Finite Group Databasebeta

← #13087504

Subgroups — #13087504

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
1C3×D4×D15C211yesmaximal, characteristic, complemented
2#10624201C211yesmaximal, characteristic, complemented
3#10623909C211yesmaximal, characteristic, complemented
4C3×C5×#214C211yesmaximal, characteristic, complemented
5#10623801C211yesmaximal, characteristic, complemented
6#10623799C211yesmaximal, characteristic, complemented
7#10623795C211yesmaximal, characteristic, complemented
8#91803—31nomaximal
9#91491—31nomaximal
10C22×C3×D15—21no—
11C32×C5×D4C2211yescharacteristic
12C4⋊(C3⋊D15)C2211yescharacteristic, complemented
13C3×#1220—21no—
14C3×D60C2211yescharacteristic
15C3×C4×D15C2211yescharacteristic
16C3×C5×Dic6C2211yescharacteristic, complemented
17#67151—21no—
18#67149—21no—
19C3⋊(C3⋊#168)C2211yescharacteristic, complemented
20C3×C5×C3⋊C8C2211yescharacteristic, complemented
21#63934—51nomaximal, Hall
22#6821—31no—
23D4×D15D311yescharacteristic, complemented
24C3×D4×D5—31no—
25#6773—31no—
26C3×C5×SD16—31no—
27#6717—31no—
28#6715—31no—
29#6715—31no—
30#6715—61no—
31D120—31no—
32#6705—31no—
33C5×#214D311yescharacteristic, complemented
34#6655—31no—
35#6653—31no—
36#6649—31no—
37C22×C32×C5—21no—
38C2×C3⋊D15—21no—
39C2×C3×D15—21no—
40C2×C3×D15—42no—
41C2×C3×D15D411yescharacteristic
42C32×C4×C5C2311yescharacteristic
43C3×Dic15D411yescharacteristic
44C3×C5×Dic3—21no—
45#4060—91noHall
46C3×D3×D4—51no—
47#3799—51no—
48#3753—51no—
49C3×#214D511yescharacteristic, complemented
50#3715—51no—
51#3714—51no—
52#3711—51no—
53C22×D15—21no—
54C22×C3×D5—61no—
55C3×C5×Q8—31no—
56C3×C5×D4—21no—
57C3×C5×D4C2×D311yescharacteristic
58C3×C5×D4C2×D311yescharacteristic
59#1220—21no—
60D60—31no—
61D60—31no—
62D60—61no—
63D60—61no—
64D60C2×D311yescharacteristic
65C4×D15—61no—
66C4×D15C2×D311yescharacteristic
67C5×Dic6C2×D311yescharacteristic, complemented
68C3×C22⋊D5—61no—
69C3×D20—31no—
70C3×C4×D5—31no—
71#1202—61no—
72#1200—61no—
73C3×C5×C8—31no—
74C3⋊#168—31no—
75C3⋊#168—31no—
76C3⋊#168—61no—
77C5×C3⋊C8C2×D311yescharacteristic, complemented
78#932—151no—
79#914—151no—
80C2×C32×C5—41no—
81C2×C32×C5C2×D411yescharacteristic
82C3⋊D15—41no—
83C3×D15—21no—
84C3×D15—41no—
85#713—91no—
86D4×D5—31no—
87C5×SD16—31no—
88#688—91no—
89#686—91no—
90D40—91no—
91#676—91no—
92C22×C3×D3—101no—
93C32×D4C2×D511yescharacteristic
94C4⋊(C3⋊D3)—51no—
95C3×#67—101no—
96C3×D12—51no—
97C3×C4×D3—51no—
98C3×Dic6C2×D511yescharacteristic, complemented
99#626—101no—
100#624—101no—
101C3⋊(C3⋊C8)—51no—
102C3×C3⋊C8C2×D511yescharacteristic, complemented
103C22×C3×C5—21no—
104C22×C3×C5—21no—
105C22×C3×C5—41no—
106C2×D15—21no—
107C2×D15—42no—
108C2×D15—61no—
109C2×D15—61no—
110C2×D15—121no—
111C2×D15#6711yescharacteristic
112C2×C3×D5—31no—
113C2×C3×D5—61no—
114C2×C3×D5—122no—
115C3×C4×C5—21no—
116C3×C4×C5—61no—
117C3×C4×C5C22×D311yescharacteristic
118C3×C4×C5C22×D311yescharacteristic
119Dic15#6711yescharacteristic
120C3×Dic5—31no—
121C5×Dic3—21no—
122C2×C3×D4—151no—
123#239—151no—
124D3×D4—51no—
125#235—151no—
126C3×SD16—31no—
127#215—151no—
128#214—151no—
129#214D1511yescharacteristic, complemented
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×D4D3211yescharacteristic
141C22⋊D5—61no—
142D20—31no—
143D20—91no—
144D20—181no—
145C4×D5—31no—
146C4×D5—181no—
147C5×C8—31no—
148#168—91no—
149C22×C32—21no—
150C2×C3⋊D3—101no—
151C2×C3×D3—51no—
152C2×C3×D3—101no—
153C2×C3×D3—202no—
154C32×C4C22×D511yescharacteristic
155C3×Dic3—21no—
