Subgroups — C₅×#65128 #13102102

Order 1600.

266 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#676	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₈×D₅	C₂	1	1	yes	maximal, characteristic
3	C₅×#4018	C₂	1	1	yes	maximal, characteristic
4	C₅²×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#3946	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#3946	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×C₄⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
9	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
11	C₄²×C₅²	C₂²	1	1	yes	characteristic
12	C₅×#685	C₄	1	1	yes	characteristic
13	C₅×#685	C₄	1	1	yes	characteristic
14	C₅×#683	C₄	1	1	yes	characteristic
15	C₅×#683	C₄	1	1	yes	characteristic
16	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
17	C₂×C₅×#168	C₂²	1	1	yes	characteristic
18	C₂×C₅×#168	C₂²	1	1	yes	characteristic
19	C₅×#676	—	2	2	no	—
20	C₅×C₈×D₅	—	2	2	no	—
21	C₅×#424	—	5	1	no	maximal
22	#65128	C₅	1	1	yes	maximal, characteristic, direct product factor
23	C₂²×C₅×D₅	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
27	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
28	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
29	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
30	C₄×C₅×D₅	D₄	1	2	yes	—
31	C₄×C₅×D₅	Q₈	1	2	yes	—
32	C₅²×C₈	—	2	1	no	—
33	C₅²×C₈	—	2	1	no	—
34	C₅×#168	—	2	1	no	—
35	C₅×#168	—	2	1	no	—
36	C₂×C₅×M₁₆	—	5	1	no	—
37	C₂²×C₅×C₈	—	5	1	no	—
38	C₅×#117	—	5	1	no	—
39	C₂×#676	C₂×C₅	1	1	yes	characteristic
40	C₂×C₈×D₅	C₂×C₅	1	1	yes	characteristic
41	#4018	C₂×C₅	1	1	yes	characteristic
42	C₅×C₄⋊C₈	—	2	2	no	—
43	C₅×C₄⋊C₈	—	5	1	no	—
44	C₅×C₄⋊C₈	—	5	1	no	—
45	C₅×C₄⋊C₈	—	5	1	no	—
46	C₅×C₄⋊C₈	D₅	1	1	yes	characteristic, complemented
47	C₅×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
48	#3946	C₂×C₅	1	1	yes	characteristic, complemented
49	#3946	C₂×C₅	1	1	yes	characteristic, complemented
50	C₄⋊#168	C₂×C₅	1	1	yes	characteristic, complemented
51	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₅×D₅	—	2	1	no	—
53	C₂×C₅×D₅	—	2	1	no	—
54	C₂×C₅×D₅	C₄⋊C₄	1	2	yes	—
55	C₄×C₅²	—	2	1	no	—
56	C₄×C₅²	—	2	1	no	—
57	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
58	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
59	C₅×Dic₅	—	2	1	no	—
60	C₅×Dic₅	—	2	1	no	—
61	C₅×Dic₅	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₅	—	5	1	no	—
63	C₂×C₄×D₅	C₂²×C₅	1	1	yes	characteristic
64	C₅×M₁₆	—	10	2	no	—
65	C₂×C₅×C₈	—	2	2	no	—
66	C₂×C₅×C₈	—	2	2	no	—
67	C₂×C₅×C₈	—	5	1	no	—
68	C₂×C₅×C₈	—	5	1	no	—
69	C₂×C₅×C₈	—	10	2	no	—
70	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic
71	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic
