Subgroups — C₂×D₅×D₅⋊C₈ #13107596

Order 1600.

703 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₅²	C₂	1	1	yes	maximal, characteristic
2	C₂×D₅⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×D₅⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#87827	C₂	1	2	yes	maximal, complemented
5	C₂×D₅×#170	C₂	1	2	yes	maximal, complemented
6	D₅×D₅⋊C₈	C₂	1	8	yes	maximal, direct product factor
7	C₂²×D₅²	C₄	1	1	yes	characteristic
8	C₂×C₄×C₅⋊D₅	C₄	1	1	yes	characteristic
9	C₂×C₄×C₅×D₅	C₄	1	1	yes	characteristic
10	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
11	C₂×#6036	C₂²	1	1	yes	characteristic
12	C₂×D₅×Dic₅	C₄	1	1	yes	characteristic
13	C₂×D₅×Dic₅	C₂²	1	1	yes	characteristic
14	C₄×D₅²	C₄	1	4	yes	—
15	C₄×D₅²	C₂²	1	4	yes	—
16	C₂×#6032	C₂²	1	2	yes	complemented
17	D₅⋊#168	C₂²	1	4	yes	complemented
18	C₂×C₅×#170	C₂²	1	2	yes	complemented
19	C₅×D₅⋊C₈	C₂²	1	4	yes	complemented
20	#87827	C₂²	1	8	yes	complemented
21	D₅×#170	C₂²	1	8	yes	complemented
22	C₂²×D₅⋊C₈	—	5	1	no	maximal
23	C₂²×C₈×D₅	—	5	1	no	maximal
24	C₂²×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×D₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅×D₅	C₂×C₄	1	1	yes	characteristic
27	C₂×D₅²	C₈	1	8	yes	complemented
28	C₂×D₅²	C₂×C₄	1	4	yes	—
29	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
30	C₂×#1039	C₂×C₄	1	1	yes	characteristic
31	C₄×C₅⋊D₅	C₂×C₄	1	4	yes	—
32	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
34	C₄×C₅×D₅	C₂×C₄	1	2	yes	—
35	C₄×C₅×D₅	C₂×C₄	1	4	yes	—
36	C₄×C₅×D₅	C₂³	1	2	yes	—
37	#6036	C₂×C₄	1	2	yes	—
38	#6036	C₂³	1	2	yes	—
39	D₅×Dic₅	C₂×C₄	1	2	yes	—
40	D₅×Dic₅	C₂×C₄	1	4	yes	—
41	D₅×Dic₅	C₂³	1	2	yes	—
42	#6032	C₂³	1	4	yes	complemented
43	C₅×#170	C₂³	1	4	yes	complemented
44	C₂²×C₄×D₅	—	5	1	no	—
45	C₂²×C₄×D₅	—	5	1	no	—
46	C₂²×#170	—	5	2	no	—
47	C₂×D₅⋊C₈	—	5	1	no	—
48	C₂×D₅⋊C₈	—	5	2	no	—
49	C₂×D₅⋊C₈	—	5	8	no	—
50	C₂×D₅⋊C₈	D₅	1	1	yes	characteristic, direct product factor
51	C₂²×C₅×C₈	—	5	1	no	—
52	C₂²×#168	—	5	1	no	—
53	C₂×C₈×D₅	—	5	8	no	—
54	C₂×C₈×D₅	—	5	2	no	—
55	C₂×C₈×D₅	—	5	2	no	—
56	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₅⋊D₅	C₂×C₈	1	4	yes	complemented
58	C₂×C₅⋊D₅	C₂²×C₄	1	2	yes	—
59	C₂×C₅×D₅	C₂×C₈	1	4	yes	complemented
60	C₂×C₅×D₅	C₂×C₈	1	4	yes	complemented
61	C₂×C₅×D₅	C₂²×C₄	1	2	yes	—
62	C₂×C₅×D₅	C₂²×C₄	1	2	yes	—
