Subgroups — C₅×#82013 #14478497

Order 1920.

740 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5752	C₂	1	2	yes	maximal, complemented
2	C₃×C₅×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4908	C₂	1	6	yes	maximal, complemented
4	C₅×#4904	C₂	1	3	yes	maximal, complemented
5	C₂×C₅×D₄₈	C₂	1	3	yes	maximal, complemented
6	C₅×#3415	—	3	1	no	maximal, Hall
7	C₅×#1009	—	2	2	no	—
8	C₃×C₅×#131	C₂²	1	1	yes	characteristic
9	C₅×#908	—	2	6	no	—
10	C₅×#904	C₂²	1	6	yes	complemented
11	C₂×C₅×D₂₄	C₂²	1	6	yes	complemented
12	C₃×C₅×M₃₂	C₂²	1	3	yes	complemented
13	C₂×C₃×C₅×C₁₆	C₂²	1	3	yes	complemented
14	C₅×Dic₂₄	C₂²	1	1	yes	characteristic, complemented
15	C₅×#800	C₂²	1	6	yes	complemented
16	C₅×D₄₈	C₂²	1	3	yes	complemented
17	C₅×D₄₈	C₂²	1	6	yes	complemented
18	#82013	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#576	—	3	2	no	—
20	C₅×#509	—	3	6	no	—
21	C₅×#508	—	3	3	no	—
22	C₂×C₅×D₁₆	—	3	3	no	—
23	C₅×#504	D₃	1	1	yes	characteristic, complemented
24	C₃×C₅×#41	D₄	1	1	yes	characteristic
25	C₅×#239	—	4	2	no	—
26	C₅×D₃×D₄	—	4	6	no	—
27	C₅×#235	—	2	6	no	—
28	C₂×C₅×D₁₂	—	2	6	no	—
29	C₃×C₅×M₁₆	D₄	1	3	yes	—
30	C₂×C₃×C₅×C₈	C₂³	1	3	yes	—
31	C₅×Dic₁₂	C₂³	1	2	yes	—
32	C₅×D₂₄	—	2	6	no	—
33	C₅×D₂₄	C₂³	1	6	yes	—
34	C₅×#204	—	2	6	no	—
35	C₃×C₅×C₁₆	C₂³	1	1	yes	characteristic
36	C₃×C₅×C₁₆	C₂³	1	3	yes	—
37	#5752	C₂×C₅	1	2	yes	complemented
38	C₃×#504	C₂×C₅	1	1	yes	characteristic, complemented
39	#4908	C₂×C₅	1	6	yes	complemented
40	#4904	C₂×C₅	1	3	yes	complemented
41	C₂×D₄₈	C₂×C₅	1	3	yes	complemented
42	C₅×H(5,𝔽₂)	—	6	2	no	—
43	C₅×AGL(1,ℤ/8)	—	6	6	no	—
44	C₅×#135	—	3	6	no	—
45	C₂×C₅×D₈	—	3	6	no	—
46	C₅×#131	C₂×D₃	1	1	yes	characteristic
47	C₅×Q₃₂	—	3	1	no	—
48	C₅×SD₃₂	—	3	6	no	—
49	C₅×D₁₆	—	3	3	no	—
50	C₅×D₁₆	—	3	6	no	—
51	C₅×M₃₂	C₂×D₃	1	3	yes	complemented
52	C₂×C₅×C₁₆	C₂×D₃	1	3	yes	complemented
53	#3415	—	3	1	no	Sylow
54	C₂²×C₅×D₃	—	4	6	no	—
55	C₃×C₅×Q₈	D₈	1	1	yes	characteristic
56	C₃×C₅×D₄	D₈	1	3	yes	—
57	C₂×C₃×C₄×C₅	C₂×D₄	1	3	yes	—
58	C₅×#67	—	4	6	no	—
59	C₅×D₁₂	—	2	6	no	—
60	C₅×D₁₂	—	4	6	no	—
61	C₄×C₅×D₃	—	4	6	no	—
62	C₅×Dic₆	—	2	2	no	—
63	C₃×C₅×C₈	C₂×D₄	1	3	yes	—
64	C₃×C₅×C₈	C₂⁴	1	1	yes	characteristic
65	#1009	—	2	2	no	—
66	C₃×#131	C₂²×C₅	1	1	yes	characteristic
67	#908	—	2	6	no	—
68	#904	C₂²×C₅	1	6	yes	complemented
69	C₂×D₂₄	C₂²×C₅	1	6	yes	complemented
70	C₃×M₃₂	C₂²×C₅	1	3	yes	complemented
71	C₂×C₃×C₁₆	C₂²×C₅	1	3	yes	complemented
72	Dic₂₄	C₂²×C₅	1	1	yes	characteristic, complemented
73	#800	C₂²×C₅	1	6	yes	complemented
74	D₄₈	C₂²×C₅	1	3	yes	complemented
75	D₄₈	C₂²×C₅	1	6	yes	complemented
76	C₅×#41	—	6	6	no	—
77	C₅×#41	—	12	2	no	—
78	C₅×#41	D₁₂	1	1	yes	characteristic
79	C₂×C₅×D₄	—	6	6	no	—
80	C₂×C₅×D₄	—	12	6	no	—
81	C₅×Q₁₆	—	3	2	no	—
82	C₅×SD₁₆	—	6	6	no	—
83	C₅×D₈	—	3	6	no	—
84	C₅×D₈	—	6	6	no	—
85	C₅×M₁₆	D₁₂	1	3	yes	—
86	C₂×C₅×C₈	C₂²×D₃	1	3	yes	—
87	C₅×C₁₆	C₂²×D₃	1	1	yes	characteristic
88	C₅×C₁₆	C₂²×D₃	1	3	yes	—
89	#576	—	3	2	no	—
90	#509	—	3	6	no	—
91	#508	—	3	3	no	—
92	C₂×D₁₆	—	3	3	no	—
93	#504	C₅×D₃	1	1	yes	characteristic, complemented
94	C₂²×C₃×C₅	C₂×D₈	1	3	yes	—
95	C₂×C₅×D₃	—	4	6	no	—
96	C₂×C₅×D₃	—	16	6	no	—
97	C₃×C₄×C₅	C₂×D₈	1	3	yes	—
98	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
