Finite Group Databasebeta

← #13085768

Subgroups — #13085768

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
1C5×#3799C211yesmaximal, characteristic, complemented
2C3×D3×D20C211yesmaximal, characteristic, complemented
3#10623821C211yesmaximal, characteristic, complemented
4#10623801C211yesmaximal, characteristic, complemented
5#10623796C211yesmaximal, characteristic, complemented
6#10623757C211yesmaximal, characteristic, complemented
7C3×#6655C211yesmaximal, characteristic, complemented
8#91504—31nomaximal
9#91298—31nomaximal
10C2×C3×D3×D5—21no—
11C5×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
12C3×D60C2211yescharacteristic
13C3×C4×C5×D3C2211yescharacteristic
14C3×C5×Dic6C2211yescharacteristic, complemented
15C32×D20C2211yescharacteristic
16C5×#626—21no—
17C5×#624—21no—
18C3×#1202—21no—
19C3⋊(C3⋊#168)C2211yescharacteristic, complemented
20C3×C3⋊#168C2211yescharacteristic, complemented
21#63934—51nomaximal, Hall
22C5×#239—31no—
23C5×#235—31no—
24C2×C3×D20—31no—
25D3×D20D311yescharacteristic, complemented
26#6717—31no—
27#6710—31no—
28C3×#688—31no—
29#6655—31no—
30#6655D311yescharacteristic, complemented
31#6654—31no—
32#6652—31no—
33#6652—31no—
34#6652—31no—
35#6652—61no—
36#6650—31no—
37C2×C5×C3⋊D3—21no—
38C2×C3×D15—21no—
39C2×C3×C5×D3D411yescharacteristic
40C2×C32×D5—21no—
41C3×D3×D5—42no—
42C32×C4×C5C2311yescharacteristic
43C3×C5×Dic3—21no—
44C3×C5×Dic3D411yescharacteristic
45#4095—91noHall
46C3×D3×D4—51no—
47#3799D511yescharacteristic, complemented
48#3753—51no—
49C3×#214—51no—
50#3715—51no—
51#3714—51no—
52#3711—51no—
53C22×C3×D5—61no—
54C2×D3×D5—21no—
55C3×C5×Q8—31no—
56C2×C3×C4×C5—31no—
57D60C2×D311yescharacteristic
58C5×#67—61no—
59C5×D12—31no—
60C5×D12—31no—
61C5×D12—61no—
62C5×D12—61no—
63C4×C5×D3—61no—
64C4×C5×D3—61no—
65C4×C5×D3C2×D311yescharacteristic
66C5×Dic6C2×D311yescharacteristic, complemented
67C3×D20—21no—
68C3×D20—31no—
69C3×D20—61no—
70C3×D20C2×D311yescharacteristic
71C3×D20C2×D311yescharacteristic
72#1202—21no—
73C3⋊#168—31no—
74C3⋊#168—31no—
75C3⋊#168—61no—
76C3⋊#168C2×D311yescharacteristic, complemented
77C3×#168—31no—
78#932—151no—
79#914—151no—
80C2×C32×C5C2×D411yescharacteristic
81C5×C3⋊D3—41no—
82C3×D15—41no—
83C3×C5×D3—21no—
84C32×D5—41no—
85C5×#41—91no—
86C2×D20—31no—
87#688—31no—
88#688—91no—
89#686—91no—
90#686—91no—
91#681—91no—
92C22×C3×D3—101no—
93C32×D4—51no—
94C4⋊(C3⋊D3)C2×D511yescharacteristic, complemented
95C3×#67—101no—
96C3×D12—51no—
97C3×C4×D3C2×D511yescharacteristic
98C3×Dic6C2×D511yescharacteristic, complemented
99#626—21no—
100#624—21no—
101C3⋊(C3⋊C8)—51no—
102C3×C3⋊C8—51no—
103C22×C3×C5—31no—
104C2×D15—21no—
105C2×C5×D3—61no—
106C2×C5×D3—61no—
107C2×C5×D3—121no—
108C2×C5×D3#6711yescharacteristic
109C2×C3×D5—21no—
110C2×C3×D5—21no—
111C2×C3×D5—41no—
112C2×C3×D5—61no—
113C2×C3×D5—122no—
114D3×D5—42no—
115C3×C4×C5—21no—
116C3×C4×C5—31no—
117C3×C4×C5—61no—
118C3×C4×C5C22×D311yescharacteristic
119C3×C4×C5C22×D311yescharacteristic
120C5×Dic3—21no—
121C5×Dic3#6711yescharacteristic
122C2×C3×D4—151no—
123#239—31no—
124D3×D4—51no—
125#235—31no—
126C3×SD16—151no—
127#215—151no—
128#214—51no—
129#214—151no—
130#213—151no—
131#213—151no—
132#213—151no—
133#213—301no—
134#208—151no—
135D24—151no—
136ΓL(1,𝔽25)—151no—
137C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
138C22×D5—61no—
139C5×Q8—31no—
140C5×D4—91no—
141C5×D4—181no—
142C2×C4×C5—31no—
143C2×C4×C5—181no—
144D20—31no—
145D20—61no—
146D20D3211yescharacteristic
147#168—31no—
148#168—91no—
149C22×C32—101no—
150C2×C3⋊D3—21no—
