Subgroups — C₅×#66105 #13106925

Order 1600.

558 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#4150	C₂	1	2	yes	maximal, complemented
2	C₅²×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4097	C₂	1	2	yes	maximal, complemented
4	C₅×D₅×Q₁₆	C₂	1	2	yes	maximal, complemented
5	C₅×#4061	C₂	1	4	yes	maximal, complemented
6	C₅×#4058	C₂	1	2	yes	maximal, complemented
7	C₂×C₅×Dic₂₀	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#4047	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅²×#41	C₂²	1	2	yes	—
10	C₅×Q₈×D₅	C₂²	1	4	yes	—
11	C₅×#711	—	2	4	no	—
12	C₅×#711	C₂²	1	4	yes	—
13	C₅×#709	—	2	2	no	—
14	C₅×#709	C₂²	1	1	yes	characteristic
15	C₂×C₅×Dic₁₀	—	2	2	no	—
16	C₂×C₅×Dic₁₀	C₂²	1	2	yes	—
17	C₅²×Q₁₆	C₂²	1	1	yes	characteristic
18	C₅²×SD₁₆	C₂²	1	2	yes	—
19	C₅²×D₈	C₂²	1	1	yes	characteristic
20	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
21	C₅×#689	C₂²	1	4	yes	—
22	C₅×#687	C₂²	1	4	yes	—
23	C₅×#681	C₂²	1	1	yes	characteristic
24	C₅×Dic₂₀	C₂²	1	2	yes	—
25	C₅×Dic₂₀	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	#66105	C₅	1	1	yes	maximal, characteristic, direct product factor
30	C₅²×Q₈	C₂³	1	2	yes	—
31	C₅²×D₄	—	2	2	no	—
32	C₅²×D₄	C₂³	1	2	yes	—
33	C₂×C₄×C₅²	—	2	2	no	—
34	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
35	C₅×C₂²⋊D₅	—	2	4	no	—
36	C₅×C₂²⋊D₅	D₄	1	2	yes	—
37	C₂×C₅×Dic₅	—	2	2	no	—
38	C₂×C₅×Dic₅	—	2	4	no	—
39	C₅×D₂₀	D₄	1	1	yes	characteristic
40	C₄×C₅×D₅	—	2	4	no	—
41	C₄×C₅×D₅	C₂³	1	2	yes	—
42	C₅×Dic₁₀	—	2	2	no	—
43	C₅×Dic₁₀	—	2	4	no	—
44	C₅×Dic₁₀	D₄	1	1	yes	characteristic
45	C₅×Dic₁₀	C₂³	1	4	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	#4150	C₂×C₅	1	2	yes	complemented
51	C₅×#137	—	5	2	no	—
52	C₅×#137	—	5	4	no	—
53	C₅×#135	—	2	2	no	—
54	C₅×#135	—	5	2	no	—
55	C₅×#135	D₅	1	1	yes	characteristic, complemented
56	C₅×#135	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂×C₅×Q₁₆	—	5	2	no	—
58	C₂×C₅×Q₁₆	—	5	1	no	—
59	C₅×#131	—	5	1	no	—
60	#4097	C₂×C₅	1	2	yes	complemented
61	D₅×Q₁₆	C₂×C₅	1	2	yes	complemented
62	#4061	C₂×C₅	1	4	yes	complemented
63	#4058	C₂×C₅	1	2	yes	complemented
64	C₂×Dic₂₀	C₂×C₅	1	1	yes	characteristic, complemented
65	#4047	C₂×C₅	1	1	yes	characteristic, complemented
66	C₂²×C₅²	—	2	2	no	—
67	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
68	C₂×C₅×D₅	C₂×D₄	1	2	yes	—
69	C₄×C₅²	—	2	2	no	—
70	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
71	C₄×C₅²	C₂⁴	1	1	yes	characteristic
72	C₅×Dic₅	—	2	4	no	—
73	C₅×Dic₅	C₂×D₄	1	2	yes	—
74	C₅×#41	—	2	4	no	—
75	C₅×#41	—	5	1	no	—
76	C₅×#41	—	5	4	no	—
77	C₅×#41	—	10	2	no	—
78	C₅×#41	—	10	4	no	—
