Subgroups — C₅×C₄⋊C₄×AGL(1,𝔽₅) #13104470

Order 1600.

401 conjugacy classes of subgroups.

The subgroup lattice has 238 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	1	yes	maximal, characteristic
2	C₂×C₄×C₅×AGL(1,𝔽₅)	C₂	1	2	yes	maximal
3	C₂×C₄×C₅×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
4	C₅×D₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₅×(C₂×AGL(1,𝔽₅))⋊C₄	C₂	1	2	yes	maximal
6	C₂²×C₅×AGL(1,𝔽₅)	C₂²	1	2	yes	—
7	C₂²×C₅×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
8	C₂²×C₅×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₅×D₅	C₂²	1	2	yes	—
10	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
11	C₅×C₄⋊AGL(1,𝔽₅)	C₄	1	4	yes	complemented
12	C₄×C₅×AGL(1,𝔽₅)	—	2	4	no	—
13	C₄×C₅×AGL(1,𝔽₅)	C₄	1	4	yes	complemented
14	C₅²×C₄⋊C₄	C₄	1	1	yes	characteristic, complemented
15	C₅×C₄⋊Dic₅	C₄	1	1	yes	characteristic, complemented
16	C₅×#683	C₄	1	2	yes	complemented
17	C₄⋊C₄×AGL(1,𝔽₅)	C₅	1	1	yes	maximal, characteristic, direct product factor
18	C₄×C₅×C₄⋊C₄	—	5	1	no	maximal
19	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
20	C₂×C₅×AGL(1,𝔽₅)	—	2	2	no	—
21	C₂×C₅×AGL(1,𝔽₅)	—	2	1	no	—
22	C₂×C₅×AGL(1,𝔽₅)	—	2	1	no	—
23	C₂×C₅×AGL(1,𝔽₅)	—	2	2	no	—
24	C₂×C₅×AGL(1,𝔽₅)	C₂×C₄	1	2	yes	—
25	C₂×C₅×AGL(1,𝔽₅)	C₂×C₄	1	4	yes	—
26	C₂×C₅×AGL(1,𝔽₅)	C₂×C₄	1	2	yes	—
27	C₂×C₅×AGL(1,𝔽₅)	D₄	1	2	yes	—
28	C₂×C₅×AGL(1,𝔽₅)	Q₈	1	2	yes	—
29	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
30	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
31	C₂×C₅×Dic₅	C₂×C₄	1	2	yes	—
32	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
33	C₄×C₅×D₅	—	2	2	no	—
34	C₄×C₅×D₅	—	2	2	no	—
35	C₄×C₅×D₅	C₂×C₄	1	2	yes	—
36	C₄×C₅×D₅	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊AGL(1,𝔽₅)	C₂×C₅	1	1	yes	characteristic
38	C₂×C₄×AGL(1,𝔽₅)	C₂×C₅	1	2	yes	—
39	C₂×C₄×AGL(1,𝔽₅)	C₂×C₅	1	1	yes	characteristic
40	C₂×C₅×C₄⋊C₄	—	5	1	no	—
41	C₂×C₅×C₄⋊C₄	—	5	1	no	—
42	C₂×C₄²×C₅	—	5	1	no	—
43	C₂×C₄²×C₅	—	5	2	no	—
44	D₅×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
45	(C₂×AGL(1,𝔽₅))⋊C₄	C₂×C₅	1	2	yes	—
46	C₅×#95	—	5	2	no	—
47	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₅×D₅	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₅×D₅	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₅×D₅	C₂×D₄	1	1	yes	characteristic
