Subgroups — C₅×#82015 #14478625

Order 1920.

700 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5753	C₂	1	2	yes	maximal, complemented
2	C₃×C₅×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4909	C₂	1	6	yes	maximal, complemented
4	C₂×C₅×Dic₂₄	C₂	1	3	yes	maximal, complemented
5	C₅×#4904	C₂	1	3	yes	maximal, complemented
6	C₅×#3417	—	3	1	no	maximal, Hall
7	C₅×#1010	—	2	2	no	—
8	C₃×C₅×#131	C₂²	1	1	yes	characteristic
9	C₅×#909	—	2	6	no	—
10	C₂×C₅×Dic₁₂	C₂²	1	6	yes	—
11	C₅×#904	C₂²	1	6	yes	—
12	C₃×C₅×M₃₂	C₂²	1	3	yes	—
13	C₂×C₃×C₅×C₁₆	C₂²	1	3	yes	—
14	C₅×Dic₂₄	C₂²	1	3	yes	—
15	C₅×Dic₂₄	C₂²	1	6	yes	—
16	C₅×#800	C₂²	1	6	yes	—
17	C₅×D₄₈	C₂²	1	1	yes	characteristic
18	#82015	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#578	—	3	2	no	—
20	C₅×#510	—	3	6	no	—
21	C₅×#508	—	3	3	no	—
22	C₂×C₅×Q₃₂	—	3	3	no	—
23	C₅×#504	D₃	1	1	yes	characteristic, complemented
24	C₃×C₅×#41	D₄	1	1	yes	characteristic
25	C₅×D₃×Q₈	—	4	2	no	—
26	C₅×#237	—	4	6	no	—
27	C₅×#235	—	2	6	no	—
28	C₂×C₅×Dic₆	—	2	6	no	—
29	C₃×C₅×M₁₆	D₄	1	3	yes	—
30	C₂×C₃×C₅×C₈	C₂³	1	3	yes	—
31	C₅×Dic₁₂	—	2	6	no	—
32	C₅×Dic₁₂	C₂³	1	6	yes	—
33	C₅×D₂₄	C₂³	1	2	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	#5753	C₂×C₅	1	2	yes	complemented
38	C₃×#504	C₂×C₅	1	1	yes	characteristic, complemented
39	#4909	C₂×C₅	1	6	yes	complemented
40	C₂×Dic₂₄	C₂×C₅	1	3	yes	complemented
41	#4904	C₂×C₅	1	3	yes	complemented
42	C₅×2₋¹⁺⁴	—	6	2	no	—
43	C₅×#137	—	6	6	no	—
44	C₅×#135	—	3	6	no	—
45	C₂×C₅×Q₁₆	—	3	6	no	—
46	C₅×#131	C₂×D₃	1	1	yes	characteristic
47	C₅×Q₃₂	—	3	3	no	—
48	C₅×Q₃₂	—	3	6	no	—
49	C₅×SD₃₂	—	3	6	no	—
50	C₅×D₁₆	—	3	1	no	—
51	C₅×M₃₂	C₂×D₃	1	3	yes	—
52	C₂×C₅×C₁₆	C₂×D₃	1	3	yes	—
53	#3417	—	3	1	no	Sylow
54	C₃×C₅×Q₈	D₈	1	1	yes	characteristic
55	C₃×C₅×D₄	D₈	1	3	yes	—
56	C₂×C₃×C₄×C₅	C₂×D₄	1	3	yes	—
57	C₅×#67	—	4	6	no	—
58	C₂×C₅×Dic₃	—	4	6	no	—
59	C₅×D₁₂	—	2	2	no	—
60	C₄×C₅×D₃	—	4	6	no	—
61	C₅×Dic₆	—	2	6	no	—
62	C₅×Dic₆	—	4	6	no	—
63	C₃×C₅×C₈	C₂×D₄	1	3	yes	—
64	C₃×C₅×C₈	C₂⁴	1	1	yes	characteristic
65	#1010	—	2	2	no	—
66	C₃×#131	C₂²×C₅	1	1	yes	characteristic
67	#909	—	2	6	no	—
68	C₂×Dic₁₂	C₂²×C₅	1	6	yes	—
