Finite Group Databasebeta

← #72218243

Subgroups — #72218243

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
1ΓL(1,𝔽8)×#711C211yesmaximal, characteristic, complemented
2D5×C4⋊AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
3#10587903⋊C3C211yesmaximal, characteristic, complemented
4C5×#1141⋊C3C211yesmaximal, characteristic, complemented
5#10587849⋊C3C211yesmaximal, characteristic, complemented
6#10587844⋊C3C211yesmaximal, characteristic, complemented
7#10587839⋊C3C211yesmaximal, characteristic, complemented
8#11764928C311yesmaximal, characteristic, complemented, Hall
9C2×D5×AGL(1,𝔽7)—21no—
10C5×D4×ΓL(1,𝔽8)C2211yescharacteristic, complemented
11C4⋊(C5⋊AGL(1,𝔽7))C2211yescharacteristic
12ΓL(1,𝔽8)×C22⋊D5—21no—
13C2×Dic5×ΓL(1,𝔽8)—21no—
14C4×D5×ΓL(1,𝔽8)C2211yescharacteristic
15ΓL(1,𝔽8)×Dic10C2211yescharacteristic, complemented
16C5×C4⋊AGL(1,𝔽7)C2211yescharacteristic
17#63306⋊C3—21no—
18ΓL(1,𝔽8)⋊#168C2211yescharacteristic, complemented
19C5×ΓL(1,𝔽8)⋊C8C2211yescharacteristic, complemented
20#6512⋊C3—51nomaximal, Hall
21C7×#711C2×C311yescharacteristic, complemented
22D5×D28C2×C311yescharacteristic, complemented
23#10587903C2×C311yescharacteristic, complemented
24C5×#1141C2×C311yescharacteristic, complemented
25#10587849C2×C311yescharacteristic, complemented
26#10587844C2×C311yescharacteristic, complemented
27#10587839C2×C311yescharacteristic, complemented
28C3×#4057—71nomaximal, Hall
29C22×C5×ΓL(1,𝔽8)—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)—21no—
36Dic5×ΓL(1,𝔽8)D411yescharacteristic
37#41×ΓL(1,𝔽8)—51no—
38C2×C4⋊AGL(1,𝔽7)—51no—
39#1143⋊C3—51no—
40#1143⋊C3—51no—
41#1141⋊C3—51no—
42#1141⋊C3D511yescharacteristic, complemented
43#1136⋊C3—51no—
44C2×D5×D7—21no—
45C5×C7×D4C22×C311yescharacteristic, complemented
46D140C22×C311yescharacteristic
47C7×C22⋊D5—21no—
48C2×C7×Dic5—21no—
49C4×C7×D5C22×C311yescharacteristic
50C7×Dic10C22×C311yescharacteristic, complemented
51C5×D28C22×C311yescharacteristic
52#63306—21no—
53C7⋊#168C22×C311yescharacteristic, complemented
54C5×C7⋊C8C22×C311yescharacteristic, complemented
55C3×#711—71no—
56C3×D4×D5—71no—
57C3×C5×D8—71no—
58C3×#687—71no—
59C3×#686—71no—
60C3×#677—71no—
61C3×#676—71no—
62#6512—51noHall
63C2×C5×ΓL(1,𝔽8)—41no—
64C2×C5×ΓL(1,𝔽8)C2×D411yescharacteristic
65C5⋊AGL(1,𝔽7)—41no—
66D5×ΓL(1,𝔽8)—21no—
67C5×AGL(1,𝔽7)—41no—
68C22×AGL(1,𝔽7)—101no—
69Q8×ΓL(1,𝔽8)—51no—
70D4×ΓL(1,𝔽8)—101no—
71D4×ΓL(1,𝔽8)C2×D511yescharacteristic, complemented
72C2×C4×ΓL(1,𝔽8)—51no—
73C2×C4×ΓL(1,𝔽8)—101no—
74C4⋊AGL(1,𝔽7)—51no—
75C4⋊AGL(1,𝔽7)—101no—
76C4⋊AGL(1,𝔽7)C2×D511yescharacteristic
77ΓL(1,𝔽8)⋊C8—51no—
78ΓL(1,𝔽8)⋊C8C2×D511yescharacteristic, complemented
79#4057—71noHall
