Finite Group Databasebeta

← #72217995

Subgroups — #72217995

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#10587989⋊C3C211yesmaximal, characteristic, complemented
2D4×C5⋊AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
3C5×D8×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
4#10587904⋊C3C211yesmaximal, characteristic, complemented
5#10587903⋊C3C211yesmaximal, characteristic, complemented
6#10587894⋊C3C211yesmaximal, characteristic, complemented
7#10587893⋊C3C211yesmaximal, characteristic, complemented
8#11765241C311yesmaximal, characteristic, complemented, Hall
9C22×C5⋊AGL(1,𝔽7)—21no—
10C5×D4×ΓL(1,𝔽8)C2211yescharacteristic
11C5×D4×ΓL(1,𝔽8)C2211yescharacteristic, complemented
12(C22⋊D35)⋊C3—21no—
13(C22⋊D35)⋊C3—21no—
14C2×ΓL(1,𝔽8)⋊Dic5—21no—
15C4⋊(C5⋊AGL(1,𝔽7))C2211yescharacteristic
16C4×C5⋊AGL(1,𝔽7)C2211yescharacteristic
17Dic70⋊C3C2211yescharacteristic, complemented
18C5×C8×ΓL(1,𝔽8)C2211yescharacteristic, complemented
19ΓL(1,𝔽8)⋊#168C2211yescharacteristic, complemented
20#6474⋊C3—51nomaximal, Hall
21#10587989C2×C311yescharacteristic, complemented
22D4×D35C2×C311yescharacteristic, complemented
23C5×C7×D8C2×C311yescharacteristic, complemented
24#10587904C2×C311yescharacteristic, complemented
25#10587903C2×C311yescharacteristic, complemented
26#10587894C2×C311yescharacteristic, complemented
27#10587893C2×C311yescharacteristic, complemented
28C3×#4057—71nomaximal, Hall
29C22×C5×ΓL(1,𝔽8)—21no—
30C22×C5×ΓL(1,𝔽8)—21no—
31C2×C5⋊AGL(1,𝔽7)—21no—
32C2×C5⋊AGL(1,𝔽7)—42no—
33C2×C5⋊AGL(1,𝔽7)D411yescharacteristic
34C4×C5×ΓL(1,𝔽8)C2311yescharacteristic
35ΓL(1,𝔽8)⋊Dic5—21no—
36ΓL(1,𝔽8)⋊Dic5D411yescharacteristic
37#1159⋊C3—51no—
38D4×AGL(1,𝔽7)—51no—
39D8×ΓL(1,𝔽8)D511yescharacteristic, complemented
40#1142⋊C3—51no—
41#1141⋊C3—51no—
42#1132⋊C3—51no—
43#1131⋊C3—51no—
44C22×D35—21no—
45C5×C7×D4C22×C311yescharacteristic
46C5×C7×D4C22×C311yescharacteristic, complemented
47C22⋊D35—21no—
48C22⋊D35—21no—
49C2×Dic35—21no—
50D140C22×C311yescharacteristic
51C4×D35C22×C311yescharacteristic
52Dic70C22×C311yescharacteristic, complemented
53C5×C7×C8C22×C311yescharacteristic, complemented
54C7⋊#168C22×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#6474—51noHall
63C2×C5×ΓL(1,𝔽8)—41no—
64C2×C5×ΓL(1,𝔽8)—41no—
65C2×C5×ΓL(1,𝔽8)C2×D411yescharacteristic
66C5⋊AGL(1,𝔽7)—21no—
67C5⋊AGL(1,𝔽7)—41no—
68C22×AGL(1,𝔽7)—101no—
69D4×ΓL(1,𝔽8)C2×D511yescharacteristic
70D4×ΓL(1,𝔽8)C2×D511yescharacteristic, complemented
71(C22⋊D7)⋊C3—101no—
72(C22⋊D7)⋊C3—101no—
73C2×ΓL(1,𝔽8)⋊C4—101no—
74C4⋊AGL(1,𝔽7)—51no—
75C4×AGL(1,𝔽7)—51no—
76ΓL(1,𝔽8)⋊Q8—51no—
77C8×ΓL(1,𝔽8)C2×D511yescharacteristic, complemented
78ΓL(1,𝔽8)⋊C8—51no—
79#4057—71noHall
80C22×C5×C7—21no—
81C22×C5×C7—21no—
82C2×D35—21no—
83C2×D35—42no—
84C2×D35C3×D411yescharacteristic
85C4×C5×C7C23×C311yescharacteristic
86Dic35—21no—
87Dic35C3×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—
99#1159—51no—
100D4×D7—51no—
101C7×D8C3×D511yescharacteristic, complemented
102#1142—51no—
103#1141—51no—
104#1132—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—
109C22×ΓL(1,𝔽8)—21no—
110C2×AGL(1,𝔽7)—51no—
111C2×AGL(1,𝔽7)—101no—
112C2×AGL(1,𝔽7)—202no—
113C4×ΓL(1,𝔽8)C22×D511yescharacteristic
114ΓL(1,𝔽8)⋊C4—51no—
115ΓL(1,𝔽8)⋊C4—101no—
116#711—71no—
117D4×D5—71no—
118C5×D8AGL(1,𝔽7)11yescharacteristic, complemented
119#687—71no—
120#686—71no—
121#677—71no—
122#676—71no—
123C2×C5×C7—41no—
124C2×C5×C7—41no—
125C2×C5×C7C2×C3×D411yescharacteristic
126D35—21no—
127D35—41no—
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×D4C2×C3×D511yescharacteristic
138C7×D4C2×C3×D511yescharacteristic, complemented
139C22⋊D7—101no—
140C22⋊D7—101no—
141C2×Dic7—101no—
142D28—51no—
143C4×D7—51no—
144Dic14—51no—
145C7×C8C2×C3×D511yescharacteristic, complemented
146C7⋊C8—51no—
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)—41no—
156C2×ΓL(1,𝔽8)D4×D511yescharacteristic
157AGL(1,𝔽7)—101no—
158AGL(1,𝔽7)—201no—
159C22×D5—141no—
160C5×D4C2×AGL(1,𝔽7)11yescharacteristic
161C5×D4C2×AGL(1,𝔽7)11yescharacteristic, complemented
162C22⋊D5—141no—
163C22⋊D5—141no—
164C2×Dic5—141no—
165D20—71no—
166C4×D5—71no—
167Dic10—71no—
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—
174C2×C3×C5—281no—
175C3×D5—141no—
176C3×D5—281no—
177C22×C7—21no—
178C22×C7—21no—
179C2×D7—51no—
180C2×D7—101no—
181C2×D7—202no—
182C4×C7C22×C3×D511yescharacteristic
183Dic7—51no—
184Dic7—101no—
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—21no—
199C2×D5—71no—
200C2×D5—141no—
201C2×D5—282no—
202C4×C5C22×AGL(1,𝔽7)11yescharacteristic
203Dic5—71no—
204Dic5—141no—
205#41—351no—
206C2×D4—351no—
207SD16—351no—
208SD16—351no—
209D8—351no—
210D8C5⋊AGL(1,𝔽7)11yescharacteristic, complemented
211M16—351no—
212C3×C5—71noHall
213C2×C7—41no—
214C2×C7—41no—
215C2×C7C3×D4×D511yescharacteristic
216D7—101no—
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—41no—
228C2×C5D4×AGL(1,𝔽7)11yescharacteristic
229D5—141no—
230D5—281no—
231C23—701no—
232Q8—351no—
233D4—351no—
234D4—701no—
235D4—701no—
236D4C2×C5⋊AGL(1,𝔽7)11yescharacteristic
237D4C2×C5⋊AGL(1,𝔽7)11yescharacteristic, complemented
238C2×C4—351no—
239C2×C4—701no—
240C8—351no—
241C8C2×C5⋊AGL(1,𝔽7)11yescharacteristic, complemented
242C7C3×#405711yescharacteristic, complemented, Sylow
243C2×C3—71no—
244C2×C3—281no—
245C2×C3—281no—
246C2×C3—701no—
247C2×C3—1401no—
248C5#6474⋊C311yescharacteristic, complemented, Sylow
249C22—21no—
250C22—21no—
251C22—351no—
252C22—701no—
253C22—1402no—
254C4—351no—
255C4—701no—
256C4C22×C5⋊AGL(1,𝔽7)11yescharacteristic
257C3—71noSylow
258C2—41no—
259C2—41no—
260C2—701no—
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×D8C2×C3
socleC2×C5×C7C2×C3×D4
hypercenterD8C5⋊AGL(1,𝔽7)
hypocenterC5×C7C3×AGL(1,ℤ/8)
LCSG ▷ C4×C5×C7 ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C2 ◁ C4 ◁ D8