69	#904	C₂²×C₅	1	6	yes	—
70	C₃×M₃₂	C₂²×C₅	1	3	yes	—
71	C₂×C₃×C₁₆	C₂²×C₅	1	3	yes	—
72	Dic₂₄	C₂²×C₅	1	3	yes	—
73	Dic₂₄	C₂²×C₅	1	6	yes	—
74	#800	C₂²×C₅	1	6	yes	—
75	D₄₈	C₂²×C₅	1	1	yes	characteristic
76	C₅×#41	—	6	6	no	—
77	C₅×#41	—	12	6	no	—
78	C₅×#41	D₁₂	1	1	yes	characteristic
79	C₂×C₅×Q₈	—	6	6	no	—
80	C₂×C₅×Q₈	—	12	2	no	—
81	C₅×Q₁₆	—	3	6	no	—
82	C₅×Q₁₆	—	6	6	no	—
83	C₅×SD₁₆	—	6	6	no	—
84	C₅×D₈	—	3	2	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	#578	—	3	2	no	—
90	#510	—	3	6	no	—
91	#508	—	3	3	no	—
92	C₂×Q₃₂	—	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	2	no	—
96	C₃×C₄×C₅	C₂×D₈	1	3	yes	—
97	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
98	C₅×Dic₃	—	4	6	no	—
99	C₃×#41	C₅×D₄	1	1	yes	characteristic
100	D₃×Q₈	—	4	2	no	—
101	#237	—	4	6	no	—
102	#235	—	2	6	no	—
103	C₂×Dic₆	—	2	6	no	—
104	C₃×M₁₆	C₅×D₄	1	3	yes	—
105	C₂×C₃×C₈	C₂³×C₅	1	3	yes	—
106	Dic₁₂	—	2	6	no	—
107	Dic₁₂	C₂³×C₅	1	6	yes	—
108	D₂₄	C₂³×C₅	1	2	yes	—
109	#204	—	2	6	no	—
110	C₃×C₁₆	C₂³×C₅	1	1	yes	characteristic
111	C₃×C₁₆	C₂³×C₅	1	3	yes	—
112	C₅×Q₈	—	6	6	no	—
113	C₅×Q₈	—	12	6	no	—
114	C₅×Q₈	D₂₄	1	1	yes	characteristic
115	C₅×D₄	—	6	2	no	—
116	C₅×D₄	—	12	6	no	—
117	C₅×D₄	D₂₄	1	3	yes	—
118	C₂×C₄×C₅	—	12	6	no	—
119	C₂×C₄×C₅	—	12	6	no	—
120	C₂×C₄×C₅	C₂×D₁₂	1	3	yes	—
121	C₅×C₈	C₂×D₁₂	1	3	yes	—
122	C₅×C₈	C₂³×D₃	1	1	yes	characteristic
123	2₋¹⁺⁴	—	6	2	no	—
124	#137	—	6	6	no	—
125	#135	—	3	6	no	—
126	C₂×Q₁₆	—	3	6	no	—
127	#131	C₂×C₅×D₃	1	1	yes	characteristic
128	Q₃₂	—	3	3	no	—
129	Q₃₂	—	3	6	no	—
130	SD₃₂	—	3	6	no	—
131	D₁₆	—	3	1	no	—
132	M₃₂	C₂×C₅×D₃	1	3	yes	—
133	C₂×C₁₆	C₂×C₅×D₃	1	3	yes	—
134	C₂×C₃×C₅	—	2	3	no	—
135	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
136	C₅×D₃	—	8	2	no	—
137	C₃×Q₈	C₅×D₈	1	1	yes	characteristic
138	C₃×D₄	C₅×D₈	1	3	yes	—
139	C₂×C₃×C₄	C₂×C₅×D₄	1	3	yes	—
140	#67	—	4	6	no	—
141	C₂×Dic₃	—	4	6	no	—
142	D₁₂	—	2	2	no	—
143	C₄×D₃	—	4	6	no	—
144	Dic₆	—	2	6	no	—
145	Dic₆	—	4	6	no	—
146	C₃×C₈	C₂×C₅×D₄	1	3	yes	—
147	C₃×C₈	C₂⁴×C₅	1	1	yes	characteristic
148	C₂²×C₅	—	12	2	no	—
149	C₂²×C₅	C₂×D₂₄	1	3	yes	—
150	C₄×C₅	—	12	6	no	—
151	C₄×C₅	C₂×D₂₄	1	3	yes	—
