Subgroups — C₃×C₄⋊C₄×AGL(1,𝔽₅) #11752243

Order 960.

388 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄⋊AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×AGL(1,𝔽₅)	C₂	1	2	yes	maximal
3	C₂×C₃×C₄×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
4	C₃×D₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₃×(C₂×AGL(1,𝔽₅))⋊C₄	C₂	1	2	yes	maximal
6	C₄⋊C₄×AGL(1,𝔽₅)	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×AGL(1,𝔽₅)	C₂²	1	2	yes	—
8	C₂²×C₃×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
9	C₂²×C₃×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
11	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
12	C₃×C₄⋊AGL(1,𝔽₅)	C₄	1	4	yes	complemented
13	C₃×C₄×AGL(1,𝔽₅)	—	2	4	no	—
14	C₃×C₄×AGL(1,𝔽₅)	C₄	1	4	yes	complemented
15	C₃×C₅×C₄⋊C₄	C₄	1	1	yes	characteristic, complemented
16	C₃×C₄⋊Dic₅	C₄	1	1	yes	characteristic, complemented
17	C₃×#683	C₄	1	2	yes	complemented
18	C₃×C₄×C₄⋊C₄	—	5	1	no	maximal, Hall
19	C₂×C₄⋊AGL(1,𝔽₅)	C₂×C₃	1	1	yes	characteristic
20	C₂×C₄×AGL(1,𝔽₅)	C₂×C₃	1	2	yes	—
21	C₂×C₄×AGL(1,𝔽₅)	C₂×C₃	1	1	yes	characteristic
22	D₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
23	(C₂×AGL(1,𝔽₅))⋊C₄	C₂×C₃	1	2	yes	—
24	C₂²×C₃×D₅	C₂³	1	1	yes	characteristic
25	C₂×C₃×AGL(1,𝔽₅)	—	2	2	no	—
26	C₂×C₃×AGL(1,𝔽₅)	—	2	1	no	—
27	C₂×C₃×AGL(1,𝔽₅)	—	2	1	no	—
28	C₂×C₃×AGL(1,𝔽₅)	—	2	2	no	—
29	C₂×C₃×AGL(1,𝔽₅)	C₂×C₄	1	2	yes	—
30	C₂×C₃×AGL(1,𝔽₅)	C₂×C₄	1	4	yes	—
31	C₂×C₃×AGL(1,𝔽₅)	C₂×C₄	1	2	yes	—
32	C₂×C₃×AGL(1,𝔽₅)	D₄	1	2	yes	—
33	C₂×C₃×AGL(1,𝔽₅)	Q₈	1	2	yes	—
34	C₂×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
35	C₂×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×Dic₅	C₂×C₄	1	2	yes	—
37	C₂×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
38	C₃×C₄×D₅	—	2	2	no	—
39	C₃×C₄×D₅	—	2	2	no	—
40	C₃×C₄×D₅	C₂×C₄	1	2	yes	—
41	C₃×C₄×D₅	C₂×C₄	1	2	yes	—
42	C₂×C₃×C₄⋊C₄	—	5	1	no	—
43	C₂×C₃×C₄⋊C₄	—	5	1	no	—
44	C₂×C₃×C₄²	—	5	1	no	—
45	C₂×C₃×C₄²	—	5	2	no	—
46	C₃×#95	—	5	2	no	—
47	C₂²×AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
48	C₂²×AGL(1,𝔽₅)	C₂²×C₃	1	1	yes	characteristic
49	C₂²×AGL(1,𝔽₅)	C₂²×C₃	1	1	yes	characteristic
50	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
51	C₂×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
52	C₄⋊AGL(1,𝔽₅)	C₃×C₄	1	4	yes	complemented
53	C₄×AGL(1,𝔽₅)	—	2	4	no	—
54	C₄×AGL(1,𝔽₅)	C₃×C₄	1	4	yes	complemented
