Finite Group Databasebeta

← #14438629

Subgroups — #14438629

Order 1920.

848 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756475C211yesmaximal, characteristic
2#11752562C211yesmaximal, characteristic, complemented
3C5×#5070C211yesmaximal, characteristic, complemented
4C3×#65359C211yesmaximal, characteristic, complemented
5#11749117C214yesmaximal, complemented
6#11748959C214yesmaximal, complemented
7C4×#6651C211yesmaximal, characteristic, complemented
8#11746930C211yesmaximal, characteristic, complemented
9#11746921C211yesmaximal, characteristic, complemented
10#10613296—31nomaximal, Hall
11C2×#6819C2212yes—
12C2×D15⋊Q8—21no—
13C3×C5×#127C2211yescharacteristic
14#91827—21no—
15#91827—21no—
16#91825C2212yes—
17D4×Dic15—21no—
18D4×Dic15—21no—
19C42×D15C2211yescharacteristic
20#91758C2211yescharacteristic
21C5×#869C2211yescharacteristic
22C3×#4015C2211yescharacteristic
23#91470—21no—
24#91470C2212yes—
25#91437—21no—
26#91437C2212yes—
27#91385—21no—
28C2×#6651C2212yes—
29C5×#832C2214yes—
30C4×C5×C3⋊C8C2211yescharacteristic
31C3×#3964C2214yes—
32C3×C4×#168C2211yescharacteristic
33#90975C2214yes—
34#90974C2214yes—
35#90954C2212yes—
36#81565—51nomaximal, Hall
37D5×C4⋊Q8—31no—
38#66050—31no—
39C5×#486—31no—
40#65382—31no—
41#65359D311yescharacteristic, complemented
42#65060—34no—
43#65054—34no—
44C4×#676—31no—
45#64970—31no—
46C2×C3×C5×D4—21no—
47C2×C3×C5×D4—21no—
48C2×C3×C5×D4C2312yes—
49C2×#1220—22no—
50C22×Dic15—22no—
51#6819—28no—
52C2×C4×D15D411yescharacteristic
53C2×C4×D15D411yescharacteristic
54C2×C4×D15C2311yescharacteristic
55C2×Dic30C2312yes—
56C2×C5×Dic6—21no—
57C2×C3×Dic10—21no—
58C2×#1204—22no—
59C2×#1200—22no—
60D15⋊Q8—44no—
61C3×C42×C5C2311yescharacteristic
62#6719—22no—
63#6719—22no—
64C4⋊Dic15—21no—
65C4⋊Dic15—21no—
66C4×Dic15D411yescharacteristic
67C4×Dic15D411yescharacteristic
68C4×Dic15C2311yescharacteristic
69C5×C4⋊Dic3—21no—
70C5×C4⋊Dic3C2312yes—
71C2×C5×C3⋊C8C2312yes—
72C3×C4⋊Dic5—21no—
73C3×C4⋊Dic5C2312yes—
74C2×C3×#168C2312yes—
75#6669—22no—
76#6669—22no—
77#6651—24no—
78D3×C4⋊Q8—51no—
79#5716—51no—
80C3×#486—51no—
81#5093—51no—
82#5070D511yescharacteristic, complemented
83#4771—54no—
84#4765—54no—
85C4×ΓL(1,𝔽25)—51no—
86#4681—51no—
87C2×Q8×D5—61no—
88C2×#711—32no—
89C5×C4⋊Q8—31no—
90C5×#127C2×D311yescharacteristic
91#4090—61no—
92#4084—61no—
93#4084—61no—
94#4082—32no—
95D4×Dic5—61no—
96D4×Dic5—61no—
97C2×#676—32no—
98D5×C4⋊C4—32no—
99D5×C4⋊C4—61no—
100#4034—32no—
101#4034—61no—
102C42×D5—31no—
103#4015—31no—
104#4015C2×D311yescharacteristic
