Subgroups — C₅×#41×AGL(1,𝔽₅) #13107732

Order 1600.

640 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×D₅×#41	C₂	1	1	yes	maximal, characteristic
2	C₅×Q₈×AGL(1,𝔽₅)	C₂	1	2	yes	maximal, complemented
3	C₅×D₄×AGL(1,𝔽₅)	C₂	1	6	yes	maximal, complemented
4	C₅×#4130	C₂	1	3	yes	maximal, complemented
5	C₂×C₄×C₅×AGL(1,𝔽₅)	C₂	1	3	yes	maximal, complemented
6	C₂²×C₅×AGL(1,𝔽₅)	C₂²	1	6	yes	—
7	C₅²×#41	C₄	1	1	yes	characteristic, complemented
8	C₅×#713	C₄	1	1	yes	characteristic, complemented
9	C₅×Q₈×D₅	C₂²	1	1	yes	characteristic
10	C₅×#711	C₄	1	3	yes	complemented
11	C₅×D₄×D₅	C₂²	1	3	yes	—
12	C₅×#709	C₄	1	3	yes	complemented
13	C₂×C₄×C₅×D₅	C₂²	1	3	yes	—
14	C₅×C₂²⋊AGL(1,𝔽₅)	C₂²	1	6	yes	—
15	C₅×C₄⋊AGL(1,𝔽₅)	C₂²	1	6	yes	—
16	C₄×C₅×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
17	C₄×C₅×AGL(1,𝔽₅)	C₂²	1	3	yes	—
18	C₄×C₅×AGL(1,𝔽₅)	C₂²	1	6	yes	—
19	#41×AGL(1,𝔽₅)	C₅	1	1	yes	maximal, characteristic, direct product factor
20	C₄×C₅×#41	—	5	1	no	maximal
21	C₂²×C₅×D₅	C₂³	1	3	yes	—
22	C₂×C₅×AGL(1,𝔽₅)	—	2	12	no	—
23	C₂×C₅×AGL(1,𝔽₅)	C₂³	1	6	yes	—
24	C₂×C₅×AGL(1,𝔽₅)	C₂³	1	2	yes	—
25	C₅²×Q₈	C₂×C₄	1	1	yes	characteristic, complemented
26	C₅²×D₄	C₂×C₄	1	3	yes	complemented
27	C₂×C₄×C₅²	C₂×C₄	1	3	yes	complemented
28	C₅×C₂²⋊D₅	C₂×C₄	1	6	yes	complemented
29	C₂×C₅×Dic₅	C₂×C₄	1	3	yes	complemented
30	C₅×D₂₀	C₂×C₄	1	3	yes	complemented
31	C₄×C₅×D₅	C₂×C₄	1	3	yes	complemented
32	C₄×C₅×D₅	C₂×C₄	1	3	yes	complemented
33	C₄×C₅×D₅	C₂³	1	1	yes	characteristic
34	C₄×C₅×D₅	C₂³	1	3	yes	—
35	C₅×Dic₁₀	C₂×C₄	1	3	yes	complemented
36	C₂×C₅×#41	—	5	1	no	—
37	D₅×#41	C₂×C₅	1	1	yes	characteristic
38	Q₈×AGL(1,𝔽₅)	C₂×C₅	1	2	yes	complemented
39	D₄×AGL(1,𝔽₅)	C₂×C₅	1	6	yes	complemented
40	#4130	C₂×C₅	1	3	yes	complemented
41	C₂×C₄×AGL(1,𝔽₅)	C₂×C₅	1	3	yes	complemented
42	C₄×C₅×Q₈	—	5	2	no	—
43	C₄×C₅×D₄	—	5	6	no	—
44	C₅×#117	—	5	3	no	—
45	C₂×C₄²×C₅	—	5	3	no	—
46	C₂²×C₅²	C₂²×C₄	1	3	yes	—
47	C₂×C₅×D₅	—	2	6	no	—
48	C₂×C₅×D₅	C₂²×C₄	1	3	yes	—
49	C₂×C₅×D₅	C₂⁴	1	1	yes	characteristic
50	C₅×AGL(1,𝔽₅)	—	2	6	no	—
51	C₅×AGL(1,𝔽₅)	#41	1	4	yes	direct product factor
52	C₄×C₅²	C₂²×C₄	1	3	yes	—
53	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
54	C₅×Dic₅	C₂²×C₄	1	3	yes	—
55	C₅×Dic₅	C₂²×C₄	1	1	yes	characteristic
56	C₂²×AGL(1,𝔽₅)	C₂²×C₅	1	6	yes	—
57	C₅×#41	—	4	1	no	—
58	C₅×#41	—	5	1	no	—
59	C₅×#41	—	5	3	no	—
60	C₅×#41	—	5	3	no	—
61	C₅×#41	C₄×C₅	1	1	yes	characteristic, complemented
