Finite Group Databasebeta

← #72218065

Subgroups — #72218065

Order 3360.

272 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 65 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
1#13132582C211yesmaximal, characteristic, complemented
2#10587950⋊C3C211yesmaximal, characteristic, complemented
3ΓL(1,𝔽8)×#688C211yesmaximal, characteristic, complemented
4C5×#1143⋊C3C211yesmaximal, characteristic, complemented
5#10587843⋊C3C211yesmaximal, characteristic, complemented
6#10587842⋊C3C211yesmaximal, characteristic, complemented
7#10587840⋊C3C211yesmaximal, characteristic, complemented
8#11764954C311yesmaximal, 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, complemented
13C4×C5⋊AGL(1,𝔽7)—21no—
14C4×C5⋊AGL(1,𝔽7)C2211yescharacteristic
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
20#6477⋊C3—51nomaximal, Hall
21#10587991C2×C311yescharacteristic, complemented
22#10587950C2×C311yescharacteristic, complemented
23C7×#688C2×C311yescharacteristic, complemented
24C5×#1143C2×C311yescharacteristic, complemented
25#10587843C2×C311yescharacteristic, complemented
26#10587842C2×C311yescharacteristic, complemented
27#10587840C2×C311yescharacteristic, complemented
28C3×#4060—71nomaximal, Hall
29C2×C5⋊AGL(1,𝔽7)—21no—
30C2×C5⋊AGL(1,𝔽7)D411yescharacteristic
31C2×D5×ΓL(1,𝔽8)—21no—
32C2×C5×AGL(1,𝔽7)—21no—
33D5×AGL(1,𝔽7)—42no—
34C4×C5×ΓL(1,𝔽8)—21no—
35C4×C5×ΓL(1,𝔽8)C2311yescharacteristic
36ΓL(1,𝔽8)⋊Dic5D411yescharacteristic
37#66659—51no—
38D4×AGL(1,𝔽7)—51no—
39SD16×ΓL(1,𝔽8)—51no—
40#1143⋊C3D511yescharacteristic, complemented
41#1141⋊C3—51no—
42D56⋊C3—51no—
43#1131⋊C3—51no—
44C2×D5×D7—21no—
45C5×C7×Q8C22×C311yescharacteristic, complemented
46D140—21no—
47D140C22×C311yescharacteristic, complemented
48C4×D35—21no—
49C4×D35C22×C311yescharacteristic
50C7×D20C22×C311yescharacteristic
51C5×D28C22×C311yescharacteristic
52#63305—21no—
53C5×C7⋊C8C22×C311yescharacteristic, complemented
54C7×#168C22×C311yescharacteristic, complemented
55C3×#713—71no—
56C3×D4×D5—71no—
57C3×C5×SD16—71no—
58C3×#688—71no—
59C3×#686—71no—
60C3×D40—71no—
61C3×#676—71no—
62#6477—51noHall
63C2×C5×ΓL(1,𝔽8)C2×D411yescharacteristic
64C5⋊AGL(1,𝔽7)—21no—
65C5⋊AGL(1,𝔽7)—41no—
66D5×ΓL(1,𝔽8)—41no—
67C5×AGL(1,𝔽7)—41no—
68C22×AGL(1,𝔽7)—101no—
69Q8×ΓL(1,𝔽8)C2×D511yescharacteristic, complemented
70D4×ΓL(1,𝔽8)—51no—
71(C22⋊D7)⋊C3—101no—
72C4⋊AGL(1,𝔽7)—51no—
73C4⋊AGL(1,𝔽7)—101no—
74C4⋊AGL(1,𝔽7)C2×D511yescharacteristic
75C4×AGL(1,𝔽7)—51no—
76C4×AGL(1,𝔽7)—101no—
77C8×ΓL(1,𝔽8)—51no—
78ΓL(1,𝔽8)⋊C8C2×D511yescharacteristic, complemented
79#4060—71noHall
80C2×D35—21no—
