Subgroups — C₂×C₅×#4088 #13106965

Order 1600.

672 conjugacy classes of subgroups.

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

	subgroup	G/H
center	C₂²×C₅	C₂×C₂²⋊D₅
commutator, cosocle	C₄×C₅	C₂⁴×C₅
Frattini	C₄	C₂³×C₅×D₅
Fitting	C₂²×C₅²×Q₈	C₂
socle	C₂²×C₅²	C₂×D₄
prime radical	C₂×C₅×#709	C₂
hypercenter	C₂²×C₅×Q₈	D₅
hypocenter	C₅	C₂×C₅×#137
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×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. C₂×C₅×#689	C₂	semidirect
4. C₅×#4088	C₂	direct (N × H)
5. C₂×C₅×#688	C₂	semidirect
6. C₂×C₅×#681	C₂	semidirect
10. C₂×C₅²×Q₈	C₂²	semidirect
19. C₅×#689	C₂²	semidirect
20. C₅×#688	C₂²	semidirect
21. C₅×#681	C₂²	semidirect
24. C₂×#4088	C₅	direct (N × H)
46. C₂²×C₅×Q₈	D₅	semidirect
47. C₂²×C₅×Q₈	C₂×C₅	semidirect
53. C₂×#689	C₂×C₅	semidirect
54. #4088	C₂×C₅	direct (N × H)
55. C₂×#688	C₂×C₅	semidirect
56. C₂×#681	C₂×C₅	semidirect
79. C₂×C₅×Q₈	C₂×D₅	semidirect
81. C₂×C₅×Q₈	C₂²×C₅	semidirect
101. #689	C₂²×C₅	semidirect
102. #688	C₂²×C₅	semidirect
103. #681	C₂²×C₅	semidirect
107. C₂×C₅²	#137	semidirect
159. C₂²×Q₈	C₅×D₅	semidirect
164. C₅²	C₂×#137	semidirect
198. C₂×Q₈	C₂×C₅×D₅	semidirect
214. C₂×C₅	#4088	direct (N × H)
215. C₂×C₅	C₅×#137	semidirect
239. C₅	C₂×#4088	direct (N × H)
240. C₅	C₂×C₅×#137	semidirect
253. C₂	C₅×#4088	direct (N × H)

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
C₂×C₅×#689	2
C₅×#4088	2
C₂×C₅×#688	2
C₂×C₅×#681	2
C₂×C₅×#137	5
C₂×#4088	5

Subgroups — C2×C5×#4088 #13106965

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4088 #13106965