Subgroups — C₂×C₅×#4054 #13106900

Order 1600.

664 conjugacy classes of subgroups.

The subgroup lattice has 264 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, complemented
2	C₂²×C₅×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅²×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4054	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×D₄₀	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#677	C₂	1	2	yes	maximal, complemented
7	C₂³×C₅×D₅	—	2	1	no	—
8	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
9	C₂×C₅×C₂²⋊D₅	—	2	1	no	—
10	C₅×#709	—	2	2	no	—
11	C₅×#709	C₂²	1	4	yes	complemented
12	C₂×C₅×D₂₀	—	2	2	no	—
13	C₂×C₅×D₂₀	—	2	1	no	—
14	C₂×C₅×D₂₀	C₂²	1	2	yes	—
15	C₂×C₅×D₂₀	C₂²	1	4	yes	complemented
16	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₄×C₅×D₅	—	2	1	no	—
18	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic, complemented
19	C₅²×M₁₆	C₂²	1	4	yes	complemented
20	C₂×C₅²×C₈	C₂²	1	2	yes	complemented
21	C₅×D₄₀	C₂²	1	8	yes	complemented
22	C₅×#677	C₂²	1	8	yes	complemented
23	C₂×C₅×AGL(1,ℤ/8)	—	5	1	no	maximal
24	C₂×#4054	C₅	1	1	yes	maximal, characteristic, direct product factor
25	C₂³×C₅²	D₄	1	1	yes	characteristic
26	C₂²×C₅×D₅	—	2	1	no	—
27	C₂²×C₅×D₅	—	2	2	no	—
28	C₂²×C₅×D₅	—	2	4	no	—
29	C₂²×C₅×D₅	—	4	4	no	—
30	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
31	C₂×C₄×C₅²	D₄	1	2	yes	—
32	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₅²	C₂³	1	2	yes	—
34	C₅×C₂²⋊D₅	—	2	4	no	—
35	C₂×C₅×Dic₅	—	2	1	no	—
36	C₅×D₂₀	—	2	2	no	—
37	C₅×D₂₀	—	2	4	no	—
38	C₅×D₂₀	—	2	1	no	—
39	C₅×D₂₀	C₂³	1	4	yes	—
40	C₅×D₂₀	C₂³	1	2	yes	complemented
41	C₄×C₅×D₅	—	2	4	no	—
42	C₅×Dic₁₀	—	2	1	no	—
43	C₅×Dic₁₀	C₂³	1	2	yes	complemented
44	C₅²×C₈	C₂³	1	4	yes	complemented
45	C₂×C₅×#41	—	5	1	no	—
46	C₂²×C₅×D₄	—	5	1	no	—
47	C₂×#709	C₂×C₅	1	1	yes	characteristic, complemented
48	C₂²×D₂₀	C₂×C₅	1	1	yes	characteristic, complemented
49	C₅×AGL(1,ℤ/8)	—	5	8	no	—
50	C₂×C₅×SD₁₆	—	5	2	no	—
51	C₂×C₅×D₈	—	5	2	no	—
52	C₂×C₅×M₁₆	—	2	2	no	—
53	C₂×C₅×M₁₆	D₅	1	1	yes	characteristic, complemented
54	C₂×C₅×M₁₆	C₂×C₅	1	1	yes	characteristic, complemented
55	#4054	C₂×C₅	1	8	yes	direct product factor
56	C₂×D₄₀	C₂×C₅	1	2	yes	complemented
57	C₂×#677	C₂×C₅	1	2	yes	complemented
58	C₂²×C₅²	—	2	2	no	—
59	C₂²×C₅²	C₂×D₄	1	2	yes	—
60	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
61	C₂×C₅×D₅	—	2	2	no	—
62	C₂×C₅×D₅	—	2	4	no	—
63	C₂×C₅×D₅	—	4	8	no	—
64	C₂×C₅×D₅	—	4	2	no	—
65	C₂×C₅×D₅	—	4	4	no	—
66	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