81C2×D35C3×D411yescharacteristic
82C2×C7×D5—21no—
83C2×C5×D7—21no—
84D5×D7—42no—
85C4×C5×C7—21no—
86C4×C5×C7C23×C311yescharacteristic
87Dic35C3×D411yescharacteristic
88C22×C3×D5—141no—
89C3×C5×Q8—71no—
90C3×C5×D4—71no—
91C3×C22⋊D5—141no—
92C3×D20—71no—
93C3×D20—71no—
94C3×D20—141no—
95C3×C4×D5—71no—
96C3×C4×D5—141no—
97C3×C5×C8—71no—
98C3×#168—71no—
99#1161—51no—
100D4×D7—51no—
101C7×SD16—51no—
102#1143C3×D511yescharacteristic, complemented
103#1141—51no—
104D56—51no—
105#1131—51no—
106C5×ΓL(1,𝔽8)AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
107C3×AGL(1,ℤ/8)—351noHall
108C22×ΓL(1,𝔽8)—101no—
109C2×AGL(1,𝔽7)—21no—
110C2×AGL(1,𝔽7)—51no—
111C2×AGL(1,𝔽7)—101no—
112C2×AGL(1,𝔽7)—202no—
113C4×ΓL(1,𝔽8)—21no—
114C4×ΓL(1,𝔽8)C22×D511yescharacteristic
115ΓL(1,𝔽8)⋊C4—51no—
116#713—71no—
117D4×D5—71no—
118C5×SD16—71no—
119#688AGL(1,𝔽7)11yescharacteristic, complemented
120#686—71no—
121D40—71no—
122#676—71no—
123C2×C5×C7C2×C3×D411yescharacteristic
124D35—21no—
125D35—41no—
126C5×D7—41no—
127C7×D5—41no—
128C22×C3×C5—141no—
129C2×C3×D5—71no—
130C2×C3×D5—141no—
131C2×C3×D5—141no—
132C2×C3×D5—282no—
133C3×C4×C5—71no—
134C3×C4×C5—141no—
135C3×Dic5—71no—
136C22×D7—101no—
137C7×Q8C2×C3×D511yescharacteristic, complemented
138C7×D4—51no—
139C22⋊D7—101no—
140D28—51no—
141D28—101no—
142D28C2×C3×D511yescharacteristic
143C4×D7—51no—
144C4×D7—101no—
145C7×C8—51no—
146C7⋊C8C2×C3×D511yescharacteristic, complemented
147C3×#41—351no—
148C2×C3×D4—351no—
149C3×SD16—71no—
150C3×SD16—351no—
151C3×D8—351no—
152C3×D8—351no—
153C3×M16—351no—
154C2×ΓL(1,𝔽8)—201no—
155C2×ΓL(1,𝔽8)D4×D511yescharacteristic
156AGL(1,𝔽7)—41no—
157AGL(1,𝔽7)—101no—
158AGL(1,𝔽7)—201no—
159C22×D5—141no—
160C5×Q8C2×AGL(1,𝔽7)11yescharacteristic, complemented
161C5×D4—71no—
162C22⋊D5—141no—
163D20—71no—
164D20—141no—
165D20C2×AGL(1,𝔽7)11yescharacteristic
166C4×D5—71no—
167C4×D5—141no—
168C5×C8—71no—
169#168C2×AGL(1,𝔽7)11yescharacteristic, complemented
170C5×C7C3×AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
171AGL(1,ℤ/8)—351noSylow
172C2×C3×C5—71no—
173C2×C3×C5—281no—
174C3×D5—141no—
175C3×D5—281no—
176C3×D5—281no—
177C22×C7—101no—
178C2×D7—21no—
179C2×D7—51no—
180C2×D7—101no—
181C2×D7—202no—
182C4×C7—21no—
183C4×C7C22×C3×D511yescharacteristic
184Dic7—51no—
185C23×C3—701no—
186C3×Q8—71no—
187C3×D4—71no—
188C3×D4—351no—
189C3×D4—351no—
190C3×D4—701no—
191C3×D4—701no—
192C2×C3×C4—351no—
193C2×C3×C4—701no—
194C3×C8—71no—
195C3×C8—351no—
196ΓL(1,𝔽8)#406011yescharacteristic, complemented, Hall
197C22×C5—141no—
198C2×D5—21no—
199C2×D5—71no—
200C2×D5—141no—
201C2×D5—282no—
202C4×C5—21no—
