Finite Group Databasebeta

← D15×#204 #13087260

Subgroups — D15×#204 #13087260

Order 1440.

292 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 66 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
1D12×D15C211yesmaximal, characteristic
2Dic6×D15C211yesmaximal, characteristic, complemented
3#10623900C211yesmaximal, characteristic, complemented
4C3×C8×D15C211yesmaximal, characteristic, complemented
5C3×C5×#204C211yesmaximal, characteristic, complemented
6#10623806C211yesmaximal, characteristic, complemented
7#10623805C211yesmaximal, characteristic, complemented
8SD16×D15—31nomaximal
9D5×#204—31nomaximal
10C2×D3×D15—21no—
11C4⋊(C3⋊D15)C2211yescharacteristic
12#67180C2211yescharacteristic, complemented
13C3×C4×D15C2211yescharacteristic
14C3×C5×D12C2211yescharacteristic
15C3×C5×Dic6C2211yescharacteristic, complemented
16#67153—21no—
17#67152—21no—
18Dic3×D15—21no—
19C32×C5×C8C2211yescharacteristic, complemented
20C3×C3⋊#168C2211yescharacteristic, complemented
21D3×#204—51nomaximal, Hall
22Q8×D15—31no—
23D4×D15—31no—
24D5×D12—31no—
25D5×Dic6—31no—
26C3×C5×SD16—31no—
27#6717—31no—
28#6716—31no—
29#6706—31no—
30#6706—31no—
31#6706—61no—
32C8×D15D311yescharacteristic, complemented
33C5×#204D311yescharacteristic, complemented
34C3×C8×D5—31no—
35#6660—31no—
36#6659—31no—
37C2×C3⋊D15—21no—
38C2×C3×D15D411yescharacteristic
39C2×C3×C5×D3—21no—
40D3×D15—42no—
41C32×C4×C5C2311yescharacteristic
42C3⋊Dic15—21no—
43C3×Dic15D411yescharacteristic
44C3×C5×Dic3—21no—
45D5×SD16—91noHall
46D3×D12—51no—
47D3×Dic6—51no—
48#3744—51no—
49C3×#204D511yescharacteristic, complemented
50C3×D3×C8—51no—
51#3717—51no—
52#3716—51no—
53C22×D15—61no—
54C2×D3×D5—61no—
55C3×C5×Q8—31no—
56C3×C5×D4—31no—
57#1220—61no—
58D60—31no—
59D60—31no—
60D60—61no—
61C4×D15—61no—
62C4×D15C2×D311yescharacteristic
63Dic30—31no—
64Dic30—31no—
65Dic30—61no—
66Dic30—61no—
67C5×D12C2×D311yescharacteristic
68C5×Dic6C2×D311yescharacteristic, complemented
69C3×C4×D5—31no—
70#1204—61no—
71#1203—61no—
72D5×Dic3—61no—
73C3×C5×C8—21no—
74C3×C5×C8C2×D311yescharacteristic, complemented
75C3×C5×C8C2×D311yescharacteristic, complemented
76C3⋊#168C2×D311yescharacteristic, complemented
77C3×#168—31no—
78D3×SD16—151no—
79C2×#204—151no—
80C2×C32×C5C2×D411yescharacteristic
81C3⋊D15—41no—
82C3×D15SD1612yescomplemented
83C3×C5×D3—41no—
84Q8×D5—91no—
85D4×D5—91no—
86C5×SD16—31no—
87#688—91no—
88#687—91no—
89#677—91no—
90C8×D5—31no—
91C2×D32—101no—
92C4⋊(C3⋊D3)—51no—
93#634—51no—
94C3×D12C2×D511yescharacteristic
95C3×C4×D3—51no—
96C3×Dic6C2×D511yescharacteristic, complemented
97#627—101no—
98#626—101no—
99D3×Dic3—101no—
100C32×C8C2×D511yescharacteristic, complemented
101C3×C3⋊C8—51no—
102C22×C3×C5—61no—
103C2×D15—61no—
104C2×D15—61no—
105C2×D15—121no—
106C2×D15—122no—
107C2×D15D1211yescharacteristic
108C2×C5×D3—21no—
109C2×C3×D5—31no—
110D3×D5—122no—
111C3×C4×C5—21no—
112C3×C4×C5—61no—
113C3×C4×C5C22×D311yescharacteristic
114C3×C4×C5C22×D311yescharacteristic
115Dic15—61no—
116Dic15—61no—
117Dic15—121no—
118Dic15D1211yescharacteristic
119C3×Dic5—31no—
120C5×Dic3—21no—
121D3×Q8—151no—
122D3×D4—151no—
123C2×D12—151no—
124C2×Dic6—151no—
125C3×SD16—31no—
126C2×C3×C8—151no—
127#215—151no—
128#214—151no—
129#204—151no—
130#204—151no—
131#204—151no—
132#204—151no—
133#204—301no—
134#204D1511yescharacteristic, direct product factor
135D3×C8—51no—
136C32×C5C2×SD1611yescharacteristic, complemented, Hall
137C22×D5—181no—
138C5×Q8—31no—
139C5×D4—31no—
140C22⋊D5—181no—
141D20—91no—
142C4×D5—31no—
143C4×D5—181no—
144Dic10—91no—
145Dic10—181no—
146C5×C8D3211yescharacteristic, complemented
147#168—31no—
148C2×C3⋊D3—101no—
149C2×C3×D3—21no—
150C2×C3×D3—51no—
151D32—202no—
152C32×C4C22×D511yescharacteristic
153C3⋊Dic3—101no—
154C3×Dic3—21no—
