Subgroups — C₂×C₅×#4055 #13106901

Order 1600.

584 conjugacy classes of subgroups.

The subgroup lattice has 248 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
2	C₂²×C₅×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅²×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4055	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×Dic₂₀	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#677	C₂	1	2	yes	maximal, complemented
7	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
8	C₂×C₅×C₂²⋊D₅	—	2	1	no	—
9	C₂²×C₅×Dic₅	—	2	1	no	—
10	C₅×#709	—	2	2	no	—
11	C₅×#709	C₂²	1	4	yes	—
12	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×D₅	—	2	1	no	—
14	C₂×C₅×Dic₁₀	—	2	2	no	—
15	C₂×C₅×Dic₁₀	—	2	1	no	—
16	C₂×C₅×Dic₁₀	C₂²	1	2	yes	—
17	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic
18	C₂×C₅×Dic₁₀	C₂²	1	4	yes	complemented
19	C₅²×M₁₆	C₂²	1	4	yes	complemented
20	C₂×C₅²×C₈	C₂²	1	2	yes	—
21	C₅×Dic₂₀	C₂²	1	8	yes	complemented
22	C₅×#677	C₂²	1	8	yes	complemented
23	C₂×C₅×#137	—	5	1	no	maximal
24	C₂×#4055	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	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
28	C₂×C₄×C₅²	D₄	1	2	yes	—
29	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₅²	C₂³	1	2	yes	—
31	C₅×C₂²⋊D₅	—	2	4	no	—
32	C₂×C₅×Dic₅	—	2	2	no	—
33	C₂×C₅×Dic₅	—	2	4	no	—
34	C₂×C₅×Dic₅	—	2	1	no	—
35	C₅×D₂₀	—	2	1	no	—
36	C₅×D₂₀	C₂³	1	2	yes	—
37	C₄×C₅×D₅	—	2	4	no	—
38	C₅×Dic₁₀	—	2	2	no	—
39	C₅×Dic₁₀	—	2	4	no	—
40	C₅×Dic₁₀	—	2	1	no	—
41	C₅×Dic₁₀	C₂³	1	4	yes	—
42	C₅×Dic₁₀	C₂³	1	2	yes	—
43	C₅²×C₈	C₂³	1	4	yes	—
44	C₂×C₅×#41	—	5	1	no	—
45	C₂²×C₅×Q₈	—	5	1	no	—
46	C₂×#709	C₂×C₅	1	1	yes	characteristic
47	C₂²×Dic₁₀	C₂×C₅	1	1	yes	characteristic, complemented
48	C₅×#137	—	5	8	no	—
49	C₂×C₅×Q₁₆	—	5	2	no	—
50	C₂×C₅×SD₁₆	—	5	2	no	—
51	C₂×C₅×M₁₆	—	2	2	no	—
52	C₂×C₅×M₁₆	D₅	1	1	yes	characteristic, complemented
53	C₂×C₅×M₁₆	C₂×C₅	1	1	yes	characteristic, complemented
54	#4055	C₂×C₅	1	8	yes	direct product factor
55	C₂×Dic₂₀	C₂×C₅	1	2	yes	complemented
56	C₂×#677	C₂×C₅	1	2	yes	complemented
57	C₂²×C₅²	—	2	2	no	—
58	C₂²×C₅²	C₂×D₄	1	2	yes	—
59	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
60	C₂×C₅×D₅	—	2	2	no	—
61	C₂×C₅×D₅	—	4	2	no	—
62	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
63	C₄×C₅²	C₂×D₄	1	2	yes	—
64	C₄×C₅²	C₂⁴	1	1	yes	characteristic
65	C₅×Dic₅	—	2	4	no	—
66	C₅×Dic₅	—	2	2	no	—
67	C₅×#41	—	5	4	no	—
68	C₅×#41	—	10	2	no	—
69	C₂×C₅×Q₈	—	5	2	no	—
70	C₂×C₅×Q₈	—	5	1	no	—
71	C₂×C₅×Q₈	—	5	4	no	—
72	C₂×C₅×Q₈	—	10	2	no	—
73	C₂×C₅×Q₈	—	10	1	no	—
74	C₂×C₅×D₄	—	5	1	no	—
75	C₂×C₅×D₄	—	10	1	no	—
76	C₂²×C₄×C₅	—	2	2	no	—
77	C₂²×C₄×C₅	—	10	1	no	—
78	C₂²×C₄×C₅	—	10	1	no	—
79	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
81	C₂×C₂²⋊D₅	—	2	1	no	—
82	C₂²×Dic₅	—	2	1	no	—
83	#709	—	2	2	no	—
84	#709	C₂²×C₅	1	4	yes	—
85	C₂×D₂₀	C₂²×C₅	1	1	yes	characteristic
86	C₂×C₄×D₅	—	2	1	no	—
87	C₂×Dic₁₀	—	2	2	no	—
88	C₂×Dic₁₀	—	2	1	no	—
