Subgroups — C₅×#65421 #13102392

Order 1600.

368 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#709	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅²×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#681	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#3956	C₂	1	2	yes	maximal, complemented
5	C₅×#3955	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
7	C₂×C₅×C₂²⋊D₅	C₄	1	2	yes	—
8	C₅×#709	—	2	1	no	—
9	C₅×#709	—	2	1	no	—
10	C₅×#709	—	2	1	no	—
11	C₅×#709	—	2	1	no	—
12	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×D₅	C₄	1	2	yes	—
14	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic
15	C₅²×M₁₆	—	2	1	no	—
16	C₅²×M₁₆	C₂²	1	2	yes	—
17	C₂×C₅²×C₈	—	2	1	no	—
18	C₅×#681	—	2	1	no	—
19	C₅×#681	C₂²	1	2	yes	—
20	C₂×C₅×#168	—	2	1	no	—
21	C₅×#413	—	5	1	no	maximal
22	#65421	C₅	1	1	yes	maximal, characteristic, direct product factor
23	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅×D₅	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
27	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
28	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
29	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
30	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
31	C₅×C₂²⋊D₅	—	2	2	no	—
32	C₅×C₂²⋊D₅	—	2	2	no	—
33	C₂×C₅×Dic₅	C₂×C₄	1	2	yes	—
34	C₅×D₂₀	—	2	1	no	—
35	C₅×D₂₀	—	2	1	no	—
36	C₄×C₅×D₅	—	2	2	no	—
37	C₄×C₅×D₅	—	2	2	no	—
38	C₅×Dic₁₀	—	2	1	no	—
39	C₅×Dic₁₀	—	2	1	no	—
40	C₅²×C₈	—	2	2	no	—
41	C₅×#168	—	2	2	no	—
42	C₂×C₅×#41	—	5	1	no	—
43	C₂×#709	C₂×C₅	1	1	yes	characteristic
44	C₂×C₅×M₁₆	—	2	2	no	—
45	C₂×C₅×M₁₆	—	5	1	no	—
46	C₂×C₅×M₁₆	D₅	1	1	yes	characteristic, complemented
47	C₂×C₅×M₁₆	C₂×C₅	1	1	yes	characteristic, complemented
48	C₂×#681	C₂×C₅	1	1	yes	characteristic, complemented
49	C₅×#101	—	5	2	no	—
50	C₅×#100	—	5	2	no	—
51	#3956	C₂×C₅	1	2	yes	complemented
52	#3955	C₂×C₅	1	2	yes	complemented
53	C₂²×C₅²	—	4	1	no	—
54	C₂²×C₅²	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₅²	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₅×D₅	—	2	2	no	—
58	C₂×C₅×D₅	—	4	2	no	—
59	C₄×C₅²	C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₅²	C₂²⋊C₄	1	1	yes	characteristic
61	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
62	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
63	C₅×Dic₅	—	2	2	no	—
64	C₅×#41	—	10	1	no	—
65	C₅×#41	—	10	1	no	—
66	C₅×#41	—	10	1	no	—
67	C₅×#41	—	10	1	no	—
68	C₂×C₅×Q₈	—	5	1	no	—
69	C₂×C₅×D₄	—	5	2	no	—
70	C₂×C₅×D₄	—	5	1	no	—
71	C₂²×C₄×C₅	—	2	2	no	—
72	C₂²×C₄×C₅	—	5	2	no	—
73	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
75	C₂×C₂²⋊D₅	C₄×C₅	1	2	yes	—
76	#709	—	2	1	no	—
77	#709	—	2	1	no	—
78	#709	—	2	1	no	—
79	#709	—	2	1	no	—
80	C₂×D₂₀	C₂²×C₅	1	1	yes	characteristic
81	C₂×C₄×D₅	C₄×C₅	1	2	yes	—
82	C₂×Dic₁₀	C₂²×C₅	1	1	yes	characteristic
83	C₅×M₁₆	—	2	4	no	—
84	C₅×M₁₆	—	2	1	no	—
85	C₅×M₁₆	—	2	1	no	—
86	C₅×M₁₆	—	2	4	no	—