79	C₅×#41	C₂×D₅	1	2	yes	—
80	C₅×#41	C₂²×C₅	1	2	yes	—
81	C₂×C₅×Q₈	—	5	4	no	—
82	C₂×C₅×Q₈	—	5	2	no	—
83	C₂×C₅×Q₈	—	10	2	no	—
84	Q₈×D₅	C₂²×C₅	1	4	yes	—
85	#711	—	2	4	no	—
86	#711	C₂²×C₅	1	4	yes	—
87	#709	—	2	2	no	—
88	#709	C₂²×C₅	1	1	yes	characteristic
89	C₂×Dic₁₀	—	2	2	no	—
90	C₂×Dic₁₀	C₂²×C₅	1	2	yes	—
91	C₅×Q₁₆	—	2	2	no	—
92	C₅×Q₁₆	—	5	4	no	—
93	C₅×Q₁₆	—	5	2	no	—
94	C₅×Q₁₆	—	5	2	no	—
95	C₅×Q₁₆	C₂×D₅	1	1	yes	characteristic
96	C₅×Q₁₆	C₂²×C₅	1	1	yes	characteristic
97	C₅×SD₁₆	—	2	4	no	—
98	C₅×SD₁₆	—	5	4	no	—
99	C₅×SD₁₆	C₂×D₅	1	2	yes	—
100	C₅×SD₁₆	C₂²×C₅	1	2	yes	—
101	C₅×D₈	—	2	2	no	—
102	C₅×D₈	C₂×D₅	1	1	yes	characteristic
103	C₅×D₈	C₂²×C₅	1	1	yes	characteristic
104	C₅×M₁₆	—	5	1	no	—
105	C₅×M₁₆	—	5	2	no	—
106	C₂×C₅×C₈	—	2	2	no	—
107	C₂×C₅×C₈	—	5	2	no	—
108	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic
109	C₂×C₅×C₈	C₂²×C₅	1	1	yes	characteristic
110	#689	C₂²×C₅	1	4	yes	—
111	#687	C₂²×C₅	1	4	yes	—
112	#681	C₂²×C₅	1	1	yes	characteristic
113	Dic₂₀	C₂²×C₅	1	2	yes	—
114	Dic₂₀	C₂²×C₅	1	2	yes	—
115	#676	C₂²×C₅	1	2	yes	—
116	C₈×D₅	C₂²×C₅	1	2	yes	—
117	#578	—	5	1	no	Sylow
118	C₂×C₅²	—	2	1	no	—
119	C₂×C₅²	—	4	2	no	—
120	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
121	C₅×D₅	—	2	2	no	—
122	C₅×Q₈	—	2	4	no	—
123	C₅×Q₈	—	5	1	no	—
124	C₅×Q₈	—	5	4	no	—
125	C₅×Q₈	—	10	4	no	—
126	C₅×Q₈	—	10	2	no	—
127	C₅×Q₈	C₂²×D₅	1	2	yes	—
128	C₅×Q₈	C₂³×C₅	1	2	yes	—
129	C₅×D₄	—	2	2	no	—
130	C₅×D₄	—	2	2	no	—
131	C₅×D₄	—	2	4	no	—
132	C₅×D₄	—	4	4	no	—
133	C₅×D₄	—	5	2	no	—
134	C₅×D₄	—	5	1	no	—
135	C₅×D₄	—	10	4	no	—
136	C₅×D₄	C₂²×D₅	1	2	yes	—
137	C₅×D₄	C₂³×C₅	1	2	yes	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	—	2	2	no	—
140	C₂×C₄×C₅	—	2	2	no	—
141	C₂×C₄×C₅	—	4	4	no	—
142	C₂×C₄×C₅	—	5	2	no	—
143	C₂×C₄×C₅	—	10	4	no	—
144	C₂×C₄×C₅	—	10	4	no	—
145	C₂×C₄×C₅	—	10	2	no	—
146	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
148	C₂²⋊D₅	—	2	4	no	—
149	C₂²⋊D₅	C₅×D₄	1	2	yes	—
150	C₂×Dic₅	—	2	2	no	—
151	C₂×Dic₅	—	2	4	no	—
152	D₂₀	C₅×D₄	1	1	yes	characteristic
153	C₄×D₅	—	2	4	no	—
154	C₄×D₅	C₂³×C₅	1	2	yes	—
155	Dic₁₀	—	2	2	no	—
156	Dic₁₀	—	2	4	no	—
157	Dic₁₀	C₅×D₄	1	1	yes	characteristic
158	Dic₁₀	C₂³×C₅	1	4	yes	—
159	C₅×C₈	—	2	2	no	—
160	C₅×C₈	—	2	2	no	—
161	C₅×C₈	—	5	2	no	—
162	C₅×C₈	C₂²×D₅	1	1	yes	characteristic
163	C₅×C₈	C₂²×D₅	1	1	yes	characteristic
164	C₅×C₈	C₂³×C₅	1	1	yes	characteristic
165	C₅×C₈	C₂³×C₅	1	1	yes	characteristic
166	#168	C₂³×C₅	1	2	yes	—
167	2₋¹⁺⁴	—	5	2	no	—
168	#137	—	5	2	no	—
169	#137	—	5	4	no	—
170	#135	—	5	2	no	—
171	#135	C₅×D₅	1	1	yes	characteristic, complemented
172	C₂×Q₁₆	—	5	2	no	—
173	C₂×Q₁₆	—	5	1	no	—