203C4×C5C22×AGL(1,𝔽7)11yescharacteristic
204Dic5—71no—
205#41—351no—
206C2×D4—351no—
207SD16—51no—
208SD16—71no—
209D8—351no—
210D8—351no—
211M16—351no—
212C3×C5—71noHall
213C2×C7—201no—
214C2×C7C3×D4×D511yescharacteristic
215D7—41no—
216D7—101no—
217D7—201no—
218C22×C3—141no—
219C22×C3—351no—
220C22×C3—701no—
221C22×C3—701no—
222C22×C3—1402no—
223C3×C4—71no—
224C3×C4—141no—
225C3×C4—351no—
226C2×C5—281no—
227C2×C5D4×AGL(1,𝔽7)11yescharacteristic
228D5—41no—
229D5—141no—
230D5—281no—
231C23—701no—
232Q8D5×AGL(1,𝔽7)11yescharacteristic, complemented
233D4—51no—
234D4—71no—
235D4—351no—
236D4—701no—
237D4—701no—
238C2×C4—351no—
239C2×C4—701no—
240C8—51no—
241C8—71no—
242C7C3×#406011yescharacteristic, complemented, Sylow
243C2×C3—71no—
244C2×C3—281no—
245C2×C3—701no—
246C2×C3—1401no—
247C2×C3—1401no—
248C5#6477⋊C311yescharacteristic, complemented, Sylow
249C22—101no—
250C22—141no—
251C22—351no—
252C22—701no—
253C22—1402no—
254C4—21no—
255C4—351no—
256C4C2×D5×AGL(1,𝔽7)11yescharacteristic
257C3—71noSylow
258C2—201no—
259C2—281no—
260C2—701no—
261C2—1401no—
262C2#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
hypercenterQ8D5×AGL(1,𝔽7)
hypocenterC5×C7C3×AGL(1,ℤ/8)
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. #13132582C2semidirect
2. #10587950⋊C3C2semidirect
3. ΓL(1,𝔽8)×#688C2semidirect
4. C5×#1143⋊C3C2semidirect
5. #10587843⋊C3C2semidirect
6. #10587842⋊C3C2semidirect
7. #10587840⋊C3C2semidirect
8. #11764954C3semidirect
10. C5×Q8×ΓL(1,𝔽8)C22semidirect
12. C4⋊(C5⋊AGL(1,𝔽7))C22semidirect
18. ΓL(1,𝔽8)×#168C22semidirect
19. C5×ΓL(1,𝔽8)⋊C8C22semidirect
21. #10587991C2×C3semidirect
22. #10587950C2×C3semidirect
23. C7×#688C2×C3semidirect
24. C5×#1143C2×C3semidirect
25. #10587843C2×C3semidirect
26. #10587842C2×C3semidirect
27. #10587840C2×C3semidirect
40. #1143⋊C3D5semidirect
45. C5×C7×Q8C22×C3semidirect
47. D140C22×C3semidirect
53. C5×C7⋊C8C22×C3semidirect
54. C7×#168C22×C3semidirect
69. Q8×ΓL(1,𝔽8)C2×D5semidirect
78. ΓL(1,𝔽8)⋊C8C2×D5semidirect
102. #1143C3×D5semidirect
106. C5×ΓL(1,𝔽8)AGL(1,ℤ/8)semidirect
119. #688AGL(1,𝔽7)semidirect
137. C7×Q8C2×C3×D5semidirect
146. C7⋊C8C2×C3×D5semidirect
160. C5×Q8C2×AGL(1,𝔽7)semidirect
169. #168C2×AGL(1,𝔽7)semidirect
170. C5×C7C3×AGL(1,ℤ/8)semidirect
196. ΓL(1,𝔽8)#4060semidirect
232. Q8D5×AGL(1,𝔽7)semidirect
242. C7C3×#4060semidirect
248. C5#6477⋊C3semidirect

Maximal subgroups

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

subgroupindex
#131325822
#10587950⋊C32
ΓL(1,𝔽8)×#6882
C5×#1143⋊C32
#10587843⋊C32
#10587842⋊C32
#10587840⋊C32
#117649543
#6477⋊C35
C3×#40607