Finite Group Databasebeta

← #14438628

Subgroups — #14438628

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
1C3×#66050C211yesmaximal, characteristic
2D5×#869C211yesmaximal, characteristic, complemented
3#11751171C211yesmaximal, characteristic, complemented
4C5×#5070C211yesmaximal, characteristic, complemented
5#11749105C214yesmaximal, complemented
6#11748951C214yesmaximal, complemented
7C4×#6649C211yesmaximal, characteristic, complemented
8#11746924C211yesmaximal, characteristic, complemented
9#11746920C211yesmaximal, characteristic, complemented
10#10613296—31nomaximal, Hall
11C2×C3×#711C2212yes—
12C2×D5×Dic6—21no—
13C3×C5×#127C2211yescharacteristic
14#91758C2211yescharacteristic
15C5×#869C2211yescharacteristic
16C3×#4084—21no—
17C3×#4084—21no—
18C3×#4082C2212yes—
19C3×D4×Dic5—21no—
20C3×D4×Dic5—21no—
21C3×C42×D5C2211yescharacteristic
22C3×#4015C2211yescharacteristic
23#91468—21no—
24#91468C2212yes—
25D5×C4⋊Dic3—21no—
26D5×C4⋊Dic3C2212yes—
27#91376—21no—
28C2×#6649C2212yes—
29#91116C2214yes—
30C4×C3⋊#168C2211yescharacteristic
31C5×#832C2214yes—
32C4×C5×C3⋊C8C2211yescharacteristic
33#90971C2214yes—
34#90970C2214yes—
35#90952C2212yes—
36#83283—51nomaximal, Hall
37D5×C4⋊Q8—31no—
38#66050D311yescharacteristic
39C5×#486—31no—
40#65382—31no—
41#65359—31no—
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×Dic30—21no—
50C2×C5×Dic6—21no—
51C2×C3×C22⋊D5—22no—
52C22×C3×Dic5—22no—
53C3×#711—28no—
54C2×C3×C4×D5D411yescharacteristic
55C2×C3×C4×D5D411yescharacteristic
56C2×C3×C4×D5C2311yescharacteristic
57C2×C3×Dic10C2312yes—
58C2×#1204—22no—
59C2×D5×Dic3—22no—
60D5×Dic6—44no—
61C3×C42×C5C2311yescharacteristic
62C4⋊Dic15—21no—
63C4⋊Dic15C2312yes—
64C2×C3⋊#168C2312yes—
65C5×C4⋊Dic3—21no—
66C5×C4⋊Dic3C2312yes—
67C2×C5×C3⋊C8C2312yes—
68C3×C22⋊Dic5—22no—
69C3×C22⋊Dic5—22no—
70C3×C4⋊Dic5—21no—
71C3×C4⋊Dic5—21no—
72C3×C4×Dic5D411yescharacteristic
73C3×C4×Dic5D411yescharacteristic
74C3×C4×Dic5C2311yescharacteristic
75#6668—22no—
76#6668—22no—
77#6649—24no—
78C3×#532—51no—
79C2×#869—51no—
80#5232—54no—
81#5079—51no—
82#5079—51no—
83#5070—51no—
84#5070D511yescharacteristic, complemented
85#5055—54no—
86C4×#208—51no—
87C2×Q8×D5—61no—
88C2×#711C2×D312yes—
89C5×C4⋊Q8—31no—
90C5×#127C2×D311yescharacteristic
91#4090—61no—
92#4084—21no—
93#4084—21no—
94#4082C2×D312yes—
95D4×Dic5—21no—
96D4×Dic5—21no—
97C2×#676—32no—
98D5×C4⋊C4—32no—
99D5×C4⋊C4—61no—
100#4034—32no—
101#4034—61no—
102C42×D5C2×D311yescharacteristic
103#4015—31no—