105C5×#102—34no—
106C4×C5×C8—31no—
107#3964C2×D314yes—
108#3948—34no—
109#3947—32no—
110#3942—34no—
111C4×#168C2×D311yescharacteristic
112#3141—151noSylow
113C23×C3×C5—22no—
114C22×D15C2×D411yescharacteristic
115C3×C5×D4—24no—
116C3×C5×D4—42no—
117C3×C5×D4—42no—
118C2×C3×C4×C5C2×D411yescharacteristic
119C2×C3×C4×C5C2×D411yescharacteristic
120C2×C3×C4×C5C2411yescharacteristic
121#1220—44no—
122C2×Dic15—22no—
123C2×Dic15—44no—
124C2×Dic15C2×D411yescharacteristic
125C2×Dic15C2×D411yescharacteristic
126C2×Dic15C2×D411yescharacteristic
127C4×D15—22no—
128C4×D15—22no—
129C4×D15#4114yes—
130Dic30—24no—
131C2×C5×Dic3—22no—
132C5×Dic6—42no—
133C2×C3×Dic5—22no—
134C3×Dic10—42no—
135#1204—44no—
136#1200—44no—
137C3×#168—22no—
138C5×C3⋊C8—22no—
139C2×D3×Q8—101no—
140C2×#237—52no—
141C3×C4⋊Q8—51no—
142C3×#127C2×D511yescharacteristic
143#944—101no—
144#938—101no—
145#938—101no—
146#936—52no—
147D4×Dic3—101no—
148D4×Dic3—101no—
149C2×ΓL(1,𝔽25)—52no—
150D3×C4⋊C4—52no—
151D3×C4⋊C4—101no—
152#888—52no—
153#888—101no—
154C42×D3—51no—
155#869—51no—
156#869C2×D511yescharacteristic
157C3×#102—54no—
158C3×C4×C8—51no—
159#832C2×D514yes—
160#816—54no—
161#815—52no—
162#810—54no—
163C4×C3⋊C8C2×D511yescharacteristic
164C2×C5×Q8—61no—
165C2×C5×D4—21no—
166C2×C5×D4—21no—
167C2×C5×D4C22×D312yes—
168C2×C22⋊D5—62no—
169C22×Dic5—62no—
170Q8×D5—124no—
171#711—68no—
172C2×C4×D5—31no—
173C2×C4×D5—31no—
174C2×C4×D5—31no—
175C2×C4×D5—62no—
176C2×Dic10—21no—
177C2×Dic10—32no—
178C2×Dic10—62no—
179C2×C5×C8—32no—
180C5×C4⋊C4—32no—
181C5×C4⋊C4—61no—
182C42×C5C22×D311yescharacteristic
183C22⋊Dic5—62no—
184C22⋊Dic5—62no—
185C4⋊Dic5—21no—
186C4⋊Dic5—61no—
187C4⋊Dic5—61no—
188C4⋊Dic5C22×D312yes—
189#683—62no—
190#683—62no—
191C4×Dic5—31no—
192C4×Dic5—31no—
193C4×Dic5—31no—
194C2×#168C22×D312yes—
195#676—64no—
196#532—151no—
197C2×C4⋊Q8—151no—
198#489—154no—
199#487—151no—
200#487—151no—
201#486—31no—
202#486—51no—
203#417—154no—
204C4×M16—151no—
205C22×C3×C5—42no—
206C22×C3×C5—42no—
207C22×C3×C5—42no—
208C22×C3×C5C22×D411yescharacteristic
209C2×D15—21no—
210C2×D15—21no—
211C2×D15#12412yes—
212C3×C4×C5AGL(1,ℤ/8)12yes—
213C3×C4×C5#13712yes—
214C3×C4×C5C2×#4112yes—
215Dic15—21no—
216Dic15—21no—
217Dic15—42no—
218Dic15#12412yes—
219C3×Dic5—42no—
220C5×Dic3—42no—
221C2×C3×Q8—101no—
222C2×C3×D4—21no—
223C2×C3×D4—21no—
224C2×C3×D4C22×D512yes—
225C2×#67—102no—
226C22×Dic3—102no—
227D3×Q8—204no—
228#237—108no—
229C2×C4×D3—51no—
230C2×C4×D3—51no—
231C2×C4×D3—51no—
232C2×C4×D3—102no—
233C2×Dic6—21no—
