Finite Group Databasebeta

← #13085772

Subgroups — #13085772

Order 1440.

292 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 62 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#10624193C211yesmaximal, characteristic, complemented
2C3×D3×D20C211yesmaximal, characteristic, complemented
3#10623822C211yesmaximal, characteristic, complemented
4#10623803C211yesmaximal, characteristic, complemented
5C5×#3711C211yesmaximal, characteristic, complemented
6#10623762C211yesmaximal, characteristic, complemented
7C3×#6653C211yesmaximal, characteristic, complemented
8#91312—31nomaximal
9#91254—31nomaximal
10C2×C3×D3×D5—21no—
11#67180C2211yescharacteristic, complemented
12C3×D60C2211yescharacteristic
13C3×D60C2211yescharacteristic, complemented
14C3×C4×C5×D3C2211yescharacteristic
15C32×D20C2211yescharacteristic
16#67150—21no—
17Dic3×D15—21no—
18C3×#1202—21no—
19C5×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
20C3×C5×C3⋊C8C2211yescharacteristic, complemented
21#63932—51nomaximal, Hall
22#6817—31no—
23C2×C3×D20—31no—
24D3×D20D311yescharacteristic, complemented
25#6765—31no—
26#6706—31no—
27C5×#208—31no—
28C5×ΓL(1,𝔽25)—31no—
29C3×D40—31no—
30#6657—31no—
31#6657—31no—
32#6657—31no—
33#6657—61no—
34#6653—31no—
35#6653—31no—
36#6653D311yescharacteristic, complemented
37C2×C3×D15—21no—
38C2×C3×D15—21no—
39C2×C3×C5×D3D411yescharacteristic
40C2×C32×D5—21no—
41C3×D3×D5—42no—
42C32×C4×C5C2311yescharacteristic
43C3⋊Dic15—21no—
44C3×C5×Dic3D411yescharacteristic
45#4054—91noHall
46C3×D3×D4—51no—
47#3795—51no—
48#3754—51no—
49C3×#213—51no—
50#3716—51no—
51#3713—51no—
52#3711D511yescharacteristic, complemented
53C22×C3×D5—61no—
54C2×D3×D5—21no—
55C2×C3×C4×C5—31no—
56#1220—61no—
57D60C2×D311yescharacteristic
58D60C2×D311yescharacteristic, complemented
59C4×D15—61no—
60Dic30—31no—
61Dic30—31no—
62Dic30—61no—
63C4×C5×D3C2×D311yescharacteristic
64C3×D20—21no—
65C3×D20—31no—
66C3×D20—31no—
67C3×D20—61no—
68C3×D20C2×D311yescharacteristic
69C3×D20C2×D311yescharacteristic
70#1202—21no—
71#1201—61no—
72D5×Dic3—61no—
73C3×C5×C8—31no—
74C5×C3⋊C8—31no—
75C5×C3⋊C8—31no—
76C5×C3⋊C8—61no—
77C5×C3⋊C8C2×D311yescharacteristic, complemented
78#932—151no—
79#911—151no—
80C2×C32×C5C2×D411yescharacteristic
81C3×D15—41no—
82C3×D15—41no—
83C3×C5×D3—21no—
84C32×D5—41no—
85#709—91no—
86C2×D20—31no—
87C5×M16—91no—
88D40—31no—
89D40—91no—
90#677—91no—
91#677—91no—
92C22×C3×D3—101no—
93C32×D4—51no—
94#634—51no—
95C3×#67—101no—
96C3×D12—51no—
97C3×D12—51no—
98C3×C4×D3C2×D511yescharacteristic
99#625—101no—
100D3×Dic3—101no—
101C3⋊(C3⋊C8)C2×D511yescharacteristic, complemented
102C3×C3⋊C8C2×D511yescharacteristic, complemented
103C22×C3×C5—31no—
104C2×D15—21no—
105C2×D15—21no—
106C2×C5×D3#6711yescharacteristic
107C2×C3×D5—21no—
108C2×C3×D5—21no—
109C2×C3×D5—41no—
110C2×C3×D5—61no—
111C2×C3×D5—61no—
112C2×C3×D5—122no—
113D3×D5—42no—
114C3×C4×C5—21no—
115C3×C4×C5—31no—
116C3×C4×C5C22×D311yescharacteristic
117C3×C4×C5C22×D311yescharacteristic
118Dic15—61no—
119Dic15—61no—
120Dic15—121no—
121C5×Dic3#6711yescharacteristic
122C2×C3×D4—151no—
123#237—151no—
124D3×D4—51no—
125#235—151no—
126C3×D8—151no—
127#214—151no—
128#214—151no—
129#214—151no—
130#214—301no—
131#213—51no—
132#213—151no—
133#213—151no—
134#208—31no—
135#204—151no—
136ΓL(1,𝔽25)—31no—
137C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
138C22×D5—61no—
139C2×C4×C5—31no—
140C22⋊D5—181no—
141D20—31no—
142D20—31no—
143D20—61no—
144D20D3211yescharacteristic
145C4×D5—181no—
146Dic10—91no—
147C5×C8—31no—
148C5×C8—91no—
149C22×C32—101no—
150C2×C3×D3—101no—
151C2×C3×D3—101no—
152C2×C3×D3—202no—
