Finite Group Databasebeta

← #14299730

Subgroups — #14299730

Order 1920.

604 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11754509C211yesmaximal, characteristic
2#11751076C211yesmaximal, characteristic, complemented
3C5×#4964C211yesmaximal, characteristic, complemented
4C3×#65255C211yesmaximal, characteristic
5C2×#90976C211yesmaximal, characteristic
6#11745987C211yesmaximal, characteristic, complemented
7#11745975C211yesmaximal, characteristic, complemented
8#10601473—31nomaximal, Hall
9C2×C3×#709C2211yescharacteristic
10C2×C3×C5×C4⋊C4C2211yescharacteristic
11C2×C4⋊Dic15C2211yescharacteristic
12C2×#6712—22no—
13C2×C4×Dic15C2211yescharacteristic
14C5×#926C2211yescharacteristic
15C2×C5×#208C2211yescharacteristic
16C2×C3×#681C2211yescharacteristic
17#91559—22no—
18Dic3×D20—21no—
19D5×C4⋊Dic3—21no—
20#91358—21no—
21Dic3×Dic10—21no—
22C5×#808—21no—
23C5×#807—21no—
24C3×#3942—21no—
25C3×#3941—21no—
26#90976C414yescomplemented
27#70297—51nomaximal, Hall
28#65857—31no—
29C5×#428—31no—
30C2×#3957—31no—
31#65265—31no—
32#65255D311yescharacteristic
33#64774—31no—
34#64752—31no—
35C22×C3×C4×C5—21no—
36C22×C3×C4×C5C2311yescharacteristic
37C22×Dic15—21no—
38C22×Dic15D412yes—
39C2×C3×C22⋊D5—21no—
40C3×#709—21no—
41C3×#709—21no—
42C3×#709C2×C412yes—
43C3×#709Q812yes—
44C2×C3×D20D411yescharacteristic
45C2×C3×C4×D5—21no—
46C2×C3×Dic10D411yescharacteristic
47C2×D5×Dic3—22no—
48C3×C5×C4⋊C4—21no—
49C3×C5×C4⋊C4—21no—
50C4⋊Dic15—21no—
51C4⋊Dic15—21no—
52#6712—44no—
53C4×Dic15—21no—
54C4×Dic15C2×C412yescomplemented
55C5×#217—21no—
56C5×C4⋊Dic3D411yescharacteristic
57C5×C4⋊Dic3D411yescharacteristic
58C4×C5×Dic3—21no—
59C5×#208—21no—
60C5×#208C2×C412yescomplemented
61C2×C5×C3⋊C8—21no—
62C3×#681—21no—
63C3×#681—21no—
64C2×C3×#168—21no—
65#6668—22no—
66#6665—22no—
67Dic3×Dic5—22no—
68#5827—51no—
69C3×#417—51no—
70C2×#837—51no—
71#4975—51no—
72#4964D511yescharacteristic, complemented
73#4555—51no—
74#4554—51no—
75C2×#709C2×D311yescharacteristic
76C2×C5×M16—31no—
77C5×#117—31no—
78C2×C5×C4⋊C4C2×D311yescharacteristic
79C2×C4⋊Dic5—31no—
80C2×#683—62no—
81C2×C4×Dic5—31no—
82C2×#681C2×D311yescharacteristic
83#4038—61no—
84D5×C4⋊C4—61no—
85#4027—62no—
86C4×D20—61no—
87C4×Dic10—61no—
88C5×#107—61no—
89C5×#106—61no—
90#3957—34no—
91#3942—21no—
92#3941—21no—
93#1859—151noSylow
94C23×C3×C5C2×D411yescharacteristic
95C22×C3×D5—21no—
96C2×C3×C4×C5—42no—
97C2×C3×C4×C5—41no—
98C2×C3×C4×C5C22×C411yescharacteristic
99C2×C3×C4×C5C2×D411yescharacteristic
100C2×C3×C4×C5C2×D411yescharacteristic
101C2×C3×C4×C5C2×Q811yescharacteristic
102C2×C3×C4×C5#4111yescharacteristic
103C2×C3×C4×C5#4111yescharacteristic
104C2×Dic15—22no—
105C2×Dic15—22no—
106C2×Dic15—22no—
