Finite Group Databasebeta

← #72218058

Subgroups — #72218058

Order 3360.

272 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 68 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
2D4×C5⋊AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
3C5×SD16×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
4#10587905⋊C3C211yesmaximal, characteristic, complemented
5#10587903⋊C3C211yesmaximal, characteristic, complemented
6D280⋊C3C211yesmaximal, characteristic, complemented
7#10587893⋊C3C211yesmaximal, characteristic, complemented
8#11765244C311yesmaximal, characteristic, complemented, Hall
9C22×C5⋊AGL(1,𝔽7)—21no—
10C5×Q8×ΓL(1,𝔽8)C2211yescharacteristic, complemented
11C5×D4×ΓL(1,𝔽8)C2211yescharacteristic
12(C22⋊D35)⋊C3—21no—
13C4⋊(C5⋊AGL(1,𝔽7))—21no—
14C4⋊(C5⋊AGL(1,𝔽7))C2211yescharacteristic
15C4⋊(C5⋊AGL(1,𝔽7))C2211yescharacteristic, complemented
16C4×C5⋊AGL(1,𝔽7)—21no—
17C4×C5⋊AGL(1,𝔽7)C2211yescharacteristic
18C5×C8×ΓL(1,𝔽8)C2211yescharacteristic, complemented
19ΓL(1,𝔽8)⋊#168C2211yescharacteristic, complemented
20#6477⋊C3—51nomaximal, Hall
21#10587991C2×C311yescharacteristic, complemented
22D4×D35C2×C311yescharacteristic, complemented
23C5×C7×SD16C2×C311yescharacteristic, complemented
24#10587905C2×C311yescharacteristic, complemented
25#10587903C2×C311yescharacteristic, complemented
26D280C2×C311yescharacteristic, complemented
27#10587893C2×C311yescharacteristic, complemented
28C3×#4060—71nomaximal, Hall
29C22×C5×ΓL(1,𝔽8)—21no—
30C2×C5⋊AGL(1,𝔽7)—21no—
31C2×C5⋊AGL(1,𝔽7)—21no—
32C2×C5⋊AGL(1,𝔽7)—42no—
33C2×C5⋊AGL(1,𝔽7)D411yescharacteristic
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)D511yescharacteristic, complemented
40#1143⋊C3—51no—
41#1141⋊C3—51no—
42D56⋊C3—51no—
43#1131⋊C3—51no—
44C22×D35—21no—
45C5×C7×Q8C22×C311yescharacteristic, complemented
46C5×C7×D4C22×C311yescharacteristic
47C22⋊D35—21no—
48D140—21no—
49D140C22×C311yescharacteristic
50D140C22×C311yescharacteristic, complemented
51C4×D35—21no—
52C4×D35C22×C311yescharacteristic
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)—41no—
64C2×C5×ΓL(1,𝔽8)C2×D411yescharacteristic
65C5⋊AGL(1,𝔽7)—21no—
66C5⋊AGL(1,𝔽7)—41no—
67C5⋊AGL(1,𝔽7)—41no—
68C22×AGL(1,𝔽7)—101no—
69Q8×ΓL(1,𝔽8)C2×D511yescharacteristic, complemented
70D4×ΓL(1,𝔽8)C2×D511yescharacteristic
71(C22⋊D7)⋊C3—101no—
72C4⋊AGL(1,𝔽7)—51no—
73C4⋊AGL(1,𝔽7)—51no—
74C4⋊AGL(1,𝔽7)—101no—
75C4×AGL(1,𝔽7)—51no—
76C4×AGL(1,𝔽7)—101no—
77C8×ΓL(1,𝔽8)C2×D511yescharacteristic, complemented
78ΓL(1,𝔽8)⋊C8—51no—
79#4060—71noHall
80C22×C5×C7—21no—
81C2×D35—21no—
82C2×D35—21no—
83C2×D35—42no—
84C2×D35C3×D411yescharacteristic
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×SD16C3×D511yescharacteristic, complemented
102#1143—51no—
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)—21no—
109C2×AGL(1,𝔽7)—51no—
110C2×AGL(1,𝔽7)—101no—
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×SD16AGL(1,𝔽7)11yescharacteristic, complemented
119#688—71no—
120#686—71no—
121D40—71no—
122#676—71no—
123C2×C5×C7—41no—
124C2×C5×C7C2×C3×D411yescharacteristic
125D35—21no—
126D35—41no—
127D35—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×D4C2×C3×D511yescharacteristic
139C22⋊D7—101no—
140D28—51no—
141D28—51no—
142D28—101no—
143C4×D7—51no—
144C4×D7—101no—
145C7×C8C2×C3×D511yescharacteristic, complemented
146C7⋊C8—51no—
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)—41no—
155C2×ΓL(1,𝔽8)D4×D511yescharacteristic
156AGL(1,𝔽7)—101no—
157AGL(1,𝔽7)—201no—
158AGL(1,𝔽7)—201no—
159C22×D5—141no—
160C5×Q8C2×AGL(1,𝔽7)11yescharacteristic, complemented
161C5×D4C2×AGL(1,𝔽7)11yescharacteristic
162C22⋊D5—141no—
163D20—71no—
164D20—71no—
165D20—141no—
166C4×D5—71no—
167C4×D5—141no—
168C5×C8C2×AGL(1,𝔽7)11yescharacteristic, complemented
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—
174C3×D5—141no—
175C3×D5—281no—
176C3×D5—281no—
177C22×C7—21no—
178C2×D7—51no—
179C2×D7—101no—
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—21no—
198C2×D5—71no—
199C2×D5—141no—
200C2×D5—141no—
201C2×D5—282no—
202C4×C5—21no—
203C4×C5C22×AGL(1,𝔽7)11yescharacteristic
204Dic5—71no—
205#41—351no—
206C2×D4—351no—
207SD16—351no—
208SD16C5⋊AGL(1,𝔽7)11yescharacteristic, complemented
209D8—351no—
210D8—351no—
211M16—351no—
212C3×C5—71noHall
213C2×C7—41no—
214C2×C7C3×D4×D511yescharacteristic
215D7—101no—
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—41no—
227C2×C5D4×AGL(1,𝔽7)11yescharacteristic
228D5—141no—
229D5—281no—
230D5—281no—
231C23—701no—
232Q8C2×C5⋊AGL(1,𝔽7)11yescharacteristic, complemented
233D4—351no—
234D4—351no—
235D4—701no—
236D4—701no—
237D4C2×C5⋊AGL(1,𝔽7)11yescharacteristic
238C2×C4—351no—
239C2×C4—701no—
240C8—351no—
241C8C2×C5⋊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#6477⋊C311yescharacteristic, complemented, Sylow
249C22—21no—
250C22—351no—
251C22—701no—
252C22—701no—
253C22—1402no—
254C4—21no—
255C4—351no—
256C4C22×C5⋊AGL(1,𝔽7)11yescharacteristic
257C3—71noSylow
258C2—41no—
259C2—701no—
260C2—1401no—
261C2—1401no—
262C2D4×C5⋊AGL(1,𝔽7)11yescharacteristic
subgroupG/H
centerC2D4×C5⋊AGL(1,𝔽7)
commutator, cosocleC4×C5×C7C23×C3
FrattiniC4C22×C5⋊AGL(1,𝔽7)
FittingC5×C7×SD16C2×C3
socleC2×C5×C7C2×C3×D4
hypercenterSD16C5⋊AGL(1,𝔽7)
hypocenterC5×C7C3×AGL(1,ℤ/8)
LCSG ▷ C4×C5×C7 ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C2 ◁ C4 ◁ SD16

