Finite Group Databasebeta

← #13085796

Subgroups — #13085796

Order 1440.

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

normal subgroups only

This group has 294 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
1D3×D60C211yesmaximal, characteristic, complemented
2C5×#3799C211yesmaximal, characteristic, complemented
3#10623821C211yesmaximal, characteristic, complemented
4#10623806C211yesmaximal, characteristic, complemented
5#10623759C211yesmaximal, characteristic, complemented
6C3×#6660C211yesmaximal, characteristic, complemented
7C3×#6650C211yesmaximal, characteristic, complemented
8#91504—31nomaximal
9#91305—31nomaximal
10C2×D3×D15—21no—
11C4⋊(C3⋊D15)C2211yescharacteristic
12C5×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
13C3×D60C2211yescharacteristic
14C3×C4×C5×D3C2211yescharacteristic
15C3×C5×Dic6C2211yescharacteristic, complemented
16#67151—21no—
17C5×#626—21no—
18C5×#624—21no—
19C3×C3⋊#168C2211yescharacteristic, complemented
20C32×#168C2211yescharacteristic, complemented
21#63794—51nomaximal, Hall
22C2×D60—31no—
23C5×#239—31no—
24C5×#235—31no—
25D3×D20—31no—
26#6717—31no—
27C3×#688—31no—
28C3×#681—31no—
29#6660—31no—
30#6660D311yescharacteristic, complemented
31#6654—31no—
32#6654—31no—
33#6654—31no—
34#6654—61no—
35#6652—31no—
36#6650D311yescharacteristic, complemented
37C2×C3⋊D15—21no—
38C2×C5×C3⋊D3—21no—
39C2×C3×D15—21no—
40C2×C3×C5×D3D411yescharacteristic
41D3×D15—42no—
42C32×C4×C5C2311yescharacteristic
43C3×C5×Dic3—21no—
44C3×C5×Dic3D411yescharacteristic
45#4095—91noHall
46D3×D12—51no—
47#3799D511yescharacteristic, complemented
48#3745—51no—
49C3×#204—51no—
50C3×ΓL(1,𝔽25)—51no—
51#3717—51no—
52#3714—51no—
53C22×D15—61no—
54C2×D3×D5—61no—
55C3×C5×Q8—31no—
56C2×C3×C4×C5—31no—
57D60—31no—
58D60—31no—
59D60—61no—
60D60—61no—
61D60C2×D311yescharacteristic
62C5×#67—61no—
63C5×D12—31no—
64C5×D12—31no—
65C5×D12—61no—
66C5×D12—61no—
67C4×C5×D3—61no—
68C4×C5×D3—61no—
69C4×C5×D3C2×D311yescharacteristic
70C5×Dic6C2×D311yescharacteristic, complemented
71C3×D20—31no—
72#1202—61no—
73C3⋊#168C2×D311yescharacteristic, complemented
74C3×#168—21no—
75C3×#168—31no—
76C3×#168C2×D311yescharacteristic, complemented
77C3×#168C2×D311yescharacteristic, complemented
78#914—151no—
79#908—151no—
80C2×C32×C5C2×D411yescharacteristic
81C3⋊D15—41no—
82C5×C3⋊D3—41no—
83C3×D15—41no—
84C3×C5×D3—21no—
85C5×#41—91no—
86C2×D20—91no—
87#688—31no—
88#688—91no—
89#686—91no—
90#686—91no—
91#681—31no—
92C2×D32—101no—
93C4⋊(C3⋊D3)—51no—
94C4⋊(C3⋊D3)C2×D511yescharacteristic, complemented
95C3×D12—51no—
96C3×C4×D3C2×D511yescharacteristic
97C3×Dic6C2×D511yescharacteristic, complemented
98#626—21no—
99#626—101no—
100#624—21no—
101C32×C8—51no—
102C3×C3⋊C8—51no—
103C22×C3×C5—31no—
104C2×D15—21no—
105C2×D15—61no—
106C2×D15—61no—
107C2×D15—121no—
108C2×D15—122no—
109C2×C5×D3—61no—
110C2×C5×D3—61no—
111C2×C5×D3—121no—
112C2×C5×D3D1211yescharacteristic
113C2×C3×D5—61no—
114D3×D5—122no—
115C3×C4×C5—21no—
116C3×C4×C5—31no—
117C3×C4×C5—61no—
118C3×C4×C5C22×D311yescharacteristic
119C3×C4×C5C22×D311yescharacteristic
120C5×Dic3—21no—
121C5×Dic3D1211yescharacteristic
122#239—31no—
123D3×D4—151no—
124#235—31no—
125C2×D12—151no—
126C3×SD16—151no—
127C3×M16—151no—
128#215—151no—
129#213—151no—
130D24—151no—
131D24—151no—
132D24—151no—
133D24—301no—
134#204—51no—
135#204—151no—
136ΓL(1,𝔽25)—51no—
137C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
138C22×D5—181no—
139C5×Q8—31no—
140C5×D4—91no—
141C5×D4—181no—
142C2×C4×C5—31no—
143C2×C4×C5—181no—
144D20—31no—
145D20—91no—
146D20—181no—
147#168—31no—
148#168D3211yescharacteristic, complemented