174	#131	—	5	1	no	—
175	C₅²	#578	1	1	yes	characteristic, complemented, Sylow
176	C₂²×C₅	—	2	2	no	—
177	C₂²×C₅	—	2	2	no	—
178	C₂²×C₅	—	2	2	no	—
179	C₂²×C₅	—	4	4	no	—
180	C₂²×C₅	—	5	2	no	—
181	C₂²×C₅	D₄×D₅	1	1	yes	characteristic
182	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
183	C₂×D₅	C₂×C₅×D₄	1	2	yes	—
184	C₄×C₅	—	2	2	no	—
185	C₄×C₅	—	2	2	no	—
186	C₄×C₅	—	2	2	no	—
187	C₄×C₅	—	2	2	no	—
188	C₄×C₅	—	4	4	no	—
189	C₄×C₅	—	5	2	no	—
190	C₄×C₅	—	10	4	no	—
191	C₄×C₅	D₄×D₅	1	1	yes	characteristic
192	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
193	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
194	C₄×C₅	C₂⁴×C₅	1	1	yes	characteristic
195	Dic₅	—	2	4	no	—
196	Dic₅	C₂×C₅×D₄	1	2	yes	—
197	#41	—	5	1	no	—
198	#41	—	5	4	no	—
199	#41	—	10	2	no	—
200	#41	—	10	4	no	—
201	#41	C₂×C₅×D₅	1	2	yes	—
202	C₂×Q₈	—	5	4	no	—
203	C₂×Q₈	—	5	2	no	—
204	C₂×Q₈	—	10	2	no	—
205	Q₁₆	—	5	4	no	—
206	Q₁₆	—	5	2	no	—
207	Q₁₆	—	5	2	no	—
208	Q₁₆	C₂×C₅×D₅	1	1	yes	characteristic
209	SD₁₆	—	5	4	no	—
210	SD₁₆	C₂×C₅×D₅	1	2	yes	—
211	D₈	C₂×C₅×D₅	1	1	yes	characteristic
212	M₁₆	—	5	1	no	—
213	M₁₆	—	5	2	no	—
214	C₂×C₈	—	5	2	no	—
215	C₂×C₈	C₂×C₅×D₅	1	1	yes	characteristic
216	C₂×C₅	—	2	1	no	—
217	C₂×C₅	—	2	1	no	—
218	C₂×C₅	—	2	2	no	—
219	C₂×C₅	—	4	2	no	—
220	C₂×C₅	—	4	2	no	—
221	C₂×C₅	—	4	2	no	—
222	C₂×C₅	—	8	4	no	—
223	C₂×C₅	—	10	2	no	—
224	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
225	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
226	D₅	—	2	2	no	—
227	Q₈	—	5	1	no	—
228	Q₈	—	5	4	no	—
229	Q₈	—	10	4	no	—
230	Q₈	—	10	2	no	—
231	Q₈	C₂²×C₅×D₅	1	2	yes	—
232	D₄	—	2	2	no	—
233	D₄	—	5	2	no	—
234	D₄	—	5	1	no	—
235	D₄	—	10	4	no	—
236	D₄	C₂²×C₅×D₅	1	2	yes	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	5	2	no	—
239	C₂×C₄	—	10	4	no	—
240	C₂×C₄	—	10	4	no	—
241	C₂×C₄	—	10	2	no	—
242	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
243	C₈	—	5	2	no	—
244	C₈	C₂²×C₅×D₅	1	1	yes	characteristic
245	C₈	C₂²×C₅×D₅	1	1	yes	characteristic
246	C₅	—	2	2	no	—
247	C₅	#66105	1	1	yes	characteristic, direct product factor
248	C₅	C₅×#578	1	1	yes	characteristic, complemented
249	C₂²	—	2	2	no	—
250	C₂²	—	5	2	no	—
251	C₂²	C₅×D₄×D₅	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	5	2	no	—
254	C₄	—	10	4	no	—
255	C₄	C₅×D₄×D₅	1	1	yes	characteristic
256	C₄	C₂³×C₅×D₅	1	1	yes	characteristic
257	C₂	—	2	1	no	—
258	C₂	—	4	2	no	—
259	C₂	—	10	2	no	—
260	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	C₅²×#135	C₂
socle	C₂×C₅²	C₂²×D₄
hypercenter	C₅×#135	D₅
hypocenter	C₅	C₅×#578
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#135