104#4015C2×D311yescharacteristic
105C5×#102—34no—
106C4×C5×C8—31no—
107#3964—34no—
108#3948—34no—
109#3947—32no—
110#3942—34no—
111C4×#168—31no—
112#3141—151noSylow
113C23×C3×C5—22no—
114C22×C3×D5C2×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
121C2×Dic15—22no—
122Dic30—42no—
123C2×C5×Dic3—22no—
124C5×Dic6—42no—
125C3×C22⋊D5—44no—
126C2×C3×Dic5—22no—
127C2×C3×Dic5—44no—
128C2×C3×Dic5C2×D411yescharacteristic
129C2×C3×Dic5C2×D411yescharacteristic
130C2×C3×Dic5C2×D411yescharacteristic
131C3×C4×D5—22no—
132C3×C4×D5—22no—
133C3×C4×D5#4114yes—
134C3×Dic10—24no—
135#1204—44no—
136D5×Dic3—44no—
137C3⋊#168—22no—
138C5×C3⋊C8—22no—
139C2×C3×#41—52no—
140C22×Dic6—101no—
141C3×C4⋊Q8—51no—
142C3×#127C2×D511yescharacteristic
143C3×#124—52no—
144C3×#121—101no—
145C3×#121—101no—
146C3×C4×D4—101no—
147C3×C4×D4—101no—
148C2×C3×C42—51no—
149C2×C4⋊Dic3—52no—
150C2×C4⋊Dic3—101no—
151C2×#208—52no—
152#869—51no—
153#869—52no—
154#869—101no—
155#869—101no—
156#869C2×D511yescharacteristic
157#835—54no—
158#835—54no—
159#832—54no—
160#832C2×D514yes—
161#803—52no—
162C4×C3⋊C8—51no—
163C4×C3⋊C8C2×D511yescharacteristic
164C2×C5×Q8—61no—
165C2×C5×D4—21no—
166C2×C5×D4—21no—
167C2×C5×D4C22×D312yes—
168C2×C22⋊D5—22no—
169C22×Dic5—22no—
170Q8×D5—124no—
171#711—28no—
172C2×C4×D5—62no—
173C2×C4×D5#6711yescharacteristic
174C2×C4×D5#6711yescharacteristic
175C2×C4×D5C22×D311yescharacteristic
176C2×Dic10—61no—
177C2×Dic10—62no—
178C2×Dic10C22×D312yes—
179C2×C5×C8—32no—
180C5×C4⋊C4—32no—
181C5×C4⋊C4—61no—
182C42×C5C22×D311yescharacteristic
183C22⋊Dic5—22no—
184C22⋊Dic5—22no—
185C4⋊Dic5—21no—
186C4⋊Dic5—21no—
187C4⋊Dic5—32no—
188C4⋊Dic5—61no—
189#683—62no—
190#683—62no—
191C4×Dic5#6711yescharacteristic
192C4×Dic5#6711yescharacteristic
193C4×Dic5C22×D311yescharacteristic
194C2×#168—32no—
195#676—64no—
196#532—51no—
197C2×C4⋊Q8—151no—
198#489—154no—
199#487—151no—
200#487—151no—
201#486—31no—
202#486—151no—
203#417—154no—
204C4×M16—151no—
205C22×C3×C5—42no—
206C22×C3×C5—42no—
207C22×C3×C5—42no—
208C22×C3×C5C22×D411yescharacteristic
209C2×C3×D5—21no—
210C2×C3×D5—21no—
211C2×C3×D5#12412yes—
212C3×C4×C5AGL(1,ℤ/8)12yes—
213C3×C4×C5#13712yes—
214C3×C4×C5C2×#4112yes—
215Dic15—42no—
216C3×Dic5—21no—
217C3×Dic5—21no—
218C3×Dic5—42no—
219C3×Dic5#12412yes—
220C5×Dic3—42no—
221C3×#41—108no—
222C2×C3×Q8—52no—
223C2×C3×D4—21no—
224C2×C3×D4—21no—
225C2×C3×D4—102no—
226C2×C3×D4C22×D512yes—
227C22×C3×C4—51no—
228C22×C3×C4—51no—
229C22×C3×C4—51no—
230C22×C3×C4—102no—
231C22×Dic3—102no—
