Subgroups — C₅×#65913 #13106733

Order 1600.

526 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#4092	C₂	1	2	yes	maximal, complemented
3	C₅×#4042	C₂	1	4	yes	maximal, complemented
4	C₅×D₅×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₄×C₅×D₂₀	C₂	1	1	yes	maximal, characteristic
6	C₄×C₅×D₂₀	C₂	1	2	yes	maximal
7	C₅×#4016	C₂	1	2	yes	maximal, complemented
8	C₅×#4015	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×D₅	C₂²	1	2	yes	—
12	C₂×C₄×C₅×D₅	C₂²	1	4	yes	—
13	C₂×C₅×Dic₁₀	C₂²	1	2	yes	—
14	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₅²×C₄⋊C₄	C₂²	1	2	yes	—
16	C₄²×C₅²	C₂²	1	1	yes	characteristic
17	C₄²×C₅²	C₂²	1	2	yes	—
18	C₅×#685	C₂²	1	2	yes	—
19	C₅×#685	C₂²	1	4	yes	—
20	C₅×C₄⋊Dic₅	C₂²	1	2	yes	—
21	C₅×C₄⋊Dic₅	C₂²	1	2	yes	—
22	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
23	C₅×#683	C₂²	1	4	yes	—
24	C₅×#683	C₂²	1	4	yes	—
25	C₅×#554	—	5	1	no	maximal
26	#65913	C₅	1	1	yes	maximal, characteristic, direct product factor
27	C₂²×C₅×D₅	C₂³	1	2	yes	—
28	C₅²×Q₈	—	2	2	no	—
29	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₅²	C₂³	1	2	yes	—
31	C₂×C₄×C₅²	C₂³	1	2	yes	—
32	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
34	C₂×C₅×Dic₅	C₂³	1	4	yes	—
35	C₂×C₅×Dic₅	C₂³	1	2	yes	—
36	C₅×D₂₀	Q₈	1	4	yes	complemented
37	C₄×C₅×D₅	—	2	4	no	—
38	C₄×C₅×D₅	—	4	4	no	—
39	C₅×Dic₁₀	—	4	2	no	—
40	C₅×C₄⋊Q₈	—	5	1	no	—
41	C₅×#125	—	5	2	no	—
42	C₅×C₂²⋊Q₈	—	5	2	no	—
43	C₅×C₂²⋊Q₈	—	5	4	no	—
44	C₄×C₅×Q₈	—	2	2	no	—
45	C₄×C₅×Q₈	D₅	1	1	yes	characteristic, complemented
46	C₄×C₅×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
47	C₄×C₅×D₄	—	5	2	no	—
48	C₄×C₅×D₄	—	5	1	no	—
49	C₂×C₅×C₄⋊C₄	—	5	2	no	—
50	#4092	C₂×C₅	1	2	yes	complemented
51	#4042	C₂×C₅	1	4	yes	complemented
52	D₅×C₄⋊C₄	C₂×C₅	1	2	yes	complemented
53	C₄×D₂₀	C₂×C₅	1	2	yes	—
54	C₄×D₂₀	C₂×C₅	1	1	yes	characteristic
55	#4016	C₂×C₅	1	2	yes	complemented
56	#4015	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
58	C₂×C₅×D₅	—	2	2	no	—
59	C₂×C₅×D₅	—	2	2	no	—
60	C₂×C₅×D₅	C₂×Q₈	1	4	yes	—
61	C₄×C₅²	—	2	2	no	—
62	C₄×C₅²	—	2	1	no	—
63	C₄×C₅²	—	2	2	no	—
64	C₄×C₅²	C₂×Q₈	1	2	yes	—
65	C₄×C₅²	#41	1	2	yes	—
66	C₅×Dic₅	—	2	2	no	—
67	C₅×Dic₅	—	2	4	no	—
68	C₂×C₅×Q₈	—	2	2	no	—
69	C₂×C₅×Q₈	—	5	2	no	—
70	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic
71	C₂×C₅×Q₈	C₂²×C₅	1	1	yes	characteristic
72	C₂×C₅×D₄	—	5	1	no	—
73	C₂²×C₄×C₅	—	5	2	no	—
74	C₂²×C₄×C₅	—	5	4	no	—
75	C₂×D₂₀	C₂²×C₅	1	1	yes	characteristic
76	C₂×C₄×D₅	C₂²×C₅	1	2	yes	—
77	C₂×C₄×D₅	C₂²×C₅	1	4	yes	—
78	C₂×Dic₁₀	C₂²×C₅	1	2	yes	—
79	C₅×C₄⋊C₄	—	2	2	no	—
80	C₅×C₄⋊C₄	—	2	4	no	—
81	C₅×C₄⋊C₄	—	5	4	no	—
82	C₅×C₄⋊C₄	—	5	2	no	—
83	C₅×C₄⋊C₄	—	5	1	no	—
84	C₅×C₄⋊C₄	—	5	2	no	—
85	C₅×C₄⋊C₄	—	5	4	no	—
86	C₅×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
87	C₅×C₄⋊C₄	C₂×D₅	1	2	yes	—
88	C₅×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
89	C₅×C₄⋊C₄	C₂²×C₅	1	2	yes	—
90	C₅×C₂²⋊C₄	—	5	2	no	—
91	C₅×C₂²⋊C₄	—	5	4	no	—
92	C₄²×C₅	—	2	4	no	—
93	C₄²×C₅	—	2	2	no	—
94	C₄²×C₅	C₂×D₅	1	2	yes	—
95	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
96	C₄²×C₅	C₂²×C₅	1	1	yes	characteristic
97	C₄²×C₅	C₂²×C₅	1	2	yes	—
98	#685	C₂²×C₅	1	2	yes	—
99	#685	C₂²×C₅	1	4	yes	—
100	C₄⋊Dic₅	C₂²×C₅	1	2	yes	—
101	C₄⋊Dic₅	C₂²×C₅	1	2	yes	—
102	C₄⋊Dic₅	C₂²×C₅	1	1	yes	characteristic
103	#683	C₂²×C₅	1	4	yes	—
104	#683	C₂²×C₅	1	4	yes	—
105	#554	—	5	1	no	Sylow
106	C₂×C₅²	C₂²×Q₈	1	1	yes	characteristic
107	C₂×C₅²	C₂×#41	1	1	yes	characteristic
108	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
109	C₅×D₅	—	2	4	no	—
110	C₂³×C₅	—	5	2	no	—
111	C₂²×D₅	C₂³×C₅	1	2	yes	—
112	C₅×Q₈	—	2	8	no	—
113	C₅×Q₈	—	2	2	no	—
114	C₅×Q₈	—	2	2	no	—
115	C₅×Q₈	—	20	2	no	—
