Finite Group Databasebeta

← #13088955

Subgroups — #13088955

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#10624219C211yesmaximal, characteristic, complemented
2#10624199C211yesmaximal, characteristic, complemented
3#10624196C211yesmaximal, characteristic, complemented
4C5×#3795C211yesmaximal, characteristic, complemented
5#10624153C211yesmaximal, characteristic, complemented
6C3×#6768C211yesmaximal, characteristic, complemented
7C3×D3×Dic10C211yesmaximal, characteristic, complemented
8#10624055C212yesmaximal, complemented
9#10624050C212yesmaximal, complemented
10#10624048C212yesmaximal, complemented
11D3×#1204C212yesmaximal, complemented
12#92020—31nomaximal
13#92004—31nomaximal
14C4×C3⋊D15C2211yescharacteristic
15C5×#634C2211yescharacteristic
16C3×C4×D15C2211yescharacteristic
17C3×Dic30C2211yescharacteristic
18C3×C5×D12C2211yescharacteristic
19C3×C4×C5×D3C2211yescharacteristic
20C32×Dic10C2211yescharacteristic
21#67158C2212yes—
22#67154C2212yes—
23#67153C2211yescharacteristic
24#67152C2211yescharacteristic
25#67152C2212yes—
26#67151C2211yescharacteristic
27#67150C2211yescharacteristic
28D3×Dic15C2212yes—
29C5×#625C2212yes—
30C5×D3×Dic3C2212yes—
31#67141C2212yes—
32#67137C2212yes—
33C3×#1204C2212yes—
34C3×#1203C2212yes—
35C3×D3×Dic5C2212yes—
36C3×D3×Dic5C2212yes—
37#64315—51nomaximal, Hall
38#6819—31no—
39#6817—31no—
40C5×#237—31no—
41C5×#235—31no—
42C3×#711—31no—
43C2×C3×Dic10—31no—
44C2×#1204—32no—
45#6782—32no—
46#6782—32no—
47#6782—32no—
48#6780—32no—
49#6779—32no—
50#6771—31no—
51D15⋊Q8—31no—
52D15⋊Q8—31no—
53D15⋊Q8—31no—
54D15⋊Q8—61no—
55#6769—32no—
56#6768—31no—
57#6768D311yescharacteristic, complemented
58D3×Dic10—32no—
59D3×Dic10D311yescharacteristic, complemented
60D5×Dic6—31no—
61C2×C3⋊D15C2311yescharacteristic
62C2×C3×D15C2311yescharacteristic
63C2×C3×C5×D3C2311yescharacteristic
64C2×C3×C5×D3C2312yes—
65C32×C4×C5C2311yescharacteristic
66C3⋊Dic15C2311yescharacteristic
67C5×C3⋊Dic3C2312yes—
68C3×Dic15C2311yescharacteristic
69C3×Dic15C2312yes—
70C3×C5×Dic3C2311yescharacteristic
71C32×Dic5C2312yes—
72#4150—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—52no—
81#3795D511yescharacteristic, complemented
82D3×Dic6—51no—
83D3×Dic6—52no—
84C3×C5×D4—31no—
85C2×C3×C4×C5—31no—
86#1220—31no—
87#1220—32no—
88#1220—31no—
89C2×Dic15—32no—
90D60—31no—
91C4×D15—31no—
92C4×D15—31no—
93C4×D15—61no—
94C4×D15C2×D311yescharacteristic
95Dic30—31no—
96Dic30C2×D311yescharacteristic
97C5×#67—32no—
98C5×#67—32no—
99C2×C5×Dic3—32no—
100C5×D12C2×D311yescharacteristic
101C4×C5×D3—32no—
102C4×C5×D3C2×D311yescharacteristic
103C5×Dic6—31no—
104C5×Dic6—31no—
105C5×Dic6—61no—
106C3×C22⋊D5—32no—
107C2×C3×Dic5—32no—
108C2×C3×Dic5—32no—
109C3×C4×D5—31no—
110C3×Dic10—21no—
111C3×Dic10—31no—
112C3×Dic10—32no—
113C3×Dic10C2×D311yescharacteristic
114C3×Dic10C2×D311yescharacteristic
115#1204—32no—
116#1204—32no—
117#1204—32no—
118#1204—32no—
119#1204—31no—
120#1204—62no—
121#1204C2×D312yes—
122#1203—32no—
123#1203—31no—
124#1203C2×D312yes—
125#1202—31no—
126#1201—31no—
127#1200—32no—
