Finite Group Databasebeta

← #14471505

Subgroups — #14471505

Order 1920.

740 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756480C211yesmaximal, characteristic, complemented
2D4×Dic30C211yesmaximal, characteristic
3C3×C5×#461C211yesmaximal, characteristic
4#11751263C212yesmaximal
5SD16×Dic15C212yesmaximal
6#11751180C211yesmaximal, characteristic, complemented
7C4×#6716C211yesmaximal, characteristic
8#11750952C211yesmaximal, characteristic, complemented
9#11750931C211yesmaximal, characteristic, complemented
10#11750893C212yesmaximal, complemented
11#11750892C212yesmaximal, complemented
12#10612145—31nomaximal, Hall
13C22×Dic30—21no—
14C2×C3×C5×SD16C2212yes—
15C3×C5×C4⋊Q8C2211yescharacteristic
16C3×C4×C5×D4C2211yescharacteristic
17Q8×Dic15C2212yes—
18#91833C2212yes—
19D4×Dic15C2212yes—
20C2×#6716C2211yescharacteristic
21#91813—21no—
22#91779—21no—
23#91778—22no—
24#91777C2212yes—
25#91776—21no—
26#91767—22no—
27#91758C2211yescharacteristic
28C4×Dic30C2211yescharacteristic
29C4×Dic30C2211yescharacteristic
30C3×C5×C4⋊C8C2211yescharacteristic
31C3×C5×#103C2212yes—
32#91119C2212yes—
33#91119C2212yes—
34#91116C2211yescharacteristic
35#91101C2212yes—
36#91100C2212yes—
37#91098C2212yes—
38C8×Dic15C2212yes—
39#91094C2211yescharacteristic
40#91092C2211yescharacteristic
41C4×C3⋊#168C2211yescharacteristic
42#80414—51nomaximal, Hall
43#66055—31no—
44D4×Dic10—31no—
45C5×#461D311yescharacteristic
46#65451—32no—
47SD16×Dic5—32no—
48#65370—31no—
49C4×#687—31no—
50#65140—31no—
51#65119—31no—
52#65081—32no—
53#65080—32no—
54C2×C3×C5×Q8C2312yes—
55C2×C3×C5×D4C2311yescharacteristic
56C22×C3×C4×C5—21no—
57C22×Dic15—22no—
58C2×Dic30—21no—
59C2×Dic30—22no—
60C2×Dic30—44no—
61C2×Dic30D411yescharacteristic
62C2×Dic30C2311yescharacteristic
63C3×C5×SD16—24no—
64C2×C3×C5×C8C2312yes—
65C3×C5×C4⋊C4—21no—
66C3×C5×C4⋊C4—21no—
67C3×C5×C4⋊C4C2311yescharacteristic
68C3×C5×C22⋊C4—21no—
69C3×C42×C5C2311yescharacteristic
70#6719—22no—
71#6716—22no—
72C4⋊Dic15—21no—
73C4⋊Dic15—22no—
74C4⋊Dic15D411yescharacteristic
75C4⋊Dic15C2311yescharacteristic
76C4⋊Dic15C2312yes—
77#6712—21no—
78#6712—22no—
79#6712—22no—
80#6712—22no—
81#6712D412yes—
82C4×Dic15—21no—
83C4×Dic15—22no—
84C4×Dic15C2312yes—
85C2×C3⋊#168C2311yescharacteristic
86C2×C3⋊#168C2311yescharacteristic
87#5721—51no—
88D4×Dic6—51no—
89C3×#461D511yescharacteristic
90#5162—52no—
91Dic3×SD16—52no—
92#5079—51no—
93C4×#214—51no—
94#4851—51no—
95#4830—51no—
96#4792—52no—
97#4791—52no—
98C22×Dic10—61no—
99C2×C5×SD16C2×D312yes—
100C5×C4⋊Q8C2×D311yescharacteristic
101C4×C5×D4C2×D311yescharacteristic
102Q8×Dic5—32no—
103#4090—32no—
104D4×Dic5—32no—
105C2×#687—31no—
