Finite Group Databasebeta

← C3×#10599685 #14297631

Subgroups — C3×#10599685 #14297631

Order 1920.

744 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4039C211yesmaximal, characteristic
2C2×C3×#3953C211yesmaximal, characteristic
3C2×C3×#3946C211yesmaximal, characteristic, complemented
4C3×#65265C211yesmaximal, characteristic, complemented
5C2×C3×#3941C211yesmaximal, characteristic
6C3×C5×#340C211yesmaximal, characteristic, complemented
7C3×#64749C211yesmaximal, characteristic, complemented
8#10599685C311yesmaximal, characteristic, direct product factor, Hall
9C22×C3×D20C2211yescharacteristic
10C22×C3×C4×D5—21no—
11C22×C3×C5×C8C2211yescharacteristic
12C2×C3×C5×C4⋊C4C2211yescharacteristic
13C2×C3×C5×C4⋊C4C2211yescharacteristic
14C2×C3×#685—21no—
15C2×C3×C4⋊Dic5C2211yescharacteristic
16C2×C3×#683—21no—
17C2×C3×#683—21no—
18C2×C3×C4×Dic5C2211yescharacteristic
19C22×C3×#168C2211yescharacteristic
20C3×#4039—22no—
21C3×#4039C414yescomplemented
22C3×#3953—22no—
23C3×#3946—22no—
24C3×#3941—22no—
25C3×#1802—51nomaximal, Hall
26C2×#4039C2×C311yescharacteristic
27C2×#3953C2×C311yescharacteristic
28C2×#3946C2×C311yescharacteristic, complemented
29#65265C2×C311yescharacteristic, complemented
30C2×#3941C2×C311yescharacteristic
31C5×#340C2×C311yescharacteristic, complemented
32#64749C2×C311yescharacteristic, complemented
33C23×C3×D5—21no—
34C22×C3×C4×C5—21no—
35C22×C3×C4×C5—21no—
36C22×C3×C4×C5C2311yescharacteristic
37C22×C3×Dic5—21no—
38C22×C3×Dic5D411yescharacteristic
39C22×C3×Dic5D411yescharacteristic
40C2×C3×D20—21no—
41C2×C3×D20—21no—
42C2×C3×D20—21no—
43C2×C3×D20C2×C412yes—
44C2×C3×D20D412yes—
45C2×C3×D20Q812yes—
46C2×C3×C4×D5—24no—
47C2×C3×C4×D5—42no—
48C2×C3×C4×D5—42no—
49C2×C3×C5×C8—21no—
50C2×C3×C5×C8—21no—
51C2×C3×C5×C8—21no—
52C3×C5×C4⋊C4—21no—
53C3×C5×C4⋊C4—21no—
54C3×C5×C4⋊C4—21no—
55C3×C5×C4⋊C4C2×C412yes—
56C3×#685—24no—
57C3×C4⋊Dic5—21no—
58C3×C4⋊Dic5—21no—
59C3×#683—42no—
60C3×#683—42no—
61C3×C4×Dic5C2×C412yes—
62C3×C4×Dic5D412yes—
63C2×C3×#168—21no—
64C2×C3×#168—21no—
65C2×C3×#168—21no—
66C2×C3×C4×D4—51no—
67C2×C3×C4⋊C8—51no—
68C2×C3×#102—51no—
69C2×C3×#102—51no—
70C3×#384—51no—
71C3×#340—51no—
72C3×#340D511yescharacteristic, complemented
73C22×D20C22×C311yescharacteristic
74C22×C4×D5—21no—
75C22×C5×C8C22×C311yescharacteristic
76C2×C5×C4⋊C4C22×C311yescharacteristic
77C2×C5×C4⋊C4C22×C311yescharacteristic
78C2×#685—21no—
79C2×C4⋊Dic5C22×C311yescharacteristic
80C2×#683—21no—
81C2×#683—21no—
82C2×C4×Dic5C22×C311yescharacteristic
83C22×#168C22×C311yescharacteristic
84#4039—22no—
85#4039C3×C414yescomplemented
86#3953—22no—