55	C₅×C₄⋊C₄	C₃×C₄	1	1	yes	characteristic, complemented
56	C₄⋊Dic₅	C₃×C₄	1	1	yes	characteristic, complemented
57	#683	C₃×C₄	1	2	yes	complemented
58	C₄×C₄⋊C₄	—	5	1	no	Sylow
59	C₂²×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₃×D₅	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃×D₅	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃×D₅	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×D₅	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₃×D₅	#41	1	1	yes	characteristic
65	C₂×C₃×D₅	#41	1	1	yes	characteristic
66	C₃×AGL(1,𝔽₅)	—	2	2	no	—
67	C₃×AGL(1,𝔽₅)	—	2	4	no	—
68	C₃×AGL(1,𝔽₅)	C₄⋊C₄	1	4	yes	direct product factor
69	C₃×C₄×C₅	—	2	2	no	—
70	C₃×C₄×C₅	C₄²	1	2	yes	complemented
71	C₃×Dic₅	—	2	2	no	—
72	C₃×Dic₅	C₄²	1	2	yes	complemented
73	C₂²×C₃×C₄	—	5	1	no	—
74	C₂²×C₃×C₄	—	5	1	no	—
75	C₂²×C₃×C₄	—	5	1	no	—
76	C₂²×C₃×C₄	—	5	2	no	—
77	C₂²×C₃×C₄	—	5	2	no	—
78	C₃×C₄⋊C₄	—	5	4	no	—
79	C₃×C₄⋊C₄	—	5	2	no	—
80	C₃×C₄⋊C₄	—	5	1	no	—
81	C₃×C₄⋊C₄	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
82	C₃×C₄²	—	5	4	no	—
83	C₃×C₄²	—	10	4	no	—
84	C₂²×D₅	C₂³×C₃	1	1	yes	characteristic
85	C₂×AGL(1,𝔽₅)	—	2	2	no	—
86	C₂×AGL(1,𝔽₅)	—	2	1	no	—
87	C₂×AGL(1,𝔽₅)	—	2	1	no	—
88	C₂×AGL(1,𝔽₅)	—	2	2	no	—
89	C₂×AGL(1,𝔽₅)	C₂×C₃×C₄	1	2	yes	—
90	C₂×AGL(1,𝔽₅)	C₂×C₃×C₄	1	4	yes	—
91	C₂×AGL(1,𝔽₅)	C₂×C₃×C₄	1	2	yes	—
92	C₂×AGL(1,𝔽₅)	C₃×D₄	1	2	yes	—
93	C₂×AGL(1,𝔽₅)	C₃×Q₈	1	2	yes	—
94	C₂×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
95	C₂×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
96	C₂×Dic₅	C₂×C₃×C₄	1	2	yes	—
97	C₂×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
98	C₄×D₅	—	2	2	no	—
99	C₄×D₅	—	2	2	no	—
100	C₄×D₅	C₂×C₃×C₄	1	2	yes	—
101	C₄×D₅	C₂×C₃×C₄	1	2	yes	—
102	C₂×C₄⋊C₄	—	5	1	no	—
103	C₂×C₄⋊C₄	—	5	1	no	—
104	C₂×C₄²	—	5	1	no	—
105	C₂×C₄²	—	5	2	no	—
106	#95	—	5	2	no	—
107	C₂×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₃×C₅	C₄×D₄	1	1	yes	characteristic
109	C₂×C₃×C₅	C₄×Q₈	1	1	yes	characteristic
110	C₃×D₅	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₃×D₅	#117	1	1	yes	characteristic
112	C₃×D₅	C₄×D₄	1	1	yes	characteristic
113	C₃×D₅	C₄×Q₈	1	1	yes	characteristic
114	C₂³×C₃	—	5	1	no	—
115	C₂×C₃×C₄	—	5	2	no	—
116	C₂×C₃×C₄	—	5	2	no	—
117	C₂×C₃×C₄	—	5	2	no	—
118	C₂×C₃×C₄	—	5	1	no	—
119	C₂×C₃×C₄	—	5	2	no	—
120	C₂×C₃×C₄	—	5	2	no	—
121	C₂×C₃×C₄	—	5	2	no	—
122	C₂×C₃×C₄	—	5	4	no	—
123	C₂×C₃×C₄	—	5	2	no	—