89	C₂×Dic₁₀	C₂²×C₅	1	2	yes	—
90	C₂×Dic₁₀	C₂²×C₅	1	1	yes	characteristic
91	C₂×Dic₁₀	C₂²×C₅	1	4	yes	complemented
92	C₅×Q₁₆	—	5	8	no	—
93	C₅×SD₁₆	—	5	8	no	—
94	C₅×M₁₆	—	2	8	no	—
95	C₅×M₁₆	C₂×D₅	1	4	yes	complemented
96	C₅×M₁₆	C₂²×C₅	1	4	yes	complemented
97	C₂×C₅×C₈	—	2	4	no	—
98	C₂×C₅×C₈	C₂×D₅	1	2	yes	—
99	C₂×C₅×C₈	C₂²×C₅	1	2	yes	—
100	Dic₂₀	C₂²×C₅	1	8	yes	complemented
101	#677	C₂²×C₅	1	8	yes	complemented
102	C₂×#137	—	5	1	no	Sylow
103	C₂×C₅²	—	2	2	no	—
104	C₂×C₅²	#137	1	2	yes	complemented
105	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
106	C₅×D₅	—	4	2	no	—
107	C₂³×C₅	—	2	2	no	—
108	C₂³×C₅	—	10	1	no	—
109	C₂³×C₅	D₂₀	1	1	yes	characteristic
110	C₂³×C₅	C₅×D₄	1	1	yes	characteristic
111	C₂²×D₅	—	2	1	no	—
112	C₅×Q₈	—	5	4	no	—
113	C₅×Q₈	—	5	2	no	—
114	C₅×Q₈	—	10	4	no	—
115	C₅×Q₈	—	10	2	no	—
116	C₅×Q₈	—	10	1	no	—
117	C₅×D₄	—	5	2	no	—
118	C₅×D₄	—	10	4	no	—
119	C₅×D₄	—	10	1	no	—
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	2	4	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	4	no	—
124	C₂×C₄×C₅	—	10	4	no	—
125	C₂×C₄×C₅	—	10	1	no	—
126	C₂×C₄×C₅	—	10	2	no	—
127	C₂×C₄×C₅	—	10	4	no	—
128	C₂×C₄×C₅	D₂₀	1	2	yes	—
129	C₂×C₄×C₅	D₂₀	1	1	yes	characteristic
130	C₂×C₄×C₅	C₅×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅	C₅×D₄	1	2	yes	—
132	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
133	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
136	C₂²⋊D₅	—	2	4	no	—
137	C₂×Dic₅	—	2	2	no	—
138	C₂×Dic₅	—	2	4	no	—
139	C₂×Dic₅	—	2	1	no	—
140	D₂₀	—	2	1	no	—
141	D₂₀	C₂³×C₅	1	2	yes	—
142	C₄×D₅	—	2	4	no	—
143	Dic₁₀	—	2	2	no	—
144	Dic₁₀	—	2	4	no	—
145	Dic₁₀	—	2	1	no	—
146	Dic₁₀	C₂³×C₅	1	4	yes	—
147	Dic₁₀	C₂³×C₅	1	2	yes	—
148	C₅×C₈	—	2	8	no	—
149	C₅×C₈	C₂²×D₅	1	4	yes	—
150	C₅×C₈	C₂³×C₅	1	4	yes	—
151	C₂×#41	—	5	1	no	—
152	C₂²×Q₈	—	5	1	no	—
153	#137	—	5	8	no	—
154	C₂×Q₁₆	—	5	2	no	—
155	C₂×SD₁₆	—	5	2	no	—
156	C₂×M₁₆	C₅×D₅	1	1	yes	characteristic, complemented
157	C₅²	C₂×#137	1	1	yes	characteristic, complemented, Sylow
158	C₂²×C₅	—	2	2	no	—
159	C₂²×C₅	—	2	2	no	—
160	C₂²×C₅	—	2	2	no	—
161	C₂²×C₅	—	2	4	no	—
162	C₂²×C₅	—	4	4	no	—
163	C₂²×C₅	—	10	2	no	—
164	C₂²×C₅	—	20	2	no	—
165	C₂²×C₅	C₂×D₂₀	1	2	yes	—
166	C₂²×C₅	C₂×D₂₀	1	1	yes	characteristic
167	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
168	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
169	C₂×D₅	—	2	2	no	—
170	C₂×D₅	—	4	2	no	—
171	C₄×C₅	—	2	4	no	—
172	C₄×C₅	—	2	2	no	—
173	C₄×C₅	—	2	2	no	—
174	C₄×C₅	—	10	2	no	—
175	C₄×C₅	—	10	4	no	—
176	C₄×C₅	C₂×D₂₀	1	2	yes	—
177	C₄×C₅	C₂×D₂₀	1	1	yes	characteristic
178	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
179	C₄×C₅	C₂×C₅×D₄	1	2	yes	—
180	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
181	C₄×C₅	C₂⁴×C₅	1	1	yes	characteristic
182	Dic₅	—	2	4	no	—
183	Dic₅	—	2	2	no	—
184	#41	—	5	4	no	—
185	#41	—	10	2	no	—
186	C₂×Q₈	—	5	2	no	—
187	C₂×Q₈	—	5	1	no	—
188	C₂×Q₈	—	5	4	no	—
189	C₂×Q₈	—	10	1	no	—
190	C₂×Q₈	—	10	2	no	—