87#3946—22no—
88#3941—22no—
89#1802—51noSylow
90C23×C3×C5C2×D411yescharacteristic
91C22×C3×D5—22no—
92C22×C3×D5—22no—
93C22×C3×D5—22no—
94C22×C3×D5—41no—
95C22×C3×D5—41no—
96C22×C3×D5—41no—
97C22×C3×D5—41no—
98C2×C3×C4×C5—22no—
99C2×C3×C4×C5—41no—
100C2×C3×C4×C5—41no—
101C2×C3×C4×C5—41no—
102C2×C3×C4×C5—41no—
103C2×C3×C4×C5—41no—
104C2×C3×C4×C5C22×C411yescharacteristic
105C2×C3×C4×C5C2×D411yescharacteristic
106C2×C3×C4×C5C2×D411yescharacteristic
107C2×C3×C4×C5C2×Q811yescharacteristic
108C2×C3×C4×C5#4111yescharacteristic
109C2×C3×C4×C5#4111yescharacteristic
110C2×C3×Dic5—21no—
111C2×C3×Dic5—21no—
112C2×C3×Dic5—41no—
113C2×C3×Dic5—41no—
114C2×C3×Dic5—41no—
115C2×C3×Dic5C22⋊C412yes—
116C2×C3×Dic5C22⋊C412yes—
117C2×C3×Dic5D812yes—
118C2×C3×Dic5SD1612yes—
119C3×D20—22no—
120C3×D20—22no—
121C3×D20—22no—
122C3×D20C4⋊C414yescomplemented
123C3×C4×D5—48no—
124C3×C5×C8—41no—
125C3×#168—41no—
126C22×C3×D4—51no—
127C23×C3×C4—101no—
128C22×C3×C8—51no—
129C22×C3×C8C2×D511yescharacteristic
130C3×C4×D4—54no—
131C3×C4×D4—102no—
132C2×C3×C4⋊C4—51no—
133C2×C3×C4⋊C4—101no—
134C2×C3×C4⋊C4—101no—
135C2×C3×C4⋊C4C2×D511yescharacteristic
136C2×C3×C4⋊C4C2×D511yescharacteristic
137C2×C3×C22⋊C4—101no—
138C2×C3×C42—51no—
139C3×C4⋊C8—102no—
140C3×#102—102no—
141C3×#102—102no—
142C23×D5—21no—
143C22×C4×C5—21no—
144C22×C4×C5—21no—
145C22×C4×C5C23×C311yescharacteristic
146C22×Dic5—21no—
147C22×Dic5C3×D411yescharacteristic
148C22×Dic5C3×D411yescharacteristic
149C2×D20—21no—
150C2×D20—21no—
151C2×D20—21no—
152C2×D20C2×C3×C412yes—
153C2×D20C3×D412yes—
154C2×D20C3×Q812yes—
155C2×C4×D5—24no—
156C2×C4×D5—42no—
157C2×C4×D5—42no—
158C2×C5×C8—21no—
159C2×C5×C8—21no—
160C2×C5×C8—21no—
161C5×C4⋊C4—21no—
162C5×C4⋊C4—21no—
163C5×C4⋊C4—21no—
164C5×C4⋊C4C2×C3×C412yes—
165#685—24no—
166C4⋊Dic5—21no—
167C4⋊Dic5—21no—
168#683—42no—
169#683—42no—
170C4×Dic5C2×C3×C412yes—
171C4×Dic5C3×D412yes—
172C2×#168—21no—
173C2×#168—21no—
174C2×#168—21no—
175C2×C4×D4—51no—
176C2×C4⋊C8—51no—
177C2×#102—51no—
178C2×#102—51no—
179#384—51no—
180#340—51no—
181#340C3×D511yescharacteristic, complemented
182C22×C3×C5C2×C22⋊C411yescharacteristic
183C22×C3×C5#12111yescharacteristic
184C22×C3×C5C22⋊Q811yescharacteristic
185C22×C3×C5C2×D811yescharacteristic
186C22×C3×C5C2×SD1611yescharacteristic
187C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
188C22×C3×C5#13711yescharacteristic
189C2×C3×D5—24no—
190C2×C3×D5—42no—
191C2×C3×D5—42no—
192C2×C3×D5—42no—
193C2×C3×D5—42no—
194C2×C3×D5—42no—
195C2×C3×D5—42no—
196C3×C4×C5—42no—
197C3×C4×C5—81no—
198C3×C4×C5C2×C4⋊C411yescharacteristic
199C3×C4×C5#11711yescharacteristic
