Finite Group Databasebeta

← #13088946

Subgroups — #13088946

Order 1440.

624 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10624219C211yesmaximal, characteristic, complemented
2#10624201C211yesmaximal, characteristic, complemented
3#10624197C211yesmaximal, characteristic, complemented
4C5×D3×Dic6C211yesmaximal, characteristic, complemented
5#10624156C211yesmaximal, characteristic, complemented
6C3×#6771C211yesmaximal, characteristic, complemented
7C3×D5×Dic6C211yesmaximal, characteristic, complemented
8#10624055C212yesmaximal, complemented
9#10624044C212yesmaximal, complemented
10#10624041C212yesmaximal, complemented
11#10624040C212yesmaximal, complemented
12#92029—31nomaximal
13#92000—31nomaximal
14C4⋊(C3⋊D15)C2211yescharacteristic
15C5×#634C2211yescharacteristic
16C3×C4×D15C2211yescharacteristic
17C3×Dic30C2211yescharacteristic
18C3×C4×C5×D3C2211yescharacteristic
19C3×C5×Dic6C2211yescharacteristic
20C32×C4×D5C2211yescharacteristic
21#67158C2212yes—
22#67154C2212yes—
23#67152C2212yes—
24#67151C2212yes—
25#67149C2212yes—
26#67149C2212yes—
27C5×#627C2212yes—
28C5×D3×Dic3C2212yes—
29#67139C2212yes—
30#67137C2212yes—
31C3×#1204C2211yescharacteristic
32C3×#1204C2212yes—
33C3×#1203C2211yescharacteristic
34C3×#1202C2211yescharacteristic
35C3×#1201C2211yescharacteristic
36C3×D5×Dic3C2212yes—
37#64297—51nomaximal, Hall
38#6821—31no—
39#6817—31no—
40C5×D3×Q8—31no—
41C2×C5×Dic6—31no—
42C3×Q8×D5—31no—
43C3×#709—31no—
44#6779—32no—
45#6779—32no—
46#6773—31no—
47#6773—31no—
48#6773—31no—
49#6773—32no—
50#6773—61no—
51#6771—32no—
52#6771D311yescharacteristic, complemented
53D15⋊Q8—31no—
54D15⋊Q8—32no—
55D15⋊Q8—32no—
56#6769—31no—
57#6769—32no—
58#6766—32no—
59D5×Dic6—31no—
60D5×Dic6D311yescharacteristic, complemented
61C2×C3⋊D15C2312yes—
62C2×C3×D15C2311yescharacteristic
63C2×C3×C5×D3C2311yescharacteristic
64C2×C32×D5C2311yescharacteristic
65C32×C4×C5C2311yescharacteristic
66C5×C3⋊Dic3C2312yes—
67C3×Dic15C2312yes—
68C3×Dic15C2311yescharacteristic
69C3×C5×Dic3C2312yes—
70C3×C5×Dic3C2311yescharacteristic
71C32×Dic5C2311yescharacteristic
72#4147—91noHall
73#3828—51no—
74C3×#235—51no—
75C2×C3×Dic6—51no—
76#3804—52no—
77#3804—52no—
78#3799—51no—
79#3799—51no—
80#3797—52no—
81#3796—52no—
82D3×Dic6—51no—
83D3×Dic6D511yescharacteristic, complemented
84C3×C5×Q8—31no—
85C2×C3×C4×C5—31no—
86#1220—32no—
87D60—32no—
88D60—31no—
89D60—31no—
90D60—61no—
91C4×D15—32no—
92C4×D15—32no—
93C4×D15C2×D311yescharacteristic
94Dic30C2×D311yescharacteristic
