Finite Group Databasebeta

← #13088962

Subgroups — #13088962

Order 1440.

622 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10624221C211yesmaximal, characteristic, complemented
2#10624199C211yesmaximal, characteristic, complemented
3#10624194C211yesmaximal, characteristic, complemented
4C5×D3×Dic6C211yesmaximal, characteristic, complemented
5#10624153C211yesmaximal, characteristic, complemented
6C3×D15⋊Q8C211yesmaximal, characteristic, complemented
7C3×#6769C211yesmaximal, characteristic, complemented
8#10624055C212yesmaximal, complemented
9#10624053C212yesmaximal, complemented
10#10624052C212yesmaximal, complemented
11#10624038C212yesmaximal, complemented
12#92029—31nomaximal
13#92000—31nomaximal
14C4×C3⋊D15C2211yescharacteristic
15C5×#634C2211yescharacteristic
16C3×D60C2211yescharacteristic
17C3×C4×D15C2211yescharacteristic
18C3×C4×C5×D3C2211yescharacteristic
19C3×C5×Dic6C2211yescharacteristic
20C32×Dic10C2211yescharacteristic
21#67157C2212yes—
22#67154C2212yes—
23#67153C2211yescharacteristic
24#67152C2211yescharacteristic
25#67151C2211yescharacteristic
26#67151C2212yes—
27#67150C2211yescharacteristic
28Dic3×D15C2212yes—
29C5×#627C2212yes—
30C5×D3×Dic3C2212yes—
31#67141C2212yes—
32#67137C2212yes—
33C3×#1204C2212yes—
34C3×#1203C2212yes—
35C3×#1200C2212yes—
36C3×#1200C2212yes—
37#64315—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—32no—
48#6773—32no—
49#6771—31no—
50#6771—32no—
51D15⋊Q8—31no—
52D15⋊Q8—31no—
53D15⋊Q8—32no—
54D15⋊Q8—61no—
55D15⋊Q8D311yescharacteristic, complemented
56#6769—32no—
57#6769D311yescharacteristic, complemented
58#6766—31no—
59#6766—31no—
60D5×Dic6—32no—
61C2×C3⋊D15C2311yescharacteristic
62C2×C3×D15C2311yescharacteristic
63C2×C3×D15C2312yes—
64C2×C3×C5×D3C2311yescharacteristic
65C32×C4×C5C2311yescharacteristic
66C3⋊Dic15C2311yescharacteristic
67C5×C3⋊Dic3C2312yes—
68C3×Dic15C2311yescharacteristic
69C3×C5×Dic3C2311yescharacteristic
70C3×C5×Dic3C2312yes—
71C32×Dic5C2312yes—
72#4147—91noHall
73Q8×C3⋊D3—51no—
74C3×#239—51no—
75C3×D3×Q8—51no—
76#3798—51no—
77#3798—52no—
78#3796—51no—
79#3796—52no—
80#3795—51no—
81#3795—52no—
82D3×Dic6—52no—
83D3×Dic6D511yescharacteristic, complemented
84C3×C5×Q8—31no—
85C2×C3×C4×C5—31no—
86#1220—31no—
87#1220—31no—
88D60—32no—
89D60—31no—
90D60C2×D311yescharacteristic
91C4×D15—32no—
92C4×D15—31no—
93C4×D15—31no—
94C4×D15—61no—
95C4×D15C2×D311yescharacteristic
96Dic30—31no—
97C2×C5×Dic3—32no—
98C4×C5×D3—32no—
99C4×C5×D3C2×D311yescharacteristic
100C5×Dic6—31no—
101C5×Dic6—31no—
102C5×Dic6—32no—
103C5×Dic6—32no—
104C5×Dic6—61no—
105C5×Dic6C2×D311yescharacteristic
106C3×C22⋊D5—32no—
107C3×D20—31no—
108C3×C4×D5—31no—
109C3×C4×D5—32no—
110C3×C4×D5—32no—
111C3×Dic10—21no—
112C3×Dic10—32no—
113C3×Dic10C2×D311yescharacteristic
114C3×Dic10C2×D311yescharacteristic
115#1204—32no—
116#1204—32no—
117#1204—31no—
118#1204—62no—
119#1204C2×D312yes—
120#1203—31no—
121#1203C2×D312yes—
122#1202—32no—
123#1202—32no—
124#1202—32no—
125#1202—31no—
126#1201—31no—
127#1200—32no—
128#1200—32no—
129#1200—32no—
130#1200—62no—
