Finite Group Databasebeta

← #13089410

Subgroups — #13089410

Order 1440.

704 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 143 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×#3828C211yesmaximal, characteristic, complemented
2C3×#6817C211yesmaximal, characteristic, complemented
3C2×C3×D60C211yesmaximal, characteristic, complemented
4#10624235C214yesmaximal, complemented
5#10624227C212yesmaximal, complemented
6#10624226C212yesmaximal, complemented
7#10624221C212yesmaximal, complemented
8#10624220C212yesmaximal, complemented
9#92019—31nomaximal
10#92017—31nomaximal
11C22×C3×D15C2212yescomplemented
12C2×(C3⋊D3)⋊D5C2214yescomplemented
13C2×C32×C4×C5C2211yescharacteristic, complemented
14C5×#638C2212yescomplemented
15C5×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
16C4×C5×C3⋊D3C2212yescomplemented
17C5×#634C2211yescharacteristic, complemented
18C3×#1220C2212yescomplemented
19C3×D60C2212yescomplemented
20C3×D60C2212yescomplemented
21C3×D60C2211yescharacteristic, complemented
22C3×C4×D15C2212yescomplemented
23C3×Dic30C2211yescharacteristic, complemented
24#67157C2214yescomplemented
25#67156C2214yescomplemented
26#67154C2214yescomplemented
27#64307—51nomaximal, Hall
28#6817D311yescharacteristic, complemented
29C2×D60D311yescharacteristic, complemented
30C5×#235—31no—
31C5×#235—31no—
32C5×#235—61no—
33C3×#709—31no—
34C2×C3×D20—31no—
35#6790—34no—
36D5×#67—34no—
37#6777—32no—
38D3×D20—32no—
39D3×D20—32no—
40D5×D12—32no—
41#6773—32no—
42#6769—32no—
43#6766—32no—
44#6765—32no—
45C22×C32×C5C2311yescharacteristic
46C2×C5×C3⋊D3C2312yes—
47C2×C3×D15—42no—
48C2×C3×D15C2314yes—
49C2×C3×D15C2312yes—
50(C3⋊D3)⋊D5—44no—
51C32×C4×C5C2311yescharacteristic
52C32×C4×C5C2311yescharacteristic
53C5×C3⋊Dic3C2312yes—
54C3×Dic15C2312yes—
55#4149—91noHall
56#3828D511yescharacteristic, complemented
57C3×#235—51no—
58C2×C3×D12—51no—
59D3×#67—54no—
60#3802—52no—
61D3×D12—52no—
62#3796—52no—
63#3795—52no—
64C22×D15C2×D312yescomplemented
65C22×C3×D5—32no—
66C2×D3×D5—34no—
67C2×D3×D5—34no—
68C2×C3×C4×C5—21no—
69C2×C3×C4×C5C2×D311yescharacteristic, complemented
70C2×C3×C4×C5C2×D311yescharacteristic, complemented
71#1220C2×D312yescomplemented
72D60C2×D312yescomplemented
73D60C2×D312yescomplemented
74D60C2×D311yescharacteristic, complemented
75C4×D15C2×D312yescomplemented
76Dic30C2×D311yescharacteristic, complemented
77C5×#67—32no—
78C5×#67—32no—
79C5×#67—62no—
80C5×D12—31no—
81C5×D12—31no—
82C5×D12—61no—
83C4×C5×D3—32no—
84C4×C5×D3—32no—
85C4×C5×D3—62no—
86C5×Dic6—31no—
87C5×Dic6—31no—
88C5×Dic6—61no—
89C3×C22⋊D5—32no—
90C3×D20—32no—
91C3×D20—32no—
92C3×D20—31no—
93C3×C4×D5—32no—
94C3×Dic10—31no—
95#1203—34no—
96#1202—34no—
97#1202—34no—
98#1201—34no—
99#1200—34no—
100D5×Dic3—34no—
101#1009—151no—
102#1004—151no—
103C2×C32×C5—21no—
104C2×C32×C5C2411yescharacteristic
105C5×C3⋊D3—22no—
106C3×D15—22no—
107C3×D15—24no—
108C5×#41—91no—
109#713—92no—
110D4×D5—92no—
111D4×D5—94no—
112#709—31no—
113#709—92no—
114C2×D20—31no—
115C2×D20—92no—
116C22×C3×D3—52no—
117C2×D32—54no—
118C2×C32×C4C2×D511yescharacteristic, complemented
119#638C2×D512yescomplemented
120C4⋊(C3⋊D3)C2×D511yescharacteristic, complemented
121C4×C3⋊D3C2×D512yescomplemented
122#634C2×D511yescharacteristic, complemented