95C2×C5×Dic3—32no—
96C4×C5×D3—32no—
97C4×C5×D3C2×D311yescharacteristic
98C5×Dic6—31no—
99C5×Dic6—31no—
100C5×Dic6—32no—
101C5×Dic6—32no—
102C5×Dic6—61no—
103C5×Dic6C2×D311yescharacteristic
104C3×C22⋊D5—31no—
105C3×C22⋊D5—31no—
106C3×D20—31no—
107C3×C4×D5—21no—
108C3×C4×D5—32no—
109C3×C4×D5—31no—
110C3×C4×D5C2×D311yescharacteristic
111C3×C4×D5C2×D311yescharacteristic
112C3×Dic10—31no—
113C3×Dic10—32no—
114C3×Dic10—31no—
115#1204—32no—
116#1204—32no—
117#1204C2×D312yes—
118#1204C2×D311yescharacteristic
119#1203—32no—
120#1203C2×D311yescharacteristic
121#1202—32no—
122#1202—32no—
123#1202—32no—
124#1202—62no—
125#1202C2×D311yescharacteristic
126#1201C2×D311yescharacteristic
127#1200—32no—
128#1200—32no—
129#1200—32no—
130#1200—32no—
131#1200—32no—
132#1200—62no—
133D5×Dic3—32no—
134D5×Dic3C2×D312yes—
135#1010—151no—
136#1007—151no—
137C2×C32×C5C2411yescharacteristic
138C3⋊D15—22no—
139C3×D15—21no—
140C3×C5×D3—21no—
141C32×D5—21no—
142C2×C5×Q8—91no—
143#713—92no—
144#713—91no—
145#713—91no—
146Q8×D5—31no—
147Q8×D5—92no—
148Q8×D5—91no—
149#709—31no—
150#709—92no—
151#709—91no—
152#709—92no—
153C2×C32×C4—51no—
154#638—52no—
155C4⋊(C3⋊D3)—51no—
156C4×C3⋊D3—52no—
157#634C2×D511yescharacteristic
158C3×#67—51no—
159C3×#67—51no—
160C2×C3×Dic3—52no—
161C3×D12—51no—
162C3×C4×D3—51no—
163C3×C4×D3C2×D511yescharacteristic
164C3×Dic6—51no—
165C3×Dic6—52no—
166C3×Dic6—51no—
167C3×Dic6C2×D511yescharacteristic
168#627—52no—
169#627C2×D512yes—
170#626—52no—
171#626—52no—
172#624—52no—
173#624—52no—
174D3×Dic3—52no—
175D3×Dic3C2×D512yes—
176C22×C3×C5—31no—
177C2×D15—32no—
178C2×D15—32no—
179C2×D15—62no—
180C2×D15C22×D311yescharacteristic
181C2×C5×D3C22×D311yescharacteristic
182C2×C3×D5—21no—
183C2×C3×D5—31no—
184C2×C3×D5C22×D311yescharacteristic
185C2×C3×D5C22×D311yescharacteristic
186C3×C4×C5—21no—
187C3×C4×C5—32no—
188C3×C4×C5—31no—
189C3×C4×C5C22×D311yescharacteristic
190C3×C4×C5C22×D311yescharacteristic
191Dic15C22×D312yes—
192Dic15C22×D311yescharacteristic
193C3×Dic5—21no—
194C3×Dic5—32no—
195C3×Dic5—31no—
196C3×Dic5C22×D311yescharacteristic
197C3×Dic5C22×D311yescharacteristic
198C5×Dic3—32no—
199C5×Dic3—32no—
200C5×Dic3—62no—
201C5×Dic3C22×D312yes—
202C5×Dic3C22×D311yescharacteristic
203C3×#41—151no—
204C2×C3×Q8—151no—
205#239—152no—
206#239—151no—
207#239—151no—
208D3×Q8—31no—
209D3×Q8—152no—
210D3×Q8—152no—
211D3×Q8—151no—
212#237—152no—
213#237—152no—
214#237—152no—
215#235—51no—
216#235—152no—
217#235—151no—
218#235—151no—
219#235—151no—
220#235—152no—
