Subgroups — C₅×#68075 #14280709

Order 1920.

428 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×ΓL(1,𝔽₂₅)	C₂	1	2	yes	maximal
2	C₄×C₅×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#322	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4454	C₂	1	2	yes	maximal, complemented
5	C₅×C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1455	—	3	1	no	maximal, Hall
7	C₂×C₅×ΓL(1,𝔽₂₅)	C₄	1	2	yes	—
8	C₂×C₅×ΓL(1,𝔽₂₅)	C₄	1	4	yes	—
9	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
10	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
12	C₅×#815	C₄	1	2	yes	—
13	C₅×#815	C₄	1	4	yes	—
14	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	—
15	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	#68075	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₄×C₅×M₁₆	—	3	2	no	—
18	C₄×C₅×M₁₆	—	3	1	no	—
19	C₅×#322	—	3	2	no	—
20	C₅×#322	D₃	1	1	yes	characteristic, complemented
21	C₅×C₈²	—	3	1	no	—
22	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₅×D₃	C₂×C₄	1	2	yes	—
24	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
25	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
26	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
27	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₅×Dic₃	C₂×C₄	1	2	yes	—
29	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
30	C₂×C₅×C₃⋊C₈	C₂×C₄	1	4	yes	—
31	C₅×ΓL(1,𝔽₂₅)	—	2	4	no	—
32	C₅×ΓL(1,𝔽₂₅)	—	2	8	no	—
33	C₄×ΓL(1,𝔽₂₅)	C₂×C₅	1	2	yes	—
34	C₄×ΓL(1,𝔽₂₅)	C₂×C₅	1	1	yes	characteristic
35	C₃×#322	C₂×C₅	1	1	yes	characteristic, complemented
36	#4454	C₂×C₅	1	2	yes	complemented
37	C₈×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
38	C₂×C₅×M₁₆	—	3	4	no	—
39	C₂×C₅×M₁₆	—	3	2	no	—
40	C₂×C₄²×C₅	—	3	1	no	—
41	C₅×#97	—	3	4	no	—
42	C₅×#97	—	3	2	no	—
43	C₄×C₅×C₈	—	3	2	no	—
44	C₄×C₅×C₈	—	3	1	no	—
45	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
46	C₄×C₅×C₈	C₂×D₃	1	2	yes	—
47	#1455	—	3	1	no	Sylow
48	C₂²×C₅×D₃	C₄²	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
51	C₂×C₅×Dic₃	C₄²	1	1	yes	characteristic
52	C₂×C₅×Dic₃	C₄²	1	2	yes	—
53	C₄×C₅×D₃	—	2	2	no	—
54	C₄×C₅×D₃	—	2	4	no	—
55	C₃×C₅×C₈	—	2	4	no	—
56	C₃×C₅×C₈	M₁₆	1	4	yes	complemented
57	C₅×C₃⋊C₈	—	2	2	no	—
58	C₅×C₃⋊C₈	M₁₆	1	8	yes	complemented
59	C₂×ΓL(1,𝔽₂₅)	C₄×C₅	1	2	yes	—
60	C₂×ΓL(1,𝔽₂₅)	C₄×C₅	1	4	yes	—
61	C₄²×D₃	C₂²×C₅	1	1	yes	characteristic
62	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
63	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
64	#815	C₄×C₅	1	2	yes	—
65	#815	C₄×C₅	1	4	yes	—
66	C₄×C₃⋊C₈	C₂²×C₅	1	2	yes	—
67	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
68	C₂²×C₄×C₅	—	3	1	no	—
69	C₂²×C₄×C₅	—	3	2	no	—
70	C₅×M₁₆	—	6	8	no	—
71	C₅×M₁₆	—	6	4	no	—
72	C₂×C₅×C₈	—	3	4	no	—
73	C₂×C₅×C₈	—	3	2	no	—
74	C₂×C₅×C₈	C₄×D₃	1	4	yes	—
75	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
76	C₄²×C₅	—	3	1	no	—
77	C₄²×C₅	—	3	2	no	—
78	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
79	C₄×M₁₆	—	3	2	no	—
80	C₄×M₁₆	—	3	1	no	—
81	#322	—	3	2	no	—
82	#322	C₅×D₃	1	1	yes	characteristic, complemented
83	C₈²	—	3	1	no	—
84	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
85	C₂×C₅×D₃	—	2	1	no	—
86	C₂×C₅×D₃	—	2	2	no	—
87	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
88	C₃×C₄×C₅	C₂×M₁₆	1	4	yes	—
89	C₅×Dic₃	—	2	1	no	—
90	C₅×Dic₃	—	2	2	no	—
91	C₂×C₄×D₃	C₂×C₄×C₅	1	1	yes	characteristic
92	C₂×C₄×D₃	C₂×C₄×C₅	1	2	yes	—
93	C₂×C₃×C₈	C₂×C₄×C₅	1	4	yes	—
94	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
95	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
96	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
97	C₄×Dic₃	C₂×C₄×C₅	1	2	yes	—
98	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
99	C₂×C₃⋊C₈	C₂×C₄×C₅	1	4	yes	—
100	ΓL(1,𝔽₂₅)	—	2	4	no	—
101	ΓL(1,𝔽₂₅)	—	2	8	no	—
102	C₂³×C₅	—	3	1	no	—
103	C₂×C₄×C₅	—	3	1	no	—
104	C₂×C₄×C₅	—	3	2	no	—
105	C₂×C₄×C₅	—	6	2	no	—
106	C₂×C₄×C₅	—	6	4	no	—
107	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
109	C₅×C₈	—	2	4	no	—
110	C₅×C₈	—	3	8	no	—
111	C₅×C₈	—	6	2	no	—
112	C₅×C₈	ΓL(1,𝔽₂₅)	1	4	yes	complemented
113	C₂×M₁₆	—	3	4	no	—
114	C₂×M₁₆	—	3	2	no	—
115	C₂×C₄²	—	3	1	no	—
116	#97	—	3	4	no	—
117	#97	—	3	2	no	—
118	C₄×C₈	—	3	2	no	—
119	C₄×C₈	—	3	1	no	—
120	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
121	C₄×C₈	C₂×C₅×D₃	1	2	yes	—
122	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
123	C₂×C₃×C₅	C₄×M₁₆	1	2	yes	—
124	C₅×D₃	—	4	1	no	—
125	C₂²×D₃	C₄²×C₅	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
127	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
128	C₂×Dic₃	C₄²×C₅	1	1	yes	characteristic
129	C₂×Dic₃	C₄²×C₅	1	2	yes	—
130	C₄×D₃	—	2	2	no	—
131	C₄×D₃	—	2	4	no	—
132	C₃×C₈	—	2	4	no	—
133	C₃×C₈	C₅×M₁₆	1	4	yes	complemented
134	C₃⋊C₈	—	2	2	no	—
135	C₃⋊C₈	C₅×M₁₆	1	8	yes	complemented
136	C₂²×C₅	—	6	1	no	—
137	C₂²×C₅	—	6	2	no	—
138	C₂²×C₅	C₄²×D₃	1	1	yes	characteristic
139	C₄×C₅	—	6	1	no	—
140	C₄×C₅	—	6	2	no	—
141	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
142	C₄×C₅	D₃×M₁₆	1	4	yes	—
143	C₂²×C₄	—	3	1	no	—
144	C₂²×C₄	—	3	2	no	—
145	M₁₆	—	6	8	no	—
146	M₁₆	—	6	4	no	—
147	C₂×C₈	—	3	4	no	—
148	C₂×C₈	—	3	2	no	—
149	C₂×C₈	C₄×C₅×D₃	1	4	yes	—
150	C₂×C₈	C₄×C₅×D₃	1	2	yes	—
151	C₄²	—	3	1	no	—
152	C₄²	—	3	2	no	—
153	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
154	C₃×C₅	#1455	1	1	yes	characteristic, complemented, Hall
155	C₂²×C₃	C₂×C₄²×C₅	1	1	yes	characteristic
156	C₂×D₃	—	2	1	no	—
157	C₂×D₃	—	2	2	no	—
158	C₃×C₄	C₂×C₅×M₁₆	1	2	yes	—
159	C₃×C₄	C₂×C₅×M₁₆	1	4	yes	—
160	Dic₃	—	2	1	no	—
161	Dic₃	—	2	2	no	—
162	C₂×C₅	—	12	1	no	—
163	C₂×C₅	C₄×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
164	C₂×C₅	#4707	1	2	yes	—
165	C₂³	—	3	1	no	—
166	C₂×C₄	—	3	1	no	—
167	C₂×C₄	—	3	2	no	—
168	C₂×C₄	—	6	2	no	—
169	C₂×C₄	—	6	4	no	—
170	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
172	C₈	—	2	4	no	—
173	C₈	—	3	8	no	—
174	C₈	—	6	2	no	—
175	C₈	C₅×ΓL(1,𝔽₂₅)	1	4	yes	complemented
176	C₂×C₃	C₄×C₅×M₁₆	1	1	yes	characteristic
177	C₂×C₃	C₄×C₅×M₁₆	1	2	yes	—
178	D₃	—	4	1	no	—
179	C₅	#68075	1	1	yes	characteristic, direct product factor, Sylow
180	C₂²	—	6	1	no	—
181	C₂²	—	6	2	no	—
182	C₂²	C₄²×C₅×D₃	1	1	yes	characteristic
183	C₄	—	6	1	no	—
184	C₄	—	6	2	no	—
185	C₄	C₂×C₅×ΓL(1,𝔽₂₅)	1	2	yes	—
186	C₄	C₅×D₃×M₁₆	1	4	yes	—
187	C₃	C₅×#1455	1	1	yes	characteristic, complemented, Sylow
188	C₂	—	12	1	no	—
189	C₂	C₄×C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
190	C₂	C₅×#4707	1	2	yes	—