123C3×#67—52no—
124C3×D12—52no—
125C3×D12—52no—
126C3×D12—51no—
127C3×C4×D3—52no—
128C3×Dic6—51no—
129#626—54no—
130#625—54no—
131D3×Dic3—54no—
132C22×C3×C5—21no—
133C22×C3×C5C22×D311yescharacteristic
134C22×C3×C5C22×D311yescharacteristic
135C2×D15—42no—
136C2×D15C22×D314yes—
137C2×D15C22×D312yes—
138C2×C5×D3—32no—
139C2×C5×D3—32no—
140C2×C5×D3—62no—
141C2×C3×D5—34no—
142C2×C3×D5—32no—
143C2×C3×D5—122no—
144D3×D5—124no—
145D3×D5—124no—
146C3×C4×C5—21no—
147C3×C4×C5—21no—
148C3×C4×C5C22×D311yescharacteristic
149C3×C4×C5C22×D311yescharacteristic
150C3×C4×C5C22×D311yescharacteristic
151C3×C4×C5C22×D311yescharacteristic
152Dic15C22×D312yes—
153C3×Dic5—32no—
154C5×Dic3—32no—
155C5×Dic3—32no—
156C5×Dic3—62no—
157C3×#41—151no—
158C2×C3×D4—151no—
159C2×#67—154no—
160#239—152no—
161#237—152no—
162#237—152no—
163D3×D4—152no—
164D3×D4—152no—
165D3×D4—152no—
166D3×D4—154no—
167#235—31no—
168#235—31no—
169#235—51no—
170#235—61no—
171#235—152no—
172C2×D12—51no—
173C2×D12—152no—
174C32×C5H(5,𝔽2)11yescharacteristic, complemented, Hall
175C22×D5—32no—
176C22×D5—94no—
177C5×Q8—91no—
178C5×D4—92no—
179C5×D4—91no—
180C2×C4×C5—92no—
181C2×C4×C5D3211yescharacteristic, complemented
182C22⋊D5—32no—
183C22⋊D5—94no—
184D20—32no—
185D20—32no—
186D20—31no—
187D20—94no—
188C4×D5—32no—
189C4×D5—94no—
190Dic10—31no—
191C22×C32C22×D511yescharacteristic
192C2×C3⋊D3C22×D512yes—
193C2×C3×D3—54no—
194C2×C3×D3—52no—
195C2×C3×D3—202no—
196D32—204no—
197C32×C4C22×D511yescharacteristic
198C32×C4C22×D511yescharacteristic
199C3⋊Dic3C22×D512yes—
200C3×Dic3—52no—
201H(5,𝔽2)—451noSylow
202C2×C3×C5—21no—
203C2×C3×C5—21no—
204C2×C3×C5—21no—
205C2×C3×C5—41no—
206C2×C3×C5C23×D311yescharacteristic
207C2×C3×C5C23×D311yescharacteristic
208D15—24no—
209D15—22no—
210C3×D5—62no—
211C3×D5—64no—
212C5×D3—62no—
213C5×D3—62no—
214C5×D3—122no—
215C23×C3—152no—
216C22×D3—52no—
217C22×D3—154no—
218C22×D3—154no—
219C3×Q8—151no—
220C3×D4—152no—
221C3×D4—151no—
222C3×D4—152no—
223C3×D4—152no—
224C2×C3×C4—21no—
225C2×C3×C4—152no—
226C2×C3×C4D3×D511yescharacteristic, complemented
227C2×C3×C4D3×D511yescharacteristic, complemented
228#67—32no—
229#67—32no—
230#67—52no—
231#67—62no—
232#67—154no—
233#67—154no—
234#67—154no—
235C2×Dic3—154no—
236D12—31no—
237D12—31no—
238D12—52no—
239D12—52no—
240D12—51no—
241D12—61no—
242D12—154no—
243C4×D3—32no—
244C4×D3—32no—
245C4×D3—52no—
246C4×D3—62no—
247C4×D3—154no—
248Dic6—31no—
249Dic6—31no—
250Dic6—51no—
251Dic6—61no—
252C22×C5—92no—
253C22×C5C2×D3211yescharacteristic
254C2×D5—32no—
255C2×D5—34no—
256C2×D5—122no—
257C2×D5—364no—
258C4×C5—92no—
259C4×C5C2×D3211yescharacteristic
260C4×C5C2×D3211yescharacteristic
261Dic5—32no—
262C2×C32—21no—
263C2×C32C23×D511yescharacteristic
264C3⋊D3—22no—
265C3×D3—102no—
266C3×D3—104no—
267#41—91no—
268#41—151no—
269#41—452no—
270#41—452no—
271C2×D4—151no—
272C2×D4—452no—
273C2×D4—452no—
274C2×D4—454no—
275C3×C5—21no—
276C3×C5#100411yescharacteristic, complemented
277C3×C5#100911yescharacteristic, complemented
278C22×C3—21no—
279C22×C3—154no—
280C22×C3—152no—
281C22×C3—602no—
282C22×C3C2×D3×D511yescharacteristic
283C22×C3C2×D3×D511yescharacteristic
284C2×D3—32no—
285C2×D3—32no—
286C2×D3—52no—
287C2×D3—54no—
288C2×D3—62no—
