Finite Group Databasebeta

← D3×#65733 #14417981

Subgroups — D3×#65733 #14417981

Order 1920.

928 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#4131C212yesmaximal, complemented
2D3×#4058C211yesmaximal, characteristic
3#11752392C211yesmaximal, characteristic, complemented
4C3×#65733C211yesmaximal, characteristic, complemented
5#11751956C212yesmaximal, complemented
6#11751952C212yesmaximal, complemented
7D3×#4008C212yesmaximal, complemented
8#11751687C211yesmaximal, characteristic, complemented
9#11751682C211yesmaximal, characteristic, complemented
10D3×#3995C211yesmaximal, characteristic, complemented
11D3×#3992C211yesmaximal, characteristic, complemented
12C2×#65733—31nomaximal, Hall
13D3×#711C2212yes—
14#91990C2212yescomplemented
15C3×#4131C2212yescomplemented
16#91929—22no—
17D3×#704—22no—
18#91927—22no—
19C2×D3×#170—22no—
20C4×D3×AGL(1,𝔽5)C2211yescharacteristic
21#91913C2212yescomplemented
22D3×#700C2212yes—
23#91911C2212yescomplemented
24D3×D5⋊C8C2211yescharacteristic
25#91801C411yescharacteristic, complemented
26C5×D3×D8C411yescharacteristic, complemented
27C3×#4058C2211yescharacteristic
28#91499C412yescomplemented
29#91490C2212yes—
30D3×#687C412yescomplemented
31#91279C411yescharacteristic, complemented
32#91270C2211yescharacteristic
33D3×Dic20C411yescharacteristic, complemented
34D3×C8×D5C2211yescharacteristic
35#91237C2212yescomplemented
36#91224C2211yescharacteristic, complemented
37#91221C2211yescharacteristic, complemented
38C3×#4008C2212yescomplemented
39C3×#3995C2211yescharacteristic, complemented
40C3×#3992C2211yescharacteristic, complemented
41#91161C2212yescomplemented
42#91156C2212yescomplemented
43#91155C2212yescomplemented
44#91150C2211yescharacteristic, complemented
45#91149C2211yescharacteristic, complemented
46D3×#444—51nomaximal, Hall
47C2×#4131—32no—
48C2×#4058—31no—
49C2×#4008—32no—
50#65733—31no—
51#65733—32no—
52#65733—31no—
53#65733—31no—
54#65733—32no—
55#65733D311yescharacteristic, direct product factor
56C2×#3995—31no—
57C2×#3992—31no—
58C2×D3×AGL(1,𝔽5)—21no—
59#6819C2×C412yes—
60C5×D3×D4C2×C412yes—
61C3×#711C2312yes—
62D3×C22⋊D5—22no—
63#6782—22no—
64C2×D3×Dic5—22no—
65#6780—22no—
66C4×D3×D5C2311yescharacteristic
67#6772C2312yes—
68#6768C2×C412yes—
69D3×Dic10C2×C412yes—
70#6762—22no—
71C2×C3⋊#170—22no—
72C4×ΓL(1,𝔽16)C2311yescharacteristic
73#6758C2312yes—
74D5⋊(C3⋊C8)C2311yescharacteristic
75C3×#704—22no—
76C2×C3×#170—22no—
77C3×C4×AGL(1,𝔽5)C2311yescharacteristic
78C3×#700C2312yes—
79C3×D5⋊C8C2311yescharacteristic
80#6740—22no—
81#6739—22no—
82#6739—22no—
83#6738D412yes—
84D3×#170—22no—
85D3×#170—22no—
86D3×#170—22no—
87D3×#170D412yes—
88Dic3×AGL(1,𝔽5)—21no—
89C3×C5×D8C2×C411yescharacteristic, complemented
90#6716C2×C412yescomplemented
91Dic60C2×C411yescharacteristic, complemented
92C8×D15C2×C411yescharacteristic
93C5×#213C2×C412yescomplemented
94C5×D24C2×C411yescharacteristic, complemented
95C5×D3×C8C2×C411yescharacteristic
96C3×#687C2×C412yescomplemented
97C3×Dic20C2×C411yescharacteristic, complemented
98C3×C8×D5C2311yescharacteristic
99#6662C2×C412yescomplemented
100#6659C2×C412yescomplemented
101#6656C2×C412yescomplemented
102#6648C2×C411yescharacteristic