80C22×C5×C7—21no—
81C2×D35—21no—
82C2×C7×D5C3×D411yescharacteristic
83C2×C5×D7—21no—
84D5×D7—42no—
85C4×C5×C7C23×C311yescharacteristic
86C7×Dic5—21no—
87C7×Dic5C3×D411yescharacteristic
88C22×C3×D5—141no—
89C3×C5×D4—71no—
90C3×C5×D4—71no—
91C3×C22⋊D5—141no—
92C3×C22⋊D5—141no—
93C2×C3×Dic5—141no—
94C3×D20—71no—
95C3×C4×D5—71no—
96C3×Dic10—71no—
97C3×C5×C8—71no—
98C3×#168—71no—
99C7×#41—51no—
100C2×D28—51no—
101#1143—51no—
102#1143—51no—
103#1141—51no—
104#1141C3×D511yescharacteristic, complemented
105#1136—51no—
106C5×ΓL(1,𝔽8)AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
107C3×AGL(1,ℤ/8)—351noHall
108C22×ΓL(1,𝔽8)—21no—
109C22×ΓL(1,𝔽8)—51no—
110C2×AGL(1,𝔽7)—21no—
111C2×AGL(1,𝔽7)—101no—
112C2×AGL(1,𝔽7)—202no—
113C4×ΓL(1,𝔽8)—51no—
114C4×ΓL(1,𝔽8)—101no—
115C4×ΓL(1,𝔽8)C22×D511yescharacteristic
116#711AGL(1,𝔽7)11yescharacteristic, complemented
117D4×D5—71no—
118C5×D8—71no—
119#687—71no—
120#686—71no—
121#677—71no—
122#676—71no—
123C2×C5×C7—41no—
124C2×C5×C7C2×C3×D411yescharacteristic
125D35—41no—
126C5×D7—41no—
127C7×D5—21no—
128C22×C3×C5—141no—
129C22×C3×C5—141no—
130C2×C3×D5—71no—
131C2×C3×D5—141no—
132C2×C3×D5—282no—
133C3×C4×C5—71no—
134C3×Dic5—71no—
135C3×Dic5—141no—
136C22×D7—101no—
137C7×Q8—51no—
138C7×D4—101no—
139C7×D4C2×C3×D511yescharacteristic, complemented
140C2×C4×C7—51no—
141C2×C4×C7—101no—
142D28—51no—
143D28—101no—
144D28C2×C3×D511yescharacteristic
145C7⋊C8—51no—
146C7⋊C8C2×C3×D511yescharacteristic, complemented
147C3×#41—351no—
148C2×C3×D4—351no—
149C3×SD16—351no—
150C3×SD16—351no—
151C3×D8—71no—
152C3×D8—351no—
153C3×M16—351no—
154C2×ΓL(1,𝔽8)—41no—
155C2×ΓL(1,𝔽8)—101no—
156C2×ΓL(1,𝔽8)D4×D511yescharacteristic
157AGL(1,𝔽7)—41no—
158AGL(1,𝔽7)—201no—
159C22×D5—141no—
160C5×D4—71no—
161C5×D4C2×AGL(1,𝔽7)11yescharacteristic, complemented
162C22⋊D5—21no—
163C22⋊D5—141no—
164C2×Dic5—21no—
165D20—71no—
166C4×D5C2×AGL(1,𝔽7)11yescharacteristic
167Dic10C2×AGL(1,𝔽7)11yescharacteristic, complemented
168C5×C8—71no—
169#168—71no—
170C5×C7C3×AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
171AGL(1,ℤ/8)—351noSylow
172C2×C3×C5—71no—
173C2×C3×C5—281no—
174C2×C3×C5—281no—
175C3×D5—141no—
176C3×D5—281no—
177C22×C7—21no—
178C22×C7—51no—
179C2×D7—21no—
180C2×D7—101no—
181C2×D7—202no—
182C4×C7—51no—
183C4×C7—101no—
184C4×C7C22×C3×D511yescharacteristic
185C23×C3—701no—
186C3×Q8—351no—
187C3×D4—71no—
188C3×D4—71no—
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)#405711yescharacteristic, complemented, Hall
197C22×C5—21no—
198C22×C5—141no—
199C2×D5—141no—
200C2×D5—282no—
201C2×D5(C22⋊D7)⋊C311yescharacteristic
202C4×C5C22×AGL(1,𝔽7)11yescharacteristic
203Dic5—21no—
204Dic5(C22⋊D7)⋊C311yescharacteristic
