Finite Group Databasebeta

← #14438638

Subgroups — #14438638

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#11756484C211yesmaximal, characteristic
2#11752562C211yesmaximal, characteristic, complemented
3C5×#5079C211yesmaximal, characteristic, complemented
4C3×#65370C211yesmaximal, characteristic, complemented
5#11749363C214yesmaximal, complemented
6#11749205C214yesmaximal, complemented
7C4×#6651C211yesmaximal, characteristic, complemented
8#11746912C211yesmaximal, characteristic, complemented
9#11746910C211yesmaximal, characteristic, complemented
10#10613332—31nomaximal, Hall
11C2×#6821C2212yes—
12C2×D15⋊Q8—21no—
13C3×C5×C4⋊Q8C2211yescharacteristic
14#91836C2212yes—
15#91784—21no—
16#91784—21no—
17#91782—21no—
18#91782—21no—
19#91763C2211yescharacteristic
20C42×D15C2211yescharacteristic
21C5×#869C2211yescharacteristic
22C3×#4015C2211yescharacteristic
23#91470—21no—
24#91470C2212yes—
25#91437—21no—
26#91437C2212yes—
27#91385—21no—
28C2×#6651C2212yes—
29C5×#835C2214yes—
30C4×C5×C3⋊C8C2211yescharacteristic
31C3×#3967C2214yes—
32C3×C4×#168C2211yescharacteristic
33#90969C2214yes—
34#90968C2214yes—
35#90954C2212yes—
36#81601—51nomaximal, Hall
37#66059—31no—
38D5×C4⋊Q8—31no—
39C5×#487—31no—
40#65377—31no—
41#65370D311yescharacteristic, complemented
42#65092—34no—
43#65084—34no—
44#64983—31no—
45C4×#676—31no—
46C2×C3×C5×Q8C2312yes—
47#6821—28no—
48C2×D60—22no—
49C2×D60—21no—
50C2×D60—21no—
51C2×D60C2312yes—
52C2×C4×D15—22no—
53C2×C4×D15D411yescharacteristic
54C2×C4×D15D411yescharacteristic
55C2×C4×D15C2311yescharacteristic
56C2×C5×Dic6—21no—
57C2×C3×Dic10—21no—
58C2×#1204—22no—
59C2×#1200—22no—
60D15⋊Q8—44no—
61C3×C5×C4⋊C4—21no—
62C3×C5×C4⋊C4—21no—
63C3×C42×C5C2311yescharacteristic
64#6714—22no—
65#6714—22no—
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—
78#5725—51no—
79D3×C4⋊Q8—51no—
80C3×#487—51no—
81#5088—51no—
82#5079D511yescharacteristic, complemented
83#4803—54no—
84#4794—54no—
85#4694—51no—
86C4×ΓL(1,𝔽25)—51no—
87C2×#713—32no—
88C2×Q8×D5—61no—
89C5×C4⋊Q8—31no—
90C5×C4⋊Q8C2×D311yescharacteristic
91#4093—32no—
92#4090—61no—
93C2×#676—32no—
94#4041—61no—
95#4041—61no—
96#4039—61no—
97#4039—61no—
98D5×C4⋊C4—32no—
99D5×C4⋊C4—61no—
100#4034—32no—
101#4034—61no—
102#4020—31no—
103C42×D5—31no—
104#4015C2×D311yescharacteristic
105C5×#103—34no—
106C4×C5×C8—31no—
107#3967C2×D314yes—
108#3953—34no—
109#3947—32no—
110#3941—34no—
111C4×#168C2×D311yescharacteristic
112#3141—151noSylow
113C22×D15—22no—
114C22×D15C2×D411yescharacteristic
115C3×C5×Q8—24no—
116C2×C3×C4×C5—22no—
117C2×C3×C4×C5C2×D411yescharacteristic
118C2×C3×C4×C5C2×D411yescharacteristic
119C2×C3×C4×C5C2411yescharacteristic
120C2×Dic15C2×D411yescharacteristic
121C2×Dic15C2×D411yescharacteristic
122C2×Dic15C2×D411yescharacteristic
123D60—24no—
124D60—44no—
125D60—42no—
126D60—42no—
127C4×D15—22no—
128C4×D15—22no—
129C4×D15—44no—
130C4×D15#4114yes—
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×#239—52no—
140C2×D3×Q8—101no—
141C3×C4⋊Q8—51no—