	subgroup	G/H
center	C₄²×C₅	C₂²×D₃
commutator	C₂²×C₃	C₂×C₄²×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#322	C₂
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂²×C₅×D₃	C₄²
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#322	D₃
hypocenter	C₃	C₅×#1455
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₄²×C₅ ◁ C₅×#322

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#322	C₂	semidirect
4. C₅×#4454	C₂	semidirect
5. C₅×C₈×C₃⋊C₈	C₂	semidirect
16. #68075	C₅	direct (N × H)
20. C₅×#322	D₃	semidirect
35. C₃×#322	C₂×C₅	semidirect
36. #4454	C₂×C₅	semidirect
37. C₈×C₃⋊C₈	C₂×C₅	semidirect
56. C₃×C₅×C₈	M₁₆	semidirect
58. C₅×C₃⋊C₈	M₁₆	semidirect
82. #322	C₅×D₃	semidirect
112. C₅×C₈	ΓL(1,𝔽₂₅)	semidirect
133. C₃×C₈	C₅×M₁₆	semidirect
135. C₃⋊C₈	C₅×M₁₆	semidirect
154. C₃×C₅	#1455	semidirect
175. C₈	C₅×ΓL(1,𝔽₂₅)	semidirect
179. C₅	#68075	direct (N × H)
187. C₃	C₅×#1455	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×ΓL(1,𝔽₂₅)	2
C₃×C₅×#322	2
C₅×#4454	2
C₅×C₈×C₃⋊C₈	2
C₅×#1455	3
#68075	5

Subgroups — C5×#68075 #14280709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68075 #14280709