Finite Group Databasebeta

← C3×#91500 #13086648

Subgroups — C3×#91500 #13086648

Order 1440.

292 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 76 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×D4×D15C211yesmaximal, characteristic, complemented
2C3×#6768C211yesmaximal, characteristic, complemented
3C3×C5×#213C211yesmaximal, characteristic, complemented
4C32×#687C211yesmaximal, characteristic, complemented
5C3×#6658C211yesmaximal, characteristic, complemented
6C3×#6654C211yesmaximal, characteristic, complemented
7C3×#6651C211yesmaximal, characteristic, complemented
8C3×#4057—31nomaximal
9#91500C311yesmaximal, characteristic, direct product factor
10C22×C3×D15—21no—
11C32×C5×D4C2211yescharacteristic
12C3×#1220—21no—
13C3×D60C2211yescharacteristic
14C3×C4×D15C2211yescharacteristic
15C3×C5×D12C2211yescharacteristic, complemented
16C32×Dic10C2211yescharacteristic, complemented
17C3×#1203—21no—
18C3×D3×Dic5—21no—
19C3×C5×C3⋊C8C2211yescharacteristic, complemented
20C32×#168C2211yescharacteristic, complemented
21C3×#914—51nomaximal, Hall
22D4×D15C2×C311yescharacteristic, complemented
23C3×#711—31no—
24C3×D4×D5—31no—
25#6768C2×C311yescharacteristic, complemented
26C3×C5×D8—31no—
27C5×#213C2×C311yescharacteristic, complemented
28C3×#687—21no—
29C3×#687D311yescharacteristic, complemented
30C3×#687C2×C311yescharacteristic, complemented
31C3×#686—31no—
32C3×#677—31no—
33C3×#676—31no—
34#6658C2×C311yescharacteristic, complemented
35#6654C2×C311yescharacteristic, complemented
36#6651C2×C311yescharacteristic, complemented
37C22×C32×C5—21no—
38C2×C3×D15—21no—
39C2×C3×D15—42no—
40C2×C3×D15D411yescharacteristic
41C2×C3×C5×D3—21no—
42C32×C4×C5C2311yescharacteristic
43C3×Dic15D411yescharacteristic
44C32×Dic5—21no—
45#4057—31noHall
46C3×#239—51no—
47C3×D3×D4—51no—
48C32×SD16—51no—
49C3×#215—51no—
50C3×#213D511yescharacteristic, complemented
51C3×D24—51no—
52C3×ΓL(1,𝔽25)—51no—
53C22×D15—21no—
54C22×C3×D5—61no—
55C3×C5×D4—21no—
56C3×C5×D4—31no—
57C3×C5×D4C2×D311yescharacteristic
58C3×C5×D4C22×C311yescharacteristic
59#1220—21no—
60D60C22×C311yescharacteristic
61C4×D15C22×C311yescharacteristic
62C5×D12C22×C311yescharacteristic, complemented
63C3×C22⋊D5—61no—
64C3×C22⋊D5—61no—
65C2×C3×Dic5—61no—
66C3×D20—31no—
67C3×C4×D5—31no—
68C3×Dic10—21no—
69C3×Dic10C2×D311yescharacteristic, complemented
70C3×Dic10C22×C311yescharacteristic, complemented
71#1203—21no—
72D3×Dic5—21no—
73C3×C5×C8—31no—
74C3×#168—21no—
75C3×#168C2×D311yescharacteristic, complemented
76C3×#168C22×C311yescharacteristic, complemented
77C5×C3⋊C8C22×C311yescharacteristic, complemented
78C3×AGL(1,ℤ/8)—151no—
79#914—51no—
80C2×C32×C5—41no—
81C2×C32×C5C2×D411yescharacteristic
82C3×D15—21no—
83C3×D15—41no—