103D3×#168C2×C411yescharacteristic
104D5×C3⋊C8C2311yescharacteristic
105D3×#135—51no—
106D3×#131—52no—
107C3×#444—51no—
108#5197—51no—
109#5145—52no—
110#4894—51no—
111D3×#108—51no—
112#4823—52no—
113D3×#104—52no—
114#4717—51no—
115D3×#97—51no—
116C2×#711—32no—
117C2×#704—62no—
118C22×#170—62no—
119#4131—32no—
120#4131—32no—
121#4131—32no—
122#4131—62no—
123#4131—62no—
124#4131C2×D312yescomplemented
125C2×C4×AGL(1,𝔽5)—31no—
126C2×#700—32no—
127C2×D5⋊C8—31no—
128C2×C5×D8—31no—
129C2×#687—32no—
130#4058—32no—
131#4058—31no—
132#4058—32no—
133#4058—31no—
134#4058—31no—
135#4058C2×D311yescharacteristic
136C2×Dic20—31no—
137C2×C8×D5—31no—
138#4008—32no—
139#4008—32no—
140#4008—32no—
141#4008C2×D312yescomplemented
142#3995—32no—
143#3995—31no—
144#3995C2×D311yescharacteristic, complemented
145#3992—31no—
146#3992—31no—
147#3992—31no—
148#3992C2×D311yescharacteristic, complemented
149C2×#444—151noSylow
150C22×C5×D3—22no—
151C2×D3×D5#4111yescharacteristic
152C2×ΓL(1,𝔽16)—21no—
153C2×C3×AGL(1,𝔽5)—21no—
154D3×AGL(1,𝔽5)—42no—
155C3×C5×D4C22×C412yes—
156#1220—22no—
157C2×Dic15—22no—
158C4×D15C22×C411yescharacteristic
159Dic30C22×C412yes—
160C5×#67—22no—
161C5×D12C22×C412yes—
162C4×C5×D3C22×C411yescharacteristic
163C3×C22⋊D5—22no—
164C2×C3×Dic5—22no—
165C3×C4×D5C2411yescharacteristic
166C3×Dic10C22×C412yes—
167#1204—22no—
168#1203—22no—
169#1201—22no—
170#1200C2×D411yescharacteristic
171D3×Dic5—22no—
172D3×Dic5—22no—
173D3×Dic5—22no—
174D3×Dic5C2×D411yescharacteristic
175D5×Dic3#4111yescharacteristic
176C3⋊#170—22no—
177C3⋊#170C2×D412yes—
178C3×#170—22no—
179C3×#170C2×D412yes—
180C3×C5×C8C22×C411yescharacteristic
181C3⋊#168C22×C411yescharacteristic
182C3×#168C22×C411yescharacteristic
183C5×C3⋊C8C22×C411yescharacteristic
184D3×#41—52no—
185C3×#135—51no—
186C3×#131—52no—
187#950—52no—
188#948—52no—
189#919—51no—
190D3×Q16—51no—
191#916—52no—
192D3×SD16—52no—
193#912—51no—
194D3×D8AGL(1,𝔽5)11yescharacteristic, complemented
195#907—52no—
196#907—102no—
197D3×M16—52no—
198D3×M16—102no—
199#904—51no—
200#901—52no—
201#901—102no—
202C2×D3×C8—51no—
203C2×D3×C8—51no—
204C2×D3×C8—102no—
205C42×D3—51no—
206C3×#108—51no—
207C3×#104—52no—
208C3×#97—51no—
209#837—52no—
210#825—52no—
211#822—52no—
212#819—51no—
213#815—51no—
214#815—51no—
215#805—52no—
216#803—51no—
217C22×AGL(1,𝔽5)—61no—
218C2×C5×D4—32no—
219C2×C22⋊D5—62no—
220C22×Dic5—62no—
221#711—32no—
222#711—32no—
223#711—32no—
224#711—62no—
225#711—62no—
226#711C22×D312yes—
227C2×C4×D5—31no—
228C2×Dic10—32no—
229#704—22no—
230#704—62no—
231#704—62no—
232#704—62no—
233C2×#170—22no—
234C2×#170—32no—
235C2×#170—62no—
236C2×#170—62no—
237C2×#170—62no—
238C2×#170—62no—
239C4×AGL(1,𝔽5)—31no—
240C4×AGL(1,𝔽5)—61no—
241C4×AGL(1,𝔽5)C22×D311yescharacteristic
242#700—32no—
243#700—62no—
244#700C22×D312yes—
245D5⋊C8—31no—
246D5⋊C8—32no—
247D5⋊C8C22×D311yescharacteristic
248C5×D8—32no—
249C5×D8—31no—
250C5×D8C4×D311yescharacteristic, complemented
251C2×C5×C8—31no—
252#687—32no—
253#687—32no—
254#687—32no—
255#687C4×D312yescomplemented
256C2×#168—31no—
257Dic20—32no—
258Dic20—31no—
259Dic20C4×D311yescharacteristic, complemented
260C8×D5—31no—
261C8×D5—31no—
