Finite Group Databasebeta

← AGL(1,𝔽7)×#687 #72218155

Subgroups — AGL(1,𝔽7)×#687 #72218155

Order 3360.

272 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C5×D4×AGL(1,𝔽7)C211yesmaximal, characteristic
2Dic10×AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
3#10587904⋊C3C211yesmaximal, characteristic, complemented
4ΓL(1,𝔽8)×#687C211yesmaximal, characteristic, complemented
5#10587846⋊C3C211yesmaximal, characteristic, complemented
6#10587844⋊C3C211yesmaximal, characteristic, complemented
7#168×AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
8D7×#687C311yesmaximal, characteristic, complemented, Hall
9C22×C5×AGL(1,𝔽7)—21no—
10C5×D4×ΓL(1,𝔽8)C2211yescharacteristic
11Dic70⋊C3C2211yescharacteristic, complemented
12ΓL(1,𝔽8)×Dic10C2211yescharacteristic, complemented
13C5×(C22⋊D7)⋊C3—21no—
14C5×C4⋊AGL(1,𝔽7)C2211yescharacteristic
15C4×C5×AGL(1,𝔽7)C2211yescharacteristic
16#63308⋊C3—21no—
17Dic5×AGL(1,𝔽7)—21no—
18ΓL(1,𝔽8)⋊#168C2211yescharacteristic, complemented
19ΓL(1,𝔽8)×#168C2211yescharacteristic, complemented
20SD16×AGL(1,𝔽7)—51nomaximal, Hall
21C5×D4×D7C2×C311yescharacteristic
22D7×Dic10C2×C311yescharacteristic, complemented
23#10587904C2×C311yescharacteristic, complemented
24C7×#687C2×C311yescharacteristic, complemented
25#10587846C2×C311yescharacteristic, complemented
26#10587844C2×C311yescharacteristic, complemented
27D7×#168C2×C311yescharacteristic, complemented
28C2×C3×#687—71nomaximal, Hall
29C22×C5×ΓL(1,𝔽8)—21no—
30C2×C5×AGL(1,𝔽7)—21no—
31C2×C5×AGL(1,𝔽7)—42no—
32C2×C5×AGL(1,𝔽7)D411yescharacteristic
33C4×C5×ΓL(1,𝔽8)C2311yescharacteristic
34ΓL(1,𝔽8)⋊Dic5—21no—
35Dic5×ΓL(1,𝔽8)—21no—
36C5×ΓL(1,𝔽8)⋊C4D411yescharacteristic
37Q8×AGL(1,𝔽7)—51no—
38D4×AGL(1,𝔽7)D511yescharacteristic
39SD16×ΓL(1,𝔽8)—51no—
40#1143⋊C3—51no—
41#1142⋊C3—51no—
42#1132⋊C3—51no—
43C8×AGL(1,𝔽7)—51no—
44C22×C5×D7—21no—
45C5×C7×D4C22×C311yescharacteristic
46Dic70C22×C311yescharacteristic, complemented
47C7×Dic10C22×C311yescharacteristic, complemented
48C5×C22⋊D7—21no—
49C5×D28C22×C311yescharacteristic
50C4×C5×D7C22×C311yescharacteristic
51#63308—21no—
52D7×Dic5—21no—
53C7⋊#168C22×C311yescharacteristic, complemented
54C7×#168C22×C311yescharacteristic, complemented
55C2×C3×C5×D4—71no—
56C2×C3×Dic10—71no—
57C3×#687—71no—
58C3×#687—71no—
59C3×#687—71no—
60C3×#687—71no—
61C2×C3×#168—71no—
62D7×SD16—51noHall
63C2×C5×ΓL(1,𝔽8)—41no—
64C2×C5×ΓL(1,𝔽8)C2×D411yescharacteristic
65C5×AGL(1,𝔽7)—41no—
66C5×AGL(1,𝔽7)SD1612yescomplemented
67C22×AGL(1,𝔽7)—21no—
68Q8×ΓL(1,𝔽8)—51no—
69D4×ΓL(1,𝔽8)C2×D511yescharacteristic
70(C22⋊D7)⋊C3—21no—
71C4⋊AGL(1,𝔽7)C2×D511yescharacteristic
72C4×AGL(1,𝔽7)—101no—
73C4×AGL(1,𝔽7)C2×D511yescharacteristic
74ΓL(1,𝔽8)⋊Q8—51no—
