Finite Group Databasebeta

← #72218045

Subgroups — #72218045

Order 3360.

272 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 67 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×C4⋊AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
2#10587942⋊C3C211yesmaximal, characteristic, complemented
3D280⋊C3C211yesmaximal, characteristic, complemented
4ΓL(1,𝔽8)×#676C211yesmaximal, characteristic, complemented
5C5×#1132⋊C3C211yesmaximal, characteristic, complemented
6#10587847⋊C3C211yesmaximal, characteristic, complemented
7#10587842⋊C3C211yesmaximal, characteristic, complemented
8#11764700C311yesmaximal, characteristic, complemented, Hall
9C2×D5×AGL(1,𝔽7)—21no—
10C4⋊(C5⋊AGL(1,𝔽7))C2211yescharacteristic
11C4⋊(C5⋊AGL(1,𝔽7))C2211yescharacteristic, complemented
12C4×D5×ΓL(1,𝔽8)C2211yescharacteristic
13C5×C4⋊AGL(1,𝔽7)C2211yescharacteristic
14C5×ΓL(1,𝔽8)⋊Q8C2211yescharacteristic, complemented
15#63307⋊C3—21no—
16#63306⋊C3—21no—
17#63304⋊C3—21no—
18C5×C8×ΓL(1,𝔽8)C2211yescharacteristic, complemented
19ΓL(1,𝔽8)×#168C2211yescharacteristic, complemented
20#6471⋊C3—51nomaximal, Hall
21D5×D28C2×C311yescharacteristic, complemented
22#10587942C2×C311yescharacteristic, complemented
23D280C2×C311yescharacteristic, complemented
24C7×#676C2×C311yescharacteristic, complemented
25C5×#1132C2×C311yescharacteristic, complemented
26#10587847C2×C311yescharacteristic, complemented
27#10587842C2×C311yescharacteristic, complemented
28C3×#4060—71nomaximal, Hall
29C2×C5⋊AGL(1,𝔽7)—21no—
30C2×C5⋊AGL(1,𝔽7)—21no—
31C2×D5×ΓL(1,𝔽8)D411yescharacteristic
32C2×C5×AGL(1,𝔽7)—21no—
33D5×AGL(1,𝔽7)—42no—
34C4×C5×ΓL(1,𝔽8)C2311yescharacteristic
35Dic5×ΓL(1,𝔽8)D411yescharacteristic
36C5×ΓL(1,𝔽8)⋊C4—21no—
37#1157⋊C3—51no—
38C2×C4⋊AGL(1,𝔽7)—51no—
39M16×ΓL(1,𝔽8)—51no—
40D56⋊C3—51no—
41D56⋊C3—51no—
42#1132⋊C3—51no—
43#1132⋊C3D511yescharacteristic, complemented
44C2×D5×D7—21no—
45D140C22×C311yescharacteristic
46D140C22×C311yescharacteristic, complemented
47C4×C7×D5C22×C311yescharacteristic
48C5×D28C22×C311yescharacteristic
49C5×Dic14C22×C311yescharacteristic, complemented
50#63307—21no—
51#63306—21no—
52#63304—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#6471—51noHall
63C2×C5×ΓL(1,𝔽8)C2×D411yescharacteristic
64C5⋊AGL(1,𝔽7)—41no—
65C5⋊AGL(1,𝔽7)—41no—
66D5×ΓL(1,𝔽8)—21no—
67C5×AGL(1,𝔽7)—41no—
68C22×AGL(1,𝔽7)—101no—
69C2×C4×ΓL(1,𝔽8)—51no—
70(C22⋊D7)⋊C3—101no—
71C4⋊AGL(1,𝔽7)—51no—
72C4⋊AGL(1,𝔽7)—51no—
73C4⋊AGL(1,𝔽7)—101no—
74C4⋊AGL(1,𝔽7)C2×D511yescharacteristic
75C4×AGL(1,𝔽7)—101no—
76ΓL(1,𝔽8)⋊Q8C2×D511yescharacteristic, complemented
77C8×ΓL(1,𝔽8)—51no—
78C8×ΓL(1,𝔽8)C2×D511yescharacteristic, complemented
79#4060—71noHall
80C2×D35—21no—
81C2×D35—21no—
82C2×C7×D5C3×D411yescharacteristic
83C2×C5×D7—21no—
84D5×D7—42no—
85C4×C5×C7C23×C311yescharacteristic
86C5×Dic7—21no—
87C7×Dic5C3×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#1157—51no—
100C2×D28—51no—
101C7×M16—51no—
102D56—51no—
103D56—51no—
104#1132—51no—
105#1132C3×D511yescharacteristic, complemented
106C5×ΓL(1,𝔽8)AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
107C3×AGL(1,ℤ/8)—351noHall
108C22×ΓL(1,𝔽8)—51no—
109C2×AGL(1,𝔽7)—21no—
110C2×AGL(1,𝔽7)—101no—
111C2×AGL(1,𝔽7)—101no—
112C2×AGL(1,𝔽7)—202no—
113C4×ΓL(1,𝔽8)—51no—
114C4×ΓL(1,𝔽8)C22×D511yescharacteristic
115ΓL(1,𝔽8)⋊C4—21no—
116#713—71no—
117D4×D5—71no—
118C5×SD16—71no—
119#688—71no—
120#686—71no—
121D40—71no—
122#676AGL(1,𝔽7)11yescharacteristic, complemented
123C2×C5×C7C2×C3×D411yescharacteristic
124D35—41no—
125D35—41no—
126C5×D7—41no—
127C7×D5—21no—
