Finite Group Databasebeta

← #13088951

Subgroups — #13088951

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#10624200C211yesmaximal, characteristic, complemented
3D3×Dic30C211yesmaximal, characteristic, complemented
4C5×#3795C211yesmaximal, characteristic, complemented
5D5×#634C211yesmaximal, characteristic, complemented
6C3×#6772C211yesmaximal, characteristic, complemented
7C3×#6771C211yesmaximal, characteristic, complemented
8#10624050C212yesmaximal, complemented
9#10624046C212yesmaximal, complemented
10#10624045C212yesmaximal, complemented
11D3×#1204C212yesmaximal, complemented
12#92020—31nomaximal
13#92001—31nomaximal
14#67180C2211yescharacteristic
15C5×#634C2211yescharacteristic
16C3×C4×D15C2211yescharacteristic
17C3×Dic30C2211yescharacteristic
18C3×C5×D12C2211yescharacteristic
19C3×C4×C5×D3C2211yescharacteristic
20C32×C4×D5C2211yescharacteristic
21#67158C2212yes—
22#67154C2212yes—
23#67153C2212yes—
24#67150C2212yes—
25D3×Dic15C2212yes—
26D3×Dic15C2212yes—
27C5×#625C2212yes—
28C5×D3×Dic3C2212yes—
29#67141C2212yes—
30D5×C3⋊Dic3C2212yes—
31C3×#1204C2211yescharacteristic
32C3×#1203C2211yescharacteristic
33C3×#1202C2211yescharacteristic
34C3×#1201C2212yes—
35C3×#1201C2211yescharacteristic
36C3×D3×Dic5C2212yes—
37#64298—51nomaximal, Hall
38#6819—31no—
39C2×Dic30—31no—
40C5×#237—31no—
41C5×#235—31no—
42C3×#711—31no—
43C3×#709—31no—
44C2×#1204—32no—
45#6782—32no—
46#6780—32no—
47#6780—32no—
48#6780—32no—
49#6779—32no—
50#6772—31no—
51#6772D311yescharacteristic, complemented
52#6771D311yescharacteristic, complemented
53D15⋊Q8—31no—
54D3×Dic10—31no—
55D3×Dic10—32no—
56#6765—32no—
57D5×Dic6—31no—
58D5×Dic6—31no—
59D5×Dic6—31no—
60D5×Dic6—61no—
61C2×C3×D15C2311yescharacteristic
62C2×C3×C5×D3C2312yes—
63C2×C3×C5×D3C2311yescharacteristic
64C2×C32×D5C2311yescharacteristic
65C32×C4×C5C2311yescharacteristic
66C3⋊Dic15C2312yes—
67C5×C3⋊Dic3C2312yes—
68C3×Dic15C2312yes—
69C3×Dic15C2311yescharacteristic
70C3×C5×Dic3C2311yescharacteristic
71C32×Dic5C2311yescharacteristic
72#4150—91noHall
73C2×#634—51no—
74C3×#235—51no—
75C3×#235—51no—
76#3805—52no—
77#3805—52no—
78#3795—51no—
79#3795—52no—
80#3795D511yescharacteristic, complemented
81D3×Dic6—51no—
82D3×Dic6—51no—
83D3×Dic6—52no—
84C3×C5×D4—31no—
85C2×C3×C4×C5—31no—
86#1220—32no—
87C2×Dic15—32no—
88C2×Dic15—32no—
89C4×D15C2×D311yescharacteristic
90Dic30—31no—
91Dic30—31no—
92Dic30—32no—
93Dic30—61no—
94Dic30C2×D311yescharacteristic
95C5×#67—32no—
96C5×#67—32no—
97C2×C5×Dic3—32no—
98C5×D12C2×D311yescharacteristic
99C4×C5×D3—32no—
100C4×C5×D3C2×D311yescharacteristic
101C5×Dic6—31no—
102C5×Dic6—31no—
103C5×Dic6—61no—
104C3×C22⋊D5—32no—
105C3×C22⋊D5—31no—
106C3×C22⋊D5—31no—
107C2×C3×Dic5—32no—
108C3×D20—31no—
109C3×C4×D5—21no—
110C3×C4×D5—31no—
111C3×C4×D5C2×D311yescharacteristic
112C3×C4×D5C2×D311yescharacteristic
113C3×Dic10—31no—
114C3×Dic10—31no—
115#1204—32no—
116#1204—32no—
117#1204—32no—
118#1204—32no—
119#1204—32no—
120#1204—62no—
121#1204C2×D311yescharacteristic
122#1203C2×D311yescharacteristic
123#1202C2×D311yescharacteristic
124#1201—32no—
125#1201C2×D312yes—
126#1201C2×D311yescharacteristic
127#1200—32no—
128D3×Dic5—32no—