75ΓL(1,𝔽8)⋊Q8—101no—
76C8×ΓL(1,𝔽8)—51no—
77ΓL(1,𝔽8)⋊C8—51no—
78C2×#687—71noHall
79C22×C5×C7—21no—
80C2×C5×D7—21no—
81C2×C5×D7—42no—
82C2×C5×D7C3×D411yescharacteristic
83C4×C5×C7C23×C311yescharacteristic
84Dic35—21no—
85C5×Dic7C3×D411yescharacteristic
86C7×Dic5—21no—
87C23×C3×C5—141no—
88C3×C5×D4—71no—
89C3×C5×D4—71no—
90C3×C5×D4—141no—
91C2×C3×C4×C5—71no—
92C2×C3×Dic5—141no—
93C3×Dic10—71no—
94C3×Dic10—71no—
95C3×Dic10—141no—
96C3×#168—71no—
97C3×#168—71no—
98Q8×D7—51no—
99D4×D7C3×D511yescharacteristic
100C7×SD16—51no—
101#1143—51no—
102#1142—51no—
103#1132—51no—
104C8×D7—51no—
105C5×ΓL(1,𝔽8)C2×SD1611yescharacteristic, complemented, Hall
106C2×C3×SD16—351noHall
107C22×ΓL(1,𝔽8)—21no—
108C2×AGL(1,𝔽7)—21no—
109C2×AGL(1,𝔽7)—42no—
110C2×AGL(1,𝔽7)C22⋊D511yescharacteristic
111C4×ΓL(1,𝔽8)—101no—
112C4×ΓL(1,𝔽8)C22×D511yescharacteristic
113ΓL(1,𝔽8)⋊C4—101no—
114ΓL(1,𝔽8)⋊C4C22⋊D511yescharacteristic
115C2×C5×D4—71no—
116C2×Dic10—71no—
117#687—71no—
118#687—71no—
119#687—71no—
120#687AGL(1,𝔽7)11yescharacteristic, direct product factor
121C2×#168—71no—
122C2×C5×C7—41no—
123C2×C5×C7C2×C3×D411yescharacteristic
124C5×D7—41no—
125C5×D7C3×SD1612yescomplemented
126C22×C3×C5—71no—
127C22×C3×C5—141no—
128C22×C3×C5—141no—
129C22×C3×C5—282no—
130C3×C4×C5—71no—
131C3×C4×C5—71no—
132C3×Dic5—141no—
133C3×Dic5—141no—
134C22×D7—21no—
135C7×Q8—51no—
136C7×D4C2×C3×D511yescharacteristic
137C22⋊D7—21no—
138D28C2×C3×D511yescharacteristic
139C4×D7—101no—
140C4×D7C2×C3×D511yescharacteristic
141Dic14—51no—
142Dic14—101no—
143C7×C8—51no—
144C7⋊C8—51no—
145C2×C3×Q8—351no—
146C2×C3×D4—71no—
147C3×SD16—351no—
148C3×SD16—351no—
149C3×SD16—351no—
150C3×SD16—351no—
151C2×C3×C8—351no—
152C2×ΓL(1,𝔽8)—41no—
153C2×ΓL(1,𝔽8)C2×C22⋊D511yescharacteristic
154AGL(1,𝔽7)—41no—
155AGL(1,𝔽7)#68712yesdirect product factor
156C23×C5—141no—
157C5×D4—71no—
158C5×D4—141no—
159C5×D4C2×AGL(1,𝔽7)11yescharacteristic
160C2×C4×C5—71no—
161C2×Dic5—141no—
162Dic10—71no—
163Dic10—141no—
164Dic10C2×AGL(1,𝔽7)11yescharacteristic, complemented
165#168—71no—
166#168C2×AGL(1,𝔽7)11yescharacteristic, complemented
167C5×C7C2×C3×SD1611yescharacteristic, complemented, Hall
168C2×SD16—351noSylow
169C2×C3×C5—72no—
170C2×C3×C5—71no—
171C2×C3×C5—281no—
172C2×C3×C5—281no—
173C22×C7—21no—
174C2×D7—21no—
175C2×D7—42no—
176C2×D7C3×C22⋊D511yescharacteristic
177C4×C7—101no—
178C4×C7C22×C3×D511yescharacteristic
179Dic7—101no—
180Dic7C3×C22⋊D511yescharacteristic
181C23×C3—141no—
182C3×Q8—351no—
183C3×Q8—351no—
184C3×Q8—701no—
185C3×D4—71no—
186C3×D4—71no—
187C3×D4—141no—
188C2×C3×C4—71no—
189C2×C3×C4—701no—
190C3×C8—351no—
191C3×C8—351no—
192ΓL(1,𝔽8)C2×#68711yescharacteristic, complemented, Hall
