Finite Group Databasebeta

← #13088960

Subgroups — #13088960

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#10624223C211yesmaximal, characteristic, complemented
2#10624200C211yesmaximal, characteristic, complemented
3Dic6×D15C211yesmaximal, characteristic, complemented
4C5×#3796C211yesmaximal, characteristic, complemented
5C3⋊D3×Dic10C211yesmaximal, characteristic, complemented
6C3×D15⋊Q8C211yesmaximal, characteristic, complemented
7C3×#6768C211yesmaximal, characteristic, complemented
8#10624056C212yesmaximal, complemented
9#10624053C212yesmaximal, complemented
10#10624049C212yesmaximal, complemented
11D3×#1204C212yesmaximal, complemented
12#92030—31nomaximal
13#92020—31nomaximal
14#67180C2211yescharacteristic
15C4×C5×C3⋊D3C2211yescharacteristic
16C3×C4×D15C2211yescharacteristic
17C3×C4×D15C2211yescharacteristic
18C3×C5×D12C2211yescharacteristic
19C3×C5×Dic6C2211yescharacteristic
20C32×Dic10C2211yescharacteristic
21#67158C2211yescharacteristic
22#67157C2211yescharacteristic
23#67156C2211yescharacteristic
24#67156C2211yescharacteristic
25#67153C2212yes—
26#67150C2212yes—
27D3×Dic15C2212yes—
28Dic3×D15C2212yes—
29C5×#626C2212yes—
30C5×D3×Dic3C2212yes—
31#67141C2212yes—
32C3⋊D3×Dic5C2212yes—
33C3×#1204C2212yes—
34C3×#1203C2212yes—
35C3×#1200C2212yes—
36C3×D3×Dic5C2212yes—
37#64315—51nomaximal, Hall
38Q8×D15—31no—
39#6819—31no—
40C5×#239—31no—
41C5×#237—31no—
42C3×Q8×D5—31no—
43C3×#711—31no—
44C2×#1204—32no—
45#6782—32no—
46#6780—32no—
47#6779—32no—
48#6772—31no—
49#6772—32no—
50#6771—31no—
51#6771—31no—
52#6771—32no—
53D15⋊Q8D311yescharacteristic, complemented
54#6768—32no—
55#6768D311yescharacteristic, complemented
56D3×Dic10—31no—
57D3×Dic10—31no—
58D3×Dic10—32no—
59D3×Dic10—61no—
60D5×Dic6—31no—
61C2×C5×C3⋊D3C2311yescharacteristic
62C2×C3×D15C2311yescharacteristic
63C2×C3×D15C2311yescharacteristic
64C2×C3×C5×D3C2312yes—
65C32×C4×C5C2311yescharacteristic
66C3⋊Dic15C2312yes—
67C5×C3⋊Dic3C2311yescharacteristic
68C3×Dic15C2311yescharacteristic
69C3×Dic15C2311yescharacteristic
70C3×C5×Dic3C2312yes—
71C32×Dic5C2312yes—
72#4150—91noHall
73Q8×C3⋊D3—51no—
74C3×#239—51no—
75C3×D3×Q8—51no—
76#3798—51no—
77#3798—52no—
78#3796—52no—
79#3796D511yescharacteristic, complemented
80#3795—51no—
81#3795—52no—
82D3×Dic6—51no—
83D3×Dic6—52no—
84C3×C5×Q8—31no—
85C3×C5×D4—31no—
86#1220—32no—
87C2×Dic15—32no—
88C4×D15—32no—
89C4×D15C2×D311yescharacteristic
90C4×D15C2×D311yescharacteristic
91Dic30—31no—
92Dic30—31no—
93Dic30—32no—
94Dic30—61no—
95C5×#67—32no—
96C2×C5×Dic3—32no—
97C5×D12—32no—
98C5×D12C2×D311yescharacteristic
99C4×C5×D3—32no—
100C4×C5×D3—31no—
101C4×C5×D3—31no—
102C4×C5×D3—61no—
103C5×Dic6C2×D311yescharacteristic
104C3×C22⋊D5—32no—
105C2×C3×Dic5—32no—
106C3×C4×D5—31no—
107C3×C4×D5—31no—
108C3×C4×D5—32no—
109C3×Dic10—21no—
110C3×Dic10—32no—
111C3×Dic10C2×D311yescharacteristic
112C3×Dic10C2×D311yescharacteristic
113#1204—31no—
114#1204—31no—
115#1204—32no—
116#1204—32no—
117#1204—32no—
118#1204—62no—
119#1204C2×D312yes—
120#1203—31no—
121#1203—31no—
122#1203C2×D312yes—
123#1202—31no—
124#1202—31no—
125#1201—32no—
126#1201—31no—
127#1201—31no—
128#1200C2×D312yes—
129D3×Dic5—32no—
