Finite Group Databasebeta

← C3×#10602677 #14287904

Subgroups — C3×#10602677 #14287904

Order 1920.

640 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#538C211yesmaximal, characteristic, complemented
2C3×#65335C211yesmaximal, characteristic, complemented
3C3×#65325C211yesmaximal, characteristic, complemented
4C3×#65249C211yesmaximal, characteristic, complemented
5C3×#64758C211yesmaximal, characteristic, complemented
6C3×#64757C211yesmaximal, characteristic, complemented
7C3×#64745C211yesmaximal, characteristic, complemented
8#10602677C311yesmaximal, characteristic, direct product factor, Hall
9C22×C3×C22⋊D5—21no—
10C22×C3×D20C2211yescharacteristic
11C3×C5×#126—21no—
12C3×C5×#124—21no—
13C3×C5×#123—21no—
14C3×C5×C22⋊Q8C2211yescharacteristic
15C3×C5×#121—21no—
16C3×C5×#121C2211yescharacteristic
17C3×C5×#120—21no—
18C3×C4×C5×D4—21no—
19C2×C3×C5×C22⋊C4C2211yescharacteristic
20C2×C3×#688—21no—
21C2×C3×#686—21no—
22C2×C3×#685—21no—
23C3×#3963C2211yescharacteristic
24C3×C22⋊#168C2211yescharacteristic
25C3×C22⋊#168C2211yescharacteristic
26C3×#3941—21no—
27C3×#3941—21no—
28C3×#1600—51nomaximal, Hall
29C5×#538C2×C311yescharacteristic, complemented
30#65335C2×C311yescharacteristic, complemented
31#65325C2×C311yescharacteristic, complemented
32#65249C2×C311yescharacteristic, complemented
33#64758C2×C311yescharacteristic, complemented
34#64757C2×C311yescharacteristic, complemented
35#64745C2×C311yescharacteristic, complemented
36C24×C3×C5D411yescharacteristic
37C23×C3×D5—21no—
38C2×C3×C5×Q8D411yescharacteristic, complemented
39C2×C3×C5×D4—21no—
40C2×C3×C5×D4—21no—
41C2×C3×C5×D4—21no—
42C2×C3×C5×D4D411yescharacteristic, complemented
43C22×C3×C4×C5—21no—
44C22×C3×C4×C5D411yescharacteristic
45C22×C3×C4×C5C2311yescharacteristic
46C2×C3×C22⋊D5—42no—
47C2×C3×C22⋊D5—42no—
48C2×C3×C22⋊D5—42no—
49C22×C3×Dic5—21no—
50C2×C3×D20—21no—
51C2×C3×D20—21no—
52C2×C3×D20—42no—
53C3×C5×C4⋊C4—21no—
54C3×C5×C4⋊C4—21no—
55C3×C5×C4⋊C4D411yescharacteristic, complemented
56C3×C5×C4⋊C4D411yescharacteristic, complemented
57C3×C5×C22⋊C4—21no—
58C3×C5×C22⋊C4—21no—
59C3×C5×C22⋊C4—21no—
60C3×C5×C22⋊C4—21no—
61C3×C5×C22⋊C4—21no—
62C3×C5×C22⋊C4—21no—
63C3×C5×C22⋊C4—21no—
64C3×C42×C5—21no—
65C3×#688—42no—
66C3×#686—42no—
67C3×#685—42no—
68C2×C3×#168—21no—
69C2×C3×#168—21no—
70C3×#538D511yescharacteristic, complemented
71C3×#450—51no—
72C3×#447—51no—
73C3×#392—51no—
74C3×#328—51no—
75C3×#327—51no—
76C3×#323—51no—
77C22×C22⋊D5—21no—
78C22×D20C22×C311yescharacteristic
79C5×#126—21no—
80C5×#124—21no—
81C5×#123—21no—
82C5×C22⋊Q8C22×C311yescharacteristic
83C5×#121—21no—
84C5×#121C22×C311yescharacteristic
