Finite Group Databasebeta

← #14471517

Subgroups — #14471517

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#11756486C211yesmaximal, characteristic
2#11756326C211yesmaximal, characteristic, complemented
3C3×C5×#475C211yesmaximal, characteristic, complemented
4#11751194C211yesmaximal, characteristic, complemented
5C4×#6717C211yesmaximal, characteristic
6#11750955C212yesmaximal, complemented
7#11750953C212yesmaximal
8#11750942C211yesmaximal, characteristic
9#11750931C211yesmaximal, characteristic, complemented
10#11750919C212yesmaximal
11#11750893C212yesmaximal, complemented
12#10612768—31nomaximal, Hall
13C2×Q8×D15—21no—
14C3×C5×C4⋊Q8C2211yescharacteristic
15C3×C4×C5×Q8C2211yescharacteristic
16#91835—21no—
17#91833C2212yes—
18C2×#6717C2211yescharacteristic
19C2×#6706C2212yes—
20#91785—22no—
21#91782C2212yes—
22#91778—22no—
23#91777C2212yes—
24#91776C2212yes—
25C4×D60C2211yescharacteristic
26#91759—21no—
27#91758C2211yescharacteristic
28C4×Dic30—21no—
29C4×Dic30C2211yescharacteristic
30C3×C5×#106C2212yes—
31C3×C5×C4⋊C8C2211yescharacteristic
32C3×C5×#103C2212yes—
33#91119C2211yescharacteristic
34#91100C2212yes—
35#91098C2212yes—
36C8×Dic15C2212yes—
37#91094C2212yes—
38#91094C2212yes—
39#91093C2211yescharacteristic
40#91092C2211yescharacteristic
41C4×C3⋊#168C2211yescharacteristic
42#81037—51nomaximal, Hall
43#66061—31no—
44#65901—31no—
45C5×#475D311yescharacteristic, complemented
46#65382—31no—
47C4×#688—31no—
48#65143—32no—
49#65141—32no—
50#65130—31no—
51#65119—31no—
52#65107—32no—
53#65081—32no—
54C2×C3×C5×Q8—21no—
55C2×C3×C5×Q8C2311yescharacteristic
56Q8×D15—44no—
57C2×D60C2311yescharacteristic
58C2×C4×D15—21no—
59C2×C4×D15—22no—
60C2×Dic30—22no—
61C2×Dic30D411yescharacteristic
62C2×Dic30C2312yes—
63C2×C3×C5×C8C2312yes—
64C3×C5×C4⋊C4—21no—
65C3×C5×C4⋊C4—21no—
66C3×C5×C4⋊C4C2311yescharacteristic
67C3×C5×C4⋊C4C2312yes—
68C3×C42×C5—21no—
69C3×C42×C5C2311yescharacteristic
70#6717—22no—
71#6714—21no—
72#6714—22no—
73C4⋊Dic15—21no—
74C4⋊Dic15—21no—
75C4⋊Dic15D411yescharacteristic
76C4⋊Dic15C2311yescharacteristic
77#6712—22no—
78#6712—22no—
79#6712—22no—
80#6712—22no—
81#6712D412yes—
82C4×Dic15—22no—
83C4×Dic15C2312yes—
84C2×C3⋊#168C2311yescharacteristic
85C2×C3⋊#168C2311yescharacteristic
86#6706—24no—
87#5727—51no—
88#5567—51no—
89C3×#475D511yescharacteristic, complemented
90#5093—51no—
91C4×#215—51no—
92#4854—52no—
93#4852—52no—
94#4841—51no—
95#4830—51no—
96#4818—52no—
97#4792—52no—
98C2×Q8×D5—61no—
99C5×C4⋊Q8C2×D311yescharacteristic
100C4×C5×Q8C2×D311yescharacteristic
101#4092—61no—
102#4090—32no—
103C2×#688—31no—
104C2×#677—32no—
105#4042—62no—