62	C₅×#41	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
63	C₂×C₅×Q₈	—	5	1	no	—
64	C₂×C₅×D₄	—	5	3	no	—
65	C₂²×C₄×C₅	—	5	3	no	—
66	C₂²×C₄×C₅	—	5	6	no	—
67	#713	C₄×C₅	1	1	yes	characteristic, complemented
68	Q₈×D₅	C₂²×C₅	1	1	yes	characteristic
69	#711	C₄×C₅	1	3	yes	complemented
70	D₄×D₅	C₂²×C₅	1	3	yes	—
71	#709	C₄×C₅	1	3	yes	complemented
72	C₂×C₄×D₅	C₂²×C₅	1	3	yes	—
73	C₂²⋊AGL(1,𝔽₅)	C₂²×C₅	1	6	yes	—
74	C₄⋊AGL(1,𝔽₅)	C₂²×C₅	1	6	yes	—
75	C₄×AGL(1,𝔽₅)	C₂²×C₅	1	1	yes	characteristic
76	C₄×AGL(1,𝔽₅)	C₂²×C₅	1	3	yes	—
77	C₄×AGL(1,𝔽₅)	C₂²×C₅	1	6	yes	—
78	C₅×C₄⋊C₄	—	5	6	no	—
79	C₅×C₂²⋊C₄	—	5	6	no	—
80	C₄²×C₅	—	5	1	no	—
81	C₄²×C₅	—	5	3	no	—
82	C₄²×C₅	—	5	6	no	—
83	C₄×#41	—	5	1	no	Sylow
84	C₂×C₅²	—	2	3	no	—
85	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
86	C₅×D₅	—	2	3	no	—
87	C₅×D₅	C₂×#41	1	2	yes	—
88	C₂³×C₅	—	5	3	no	—
89	C₂²×D₅	C₂³×C₅	1	3	yes	—
90	C₂×AGL(1,𝔽₅)	—	2	12	no	—
91	C₂×AGL(1,𝔽₅)	C₂³×C₅	1	6	yes	—
92	C₂×AGL(1,𝔽₅)	C₂³×C₅	1	2	yes	—
93	C₅×Q₈	—	4	1	no	—
94	C₅×Q₈	—	5	3	no	—
95	C₅×Q₈	C₂×C₄×C₅	1	1	yes	characteristic, complemented
96	C₅×Q₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
97	C₅×D₄	—	4	3	no	—
98	C₅×D₄	—	5	6	no	—
99	C₅×D₄	—	5	3	no	—
100	C₅×D₄	C₂×C₄×C₅	1	3	yes	complemented
101	C₅×D₄	C₂×AGL(1,𝔽₅)	1	3	yes	complemented
102	C₂×C₄×C₅	—	4	3	no	—
103	C₂×C₄×C₅	—	5	2	no	—
104	C₂×C₄×C₅	—	5	3	no	—
105	C₂×C₄×C₅	—	5	3	no	—
106	C₂×C₄×C₅	—	5	3	no	—
107	C₂×C₄×C₅	—	5	3	no	—
108	C₂×C₄×C₅	—	5	1	no	—
109	C₂×C₄×C₅	—	5	6	no	—
110	C₂×C₄×C₅	—	10	12	no	—
111	C₂×C₄×C₅	C₂×C₄×C₅	1	3	yes	complemented
112	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	3	yes	complemented
113	C₂²⋊D₅	C₂×C₄×C₅	1	6	yes	complemented
114	C₂×Dic₅	C₂×C₄×C₅	1	3	yes	complemented
115	D₂₀	C₂×C₄×C₅	1	3	yes	complemented
116	C₄×D₅	C₂×C₄×C₅	1	3	yes	complemented
117	C₄×D₅	C₂×C₄×C₅	1	3	yes	complemented
118	C₄×D₅	C₂³×C₅	1	1	yes	characteristic
119	C₄×D₅	C₂³×C₅	1	3	yes	—
120	Dic₁₀	C₂×C₄×C₅	1	3	yes	complemented
121	C₂×#41	—	5	1	no	—
122	C₄×Q₈	—	5	2	no	—
123	C₄×D₄	—	5	6	no	—
124	#117	—	5	3	no	—
125	C₂×C₄²	—	5	3	no	—
126	C₅²	C₄×#41	1	1	yes	characteristic, complemented, Sylow
127	C₂²×C₅	—	4	3	no	—
128	C₂²×C₅	—	5	1	no	—
129	C₂²×C₅	—	5	3	no	—
130	C₂²×C₅	—	10	6	no	—
131	C₂²×C₅	C₂²×C₄×C₅	1	3	yes	—
132	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	3	yes	—
133	C₂×D₅	—	2	6	no	—
134	C₂×D₅	C₂²×C₄×C₅	1	3	yes	—