142C3×C4⋊Q8C2×D511yescharacteristic
143#947—52no—
144#944—101no—
145C2×ΓL(1,𝔽25)—52no—
146#895—101no—
147#895—101no—
148#893—101no—
149#893—101no—
150D3×C4⋊C4—52no—
151D3×C4⋊C4—101no—
152#888—52no—
153#888—101no—
154#874—51no—
155C42×D3—51no—
156#869C2×D511yescharacteristic
157C3×#103—54no—
158C3×C4×C8—51no—
159#835C2×D514yes—
160#821—54no—
161#815—52no—
162#809—54no—
163C4×C3⋊C8C2×D511yescharacteristic
164C2×C5×Q8—61no—
165C2×C5×Q8C22×D312yes—
166#713—68no—
167Q8×D5—124no—
168C2×D20—32no—
169C2×D20—62no—
170C2×D20—61no—
171C2×D20—61no—
172C2×C4×D5—31no—
173C2×C4×D5—31no—
174C2×C4×D5—31no—
175C2×C4×D5—62no—
176C2×C4×D5—62no—
177C2×Dic10—21no—
178C2×Dic10—62no—
179C2×C5×C8—32no—
180C5×C4⋊C4—21no—
181C5×C4⋊C4—21no—
182C5×C4⋊C4—32no—
183C5×C4⋊C4—61no—
184C42×C5C22×D311yescharacteristic
185#685—62no—
186#685—62no—
187C4⋊Dic5—21no—
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—31no—
200#487—51no—
201#486—151no—
202#486—151no—
203#417—154no—
204C4×M16—151no—
205C22×C3×C5C22×D411yescharacteristic
206C2×D15—21no—
207C2×D15—21no—
208C2×D15—42no—
209C2×D15—42no—
210C2×D15—42no—
211C2×D15#12412yes—
212C3×C4×C5—42no—
213C3×C4×C5AGL(1,ℤ/8)12yes—
214C3×C4×C5#13712yes—
215C3×C4×C5C2×#4112yes—
216Dic15—21no—
217Dic15—21no—
218Dic15#12412yes—
219C3×Dic5—42no—
220C5×Dic3—42no—
221C2×C3×Q8—101no—
222C2×C3×Q8C22×D512yes—
223#239—108no—
224D3×Q8—204no—
225C2×D12—52no—
226C2×D12—102no—
227C2×D12—101no—
228C2×D12—101no—
229C2×C4×D3—51no—
230C2×C4×D3—51no—
231C2×C4×D3—51no—
232C2×C4×D3—102no—
233C2×C4×D3—102no—
234C2×Dic6—21no—
235C2×Dic6—102no—
236C2×C3×C8—52no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—52no—
240C3×C4⋊C4—101no—
241C3×C42C22×D511yescharacteristic
242(C2×D3)⋊C4—102no—
243(C2×D3)⋊C4—102no—
244C4⋊Dic3—21no—
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—
253C22×D5—31no—
254C22×D5—62no—
255C5×Q8—24no—
256C5×Q8—122no—
257C2×C4×C5—22no—
258C2×C4×C5—62no—
259C2×C4×C5D3×D411yescharacteristic
260C2×C4×C5D3×D411yescharacteristic
261C2×C4×C5C23×D311yescharacteristic
262C2×Dic5—22no—
263C2×Dic5—31no—
264C2×Dic5—31no—
265C2×Dic5—31no—
266D20—64no—
267D20—124no—
268D20—122no—
269D20—122no—
270C4×D5—34no—
271C4×D5—62no—
272C4×D5—62no—
273C4×D5—124no—
274C4×D5—124no—
275Dic10—42no—
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—301no—
286C4⋊Q8—301no—
287C4⋊Q8D3×D511yescharacteristic
288#127—151no—
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—34no—
298#103—54no—
299#102—154no—
300#102—154no—
301#97—152no—
302C4×C8—31no—
303C4×C8—51no—
304C2×C3×C5C2×#12411yescharacteristic
305C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
306C2×C3×C5C2×#13711yescharacteristic
307D15—22no—
308D15—82no—
309C22×D3—51no—
310C22×D3—102no—
311C3×Q8—24no—
312C3×Q8—202no—
313C2×C3×C4—22no—
314C2×C3×C4—102no—
315C2×C3×C4D4×D511yescharacteristic
316C2×C3×C4D4×D511yescharacteristic
317C2×C3×C4C23×D511yescharacteristic
318C2×Dic3—22no—
319C2×Dic3—51no—
320C2×Dic3—51no—
321C2×Dic3—51no—
322D12—104no—
323D12—204no—
324D12—202no—
325D12—202no—
