Subgroups — C₂×C₅×#4078 #13106953

Order 1600.

832 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅²×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#709	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#687	C₂	1	2	yes	maximal, complemented
4	C₅×#4078	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×#686	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#681	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴×C₅²	—	2	1	no	—
8	C₂×C₅²×D₄	—	2	1	no	—
9	C₂×C₅²×D₄	—	2	2	no	—
10	C₂×C₅²×D₄	C₂²	1	2	yes	—
11	C₂×C₅²×D₄	C₂²	1	4	yes	complemented
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	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	C₅×#687	C₂²	1	8	yes	complemented
20	C₅×#686	C₂²	1	8	yes	complemented
21	C₅×#681	C₂²	1	4	yes	complemented
22	C₂×C₅×#168	C₂²	1	2	yes	complemented
23	C₂×C₅×AGL(1,ℤ/8)	—	5	1	no	maximal
24	C₂×#4078	C₅	1	1	yes	maximal, characteristic, direct product factor
25	C₂³×C₅²	—	2	2	no	—
26	C₂³×C₅²	—	2	4	no	—
27	C₂³×C₅²	—	4	4	no	—
28	C₂³×C₅²	D₄	1	1	yes	characteristic
29	C₂²×C₅×D₅	—	2	1	no	—
30	C₅²×D₄	—	2	2	no	—
31	C₅²×D₄	—	2	4	no	—
32	C₅²×D₄	C₂³	1	4	yes	—
33	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
34	C₂×C₄×C₅²	D₄	1	2	yes	—
35	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₅²	C₂³	1	2	yes	—
37	C₅×C₂²⋊D₅	—	2	4	no	—
38	C₂×C₅×Dic₅	—	2	1	no	—
39	C₅×D₂₀	—	2	1	no	—
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	C₅×#168	C₂³	1	4	yes	complemented
45	C₂×C₅×#41	—	5	1	no	—
46	C₂²×C₅×D₄	—	2	2	no	—
47	C₂²×C₅×D₄	D₅	1	1	yes	characteristic, complemented
48	C₂²×C₅×D₄	C₂×C₅	1	1	yes	characteristic, complemented
49	C₂×#709	C₂×C₅	1	1	yes	characteristic, complemented
50	C₅×AGL(1,ℤ/8)	—	5	8	no	—
51	C₂×C₅×SD₁₆	—	5	2	no	—
52	C₂×C₅×D₈	—	5	2	no	—
53	C₂×C₅×M₁₆	—	5	1	no	—
54	C₂×#687	C₂×C₅	1	2	yes	complemented
55	#4078	C₂×C₅	1	8	yes	direct product factor
56	C₂×#686	C₂×C₅	1	2	yes	complemented
57	C₂×#681	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂²×C₅²	—	2	2	no	—
59	C₂²×C₅²	—	2	4	no	—
60	C₂²×C₅²	—	4	8	no	—
61	C₂²×C₅²	—	4	4	no	—
62	C₂²×C₅²	C₂×D₄	1	2	yes	—
63	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
64	C₂×C₅×D₅	—	2	2	no	—
65	C₂×C₅×D₅	—	4	2	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	—	2	4	no	—
71	C₂⁴×C₅	—	2	1	no	—
72	C₂⁴×C₅	—	2	1	no	—
73	C₅×#41	—	5	4	no	—
74	C₅×#41	—	10	2	no	—
75	C₂×C₅×Q₈	—	5	1	no	—
76	C₂×C₅×D₄	—	2	4	no	—
77	C₂×C₅×D₄	—	2	8	no	—
78	C₂×C₅×D₄	—	2	2	no	—
79	C₂×C₅×D₄	—	2	1	no	—
80	C₂×C₅×D₄	—	2	1	no	—