221#235—151no—
222#235—301no—
223C2×Dic6—31no—
224C2×Dic6—51no—
225C2×Dic6—151no—
226C32×C52−1+411yescharacteristic, complemented, Hall
227C5×Q8—31no—
228C5×Q8—92no—
229C5×Q8—91no—
230C2×C4×C5—31no—
231C2×C4×C5—92no—
232C22⋊D5—31no—
233C22⋊D5—31no—
234C22⋊D5—92no—
235D20—31no—
236D20—92no—
237D20—92no—
238D20—91no—
239C4×D5—32no—
240C4×D5—31no—
241C4×D5—92no—
242C4×D5—92no—
243C4×D5—92no—
244C4×D5—92no—
245C4×D5D3211yescharacteristic
246Dic10—31no—
247Dic10—31no—
248Dic10—32no—
249Dic10—92no—
250C22×C32—51no—
251C2×C3⋊D3—52no—
252C2×C3×D3—51no—
253C2×C3×D3C22×D511yescharacteristic
254C32×C4—51no—
255C32×C4C22×D511yescharacteristic
256C3⋊Dic3C22×D512yes—
257C3×Dic3—52no—
258C3×Dic3—51no—
259C3×Dic3C22×D512yes—
260C3×Dic3C22×D511yescharacteristic
2612−1+4—451noSylow
262C2×C3×C5—21no—
263C2×C3×C5—61no—
264C2×C3×C5C23×D311yescharacteristic
265C2×C3×C5C23×D311yescharacteristic
266D15—21no—
267D15—62no—
268D15—62no—
269D15—122no—
270C3×D5—21no—
271C3×D5—21no—
272C3×D5—41no—
273C3×D5—61no—
274C5×D3—21no—
275C3×Q8—31no—
276C3×Q8—151no—
277C3×Q8—152no—
278C3×Q8—151no—
279C3×D4—151no—
280C3×D4—151no—
281C3×D4—151no—
282C2×C3×C4—31no—
283C2×C3×C4—51no—
284C2×C3×C4—51no—
285C2×C3×C4—101no—
286C2×C3×C4—152no—
287C2×C3×C4—151no—
288#67—51no—
289#67—51no—
290#67—152no—
291#67—152no—
292#67—152no—
293#67—152no—
294#67—302no—
295C2×Dic3—32no—
296C2×Dic3—52no—
297C2×Dic3—152no—
298D12—51no—
299D12—152no—
300D12—152no—
301D12—151no—
302D12—151no—
303D12—301no—
304C4×D3—32no—
305C4×D3—51no—
306C4×D3—152no—
307C4×D3—152no—
308C4×D3—152no—
309C4×D3—152no—
310C4×D3—152no—
311C4×D3—152no—
312C4×D3—152no—
313C4×D3—302no—
314C4×D3D3×D511yescharacteristic
315Dic6—31no—
316Dic6—31no—
317Dic6—32no—
318Dic6—32no—
319Dic6—51no—
320Dic6—52no—
321Dic6—51no—
322Dic6—61no—
323Dic6—152no—
324Dic6—152no—
325Dic6D3×D511yescharacteristic
326C22×C5—31no—
327C2×D5—31no—
328C2×D5—92no—
329C2×D5C2×D3211yescharacteristic
330C4×C5—32no—
331C4×C5—31no—
332C4×C5—92no—
333C4×C5C2×D3211yescharacteristic
334Dic5—32no—
335Dic5—31no—
336Dic5C2×D3211yescharacteristic
337C2×C32—101no—
338C2×C32C23×D511yescharacteristic
339C3⋊D3—102no—
340C3×D3—21no—
341C3×D3—101no—
342#41—151no—
343#41—451no—
344#41—451no—
345#41—452no—
346#41—452no—
347#41—452no—
348#41—451no—
349C2×Q8—91no—
350C2×Q8—151no—
351C2×Q8—451no—
352C2×Q8—452no—
353C3×C5—21no—
354C3×C5#100711yescharacteristic, complemented
355C3×C5#101011yescharacteristic, complemented
356C22×C3—31no—
357C22×C3—51no—
358C22×C3—51no—
359C22×C3—101no—
