Subgroups — C₅×#66150 #13106970

Order 1600.

696 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#709	C₂	1	1	yes	maximal, characteristic
3	C₅×#4093	C₂	1	2	yes	maximal, complemented
4	C₅×#4092	C₂	1	4	yes	maximal, complemented
5	C₅×#4090	C₂	1	2	yes	maximal, complemented
6	C₅×#4075	C₂	1	4	yes	maximal, complemented
7	C₅×#4072	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅²×Q₈	—	2	4	no	—
9	C₂×C₅²×Q₈	C₂²	1	2	yes	—
10	C₂×C₅²×Q₈	C₂²	1	2	yes	—
11	C₂²×C₄×C₅²	C₂²	1	2	yes	—
12	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
13	C₂×C₅×C₂²⋊D₅	C₂²	1	2	yes	—
14	C₅×#709	—	2	4	no	—
15	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×D₅	C₂²	1	2	yes	—
17	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic
18	C₅×C₂²⋊Dic₅	C₂²	1	2	yes	—
19	C₅×#685	C₂²	1	8	yes	—
20	C₅×C₄⋊Dic₅	C₂²	1	2	yes	—
21	C₅×#683	C₂²	1	8	yes	—
22	C₄×C₅×Dic₅	C₂²	1	2	yes	—
23	C₅×#536	—	5	1	no	maximal
24	#66150	C₅	1	1	yes	maximal, characteristic, direct product factor
25	C₂³×C₅²	C₂³	1	1	yes	characteristic
26	C₂²×C₅×D₅	C₂³	1	2	yes	—
27	C₅²×Q₈	—	2	4	no	—
28	C₅²×Q₈	—	2	4	no	—
29	C₂×C₄×C₅²	—	2	4	no	—
30	C₂×C₄×C₅²	D₄	1	4	yes	—
31	C₂×C₄×C₅²	C₂³	1	4	yes	—
32	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
34	C₅×C₂²⋊D₅	—	2	4	no	—
35	C₂×C₅×Dic₅	C₂³	1	2	yes	—
36	C₂×C₅×Dic₅	C₂³	1	4	yes	—
37	C₅×D₂₀	—	2	2	no	—
38	C₄×C₅×D₅	—	2	4	no	—
39	C₅×Dic₁₀	—	2	2	no	—
40	C₂×C₅×#41	—	5	1	no	—
41	C₂²×C₅×Q₈	—	2	2	no	—
42	C₂²×C₅×Q₈	D₅	1	1	yes	characteristic, complemented
43	C₂²×C₅×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
44	C₂×#709	C₂×C₅	1	1	yes	characteristic
45	C₅×C₄⋊Q₈	—	5	2	no	—
46	C₅×#124	—	5	2	no	—
47	C₅×#123	—	5	4	no	—
48	C₅×C₂²⋊Q₈	—	5	4	no	—
49	C₅×#117	—	5	1	no	—
50	#4093	C₂×C₅	1	2	yes	complemented
51	#4092	C₂×C₅	1	4	yes	complemented
52	#4090	C₂×C₅	1	2	yes	complemented
53	#4075	C₂×C₅	1	4	yes	complemented
54	#4072	C₂×C₅	1	1	yes	characteristic, complemented
55	C₂²×C₅²	—	2	2	no	—
56	C₂²×C₅²	C₂×D₄	1	2	yes	—
57	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
58	C₂×C₅×D₅	—	2	2	no	—
59	C₂×C₅×D₅	—	2	4	no	—
60	C₄×C₅²	—	2	4	no	—
61	C₄×C₅²	C₂×D₄	1	2	yes	—
62	C₄×C₅²	C₂×D₄	1	2	yes	—
63	C₅×Dic₅	—	2	2	no	—
64	C₅×Dic₅	—	2	4	no	—
65	C₅×#41	—	10	4	no	—
66	C₂×C₅×Q₈	—	2	16	no	—
67	C₂×C₅×Q₈	—	2	4	no	—
68	C₂×C₅×Q₈	—	2	4	no	—
69	C₂×C₅×Q₈	—	2	4	no	—
70	C₂×C₅×Q₈	—	2	4	no	—
71	C₂×C₅×Q₈	—	5	1	no	—
72	C₂×C₅×Q₈	C₂×D₅	1	2	yes	—
73	C₂×C₅×Q₈	C₂×D₅	1	2	yes	—
74	C₂×C₅×Q₈	C₂²×C₅	1	2	yes	—
75	C₂×C₅×Q₈	C₂²×C₅	1	2	yes	—
76	C₂×C₅×D₄	—	5	2	no	—
77	C₂×C₅×D₄	—	5	1	no	—
78	C₂²×C₄×C₅	—	2	4	no	—
79	C₂²×C₄×C₅	—	2	2	no	—
80	C₂²×C₄×C₅	—	5	2	no	—
81	C₂²×C₄×C₅	C₂×D₅	1	2	yes	—