232C2×Dic6—21no—
233C2×Dic6—102no—
234C2×Dic6—101no—
235C2×Dic6—204no—
236C3×C4⋊C4—101no—
237C3×C4⋊C4—101no—
238C3×C22⋊C4—102no—
239C3×C22⋊C4—102no—
240C3×C42—51no—
241C3×C42—51no—
242C3×C42—51no—
243C3×C42C22×D511yescharacteristic
244C4⋊Dic3—21no—
245C4⋊Dic3—52no—
246C4⋊Dic3—102no—
247C4⋊Dic3—102no—
248C4⋊Dic3—101no—
249C4⋊Dic3C22×D512yes—
250#208—104no—
251C2×C3⋊C8—52no—
252C2×C3⋊C8C22×D512yes—
253C23×C5—22no—
254C22×D5C2×#6711yescharacteristic
255C5×Q8—122no—
256C5×D4—24no—
257C5×D4—42no—
258C5×D4—42no—
259C2×C4×C5—62no—
260C2×C4×C5C2×#6711yescharacteristic
261C2×C4×C5C2×#6711yescharacteristic
262C2×C4×C5C23×D311yescharacteristic
263C22⋊D5—44no—
264C2×Dic5—22no—
265C2×Dic5—44no—
266C2×Dic5—62no—
267C2×Dic5C2×#6711yescharacteristic
268C2×Dic5C2×#6711yescharacteristic
269C2×Dic5C2×#6711yescharacteristic
270C4×D5—22no—
271C4×D5—22no—
272C4×D5—124no—
273C4×D5#23714yes—
274Dic10—24no—
275Dic10—124no—
276Dic10—122no—
277C5×C8—62no—
278#168—62no—
279C2×#41—52no—
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—52no—
290#121—101no—
291#121—101no—
292C4×D4—101no—
293C4×D4—101no—
294C2×C4⋊C4—152no—
295C2×C4⋊C4—301no—
296C2×C42—51no—
297#103—154no—
298#103—154no—
299#102—34no—
300#102—154no—
301#97—152no—
302C4×C8—31no—
303C4×C8—151no—
304C2×C3×C5—82no—
305C2×C3×C5C2×#12411yescharacteristic
306C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
307C2×C3×C5C2×#13711yescharacteristic
308C3×D5—22no—
309C23×C3—22no—
310C23×C3—51no—
311C3×Q8—104no—
312C3×D4—24no—
313C3×D4—42no—
314C3×D4—42no—
315C3×D4—204no—
316C2×C3×C4—51no—
317C2×C3×C4—54no—
318C2×C3×C4—51no—
319C2×C3×C4—51no—
320C2×C3×C4—102no—
321C2×C3×C4—102no—
322C2×C3×C4—102no—
323C2×C3×C4—204no—
324C2×C3×C4D4×D511yescharacteristic
325C2×C3×C4D4×D511yescharacteristic
326C2×C3×C4C23×D511yescharacteristic
327C2×Dic3—22no—
328C2×Dic3—102no—
329C2×Dic3—204no—
330Dic6—42no—
331Dic6—204no—
332Dic6—202no—
333C3⋊C8—22no—
334C3⋊C8—102no—
335C22×C5—42no—
336C22×C5—42no—
337C22×C5—42no—
338C22×C5C22×#6711yescharacteristic
339C2×D5—21no—
340C2×D5—21no—
341C2×D5#93612yes—
342C4×C5—122no—
343C4×C5#94912yes—
344C4×C5#95112yes—
345C4×C5C2×#23712yes—
346Dic5—21no—
347Dic5—21no—
348Dic5—42no—
349Dic5—122no—
350Dic5#93612yes—
351#41—108no—
352C2×Q8—52no—
353C2×Q8—61no—
354C2×Q8—302no—
355C2×Q8—301no—
356C2×Q8—604no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—102no—
360C2×D4C2×D3×D512yes—
361C22×C4—51no—
362C22×C4—51no—
363C22×C4—51no—
364C22×C4—102no—
365C22×C4—302no—
366M16—304no—
367C2×C8—32no—
