Subgroups — C₅×#65274 #13102240

Order 1600.

432 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×C₄⋊Dic₅	C₂	1	2	yes	maximal
4	C₂×C₄×C₅×Dic₅	C₂	1	1	yes	maximal, characteristic
5	C₅×#3963	C₂	1	2	yes	maximal
6	C₂²×C₄×C₅²	C₂²	1	2	yes	—
7	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
8	C₂²×C₅×Dic₅	C₂²	1	2	yes	—
9	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
10	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
11	C₅²×C₄⋊C₄	C₄	1	4	yes	complemented
12	C₅×C₄⋊Dic₅	—	2	2	no	—
13	C₅×C₄⋊Dic₅	—	2	4	no	—
14	C₄×C₅×Dic₅	—	2	2	no	—
15	C₅×#389	—	5	1	no	maximal
16	#65274	C₅	1	1	yes	maximal, characteristic, direct product factor
17	C₂³×C₅²	C₂³	1	1	yes	characteristic
18	C₂×C₄×C₅²	—	2	2	no	—
19	C₂×C₄×C₅²	—	2	2	no	—
20	C₂×C₄×C₅²	C₂×C₄	1	4	yes	—
21	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₅²	D₄	1	2	yes	—
23	C₂×C₄×C₅²	Q₈	1	2	yes	—
24	C₂×C₅×Dic₅	—	2	1	no	—
25	C₂×C₅×Dic₅	—	2	1	no	—
26	C₂×C₅×Dic₅	—	2	1	no	—
27	C₂×C₅×Dic₅	—	2	1	no	—
28	C₂×C₅×Dic₅	—	2	2	no	—
29	C₂×C₅×Dic₅	—	2	2	no	—
30	C₂×C₅×Dic₅	—	2	2	no	—
31	C₂×C₅×Dic₅	D₄	1	2	yes	—
32	C₂×C₅×Dic₅	Q₈	1	2	yes	—
33	C₂×C₅×C₄⋊C₄	—	2	2	no	—
34	C₂×C₅×C₄⋊C₄	—	5	2	no	—
35	C₂×C₅×C₄⋊C₄	—	5	1	no	—
36	C₂×C₅×C₄⋊C₄	D₅	1	1	yes	characteristic
37	C₂×C₅×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
38	C₂×C₄²×C₅	—	5	1	no	—
39	C₂×C₄⋊Dic₅	C₂×C₅	1	1	yes	characteristic
40	C₂×C₄⋊Dic₅	C₂×C₅	1	2	yes	—
41	C₂×C₄×Dic₅	C₂×C₅	1	1	yes	characteristic
42	C₅×#95	—	5	2	no	—
43	#3963	C₂×C₅	1	2	yes	—
44	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₅²	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₅²	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₅²	#41	1	1	yes	characteristic
50	C₂²×C₅²	#41	1	1	yes	characteristic
51	C₄×C₅²	—	2	4	no	—
52	C₄×C₅²	C₄⋊C₄	1	4	yes	complemented
53	C₅×Dic₅	—	2	2	no	—
54	C₅×Dic₅	—	2	2	no	—
55	C₅×Dic₅	—	4	2	no	—
56	C₂²×C₄×C₅	—	2	2	no	—
57	C₂²×C₄×C₅	—	2	4	no	—
58	C₂²×C₄×C₅	—	5	1	no	—
59	C₂²×C₄×C₅	—	5	1	no	—
60	C₂²×C₄×C₅	—	5	2	no	—
61	C₂²×C₄×C₅	C₂×D₅	1	2	yes	—
62	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂²×C₅	1	2	yes	—
64	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
65	C₂²×Dic₅	C₂²×C₅	1	2	yes	—
66	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
67	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
68	C₅×C₄⋊C₄	—	2	8	no	—
69	C₅×C₄⋊C₄	—	10	4	no	—
70	C₅×C₄⋊C₄	—	10	2	no	—
71	C₅×C₄⋊C₄	Dic₅	1	4	yes	complemented
72	C₅×C₄⋊C₄	C₄×C₅	1	4	yes	complemented
73	C₄²×C₅	—	10	2	no	—
74	C₄⋊Dic₅	—	2	2	no	—
75	C₄⋊Dic₅	—	2	4	no	—
76	C₄×Dic₅	—	2	2	no	—
77	#389	—	5	1	no	Sylow
78	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₅²	C₄×D₄	1	1	yes	characteristic
80	C₂×C₅²	C₄×Q₈	1	1	yes	characteristic
81	C₂×C₅²	#121	1	1	yes	characteristic
82	C₂×C₅²	C₂²⋊Q₈	1	1	yes	characteristic