106#4070—61no—
107#4036—61no—
108#4035—62no—
109#4034—32no—
110#4033—61no—
111#4024—62no—
112#4015—31no—
113C4×Dic10—31no—
114C4×Dic10—31no—
115C5×C4⋊C8C2×D311yescharacteristic
116C5×#103C2×D312yes—
117#3967—32no—
118#3967—32no—
119#3964—31no—
120#3949—32no—
121#3948—32no—
122#3946—32no—
123C8×Dic5—32no—
124#3942—31no—
125#3940—31no—
126C4×#168—31no—
127#3359—151noSylow
128C23×C3×C5—21no—
129C3×C5×Q8—22no—
130C3×C5×Q8—22no—
131C3×C5×D4—21no—
132C3×C5×D4#4112yes—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5—21no—
135C2×C3×C4×C5—42no—
136C2×C3×C4×C5C2×D411yescharacteristic
137C2×C3×C4×C5C2×D411yescharacteristic
138C2×C3×C4×C5C2411yescharacteristic
139C2×Dic15—21no—
140C2×Dic15—21no—
141C2×Dic15—22no—
142C2×Dic15—44no—
143C2×Dic15C2×D412yes—
144C2×Dic15C2×D412yes—
145Dic30—21no—
146Dic30—21no—
147Dic30—42no—
148Dic30—44no—
149Dic30SD1614yescomplemented
150C3×C5×C8—22no—
151C3⋊#168—21no—
152C3⋊#168—21no—
153C22×Dic6—101no—
154C2×C3×SD16C2×D512yes—
155C3×C4⋊Q8C2×D511yescharacteristic
156C3×C4×D4C2×D511yescharacteristic
157Q8×Dic3—52no—
158#944—52no—
159D4×Dic3—52no—
160C2×#214—51no—
161#924—101no—
162#890—101no—
163#889—102no—
164#888—52no—
165#887—101no—
166#878—102no—
167#869—51no—
168C4×Dic6—51no—
169C4×Dic6—51no—
170C3×C4⋊C8C2×D511yescharacteristic
171C3×#103C2×D512yes—
172#835—52no—
173#835—52no—
174#832—51no—
175#817—52no—
176#816—52no—
177Dic3⋊C8—52no—
178C8×Dic3—52no—
179#810—51no—
180#808—51no—
181C4×C3⋊C8—51no—
182C2×C5×Q8C22×D312yes—
183C2×C5×D4C22×D311yescharacteristic
184C22×C4×C5—21no—
185C22×Dic5—62no—
186C2×Dic10—31no—
187C2×Dic10—31no—
188C2×Dic10—62no—
189C2×Dic10—61no—
190C2×Dic10—124no—
191C5×SD16—24no—
192C2×C5×C8C22×D312yes—
193C5×C4⋊C4—21no—
194C5×C4⋊C4—21no—
195C5×C4⋊C4C22×D311yescharacteristic
196C5×C22⋊C4—21no—
197C42×C5C22×D311yescharacteristic
198C22⋊Dic5—62no—
199#687—62no—
200C4⋊Dic5—31no—
201C4⋊Dic5—31no—
202C4⋊Dic5—32no—
203C4⋊Dic5—62no—
204C4⋊Dic5—61no—
205#683—32no—
206#683—62no—
207#683—62no—
208#683—62no—
209#683—61no—
210C4×Dic5—32no—
211C4×Dic5—62no—
212C4×Dic5—61no—
213C2×#168—31no—
214C2×#168—31no—
215#557—151no—
216D4×Q8—151no—
217#487—151no—
218#487—152no—
219#475—151no—
220#475—152no—
221#461—152no—
222#461D1511yescharacteristic
223C8×Q8—151no—
224C4×SD16—152no—
225C4×SD16—151no—
226C22×C3×C5—22no—
227C22×C3×C5—41no—
228C22×C3×C5—41no—
229C22×C3×C5C22×D411yescharacteristic
230C3×C4×C5—21no—
231C3×C4×C5—42no—
232C3×C4×C5—41no—
233C3×C4×C5C2×SD1612yes—
234C3×C4×C5C2×#4111yescharacteristic
235C3×C4×C52−1+411yescharacteristic
236Dic15—22no—
237Dic15—41no—
238Dic15—42no—
239Dic15—41no—
240Dic15C2×SD1614yes—
241C2×C3×Q8C22×D512yes—
