Finite Group Databasebeta

← C3×#10599463 #14299947

Subgroups — C3×#10599463 #14299947

Order 1920.

724 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D20C211yesmaximal, characteristic
2C2×C3×#3953C211yesmaximal, characteristic
3C2×C3×#3953C211yesmaximal, characteristic
4C2×C3×#3950C211yesmaximal, characteristic, complemented
5C3×#65217C211yesmaximal, characteristic, complemented
6C3×C5×#336C211yesmaximal, characteristic, complemented
7C3×#64771C211yesmaximal, characteristic, complemented
8#10599463C311yesmaximal, characteristic, direct product factor, Hall
9C22×C3×D20C2211yescharacteristic
10C22×C3×C4×D5—21no—
11C22×C3×C5×C8C2211yescharacteristic
12C22×C3×C5×C8C2211yescharacteristic
13C2×C3×C42×C5C2211yescharacteristic
14C2×C3×#685—21no—
15C2×C3×C4⋊Dic5C2211yescharacteristic
16C2×C3×C4⋊Dic5C2211yescharacteristic
17C2×C3×C4⋊Dic5C2211yescharacteristic
18C2×C3×#683—21no—
19C3×C4×D20—22no—
20C3×C4×D20—22no—
21C3×#3963—21no—
22C3×#3953—22no—
23C3×#3953C414yescomplemented
24C3×#3950—22no—
25C3×#1862—51nomaximal, Hall
26C2×C4×D20C2×C311yescharacteristic
27C2×#3953C2×C311yescharacteristic
28C2×#3953C2×C311yescharacteristic
29C2×#3950C2×C311yescharacteristic, complemented
30#65217C2×C311yescharacteristic, complemented
31C5×#336C2×C311yescharacteristic, complemented
32#64771C2×C311yescharacteristic, complemented
33C23×C3×D5—21no—
34C22×C3×C4×C5D411yescharacteristic
35C22×C3×C4×C5D411yescharacteristic
36C22×C3×C4×C5C2311yescharacteristic
37C22×C3×Dic5—21no—
38C22×C3×Dic5—21no—
39C22×C3×Dic5—21no—
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—
52C2×C3×C5×C8—21no—
53C2×C3×C5×C8—21no—
54C2×C3×C5×C8C2×C412yescomplemented
55C3×C42×C5—21no—
56C3×C42×C5—21no—
57C3×#685—24no—
58C3×C4⋊Dic5—21no—
59C3×C4⋊Dic5—21no—
60C3×C4⋊Dic5—21no—
61C3×C4⋊Dic5—21no—
62C3×C4⋊Dic5C2×C412yescomplemented
63C3×C4⋊Dic5D412yes—
64C3×#683—42no—
65C2×C3×C4×D4—51no—
66C2×C3×#107—51no—
67C2×C3×#102—51no—
68C2×C3×#102—51no—
69C3×#389—51no—
70C3×#340—51no—
71C3×#336D511yescharacteristic, complemented
72C22×D20C22×C311yescharacteristic
73C22×C4×D5—21no—
74C22×C5×C8C22×C311yescharacteristic
75C22×C5×C8C22×C311yescharacteristic
76C2×C42×C5C22×C311yescharacteristic
77C2×#685—21no—
78C2×C4⋊Dic5C22×C311yescharacteristic
79C2×C4⋊Dic5C22×C311yescharacteristic
80C2×C4⋊Dic5C22×C311yescharacteristic
81C2×#683—21no—
82C4×D20—22no—
83C4×D20—22no—
84#3963—21no—
85#3953—22no—
86#3953C3×C414yescomplemented
87#3950—22no—
88#1862—51noSylow
89C23×C3×C5C2×D411yescharacteristic
90C22×C3×D5—22no—
91C22×C3×D5—22no—
92C22×C3×D5—22no—
93C22×C3×D5—41no—
