Finite Group Databasebeta

← C3×#10601107 #14302398

Subgroups — C3×#10601107 #14302398

Order 1920.

692 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66132C211yesmaximal, characteristic, complemented
2C2×C3×#4078C211yesmaximal, characteristic
3C3×C5×#418C211yesmaximal, characteristic, complemented
4C3×#65428C211yesmaximal, characteristic, complemented
5C3×#65295C211yesmaximal, characteristic, complemented
6C3×#64779C211yesmaximal, characteristic, complemented
7C3×#64751C211yesmaximal, characteristic, complemented
8#10601107C311yesmaximal, characteristic, direct product factor, Hall
9C22×C3×C5×D4C2211yescharacteristic
10C2×C3×#709C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12C3×C5×#117C2211yescharacteristic
13C2×C3×C22⋊Dic5—22no—
14C3×D4×Dic5—22no—
15C3×D4×Dic5—22no—
16C2×C3×#687C412yescomplemented
17C3×#4078—22no—
18C3×#4078—22no—
19C2×C3×#686C412yescomplemented
20C3×#4072C2211yescharacteristic
21C3×#4072C2211yescharacteristic
22C2×C3×#681C2211yescharacteristic
23C3×C5×#102—22no—
24C3×#3957—22no—
25C3×#3938—22no—
26C3×#1904—51nomaximal, Hall
27#66132C2×C311yescharacteristic, complemented
28C2×#4078C2×C311yescharacteristic
29C5×#418C2×C311yescharacteristic, complemented
30#65428C2×C311yescharacteristic, complemented
31#65295C2×C311yescharacteristic, complemented
32#64779C2×C311yescharacteristic, complemented
33#64751C2×C311yescharacteristic, complemented
34C24×C3×C5—21no—
35C2×C3×C5×D4—21no—
36C2×C3×C5×D4—21no—
37C2×C3×C5×D4—21no—
38C2×C3×C5×D4C2×C412yes—
39C2×C3×C5×D4D412yes—
40C2×C3×C5×D4D412yes—
41C22×C3×C4×C5C2311yescharacteristic
42C2×C3×C22⋊D5—21no—
43C22×C3×Dic5—24no—
44C3×#709—21no—
45C3×#709—21no—
46C3×#709—21no—
47C3×#709—21no—
48C2×C3×D20C2×C411yescharacteristic, complemented
49C2×C3×C4×D5—21no—
50C2×C3×Dic10C2×C411yescharacteristic, complemented
51C3×C5×M16—21no—
52C3×C5×M16—21no—
53C2×C3×C5×C8—21no—
54C3×C5×C4⋊C4—21no—
55C3×C5×C22⋊C4—21no—
56C3×C42×C5—21no—
57C3×C22⋊Dic5—24no—
58C3×C22⋊Dic5—21no—
59C3×C22⋊Dic5D412yes—
60C3×#687—24no—
61C3×#686—24no—
62C3×C4⋊Dic5—21no—
63C3×C4⋊Dic5—21no—
64C3×C4×Dic5—21no—
65C3×C4×Dic5D412yes—
66C3×#681—21no—
67C3×#681—21no—
68C2×C3×#168C2×C412yescomplemented
69C2×C3×AGL(1,ℤ/8)—51no—
70C3×#518—51no—
71C3×#421—51no—
72C3×#418D511yescharacteristic, complemented
73C3×#410—51no—
74C3×#344—51no—
75C3×H(3,ℤ/4)—51no—
76C22×C5×D4C22×C311yescharacteristic
77C2×#709C22×C311yescharacteristic
78C2×C5×M16C22×C311yescharacteristic
79C5×#117C22×C311yescharacteristic
80C2×C22⋊Dic5—22no—
81D4×Dic5—22no—
82D4×Dic5—22no—
83C2×#687C3×C412yescomplemented
84#4078—22no—
85#4078—22no—
86C2×#686C3×C412yescomplemented
87#4072C22×C311yescharacteristic
88#4072C22×C311yescharacteristic
89C2×#681C22×C311yescharacteristic
90C5×#102—22no—
91#3957—22no—
92#3938—22no—
93#1904—51noSylow
94C23×C3×C5—22no—
95C23×C3×C5—22no—
96C23×C3×C5—22no—
97C23×C3×C5—82no—
98C23×C3×C5C2×D411yescharacteristic
99C22×C3×D5—21no—
100C3×C5×D4—22no—
101C3×C5×D4—22no—
102C3×C5×D4—22no—
103C3×C5×D4C22⋊C414yes—
104C2×C3×C4×C5—41no—
105C2×C3×C4×C5C22×C411yescharacteristic
106C2×C3×C4×C5C2×D411yescharacteristic
107C2×C3×C4×C5C2×D411yescharacteristic
108C2×C3×C4×C5C2×D411yescharacteristic
109C2×C3×C4×C5#4111yescharacteristic
110C2×C3×C4×C5#4111yescharacteristic
111C3×C22⋊D5—41no—
112C3×C22⋊D5—41no—
113C2×C3×Dic5—21no—
114C2×C3×Dic5—22no—
115C2×C3×Dic5—22no—
116C2×C3×Dic5—84no—
117C3×D20—21no—
118C3×D20—21no—
119C3×C4×D5—41no—
120C3×C4×D5—41no—
121C3×Dic10—21no—
122C3×Dic10—21no—
123C3×C5×C8—41no—
124C3×#168—22no—
125C2×C3×#41—51no—
126C22×C3×D4C2×D511yescharacteristic
127C3×AGL(1,ℤ/8)—102no—
128C3×AGL(1,ℤ/8)—102no—
