Finite Group Databasebeta

← #13088947

Subgroups — #13088947

Order 1440.

624 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10624222C211yesmaximal, characteristic, complemented
2#10624198C211yesmaximal, characteristic, complemented
3#10624196C211yesmaximal, characteristic, complemented
4C5×#3796C211yesmaximal, characteristic, complemented
5#10624154C211yesmaximal, characteristic, complemented
6C3×#6772C211yesmaximal, characteristic, complemented
7C3×D5×Dic6C211yesmaximal, characteristic, complemented
8#10624051C212yesmaximal, complemented
9#10624043C212yesmaximal, complemented
10#10624041C212yesmaximal, complemented
11D3×#1204C212yesmaximal, complemented
12#92030—31nomaximal
13#92020—31nomaximal
14C4×C3⋊D15C2211yescharacteristic
15C4×C5×C3⋊D3C2211yescharacteristic
16C3×Dic30C2211yescharacteristic
17C3×Dic30C2211yescharacteristic
18C3×C5×D12C2211yescharacteristic
19C3×C5×Dic6C2211yescharacteristic
20C32×C4×D5C2211yescharacteristic
21#67158C2212yes—
22#67155C2212yes—
23#67153C2212yes—
24#67152C2212yes—
25#67149C2212yes—
26D3×Dic15C2212yes—
27C5×#626C2212yes—
28C5×D3×Dic3C2212yes—
29#67141C2211yescharacteristic
30#67140C2211yescharacteristic
31#67139C2211yescharacteristic
32#67138C2211yescharacteristic
33C3×#1204C2212yes—
34C3×#1201C2212yes—
35C3×D3×Dic5C2212yes—
36C3×D5×Dic3C2212yes—
37#64297—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—32no—
49#6772D311yescharacteristic, complemented
50#6771—31no—
51#6771—31no—
52#6771—32no—
53#6771—61no—
54D15⋊Q8—31no—
55#6768—31no—
56#6768—32no—
57D3×Dic10—31no—
58D3×Dic10—31no—
59D3×Dic10—32no—
60D5×Dic6D311yescharacteristic, complemented
61C2×C3⋊D15C2311yescharacteristic
62C2×C5×C3⋊D3C2311yescharacteristic
63C2×C3×C5×D3C2312yes—
64C2×C32×D5C2311yescharacteristic
65C32×C4×C5C2311yescharacteristic
66C3⋊Dic15C2311yescharacteristic
67C5×C3⋊Dic3C2311yescharacteristic
68C3×Dic15C2312yes—
69C3×Dic15C2312yes—
70C3×C5×Dic3C2312yes—
71C32×Dic5C2311yescharacteristic
72#4150—91noHall
73#3828—51no—
74C3×#235—51no—
75C2×C3×Dic6—51no—
76#3804—52no—
77#3804—52no—
78#3799—52no—
79#3797—51no—
80#3797—51no—
81#3796—51no—
82#3796D511yescharacteristic, complemented
83D3×Dic6—52no—
84C3×C5×Q8—31no—
85C3×C5×D4—31no—
86#1220—32no—
87C2×Dic15—32no—
88C4×D15—31no—
89C4×D15—31no—
90C4×D15—32no—
91C4×D15—61no—
92Dic30—32no—
93Dic30C2×D311yescharacteristic
94Dic30C2×D311yescharacteristic
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—21no—
107C3×C4×D5—32no—
108C3×C4×D5C2×D311yescharacteristic
109C3×C4×D5C2×D311yescharacteristic
110C3×Dic10—31no—
111C3×Dic10—31no—
112C3×Dic10—32no—
113#1204—32no—
114#1204—32no—
115#1204—31no—
116#1204—31no—
117#1204—32no—
118#1204—61no—
119#1204C2×D312yes—
120#1203—32no—
121#1203—31no—
122#1203—31no—
123#1203—61no—
124#1202—31no—
125#1202—31no—
126#1202—61no—
127#1201—31no—
128#1201—31no—
129#1201—61no—
130#1201C2×D312yes—
131#1200—32no—
132D3×Dic5—32no—
133D3×Dic5—32no—
134D3×Dic5—32no—
135D3×Dic5C2×D312yes—
136D5×Dic3C2×D312yes—
137#1010—151no—
138#1007—151no—
139C2×C32×C5C2411yescharacteristic
140C3⋊D15—21no—
141C5×C3⋊D3—21no—
142C3×C5×D3—22no—
143C32×D5—21no—
144C5×#41—91no—
145Q8×D5—31no—
146Q8×D5—91no—
147#711—31no—
