Finite Group Databasebeta

← AGL(1,𝔽5)×H(3,𝔽4) #13044196

Subgroups — AGL(1,𝔽5)×H(3,𝔽4) #13044196

Order 1280.

2664 conjugacy classes of subgroups; only the 776 normal subgroups are shown.

normal subgroups only

This group has 2664 conjugacy classes of subgroups — too many to draw. Only its 776 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D5×H(3,𝔽4)C211yesmaximal, characteristic
2#10617328C219yesmaximal, complemented
3AGL(1,𝔽5)×#124C219yesmaximal, complemented
4#10617224C216yesmaximal, complemented
5AGL(1,𝔽5)×#120C216yesmaximal, complemented
6C24×AGL(1,𝔽5)C2212yescomplemented
7C2×Q8×AGL(1,𝔽5)C2213yescomplemented
8C2×D4×AGL(1,𝔽5)C2219yes—
9C5×H(3,𝔽4)C411yescharacteristic, complemented
10#66017C419yescomplemented
11D5×#124C2219yes—
12#65928C416yescomplemented
13D5×#120C2216yes—
14#65800C2216yescomplemented
15(C2×AGL(1,𝔽5))⋊Q8C2219yescomplemented
16#65780C2219yes—
17#65779C22118yes—
18#65700C22136yescomplemented
19#65699C22118yescomplemented
20C22⋊C4×AGL(1,𝔽5)C22118yescomplemented
21#65689C2219yescomplemented
22C42×AGL(1,𝔽5)C2213yescomplemented
23C4×H(3,𝔽4)—51nomaximal, Sylow
24C24×D5C2312yes—
25C23×AGL(1,𝔽5)—412no—
26C23×AGL(1,𝔽5)C2316yes—
27C23×AGL(1,𝔽5)C2316yes—
28C2×Q8×D5C2313yes—
29C2×D4×D5C2319yes—
30C2×C22⋊AGL(1,𝔽5)C23118yes—
31C2×C22⋊AGL(1,𝔽5)C23118yes—
32Q8×AGL(1,𝔽5)—46no—
33D4×AGL(1,𝔽5)—418no—
34C2×C4⋊AGL(1,𝔽5)C2319yes—
35C2×C4⋊AGL(1,𝔽5)C2319yes—
36C2×C4×AGL(1,𝔽5)C2319yes—
37C2×C4×AGL(1,𝔽5)C2319yes—
38C5×#124C2×C419yescomplemented
39C5×#120C2×C416yescomplemented
40#4099C2×C416yescomplemented
41#4093C2×C419yescomplemented
42#4083C2×C4118yescomplemented
43#4082C2×C419yescomplemented
44#4031C2×C4136yescomplemented
45#4028C2×C4118yescomplemented
46D5×C22⋊C4C23118yes—
47#4021C2×C419yescomplemented
48C42×D5C2313yes—
49(C2×AGL(1,𝔽5))⋊C4C23136yes—
50C2×H(3,𝔽4)—51no—
51#2379—59no—
52#2375—56no—
53C4×#124—59no—
54C4×#120—56no—
55C24×C5C22×C412yescomplemented
56C23×D5—46no—
57C23×D5—46no—
58C23×D5C22×C416yescomplemented
59C23×D5C2416yes—
60C22×AGL(1,𝔽5)—236no—
61C22×AGL(1,𝔽5)—418no—
62C22×AGL(1,𝔽5)—432no—
63C22×AGL(1,𝔽5)C2419yes—
64C22×AGL(1,𝔽5)C2416yes—
65C22×AGL(1,𝔽5)C2411yescharacteristic
66C2×C5×Q8C22×C413yescomplemented
67C2×C5×D4C22×C419yes—
68C2×C22⋊D5C22×C4118yes—
69Q8×D5—43no—
70Q8×D5—43no—
71D4×D5—49no—
72D4×D5—49no—
73C2×D20C22×C419yes—
74C2×C4×D5C2419yes—
75C2×Dic10C22×C419yescomplemented
76C22⋊AGL(1,𝔽5)—418no—
77C22⋊AGL(1,𝔽5)—418no—
78C4⋊AGL(1,𝔽5)—49no—
