Finite Group Databasebeta

← #13089367

Subgroups — #13089367

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#10624285C211yesmaximal, characteristic, complemented
2C3×#6819C211yesmaximal, characteristic, complemented
3C3×C5×#237C211yesmaximal, characteristic, complemented
4C2×#67153C212yesmaximal, complemented
5#10624210C212yesmaximal, complemented
6#10624208C212yesmaximal, complemented
7#10624207C212yesmaximal, complemented
8#10624199C211yesmaximal, characteristic, complemented
9#10624198C211yesmaximal, characteristic, complemented
10D3×Dic30C211yesmaximal, characteristic, complemented
11Dic6×D15C211yesmaximal, characteristic, complemented
12#92101—31nomaximal
13#92020—31nomaximal
14C32×C5×D4C2211yescharacteristic
15#67184C2212yes—
16C2×C3⋊Dic15C2212yes—
17C4×C3⋊D15C2211yescharacteristic
18#67180C2211yescharacteristic
19C3×#1220C2212yes—
20C2×C3×Dic15C2212yes—
21C3×C4×D15C2211yescharacteristic
22C3×Dic30C2211yescharacteristic
23C3×C5×#67C2212yes—
24C2×C3×C5×Dic3C2212yes—
25C3×C4×C5×D3C2211yescharacteristic
26C3×C5×Dic6C2211yescharacteristic
27#67153C2212yes—
28#67153C2212yes—
29#67153C2212yes—
30#67153C2211yescharacteristic
31#67152C2211yescharacteristic
32#67151C2211yescharacteristic
33#67150C2211yescharacteristic
34#67149C2212yes—
35D3×Dic15C2212yes—
36Dic3×D15C2212yes—
37#64309—51nomaximal, Hall
38C3×C5×#41—31no—
39Q8×D15—31no—
40Q8×D15—31no—
41#6819—31no—
42#6819—31no—
43#6819—32no—
44#6819—32no—
45#6819—61no—
46#6819D311yescharacteristic, complemented
47#6817—32no—
48#6817—31no—
49C2×Dic30—32no—
50C2×Dic30—31no—
51C5×#237D311yescharacteristic, complemented
52C3×#711—31no—
53C2×#1204—32no—
54#6782—32no—
55#6780—32no—
56#6779—32no—
57#6771—31no—
58D15⋊Q8—31no—
59D3×Dic10—31no—
60D5×Dic6—31no—
61C22×C32×C5C2312yes—
62C2×C3⋊D15C2311yescharacteristic
63C2×C3×D15C2311yescharacteristic
64C2×C3×C5×D3C2311yescharacteristic
65C32×C4×C5C2311yescharacteristic
66C3⋊Dic15C2312yes—
67C3⋊Dic15C2311yescharacteristic
68C3×Dic15C2312yes—
69C3×Dic15C2311yescharacteristic
70C3×C5×Dic3C2312yes—
71C3×C5×Dic3C2311yescharacteristic
72#4150—91noHall
73#3830—51no—
74C3×#237—51no—
75C3×#237D511yescharacteristic, complemented
76C2×#627—52no—
77#3805—52no—
78#3804—52no—
79#3804—52no—
80#3798—51no—
81#3797—51no—
82D3×Dic6—51no—
83D3×Dic6—51no—
84C3×C5×Q8—31no—
85C3×C5×D4—21no—
86C3×C5×D4—32no—
87C3×C5×D4C2×D311yescharacteristic
88C3×C5×D4C2×D311yescharacteristic
89C2×C3×C4×C5—32no—
90C2×C3×C4×C5—31no—
91#1220—32no—
92#1220—32no—
93#1220—31no—
94#1220—31no—