129C2×C3×SD16—52no—
130C2×C3×D8—52no—
131C2×C3×M16—51no—
132C2×C3×M16C2×D511yescharacteristic
133C3×C4×D4—102no—
134C3×C4×D4—102no—
135C3×#117—51no—
136C3×#117—51no—
137C3×#117C2×D511yescharacteristic
138C2×C3×C22⋊C4—102no—
139C3×#104—102no—
140C3×#102—22no—
141C3×#99—102no—
142C24×C5—21no—
143C2×C5×D4—21no—
144C2×C5×D4—21no—
145C2×C5×D4—21no—
146C2×C5×D4C2×C3×C412yes—
147C2×C5×D4C3×D412yes—
148C2×C5×D4C3×D412yes—
149C22×C4×C5C23×C311yescharacteristic
150C2×C22⋊D5—21no—
151C22×Dic5—24no—
152#709—21no—
153#709—21no—
154#709—21no—
155#709—21no—
156C2×D20C2×C3×C411yescharacteristic, complemented
157C2×C4×D5—21no—
158C2×Dic10C2×C3×C411yescharacteristic, complemented
159C5×M16—21no—
160C5×M16—21no—
161C2×C5×C8—21no—
162C5×C4⋊C4—21no—
163C5×C22⋊C4—21no—
164C42×C5—21no—
165C22⋊Dic5—24no—
166C22⋊Dic5—21no—
167C22⋊Dic5C3×D412yes—
168#687—24no—
169#686—24no—
170C4⋊Dic5—21no—
171C4⋊Dic5—21no—
172C4×Dic5—21no—
173C4×Dic5C3×D412yes—
174#681—21no—
175#681—21no—
176C2×#168C2×C3×C412yescomplemented
177C2×AGL(1,ℤ/8)—51no—
178#518—51no—
179#421—51no—
180#418C3×D511yescharacteristic, complemented
181#410—51no—
182#344—51no—
183H(3,ℤ/4)—51no—
184C22×C3×C5—24no—
185C22×C3×C5—41no—
186C22×C3×C5—44no—
187C22×C3×C5—82no—
188C22×C3×C5—82no—
189C22×C3×C5C4×D411yescharacteristic
190C22×C3×C5#12011yescharacteristic
191C22×C3×C5#12111yescharacteristic
192C2×C3×D5—41no—
193C2×C3×D5—81no—
194C3×C4×C5—81no—
195C3×C4×C5C2×C22⋊C411yescharacteristic
196C3×C4×C5C4×D411yescharacteristic
197C3×C4×C5#12111yescharacteristic
198C3×C4×C5#12311yescharacteristic
199C3×Dic5—42no—
200C3×Dic5—42no—
201C3×Dic5—41no—
202C24×C3—21no—
203C3×#41—101no—
204C3×#41—101no—
205C3×#41—101no—
206C3×#41—101no—
207C2×C3×Q8—51no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4—51no—
212C2×C3×D4—101no—
213C2×C3×D4C4×D512yes—
214C2×C3×D4D2012yes—
215C2×C3×D4C22⋊D512yes—
216C22×C3×C4—104no—
217C22×C3×C4—101no—
218C22×C3×C4C22×D511yescharacteristic
219C3×SD16—104no—
220C3×D8—104no—
221C3×M16—21no—
222C3×M16—21no—
223C3×M16—101no—
224C3×M16—101no—
225C2×C3×C8—21no—
226C2×C3×C8—52no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—101no—
229C3×C4⋊C4—101no—
230C3×C22⋊C4—21no—
231C3×C22⋊C4—52no—
232C3×C22⋊C4—104no—
233C3×C22⋊C4—101no—
234C3×C42—21no—
235C3×C42—52no—
236C3×C42—101no—
237C23×C5—22no—
238C23×C5—22no—
239C23×C5—22no—
240C23×C5—82no—
241C23×C5C2×C3×D411yescharacteristic
242C22×D5—21no—
243C5×D4—22no—
244C5×D4—22no—
245C5×D4—22no—
246C5×D4C3×C22⋊C414yes—
247C2×C4×C5—41no—
248C2×C4×C5C22×C3×C411yescharacteristic
249C2×C4×C5C2×C3×D411yescharacteristic
250C2×C4×C5C2×C3×D411yescharacteristic
251C2×C4×C5C2×C3×D411yescharacteristic
252C2×C4×C5C3×#4111yescharacteristic
253C2×C4×C5C3×#4111yescharacteristic
254C22⋊D5—41no—
255C22⋊D5—41no—
256C2×Dic5—21no—
257C2×Dic5—22no—
258C2×Dic5—22no—
259C2×Dic5—84no—
260D20—21no—
261D20—21no—
262C4×D5—41no—
263C4×D5—41no—
264Dic10—21no—
265Dic10—21no—
266C5×C8—41no—
267#168—22no—
268C2×#41—51no—
269C22×D4C2×C3×D511yescharacteristic
270AGL(1,ℤ/8)—102no—
271AGL(1,ℤ/8)—102no—
272C2×SD16—52no—
273C2×D8—52no—
274C2×M16—51no—
275C2×M16C2×C3×D511yescharacteristic
276C4×D4—102no—
277C4×D4—102no—
278#117—51no—
279#117—51no—
280#117C2×C3×D511yescharacteristic
281C2×C22⋊C4—102no—
282#104—102no—
283#102—22no—
284#99—102no—
285C2×C3×C5—21no—
286C2×C3×C5—21no—
287C2×C3×C5—21no—
288C2×C3×C5—44no—
289C2×C3×C5#38611yescharacteristic
290C3×D5—81no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—82no—
295C23×C3—101no—
296C23×C3D4×D511yescharacteristic
297C3×Q8—101no—
298C3×Q8—101no—
299C3×D4—22no—
300C3×D4—22no—
301C3×D4—22no—
302C3×D4—101no—
