Finite Group Databasebeta

← #13085770

Subgroups — #13085770

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
1C5×D3×D12C211yesmaximal, characteristic, complemented
2C3×#6769C211yesmaximal, characteristic, complemented
3#10623821C211yesmaximal, characteristic, complemented
4#10623798C211yesmaximal, characteristic, complemented
5#10623796C211yesmaximal, characteristic, complemented
6#10623761C211yesmaximal, characteristic, complemented
7C3×#6656C211yesmaximal, characteristic, complemented
8#91496—31nomaximal
9#91299—31nomaximal
10C2×C5×D32—21no—
11C5×C4⋊(C3⋊D3)C2211yescharacteristic
12C3×D60C2211yescharacteristic, complemented
13C3×C5×D12C2211yescharacteristic
14C3×C4×C5×D3C2211yescharacteristic
15C32×Dic10C2211yescharacteristic, complemented
16C5×#626—21no—
17C3×#1203—21no—
18C3×#1200—21no—
19C3⋊(C3⋊#168)C2211yescharacteristic, complemented
20C3×C3⋊#168C2211yescharacteristic, complemented
21#63939—51nomaximal, Hall
22C5×D3×D4—31no—
23C2×C5×D12—31no—
24C3×#709—31no—
25#6769D311yescharacteristic, complemented
26#6715—31no—
27#6710—31no—
28C3×#687—31no—
29#6656—31no—
30#6656—31no—
31#6656—61no—
32#6656D311yescharacteristic, complemented
33#6654—31no—
34#6652—31no—
35#6652—31no—
36#6650—31no—
37C2×C5×C3⋊D3—21no—
38C2×C3×D15—21no—
39C2×C3×C5×D3—21no—
40C2×C3×C5×D3D411yescharacteristic
41C5×D32—42no—
42C32×C4×C5C2311yescharacteristic
43C3×C5×Dic3D411yescharacteristic
44C32×Dic5—21no—
45#4078—91noHall
46C3×#239—51no—
47D3×D12D511yescharacteristic, complemented
48#3755—51no—
49C3×#215—51no—
50#3714—51no—
51#3713—51no—
52#3711—51no—
53C22×C5×D3—61no—
54C22×C5×D3—61no—
55C3×C5×D4—31no—
56C2×C3×C4×C5—31no—
57D60C2×D311yescharacteristic, complemented
58C5×#67—61no—
59C5×D12—31no—
60C5×D12—31no—
61C5×D12—61no—
62C5×D12—61no—
63C5×D12C2×D311yescharacteristic
64C4×C5×D3C2×D311yescharacteristic
65C3×C22⋊D5—61no—
66C3×D20—31no—
67C3×C4×D5—61no—
68C3×Dic10—21no—
69C3×Dic10C2×D311yescharacteristic, complemented
70C3×Dic10C2×D311yescharacteristic, complemented
71#1203—21no—
72#1200—21no—
73C3⋊#168—31no—
74C3⋊#168—31no—
75C3⋊#168—61no—
76C3⋊#168C2×D311yescharacteristic, complemented
77C3×#168—31no—
78#949—151no—
79#914—151no—
80C2×C32×C5C2×D411yescharacteristic
81C5×C3⋊D3—41no—
82C3×D15—41no—
83C3×C5×D3—21no—
84C3×C5×D3—41no—
85C2×C5×D4—91no—
86#709—31no—
87#687—31no—
88#687—91no—
89#686—91no—
90#686—91no—
91#681—91no—
92C2×D32—21no—
93C32×Q8—51no—
94C4⋊(C3⋊D3)C2×D511yescharacteristic
95C3×D12—51no—
96C3×D12—101no—
97C3×D12C2×D511yescharacteristic
98C3×C4×D3—101no—
99C3×C4×D3C2×D511yescharacteristic
100#626—21no—
101C3⋊(C3⋊C8)—51no—
102C3×C3⋊C8—51no—
103C22×C3×C5—31no—
104C22×C3×C5—61no—
105C2×D15—21no—
106C2×C5×D3—21no—
107C2×C5×D3—61no—
108C2×C5×D3—61no—
109C2×C5×D3—121no—
110C2×C5×D3—122no—
111C2×C5×D3—122no—
112C2×C5×D3#6711yescharacteristic
113C2×C3×D5—61no—
114C3×C4×C5—21no—
115C3×C4×C5—31no—
116C3×C4×C5C22×D311yescharacteristic
117C3×C4×C5C22×D311yescharacteristic
118C3×Dic5—21no—
119C3×Dic5—21no—
120C3×Dic5—41no—
121C5×Dic3#6711yescharacteristic
122C3×#41—151no—
123#239—51no—
124D3×D4—31no—
125C2×D12—31no—
126C3×SD16—151no—
127#215—51no—
128#215—151no—
129#215—151no—
130#215—301no—
131#213—151no—
132#213—151no—
133#213—151no—
134#208—151no—
135D24—151no—
136ΓL(1,𝔽25)—151no—
137C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
138C23×C5—181no—
139C5×D4—31no—
140C5×D4—91no—
141C5×D4—181no—
142C2×C4×C5—31no—
143C22⋊D5—61no—
144D20—31no—
145C4×D5—61no—
146Dic10D3211yescharacteristic, complemented
