Finite Group Databasebeta

← #13087495

Subgroups — #13087495

Order 1440.

312 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 60 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#10624285C211yesmaximal, characteristic, complemented
2#10624205C211yesmaximal, characteristic, complemented
3C3×#6715C211yesmaximal, characteristic, complemented
4C3×C5×#213C211yesmaximal, characteristic, complemented
5#10623805C211yesmaximal, characteristic, complemented
6#10623803C211yesmaximal, characteristic, complemented
7#10623797C211yesmaximal, characteristic, complemented
8#91800—31nomaximal
9#91482—31nomaximal
10C2×D3×D15—21no—
11C32×C5×D4C2211yescharacteristic, complemented
12#67184—21no—
13C2×C3⋊Dic15—21no—
14C4×C3⋊D15C2211yescharacteristic
15#67180C2211yescharacteristic, complemented
16C3×D60C2211yescharacteristic
17C3×C5×D12C2211yescharacteristic
18#67150—21no—
19C3×C3⋊#168C2211yescharacteristic, complemented
20C3×C5×C3⋊C8C2211yescharacteristic, complemented
21#63927—51nomaximal, Hall
22#6819—31no—
23#6819—31no—
24#6819—61no—
25D4×D15—31no—
26#6777—31no—
27C3×C5×D8—31no—
28#6716—31no—
29#6715D311yescharacteristic, complemented
30#6706—31no—
31#6705—31no—
32C5×#213D311yescharacteristic, complemented
33C3×#686—31no—
34#6659—31no—
35#6657—31no—
36#6651—31no—
37C22×C32×C5—21no—
38C2×C3⋊D15D411yescharacteristic
39C2×C3×D15—21no—
40C2×C3×C5×D3—21no—
41D3×D15—42no—
42C32×C4×C5C2311yescharacteristic
43C3⋊Dic15—21no—
44C3⋊Dic15D411yescharacteristic
45#4057—91noHall
46#3830—51no—
47#3802—51no—
48C3×#213—51no—
49C3×#213D511yescharacteristic, complemented
50#3716—51no—
51#3716—51no—
52#3712—51no—
53C22×D15—61no—
54C2×D3×D5—61no—
55C3×C5×D4—21no—
56C3×C5×D4—31no—
57C3×C5×D4C2×D311yescharacteristic, complemented
58C3×C5×D4C2×D311yescharacteristic, complemented
59#1220—61no—
60#1220—61no—
61#1220—61no—
62#1220—62no—
63C2×Dic15—61no—
64C2×Dic15—61no—
65C2×Dic15—62no—
66D60C2×D311yescharacteristic
67C4×D15—31no—
68C4×D15—31no—
69C4×D15—61no—
70Dic30—31no—
71Dic30—31no—
72Dic30—61no—
73C5×D12C2×D311yescharacteristic
74C3×D20—31no—
75#1201—61no—
76C3×C5×C8—31no—
77C3⋊#168C2×D311yescharacteristic, complemented
78C3×#168—31no—
79C5×C3⋊C8C2×D311yescharacteristic, complemented
80#911—151no—
81#911—151no—
82C2×C32×C5—41no—
83C2×C32×C5C2×D411yescharacteristic
84C3⋊D15—21no—
85C3×D15—41no—
86C3×C5×D3—41no—
87#711—91no—
88D4×D5—91no—
89C5×D8—31no—
90#687—91no—
91#686—31no—
92#677—91no—
93#676—91no—
94C2×D32—101no—
95C32×D4C2×D511yescharacteristic, complemented
96#638—101no—
97C2×C3⋊Dic3—101no—
98C4×C3⋊D3—51no—
99#634—51no—
100C3×D12—51no—
101C3×D12C2×D511yescharacteristic
102#625—101no—
103C3×C3⋊C8—51no—
104C3×C3⋊C8C2×D511yescharacteristic, complemented
105C22×C3×C5—22no—
106C22×C3×C5—21no—
107C22×C3×C5—21no—
108C22×C3×C5—61no—
109C2×D15—21no—
110C2×D15—31no—
111C2×D15—31no—
112C2×D15—61no—
113C2×D15—122no—
114C2×C5×D3—21no—
115C2×C3×D5—61no—
116D3×D5—122no—
117C3×C4×C5—21no—
118C3×C4×C5C22×D311yescharacteristic
119C3×C4×C5C22×D311yescharacteristic
120Dic15—31no—
121Dic15—31no—
122Dic15—61no—
123Dic15—62no—
124Dic15—61no—
125Dic15—61no—
126#237—151no—
127#237—151no—
128#237—301no—
129D3×D4—151no—
130D3×D4—151no—
131C3×D8—31no—
132C3×D8—151no—
133#214—151no—
134#214—151no—
135#213—51no—
136#213D1511yescharacteristic, complemented
137#204—151no—
138#204—151no—
139ΓL(1,𝔽25)—151no—
140ΓL(1,𝔽25)—151no—
141C32×C5AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
142C22×D5—181no—
143C5×D4—31no—
144C5×D4D3211yescharacteristic, complemented
145C22⋊D5—181no—
146C22⋊D5—181no—
147C2×Dic5—181no—
148D20—31no—
149C4×D5—91no—
150Dic10—91no—
151C5×C8—31no—
152#168—31no—
153C22×C32—21no—
154C2×C3⋊D3—51no—
155C2×C3×D3—21no—
156C2×C3×D3—101no—
157D32—202no—
158C32×C4C22×D511yescharacteristic
159C3⋊Dic3—51no—
160C3⋊Dic3—101no—
161AGL(1,ℤ/8)—451noSylow