85C5×#120—21no—
86C4×C5×D4—21no—
87C2×C5×C22⋊C4C22×C311yescharacteristic
88C2×#688—21no—
89C2×#686—21no—
90C2×#685—21no—
91#3963C22×C311yescharacteristic
92C22⋊#168C22×C311yescharacteristic
93C22⋊#168C22×C311yescharacteristic
94#3941—21no—
95#3941—21no—
96#1600—51noSylow
97C23×C3×C5—21no—
98C23×C3×C5—21no—
99C23×C3×C5—21no—
100C23×C3×C5—21no—
101C23×C3×C5—41no—
102C23×C3×C5—41no—
103C23×C3×C5C2×D411yescharacteristic
104C22×C3×D5—22no—
105C22×C3×D5—41no—
106C22×C3×D5—41no—
107C22×C3×D5—41no—
108C22×C3×D5—41no—
109C22×C3×D5—41no—
110C22×C3×D5—41no—
111C3×C5×Q8—41no—
112C3×C5×D4—41no—
113C3×C5×D4—41no—
114C3×C5×D4—41no—
115C3×C5×D4—81no—
116C3×C5×D4—81no—
117C2×C3×C4×C5—21no—
118C2×C3×C4×C5—21no—
119C2×C3×C4×C5—21no—
120C2×C3×C4×C5—21no—
121C2×C3×C4×C5—41no—
122C2×C3×C4×C5—41no—
123C2×C3×C4×C5—81no—
124C2×C3×C4×C5C2×D411yescharacteristic
125C2×C3×C4×C5C2×D411yescharacteristic
126C3×C22⋊D5—48no—
127C2×C3×Dic5—22no—
128C2×C3×Dic5—41no—
129C2×C3×Dic5—41no—
130C3×D20—22no—
131C3×D20—22no—
132C3×D20—41no—
133C3×D20—41no—
134C3×#168—41no—
135C3×#168—41no—
136C22×C3×D4—51no—
137C22×C3×D4—101no—
138C2×C3×SD16—101no—
139C2×C3×D8—101no—
140C3×#126—21no—
141C3×#124—21no—
142C3×#123—21no—
143C3×C22⋊Q8C2×D511yescharacteristic
144C3×#121—21no—
145C3×#121C2×D511yescharacteristic
146C3×#120—21no—
147C3×C4×D4—21no—
148C2×C3×C22⋊C4—101no—
149C2×C3×C22⋊C4C2×D511yescharacteristic
150C3×#102—101no—
151C3×#102—101no—
152C3×C22⋊C8—51no—
153C3×C22⋊C8—51no—
154C3×#95—51no—
155C24×C5C3×D411yescharacteristic
156C23×D5—21no—
157C2×C5×Q8C3×D411yescharacteristic, complemented
158C2×C5×D4—21no—
159C2×C5×D4—21no—
160C2×C5×D4—21no—
161C2×C5×D4C3×D411yescharacteristic, complemented
162C22×C4×C5—21no—
163C22×C4×C5C3×D411yescharacteristic
164C22×C4×C5C23×C311yescharacteristic
165C2×C22⋊D5—42no—
166C2×C22⋊D5—42no—
167C2×C22⋊D5—42no—
168C22×Dic5—21no—
169C2×D20—21no—
170C2×D20—21no—
171C2×D20—42no—
172C5×C4⋊C4—21no—
173C5×C4⋊C4—21no—
174C5×C4⋊C4C3×D411yescharacteristic, complemented
175C5×C4⋊C4C3×D411yescharacteristic, complemented
176C5×C22⋊C4—21no—
177C5×C22⋊C4—21no—
178C5×C22⋊C4—21no—
179C5×C22⋊C4—21no—
180C5×C22⋊C4—21no—
181C5×C22⋊C4—21no—
182C5×C22⋊C4—21no—
183C42×C5—21no—
184#688—42no—
185#686—42no—
186#685—42no—
187C2×#168—21no—
188C2×#168—21no—
189#538C3×D511yescharacteristic, complemented
190#450—51no—
191#447—51no—
192#392—51no—
193#328—51no—
194#327—51no—
195#323—51no—
196C22×C3×C5—21no—
197C22×C3×C5—21no—
198C22×C3×C5—42no—
199C22×C3×C5—41no—
200C22×C3×C5—41no—
201C22×C3×C5—41no—
202C22×C3×C5—41no—
203C22×C3×C5—41no—
204C22×C3×C5—41no—
205C22×C3×C5—81no—
206C22×C3×C5#12011yescharacteristic