124	C₂×C₃×C₄	—	10	2	no	—
125	C₂×C₃×C₄	—	10	2	no	—
126	C₂×C₃×C₄	—	10	2	no	—
127	C₂×C₃×C₄	—	10	1	no	—
128	C₂×C₃×C₄	—	10	1	no	—
129	C₂×C₃×C₄	—	10	2	no	—
130	C₂×C₃×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
131	C₂×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
132	C₂²×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
133	C₂×D₅	C₂²×C₃×C₄	1	1	yes	characteristic
134	C₂×D₅	C₂²×C₃×C₄	1	1	yes	characteristic
135	C₂×D₅	C₂×C₃×D₄	1	1	yes	characteristic
136	C₂×D₅	C₂×C₃×Q₈	1	1	yes	characteristic
137	C₂×D₅	C₃×#41	1	1	yes	characteristic
138	C₂×D₅	C₃×#41	1	1	yes	characteristic
139	AGL(1,𝔽₅)	—	2	2	no	—
140	AGL(1,𝔽₅)	—	2	4	no	—
141	AGL(1,𝔽₅)	C₃×C₄⋊C₄	1	4	yes	direct product factor
142	C₄×C₅	—	2	2	no	—
143	C₄×C₅	C₃×C₄²	1	2	yes	complemented
144	Dic₅	—	2	2	no	—
145	Dic₅	C₃×C₄²	1	2	yes	complemented
146	C₂²×C₄	—	5	1	no	—
147	C₂²×C₄	—	5	1	no	—
148	C₂²×C₄	—	5	1	no	—
149	C₂²×C₄	—	5	2	no	—
150	C₂²×C₄	—	5	2	no	—
151	C₄⋊C₄	—	5	4	no	—
152	C₄⋊C₄	—	5	2	no	—
153	C₄⋊C₄	—	5	1	no	—
154	C₄⋊C₄	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
155	C₄²	—	5	4	no	—
156	C₄²	—	10	4	no	—
157	C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
158	C₂²×C₃	—	5	1	no	—
159	C₂²×C₃	—	5	1	no	—
160	C₂²×C₃	—	5	1	no	—
161	C₂²×C₃	—	5	1	no	—
162	C₂²×C₃	—	5	1	no	—
163	C₂²×C₃	—	5	1	no	—
164	C₂²×C₃	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
165	C₃×C₄	—	2	2	no	—
166	C₃×C₄	—	5	4	no	—
167	C₃×C₄	—	5	2	no	—
168	C₃×C₄	—	10	4	no	—
169	C₃×C₄	—	10	2	no	—
170	C₃×C₄	—	10	2	no	—
171	C₃×C₄	C₄×AGL(1,𝔽₅)	1	2	yes	complemented
172	C₂×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
173	C₂×C₅	C₃×C₄×D₄	1	1	yes	characteristic
174	C₂×C₅	C₃×C₄×Q₈	1	1	yes	characteristic
175	D₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
176	D₅	C₃×#117	1	1	yes	characteristic
177	D₅	C₃×C₄×D₄	1	1	yes	characteristic
178	D₅	C₃×C₄×Q₈	1	1	yes	characteristic
179	C₂³	—	5	1	no	—
180	C₂×C₄	—	5	2	no	—
181	C₂×C₄	—	5	2	no	—
182	C₂×C₄	—	5	2	no	—
183	C₂×C₄	—	5	1	no	—
184	C₂×C₄	—	5	2	no	—
185	C₂×C₄	—	5	2	no	—
186	C₂×C₄	—	5	2	no	—
187	C₂×C₄	—	5	4	no	—
188	C₂×C₄	—	5	2	no	—
189	C₂×C₄	—	10	2	no	—
190	C₂×C₄	—	10	2	no	—
191	C₂×C₄	—	10	2	no	—
192	C₂×C₄	—	10	1	no	—
193	C₂×C₄	—	10	1	no	—
194	C₂×C₄	—	10	2	no	—
195	C₂×C₄	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
196	C₂×C₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
197	C₂×C₃	—	5	1	no	—
198	C₂×C₃	—	5	1	no	—