67	C₄×C₅²	C₂×D₄	1	2	yes	—
68	C₄×C₅²	C₂⁴	1	1	yes	characteristic
69	C₅×Dic₅	—	2	2	no	—
70	C₂⁴×C₅	—	10	1	no	—
71	C₂³×D₅	—	2	1	no	—
72	C₅×#41	—	5	4	no	—
73	C₅×#41	—	10	2	no	—
74	C₂×C₅×Q₈	—	5	1	no	—
75	C₂×C₅×D₄	—	5	1	no	—
76	C₂×C₅×D₄	—	5	2	no	—
77	C₂×C₅×D₄	—	5	4	no	—
78	C₂×C₅×D₄	—	10	1	no	—
79	C₂×C₅×D₄	—	10	2	no	—
80	C₂×C₅×D₄	—	10	1	no	—
81	C₂²×C₄×C₅	—	2	2	no	—
82	C₂²×C₄×C₅	—	10	1	no	—
83	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
85	C₂×C₂²⋊D₅	—	2	1	no	—
86	#709	—	2	2	no	—
87	#709	C₂²×C₅	1	4	yes	complemented
88	C₂×D₂₀	—	2	2	no	—
89	C₂×D₂₀	—	2	1	no	—
90	C₂×D₂₀	C₂²×C₅	1	2	yes	—
91	C₂×D₂₀	C₂²×C₅	1	4	yes	complemented
92	C₂×D₂₀	C₂²×C₅	1	1	yes	characteristic, complemented
93	C₂×C₄×D₅	—	2	1	no	—
94	C₂×Dic₁₀	C₂²×C₅	1	1	yes	characteristic, complemented
95	C₅×SD₁₆	—	5	8	no	—
96	C₅×D₈	—	5	8	no	—
97	C₅×M₁₆	—	2	8	no	—
98	C₅×M₁₆	C₂×D₅	1	4	yes	complemented
99	C₅×M₁₆	C₂²×C₅	1	4	yes	complemented
100	C₂×C₅×C₈	—	2	4	no	—
101	C₂×C₅×C₈	C₂×D₅	1	2	yes	complemented
102	C₂×C₅×C₈	C₂²×C₅	1	2	yes	complemented
103	D₄₀	C₂²×C₅	1	8	yes	complemented
104	#677	C₂²×C₅	1	8	yes	complemented
105	C₂×AGL(1,ℤ/8)	—	5	1	no	Sylow
106	C₂×C₅²	—	2	2	no	—
107	C₂×C₅²	AGL(1,ℤ/8)	1	2	yes	complemented
108	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
109	C₅×D₅	—	4	2	no	—
110	C₅×D₅	—	4	4	no	—
111	C₂³×C₅	—	2	2	no	—
112	C₂³×C₅	—	10	4	no	—
113	C₂³×C₅	—	10	1	no	—
114	C₂³×C₅	—	10	2	no	—
115	C₂³×C₅	—	20	4	no	—
116	C₂³×C₅	D₂₀	1	1	yes	characteristic
117	C₂³×C₅	C₅×D₄	1	1	yes	characteristic
118	C₂²×D₅	—	2	1	no	—
119	C₂²×D₅	—	2	2	no	—
120	C₂²×D₅	—	2	4	no	—
121	C₂²×D₅	—	4	4	no	—
122	C₅×Q₈	—	5	2	no	—
123	C₅×Q₈	—	10	1	no	—
124	C₅×D₄	—	5	4	no	—
125	C₅×D₄	—	5	2	no	—
126	C₅×D₄	—	10	4	no	—
127	C₅×D₄	—	10	4	no	—
128	C₅×D₄	—	10	2	no	—
129	C₅×D₄	—	10	1	no	—
130	C₂×C₄×C₅	—	2	2	no	—
131	C₂×C₄×C₅	—	2	4	no	—
132	C₂×C₄×C₅	—	2	2	no	—
133	C₂×C₄×C₅	—	2	4	no	—
134	C₂×C₄×C₅	—	10	1	no	—
135	C₂×C₄×C₅	—	10	4	no	—
136	C₂×C₄×C₅	D₂₀	1	2	yes	—
137	C₂×C₄×C₅	D₂₀	1	1	yes	characteristic
138	C₂×C₄×C₅	C₅×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅	C₅×D₄	1	2	yes	—
140	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
141	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
144	C₂²⋊D₅	—	2	4	no	—
145	C₂×Dic₅	—	2	1	no	—
146	D₂₀	—	2	2	no	—
147	D₂₀	—	2	4	no	—
148	D₂₀	—	2	1	no	—
149	D₂₀	C₂³×C₅	1	4	yes	—
150	D₂₀	C₂³×C₅	1	2	yes	complemented
151	C₄×D₅	—	2	4	no	—
152	Dic₁₀	—	2	1	no	—
153	Dic₁₀	C₂³×C₅	1	2	yes	complemented
154	C₅×C₈	—	2	8	no	—
155	C₅×C₈	C₂²×D₅	1	4	yes	complemented
156	C₅×C₈	C₂³×C₅	1	4	yes	complemented