84C3×C5×D3—41no—
85#711—31no—
86D4×D5—31no—
87C5×D8—31no—
88#687C3×D311yescharacteristic, complemented
89#686—31no—
90#677—31no—
91#676—31no—
92C22×C3×D3—101no—
93C32×Q8—51no—
94C32×D4C2×D511yescharacteristic
95C3×#67—101no—
96C3×D12—51no—
97C3×D12—101no—
98C3×D12C2×D511yescharacteristic, complemented
99C3×C4×D3—51no—
100C3×C4×D3—101no—
101C32×C8—51no—
102C3×C3⋊C8C2×D511yescharacteristic, complemented
103C22×C3×C5—21no—
104C22×C3×C5—21no—
105C22×C3×C5—41no—
106C22×C3×C5—61no—
107C2×D15—21no—
108C2×D15—42no—
109C2×D15C3×D411yescharacteristic
110C2×C5×D3—21no—
111C2×C3×D5—31no—
112C2×C3×D5—61no—
113C2×C3×D5—122no—
114C3×C4×C5—21no—
115C3×C4×C5C22×D311yescharacteristic
116C3×C4×C5C23×C311yescharacteristic
117Dic15C3×D411yescharacteristic
118C3×Dic5—21no—
119C3×Dic5—21no—
120C3×Dic5—31no—
121C3×Dic5—41no—
122C3×#41—151no—
123C2×C3×D4—151no—
124#239—51no—
125D3×D4—51no—
126C3×SD16—51no—
127C3×SD16—51no—
128C3×SD16—101no—
129C3×SD16—151no—
130C3×D8—31no—
131C3×D8—151no—
132C3×M16—151no—
133#215—51no—
134#213C3×D511yescharacteristic, complemented
135D24—51no—
136ΓL(1,𝔽25)—51no—
137C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
138C22×D5—61no—
139C5×D4—31no—
140C5×D4C2×C3×D311yescharacteristic
141C22⋊D5—61no—
142C22⋊D5—61no—
143C2×Dic5—61no—
144D20—31no—
145C4×D5—31no—
146Dic10C2×C3×D311yescharacteristic, complemented
147C5×C8—31no—
148#168C2×C3×D311yescharacteristic, complemented
149C22×C32—21no—
150C2×C3×D3—21no—
151C2×C3×D3—51no—
152C2×C3×D3—101no—
153C2×C3×D3—202no—
154C32×C4—101no—
155C32×C4C22×D511yescharacteristic
156C3×Dic3—51no—
157AGL(1,ℤ/8)—151noSylow
158C2×C3×C5—21no—
159C2×C3×C5—41no—
160C2×C3×C5—41no—
161C2×C3×C5—81no—
162C2×C3×C5—121no—
163C2×C3×C5D3×D411yescharacteristic
164C2×C3×C5C2×C3×D411yescharacteristic
165D15—21no—
166D15—41no—
167C3×D5—61no—
168C3×D5—121no—
169C5×D3—41no—
170C23×C3—301no—
171C22×D3—101no—
172C3×Q8—51no—
173C3×Q8—51no—
174C3×Q8—101no—
175C3×D4—21no—
176C3×D4—31no—
177C3×D4—151no—
178C3×D4—301no—
179C3×D4—301no—
180C3×D4D3×D511yescharacteristic
181C3×D4C2×C3×D511yescharacteristic
182C2×C3×C4—151no—
183C2×C3×C4—301no—
184#67—101no—
185D12—51no—
186D12—101no—
187D12C2×C3×D511yescharacteristic, complemented
188C4×D3—51no—
189C4×D3—101no—
190C3×C8—31no—
191C3×C8—51no—
192C3×C8—51no—
193C3×C8—101no—
194C3⋊C8C2×C3×D511yescharacteristic, complemented
195C22×C5—21no—
196C22×C5—61no—
197C2×D5—31no—
198C2×D5—61no—
199C2×D5—122no—
200C4×C5C22×C3×D311yescharacteristic
201Dic5—21no—
202Dic5—31no—
203C2×C32—41no—
204C2×C32D4×D511yescharacteristic
205C3×D3—41no—
206C3×D3—101no—
207C3×D3—201no—
208#41—151no—
209C2×D4—151no—
210SD16—51no—
211SD16—151no—
212D8—31no—
213D8—151no—
214M16—151no—
215C3×C5—21no—
216C3×C5#91411yescharacteristic, complemented