82	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂²×C₅	1	2	yes	—
84	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
85	C₂×C₂²⋊D₅	C₂²×C₅	1	2	yes	—
86	#709	—	2	4	no	—
87	C₂×D₂₀	C₂²×C₅	1	1	yes	characteristic
88	C₂×C₄×D₅	C₂²×C₅	1	2	yes	—
89	C₂×Dic₁₀	C₂²×C₅	1	1	yes	characteristic
90	C₅×C₄⋊C₄	—	5	2	no	—
91	C₅×C₄⋊C₄	—	5	8	no	—
92	C₅×C₂²⋊C₄	—	5	2	no	—
93	C₅×C₂²⋊C₄	—	5	8	no	—
94	C₄²×C₅	—	5	2	no	—
95	C₂²⋊Dic₅	C₂²×C₅	1	2	yes	—
96	#685	C₂²×C₅	1	8	yes	—
97	C₄⋊Dic₅	C₂²×C₅	1	2	yes	—
98	#683	C₂²×C₅	1	8	yes	—
99	C₄×Dic₅	C₂²×C₅	1	2	yes	—
100	#536	—	5	1	no	Sylow
101	C₂×C₅²	—	2	2	no	—
102	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₅²	2₋¹⁺⁴	1	2	yes	—
104	C₅×D₅	—	4	2	no	—
105	C₂³×C₅	—	2	2	no	—
106	C₂³×C₅	—	5	2	no	—
107	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
108	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
109	C₂²×D₅	C₂³×C₅	1	2	yes	—
110	C₅×Q₈	—	2	16	no	—
111	C₅×Q₈	—	2	16	no	—
112	C₅×Q₈	—	2	4	no	—
113	C₅×Q₈	—	2	4	no	—
114	C₅×Q₈	—	2	4	no	—
115	C₅×Q₈	—	2	4	no	—
116	C₅×Q₈	—	10	2	no	—
117	C₅×D₄	—	10	4	no	—
118	C₅×D₄	—	10	2	no	—
119	C₂×C₄×C₅	—	2	16	no	—
120	C₂×C₄×C₅	—	2	4	no	—
121	C₂×C₄×C₅	—	2	4	no	—
122	C₂×C₄×C₅	—	2	8	no	—
123	C₂×C₄×C₅	—	2	8	no	—
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	5	4	no	—
127	C₂×C₄×C₅	—	5	2	no	—
128	C₂×C₄×C₅	—	10	4	no	—
129	C₂×C₄×C₅	C₂²⋊D₅	1	4	yes	—
130	C₂×C₄×C₅	C₅×D₄	1	4	yes	—
131	C₂×C₄×C₅	C₂²×D₅	1	4	yes	—
132	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂³×C₅	1	4	yes	—
135	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
137	C₂²⋊D₅	—	2	4	no	—
138	C₂×Dic₅	C₂³×C₅	1	4	yes	—
139	C₂×Dic₅	C₂³×C₅	1	2	yes	—
140	D₂₀	—	2	2	no	—
141	C₄×D₅	—	2	4	no	—
142	Dic₁₀	—	2	2	no	—
143	C₂×#41	—	5	1	no	—
144	C₂²×Q₈	C₅×D₅	1	1	yes	characteristic, complemented
145	C₄⋊Q₈	—	5	2	no	—
146	#124	—	5	2	no	—
147	#123	—	5	4	no	—
148	C₂²⋊Q₈	—	5	4	no	—
149	#117	—	5	1	no	—
150	C₅²	#536	1	1	yes	characteristic, complemented, Sylow
151	C₂²×C₅	—	2	8	no	—
152	C₂²×C₅	—	2	2	no	—
153	C₂²×C₅	—	2	2	no	—
154	C₂²×C₅	—	2	4	no	—
155	C₂²×C₅	—	2	2	no	—
156	C₂²×C₅	—	10	4	no	—
157	C₂²×C₅	—	10	2	no	—
158	C₂²×C₅	C₂×C₂²⋊D₅	1	2	yes	—
159	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
160	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
161	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
162	C₂×D₅	—	2	2	no	—
163	C₂×D₅	—	2	4	no	—
164	C₄×C₅	—	2	16	no	—
165	C₄×C₅	—	2	4	no	—
166	C₄×C₅	—	2	4	no	—
167	C₄×C₅	—	2	4	no	—
168	C₄×C₅	—	2	4	no	—
169	C₄×C₅	—	10	4	no	—
170	C₄×C₅	—	10	2	no	—
171	C₄×C₅	C₂×C₂²⋊D₅	1	2	yes	—
172	C₄×C₅	C₂×C₂²⋊D₅	1	2	yes	—
173	C₄×C₅	C₂×C₅×D₄	1	2	yes	—
174	C₄×C₅	C₂×C₅×D₄	1	2	yes	—
175	Dic₅	—	2	2	no	—
176	Dic₅	—	2	4	no	—
177	#41	—	10	4	no	—