72	C₂×C₅×C₈	C₂²×C₅	1	1	yes	characteristic
73	C₂×C₅×C₈	C₂²×C₅	1	1	yes	characteristic
74	C₅×C₄⋊C₄	—	5	1	no	—
75	C₅×C₄⋊C₄	—	5	1	no	—
76	C₅×C₂²⋊C₄	—	5	1	no	—
77	C₅×C₂²⋊C₄	—	5	1	no	—
78	C₄²×C₅	—	2	2	no	—
79	C₄²×C₅	—	5	1	no	—
80	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
81	C₄²×C₅	C₂²×C₅	1	1	yes	characteristic
82	#685	C₄×C₅	1	1	yes	characteristic
83	#685	C₄×C₅	1	1	yes	characteristic
84	#683	C₄×C₅	1	1	yes	characteristic
85	#683	C₄×C₅	1	1	yes	characteristic
86	C₄×Dic₅	C₂²×C₅	1	1	yes	characteristic
87	C₂×#168	C₂²×C₅	1	1	yes	characteristic
88	C₂×#168	C₂²×C₅	1	1	yes	characteristic
89	#676	—	2	2	no	—
90	C₈×D₅	—	2	2	no	—
91	#424	—	5	1	no	Sylow
92	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₅²	#131	1	1	yes	characteristic
94	C₂×C₅²	#131	1	1	yes	characteristic
95	C₅×D₅	—	2	2	no	—
96	C₂³×C₅	—	5	1	no	—
97	C₂²×D₅	C₂×C₄×C₅	1	1	yes	characteristic
98	C₂×C₄×C₅	—	2	2	no	—
99	C₂×C₄×C₅	—	2	2	no	—
100	C₂×C₄×C₅	—	2	2	no	—
101	C₂×C₄×C₅	—	5	1	no	—
102	C₂×C₄×C₅	—	5	2	no	—
103	C₂×C₄×C₅	—	5	2	no	—
104	C₂×C₄×C₅	—	5	1	no	—
105	C₂×C₄×C₅	—	5	1	no	—
106	C₂×C₄×C₅	C₄×D₅	1	1	yes	characteristic
107	C₂×C₄×C₅	C₄×D₅	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂×C₄×C₅	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂×C₄×C₅	1	1	yes	characteristic
110	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
111	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
112	C₂×Dic₅	C₂×C₄×C₅	1	1	yes	characteristic
113	C₂×Dic₅	C₂×C₄×C₅	1	1	yes	characteristic
114	C₂×Dic₅	C₂×C₄×C₅	1	1	yes	characteristic
115	C₄×D₅	C₅×D₄	1	2	yes	—
116	C₄×D₅	C₅×Q₈	1	2	yes	—
117	C₅×C₈	—	2	1	no	—
118	C₅×C₈	—	2	1	no	—
119	C₅×C₈	—	2	1	no	—
120	C₅×C₈	—	2	1	no	—
121	C₅×C₈	—	4	2	no	—
122	C₅×C₈	—	4	2	no	—
123	C₅×C₈	—	10	1	no	—
124	C₅×C₈	—	10	1	no	—
125	#168	—	2	1	no	—
126	#168	—	2	1	no	—
127	C₂×M₁₆	—	5	1	no	—
128	C₂²×C₈	—	5	1	no	—
129	#117	—	5	1	no	—
130	C₄⋊C₈	—	5	1	no	—
131	C₄⋊C₈	—	5	1	no	—
132	C₄⋊C₈	—	5	1	no	—
133	C₄⋊C₈	C₅×D₅	1	1	yes	characteristic, complemented
134	C₅²	#424	1	1	yes	characteristic, complemented, Sylow
135	C₂²×C₅	—	2	2	no	—
136	C₂²×C₅	—	5	2	no	—
137	C₂²×C₅	—	10	1	no	—
138	C₂²×C₅	—	10	1	no	—
139	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
140	C₂²×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
141	C₂×D₅	—	2	1	no	—
142	C₂×D₅	—	2	1	no	—
143	C₂×D₅	C₅×C₄⋊C₄	1	2	yes	—
144	C₄×C₅	—	2	4	no	—
145	C₄×C₅	—	2	1	no	—
146	C₄×C₅	—	2	1	no	—
147	C₄×C₅	—	2	1	no	—
148	C₄×C₅	—	2	1	no	—
149	C₄×C₅	—	2	2	no	—
150	C₄×C₅	—	2	2	no	—
151	C₄×C₅	—	4	2	no	—
152	C₄×C₅	—	5	2	no	—
153	C₄×C₅	—	10	1	no	—
154	C₄×C₅	—	10	1	no	—
155	C₄×C₅	D₄×D₅	1	1	yes	characteristic
156	C₄×C₅	Q₈×D₅	1	1	yes	characteristic
157	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