360C22×C3—151no—
361C2×D3—51no—
362C2×D3—152no—
363C2×D3—152no—
364C2×D3—302no—
365C2×D3C2×D3×D511yescharacteristic
366C3×C4—21no—
367C3×C4—31no—
368C3×C4—32no—
369C3×C4—51no—
370C3×C4—51no—
371C3×C4—101no—
372C3×C4—151no—
373C3×C4—152no—
374C3×C4C2×D3×D511yescharacteristic
375C3×C4C2×D3×D511yescharacteristic
376Dic3—32no—
377Dic3—32no—
378Dic3—52no—
379Dic3—51no—
380Dic3—62no—
381Dic3C2×D3×D511yescharacteristic
382Dic3C2×D3×D512yes—
383C2×C5—61no—
384C2×C5C22×D3211yescharacteristic
385D5—21no—
386D5—61no—
387D5—182no—
388C32#414711yescharacteristic, complemented, Sylow
389Q8—31no—
390Q8—92no—
391Q8—91no—
392Q8—152no—
393Q8—151no—
394Q8—151no—
395Q8—452no—
396D4—151no—
397D4—151no—
398D4—151no—
399D4—452no—
400D4—452no—
401D4—452no—
402D4—451no—
403C2×C4—31no—
404C2×C4—51no—
405C2×C4—92no—
406C2×C4—152no—
407C2×C4—151no—
408C2×C4—452no—
409C2×C4—452no—
410C2×C4—452no—
411C2×C4—452no—
412C2×C3—21no—
413C2×C3—61no—
414C2×C3—101no—
415C2×C3—101no—
416C2×C3—201no—
417C2×C3—301no—
418C2×C3C22×D3×D511yescharacteristic
419C2×C3C22×D3×D511yescharacteristic
420D3—21no—
421D3—101no—
422D3—302no—
423D3—302no—
424D3—602no—
425C5#6429711yescharacteristic, complemented, Sylow
426C22—31no—
427C22—51no—
428C22—151no—
429C22—452no—
430C4—31no—
431C4—32no—
432C4—51no—
433C4—92no—
434C4—151no—
435C4—152no—
436C4D32×D511yescharacteristic
437C3—21no—
438C3#9200011yescharacteristic, complemented
439C3#9202911yescharacteristic, complemented
440C2—61no—
441C2—101no—
442C2—301no—
443C2—902no—
444C2C2×D32×D511yescharacteristic
subgroupG/H
center, FrattiniC2C2×D32×D5
commutator, socle, cosocleC2×C32×C5C24
FittingC32×C4×C5C23
hypercenterC4D32×D5
hypocenterC32×C52−1+4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #10624219C2semidirect
2. #10624201C2semidirect
3. #10624197C2semidirect
4. C5×D3×Dic6C2semidirect
5. #10624156C2semidirect
6. C3×#6771C2semidirect
7. C3×D5×Dic6C2semidirect
8. #10624055C2semidirect
9. #10624044C2semidirect
10. #10624041C2semidirect
11. #10624040C2semidirect
52. #6771D3semidirect
60. D5×Dic6D3semidirect
83. D3×Dic6D5semidirect
226. C32×C52−1+4semidirect
354. C3×C5#1007semidirect
355. C3×C5#1010semidirect
388. C32#4147semidirect
425. C5#64297semidirect
438. C3#92000semidirect
439. C3#92029semidirect

Maximal subgroups

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

subgroupindex
#106242192
#106242012
#106241972
C5×D3×Dic62
#106241562
C3×#67712
C3×D5×Dic62
#106240552
#106240442
#106240412
#106240402
#920293
#920003
#642975