368C2×C8—152no—
369C4⋊C4—32no—
370C4⋊C4—61no—
371C4⋊C4—101no—
372C4⋊C4—101no—
373C4⋊C4—152no—
374C4⋊C4—302no—
375C4⋊C4—302no—
376C4⋊C4—301no—
377C22⋊C4—102no—
378C22⋊C4—102no—
379C42—51no—
380C42—51no—
381C42—51no—
382C42C2×D3×D511yescharacteristic
383C3×C5#314111yescharacteristic, complemented, Hall
384C22×C3—42no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3—52no—
388C22×C3—101no—
389C22×C3—101no—
390C22×C3C2×D4×D511yescharacteristic
391C3×C4—52no—
392C3×C4—101no—
393C3×C4—101no—
394C3×C4—202no—
395C3×C4#405712yes—
396C3×C4#406112yes—
397C3×C4D5×#4112yes—
398Dic3—42no—
399Dic3—202no—
400C2×C5—82no—
401C2×C5C2×#93611yescharacteristic
402C2×C5C2×#94911yescharacteristic
403C2×C5C2×#95111yescharacteristic
404D5—22no—
405C23—22no—
406C23—51no—
407Q8—104no—
408Q8—122no—
409Q8—604no—
410Q8—602no—
411D4—24no—
412D4—42no—
413D4—42no—
414D4—204no—
415C2×C4—51no—
416C2×C4—54no—
417C2×C4—51no—
418C2×C4—51no—
419C2×C4—62no—
420C2×C4—102no—
421C2×C4—102no—
422C2×C4—102no—
423C2×C4—204no—
424C2×C4—302no—
425C2×C4—604no—
426C2×C4D5×#6711yescharacteristic
427C2×C4D5×#6711yescharacteristic
428C2×C4C22×D3×D511yescharacteristic
429C8—62no—
430C8—302no—
431C2×C3—82no—
432C2×C3—102no—
433C2×C3D5×#12411yescharacteristic
434C2×C3C2×#405711yescharacteristic
435C2×C3C2×#406111yescharacteristic
436C5#8328311yescharacteristic, complemented, Sylow
437C22—42no—
438C22—42no—
439C22—42no—
440C22—52no—
441C22—101no—
442C22—101no—
443C22C2×D5×#6711yescharacteristic
444C4—52no—
445C4—101no—
446C4—101no—
447C4—122no—
448C4—202no—
449C4—602no—
450C4#9147812yes—
451C4#9148412yes—
452C4D5×#23712yes—
453C3#1061329611yescharacteristic, complemented, Sylow
454C2—82no—
455C2—102no—
456C2C2×#9147811yescharacteristic
457C2C2×#9148411yescharacteristic
458C2D5×#93611yescharacteristic
subgroupG/H
centerC22C2×D5×#67
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#127C22
socleC22×C3×C5C22×D4
prime radicalC3×#66050C2
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. D5×#869C2semidirect
3. #11751171C2semidirect
4. C5×#5070C2semidirect
5. #11749105C2semidirect
6. #11748951C2semidirect
7. C4×#6649C2semidirect
8. #11746924C2semidirect
9. #11746920C2semidirect
84. #5070D5semidirect
383. C3×C5#3141semidirect
436. C5#83283semidirect
453. C3#10613296semidirect

Maximal subgroups

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

subgroupindex
C3×#660502
D5×#8692
#117511712
C5×#50702
#117491052
#117489512
C4×#66492
#117469242
#117469202
#106132963
#832835