205#41—51no—
206C2×D4—351no—
207SD16—351no—
208SD16—351no—
209D8—71no—
210D8—351no—
211M16—351no—
212C3×C5—71noHall
213C2×C7—41no—
214C2×C7—101no—
215C2×C7C3×D4×D511yescharacteristic
216D7—41no—
217D7—201no—
218C22×C3—141no—
219C22×C3—141no—
220C22×C3—351no—
221C22×C3—701no—
222C22×C3—1402no—
223C3×C4—71no—
224C3×C4—351no—
225C3×C4—701no—
226C2×C5—41no—
227C2×C5—281no—
228C2×C5C2×(C22⋊D7)⋊C311yescharacteristic
229D5—21no—
230D5—281no—
231C23—701no—
232Q8—51no—
233D4—71no—
234D4—101no—
235D4—351no—
236D4—701no—
237D4D5×AGL(1,𝔽7)11yescharacteristic, complemented
238C2×C4—51no—
239C2×C4—101no—
240C8—71no—
241C8—351no—
242C7C3×#405711yescharacteristic, complemented, Sylow
243C2×C3—71no—
244C2×C3—281no—
245C2×C3—281no—
246C2×C3—701no—
247C2×C3—1401no—
248C5#6512⋊C311yescharacteristic, complemented, Sylow
249C22—21no—
250C22—51no—
251C22—141no—
252C22—701no—
253C22—1402no—
254C4—51no—
255C4—101no—
256C4C2×D5×AGL(1,𝔽7)11yescharacteristic
257C3—71noSylow
258C2—41no—
259C2—101no—
260C2—281no—
261C2—1401no—
262C2D5×(C22⋊D7)⋊C311yescharacteristic
subgroupG/H
centerC2D5×(C22⋊D7)⋊C3
commutator, cosocleC4×C5×C7C23×C3
FrattiniC4C2×D5×AGL(1,𝔽7)
FittingC5×C7×D4C22×C3
socleC2×C5×C7C2×C3×D4
hypercenterD4D5×AGL(1,𝔽7)
hypocenterC5×C7C3×AGL(1,ℤ/8)
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
1. ΓL(1,𝔽8)×#711C2semidirect
2. D5×C4⋊AGL(1,𝔽7)C2semidirect
3. #10587903⋊C3C2semidirect
4. C5×#1141⋊C3C2semidirect
5. #10587849⋊C3C2semidirect
6. #10587844⋊C3C2semidirect
7. #10587839⋊C3C2semidirect
8. #11764928C3semidirect
10. C5×D4×ΓL(1,𝔽8)C22semidirect
15. ΓL(1,𝔽8)×Dic10C22semidirect
18. ΓL(1,𝔽8)⋊#168C22semidirect
19. C5×ΓL(1,𝔽8)⋊C8C22semidirect
21. C7×#711C2×C3semidirect
22. D5×D28C2×C3semidirect
23. #10587903C2×C3semidirect
24. C5×#1141C2×C3semidirect
25. #10587849C2×C3semidirect
26. #10587844C2×C3semidirect
27. #10587839C2×C3semidirect
42. #1141⋊C3D5semidirect
45. C5×C7×D4C22×C3semidirect
50. C7×Dic10C22×C3semidirect
53. C7⋊#168C22×C3semidirect
54. C5×C7⋊C8C22×C3semidirect
71. D4×ΓL(1,𝔽8)C2×D5semidirect
78. ΓL(1,𝔽8)⋊C8C2×D5semidirect
104. #1141C3×D5semidirect
106. C5×ΓL(1,𝔽8)AGL(1,ℤ/8)semidirect
116. #711AGL(1,𝔽7)semidirect
139. C7×D4C2×C3×D5semidirect
146. C7⋊C8C2×C3×D5semidirect
161. C5×D4C2×AGL(1,𝔽7)semidirect
167. Dic10C2×AGL(1,𝔽7)semidirect
170. C5×C7C3×AGL(1,ℤ/8)semidirect
196. ΓL(1,𝔽8)#4057semidirect
237. D4D5×AGL(1,𝔽7)semidirect
242. C7C3×#4057semidirect
248. C5#6512⋊C3semidirect

Maximal subgroups

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

subgroupindex
ΓL(1,𝔽8)×#7112
D5×C4⋊AGL(1,𝔽7)2
#10587903⋊C32
C5×#1141⋊C32
#10587849⋊C32
#10587844⋊C32
#10587839⋊C32
#117649283
#6512⋊C35
C3×#40577