63	D₅²	C₂×C₈	1	16	yes	complemented
64	C₄×C₅²	C₂²×C₄	1	2	yes	—
65	#1039	C₂²×C₄	1	2	yes	—
66	C₅×Dic₅	C₂²×C₄	1	1	yes	characteristic
67	C₅×Dic₅	C₂²×C₄	1	2	yes	—
68	C₅×Dic₅	C₂⁴	1	1	yes	characteristic
69	C₂³×D₅	—	5	1	no	—
70	C₂³×D₅	—	5	1	no	—
71	C₂²×C₄×C₅	—	5	1	no	—
72	C₂²×C₄×C₅	—	5	1	no	—
73	C₂²×Dic₅	—	5	1	no	—
74	C₂²×Dic₅	—	5	1	no	—
75	C₂×C₄×D₅	—	5	1	no	—
76	C₂×C₄×D₅	—	5	1	no	—
77	C₂×C₄×D₅	—	5	1	no	—
78	C₂×C₄×D₅	—	5	1	no	—
79	C₂×C₄×D₅	—	5	1	no	—
80	C₂×C₄×D₅	—	5	1	no	—
81	C₂×C₄×D₅	—	5	4	no	—
82	C₂×C₄×D₅	—	5	4	no	—
83	C₂×C₄×D₅	—	5	4	no	—
84	C₂×C₄×D₅	—	5	4	no	—
85	C₂×C₄×D₅	—	20	1	no	—
86	C₂×C₄×D₅	AGL(1,𝔽₅)	1	1	yes	characteristic
87	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
88	C₂×#170	—	5	2	no	—
89	C₂×#170	—	5	8	no	—
90	C₂×#170	C₂×D₅	1	2	yes	complemented
91	D₅⋊C₈	—	5	4	no	—
92	D₅⋊C₈	—	5	8	no	—
93	D₅⋊C₈	C₂×D₅	1	4	yes	direct product factor
94	C₂×C₅×C₈	—	5	4	no	—
95	C₂×C₅×C₈	—	5	2	no	—
96	C₂×#168	—	5	4	no	—
97	C₂×#168	—	5	2	no	—
98	C₈×D₅	—	5	8	no	—
99	C₈×D₅	—	5	8	no	—
100	C₂³×C₈	—	25	1	no	Sylow
101	C₂×C₅²	C₂²×C₈	1	2	yes	complemented
102	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
103	C₅⋊D₅	C₂²×C₈	1	4	yes	complemented
104	C₅×D₅	C₂²×C₈	1	4	yes	complemented
105	C₅×D₅	C₂²×C₈	1	4	yes	complemented
106	C₂³×C₅	—	5	1	no	—
107	C₂³×C₅	—	5	1	no	—
108	C₂²×D₅	—	5	1	no	—
109	C₂²×D₅	—	5	1	no	—
110	C₂²×D₅	—	5	8	no	—
111	C₂²×D₅	—	5	8	no	—
112	C₂²×D₅	—	5	4	no	—
113	C₂²×D₅	—	5	4	no	—
114	C₂²×D₅	—	20	1	no	—
115	C₂²×D₅	C₄×D₅	1	1	yes	characteristic
116	C₂²×D₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
117	C₂×C₄×C₅	—	4	1	no	—
118	C₂×C₄×C₅	—	5	1	no	—
119	C₂×C₄×C₅	—	5	1	no	—
120	C₂×C₄×C₅	—	5	4	no	—
121	C₂×C₄×C₅	—	5	2	no	—
122	C₂×C₄×C₅	—	5	2	no	—
123	C₂×C₄×C₅	C₄×D₅	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
125	C₂×Dic₅	—	5	1	no	—
126	C₂×Dic₅	—	5	1	no	—
127	C₂×Dic₅	—	5	2	no	—
128	C₂×Dic₅	—	5	4	no	—
129	C₂×Dic₅	—	5	2	no	—
130	C₂×Dic₅	—	20	1	no	—
131	C₂×Dic₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
132	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
133	C₄×D₅	—	5	4	no	—
134	C₄×D₅	—	5	2	no	—
135	C₄×D₅	—	5	2	no	—
136	C₄×D₅	—	5	4	no	—
137	C₄×D₅	—	5	4	no	—
138	C₄×D₅	—	5	2	no	—
139	C₄×D₅	—	5	2	no	—
140	C₄×D₅	—	5	2	no	—
141	C₄×D₅	—	5	2	no	—
142	C₄×D₅	—	20	4	no	—
143	C₄×D₅	C₄×D₅	1	2	yes	—
144	C₄×D₅	C₂×AGL(1,𝔽₅)	1	4	yes	—
145	C₄×D₅	C₂²×D₅	1	2	yes	—