148#711—92no—
149#711—91no—
150#711—92no—
151#709—92no—
152#709—91no—
153C2×Dic10—91no—
154C2×Dic10—92no—
155C2×C32×C4—51no—
156#638—51no—
157#638—51no—
158C4⋊(C3⋊D3)—51no—
159C4×C3⋊D3—51no—
160C4×C3⋊D3C2×D511yescharacteristic
161#634—51no—
162C3×#67—52no—
163C2×C3×Dic3—52no—
164C3×D12C2×D511yescharacteristic
165C3×C4×D3—52no—
166C3×Dic6—51no—
167C3×Dic6—51no—
168C3×Dic6—52no—
169C3×Dic6C2×D511yescharacteristic
170#627—52no—
171#627—52no—
172#626—52no—
173#626C2×D512yes—
174#624—52no—
175#624—52no—
176D3×Dic3—52no—
177D3×Dic3C2×D512yes—
178C22×C3×C5—32no—
179C2×D15—31no—
180C2×D15—31no—
181C2×D15—61no—
182C2×C5×D3—31no—
183C2×C5×D3—31no—
184C2×C5×D3—61no—
185C2×C5×D3C22×D312yes—
186C2×C3×D5—21no—
187C2×C3×D5C22×D311yescharacteristic
188C2×C3×D5C22×D311yescharacteristic
189C3×C4×C5—21no—
190C3×C4×C5—32no—
191C3×C4×C5C22×D311yescharacteristic
192C3×C4×C5C22×D311yescharacteristic
193Dic15—31no—
194Dic15—31no—
195Dic15—61no—
196Dic15C22×D312yes—
197Dic15C22×D312yes—
198C3×Dic5—21no—
199C3×Dic5—32no—
200C3×Dic5—32no—
201C3×Dic5C22×D311yescharacteristic
202C3×Dic5C22×D311yescharacteristic
203C5×Dic3—31no—
204C5×Dic3—31no—
205C5×Dic3—61no—
206C5×Dic3C22×D312yes—
207C3×#41—151no—
208C2×C3×Q8—151no—
209#239—31no—
210#239—152no—
211#239—151no—
212D3×Q8—151no—
213D3×Q8—151no—
214D3×Q8—151no—
215D3×Q8—152no—
216D3×Q8—151no—
217#237—31no—
218#237—151no—
219#237—152no—
220#237—152no—
221#235—51no—
222#235—152no—
223#235—152no—
224#235—151no—
225#235—151no—
226#235—152no—
227#235—301no—
228C2×Dic6—51no—
229C2×Dic6—152no—
230C32×C52−1+411yescharacteristic, complemented, Hall
231C5×Q8—31no—
232C5×D4—31no—
233C5×D4—92no—
234C2×C4×C5—92no—
235C2×C4×C5—91no—
236C22⋊D5—32no—
237C22⋊D5—92no—
238C22⋊D5—91no—
239C22⋊D5—91no—
240C2×Dic5—32no—
241C2×Dic5—92no—
242C2×Dic5—92no—
243D20—91no—
244C4×D5—32no—
245C4×D5—92no—
246C4×D5—91no—
247C4×D5D3211yescharacteristic
248Dic10—31no—
249Dic10—31no—
250Dic10—32no—
251Dic10—92no—
252Dic10—92no—
253Dic10—91no—
254C22×C32—51no—
255C2×C3⋊D3—51no—
256C2×C3⋊D3C22×D511yescharacteristic
257C2×C3×D3C22×D512yes—
258C32×C4—51no—
259C32×C4C22×D511yescharacteristic
260C3⋊Dic3—51no—
261C3⋊Dic3C22×D511yescharacteristic
262C3×Dic3—52no—
263C3×Dic3—52no—
264C3×Dic3C22×D512yes—
2652−1+4—451noSylow
266C2×C3×C5—21no—
267C2×C3×C5—62no—
268C2×C3×C5C23×D311yescharacteristic
269C2×C3×C5C23×D311yescharacteristic
270D15—61no—
271D15—61no—
272D15—121no—
273C3×D5—21no—
274C3×D5—21no—
275C3×D5—41no—
276C5×D3—22no—
277C5×D3—61no—
278C5×D3—61no—
279C5×D3—121no—
280C3×Q8—31no—
281C3×Q8—152no—
282C3×Q8—151no—
283C3×Q8—151no—
284C3×D4—31no—
285C3×D4—152no—
286C2×C3×C4—51no—
287C2×C3×C4—51no—
288C2×C3×C4—101no—
289C2×C3×C4—152no—
290C2×C3×C4—152no—
291#67—32no—
292#67—52no—
293#67—151no—
294#67—151no—
295#67—152no—
296#67—151no—
297#67—151no—
298#67—301no—
299#67—301no—
300C2×Dic3—32no—
301C2×Dic3—52no—
302C2×Dic3—152no—
303D12—32no—
304D12—151no—
305D12—151no—
306D12—152no—
307D12—301no—
308D12D3×D511yescharacteristic
309C4×D3—32no—
310C4×D3—31no—
311C4×D3—31no—
312C4×D3—52no—
313C4×D3—61no—
314C4×D3—152no—
315C4×D3—152no—
316C4×D3—152no—
317C4×D3—152no—
318C4×D3—151no—
319C4×D3—151no—
320C4×D3—152no—
321C4×D3—301no—