99	C₅×Dic₃	—	4	2	no	—
100	C₃×#41	C₅×D₄	1	1	yes	characteristic
101	#239	—	4	2	no	—
102	D₃×D₄	—	4	6	no	—
103	#235	—	2	6	no	—
104	C₂×D₁₂	—	2	6	no	—
105	C₃×M₁₆	C₅×D₄	1	3	yes	—
106	C₂×C₃×C₈	C₂³×C₅	1	3	yes	—
107	Dic₁₂	C₂³×C₅	1	2	yes	—
108	D₂₄	—	2	6	no	—
109	D₂₄	C₂³×C₅	1	6	yes	—
110	#204	—	2	6	no	—
111	C₃×C₁₆	C₂³×C₅	1	1	yes	characteristic
112	C₃×C₁₆	C₂³×C₅	1	3	yes	—
113	C₂³×C₅	—	12	6	no	—
114	C₅×Q₈	—	6	2	no	—
115	C₅×Q₈	D₂₄	1	1	yes	characteristic
116	C₅×D₄	—	6	6	no	—
117	C₅×D₄	—	12	6	no	—
118	C₅×D₄	—	12	6	no	—
119	C₅×D₄	D₂₄	1	3	yes	—
120	C₂×C₄×C₅	—	12	6	no	—
121	C₂×C₄×C₅	C₂×D₁₂	1	3	yes	—
122	C₅×C₈	C₂×D₁₂	1	3	yes	—
123	C₅×C₈	C₂³×D₃	1	1	yes	characteristic
124	H(5,𝔽₂)	—	6	2	no	—
125	AGL(1,ℤ/8)	—	6	6	no	—
126	#135	—	3	6	no	—
127	C₂×D₈	—	3	6	no	—
128	#131	C₂×C₅×D₃	1	1	yes	characteristic
129	Q₃₂	—	3	1	no	—
130	SD₃₂	—	3	6	no	—
131	D₁₆	—	3	3	no	—
132	D₁₆	—	3	6	no	—
133	M₃₂	C₂×C₅×D₃	1	3	yes	complemented
134	C₂×C₁₆	C₂×C₅×D₃	1	3	yes	complemented
135	C₂×C₃×C₅	—	2	3	no	—
136	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
137	C₅×D₃	—	8	6	no	—
138	C₂²×D₃	—	4	6	no	—
139	C₃×Q₈	C₅×D₈	1	1	yes	characteristic
140	C₃×D₄	C₅×D₈	1	3	yes	—
141	C₂×C₃×C₄	C₂×C₅×D₄	1	3	yes	—
142	#67	—	4	6	no	—
143	D₁₂	—	2	6	no	—
144	D₁₂	—	4	6	no	—
145	C₄×D₃	—	4	6	no	—
146	Dic₆	—	2	2	no	—
147	C₃×C₈	C₂×C₅×D₄	1	3	yes	—
148	C₃×C₈	C₂⁴×C₅	1	1	yes	characteristic
149	C₂²×C₅	—	12	6	no	—
150	C₂²×C₅	—	48	6	no	—
151	C₂²×C₅	C₂×D₂₄	1	3	yes	—
152	C₄×C₅	—	12	2	no	—
153	C₄×C₅	C₂×D₂₄	1	3	yes	—
154	C₄×C₅	C₂²×D₁₂	1	1	yes	characteristic
155	#41	—	6	6	no	—
156	#41	—	12	2	no	—
157	#41	C₅×D₁₂	1	1	yes	characteristic
158	C₂×D₄	—	6	6	no	—
159	C₂×D₄	—	12	6	no	—
160	Q₁₆	—	3	2	no	—
161	SD₁₆	—	6	6	no	—
162	D₈	—	3	6	no	—
163	D₈	—	6	6	no	—
164	M₁₆	C₅×D₁₂	1	3	yes	—
165	C₂×C₈	C₂²×C₅×D₃	1	3	yes	—
166	C₁₆	C₂²×C₅×D₃	1	1	yes	characteristic
167	C₁₆	C₂²×C₅×D₃	1	3	yes	—
168	C₃×C₅	#3415	1	1	yes	characteristic, complemented, Hall
169	C₂²×C₃	C₂×C₅×D₈	1	3	yes	—
170	C₂×D₃	—	4	6	no	—
171	C₂×D₃	—	16	6	no	—
172	C₃×C₄	C₂×C₅×D₈	1	3	yes	—
173	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
174	Dic₃	—	4	2	no	—
175	C₂×C₅	—	2	3	no	—
176	C₂×C₅	—	24	6	no	—
177	C₂×C₅	C₂²×D₂₄	1	1	yes	characteristic
178	C₂³	—	12	6	no	—
179	Q₈	—	6	2	no	—
180	Q₈	C₅×D₂₄	1	1	yes	characteristic
181	D₄	—	6	6	no	—
182	D₄	—	12	6	no	—
183	D₄	—	12	6	no	—
184	D₄	C₅×D₂₄	1	3	yes	—
185	C₂×C₄	—	12	6	no	—
186	C₂×C₄	C₂×C₅×D₁₂	1	3	yes	—
187	C₈	C₂×C₅×D₁₂	1	3	yes	—
188	C₈	C₂³×C₅×D₃	1	1	yes	characteristic
189	C₂×C₃	—	2	3	no	—
190	C₂×C₃	C₂²×C₅×D₈	1	1	yes	characteristic
191	D₃	—	8	6	no	—
192	C₅	#82013	1	1	yes	characteristic, direct product factor, Sylow
193	C₂²	—	12	6	no	—
194	C₂²	—	48	6	no	—
195	C₂²	C₂×C₅×D₂₄	1	3	yes	—
196	C₄	—	12	2	no	—
197	C₄	C₂×C₅×D₂₄	1	3	yes	—
198	C₄	C₂²×C₅×D₁₂	1	1	yes	characteristic
199	C₃	C₅×#3415	1	1	yes	characteristic, complemented, Sylow
200	C₂	—	2	3	no	—
201	C₂	—	24	6	no	—
202	C₂	C₂²×C₅×D₂₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	C₂²×D₂₄
commutator, cosocle	C₃×C₈	C₂⁴×C₅
Frattini	C₈	C₂³×C₅×D₃
Fitting	C₃×C₅×#504	C₂
socle	C₂×C₃×C₅	C₂²×D₈
hypercenter	C₅×#504	D₃
hypocenter	C₃	C₅×#3415
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₅×#41 ◁ C₅×#131 ◁ C₅×#504