94C22×C3×D5—41no—
95C22×C3×D5—41no—
96C22×C3×D5—41no—
97C2×C3×C4×C5—21no—
98C2×C3×C4×C5—21no—
99C2×C3×C4×C5—21no—
100C2×C3×C4×C5—21no—
101C2×C3×C4×C5—21no—
102C2×C3×C4×C5D812yes—
103C2×C3×C4×C5C22×C411yescharacteristic
104C2×C3×C4×C5C2×D411yescharacteristic
105C2×C3×C4×C5C2×D411yescharacteristic
106C2×C3×C4×C5C2×Q811yescharacteristic
107C2×C3×C4×C5#4111yescharacteristic
108C2×C3×C4×C5#4111yescharacteristic
109C2×C3×Dic5—22no—
110C2×C3×Dic5—22no—
111C2×C3×Dic5—41no—
112C2×C3×Dic5—41no—
113C2×C3×Dic5—41no—
114C2×C3×Dic5—41no—
115C2×C3×Dic5—41no—
116C2×C3×Dic5—41no—
117C2×C3×Dic5—41no—
118C3×D20—22no—
119C3×D20—22no—
120C3×D20—22no—
121C3×D20C4⋊C414yescomplemented
122C3×C4×D5—48no—
123C3×C5×C8—22no—
124C3×C5×C8—41no—
125C22×C3×D4—51no—
126C23×C3×C4—101no—
127C22×C3×C8C2×D511yescharacteristic
128C22×C3×C8C2×D511yescharacteristic
129C3×C4×D4—102no—
130C3×C4×D4—102no—
131C2×C3×C4⋊C4—51no—
132C2×C3×C4⋊C4—51no—
133C2×C3×C4⋊C4—51no—
134C2×C3×C4⋊C4—101no—
135C2×C3×C22⋊C4—101no—
136C2×C3×C42C2×D511yescharacteristic
137C3×#107—102no—
138C3×#102—54no—
139C3×#102—102no—
140C3×#95—101no—
141C23×D5—21no—
142C22×C4×C5C3×D411yescharacteristic
143C22×C4×C5C3×D411yescharacteristic
144C22×C4×C5C23×C311yescharacteristic
145C22×Dic5—21no—
146C22×Dic5—21no—
147C22×Dic5—21no—
148C2×D20—21no—
149C2×D20—21no—
150C2×D20—21no—
151C2×D20C2×C3×C412yes—
152C2×D20C3×D412yes—
153C2×D20C3×Q812yes—
154C2×C4×D5—24no—
155C2×C4×D5—42no—
156C2×C4×D5—42no—
157C2×C5×C8—21no—
158C2×C5×C8—21no—
159C2×C5×C8—21no—
160C2×C5×C8—21no—
161C2×C5×C8—21no—
162C2×C5×C8C2×C3×C412yescomplemented
163C42×C5—21no—
164C42×C5—21no—
165#685—24no—
166C4⋊Dic5—21no—
167C4⋊Dic5—21no—
168C4⋊Dic5—21no—
169C4⋊Dic5—21no—
170C4⋊Dic5C2×C3×C412yescomplemented
171C4⋊Dic5C3×D412yes—
172#683—42no—
173C2×C4×D4—51no—
174C2×#107—51no—
175C2×#102—51no—
176C2×#102—51no—
177#389—51no—
178#340—51no—
179#336C3×D511yescharacteristic, complemented
180C22×C3×C5C4×D411yescharacteristic
181C22×C3×C5#12011yescharacteristic
182C22×C3×C5C22⋊Q811yescharacteristic
183C22×C3×C5C2×D811yescharacteristic
184C22×C3×C5#13511yescharacteristic
185C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
186C22×C3×C5#13711yescharacteristic
187C2×C3×D5—24no—
188C2×C3×D5—42no—
189C2×C3×D5—42no—
190C2×C3×D5—42no—
191C2×C3×D5—42no—
192C2×C3×D5—42no—
193C2×C3×D5—42no—
194C3×C4×C5—22no—
195C3×C4×C5—41no—
196C3×C4×C5C2×C4⋊C411yescharacteristic
197C3×C4×C5C4×D411yescharacteristic
198C3×C4×C5C22⋊Q811yescharacteristic
199C3×C4×C5#12311yescharacteristic
200C3×Dic5—42no—
201C3×Dic5—42no—
202C3×Dic5—81no—
203C24×C3—101no—
