Finite Group Databasebeta

← C3×#91249 #13086397

Subgroups — C3×#91249 #13086397

Order 1440.

292 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 80 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
1C3×D5×D12C211yesmaximal, characteristic, complemented
2C3×#6773C211yesmaximal, characteristic, complemented
3C3×D120C211yesmaximal, characteristic, complemented
4C3×C5×#204C211yesmaximal, characteristic, complemented
5C32×#676C211yesmaximal, characteristic, complemented
6C3×#6660C211yesmaximal, characteristic, complemented
7C3×#6654C211yesmaximal, characteristic, complemented
8C3×#4060—31nomaximal
9#91249C311yesmaximal, characteristic, direct product factor
10C2×C3×D3×D5—21no—
11C3×D60C2211yescharacteristic
12C3×D60C2211yescharacteristic, complemented
13C3×C5×D12C2211yescharacteristic
14C3×C5×Dic6C2211yescharacteristic, complemented
15C32×C4×D5C2211yescharacteristic
16C3×#1203—21no—
17C3×#1202—21no—
18C3×#1200—21no—
19C32×C5×C8C2211yescharacteristic, complemented
20C32×#168C2211yescharacteristic, complemented
21C3×#908—51nomaximal, Hall
22C3×#713—31no—
23C3×D4×D5—31no—
24D5×D12C2×C311yescharacteristic, complemented
25#6773C2×C311yescharacteristic, complemented
26C3×C5×SD16—31no—
27D120C2×C311yescharacteristic, complemented
28C5×#204C2×C311yescharacteristic, complemented
29C3×#688—31no—
30C3×#686—31no—
31C3×D40—31no—
32C3×#676—21no—
33C3×#676D311yescharacteristic, complemented
34C3×#676C2×C311yescharacteristic, complemented
35#6660C2×C311yescharacteristic, complemented
36#6654C2×C311yescharacteristic, complemented
37C2×C3×D15—21no—
38C2×C3×D15—21no—
39C2×C3×C5×D3—21no—
40C2×C32×D5D411yescharacteristic
41C3×D3×D5—42no—
42C32×C4×C5C2311yescharacteristic
43C3×C5×Dic3—21no—
44C32×Dic5D411yescharacteristic
45#4060—31noHall
46C3×#235—51no—
47C2×C3×D12—51no—
48C32×M16—51no—
49C3×D24—51no—
50C3×D24—51no—
51C3×#204—51no—
52C3×#204D511yescharacteristic, complemented
53C22×C3×D5—61no—
54C2×D3×D5—21no—
55C3×C5×Q8—31no—
56C3×C5×D4—31no—
57D60C22×C311yescharacteristic
58D60C22×C311yescharacteristic, complemented
59C5×D12C22×C311yescharacteristic
60C5×Dic6C22×C311yescharacteristic, complemented
61C3×C22⋊D5—61no—
62C3×D20—31no—
63C3×D20—31no—
64C3×D20—61no—
65C3×C4×D5—21no—
66C3×C4×D5—61no—
67C3×C4×D5C2×D311yescharacteristic
68C3×C4×D5C22×C311yescharacteristic
69#1203—21no—
70#1202—21no—
71#1200—21no—
72C3×C5×C8—21no—
73C3×C5×C8C2×D311yescharacteristic, complemented
74C3×C5×C8C22×C311yescharacteristic, complemented
75C3×#168—21no—
76C3×#168C2×D311yescharacteristic, complemented
77C3×#168C22×C311yescharacteristic, complemented
78C3×AGL(1,ℤ/8)—151no—
79#908—51no—
80C2×C32×C5C2×D411yescharacteristic
81C3×D15—41no—
82C3×D15—41no—
83C3×C5×D3—41no—
84C32×D5—21no—
85#713—31no—
86D4×D5—31no—
87C5×SD16—31no—
88#688—31no—
89#686—31no—
90D40—31no—
91#676C3×D311yescharacteristic, complemented
92C22×C3×D3—101no—
93C2×C32×C4—51no—
94C3×#67—101no—
95C3×D12—51no—
96C3×D12—51no—
97C3×D12—101no—
98C3×D12C2×D511yescharacteristic
99C3×C4×D3—101no—
100C3×Dic6C2×D511yescharacteristic, complemented
101C32×C8—51no—
102C32×C8C2×D511yescharacteristic, complemented
103C22×C3×C5—61no—
104C2×D15—21no—
105C2×D15—21no—
106C2×C5×D3—21no—
107C2×C3×D5—21no—
108C2×C3×D5—61no—
109C2×C3×D5—61no—
110C2×C3×D5—122no—
111C2×C3×D5D1211yescharacteristic
112C2×C3×D5C3×D411yescharacteristic
113D3×D5—42no—
114C3×C4×C5—21no—
115C3×C4×C5—61no—
116C3×C4×C5C22×D311yescharacteristic
117C3×C4×C5C23×C311yescharacteristic
118C3×Dic5—21no—
119C3×Dic5D1211yescharacteristic
120C3×Dic5C3×D411yescharacteristic
