Subgroups — C₅×#66100 #13106920

Order 1600.

556 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#4147	C₂	1	2	yes	maximal, complemented
2	C₂×C₅²×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4088	C₂	1	2	yes	maximal, complemented
4	C₅×#4065	C₂	1	2	yes	maximal, complemented
5	C₅×#4064	C₂	1	4	yes	maximal, complemented
6	C₅×D₅×Q₁₆	C₂	1	2	yes	maximal, complemented
7	C₅×#4050	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#4047	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅²×Q₈	C₂²	1	2	yes	—
10	C₅×#713	—	2	2	no	—
11	C₅×#713	C₂²	1	4	yes	—
12	C₅×Q₈×D₅	—	2	2	no	—
13	C₅×Q₈×D₅	C₂²	1	4	yes	—
14	C₅×#709	—	2	4	no	—
15	C₅×#709	C₂²	1	1	yes	characteristic
16	C₅×#709	C₂²	1	2	yes	—
17	C₅²×Q₁₆	C₂²	1	2	yes	—
18	C₅²×Q₁₆	C₂²	1	2	yes	—
19	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
20	C₅×#689	C₂²	1	4	yes	—
21	C₅×#688	C₂²	1	4	yes	—
22	C₅×#681	C₂²	1	1	yes	characteristic
23	C₅×Dic₂₀	C₂²	1	1	yes	characteristic
24	C₅×D₄₀	C₂²	1	1	yes	characteristic
25	C₅×#677	C₂²	1	2	yes	—
26	C₅×#676	C₂²	1	2	yes	—
27	C₅×C₈×D₅	C₂²	1	2	yes	—
28	C₅×#578	—	5	1	no	maximal
29	#66100	C₅	1	1	yes	maximal, characteristic, direct product factor
30	C₅²×Q₈	—	2	2	no	—
31	C₅²×Q₈	C₂³	1	4	yes	—
32	C₂×C₄×C₅²	—	2	2	no	—
33	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
34	C₅×C₂²⋊D₅	—	2	2	no	—
35	C₅×C₂²⋊D₅	D₄	1	2	yes	—
36	C₅×D₂₀	—	2	4	no	—
37	C₅×D₂₀	D₄	1	1	yes	characteristic
38	C₅×D₂₀	C₂³	1	2	yes	—
39	C₄×C₅×D₅	—	2	4	no	—
40	C₄×C₅×D₅	—	2	2	no	—
41	C₄×C₅×D₅	—	2	4	no	—
42	C₄×C₅×D₅	C₂³	1	2	yes	—
43	C₅×Dic₁₀	—	2	4	no	—
44	C₅×Dic₁₀	D₄	1	1	yes	characteristic
45	C₅×Dic₁₀	C₂³	1	2	yes	—
46	C₅²×C₈	C₂³	1	1	yes	characteristic
47	C₅²×C₈	C₂³	1	1	yes	characteristic
48	C₅×#168	C₂³	1	2	yes	—
49	C₅×2₋¹⁺⁴	—	5	2	no	—
50	#4147	C₂×C₅	1	2	yes	complemented
51	C₅×#137	—	5	2	no	—
52	C₅×#137	—	5	4	no	—
53	C₅×#135	—	5	2	no	—
54	C₅×#135	—	5	1	no	—
55	C₂×C₅×Q₁₆	—	2	2	no	—
56	C₂×C₅×Q₁₆	—	5	2	no	—
57	C₂×C₅×Q₁₆	D₅	1	1	yes	characteristic, complemented
58	C₂×C₅×Q₁₆	C₂×C₅	1	1	yes	characteristic, complemented
59	C₅×#131	—	5	1	no	—
60	#4088	C₂×C₅	1	2	yes	complemented
61	#4065	C₂×C₅	1	2	yes	complemented
62	#4064	C₂×C₅	1	4	yes	complemented
