Subgroups — C₄×C₅×C₄⋊AGL(1,𝔽₅) #13104447

Order 1600.

403 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₄²×C₅
Frattini	C₂²	C₂²×C₅×AGL(1,𝔽₅)
Fitting	C₄²×C₅²	C₄
socle	C₂²×C₅²	C₂²×C₄
prime radical	C₂²×C₅×D₅	C₂³
cosocle	C₂²×D₅	C₂³×C₅
hypercenter	C₄²×C₅	AGL(1,𝔽₅)
hypocenter	C₅	C₄×C₅×C₄⋊C₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₄²×C₅

Complements

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

N	H ≅ G/N	type
9. C₅×C₄⋊AGL(1,𝔽₅)	C₄	direct (N × H)
11. C₄²×C₅²	C₄	semidirect
12. C₄×C₅×Dic₅	C₄	semidirect
13. C₄×C₅×Dic₅	C₄	semidirect
14. C₄×C₅×Dic₅	C₄	semidirect
15. C₄×C₄⋊AGL(1,𝔽₅)	C₅	direct (N × H)
46. C₄×C₅²	C₄²	semidirect
47. C₄×C₅²	C₄⋊C₄	semidirect
49. C₅×Dic₅	C₄²	semidirect
50. C₅×Dic₅	C₄⋊C₄	semidirect
59. C₄⋊AGL(1,𝔽₅)	C₄×C₅	direct (N × H)
67. C₄²×C₅	C₄×C₅	semidirect
68. C₄²×C₅	AGL(1,𝔽₅)	semidirect
69. C₄×Dic₅	C₄×C₅	semidirect
70. C₄×Dic₅	C₄×C₅	semidirect
71. C₄×Dic₅	C₄×C₅	semidirect
113. C₅²	C₄×C₄⋊C₄	semidirect
135. C₄×C₅	C₄²×C₅	semidirect
136. C₄×C₅	C₅×C₄⋊C₄	semidirect
137. C₄×C₅	C₄×AGL(1,𝔽₅)	semidirect
138. C₄×C₅	C₄⋊AGL(1,𝔽₅)	direct (N × H)
140. Dic₅	C₄²×C₅	semidirect
141. Dic₅	C₅×C₄⋊C₄	semidirect
151. C₄²	C₅×AGL(1,𝔽₅)	semidirect
180. C₅	C₄×C₅×C₄⋊C₄	semidirect
181. C₅	C₄×C₄⋊AGL(1,𝔽₅)	direct (N × H)
192. C₄	C₄×C₅×AGL(1,𝔽₅)	semidirect
193. C₄	C₅×C₄⋊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
C₂×C₅×C₄⋊AGL(1,𝔽₅)	2
C₂×C₄×C₅×AGL(1,𝔽₅)	2
C₄²×C₅×D₅	2
C₅×(C₂×AGL(1,𝔽₅))⋊C₄	2
C₄×C₄⋊AGL(1,𝔽₅)	5
C₄×C₅×C₄⋊C₄	5

Subgroups — C4×C5×C4⋊AGL(1,𝔽5) #13104447

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×C₄⋊AGL(1,𝔽₅) #13104447