204C2×C3×D4—52no—
205C2×C3×D4—52no—
206C2×C3×D4—52no—
207C2×C3×D4—101no—
208C2×C3×D4—101no—
209C2×C3×D4—101no—
210C22×C3×C4—104no—
211C22×C3×C4—101no—
212C22×C3×C4—101no—
213C22×C3×C4—101no—
214C22×C3×C4—202no—
215C22×C3×C4—202no—
216C22×C3×C4D2011yescharacteristic
217C22×C3×C4D2011yescharacteristic
218C22×C3×C4C22×D511yescharacteristic
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8C4×D512yescomplemented
225C3×C4⋊C4—52no—
226C3×C4⋊C4—52no—
227C3×C4⋊C4—101no—
228C3×C4⋊C4—101no—
229C3×C4⋊C4—101no—
230C3×C4⋊C4—101no—
231C3×C4⋊C4—202no—
232C3×C22⋊C4—104no—
233C3×C42—21no—
234C3×C42—21no—
235C23×C5C2×C3×D411yescharacteristic
236C22×D5—22no—
237C22×D5—22no—
238C22×D5—22no—
239C22×D5—41no—
240C22×D5—41no—
241C22×D5—41no—
242C22×D5—41no—
243C2×C4×C5—21no—
244C2×C4×C5—21no—
245C2×C4×C5—21no—
246C2×C4×C5—21no—
247C2×C4×C5—21no—
248C2×C4×C5C3×D812yes—
249C2×C4×C5C22×C3×C411yescharacteristic
250C2×C4×C5C2×C3×D411yescharacteristic
251C2×C4×C5C2×C3×D411yescharacteristic
252C2×C4×C5C2×C3×Q811yescharacteristic
253C2×C4×C5C3×#4111yescharacteristic
254C2×C4×C5C3×#4111yescharacteristic
255C2×Dic5—22no—
256C2×Dic5—22no—
257C2×Dic5—41no—
258C2×Dic5—41no—
259C2×Dic5—41no—
260C2×Dic5—41no—
261C2×Dic5—41no—
262C2×Dic5—41no—
263C2×Dic5—41no—
264D20—22no—
265D20—22no—
266D20—22no—
267D20C3×C4⋊C414yescomplemented
268C4×D5—48no—
269C5×C8—22no—
270C5×C8—41no—
271C22×D4—51no—
272C23×C4—101no—
273C22×C8C2×C3×D511yescharacteristic
274C22×C8C2×C3×D511yescharacteristic
275C4×D4—102no—
276C4×D4—102no—
277C2×C4⋊C4—51no—
278C2×C4⋊C4—51no—
279C2×C4⋊C4—51no—
280C2×C4⋊C4—101no—
281C2×C22⋊C4—101no—
282C2×C42C2×C3×D511yescharacteristic
283#107—102no—
284#102—54no—
285#102—102no—
286#95—101no—
287C2×C3×C5#38511yescharacteristic
288C2×C3×C5C4×D811yescharacteristic
289C2×C3×C5#44011yescharacteristic
290C2×C3×C5#44711yescharacteristic
291C2×C3×C5#45211yescharacteristic
292C2×C3×C5#47411yescharacteristic
293C2×C3×C5#47811yescharacteristic
294C3×D5—44no—
295C23×C3—102no—
296C23×C3—102no—
297C23×C3—102no—
298C23×C3—201no—
299C23×C3—201no—
300C23×C3—201no—
301C23×C3—201no—
302C23×C3C2×D2011yescharacteristic
303C3×D4—54no—
304C3×D4—102no—
305C3×D4—102no—
306C3×D4—102no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—102no—
313C2×C3×C4—102no—
314C2×C3×C4—208no—
315C2×C3×C4—201no—
316C2×C3×C4—201no—
317C2×C3×C4—201no—
318C2×C3×C4—201no—
319C2×C3×C4—201no—
320C2×C3×C4—201no—
321C2×C3×C4—201no—
322C2×C3×C4D4012yes—
323C2×C3×C4C2×C4×D511yescharacteristic
324C2×C3×C4#70911yescharacteristic
325C2×C3×C4D4×D511yescharacteristic