262C8×D5—31no—
263C8×D5C22×D311yescharacteristic
264C2×#135—151no—
265C2×#131—152no—
266#444—51no—
267#444—151no—
268#444—152no—
269#444—151no—
270#444—152no—
271#444—151no—
272C2×#108—151no—
273C2×#104—152no—
274C2×#97—151no—
275C22×C3×C5—22no—
276C2×D15C4×D411yescharacteristic
277C2×C5×D3—22no—
278C2×C5×D3—44no—
279C2×C5×D3C4×D411yescharacteristic
280C2×C3×D5C2×#4111yescharacteristic
281D3×D5—22no—
282ΓL(1,𝔽16)—41no—
283C3×AGL(1,𝔽5)—41no—
284C3×C4×C5C23×C411yescharacteristic
285Dic15—22no—
286Dic15C4×D411yescharacteristic
287C3×Dic5—22no—
288C3×Dic5C22×D411yescharacteristic
289C5×Dic3C4×D411yescharacteristic
290C3×#41—52no—
291#239—52no—
292D3×Q8—52no—
293#237—52no—
294#237—102no—
295D3×D4—102no—
296D3×D4C2×AGL(1,𝔽5)12yes—
297#235—52no—
298#235—102no—
299C2×C4×D3—51no—
300C2×C4×D3—102no—
301C2×C4×D3—101no—
302C3×Q16—51no—
303C3×SD16—52no—
304C3×D8C2×AGL(1,𝔽5)11yescharacteristic, complemented
305C3×M16—52no—
306C3×M16—102no—
307C2×C3×C8—51no—
308C2×C3×C8—51no—
309C2×C3×C8—102no—
310C3×C42—51no—
311#216—52no—
312#215—52no—
313#214—52no—
314#213C2×AGL(1,𝔽5)12yescomplemented
315C4×Dic3—51no—
316C4×Dic3—101no—
317#208—52no—
318#208—102no—
319C2×C3⋊C8—51no—
320C2×C3⋊C8—51no—
321C2×C3⋊C8—102no—
322Dic12—51no—
323D24C2×AGL(1,𝔽5)11yescharacteristic, complemented
324#204—52no—
325ΓL(1,𝔽25)—102no—
326ΓL(1,𝔽25)—102no—
327ΓL(1,𝔽25)—102no—
328D3×C8—52no—
329D3×C8—52no—
330D3×C8—51no—
331D3×C8—51no—
332D3×C8—51no—
333D3×C8—102no—
334D3×C8—102no—
335D3×C8—102no—
336D3×C8C2×AGL(1,𝔽5)11yescharacteristic
337C23×C5—62no—
338C22×D5—31no—
339C2×AGL(1,𝔽5)—21no—
340C2×AGL(1,𝔽5)—61no—
341C2×AGL(1,𝔽5)—122no—
342C5×D4—32no—
343C5×D4—62no—
344C5×D4C2×C4×D312yes—
345C2×C4×C5—31no—
346C22⋊D5—22no—
347C22⋊D5—62no—
348C22⋊D5—62no—
349C22⋊D5—62no—
350C2×Dic5—22no—
351C2×Dic5—31no—
352C2×Dic5—62no—
353C2×Dic5—62no—
354C2×Dic5—62no—
355C2×Dic5—62no—
356C4×D5—31no—
357C4×D5—31no—
358C4×D5—31no—
359C4×D5C23×D311yescharacteristic
360Dic10—32no—
361Dic10—62no—
362Dic10C2×C4×D312yes—
363#170—22no—
364#170—32no—
365#170—62no—
366#170D3×D412yes—
367C5×C8—31no—
368C5×C8C2×C4×D311yescharacteristic
369#168—31no—
370#168C2×C4×D311yescharacteristic
371C2×#41—152no—
372#135—51no—
373#135—151no—
374#135—152no—
375#135—151no—
376#135—152no—
377#135—151no—
378C2×Q16—151no—
379C2×SD16—152no—
380C2×D8—31no—
381#131—52no—
382#131—152no—
383#131—152no—
384#131—152no—
385#131—302no—
386#131—302no—
387C2×M16—152no—
388C2×M16—302no—
389C22×C8—151no—
390C22×C8—151no—
391C22×C8—302no—
392C2×C42—151no—
393#108—51no—
394#108—151no—
395#108—152no—
396#104—52no—
397#104—152no—
398#104—152no—
399#104—152no—
400#97—51no—
401#97—151no—
402#97—151no—
403#97—151no—
404C2×C3×C5—42no—
405C2×C3×C5C2×C4×D411yescharacteristic
406D15—21no—
407C3×D5—21no—
408C5×D3—42no—
409C5×D3#44412yescomplemented
410C22×D3—22no—
411C22×D3—51no—
412C3×Q8—52no—
413C3×D4—102no—
414C3×D4C22×AGL(1,𝔽5)12yes—
415C2×C3×C4—51no—
416C2×C3×C4—102no—
417C2×C3×C4—101no—
418#67—22no—
419#67—102no—
420#67—102no—
421C2×Dic3—51no—
422C2×Dic3—102no—
423C2×Dic3—101no—
424D12—102no—
425D12C22×AGL(1,𝔽5)12yes—
426C4×D3—51no—
427C4×D3—51no—
428C4×D3—51no—
429C4×D3—102no—
430C4×D3—102no—