289C2×D3—202no—
290C2×D3—604no—
291C2×D3—604no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—152no—
295C3×C4C2×D3×D511yescharacteristic
296C3×C4C2×D3×D511yescharacteristic
297C3×C4C2×D3×D511yescharacteristic
298C3×C4C2×D3×D511yescharacteristic
299Dic3—32no—
300Dic3—32no—
301Dic3—52no—
302Dic3—62no—
303C2×C5—21no—
304C2×C5—182no—
305C2×C5C22×D3211yescharacteristic
306D5—64no—
307D5—62no—
308C32#414911yescharacteristic, complemented, Sylow
309C23—152no—
310C23—454no—
311Q8—91no—
312Q8—151no—
313D4—92no—
314D4—91no—
315D4—152no—
316D4—152no—
317D4—152no—
318D4—151no—
319D4—454no—
320D4—454no—
321C2×C4—92no—
322C2×C4—152no—
323C2×C4—454no—
324C2×C4(C3⋊D3)⋊D511yescharacteristic, complemented
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—41no—
329C2×C3—302no—
330C2×C3—304no—
331C2×C3C22×D3×D511yescharacteristic
332C2×C3C22×D3×D511yescharacteristic
333D3—62no—
334D3—62no—
335D3—104no—
336D3—102no—
337D3—122no—
338C5#6430711yescharacteristic, complemented, Sylow
339C22—92no—
340C22—152no—
341C22—154no—
342C22—602no—
343C22—1804no—
344C22C2×(C3⋊D3)⋊D511yescharacteristic
345C4—92no—
346C4—152no—
347C4C2×(C3⋊D3)⋊D511yescharacteristic
348C4C2×(C3⋊D3)⋊D511yescharacteristic
349C3—21no—
350C3#9201711yescharacteristic, complemented
351C3#9201911yescharacteristic, complemented
352C2—21no—
353C2—182no—
354C2—304no—
355C2—302no—
356C2C22×(C3⋊D3)⋊D511yescharacteristic
subgroupG/H
center, FrattiniC2C22×(C3⋊D3)⋊D5
commutator, socle, cosocleC2×C32×C5C24
FittingC2×C32×C4×C5C22
hypercenterC2×C4(C3⋊D3)⋊D5
hypocenterC32×C5H(5,𝔽2)
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C2×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C5×#3828C2semidirect
2. C3×#6817C2semidirect
3. C2×C3×D60C2semidirect
4. #10624235C2semidirect
5. #10624227C2semidirect
6. #10624226C2semidirect
7. #10624221C2semidirect
8. #10624220C2semidirect
11. C22×C3×D15C22semidirect
12. C2×(C3⋊D3)⋊D5C22semidirect
13. C2×C32×C4×C5C22semidirect
14. C5×#638C22semidirect
15. C5×C4⋊(C3⋊D3)C22semidirect
16. C4×C5×C3⋊D3C22semidirect
17. C5×#634C22semidirect
18. C3×#1220C22semidirect
19. C3×D60C22semidirect
20. C3×D60C22semidirect
21. C3×D60C22semidirect
22. C3×C4×D15C22semidirect
23. C3×Dic30C22semidirect
24. #67157C22semidirect
25. #67156C22semidirect
26. #67154C22semidirect
28. #6817D3semidirect
29. C2×D60D3semidirect
56. #3828D5semidirect
64. C22×D15C2×D3semidirect
69. C2×C3×C4×C5C2×D3semidirect
70. C2×C3×C4×C5C2×D3semidirect
71. #1220C2×D3semidirect
72. D60C2×D3semidirect
73. D60C2×D3semidirect
74. D60C2×D3semidirect
75. C4×D15C2×D3semidirect
76. Dic30C2×D3semidirect
118. C2×C32×C4C2×D5semidirect
119. #638C2×D5semidirect
120. C4⋊(C3⋊D3)C2×D5semidirect
121. C4×C3⋊D3C2×D5semidirect
122. #634C2×D5semidirect
174. C32×C5H(5,𝔽2)semidirect
181. C2×C4×C5D32semidirect
226. C2×C3×C4D3×D5semidirect
227. C2×C3×C4D3×D5semidirect
276. C3×C5#1004semidirect
277. C3×C5#1009semidirect
308. C32#4149semidirect
324. C2×C4(C3⋊D3)⋊D5semidirect
338. C5#64307semidirect
350. C3#92017semidirect
351. C3#92019semidirect

Maximal subgroups

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

subgroupindex
C5×#38282
C3×#68172
C2×C3×D602
#106242352
#106242272
#106242262
#106242212
#106242202
#920193
#920173
#643075