200C3×C4×C5#12111yescharacteristic
201C3×C4×C5C22⋊Q811yescharacteristic
202C3×Dic5—22no—
203C3×Dic5—81no—
204C3×Dic5#10214yescomplemented
205C24×C3—101no—
206C2×C3×D4—52no—
207C2×C3×D4—52no—
208C2×C3×D4—52no—
209C2×C3×D4—101no—
210C2×C3×D4—101no—
211C2×C3×D4—101no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—51no—
215C22×C3×C4—51no—
216C22×C3×C4—104no—
217C22×C3×C4—101no—
218C22×C3×C4—202no—
219C22×C3×C4—202no—
220C22×C3×C4C22×D511yescharacteristic
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—101no—
225C2×C3×C8—101no—
226C2×C3×C8—101no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—101no—
231C3×C4⋊C4—101no—
232C3×C4⋊C4—202no—
233C3×C4⋊C4—202no—
234C3×C4⋊C4C4×D512yes—
235C3×C22⋊C4—104no—
236C3×C42—52no—
237C3×C42—52no—
238C23×C5C2×C3×D411yescharacteristic
239C22×D5—22no—
240C22×D5—22no—
241C22×D5—22no—
242C22×D5—41no—
243C22×D5—41no—
244C22×D5—41no—
245C22×D5—41no—
246C2×C4×C5—22no—
247C2×C4×C5—41no—
248C2×C4×C5—41no—
249C2×C4×C5—41no—
250C2×C4×C5—41no—
251C2×C4×C5—41no—
252C2×C4×C5C22×C3×C411yescharacteristic
253C2×C4×C5C2×C3×D411yescharacteristic
254C2×C4×C5C2×C3×D411yescharacteristic
255C2×C4×C5C2×C3×Q811yescharacteristic
256C2×C4×C5C3×#4111yescharacteristic
257C2×C4×C5C3×#4111yescharacteristic
258C2×Dic5—21no—
259C2×Dic5—21no—
260C2×Dic5—41no—
261C2×Dic5—41no—
262C2×Dic5—41no—
263C2×Dic5C3×C22⋊C412yes—
264C2×Dic5C3×C22⋊C412yes—
265C2×Dic5C3×D812yes—
266C2×Dic5C3×SD1612yes—
267D20—22no—
268D20—22no—
269D20—22no—
270D20C3×C4⋊C414yescomplemented
271C4×D5—48no—
272C5×C8—41no—
273#168—41no—
274C22×D4—51no—
275C23×C4—101no—
276C22×C8—51no—
277C22×C8C2×C3×D511yescharacteristic
278C4×D4—54no—
279C4×D4—102no—
280C2×C4⋊C4—51no—
281C2×C4⋊C4—101no—
282C2×C4⋊C4—101no—
283C2×C4⋊C4C2×C3×D511yescharacteristic
284C2×C4⋊C4C2×C3×D511yescharacteristic
285C2×C22⋊C4—101no—
286C2×C42—51no—
287C4⋊C8—102no—
288#102—102no—
289#102—102no—
290C2×C3×C5#38011yescharacteristic
291C2×C3×C5C2×#10211yescharacteristic
292C2×C3×C5#41711yescharacteristic
293C2×C3×C5#45911yescharacteristic
294C2×C3×C5#46111yescharacteristic
295C2×C3×C5#47411yescharacteristic
296C2×C3×C5#47611yescharacteristic
297C3×D5—44no—
298C23×C3—102no—
299C23×C3—102no—
300C23×C3—102no—
301C23×C3—201no—
302C23×C3—201no—
303C23×C3—201no—
304C23×C3—201no—
305C23×C3D4×D511yescharacteristic
306C3×D4—54no—
307C3×D4—102no—
308C3×D4—102no—
309C3×D4—102no—
310C2×C3×C4—22no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—52no—
317C2×C3×C4—52no—
318C2×C3×C4—52no—
319C2×C3×C4—52no—
320C2×C3×C4—101no—
321C2×C3×C4—101no—
322C2×C3×C4—208no—
323C2×C3×C4—201no—
324C2×C3×C4—201no—