106#4039—32no—
107#4035—62no—
108#4034—32no—
109#4033—32no—
110C4×D20—31no—
111#4016—61no—
112#4015—31no—
113C4×Dic10—31no—
114C4×Dic10—61no—
115C5×#106C2×D312yes—
116C5×C4⋊C8C2×D311yescharacteristic
117C5×#103C2×D312yes—
118#3967—31no—
119#3948—32no—
120#3946—32no—
121C8×Dic5—32no—
122#3942—32no—
123#3942—32no—
124#3941—31no—
125#3940—31no—
126C4×#168—31no—
127#3359—151noSylow
128C22×D15—21no—
129C3×C5×Q8—21no—
130C3×C5×Q8—21no—
131C3×C5×Q8—42no—
132C2×C3×C4×C5—21no—
133C2×C3×C4×C5—21no—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5C2×D411yescharacteristic
136C2×C3×C4×C5C2×D411yescharacteristic
137C2×C3×C4×C5C2411yescharacteristic
138C2×Dic15—22no—
139C2×Dic15—21no—
140C2×Dic15C2×D412yes—
141C2×Dic15C2×D412yes—
142D60—21no—
143D60#4112yes—
144C4×D15—42no—
145C4×D15—44no—
146Dic30—22no—
147Dic30—22no—
148Dic30—44no—
149Dic30SD1614yescomplemented
150C3×C5×C8—22no—
151C3⋊#168—21no—
152C3⋊#168—21no—
153C2×D3×Q8—101no—
154C3×C4⋊Q8C2×D511yescharacteristic
155C3×C4×Q8C2×D511yescharacteristic
156#946—101no—
157#944—52no—
158C2×#215—51no—
159C2×#204—52no—
160#896—102no—
161#893—52no—
162#889—102no—
163#888—52no—
164#887—52no—
165C4×D12—51no—
166#870—101no—
167#869—51no—
168C4×Dic6—51no—
169C4×Dic6—101no—
170C3×#106C2×D512yes—
171C3×C4⋊C8C2×D511yescharacteristic
172C3×#103C2×D512yes—
173#835—51no—
174#816—52no—
175Dic3⋊C8—52no—
176C8×Dic3—52no—
177#810—52no—
178#810—52no—
179#809—51no—
180#808—51no—
181C4×C3⋊C8—51no—
182C2×C5×Q8—21no—
183C2×C5×Q8C22×D311yescharacteristic
184Q8×D5—124no—
185C2×D20—31no—
186C2×C4×D5—62no—
187C2×C4×D5—61no—
188C2×Dic10—31no—
189C2×Dic10—32no—
190C2×Dic10—62no—
191C2×C5×C8C22×D312yes—
192C5×C4⋊C4—21no—
193C5×C4⋊C4—21no—
194C5×C4⋊C4C22×D311yescharacteristic
195C5×C4⋊C4C22×D312yes—
196C42×C5—21no—
197C42×C5C22×D311yescharacteristic
198#688—62no—
199#685—62no—
200#685—61no—
201C4⋊Dic5—31no—
202C4⋊Dic5—31no—
203C4⋊Dic5—61no—
204C4⋊Dic5—61no—
205#683—32no—
206#683—62no—
207#683—62no—
208#683—62no—
209#683—62no—
210C4×Dic5—32no—
211C4×Dic5—62no—
212C2×#168—31no—
213C2×#168—31no—
214#677—64no—
215#557—151no—
216D4×Q8—151no—
217#487—151no—
218#487—152no—
219#475—152no—
220#475D1511yescharacteristic, complemented
221#461—151no—
222#461—152no—
223C8×Q8—151no—
224C4×SD16—151no—
225C4×SD16—152no—
226C22×C3×C5C22×D411yescharacteristic
227C2×D15—22no—
228C2×D15—41no—
229C2×D15—41no—
230C3×C4×C5—21no—
231C3×C4×C5—41no—
232C3×C4×C5—42no—
233C3×C4×C5—41no—
234C3×C4×C5C2×SD1612yes—
235C3×C4×C5C2×#4111yescharacteristic
236C3×C4×C52−1+411yescharacteristic
237Dic15—22no—
238Dic15—42no—
239Dic15—41no—
240Dic15C2×SD1614yes—
241C2×C3×Q8—21no—