130D3×Dic5—32no—
131D3×Dic5—32no—
132D3×Dic5—62no—
133D3×Dic5C2×D312yes—
134D5×Dic3—32no—
135#1010—151no—
136#1007—151no—
137C2×C32×C5C2411yescharacteristic
138C5×C3⋊D3—21no—
139C3×D15—21no—
140C3×D15—21no—
141C3×C5×D3—22no—
142C5×#41—91no—
143Q8×D5—31no—
144Q8×D5—91no—
145#711—31no—
146#711—92no—
147#711—92no—
148#711—91no—
149#709—91no—
150#709—92no—
151C2×Dic10—92no—
152C2×Dic10—91no—
153C32×Q8—51no—
154C4×C3⋊D3—52no—
155C4×C3⋊D3C2×D511yescharacteristic
156#634—51no—
157#634—52no—
158C3×D12—52no—
159C3×D12C2×D511yescharacteristic
160C3×C4×D3—52no—
161C3×C4×D3—51no—
162C3×C4×D3—51no—
163C3×C4×D3—52no—
164C3×Dic6—52no—
165C3×Dic6C2×D511yescharacteristic
166#627—51no—
167#627—52no—
168#626—51no—
169#626—51no—
170#626C2×D512yes—
171#625—52no—
172#625—51no—
173D3×Dic3—52no—
174D3×Dic3—52no—
175D3×Dic3C2×D512yes—
176C22×C3×C5—32no—
177C2×D15C22×D311yescharacteristic
178C2×D15C22×D311yescharacteristic
179C2×C5×D3—31no—
180C2×C5×D3—31no—
181C2×C5×D3—61no—
182C2×C5×D3C22×D312yes—
183C2×C3×D5—31no—
184C2×C3×D5—31no—
185C3×C4×C5—21no—
186C3×C4×C5—32no—
187C3×C4×C5C22×D311yescharacteristic
188C3×C4×C5C22×D311yescharacteristic
189Dic15—32no—
190Dic15—32no—
191Dic15—62no—
192Dic15C22×D311yescharacteristic
193Dic15C22×D311yescharacteristic
194C3×Dic5—22no—
195C3×Dic5—31no—
196C3×Dic5—31no—
197C3×Dic5C22×D312yes—
198C3×Dic5C22×D312yes—
199C5×Dic3—31no—
200C5×Dic3—31no—
201C5×Dic3—61no—
202C5×Dic3C22×D312yes—
203C3×#41—151no—
204C2×C3×Q8—151no—
205#239—31no—
206#239—51no—
207#239—152no—
208D3×Q8—51no—
209D3×Q8—152no—
210D3×Q8—151no—
211D3×Q8—151no—
212D3×Q8—151no—
213D3×Q8—301no—
214#237—31no—
215#237—152no—
216#237—152no—
217#237—151no—
218#235—151no—
219#235—151no—
220#235—152no—
221#235—152no—
222#235—152no—
223#235—151no—
224C2×Dic6—152no—
225C2×Dic6—151no—
226C32×C52−1+411yescharacteristic, complemented, Hall
227C5×Q8—31no—
228C5×D4—31no—
229C5×D4—92no—
230C2×C4×C5—92no—
231C2×C4×C5—91no—
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—31no—
242C4×D5—32no—
243C4×D5—92no—
244Dic10—32no—
245Dic10—91no—
246Dic10—92no—
247Dic10—91no—
248Dic10—92no—
249Dic10D3211yescharacteristic
250C2×C3⋊D3C22×D511yescharacteristic
251C2×C3×D3—51no—
252C2×C3×D3—51no—
253C2×C3×D3C22×D512yes—
254C32×C4—52no—
255C32×C4C22×D511yescharacteristic
256C3⋊Dic3—52no—
257C3⋊Dic3C22×D511yescharacteristic
258C3×Dic3—51no—
259C3×Dic3—51no—
260C3×Dic3C22×D512yes—
2612−1+4—451noSylow
262C2×C3×C5—21no—
263C2×C3×C5—62no—
264C2×C3×C5C23×D311yescharacteristic
265C2×C3×C5C23×D311yescharacteristic
266D15—21no—
267D15—21no—
268C3×D5—61no—
269C3×D5—61no—
270C5×D3—22no—
271C5×D3—62no—
272C5×D3—61no—
273C5×D3—61no—
274C3×Q8—31no—
275C3×Q8—51no—
276C3×Q8—51no—
277C3×Q8—101no—
278C3×Q8—152no—
279C3×D4—31no—
280C3×D4—152no—
281C2×C3×C4—152no—
282C2×C3×C4—152no—
283C2×C3×C4—151no—
284C2×C3×C4—151no—
285#67—32no—
286#67—151no—
287#67—151no—
288#67—151no—
289#67—151no—
290#67—152no—
291#67—152no—
292C2×Dic3—32no—
293C2×Dic3—152no—
294C2×Dic3—152no—
295D12—32no—
296D12—52no—
297D12—151no—
298D12—151no—
299D12D3×D511yescharacteristic
300C4×D3—32no—
301C4×D3—31no—
302C4×D3—31no—
303C4×D3—52no—
304C4×D3—51no—
305C4×D3—51no—
306C4×D3—52no—
307C4×D3—61no—