146	#170	—	5	4	no	—
147	#170	C₂²×D₅	1	4	yes	complemented
148	C₅×C₈	—	5	4	no	—
149	#168	—	5	4	no	—
150	C₂³×C₄	—	25	1	no	—
151	C₂²×C₈	—	5	1	no	—
152	C₂²×C₈	—	25	8	no	—
153	C₂²×C₈	—	25	2	no	—
154	C₂²×C₈	—	25	2	no	—
155	C₂²×C₈	—	25	1	no	—
156	C₅²	C₂³×C₈	1	1	yes	characteristic, complemented, Sylow
157	C₂²×C₅	—	4	1	no	—
158	C₂²×C₅	—	5	4	no	—
159	C₂²×C₅	—	5	2	no	—
160	C₂²×C₅	—	5	2	no	—
161	C₂²×C₅	—	5	4	no	—
162	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
163	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
164	C₂×D₅	—	5	4	no	—
165	C₂×D₅	—	5	2	no	—
166	C₂×D₅	—	5	2	no	—
167	C₂×D₅	—	5	4	no	—
168	C₂×D₅	—	5	16	no	—
169	C₂×D₅	—	5	16	no	—
170	C₂×D₅	—	20	4	no	—
171	C₂×D₅	—	20	2	no	—
172	C₂×D₅	C₈×D₅	1	4	yes	complemented
173	C₂×D₅	D₅⋊C₈	1	4	yes	direct product factor
174	C₂×D₅	C₂×C₄×D₅	1	2	yes	—
175	C₂×D₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
176	C₄×C₅	—	4	2	no	—
177	C₄×C₅	—	5	2	no	—
178	C₄×C₅	—	5	1	no	—
179	C₄×C₅	—	5	1	no	—
180	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
181	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
182	Dic₅	—	5	2	no	—
183	Dic₅	—	5	2	no	—
184	Dic₅	—	20	2	no	—
185	Dic₅	C₂×C₄×D₅	1	1	yes	characteristic
186	Dic₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
187	Dic₅	C₂³×D₅	1	1	yes	characteristic
188	C₂⁴	—	25	1	no	—
189	C₂²×C₄	—	5	1	no	—
190	C₂²×C₄	—	5	1	no	—
191	C₂²×C₄	—	25	1	no	—
192	C₂²×C₄	—	25	4	no	—
193	C₂²×C₄	—	25	4	no	—
194	C₂²×C₄	—	25	1	no	—
195	C₂²×C₄	—	25	1	no	—
196	C₂²×C₄	—	25	1	no	—
197	C₂×C₈	—	5	4	no	—
198	C₂×C₈	—	5	2	no	—
199	C₂×C₈	—	25	8	no	—
200	C₂×C₈	—	25	8	no	—
201	C₂×C₈	—	25	4	no	—
202	C₂×C₈	—	25	2	no	—
203	C₂×C₅	—	4	1	no	—
204	C₂×C₅	—	4	2	no	—
205	C₂×C₅	—	5	4	no	—
206	C₂×C₅	—	5	4	no	—
207	C₂×C₅	C₂×C₈×D₅	1	2	yes	complemented
208	C₂×C₅	C₂×D₅⋊C₈	1	2	yes	complemented
209	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
210	C₂×C₅	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
211	D₅	—	5	4	no	—
212	D₅	—	5	4	no	—
213	D₅	—	20	4	no	—
214	D₅	C₂×C₈×D₅	1	4	yes	complemented
215	D₅	C₂×D₅⋊C₈	1	4	yes	direct product factor
216	C₂³	—	5	1	no	—
217	C₂³	—	5	1	no	—
218	C₂³	—	25	4	no	—
219	C₂³	—	25	8	no	—
220	C₂³	—	25	1	no	—
221	C₂×C₄	—	5	1	no	—
222	C₂×C₄	—	5	1	no	—
223	C₂×C₄	—	5	2	no	—
224	C₂×C₄	—	5	2	no	—
225	C₂×C₄	—	5	4	no	—
226	C₂×C₄	—	25	2	no	—
227	C₂×C₄	—	25	2	no	—