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—
137C2×C4×C7—51no—
138C22⋊D7—101no—
139D28—51no—
140D28—51no—
141D28—101no—
142D28C2×C3×D511yescharacteristic
143C4×D7—101no—
144Dic14C2×C3×D511yescharacteristic, complemented
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)—101no—
155C2×ΓL(1,𝔽8)D4×D511yescharacteristic
156AGL(1,𝔽7)—41no—
157AGL(1,𝔽7)—201no—
158AGL(1,𝔽7)—201no—
159C22×D5—141no—
160C5×Q8—71no—
161C5×D4—71no—
162C22⋊D5—141no—
163D20—71no—
164D20—71no—
165D20—141no—
166C4×D5—141no—
167C4×D5C2×AGL(1,𝔽7)11yescharacteristic
168C5×C8C2×AGL(1,𝔽7)11yescharacteristic, complemented
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—51no—
178C2×D7—21no—
179C2×D7—101no—
180C2×D7—101no—
181C2×D7—202no—
182C4×C7—51no—
183C4×C7C22×C3×D511yescharacteristic
184Dic7—21no—
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—141no—
199C2×D5—141no—
200C2×D5—282no—
201C2×D5C4⋊AGL(1,𝔽7)11yescharacteristic
202C4×C5—141no—
203C4×C5C22×AGL(1,𝔽7)11yescharacteristic
204Dic5C4⋊AGL(1,𝔽7)11yescharacteristic
205#41—351no—
206C2×D4—351no—
207SD16—71no—
208SD16—351no—
209D8—351no—
210D8—351no—
211M16—51no—
212C3×C5—71noHall
213C2×C7—101no—
214C2×C7C3×D4×D511yescharacteristic
215D7—41no—
216D7—201no—
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×C5C2×C4⋊AGL(1,𝔽7)11yescharacteristic
228D5—21no—
229D5—281no—
230D5—281no—
231C23—701no—
232Q8—71no—
233D4—71no—
234D4—351no—
235D4—351no—
236D4—701no—
237D4—701no—
238C2×C4—51no—
239C2×C4—701no—
240C8—51no—
241C8D5×AGL(1,𝔽7)11yescharacteristic, complemented
242C7C3×#406011yescharacteristic, complemented, Sylow
243C2×C3—71no—
244C2×C3—281no—
245C2×C3—701no—
246C2×C3—1401no—
247C2×C3—1401no—
248C5#6471⋊C311yescharacteristic, complemented, Sylow
249C22—51no—
250C22—141no—
251C22—701no—
252C22—701no—
253C22—1402no—
254C4—51no—
255C4—141no—
256C4C2×D5×AGL(1,𝔽7)11yescharacteristic
257C3—71noSylow
258C2—101no—
259C2—281no—
260C2—1401no—
261C2—1401no—
262C2D5×C4⋊AGL(1,𝔽7)11yescharacteristic
subgroupG/H
centerC2D5×C4⋊AGL(1,𝔽7)
commutator, cosocleC4×C5×C7C23×C3
FrattiniC4C2×D5×AGL(1,𝔽7)
FittingC5×C7×C8C22×C3
socleC2×C5×C7C2×C3×D4
hypercenterC8D5×AGL(1,𝔽7)
hypocenterC5×C7C3×AGL(1,ℤ/8)
LCSG ▷ C4×C5×C7 ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. D5×C4⋊AGL(1,𝔽7)C2semidirect
2. #10587942⋊C3C2semidirect
3. D280⋊C3C2semidirect
4. ΓL(1,𝔽8)×#676C2semidirect
5. C5×#1132⋊C3C2semidirect
6. #10587847⋊C3C2semidirect
7. #10587842⋊C3C2semidirect
8. #11764700C3semidirect
11. C4⋊(C5⋊AGL(1,𝔽7))C22semidirect
14. C5×ΓL(1,𝔽8)⋊Q8C22semidirect
18. C5×C8×ΓL(1,𝔽8)C22semidirect
19. ΓL(1,𝔽8)×#168C22semidirect
21. D5×D28C2×C3semidirect
22. #10587942C2×C3semidirect
23. D280C2×C3semidirect
24. C7×#676C2×C3semidirect
25. C5×#1132C2×C3semidirect
26. #10587847C2×C3semidirect
27. #10587842C2×C3semidirect
43. #1132⋊C3D5semidirect
46. D140C22×C3semidirect
49. C5×Dic14C22×C3semidirect
53. C5×C7×C8C22×C3semidirect
54. C7×#168C22×C3semidirect
76. ΓL(1,𝔽8)⋊Q8C2×D5semidirect
78. C8×ΓL(1,𝔽8)C2×D5semidirect
105. #1132C3×D5semidirect
106. C5×ΓL(1,𝔽8)AGL(1,ℤ/8)semidirect
122. #676AGL(1,𝔽7)semidirect
144. Dic14C2×C3×D5semidirect
146. C7×C8C2×C3×D5semidirect
168. C5×C8C2×AGL(1,𝔽7)semidirect
169. #168C2×AGL(1,𝔽7)semidirect
170. C5×C7C3×AGL(1,ℤ/8)semidirect
196. ΓL(1,𝔽8)#4060semidirect
241. C8D5×AGL(1,𝔽7)semidirect
242. C7C3×#4060semidirect
248. C5#6471⋊C3semidirect

Maximal subgroups

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

subgroupindex
D5×C4⋊AGL(1,𝔽7)2
#10587942⋊C32
D280⋊C32
ΓL(1,𝔽8)×#6762
C5×#1132⋊C32
#10587847⋊C32
#10587842⋊C32
#117647003
#6471⋊C35
C3×#40607