308C4×D3—152no—
309C4×D3—152no—
310C4×D3—152no—
311C4×D3—152no—
312C4×D3—302no—
313Dic6—52no—
314Dic6—151no—
315Dic6—151no—
316Dic6—152no—
317Dic6—152no—
318Dic6—152no—
319Dic6—151no—
320Dic6—151no—
321Dic6—152no—
322Dic6—302no—
323Dic6—301no—
324Dic6D3×D511yescharacteristic
325C22×C5—32no—
326C22×C5—91no—
327C2×D5—31no—
328C2×D5—31no—
329C4×C5—32no—
330C4×C5—91no—
331C4×C5C2×D3211yescharacteristic
332Dic5—31no—
333Dic5—31no—
334Dic5—92no—
335Dic5C2×D3212yes—
336C2×C32C23×D511yescharacteristic
337C3⋊D3—21no—
338C3×D3—22no—
339C3×D3—101no—
340C3×D3—101no—
341#41—91no—
342#41—151no—
343#41—451no—
344#41—452no—
345#41—452no—
346#41—451no—
347#41—452no—
348C2×Q8—151no—
349C2×Q8—452no—
350C2×Q8—451no—
351C2×Q8—451no—
352C3×C5—21no—
353C3×C5#100711yescharacteristic, complemented
354C3×C5#101011yescharacteristic, complemented
355C22×C3—32no—
356C22×C3—151no—
357C22×C3—151no—
358C2×D3—31no—
359C2×D3—31no—
360C2×D3—51no—
361C2×D3—51no—
362C2×D3—61no—
363C2×D3C2×D3×D512yes—
364C3×C4—21no—
365C3×C4—32no—
366C3×C4—52no—
367C3×C4—52no—
368C3×C4—102no—
369C3×C4—151no—
370C3×C4—151no—
371C3×C4C2×D3×D511yescharacteristic
372C3×C4C2×D3×D511yescharacteristic
373Dic3—31no—
374Dic3—31no—
375Dic3—51no—
376Dic3—51no—
377Dic3—61no—
378Dic3—152no—
379Dic3—152no—
380Dic3—302no—
381Dic3C2×D3×D512yes—
382C2×C5—62no—
383C2×C5—181no—
384C2×C5C22×D3211yescharacteristic
385D5—61no—
386D5—61no—
387C32#415011yescharacteristic, complemented, Sylow
388Q8—31no—
389Q8—51no—
390Q8—152no—
391Q8—452no—
392Q8—452no—
393Q8—451no—
394Q8—451no—
395D4—31no—
396D4—92no—
397D4—152no—
398D4—451no—
399D4—451no—
400D4—451no—
401D4—452no—
402C2×C4—92no—
403C2×C4—91no—
404C2×C4—152no—
405C2×C4—152no—
406C2×C4—151no—
407C2×C4—151no—
408C2×C4—452no—
409C2×C4—452no—
410C2×C4—452no—
411C2×C3—21no—
412C2×C3—62no—
413C2×C3—301no—
414C2×C3—301no—
415C2×C3C22×D3×D511yescharacteristic
416C2×C3C22×D3×D511yescharacteristic
417D3—22no—
418D3—62no—
419D3—61no—
420D3—61no—
421D3—101no—
422D3—101no—
423C5#6431511yescharacteristic, complemented, Sylow
424C22—32no—
425C22—91no—
426C22—151no—
427C22—151no—
428C4—32no—
429C4—52no—
430C4—91no—
431C4—151no—
432C4—151no—
433C4—452no—
434C4D32×D511yescharacteristic
435C3—21no—
436C3#9202011yescharacteristic, complemented
437C3#9203011yescharacteristic, complemented
438C2—62no—
439C2—181no—
440C2—301no—
441C2—301no—
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. #10624223C2semidirect
2. #10624200C2semidirect
3. Dic6×D15C2semidirect
4. C5×#3796C2semidirect
5. C3⋊D3×Dic10C2semidirect
6. C3×D15⋊Q8C2semidirect
7. C3×#6768C2semidirect
8. #10624056C2semidirect
9. #10624053C2semidirect
10. #10624049C2semidirect
11. D3×#1204C2semidirect
53. D15⋊Q8D3semidirect
55. #6768D3semidirect
79. #3796D5semidirect
226. C32×C52−1+4semidirect
353. C3×C5#1007semidirect
354. C3×C5#1010semidirect
387. C32#4150semidirect
423. C5#64315semidirect
436. C3#92020semidirect
437. C3#92030semidirect

Maximal subgroups

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

subgroupindex
#106242232
#106242002
Dic6×D152
C5×#37962
C3⋊D3×Dic102
C3×D15⋊Q82
C3×#67682
#106240562
#106240532
#106240492
D3×#12042
#920303
#920203
#643155