63	D₅×Q₁₆	C₂×C₅	1	2	yes	complemented
64	#4050	C₂×C₅	1	1	yes	characteristic, complemented
65	#4047	C₂×C₅	1	1	yes	characteristic, complemented
66	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
67	C₂×C₅×D₅	—	2	2	no	—
68	C₂×C₅×D₅	C₂×D₄	1	2	yes	—
69	C₄×C₅²	—	2	4	no	—
70	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
71	C₄×C₅²	C₂⁴	1	1	yes	characteristic
72	C₅×Dic₅	—	2	2	no	—
73	C₅×Dic₅	C₂×D₄	1	2	yes	—
74	C₅×#41	—	5	1	no	—
75	C₅×#41	—	5	4	no	—
76	C₅×#41	—	5	2	no	—
77	C₅×#41	—	10	4	no	—
78	C₅×#41	—	10	2	no	—
79	C₂×C₅×Q₈	—	2	4	no	—
80	C₂×C₅×Q₈	—	5	4	no	—
81	C₂×C₅×Q₈	—	10	2	no	—
82	C₂×C₅×Q₈	C₂×D₅	1	2	yes	—
83	C₂×C₅×Q₈	C₂²×C₅	1	2	yes	—
84	#713	—	2	2	no	—
85	#713	C₂²×C₅	1	4	yes	—
86	Q₈×D₅	—	2	2	no	—
87	Q₈×D₅	C₂²×C₅	1	4	yes	—
88	#709	—	2	4	no	—
89	#709	C₂²×C₅	1	1	yes	characteristic
90	#709	C₂²×C₅	1	2	yes	—
91	C₅×Q₁₆	—	2	4	no	—
92	C₅×Q₁₆	—	2	4	no	—
93	C₅×Q₁₆	—	5	4	no	—
94	C₅×Q₁₆	—	5	1	no	—
95	C₅×Q₁₆	C₂×D₅	1	2	yes	—
96	C₅×Q₁₆	C₂×D₅	1	2	yes	—
97	C₅×Q₁₆	C₂²×C₅	1	2	yes	—
98	C₅×Q₁₆	C₂²×C₅	1	2	yes	—
99	C₅×SD₁₆	—	5	4	no	—
100	C₅×SD₁₆	—	5	2	no	—
101	C₅×D₈	—	5	1	no	—
102	C₅×M₁₆	—	5	1	no	—
103	C₅×M₁₆	—	5	2	no	—
104	C₂×C₅×C₈	—	2	2	no	—
105	C₂×C₅×C₈	—	5	2	no	—
106	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic
107	C₂×C₅×C₈	C₂²×C₅	1	1	yes	characteristic
108	#689	C₂²×C₅	1	4	yes	—
109	#688	C₂²×C₅	1	4	yes	—
110	#681	C₂²×C₅	1	1	yes	characteristic
111	Dic₂₀	C₂²×C₅	1	1	yes	characteristic
112	D₄₀	C₂²×C₅	1	1	yes	characteristic
113	#677	C₂²×C₅	1	2	yes	—
114	#676	C₂²×C₅	1	2	yes	—
115	C₈×D₅	C₂²×C₅	1	2	yes	—
116	#578	—	5	1	no	Sylow
117	C₂×C₅²	—	2	1	no	—
118	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
119	C₅×D₅	—	2	2	no	—
120	C₅×D₅	—	4	2	no	—
121	C₅×Q₈	—	2	2	no	—
122	C₅×Q₈	—	2	2	no	—
123	C₅×Q₈	—	2	8	no	—
124	C₅×Q₈	—	4	4	no	—
125	C₅×Q₈	—	5	1	no	—
126	C₅×Q₈	—	5	2	no	—
127	C₅×Q₈	—	10	4	no	—
128	C₅×Q₈	C₂²×D₅	1	4	yes	—
129	C₅×Q₈	C₂³×C₅	1	4	yes	—
130	C₅×D₄	—	5	2	no	—
131	C₅×D₄	—	5	1	no	—
132	C₅×D₄	—	5	2	no	—
133	C₅×D₄	—	10	4	no	—
134	C₅×D₄	—	10	2	no	—
135	C₂×C₄×C₅	—	2	2	no	—
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	—	4	4	no	—
139	C₂×C₄×C₅	—	5	2	no	—
140	C₂×C₄×C₅	—	10	4	no	—
141	C₂×C₄×C₅	—	10	4	no	—