303C3×D4—101no—
304C3×D4—201no—
305C3×D4—201no—
306C3×D4#68514yes—
307C2×C3×C4—41no—
308C2×C3×C4—101no—
309C2×C3×C4—102no—
310C2×C3×C4—102no—
311C2×C3×C4—201no—
312C2×C3×C4—201no—
313C2×C3×C4—404no—
314C2×C3×C4C2×C4×D511yescharacteristic
315C2×C3×C4C2×D2011yescharacteristic
316C2×C3×C4D4×D511yescharacteristic
317C2×C3×C4#71111yescharacteristic
318C2×C3×C4#71111yescharacteristic
319C2×C3×C4C2×C22⋊D511yescharacteristic
320C3×C8—41no—
321C3×C8—102no—
322C22×C5—24no—
323C22×C5—41no—
324C22×C5—44no—
325C22×C5—82no—
326C22×C5—82no—
327C22×C5C3×C4×D411yescharacteristic
328C22×C5C3×#12011yescharacteristic
329C22×C5C3×#12111yescharacteristic
330C2×D5—41no—
331C2×D5—81no—
332C4×C5—81no—
333C4×C5C2×C3×C22⋊C411yescharacteristic
334C4×C5C3×C4×D411yescharacteristic
335C4×C5C3×#12111yescharacteristic
336C4×C5C3×#12311yescharacteristic
337Dic5—42no—
338Dic5—42no—
339Dic5—41no—
340C24—21no—
341#41—101no—
342#41—101no—
343#41—101no—
344#41—101no—
345C2×Q8—51no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—51no—
350C2×D4—101no—
351C2×D4C3×C4×D512yes—
352C2×D4C3×D2012yes—
353C2×D4C3×C22⋊D512yes—
354C22×C4—104no—
355C22×C4—101no—
356C22×C4C22×C3×D511yescharacteristic
357SD16—104no—
358D8—104no—
359M16—21no—
360M16—21no—
361M16—101no—
362M16—101no—
363C2×C8—21no—
364C2×C8—52no—
365C4⋊C4—21no—
366C4⋊C4—101no—
367C4⋊C4—101no—
368C22⋊C4—21no—
369C22⋊C4—52no—
370C22⋊C4—104no—
371C22⋊C4—101no—
372C42—21no—
373C42—52no—
374C42—101no—
375C3×C5#190411yescharacteristic, complemented, Hall
376C22×C3—24no—
377C22×C3—41no—
378C22×C3—44no—
379C22×C3—82no—
380C22×C3—82no—
381C22×C3—201no—
382C22×C3—401no—
383C22×C3#402711yescharacteristic
384C22×C3#402811yescharacteristic
385C22×C3#408511yescharacteristic
386C3×C4—81no—
387C3×C4—202no—
388C3×C4—202no—
389C3×C4—201no—
390C3×C4#402711yescharacteristic
391C3×C4#403211yescharacteristic
392C3×C4C2×#68511yescharacteristic
393C3×C4#408511yescharacteristic
394C2×C5—21no—
395C2×C5—21no—
396C2×C5—21no—
397C2×C5—44no—
398C2×C5C3×#38611yescharacteristic
399D5—81no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—82no—
404C23—101no—
405C23C3×D4×D511yescharacteristic
406Q8—101no—
407Q8—101no—
408D4—22no—
409D4—22no—
410D4—22no—
411D4—101no—
412D4—101no—
413D4—201no—
414D4—201no—
415D4C3×#68514yes—
416C2×C4—41no—
417C2×C4—101no—
418C2×C4—102no—
419C2×C4—102no—
420C2×C4—201no—
421C2×C4—201no—
422C2×C4—404no—
423C2×C4C2×C3×C4×D511yescharacteristic
424C2×C4C2×C3×D2011yescharacteristic
425C2×C4C3×D4×D511yescharacteristic
426C2×C4C3×#71111yescharacteristic
427C2×C4C3×#71111yescharacteristic
428C2×C4C2×C3×C22⋊D511yescharacteristic
429C8—41no—
430C8—102no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—44no—
435C2×C3—401no—
436C2×C3#6524711yescharacteristic
437C5C3×#190411yescharacteristic, complemented, Sylow
438C22—24no—
439C22—41no—
440C22—44no—
441C22—82no—
442C22—82no—
443C22—201no—
444C22—401no—
445C22C3×#402711yescharacteristic
446C22C3×#402811yescharacteristic
447C22C3×#408511yescharacteristic
448C4—81no—
449C4—202no—
450C4—202no—
451C4—201no—
452C4C3×#402711yescharacteristic
453C4C3×#403211yescharacteristic
454C4C2×C3×#68511yescharacteristic
455C4C3×#408511yescharacteristic
456C3#1060110711yescharacteristic, direct product factor, Sylow
457C2—21no—
458C2—21no—
459C2—21no—
460C2—44no—
461C2—401no—
462C2C3×#6524711yescharacteristic
subgroupG/H
centerC2×C3#65247
commutatorC2×C4×C5C22×C3×C4
FrattiniC22×C4C22×C3×D5
FittingC3×C5×#418C2
socleC2×C3×C5#386
prime radicalC2×C3×#4078C2
cosocleC22×C4×C5C23×C3
hypercenterC3×#418D5
hypocenterC5C3×#1904
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#418

