Finite Group Databasebeta

← #14424539

Subgroups — #14424539

Order 1920.

1400 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#92101C211yesmaximal, characteristic
2C2×C3×C5×#137C211yesmaximal, characteristic, complemented
3#11756575C211yesmaximal, characteristic, complemented
4C22×#6718C211yesmaximal, characteristic, complemented
5#11751334C211yesmaximal, characteristic, complemented
6#11751287C214yesmaximal, complemented
7#11751274C214yesmaximal, complemented
8#11751230C211yesmaximal, characteristic, complemented
9#11751221C211yesmaximal, characteristic, complemented
10#10615714—31nomaximal, Hall
11C2×C3×C5×#41C2211yescharacteristic
12C22×C3×C5×Q8C2211yescharacteristic
13#92101—24no—
14#92101—24no—
15C2×Q8×D15C2212yes—
16C2×#6819—22no—
17C2×#6819C2212yes—
18C2×#6817—21no—
19C2×#6817C2211yescharacteristic
20C22×Dic30—21no—
21C22×Dic30C2211yescharacteristic
22C3×C5×#137—24no—
23C2×C3×C5×Q16C2212yes—
24C2×C3×C5×SD16C2212yes—
25C2×C3×C5×M16C2211yescharacteristic
26#91836—21no—
27#91835C2214yes—
28#91833—21no—
29C2×#6718—24no—
30C2×#6718C2214yes—
31#91818—22no—
32#91815C2211yescharacteristic
33C22×C3⋊#168C2211yescharacteristic
34#91119C2212yes—
35#91116C2212yes—
36#91104C2214yes—
37#91100C2214yes—
38#83983—51nomaximal, Hall
39C2×#4150—31no—
40C2×C5×#137D311yescharacteristic, complemented
41#66150—31no—
42C22×#689—31no—
43#65522—31no—
44#65475—34no—
45#65462—34no—
46#65418—31no—
47#65409—31no—
48C3×C5×#41—22no—
49C3×C5×#41D414yes—
50C2×C3×C5×Q8—21no—
51C2×C3×C5×Q8—22no—
52C2×C3×C5×Q8—22no—
53C2×C3×C5×Q8C2312yes—
54C2×C3×C5×Q8C2311yescharacteristic
55C2×C3×C5×D4—21no—
56C2×C3×C5×D4C2311yescharacteristic
57C22×C3×C4×C5—21no—
58C22×C3×C4×C5—21no—
59C22×C3×C4×C5C2311yescharacteristic
60C2×#1220—22no—
61C2×#1220D412yes—
62C22×Dic15—21no—
63C22×Dic15—22no—
64Q8×D15—24no—
65Q8×D15—24no—
66#6819—216no—
67#6819—24no—
68#6819—24no—
69#6817—28no—
70#6817—22no—
71#6817—22no—
72C2×D60D411yescharacteristic
73C2×C4×D15—22no—
74C2×C4×D15C2312yes—
75C2×Dic30—28no—
76C2×Dic30—21no—
77C2×Dic30—22no—
78C2×Dic30—22no—
79C2×Dic30D411yescharacteristic
80C2×Dic30D414yes—
81C2×Dic30C2312yes—
82C3×C5×Q16—24no—
83C3×C5×SD16—24no—
84C3×C5×M16—22no—
85C2×C3×C5×C8C2312yes—
86#6719—21no—
87#6718—28no—
88#6714—24no—
89C4⋊Dic15C2312yes—
90#6712—24no—
91C4×Dic15—21no—
92C2×C3⋊#168—22no—
93C2×C3⋊#168C2312yes—
94C2×#1010—51no—
95C2×C3×#137D511yescharacteristic, complemented
96#5816—51no—
97C22×#216—51no—
98#5233—51no—
99#5186—54no—
100#5173—54no—
101#5129—51no—
102#5120—51no—
103C2×C5×#41C2×D311yescharacteristic
104C22×C5×Q8C2×D311yescharacteristic
105#4150—64no—
106#4150—64no—
107C2×Q8×D5—32no—
108C2×#711—32no—
109C2×#711—62no—
110C2×#709—31no—
111C2×#709—61no—
112C22×Dic10—31no—
113C22×Dic10—61no—
114C5×#137—24no—
115C2×C5×Q16C2×D312yes—
116C2×C5×SD16C2×D312yes—
117C2×C5×M16C2×D311yescharacteristic
118#4093—61no—
119#4092—34no—
120#4090—61no—
121C2×#689—34no—
122C2×#689—64no—
123#4075—62no—
124#4072—31no—
125C22×#168—31no—
126#3967—32no—
127#3964—32no—
128#3952—34no—