83	C₂×C₅²	#125	1	1	yes	characteristic
84	C₂×C₅²	C₄⋊Q₈	1	1	yes	characteristic
85	C₂³×C₅	—	2	2	no	—
86	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
87	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
88	C₂×C₄×C₅	—	2	2	no	—
89	C₂×C₄×C₅	—	2	2	no	—
90	C₂×C₄×C₅	—	2	2	no	—
91	C₂×C₄×C₅	—	2	2	no	—
92	C₂×C₄×C₅	—	2	4	no	—
93	C₂×C₄×C₅	—	2	4	no	—
94	C₂×C₄×C₅	—	2	4	no	—
95	C₂×C₄×C₅	—	2	8	no	—
96	C₂×C₄×C₅	—	4	4	no	—
97	C₂×C₄×C₅	—	4	4	no	—
98	C₂×C₄×C₅	—	5	2	no	—
99	C₂×C₄×C₅	—	5	2	no	—
100	C₂×C₄×C₅	—	10	1	no	—
101	C₂×C₄×C₅	—	10	1	no	—
102	C₂×C₄×C₅	—	10	1	no	—
103	C₂×C₄×C₅	—	10	1	no	—
104	C₂×C₄×C₅	—	10	2	no	—
105	C₂×C₄×C₅	—	10	2	no	—
106	C₂×C₄×C₅	—	10	2	no	—
107	C₂×C₄×C₅	Dic₁₀	1	2	yes	—
108	C₂×C₄×C₅	D₂₀	1	2	yes	—
109	C₂×C₄×C₅	C₂×Dic₅	1	4	yes	—
110	C₂×C₄×C₅	C₂×Dic₅	1	2	yes	—
111	C₂×C₄×C₅	C₂×C₄×C₅	1	4	yes	—
112	C₂×C₄×C₅	C₂×C₄×C₅	1	2	yes	—
113	C₂×C₄×C₅	C₅×D₄	1	2	yes	—
114	C₂×C₄×C₅	C₅×Q₈	1	2	yes	—
115	C₂×Dic₅	—	2	1	no	—
116	C₂×Dic₅	—	2	1	no	—
117	C₂×Dic₅	—	2	1	no	—
118	C₂×Dic₅	—	2	1	no	—
119	C₂×Dic₅	—	2	2	no	—
120	C₂×Dic₅	—	2	2	no	—
121	C₂×Dic₅	—	2	2	no	—
122	C₂×Dic₅	C₅×D₄	1	2	yes	—
123	C₂×Dic₅	C₅×Q₈	1	2	yes	—
124	C₂×C₄⋊C₄	—	5	2	no	—
125	C₂×C₄⋊C₄	—	5	1	no	—
126	C₂×C₄⋊C₄	C₅×D₅	1	1	yes	characteristic
127	C₂×C₄²	—	5	1	no	—
128	#95	—	5	2	no	—
129	C₅²	#389	1	1	yes	characteristic, complemented, Sylow
130	C₂²×C₅	—	2	2	no	—
131	C₂²×C₅	—	2	2	no	—
132	C₂²×C₅	—	2	2	no	—
133	C₂²×C₅	—	2	2	no	—
134	C₂²×C₅	—	2	2	no	—
135	C₂²×C₅	—	2	2	no	—
136	C₂²×C₅	—	2	2	no	—
137	C₂²×C₅	C₂×Dic₁₀	1	1	yes	characteristic
138	C₂²×C₅	C₂×D₂₀	1	1	yes	characteristic
139	C₂²×C₅	D₄×D₅	1	1	yes	characteristic
140	C₂²×C₅	#711	1	1	yes	characteristic
141	C₂²×C₅	Q₈×D₅	1	1	yes	characteristic
142	C₂²×C₅	#713	1	1	yes	characteristic
143	C₂²×C₅	C₂²×Dic₅	1	1	yes	characteristic
144	C₂²×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
145	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
146	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
147	C₂²×C₅	C₂×C₅×Q₈	1	1	yes	characteristic
148	C₂²×C₅	C₂×C₅×Q₈	1	1	yes	characteristic
149	C₂²×C₅	C₅×#41	1	1	yes	characteristic
150	C₂²×C₅	C₅×#41	1	1	yes	characteristic
151	C₄×C₅	—	2	4	no	—
152	C₄×C₅	—	2	4	no	—
153	C₄×C₅	—	2	8	no	—
154	C₄×C₅	—	4	8	no	—
155	C₄×C₅	—	10	2	no	—
156	C₄×C₅	—	10	2	no	—
157	C₄×C₅	—	20	2	no	—
158	C₄×C₅	C₄⋊Dic₅	1	4	yes	complemented
159	C₄×C₅	C₅×C₄⋊C₄	1	4	yes	complemented
160	Dic₅	—	2	2	no	—
161	Dic₅	—	2	2	no	—
162	Dic₅	—	4	2	no	—
163	C₂²×C₄	—	5	1	no	—
164	C₂²×C₄	—	5	1	no	—
165	C₂²×C₄	—	5	2	no	—
166	C₂²×C₄	C₂×C₅×D₅	1	2	yes	—
167	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
168	C₄⋊C₄	—	10	4	no	—
169	C₄⋊C₄	—	10	2	no	—
170	C₄⋊C₄	C₅×Dic₅	1	4	yes	complemented
171	C₄²	—	10	2	no	—
172	C₂×C₅	—	2	2	no	—
173	C₂×C₅	—	2	2	no	—
174	C₂×C₅	—	2	2	no	—
175	C₂×C₅	—	2	2	no	—
176	C₂×C₅	—	2	2	no	—
