Finite Group Databasebeta

← C2×#11751300 #14429375

Subgroups — C2×#11751300 #14429375

Order 1920.

976 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
2C2×#91838C211yesmaximal, characteristic, complemented
3C2×#91837C211yesmaximal, characteristic, complemented
4C22×#6715C211yesmaximal, characteristic, complemented
5C2×#91117C211yesmaximal, characteristic, complemented
6#11751300C218yesmaximal, direct product factor
7C2×#91107C211yesmaximal, characteristic, complemented
8C2×#91106C211yesmaximal, characteristic, complemented
9C2×#65488—31nomaximal, Hall
10C2×C3×C5×#41C2211yescharacteristic
11C22×C3×C5×D4C2211yescharacteristic
12C22×D60C2211yescharacteristic
13C3×C5×AGL(1,ℤ/8)—22no—
14C3×C5×AGL(1,ℤ/8)C2214yescomplemented
15C2×C3×C5×SD16—21no—
16C2×C3×C5×D8—21no—
17C2×C3×C5×M16C2211yescharacteristic
18#91838—22no—
19#91838C2214yescomplemented
20#91837—22no—
21#91837C2214yescomplemented
22C2×#6717—21no—
23C2×#6715—21no—
24C2×#6715—21no—
25C2×#6715—21no—
26C2×#6715—22no—
27C2×#6715C2214yescomplemented
28C2×#6710—21no—
29C2×#6710C2211yescharacteristic
30C2×#6710C2211yescharacteristic
31C22×C3⋊#168—21no—
32C22×C3⋊#168C2211yescharacteristic
33#91117C2214yescomplemented
34#91107C2214yescomplemented
35#91106C2214yescomplemented
36C2×#5199—51nomaximal, Hall
37C2×C5×AGL(1,ℤ/8)D311yescharacteristic, complemented
38C2×#4095—31no—
39C2×#4094—31no—
40C22×#686—31no—
41C2×#3965—31no—
42#65488—38no—
43C2×#3955—31no—
44C2×#3954—31no—
45C24×C3×C5—21no—
46C23×D15—21no—
47C3×C5×#41—21no—
48C3×C5×#41—21no—
49C3×C5×#41D412yes—
50C3×C5×#41C2312yes—
51C2×C3×C5×Q8D411yescharacteristic
52C2×C3×C5×D4—21no—
53C2×C3×C5×D4—21no—
54C2×C3×C5×D4—21no—
55C2×C3×C5×D4—21no—
56C2×C3×C5×D4—21no—
57C2×C3×C5×D4—21no—
58C2×C3×C5×D4D411yescharacteristic
59C2×C3×C5×D4C2312yes—
60C22×C3×C4×C5—21no—
61C22×C3×C4×C5C2311yescharacteristic
62C2×D60—21no—
63C2×D60—21no—
64C2×D60—21no—
65C2×D60—21no—
66C2×D60—21no—
67C2×D60C2312yes—
68C3×C5×SD16—24no—
69C3×C5×D8—24no—
70C3×C5×M16—21no—
71C3×C5×M16C2312yes—
72C2×C3×C5×C8—21no—
73#6717—24no—
74#6715—24no—
75#6715—24no—
76#6715—24no—
77#6710—21no—
78#6710—21no—
79#6710—24no—
80#6710C2312yes—
81#6710C2312yes—
82C2×C3⋊#168—21no—
83C2×C3⋊#168—21no—
84C2×C3⋊#168—24no—
85C2×C3⋊#168D411yescharacteristic
86C2×C3⋊#168D411yescharacteristic
87C2×C3⋊#168D412yes—
88C2×C3⋊#168C2312yes—
89C2×C3×AGL(1,ℤ/8)D511yescharacteristic, complemented
90C2×#949—51no—
91C2×#948—51no—
92C22×#213—51no—
93C2×#833—51no—
94#5199—58no—
95C2×#823—51no—
96C2×#822—51no—
97C2×C5×#41C2×D311yescharacteristic
98C22×C5×D4C2×D311yescharacteristic
99C22×D20—31no—
100C5×AGL(1,ℤ/8)—22no—
101C5×AGL(1,ℤ/8)C2×D314yescomplemented