147#168—31no—
148#168—91no—
149C2×C3⋊D3—21no—
150C2×C3×D3—21no—
151C2×C3×D3—101no—
152C2×C3×D3C22⋊D511yescharacteristic
153D32—42no—
154C32×C4—101no—
155C32×C4C22×D511yescharacteristic
156C3×Dic3C22⋊D511yescharacteristic
157AGL(1,ℤ/8)—451noSylow
158C2×C3×C5—21no—
159C2×C3×C5—61no—
160C2×C3×C5—121no—
161C2×C3×C5D3×D411yescharacteristic
162C2×C3×C5C2×#6711yescharacteristic
163D15—41no—
164C3×D5—121no—
165C5×D3—21no—
166C5×D3—41no—
167C5×D3—121no—
168C5×D3—121no—
169C5×D3—241no—
170C22×D3—61no—
171C22×D3—61no—
172C3×Q8—51no—
173C3×Q8—51no—
174C3×Q8—101no—
175C3×D4—31no—
176C3×D4—151no—
177C3×D4—301no—
178C2×C3×C4—31no—
179C2×C3×C4—301no—
180#67—61no—
181D12—31no—
182D12—31no—
183D12—51no—
184D12—61no—
185D12—61no—
186D12—101no—
187D12D3×D511yescharacteristic
188C4×D3—101no—
189C4×D3D3×D511yescharacteristic
190C3×C8—151no—
191C3⋊C8—51no—
192C3⋊C8—151no—
193C3⋊C8—151no—
194C3⋊C8—301no—
195C22×C5—31no—
196C22×C5—61no—
197C22×C5—181no—
198C22×C5—362no—
199C2×D5—61no—
200C4×C5—31no—
201C4×C5C2×D3211yescharacteristic
202Dic5—21no—
203C2×C32C2×C22⋊D511yescharacteristic
204C3⋊D3—41no—
205C3×D3—21no—
206C3×D3—41no—
207C3×D3—201no—
208#41—151no—
209C2×D4—91no—
210SD16—151no—
211SD16—451no—
212D8—451no—
213D8—451no—
214M16—451no—
215C3×C5—21no—
216C3×C5#91411yescharacteristic, complemented
217C3×C5#94911yescharacteristic, complemented
218C22×C3—31no—
219C22×C3—61no—
220C22×C3—301no—
221C2×D3—21no—
222C2×D3—61no—
223C2×D3—61no—
224C2×D3—101no—
225C2×D3—121no—
226C2×D3—122no—
227C2×D3—122no—
228C2×D3#120111yescharacteristic
229C3×C4—21no—
230C3×C4—31no—
231C3×C4—101no—
232C3×C4—101no—
233C3×C4—201no—
234C3×C4C2×D3×D511yescharacteristic
235C3×C4C2×D3×D511yescharacteristic
236Dic3#120111yescharacteristic
237C2×C5—61no—
238C2×C5—121no—
239C2×C5—361no—
240C2×C5D3×#6711yescharacteristic
241D5—121no—
242C32#407811yescharacteristic, complemented, Sylow
243C23—181no—
244Q8—51no—
245D4—31no—
246D4—91no—
247D4—151no—
248D4—181no—
249D4—301no—
250C2×C4—31no—
251C2×C4—301no—
252C8—151no—
253C8—451no—
254C2×C3—21no—
255C2×C3—61no—
256C2×C3—121no—
257C2×C3—601no—
258C2×C3C2×#120111yescharacteristic
259C2×C3D3×C22⋊D511yescharacteristic
260D3—21no—
261D3—41no—
262D3—121no—
263D3—121no—
264D3—201no—
265D3—241no—
266C5#6393911yescharacteristic, complemented, Sylow
267C22—31no—
268C22—61no—
269C22—181no—
270C22—301no—
271C22—362no—
272C4—31no—
273C4—101no—
274C4D32×D511yescharacteristic
275C3—21no—
276C3#9129911yescharacteristic, complemented
277C3#9149611yescharacteristic, complemented
278C2—61no—
279C2—121no—
280C2—361no—
281C2—601no—
282C2D3×#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×D3×D12C2semidirect
2. C3×#6769C2semidirect
3. #10623821C2semidirect
4. #10623798C2semidirect
5. #10623796C2semidirect
6. #10623761C2semidirect
7. C3×#6656C2semidirect
12. C3×D60C22semidirect
15. C32×Dic10C22semidirect
19. C3⋊(C3⋊#168)C22semidirect
20. C3×C3⋊#168C22semidirect
25. #6769D3semidirect
32. #6656D3semidirect
47. D3×D12D5semidirect
57. D60C2×D3semidirect
69. C3×Dic10C2×D3semidirect
70. C3×Dic10C2×D3semidirect
76. C3⋊#168C2×D3semidirect
137. C32×C5AGL(1,ℤ/8)semidirect
146. Dic10D32semidirect
216. C3×C5#914semidirect
217. C3×C5#949semidirect
242. C32#4078semidirect
266. C5#63939semidirect
276. C3#91299semidirect
277. C3#91496semidirect

Maximal subgroups

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

subgroupindex
C5×D3×D122
C3×#67692
#106238212
#106237982
#106237962
#106237612
C3×#66562
#914963
#912993
#639395