116	C₅×D₄	—	5	4	no	—
117	C₂×C₄×C₅	—	2	2	no	—
118	C₂×C₄×C₅	—	2	4	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	2	4	no	—
122	C₂×C₄×C₅	—	5	2	no	—
123	C₂×C₄×C₅	—	5	4	no	—
124	C₂×C₄×C₅	—	10	4	no	—
125	C₂×C₄×C₅	—	20	4	no	—
126	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
127	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
129	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
132	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
133	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
136	C₂×Dic₅	C₂³×C₅	1	2	yes	—
137	C₂×Dic₅	C₂³×C₅	1	4	yes	—
138	D₂₀	C₅×Q₈	1	4	yes	complemented
139	C₄×D₅	—	2	4	no	—
140	C₄×D₅	—	4	4	no	—
141	Dic₁₀	—	4	2	no	—
142	C₄⋊Q₈	—	5	1	no	—
143	#125	—	5	2	no	—
144	C₂²⋊Q₈	—	5	2	no	—
145	C₂²⋊Q₈	—	5	4	no	—
146	C₄×Q₈	C₅×D₅	1	1	yes	characteristic, complemented
147	C₄×D₄	—	5	2	no	—
148	C₄×D₄	—	5	1	no	—
149	C₂×C₄⋊C₄	—	5	2	no	—
150	C₅²	#554	1	1	yes	characteristic, complemented, Sylow
151	C₂²×C₅	—	2	2	no	—
152	C₂²×C₅	—	5	4	no	—
153	C₂²×C₅	—	10	2	no	—
154	C₂²×C₅	—	10	2	no	—
155	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
156	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
157	C₂×D₅	—	2	2	no	—
158	C₂×D₅	—	2	2	no	—
159	C₂×D₅	C₂×C₅×Q₈	1	4	yes	—
160	C₄×C₅	—	2	8	no	—
161	C₄×C₅	—	2	2	no	—
162	C₄×C₅	—	2	2	no	—
163	C₄×C₅	—	2	1	no	—
164	C₄×C₅	—	2	2	no	—
165	C₄×C₅	—	2	1	no	—
166	C₄×C₅	—	2	2	no	—
167	C₄×C₅	—	2	4	no	—
168	C₄×C₅	—	2	4	no	—
169	C₄×C₅	—	4	4	no	—
170	C₄×C₅	—	4	2	no	—
171	C₄×C₅	—	10	2	no	—
172	C₄×C₅	—	10	4	no	—
173	C₄×C₅	#709	1	2	yes	—
174	C₄×C₅	Q₈×D₅	1	2	yes	—
175	C₄×C₅	C₂×C₅×Q₈	1	2	yes	—
176	C₄×C₅	C₅×#41	1	2	yes	—
177	Dic₅	—	2	2	no	—
178	Dic₅	—	2	4	no	—
179	C₂×Q₈	—	5	2	no	—
180	C₂×Q₈	C₂×C₅×D₅	1	1	yes	characteristic
181	C₂×D₄	—	5	1	no	—
182	C₂²×C₄	—	5	2	no	—
183	C₂²×C₄	—	5	4	no	—
184	C₄⋊C₄	—	5	4	no	—
185	C₄⋊C₄	—	5	2	no	—
186	C₄⋊C₄	—	5	1	no	—
187	C₄⋊C₄	—	5	2	no	—
188	C₄⋊C₄	—	5	4	no	—
189	C₄⋊C₄	C₂×C₅×D₅	1	1	yes	characteristic
190	C₄⋊C₄	C₂×C₅×D₅	1	2	yes	—
191	C₂²⋊C₄	—	5	2	no	—
192	C₂²⋊C₄	—	5	4	no	—
193	C₄²	C₂×C₅×D₅	1	2	yes	—
194	C₄²	C₂×C₅×D₅	1	1	yes	characteristic
195	C₂×C₅	—	2	2	no	—
196	C₂×C₅	—	2	2	no	—
197	C₂×C₅	—	2	2	no	—
198	C₂×C₅	—	10	4	no	—
199	C₂×C₅	C₂×#709	1	1	yes	characteristic
200	C₂×C₅	C₂×Q₈×D₅	1	1	yes	characteristic
201	C₂×C₅	#4149	1	1	yes	characteristic
202	C₂×C₅	C₂²×C₅×Q₈	1	1	yes	characteristic
203	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
204	C₂×C₅	C₅×H(5,𝔽₂)	1	1	yes	characteristic
205	D₅	—	2	4	no	—
206	C₂³	—	5	2	no	—
207	Q₈	—	2	2	no	—
208	Q₈	—	20	2	no	—
209	D₄	—	5	4	no	—
210	C₂×C₄	—	5	2	no	—
211	C₂×C₄	—	5	4	no	—
212	C₂×C₄	—	10	4	no	—
213	C₂×C₄	—	20	4	no	—
214	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
215	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
216	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
217	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
218	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
219	C₅	—	2	2	no	—
220	C₅	#65913	1	1	yes	characteristic, direct product factor
221	C₅	C₅×#554	1	1	yes	characteristic, complemented
222	C₂²	—	5	4	no	—
223	C₂²	—	10	2	no	—
224	C₂²	—	10	2	no	—
225	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	2	2	no	—
228	C₄	—	2	1	no	—
229	C₄	—	10	2	no	—
230	C₄	—	10	4	no	—
231	C₄	C₅×#709	1	2	yes	—
232	C₄	C₅×Q₈×D₅	1	2	yes	—
233	C₂	—	10	4	no	—
234	C₂	C₂×C₅×#709	1	1	yes	characteristic
235	C₂	C₂×C₅×Q₈×D₅	1	1	yes	characteristic
236	C₂	C₅×#4149	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting	C₄×C₅²×Q₈	C₂
socle	C₂²×C₅²	C₂⁴
prime radical	C₂×C₅×D₂₀	C₂²
hypercenter	C₄×C₅×Q₈	D₅
hypocenter	C₅	C₅×#554
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅ ◁ C₄×C₅×Q₈