95#1220—62no—
96#1220C2×D312yes—
97C2×Dic15—32no—
98C2×Dic15—32no—
99C2×Dic15—32no—
100C2×Dic15—62no—
101C2×Dic15C2×D312yes—
102D60—31no—
103C4×D15—32no—
104C4×D15—31no—
105C4×D15—31no—
106C4×D15—32no—
107C4×D15—61no—
108C4×D15C2×D311yescharacteristic
109Dic30—31no—
110Dic30—31no—
111Dic30—31no—
112Dic30—32no—
113Dic30—32no—
114Dic30—32no—
115Dic30—61no—
116Dic30C2×D311yescharacteristic
117C5×#67C2×D312yes—
118C2×C5×Dic3C2×D312yes—
119C4×C5×D3C2×D311yescharacteristic
120C5×Dic6C2×D311yescharacteristic
121C3×C22⋊D5—32no—
122C2×C3×Dic5—32no—
123C3×C4×D5—31no—
124C3×Dic10—31no—
125#1204—32no—
126#1204—32no—
127#1204—32no—
128#1204—31no—
129#1203—31no—
130#1202—31no—
131#1201—31no—
132#1200—32no—
133D3×Dic5—32no—
134D5×Dic3—32no—
135#1010—151no—
136#1010—151no—
137C2×C32×C5—22no—
138C2×C32×C5C2411yescharacteristic
139C3⋊D15—21no—
140C3×D15—21no—
141C3×C5×D3—21no—
142C5×#41—31no—
143Q8×D5—91no—
144Q8×D5—91no—
145#711—31no—
146#711—92no—
147#711—92no—
148#711—91no—
149#709—91no—
150#709—92no—
151C2×Dic10—91no—
152C2×Dic10—92no—
153C32×D4C2×D511yescharacteristic
154#638—52no—
155C2×C3⋊Dic3—52no—
156C4×C3⋊D3—51no—
157#634—51no—
158C3×#67—52no—
159C3×#67C2×D512yes—
160C2×C3×Dic3—52no—
161C2×C3×Dic3C2×D512yes—
162C3×C4×D3—51no—
163C3×C4×D3C2×D511yescharacteristic
164C3×Dic6—51no—
165C3×Dic6C2×D511yescharacteristic
166#627—52no—
167#627—52no—
168#627—52no—
169#627—51no—
170#626—51no—
171#626—51no—
172#625—51no—
173#624—52no—
174D3×Dic3—52no—
175D3×Dic3—52no—
176C22×C3×C5—22no—
177C22×C3×C5—31no—
178C22×C3×C5C22×D312yes—
179C22×C3×C5C22×D312yes—
180C2×D15—31no—
181C2×D15—31no—
182C2×D15—61no—
183C2×D15C22×D311yescharacteristic
184C2×C5×D3C22×D311yescharacteristic
185C2×C3×D5—31no—
186C3×C4×C5—21no—
187C3×C4×C5—31no—
188C3×C4×C5—32no—
189C3×C4×C5C22×D311yescharacteristic
190C3×C4×C5C22×D311yescharacteristic
191Dic15—32no—
192Dic15—31no—
193Dic15—32no—
194Dic15—31no—
195Dic15—62no—
196Dic15—61no—
197Dic15C22×D311yescharacteristic
198Dic15C22×D312yes—
199C3×Dic5—32no—
200C3×Dic5—31no—
201C5×Dic3C22×D312yes—
202C5×Dic3C22×D311yescharacteristic
203C3×#41—31no—
204C3×#41—151no—
205D3×Q8—151no—
206D3×Q8—151no—
207D3×Q8—151no—
208D3×Q8—151no—
209#237—51no—
210#237—152no—
211#237—152no—
212#237—151no—
213#237—151no—
214#237—152no—
215#237—152no—
216#237—301no—
217#237D1511yescharacteristic, complemented
218#235—151no—
219#235—152no—
220#235—152no—
221#235—151no—
222C2×Dic6—151no—
223C2×Dic6—152no—
224C2×Dic6—152no—
225C2×Dic6—151no—
226C32×C52−1+411yescharacteristic, complemented, Hall
