Finite Group Databasebeta

← C5×#6474⋊C3 #72217992

Subgroups — C5×#6474⋊C3 #72217992

Order 3360.

272 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C5×#1159⋊C3C211yesmaximal, characteristic, complemented
2C5×D4×AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
3C5×D8×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
4C5×#1142⋊C3C211yesmaximal, characteristic, complemented
5C5×#1141⋊C3C211yesmaximal, characteristic, complemented
6C5×#1132⋊C3C211yesmaximal, characteristic, complemented
7C5×#1131⋊C3C211yesmaximal, characteristic, complemented
8C5×#6474C311yesmaximal, characteristic, complemented, Hall
9C22×C5×AGL(1,𝔽7)—21no—
10C5×D4×ΓL(1,𝔽8)C2211yescharacteristic
11C5×D4×ΓL(1,𝔽8)C2211yescharacteristic, complemented
12C5×(C22⋊D7)⋊C3—21no—
13C5×(C22⋊D7)⋊C3—21no—
14C2×C5×ΓL(1,𝔽8)⋊C4—21no—
15C5×C4⋊AGL(1,𝔽7)C2211yescharacteristic
16C4×C5×AGL(1,𝔽7)C2211yescharacteristic
17C5×ΓL(1,𝔽8)⋊Q8C2211yescharacteristic, complemented
18C5×C8×ΓL(1,𝔽8)C2211yescharacteristic, complemented
19C5×ΓL(1,𝔽8)⋊C8C2211yescharacteristic, complemented
20#6474⋊C3C511yesmaximal, characteristic, direct product factor, Hall
21C5×#1159C2×C311yescharacteristic, complemented
22C5×D4×D7C2×C311yescharacteristic, complemented
23C5×C7×D8C2×C311yescharacteristic, complemented
24C5×#1142C2×C311yescharacteristic, complemented
25C5×#1141C2×C311yescharacteristic, complemented
26C5×#1132C2×C311yescharacteristic, complemented
27C5×#1131C2×C311yescharacteristic, complemented
28C3×C5×AGL(1,ℤ/8)—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
35C5×ΓL(1,𝔽8)⋊C4—21no—
36C5×ΓL(1,𝔽8)⋊C4D411yescharacteristic
37#1159⋊C3C2×C511yescharacteristic, complemented
38D4×AGL(1,𝔽7)C2×C511yescharacteristic, complemented
39D8×ΓL(1,𝔽8)C2×C511yescharacteristic, complemented
40#1142⋊C3C2×C511yescharacteristic, complemented
41#1141⋊C3C2×C511yescharacteristic, complemented
42#1132⋊C3C2×C511yescharacteristic, complemented
43#1131⋊C3C2×C511yescharacteristic, complemented
44C22×C5×D7—21no—
45C5×C7×D4C22×C311yescharacteristic
46C5×C7×D4C22×C311yescharacteristic, complemented
47C5×C22⋊D7—21no—
48C5×C22⋊D7—21no—
49C2×C5×Dic7—21no—
50C5×D28C22×C311yescharacteristic
51C4×C5×D7C22×C311yescharacteristic
52C5×Dic14C22×C311yescharacteristic, complemented
53C5×C7×C8C22×C311yescharacteristic, complemented
54C5×C7⋊C8C22×C311yescharacteristic, complemented
55C3×C5×#41—71no—
56C2×C3×C5×D4—71no—
57C3×C5×SD16—71no—
58C3×C5×SD16—71no—
59C3×C5×D8—71no—
60C3×C5×D8—71no—
61C3×C5×M16—71no—
62#6474C3×C511yescharacteristic, complemented, Hall