149C2×C3⋊D3—21no—
150C2×C3⋊D3—101no—
151C2×C3×D3—101no—
152C2×C3×D3C22⋊D511yescharacteristic
153D32—202no—
154C32×C4C22×D511yescharacteristic
155C3×Dic3—21no—
156C3×Dic3C22⋊D511yescharacteristic
157AGL(1,ℤ/8)—451noSylow
158C2×C3×C5—21no—
159C2×C3×C5—61no—
160C2×C3×C5C2×D1211yescharacteristic
161C2×C3×C5D3×D411yescharacteristic
162D15—41no—
163D15—121no—
164D15—121no—
165D15—241no—
166C3×D5—121no—
167C5×D3—21no—
168C5×D3—121no—
169C5×D3—121no—
170C5×D3—241no—
171C22×D3—301no—
172C22×D3—301no—
173C3×Q8—31no—
174C3×D4—151no—
175C2×C3×C4—31no—
176#67—61no—
177#67—301no—
178D12—31no—
179D12—31no—
180D12—51no—
181D12—61no—
182D12—61no—
183D12—151no—
184D12—151no—
185D12—301no—
186D12—301no—
187C4×D3—61no—
188C4×D3—61no—
189C4×D3D3×D511yescharacteristic
190Dic6D3×D511yescharacteristic, complemented
191C3×C8—51no—
192C3×C8—51no—
193C3×C8—101no—
194C3×C8—151no—
195C3⋊C8—51no—
196C22×C5—31no—
197C22×C5—181no—
198C2×D5—61no—
199C2×D5—181no—
200C2×D5—362no—
201C4×C5—31no—
202C4×C5—61no—
203C4×C5C2×D3211yescharacteristic
204C2×C32C2×C22⋊D511yescharacteristic
205C3⋊D3—41no—
206C3⋊D3—201no—
207C3×D3—21no—
208C3×D3—201no—
209#41—91no—
210C2×D4—451no—
211SD16—151no—
212SD16—451no—
213D8—451no—
214D8—451no—
215M16—151no—
216C3×C5—21no—
217C3×C5#90811yescharacteristic, complemented
218C3×C5#91411yescharacteristic, complemented
219C22×C3—31no—
220C22×C3—301no—
221C2×D3—61no—
222C2×D3—61no—
223C2×D3—101no—
224C2×D3—121no—
225C2×D3—301no—
226C2×D3—301no—
227C2×D3—601no—
228C2×D3—602no—
229C2×D3—602no—
230C2×D3#120311yescharacteristic
231C3×C4—21no—
232C3×C4—31no—
233C3×C4—61no—
234C3×C4C2×D3×D511yescharacteristic
235C3×C4C2×D3×D511yescharacteristic
236Dic3—21no—
237Dic3#120311yescharacteristic
238C2×C5—61no—
239C2×C5—361no—
240C2×C5D3×D1211yescharacteristic
241D5—121no—
242D5—361no—
243C32#409511yescharacteristic, complemented, Sylow
244C23—901no—
245Q8—31no—
246D4—91no—
247D4—151no—
248D4—181no—
249D4—451no—
250D4—901no—
251C2×C4—31no—
252C2×C4—181no—
253C8—51no—
254C8—151no—
255C2×C3—21no—
256C2×C3—61no—
257C2×C3—601no—
258C2×C3C2×#120311yescharacteristic
259C2×C3D3×C22⋊D511yescharacteristic
260D3—21no—
261D3—121no—
262D3—121no—
263D3—201no—
264D3—241no—
265D3—601no—
266D3—601no—
267D3—1201no—
268C5#6379411yescharacteristic, complemented, Sylow
269C22—31no—
270C22—181no—
271C22—301no—
272C22—901no—
273C22—1802no—
274C4—31no—
275C4—61no—
276C4D32×D511yescharacteristic
277C3—21no—
278C3#9130511yescharacteristic, complemented
279C3#9150411yescharacteristic, complemented
280C2—61no—
281C2—361no—
282C2—601no—
283C2—1801no—
284C2D3×#120311yescharacteristic
subgroupG/H
centerC2D3×#1203
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. D3×D60C2semidirect
2. C5×#3799C2semidirect
3. #10623821C2semidirect
4. #10623806C2semidirect
5. #10623759C2semidirect
6. C3×#6660C2semidirect
7. C3×#6650C2semidirect
12. C5×C4⋊(C3⋊D3)C22semidirect
15. C3×C5×Dic6C22semidirect
19. C3×C3⋊#168C22semidirect
20. C32×#168C22semidirect
30. #6660D3semidirect
36. #6650D3semidirect
47. #3799D5semidirect
70. C5×Dic6C2×D3semidirect
73. C3⋊#168C2×D3semidirect
76. C3×#168C2×D3semidirect
77. C3×#168C2×D3semidirect
94. C4⋊(C3⋊D3)C2×D5semidirect
97. C3×Dic6C2×D5semidirect
137. C32×C5AGL(1,ℤ/8)semidirect
148. #168D32semidirect
190. Dic6D3×D5semidirect
217. C3×C5#908semidirect
218. C3×C5#914semidirect
243. C32#4095semidirect
268. C5#63794semidirect
278. C3#91305semidirect
279. C3#91504semidirect

Maximal subgroups

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

subgroupindex
D3×D602
C5×#37992
#106238212
#106238062
#106237592
C3×#66602
C3×#66502
#915043
#913053
#637945