234C2×Dic6—52no—
235C2×Dic6—102no—
236C2×C3×C8—52no—
237C3×C4⋊C4—52no—
238C3×C4⋊C4—101no—
239C3×C42C22×D511yescharacteristic
240#217—102no—
241#217—102no—
242C4⋊Dic3—21no—
243C4⋊Dic3—101no—
244C4⋊Dic3—101no—
245C4⋊Dic3C22×D512yes—
246Dic3⋊C4—102no—
247Dic3⋊C4—102no—
248C4×Dic3—51no—
249C4×Dic3—51no—
250C4×Dic3—51no—
251C2×C3⋊C8C22×D512yes—
252ΓL(1,𝔽25)—104no—
253C23×C5—22no—
254C22×D5—31no—
255C5×Q8—122no—
256C5×D4—24no—
257C5×D4—42no—
258C5×D4—42no—
259C2×C4×C5—62no—
260C2×C4×C5D3×D411yescharacteristic
261C2×C4×C5D3×D411yescharacteristic
262C2×C4×C5C23×D311yescharacteristic
263C22⋊D5—124no—
264C2×Dic5—22no—
265C2×Dic5—31no—
266C2×Dic5—31no—
267C2×Dic5—31no—
268C2×Dic5—62no—
269C2×Dic5—124no—
270C4×D5—34no—
271C4×D5—62no—
272C4×D5—62no—
273C4×D5—124no—
274Dic10—42no—
275Dic10—64no—
276Dic10—124no—
277C5×C8—62no—
278#168—22no—
279C2×#41—152no—
280C22×Q8—301no—
281C2×M16—152no—
282C4⋊Q8—31no—
283C4⋊Q8—51no—
284C4⋊Q8—152no—
285C4⋊Q8—151no—
286C4⋊Q8—301no—
287C4⋊Q8—301no—
288#127D3×D511yescharacteristic
289#124—152no—
290#121—301no—
291#121—301no—
292C4×D4—301no—
293C4×D4—301no—
294C2×C4⋊C4—152no—
295C2×C4⋊C4—301no—
296C2×C42—151no—
297#103—154no—
298#103—154no—
299#102—34no—
300#102—54no—
301#97—152no—
302C4×C8—31no—
303C4×C8—51no—
304C2×C3×C5—82no—
305C2×C3×C5C2×#12411yescharacteristic
306C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
307C2×C3×C5C2×#13711yescharacteristic
308D15—22no—
309C23×C3—22no—
310C22×D3—51no—
311C3×Q8—202no—
312C3×D4—24no—
313C3×D4—42no—
314C3×D4—42no—
315C2×C3×C4—102no—
316C2×C3×C4D4×D511yescharacteristic
317C2×C3×C4D4×D511yescharacteristic
318C2×C3×C4C23×D511yescharacteristic
319#67—204no—
320C2×Dic3—22no—
321C2×Dic3—51no—
322C2×Dic3—51no—
323C2×Dic3—51no—
324C2×Dic3—102no—
325C2×Dic3—204no—
326C4×D3—54no—
327C4×D3—102no—
328C4×D3—102no—
329C4×D3—204no—
330Dic6—42no—
331Dic6—104no—
332Dic6—204no—
333C3×C8—102no—
334C3⋊C8—22no—
335C22×C5—42no—
336C22×C5—42no—
337C22×C5—42no—
338C22×C5C2×D3×D411yescharacteristic
339C2×D5—32no—
340C2×D5—61no—
341C2×D5—61no—
342C4×C5—122no—
343C4×C5#91112yes—
344C4×C5#91512yes—
345C4×C5D3×#4112yes—
346Dic5—32no—
347Dic5—42no—
348Dic5—61no—
349Dic5—61no—
350Dic5—122no—
351#41—308no—
352C2×Q8—61no—
353C2×Q8—101no—
354C2×Q8—152no—
355C2×Q8—302no—
356C2×Q8—604no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—302no—
360C2×D4C2×D3×D512yes—
361C22×C4—151no—
362C22×C4—151no—
363C22×C4—151no—
364C22×C4—302no—
365C22×C4—302no—
366M16—304no—
367C2×C8—32no—
368C2×C8—52no—
369C4⋊C4—32no—