107C2×Dic15—22no—
108C2×Dic15—42no—
109C2×C5×Dic3—21no—
110C2×C5×Dic3—21no—
111C3×C22⋊D5—22no—
112C3×C22⋊D5—22no—
113C2×C3×Dic5—21no—
114C3×D20—21no—
115C3×D20C4⋊C412yescomplemented
116C3×C4×D5—22no—
117C3×C4×D5—22no—
118C3×Dic10—21no—
119C3×Dic10C4⋊C412yescomplemented
120D5×Dic3—82no—
121C3×#168—41no—
122C5×C3⋊C8—22no—
123C2×C3×#41—51no—
124C2×C3×M16—51no—
125C2×C3×C4⋊C4C2×D511yescharacteristic
126Q8×Dic3—101no—
127D4×Dic3—101no—
128D4×Dic3—102no—
129#926—101no—
130#926C2×D511yescharacteristic
131C2×C4⋊Dic3—51no—
132C2×C4⋊Dic3—101no—
133C2×Dic3⋊C4—102no—
134C2×C4×Dic3—51no—
135C2×#208C2×D511yescharacteristic
136C3×#103—101no—
137C3×#102—101no—
138#837—54no—
139#808—21no—
140#807—21no—
141C22×C4×C5—21no—
142C22×C4×C5C22×D311yescharacteristic
143C2×C22⋊D5—21no—
144C22×Dic5—32no—
145C22×Dic5—61no—
146#709—21no—
147#709—21no—
148#709Dic612yes—
149#709C4×D312yes—
150C2×D20#6711yescharacteristic
151C2×C4×D5—21no—
152C2×C4×D5—62no—
153C2×Dic10#6711yescharacteristic
154C5×M16—32no—
155C5×M16—61no—
156C2×C5×C8—61no—
157C5×C4⋊C4—21no—
158C5×C4⋊C4—21no—
159C5×C4⋊C4—31no—
160C5×C4⋊C4—31no—
161C5×C22⋊C4—61no—
162C42×C5—61no—
163#685—62no—
164C4⋊Dic5—61no—
165C4⋊Dic5—61no—
166#683—62no—
167#683—124no—
168C4×Dic5—32no—
169C4×Dic5—62no—
170C4×Dic5—61no—
171#681—21no—
172#681—21no—
173C2×#168—21no—
174#520—151no—
175#428—31no—
176C2×#104—151no—
177#417—51no—
178#384—151no—
179#343—151no—
180#339—151no—
181C22×C3×C5—21no—
182C22×C3×C5—21no—
183C22×C3×C5C4×D411yescharacteristic
184C22×C3×C5#12011yescharacteristic
185C22×C3×C5C22⋊Q811yescharacteristic
186C2×C3×D5—22no—
187C2×C3×D5—41no—
188C2×C3×D5—41no—
189C3×C4×C5—81no—
190C3×C4×C5C2×C4⋊C411yescharacteristic
191C3×C4×C5C4×D411yescharacteristic
192C3×C4×C5C22⋊Q811yescharacteristic
193C3×C4×C5#12311yescharacteristic
194Dic15—24no—
195Dic15—42no—
196C3×Dic5—22no—
197C5×Dic3—41no—
198C5×Dic3—41no—
199C3×#41—52no—
200C3×#41—52no—
201C3×#41—101no—
202C3×#41—101no—
203C2×C3×Q8—51no—
204C2×C3×D4—51no—
205C2×C3×D4—101no—
206C22×C3×C4—21no—
207C22×C3×C4—101no—
208C22×C3×C4C22×D511yescharacteristic
209C22×Dic3—52no—
210C22×Dic3—102no—
211C22×Dic3—101no—
212C3×M16—101no—
213C3×M16—101no—
214C2×C3×C8—101no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217#217—21no—
218#217—102no—
219C4⋊Dic3—102no—
220C4⋊Dic3—101no—
221C4⋊Dic3—101no—
222C4⋊Dic3C22⋊D511yescharacteristic
223C4⋊Dic3C22⋊D511yescharacteristic
224Dic3⋊C4—204no—
225C4×Dic3—21no—
226C4×Dic3—52no—
227C4×Dic3—102no—
228C4×Dic3—101no—
229#208—21no—
230#208C4×D512yescomplemented
231C2×C3⋊C8—21no—
232C23×C5D3×D411yescharacteristic
233C22×D5—21no—
234C2×C4×C5—42no—
235C2×C4×C5—41no—
236C2×C4×C5—61no—
237C2×C4×C5—61no—
238C2×C4×C5C2×Dic611yescharacteristic