129#3948—34no—
130#3011—151noSylow
131C23×C3×C5—21no—
132C23×C3×C5C2×D411yescharacteristic
133C22×D15C2×D412yes—
134C3×C5×Q8—21no—
135C3×C5×Q8—22no—
136C3×C5×Q8—22no—
137C3×C5×Q8—42no—
138C3×C5×Q8C2×D412yes—
139C3×C5×D4—24no—
140C3×C5×D4—21no—
141C3×C5×D4C2×D412yes—
142C2×C3×C4×C5—21no—
143C2×C3×C4×C5—24no—
144C2×C3×C4×C5—22no—
145C2×C3×C4×C5—42no—
146C2×C3×C4×C5C2×D411yescharacteristic
147C2×C3×C4×C5C2×D412yes—
148C2×C3×C4×C5C2×D412yes—
149C2×C3×C4×C5C2411yescharacteristic
150#1220—28no—
151#1220—24no—
152C2×Dic15—28no—
153C2×Dic15—24no—
154C2×Dic15—22no—
155C2×Dic15—22no—
156C2×Dic15C2×D412yes—
157D60—22no—
158C4×D15—28no—
159C4×D15—22no—
160C4×D15—22no—
161Dic30—28no—
162Dic30—22no—
163Dic30—24no—
164Dic30—22no—
165Dic30C2×D414yes—
166C3×C5×C8—22no—
167C3⋊#168—22no—
168C2×C3×#41C2×D511yescharacteristic
169C22×C3×Q8C2×D511yescharacteristic
170#1010—104no—
171#1010—104no—
172C2×D3×Q8—52no—
173C2×#237—52no—
174C2×#237—102no—
175C2×#235—51no—
176C2×#235—101no—
177C22×Dic6—51no—
178C22×Dic6—101no—
179C3×#137—24no—
180C2×C3×Q16C2×D512yes—
181C2×C3×SD16C2×D512yes—
182C2×C3×M16C2×D511yescharacteristic
183#947—101no—
184#946—54no—
185#944—101no—
186C2×#216—54no—
187C2×#216—104no—
188#929—102no—
189#926—51no—
190C22×C3⋊C8—51no—
191#835—52no—
192#832—52no—
193(C2×D3)⋊C8—54no—
194#816—54no—
195C5×#41—22no—
196C5×#41#6714yes—
197C2×C5×Q8—21no—
198C2×C5×Q8—22no—
199C2×C5×Q8—22no—
200C2×C5×Q8C22×D312yes—
201C2×C5×Q8C22×D311yescharacteristic
202C2×C5×D4—21no—
203C2×C5×D4C22×D311yescharacteristic
204C22×C4×C5—21no—
205C22×C4×C5—21no—
206C22×C4×C5C22×D311yescharacteristic
207C2×C22⋊D5—32no—
208C2×C22⋊D5—62no—
209C22×Dic5—62no—
210C22×Dic5—61no—
211Q8×D5—64no—
212Q8×D5—64no—
213#711—64no—
214#711—616no—
215#711—64no—
216#709—68no—
217#709—62no—
218#709—62no—
219C2×D20—31no—
220C2×C4×D5—32no—
221C2×C4×D5—62no—
222C2×Dic10—34no—
223C2×Dic10—32no—
224C2×Dic10—31no—
225C2×Dic10—62no—
226C2×Dic10—68no—
227C2×Dic10—61no—
228C2×Dic10—62no—
229C5×Q16—24no—
230C5×SD16—24no—
231C5×M16—22no—
232C2×C5×C8C22×D312yes—
233C22⋊Dic5—61no—
234#689—68no—
235#685—64no—
236C4⋊Dic5—32no—
237#683—64no—
238C4×Dic5—61no—
239C2×#168—32no—
240C2×#168—62no—
241C2×2−1+4—151no—
242C2×#137D1511yescharacteristic, complemented
243C22×Q16—151no—
244#536—151no—
245#452—154no—
246#451—154no—
247#419—151no—
248#416—151no—
249#408—151no—
250C22×C3×C5—22no—
251C22×C3×C5—22no—
252C22×C3×C5—42no—
253C22×C3×C5#12012yes—
254C22×C3×C5C22×D411yescharacteristic
255C2×D15—22no—
256C2×D15—24no—
257C3×C4×C5—22no—
258C3×C4×C5—42no—
259C3×C4×C5#12012yes—
260C3×C4×C5C22×D411yescharacteristic
261C3×C4×C5C22×D411yescharacteristic
262Dic15—22no—
263Dic15—24no—
264Dic15—42no—
265C3×#41—22no—
266C3×#41C22⋊D514yes—
267C2×C3×Q8—21no—
268C2×C3×Q8—22no—
269C2×C3×Q8—22no—
270C2×C3×Q8C22×D512yes—
271C2×C3×Q8C22×D511yescharacteristic
272C2×C3×D4—21no—
273C2×C3×D4C22×D511yescharacteristic
274C22×C3×C4—21no—
275C22×C3×C4—21no—
276C22×C3×C4C22×D511yescharacteristic
277C2×#67—52no—
278C2×#67—102no—
279C22×Dic3—102no—
280C22×Dic3—101no—
