Finite Group Databasebeta

← #13087730

Subgroups — #13087730

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
1C3×D4×D15C211yesmaximal, characteristic
2#10624219C211yesmaximal, characteristic, complemented
3C5×#3754C211yesmaximal, characteristic, complemented
4C3×#6716C211yesmaximal, characteristic, complemented
5#10623825C211yesmaximal, characteristic, complemented
6#10623823C211yesmaximal, characteristic, complemented
7#10623817C211yesmaximal, characteristic, complemented
8#91487—31nomaximal
9D5×#214—31nomaximal
10C22×C3×D15—21no—
11C32×C5×D4C2211yescharacteristic
12C5×#634C2211yescharacteristic, complemented
13C3×#1220—21no—
14C3×D60C2211yescharacteristic
15C3×C4×D15C2211yescharacteristic
16C3×Dic30C2211yescharacteristic, complemented
17#67158—21no—
18#67154—21no—
19C5×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
20C3×C3⋊#168C2211yescharacteristic, complemented
21D3×#214—51nomaximal, Hall
22D4×D15D311yescharacteristic
23C3×D4×D5—31no—
24D15⋊Q8—31no—
25D5×Dic6—31no—
26#6716D311yescharacteristic, complemented
27C5×#214—31no—
28C5×#214—31no—
29C5×#214—61no—
30C3×#687—31no—
31#6660—31no—
32#6658—31no—
33#6657—31no—
34#6655—31no—
35#6648—31no—
36D5×C3⋊C8—31no—
37C22×C32×C5—21no—
38C2×C3×D15—21no—
39C2×C3×D15—42no—
40C2×C3×D15D411yescharacteristic
41C32×C4×C5C2311yescharacteristic
42C5×C3⋊Dic3—21no—
43C3×Dic15—21no—
44C3×Dic15D411yescharacteristic
45D5×SD16—91noHall
46C3×D3×D4—51no—
47D3×Dic6—51no—
48#3754D511yescharacteristic, complemented
49C3×#214—51no—
50#3716—51no—
51#3715—51no—
52D3×C3⋊C8—51no—
53C22×D15—21no—
54C22×C3×D5—61no—
55C3×C5×D4—21no—
56C3×C5×D4C2×D311yescharacteristic
57C3×C5×D4C2×D311yescharacteristic
58#1220—21no—
59D60C2×D311yescharacteristic
60C4×D15C2×D311yescharacteristic
61Dic30C2×D311yescharacteristic, complemented
62C5×Dic6—31no—
63C5×Dic6—31no—
64C5×Dic6—61no—
65C3×C22⋊D5—61no—
66C3×D20—31no—
67C3×C4×D5—31no—
68C3×Dic10—31no—
69#1204—61no—
70#1204—61no—
71#1200—61no—
72D5×Dic3—61no—
73C3⋊#168C2×D311yescharacteristic, complemented
74C3×#168—31no—
75C5×C3⋊C8—31no—
76C5×C3⋊C8—31no—
77C5×C3⋊C8—61no—
78C2×#214—151no—
79D3×SD16—151no—
80C2×C32×C5—41no—
81C2×C32×C5C2×D411yescharacteristic
82C3×D15—41no—
83C3×D15SD1612yescomplemented
84Q8×D5—91no—
85D4×D5—31no—
86C5×SD16—91no—
87#688—91no—
88#687—31no—
89#677—91no—
90C8×D5—91no—
91C22×C3×D3—101no—
92C32×D4C2×D511yescharacteristic
93#634C2×D511yescharacteristic, complemented
94C3×#67—101no—
95C3×D12—51no—
96C3×C4×D3—51no—
97C3×Dic6—51no—
98#627—101no—
99D3×Dic3—101no—
100C3⋊(C3⋊C8)C2×D511yescharacteristic, complemented
101C3×C3⋊C8—51no—
102C22×C3×C5—21no—
103C22×C3×C5—21no—
104C22×C3×C5—41no—
105C2×D15—21no—
106C2×D15—42no—
107C2×D15#6711yescharacteristic
108C2×C3×D5—31no—
109C2×C3×D5—61no—
110C2×C3×D5—122no—
111C3×C4×C5—21no—
112C3×C4×C5C22×D311yescharacteristic
113C3×C4×C5C22×D311yescharacteristic
114Dic15—21no—
115Dic15#6711yescharacteristic
116C3×Dic5—31no—
117C3×Dic5—61no—
118C5×Dic3—61no—
119C5×Dic3—61no—
120C5×Dic3—121no—
121C2×C3×D4—151no—
122D3×Q8—151no—
123D3×D4—51no—
124C2×Dic6—151no—
125C3×SD16—151no—
126#215—151no—
127#214—31no—
128#214—31no—
129#214—51no—
130#214—61no—
131#214—151no—
132#214—151no—
133C2×C3⋊C8—151no—
134#204—151no—
135D3×C8—151no—
136C32×C5C2×SD1611yescharacteristic, complemented, Hall
137C22×D5—61no—
138C5×Q8—91no—
139C5×D4D3211yescharacteristic
140C22⋊D5—61no—
141D20—31no—
142C4×D5—31no—
143C4×D5—181no—
144Dic10—31no—
145Dic10—181no—