155C3×Dic3—51no—
156C2×SD16—451noSylow
157C2×C3×C5—21no—
158C2×C3×C5—121no—
159C2×C3×C5C2×D1211yescharacteristic
160C2×C3×C5D3×D411yescharacteristic
161D15—121no—
162D15—121no—
163D15—241no—
164D15#20412yesdirect product factor
165C3×D5—32no—
166C5×D3—41no—
167C22×D3—301no—
168C22×D3—301no—
169C3×Q8—31no—
170C3×D4—31no—
171C2×C3×C4—151no—
172#67—301no—
173C2×Dic3—301no—
174D12—151no—
175D12—151no—
176D12—301no—
177D12—301no—
178D12C2×D1511yescharacteristic
179C4×D3—51no—
180C4×D3—301no—
181Dic6—151no—
182Dic6—151no—
183Dic6—301no—
184Dic6—301no—
185Dic6—301no—
186Dic6C2×D1511yescharacteristic, complemented
187C3×C8—21no—
188C3×C8—151no—
189C3×C8D3×D511yescharacteristic, complemented
190C3×C8C2×D1511yescharacteristic, complemented
191C3⋊C8—51no—
192C22×C5—61no—
193C2×D5—31no—
194C2×D5—181no—
195C2×D5—362no—
196C4×C5—61no—
197C4×C5C2×D3211yescharacteristic
198Dic5—31no—
199Dic5—181no—
200C2×C32D4×D511yescharacteristic
201C3⋊D3—201no—
202C3×D3—41no—
203C3×D3—52no—
204C2×Q8—451no—
205C2×D4—451no—
206SD16—31no—
207SD16—451no—
208SD16—451no—
209SD16—451no—
210C2×C8—151no—
211C3×C5—21no—
212C3×C5C2×#20411yescharacteristic, complemented
213C3×C5D3×SD1611yescharacteristic, complemented
214C22×C3—61no—
215C22×C3—151no—
216C2×D3—21no—
217C2×D3—51no—
218C2×D3—301no—
219C2×D3—301no—
220C2×D3—601no—
221C2×D3—602no—
222C2×D3—602no—
223C3×C4—21no—
224C3×C4—61no—
225C3×C4—151no—
226C3×C4C2×D3×D511yescharacteristic
227C3×C4C22×D1511yescharacteristic
228Dic3—21no—
229Dic3—51no—
230Dic3—301no—
231Dic3—301no—
232Dic3—601no—
233C2×C5—121no—
234C2×C5D3×D1211yescharacteristic
235D5—32no—
236D5—361no—
237C32D5×SD1611yescharacteristic, complemented, Sylow
238C23—901no—
239Q8—31no—
240Q8—451no—
241Q8—901no—
242D4—31no—
243D4—451no—
244D4—901no—
245C2×C4—151no—
246C2×C4—901no—
247C8—151no—
248C8D3×D1511yescharacteristic, complemented
249C2×C3—21no—
250C2×C3—121no—
251C2×C3—152no—
252C2×C3D5×D1211yescharacteristic
253C2×C3D4×D1511yescharacteristic
254D3—41no—
255D3—52no—
256D3—601no—
257D3—601no—
258D3—1201no—
259C5D3×#20411yescharacteristic, complemented, Sylow
260C22—61no—
261C22—151no—
262C22—901no—
263C22—1802no—
264C4—61no—
265C4—151no—
266C4—901no—
267C4C2×D3×D1511yescharacteristic
268C3—21no—
269C3D5×#20411yescharacteristic, complemented
270C3SD16×D1511yescharacteristic, complemented
271C2—121no—
272C2—152no—
273C2—1801no—
274C2D12×D1511yescharacteristic
subgroupG/H
centerC2D12×D15
commutator, cosocleC32×C4×C5C23
FrattiniC4C2×D3×D15
FittingC32×C5×C8C22
socleC2×C32×C5C2×D4
prime radicalD12×D15C2
hypercenterC8D3×D15
hypocenterC32×C5C2×SD16
LCSG ▷ C32×C4×C5 ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
2. Dic6×D15C2semidirect
3. #10623900C2semidirect
4. C3×C8×D15C2semidirect
5. C3×C5×#204C2semidirect
6. #10623806C2semidirect
7. #10623805C2semidirect
12. #67180C22semidirect
15. C3×C5×Dic6C22semidirect
19. C32×C5×C8C22semidirect
20. C3×C3⋊#168C22semidirect
32. C8×D15D3semidirect
33. C5×#204D3semidirect
49. C3×#204D5semidirect
68. C5×Dic6C2×D3semidirect
74. C3×C5×C8C2×D3semidirect
75. C3×C5×C8C2×D3semidirect
76. C3⋊#168C2×D3semidirect
82. C3×D15SD16semidirect
96. C3×Dic6C2×D5semidirect
100. C32×C8C2×D5semidirect
134. #204D15direct (N × H)
136. C32×C5C2×SD16semidirect
146. C5×C8D32semidirect
164. D15#204direct (N × H)
186. Dic6C2×D15semidirect
189. C3×C8D3×D5semidirect
190. C3×C8C2×D15semidirect
212. C3×C5C2×#204semidirect
213. C3×C5D3×SD16semidirect
237. C32D5×SD16semidirect
248. C8D3×D15semidirect
259. C5D3×#204semidirect
269. C3D5×#204semidirect
270. C3SD16×D15semidirect

Maximal subgroups

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

subgroupindex
D12×D152
Dic6×D152
#106239002
C3×C8×D152
C3×C5×#2042
#106238062
#106238052
SD16×D153
D5×#2043
D3×#2045