431C4×D3—102no—
432C4×D3—202no—
433C4×D3C22×AGL(1,𝔽5)11yescharacteristic
434Dic6—52no—
435Dic6—102no—
436C3×C8—52no—
437C3×C8—51no—
438C3×C8—102no—
439C3×C8C22×AGL(1,𝔽5)11yescharacteristic
440C3⋊C8—52no—
441C3⋊C8—51no—
442C3⋊C8—102no—
443C3⋊C8C22×AGL(1,𝔽5)11yescharacteristic
444C22×C5—22no—
445C22×C5—31no—
446C22×C5—62no—
447C22×C5—124no—
448C2×D5—31no—
449C2×D5—62no—
450C2×D5D3×#4111yescharacteristic
451AGL(1,𝔽5)—41no—
452AGL(1,𝔽5)—121no—
453C4×C5—31no—
454C4×C5C22×C4×D311yescharacteristic
455Dic5—22no—
456Dic5—31no—
457Dic5—62no—
458Dic5C2×D3×D411yescharacteristic
459#41—52no—
460#41—152no—
461#41—152no—
462#41—152no—
463#41—302no—
464#41—302no—
465C2×Q8—152no—
466C2×D4—32no—
467C2×D4—302no—
468C22×C4—151no—
469C22×C4—302no—
470C22×C4—301no—
471Q16—51no—
472Q16—152no—
473Q16—151no—
474SD16—52no—
475SD16—152no—
476SD16—152no—
477SD16—152no—
478D8—32no—
479D8—31no—
480D8D3×AGL(1,𝔽5)11yescharacteristic, complemented
481M16—52no—
482M16—102no—
483M16—152no—
484M16—302no—
485M16—302no—
486M16—302no—
487M16—302no—
488C2×C8—31no—
489C2×C8—51no—
490C2×C8—51no—
491C2×C8—102no—
492C2×C8—151no—
493C2×C8—151no—
494C2×C8—151no—
495C2×C8—152no—
496C2×C8—152no—
497C2×C8—151no—
498C2×C8—151no—
499C2×C8—302no—
500C2×C8—302no—
501C2×C8—302no—
502C2×C8—302no—
503C42—51no—
504C42—151no—
505C42—301no—
506C3×C5C2×#44411yescharacteristic, complemented, Hall
507C22×C3—22no—
508C22×C3—51no—
509C2×D3—22no—
510C2×D3—44no—
511C2×D3—51no—
512C2×D3—102no—
513C2×D3D4×AGL(1,𝔽5)11yescharacteristic
514C3×C4—51no—
515C3×C4—102no—
516C3×C4—201no—
517C3×C4C23×AGL(1,𝔽5)11yescharacteristic
518Dic3—51no—
519Dic3—102no—
520Dic3—201no—
521Dic3D4×AGL(1,𝔽5)11yescharacteristic
522C2×C5—32no—
523C2×C5—42no—
524C2×C5—122no—
525C2×C5C4×D3×D411yescharacteristic
526D5—21no—
527D5—61no—
528C23—62no—
529C23—151no—
530Q8—52no—
531Q8—152no—
532Q8—302no—
533D4—32no—
534D4—62no—
535D4—102no—
536D4—302no—
537D4—302no—
538D4—302no—
539D4C2×D3×AGL(1,𝔽5)12yes—
540C2×C4—31no—
541C2×C4—51no—
542C2×C4—102no—
543C2×C4—101no—
544C2×C4—151no—
545C2×C4—151no—
546C2×C4—151no—
547C2×C4—151no—
548C2×C4—302no—
549C2×C4—302no—
550C2×C4—302no—
551C2×C4—302no—
552C2×C4—301no—
553C2×C4—602no—
554C8—31no—
555C8—52no—
556C8—51no—
557C8—102no—
558C8—152no—
559C8—151no—
560C8—302no—
561C8C2×D3×AGL(1,𝔽5)11yescharacteristic
562C2×C3—42no—
563C2×C3—101no—
564C2×C3C2×D4×AGL(1,𝔽5)11yescharacteristic
565D3—42no—
566D3—101no—
567D3#6573312yesdirect product factor
568C5D3×#44411yescharacteristic, complemented, Sylow
569C22—22no—
570C22—31no—
571C22—51no—
572C22—62no—
573C22—124no—
574C22—151no—
575C22—302no—
576C4—31no—
577C4—51no—
578C4—102no—
579C4—151no—
580C4—201no—
581C4—302no—
582C4—601no—
583C4C22×D3×AGL(1,𝔽5)11yescharacteristic
584C3C2×#6573311yescharacteristic, complemented, Sylow
585C2—32no—
586C2—42no—
587C2—101no—
588C2—122no—
589C2—301no—
590C2D3×D4×AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC2D3×D4×AGL(1,𝔽5)
commutatorC3×C4×C5C23×C4
FrattiniC4C22×D3×AGL(1,𝔽5)
FittingC3×C5×D8C2×C4
socleC2×C3×C5C2×C4×D4
prime radicalD3×#4058C2
cosocleC3×C4×D5C24
hypercenterD8D3×AGL(1,𝔽5)
hypocenterC3×C5C2×#444
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C4 ◁ D8