207C22×C3×C5#13511yescharacteristic
208C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
209C2×C3×D5—42no—
210C2×C3×D5—42no—
211C2×C3×D5—42no—
212C2×C3×D5—81no—
213C2×C3×D5—81no—
214C2×C3×D5—81no—
215C2×C3×D5—81no—
216C3×C4×C5—21no—
217C3×C4×C5—21no—
218C3×C4×C5—41no—
219C3×C4×C5—41no—
220C3×C4×C5—41no—
221C3×C4×C5—41no—
222C3×Dic5—42no—
223C24×C3—101no—
224C24×C3C22⋊D511yescharacteristic
225C2×C3×Q8C22⋊D511yescharacteristic, complemented
226C2×C3×D4—21no—
227C2×C3×D4—21no—
228C2×C3×D4—21no—
229C2×C3×D4—101no—
230C2×C3×D4—101no—
231C2×C3×D4—202no—
232C2×C3×D4—202no—
233C2×C3×D4—202no—
234C2×C3×D4—202no—
235C2×C3×D4C22⋊D511yescharacteristic, complemented
236C22×C3×C4—21no—
237C22×C3×C4—101no—
238C22×C3×C4C22⋊D511yescharacteristic
239C22×C3×C4C22×D511yescharacteristic
240C3×SD16—202no—
241C3×D8—202no—
242C2×C3×C8—101no—
243C2×C3×C8—101no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4C22⋊D511yescharacteristic, complemented
247C3×C4⋊C4C22⋊D511yescharacteristic, complemented
248C3×C22⋊C4—21no—
249C3×C22⋊C4—21no—
250C3×C22⋊C4—21no—
251C3×C22⋊C4—21no—
252C3×C22⋊C4—21no—
253C3×C22⋊C4—21no—
254C3×C22⋊C4—21no—
255C3×C22⋊C4—202no—
256C3×C42—21no—
257C23×C5—21no—
258C23×C5—21no—
259C23×C5—21no—
260C23×C5—21no—
261C23×C5—41no—
262C23×C5—41no—
263C23×C5C2×C3×D411yescharacteristic
264C22×D5—22no—
265C22×D5—41no—
266C22×D5—41no—
267C22×D5—41no—
268C22×D5—41no—
269C22×D5—41no—
270C22×D5—41no—
271C5×Q8—41no—
272C5×D4—41no—
273C5×D4—41no—
274C5×D4—41no—
275C5×D4—81no—
276C5×D4—81no—
277C2×C4×C5—21no—
278C2×C4×C5—21no—
279C2×C4×C5—21no—
280C2×C4×C5—21no—
281C2×C4×C5—41no—
282C2×C4×C5—41no—
283C2×C4×C5—81no—
284C2×C4×C5C2×C3×D411yescharacteristic
285C2×C4×C5C2×C3×D411yescharacteristic
286C22⋊D5—48no—
287C2×Dic5—22no—
288C2×Dic5—41no—
289C2×Dic5—41no—
290D20—22no—
291D20—22no—
292D20—41no—
293D20—41no—
294#168—41no—
295#168—41no—
296C22×D4—51no—
297C22×D4—101no—
298C2×SD16—101no—
299C2×D8—101no—
300#126—21no—
301#124—21no—
302#123—21no—
303C22⋊Q8C2×C3×D511yescharacteristic
304#121—21no—
305#121C2×C3×D511yescharacteristic
306#120—21no—
307C4×D4—21no—
308C2×C22⋊C4—101no—
309C2×C22⋊C4C2×C3×D511yescharacteristic
310#102—101no—
311#102—101no—
312C22⋊C8—51no—
313C22⋊C8—51no—
314#95—51no—
315C2×C3×C5—21no—
316C2×C3×C5—21no—
317C2×C3×C5—81no—
318C2×C3×C5—81no—
319C2×C3×C5#44911yescharacteristic
320C2×C3×C5#45311yescharacteristic
321C2×C3×C5B(4,𝔽2)11yescharacteristic
322C3×D5—82no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—102no—
330C23×C3—201no—
331C23×C3—201no—
332C23×C3—201no—
333C23×C3—201no—
334C23×C3—201no—
335C23×C3—201no—
336C23×C3C2×C22⋊D511yescharacteristic