152	C₄×C₅	C₂²×D₁₂	1	1	yes	characteristic
153	#41	—	6	6	no	—
154	#41	—	12	6	no	—
155	#41	C₅×D₁₂	1	1	yes	characteristic
156	C₂×Q₈	—	6	6	no	—
157	C₂×Q₈	—	12	2	no	—
158	Q₁₆	—	3	6	no	—
159	Q₁₆	—	6	6	no	—
160	SD₁₆	—	6	6	no	—
161	D₈	—	3	2	no	—
162	M₁₆	C₅×D₁₂	1	3	yes	—
163	C₂×C₈	C₂²×C₅×D₃	1	3	yes	—
164	C₁₆	C₂²×C₅×D₃	1	1	yes	characteristic
165	C₁₆	C₂²×C₅×D₃	1	3	yes	—
166	C₃×C₅	#3417	1	1	yes	characteristic, complemented, Hall
167	C₂²×C₃	C₂×C₅×D₈	1	3	yes	—
168	C₂×D₃	—	4	2	no	—
169	C₃×C₄	C₂×C₅×D₈	1	3	yes	—
170	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
171	Dic₃	—	4	6	no	—
172	C₂×C₅	—	2	3	no	—
173	C₂×C₅	—	24	2	no	—
174	C₂×C₅	C₂²×D₂₄	1	1	yes	characteristic
175	Q₈	—	6	6	no	—
176	Q₈	—	12	6	no	—
177	Q₈	C₅×D₂₄	1	1	yes	characteristic
178	D₄	—	6	2	no	—
179	D₄	—	12	6	no	—
180	D₄	C₅×D₂₄	1	3	yes	—
181	C₂×C₄	—	12	6	no	—
182	C₂×C₄	—	12	6	no	—
183	C₂×C₄	C₂×C₅×D₁₂	1	3	yes	—
184	C₈	C₂×C₅×D₁₂	1	3	yes	—
185	C₈	C₂³×C₅×D₃	1	1	yes	characteristic
186	C₂×C₃	—	2	3	no	—
187	C₂×C₃	C₂²×C₅×D₈	1	1	yes	characteristic
188	D₃	—	8	2	no	—
189	C₅	#82015	1	1	yes	characteristic, direct product factor, Sylow
190	C₂²	—	12	2	no	—
191	C₂²	C₂×C₅×D₂₄	1	3	yes	—
192	C₄	—	12	6	no	—
193	C₄	C₂×C₅×D₂₄	1	3	yes	—
194	C₄	C₂²×C₅×D₁₂	1	1	yes	characteristic
195	C₃	C₅×#3417	1	1	yes	characteristic, complemented, Sylow
196	C₂	—	2	3	no	—
197	C₂	—	24	2	no	—
198	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×#3417
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×#5753	C₂	semidirect
2. C₃×C₅×#504	C₂	semidirect
3. C₅×#4909	C₂	semidirect
4. C₂×C₅×Dic₂₄	C₂	semidirect
5. C₅×#4904	C₂	semidirect
18. #82015	C₅	direct (N × H)
23. C₅×#504	D₃	semidirect
37. #5753	C₂×C₅	semidirect
38. C₃×#504	C₂×C₅	semidirect
39. #4909	C₂×C₅	semidirect
40. C₂×Dic₂₄	C₂×C₅	semidirect
41. #4904	C₂×C₅	semidirect
93. #504	C₅×D₃	semidirect
166. C₃×C₅	#3417	semidirect
189. C₅	#82015	direct (N × H)
195. C₃	C₅×#3417	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5753	2
C₃×C₅×#504	2
C₅×#4909	2
C₂×C₅×Dic₂₄	2
C₅×#4904	2
C₅×#3417	3
#82015	5

Subgroups — C5×#82015 #14478625

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#82015 #14478625