227C5×Q8—31no—
228C5×D4—32no—
229C5×D4D3211yescharacteristic
230C2×C4×C5—32no—
231C2×C4×C5—31no—
232C22⋊D5—32no—
233C22⋊D5—91no—
234C22⋊D5—91no—
235C22⋊D5—92no—
236C2×Dic5—32no—
237C2×Dic5—92no—
238C2×Dic5—92no—
239D20—91no—
240C4×D5—31no—
241C4×D5—92no—
242C4×D5—92no—
243C4×D5—91no—
244Dic10—31no—
245Dic10—92no—
246Dic10—91no—
247Dic10—92no—
248Dic10—92no—
249Dic10—91no—
250C22×C32C22×D512yes—
251C2×C3⋊D3—51no—
252C2×C3×D3—51no—
253C2×C3×D3C22×D511yescharacteristic
254C32×C4C22×D511yescharacteristic
255C3⋊Dic3—52no—
256C3⋊Dic3—51no—
257C3×Dic3—52no—
258C3×Dic3—51no—
259C3×Dic3C22×D512yes—
260C3×Dic3C22×D511yescharacteristic
2612−1+4—451noSylow
262C2×C3×C5—22no—
263C2×C3×C5—22no—
264C2×C3×C5—21no—
265C2×C3×C5—42no—
266C2×C3×C5—61no—
267C2×C3×C5C23×D311yescharacteristic
268C2×C3×C5C23×D311yescharacteristic
269D15—21no—
270D15—61no—
271D15—61no—
272D15—121no—
273C3×D5—61no—
274C5×D3—21no—
275C3×Q8—31no—
276C3×Q8—151no—
277C3×D4—21no—
278C3×D4—32no—
279C3×D4—152no—
280C3×D4D3×D511yescharacteristic
281C3×D4C2×D1511yescharacteristic
282C2×C3×C4—32no—
283C2×C3×C4—31no—
284C2×C3×C4—152no—
285C2×C3×C4—151no—
286#67—52no—
287#67—151no—
288#67—151no—
289#67—152no—
290#67—152no—
291#67—151no—
292#67—151no—
293#67—302no—
294#67C2×D1512yes—
295C2×Dic3—52no—
296C2×Dic3—152no—
297C2×Dic3—152no—
298C2×Dic3—152no—
299C2×Dic3—152no—
300C2×Dic3—302no—
301C2×Dic3C2×D1512yes—
302D12—151no—
303D12—151no—
304C4×D3—51no—
305C4×D3—152no—
306C4×D3—152no—
307C4×D3—152no—
308C4×D3—151no—
309C4×D3—151no—
310C4×D3—152no—
311C4×D3—301no—
312C4×D3C2×D1511yescharacteristic
313Dic6—51no—
314Dic6—152no—
315Dic6—151no—
316Dic6—152no—
317Dic6—152no—
318Dic6—151no—
319Dic6—151no—
320Dic6—151no—
321Dic6—152no—
322Dic6—152no—
323Dic6—152no—
324Dic6—301no—
325Dic6C2×D1511yescharacteristic
326C22×C5—31no—
327C22×C5C2×D3212yes—
328C2×D5—31no—
329C2×D5—91no—
330C4×C5—31no—
331C4×C5—32no—
332C4×C5C2×D3211yescharacteristic
333Dic5—32no—
334Dic5—31no—
335Dic5—92no—
336Dic5—91no—
337C2×C32—22no—
338C2×C32C23×D511yescharacteristic
339C3⋊D3—101no—
340C3×D3—21no—
341C3×D3—101no—
342#41—31no—
343#41—151no—
344#41—451no—
345#41—452no—
346#41—452no—
347#41—452no—
348#41—451no—
349C2×Q8—452no—
350C2×Q8—451no—
351C2×Q8—451no—
352C2×Q8—451no—
353C3×C5—21no—
354C3×C5#101011yescharacteristic, complemented
355C3×C5#101011yescharacteristic, complemented
356C22×C3—22no—
357C22×C3—31no—
358C22×C3—151no—
359C22×C3C2×D3×D512yes—
360C22×C3C22×D1512yes—
361C2×D3—51no—
362C2×D3—151no—