79C4⋊AGL(1,𝔽5)—49no—
80C4×AGL(1,𝔽5)—218no—
81C4×AGL(1,𝔽5)—49no—
82C5×C22⋊C4C22×C4118yescomplemented
83C42×C5C22×C413yescomplemented
84C22⋊Dic5C22×C4118yescomplemented
85#685C22×C4136yescomplemented
86C4×Dic5C22×C419yescomplemented
87C24×C4—52no—
88H(3,𝔽4)—59no—
89H(3,𝔽4)—56no—
90H(3,𝔽4)AGL(1,𝔽5)11yescharacteristic, direct product factor
91C2×#124—59no—
92C2×#120—56no—
93C2×C4×Q8—53no—
94C2×C4×D4—59no—
95#391—59no—
96#390—59no—
97#388—536no—
98#386—518no—
99#385—518no—
100#382—59no—
101#379—56no—
102C4×C22⋊C4—518no—
103C43—53no—
104C23×C5—46no—
105C23×C5C23×C416yes—
106C22×D5—218no—
107C22×D5—218no—
108C22×D5—418no—
109C22×D5—48no—
110C22×D5—424no—
111C22×D5C23×C416yes—
112C22×D5C2511yescharacteristic
113C2×AGL(1,𝔽5)—29no—
114C2×AGL(1,𝔽5)—218no—
115C2×AGL(1,𝔽5)—218no—
116C2×AGL(1,𝔽5)—424no—
117C2×AGL(1,𝔽5)—424no—
118C2×AGL(1,𝔽5)H(5,𝔽2)16yes—
119C5×Q8—43no—
120C5×D4—49no—
121C2×C4×C5C23×C419yes—
122C22⋊D5—418no—
123C2×Dic5C23×C419yes—
124D20—49no—
125C4×D5—29no—
126C4×D5—29no—
127Dic10—49no—
128C25—52no—
129C22×Q8—53no—
130C22×D4—59no—
131C23×C4—56no—
132C23×C4—56no—
133C23×C4—2012no—
134#124—59no—
135#124—59no—
136#124—536no—
137#124—59no—
138#124C2×AGL(1,𝔽5)19yescomplemented
139#120—56no—
140#120—518no—
141#120—518no—
142#120C2×AGL(1,𝔽5)16yescomplemented
143C4×Q8—206no—
144C4×D4—2018no—
145C2×C4⋊C4—59no—
146C2×C4⋊C4—59no—
147C2×C22⋊C4—518no—
148C2×C22⋊C4—518no—
149C2×C22⋊C4—518no—
150C2×C42—53no—
151C2×C42—59no—
152C2×C42—59no—
153#95—536no—
154C22×C5—218no—
155C22×C5—48no—
156C22×C5C24×C411yescharacteristic
157C2×D5—218no—
158C2×D5—424no—
159C2×D5—46no—
160C2×D5—418no—
161C2×D5C2×H(5,𝔽2)13yes—
162C2×D5#58513yes—
163AGL(1,𝔽5)—46no—
164AGL(1,𝔽5)—49no—
165AGL(1,𝔽5)H(3,𝔽4)14yesdirect product factor
166C4×C5—29no—
167Dic5—29no—
168C24—56no—
169C24—56no—
170C24—206no—
171C24—206no—
172C24C22×AGL(1,𝔽5)12yescomplemented
173C2×Q8—59no—
174C2×Q8—203no—
175C2×Q8—203no—
176C2×Q8C22×AGL(1,𝔽5)13yescomplemented
177C2×D4—518no—
178C2×D4—59no—
179C2×D4—209no—
180C2×D4—209no—
181C2×D4C22×AGL(1,𝔽5)19yes—
182C22×C4—51no—
183C22×C4—59no—
184C22×C4—56no—
185C22×C4—59no—
186C22×C4—1036no—
187C22×C4—2018no—
188C22×C4—2032no—
189C4⋊C4—209no—
190C4⋊C4—209no—
191C22⋊C4—518no—
192C22⋊C4—536no—
193C22⋊C4—2018no—
194C22⋊C4—2018no—
195C22⋊C4C22×AGL(1,𝔽5)118yescomplemented
196C42—59no—
197C42—1018no—
198C42—209no—
199C42C22×AGL(1,𝔽5)13yescomplemented
200C2×C5—46no—
201C2×C5C4×H(5,𝔽2)13yes—
