Subgroups — C₂×C₅×(C₂×AGL(1,𝔽₅))⋊C₄ #13104522

Order 1600.

687 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂³×C₅	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₄²×C₅
Frattini	C₂²	C₂²×C₅×AGL(1,𝔽₅)
Fitting	C₂²×C₄×C₅²	C₄
socle	C₂³×C₅²	C₂×C₄
prime radical	C₂³×C₅×D₅	C₂²
cosocle	C₂²×D₅	C₂³×C₅
hypercenter	C₂²×C₄×C₅	AGL(1,𝔽₅)
hypocenter	C₅	C₂×C₅×#95
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄×C₅

Complements

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

N	H ≅ G/N	type
3. C₅×(C₂×AGL(1,𝔽₅))⋊C₄	C₂	direct (N × H)
7. C₂²×C₅×AGL(1,𝔽₅)	C₄	semidirect
9. C₂²×C₄×C₅²	C₄	semidirect
10. C₂²×C₅×Dic₅	C₄	semidirect
12. C₂×C₄×C₅×D₅	C₄	semidirect
14. C₂×(C₂×AGL(1,𝔽₅))⋊C₄	C₅	direct (N × H)
26. C₂×C₅×AGL(1,𝔽₅)	C₂×C₄	semidirect
28. C₂×C₄×C₅²	C₂×C₄	semidirect
30. C₂×C₅×Dic₅	C₂×C₄	semidirect
37. (C₂×AGL(1,𝔽₅))⋊C₄	C₂×C₅	direct (N × H)
59. C₂²×AGL(1,𝔽₅)	C₄×C₅	semidirect
70. C₂²×C₄×C₅	C₄×C₅	semidirect
71. C₂²×C₄×C₅	AGL(1,𝔽₅)	semidirect
72. C₂²×Dic₅	C₄×C₅	semidirect
74. C₂×C₄×D₅	C₄×C₅	semidirect
77. C₂×C₅²	#95	semidirect
81. C₅×D₅	#95	semidirect
102. C₂×AGL(1,𝔽₅)	C₂×C₄×C₅	semidirect
115. C₂×C₄×C₅	C₂×C₄×C₅	semidirect
116. C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	semidirect
118. C₂×Dic₅	C₂×C₄×C₅	semidirect
124. C₅²	C₂×#95	semidirect
171. C₂²×C₄	C₅×AGL(1,𝔽₅)	semidirect
179. C₂×C₅	C₅×#95	semidirect
180. C₂×C₅	(C₂×AGL(1,𝔽₅))⋊C₄	direct (N × H)
187. D₅	C₅×#95	semidirect
206. C₂×C₄	C₂×C₅×AGL(1,𝔽₅)	semidirect
208. C₅	C₂×C₅×#95	semidirect
209. C₅	C₂×(C₂×AGL(1,𝔽₅))⋊C₄	direct (N × H)
229. C₂	C₅×(C₂×AGL(1,𝔽₅))⋊C₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₅×AGL(1,𝔽₅)	2
C₂²×C₄×C₅×D₅	2
C₅×(C₂×AGL(1,𝔽₅))⋊C₄	2
C₂×(C₂×AGL(1,𝔽₅))⋊C₄	5
C₂×C₅×#95	5

Subgroups — C2×C5×(C2×AGL(1,𝔽5))⋊C4 #13104522

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×(C₂×AGL(1,𝔽₅))⋊C₄ #13104522