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)—21no—
69D4×ΓL(1,𝔽8)C22×C511yescharacteristic
70D4×ΓL(1,𝔽8)C22×C511yescharacteristic, complemented
71(C22⋊D7)⋊C3—21no—
72(C22⋊D7)⋊C3—21no—
73C2×ΓL(1,𝔽8)⋊C4—21no—
74C4⋊AGL(1,𝔽7)C22×C511yescharacteristic
75C4×AGL(1,𝔽7)C22×C511yescharacteristic
76ΓL(1,𝔽8)⋊Q8C22×C511yescharacteristic, complemented
77C8×ΓL(1,𝔽8)C22×C511yescharacteristic, complemented
78ΓL(1,𝔽8)⋊C8C22×C511yescharacteristic, complemented
79C5×AGL(1,ℤ/8)—71noHall
80C22×C5×C7—21no—
81C22×C5×C7—21no—
82C2×C5×D7—21no—
83C2×C5×D7—42no—
84C2×C5×D7C3×D411yescharacteristic
85C4×C5×C7C23×C311yescharacteristic
86C5×Dic7—21no—
87C5×Dic7C3×D411yescharacteristic
88C23×C3×C5—141no—
89C3×C5×Q8—71no—
90C3×C5×D4—71no—
91C3×C5×D4—71no—
92C3×C5×D4—71no—
93C3×C5×D4—141no—
94C3×C5×D4—141no—
95C2×C3×C4×C5—71no—
96C2×C3×C4×C5—141no—
97C3×C5×C8—71no—
98C3×C5×C8—71no—
99#1159C2×C3×C511yescharacteristic, complemented
100D4×D7C2×C3×C511yescharacteristic, complemented
101C7×D8C2×C3×C511yescharacteristic, complemented
102#1142C2×C3×C511yescharacteristic, complemented
103#1141C2×C3×C511yescharacteristic, complemented
104#1132C2×C3×C511yescharacteristic, complemented
105#1131C2×C3×C511yescharacteristic, complemented
106C5×ΓL(1,𝔽8)AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
107C3×AGL(1,ℤ/8)—71noHall
108C22×ΓL(1,𝔽8)—21no—
109C22×ΓL(1,𝔽8)—21no—
110C2×AGL(1,𝔽7)—21no—
111C2×AGL(1,𝔽7)—42no—
112C2×AGL(1,𝔽7)C5×D411yescharacteristic
113C4×ΓL(1,𝔽8)C23×C511yescharacteristic
114ΓL(1,𝔽8)⋊C4—21no—
115ΓL(1,𝔽8)⋊C4C5×D411yescharacteristic
116C5×#41—71no—
117C2×C5×D4—71no—
118C5×SD16—71no—
119C5×SD16—71no—
120C5×D8—71no—
121C5×D8AGL(1,𝔽7)11yescharacteristic, complemented
122C5×M16—71no—
123C2×C5×C7—41no—
124C2×C5×C7—41no—
125C2×C5×C7C2×C3×D411yescharacteristic
126C5×D7—21no—
127C5×D7—41no—
128C22×C3×C5—71no—
129C22×C3×C5—141no—
130C22×C3×C5—141no—
131C22×C3×C5—141no—
132C22×C3×C5—282no—
133C3×C4×C5—71no—
134C3×C4×C5—71no—
135C3×C4×C5—141no—
136C22×D7—21no—
137C7×D4C22×C3×C511yescharacteristic
138C7×D4C22×C3×C511yescharacteristic, complemented
139C22⋊D7—21no—
140C22⋊D7—21no—
141C2×Dic7—21no—
142D28C22×C3×C511yescharacteristic
143C4×D7C22×C3×C511yescharacteristic
144Dic14C22×C3×C511yescharacteristic, complemented
145C7×C8C22×C3×C511yescharacteristic, complemented
146C7⋊C8C22×C3×C511yescharacteristic, complemented
147C3×#41—71no—
148C2×C3×D4—71no—
149C3×SD16—71no—
150C3×SD16—71no—
151C3×D8—71no—
152C3×D8—71no—
153C3×M16—71no—
154C2×ΓL(1,𝔽8)—41no—
155C2×ΓL(1,𝔽8)—41no—
156C2×ΓL(1,𝔽8)C2×C5×D411yescharacteristic