326C2×C3×C4D4×D511yescharacteristic
327C2×C3×C4#71111yescharacteristic
328C2×C3×C4Q8×D511yescharacteristic
329C3×C8—22no—
330C3×C8—41no—
331C22×C5C3×C4×D411yescharacteristic
332C22×C5C3×#12011yescharacteristic
333C22×C5C3×C22⋊Q811yescharacteristic
334C22×C5C2×C3×D811yescharacteristic
335C22×C5C3×#13511yescharacteristic
336C22×C5C3×AGL(1,ℤ/8)11yescharacteristic
337C22×C5C3×#13711yescharacteristic
338C2×D5—24no—
339C2×D5—42no—
340C2×D5—42no—
341C2×D5—42no—
342C2×D5—42no—
343C2×D5—42no—
344C2×D5—42no—
345C4×C5—22no—
346C4×C5—41no—
347C4×C5C2×C3×C4⋊C411yescharacteristic
348C4×C5C3×C4×D411yescharacteristic
349C4×C5C3×C22⋊Q811yescharacteristic
350C4×C5C3×#12311yescharacteristic
351Dic5—42no—
352Dic5—42no—
353Dic5—81no—
354C24—101no—
355C2×D4—52no—
356C2×D4—52no—
357C2×D4—52no—
358C2×D4—101no—
359C2×D4—101no—
360C2×D4—101no—
361C22×C4—104no—
362C22×C4—101no—
363C22×C4—101no—
364C22×C4—101no—
365C22×C4—202no—
366C22×C4—202no—
367C22×C4C3×D2011yescharacteristic
368C22×C4C3×D2011yescharacteristic
369C22×C4C22×C3×D511yescharacteristic
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8C3×C4×D512yescomplemented
376C4⋊C4—52no—
377C4⋊C4—52no—
378C4⋊C4—101no—
379C4⋊C4—101no—
380C4⋊C4—101no—
381C4⋊C4—101no—
382C4⋊C4—202no—
383C22⋊C4—104no—
384C42—21no—
385C42—21no—
386C3×C5#186211yescharacteristic, complemented, Hall
387C22×C3—104no—
388C22×C3—202no—
389C22×C3—202no—
390C22×C3—202no—
391C22×C3—202no—
392C22×C3—202no—
393C22×C3—202no—
394C22×C3C4×D2011yescharacteristic
395C22×C3#402811yescharacteristic
396C22×C3#404311yescharacteristic
397C22×C3C2×D4011yescharacteristic
398C22×C3#405011yescharacteristic
399C22×C3#405411yescharacteristic
400C22×C3#405511yescharacteristic
401C3×C4—22no—
402C3×C4—41no—
403C3×C4—202no—
404C3×C4—202no—
405C3×C4—401no—
406C3×C4#402711yescharacteristic
407C3×C4#402911yescharacteristic
408C3×C4D5×C4⋊C411yescharacteristic
409C3×C4#404211yescharacteristic
410C2×C5C3×#38511yescharacteristic
411C2×C5C3×C4×D811yescharacteristic
412C2×C5C3×#44011yescharacteristic
413C2×C5C3×#44711yescharacteristic
414C2×C5C3×#45211yescharacteristic
415C2×C5C3×#47411yescharacteristic
416C2×C5C3×#47811yescharacteristic
417D5—44no—
418C23—102no—
419C23—102no—
420C23—102no—
421C23—201no—
422C23—201no—
423C23—201no—
424C23—201no—
425C23C2×C3×D2011yescharacteristic
426D4—54no—
427D4—102no—
428D4—102no—
429D4—102no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—102no—
436C2×C4—102no—
437C2×C4—208no—
438C2×C4—201no—
439C2×C4—201no—
440C2×C4—201no—
441C2×C4—201no—
442C2×C4—201no—
443C2×C4—201no—
444C2×C4—201no—
445C2×C4C3×D4012yes—