129D3×Dic5—32no—
130D3×Dic5C2×D312yes—
131D5×Dic3—32no—
132D5×Dic3—32no—
133D5×Dic3—32no—
134D5×Dic3—62no—
135#1010—151no—
136#1010—151no—
137C2×C32×C5C2411yescharacteristic
138C3×D15—21no—
139C3×C5×D3—22no—
140C3×C5×D3—21no—
141C32×D5—21no—
142C5×#41—91no—
143Q8×D5—91no—
144Q8×D5—91no—
145#711—31no—
146#711—91no—
147#711—92no—
148#711—92no—
149#709—31no—
150#709—92no—
151C2×Dic10—92no—
152C2×Dic10—91no—
153C2×C32×C4—51no—
154C2×C3⋊Dic3—52no—
155#634—51no—
156#634—52no—
157#634C2×D511yescharacteristic
158C3×#67—51no—
159C3×#67—52no—
160C3×#67—51no—
161C3×D12—51no—
162C3×D12C2×D511yescharacteristic
163C3×C4×D3—52no—
164C3×C4×D3—51no—
165C3×C4×D3C2×D511yescharacteristic
166C3×Dic6—51no—
167C3×Dic6—51no—
168#627—52no—
169#627—52no—
170#625—52no—
171#625C2×D512yes—
172D3×Dic3—52no—
173D3×Dic3—52no—
174D3×Dic3—52no—
175D3×Dic3C2×D512yes—
176C22×C3×C5—32no—
177C22×C3×C5—31no—
178C2×D15C22×D311yescharacteristic
179C2×C5×D3C22×D311yescharacteristic
180C2×C5×D3C22×D312yes—
181C2×C3×D5—21no—
182C2×C3×D5—31no—
183C2×C3×D5C22×D311yescharacteristic
184C2×C3×D5C22×D311yescharacteristic
185C3×C4×C5—21no—
186C3×C4×C5—31no—
187C3×C4×C5C22×D311yescharacteristic
188C3×C4×C5C22×D311yescharacteristic
189Dic15—32no—
190Dic15—32no—
191Dic15—62no—
192Dic15C22×D312yes—
193Dic15C22×D311yescharacteristic
194C3×Dic5—21no—
195C3×Dic5—31no—
196C3×Dic5—32no—
197C3×Dic5C22×D311yescharacteristic
198C3×Dic5C22×D311yescharacteristic
199C5×Dic3—32no—
200C5×Dic3—32no—
201C5×Dic3—62no—
202C5×Dic3C22×D311yescharacteristic
203C3×#41—151no—
204C3×#41—151no—
205D3×Q8—151no—
206D3×Q8—152no—
207D3×Q8—151no—
208#237—31no—
209#237—152no—
210#237—152no—
211#237—151no—
212#237—152no—
213#237—152no—
214#237—152no—
215#235—31no—
216#235—51no—
217#235—51no—
218#235—152no—
219#235—151no—
220C2×Dic6—151no—
221C2×Dic6—151no—
222C2×Dic6—151no—
223C2×Dic6—152no—
224C2×Dic6—151no—
225C2×Dic6—301no—
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—31no—
233C22⋊D5—32no—
234C22⋊D5—31no—
235C22⋊D5—92no—
236C2×Dic5—32no—
237C2×Dic5—92no—
238C2×Dic5—92no—
239D20—31no—
240C4×D5—31no—
241C4×D5—92no—
242C4×D5—92no—
243C4×D5D3211yescharacteristic
244Dic10—31no—
245Dic10—31no—
246Dic10—92no—
247Dic10—92no—
248Dic10—91no—
249Dic10—92no—
250C22×C32—51no—
251C2×C3×D3—51no—
252C2×C3×D3C22×D511yescharacteristic
253C2×C3×D3C22×D512yes—
254C32×C4—51no—
255C32×C4C22×D511yescharacteristic
256C3⋊Dic3—52no—
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—
268C3×D5—22no—
269C3×D5—21no—
270C3×D5—21no—
271C3×D5—61no—
272C5×D3—21no—
273C5×D3—22no—
274C3×Q8—151no—
275C3×Q8—151no—
276C3×D4—31no—
277C3×D4—151no—
278C3×D4—151no—
279C3×D4—152no—
280C3×D4—151no—
281C2×C3×C4—31no—
282C2×C3×C4—51no—
283C2×C3×C4—51no—
284C2×C3×C4—101no—
285C2×C3×C4—152no—
286C2×C3×C4—151no—
287#67—32no—
288#67—32no—
289#67—52no—
290#67—51no—
291#67—51no—
292#67—152no—
293#67—152no—
294C2×Dic3—32no—
295C2×Dic3—152no—
296C2×Dic3—152no—
297C2×Dic3—152no—
298C2×Dic3—152no—
299C2×Dic3—152no—
300C2×Dic3—302no—
301D12—51no—
302D12D3×D511yescharacteristic
303C4×D3—32no—
304C4×D3—52no—
305C4×D3—51no—
306C4×D3—152no—
307C4×D3—152no—
308C4×D3—152no—
309C4×D3D3×D511yescharacteristic