157AGL(1,𝔽7)—21no—
158AGL(1,𝔽7)—41no—
159C23×C5—141no—
160C5×Q8—71no—
161C5×D4—71no—
162C5×D4—141no—
163C5×D4—141no—
164C5×D4C2×AGL(1,𝔽7)11yescharacteristic
165C5×D4C2×AGL(1,𝔽7)11yescharacteristic, complemented
166C2×C4×C5—71no—
167C2×C4×C5—141no—
168C5×C8—71no—
169C5×C8C2×AGL(1,𝔽7)11yescharacteristic, complemented
170C5×C7C3×AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
171AGL(1,ℤ/8)—71noSylow
172C2×C3×C5—71no—
173C2×C3×C5—141no—
174C2×C3×C5—281no—
175C2×C3×C5—281no—
176C2×C3×C5—281no—
177C22×C7—21no—
178C22×C7—21no—
179C2×D7—21no—
180C2×D7—42no—
181C2×D7C3×C5×D411yescharacteristic
182C4×C7C23×C3×C511yescharacteristic
183Dic7—21no—
184Dic7C3×C5×D411yescharacteristic
185C23×C3—141no—
186C3×Q8—71no—
187C3×D4—71no—
188C3×D4—71no—
189C3×D4—71no—
190C3×D4—141no—
191C3×D4—141no—
192C2×C3×C4—71no—
193C2×C3×C4—141no—
194C3×C8—71no—
195C3×C8—71no—
196ΓL(1,𝔽8)C5×AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
197C22×C5—21no—
198C22×C5—21no—
199C22×C5—71no—
200C22×C5—141no—
201C22×C5—282no—
202C4×C5—71no—
203C4×C5—141no—
204C4×C5C22×AGL(1,𝔽7)11yescharacteristic
205#41—71no—
206C2×D4—71no—
207SD16—71no—
208SD16—71no—
209D8—71no—
210D8C5×AGL(1,𝔽7)11yescharacteristic, complemented
211M16—71no—
212C3×C5—71noHall
213C2×C7—41no—
214C2×C7—41no—
215C2×C7C2×C3×C5×D411yescharacteristic
216D7—21no—
217D7—41no—
218C22×C3—71no—
219C22×C3—141no—
220C22×C3—141no—
221C22×C3—141no—
222C22×C3—282no—
223C3×C4—71no—
224C3×C4—71no—
225C3×C4—141no—
226C2×C5—41no—
227C2×C5—41no—
228C2×C5—141no—
229C2×C5—281no—
230C2×C5D4×AGL(1,𝔽7)11yescharacteristic
231C23—141no—
232Q8—71no—
233D4—71no—
234D4—141no—
235D4—141no—
236D4C2×C5×AGL(1,𝔽7)11yescharacteristic
237D4C2×C5×AGL(1,𝔽7)11yescharacteristic, complemented
238C2×C4—71no—
239C2×C4—141no—
240C8—71no—
241C8C2×C5×AGL(1,𝔽7)11yescharacteristic, complemented
242C7C3×C5×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
243C2×C3—71no—
244C2×C3—141no—
245C2×C3—281no—
246C2×C3—281no—
247C2×C3—281no—
248C5#6474⋊C311yescharacteristic, direct product factor, Sylow
249C22—21no—
250C22—21no—
251C22—71no—
252C22—141no—
253C22—282no—
254C4—71no—
255C4—141no—
256C4C22×C5×AGL(1,𝔽7)11yescharacteristic
257C3—71noSylow
258C2—41no—
259C2—41no—
260C2—141no—
261C2—281no—
262C2C5×D4×AGL(1,𝔽7)11yescharacteristic
subgroupG/H
centerC2×C5D4×AGL(1,𝔽7)
commutator, cosocleC4×C7C23×C3×C5
FrattiniC4C22×C5×AGL(1,𝔽7)
FittingC5×C7×D8C2×C3
socleC2×C5×C7C2×C3×D4
hypercenterC5×D8AGL(1,𝔽7)
hypocenterC7C3×C5×AGL(1,ℤ/8)
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C5 ◁ C4×C5 ◁ C5×D8