370C4⋊C4—52no—
371C4⋊C4—61no—
372C4⋊C4—101no—
373C4⋊C4—302no—
374C4⋊C4—302no—
375C4⋊C4—301no—
376C4⋊C4—301no—
377C22⋊C4—302no—
378C22⋊C4—302no—
379C42—151no—
380C42—151no—
381C42—151no—
382C42C2×D3×D511yescharacteristic
383C3×C5#314111yescharacteristic, complemented, Hall
384C22×C3—42no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3C2×D4×D511yescharacteristic
388C2×D3—52no—
389C2×D3—101no—
390C2×D3—101no—
391C3×C4—202no—
392C3×C4#405712yes—
393C3×C4#406112yes—
394C3×C4D5×#4112yes—
395Dic3—42no—
396Dic3—52no—
397Dic3—101no—
398Dic3—101no—
399Dic3—202no—
400C2×C5—82no—
401C2×C5D3×#12411yescharacteristic
402C2×C5C2×#91111yescharacteristic
403C2×C5C2×#91511yescharacteristic
404D5—62no—
405C23—22no—
406C23—151no—
407Q8—122no—
408Q8—202no—
409Q8—304no—
410Q8—604no—
411D4—24no—
412D4—42no—
413D4—42no—
414D4—604no—
415C2×C4—62no—
416C2×C4—102no—
417C2×C4—151no—
418C2×C4—154no—
419C2×C4—151no—
420C2×C4—151no—
421C2×C4—302no—
422C2×C4—302no—
423C2×C4—302no—
424C2×C4—604no—
425C2×C4—604no—
426C2×C4#679011yescharacteristic
427C2×C4#679011yescharacteristic
428C2×C4C22×D3×D511yescharacteristic
429C8—62no—
430C8—102no—
431C2×C3—82no—
432C2×C3D5×#12411yescharacteristic
433C2×C3C2×#405711yescharacteristic
434C2×C3C2×#406111yescharacteristic
435D3—102no—
436C5#8156511yescharacteristic, complemented, Sylow
437C22—42no—
438C22—42no—
439C22—42no—
440C22—152no—
441C22—301no—
442C22—301no—
443C22C2×#679011yescharacteristic
444C4—122no—
445C4—152no—
446C4—202no—
447C4—301no—
448C4—301no—
449C4—602no—
450C4#9148212yes—
451C4#9148812yes—
452C4#9202412yes—
453C3#1061329611yescharacteristic, complemented, Sylow
454C2—82no—
455C2—302no—
456C2C2×#9148211yescharacteristic
457C2C2×#9148811yescharacteristic
458C2#1175483111yescharacteristic
subgroupG/H
centerC22C2×#6790
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#127C22
socleC22×C3×C5C22×D4
prime radical#11756475C2
hypercenter#127D3×D5
hypocenterC3×C5#3141
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
2. #11752562C2semidirect
3. C5×#5070C2semidirect
4. C3×#65359C2semidirect
5. #11749117C2semidirect
6. #11748959C2semidirect
7. C4×#6651C2semidirect
8. #11746930C2semidirect
9. #11746921C2semidirect
41. #65359D3semidirect
82. #5070D5semidirect
383. C3×C5#3141semidirect
436. C5#81565semidirect
453. C3#10613296semidirect

Maximal subgroups

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

subgroupindex
#117564752
#117525622
C5×#50702
C3×#653592
#117491172
#117489592
C4×#66512
#117469302
#117469212
#106132963
#815655