177	C₂×C₅	—	2	2	no	—
178	C₂×C₅	—	2	2	no	—
179	C₂×C₅	#4034	1	1	yes	characteristic
180	C₂×C₅	#4036	1	1	yes	characteristic
181	C₂×C₅	#4041	1	1	yes	characteristic
182	C₂×C₅	#4043	1	1	yes	characteristic
183	C₂×C₅	C₂×C₄⋊Dic₅	1	1	yes	characteristic
184	C₂×C₅	D₄×Dic₅	1	1	yes	characteristic
185	C₂×C₅	Q₈×Dic₅	1	1	yes	characteristic
186	C₂×C₅	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
187	C₂×C₅	C₄×C₅×D₄	1	1	yes	characteristic
188	C₂×C₅	C₄×C₅×Q₈	1	1	yes	characteristic
189	C₂×C₅	C₅×#121	1	1	yes	characteristic
190	C₂×C₅	C₅×C₂²⋊Q₈	1	1	yes	characteristic
191	C₂×C₅	C₅×#125	1	1	yes	characteristic
192	C₂×C₅	C₅×C₄⋊Q₈	1	1	yes	characteristic
193	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	5	2	no	—
197	C₂×C₄	—	5	2	no	—
198	C₂×C₄	—	10	1	no	—
199	C₂×C₄	—	10	1	no	—
200	C₂×C₄	—	10	1	no	—
201	C₂×C₄	—	10	1	no	—
202	C₂×C₄	—	10	2	no	—
203	C₂×C₄	—	10	2	no	—
204	C₂×C₄	—	10	2	no	—
205	C₂×C₄	C₅×Dic₁₀	1	2	yes	—
206	C₂×C₄	C₅×D₂₀	1	2	yes	—
207	C₂×C₄	C₂×C₅×Dic₅	1	4	yes	—
208	C₂×C₄	C₂×C₅×Dic₅	1	2	yes	—
209	C₅	—	2	2	no	—
210	C₅	#65274	1	1	yes	characteristic, direct product factor
211	C₅	C₅×#389	1	1	yes	characteristic, complemented
212	C₂²	C₂×C₅×Dic₁₀	1	1	yes	characteristic
213	C₂²	C₂×C₅×D₂₀	1	1	yes	characteristic
214	C₂²	C₅×D₄×D₅	1	1	yes	characteristic
215	C₂²	C₅×#711	1	1	yes	characteristic
216	C₂²	C₅×Q₈×D₅	1	1	yes	characteristic
217	C₂²	C₅×#713	1	1	yes	characteristic
218	C₂²	C₂²×C₅×Dic₅	1	1	yes	characteristic
219	C₄	—	2	4	no	—
220	C₄	—	10	2	no	—
221	C₄	—	10	2	no	—
222	C₄	—	20	2	no	—
223	C₄	C₅×C₄⋊Dic₅	1	4	yes	complemented
224	C₂	C₅×#4034	1	1	yes	characteristic
225	C₂	C₅×#4036	1	1	yes	characteristic
226	C₂	C₅×#4041	1	1	yes	characteristic
227	C₂	C₅×#4043	1	1	yes	characteristic
228	C₂	C₂×C₅×C₄⋊Dic₅	1	1	yes	characteristic
229	C₂	C₅×D₄×Dic₅	1	1	yes	characteristic
230	C₂	C₅×Q₈×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂²×D₅
commutator	C₂²×C₅	C₂²×C₄×C₅
Frattini	C₂³	C₂²×C₅×D₅
Fitting	C₂×C₅²×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₅²	C₂³
cosocle	C₂³×C₅	C₂³×C₅
hypercenter	C₂×C₅×C₄⋊C₄	D₅
hypocenter	C₅	C₅×#389
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₅×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
11. C₅²×C₄⋊C₄	C₄	semidirect
16. #65274	C₅	direct (N × H)
52. C₄×C₅²	C₄⋊C₄	semidirect
71. C₅×C₄⋊C₄	Dic₅	semidirect
72. C₅×C₄⋊C₄	C₄×C₅	semidirect
129. C₅²	#389	semidirect
158. C₄×C₅	C₄⋊Dic₅	semidirect
159. C₄×C₅	C₅×C₄⋊C₄	semidirect
170. C₄⋊C₄	C₅×Dic₅	semidirect
210. C₅	#65274	direct (N × H)
211. C₅	C₅×#389	semidirect
223. C₄	C₅×C₄⋊Dic₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅²×C₄⋊C₄	2
C₂×C₅×C₄⋊Dic₅	2
C₂×C₄×C₅×Dic₅	2
C₅×#3963	2
C₅×#389	5
#65274	5

Subgroups — C5×#65274 #13102240

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65274 #13102240