Finite Group Databasebeta

← #13087594

Subgroups — #13087594

Order 1440.

292 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 63 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
1#10624226C211yesmaximal, characteristic
2#10624219C211yesmaximal, characteristic, complemented
3C5×#3744C211yesmaximal, characteristic, complemented
4C3×#6706C211yesmaximal, characteristic, complemented
5C3×C8×D15C211yesmaximal, characteristic, complemented
6#10623824C211yesmaximal, characteristic, complemented
7#10623823C211yesmaximal, characteristic, complemented
8#91255—31nomaximal
9D5×#204—31nomaximal
10C2×(C3⋊D3)⋊D5—21no—
11C5×C4⋊(C3⋊D3)C2211yescharacteristic
12C5×#634C2211yescharacteristic, complemented
13C3×D60C2211yescharacteristic
14C3×C4×D15C2211yescharacteristic
15C3×Dic30C2211yescharacteristic, complemented
16#67158—21no—
17#67156—21no—
18#67154—21no—
19C32×C5×C8C2211yescharacteristic, complemented
20C3×C3⋊#168C2211yescharacteristic, complemented
21D3×#204—51nomaximal, Hall
22#6777—31no—
23D5×D12—31no—
24D15⋊Q8—31no—
25D5×Dic6—31no—
26#6706D311yescharacteristic, complemented
27C8×D15D311yescharacteristic, complemented
28C5×#204—31no—
29C5×#204—31no—
30C5×#204—61no—
31C3×#677—31no—
32C3×C8×D5—31no—
33#6660—31no—
34#6659—31no—
35#6656—31no—
36#6655—31no—
37C2×C5×C3⋊D3—21no—
38C2×C3×D15—21no—
39C2×C3×D15D411yescharacteristic
40(C3⋊D3)⋊D5—42no—
41C32×C4×C5C2311yescharacteristic
42C5×C3⋊Dic3—21no—
43C3×Dic15—21no—
44C3×Dic15D411yescharacteristic
45D5×SD16—91noHall
46D3×D12—51no—
47D3×Dic6—51no—
48#3744D511yescharacteristic, complemented
49C3×#204—51no—
50C3×D3×C8—51no—
51#3717—51no—
52#3716—51no—
53C2×D3×D5—61no—
54C2×D3×D5—61no—
55D60C2×D311yescharacteristic
56C4×D15C2×D311yescharacteristic
57Dic30C2×D311yescharacteristic, complemented
58C5×D12—31no—
59C5×D12—31no—
60C5×D12—61no—
61C5×Dic6—31no—
62C5×Dic6—31no—
63C5×Dic6—61no—
64C3×D20—31no—
65C3×C4×D5—31no—
66C3×Dic10—31no—
67#1204—61no—
68#1204—61no—
69#1203—61no—
70#1201—61no—
71#1200—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
81C5×C3⋊D3—41no—
82C3×D15—41no—
83C3×D15SD1612yescomplemented
84Q8×D5—91no—
85D4×D5—91no—
86C5×SD16—91no—
87#688—91no—
88#687—91no—
89#677—31no—
90C8×D5—31no—
91C2×D32—101no—
92C4⋊(C3⋊D3)C2×D511yescharacteristic
93#634C2×D511yescharacteristic, complemented
94C3×D12—51no—
95C3×C4×D3—51no—
96C3×Dic6—51no—
97#627—101no—
98#626—101no—
99D3×Dic3—101no—
100C32×C8C2×D511yescharacteristic, complemented
101C3×C3⋊C8—51no—
102C2×D15—21no—
103C2×D15D1211yescharacteristic
104C2×C5×D3—61no—
105C2×C5×D3—61no—
106C2×C5×D3—121no—
107C2×C3×D5—31no—
108C2×C3×D5—61no—
109D3×D5—122no—
110D3×D5—122no—
111C3×C4×C5—21no—
112C3×C4×C5C22×D311yescharacteristic
113C3×C4×C5C22×D311yescharacteristic
114Dic15—21no—
115Dic15D1211yescharacteristic
116C3×Dic5—31no—
117C3×Dic5—61no—
118C5×Dic3—61no—
119C5×Dic3—61no—
120C5×Dic3—121no—
121D3×Q8—151no—
122D3×D4—151no—
123C2×D12—151no—
124C2×Dic6—151no—
125C3×SD16—151no—
126C2×C3×C8—151no—
127#215—151no—
128#214—151no—
129#204—31no—
130#204—31no—
131#204—51no—
132#204—61no—
133#204—151no—
134#204—151no—
135D3×C8—51no—
136C32×C5C2×SD1611yescharacteristic, complemented, Hall
137C22×D5—181no—
138C5×Q8—91no—
139C5×D4—91no—
140C22⋊D5—181no—
141D20—31no—
142C4×D5—31no—
143C4×D5—181no—
144Dic10—31no—
145Dic10—181no—
146C5×C8D3211yescharacteristic, complemented
147#168—31no—
148C2×C3⋊D3—21no—
149C2×C3×D3—51no—
150C2×C3×D3—101no—
151D32—202no—
152C32×C4C22×D511yescharacteristic
153C3⋊Dic3—21no—
154C3×Dic3—51no—
155C3×Dic3—101no—
156C2×SD16—451noSylow