51	C₂×C₅×D₅	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₅×D₅	#41	1	1	yes	characteristic
53	C₂×C₅×D₅	#41	1	1	yes	characteristic
54	C₅×AGL(1,𝔽₅)	—	2	2	no	—
55	C₅×AGL(1,𝔽₅)	—	2	4	no	—
56	C₅×AGL(1,𝔽₅)	C₄⋊C₄	1	4	yes	direct product factor
57	C₄×C₅²	—	2	2	no	—
58	C₄×C₅²	C₄²	1	2	yes	complemented
59	C₅×Dic₅	—	2	2	no	—
60	C₅×Dic₅	C₄²	1	2	yes	complemented
61	C₂²×AGL(1,𝔽₅)	C₂²×C₅	1	2	yes	—
62	C₂²×AGL(1,𝔽₅)	C₂²×C₅	1	1	yes	characteristic
63	C₂²×AGL(1,𝔽₅)	C₂²×C₅	1	1	yes	characteristic
64	C₂²×C₄×C₅	—	5	1	no	—
65	C₂²×C₄×C₅	—	5	1	no	—
66	C₂²×C₄×C₅	—	5	2	no	—
67	C₂²×C₄×C₅	—	5	1	no	—
68	C₂²×C₄×C₅	—	5	2	no	—
69	C₂×C₄×D₅	C₂²×C₅	1	2	yes	—
70	C₂×C₄×D₅	C₂²×C₅	1	1	yes	characteristic
71	C₄⋊AGL(1,𝔽₅)	C₄×C₅	1	4	yes	complemented
72	C₄×AGL(1,𝔽₅)	—	2	4	no	—
73	C₄×AGL(1,𝔽₅)	C₄×C₅	1	4	yes	complemented
74	C₅×C₄⋊C₄	—	4	1	no	—
75	C₅×C₄⋊C₄	—	5	4	no	—
76	C₅×C₄⋊C₄	—	5	2	no	—
77	C₅×C₄⋊C₄	—	5	1	no	—
78	C₅×C₄⋊C₄	C₄×C₅	1	1	yes	characteristic, complemented
79	C₅×C₄⋊C₄	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
80	C₄²×C₅	—	5	4	no	—
81	C₄²×C₅	—	10	4	no	—
82	C₄⋊Dic₅	C₄×C₅	1	1	yes	characteristic, complemented
83	#683	C₄×C₅	1	2	yes	complemented
84	C₄×C₄⋊C₄	—	5	1	no	Sylow
85	C₂×C₅²	C₂×C₄²	1	1	yes	characteristic
86	C₂×C₅²	C₄×D₄	1	1	yes	characteristic
87	C₂×C₅²	C₄×Q₈	1	1	yes	characteristic
88	C₅×D₅	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₅×D₅	#117	1	1	yes	characteristic
90	C₅×D₅	C₄×D₄	1	1	yes	characteristic
91	C₅×D₅	C₄×Q₈	1	1	yes	characteristic
92	C₂³×C₅	—	5	1	no	—
93	C₂²×D₅	C₂³×C₅	1	1	yes	characteristic
94	C₂×AGL(1,𝔽₅)	—	2	2	no	—
95	C₂×AGL(1,𝔽₅)	—	2	1	no	—
96	C₂×AGL(1,𝔽₅)	—	2	1	no	—
97	C₂×AGL(1,𝔽₅)	—	2	2	no	—
98	C₂×AGL(1,𝔽₅)	C₂×C₄×C₅	1	2	yes	—
99	C₂×AGL(1,𝔽₅)	C₂×C₄×C₅	1	4	yes	—
100	C₂×AGL(1,𝔽₅)	C₂×C₄×C₅	1	2	yes	—
101	C₂×AGL(1,𝔽₅)	C₅×D₄	1	2	yes	—
102	C₂×AGL(1,𝔽₅)	C₅×Q₈	1	2	yes	—
103	C₂×C₄×C₅	—	4	1	no	—
104	C₂×C₄×C₅	—	4	2	no	—
105	C₂×C₄×C₅	—	5	2	no	—
106	C₂×C₄×C₅	—	5	2	no	—
107	C₂×C₄×C₅	—	5	2	no	—
108	C₂×C₄×C₅	—	5	1	no	—
109	C₂×C₄×C₅	—	5	2	no	—
110	C₂×C₄×C₅	—	5	2	no	—
111	C₂×C₄×C₅	—	5	2	no	—
112	C₂×C₄×C₅	—	5	4	no	—
113	C₂×C₄×C₅	—	5	2	no	—
114	C₂×C₄×C₅	—	10	2	no	—
115	C₂×C₄×C₅	—	10	2	no	—
116	C₂×C₄×C₅	—	10	2	no	—
117	C₂×C₄×C₅	—	10	1	no	—
118	C₂×C₄×C₅	—	10	2	no	—
119	C₂×C₄×C₅	—	10	1	no	—
120	C₂×C₄×C₅	C₂×C₄×C₅	1	2	yes	—