310Dic6—31no—
311Dic6—31no—
312Dic6—51no—
313Dic6—51no—
314Dic6—61no—
315Dic6—152no—
316Dic6—152no—
317Dic6—152no—
318Dic6—152no—
319Dic6—152no—
320Dic6—151no—
321Dic6—151no—
322Dic6—152no—
323Dic6—302no—
324Dic6—301no—
325C22×C5—32no—
326C22×C5—31no—
327C2×D5—31no—
328C2×D5C2×D3211yescharacteristic
329C4×C5—31no—
330C4×C5—92no—
331C4×C5C2×D3211yescharacteristic
332Dic5—31no—
333Dic5—32no—
334Dic5—92no—
335Dic5C2×D3211yescharacteristic
336C2×C32—101no—
337C2×C32C23×D511yescharacteristic
338C3×D3—21no—
339C3×D3—22no—
340C3×D3—101no—
341#41—91no—
342#41—151no—
343#41—151no—
344#41—452no—
345#41—451no—
346#41—452no—
347#41—452no—
348C2×Q8—451no—
349C2×Q8—451no—
350C2×Q8—452no—
351C2×Q8—451no—
352C3×C5—21no—
353C3×C5#101011yescharacteristic, complemented
354C3×C5#101011yescharacteristic, complemented
355C22×C3—32no—
356C22×C3—31no—
357C22×C3—51no—
358C22×C3—51no—
359C22×C3—101no—
360C22×C3—151no—
361C2×D3—51no—
362C2×D3C2×D3×D512yes—
363C2×D3C2×D3×D511yescharacteristic
364C3×C4—21no—
365C3×C4—31no—
366C3×C4—51no—
367C3×C4—51no—
368C3×C4—101no—
369C3×C4—152no—
370C3×C4—151no—
371C3×C4C2×D3×D511yescharacteristic
372C3×C4C2×D3×D511yescharacteristic
373Dic3—32no—
374Dic3—32no—
375Dic3—51no—
376Dic3—52no—
377Dic3—62no—
378Dic3—152no—
379Dic3—152no—
380Dic3—302no—
381Dic3C2×D3×D511yescharacteristic
382C2×C5—62no—
383C2×C5—61no—
384C2×C5C22×D3211yescharacteristic
385D5—21no—
386D5—61no—
387C32#415011yescharacteristic, complemented, Sylow
388Q8—91no—
389Q8—151no—
390Q8—151no—
391Q8—452no—
392Q8—452no—
393Q8—452no—
394Q8—451no—
395D4—31no—
396D4—92no—
397D4—151no—
398D4—151no—
399D4—151no—
400D4—152no—
401D4—452no—
402C2×C4—31no—
403C2×C4—51no—
404C2×C4—92no—
405C2×C4—152no—
406C2×C4—151no—
407C2×C4—452no—
408C2×C4—452no—
409C2×C4—452no—
410C2×C4—452no—
411C2×C3—21no—
412C2×C3—62no—
413C2×C3—61no—
414C2×C3—102no—
415C2×C3—101no—
416C2×C3—101no—
417C2×C3—301no—
418C2×C3C22×D3×D511yescharacteristic
419C2×C3C22×D3×D511yescharacteristic
420D3—21no—
421D3—22no—
422D3—101no—
423C5#6429811yescharacteristic, complemented, Sylow
424C22—31no—
425C22—32no—
426C22—51no—
427C22—151no—
428C4—31no—
429C4—51no—
430C4—92no—
431C4—152no—
432C4—151no—
433C4—452no—
434C4D32×D511yescharacteristic
435C3—21no—
436C3#9200111yescharacteristic, complemented
437C3#9202011yescharacteristic, complemented
438C2—61no—
439C2—62no—
440C2—101no—
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. #10624219C2semidirect
2. #10624200C2semidirect
3. D3×Dic30C2semidirect
4. C5×#3795C2semidirect
5. D5×#634C2semidirect
6. C3×#6772C2semidirect
7. C3×#6771C2semidirect
8. #10624050C2semidirect
9. #10624046C2semidirect
10. #10624045C2semidirect
11. D3×#1204C2semidirect
51. #6772D3semidirect
52. #6771D3semidirect
80. #3795D5semidirect
226. C32×C52−1+4semidirect
353. C3×C5#1010semidirect
354. C3×C5#1010semidirect
387. C32#4150semidirect
423. C5#64298semidirect
436. C3#92001semidirect
437. C3#92020semidirect

Maximal subgroups

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

subgroupindex
#106242192
#106242002
D3×Dic302
C5×#37952
D5×#6342
C3×#67722
C3×#67712
#106240502
#106240462
#106240452
D3×#12042
#920203
#920013
#642985