Complements

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

N	H ≅ G/N	type
1. C₄×C₅²×Q₈	C₂	semidirect
2. C₅×#4092	C₂	semidirect
3. C₅×#4042	C₂	semidirect
4. C₅×D₅×C₄⋊C₄	C₂	semidirect
7. C₅×#4016	C₂	semidirect
8. C₅×#4015	C₂	semidirect
26. #65913	C₅	direct (N × H)
36. C₅×D₂₀	Q₈	semidirect
45. C₄×C₅×Q₈	D₅	semidirect
46. C₄×C₅×Q₈	C₂×C₅	semidirect
50. #4092	C₂×C₅	semidirect
51. #4042	C₂×C₅	semidirect
52. D₅×C₄⋊C₄	C₂×C₅	semidirect
55. #4016	C₂×C₅	semidirect
56. #4015	C₂×C₅	semidirect
138. D₂₀	C₅×Q₈	semidirect
146. C₄×Q₈	C₅×D₅	semidirect
150. C₅²	#554	semidirect
220. C₅	#65913	direct (N × H)
221. C₅	C₅×#554	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅²×Q₈	2
C₅×#4092	2
C₅×#4042	2
C₅×D₅×C₄⋊C₄	2
C₄×C₅×D₂₀	2
C₅×#4016	2
C₅×#4015	2
C₅×#554	5
#65913	5

Subgroups — C5×#65913 #13106733

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65913 #13106733