337C3×Q8—41no—
338C3×D4—41no—
339C3×D4—41no—
340C3×D4—41no—
341C3×D4—81no—
342C3×D4—81no—
343C3×D4—102no—
344C3×D4—102no—
345C3×D4—201no—
346C3×D4—201no—
347C3×D4—208no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—81no—
355C2×C3×C4—102no—
356C2×C3×C4—201no—
357C2×C3×C4—201no—
358C2×C3×C4C2×C22⋊D511yescharacteristic
359C2×C3×C4C2×C22⋊D511yescharacteristic
360C3×C8—201no—
361C3×C8—201no—
362C22×C5—21no—
363C22×C5—21no—
364C22×C5—42no—
365C22×C5—41no—
366C22×C5—41no—
367C22×C5—41no—
368C22×C5—41no—
369C22×C5—41no—
370C22×C5—41no—
371C22×C5—81no—
372C22×C5C3×#12011yescharacteristic
373C22×C5C3×#13511yescharacteristic
374C22×C5C3×AGL(1,ℤ/8)11yescharacteristic
375C2×D5—42no—
376C2×D5—42no—
377C2×D5—42no—
378C2×D5—81no—
379C2×D5—81no—
380C2×D5—81no—
381C2×D5—81no—
382C4×C5—21no—
383C4×C5—21no—
384C4×C5—41no—
385C4×C5—41no—
386C4×C5—41no—
387C4×C5—41no—
388Dic5—42no—
389C24—101no—
390C24C3×C22⋊D511yescharacteristic
391C2×Q8C3×C22⋊D511yescharacteristic, complemented
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—101no—
396C2×D4—101no—
397C2×D4—202no—
398C2×D4—202no—
399C2×D4—202no—
400C2×D4—202no—
401C2×D4C3×C22⋊D511yescharacteristic, complemented
402C22×C4—21no—
403C22×C4—101no—
404C22×C4C3×C22⋊D511yescharacteristic
405C22×C4C22×C3×D511yescharacteristic
406SD16—202no—
407D8—202no—
408C2×C8—101no—
409C2×C8—101no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4C3×C22⋊D511yescharacteristic, complemented
413C4⋊C4C3×C22⋊D511yescharacteristic, complemented
414C22⋊C4—21no—
415C22⋊C4—21no—
416C22⋊C4—21no—
417C22⋊C4—21no—
418C22⋊C4—21no—
419C22⋊C4—21no—
420C22⋊C4—21no—
421C22⋊C4—202no—
422C42—21no—
423C3×C5#160011yescharacteristic, complemented, Hall
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—42no—
427C22×C3—41no—
428C22×C3—41no—
429C22×C3—41no—
430C22×C3—41no—
431C22×C3—41no—
432C22×C3—41no—
433C22×C3—81no—
434C22×C3—202no—
435C22×C3—202no—
436C22×C3—202no—
437C22×C3—401no—
438C22×C3—401no—
439C22×C3—401no—
440C22×C3—401no—
441C22×C3#409511yescharacteristic
442C22×C3#409611yescharacteristic
443C22×C3#409911yescharacteristic
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—41no—
447C3×C4—41no—
448C3×C4—41no—
449C3×C4—41no—
450C3×C4—202no—
451C2×C5—21no—
452C2×C5—21no—
453C2×C5—81no—
454C2×C5—81no—
455C2×C5C3×#44911yescharacteristic
456C2×C5C3×#45311yescharacteristic
457C2×C5C3×B(4,𝔽2)11yescharacteristic
458D5—82no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—41no—
464C23—41no—
465C23—102no—
466C23—201no—
467C23—201no—
468C23—201no—
469C23—201no—
470C23—201no—
471C23—201no—
472C23C2×C3×C22⋊D511yescharacteristic
473Q8—41no—
474D4—41no—