363C2×D3—151no—
364C2×D3—301no—
365C2×D3C22×D1511yescharacteristic
366C3×C4—21no—
367C3×C4—31no—
368C3×C4—32no—
369C3×C4—151no—
370C3×C4—152no—
371C3×C4C2×D3×D511yescharacteristic
372C3×C4C22×D1511yescharacteristic
373Dic3—52no—
374Dic3—51no—
375Dic3—151no—
376Dic3—152no—
377Dic3—152no—
378Dic3—151no—
379Dic3—302no—
380Dic3—301no—
381Dic3C22×D1511yescharacteristic
382Dic3C22×D1512yes—
383C2×C5—22no—
384C2×C5—61no—
385C2×C5C22×D3211yescharacteristic
386D5—61no—
387D5—181no—
388C32#415011yescharacteristic, complemented, Sylow
389Q8—31no—
390Q8—151no—
391Q8—451no—
392Q8—452no—
393Q8—452no—
394Q8—452no—
395Q8—451no—
396D4—32no—
397D4—152no—
398D4—451no—
399D4—451no—
400D4—451no—
401D4—452no—
402D4D3×D1511yescharacteristic
403C2×C4—32no—
404C2×C4—31no—
405C2×C4—152no—
406C2×C4—151no—
407C2×C4—452no—
408C2×C4—452no—
409C2×C4—452no—
410C2×C4—452no—
411C2×C4—451no—
412C2×C3—22no—
413C2×C3—22no—
414C2×C3—21no—
415C2×C3—42no—
416C2×C3—61no—
417C2×C3—301no—
418C2×C3C22×D3×D511yescharacteristic
419C2×C3C23×D1511yescharacteristic
420D3—21no—
421D3—101no—
422D3—301no—
423D3—301no—
424D3—601no—
425C5#6430911yescharacteristic, complemented, Sylow
426C22—31no—
427C22—151no—
428C22—451no—
429C22C2×D3×D1512yes—
430C4—31no—
431C4—32no—
432C4—151no—
433C4—152no—
434C4—452no—
435C4—451no—
436C4C2×D3×D1511yescharacteristic
437C3—21no—
438C3#9202011yescharacteristic, complemented
439C3#9210111yescharacteristic, complemented
440C2—22no—
441C2—61no—
442C2—301no—
443C2—901no—
444C2C22×D3×D1511yescharacteristic
subgroupG/H
center, FrattiniC2C22×D3×D15
commutator, socle, cosocleC2×C32×C5C24
FittingC32×C5×D4C22
hypercenterD4D3×D15
hypocenterC32×C52−1+4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
1. #10624285C2semidirect
2. C3×#6819C2semidirect
3. C3×C5×#237C2semidirect
4. C2×#67153C2semidirect
5. #10624210C2semidirect
6. #10624208C2semidirect
7. #10624207C2semidirect
8. #10624199C2semidirect
9. #10624198C2semidirect
10. D3×Dic30C2semidirect
11. Dic6×D15C2semidirect
46. #6819D3semidirect
51. C5×#237D3semidirect
75. C3×#237D5semidirect
217. #237D15semidirect
226. C32×C52−1+4semidirect
354. C3×C5#1010semidirect
355. C3×C5#1010semidirect
388. C32#4150semidirect
425. C5#64309semidirect
438. C3#92020semidirect
439. C3#92101semidirect

Maximal subgroups

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

subgroupindex
#106242852
C3×#68192
C3×C5×#2372
C2×#671532
#106242102
#106242082
#106242072
#106241992
#106241982
D3×Dic302
Dic6×D152
#921013
#920203
#643095