228	C₂×C₄	—	25	2	no	—
229	C₂×C₄	—	25	2	no	—
230	C₂×C₄	—	25	4	no	—
231	C₂×C₄	—	25	1	no	—
232	C₂×C₄	—	25	4	no	—
233	C₂×C₄	D₅×AGL(1,𝔽₅)	1	1	yes	characteristic
234	C₈	—	5	4	no	—
235	C₈	—	25	4	no	—
236	C₅	—	4	1	no	—
237	C₅	C₂²×C₈×D₅	1	1	yes	characteristic, complemented
238	C₅	C₂²×D₅⋊C₈	1	1	yes	characteristic, complemented
239	C₂²	—	5	2	no	—
240	C₂²	—	5	2	no	—
241	C₂²	—	5	4	no	—
242	C₂²	—	5	4	no	—
243	C₂²	—	25	16	no	—
244	C₂²	—	25	2	no	—
245	C₂²	—	25	4	no	—
246	C₂²	C₂×D₅×AGL(1,𝔽₅)	1	1	yes	characteristic
247	C₄	—	5	1	no	—
248	C₄	—	5	1	no	—
249	C₄	—	5	2	no	—
250	C₄	—	25	2	no	—
251	C₄	C₂×D₅×AGL(1,𝔽₅)	1	2	yes	—
252	C₂	—	5	4	no	—
253	C₂	—	5	4	no	—
254	C₂	—	25	4	no	—
255	C₂	D₅×D₅⋊C₈	1	2	yes	direct product factor
256	C₂	C₂²×D₅×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄	D₅×AGL(1,𝔽₅)
commutator, hypocenter	C₅²	C₂³×C₈
Frattini	C₂	C₂²×D₅×AGL(1,𝔽₅)
Fitting	C₂×C₄×C₅²	C₂×C₄
socle	C₂²×C₅²	C₂²×C₄
prime radical	C₂²×D₅²	C₄
cosocle	C₅×Dic₅	C₂⁴
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×D₅⋊#168	C₂	semidirect
3. C₂×C₅×D₅⋊C₈	C₂	semidirect
4. C₂×#87827	C₂	semidirect
5. C₂×D₅×#170	C₂	semidirect
6. D₅×D₅⋊C₈	C₂	direct (N × H)
16. C₂×#6032	C₂²	semidirect
17. D₅⋊#168	C₂²	semidirect
18. C₂×C₅×#170	C₂²	semidirect
19. C₅×D₅⋊C₈	C₂²	semidirect
20. #87827	C₂²	semidirect
21. D₅×#170	C₂²	semidirect
27. C₂×D₅²	C₈	semidirect
42. #6032	C₂³	semidirect
43. C₅×#170	C₂³	semidirect
50. C₂×D₅⋊C₈	D₅	direct (N × H)
57. C₂×C₅⋊D₅	C₂×C₈	semidirect
59. C₂×C₅×D₅	C₂×C₈	semidirect
60. C₂×C₅×D₅	C₂×C₈	semidirect
63. D₅²	C₂×C₈	semidirect
90. C₂×#170	C₂×D₅	semidirect
93. D₅⋊C₈	C₂×D₅	direct (N × H)
101. C₂×C₅²	C₂²×C₈	semidirect
103. C₅⋊D₅	C₂²×C₈	semidirect
104. C₅×D₅	C₂²×C₈	semidirect
105. C₅×D₅	C₂²×C₈	semidirect
147. #170	C₂²×D₅	semidirect
156. C₅²	C₂³×C₈	semidirect
172. C₂×D₅	C₈×D₅	semidirect
173. C₂×D₅	D₅⋊C₈	direct (N × H)
207. C₂×C₅	C₂×C₈×D₅	semidirect
208. C₂×C₅	C₂×D₅⋊C₈	semidirect
214. D₅	C₂×C₈×D₅	semidirect
215. D₅	C₂×D₅⋊C₈	direct (N × H)
237. C₅	C₂²×C₈×D₅	semidirect
238. C₅	C₂²×D₅⋊C₈	semidirect
255. C₂	D₅×D₅⋊C₈	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×C₄×D₅²	2
C₂×D₅⋊#168	2
C₂×C₅×D₅⋊C₈	2
C₂×#87827	2
C₂×D₅×#170	2
D₅×D₅⋊C₈	2
C₂²×D₅⋊C₈	5
C₂²×C₈×D₅	5

Subgroups — C2×D5×D5⋊C8 #13107596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₅×D₅⋊C₈ #13107596