81	C₂×C₅×D₄	—	2	2	no	—
82	C₂×C₅×D₄	—	2	4	no	—
83	C₂×C₅×D₄	—	2	8	no	—
84	C₂×C₅×D₄	—	5	1	no	—
85	C₂×C₅×D₄	—	10	1	no	—
86	C₂×C₅×D₄	C₂×D₅	1	2	yes	—
87	C₂×C₅×D₄	C₂×D₅	1	4	yes	complemented
88	C₂×C₅×D₄	C₂²×C₅	1	2	yes	—
89	C₂×C₅×D₄	C₂²×C₅	1	4	yes	complemented
90	C₂²×C₄×C₅	—	2	2	no	—
91	C₂²×C₄×C₅	—	10	1	no	—
92	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
94	C₂×C₂²⋊D₅	—	2	1	no	—
95	#709	—	2	2	no	—
96	#709	C₂²×C₅	1	4	yes	complemented
97	C₂×D₂₀	C₂²×C₅	1	1	yes	characteristic, complemented
98	C₂×C₄×D₅	—	2	1	no	—
99	C₂×Dic₁₀	C₂²×C₅	1	1	yes	characteristic, complemented
100	C₅×SD₁₆	—	5	8	no	—
101	C₅×D₈	—	5	8	no	—
102	C₅×M₁₆	—	5	4	no	—
103	C₂×C₅×C₈	—	5	2	no	—
104	#687	C₂²×C₅	1	8	yes	complemented
105	#686	C₂²×C₅	1	8	yes	complemented
106	#681	C₂²×C₅	1	4	yes	complemented
107	C₂×#168	C₂²×C₅	1	2	yes	complemented
108	C₂×AGL(1,ℤ/8)	—	5	1	no	Sylow
109	C₂×C₅²	—	2	2	no	—
110	C₂×C₅²	—	4	4	no	—
111	C₂×C₅²	AGL(1,ℤ/8)	1	2	yes	complemented
112	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
113	C₅×D₅	—	4	2	no	—
114	C₂³×C₅	—	2	8	no	—
115	C₂³×C₅	—	2	16	no	—
116	C₂³×C₅	—	2	4	no	—
117	C₂³×C₅	—	2	2	no	—
118	C₂³×C₅	—	2	2	no	—
119	C₂³×C₅	—	2	4	no	—
120	C₂³×C₅	—	2	2	no	—
121	C₂³×C₅	—	4	16	no	—
122	C₂³×C₅	—	4	4	no	—
123	C₂³×C₅	—	4	4	no	—
124	C₂³×C₅	—	10	1	no	—
125	C₂³×C₅	C₂²⋊D₅	1	1	yes	characteristic
126	C₂³×C₅	C₅×D₄	1	1	yes	characteristic
127	C₂²×D₅	—	2	1	no	—
128	C₅×Q₈	—	5	2	no	—
129	C₅×Q₈	—	10	1	no	—
130	C₅×D₄	—	2	8	no	—
131	C₅×D₄	—	2	16	no	—
132	C₅×D₄	—	2	2	no	—
133	C₅×D₄	—	2	4	no	—
134	C₅×D₄	—	2	2	no	—
135	C₅×D₄	—	2	4	no	—
136	C₅×D₄	—	2	8	no	—
137	C₅×D₄	—	5	2	no	—
138	C₅×D₄	—	10	4	no	—
139	C₅×D₄	—	10	1	no	—
140	C₅×D₄	C₂²×D₅	1	4	yes	—
141	C₅×D₄	C₂³×C₅	1	4	yes	—
142	C₂×C₄×C₅	—	2	2	no	—
143	C₂×C₄×C₅	—	2	4	no	—
144	C₂×C₄×C₅	—	2	2	no	—
145	C₂×C₄×C₅	—	2	4	no	—
146	C₂×C₄×C₅	—	10	1	no	—
147	C₂×C₄×C₅	—	10	4	no	—
148	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
149	C₂×C₄×C₅	C₂²⋊D₅	1	1	yes	characteristic
150	C₂×C₄×C₅	C₅×D₄	1	1	yes	characteristic
151	C₂×C₄×C₅	C₅×D₄	1	2	yes	—
152	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
153	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
156	C₂²⋊D₅	—	2	4	no	—
157	C₂×Dic₅	—	2	1	no	—
158	D₂₀	—	2	1	no	—
159	D₂₀	C₂³×C₅	1	2	yes	complemented
160	C₄×D₅	—	2	4	no	—
161	Dic₁₀	—	2	1	no	—
162	Dic₁₀	C₂³×C₅	1	2	yes	complemented
163	C₅×C₈	—	5	4	no	—
164	#168	C₂³×C₅	1	4	yes	complemented
165	C₂×#41	—	5	1	no	—
166	C₂²×D₄	C₅×D₅	1	1	yes	characteristic, complemented