158	C₄×C₅	C₂×C₅×Q₈	1	1	yes	characteristic
159	Dic₅	—	2	1	no	—
160	Dic₅	—	2	1	no	—
161	Dic₅	C₅×C₄⋊C₄	1	2	yes	—
162	C₂²×C₄	—	5	1	no	—
163	M₁₆	—	10	2	no	—
164	C₂×C₈	—	5	1	no	—
165	C₂×C₈	—	5	1	no	—
166	C₂×C₈	—	10	2	no	—
167	C₂×C₈	C₂×C₅×D₅	1	1	yes	characteristic
168	C₂×C₈	C₂×C₅×D₅	1	1	yes	characteristic
169	C₄⋊C₄	—	5	1	no	—
170	C₄⋊C₄	—	5	1	no	—
171	C₂²⋊C₄	—	5	1	no	—
172	C₂²⋊C₄	—	5	1	no	—
173	C₄²	—	5	1	no	—
174	C₄²	C₂×C₅×D₅	1	1	yes	characteristic
175	C₂×C₅	—	2	2	no	—
176	C₂×C₅	—	2	2	no	—
177	C₂×C₅	—	2	2	no	—
178	C₂×C₅	—	10	2	no	—
179	C₂×C₅	D₅×C₄⋊C₄	1	1	yes	characteristic
180	C₂×C₅	#4047	1	1	yes	characteristic
181	C₂×C₅	#4053	1	1	yes	characteristic
182	C₂×C₅	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
183	C₂×C₅	C₅×#131	1	1	yes	characteristic
184	C₂×C₅	C₅×#131	1	1	yes	characteristic
185	D₅	—	2	2	no	—
186	C₂³	—	5	1	no	—
187	C₂×C₄	—	5	1	no	—
188	C₂×C₄	—	5	2	no	—
189	C₂×C₄	—	5	2	no	—
190	C₂×C₄	—	5	1	no	—
191	C₂×C₄	—	5	1	no	—
192	C₂×C₄	C₄×C₅×D₅	1	1	yes	characteristic
193	C₂×C₄	C₄×C₅×D₅	1	1	yes	characteristic
194	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
195	C₈	—	2	1	no	—
196	C₈	—	2	1	no	—
197	C₈	—	10	1	no	—
198	C₈	—	10	1	no	—
199	C₅	—	2	2	no	—
200	C₅	#65128	1	1	yes	characteristic, direct product factor
201	C₅	C₅×#424	1	1	yes	characteristic, complemented
202	C₂²	—	5	2	no	—
203	C₂²	—	10	1	no	—
204	C₂²	—	10	1	no	—
205	C₂²	C₂×C₄×C₅×D₅	1	1	yes	characteristic
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	5	2	no	—
209	C₄	—	10	1	no	—
210	C₄	—	10	1	no	—
211	C₄	C₅×D₄×D₅	1	1	yes	characteristic
212	C₄	C₅×Q₈×D₅	1	1	yes	characteristic
213	C₂	—	10	2	no	—
214	C₂	C₅×D₅×C₄⋊C₄	1	1	yes	characteristic
215	C₂	C₅×#4047	1	1	yes	characteristic
216	C₂	C₅×#4053	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₅
commutator	C₂²×C₅	C₂²×C₄×C₅
Frattini	C₂×C₄	C₂²×C₅×D₅
Fitting	C₅²×C₄⋊C₈	C₂
socle	C₂²×C₅²	C₂²×C₄
prime radical	C₂²×C₅×D₅	C₂×C₄
cosocle	C₂×C₄×C₅	C₂³×C₅
hypercenter	C₅×C₄⋊C₈	D₅
hypocenter	C₅	C₅×#424
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₅×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
4. C₅²×C₄⋊C₈	C₂	semidirect
5. C₅×#3946	C₂	semidirect
6. C₅×#3946	C₂	semidirect
7. C₅×C₄⋊#168	C₂	semidirect
22. #65128	C₅	direct (N × H)
46. C₅×C₄⋊C₈	D₅	semidirect
47. C₅×C₄⋊C₈	C₂×C₅	semidirect
48. #3946	C₂×C₅	semidirect
49. #3946	C₂×C₅	semidirect
50. C₄⋊#168	C₂×C₅	semidirect
133. C₄⋊C₈	C₅×D₅	semidirect
134. C₅²	#424	semidirect
200. C₅	#65128	direct (N × H)
201. C₅	C₅×#424	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#676	2
C₂×C₅×C₈×D₅	2
C₅×#4018	2
C₅²×C₄⋊C₈	2
C₅×#3946	2
C₅×C₄⋊#168	2
C₅×#424	5
#65128	5

Subgroups — C5×#65128 #13102102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65128 #13102102