87	C₅×M₁₆	—	5	2	no	—
88	C₅×M₁₆	—	10	1	no	—
89	C₅×M₁₆	C₂×D₅	1	2	yes	—
90	C₅×M₁₆	C₂²×C₅	1	2	yes	—
91	C₂×C₅×C₈	—	2	4	no	—
92	C₂×C₅×C₈	—	2	1	no	—
93	C₂×C₅×C₈	—	2	1	no	—
94	C₂×C₅×C₈	—	10	1	no	—
95	#681	—	2	1	no	—
96	#681	C₂²×C₅	1	2	yes	—
97	C₂×#168	—	2	1	no	—
98	#413	—	5	1	no	Sylow
99	C₂×C₅²	—	2	1	no	—
100	C₂×C₅²	—	2	1	no	—
101	C₂×C₅²	—	2	1	no	—
102	C₂×C₅²	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₅×D₅	—	4	2	no	—
104	C₂³×C₅	—	2	2	no	—
105	C₂³×C₅	—	5	2	no	—
106	C₂³×C₅	C₄×D₅	1	1	yes	characteristic
107	C₂³×C₅	C₂×C₄×C₅	1	1	yes	characteristic
108	C₂²×D₅	C₂×C₄×C₅	1	2	yes	—
109	C₅×Q₈	—	10	1	no	—
110	C₅×Q₈	—	10	1	no	—
111	C₅×D₄	—	10	2	no	—
112	C₅×D₄	—	10	2	no	—
113	C₅×D₄	—	10	1	no	—
114	C₅×D₄	—	10	1	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	—	2	2	no	—
118	C₂×C₄×C₅	—	2	2	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	5	2	no	—
122	C₂×C₄×C₅	—	10	2	no	—
123	C₂×C₄×C₅	—	10	2	no	—
124	C₂×C₄×C₅	C₄×D₅	1	1	yes	characteristic
125	C₂×C₄×C₅	D₂₀	1	1	yes	characteristic
126	C₂×C₄×C₅	D₂₀	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂²⋊D₅	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂²⋊D₅	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂×C₄×C₅	1	1	yes	characteristic
130	C₂×C₄×C₅	C₅×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅	C₅×D₄	1	1	yes	characteristic
132	C₂×C₄×C₅	C₅×D₄	1	1	yes	characteristic
133	C₂×C₄×C₅	C₅×D₄	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
136	C₂²⋊D₅	—	2	2	no	—
137	C₂²⋊D₅	—	2	2	no	—
138	C₂×Dic₅	C₂×C₄×C₅	1	2	yes	—
139	D₂₀	—	2	1	no	—
140	D₂₀	—	2	1	no	—
141	C₄×D₅	—	2	2	no	—
142	C₄×D₅	—	2	2	no	—
143	Dic₁₀	—	2	1	no	—
144	Dic₁₀	—	2	1	no	—
145	C₅×C₈	—	2	8	no	—
146	C₅×C₈	—	2	2	no	—
147	C₅×C₈	—	2	2	no	—
148	C₅×C₈	—	10	2	no	—
149	#168	—	2	2	no	—
150	C₂×#41	—	5	1	no	—
151	C₂×M₁₆	—	5	1	no	—
152	C₂×M₁₆	C₅×D₅	1	1	yes	characteristic, complemented
153	#101	—	5	2	no	—
154	#100	—	5	2	no	—
155	C₅²	#413	1	1	yes	characteristic, complemented, Sylow
156	C₂²×C₅	—	2	2	no	—
157	C₂²×C₅	—	2	2	no	—
158	C₂²×C₅	—	2	2	no	—
159	C₂²×C₅	—	4	1	no	—
160	C₂²×C₅	—	4	1	no	—
161	C₂²×C₅	—	4	4	no	—
162	C₂²×C₅	—	10	2	no	—
163	C₂²×C₅	—	20	2	no	—
164	C₂²×C₅	#685	1	1	yes	characteristic
165	C₂²×C₅	#685	1	1	yes	characteristic
166	C₂²×C₅	C₅×C₂²⋊C₄	1	1	yes	characteristic
167	C₂²×C₅	C₅×C₂²⋊C₄	1	1	yes	characteristic
168	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
169	C₂²×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
170	C₂×D₅	—	2	2	no	—
171	C₂×D₅	—	4	2	no	—
172	C₄×C₅	—	2	2	no	—
173	C₄×C₅	—	2	2	no	—
174	C₄×C₅	—	2	2	no	—
175	C₄×C₅	—	2	2	no	—
176	C₄×C₅	—	10	2	no	—
177	C₄×C₅	#685	1	1	yes	characteristic
178	C₄×C₅	#685	1	1	yes	characteristic
179	C₄×C₅	C₅×C₂²⋊C₄	1	1	yes	characteristic
180	C₄×C₅	C₅×C₂²⋊C₄	1	1	yes	characteristic
181	C₄×C₅	C₂×D₂₀	1	1	yes	characteristic
182	C₄×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
183	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