153C2×C3×D3D2011yescharacteristic
154C32×C4C22×D511yescharacteristic
155C3⋊Dic3—101no—
156C3×Dic3D2011yescharacteristic
157AGL(1,ℤ/8)—451noSylow
158C2×C3×C5—21no—
159C2×C3×C5—61no—
160C2×C3×C5D3×D411yescharacteristic
161C2×C3×C5C2×#6711yescharacteristic
162D15—41no—
163D15—41no—
164C3×D5—41no—
165C3×D5—41no—
166C3×D5—81no—
167C3×D5—121no—
168C3×D5—121no—
169C5×D3—21no—
170C23×C3—301no—
171C22×D3—101no—
172C3×D4—51no—
173C3×D4—51no—
174C3×D4—101no—
175C3×D4—151no—
176C3×D4—151no—
177C3×D4—301no—
178C2×C3×C4—31no—
179#67—101no—
180#67—301no—
181#67—301no—
182C2×Dic3—301no—
183D12—51no—
184D12—51no—
185C4×D3—301no—
186C4×D3D3×D511yescharacteristic
187Dic6—151no—
188Dic6—151no—
189Dic6—301no—
190C3×C8—31no—
191C3⋊C8—31no—
192C3⋊C8—31no—
193C3⋊C8—61no—
194C3⋊C8D3×D511yescharacteristic, complemented
195C22×C5—31no—
196C2×D5—21no—
197C2×D5—61no—
198C2×D5—61no—
199C2×D5—122no—
200C4×C5—31no—
201C4×C5C2×D3211yescharacteristic
202Dic5—181no—
203C2×C32—201no—
204C2×C32C2×D2011yescharacteristic
205C3×D3—21no—
206C3×D3—201no—
207C3×D3—201no—
208#41—451no—
209C2×D4—151no—
210SD16—451no—
211SD16—451no—
212D8—151no—
213D8—451no—
214M16—91no—
215C3×C5—21no—
216C3×C5#91111yescharacteristic, complemented
217C3×C5#93211yescharacteristic, complemented
218C22×C3—31no—
219C22×C3—101no—
220C22×C3—101no—
221C22×C3—201no—
222C22×C3—301no—
223C22×C3—301no—
224C22×C3—602no—
225C2×D3—101no—
226C2×D3—101no—
227C2×D3—202no—
228C2×D3#120211yescharacteristic
229C3×C4—21no—
230C3×C4—31no—
231C3×C4C2×D3×D511yescharacteristic
232C3×C4C2×D3×D511yescharacteristic
233Dic3—301no—
234Dic3—301no—
235Dic3—601no—
236Dic3#120211yescharacteristic
237C2×C5—61no—
238C2×C5D3×#6711yescharacteristic
239D5—41no—
240D5—121no—
241D5—121no—
242C32#405411yescharacteristic, complemented, Sylow
243C23—301no—
244Q8—451no—
245D4—51no—
246D4—151no—
247D4—151no—
248D4—301no—
249D4—901no—
250C2×C4—31no—
251C2×C4—901no—
252C8—31no—
253C8—91no—
254C2×C3—21no—
255C2×C3—61no—
256C2×C3—201no—
257C2×C3—201no—
258C2×C3—401no—
259C2×C3—601no—
260C2×C3—601no—
261C2×C3D3×D2011yescharacteristic
262C2×C3C2×#120211yescharacteristic
263D3—21no—
264D3—201no—
265D3—201no—
266C5#6393211yescharacteristic, complemented, Sylow
267C22—31no—
268C22—101no—
269C22—301no—
270C22—301no—
271C22—602no—
272C4—31no—
273C4—901no—
274C4D32×D511yescharacteristic
275C3—21no—
276C3#9125411yescharacteristic, complemented
277C3#9131211yescharacteristic, complemented
278C2—61no—
279C2—201no—
280C2—601no—
281C2—601no—
282C2D3×#120211yescharacteristic
subgroupG/H
centerC2D3×#1202
commutator, Fitting, cosocleC32×C4×C5C23
Frattini, hypercenterC4D32×D5
socleC2×C32×C5C2×D4
hypocenterC32×C5AGL(1,ℤ/8)
LCSG ▷ C32×C4×C5 ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #10624193C2semidirect
2. C3×D3×D20C2semidirect
3. #10623822C2semidirect
4. #10623803C2semidirect
5. C5×#3711C2semidirect
6. #10623762C2semidirect
7. C3×#6653C2semidirect
11. #67180C22semidirect
13. C3×D60C22semidirect
19. C5×C3⋊(C3⋊C8)C22semidirect
20. C3×C5×C3⋊C8C22semidirect
24. D3×D20D3semidirect
36. #6653D3semidirect
52. #3711D5semidirect
58. D60C2×D3semidirect
77. C5×C3⋊C8C2×D3semidirect
101. C3⋊(C3⋊C8)C2×D5semidirect
102. C3×C3⋊C8C2×D5semidirect
137. C32×C5AGL(1,ℤ/8)semidirect
194. C3⋊C8D3×D5semidirect
216. C3×C5#911semidirect
217. C3×C5#932semidirect
242. C32#4054semidirect
266. C5#63932semidirect
276. C3#91254semidirect
277. C3#91312semidirect

Maximal subgroups

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

subgroupindex
#106241932
C3×D3×D202
#106238222
#106238032
C5×#37112
#106237622
C3×#66532
#913123
#912543
#639325