135	C₂×D₅	C₂⁴×C₅	1	1	yes	characteristic
136	AGL(1,𝔽₅)	—	2	6	no	—
137	AGL(1,𝔽₅)	C₅×#41	1	4	yes	direct product factor
138	C₄×C₅	—	4	1	no	—
139	C₄×C₅	—	4	3	no	—
140	C₄×C₅	—	5	4	no	—
141	C₄×C₅	—	5	1	no	—
142	C₄×C₅	—	5	3	no	—
143	C₄×C₅	—	10	6	no	—
144	C₄×C₅	C₂²×C₄×C₅	1	3	yes	—
145	C₄×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
146	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	3	yes	—
147	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
148	Dic₅	C₂²×C₄×C₅	1	3	yes	—
149	Dic₅	C₂²×C₄×C₅	1	1	yes	characteristic
150	#41	—	5	1	no	—
151	#41	—	5	3	no	—
152	#41	—	5	3	no	—
153	#41	C₅×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
154	C₂×Q₈	—	5	1	no	—
155	C₂×D₄	—	5	3	no	—
156	C₂²×C₄	—	5	3	no	—
157	C₂²×C₄	—	5	6	no	—
158	C₄⋊C₄	—	5	6	no	—
159	C₂²⋊C₄	—	5	6	no	—
160	C₄²	—	5	1	no	—
161	C₄²	—	5	3	no	—
162	C₄²	—	5	6	no	—
163	C₂×C₅	—	2	3	no	—
164	C₂×C₅	—	2	3	no	—
165	C₂×C₅	—	4	1	no	—
166	C₂×C₅	—	5	2	no	—
167	C₂×C₅	—	8	3	no	—
168	C₂×C₅	—	10	3	no	—
169	C₂×C₅	C₂³×C₄×C₅	1	1	yes	characteristic
170	C₂×C₅	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
171	D₅	—	2	3	no	—
172	D₅	C₂×C₅×#41	1	2	yes	—
173	C₂³	—	5	3	no	—
174	Q₈	—	5	3	no	—
175	Q₈	C₂×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
176	D₄	—	5	6	no	—
177	D₄	—	5	3	no	—
178	D₄	C₂×C₅×AGL(1,𝔽₅)	1	3	yes	complemented
179	C₂×C₄	—	5	2	no	—
180	C₂×C₄	—	5	3	no	—
181	C₂×C₄	—	5	1	no	—
182	C₂×C₄	—	5	3	no	—
183	C₂×C₄	—	5	3	no	—
184	C₂×C₄	—	5	3	no	—
185	C₂×C₄	—	5	6	no	—
186	C₂×C₄	—	10	12	no	—
187	C₂×C₄	C₂×C₅×AGL(1,𝔽₅)	1	3	yes	complemented
188	C₅	—	4	1	no	—
189	C₅	C₄×C₅×#41	1	1	yes	characteristic, complemented
190	C₅	#41×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
191	C₂²	—	5	1	no	—
192	C₂²	—	5	3	no	—
193	C₂²	—	10	6	no	—
194	C₂²	C₂²×C₅×AGL(1,𝔽₅)	1	3	yes	—
195	C₄	—	5	4	no	—
196	C₄	—	5	1	no	—
197	C₄	—	5	3	no	—
198	C₄	—	10	6	no	—
199	C₄	C₂²×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic
200	C₄	C₂²×C₅×AGL(1,𝔽₅)	1	3	yes	—
201	C₂	—	2	3	no	—
202	C₂	—	5	2	no	—
203	C₂	—	10	3	no	—
204	C₂	C₂³×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	C₂²×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂³×C₄×C₅
Frattini	C₂	C₂³×C₅×AGL(1,𝔽₅)
Fitting	C₅²×#41	C₄
socle	C₂×C₅²	C₂³×C₄
prime radical	C₅×D₅×#41	C₂
cosocle	C₂×D₅	C₂⁴×C₅
hypercenter	C₅×#41	AGL(1,𝔽₅)
hypocenter	C₅	C₄×C₅×#41
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄×C₅ ◁ C₅×#41

