Subgroups — C₅×#68073 #14280609

Order 1920.

380 conjugacy classes of subgroups.

The subgroup lattice has 188 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	1	yes	maximal, characteristic
2	C₅×#4685	C₂	1	2	yes	maximal
3	C₃×C₅×#322	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4455	C₂	1	2	yes	maximal, complemented
5	C₅×C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1453	—	3	1	no	maximal, Hall
7	C₂×C₅×ΓL(1,𝔽₂₅)	C₄	1	2	yes	—
8	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
9	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
10	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₅×(C₂×D₃)⋊C₈	C₄	1	4	yes	—
12	C₅×#815	C₄	1	2	yes	—
13	C₅×Dic₃⋊C₈	C₄	1	4	yes	—
14	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	—
15	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	#68073	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₅×#431	—	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	2	yes	—
28	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
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	C₄×ΓL(1,𝔽₂₅)	C₂×C₅	1	1	yes	characteristic
33	#4685	C₂×C₅	1	2	yes	—
34	C₃×#322	C₂×C₅	1	1	yes	characteristic, complemented
35	#4455	C₂×C₅	1	2	yes	complemented
36	C₈×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
37	C₂×C₅×M₁₆	—	3	2	no	—
38	C₂×C₄²×C₅	—	3	1	no	—
39	C₅×C₄⋊C₈	—	3	4	no	—
40	C₅×C₂²⋊C₈	—	3	4	no	—
41	C₅×#97	—	3	2	no	—
42	C₄×C₅×C₈	—	3	2	no	—
43	C₄×C₅×C₈	—	3	1	no	—
44	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
45	C₄×C₅×C₈	C₂×D₃	1	2	yes	—
46	#1453	—	3	1	no	Sylow
47	C₂²×C₅×D₃	C₄²	1	1	yes	characteristic
48	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
50	C₂×C₅×Dic₃	C₄²	1	1	yes	characteristic
51	C₂×C₅×Dic₃	C₄²	1	2	yes	—
52	C₄×C₅×D₃	—	2	2	no	—
53	C₄×C₅×D₃	—	2	4	no	—
54	C₃×C₅×C₈	—	2	4	no	—
55	C₃×C₅×C₈	M₁₆	1	4	yes	—
56	C₅×C₃⋊C₈	—	2	2	no	—
57	C₅×C₃⋊C₈	—	2	4	no	—
58	C₂×ΓL(1,𝔽₂₅)	C₄×C₅	1	2	yes	—
59	C₄²×D₃	C₂²×C₅	1	1	yes	characteristic
60	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
61	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
62	(C₂×D₃)⋊C₈	C₄×C₅	1	4	yes	—
63	#815	C₄×C₅	1	2	yes	—
64	Dic₃⋊C₈	C₄×C₅	1	4	yes	—
65	C₄×C₃⋊C₈	C₂²×C₅	1	2	yes	—
66	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
67	C₂²×C₄×C₅	—	3	1	no	—
68	C₂²×C₄×C₅	—	3	2	no	—
69	C₅×M₁₆	—	6	4	no	—
70	C₂×C₅×C₈	—	3	2	no	—
71	C₂×C₅×C₈	—	3	4	no	—
72	C₂×C₅×C₈	C₄×D₃	1	4	yes	—
73	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
74	C₄²×C₅	—	3	1	no	—
75	C₄²×C₅	—	3	2	no	—
76	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
77	#431	—	3	2	no	—
78	C₄×M₁₆	—	3	1	no	—
79	#322	—	3	2	no	—
80	#322	C₅×D₃	1	1	yes	characteristic, complemented
81	C₈²	—	3	1	no	—
82	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
83	C₂×C₅×D₃	—	2	1	no	—
84	C₂×C₅×D₃	—	2	2	no	—
85	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
86	C₃×C₄×C₅	#131	1	4	yes	—
87	C₅×Dic₃	—	2	1	no	—
88	C₅×Dic₃	—	2	2	no	—
89	C₂×C₄×D₃	C₂×C₄×C₅	1	2	yes	—
90	C₂×C₄×D₃	C₂×C₄×C₅	1	1	yes	characteristic
91	C₂×C₃×C₈	C₂×C₄×C₅	1	4	yes	—
92	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
93	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
94	C₄×Dic₃	C₂×C₄×C₅	1	2	yes	—
95	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
96	C₂×C₃⋊C₈	C₂×C₄×C₅	1	4	yes	—
97	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
