Finite Group Databasebeta

← C3×#91494 #13086642

Subgroups — C3×#91494 #13086642

Order 1440.

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

normal subgroups only

This group has 294 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×C5×D3×D4C211yesmaximal, characteristic, complemented
2C3×#6765C211yesmaximal, characteristic, complemented
3C3×#6716C211yesmaximal, characteristic, complemented
4C32×#686C211yesmaximal, characteristic, complemented
5C3×#6659C211yesmaximal, characteristic, complemented
6C3×#6652C211yesmaximal, characteristic, complemented
7C3×#6650C211yesmaximal, characteristic, complemented
8C3×#4078—31nomaximal
9#91494C311yesmaximal, characteristic, direct product factor
10C22×C3×C5×D3—21no—
11C32×C5×D4C2211yescharacteristic
12C3×Dic30C2211yescharacteristic, complemented
13C3×C5×#67—21no—
14C3×C5×D12C2211yescharacteristic
15C3×C4×C5×D3C2211yescharacteristic
16C32×D20C2211yescharacteristic, complemented
17C3×#1201—21no—
18C3×D5×Dic3—21no—
19C3×C3⋊#168C2211yescharacteristic, complemented
20C32×#168C2211yescharacteristic, complemented
21C3×#911—51nomaximal, Hall
22C2×C3×C5×D4—31no—
23C5×D3×D4C2×C311yescharacteristic, complemented
24C3×#709—31no—
25#6765C2×C311yescharacteristic, complemented
26#6716C2×C311yescharacteristic, complemented
27C3×#687—31no—
28C3×#687—31no—
29C3×#686—21no—
30C3×#686—31no—
31C3×#686D311yescharacteristic, complemented
32C3×#686C2×C311yescharacteristic, complemented
33C3×#681—31no—
34#6659C2×C311yescharacteristic, complemented
35#6652C2×C311yescharacteristic, complemented
36#6650C2×C311yescharacteristic, complemented
37C22×C32×C5—21no—
38C2×C3×C5×D3—21no—
39C2×C3×C5×D3—42no—
40C2×C3×C5×D3D411yescharacteristic
41C2×C32×D5—21no—
42C32×C4×C5C2311yescharacteristic
43C3×Dic15—21no—
44C3×C5×Dic3D411yescharacteristic
45#4078—31noHall
46C3×#237—51no—
47C3×D3×D4D511yescharacteristic, complemented
48C32×D8—51no—
49C3×#214—51no—
50C3×#213—51no—
51C3×#204—51no—
52C3×ΓL(1,𝔽25)—51no—
53C23×C3×C5—61no—
54C22×C5×D3—21no—
55C3×C5×D4—21no—
56C3×C5×D4—31no—
57C3×C5×D4—61no—
58C3×C5×D4C2×D311yescharacteristic
59C3×C5×D4C22×C311yescharacteristic
60C2×C3×C4×C5—31no—
61Dic30C22×C311yescharacteristic, complemented
62C5×#67—21no—
63C5×D12C22×C311yescharacteristic
64C4×C5×D3C22×C311yescharacteristic
65C3×C22⋊D5—61no—
66C3×D20—21no—
67C3×D20C2×D311yescharacteristic, complemented
68C3×D20C22×C311yescharacteristic, complemented
69C3×C4×D5—61no—
70C3×Dic10—31no—
71#1201—21no—
72D5×Dic3—21no—
73C3⋊#168C22×C311yescharacteristic, complemented
74C3×#168—21no—
75C3×#168—31no—
76C3×#168C2×D311yescharacteristic, complemented
77C3×#168C22×C311yescharacteristic, complemented
78C3×AGL(1,ℤ/8)—151no—
79#911—51no—
80C2×C32×C5—41no—
81C2×C32×C5C2×D411yescharacteristic
82C3×C5×D3—21no—
83C3×C5×D3—41no—