281D3×Q8—104no—
282D3×Q8—104no—
283#237—104no—
284#237—1016no—
285#237—104no—
286#235—108no—
287#235—102no—
288#235—102no—
289C2×D12—51no—
290C2×C4×D3—52no—
291C2×C4×D3—102no—
292C2×Dic6—54no—
293C2×Dic6—52no—
294C2×Dic6—51no—
295C2×Dic6—102no—
296C2×Dic6—108no—
297C2×Dic6—101no—
298C2×Dic6—102no—
299C3×Q16—24no—
300C3×SD16—24no—
301C3×M16—22no—
302C2×C3×C8C22×D512yes—
303#217—101no—
304#216—108no—
305(C2×D3)⋊C4—104no—
306C4⋊Dic3—52no—
307Dic3⋊C4—104no—
308C4×Dic3—101no—
309C2×C3⋊C8—52no—
310C2×C3⋊C8—102no—
311C23×C5—21no—
312C23×C5D3×D411yescharacteristic
313C22×D5—32no—
314C5×Q8—21no—
315C5×Q8—22no—
316C5×Q8—22no—
317C5×Q8—42no—
318C5×Q8C2×#6712yes—
319C5×D4—24no—
320C5×D4—21no—
321C5×D4C2×#6712yes—
322C2×C4×C5—21no—
323C2×C4×C5—24no—
324C2×C4×C5—22no—
325C2×C4×C5—42no—
326C2×C4×C5D3×D411yescharacteristic
327C2×C4×C5D3×D412yes—
328C2×C4×C5C2×#6712yes—
329C2×C4×C5C23×D311yescharacteristic
330C22⋊D5—68no—
331C22⋊D5—64no—
332C2×Dic5—32no—
333C2×Dic5—68no—
334C2×Dic5—64no—
335C2×Dic5—62no—
336C2×Dic5—62no—
337D20—62no—
338C4×D5—68no—
339C4×D5—62no—
340C4×D5—62no—
341Dic10—34no—
342Dic10—68no—
343Dic10—64no—
344Dic10—62no—
345Dic10—62no—
346C5×C8—22no—
347#168—62no—
3482−1+4—304no—
3492−1+4—304no—
350C2×#41—151no—
351C2×#41—152no—
352C2×#41—301no—
353C2×#41—302no—
354C2×#41C2×D1511yescharacteristic
355C22×Q8—152no—
356C22×Q8—151no—
357C22×Q8—301no—
358C22×Q8C2×D1511yescharacteristic
359#137—24no—
360C2×Q16—154no—
361C2×Q16—304no—
362C2×Q16C2×D1512yes—
363C2×SD16C2×D1512yes—
364C2×M16C2×D1511yescharacteristic
365C22×C8—151no—
366C4⋊Q8—301no—
367#124—301no—
368#123—302no—
369C22⋊Q8—154no—
370#117—151no—
371#103—152no—
372#103—154no—
373#102—152no—
374C22⋊C8—154no—
375C2×C3×C5—22no—
376C2×C3×C5—42no—
377C2×C3×C5C2×#12011yescharacteristic
378C2×C3×C5#57812yes—
379D15—42no—
380C23×C3—21no—
381C23×C3D4×D511yescharacteristic
382C22×D3—52no—
383C3×Q8—21no—
384C3×Q8—22no—
385C3×Q8—22no—
386C3×Q8—42no—
387C3×Q8C2×C22⋊D512yes—
388C3×D4—24no—
389C3×D4—21no—
390C3×D4C2×C22⋊D512yes—
391C2×C3×C4—21no—
392C2×C3×C4—24no—
393C2×C3×C4—22no—
394C2×C3×C4—42no—
395C2×C3×C4D4×D512yes—
396C2×C3×C4D4×D511yescharacteristic
397C2×C3×C4C2×C22⋊D512yes—
398C2×C3×C4C23×D511yescharacteristic
399#67—108no—
400#67—104no—
401C2×Dic3—52no—
402C2×Dic3—108no—
403C2×Dic3—102no—
404C2×Dic3—104no—
405C2×Dic3—102no—
406D12—102no—
407C4×D3—108no—
408C4×D3—102no—
409C4×D3—102no—
410Dic6—54no—
411Dic6—108no—
412Dic6—102no—
413Dic6—104no—
414Dic6—102no—
415C3×C8—22no—
416C3⋊C8—102no—
417C22×C5—22no—
418C22×C5—22no—
419C22×C5—42no—
420C22×C5#93712yes—
421C22×C5C2×D3×D411yescharacteristic
422C2×D5—64no—
423C2×D5—62no—
424C4×C5—22no—
425C4×C5—42no—
426C4×C5#93712yes—
427C4×C5C2×D3×D411yescharacteristic
428C4×C5C22×#6711yescharacteristic
429Dic5—64no—
430Dic5—62no—
431Dic5—122no—
432#41—22no—
433#41—308no—
434#41—304no—
435#41—3016no—
436#41—304no—
437#41—302no—
438#41—302no—
439#41#122014yes—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—151no—
444C2×Q8—154no—
445C2×Q8—152no—