Complements

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

NH ≅ G/Ntype
1. D3×#4131C2semidirect
3. #11752392C2semidirect
4. C3×#65733C2semidirect
5. #11751956C2semidirect
6. #11751952C2semidirect
7. D3×#4008C2semidirect
8. #11751687C2semidirect
9. #11751682C2semidirect
10. D3×#3995C2semidirect
11. D3×#3992C2semidirect
14. #91990C22semidirect
15. C3×#4131C22semidirect
21. #91913C22semidirect
23. #91911C22semidirect
25. #91801C4semidirect
26. C5×D3×D8C4semidirect
28. #91499C4semidirect
30. D3×#687C4semidirect
31. #91279C4semidirect
33. D3×Dic20C4semidirect
35. #91237C22semidirect
36. #91224C22semidirect
37. #91221C22semidirect
38. C3×#4008C22semidirect
39. C3×#3995C22semidirect
40. C3×#3992C22semidirect
41. #91161C22semidirect
42. #91156C22semidirect
43. #91155C22semidirect
44. #91150C22semidirect
45. #91149C22semidirect
55. #65733D3direct (N × H)
89. C3×C5×D8C2×C4semidirect
90. #6716C2×C4semidirect
91. Dic60C2×C4semidirect
93. C5×#213C2×C4semidirect
94. C5×D24C2×C4semidirect
96. C3×#687C2×C4semidirect
97. C3×Dic20C2×C4semidirect
99. #6662C2×C4semidirect
100. #6659C2×C4semidirect
101. #6656C2×C4semidirect
124. #4131C2×D3semidirect
141. #4008C2×D3semidirect
144. #3995C2×D3semidirect
148. #3992C2×D3semidirect
194. D3×D8AGL(1,𝔽5)semidirect
250. C5×D8C4×D3semidirect
255. #687C4×D3semidirect
259. Dic20C4×D3semidirect
304. C3×D8C2×AGL(1,𝔽5)semidirect
314. #213C2×AGL(1,𝔽5)semidirect
323. D24C2×AGL(1,𝔽5)semidirect
409. C5×D3#444semidirect
480. D8D3×AGL(1,𝔽5)semidirect
506. C3×C5C2×#444semidirect
567. D3#65733direct (N × H)
568. C5D3×#444semidirect
584. C3C2×#65733semidirect

Maximal subgroups

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

subgroupindex
D3×#41312
D3×#40582
#117523922
C3×#657332
#117519562
#117519522
D3×#40082
#117516872
#117516822
D3×#39952
D3×#39922
C2×#657333
D3×#4445