142	C₂×C₄×C₅	—	10	2	no	—
143	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
145	C₂²⋊D₅	—	2	2	no	—
146	C₂²⋊D₅	C₅×D₄	1	2	yes	—
147	D₂₀	—	2	4	no	—
148	D₂₀	C₅×D₄	1	1	yes	characteristic
149	D₂₀	C₂³×C₅	1	2	yes	—
150	C₄×D₅	—	2	4	no	—
151	C₄×D₅	—	2	2	no	—
152	C₄×D₅	—	2	4	no	—
153	C₄×D₅	C₂³×C₅	1	2	yes	—
154	Dic₁₀	—	2	4	no	—
155	Dic₁₀	C₅×D₄	1	1	yes	characteristic
156	Dic₁₀	C₂³×C₅	1	2	yes	—
157	C₅×C₈	—	2	2	no	—
158	C₅×C₈	—	2	2	no	—
159	C₅×C₈	—	5	2	no	—
160	C₅×C₈	C₂²×D₅	1	1	yes	characteristic
161	C₅×C₈	C₂²×D₅	1	1	yes	characteristic
162	C₅×C₈	C₂³×C₅	1	1	yes	characteristic
163	C₅×C₈	C₂³×C₅	1	1	yes	characteristic
164	#168	C₂³×C₅	1	2	yes	—
165	2₋¹⁺⁴	—	5	2	no	—
166	#137	—	5	2	no	—
167	#137	—	5	4	no	—
168	#135	—	5	2	no	—
169	#135	—	5	1	no	—
170	C₂×Q₁₆	—	5	2	no	—
171	C₂×Q₁₆	C₅×D₅	1	1	yes	characteristic, complemented
172	#131	—	5	1	no	—
173	C₅²	#578	1	1	yes	characteristic, complemented, Sylow
174	C₂²×C₅	—	2	2	no	—
175	C₂²×C₅	—	5	2	no	—
176	C₂²×C₅	—	10	2	no	—
177	C₂²×C₅	D₄×D₅	1	1	yes	characteristic
178	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
179	C₂×D₅	—	2	2	no	—
180	C₂×D₅	C₂×C₅×D₄	1	2	yes	—
181	C₄×C₅	—	2	4	no	—
182	C₄×C₅	—	2	4	no	—
183	C₄×C₅	—	2	2	no	—
184	C₄×C₅	—	2	2	no	—
185	C₄×C₅	—	4	8	no	—
186	C₄×C₅	—	5	2	no	—
187	C₄×C₅	—	10	2	no	—
188	C₄×C₅	D₄×D₅	1	1	yes	characteristic
189	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
190	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
191	C₄×C₅	C₂⁴×C₅	1	1	yes	characteristic
192	Dic₅	—	2	2	no	—
193	Dic₅	C₂×C₅×D₄	1	2	yes	—
194	#41	—	5	1	no	—
195	#41	—	5	4	no	—
196	#41	—	5	2	no	—
197	#41	—	10	4	no	—
198	#41	—	10	2	no	—
199	C₂×Q₈	—	5	4	no	—
200	C₂×Q₈	—	10	2	no	—
201	C₂×Q₈	C₂×C₅×D₅	1	2	yes	—
202	Q₁₆	—	5	4	no	—
203	Q₁₆	—	5	1	no	—
204	Q₁₆	C₂×C₅×D₅	1	2	yes	—
205	Q₁₆	C₂×C₅×D₅	1	2	yes	—
206	SD₁₆	—	5	4	no	—
207	SD₁₆	—	5	2	no	—
208	D₈	—	5	1	no	—
209	M₁₆	—	5	1	no	—
210	M₁₆	—	5	2	no	—
211	C₂×C₈	—	5	2	no	—
212	C₂×C₈	C₂×C₅×D₅	1	1	yes	characteristic
213	C₂×C₅	—	2	4	no	—
214	C₂×C₅	—	2	1	no	—
215	C₂×C₅	—	2	1	no	—
216	C₂×C₅	—	2	2	no	—
217	C₂×C₅	—	10	2	no	—
218	C₂×C₅	—	20	2	no	—
219	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