446C2×C4C2×C3×C4×D511yescharacteristic
447C2×C4C3×#70911yescharacteristic
448C2×C4C3×D4×D511yescharacteristic
449C2×C4C3×D4×D511yescharacteristic
450C2×C4C3×#71111yescharacteristic
451C2×C4C3×Q8×D511yescharacteristic
452C8—22no—
453C8—41no—
454C2×C3—204no—
455C2×C3#6495711yescharacteristic
456C2×C3C4×D4011yescharacteristic
457C2×C3#6499911yescharacteristic
458C2×C3#6502111yescharacteristic
459C2×C3#6502211yescharacteristic
460C2×C3#6513511yescharacteristic
461C2×C3#6513611yescharacteristic
462C5C3×#186211yescharacteristic, complemented, Sylow
463C22—104no—
464C22—202no—
465C22—202no—
466C22—202no—
467C22—202no—
468C22—202no—
469C22—202no—
470C22C3×C4×D2011yescharacteristic
471C22C3×#402811yescharacteristic
472C22C3×#404311yescharacteristic
473C22C2×C3×D4011yescharacteristic
474C22C3×#405011yescharacteristic
475C22C3×#405411yescharacteristic
476C22C3×#405511yescharacteristic
477C4—22no—
478C4—41no—
479C4—202no—
480C4—202no—
481C4—401no—
482C4C3×#402711yescharacteristic
483C4C3×#402911yescharacteristic
484C4C3×D5×C4⋊C411yescharacteristic
485C4C3×#404211yescharacteristic
486C3#1059946311yescharacteristic, direct product factor, Sylow
487C2—204no—
488C2C3×#6495711yescharacteristic
489C2C3×C4×D4011yescharacteristic
490C2C3×#6499911yescharacteristic
491C2C3×#6502111yescharacteristic
492C2C3×#6502211yescharacteristic
493C2C3×#6513511yescharacteristic
494C2C3×#6513611yescharacteristic
subgroupG/H
centerC23×C3C2×D20
commutatorC2×C4×C5C22×C3×C4
FrattiniC22×C4C22×C3×D5
FittingC3×C5×#336C2
socleC23×C3×C5C2×D4
prime radicalC22×C3×D20C22
cosocleC22×C4×C5C23×C3
hypercenterC3×#336D5
hypocenterC5C3×#1862
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C3×#336

Complements

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

NH ≅ G/Ntype
4. C2×C3×#3950C2semidirect
5. C3×#65217C2semidirect
6. C3×C5×#336C2semidirect
7. C3×#64771C2semidirect
8. #10599463C3direct (N × H)
23. C3×#3953C4semidirect
29. C2×#3950C2×C3semidirect
30. #65217C2×C3semidirect
31. C5×#336C2×C3semidirect
32. #64771C2×C3semidirect
54. C2×C3×C5×C8C2×C4semidirect
62. C3×C4⋊Dic5C2×C4semidirect
71. C3×#336D5semidirect
86. #3953C3×C4semidirect
121. C3×D20C4⋊C4semidirect
162. C2×C5×C8C2×C3×C4semidirect
170. C4⋊Dic5C2×C3×C4semidirect
179. #336C3×D5semidirect
224. C2×C3×C8C4×D5semidirect
267. D20C3×C4⋊C4semidirect
375. C2×C8C3×C4×D5semidirect
386. C3×C5#1862semidirect
462. C5C3×#1862semidirect
486. C3#10599463direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×D202
C2×C3×#39532
C2×C3×#39502
C3×#652172
C3×C5×#3362
C3×#647712
#105994633
C3×#18625