Finite Group Databasebeta

← #72218162

Subgroups — #72218162

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
1Q8×C5⋊AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
2#10587950⋊C3C211yesmaximal, characteristic
3ΓL(1,𝔽8)×#688C211yesmaximal, characteristic, complemented
4C5×#1143⋊C3C211yesmaximal, characteristic, complemented
5#10587848⋊C3C211yesmaximal, characteristic, complemented
6#10587846⋊C3C211yesmaximal, characteristic, complemented
7#10587837⋊C3C211yesmaximal, characteristic, complemented
8#11764953C311yesmaximal, characteristic, complemented, Hall
9C2×D5×AGL(1,𝔽7)—21no—
10C5×Q8×ΓL(1,𝔽8)C2211yescharacteristic, complemented
11C4×C5⋊AGL(1,𝔽7)—21no—
12C4×C5⋊AGL(1,𝔽7)C2211yescharacteristic
13Dic70⋊C3—21no—
14Dic70⋊C3C2211yescharacteristic, complemented
15ΓL(1,𝔽8)×D20C2211yescharacteristic
16C5×C4⋊AGL(1,𝔽7)C2211yescharacteristic
17#63305⋊C3—21no—
18ΓL(1,𝔽8)×#168C2211yescharacteristic, complemented
19C5×ΓL(1,𝔽8)⋊C8C2211yescharacteristic, complemented
20SD16×AGL(1,𝔽7)—51nomaximal, Hall
21Q8×D35C2×C311yescharacteristic, complemented
22#10587950C2×C311yescharacteristic
23C7×#688C2×C311yescharacteristic, complemented
24C5×#1143C2×C311yescharacteristic, complemented
25#10587848C2×C311yescharacteristic, complemented
26#10587846C2×C311yescharacteristic, complemented
27#10587837C2×C311yescharacteristic, complemented
28C3×D5×SD16—71nomaximal, Hall
29C2×C5⋊AGL(1,𝔽7)D411yescharacteristic
30C2×D5×ΓL(1,𝔽8)—21no—
31C2×C5×AGL(1,𝔽7)—21no—
32D5×AGL(1,𝔽7)—42no—
33C4×C5×ΓL(1,𝔽8)—21no—
34C4×C5×ΓL(1,𝔽8)C2311yescharacteristic
35ΓL(1,𝔽8)⋊Dic5—21no—
36ΓL(1,𝔽8)⋊Dic5D411yescharacteristic
37Q8×AGL(1,𝔽7)—51no—
38D4×AGL(1,𝔽7)—51no—
39SD16×ΓL(1,𝔽8)—51no—
40#1143⋊C3D511yescharacteristic, complemented
41#1142⋊C3—51no—
42#1132⋊C3—51no—
43C8×AGL(1,𝔽7)—51no—
44C2×D5×D7—21no—
45C5×C7×Q8C22×C311yescharacteristic, complemented
46C4×D35—21no—
47C4×D35C22×C311yescharacteristic
48Dic70—21no—
49Dic70C22×C311yescharacteristic, complemented
50C7×D20C22×C311yescharacteristic
51C5×D28C22×C311yescharacteristic
52#63305—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)C2×D411yescharacteristic
64C5⋊AGL(1,𝔽7)SD1612yescomplemented
65D5×ΓL(1,𝔽8)—41no—
66C5×AGL(1,𝔽7)—41no—
67C22×AGL(1,𝔽7)—101no—
68Q8×ΓL(1,𝔽8)C2×D511yescharacteristic, complemented
69D4×ΓL(1,𝔽8)—51no—
70(C22⋊D7)⋊C3—101no—
71C4⋊AGL(1,𝔽7)C2×D511yescharacteristic
72C4×AGL(1,𝔽7)—51no—
73C4×AGL(1,𝔽7)—101no—
74ΓL(1,𝔽8)⋊Q8—51no—
75ΓL(1,𝔽8)⋊Q8—101no—
76C8×ΓL(1,𝔽8)—51no—
77ΓL(1,𝔽8)⋊C8C2×D511yescharacteristic, complemented
78D5×SD16—71noHall
79C2×D35C3×D411yescharacteristic
80C2×C7×D5—21no—
81C2×C5×D7—21no—
82D5×D7—42no—
83C4×C5×C7—21no—
84C4×C5×C7C23×C311yescharacteristic
85Dic35—21no—
86Dic35C3×D411yescharacteristic
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#1143C3×D511yescharacteristic, complemented
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)—101no—
108C2×AGL(1,𝔽7)—21no—
109C2×AGL(1,𝔽7)—51no—
110C2×AGL(1,𝔽7)—202no—
111C4×ΓL(1,𝔽8)—21no—
112C4×ΓL(1,𝔽8)C22×D511yescharacteristic
113ΓL(1,𝔽8)⋊C4—51no—
114ΓL(1,𝔽8)⋊C4—101no—
115Q8×D5—71no—
116D4×D5—71no—
117C5×SD16—71no—
118#688AGL(1,𝔽7)11yescharacteristic, complemented
119#687—71no—
120#677—71no—
121C8×D5—71no—
122C2×C5×C7C2×C3×D411yescharacteristic
123D35C3×SD1612yescomplemented