157C2×C3×C5—21no—
158C2×C3×C5C2×D1211yescharacteristic
159C2×C3×C5D3×D411yescharacteristic
160D15—41no—
161D15#20412yescomplemented
162C3×D5—32no—
163C3×D5—121no—
164C5×D3—121no—
165C5×D3—121no—
166C5×D3—241no—
167C22×D3—301no—
168C22×D3—301no—
169C3×Q8—151no—
170C3×D4—151no—
171C2×C3×C4—151no—
172#67—301no—
173C2×Dic3—301no—
174D12—31no—
175D12—31no—
176D12—51no—
177D12—61no—
178D12—301no—
179C4×D3—51no—
180C4×D3—301no—
181Dic6—31no—
182Dic6—31no—
183Dic6—51no—
184Dic6—61no—
185Dic6—301no—
186Dic6—301no—
187C3×C8—21no—
188C3×C8—151no—
189C3×C8D3×D511yescharacteristic, complemented
190C3×C8D3×D511yescharacteristic, complemented
191C3⋊C8—51no—
192C22×C5—181no—
193C2×D5—31no—
194C2×D5—61no—
195C2×D5—362no—
196C4×C5—181no—
197C4×C5C2×D3211yescharacteristic
198Dic5—31no—
199Dic5—61no—
200C2×C32D4×D511yescharacteristic
201C3⋊D3—41no—
202C3×D3—52no—
203C3×D3—201no—
204C2×Q8—451no—
205C2×D4—451no—
206SD16—91no—
207SD16—151no—
208SD16—451no—
209SD16—451no—
210C2×C8—151no—
211C3×C5—21no—
212C3×C5C2×#20411yescharacteristic, complemented
213C3×C5D3×SD1611yescharacteristic, complemented
214C22×C3—151no—
215C22×C3—301no—
216C2×D3—51no—
217C2×D3—61no—
218C2×D3—61no—
219C2×D3—101no—
220C2×D3—121no—
221C2×D3—602no—
222C2×D3—602no—
223C3×C4—21no—
224C3×C4—151no—
225C3×C4—301no—
226C3×C4C2×D3×D511yescharacteristic
227C3×C4C2×D3×D511yescharacteristic
228Dic3—51no—
229Dic3—61no—
230Dic3—61no—
231Dic3—101no—
232Dic3—121no—
233C2×C5—361no—
234C2×C5D3×D1211yescharacteristic
235D5—32no—
236D5—121no—
237C32D5×SD1611yescharacteristic, complemented, Sylow
238C23—901no—
239Q8—91no—
240Q8—151no—
241Q8—901no—
242D4—91no—
243D4—151no—
244D4—901no—
245C2×C4—151no—
246C2×C4—901no—
247C8—151no—
248C8(C3⋊D3)⋊D511yescharacteristic, complemented
249C2×C3—21no—
250C2×C3—152no—
251C2×C3—601no—
252C2×C3D5×D1211yescharacteristic
253C2×C3#677711yescharacteristic
254D3—52no—
255D3—121no—
256D3—121no—
257D3—201no—
258D3—241no—
259C5D3×#20411yescharacteristic, complemented, Sylow
260C22—151no—
261C22—181no—
262C22—301no—
263C22—1802no—
264C4—151no—
265C4—181no—
266C4—301no—
267C4C2×(C3⋊D3)⋊D511yescharacteristic
268C3—21no—
269C3D5×#20411yescharacteristic, complemented
270C3#9125511yescharacteristic, complemented
271C2—152no—
272C2—361no—
273C2—601no—
274C2#1062422611yescharacteristic
subgroupG/H
centerC2#10624226
commutator, cosocleC32×C4×C5C23
FrattiniC4C2×(C3⋊D3)⋊D5
FittingC32×C5×C8C22
socleC2×C32×C5C2×D4
prime radical#10624226C2
hypercenterC8(C3⋊D3)⋊D5
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. #10624219C2semidirect
3. C5×#3744C2semidirect
4. C3×#6706C2semidirect
5. C3×C8×D15C2semidirect
6. #10623824C2semidirect
7. #10623823C2semidirect
12. C5×#634C22semidirect
15. C3×Dic30C22semidirect
19. C32×C5×C8C22semidirect
20. C3×C3⋊#168C22semidirect
26. #6706D3semidirect
27. C8×D15D3semidirect
48. #3744D5semidirect
57. Dic30C2×D3semidirect
74. C3×C5×C8C2×D3semidirect
75. C3×C5×C8C2×D3semidirect
76. C3⋊#168C2×D3semidirect
83. C3×D15SD16semidirect
93. #634C2×D5semidirect
100. C32×C8C2×D5semidirect
136. C32×C5C2×SD16semidirect
146. C5×C8D32semidirect
161. D15#204semidirect
189. C3×C8D3×D5semidirect
190. C3×C8D3×D5semidirect
212. C3×C5C2×#204semidirect
213. C3×C5D3×SD16semidirect
237. C32D5×SD16semidirect
248. C8(C3⋊D3)⋊D5semidirect
259. C5D3×#204semidirect
269. C3D5×#204semidirect
270. C3#91255semidirect

Maximal subgroups

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

subgroupindex
#106242262
#106242192
C5×#37442
C3×#67062
C3×C8×D152
#106238242
#106238232
#912553
D5×#2043
D3×#2045