157	C₂×#41	—	5	1	no	—
158	C₂²×D₄	—	5	1	no	—
159	AGL(1,ℤ/8)	—	5	8	no	—
160	C₂×SD₁₆	—	5	2	no	—
161	C₂×D₈	—	5	2	no	—
162	C₂×M₁₆	C₅×D₅	1	1	yes	characteristic, complemented
163	C₅²	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
164	C₂²×C₅	—	2	2	no	—
165	C₂²×C₅	—	2	2	no	—
166	C₂²×C₅	—	2	2	no	—
167	C₂²×C₅	—	2	4	no	—
168	C₂²×C₅	—	4	4	no	—
169	C₂²×C₅	—	10	2	no	—
170	C₂²×C₅	—	10	4	no	—
171	C₂²×C₅	—	20	8	no	—
172	C₂²×C₅	—	20	2	no	—
173	C₂²×C₅	—	20	4	no	—
174	C₂²×C₅	C₂×D₂₀	1	2	yes	—
175	C₂²×C₅	C₂×D₂₀	1	1	yes	characteristic
176	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
177	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
178	C₂×D₅	—	2	2	no	—
179	C₂×D₅	—	2	4	no	—
180	C₂×D₅	—	4	8	no	—
181	C₂×D₅	—	4	2	no	—
182	C₂×D₅	—	4	4	no	—
183	C₄×C₅	—	2	4	no	—
184	C₄×C₅	—	2	2	no	—
185	C₄×C₅	—	2	2	no	—
186	C₄×C₅	—	10	2	no	—
187	C₄×C₅	C₂×D₂₀	1	2	yes	—
188	C₄×C₅	C₂×D₂₀	1	1	yes	characteristic
189	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
190	C₄×C₅	C₂×C₅×D₄	1	2	yes	—
191	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
192	C₄×C₅	C₂⁴×C₅	1	1	yes	characteristic
193	Dic₅	—	2	2	no	—
194	C₂⁴	—	10	1	no	—
195	#41	—	5	4	no	—
196	#41	—	10	2	no	—
197	C₂×Q₈	—	5	1	no	—
198	C₂×D₄	—	5	1	no	—
199	C₂×D₄	—	5	2	no	—
200	C₂×D₄	—	5	4	no	—
201	C₂×D₄	—	10	1	no	—
202	C₂×D₄	—	10	1	no	—
203	C₂×D₄	—	10	2	no	—
204	C₂²×C₄	—	10	1	no	—
205	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
206	SD₁₆	—	5	8	no	—
207	D₈	—	5	8	no	—
208	M₁₆	C₂×C₅×D₅	1	4	yes	complemented
209	C₂×C₈	C₂×C₅×D₅	1	2	yes	complemented
210	C₂×C₅	—	2	2	no	—
211	C₂×C₅	—	2	2	no	—
212	C₂×C₅	—	2	4	no	—
213	C₂×C₅	—	2	2	no	—
214	C₂×C₅	—	4	4	no	—
215	C₂×C₅	—	20	4	no	—
216	C₂×C₅	—	20	2	no	—
217	C₂×C₅	#4054	1	2	yes	direct product factor
218	C₂×C₅	C₅×AGL(1,ℤ/8)	1	2	yes	complemented
219	C₂×C₅	C₂²×D₂₀	1	1	yes	characteristic
220	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
221	D₅	—	4	4	no	—
222	D₅	—	4	2	no	—
223	C₂³	—	10	1	no	—
224	C₂³	—	10	4	no	—
225	C₂³	—	10	2	no	—
226	C₂³	—	20	4	no	—
227	C₂³	C₅×D₂₀	1	1	yes	characteristic
228	Q₈	—	5	2	no	—
229	Q₈	—	10	1	no	—
230	D₄	—	5	2	no	—
231	D₄	—	5	4	no	—
232	D₄	—	10	4	no	—
233	D₄	—	10	4	no	—
234	D₄	—	10	2	no	—
235	D₄	—	10	1	no	—
236	C₂×C₄	—	10	1	no	—
237	C₂×C₄	—	10	4	no	—
238	C₂×C₄	C₅×D₂₀	1	1	yes	characteristic
239	C₂×C₄	C₅×D₂₀	1	2	yes	—
240	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
241	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
242	C₈	C₂²×C₅×D₅	1	4	yes	complemented
243	C₅	—	2	2	no	—
244	C₅	C₂×#4054	1	1	yes	characteristic, direct product factor