124C5×D7—41no—
125C7×D5—41no—
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×Q8C2×C3×D511yescharacteristic, complemented
136C7×D4—51no—
137C22⋊D7—101no—
138D28C2×C3×D511yescharacteristic
139C4×D7—51no—
140C4×D7—101no—
141Dic14—51no—
142Dic14—101no—
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)—201no—
153C2×ΓL(1,𝔽8)D4×D511yescharacteristic
154AGL(1,𝔽7)—41no—
155AGL(1,𝔽7)—52no—
156C22×D5—141no—
157C5×Q8C2×AGL(1,𝔽7)11yescharacteristic, complemented
158C5×D4—71no—
159C22⋊D5—141no—
160D20C2×AGL(1,𝔽7)11yescharacteristic
161C4×D5—71no—
162C4×D5—141no—
163Dic10—71no—
164Dic10—141no—
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—101no—
174C2×D7—21no—
175C2×D7—51no—
176C2×D7—202no—
177C4×C7—21no—
178C4×C7C22×C3×D511yescharacteristic
179Dic7—51no—
180Dic7—101no—
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—141no—
194C2×D5—21no—
195C2×D5—71no—
196C2×D5—282no—
197C4×C5—21no—
198C4×C5C22×AGL(1,𝔽7)11yescharacteristic
199Dic5—71no—
200Dic5—141no—
201C2×Q8—351no—
202C2×D4—351no—
203SD16—51no—
204SD16—71no—
205SD16—351no—
206SD16—351no—
207C2×C8—351no—
208C3×C5—71noHall
209C2×C7—201no—
210C2×C7C3×D4×D511yescharacteristic
211D7—41no—
212D7—52no—
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—281no—
222C2×C5D4×AGL(1,𝔽7)11yescharacteristic
223D5—41no—
224D5—72no—
225C23—701no—
226Q8—351no—
227Q8—701no—
228Q8D5×AGL(1,𝔽7)11yescharacteristic, complemented
229D4—51no—
230D4—71no—
231D4—701no—
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—101no—
243C22—141no—
244C22—351no—
245C22—1402no—
246C4—21no—
247C4—351no—
248C4—701no—
249C4C2×D5×AGL(1,𝔽7)11yescharacteristic
250C3—71noSylow
251C2—201no—
252C2—281no—
253C2—352no—
254C2#10587963⋊C311yescharacteristic
subgroupG/H
centerC2#10587963⋊C3
commutator, cosocleC4×C5×C7C23×C3
FrattiniC4C2×D5×AGL(1,𝔽7)
FittingC5×C7×Q8C22×C3
socleC2×C5×C7C2×C3×D4
prime radical#10587950⋊C3C2
hypercenterQ8D5×AGL(1,𝔽7)
hypocenterC5×C7C2×C3×SD16
LCSG ▷ C4×C5×C7 ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. Q8×C5⋊AGL(1,𝔽7)C2semidirect
3. ΓL(1,𝔽8)×#688C2semidirect
4. C5×#1143⋊C3C2semidirect
5. #10587848⋊C3C2semidirect
6. #10587846⋊C3C2semidirect
7. #10587837⋊C3C2semidirect
8. #11764953C3semidirect
10. C5×Q8×ΓL(1,𝔽8)C22semidirect
14. Dic70⋊C3C22semidirect
18. ΓL(1,𝔽8)×#168C22semidirect
19. C5×ΓL(1,𝔽8)⋊C8C22semidirect
21. Q8×D35C2×C3semidirect
23. C7×#688C2×C3semidirect
24. C5×#1143C2×C3semidirect
25. #10587848C2×C3semidirect
26. #10587846C2×C3semidirect
27. #10587837C2×C3semidirect
40. #1143⋊C3D5semidirect
45. C5×C7×Q8C22×C3semidirect
49. Dic70C22×C3semidirect
53. C5×C7⋊C8C22×C3semidirect
54. C7×#168C22×C3semidirect
64. C5⋊AGL(1,𝔽7)SD16semidirect
68. Q8×ΓL(1,𝔽8)C2×D5semidirect
77. ΓL(1,𝔽8)⋊C8C2×D5semidirect
101. #1143C3×D5semidirect
105. C5×ΓL(1,𝔽8)C2×SD16semidirect
118. #688AGL(1,𝔽7)semidirect
123. D35C3×SD16semidirect
135. C7×Q8C2×C3×D5semidirect
144. C7⋊C8C2×C3×D5semidirect
157. C5×Q8C2×AGL(1,𝔽7)semidirect
166. #168C2×AGL(1,𝔽7)semidirect
167. C5×C7C2×C3×SD16semidirect
192. ΓL(1,𝔽8)D5×SD16semidirect
228. Q8D5×AGL(1,𝔽7)semidirect
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
Q8×C5⋊AGL(1,𝔽7)2
#10587950⋊C32
ΓL(1,𝔽8)×#6882
C5×#1143⋊C32
#10587848⋊C32
#10587846⋊C32
#10587837⋊C32
#117649533
SD16×AGL(1,𝔽7)5
C3×D5×SD167