Complements

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

N	H ≅ G/N	type
1. C₅×#5752	C₂	semidirect
2. C₃×C₅×#504	C₂	semidirect
3. C₅×#4908	C₂	semidirect
4. C₅×#4904	C₂	semidirect
5. C₂×C₅×D₄₈	C₂	semidirect
10. C₅×#904	C₂²	semidirect
11. C₂×C₅×D₂₄	C₂²	semidirect
12. C₃×C₅×M₃₂	C₂²	semidirect
13. C₂×C₃×C₅×C₁₆	C₂²	semidirect
14. C₅×Dic₂₄	C₂²	semidirect
15. C₅×#800	C₂²	semidirect
16. C₅×D₄₈	C₂²	semidirect
17. C₅×D₄₈	C₂²	semidirect
18. #82013	C₅	direct (N × H)
23. C₅×#504	D₃	semidirect
37. #5752	C₂×C₅	semidirect
38. C₃×#504	C₂×C₅	semidirect
39. #4908	C₂×C₅	semidirect
40. #4904	C₂×C₅	semidirect
41. C₂×D₄₈	C₂×C₅	semidirect
51. C₅×M₃₂	C₂×D₃	semidirect
52. C₂×C₅×C₁₆	C₂×D₃	semidirect
68. #904	C₂²×C₅	semidirect
69. C₂×D₂₄	C₂²×C₅	semidirect
70. C₃×M₃₂	C₂²×C₅	semidirect
71. C₂×C₃×C₁₆	C₂²×C₅	semidirect
72. Dic₂₄	C₂²×C₅	semidirect
73. #800	C₂²×C₅	semidirect
74. D₄₈	C₂²×C₅	semidirect
75. D₄₈	C₂²×C₅	semidirect
93. #504	C₅×D₃	semidirect
133. M₃₂	C₂×C₅×D₃	semidirect
134. C₂×C₁₆	C₂×C₅×D₃	semidirect
168. C₃×C₅	#3415	semidirect
192. C₅	#82013	direct (N × H)
199. C₃	C₅×#3415	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5752	2
C₃×C₅×#504	2
C₅×#4908	2
C₅×#4904	2
C₂×C₅×D₄₈	2
C₅×#3415	3
#82013	5

Subgroups — C5×#82013 #14478497

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#82013 #14478497