Complements

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

N	H ≅ G/N	type
2. C₅×Q₈×AGL(1,𝔽₅)	C₂	semidirect
3. C₅×D₄×AGL(1,𝔽₅)	C₂	semidirect
4. C₅×#4130	C₂	semidirect
5. C₂×C₄×C₅×AGL(1,𝔽₅)	C₂	semidirect
7. C₅²×#41	C₄	semidirect
8. C₅×#713	C₄	semidirect
10. C₅×#711	C₄	semidirect
12. C₅×#709	C₄	semidirect
19. #41×AGL(1,𝔽₅)	C₅	direct (N × H)
25. C₅²×Q₈	C₂×C₄	semidirect
26. C₅²×D₄	C₂×C₄	semidirect
27. C₂×C₄×C₅²	C₂×C₄	semidirect
28. C₅×C₂²⋊D₅	C₂×C₄	semidirect
29. C₂×C₅×Dic₅	C₂×C₄	semidirect
30. C₅×D₂₀	C₂×C₄	semidirect
31. C₄×C₅×D₅	C₂×C₄	semidirect
32. C₄×C₅×D₅	C₂×C₄	semidirect
35. C₅×Dic₁₀	C₂×C₄	semidirect
38. Q₈×AGL(1,𝔽₅)	C₂×C₅	semidirect
39. D₄×AGL(1,𝔽₅)	C₂×C₅	semidirect
40. #4130	C₂×C₅	semidirect
41. C₂×C₄×AGL(1,𝔽₅)	C₂×C₅	semidirect
51. C₅×AGL(1,𝔽₅)	#41	direct (N × H)
61. C₅×#41	C₄×C₅	semidirect
62. C₅×#41	AGL(1,𝔽₅)	direct (N × H)
67. #713	C₄×C₅	semidirect
69. #711	C₄×C₅	semidirect
71. #709	C₄×C₅	semidirect
95. C₅×Q₈	C₂×C₄×C₅	semidirect
96. C₅×Q₈	C₂×AGL(1,𝔽₅)	semidirect
100. C₅×D₄	C₂×C₄×C₅	semidirect
101. C₅×D₄	C₂×AGL(1,𝔽₅)	semidirect
111. C₂×C₄×C₅	C₂×C₄×C₅	semidirect
112. C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	semidirect
113. C₂²⋊D₅	C₂×C₄×C₅	semidirect
114. C₂×Dic₅	C₂×C₄×C₅	semidirect
115. D₂₀	C₂×C₄×C₅	semidirect
116. C₄×D₅	C₂×C₄×C₅	semidirect
117. C₄×D₅	C₂×C₄×C₅	semidirect
120. Dic₁₀	C₂×C₄×C₅	semidirect
126. C₅²	C₄×#41	semidirect
137. AGL(1,𝔽₅)	C₅×#41	direct (N × H)
153. #41	C₅×AGL(1,𝔽₅)	direct (N × H)
175. Q₈	C₂×C₅×AGL(1,𝔽₅)	semidirect
178. D₄	C₂×C₅×AGL(1,𝔽₅)	semidirect
187. C₂×C₄	C₂×C₅×AGL(1,𝔽₅)	semidirect
189. C₅	C₄×C₅×#41	semidirect
190. C₅	#41×AGL(1,𝔽₅)	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅×D₅×#41	2
C₅×Q₈×AGL(1,𝔽₅)	2
C₅×D₄×AGL(1,𝔽₅)	2
C₅×#4130	2
C₂×C₄×C₅×AGL(1,𝔽₅)	2
#41×AGL(1,𝔽₅)	5
C₄×C₅×#41	5

Subgroups — C5×#41×AGL(1,𝔽5) #13107732

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#41×AGL(1,𝔽₅) #13107732