131#1200C2×D312yes—
132#1200C2×D312yes—
133D5×Dic3—32no—
134D5×Dic3—32no—
135#1010—151no—
136#1007—151no—
137C2×C32×C5C2411yescharacteristic
138C3⋊D15—21no—
139C3×D15—22no—
140C3×D15—21no—
141C3×C5×D3—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—92no—
152#709—91no—
153C32×Q8—51no—
154C4×C3⋊D3—51no—
155C4×C3⋊D3—52no—
156#634—52no—
157#634C2×D511yescharacteristic
158C3×D12—52no—
159C3×D12—51no—
160C3×C4×D3—51no—
161C3×C4×D3—52no—
162C3×C4×D3—52no—
163C3×C4×D3C2×D511yescharacteristic
164C3×Dic6—52no—
165C3×Dic6C2×D511yescharacteristic
166#627—51no—
167#627C2×D512yes—
168#626—51no—
169#626—52no—
170#626—51no—
171#625—51no—
172#625—52no—
173D3×Dic3—52no—
174D3×Dic3—52no—
175D3×Dic3C2×D512yes—
176C22×C3×C5—31no—
177C2×D15—31no—
178C2×D15—31no—
179C2×D15—61no—
180C2×D15C22×D312yes—
181C2×D15C22×D311yescharacteristic
182C2×C5×D3C22×D311yescharacteristic
183C2×C3×D5—32no—
184C2×C3×D5—31no—
185C3×C4×C5—21no—
186C3×C4×C5—32no—
187C3×C4×C5—31no—
188C3×C4×C5C22×D311yescharacteristic
189C3×C4×C5C22×D311yescharacteristic
190Dic15—31no—
191Dic15—31no—
192Dic15—61no—
193Dic15C22×D311yescharacteristic
194C3×Dic5—22no—
195C3×Dic5—31no—
196C3×Dic5C22×D312yes—
197C3×Dic5C22×D312yes—
198C5×Dic3—32no—
199C5×Dic3—32no—
200C5×Dic3—62no—
201C5×Dic3C22×D312yes—
202C5×Dic3C22×D311yescharacteristic
203C3×#41—151no—
204C2×C3×Q8—151no—
205#239—51no—
206#239—152no—
207#239—151no—
208D3×Q8—31no—
209D3×Q8—51no—
210D3×Q8—152no—
211D3×Q8—151no—
212D3×Q8—151no—
213D3×Q8—301no—
214#237—151no—
215#237—152no—
216#237—152no—
217#237—151no—
218#235—152no—
219#235—152no—
220#235—151no—
221#235—152no—
222#235—151no—
223#235—151no—
224C2×Dic6—31no—
225C2×Dic6—152no—
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—32no—
233C22⋊D5—91no—
234C22⋊D5—91no—
235D20—31no—
236D20—92no—
237D20—91no—
238D20—92no—
239C4×D5—31no—
240C4×D5—32no—
241C4×D5—32no—
242C4×D5—92no—
243C4×D5—92no—
244C4×D5—91no—
245C4×D5—92no—
246Dic10—32no—
247Dic10—91no—
248Dic10—92no—
249Dic10D3211yescharacteristic
250C2×C3⋊D3—51no—
251C2×C3×D3—52no—
252C2×C3×D3—51no—
253C2×C3×D3C22×D511yescharacteristic
254C32×C4—52no—
255C32×C4C22×D511yescharacteristic
256C3⋊Dic3—51no—
257C3⋊Dic3C22×D512yes—
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—22no—
268D15—62no—
269D15—61no—
270D15—61no—
271C3×D5—62no—
272C3×D5—61no—
273C5×D3—21no—
274C3×Q8—31no—
275C3×Q8—51no—
276C3×Q8—51no—
277C3×Q8—101no—
278C3×Q8—152no—
279C3×D4—152no—
280C3×D4—151no—
281C2×C3×C4—31no—
282C2×C3×C4—152no—
283C2×C3×C4—152no—
284C2×C3×C4—151no—
285#67—152no—
286#67—152no—
287#67—151no—
288#67—151no—
289#67—151no—
290#67—151no—
291#67—152no—
292C2×Dic3—32no—
293C2×Dic3—152no—
294C2×Dic3—152no—
295D12—52no—
296D12—51no—
297D12—151no—
298D12—152no—
299D12—151no—
300C4×D3—32no—
301C4×D3—51no—
302C4×D3—52no—
303C4×D3—52no—
304C4×D3—152no—
305C4×D3—152no—
306C4×D3—152no—
307C4×D3—152no—
308C4×D3—151no—
309C4×D3—151no—
310C4×D3—302no—
311C4×D3—301no—
312C4×D3D3×D511yescharacteristic
313Dic6—31no—
314Dic6—31no—
315Dic6—32no—
316Dic6—32no—
317Dic6—52no—
318Dic6—61no—
