄×C₅	1	4	yes	—
108	#97	—	4	4	no	—
109	C₄×C₈	—	2	1	no	—
110	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
111	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
112	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
113	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
114	C₄×C₈	C₅×Q₈	1	2	yes	—
115	#95	—	2	2	no	—
116	C₂²×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
117	C₂²×C₅	#436	1	2	yes	—
118	C₂²×C₅	#441	1	1	yes	characteristic
119	C₂²×C₅	#442	1	1	yes	characteristic
120	C₂²×C₅	#444	1	2	yes	—
121	C₄×C₅	—	2	1	no	—
122	C₄×C₅	—	2	1	no	—
123	C₄×C₅	—	8	4	no	—
124	C₄×C₅	C₂×#97	1	1	yes	characteristic
125	C₄×C₅	#405	1	1	yes	characteristic
126	C₄×C₅	#428	1	2	yes	—
127	C₄×C₅	#440	1	2	yes	—
128	C₄×C₅	#446	1	2	yes	—
129	C₂²×C₄	—	2	4	no	—
130	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
131	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
132	C₂²×C₄	C₅×#41	1	1	yes	characteristic
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	C₄²×C₅	1	2	yes	—
142	C₂×C₈	C₄²×C₅	1	2	yes	—
143	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
144	C₂×C₈	C₅×M₁₆	1	4	yes	—
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄²	—	4	2	no	—
148	C₄²	—	4	2	no	—
149	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
150	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
151	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
152	C₄²	C₅×#41	1	1	yes	characteristic
153	C₂×C₅	#1593	1	2	yes	—
154	C₂×C₅	#1594	1	2	yes	—
155	C₂×C₅	#1770	1	1	yes	characteristic
156	C₂×C₅	#1773	1	1	yes	characteristic
157	C₂×C₅	#1776	1	1	yes	characteristic
158	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
164	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₄	C₅×#117	1	1	yes	characteristic
166	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
167	C₂×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
168	C₂×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
169	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
170	C₂×C₄	C₅×#131	1	2	yes	—
171	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
172	C₂×C₄	C₅×#137	1	2	yes	—
173	C₈	—	2	2	no	—
174	C₈	—	2	4	no	—
175	C₈	—	4	2	no	—
176	C₈	—	4	4	no	—
177	C₈	C₅×#97	1	4	yes	complemented
178	C₅	#8287	1	1	yes	characteristic, direct product factor, Sylow
179	C₂²	C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
180	C₂²	C₅×#436	1	2	yes	—
181	C₂²	C₅×#441	1	1	yes	characteristic
182	C₂²	C₅×#442	1	1	yes	characteristic
183	C₂²	C₅×#444	1	2	yes	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	8	4	no	—
187	C₄	C₂×C₅×#97	1	1	yes	characteristic
188	C₄	C₅×#405	1	1	yes	characteristic
189	C₄	C₅×#428	1	2	yes	—
190	C₄	C₅×#440	1	2	yes	—
191	C₄	C₅×#446	1	2	yes	—
192	C₂	C₅×#1593	1	2	yes	—
193	C₂	C₅×#1594	1	2	yes	—
194	C₂	C₅×#1770	1	1	yes	characteristic
195	C₂	C₅×#1773	1	1	yes	characteristic
196	C₂	C₅×#1776	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle, prime radical	C₂³×C₅	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₅×#334	C₄	semidirect
16. C₅×#322	C₄	semidirect
17. #8287	C₅	direct (N × H)
75. #334	C₄×C₅	semidirect
76. #322	C₄×C₅	semidirect
96. C₅×C₈	#97	semidirect
177. C₈	C₅×#97	semidirect
178. C₅	#8287	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₅×#107	2
C₅×#1770	2
C₂×C₅×#334	2
C₂×C₅×#322	2
C₅×#1279	2
#8287	5

Subgroups — C5×#8287 #11931007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8287 #11931007