475D4—41no—
476D4—41no—
477D4—81no—
478D4—81no—
479D4—102no—
480D4—102no—
481D4—201no—
482D4—201no—
483D4—208no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—81no—
491C2×C4—102no—
492C2×C4—201no—
493C2×C4—201no—
494C2×C4C2×C3×C22⋊D511yescharacteristic
495C2×C4C2×C3×C22⋊D511yescharacteristic
496C8—201no—
497C8—201no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—81no—
501C2×C3—81no—
502C2×C3—402no—
503C2×C3#6552711yescharacteristic
504C2×C3#6553011yescharacteristic
505C2×C3#6553311yescharacteristic
506C5C3×#160011yescharacteristic, complemented, Sylow
507C22—21no—
508C22—21no—
509C22—42no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—81no—
517C22—202no—
518C22—202no—
519C22—202no—
520C22—401no—
521C22—401no—
522C22—401no—
523C22—401no—
524C22C3×#409511yescharacteristic
525C22C3×#409611yescharacteristic
526C22C3×#409911yescharacteristic
527C4—21no—
528C4—21no—
529C4—41no—
530C4—41no—
531C4—41no—
532C4—41no—
533C4—202no—
534C3#1060267711yescharacteristic, direct product factor, Sylow
535C2—21no—
536C2—21no—
537C2—81no—
538C2—81no—
539C2—402no—
540C2C3×#6552711yescharacteristic
541C2C3×#6553011yescharacteristic
542C2C3×#6553311yescharacteristic
subgroupG/H
centerC22×C3#4099
commutator, cosocleC22×C4×C5C23×C3
FrattiniC22×C4C22×C3×D5
FittingC3×C5×#538C2
socleC22×C3×C5#120
hypercenterC3×#538D5
hypocenterC5C3×#1600
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#538

Complements

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

NH ≅ G/Ntype
1. C3×C5×#538C2semidirect
2. C3×#65335C2semidirect
3. C3×#65325C2semidirect
4. C3×#65249C2semidirect
5. C3×#64758C2semidirect
6. C3×#64757C2semidirect
7. C3×#64745C2semidirect
8. #10602677C3direct (N × H)
29. C5×#538C2×C3semidirect
30. #65335C2×C3semidirect
31. #65325C2×C3semidirect
32. #65249C2×C3semidirect
33. #64758C2×C3semidirect
34. #64757C2×C3semidirect
35. #64745C2×C3semidirect
38. C2×C3×C5×Q8D4semidirect
42. C2×C3×C5×D4D4semidirect
55. C3×C5×C4⋊C4D4semidirect
56. C3×C5×C4⋊C4D4semidirect
70. C3×#538D5semidirect
157. C2×C5×Q8C3×D4semidirect
161. C2×C5×D4C3×D4semidirect
174. C5×C4⋊C4C3×D4semidirect
175. C5×C4⋊C4C3×D4semidirect
189. #538C3×D5semidirect
225. C2×C3×Q8C22⋊D5semidirect
235. C2×C3×D4C22⋊D5semidirect
246. C3×C4⋊C4C22⋊D5semidirect
247. C3×C4⋊C4C22⋊D5semidirect
391. C2×Q8C3×C22⋊D5semidirect
401. C2×D4C3×C22⋊D5semidirect
412. C4⋊C4C3×C22⋊D5semidirect
413. C4⋊C4C3×C22⋊D5semidirect
423. C3×C5#1600semidirect
506. C5C3×#1600semidirect
534. C3#10602677direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5382
C3×#653352
C3×#653252
C3×#652492
C3×#647582
C3×#647572
C3×#647452
#106026773
C3×#16005