178	C₂×Q₈	—	2	4	no	—
179	C₂×Q₈	—	5	1	no	—
180	C₂×Q₈	C₂×C₅×D₅	1	2	yes	—
181	C₂×Q₈	C₂×C₅×D₅	1	2	yes	—
182	C₂×D₄	—	5	2	no	—
183	C₂×D₄	—	5	1	no	—
184	C₂²×C₄	—	5	2	no	—
185	C₂²×C₄	C₂×C₅×D₅	1	2	yes	—
186	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
187	C₄⋊C₄	—	5	2	no	—
188	C₄⋊C₄	—	5	8	no	—
189	C₂²⋊C₄	—	5	2	no	—
190	C₂²⋊C₄	—	5	8	no	—
191	C₄²	—	5	2	no	—
192	C₂×C₅	—	2	8	no	—
193	C₂×C₅	—	2	2	no	—
194	C₂×C₅	—	2	2	no	—
195	C₂×C₅	—	2	4	no	—
196	C₂×C₅	—	2	2	no	—
197	C₂×C₅	—	20	2	no	—
198	C₂×C₅	#4147	1	2	yes	—
199	C₂×C₅	C₂²×C₂²⋊D₅	1	1	yes	characteristic
200	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
201	C₂×C₅	C₅×2₋¹⁺⁴	1	2	yes	—
202	D₅	—	4	2	no	—
203	C₂³	—	5	2	no	—
204	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
205	Q₈	—	2	4	no	—
206	Q₈	—	2	4	no	—
207	Q₈	—	10	2	no	—
208	D₄	—	10	4	no	—
209	D₄	—	10	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	5	4	no	—
212	C₂×C₄	—	5	2	no	—
213	C₂×C₄	—	10	4	no	—
214	C₂×C₄	C₅×C₂²⋊D₅	1	4	yes	—
215	C₂×C₄	C₂²×C₅×D₅	1	4	yes	—
216	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
217	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
218	C₅	—	2	2	no	—
219	C₅	#66150	1	1	yes	characteristic, direct product factor
220	C₅	C₅×#536	1	1	yes	characteristic, complemented
221	C₂²	—	2	2	no	—
222	C₂²	—	10	4	no	—
223	C₂²	—	10	2	no	—
224	C₂²	C₂×C₅×C₂²⋊D₅	1	2	yes	—
225	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
226	C₄	—	2	4	no	—
227	C₄	—	10	4	no	—
228	C₄	—	10	2	no	—
229	C₄	C₂×C₅×C₂²⋊D₅	1	2	yes	—
230	C₄	C₂×C₅×C₂²⋊D₅	1	2	yes	—
231	C₂	—	2	2	no	—
232	C₂	—	20	2	no	—
233	C₂	C₅×#4147	1	2	yes	—
234	C₂	C₂²×C₅×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting	C₂²×C₅²×Q₈	C₂
socle	C₂²×C₅²	C₂⁴
prime radical	C₂×C₅×#709	C₂
hypercenter	C₂²×C₅×Q₈	D₅
hypocenter	C₅	C₅×#536
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₅×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅²×Q₈	C₂	semidirect
3. C₅×#4093	C₂	semidirect
4. C₅×#4092	C₂	semidirect
5. C₅×#4090	C₂	semidirect
6. C₅×#4075	C₂	semidirect
7. C₅×#4072	C₂	semidirect
24. #66150	C₅	direct (N × H)
42. C₂²×C₅×Q₈	D₅	semidirect
43. C₂²×C₅×Q₈	C₂×C₅	semidirect
50. #4093	C₂×C₅	semidirect
51. #4092	C₂×C₅	semidirect
52. #4090	C₂×C₅	semidirect
53. #4075	C₂×C₅	semidirect
54. #4072	C₂×C₅	semidirect
144. C₂²×Q₈	C₅×D₅	semidirect
150. C₅²	#536	semidirect
219. C₅	#66150	direct (N × H)
220. C₅	C₅×#536	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅²×Q₈	2
C₂×C₅×#709	2
C₅×#4093	2
C₅×#4092	2
C₅×#4090	2
C₅×#4075	2
C₅×#4072	2
C₅×#536	5
#66150	5

Subgroups — C5×#66150 #13106970

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#66150 #13106970