Complements

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

NH ≅ G/Ntype
1. #10587989⋊C3C2semidirect
2. D4×C5⋊AGL(1,𝔽7)C2semidirect
3. C5×D8×ΓL(1,𝔽8)C2semidirect
4. #10587904⋊C3C2semidirect
5. #10587903⋊C3C2semidirect
6. #10587894⋊C3C2semidirect
7. #10587893⋊C3C2semidirect
8. #11765241C3semidirect
11. C5×D4×ΓL(1,𝔽8)C22semidirect
17. Dic70⋊C3C22semidirect
18. C5×C8×ΓL(1,𝔽8)C22semidirect
19. ΓL(1,𝔽8)⋊#168C22semidirect
21. #10587989C2×C3semidirect
22. D4×D35C2×C3semidirect
23. C5×C7×D8C2×C3semidirect
24. #10587904C2×C3semidirect
25. #10587903C2×C3semidirect
26. #10587894C2×C3semidirect
27. #10587893C2×C3semidirect
39. D8×ΓL(1,𝔽8)D5semidirect
46. C5×C7×D4C22×C3semidirect
52. Dic70C22×C3semidirect
53. C5×C7×C8C22×C3semidirect
54. C7⋊#168C22×C3semidirect
70. D4×ΓL(1,𝔽8)C2×D5semidirect
77. C8×ΓL(1,𝔽8)C2×D5semidirect
101. C7×D8C3×D5semidirect
106. C5×ΓL(1,𝔽8)AGL(1,ℤ/8)semidirect
118. C5×D8AGL(1,𝔽7)semidirect
138. C7×D4C2×C3×D5semidirect
145. C7×C8C2×C3×D5semidirect
161. C5×D4C2×AGL(1,𝔽7)semidirect
168. C5×C8C2×AGL(1,𝔽7)semidirect
170. C5×C7C3×AGL(1,ℤ/8)semidirect
196. ΓL(1,𝔽8)#4057semidirect
210. D8C5⋊AGL(1,𝔽7)semidirect
237. D4C2×C5⋊AGL(1,𝔽7)semidirect
241. C8C2×C5⋊AGL(1,𝔽7)semidirect
242. C7C3×#4057semidirect
248. C5#6474⋊C3semidirect

Maximal subgroups

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

subgroupindex
#10587989⋊C32
D4×C5⋊AGL(1,𝔽7)2
C5×D8×ΓL(1,𝔽8)2
#10587904⋊C32
#10587903⋊C32
#10587894⋊C32
#10587893⋊C32
#117652413
#6474⋊C35
C3×#40577