128#1200—32no—
129#1200—32no—
130#1200—62no—
131D3×Dic5—32no—
132D3×Dic5C2×D312yes—
133D3×Dic5C2×D312yes—
134D5×Dic3—32no—
135#1010—151no—
136#1007—151no—
137C2×C32×C5C2411yescharacteristic
138C3⋊D15—21no—
139C3×D15—21no—
140C3×C5×D3—21no—
141C3×C5×D3—22no—
142C5×#41—91no—
143Q8×D5—91no—
144Q8×D5—91no—
145#711—31no—
146#711—91no—
147#711—92no—
148#711—92no—
149#709—92no—
150#709—91no—
151C2×Dic10—31no—
152C2×Dic10—92no—
153C32×Q8—51no—
154C4×C3⋊D3—51no—
155C4×C3⋊D3—52no—
156#634—52no—
157#634C2×D511yescharacteristic
158C3×D12—52no—
159C3×D12C2×D511yescharacteristic
160C3×C4×D3—52no—
161C3×C4×D3—52no—
162C3×C4×D3—51no—
163C3×C4×D3C2×D511yescharacteristic
164C3×Dic6—51no—
165C3×Dic6—52no—
166#627—52no—
167#627—51no—
168#626—52no—
169#626—51no—
170#626—51no—
171#625—51no—
172#625C2×D512yes—
173D3×Dic3—52no—
174D3×Dic3—52no—
175D3×Dic3C2×D512yes—
176C22×C3×C5—31no—
177C22×C3×C5—32no—
178C2×D15—31no—
179C2×D15—31no—
180C2×D15—61no—
181C2×D15C22×D311yescharacteristic
182C2×C5×D3C22×D312yes—
183C2×C5×D3C22×D311yescharacteristic
184C2×C3×D5—31no—
185C3×C4×C5—21no—
186C3×C4×C5—31no—
187C3×C4×C5C22×D311yescharacteristic
188C3×C4×C5C22×D311yescharacteristic
189Dic15—31no—
190Dic15—31no—
191Dic15—61no—
192Dic15C22×D312yes—
193Dic15C22×D311yescharacteristic
194C3×Dic5—22no—
195C3×Dic5—32no—
196C3×Dic5—31no—
197C3×Dic5C22×D312yes—
198C3×Dic5C22×D312yes—
199C5×Dic3—32no—
200C5×Dic3—32no—
201C5×Dic3—62no—
202C5×Dic3C22×D311yescharacteristic
203C3×#41—151no—
204C2×C3×Q8—151no—
205#239—51no—
206#239—152no—
207#239—151no—
208D3×Q8—51no—
209D3×Q8—151no—
210D3×Q8—151no—
211D3×Q8—151no—
212D3×Q8—152no—
213D3×Q8—301no—
214#237—31no—
215#237—152no—
216#237—151no—
217#237—152no—
218#235—31no—
219#235—152no—
220#235—152no—
221#235—151no—
222#235—152no—
223#235—151no—
224C2×Dic6—151no—
225C2×Dic6—152no—
226C32×C52−1+411yescharacteristic, complemented, Hall
227C5×Q8—91no—
228C5×D4—31no—
229C5×D4—92no—
230C2×C4×C5—31no—
231C2×C4×C5—92no—
232C22⋊D5—32no—
233C22⋊D5—92no—
234C22⋊D5—91no—
235C22⋊D5—91no—
236C2×Dic5—32no—
237C2×Dic5—32no—
238C2×Dic5—92no—
239D20—91no—
240C4×D5—31no—
241C4×D5—92no—
242C4×D5—91no—
243C4×D5—92no—
244Dic10—31no—
245Dic10—32no—
246Dic10—92no—
247Dic10—91no—
248Dic10—92no—
249Dic10D3211yescharacteristic
250C2×C3⋊D3—51no—
251C2×C3×D3—51no—
252C2×C3×D3C22×D512yes—
253C2×C3×D3C22×D511yescharacteristic
254C32×C4—52no—
255C32×C4C22×D511yescharacteristic
256C3⋊Dic3—51no—
257C3⋊Dic3C22×D512yes—
258C3×Dic3—52no—
259C3×Dic3—51no—
260C3×Dic3C22×D511yescharacteristic
2612−1+4—451noSylow
262C2×C3×C5—21no—
263C2×C3×C5—62no—
264C2×C3×C5—61no—
265C2×C3×C5C23×D311yescharacteristic
266C2×C3×C5C23×D311yescharacteristic
267D15—21no—
268D15—62no—
269D15—61no—
270D15—61no—
271C3×D5—61no—
272C5×D3—21no—
273C5×D3—22no—
274C3×Q8—51no—
275C3×Q8—51no—
276C3×Q8—101no—
277C3×Q8—152no—
278C3×Q8—151no—
279C3×D4—31no—
280C3×D4—152no—
281C2×C3×C4—31no—
282C2×C3×C4—152no—
283C2×C3×C4—152no—
284C2×C3×C4—151no—
285#67—32no—
286#67—32no—
287#67—151no—
288#67—152no—
289#67—151no—
290#67—151no—