193C22×C5—21no—
194C22×C5—71no—
195C22×C5—141no—
196C22×C5—282no—
197C4×C5—71no—
198C4×C5C22×AGL(1,𝔽7)11yescharacteristic
199Dic5—21no—
200Dic5—141no—
201C2×Q8—351no—
202C2×D4—71no—
203SD16—51no—
204SD16—351no—
205SD16—351no—
206SD16—351no—
207C2×C8—351no—
208C3×C5—71noHall
209C2×C7—41no—
210C2×C7C2×C3×C22⋊D511yescharacteristic
211D7—41no—
212D7C3×#68712yescomplemented
213C22×C3—71no—
214C22×C3—141no—
215C22×C3—141no—
216C22×C3—282no—
217C3×C4—71no—
218C3×C4—71no—
219C3×C4—701no—
220C3×C4—701no—
221C2×C5—41no—
222C2×C5—72no—
223C2×C5—281no—
224C2×C5D4×AGL(1,𝔽7)11yescharacteristic
225C23—141no—
226Q8—51no—
227Q8—351no—
228Q8—701no—
229D4—71no—
230D4—141no—
231D4D5×AGL(1,𝔽7)11yescharacteristic
232C2×C4—71no—
233C2×C4—701no—
234C8—51no—
235C8—351no—
236C7C2×C3×#68711yescharacteristic, complemented, Sylow
237C2×C3—72no—
238C2×C3—71no—
239C2×C3—281no—
240C2×C3—281no—
241C5SD16×AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
242C22—21no—
243C22—71no—
244C22—141no—
245C22—282no—
246C4—71no—
247C4—101no—
248C4—701no—
249C4C2×D5×AGL(1,𝔽7)11yescharacteristic
250C3—71noSylow
251C2—41no—
252C2—72no—
253C2—281no—
254C2C22⋊D5×AGL(1,𝔽7)11yescharacteristic
subgroupG/H
centerC2C22⋊D5×AGL(1,𝔽7)
commutator, cosocleC4×C5×C7C23×C3
FrattiniC4C2×D5×AGL(1,𝔽7)
FittingC5×C7×D4C22×C3
socleC2×C5×C7C2×C3×D4
prime radicalC5×D4×AGL(1,𝔽7)C2
hypercenterD4D5×AGL(1,𝔽7)
hypocenterC5×C7C2×C3×SD16
LCSG ▷ C4×C5×C7 ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
2. Dic10×AGL(1,𝔽7)C2semidirect
3. #10587904⋊C3C2semidirect
4. ΓL(1,𝔽8)×#687C2semidirect
5. #10587846⋊C3C2semidirect
6. #10587844⋊C3C2semidirect
7. #168×AGL(1,𝔽7)C2semidirect
8. D7×#687C3semidirect
11. Dic70⋊C3C22semidirect
12. ΓL(1,𝔽8)×Dic10C22semidirect
18. ΓL(1,𝔽8)⋊#168C22semidirect
19. ΓL(1,𝔽8)×#168C22semidirect
22. D7×Dic10C2×C3semidirect
23. #10587904C2×C3semidirect
24. C7×#687C2×C3semidirect
25. #10587846C2×C3semidirect
26. #10587844C2×C3semidirect
27. D7×#168C2×C3semidirect
46. Dic70C22×C3semidirect
47. C7×Dic10C22×C3semidirect
53. C7⋊#168C22×C3semidirect
54. C7×#168C22×C3semidirect
66. C5×AGL(1,𝔽7)SD16semidirect
105. C5×ΓL(1,𝔽8)C2×SD16semidirect
120. #687AGL(1,𝔽7)direct (N × H)
125. C5×D7C3×SD16semidirect
155. AGL(1,𝔽7)#687direct (N × H)
164. Dic10C2×AGL(1,𝔽7)semidirect
166. #168C2×AGL(1,𝔽7)semidirect
167. C5×C7C2×C3×SD16semidirect
192. ΓL(1,𝔽8)C2×#687semidirect
212. D7C3×#687semidirect
236. C7C2×C3×#687semidirect
241. C5SD16×AGL(1,𝔽7)semidirect

Maximal subgroups

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

subgroupindex
C5×D4×AGL(1,𝔽7)2
Dic10×AGL(1,𝔽7)2
#10587904⋊C32
ΓL(1,𝔽8)×#6872
#10587846⋊C32
#10587844⋊C32
#168×AGL(1,𝔽7)2
D7×#6873
SD16×AGL(1,𝔽7)5
C2×C3×#6877