Complements

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

NH ≅ G/Ntype
1. #13132582C2semidirect
2. D4×C5⋊AGL(1,𝔽7)C2semidirect
3. C5×SD16×ΓL(1,𝔽8)C2semidirect
4. #10587905⋊C3C2semidirect
5. #10587903⋊C3C2semidirect
6. D280⋊C3C2semidirect
7. #10587893⋊C3C2semidirect
8. #11765244C3semidirect
10. C5×Q8×ΓL(1,𝔽8)C22semidirect
15. C4⋊(C5⋊AGL(1,𝔽7))C22semidirect
18. C5×C8×ΓL(1,𝔽8)C22semidirect
19. ΓL(1,𝔽8)⋊#168C22semidirect
21. #10587991C2×C3semidirect
22. D4×D35C2×C3semidirect
23. C5×C7×SD16C2×C3semidirect
24. #10587905C2×C3semidirect
25. #10587903C2×C3semidirect
26. D280C2×C3semidirect
27. #10587893C2×C3semidirect
39. SD16×ΓL(1,𝔽8)D5semidirect
45. C5×C7×Q8C22×C3semidirect
50. D140C22×C3semidirect
53. C5×C7×C8C22×C3semidirect
54. C7⋊#168C22×C3semidirect
69. Q8×ΓL(1,𝔽8)C2×D5semidirect
77. C8×ΓL(1,𝔽8)C2×D5semidirect
101. C7×SD16C3×D5semidirect
106. C5×ΓL(1,𝔽8)AGL(1,ℤ/8)semidirect
118. C5×SD16AGL(1,𝔽7)semidirect
137. C7×Q8C2×C3×D5semidirect
145. C7×C8C2×C3×D5semidirect
160. C5×Q8C2×AGL(1,𝔽7)semidirect
168. C5×C8C2×AGL(1,𝔽7)semidirect
170. C5×C7C3×AGL(1,ℤ/8)semidirect
196. ΓL(1,𝔽8)#4060semidirect
208. SD16C5⋊AGL(1,𝔽7)semidirect
232. Q8C2×C5⋊AGL(1,𝔽7)semidirect
241. C8C2×C5⋊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
D4×C5⋊AGL(1,𝔽7)2
C5×SD16×ΓL(1,𝔽8)2
#10587905⋊C32
#10587903⋊C32
D280⋊C32
#10587893⋊C32
#117652443
#6477⋊C35
C3×#40607