Complements

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

N	H ≅ G/N	type
1. C₅×#4150	C₂	semidirect
2. C₅²×#135	C₂	semidirect
3. C₅×#4097	C₂	semidirect
4. C₅×D₅×Q₁₆	C₂	semidirect
5. C₅×#4061	C₂	semidirect
6. C₅×#4058	C₂	semidirect
7. C₂×C₅×Dic₂₀	C₂	semidirect
8. C₅×#4047	C₂	semidirect
29. #66105	C₅	direct (N × H)
50. #4150	C₂×C₅	semidirect
55. C₅×#135	D₅	semidirect
56. C₅×#135	C₂×C₅	semidirect
60. #4097	C₂×C₅	semidirect
61. D₅×Q₁₆	C₂×C₅	semidirect
62. #4061	C₂×C₅	semidirect
63. #4058	C₂×C₅	semidirect
64. C₂×Dic₂₀	C₂×C₅	semidirect
65. #4047	C₂×C₅	semidirect
171. #135	C₅×D₅	semidirect
175. C₅²	#578	semidirect
247. C₅	#66105	direct (N × H)
248. 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×#4150	2
C₅²×#135	2
C₅×#4097	2
C₅×D₅×Q₁₆	2
C₅×#4061	2
C₅×#4058	2
C₂×C₅×Dic₂₀	2
C₅×#4047	2
C₅×#578	5
#66105	5

Subgroups — C5×#66105 #13106925

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#66105 #13106925