121	C₂×C₄×C₅	C₂×C₄×C₅	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
123	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
124	C₂×Dic₅	C₂×C₄×C₅	1	2	yes	—
125	C₂×Dic₅	C₂×C₄×C₅	1	1	yes	characteristic
126	C₄×D₅	—	2	2	no	—
127	C₄×D₅	—	2	2	no	—
128	C₄×D₅	C₂×C₄×C₅	1	2	yes	—
129	C₄×D₅	C₂×C₄×C₅	1	2	yes	—
130	C₂×C₄⋊C₄	—	5	1	no	—
131	C₂×C₄⋊C₄	—	5	1	no	—
132	C₂×C₄²	—	5	1	no	—
133	C₂×C₄²	—	5	2	no	—
134	#95	—	5	2	no	—
135	C₅²	C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
136	C₂²×C₅	—	4	1	no	—
137	C₂²×C₅	—	5	1	no	—
138	C₂²×C₅	—	5	1	no	—
139	C₂²×C₅	—	5	1	no	—
140	C₂²×C₅	—	5	1	no	—
141	C₂²×C₅	—	5	1	no	—
142	C₂²×C₅	—	5	1	no	—
143	C₂²×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
144	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
145	C₂×D₅	C₂²×C₄×C₅	1	1	yes	characteristic
146	C₂×D₅	C₂²×C₄×C₅	1	1	yes	characteristic
147	C₂×D₅	C₂×C₅×D₄	1	1	yes	characteristic
148	C₂×D₅	C₂×C₅×Q₈	1	1	yes	characteristic
149	C₂×D₅	C₅×#41	1	1	yes	characteristic
150	C₂×D₅	C₅×#41	1	1	yes	characteristic
151	AGL(1,𝔽₅)	—	2	2	no	—
152	AGL(1,𝔽₅)	—	2	4	no	—
153	AGL(1,𝔽₅)	C₅×C₄⋊C₄	1	4	yes	direct product factor
154	C₄×C₅	—	2	2	no	—
155	C₄×C₅	—	2	2	no	—
156	C₄×C₅	—	4	2	no	—
157	C₄×C₅	—	5	4	no	—
158	C₄×C₅	—	5	2	no	—
159	C₄×C₅	—	8	2	no	—
160	C₄×C₅	—	10	4	no	—
161	C₄×C₅	—	10	2	no	—
162	C₄×C₅	—	10	2	no	—
163	C₄×C₅	C₄²×C₅	1	2	yes	complemented
164	C₄×C₅	C₄×AGL(1,𝔽₅)	1	2	yes	complemented
165	Dic₅	—	2	2	no	—
166	Dic₅	C₄²×C₅	1	2	yes	complemented
167	C₂²×C₄	—	5	1	no	—
168	C₂²×C₄	—	5	1	no	—
169	C₂²×C₄	—	5	1	no	—
170	C₂²×C₄	—	5	2	no	—
171	C₂²×C₄	—	5	2	no	—
172	C₄⋊C₄	—	5	4	no	—
173	C₄⋊C₄	—	5	2	no	—
174	C₄⋊C₄	—	5	1	no	—
175	C₄⋊C₄	C₅×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
176	C₄²	—	5	4	no	—
177	C₄²	—	10	4	no	—
178	C₂×C₅	—	4	1	no	—
179	C₂×C₅	—	4	1	no	—
180	C₂×C₅	—	4	1	no	—
181	C₂×C₅	—	5	1	no	—
182	C₂×C₅	—	5	1	no	—
183	C₂×C₅	—	5	1	no	—
184	C₂×C₅	—	5	1	no	—
185	C₂×C₅	C₂×C₄²×C₅	1	1	yes	characteristic
186	C₂×C₅	C₄×C₅×D₄	1	1	yes	characteristic
187	C₂×C₅	C₄×C₅×Q₈	1	1	yes	characteristic
188	C₂×C₅	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
189	C₂×C₅	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
190	C₂×C₅	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
191	D₅	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
192	D₅	C₅×#117	1	1	yes	characteristic