326C4×D3—54no—
327C4×D3—102no—
328C4×D3—102no—
329C4×D3—204no—
330C4×D3—204no—
331Dic6—42no—
332Dic6—204no—
333C3×C8—102no—
334C3⋊C8—22no—
335C22×C5C2×D3×D411yescharacteristic
336C2×D5—32no—
337C2×D5—61no—
338C2×D5—61no—
339C2×D5—122no—
340C2×D5—122no—
341C2×D5—122no—
342C4×C5—42no—
343C4×C5—122no—
344C4×C5#91412yes—
345C4×C5#91812yes—
346C4×C5D3×#4112yes—
347Dic5—32no—
348Dic5—42no—
349Dic5—61no—
350Dic5—61no—
351#41—308no—
352C2×Q8—61no—
353C2×Q8—101no—
354C2×Q8—302no—
355C2×Q8—604no—
356C2×Q8C2×D3×D512yes—
357C2×D4—152no—
358C2×D4—302no—
359C2×D4—301no—
360C2×D4—301no—
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—21no—
370C4⋊C4—21no—
371C4⋊C4—32no—
372C4⋊C4—52no—
373C4⋊C4—61no—
374C4⋊C4—101no—
375C4⋊C4—302no—
376C4⋊C4—302no—
377C22⋊C4—302no—
378C22⋊C4—302no—
379C42—151no—
380C42—151no—
381C42—151no—
382C42C2×D3×D511yescharacteristic
383C3×C5#314111yescharacteristic, complemented, Hall
384C22×C3C2×D4×D511yescharacteristic
385C2×D3—52no—
386C2×D3—101no—
387C2×D3—101no—
388C2×D3—202no—
389C2×D3—202no—
390C2×D3—202no—
391C3×C4—42no—
392C3×C4—202no—
393C3×C4#406012yes—
394C3×C4#406412yes—
395C3×C4D5×#4112yes—
396Dic3—42no—
397Dic3—52no—
398Dic3—101no—
399Dic3—101no—
400C2×C5D3×#12411yescharacteristic
401C2×C5C2×#91411yescharacteristic
402C2×C5C2×#91811yescharacteristic
403D5—62no—
404D5—242no—
405C23—151no—
406C23—302no—
407Q8—24no—
408Q8—122no—
409Q8—202no—
410Q8—604no—
411D4—304no—
412D4—604no—
413D4—602no—
414D4—602no—
415C2×C4—22no—
416C2×C4—62no—
417C2×C4—102no—
418C2×C4—151no—
419C2×C4—154no—
420C2×C4—151no—
421C2×C4—151no—
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×C3D5×#12411yescharacteristic
432C2×C3C2×#406011yescharacteristic
433C2×C3C2×#406411yescharacteristic
434D3—102no—
435D3—402no—
436C5#8160111yescharacteristic, complemented, Sylow
437C22—152no—
438C22—301no—
439C22—301no—
440C22—602no—
441C22—602no—
442C22—602no—
443C22C2×#679011yescharacteristic
444C4—42no—
445C4—122no—
446C4—152no—
447C4—202no—
448C4—301no—
449C4—301no—
450C4#9150612yes—
451C4#9151212yes—
452C4#9202412yes—
453C3#1061333211yescharacteristic, complemented, Sylow
454C2—302no—
455C2—1202no—
456C2#1175483111yescharacteristic
457C2C2×#9150611yescharacteristic
458C2C2×#9151211yescharacteristic
subgroupG/H
centerC22C2×#6790
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×C4⋊Q8C22
socleC22×C3×C5C22×D4
prime radical#11756484C2
hypercenterC4⋊Q8D3×D5
hypocenterC3×C5#3141
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #11752562C2semidirect
3. C5×#5079C2semidirect
4. C3×#65370C2semidirect
5. #11749363C2semidirect
6. #11749205C2semidirect
7. C4×#6651C2semidirect
8. #11746912C2semidirect
9. #11746910C2semidirect
41. #65370D3semidirect
82. #5079D5semidirect
383. C3×C5#3141semidirect
436. C5#81601semidirect
453. C3#10613332semidirect

Maximal subgroups

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

subgroupindex
#117564842
#117525622
C5×#50792
C3×#653702
#117493632
#117492052
C4×#66512
#117469122
#117469102
#106133323
#816015