84C32×D5—41no—
85C2×C5×D4—31no—
86#709—31no—
87#687—31no—
88#687—31no—
89#686—31no—
90#686C3×D311yescharacteristic, complemented
91#681—31no—
92C22×C3×D3—21no—
93C32×D4—51no—
94C32×D4C2×D511yescharacteristic
95C3×#67—21no—
96C3×#67—101no—
97C2×C3×Dic3—101no—
98C3×D12C2×D511yescharacteristic
99C3×C4×D3C2×D511yescharacteristic
100C3×Dic6—51no—
101C32×C8—51no—
102C3×C3⋊C8—51no—
103C22×C3×C5—21no—
104C22×C3×C5—21no—
105C22×C3×C5—31no—
106C22×C3×C5—41no—
107C22×C3×C5—61no—
108C22×C3×C5—122no—
109C2×C5×D3—21no—
110C2×C5×D3—42no—
111C2×C5×D3C3×D411yescharacteristic
112C2×C3×D5—21no—
113C2×C3×D5—21no—
114C2×C3×D5—41no—
115C3×C4×C5—21no—
116C3×C4×C5—31no—
117C3×C4×C5C22×D311yescharacteristic
118C3×C4×C5C23×C311yescharacteristic
119Dic15—21no—
120C3×Dic5—61no—
121C5×Dic3C3×D411yescharacteristic
122C3×#41—151no—
123C2×C3×D4—31no—
124#237—51no—
125D3×D4C3×D511yescharacteristic, complemented
126C3×SD16—151no—
127C3×SD16—151no—
128C3×D8—51no—
129C3×D8—51no—
130C3×D8—101no—
131C3×D8—151no—
132C3×M16—151no—
133#214—51no—
134#213—51no—
135#204—51no—
136ΓL(1,𝔽25)—51no—
137C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
138C23×C5—61no—
139C5×D4—31no—
140C5×D4—61no—
141C5×D4C2×C3×D311yescharacteristic
142C2×C4×C5—31no—
143C22⋊D5—61no—
144D20C2×C3×D311yescharacteristic, complemented
145C4×D5—61no—
146Dic10—31no—
147#168—31no—
148#168C2×C3×D311yescharacteristic, complemented
149C22×C32—21no—
150C22×C32—101no—
151C2×C3×D3—21no—
152C2×C3×D3—42no—
153C2×C3×D3C22⋊D511yescharacteristic
154C32×C4C22×D511yescharacteristic
155C3×Dic3—101no—
156C3×Dic3C22⋊D511yescharacteristic
157AGL(1,ℤ/8)—151noSylow
158C2×C3×C5—21no—
159C2×C3×C5—41no—
160C2×C3×C5—41no—
161C2×C3×C5—61no—
162C2×C3×C5—81no—
163C2×C3×C5—121no—
164C2×C3×C5D3×D411yescharacteristic
165C2×C3×C5C2×C3×D411yescharacteristic
166C3×D5—41no—
167C3×D5—41no—
168C3×D5—81no—
169C5×D3—21no—
170C5×D3—41no—
171C23×C3—61no—
172C22×D3—21no—
173C3×Q8—151no—
174C3×D4—21no—
175C3×D4—31no—
176C3×D4—51no—
177C3×D4—51no—
178C3×D4—61no—
179C3×D4—101no—
180C3×D4—301no—
181C3×D4D3×D511yescharacteristic
182C3×D4C2×C3×D511yescharacteristic
183C2×C3×C4—31no—
184C2×C3×C4—301no—
185#67—21no—
186#67—101no—
187C2×Dic3—101no—
188D12C2×C3×D511yescharacteristic
189C4×D3C2×C3×D511yescharacteristic
190Dic6—51no—
191C3×C8—51no—
192C3×C8—51no—
193C3×C8—101no—
194C3×C8—151no—
195C3⋊C8—51no—
196C22×C5—21no—
197C22×C5—31no—
198C22×C5—61no—
199C22×C5—122no—
200C2×D5—21no—
201C4×C5—31no—
202C4×C5C22×C3×D311yescharacteristic
203Dic5—61no—
204C2×C32—41no—
205C2×C32—201no—
206C2×C32C2×C22⋊D511yescharacteristic
207C3×D3—21no—
208C3×D3—41no—
209#41—151no—
210C2×D4—31no—
211SD16—151no—
212SD16—151no—
213D8—51no—
214D8—151no—
215M16—151no—
216C3×C5—21no—
217C3×C5#91111yescharacteristic, complemented