151C2×C3×D3—101no—
152C2×C3×D3—202no—
153C2×C3×D3C22⋊D511yescharacteristic
154C32×C4C22×D511yescharacteristic
155C3×Dic3—21no—
156C3×Dic3C22⋊D511yescharacteristic
157AGL(1,ℤ/8)—451noSylow
158C2×C3×C5—21no—
159C2×C3×C5—61no—
160C2×C3×C5D3×D411yescharacteristic
161C2×C3×C5C2×#6711yescharacteristic
162D15—41no—
163C3×D5—41no—
164C3×D5—41no—
165C3×D5—81no—
166C3×D5—121no—
167C5×D3—21no—
168C5×D3—121no—
169C5×D3—121no—
170C5×D3—241no—
171C23×C3—301no—
172C22×D3—101no—
173C3×Q8—31no—
174C3×D4—51no—
175C3×D4—51no—
176C3×D4—101no—
177C3×D4—151no—
178C3×D4—301no—
179C2×C3×C4—31no—
180#67—61no—
181#67—101no—
182D12—31no—
183D12—31no—
184D12—51no—
185D12—61no—
186D12—61no—
187C4×D3—61no—
188C4×D3—61no—
189C4×D3D3×D511yescharacteristic
190Dic6D3×D511yescharacteristic, complemented
191C3×C8—151no—
192C3⋊C8—51no—
193C3⋊C8—151no—
194C3⋊C8—151no—
195C3⋊C8—301no—
196C22×C5—31no—
197C22×C5—181no—
198C2×D5—21no—
199C2×D5—61no—
200C2×D5—122no—
201C4×C5—31no—
202C4×C5—61no—
203C4×C5C2×D3211yescharacteristic
204C2×C32—201no—
205C2×C32C2×C22⋊D511yescharacteristic
206C3⋊D3—41no—
207C3×D3—21no—
208C3×D3—201no—
209#41—91no—
210C2×D4—151no—
211SD16—151no—
212SD16—451no—
213D8—451no—
214D8—451no—
215M16—451no—
216C3×C5—21no—
217C3×C5#91411yescharacteristic, complemented
218C3×C5#93211yescharacteristic, complemented
219C22×C3—31no—
220C22×C3—101no—
221C22×C3—101no—
222C22×C3—201no—
223C22×C3—301no—
224C22×C3—602no—
225C2×D3—61no—
226C2×D3—61no—
227C2×D3—101no—
228C2×D3—121no—
229C2×D3—202no—
230C2×D3#120111yescharacteristic
231C3×C4—21no—
232C3×C4—31no—
233C3×C4—61no—
234C3×C4C2×D3×D511yescharacteristic
235C3×C4C2×D3×D511yescharacteristic
236Dic3—21no—
237Dic3#120111yescharacteristic
238C2×C5—61no—
239C2×C5—361no—
240C2×C5D3×#6711yescharacteristic
241D5—41no—
242D5—121no—
243C32#409511yescharacteristic, complemented, Sylow
244C23—301no—
245Q8—31no—
246D4—51no—
247D4—91no—
248D4—151no—
249D4—181no—
250D4—301no—
251C2×C4—31no—
252C2×C4—181no—
253C8—151no—
254C8—451no—
255C2×C3—21no—
256C2×C3—61no—
257C2×C3—201no—
258C2×C3—201no—
259C2×C3—401no—
260C2×C3—601no—
261C2×C3C2×#120111yescharacteristic
262C2×C3D3×C22⋊D511yescharacteristic
263D3—21no—
264D3—121no—
265D3—121no—
266D3—201no—
267D3—241no—
268C5#6393411yescharacteristic, complemented, Sylow
269C22—31no—
270C22—101no—
271C22—181no—
272C22—301no—
273C22—602no—
274C4—31no—
275C4—61no—
276C4D32×D511yescharacteristic
277C3—21no—
278C3#9129811yescharacteristic, complemented
279C3#9150411yescharacteristic, complemented
280C2—61no—
281C2—201no—
282C2—361no—
283C2—601no—
284C2D3×#120111yescharacteristic
subgroupG/H
centerC2D3×#1201
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. C5×#3799C2semidirect
2. C3×D3×D20C2semidirect
3. #10623821C2semidirect
4. #10623801C2semidirect
5. #10623796C2semidirect
6. #10623757C2semidirect
7. C3×#6655C2semidirect
11. C5×C4⋊(C3⋊D3)C22semidirect
14. C3×C5×Dic6C22semidirect
19. C3⋊(C3⋊#168)C22semidirect
20. C3×C3⋊#168C22semidirect
25. D3×D20D3semidirect
30. #6655D3semidirect
47. #3799D5semidirect
66. C5×Dic6C2×D3semidirect
76. C3⋊#168C2×D3semidirect
94. C4⋊(C3⋊D3)C2×D5semidirect
98. C3×Dic6C2×D5semidirect
137. C32×C5AGL(1,ℤ/8)semidirect
190. Dic6D3×D5semidirect
217. C3×C5#914semidirect
218. C3×C5#932semidirect
243. C32#4095semidirect
268. C5#63934semidirect
278. C3#91298semidirect
279. C3#91504semidirect

Maximal subgroups

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

subgroupindex
C5×#37992
C3×D3×D202
#106238212
#106238012
#106237962
#106237572
C3×#66552
#915043
#912983
#639345