217C3×C5C3×AGL(1,ℤ/8)11yescharacteristic, complemented
218C22×C3—21no—
219C22×C3—21no—
220C22×C3—41no—
221C22×C3—61no—
222C22×C3—151no—
223C22×C3—301no—
224C22×C3—602no—
225C2×D3—21no—
226C2×D3—51no—
227C2×D3—101no—
228C2×D3—202no—
229C3×C4—21no—
230C3×C4—101no—
231C3×C4—101no—
232C3×C4—151no—
233C3×C4—201no—
234C3×C4C2×D3×D511yescharacteristic
235C3×C4C22×C3×D511yescharacteristic
236Dic3—51no—
237C2×C5—41no—
238C2×C5—121no—
239C2×C5C3×D3×D411yescharacteristic
240D5—61no—
241D5—121no—
242C32#405711yescharacteristic, complemented, Sylow
243C23—301no—
244Q8—51no—
245D4—31no—
246D4—151no—
247D4—301no—
248D4—301no—
249D4C3×D3×D511yescharacteristic
250C2×C4—151no—
251C2×C4—301no—
252C8—31no—
253C8—51no—
254C2×C3—21no—
255C2×C3—41no—
256C2×C3—41no—
257C2×C3—81no—
258C2×C3—121no—
259C2×C3—301no—
260C2×C3—601no—
261C2×C3#679011yescharacteristic
262C2×C3C3×D4×D511yescharacteristic
263D3—41no—
264D3—101no—
265D3—201no—
266C5C3×#91411yescharacteristic, complemented, Sylow
267C22—21no—
268C22—61no—
269C22—151no—
270C22—301no—
271C22—602no—
272C4—101no—
273C4—151no—
274C4C2×C3×D3×D511yescharacteristic
275C3—21no—
276C3#9150011yescharacteristic, direct product factor
277C3C3×#405711yescharacteristic, complemented
278C2—41no—
279C2—121no—
280C2—301no—
281C2—601no—
282C2C3×#679011yescharacteristic
subgroupG/H
centerC2×C3#6790
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×D4×D15C2semidirect
2. C3×#6768C2semidirect
3. C3×C5×#213C2semidirect
4. C32×#687C2semidirect
5. C3×#6658C2semidirect
6. C3×#6654C2semidirect
7. C3×#6651C2semidirect
9. #91500C3direct (N × H)
15. C3×C5×D12C22semidirect
16. C32×Dic10C22semidirect
19. C3×C5×C3⋊C8C22semidirect
20. C32×#168C22semidirect
22. D4×D15C2×C3semidirect
25. #6768C2×C3semidirect
27. C5×#213C2×C3semidirect
29. C3×#687D3semidirect
30. C3×#687C2×C3semidirect
34. #6658C2×C3semidirect
35. #6654C2×C3semidirect
36. #6651C2×C3semidirect
50. C3×#213D5semidirect
62. C5×D12C22×C3semidirect
69. C3×Dic10C2×D3semidirect
70. C3×Dic10C22×C3semidirect
75. C3×#168C2×D3semidirect
76. C3×#168C22×C3semidirect
77. C5×C3⋊C8C22×C3semidirect
88. #687C3×D3semidirect
98. C3×D12C2×D5semidirect
102. C3×C3⋊C8C2×D5semidirect
134. #213C3×D5semidirect
137. C32×C5AGL(1,ℤ/8)semidirect
146. Dic10C2×C3×D3semidirect
148. #168C2×C3×D3semidirect
187. D12C2×C3×D5semidirect
194. C3⋊C8C2×C3×D5semidirect
216. C3×C5#914semidirect
217. C3×C5C3×AGL(1,ℤ/8)semidirect
242. C32#4057semidirect
266. C5C3×#914semidirect
276. C3#91500direct (N × H)
277. C3C3×#4057semidirect

Maximal subgroups

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

subgroupindex
C3×D4×D152
C3×#67682
C3×C5×#2132
C32×#6872
C3×#66582
C3×#66542
C3×#66512
C3×#40573
#915003
C3×#9145