245	C₅	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented
246	C₂²	—	2	2	no	—
247	C₂²	—	10	2	no	—
248	C₂²	—	10	4	no	—
249	C₂²	—	20	8	no	—
250	C₂²	—	20	4	no	—
251	C₂²	—	20	2	no	—
252	C₂²	C₂×C₅×D₂₀	1	2	yes	—
253	C₂²	C₂×C₅×D₂₀	1	1	yes	characteristic
254	C₄	—	10	2	no	—
255	C₄	C₂×C₅×D₂₀	1	2	yes	—
256	C₄	C₂×C₅×D₂₀	1	1	yes	characteristic
257	C₄	C₂³×C₅×D₅	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	—	20	4	no	—
260	C₂	—	20	2	no	—
261	C₂	C₅×#4054	1	2	yes	direct product factor
262	C₂	C₂²×C₅×D₂₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×D₂₀
commutator, cosocle	C₄×C₅	C₂⁴×C₅
Frattini	C₄	C₂³×C₅×D₅
Fitting	C₂×C₅²×M₁₆	C₂
socle	C₂²×C₅²	C₂×D₄
hypercenter	C₂×C₅×M₁₆	D₅
hypocenter	C₅	C₂×C₅×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#709	C₂	semidirect
2. C₂²×C₅×D₂₀	C₂	semidirect
3. C₂×C₅²×M₁₆	C₂	semidirect
4. C₅×#4054	C₂	direct (N × H)
5. C₂×C₅×D₄₀	C₂	semidirect
6. C₂×C₅×#677	C₂	semidirect
11. C₅×#709	C₂²	semidirect
15. C₂×C₅×D₂₀	C₂²	semidirect
16. C₂×C₅×D₂₀	C₂²	semidirect
18. C₂×C₅×Dic₁₀	C₂²	semidirect
19. C₅²×M₁₆	C₂²	semidirect
20. C₂×C₅²×C₈	C₂²	semidirect
21. C₅×D₄₀	C₂²	semidirect
22. C₅×#677	C₂²	semidirect
24. C₂×#4054	C₅	direct (N × H)
40. C₅×D₂₀	C₂³	semidirect
43. C₅×Dic₁₀	C₂³	semidirect
44. C₅²×C₈	C₂³	semidirect
47. C₂×#709	C₂×C₅	semidirect
48. C₂²×D₂₀	C₂×C₅	semidirect
53. C₂×C₅×M₁₆	D₅	semidirect
54. C₂×C₅×M₁₆	C₂×C₅	semidirect
55. #4054	C₂×C₅	direct (N × H)
56. C₂×D₄₀	C₂×C₅	semidirect
57. C₂×#677	C₂×C₅	semidirect
87. #709	C₂²×C₅	semidirect
91. C₂×D₂₀	C₂²×C₅	semidirect
92. C₂×D₂₀	C₂²×C₅	semidirect
94. C₂×Dic₁₀	C₂²×C₅	semidirect
98. C₅×M₁₆	C₂×D₅	semidirect
99. C₅×M₁₆	C₂²×C₅	semidirect
101. C₂×C₅×C₈	C₂×D₅	semidirect
102. C₂×C₅×C₈	C₂²×C₅	semidirect
103. D₄₀	C₂²×C₅	semidirect
104. #677	C₂²×C₅	semidirect
107. C₂×C₅²	AGL(1,ℤ/8)	semidirect
150. D₂₀	C₂³×C₅	semidirect
153. Dic₁₀	C₂³×C₅	semidirect
155. C₅×C₈	C₂²×D₅	semidirect
156. C₅×C₈	C₂³×C₅	semidirect
162. C₂×M₁₆	C₅×D₅	semidirect
163. C₅²	C₂×AGL(1,ℤ/8)	semidirect
208. M₁₆	C₂×C₅×D₅	semidirect
209. C₂×C₈	C₂×C₅×D₅	semidirect
217. C₂×C₅	#4054	direct (N × H)
218. C₂×C₅	C₅×AGL(1,ℤ/8)	semidirect
242. C₈	C₂²×C₅×D₅	semidirect
244. C₅	C₂×#4054	direct (N × H)
245. C₅	C₂×C₅×AGL(1,ℤ/8)	semidirect
261. C₂	C₅×#4054	direct (N × H)

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
C₂²×C₅×D₂₀	2
C₂×C₅²×M₁₆	2
C₅×#4054	2
C₂×C₅×D₄₀	2
C₂×C₅×#677	2
C₂×C₅×AGL(1,ℤ/8)	5
C₂×#4054	5

Subgroups — C2×C5×#4054 #13106900

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4054 #13106900