146C5×C8—91no—
147#168—31no—
148C22×C32—21no—
149C2×C3×D3—51no—
150C2×C3×D3—101no—
151C2×C3×D3—202no—
152C32×C4C22×D511yescharacteristic
153C3⋊Dic3—21no—
154C3×Dic3—51no—
155C3×Dic3—101no—
156C2×SD16—451noSylow
157C2×C3×C5—21no—
158C2×C3×C5—41no—
159C2×C3×C5—41no—
160C2×C3×C5—81no—
161C2×C3×C5D3×D411yescharacteristic
162C2×C3×C5C2×#6711yescharacteristic
163D15—41no—
164D15#21412yescomplemented
165C3×D5—32no—
166C3×D5—121no—
167C23×C3—301no—
168C22×D3—101no—
169C3×Q8—151no—
170C3×D4—21no—
171C3×D4—151no—
172C3×D4—301no—
173C3×D4D3×D511yescharacteristic
174C3×D4D3×D511yescharacteristic
175C2×C3×C4—151no—
176#67—101no—
177C2×Dic3—301no—
178D12—51no—
179C4×D3—51no—
180C4×D3—301no—
181Dic6—31no—
182Dic6—31no—
183Dic6—51no—
184Dic6—61no—
185Dic6—301no—
186Dic6—301no—
187C3×C8—151no—
188C3⋊C8—31no—
189C3⋊C8—31no—
190C3⋊C8—51no—
191C3⋊C8—61no—
192C22×C5—21no—
193C2×D5—31no—
194C2×D5—61no—
195C2×D5—122no—
196C4×C5—181no—
197C4×C5C2×D3211yescharacteristic
198Dic5—31no—
199Dic5—61no—
200C2×C32—41no—
201C2×C32D4×D511yescharacteristic
202C3×D3—52no—
203C3×D3—201no—
204C2×Q8—451no—
205C2×D4—151no—
206SD16—91no—
207SD16—151no—
208SD16—451no—
209SD16—451no—
210C2×C8—451no—
211C3×C5—21no—
212C3×C5D3×SD1611yescharacteristic, complemented
213C3×C5C2×#21411yescharacteristic, complemented
214C22×C3—21no—
215C22×C3—21no—
216C22×C3—41no—
217C22×C3—151no—
218C22×C3—301no—
219C22×C3—602no—
220C2×D3—51no—
221C2×D3—101no—
222C2×D3—202no—
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—41no—
234C2×C5D3×#6711yescharacteristic
235D5—32no—
236D5—121no—
237C32D5×SD1611yescharacteristic, complemented, Sylow
238C23—301no—
239Q8—91no—
240Q8—151no—
241Q8—901no—
242D4—151no—
243D4—301no—
244D4(C3⋊D3)⋊D511yescharacteristic
245C2×C4—151no—
246C2×C4—901no—
247C8—91no—
248C8—151no—
249C2×C3—21no—
250C2×C3—41no—
251C2×C3—41no—
252C2×C3—81no—
253C2×C3—152no—
254C2×C3—601no—
255C2×C3D5×#6711yescharacteristic
256C2×C3#679011yescharacteristic
257D3—52no—
258D3—201no—
259C5D3×#21411yescharacteristic, complemented, Sylow
260C22—21no—
261C22—151no—
262C22—301no—
263C22—602no—
264C4—151no—
265C4—181no—
266C4—301no—
267C4C2×(C3⋊D3)⋊D511yescharacteristic
268C3—21no—
269C3D5×#21411yescharacteristic, complemented
270C3#9148711yescharacteristic, complemented
271C2—41no—
272C2—152no—
273C2—601no—
274C2#1062423511yescharacteristic
subgroupG/H
centerC2#10624235
commutator, cosocleC32×C4×C5C23
FrattiniC4C2×(C3⋊D3)⋊D5
FittingC32×C5×D4C22
socleC2×C32×C5C2×D4
prime radicalC3×D4×D15C2
hypercenterD4(C3⋊D3)⋊D5
hypocenterC32×C5C2×SD16
LCSG ▷ C32×C4×C5 ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
2. #10624219C2semidirect
3. C5×#3754C2semidirect
4. C3×#6716C2semidirect
5. #10623825C2semidirect
6. #10623823C2semidirect
7. #10623817C2semidirect
12. C5×#634C22semidirect
16. C3×Dic30C22semidirect
19. C5×C3⋊(C3⋊C8)C22semidirect
20. C3×C3⋊#168C22semidirect
26. #6716D3semidirect
48. #3754D5semidirect
61. Dic30C2×D3semidirect
73. C3⋊#168C2×D3semidirect
83. C3×D15SD16semidirect
93. #634C2×D5semidirect
100. C3⋊(C3⋊C8)C2×D5semidirect
136. C32×C5C2×SD16semidirect
164. D15#214semidirect
212. C3×C5D3×SD16semidirect
213. C3×C5C2×#214semidirect
237. C32D5×SD16semidirect
259. C5D3×#214semidirect
269. C3D5×#214semidirect
270. C3#91487semidirect

Maximal subgroups

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

subgroupindex
C3×D4×D152
#106242192
C5×#37542
C3×#67162
#106238252
#106238232
#106238172
#914873
D5×#2143
D3×#2145