184	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
185	Dic₅	—	2	2	no	—
186	#41	—	10	1	no	—
187	#41	—	10	1	no	—
188	#41	—	10	1	no	—
189	#41	—	10	1	no	—
190	C₂×Q₈	—	5	1	no	—
191	C₂×D₄	—	5	2	no	—
192	C₂×D₄	—	5	1	no	—
193	C₂²×C₄	—	5	2	no	—
194	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
195	M₁₆	—	2	1	no	—
196	M₁₆	—	5	2	no	—
197	M₁₆	—	10	1	no	—
198	M₁₆	C₂×C₅×D₅	1	2	yes	—
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	10	1	no	—
201	C₂×C₅	—	2	4	no	—
202	C₂×C₅	—	2	1	no	—
203	C₂×C₅	—	2	1	no	—
204	C₂×C₅	—	2	1	no	—
205	C₂×C₅	—	2	1	no	—
206	C₂×C₅	—	2	1	no	—
207	C₂×C₅	—	2	1	no	—
208	C₂×C₅	—	2	2	no	—
209	C₂×C₅	—	4	2	no	—
210	C₂×C₅	—	4	2	no	—
211	C₂×C₅	—	20	2	no	—
212	C₂×C₅	C₂×#685	1	1	yes	characteristic
213	C₂×C₅	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
214	D₅	—	4	2	no	—
215	C₂³	—	5	2	no	—
216	C₂³	C₄×C₅×D₅	1	1	yes	characteristic
217	Q₈	—	10	1	no	—
218	Q₈	—	10	1	no	—
219	D₄	—	10	2	no	—
220	D₄	—	10	2	no	—
221	D₄	—	10	1	no	—
222	D₄	—	10	1	no	—
223	C₂×C₄	—	5	2	no	—
224	C₂×C₄	—	10	2	no	—
225	C₂×C₄	—	10	2	no	—
226	C₂×C₄	C₄×C₅×D₅	1	1	yes	characteristic
227	C₂×C₄	C₅×D₂₀	1	1	yes	characteristic
228	C₂×C₄	C₅×D₂₀	1	1	yes	characteristic
229	C₂×C₄	C₅×C₂²⋊D₅	1	1	yes	characteristic
230	C₂×C₄	C₅×C₂²⋊D₅	1	1	yes	characteristic
231	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
232	C₈	—	2	2	no	—
233	C₈	—	10	2	no	—
234	C₅	—	2	2	no	—
235	C₅	#65421	1	1	yes	characteristic, direct product factor
236	C₅	C₅×#413	1	1	yes	characteristic, complemented
237	C₂²	—	4	1	no	—
238	C₂²	—	10	2	no	—
239	C₂²	—	20	2	no	—
240	C₂²	C₅×#685	1	1	yes	characteristic
241	C₂²	C₅×#685	1	1	yes	characteristic
242	C₂²	C₂×C₄×C₅×D₅	1	1	yes	characteristic
243	C₄	—	10	2	no	—
244	C₄	C₅×#685	1	1	yes	characteristic
245	C₄	C₅×#685	1	1	yes	characteristic
246	C₄	C₂×C₅×D₂₀	1	1	yes	characteristic
247	C₄	C₂×C₅×C₂²⋊D₅	1	1	yes	characteristic
248	C₂	—	2	1	no	—
249	C₂	—	2	1	no	—
250	C₂	—	2	1	no	—
251	C₂	—	20	2	no	—
252	C₂	C₂×C₅×#685	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	#685
commutator	C₂²×C₅	C₂²×C₄×C₅
Frattini	C₂×C₄	C₂²×C₅×D₅
Fitting	C₂×C₅²×M₁₆	C₂
socle	C₂×C₅²	C₂×C₂²⋊C₄
prime radical	C₂×C₅×#709	C₂
cosocle	C₂×C₄×C₅	C₂³×C₅
hypercenter	C₂×C₅×M₁₆	D₅
hypocenter	C₅	C₅×#413
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₅²×M₁₆	C₂	semidirect
3. C₂×C₅×#681	C₂	semidirect
4. C₅×#3956	C₂	semidirect
5. C₅×#3955	C₂	semidirect
22. #65421	C₅	direct (N × H)
46. C₂×C₅×M₁₆	D₅	semidirect
47. C₂×C₅×M₁₆	C₂×C₅	semidirect
48. C₂×#681	C₂×C₅	semidirect
51. #3956	C₂×C₅	semidirect
52. #3955	C₂×C₅	semidirect
152. C₂×M₁₆	C₅×D₅	semidirect
155. C₅²	#413	semidirect
235. C₅	#65421	direct (N × H)
236. C₅	C₅×#413	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#709	2
C₂×C₅²×M₁₆	2
C₂×C₅×#681	2
C₅×#3956	2
C₅×#3955	2
C₅×#413	5
#65421	5

Subgroups — C5×#65421 #13102392

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65421 #13102392