242C2×C3×D4C22×D511yescharacteristic
243C22×C3×C4—21no—
244C22×Dic3—102no—
245C2×Dic6—51no—
246C2×Dic6—51no—
247C2×Dic6—102no—
248C2×Dic6—101no—
249C2×Dic6—204no—
250C3×SD16—24no—
251C2×C3×C8C22×D512yes—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4C22×D511yescharacteristic
255C3×C22⋊C4—21no—
256C3×C42C22×D511yescharacteristic
257#217—102no—
258#214—102no—
259C4⋊Dic3—51no—
260C4⋊Dic3—51no—
261C4⋊Dic3—52no—
262C4⋊Dic3—102no—
263C4⋊Dic3—101no—
264Dic3⋊C4—52no—
265Dic3⋊C4—102no—
266Dic3⋊C4—102no—
267Dic3⋊C4—102no—
268Dic3⋊C4—101no—
269C4×Dic3—52no—
270C4×Dic3—102no—
271C4×Dic3—101no—
272C2×C3⋊C8—51no—
273C2×C3⋊C8—51no—
274C23×C5—21no—
275C5×Q8—22no—
276C5×Q8—22no—
277C5×D4—21no—
278C5×D4#23712yes—
279C2×C4×C5—22no—
280C2×C4×C5—21no—
281C2×C4×C5—42no—
282C2×C4×C5D3×D411yescharacteristic
283C2×C4×C5D3×D411yescharacteristic
284C2×C4×C5C23×D311yescharacteristic
285C2×Dic5—32no—
286C2×Dic5—32no—
287C2×Dic5—61no—
288C2×Dic5—61no—
289C2×Dic5—62no—
290C2×Dic5—124no—
291Dic10—34no—
292Dic10—61no—
293Dic10—61no—
294Dic10—124no—
295Dic10—122no—
296C5×C8—22no—
297#168—61no—
298#168—61no—
299C22×Q8—301no—
300C2×SD16—151no—
301C2×SD16C2×D1512yes—
302C4⋊Q8—152no—
303C4⋊Q8—152no—
304C4⋊Q8—151no—
305C4⋊Q8C2×D1511yescharacteristic
306#125—302no—
307#125—301no—
308C22⋊Q8—302no—
309C22⋊Q8—301no—
310C4×Q8—151no—
311C4×Q8—152no—
312C4×Q8—151no—
313C4×Q8—301no—
314C4×D4—152no—
315C4×D4C2×D1511yescharacteristic
316#106—152no—
317#106—151no—
318C4⋊C8—152no—
319C4⋊C8C2×D1511yescharacteristic
320#103—152no—
321#103—152no—
322#103—152no—
323#103—151no—
324#103C2×D1512yes—
325#102—151no—
326C4×C8—152no—
327C4×C8—151no—
328C2×C3×C5—42no—
329C2×C3×C5#54811yescharacteristic
330C2×C3×C5C22×SD1611yescharacteristic
331C2×C3×C5#57811yescharacteristic
332C23×C3—21no—
333C3×Q8—22no—
334C3×Q8—22no—
335C3×D4—21no—
336C3×D4#71112yes—
337C2×C3×C4—22no—
338C2×C3×C4—21no—
339C2×C3×C4—42no—
340C2×C3×C4D4×D511yescharacteristic
341C2×C3×C4D4×D511yescharacteristic
342C2×C3×C4C23×D511yescharacteristic
343C2×Dic3—52no—
344C2×Dic3—52no—
345C2×Dic3—101no—
346C2×Dic3—101no—
347C2×Dic3—102no—
348C2×Dic3—204no—
349Dic6—54no—
350Dic6—101no—
351Dic6—101no—
352Dic6—204no—
353Dic6—202no—
354C3×C8—22no—
355C3⋊C8—101no—
356C3⋊C8—101no—
357C22×C5—22no—
358C22×C5—41no—
359C22×C5—41no—
360C22×C5C2×D3×D411yescharacteristic
361C4×C5—21no—
362C4×C5—41no—
363C4×C5—42no—
364C4×C5D3×SD1612yes—
365C4×C5C2×#23711yescharacteristic
366C4×C5#101011yescharacteristic
367Dic5—34no—
368Dic5—62no—
369Dic5—122no—
370Dic5—121no—
371Dic5—121no—
372C2×Q8—151no—
373C2×Q8—151no—
374C2×Q8—302no—
375C2×Q8—301no—
376C2×Q8—604no—
377C2×Q8C22×D1512yes—
378C2×D4C22×D1511yescharacteristic