319Dic6—152no—
320Dic6—152no—
321Dic6—151no—
322Dic6—151no—
323Dic6—302no—
324Dic6D3×D511yescharacteristic
325C22×C5—31no—
326C2×D5—31no—
327C2×D5—32no—
328C2×D5—91no—
329C4×C5—32no—
330C4×C5—31no—
331C4×C5—92no—
332C4×C5C2×D3211yescharacteristic
333Dic5—31no—
334Dic5—91no—
335Dic5C2×D3212yes—
336C2×C32C23×D511yescharacteristic
337C3⋊D3—101no—
338C3×D3—21no—
339C3×D3—102no—
340C3×D3—101no—
341#41—151no—
342#41—451no—
343#41—452no—
344#41—452no—
345#41—452no—
346#41—451no—
347#41—451no—
348C2×Q8—91no—
349C2×Q8—151no—
350C2×Q8—452no—
351C2×Q8—451no—
352C3×C5—21no—
353C3×C5#100711yescharacteristic, complemented
354C3×C5#101011yescharacteristic, complemented
355C22×C3—31no—
356C22×C3—152no—
357C22×C3—151no—
358C2×D3—51no—
359C2×D3—52no—
360C2×D3—151no—
361C2×D3—151no—
362C2×D3—301no—
363C2×D3C2×D3×D511yescharacteristic
364C3×C4—21no—
365C3×C4—31no—
366C3×C4—32no—
367C3×C4—52no—
368C3×C4—52no—
369C3×C4—102no—
370C3×C4—151no—
371C3×C4C2×D3×D511yescharacteristic
372C3×C4C2×D3×D511yescharacteristic
373Dic3—32no—
374Dic3—32no—
375Dic3—51no—
376Dic3—62no—
377Dic3—151no—
378Dic3—151no—
379Dic3—301no—
380Dic3C2×D3×D511yescharacteristic
381Dic3C2×D3×D512yes—
382C2×C5—61no—
383C2×C5C22×D3211yescharacteristic
384D5—61no—
385D5—62no—
386D5—181no—
387C32#414711yescharacteristic, complemented, Sylow
388Q8—31no—
389Q8—51no—
390Q8—92no—
391Q8—91no—
392Q8—152no—
393Q8—451no—
394Q8—452no—
395D4—152no—
396D4—151no—
397D4—451no—
398D4—452no—
399D4—452no—
400D4—451no—
401D4—451no—
402C2×C4—31no—
403C2×C4—92no—
404C2×C4—152no—
405C2×C4—152no—
406C2×C4—151no—
407C2×C4—452no—
408C2×C4—452no—
409C2×C4—452no—
410C2×C4—451no—
411C2×C3—21no—
412C2×C3—61no—
413C2×C3—301no—
414C2×C3—302no—
415C2×C3C22×D3×D511yescharacteristic
416C2×C3C22×D3×D511yescharacteristic
417D3—21no—
418D3—102no—
419D3—101no—
420D3—302no—
421D3—301no—
422D3—301no—
423C5#6431511yescharacteristic, complemented, Sylow
424C22—31no—
425C22—151no—
426C22—152no—
427C22—451no—
428C4—31no—
429C4—32no—
430C4—52no—
431C4—92no—
432C4—151no—
433C4—451no—
434C4D32×D511yescharacteristic
435C3—21no—
436C3#9200011yescharacteristic, complemented
437C3#9202911yescharacteristic, complemented
438C2—61no—
439C2—301no—
440C2—302no—
441C2—901no—
442C2C2×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. #10624221C2semidirect
2. #10624199C2semidirect
3. #10624194C2semidirect
4. C5×D3×Dic6C2semidirect
5. #10624153C2semidirect
6. C3×D15⋊Q8C2semidirect
7. C3×#6769C2semidirect
8. #10624055C2semidirect
9. #10624053C2semidirect
10. #10624052C2semidirect
11. #10624038C2semidirect
55. D15⋊Q8D3semidirect
57. #6769D3semidirect
83. D3×Dic6D5semidirect
226. C32×C52−1+4semidirect
353. C3×C5#1007semidirect
354. C3×C5#1010semidirect
387. C32#4147semidirect
423. C5#64315semidirect
436. C3#92000semidirect
437. C3#92029semidirect

Maximal subgroups

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

subgroupindex
#106242212
#106241992
#106241942
C5×D3×Dic62
#106241532
C3×D15⋊Q82
C3×#67692
#106240552
#106240532
#106240522
#106240382
#920293
#920003
#643155