242C2×C3×Q8C22×D511yescharacteristic
243D3×Q8—204no—
244C2×D12—51no—
245C2×C4×D3—102no—
246C2×C4×D3—101no—
247C2×Dic6—51no—
248C2×Dic6—52no—
249C2×Dic6—102no—
250C2×C3×C8C22×D512yes—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4C22×D511yescharacteristic
254C3×C4⋊C4C22×D512yes—
255C3×C42—21no—
256C3×C42C22×D511yescharacteristic
257#215—102no—
258(C2×D3)⋊C4—102no—
259(C2×D3)⋊C4—101no—
260C4⋊Dic3—51no—
261C4⋊Dic3—51no—
262C4⋊Dic3—101no—
263C4⋊Dic3—101no—
264Dic3⋊C4—52no—
265Dic3⋊C4—102no—
266Dic3⋊C4—102no—
267Dic3⋊C4—102no—
268Dic3⋊C4—102no—
269C4×Dic3—52no—
270C4×Dic3—102no—
271C2×C3⋊C8—51no—
272C2×C3⋊C8—51no—
273#204—104no—
274C22×D5—61no—
275C5×Q8—21no—
276C5×Q8—21no—
277C5×Q8—42no—
278C2×C4×C5—21no—
279C2×C4×C5—21no—
280C2×C4×C5—22no—
281C2×C4×C5D3×D411yescharacteristic
282C2×C4×C5D3×D411yescharacteristic
283C2×C4×C5C23×D311yescharacteristic
284C2×Dic5—32no—
285C2×Dic5—32no—
286C2×Dic5—62no—
287C2×Dic5—61no—
288D20—32no—
289D20—61no—
290C4×D5—124no—
291C4×D5—122no—
292Dic10—34no—
293Dic10—62no—
294Dic10—62no—
295Dic10—124no—
296C5×C8—22no—
297#168—61no—
298#168—61no—
299C22×Q8—301no—
300C2×SD16—151no—
301C2×SD16—152no—
302C4⋊Q8—152no—
303C4⋊Q8—152no—
304C4⋊Q8—151no—
305C4⋊Q8C2×D1511yescharacteristic
306#125—301no—
307#125—302no—
308C22⋊Q8—302no—
309C22⋊Q8—301no—
310C4×Q8—151no—
311C4×Q8—152no—
312C4×Q8—301no—
313C4×Q8C2×D1511yescharacteristic
314C4×D4—152no—
315C4×D4—151no—
316#106—151no—
317#106C2×D1512yes—
318C4⋊C8—152no—
319C4⋊C8C2×D1511yescharacteristic
320#103—151no—
321#103—152no—
322#103—152no—
323#103—152no—
324#103C2×D1512yes—
325#102—151no—
326C4×C8—152no—
327C4×C8—151no—
328C2×C3×C5#54811yescharacteristic
329C2×C3×C5C22×SD1611yescharacteristic
330C2×C3×C5#57811yescharacteristic
331D15—42no—
332C22×D3—101no—
333C3×Q8—21no—
334C3×Q8—21no—
335C3×Q8—42no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—22no—
339C2×C3×C4D4×D511yescharacteristic
340C2×C3×C4D4×D511yescharacteristic
341C2×C3×C4C23×D511yescharacteristic
342C2×Dic3—52no—
343C2×Dic3—52no—
344C2×Dic3—102no—
345C2×Dic3—101no—
346D12—52no—
347D12—101no—
348C4×D3—204no—
349C4×D3—202no—
350Dic6—54no—
351Dic6—102no—
352Dic6—102no—
353Dic6—204no—
354C3×C8—22no—
355C3⋊C8—101no—
356C3⋊C8—101no—
357C22×C5C2×D3×D411yescharacteristic
358C2×D5—62no—
359C2×D5—121no—
360C2×D5—121no—
361C4×C5—21no—
362C4×C5—41no—
363C4×C5—42no—
364C4×C5—41no—
365C4×C5D3×SD1612yes—
366C4×C5#100711yescharacteristic
367C4×C5D3×#4111yescharacteristic
368Dic5—34no—
369Dic5—62no—
370Dic5—122no—
371Dic5—121no—
372C2×Q8—21no—
373C2×Q8—151no—
374C2×Q8—152no—
375C2×Q8—302no—
376C2×Q8—604no—