199	C₂×C₃	—	5	1	no	—
200	C₂×C₃	—	5	1	no	—
201	C₂×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
202	C₂×C₃	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
203	C₂×C₃	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
204	C₅	C₃×C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
205	C₂²	—	5	1	no	—
206	C₂²	—	5	1	no	—
207	C₂²	—	5	1	no	—
208	C₂²	—	5	1	no	—
209	C₂²	—	5	1	no	—
210	C₂²	—	5	1	no	—
211	C₂²	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
212	C₄	—	2	2	no	—
213	C₄	—	5	4	no	—
214	C₄	—	5	2	no	—
215	C₄	—	10	4	no	—
216	C₄	—	10	2	no	—
217	C₄	—	10	2	no	—
218	C₄	C₃×C₄×AGL(1,𝔽₅)	1	2	yes	complemented
219	C₃	C₄⋊C₄×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Sylow
220	C₂	—	5	1	no	—
221	C₂	—	5	1	no	—
222	C₂	—	5	1	no	—
223	C₂	—	5	1	no	—
224	C₂	C₂×C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
225	C₂	C₃×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
226	C₂	C₃×Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂²×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₃×C₄²
Frattini	C₂²	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×C₄⋊C₄	C₄
socle	C₂²×C₃×C₅	C₂²×C₄
prime radical	C₂²×C₃×D₅	C₂³
cosocle	C₂²×D₅	C₂³×C₃
hypercenter	C₃×C₄⋊C₄	AGL(1,𝔽₅)
hypocenter	C₅	C₃×C₄×C₄⋊C₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
6. C₄⋊C₄×AGL(1,𝔽₅)	C₃	direct (N × H)
12. C₃×C₄⋊AGL(1,𝔽₅)	C₄	semidirect
14. C₃×C₄×AGL(1,𝔽₅)	C₄	semidirect
15. C₃×C₅×C₄⋊C₄	C₄	semidirect
16. C₃×C₄⋊Dic₅	C₄	semidirect
17. C₃×#683	C₄	semidirect
52. C₄⋊AGL(1,𝔽₅)	C₃×C₄	semidirect
54. C₄×AGL(1,𝔽₅)	C₃×C₄	semidirect
55. C₅×C₄⋊C₄	C₃×C₄	semidirect
56. C₄⋊Dic₅	C₃×C₄	semidirect
57. #683	C₃×C₄	semidirect
68. C₃×AGL(1,𝔽₅)	C₄⋊C₄	direct (N × H)
70. C₃×C₄×C₅	C₄²	semidirect
72. C₃×Dic₅	C₄²	semidirect
81. C₃×C₄⋊C₄	AGL(1,𝔽₅)	direct (N × H)
141. AGL(1,𝔽₅)	C₃×C₄⋊C₄	direct (N × H)
143. C₄×C₅	C₃×C₄²	semidirect
145. Dic₅	C₃×C₄²	semidirect
154. C₄⋊C₄	C₃×AGL(1,𝔽₅)	direct (N × H)
157. C₃×C₅	C₄×C₄⋊C₄	semidirect
171. C₃×C₄	C₄×AGL(1,𝔽₅)	semidirect
204. C₅	C₃×C₄×C₄⋊C₄	semidirect
218. C₄	C₃×C₄×AGL(1,𝔽₅)	semidirect
219. C₃	C₄⋊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×C₃×C₄⋊AGL(1,𝔽₅)	2
C₂×C₃×C₄×AGL(1,𝔽₅)	2
C₃×D₅×C₄⋊C₄	2
C₃×(C₂×AGL(1,𝔽₅))⋊C₄	2
C₄⋊C₄×AGL(1,𝔽₅)	3
C₃×C₄×C₄⋊C₄	5

Subgroups — C3×C4⋊C4×AGL(1,𝔽5) #11752243

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄⋊C₄×AGL(1,𝔽₅) #11752243