Finite Group Databasebeta

← #72218156

Subgroups — #72218156

Order 3360.

272 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 69 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
1D4×C5⋊AGL(1,𝔽7)C211yesmaximal, characteristic
2(C5⋊AGL(1,𝔽7))⋊Q8C211yesmaximal, characteristic, complemented
3ΓL(1,𝔽8)×#687C211yesmaximal, characteristic, complemented
4C5×#1142⋊C3C211yesmaximal, characteristic, complemented
5#10587849⋊C3C211yesmaximal, characteristic, complemented
6#10587847⋊C3C211yesmaximal, characteristic, complemented
7#10587837⋊C3C211yesmaximal, characteristic, complemented
8#11764935C311yesmaximal, characteristic, complemented, Hall
9C22×C5⋊AGL(1,𝔽7)—21no—
10C5×D4×ΓL(1,𝔽8)C2211yescharacteristic
11(C22⋊D35)⋊C3—21no—
12C4⋊(C5⋊AGL(1,𝔽7))C2211yescharacteristic
13C4×C5⋊AGL(1,𝔽7)C2211yescharacteristic
14ΓL(1,𝔽8)×Dic10C2211yescharacteristic, complemented
15C5×ΓL(1,𝔽8)⋊Q8C2211yescharacteristic, complemented
16#63308⋊C3—21no—
17#63304⋊C3—21no—
18ΓL(1,𝔽8)×#168C2211yescharacteristic, complemented
19C5×ΓL(1,𝔽8)⋊C8C2211yescharacteristic, complemented
20SD16×AGL(1,𝔽7)—51nomaximal, Hall
21D4×D35C2×C311yescharacteristic
22D35⋊Q8C2×C311yescharacteristic, complemented
23C7×#687C2×C311yescharacteristic, complemented
24C5×#1142C2×C311yescharacteristic, complemented
25#10587849C2×C311yescharacteristic, complemented
26#10587847C2×C311yescharacteristic, complemented
27#10587837C2×C311yescharacteristic, complemented
28C3×D5×SD16—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)⋊Dic5D411yescharacteristic
35Dic5×ΓL(1,𝔽8)—21no—
36C5×ΓL(1,𝔽8)⋊C4—21no—
37Q8×AGL(1,𝔽7)—51no—
38D4×AGL(1,𝔽7)—51no—
39SD16×ΓL(1,𝔽8)—51no—
40#1143⋊C3—51no—
41#1142⋊C3D511yescharacteristic, complemented
42#1132⋊C3—51no—
43C8×AGL(1,𝔽7)—51no—
44C22×D35—21no—
45C5×C7×D4C22×C311yescharacteristic
46C22⋊D35—21no—
47D140C22×C311yescharacteristic
48C4×D35C22×C311yescharacteristic
49C7×Dic10C22×C311yescharacteristic, complemented
50C5×Dic14C22×C311yescharacteristic, complemented
51#63308—21no—
52#63304—21no—
53C5×C7⋊C8C22×C311yescharacteristic, complemented
54C7×#168C22×C311yescharacteristic, complemented
55C3×Q8×D5—71no—
56C3×D4×D5—71no—
57C3×C5×SD16—71no—
58C3×#688—71no—
59C3×#687—71no—
60C3×#677—71no—
61C3×C8×D5—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)—101no—
68Q8×ΓL(1,𝔽8)—51no—
69D4×ΓL(1,𝔽8)C2×D511yescharacteristic
70(C22⋊D7)⋊C3—101no—
71C4⋊AGL(1,𝔽7)—51no—
72C4×AGL(1,𝔽7)—51no—
73C4×AGL(1,𝔽7)—101no—
74ΓL(1,𝔽8)⋊Q8—101no—
75ΓL(1,𝔽8)⋊Q8C2×D511yescharacteristic, complemented
76C8×ΓL(1,𝔽8)—51no—