167	AGL(1,ℤ/8)	—	5	8	no	—
168	C₂×SD₁₆	—	5	2	no	—
169	C₂×D₈	—	5	2	no	—
170	C₂×M₁₆	—	5	1	no	—
171	C₅²	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
172	C₂²×C₅	—	2	8	no	—
173	C₂²×C₅	—	2	2	no	—
174	C₂²×C₅	—	2	2	no	—
175	C₂²×C₅	—	2	16	no	—
176	C₂²×C₅	—	2	4	no	—
177	C₂²×C₅	—	2	4	no	—
178	C₂²×C₅	—	2	4	no	—
179	C₂²×C₅	—	2	2	no	—
180	C₂²×C₅	—	4	32	no	—
181	C₂²×C₅	—	4	8	no	—
182	C₂²×C₅	—	4	8	no	—
183	C₂²×C₅	—	4	16	no	—
184	C₂²×C₅	—	4	4	no	—
185	C₂²×C₅	—	4	4	no	—
186	C₂²×C₅	—	10	2	no	—
187	C₂²×C₅	—	20	2	no	—
188	C₂²×C₅	C₂×C₂²⋊D₅	1	2	yes	—
189	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
190	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
191	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
192	C₂×D₅	—	2	2	no	—
193	C₂×D₅	—	4	2	no	—
194	C₄×C₅	—	2	4	no	—
195	C₄×C₅	—	2	2	no	—
196	C₄×C₅	—	2	2	no	—
197	C₄×C₅	—	10	2	no	—
198	C₄×C₅	C₂×C₂²⋊D₅	1	2	yes	—
199	C₄×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
200	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
201	C₄×C₅	C₂×C₅×D₄	1	2	yes	—
202	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
203	C₄×C₅	C₂⁴×C₅	1	1	yes	characteristic
204	Dic₅	—	2	2	no	—
205	C₂⁴	—	2	1	no	—
206	#41	—	5	4	no	—
207	#41	—	10	2	no	—
208	C₂×Q₈	—	5	1	no	—
209	C₂×D₄	—	2	1	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	5	1	no	—
212	C₂×D₄	—	10	1	no	—
213	C₂×D₄	C₂×C₅×D₅	1	2	yes	—
214	C₂×D₄	C₂×C₅×D₅	1	4	yes	complemented
215	C₂²×C₄	—	10	1	no	—
216	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
217	SD₁₆	—	5	8	no	—
218	D₈	—	5	8	no	—
219	M₁₆	—	5	4	no	—
220	C₂×C₈	—	5	2	no	—
221	C₂×C₅	—	2	8	no	—
222	C₂×C₅	—	2	2	no	—
223	C₂×C₅	—	2	2	no	—
224	C₂×C₅	—	2	4	no	—
225	C₂×C₅	—	2	2	no	—
226	C₂×C₅	—	4	16	no	—
227	C₂×C₅	—	4	4	no	—
228	C₂×C₅	—	4	4	no	—
229	C₂×C₅	—	20	2	no	—
230	C₂×C₅	#4078	1	2	yes	direct product factor
231	C₂×C₅	C₅×AGL(1,ℤ/8)	1	2	yes	complemented
232	C₂×C₅	C₂²×C₂²⋊D₅	1	1	yes	characteristic
233	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
234	D₅	—	4	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	4	no	—
237	C₂³	—	4	4	no	—
238	C₂³	—	10	1	no	—
239	C₂³	C₅×C₂²⋊D₅	1	1	yes	characteristic
240	Q₈	—	5	2	no	—
241	Q₈	—	10	1	no	—
242	D₄	—	2	2	no	—
243	D₄	—	2	4	no	—
244	D₄	—	5	2	no	—
245	D₄	—	10	4	no	—
246	D₄	—	10	1	no	—
247	D₄	C₂²×C₅×D₅	1	4	yes	—
248	C₂×C₄	—	10	1	no	—
249	C₂×C₄	—	10	4	no	—
250	C₂×C₄	C₅×C₂²⋊D₅	1	1	yes	characteristic
251	C₂×C₄	C₅×C₂²⋊D₅	1	2	yes	—
252	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