98	ΓL(1,𝔽₂₅)	—	2	4	no	—
99	C₂³×C₅	—	3	1	no	—
100	C₂×C₄×C₅	—	3	1	no	—
101	C₂×C₄×C₅	—	3	2	no	—
102	C₂×C₄×C₅	—	6	2	no	—
103	C₂×C₄×C₅	—	6	4	no	—
104	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
105	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
106	C₅×C₈	—	2	4	no	—
107	C₅×C₈	—	6	2	no	—
108	C₅×C₈	—	6	4	no	—
109	C₅×C₈	ΓL(1,𝔽₂₅)	1	4	yes	—
110	C₂×M₁₆	—	3	2	no	—
111	C₂×C₄²	—	3	1	no	—
112	C₄⋊C₈	—	3	4	no	—
113	C₂²⋊C₈	—	3	4	no	—
114	#97	—	3	2	no	—
115	C₄×C₈	—	3	2	no	—
116	C₄×C₈	—	3	1	no	—
117	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
118	C₄×C₈	C₂×C₅×D₃	1	2	yes	—
119	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
120	C₂×C₃×C₅	#405	1	2	yes	—
121	C₅×D₃	—	4	1	no	—
122	C₂²×D₃	C₄²×C₅	1	1	yes	characteristic
123	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
125	C₂×Dic₃	C₄²×C₅	1	1	yes	characteristic
126	C₂×Dic₃	C₄²×C₅	1	2	yes	—
127	C₄×D₃	—	2	2	no	—
128	C₄×D₃	—	2	4	no	—
129	C₃×C₈	—	2	4	no	—
130	C₃×C₈	C₅×M₁₆	1	4	yes	—
131	C₃⋊C₈	—	2	2	no	—
132	C₃⋊C₈	—	2	4	no	—
133	C₂²×C₅	—	6	1	no	—
134	C₂²×C₅	—	6	2	no	—
135	C₂²×C₅	C₄²×D₃	1	1	yes	characteristic
136	C₄×C₅	—	6	1	no	—
137	C₄×C₅	—	6	2	no	—
138	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
139	C₄×C₅	#907	1	4	yes	—
140	C₂²×C₄	—	3	2	no	—
141	C₂²×C₄	—	3	1	no	—
142	M₁₆	—	6	4	no	—
143	C₂×C₈	—	3	2	no	—
144	C₂×C₈	—	3	4	no	—
145	C₂×C₈	C₄×C₅×D₃	1	4	yes	—
146	C₂×C₈	C₄×C₅×D₃	1	2	yes	—
147	C₄²	—	3	1	no	—
148	C₄²	—	3	2	no	—
149	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
150	C₃×C₅	#1453	1	1	yes	characteristic, complemented, Hall
151	C₂²×C₃	C₂×C₄²×C₅	1	1	yes	characteristic
152	C₂×D₃	—	2	1	no	—
153	C₂×D₃	—	2	2	no	—
154	C₃×C₄	C₂×C₅×M₁₆	1	2	yes	—
155	C₃×C₄	C₅×#131	1	4	yes	—
156	Dic₃	—	2	1	no	—
157	Dic₃	—	2	2	no	—
158	C₂×C₅	—	12	1	no	—
159	C₂×C₅	C₄×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
160	C₂×C₅	#4708	1	2	yes	—
161	C₂³	—	3	1	no	—
162	C₂×C₄	—	3	1	no	—
163	C₂×C₄	—	3	2	no	—
164	C₂×C₄	—	6	2	no	—
165	C₂×C₄	—	6	4	no	—
166	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
167	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
168	C₈	—	2	4	no	—
169	C₈	—	6	2	no	—
170	C₈	—	6	4	no	—
171	C₈	C₅×ΓL(1,𝔽₂₅)	1	4	yes	—
172	C₂×C₃	C₄×C₅×M₁₆	1	1	yes	characteristic
173	C₂×C₃	C₅×#405	1	2	yes	—
174	D₃	—	4	1	no	—
175	C₅	#68073	1	1	yes	characteristic, direct product factor, Sylow
176	C₂²	—	6	1	no	—
177	C₂²	—	6	2	no	—
178	C₂²	C₄²×C₅×D₃	1	1	yes	characteristic
179	C₄	—	6	1	no	—
180	C₄	—	6	2	no	—
181	C₄	C₂×C₅×ΓL(1,𝔽₂₅)	1	2	yes	—
182	C₄	C₅×#907	1	4	yes	—
183	C₃	C₅×#1453	1	1	yes	characteristic, complemented, Sylow
184	C₂	—	12	1	no	—
185	C₂	C₄×C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
186	C₂	C₅×#4708	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×#1453
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×#4455	C₂	semidirect
5. C₅×C₈×C₃⋊C₈	C₂	semidirect
16. #68073	C₅	direct (N × H)
20. C₅×#322	D₃	semidirect
34. C₃×#322	C₂×C₅	semidirect
35. #4455	C₂×C₅	semidirect
36. C₈×C₃⋊C₈	C₂×C₅	semidirect
80. #322	C₅×D₃	semidirect
150. C₃×C₅	#1453	semidirect
175. C₅	#68073	direct (N × H)
183. C₃	C₅×#1453	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
C₅×#4685	2
C₃×C₅×#322	2
C₅×#4455	2
C₅×C₈×C₃⋊C₈	2
C₅×#1453	3
#68073	5

Subgroups — C5×#68073 #14280609

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68073 #14280609