220	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
221	D₅	—	2	2	no	—
222	D₅	—	4	2	no	—
223	Q₈	—	2	2	no	—
224	Q₈	—	5	1	no	—
225	Q₈	—	5	2	no	—
226	Q₈	—	10	4	no	—
227	Q₈	C₂²×C₅×D₅	1	4	yes	—
228	D₄	—	5	2	no	—
229	D₄	—	5	1	no	—
230	D₄	—	5	2	no	—
231	D₄	—	10	4	no	—
232	D₄	—	10	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	5	2	no	—
235	C₂×C₄	—	10	4	no	—
236	C₂×C₄	—	10	4	no	—
237	C₂×C₄	—	10	2	no	—
238	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
239	C₈	—	5	2	no	—
240	C₈	C₂²×C₅×D₅	1	1	yes	characteristic
241	C₈	C₂²×C₅×D₅	1	1	yes	characteristic
242	C₅	—	2	2	no	—
243	C₅	#66100	1	1	yes	characteristic, direct product factor
244	C₅	C₅×#578	1	1	yes	characteristic, complemented
245	C₂²	—	5	2	no	—
246	C₂²	—	10	2	no	—
247	C₂²	C₅×D₄×D₅	1	1	yes	characteristic
248	C₄	—	2	4	no	—
249	C₄	—	5	2	no	—
250	C₄	—	10	2	no	—
251	C₄	C₅×D₄×D₅	1	1	yes	characteristic
252	C₄	C₂³×C₅×D₅	1	1	yes	characteristic
253	C₂	—	2	1	no	—
254	C₂	—	10	2	no	—
255	C₂	—	20	2	no	—
256	C₂	C₂×C₅×D₄×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	C₂×D₄×D₅
commutator, cosocle	C₄×C₅	C₂⁴×C₅
Frattini	C₄	C₂³×C₅×D₅
Fitting	C₂×C₅²×Q₁₆	C₂
socle	C₂×C₅²	C₂²×D₄
hypercenter	C₂×C₅×Q₁₆	D₅
hypocenter	C₅	C₅×#578
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₅×#4147	C₂	semidirect
2. C₂×C₅²×Q₁₆	C₂	semidirect
3. C₅×#4088	C₂	semidirect
4. C₅×#4065	C₂	semidirect
5. C₅×#4064	C₂	semidirect
6. C₅×D₅×Q₁₆	C₂	semidirect
7. C₅×#4050	C₂	semidirect
8. C₅×#4047	C₂	semidirect
29. #66100	C₅	direct (N × H)
50. #4147	C₂×C₅	semidirect
57. C₂×C₅×Q₁₆	D₅	semidirect
58. C₂×C₅×Q₁₆	C₂×C₅	semidirect
60. #4088	C₂×C₅	semidirect
61. #4065	C₂×C₅	semidirect
62. #4064	C₂×C₅	semidirect
63. D₅×Q₁₆	C₂×C₅	semidirect
64. #4050	C₂×C₅	semidirect
65. #4047	C₂×C₅	semidirect
171. C₂×Q₁₆	C₅×D₅	semidirect
173. C₅²	#578	semidirect
243. C₅	#66100	direct (N × H)
244. C₅	C₅×#578	semidirect

Maximal subgroups

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

subgroup	index
C₅×#4147	2
C₂×C₅²×Q₁₆	2
C₅×#4088	2
C₅×#4065	2
C₅×#4064	2
C₅×D₅×Q₁₆	2
C₅×#4050	2
C₅×#4047	2
C₅×#578	5
#66100	5

Subgroups — C5×#66100 #13106920

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#66100 #13106920