379C22×C4—21no—
380C22×C4—302no—
381SD16—24no—
382SD16—302no—
383C2×C8—151no—
384C2×C8—151no—
385C2×C8C22×D1512yes—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—152no—
389C4⋊C4—151no—
390C4⋊C4—151no—
391C4⋊C4—152no—
392C4⋊C4—302no—
393C4⋊C4—302no—
394C4⋊C4—302no—
395C4⋊C4—302no—
396C4⋊C4—301no—
397C4⋊C4—301no—
398C4⋊C4C22×D1511yescharacteristic
399C22⋊C4—21no—
400C22⋊C4—302no—
401C42—152no—
402C42—302no—
403C42—301no—
404C42C22×D1511yescharacteristic
405C3×C5#335911yescharacteristic, complemented, Hall
406C22×C3—22no—
407C22×C3—41no—
408C22×C3—41no—
409C22×C3C2×D4×D511yescharacteristic
410C3×C4—21no—
411C3×C4—41no—
412C3×C4—42no—
413C3×C4D5×SD1612yes—
414C3×C4C2×#71111yescharacteristic
415C3×C4#415011yescharacteristic
416Dic3—54no—
417Dic3—102no—
418Dic3—202no—
419Dic3—201no—
420Dic3—201no—
421C2×C5—42no—
422C2×C5#559811yescharacteristic
423C2×C5C2×D3×SD1611yescharacteristic
424C2×C5#577911yescharacteristic
425C23—21no—
426Q8—22no—
427Q8—22no—
428Q8—154no—
429Q8—301no—
430Q8—301no—
431Q8—604no—
432Q8—602no—
433D4—21no—
434D4#681912yes—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—42no—
438C2×C4—152no—
439C2×C4—152no—
440C2×C4—301no—
441C2×C4—301no—
442C2×C4—302no—
443C2×C4—604no—
444C2×C4D4×D1511yescharacteristic
445C2×C4D4×D1511yescharacteristic
446C2×C4C23×D1511yescharacteristic
447C8—22no—
448C8—301no—
449C8—301no—
450C2×C3—42no—
451C2×C3#6593211yescharacteristic
452C2×C3C2×D5×SD1611yescharacteristic
453C2×C3#6611311yescharacteristic
454C5#8041411yescharacteristic, complemented, Sylow
455C22—22no—
456C22—41no—
457C22—41no—
458C22C2×D4×D1511yescharacteristic
459C4—21no—
460C4—42no—
461C4—41no—
462C4—154no—
463C4—302no—
464C4—601no—
465C4—602no—
466C4—601no—
467C4SD16×D1512yes—
468C4C2×#681911yescharacteristic
469C4#9210111yescharacteristic
470C3#1061214511yescharacteristic, complemented, Sylow
471C2—42no—
472C2#1175635711yescharacteristic
473C2C2×SD16×D1511yescharacteristic
474C2#1175653811yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#461C2
socleC22×C3×C5C22×D4
prime radicalC2×C3×C5×D4C23
hypercenter#461D15
hypocenterC3×C5#3359
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #11756480C2semidirect
6. #11751180C2semidirect
8. #11750952C2semidirect
9. #11750931C2semidirect
10. #11750893C2semidirect
11. #11750892C2semidirect
149. Dic30SD16semidirect
405. C3×C5#3359semidirect
454. C5#80414semidirect
470. C3#10612145semidirect

Maximal subgroups

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

subgroupindex
#117564802
D4×Dic302
C3×C5×#4612
#117512632
SD16×Dic152
#117511802
C4×#67162
#117509522
#117509312
#117508932
#117508922
#106121453
#804145