291#67—151no—
292C2×Dic3—32no—
293C2×Dic3—152no—
294C2×Dic3—152no—
295D12—52no—
296D12—151no—
297D12—151no—
298D12—152no—
299D12D3×D511yescharacteristic
300C4×D3—32no—
301C4×D3—52no—
302C4×D3—52no—
303C4×D3—51no—
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—51no—
316Dic6—52no—
317Dic6—61no—
318Dic6—152no—
319Dic6—152no—
320Dic6—152no—
321Dic6—152no—
322Dic6—151no—
323Dic6—151no—
324Dic6—302no—
325C22×C5—31no—
326C22×C5—32no—
327C2×D5—31no—
328C2×D5—91no—
329C4×C5—31no—
330C4×C5—92no—
331C4×C5C2×D3211yescharacteristic
332Dic5—32no—
333Dic5—31no—
334Dic5—91no—
335Dic5C2×D3212yes—
336C2×C32C23×D511yescharacteristic
337C3⋊D3—101no—
338C3×D3—21no—
339C3×D3—22no—
340C3×D3—101no—
341#41—91no—
342#41—151no—
343#41—451no—
344#41—452no—
345#41—452no—
346#41—452no—
347#41—451no—
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—31no—
357C22×C3—151no—
358C2×D3—51no—
359C2×D3—151no—
360C2×D3—151no—
361C2×D3—301no—
362C2×D3C2×D3×D512yes—
363C2×D3C2×D3×D511yescharacteristic
364C3×C4—21no—
365C3×C4—31no—
366C3×C4—52no—
367C3×C4—52no—
368C3×C4—102no—
369C3×C4—151no—
370C3×C4—152no—
371C3×C4C2×D3×D511yescharacteristic
372C3×C4C2×D3×D511yescharacteristic
373Dic3—32no—
374Dic3—32no—
375Dic3—52no—
376Dic3—51no—
377Dic3—62no—
378Dic3—151no—
379Dic3—151no—
380Dic3—301no—
381Dic3C2×D3×D511yescharacteristic
382C2×C5—61no—
383C2×C5—62no—
384C2×C5C22×D3211yescharacteristic
385D5—61no—
386D5—181no—
387C32#415011yescharacteristic, complemented, Sylow
388Q8—51no—
389Q8—91no—
390Q8—152no—
391Q8—151no—
392Q8—451no—
393Q8—452no—
394Q8—452no—
395D4—31no—
396D4—92no—
397D4—152no—
398D4—451no—
399D4—451no—
400D4—452no—
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—62no—
413C2×C3—61no—
414C2×C3—301no—
415C2×C3C22×D3×D511yescharacteristic
416C2×C3C22×D3×D511yescharacteristic
417D3—21no—
418D3—22no—
419D3—101no—
420D3—302no—
421D3—301no—
422D3—301no—
423C5#6431511yescharacteristic, complemented, Sylow
424C22—31no—
425C22—32no—
426C22—151no—
427C22—451no—
428C4—31no—
429C4—52no—
430C4—92no—
431C4—151no—
432C4—152no—
433C4—451no—
434C4D32×D511yescharacteristic
435C3—21no—
436C3#9200411yescharacteristic, complemented
437C3#9202011yescharacteristic, complemented
438C2—61no—
439C2—62no—
440C2—301no—
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. #10624219C2semidirect
2. #10624199C2semidirect
3. #10624196C2semidirect
4. C5×#3795C2semidirect
5. #10624153C2semidirect
6. C3×#6768C2semidirect
7. C3×D3×Dic10C2semidirect
8. #10624055C2semidirect
9. #10624050C2semidirect
10. #10624048C2semidirect
11. D3×#1204C2semidirect
57. #6768D3semidirect
59. D3×Dic10D3semidirect
81. #3795D5semidirect
226. C32×C52−1+4semidirect
353. C3×C5#1007semidirect
354. C3×C5#1010semidirect
387. C32#4150semidirect
423. C5#64315semidirect
436. C3#92004semidirect
437. C3#92020semidirect

Maximal subgroups

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

subgroupindex
#106242192
#106241992
#106241962
C5×#37952
#106241532
C3×#67682
C3×D3×Dic102
#106240552
#106240502
#106240482
D3×#12042
#920203
#920043
#643155