102C2×C5×SD16—21no—
103C2×C5×D8—21no—
104C2×C5×M16C2×D311yescharacteristic
105#4095—34no—
106#4095—62no—
107#4094—34no—
108#4094—62no—
109C2×#688—61no—
110C2×#686—34no—
111C2×#686—61no—
112C2×#686—61no—
113C2×#686—62no—
114C2×#686—61no—
115C2×#681—31no—
116C2×#681—31no—
117C2×#681—61no—
118C22×#168—31no—
119C22×#168—61no—
120#3965—34no—
121#3955—34no—
122#3954—34no—
123C2×#472—151noSylow
124C23×C3×C5—21no—
125C23×C3×C5—22no—
126C23×C3×C5—41no—
127C23×C3×C5—41no—
128C23×C3×C5—82no—
129C23×C3×C5#4111yescharacteristic
130C22×D15—22no—
131C22×D15—41no—
132C22×D15—41no—
133C22×D15—82no—
134C3×C5×Q8—21no—
135C3×C5×Q8C2×D412yes—
136C3×C5×D4—22no—
137C3×C5×D4—22no—
138C3×C5×D4—21no—
139C3×C5×D4—22no—
140C3×C5×D4—22no—
141C3×C5×D4—41no—
142C3×C5×D4—41no—
143C3×C5×D4C2×D412yes—
144C2×C3×C4×C5—21no—
145C2×C3×C4×C5—22no—
146C2×C3×C4×C5—22no—
147C2×C3×C4×C5C2×D411yescharacteristic
148C2×C3×C4×C5C2×D411yescharacteristic
149C2×C3×C4×C5C2×D411yescharacteristic
150C2×C3×C4×C5C2×D411yescharacteristic
151C2×C3×C4×C5#4111yescharacteristic
152C2×C3×C4×C5C2411yescharacteristic
153D60—22no—
154D60—22no—
155D60—41no—
156D60—41no—
157C3×C5×C8—22no—
158C3⋊#168—22no—
159C3⋊#168—22no—
160C3⋊#168C2×D412yes—
161C3⋊#168C2×D412yes—
162C2×C3×#41C2×D511yescharacteristic
163C22×C3×D4C2×D511yescharacteristic
164C22×D12—51no—
165C3×AGL(1,ℤ/8)—22no—
166C3×AGL(1,ℤ/8)C2×D514yescomplemented
167C2×C3×SD16—21no—
168C2×C3×D8—21no—
169C2×C3×M16C2×D511yescharacteristic
170#949—54no—
171#949—102no—
172#948—54no—
173#948—102no—
174C2×#215—101no—
175C2×#213—54no—
176C2×#213—101no—
177C2×#213—101no—
178C2×#213—102no—
179C2×#213—101no—
180C2×#208—51no—
181C2×#208—51no—
182C2×#208—101no—
183C22×C3⋊C8—51no—
184C22×C3⋊C8—101no—
185#833—54no—
186#823—54no—
187#822—54no—
188C24×C5—21no—
189C23×D5—61no—
190C5×#41—21no—
191C5×#41—21no—
192C5×#41#6712yes—
193C5×#41C22×D312yes—
194C2×C5×Q8#6711yescharacteristic
195C2×C5×D4—21no—
196C2×C5×D4—21no—
197C2×C5×D4—21no—
198C2×C5×D4—21no—
199C2×C5×D4—21no—
200C2×C5×D4—21no—
201C2×C5×D4#6711yescharacteristic
202C2×C5×D4C22×D312yes—
203C22×C4×C5—21no—
204C22×C4×C5C22×D311yescharacteristic
205C2×D20—32no—
206C2×D20—61no—
207C2×D20—61no—
208C2×D20—61no—
209C2×D20—61no—
210C2×D20—61no—
211C5×SD16—24no—
212C5×D8—24no—
213C5×M16—21no—
214C5×M16C22×D312yes—
215C2×C5×C8—21no—
216#688—64no—
217#686—64no—
218#686—64no—
219#686—64no—
220#681—32no—
221#681—32no—
222#681—64no—
223#681—61no—
224#681—61no—
225C2×#168—31no—
226C2×#168—31no—
227C2×#168—32no—
228C2×#168—32no—
229C2×#168—64no—
230C2×#168—61no—
231C2×#168—61no—
232C2×AGL(1,ℤ/8)—151no—
233C2×AGL(1,ℤ/8)D1511yescharacteristic, complemented
234C22×D8—151no—