218C3×C5C3×AGL(1,ℤ/8)11yescharacteristic, complemented
219C22×C3—21no—
220C22×C3—21no—
221C22×C3—31no—
222C22×C3—41no—
223C22×C3—61no—
224C22×C3—101no—
225C22×C3—101no—
226C22×C3—122no—
227C22×C3—201no—
228C2×D3—21no—
229C2×D3—42no—
230C2×D3C3×C22⋊D511yescharacteristic
231C3×C4—21no—
232C3×C4—31no—
233C3×C4—301no—
234C3×C4C2×D3×D511yescharacteristic
235C3×C4C22×C3×D511yescharacteristic
236Dic3—101no—
237Dic3C3×C22⋊D511yescharacteristic
238C2×C5—41no—
239C2×C5—61no—
240C2×C5—121no—
241C2×C5C3×D3×D411yescharacteristic
242D5—41no—
243C32#407811yescharacteristic, complemented, Sylow
244C23—61no—
245Q8—151no—
246D4—31no—
247D4—51no—
248D4—61no—
249D4—301no—
250D4C3×D3×D511yescharacteristic
251C2×C4—31no—
252C2×C4—301no—
253C8—51no—
254C8—151no—
255C2×C3—21no—
256C2×C3—41no—
257C2×C3—41no—
258C2×C3—61no—
259C2×C3—81no—
260C2×C3—121no—
261C2×C3—201no—
262C2×C3—201no—
263C2×C3—401no—
264C2×C3D3×C22⋊D511yescharacteristic
265C2×C3C2×C3×C22⋊D511yescharacteristic
266D3—21no—
267D3—41no—
268C5C3×#91111yescharacteristic, complemented, Sylow
269C22—21no—
270C22—31no—
271C22—61no—
272C22—101no—
273C22—122no—
274C4—31no—
275C4—301no—
276C4C2×C3×D3×D511yescharacteristic
277C3—21no—
278C3#9149411yescharacteristic, direct product factor
279C3C3×#407811yescharacteristic, complemented
280C2—41no—
281C2—61no—
282C2—121no—
283C2—201no—
284C2C3×D3×C22⋊D511yescharacteristic
subgroupG/H
centerC2×C3D3×C22⋊D5
commutator, cosocleC3×C4×C5C23×C3
FrattiniC4C2×C3×D3×D5
FittingC32×C5×D4C22
socleC2×C32×C5C2×D4
hypercenterC3×D4D3×D5
hypocenterC3×C5C3×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
1. C3×C5×D3×D4C2semidirect
2. C3×#6765C2semidirect
3. C3×#6716C2semidirect
4. C32×#686C2semidirect
5. C3×#6659C2semidirect
6. C3×#6652C2semidirect
7. C3×#6650C2semidirect
9. #91494C3direct (N × H)
12. C3×Dic30C22semidirect
16. C32×D20C22semidirect
19. C3×C3⋊#168C22semidirect
20. C32×#168C22semidirect
23. C5×D3×D4C2×C3semidirect
25. #6765C2×C3semidirect
26. #6716C2×C3semidirect
31. C3×#686D3semidirect
32. C3×#686C2×C3semidirect
34. #6659C2×C3semidirect
35. #6652C2×C3semidirect
36. #6650C2×C3semidirect
47. C3×D3×D4D5semidirect
61. Dic30C22×C3semidirect
67. C3×D20C2×D3semidirect
68. C3×D20C22×C3semidirect
73. C3⋊#168C22×C3semidirect
76. C3×#168C2×D3semidirect
77. C3×#168C22×C3semidirect
90. #686C3×D3semidirect
125. D3×D4C3×D5semidirect
137. C32×C5AGL(1,ℤ/8)semidirect
144. D20C2×C3×D3semidirect
148. #168C2×C3×D3semidirect
217. C3×C5#911semidirect
218. C3×C5C3×AGL(1,ℤ/8)semidirect
243. C32#4078semidirect
268. C5C3×#911semidirect
278. C3#91494direct (N × H)
279. C3C3×#4078semidirect

Maximal subgroups

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

subgroupindex
C3×C5×D3×D42
C3×#67652
C3×#67162
C32×#6862
C3×#66592
C3×#66522
C3×#66502
C3×#40783
#914943
C3×#9115