377C2×Q8C22×D1511yescharacteristic
378C2×D4—151no—
379C22×C4—302no—
380C22×C4—301no—
381SD16—304no—
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—302no—
392C4⋊C4—302no—
393C4⋊C4—302no—
394C4⋊C4—302no—
395C4⋊C4—301no—
396C4⋊C4—301no—
397C4⋊C4C22×D1511yescharacteristic
398C4⋊C4C22×D1512yes—
399C22⋊C4—302no—
400C22⋊C4—301no—
401C42—21no—
402C42—152no—
403C42—302no—
404C42C22×D1511yescharacteristic
405C3×C5#335911yescharacteristic, complemented, Hall
406C22×C3C2×D4×D511yescharacteristic
407C2×D3—102no—
408C2×D3—201no—
409C2×D3—201no—
410C3×C4—21no—
411C3×C4—42no—
412C3×C4—41no—
413C3×C4—41no—
414C3×C4D5×SD1612yes—
415C3×C4#414711yescharacteristic
416C3×C4D5×#4111yescharacteristic
417Dic3—54no—
418Dic3—102no—
419Dic3—201no—
420Dic3—202no—
421C2×C5#563311yescharacteristic
422C2×C5C2×D3×SD1611yescharacteristic
423C2×C5#577911yescharacteristic
424D5—122no—
425C23—301no—
426Q8—21no—
427Q8—21no—
428Q8—42no—
429Q8—154no—
430Q8—302no—
431Q8—302no—
432Q8—604no—
433D4—152no—
434D4—301no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—152no—
439C2×C4—152no—
440C2×C4—302no—
441C2×C4—301no—
442C2×C4—604no—
443C2×C4—602no—
444C2×C4D4×D1511yescharacteristic
445C2×C4D4×D1511yescharacteristic
446C2×C4C23×D1511yescharacteristic
447C8—22no—
448C8—301no—
449C8—301no—
450C2×C3#6596711yescharacteristic
451C2×C3C2×D5×SD1611yescharacteristic
452C2×C3#6611311yescharacteristic
453D3—202no—
454C5#8103711yescharacteristic, complemented, Sylow
455C22—302no—
456C22—601no—
457C22—601no—
458C22C2×D4×D1511yescharacteristic
459C4—21no—
460C4—41no—
461C4—42no—
462C4—41no—
463C4—154no—
464C4—302no—
465C4—602no—
466C4—601no—
467C4SD16×D1512yes—
468C4#9209811yescharacteristic
469C4#41×D1511yescharacteristic
470C3#1061276811yescharacteristic, complemented, Sylow
471C2—602no—
472C2#1175639211yescharacteristic
473C2C2×SD16×D1511yescharacteristic
474C2#1175653811yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#475C2
socleC22×C3×C5C22×D4
prime radicalC2×D60C23
hypercenter#475D15
hypocenterC3×C5#3359
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #11756326C2semidirect
3. C3×C5×#475C2semidirect
4. #11751194C2semidirect
6. #11750955C2semidirect
9. #11750931C2semidirect
11. #11750893C2semidirect
45. C5×#475D3semidirect
89. C3×#475D5semidirect
149. Dic30SD16semidirect
220. #475D15semidirect
405. C3×C5#3359semidirect
454. C5#81037semidirect
470. C3#10612768semidirect

Maximal subgroups

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

subgroupindex
#117564862
#117563262
C3×C5×#4752
#117511942
C4×#67172
#117509552
#117509532
#117509422
#117509312
#117509192
#117508932
#106127683
#810375