235C2×#131—151no—
236#472—158no—
237C2×#108—151no—
238C2×#100—151no—
239C2×#100—151no—
240C22×C3×C5—22no—
241C22×C3×C5—22no—
242C22×C3×C5—42no—
243C22×C3×C5—42no—
244C22×C3×C5—82no—
245C22×C3×C5—82no—
246C22×C3×C5—82no—
247C22×C3×C5#12111yescharacteristic
248C22×C3×C5#12111yescharacteristic
249C22×C3×C5C2×#4111yescharacteristic
250C2×D15—42no—
251C2×D15—82no—
252C2×D15—82no—
253C2×D15—82no—
254C3×C4×C5—22no—
255C3×C4×C5#12111yescharacteristic
256C3×C4×C5#12111yescharacteristic
257C3×C4×C5C22×D411yescharacteristic
258C3×C4×C5C22×D411yescharacteristic
259C24×C3—21no—
260C23×D3—101no—
261C3×#41—21no—
262C3×#41—21no—
263C3×#41C22⋊D512yes—
264C3×#41C22×D512yes—
265C2×C3×Q8C22⋊D511yescharacteristic
266C2×C3×D4—21no—
267C2×C3×D4—21no—
268C2×C3×D4—21no—
269C2×C3×D4—21no—
270C2×C3×D4—21no—
271C2×C3×D4—21no—
272C2×C3×D4C22⋊D511yescharacteristic
273C2×C3×D4C22×D512yes—
274C22×C3×C4—21no—
275C22×C3×C4C22×D511yescharacteristic
276C2×D12—52no—
277C2×D12—101no—
278C2×D12—101no—
279C2×D12—101no—
280C2×D12—101no—
281C2×D12—101no—
282C3×SD16—24no—
283C3×D8—24no—
284C3×M16—21no—
285C3×M16C22×D512yes—
286C2×C3×C8—21no—
287#215—104no—
288#213—104no—
289#213—104no—
290#213—104no—
291#208—52no—
292#208—52no—
293#208—104no—
294#208—101no—
295#208—101no—
296C2×C3⋊C8—51no—
297C2×C3⋊C8—51no—
298C2×C3⋊C8—52no—
299C2×C3⋊C8—52no—
300C2×C3⋊C8—104no—
301C2×C3⋊C8—101no—
302C2×C3⋊C8—101no—
303C23×C5—21no—
304C23×C5—22no—
305C23×C5—41no—
306C23×C5—41no—
307C23×C5—82no—
308C23×C5#23711yescharacteristic
309C22×D5—62no—
310C22×D5—121no—
311C22×D5—121no—
312C22×D5—242no—
313C5×Q8—21no—
314C5×Q8C2×#6712yes—
315C5×D4—22no—
316C5×D4—22no—
317C5×D4—21no—
318C5×D4—22no—
319C5×D4—22no—
320C5×D4—41no—
321C5×D4—41no—
322C5×D4C2×#6712yes—
323C2×C4×C5—21no—
324C2×C4×C5—22no—
325C2×C4×C5—22no—
326C2×C4×C5D3×D411yescharacteristic
327C2×C4×C5D3×D411yescharacteristic
328C2×C4×C5#23711yescharacteristic
329C2×C4×C5C2×#6711yescharacteristic
330C2×C4×C5C2×#6711yescharacteristic
331C2×C4×C5C23×D311yescharacteristic
332D20—62no—
333D20—62no—
334D20—121no—
335D20—121no—
336C5×C8—22no—
337#168—32no—
338#168—32no—
339#168—62no—
340#168—62no—
341C2×#41C2×D1511yescharacteristic
342C22×D4—151no—
343C22×D4C2×D1511yescharacteristic
344AGL(1,ℤ/8)—22no—
345AGL(1,ℤ/8)—154no—
346AGL(1,ℤ/8)—302no—
347AGL(1,ℤ/8)C2×D1514yescomplemented
348C2×SD16—21no—
349C2×SD16—301no—
350C2×D8—21no—
351C2×D8—154no—
352C2×D8—301no—
353C2×D8—301no—
354C2×D8—302no—
355C2×D8—301no—
356#131—154no—
357#131—302no—
358C2×M16—151no—
359C2×M16—151no—
360C2×M16—301no—
361C2×M16C2×D1511yescharacteristic
362C22×C8—151no—
363C22×C8—301no—
364#108—154no—
365#100—154no—
366#100—154no—