162C2×C3×C5—21no—
163C2×C3×C5—42no—
164C2×C3×C5—41no—
165C2×C3×C5—41no—
166C2×C3×C5—121no—
167C2×C3×C5D3×D411yescharacteristic
168C2×C3×C5D3×D411yescharacteristic
169D15—41no—
170D15—61no—
171D15—61no—
172D15—121no—
173C3×D5—121no—
174C5×D3—41no—
175C22×D3—301no—
176C22×D3—301no—
177C3×D4—21no—
178C3×D4—31no—
179C3×D4—151no—
180C3×D4D3×D511yescharacteristic, complemented
181C3×D4C2×D1511yescharacteristic, complemented
182#67—301no—
183#67—301no—
184#67—301no—
185#67—301no—
186#67—302no—
187C2×Dic3—301no—
188C2×Dic3—301no—
189C2×Dic3—302no—
190D12—51no—
191D12C2×D1511yescharacteristic
192C4×D3—151no—
193C4×D3—151no—
194C4×D3—301no—
195Dic6—151no—
196Dic6—151no—
197Dic6—301no—
198C3×C8—31no—
199C3×C8—151no—
200C3⋊C8—51no—
201C3⋊C8C2×D1511yescharacteristic, complemented
202C22×C5—21no—
203C22×C5—61no—
204C2×D5—61no—
205C2×D5—91no—
206C2×D5—362no—
207C4×C5C2×D3211yescharacteristic
208Dic5—91no—
209Dic5—181no—
210C2×C32—41no—
211C2×C32D4×D511yescharacteristic
212C3⋊D3—101no—
213C3×D3—41no—
214C3×D3—201no—
215#41—451no—
216C2×D4—451no—
217SD16—451no—
218SD16—451no—
219D8—31no—
220D8—151no—
221M16—451no—
222C3×C5—21no—
223C3×C5#91111yescharacteristic, complemented
224C3×C5#91111yescharacteristic, complemented
225C22×C3—22no—
226C22×C3—21no—
227C22×C3—21no—
228C22×C3—61no—
229C22×C3—301no—
230C2×D3—21no—
231C2×D3—101no—
232C2×D3—151no—
233C2×D3—151no—
234C2×D3—301no—
235C2×D3—602no—
236C2×D3—602no—
237C3×C4—21no—
238C3×C4C2×D3×D511yescharacteristic
239C3×C4C22×D1511yescharacteristic
240Dic3—151no—
241Dic3—151no—
242Dic3—301no—
243Dic3—302no—
244Dic3—301no—
245Dic3—301no—
246C2×C5—41no—
247C2×C5—121no—
248C2×C5#381111yescharacteristic
249D5—121no—
250D5—181no—
251C32#405711yescharacteristic, complemented, Sylow
252C23—901no—
253Q8—451no—
254D4—31no—
255D4—151no—
256D4—901no—
257D4—901no—
258D4D3×D1511yescharacteristic, complemented
259C2×C4—451no—
260C2×C4—901no—
261C8—31no—
262C8—151no—
263C2×C3—21no—
264C2×C3—42no—
265C2×C3—41no—
266C2×C3—41no—
267C2×C3—121no—
268C2×C3—601no—
269C2×C3#679011yescharacteristic
270C2×C3D4×D1511yescharacteristic
271D3—41no—
272D3—201no—
273D3—301no—
274D3—301no—
275D3—601no—
276C5#6392711yescharacteristic, complemented, Sylow
277C22—21no—
278C22—61no—
279C22—301no—
280C22—451no—
281C22—1802no—
282C4—451no—
283C4—901no—
284C4C2×D3×D1511yescharacteristic
285C3—21no—
286C3#9148211yescharacteristic, complemented
287C3#9180011yescharacteristic, complemented
288C2—41no—
289C2—121no—
290C2—601no—
291C2—901no—
292C2#1062421811yescharacteristic
subgroupG/H
centerC2#10624218
commutator, cosocleC32×C4×C5C23
FrattiniC4C2×D3×D15
FittingC32×C5×D4C22
socleC2×C32×C5C2×D4
hypercenterD4D3×D15
hypocenterC32×C5AGL(1,ℤ/8)
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
1. #10624285C2semidirect
2. #10624205C2semidirect
3. C3×#6715C2semidirect
4. C3×C5×#213C2semidirect
5. #10623805C2semidirect
6. #10623803C2semidirect
7. #10623797C2semidirect
11. C32×C5×D4C22semidirect
15. #67180C22semidirect
19. C3×C3⋊#168C22semidirect
20. C3×C5×C3⋊C8C22semidirect
29. #6715D3semidirect
32. C5×#213D3semidirect
49. C3×#213D5semidirect
57. C3×C5×D4C2×D3semidirect
58. C3×C5×D4C2×D3semidirect
77. C3⋊#168C2×D3semidirect
79. C5×C3⋊C8C2×D3semidirect
95. C32×D4C2×D5semidirect
104. C3×C3⋊C8C2×D5semidirect
136. #213D15semidirect
141. C32×C5AGL(1,ℤ/8)semidirect
144. C5×D4D32semidirect
180. C3×D4D3×D5semidirect
181. C3×D4C2×D15semidirect
201. C3⋊C8C2×D15semidirect
223. C3×C5#911semidirect
224. C3×C5#911semidirect
251. C32#4057semidirect
258. D4D3×D15semidirect
276. C5#63927semidirect
286. C3#91482semidirect
287. C3#91800semidirect

Maximal subgroups

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

subgroupindex
#106242852
#106242052
C3×#67152
C3×C5×#2132
#106238052
#106238032
#106237972
#918003
#914823
#639275