239C2×C4×C5C2×C4×D311yescharacteristic
240C2×C4×C5D3×D411yescharacteristic
241C2×C4×C5#23711yescharacteristic
242C2×C4×C5#23711yescharacteristic
243C2×C4×C5C2×#6711yescharacteristic
244C22⋊D5—22no—
245C22⋊D5—22no—
246C2×Dic5—21no—
247C2×Dic5—62no—
248C2×Dic5—62no—
249C2×Dic5—62no—
250C2×Dic5—62no—
251C2×Dic5—122no—
252D20—21no—
253D20Dic3⋊C412yescomplemented
254C4×D5—22no—
255C4×D5—22no—
256C4×D5—242no—
257Dic10—21no—
258Dic10Dic3⋊C412yescomplemented
259C5×C8—62no—
260#168—41no—
261C2×#41—51no—
262C2×M16—31no—
263C2×M16—51no—
264C4×Q8—301no—
265C4×D4—301no—
266C4×D4—302no—
267#117—31no—
268#117—301no—
269C2×C4⋊C4—151no—
270C2×C4⋊C4—301no—
271C2×C4⋊C4—302no—
272C2×C4⋊C4D3×D511yescharacteristic
273C2×C42—151no—
274#107—61no—
275#106—61no—
276#104—154no—
277#103—101no—
278#102—101no—
279C2×C3×C5—21no—
280C2×C3×C5—21no—
281C2×C3×C5#38511yescharacteristic
282C2×C3×C5#45311yescharacteristic
283C2×C3×C5#45611yescharacteristic
284C3×D5—42no—
285C23×C3—101no—
286C23×C3D4×D511yescharacteristic
287C3×Q8—52no—
288C3×Q8—101no—
289C3×D4—52no—
290C3×D4—102no—
291C3×D4—102no—
292C3×D4—101no—
293C2×C3×C4—42no—
294C2×C3×C4—41no—
295C2×C3×C4—101no—
296C2×C3×C4—102no—
297C2×C3×C4—102no—
298C2×C3×C4C2×C4×D511yescharacteristic
299C2×C3×C4#70911yescharacteristic
300C2×C3×C4D4×D511yescharacteristic
301C2×C3×C4Q8×D511yescharacteristic
302C2×C3×C4#71311yescharacteristic
303C2×C3×C4C2×C22⋊D511yescharacteristic
304C2×Dic3—21no—
305C2×Dic3—21no—
306C2×Dic3—102no—
307C2×Dic3—102no—
308C2×Dic3—102no—
309C2×Dic3—102no—
310C2×Dic3—202no—
311C2×Dic3—402no—
312C3×C8—201no—
313C3⋊C8—22no—
314C22×C5—21no—
315C22×C5—21no—
316C22×C5#87811yescharacteristic
317C22×C5#67⋊C411yescharacteristic
318C22×C5#93711yescharacteristic
319C2×D5—22no—
320C2×D5—41no—
321C2×D5—41no—
322C4×C5—81no—
323C4×C5—121no—
324C4×C5—121no—
325C4×C5#87811yescharacteristic
326C4×C5#67⋊C411yescharacteristic
327C4×C5C2×Dic3⋊C411yescharacteristic
328C4×C5#93511yescharacteristic
329Dic5—22no—
330Dic5—64no—
331Dic5—122no—
332#41—52no—
333#41—52no—
334#41—101no—
335#41—101no—
336C2×Q8—51no—
337C2×D4—51no—
338C2×D4—101no—
339C22×C4—21no—
340C22×C4—101no—
341C22×C4—152no—
342C22×C4—302no—
343C22×C4—301no—
344C22×C4C2×D3×D511yescharacteristic
345M16—32no—
346M16—61no—
347M16—101no—
348M16—101no—
349C2×C8—61no—
350C2×C8—101no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—31no—
354C4⋊C4—31no—
355C4⋊C4—302no—
356C4⋊C4—301no—
357C4⋊C4—301no—
358C4⋊C4—604no—
359C22⋊C4—61no—
360C22⋊C4—302no—
361C42—61no—
362C42—152no—
363C42—302no—
364C42—301no—
365C3×C5#185911yescharacteristic, complemented, Hall
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—102no—
369C22×C3—201no—
370C22×C3—201no—