367C2×C3×C5—21no—
368C2×C3×C5—21no—
369C2×C3×C5—42no—
370C2×C3×C5—82no—
371C2×C3×C5#47212yescomplemented
372C2×C3×C5C2×#12111yescharacteristic
373D15—82no—
374C23×C3—21no—
375C23×C3—22no—
376C23×C3—41no—
377C23×C3—41no—
378C23×C3—82no—
379C23×C3#71111yescharacteristic
380C22×D3—102no—
381C22×D3—201no—
382C22×D3—201no—
383C22×D3—402no—
384C3×Q8—21no—
385C3×Q8C2×C22⋊D512yes—
386C3×D4—22no—
387C3×D4—22no—
388C3×D4—22no—
389C3×D4—22no—
390C3×D4—21no—
391C3×D4—41no—
392C3×D4—41no—
393C3×D4C2×C22⋊D512yes—
394C2×C3×C4—21no—
395C2×C3×C4—22no—
396C2×C3×C4—22no—
397C2×C3×C4D4×D511yescharacteristic
398C2×C3×C4D4×D511yescharacteristic
399C2×C3×C4#71111yescharacteristic
400C2×C3×C4C2×C22⋊D511yescharacteristic
401C2×C3×C4C2×C22⋊D511yescharacteristic
402C2×C3×C4C23×D511yescharacteristic
403D12—102no—
404D12—102no—
405D12—201no—
406D12—201no—
407C3×C8—22no—
408C3⋊C8—52no—
409C3⋊C8—52no—
410C3⋊C8—102no—
411C3⋊C8—102no—
412C22×C5—22no—
413C22×C5—22no—
414C22×C5—42no—
415C22×C5—42no—
416C22×C5—82no—
417C22×C5—82no—
418C22×C5—82no—
419C22×C5#93911yescharacteristic
420C22×C5#93911yescharacteristic
421C22×C5C2×#23711yescharacteristic
422C2×D5—122no—
423C2×D5—242no—
424C2×D5—242no—
425C2×D5—242no—
426C4×C5—22no—
427C4×C5#93911yescharacteristic
428C4×C5#93911yescharacteristic
429C4×C5C2×D3×D411yescharacteristic
430C4×C5C22×#6711yescharacteristic
431C24—21no—
432C24—301no—
433#41—21no—
434#41—21no—
435#41#122012yes—
436#41C22×D1512yes—
437C2×Q8#122011yescharacteristic
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—152no—
445C2×D4—301no—
446C2×D4—301no—
447C2×D4—301no—
448C2×D4—301no—
449C2×D4—301no—
450C2×D4#122011yescharacteristic
451C2×D4C22×D1512yes—
452C22×C4—21no—
453C22×C4C22×D1511yescharacteristic
454SD16—24no—
455SD16—304no—
456D8—24no—
457D8—304no—
458D8—304no—
459D8—304no—
460M16—21no—
461M16—152no—
462M16—152no—
463M16—304no—
464M16—301no—
465M16—301no—
466M16C22×D1512yes—
467C2×C8—21no—
468C2×C8—151no—
469C2×C8—151no—
470C2×C8—152no—
471C2×C8—152no—
472C2×C8—304no—
473C2×C8—301no—
474C2×C8—301no—
475C3×C5C2×#47211yescharacteristic, complemented, Hall
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—42no—
479C22×C3—42no—
480C22×C3—82no—
481C22×C3—82no—
482C22×C3—82no—
483C22×C3#408511yescharacteristic
484C22×C3#408511yescharacteristic
485C22×C3C2×#71111yescharacteristic
486C2×D3—202no—
487C2×D3—402no—
488C2×D3—402no—
489C2×D3—402no—
490C3×C4—22no—
491C3×C4#408511yescharacteristic
492C3×C4#408511yescharacteristic
493C3×C4C2×D4×D511yescharacteristic
494C3×C4C22×C22⋊D511yescharacteristic
495C2×C5—21no—
496C2×C5—21no—
497C2×C5—42no—
498C2×C5—82no—
499C2×C5#519912yescomplemented
500C2×C5C2×#93911yescharacteristic