156C3×Dic3—51no—
157AGL(1,ℤ/8)—451noSylow
158C2×C3×C5—21no—
159C2×C3×C5—41no—
160C2×C3×C5—41no—
161C2×C3×C5—81no—
162C2×C3×C5D3×D411yescharacteristic
163C2×C3×C5C2×#6711yescharacteristic
164D15—21no—
165D15—41no—
166D15—121no—
167D15—121no—
168D15—241no—
169C3×D5—61no—
170C3×D5—121no—
171C23×C3—301no—
172C22×D3—101no—
173C3×Q8—31no—
174C3×D4—21no—
175C3×D4—151no—
176C3×D4—301no—
177C3×D4D3×D511yescharacteristic
178C3×D4C2×D1511yescharacteristic
179C2×C3×C4—151no—
180#67—101no—
181#67—301no—
182D12—51no—
183D12—151no—
184D12—151no—
185D12—301no—
186D12—301no—
187C4×D3—51no—
188C4×D3—301no—
189C4×D3—301no—
190Dic6C2×D1511yescharacteristic, complemented
191C3×C8—31no—
192C3⋊C8—151no—
193C3⋊C8—151no—
194C3⋊C8—301no—
195C3⋊C8C2×D1511yescharacteristic, complemented
196C22×C5—21no—
197C2×D5—31no—
198C2×D5—61no—
199C2×D5—122no—
200C2×D5—181no—
201C4×C5—61no—
202C4×C5C2×D3211yescharacteristic
203Dic5—31no—
204C2×C32—41no—
205C2×C32D4×D511yescharacteristic
206C3⋊D3—201no—
207C3×D3—101no—
208C3×D3—201no—
209#41—451no—
210C2×D4—151no—
211SD16—31no—
212SD16—451no—
213D8—451no—
214D8—451no—
215M16—451no—
216C3×C5—21no—
217C3×C5#91411yescharacteristic, complemented
218C3×C5#93211yescharacteristic, complemented
219C22×C3—21no—
220C22×C3—21no—
221C22×C3—41no—
222C22×C3—151no—
223C22×C3—301no—
224C22×C3—602no—
225C2×D3—51no—
226C2×D3—101no—
227C2×D3—202no—
228C2×D3—301no—
229C2×D3—301no—
230C2×D3—601no—
231C3×C4—21no—
232C3×C4—61no—
233C3×C4—151no—
234C3×C4C2×D3×D511yescharacteristic
235C3×C4C22×D1511yescharacteristic
236Dic3—21no—
237Dic3—51no—
238C2×C5—41no—
239C2×C5D3×#6711yescharacteristic
240D5—61no—
241D5—121no—
242D5—361no—
243C32#406011yescharacteristic, complemented, Sylow
244C23—301no—
245Q8—31no—
246D4—151no—
247D4—301no—
248D4—451no—
249D4—901no—
250D4D3×D1511yescharacteristic
251C2×C4—151no—
252C2×C4—901no—
253C8—31no—
254C8—451no—
255C2×C3—21no—
256C2×C3—41no—
257C2×C3—41no—
258C2×C3—81no—
259C2×C3—301no—
260C2×C3—601no—
261C2×C3D5×#6711yescharacteristic
262C2×C3D4×D1511yescharacteristic
263D3—101no—
264D3—201no—
265D3—601no—
266D3—601no—
267D3—1201no—
268C5#6393411yescharacteristic, complemented, Sylow
269C22—21no—
270C22—151no—
271C22—301no—
272C22—602no—
273C22—901no—
274C4—61no—
275C4—151no—
276C4C2×D3×D1511yescharacteristic
277C3—21no—
278C3#9149111yescharacteristic, complemented
279C3#9180311yescharacteristic, complemented
280C2—41no—
281C2—301no—
282C2—601no—
283C2—1801no—
284C2#67×D1511yescharacteristic
subgroupG/H
centerC2#67×D15
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. C3×D4×D15C2semidirect
2. #10624201C2semidirect
3. #10623909C2semidirect
4. C3×C5×#214C2semidirect
5. #10623801C2semidirect
6. #10623799C2semidirect
7. #10623795C2semidirect
12. C4⋊(C3⋊D15)C22semidirect
16. C3×C5×Dic6C22semidirect
19. C3⋊(C3⋊#168)C22semidirect
20. C3×C5×C3⋊C8C22semidirect
23. D4×D15D3semidirect
33. C5×#214D3semidirect
49. C3×#214D5semidirect
67. C5×Dic6C2×D3semidirect
77. C5×C3⋊C8C2×D3semidirect
98. C3×Dic6C2×D5semidirect
102. C3×C3⋊C8C2×D5semidirect
129. #214D15semidirect
137. C32×C5AGL(1,ℤ/8)semidirect
190. Dic6C2×D15semidirect
195. C3⋊C8C2×D15semidirect
217. C3×C5#914semidirect
218. C3×C5#932semidirect
243. C32#4060semidirect
268. C5#63934semidirect
278. C3#91491semidirect
279. C3#91803semidirect

Maximal subgroups

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

subgroupindex
C3×D4×D152
#106242012
#106239092
C3×C5×#2142
#106238012
#106237992
#106237952
#918033
#914913
#639345