121C5×Dic3—21no—
122C3×#41—151no—
123C2×C3×D4—151no—
124#235—51no—
125C2×D12—51no—
126C3×SD16—31no—
127C3×SD16—151no—
128C3×D8—151no—
129C3×D8—151no—
130C3×M16—51no—
131C3×M16—51no—
132C3×M16—101no—
133D24—51no—
134D24—51no—
135#204—51no—
136#204C3×D511yescharacteristic, complemented
137C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
138C22×D5—61no—
139C5×Q8—31no—
140C5×D4—31no—
141C22⋊D5—61no—
142D20—31no—
143D20—31no—
144D20—61no—
145C4×D5—61no—
146C4×D5C2×C3×D311yescharacteristic
147C5×C8C2×C3×D311yescharacteristic, complemented
148#168C2×C3×D311yescharacteristic, complemented
149C22×C32—51no—
150C2×C3×D3—21no—
151C2×C3×D3—101no—
152C2×C3×D3—101no—
153C2×C3×D3—202no—
154C32×C4—51no—
155C32×C4C22×D511yescharacteristic
156C3×Dic3—21no—
157AGL(1,ℤ/8)—151noSylow
158C2×C3×C5—21no—
159C2×C3×C5—121no—
160C2×C3×C5C2×D1211yescharacteristic
161C2×C3×C5C2×C3×D411yescharacteristic
162D15—41no—
163D15—41no—
164C3×D5—21no—
165C3×D5—21no—
166C3×D5—41no—
167C3×D5—121no—
168C3×D5—121no—
169C5×D3—41no—
170C23×C3—301no—
171C22×D3—101no—
172C3×Q8—31no—
173C3×D4—31no—
174C3×D4—151no—
175C3×D4—151no—
176C3×D4—301no—
177C3×D4—301no—
178C2×C3×C4—51no—
179C2×C3×C4—51no—
180C2×C3×C4—101no—
181C2×C3×C4—301no—
182#67—101no—
183D12—51no—
184D12—51no—
185D12—101no—
186D12C2×C3×D511yescharacteristic
187C4×D3—101no—
188Dic6C2×C3×D511yescharacteristic, complemented
189C3×C8—21no—
190C3×C8—51no—
191C3×C8—51no—
192C3×C8—101no—
193C3×C8D3×D511yescharacteristic, complemented
194C3×C8C2×C3×D511yescharacteristic, complemented
195C22×C5—61no—
196C2×D5—61no—
197C2×D5—61no—
198C2×D5—122no—
199C2×D5C3×D1211yescharacteristic
200C4×C5—61no—
201C4×C5C22×C3×D311yescharacteristic
202Dic5C3×D1211yescharacteristic
203C2×C32—101no—
204C2×C32D4×D511yescharacteristic
205C3×D3—41no—
206C3×D3—201no—
207C3×D3—201no—
208#41—151no—
209C2×D4—151no—
210SD16—31no—
211SD16—151no—
212D8—151no—
213D8—151no—
214M16—51no—
215C3×C5—21no—
216C3×C5#90811yescharacteristic, complemented
217C3×C5C3×AGL(1,ℤ/8)11yescharacteristic, complemented
218C22×C3—51no—
219C22×C3—51no—
220C22×C3—61no—
221C22×C3—101no—
222C22×C3—301no—
223C22×C3—301no—
224C22×C3—602no—
225C2×D3—21no—
226C2×D3—101no—
227C2×D3—101no—
228C2×D3—202no—
229C3×C4—21no—
230C3×C4—51no—
231C3×C4—51no—
232C3×C4—61no—
233C3×C4—101no—
234C3×C4C2×D3×D511yescharacteristic
235C3×C4C22×C3×D511yescharacteristic
236Dic3—21no—
237C2×C5—121no—
238C2×C5C2×C3×D1211yescharacteristic
239D5—21no—
240D5—121no—
241D5—121no—
242C32#406011yescharacteristic, complemented, Sylow
243C23—301no—
244Q8—31no—
245D4—31no—
246D4—151no—
247D4—151no—
248D4—301no—
249D4—301no—
250C2×C4—51no—
251C2×C4—301no—
252C8—51no—
253C8C3×D3×D511yescharacteristic, complemented
254C2×C3—21no—
255C2×C3—101no—
256C2×C3—101no—
257C2×C3—121no—
258C2×C3—201no—
259C2×C3—601no—
260C2×C3—601no—
261C2×C3D5×D1211yescharacteristic
262C2×C3C3×D4×D511yescharacteristic
263D3—41no—
264D3—201no—
265D3—201no—
266C5C3×#90811yescharacteristic, complemented, Sylow
267C22—51no—
268C22—61no—
269C22—301no—
270C22—301no—
271C22—602no—
272C4—51no—
273C4—61no—
274C4C2×C3×D3×D511yescharacteristic
275C3—21no—
276C3#9124911yescharacteristic, direct product factor
277C3C3×#406011yescharacteristic, complemented
278C2—101no—
279C2—121no—
280C2—601no—
281C2—601no—
282C2C3×D5×D1211yescharacteristic
subgroupG/H
centerC2×C3D5×D12
commutator, cosocleC3×C4×C5C23×C3
FrattiniC4C2×C3×D3×D5
FittingC32×C5×C8C22
socleC2×C32×C5C2×D4
hypercenterC3×C8D3×D5
hypocenterC3×C5C3×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8