325C2×C3×C4—201no—
326C2×C3×C4C2×C4×D511yescharacteristic
327C2×C3×C4#70911yescharacteristic
328C2×C3×C4#70911yescharacteristic
329C2×C3×C4D4×D511yescharacteristic
330C2×C3×C4D4×D511yescharacteristic
331C2×C3×C4Q8×D511yescharacteristic
332C3×C8—41no—
333C3×C8—201no—
334C22×C5C2×C3×C22⋊C411yescharacteristic
335C22×C5C3×#12111yescharacteristic
336C22×C5C3×C22⋊Q811yescharacteristic
337C22×C5C2×C3×D811yescharacteristic
338C22×C5C2×C3×SD1611yescharacteristic
339C22×C5C3×AGL(1,ℤ/8)11yescharacteristic
340C22×C5C3×#13711yescharacteristic
341C2×D5—24no—
342C2×D5—42no—
343C2×D5—42no—
344C2×D5—42no—
345C2×D5—42no—
346C2×D5—42no—
347C2×D5—42no—
348C4×C5—42no—
349C4×C5—81no—
350C4×C5C2×C3×C4⋊C411yescharacteristic
351C4×C5C3×#11711yescharacteristic
352C4×C5C3×#12111yescharacteristic
353C4×C5C3×C22⋊Q811yescharacteristic
354Dic5—22no—
355Dic5—81no—
356Dic5C3×#10214yescomplemented
357C24—101no—
358C2×D4—52no—
359C2×D4—52no—
360C2×D4—52no—
361C2×D4—101no—
362C2×D4—101no—
363C2×D4—101no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—51no—
367C22×C4—51no—
368C22×C4—104no—
369C22×C4—101no—
370C22×C4—202no—
371C22×C4—202no—
372C22×C4C22×C3×D511yescharacteristic
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—101no—
377C2×C8—101no—
378C2×C8—101no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—101no—
383C4⋊C4—101no—
384C4⋊C4—202no—
385C4⋊C4—202no—
386C4⋊C4C3×C4×D512yes—
387C22⋊C4—104no—
388C42—52no—
389C42—52no—
390C3×C5#180211yescharacteristic, complemented, Hall
391C22×C3—104no—
392C22×C3—202no—
393C22×C3—202no—
394C22×C3—202no—
395C22×C3—202no—
396C22×C3—202no—
397C22×C3—202no—
398C22×C3D5×C22⋊C411yescharacteristic
399C22×C3#403011yescharacteristic
400C22×C3#404211yescharacteristic
401C22×C3D5×D811yescharacteristic
402C22×C3D5×SD1611yescharacteristic
403C22×C3#406011yescharacteristic
404C22×C3#406411yescharacteristic
405C3×C4—42no—
406C3×C4—54no—
407C3×C4—81no—
408C3×C4—102no—
409C3×C4—401no—
410C3×C4#401811yescharacteristic
411C3×C4#403011yescharacteristic
412C3×C4D5×C4⋊C411yescharacteristic
413C3×C4#404211yescharacteristic
414C2×C5C3×#38011yescharacteristic
415C2×C5C2×C3×#10211yescharacteristic
416C2×C5C3×#41711yescharacteristic
417C2×C5C3×#45911yescharacteristic
418C2×C5C3×#46111yescharacteristic
419C2×C5C3×#47411yescharacteristic
420C2×C5C3×#47611yescharacteristic
421D5—44no—
422C23—102no—
423C23—102no—
424C23—102no—
425C23—201no—
426C23—201no—
427C23—201no—
428C23—201no—
429C23C3×D4×D511yescharacteristic
430D4—54no—
431D4—102no—
432D4—102no—
433D4—102no—
434C2×C4—22no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—52no—