193	D₅	C₄×C₅×D₄	1	1	yes	characteristic
194	D₅	C₄×C₅×Q₈	1	1	yes	characteristic
195	C₂³	—	5	1	no	—
196	C₂×C₄	—	5	2	no	—
197	C₂×C₄	—	5	2	no	—
198	C₂×C₄	—	5	2	no	—
199	C₂×C₄	—	5	1	no	—
200	C₂×C₄	—	5	2	no	—
201	C₂×C₄	—	5	2	no	—
202	C₂×C₄	—	5	2	no	—
203	C₂×C₄	—	5	4	no	—
204	C₂×C₄	—	5	2	no	—
205	C₂×C₄	—	10	2	no	—
206	C₂×C₄	—	10	2	no	—
207	C₂×C₄	—	10	2	no	—
208	C₂×C₄	—	10	1	no	—
209	C₂×C₄	—	10	1	no	—
210	C₂×C₄	—	10	2	no	—
211	C₂×C₄	C₂×C₅×AGL(1,𝔽₅)	1	2	yes	—
212	C₂×C₄	C₂×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic
213	C₅	—	4	1	no	—
214	C₅	C₄×C₅×C₄⋊C₄	1	1	yes	characteristic, complemented
215	C₅	C₄⋊C₄×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
216	C₂²	—	5	1	no	—
217	C₂²	—	5	1	no	—
218	C₂²	—	5	1	no	—
219	C₂²	—	5	1	no	—
220	C₂²	—	5	1	no	—
221	C₂²	—	5	1	no	—
222	C₂²	C₂²×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	5	4	no	—
225	C₄	—	5	2	no	—
226	C₄	—	10	4	no	—
227	C₄	—	10	2	no	—
228	C₄	—	10	2	no	—
229	C₄	C₄×C₅×AGL(1,𝔽₅)	1	2	yes	complemented
230	C₂	—	5	1	no	—
231	C₂	—	5	1	no	—
232	C₂	—	5	1	no	—
233	C₂	—	5	1	no	—
234	C₂	C₂×C₄×C₅×AGL(1,𝔽₅)	1	1	yes	characteristic
235	C₂	C₅×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
236	C₂	C₅×Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
11. C₅×C₄⋊AGL(1,𝔽₅)	C₄	semidirect
13. C₄×C₅×AGL(1,𝔽₅)	C₄	semidirect
14. C₅²×C₄⋊C₄	C₄	semidirect
15. C₅×C₄⋊Dic₅	C₄	semidirect
16. C₅×#683	C₄	semidirect
17. C₄⋊C₄×AGL(1,𝔽₅)	C₅	direct (N × H)
56. C₅×AGL(1,𝔽₅)	C₄⋊C₄	direct (N × H)
58. C₄×C₅²	C₄²	semidirect
60. C₅×Dic₅	C₄²	semidirect
71. C₄⋊AGL(1,𝔽₅)	C₄×C₅	semidirect
73. C₄×AGL(1,𝔽₅)	C₄×C₅	semidirect
78. C₅×C₄⋊C₄	C₄×C₅	semidirect
79. C₅×C₄⋊C₄	AGL(1,𝔽₅)	direct (N × H)
82. C₄⋊Dic₅	C₄×C₅	semidirect
83. #683	C₄×C₅	semidirect
135. C₅²	C₄×C₄⋊C₄	semidirect
153. AGL(1,𝔽₅)	C₅×C₄⋊C₄	direct (N × H)
163. C₄×C₅	C₄²×C₅	semidirect
164. C₄×C₅	C₄×AGL(1,𝔽₅)	semidirect
166. Dic₅	C₄²×C₅	semidirect
175. C₄⋊C₄	C₅×AGL(1,𝔽₅)	direct (N × H)
214. C₅	C₄×C₅×C₄⋊C₄	semidirect
215. C₅	C₄⋊C₄×AGL(1,𝔽₅)	direct (N × H)
229. C₄	C₄×C₅×AGL(1,𝔽₅)	semidirect

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
C₅×D₅×C₄⋊C₄	2
C₅×(C₂×AGL(1,𝔽₅))⋊C₄	2
C₄⋊C₄×AGL(1,𝔽₅)	5
C₄×C₅×C₄⋊C₄	5

Subgroups — C5×C4⋊C4×AGL(1,𝔽5) #13104470

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₄⋊C₄×AGL(1,𝔽₅) #13104470