Complements

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

NH ≅ G/Ntype
1. C3×D5×D12C2semidirect
2. C3×#6773C2semidirect
3. C3×D120C2semidirect
4. C3×C5×#204C2semidirect
5. C32×#676C2semidirect
6. C3×#6660C2semidirect
7. C3×#6654C2semidirect
9. #91249C3direct (N × H)
12. C3×D60C22semidirect
14. C3×C5×Dic6C22semidirect
19. C32×C5×C8C22semidirect
20. C32×#168C22semidirect
24. D5×D12C2×C3semidirect
25. #6773C2×C3semidirect
27. D120C2×C3semidirect
28. C5×#204C2×C3semidirect
33. C3×#676D3semidirect
34. C3×#676C2×C3semidirect
35. #6660C2×C3semidirect
36. #6654C2×C3semidirect
52. C3×#204D5semidirect
58. D60C22×C3semidirect
60. C5×Dic6C22×C3semidirect
73. C3×C5×C8C2×D3semidirect
74. C3×C5×C8C22×C3semidirect
76. C3×#168C2×D3semidirect
77. C3×#168C22×C3semidirect
91. #676C3×D3semidirect
100. C3×Dic6C2×D5semidirect
102. C32×C8C2×D5semidirect
136. #204C3×D5semidirect
137. C32×C5AGL(1,ℤ/8)semidirect
147. C5×C8C2×C3×D3semidirect
148. #168C2×C3×D3semidirect
188. Dic6C2×C3×D5semidirect
193. C3×C8D3×D5semidirect
194. C3×C8C2×C3×D5semidirect
216. C3×C5#908semidirect
217. C3×C5C3×AGL(1,ℤ/8)semidirect
242. C32#4060semidirect
253. C8C3×D3×D5semidirect
266. C5C3×#908semidirect
276. C3#91249direct (N × H)
277. C3C3×#4060semidirect

Maximal subgroups

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

subgroupindex
C3×D5×D122
C3×#67732
C3×D1202
C3×C5×#2042
C32×#6762
C3×#66602
C3×#66542
C3×#40603
#912493
C3×#9085