253	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
254	C₈	—	5	4	no	—
255	C₅	—	2	2	no	—
256	C₅	C₂×#4078	1	1	yes	characteristic, direct product factor
257	C₅	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented
258	C₂²	—	2	2	no	—
259	C₂²	—	2	4	no	—
260	C₂²	—	4	8	no	—
261	C₂²	—	4	4	no	—
262	C₂²	—	10	2	no	—
263	C₂²	—	20	2	no	—
264	C₂²	C₂×C₅×C₂²⋊D₅	1	2	yes	—
265	C₂²	C₂×C₅×C₂²⋊D₅	1	1	yes	characteristic
266	C₄	—	10	2	no	—
267	C₄	C₂×C₅×C₂²⋊D₅	1	2	yes	—
268	C₄	C₂×C₅×C₂²⋊D₅	1	1	yes	characteristic
269	C₄	C₂³×C₅×D₅	1	1	yes	characteristic
270	C₂	—	2	2	no	—
271	C₂	—	4	4	no	—
272	C₂	—	20	2	no	—
273	C₂	C₅×#4078	1	2	yes	direct product factor
274	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×D₄	C₂
socle	C₂²×C₅²	C₂×D₄
hypercenter	C₂²×C₅×D₄	D₅
hypocenter	C₅	C₂×C₅×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅²×D₄	C₂	semidirect
2. C₂×C₅×#709	C₂	semidirect
3. C₂×C₅×#687	C₂	semidirect
4. C₅×#4078	C₂	direct (N × H)
5. C₂×C₅×#686	C₂	semidirect
6. C₂×C₅×#681	C₂	semidirect
11. C₂×C₅²×D₄	C₂²	semidirect
15. C₅×#709	C₂²	semidirect
16. C₂×C₅×D₂₀	C₂²	semidirect
18. C₂×C₅×Dic₁₀	C₂²	semidirect
19. C₅×#687	C₂²	semidirect
20. C₅×#686	C₂²	semidirect
21. C₅×#681	C₂²	semidirect
22. C₂×C₅×#168	C₂²	semidirect
24. C₂×#4078	C₅	direct (N × H)
40. C₅×D₂₀	C₂³	semidirect
43. C₅×Dic₁₀	C₂³	semidirect
44. C₅×#168	C₂³	semidirect
47. C₂²×C₅×D₄	D₅	semidirect
48. C₂²×C₅×D₄	C₂×C₅	semidirect
49. C₂×#709	C₂×C₅	semidirect
54. C₂×#687	C₂×C₅	semidirect
55. #4078	C₂×C₅	direct (N × H)
56. C₂×#686	C₂×C₅	semidirect
57. C₂×#681	C₂×C₅	semidirect
87. C₂×C₅×D₄	C₂×D₅	semidirect
89. C₂×C₅×D₄	C₂²×C₅	semidirect
96. #709	C₂²×C₅	semidirect
97. C₂×D₂₀	C₂²×C₅	semidirect
99. C₂×Dic₁₀	C₂²×C₅	semidirect
104. #687	C₂²×C₅	semidirect
105. #686	C₂²×C₅	semidirect
106. #681	C₂²×C₅	semidirect
107. C₂×#168	C₂²×C₅	semidirect
111. C₂×C₅²	AGL(1,ℤ/8)	semidirect
159. D₂₀	C₂³×C₅	semidirect
162. Dic₁₀	C₂³×C₅	semidirect
164. #168	C₂³×C₅	semidirect
166. C₂²×D₄	C₅×D₅	semidirect
171. C₅²	C₂×AGL(1,ℤ/8)	semidirect
214. C₂×D₄	C₂×C₅×D₅	semidirect
230. C₂×C₅	#4078	direct (N × H)
231. C₂×C₅	C₅×AGL(1,ℤ/8)	semidirect
256. C₅	C₂×#4078	direct (N × H)
257. C₅	C₂×C₅×AGL(1,ℤ/8)	semidirect
273. C₂	C₅×#4078	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×D₄	2
C₂×C₅×#709	2
C₂×C₅×#687	2
C₅×#4078	2
C₂×C₅×#686	2
C₂×C₅×#681	2
C₂×C₅×AGL(1,ℤ/8)	5
C₂×#4078	5

Subgroups — C2×C5×#4078 #13106953

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4078 #13106953