77ΓL(1,𝔽8)⋊C8C2×D511yescharacteristic, complemented
78D5×SD16—71noHall
79C22×C5×C7—21no—
80C2×D35—21no—
81C2×D35—42no—
82C2×D35C3×D411yescharacteristic
83C4×C5×C7C23×C311yescharacteristic
84Dic35C3×D411yescharacteristic
85C5×Dic7—21no—
86C7×Dic5—21no—
87C22×C3×D5—141no—
88C3×C5×Q8—71no—
89C3×C5×D4—71no—
90C3×C22⋊D5—141no—
91C3×D20—71no—
92C3×C4×D5—71no—
93C3×C4×D5—141no—
94C3×Dic10—71no—
95C3×Dic10—141no—
96C3×C5×C8—71no—
97C3×#168—71no—
98Q8×D7—51no—
99D4×D7—51no—
100C7×SD16—51no—
101#1143—51no—
102#1142C3×D511yescharacteristic, complemented
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)—51no—
109C2×AGL(1,𝔽7)—101no—
110C2×AGL(1,𝔽7)—202no—
111C4×ΓL(1,𝔽8)—101no—
112C4×ΓL(1,𝔽8)C22×D511yescharacteristic
113ΓL(1,𝔽8)⋊C4—21no—
114ΓL(1,𝔽8)⋊C4—51no—
115Q8×D5—71no—
116D4×D5—71no—
117C5×SD16—71no—
118#688—71no—
119#687AGL(1,𝔽7)11yescharacteristic, complemented
120#677—71no—
121C8×D5—71no—
122C2×C5×C7—41no—
123C2×C5×C7C2×C3×D411yescharacteristic
124D35—41no—
125D35C3×SD1612yescomplemented
126C22×C3×C5—141no—
127C2×C3×D5—71no—
128C2×C3×D5—141no—
129C2×C3×D5—282no—
130C3×C4×C5—71no—
131C3×C4×C5—141no—
132C3×Dic5—71no—
133C3×Dic5—141no—
134C22×D7—101no—
135C7×Q8—51no—
136C7×D4C2×C3×D511yescharacteristic
137C22⋊D7—101no—
138D28—51no—
139C4×D7—51no—
140C4×D7—101no—
141Dic14—101no—
142Dic14C2×C3×D511yescharacteristic, complemented
143C7×C8—51no—
144C7⋊C8C2×C3×D511yescharacteristic, complemented
145C2×C3×Q8—351no—
146C2×C3×D4—351no—
147C3×SD16—71no—
148C3×SD16—351no—
149C3×SD16—351no—
150C3×SD16—351no—
151C2×C3×C8—351no—
152C2×ΓL(1,𝔽8)—41no—
153C2×ΓL(1,𝔽8)D4×D511yescharacteristic
154AGL(1,𝔽7)—52no—
155AGL(1,𝔽7)—201no—
156C22×D5—141no—
157C5×Q8—71no—
158C5×D4C2×AGL(1,𝔽7)11yescharacteristic
159C22⋊D5—141no—
160D20—71no—
161C4×D5—71no—
162C4×D5—141no—
163Dic10—141no—
164Dic10C2×AGL(1,𝔽7)11yescharacteristic, complemented
165C5×C8—71no—
166#168C2×AGL(1,𝔽7)11yescharacteristic, complemented
167C5×C7C2×C3×SD1611yescharacteristic, complemented, Hall
168C2×SD16—351noSylow
169C2×C3×C5—71no—
170C2×C3×C5—281no—
171C3×D5—72no—
172C3×D5—281no—
173C22×C7—21no—
174C2×D7—51no—
175C2×D7—101no—
176C2×D7—202no—
177C4×C7—101no—
178C4×C7C22×C3×D511yescharacteristic
179Dic7—21no—
180Dic7—51no—
181C23×C3—701no—
182C3×Q8—71no—
183C3×Q8—351no—
184C3×Q8—701no—
185C3×D4—71no—
186C3×D4—351no—
187C3×D4—701no—
188C2×C3×C4—351no—
189C2×C3×C4—701no—
190C3×C8—71no—
191C3×C8—351no—
192ΓL(1,𝔽8)D5×SD1611yescharacteristic, complemented, Hall
193C22×C5—21no—
194C2×D5—71no—
195C2×D5—141no—
196C2×D5—282no—
197C4×C5—141no—