371C22×C3#403911yescharacteristic
372C22×C3#404211yescharacteristic
373C22×C3#408311yescharacteristic
374C3×C4—81no—
375C3×C4—102no—
376C3×C4D5×C4⋊C411yescharacteristic
377C3×C4#404011yescharacteristic
378C3×C4C4×C22⋊D511yescharacteristic
379C3×C4#409211yescharacteristic
380Dic3—41no—
381Dic3—41no—
382Dic3—104no—
383Dic3—202no—
384C2×C5—21no—
385C2×C5—21no—
386C2×C5#494611yescharacteristic
387C2×C5#519811yescharacteristic
388C2×C5#520511yescharacteristic
389D5—42no—
390C23—101no—
391C23#679011yescharacteristic
392Q8—52no—
393Q8—101no—
394D4—52no—
395D4—102no—
396D4—102no—
397D4—101no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—61no—
401C2×C4—61no—
402C2×C4—101no—
403C2×C4—102no—
404C2×C4—102no—
405C2×C4—302no—
406C2×C4—302no—
407C2×C4—302no—
408C2×C4—302no—
409C2×C4—602no—
410C2×C4—1202no—
411C2×C4D5×Dic611yescharacteristic
412C2×C4#676511yescharacteristic
413C2×C4#676611yescharacteristic
414C2×C4C4×D3×D511yescharacteristic
415C2×C4D5×#6711yescharacteristic
416C2×C4D3×C22⋊D511yescharacteristic
417C8—62no—
418C8—201no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—202no—
422C2×C3#6527811yescharacteristic
423C2×C3#6548711yescharacteristic
424C2×C3#6549411yescharacteristic
425C5#7029711yescharacteristic, complemented, Sylow
426C22—21no—
427C22—21no—
428C22—102no—
429C22—201no—
430C22—201no—
431C22#9139611yescharacteristic
432C22#9143111yescharacteristic
433C22#9157711yescharacteristic
434C4—81no—
435C4—102no—
436C4—121no—
437C4—121no—
438C4—304no—
439C4—602no—
440C4D5×Dic3⋊C411yescharacteristic
441C4#9140711yescharacteristic
442C4#9142111yescharacteristic
443C4#9144111yescharacteristic
444C3#1060147311yescharacteristic, complemented, Sylow
445C2—21no—
446C2—21no—
447C2—202no—
448C2#1174889111yescharacteristic
449C2#1174916411yescharacteristic
450C2#1174941211yescharacteristic
subgroupG/H
centerC22#91577
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C5×C4⋊C4C22
socleC22×C3×C5#120
prime radicalC2×C3×#709C22
cosocleC22×C3×C4×C5C23
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#1859
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #11751076C2semidirect
3. C5×#4964C2semidirect
6. #11745987C2semidirect
7. #11745975C2semidirect
26. #90976C4semidirect
54. C4×Dic15C2×C4semidirect
60. C5×#208C2×C4semidirect
72. #4964D5semidirect
115. C3×D20C4⋊C4semidirect
119. C3×Dic10C4⋊C4semidirect
230. #208C4×D5semidirect
253. D20Dic3⋊C4semidirect
258. Dic10Dic3⋊C4semidirect
365. C3×C5#1859semidirect
425. C5#70297semidirect
444. C3#10601473semidirect

Maximal subgroups

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

subgroupindex
#117545092
#117510762
C5×#49642
C3×#652552
C2×#909762
#117459872
#117459752
#106014733
#702975