446C2×Q8—302no—
447C2×Q8—308no—
448C2×Q8—304no—
449C2×Q8—304no—
450C2×Q8—301no—
451C2×Q8—302no—
452C2×Q8C22×D1512yes—
453C2×Q8C22×D1511yescharacteristic
454C2×D4—21no—
455C2×D4—152no—
456C2×D4—151no—
457C2×D4—302no—
458C2×D4C22×D1511yescharacteristic
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—152no—
462C22×C4—302no—
463C22×C4—302no—
464C22×C4—301no—
465C22×C4C22×D1511yescharacteristic
466Q16—24no—
467Q16—308no—
468SD16—24no—
469M16—22no—
470C2×C8—152no—
471C2×C8—302no—
472C2×C8C22×D1512yes—
473C4⋊C4—152no—
474C4⋊C4—304no—
475C22⋊C4—304no—
476C22⋊C4—301no—
477C42—301no—
478C3×C5#301111yescharacteristic, complemented, Hall
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—42no—
482C22×C3#408312yes—
483C22×C3C2×D4×D511yescharacteristic
484C2×D3—104no—
485C2×D3—102no—
486C3×C4—22no—
487C3×C4—42no—
488C3×C4#408312yes—
489C3×C4C2×D4×D511yescharacteristic
490C3×C4C22×C22⋊D511yescharacteristic
491Dic3—104no—
492Dic3—102no—
493Dic3—202no—
494C2×C5—22no—
495C2×C5—42no—
496C2×C5#577912yes—
497C2×C5C2×#93711yescharacteristic
498D5—122no—
499C23—21no—
500C23—152no—
501C23D4×D1511yescharacteristic
502Q8—21no—
503Q8—22no—
504Q8—22no—
505Q8—42no—
506Q8—154no—
507Q8—308no—
508Q8—302no—
509Q8—304no—
510Q8—302no—
511Q8C2×#122012yes—
512D4—24no—
513D4—21no—
514D4—308no—
515D4—304no—
516D4—302no—
517D4C2×#122012yes—
518C2×C4—21no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—42no—
522C2×C4—152no—
523C2×C4—308no—
524C2×C4—308no—
525C2×C4—304no—
526C2×C4—302no—
527C2×C4—302no—
528C2×C4—302no—
529C2×C4—302no—
530C2×C4D4×D1511yescharacteristic
531C2×C4D4×D1512yes—
532C2×C4C2×#122012yes—
533C2×C4C23×D1511yescharacteristic
534C8—22no—
535C8—302no—
536C2×C3—22no—
537C2×C3—42no—
538C2×C3#6611312yes—
539C2×C3C2×#408311yescharacteristic
540D3—202no—
541C5#8398311yescharacteristic, complemented, Sylow
542C22—22no—
543C22—22no—
544C22—42no—
545C22—302no—
546C22—304no—
547C22#9182612yes—
548C22C2×D4×D1511yescharacteristic
549C4—22no—
550C4—42no—
551C4—302no—
552C4—304no—
553C4—602no—
554C4#9182612yes—
555C4C2×D4×D1511yescharacteristic
556C4C22×#122011yescharacteristic
557C3#1061571411yescharacteristic, complemented, Sylow
558C2—22no—
559C2—42no—
560C2—602no—
561C2#1175653812yes—
562C2C2×#9182611yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC2×C3×C5×#137C2
socleC22×C3×C5C22×D4
prime radicalC2×#92101C2
hypercenterC2×#137D15
hypocenterC3×C5#3011
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×C3×C5×#137C2semidirect
3. #11756575C2semidirect
4. C22×#6718C2semidirect
5. #11751334C2semidirect
6. #11751287C2semidirect
7. #11751274C2semidirect
8. #11751230C2semidirect
9. #11751221C2semidirect
40. C2×C5×#137D3semidirect
95. C2×C3×#137D5semidirect
242. C2×#137D15semidirect
478. C3×C5#3011semidirect
541. C5#83983semidirect
557. C3#10615714semidirect

Maximal subgroups

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

subgroupindex
C2×#921012
C2×C3×C5×#1372
#117565752
C22×#67182
#117513342
#117512872
#117512742
#117512302
#117512212
#106157143
#839835