441C2×C4—52no—
442C2×C4—52no—
443C2×C4—52no—
444C2×C4—101no—
445C2×C4—101no—
446C2×C4—208no—
447C2×C4—201no—
448C2×C4—201no—
449C2×C4—201no—
450C2×C4C2×C3×C4×D511yescharacteristic
451C2×C4C3×#70911yescharacteristic
452C2×C4C3×#70911yescharacteristic
453C2×C4C3×D4×D511yescharacteristic
454C2×C4C3×D4×D511yescharacteristic
455C2×C4C3×Q8×D511yescharacteristic
456C8—41no—
457C8—201no—
458C2×C3—204no—
459C2×C3#6495611yescharacteristic
460C2×C3D5×#10211yescharacteristic
461C2×C3#6507511yescharacteristic
462C2×C3#6509211yescharacteristic
463C2×C3#6510711yescharacteristic
464C2×C3#6515911yescharacteristic
465C2×C3#6517911yescharacteristic
466C5C3×#180211yescharacteristic, complemented, Sylow
467C22—104no—
468C22—202no—
469C22—202no—
470C22—202no—
471C22—202no—
472C22—202no—
473C22—202no—
474C22C3×D5×C22⋊C411yescharacteristic
475C22C3×#403011yescharacteristic
476C22C3×#404211yescharacteristic
477C22C3×D5×D811yescharacteristic
478C22C3×D5×SD1611yescharacteristic
479C22C3×#406011yescharacteristic
480C22C3×#406411yescharacteristic
481C4—42no—
482C4—54no—
483C4—81no—
484C4—102no—
485C4—401no—
486C4C3×#401811yescharacteristic
487C4C3×#403011yescharacteristic
488C4C3×D5×C4⋊C411yescharacteristic
489C4C3×#404211yescharacteristic
490C3#1059968511yescharacteristic, direct product factor, Sylow
491C2—204no—
492C2C3×#6495611yescharacteristic
493C2C3×D5×#10211yescharacteristic
494C2C3×#6507511yescharacteristic
495C2C3×#6509211yescharacteristic
496C2C3×#6510711yescharacteristic
497C2C3×#6515911yescharacteristic
498C2C3×#6517911yescharacteristic
subgroupG/H
centerC23×C3D4×D5
commutatorC2×C4×C5C22×C3×C4
FrattiniC22×C4C22×C3×D5
FittingC3×C5×#340C2
socleC23×C3×C5C2×D4
prime radicalC22×C3×D20C22
cosocleC22×C4×C5C23×C3
hypercenterC3×#340D5
hypocenterC5C3×#1802
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C3×#340

Complements

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

NH ≅ G/Ntype
3. C2×C3×#3946C2semidirect
4. C3×#65265C2semidirect
6. C3×C5×#340C2semidirect
7. C3×#64749C2semidirect
8. #10599685C3direct (N × H)
21. C3×#4039C4semidirect
28. C2×#3946C2×C3semidirect
29. #65265C2×C3semidirect
31. C5×#340C2×C3semidirect
32. #64749C2×C3semidirect
72. C3×#340D5semidirect
85. #4039C3×C4semidirect
122. C3×D20C4⋊C4semidirect
181. #340C3×D5semidirect
204. C3×Dic5#102semidirect
270. D20C3×C4⋊C4semidirect
356. Dic5C3×#102semidirect
390. C3×C5#1802semidirect
466. C5C3×#1802semidirect
490. C3#10599685direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40392
C2×C3×#39532
C2×C3×#39462
C3×#652652
C2×C3×#39412
C3×C5×#3402
C3×#647492
#105996853
C3×#18025