322Dic6—51no—
323Dic6—51no—
324Dic6—52no—
325Dic6—152no—
326Dic6—152no—
327Dic6—151no—
328Dic6—151no—
329Dic6—152no—
330Dic6—152no—
331Dic6—301no—
332Dic6D3×D511yescharacteristic
333C22×C5—32no—
334C22×C5—91no—
335C2×D5—91no—
336C2×D5C2×D3211yescharacteristic
337C4×C5—32no—
338C4×C5—91no—
339C4×C5C2×D3211yescharacteristic
340Dic5—32no—
341Dic5—32no—
342Dic5—91no—
343Dic5C2×D3211yescharacteristic
344C2×C32—101no—
345C2×C32C23×D511yescharacteristic
346C3⋊D3—21no—
347C3⋊D3—101no—
348C3×D3—22no—
349#41—91no—
350#41—151no—
351#41—451no—
352#41—452no—
353#41—452no—
354#41—452no—
355#41—451no—
356C2×Q8—151no—
357C2×Q8—452no—
358C2×Q8—451no—
359C2×Q8—451no—
360C3×C5—21no—
361C3×C5#100711yescharacteristic, complemented
362C3×C5#101011yescharacteristic, complemented
363C22×C3—32no—
364C22×C3—51no—
365C22×C3—51no—
366C22×C3—101no—
367C2×D3—31no—
368C2×D3—31no—
369C2×D3—61no—
370C2×D3—151no—
371C2×D3—151no—
372C2×D3—301no—
373C2×D3C2×D3×D512yes—
374C3×C4—21no—
375C3×C4—32no—
376C3×C4—51no—
377C3×C4—51no—
378C3×C4—101no—
379C3×C4—152no—
380C3×C4—152no—
381C3×C4C2×D3×D511yescharacteristic
382C3×C4C2×D3×D511yescharacteristic
383Dic3—31no—
384Dic3—31no—
385Dic3—52no—
386Dic3—52no—
387Dic3—61no—
388Dic3—151no—
389Dic3—151no—
390Dic3—301no—
391Dic3C2×D3×D512yes—
392C2×C5—62no—
393C2×C5—181no—
394C2×C5C22×D3211yescharacteristic
395D5—21no—
396D5—181no—
397C32#415011yescharacteristic, complemented, Sylow
398Q8—31no—
399Q8—152no—
400Q8—151no—
401Q8—151no—
402Q8—452no—
403Q8—451no—
404Q8—452no—
405D4—31no—
406D4—92no—
407D4—152no—
408D4—451no—
409D4—451no—
410D4—452no—
411D4—451no—
412C2×C4—51no—
413C2×C4—92no—
414C2×C4—91no—
415C2×C4—152no—
416C2×C4—152no—
417C2×C4—452no—
418C2×C4—452no—
419C2×C4—452no—
420C2×C4—451no—
421C2×C3—21no—
422C2×C3—62no—
423C2×C3—101no—
424C2×C3—101no—
425C2×C3—201no—
426C2×C3C22×D3×D511yescharacteristic
427C2×C3C22×D3×D511yescharacteristic
428D3—22no—
429D3—61no—
430D3—61no—
431D3—121no—
432D3—301no—
433D3—301no—
434D3—601no—
435C5#6429711yescharacteristic, complemented, Sylow
436C22—32no—
437C22—51no—
438C22—91no—
439C22—451no—
440C4—32no—
441C4—51no—
442C4—91no—
443C4—152no—
444C4—152no—
445C4—451no—
446C4D32×D511yescharacteristic
447C3—21no—
448C3#9202011yescharacteristic, complemented
449C3#9203011yescharacteristic, complemented
450C2—62no—
451C2—101no—
452C2—181no—
453C2—901no—
454C2C2×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. #10624222C2semidirect
2. #10624198C2semidirect
3. #10624196C2semidirect
4. C5×#3796C2semidirect
5. #10624154C2semidirect
6. C3×#6772C2semidirect
7. C3×D5×Dic6C2semidirect
8. #10624051C2semidirect
9. #10624043C2semidirect
10. #10624041C2semidirect
11. D3×#1204C2semidirect
49. #6772D3semidirect
60. D5×Dic6D3semidirect
82. #3796D5semidirect
230. C32×C52−1+4semidirect
361. C3×C5#1007semidirect
362. C3×C5#1010semidirect
397. C32#4150semidirect
435. C5#64297semidirect
448. C3#92020semidirect
449. C3#92030semidirect

Maximal subgroups

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

subgroupindex
#106242222
#106241982
#106241962
C5×#37962
#106241542
C3×#67722
C3×D5×Dic62
#106240512
#106240432
#106240412
D3×#12042
#920303
#920203
#642975