202D5—46no—
203D5C2×H(3,𝔽4)11yescharacteristic
204D5#352913yes—
205C23—46no—
206C23—51no—
207C23—56no—
208C23—1018no—
209C23—1018no—
210C23—2018no—
211C23—208no—
212C23—2024no—
213C23C23×AGL(1,𝔽5)16yes—
214Q8—43no—
215Q8—209no—
216D4—49no—
217D4—209no—
218D4—2018no—
219C2×C4—56no—
220C2×C4—59no—
221C2×C4—109no—
222C2×C4—109no—
223C2×C4—109no—
224C2×C4—1018no—
225C2×C4—1018no—
226C2×C4—2024no—
227C2×C4—2024no—
228C2×C4C23×AGL(1,𝔽5)19yes—
229C5C4×H(3,𝔽4)11yescharacteristic, complemented, Sylow
230C22—218no—
231C22—48no—
232C22—53no—
233C22—53no—
234C22—1018no—
235C22—206no—
236C22—2018no—
237C22—2024no—
238C22C24×AGL(1,𝔽5)11yescharacteristic
239C4—29no—
240C4—54no—
241C4—109no—
242C4—206no—
243C4—209no—
244C2—46no—
245C2—51no—
246C2—53no—
247C2—206no—
248C2AGL(1,𝔽5)×H(5,𝔽2)13yes—
subgroupG/H
center, FrattiniC22C24×AGL(1,𝔽5)
commutator, socleC22×C5C24×C4
FittingC5×H(3,𝔽4)C4
prime radicalD5×H(3,𝔽4)C2
cosocleC22×D5C25
hypercenterH(3,𝔽4)AGL(1,𝔽5)
hypocenterC5C4×H(3,𝔽4)
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
2. #10617328C2semidirect
3. AGL(1,𝔽5)×#124C2semidirect
4. #10617224C2semidirect
5. AGL(1,𝔽5)×#120C2semidirect
6. C24×AGL(1,𝔽5)C22semidirect
7. C2×Q8×AGL(1,𝔽5)C22semidirect
9. C5×H(3,𝔽4)C4semidirect
10. #66017C4semidirect
12. #65928C4semidirect
14. #65800C22semidirect
15. (C2×AGL(1,𝔽5))⋊Q8C22semidirect
18. #65700C22semidirect
19. #65699C22semidirect
20. C22⋊C4×AGL(1,𝔽5)C22semidirect
21. #65689C22semidirect
22. C42×AGL(1,𝔽5)C22semidirect
38. C5×#124C2×C4semidirect
39. C5×#120C2×C4semidirect
40. #4099C2×C4semidirect
41. #4093C2×C4semidirect
42. #4083C2×C4semidirect
43. #4082C2×C4semidirect
44. #4031C2×C4semidirect
45. #4028C2×C4semidirect
47. #4021C2×C4semidirect
55. C24×C5C22×C4semidirect
58. C23×D5C22×C4semidirect
66. C2×C5×Q8C22×C4semidirect
75. C2×Dic10C22×C4semidirect
82. C5×C22⋊C4C22×C4semidirect
83. C42×C5C22×C4semidirect
84. C22⋊Dic5C22×C4semidirect
85. #685C22×C4semidirect
86. C4×Dic5C22×C4semidirect
90. H(3,𝔽4)AGL(1,𝔽5)direct (N × H)
138. #124C2×AGL(1,𝔽5)semidirect
142. #120C2×AGL(1,𝔽5)semidirect
165. AGL(1,𝔽5)H(3,𝔽4)direct (N × H)
172. C24C22×AGL(1,𝔽5)semidirect
176. C2×Q8C22×AGL(1,𝔽5)semidirect
195. C22⋊C4C22×AGL(1,𝔽5)semidirect
199. C42C22×AGL(1,𝔽5)semidirect
229. C5C4×H(3,𝔽4)semidirect

Maximal subgroups

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

subgroupindex
D5×H(3,𝔽4)2
#106173282
AGL(1,𝔽5)×#1242
#106172242
AGL(1,𝔽5)×#1202
C4×H(3,𝔽4)5