191	C₂×D₄	—	5	1	no	—
192	C₂×D₄	—	10	1	no	—
193	C₂²×C₄	—	10	1	no	—
194	C₂²×C₄	—	10	1	no	—
195	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
196	Q₁₆	—	5	8	no	—
197	SD₁₆	—	5	8	no	—
198	M₁₆	C₂×C₅×D₅	1	4	yes	complemented
199	C₂×C₈	C₂×C₅×D₅	1	2	yes	—
200	C₂×C₅	—	2	2	no	—
201	C₂×C₅	—	2	2	no	—
202	C₂×C₅	—	2	4	no	—
203	C₂×C₅	—	2	2	no	—
204	C₂×C₅	—	4	4	no	—
205	C₂×C₅	—	20	2	no	—
206	C₂×C₅	#4055	1	2	yes	direct product factor
207	C₂×C₅	C₅×#137	1	2	yes	complemented
208	C₂×C₅	C₂²×D₂₀	1	1	yes	characteristic
209	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
210	D₅	—	4	2	no	—
211	C₂³	—	10	1	no	—
212	C₂³	C₅×D₂₀	1	1	yes	characteristic
213	Q₈	—	5	4	no	—
214	Q₈	—	5	2	no	—
215	Q₈	—	10	4	no	—
216	Q₈	—	10	2	no	—
217	Q₈	—	10	1	no	—
218	D₄	—	5	2	no	—
219	D₄	—	10	4	no	—
220	D₄	—	10	1	no	—
221	C₂×C₄	—	10	1	no	—
222	C₂×C₄	—	10	4	no	—
223	C₂×C₄	—	10	2	no	—
224	C₂×C₄	—	10	4	no	—
225	C₂×C₄	C₅×D₂₀	1	1	yes	characteristic
226	C₂×C₄	C₅×D₂₀	1	2	yes	—
227	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
228	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
229	C₈	C₂²×C₅×D₅	1	4	yes	—
230	C₅	—	2	2	no	—
231	C₅	C₂×#4055	1	1	yes	characteristic, direct product factor
232	C₅	C₂×C₅×#137	1	1	yes	characteristic, complemented
233	C₂²	—	2	2	no	—
234	C₂²	—	10	2	no	—
235	C₂²	—	20	2	no	—
236	C₂²	C₂×C₅×D₂₀	1	2	yes	—
237	C₂²	C₂×C₅×D₂₀	1	1	yes	characteristic
238	C₄	—	10	2	no	—
239	C₄	—	10	4	no	—
240	C₄	C₂×C₅×D₂₀	1	2	yes	—
241	C₄	C₂×C₅×D₂₀	1	1	yes	characteristic
242	C₄	C₂³×C₅×D₅	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	—	20	2	no	—
245	C₂	C₅×#4055	1	2	yes	direct product factor
246	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
prime radical	C₂×C₅×#709	C₂
hypercenter	C₂×C₅×M₁₆	D₅
hypocenter	C₅	C₂×C₅×#137
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
2. C₂²×C₅×Dic₁₀	C₂	semidirect
3. C₂×C₅²×M₁₆	C₂	semidirect
4. C₅×#4055	C₂	direct (N × H)
5. C₂×C₅×Dic₂₀	C₂	semidirect
6. C₂×C₅×#677	C₂	semidirect
18. C₂×C₅×Dic₁₀	C₂²	semidirect
19. C₅²×M₁₆	C₂²	semidirect
21. C₅×Dic₂₀	C₂²	semidirect
22. C₅×#677	C₂²	semidirect
24. C₂×#4055	C₅	direct (N × H)
47. C₂²×Dic₁₀	C₂×C₅	semidirect
52. C₂×C₅×M₁₆	D₅	semidirect
53. C₂×C₅×M₁₆	C₂×C₅	semidirect
54. #4055	C₂×C₅	direct (N × H)
55. C₂×Dic₂₀	C₂×C₅	semidirect
56. C₂×#677	C₂×C₅	semidirect
91. C₂×Dic₁₀	C₂²×C₅	semidirect
95. C₅×M₁₆	C₂×D₅	semidirect
96. C₅×M₁₆	C₂²×C₅	semidirect
100. Dic₂₀	C₂²×C₅	semidirect
101. #677	C₂²×C₅	semidirect
104. C₂×C₅²	#137	semidirect
156. C₂×M₁₆	C₅×D₅	semidirect
157. C₅²	C₂×#137	semidirect
198. M₁₆	C₂×C₅×D₅	semidirect
206. C₂×C₅	#4055	direct (N × H)
207. C₂×C₅	C₅×#137	semidirect
231. C₅	C₂×#4055	direct (N × H)
232. C₅	C₂×C₅×#137	semidirect
245. C₂	C₅×#4055	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×Dic₁₀	2
C₂×C₅²×M₁₆	2
C₅×#4055	2
C₂×C₅×Dic₂₀	2
C₂×C₅×#677	2
C₂×C₅×#137	5
C₂×#4055	5

Subgroups — C2×C5×#4055 #13106901

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4055 #13106901