Complements

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

NH ≅ G/Ntype
1. C5×#1159⋊C3C2semidirect
2. C5×D4×AGL(1,𝔽7)C2semidirect
3. C5×D8×ΓL(1,𝔽8)C2semidirect
4. C5×#1142⋊C3C2semidirect
5. C5×#1141⋊C3C2semidirect
6. C5×#1132⋊C3C2semidirect
7. C5×#1131⋊C3C2semidirect
8. C5×#6474C3semidirect
11. C5×D4×ΓL(1,𝔽8)C22semidirect
17. C5×ΓL(1,𝔽8)⋊Q8C22semidirect
18. C5×C8×ΓL(1,𝔽8)C22semidirect
19. C5×ΓL(1,𝔽8)⋊C8C22semidirect
20. #6474⋊C3C5direct (N × H)
21. C5×#1159C2×C3semidirect
22. C5×D4×D7C2×C3semidirect
23. C5×C7×D8C2×C3semidirect
24. C5×#1142C2×C3semidirect
25. C5×#1141C2×C3semidirect
26. C5×#1132C2×C3semidirect
27. C5×#1131C2×C3semidirect
37. #1159⋊C3C2×C5semidirect
38. D4×AGL(1,𝔽7)C2×C5semidirect
39. D8×ΓL(1,𝔽8)C2×C5semidirect
40. #1142⋊C3C2×C5semidirect
41. #1141⋊C3C2×C5semidirect
42. #1132⋊C3C2×C5semidirect
43. #1131⋊C3C2×C5semidirect
46. C5×C7×D4C22×C3semidirect
52. C5×Dic14C22×C3semidirect
53. C5×C7×C8C22×C3semidirect
54. C5×C7⋊C8C22×C3semidirect
62. #6474C3×C5semidirect
70. D4×ΓL(1,𝔽8)C22×C5semidirect
76. ΓL(1,𝔽8)⋊Q8C22×C5semidirect
77. C8×ΓL(1,𝔽8)C22×C5semidirect
78. ΓL(1,𝔽8)⋊C8C22×C5semidirect
99. #1159C2×C3×C5semidirect
100. D4×D7C2×C3×C5semidirect
101. C7×D8C2×C3×C5semidirect
102. #1142C2×C3×C5semidirect
103. #1141C2×C3×C5semidirect
104. #1132C2×C3×C5semidirect
105. #1131C2×C3×C5semidirect
106. C5×ΓL(1,𝔽8)AGL(1,ℤ/8)semidirect
121. C5×D8AGL(1,𝔽7)semidirect
138. C7×D4C22×C3×C5semidirect
144. Dic14C22×C3×C5semidirect
145. C7×C8C22×C3×C5semidirect
146. C7⋊C8C22×C3×C5semidirect
165. C5×D4C2×AGL(1,𝔽7)semidirect
169. C5×C8C2×AGL(1,𝔽7)semidirect
170. C5×C7C3×AGL(1,ℤ/8)semidirect
196. ΓL(1,𝔽8)C5×AGL(1,ℤ/8)semidirect
210. D8C5×AGL(1,𝔽7)semidirect
237. D4C2×C5×AGL(1,𝔽7)semidirect
241. C8C2×C5×AGL(1,𝔽7)semidirect
242. C7C3×C5×AGL(1,ℤ/8)semidirect
248. C5#6474⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#1159⋊C32
C5×D4×AGL(1,𝔽7)2
C5×D8×ΓL(1,𝔽8)2
C5×#1142⋊C32
C5×#1141⋊C32
C5×#1132⋊C32
C5×#1131⋊C32
C5×#64743
#6474⋊C35
C3×C5×AGL(1,ℤ/8)7