Complements

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

NH ≅ G/Ntype
1. C3×#66132C2semidirect
3. C3×C5×#418C2semidirect
4. C3×#65428C2semidirect
5. C3×#65295C2semidirect
6. C3×#64779C2semidirect
7. C3×#64751C2semidirect
8. #10601107C3direct (N × H)
16. C2×C3×#687C4semidirect
19. C2×C3×#686C4semidirect
27. #66132C2×C3semidirect
29. C5×#418C2×C3semidirect
30. #65428C2×C3semidirect
31. #65295C2×C3semidirect
32. #64779C2×C3semidirect
33. #64751C2×C3semidirect
48. C2×C3×D20C2×C4semidirect
50. C2×C3×Dic10C2×C4semidirect
68. C2×C3×#168C2×C4semidirect
72. C3×#418D5semidirect
83. C2×#687C3×C4semidirect
86. C2×#686C3×C4semidirect
156. C2×D20C2×C3×C4semidirect
158. C2×Dic10C2×C3×C4semidirect
176. C2×#168C2×C3×C4semidirect
180. #418C3×D5semidirect
375. C3×C5#1904semidirect
437. C5C3×#1904semidirect
456. C3#10601107direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#661322
C2×C3×#40782
C3×C5×#4182
C3×#654282
C3×#652952
C3×#647792
C3×#647512
#106011073
C3×#19045