198C4×C5C22×AGL(1,𝔽7)11yescharacteristic
199Dic5—21no—
200Dic5—71no—
201C2×Q8—351no—
202C2×D4—351no—
203SD16—51no—
204SD16—71no—
205SD16—351no—
206SD16—351no—
207C2×C8—351no—
208C3×C5—71noHall
209C2×C7—41no—
210C2×C7C3×D4×D511yescharacteristic
211D7—52no—
212D7—201no—
213C22×C3—141no—
214C22×C3—351no—
215C22×C3—701no—
216C22×C3—1402no—
217C3×C4—71no—
218C3×C4—141no—
219C3×C4—351no—
220C3×C4—701no—
221C2×C5—41no—
222C2×C5D4×AGL(1,𝔽7)11yescharacteristic
223D5—72no—
224D5—281no—
225C23—701no—
226Q8—51no—
227Q8—71no—
228Q8—701no—
229D4—351no—
230D4—701no—
231D4D5×AGL(1,𝔽7)11yescharacteristic
232C2×C4—351no—
233C2×C4—701no—
234C8—51no—
235C8—71no—
236C7C3×D5×SD1611yescharacteristic, complemented, Sylow
237C2×C3—71no—
238C2×C3—281no—
239C2×C3—352no—
240C2×C3—1401no—
241C5SD16×AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
242C22—21no—
243C22—351no—
244C22—701no—
245C22—1402no—
246C4—101no—
247C4—141no—
248C4—351no—
249C4C2×D5×AGL(1,𝔽7)11yescharacteristic
250C3—71noSylow
251C2—41no—
252C2—352no—
253C2—1401no—
254C2#10587963⋊C311yescharacteristic
subgroupG/H
centerC2#10587963⋊C3
commutator, cosocleC4×C5×C7C23×C3
FrattiniC4C2×D5×AGL(1,𝔽7)
FittingC5×C7×D4C22×C3
socleC2×C5×C7C2×C3×D4
prime radicalD4×C5⋊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. (C5⋊AGL(1,𝔽7))⋊Q8C2semidirect
3. ΓL(1,𝔽8)×#687C2semidirect
4. C5×#1142⋊C3C2semidirect
5. #10587849⋊C3C2semidirect
6. #10587847⋊C3C2semidirect
7. #10587837⋊C3C2semidirect
8. #11764935C3semidirect
14. ΓL(1,𝔽8)×Dic10C22semidirect
15. C5×ΓL(1,𝔽8)⋊Q8C22semidirect
18. ΓL(1,𝔽8)×#168C22semidirect
19. C5×ΓL(1,𝔽8)⋊C8C22semidirect
22. D35⋊Q8C2×C3semidirect
23. C7×#687C2×C3semidirect
24. C5×#1142C2×C3semidirect
25. #10587849C2×C3semidirect
26. #10587847C2×C3semidirect
27. #10587837C2×C3semidirect
41. #1142⋊C3D5semidirect
49. C7×Dic10C22×C3semidirect
50. C5×Dic14C22×C3semidirect
53. C5×C7⋊C8C22×C3semidirect
54. C7×#168C22×C3semidirect
66. C5⋊AGL(1,𝔽7)SD16semidirect
75. ΓL(1,𝔽8)⋊Q8C2×D5semidirect
77. ΓL(1,𝔽8)⋊C8C2×D5semidirect
102. #1142C3×D5semidirect
105. C5×ΓL(1,𝔽8)C2×SD16semidirect
119. #687AGL(1,𝔽7)semidirect
125. D35C3×SD16semidirect
142. Dic14C2×C3×D5semidirect
144. C7⋊C8C2×C3×D5semidirect
164. Dic10C2×AGL(1,𝔽7)semidirect
166. #168C2×AGL(1,𝔽7)semidirect
167. C5×C7C2×C3×SD16semidirect
192. ΓL(1,𝔽8)D5×SD16semidirect
236. C7C3×D5×SD16semidirect
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
D4×C5⋊AGL(1,𝔽7)2
(C5⋊AGL(1,𝔽7))⋊Q82
ΓL(1,𝔽8)×#6872
C5×#1142⋊C32
#10587849⋊C32
#10587847⋊C32
#10587837⋊C32
#117649353
SD16×AGL(1,𝔽7)5
C3×D5×SD167