501D5—242no—
502C23—21no—
503C23—22no—
504C23—41no—
505C23—41no—
506C23—82no—
507C23—302no—
508C23—601no—
509C23—601no—
510C23—1202no—
511C23#681911yescharacteristic
512Q8—21no—
513Q8C2×#122012yes—
514D4—22no—
515D4—22no—
516D4—21no—
517D4—22no—
518D4—22no—
519D4—41no—
520D4—41no—
521D4—302no—
522D4—302no—
523D4—601no—
524D4—601no—
525D4C2×#122012yes—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4D4×D1511yescharacteristic
530C2×C4D4×D1511yescharacteristic
531C2×C4#681911yescharacteristic
532C2×C4C2×#122011yescharacteristic
533C2×C4C2×#122011yescharacteristic
534C2×C4C23×D1511yescharacteristic
535C8—22no—
536C8—152no—
537C8—152no—
538C8—302no—
539C8—302no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—42no—
543C2×C3—82no—
544C2×C3#6548812yescomplemented
545C2×C3C2×#408511yescharacteristic
546D3—402no—
547C5C2×#519911yescharacteristic, complemented, Sylow
548C22—22no—
549C22—22no—
550C22—42no—
551C22—42no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—602no—
556C22—1202no—
557C22—1202no—
558C22—1202no—
559C22#9182811yescharacteristic
560C22#9182811yescharacteristic
561C22C2×#681911yescharacteristic
562C4—22no—
563C4#9182811yescharacteristic
564C4#9182811yescharacteristic
565C4C2×D4×D1511yescharacteristic
566C4C22×#122011yescharacteristic
567C3C2×#6548811yescharacteristic, complemented, Sylow
568C2—21no—
569C2—21no—
570C2—42no—
571C2—82no—
572C2—1202no—
573C2#1175130012yesdirect product factor
574C2C2×#9182811yescharacteristic
subgroupG/H
centerC22#91828
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC2×C3×C5×AGL(1,ℤ/8)C2
socleC22×C3×C5#121
hypercenterC2×AGL(1,ℤ/8)D15
hypocenterC3×C5C2×#472
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×AGL(1,ℤ/8)C2semidirect
2. C2×#91838C2semidirect
3. C2×#91837C2semidirect
4. C22×#6715C2semidirect
5. C2×#91117C2semidirect
6. #11751300C2direct (N × H)
7. C2×#91107C2semidirect
8. C2×#91106C2semidirect
14. C3×C5×AGL(1,ℤ/8)C22semidirect
19. #91838C22semidirect
21. #91837C22semidirect
27. C2×#6715C22semidirect
33. #91117C22semidirect
34. #91107C22semidirect
35. #91106C22semidirect
37. C2×C5×AGL(1,ℤ/8)D3semidirect
89. C2×C3×AGL(1,ℤ/8)D5semidirect
101. C5×AGL(1,ℤ/8)C2×D3semidirect
166. C3×AGL(1,ℤ/8)C2×D5semidirect
233. C2×AGL(1,ℤ/8)D15semidirect
347. AGL(1,ℤ/8)C2×D15semidirect
371. C2×C3×C5#472semidirect
475. C3×C5C2×#472semidirect
499. C2×C5#5199semidirect
544. C2×C3#65488semidirect
547. C5C2×#5199semidirect
567. C3C2×#65488semidirect
573. C2#11751300direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×AGL(1,ℤ/8)2
C2×#918382
C2×#918372
C22×#67152
C2×#911172
#117513002
C2×#911072
C2×#911062
C2×#654883
C2×#51995