Finite Group Databasebeta

← C3×#10601467 #14299567

Subgroups — C3×#10601467 #14299567

Order 1920.

648 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65855C211yesmaximal, characteristic
2C3×C5×#428C211yesmaximal, characteristic, complemented
3C2×C3×#3955C211yesmaximal, characteristic
4C3×#65277C211yesmaximal, characteristic
5C3×#65253C211yesmaximal, characteristic, complemented
6C3×#64775C211yesmaximal, characteristic, complemented
7C3×#64752C211yesmaximal, characteristic, complemented
8#10601467C311yesmaximal, characteristic, direct product factor, Hall
9C22×C3×D20C2211yescharacteristic
10C2×C3×C5×M16C2211yescharacteristic
11C3×C5×#117C2211yescharacteristic
12C2×C3×C5×C4⋊C4C2211yescharacteristic
13C2×C3×#685—22no—
14C3×#4072C2211yescharacteristic
15C2×C3×C4×Dic5C2211yescharacteristic
16C2×C3×#681C2211yescharacteristic
17C3×#4039—21no—
18C3×#4039—21no—
19C3×D5×C22⋊C4—22no—
20C3×C4×D20—21no—
21C3×C4×D20—21no—
22C3×C5×#107—21no—
23C3×C5×#106—21no—
24C3×#3955C414yescomplemented
25C3×#3940—21no—
26C3×#3939—21no—
27C3×#1856—51nomaximal, Hall
28#65855C2×C311yescharacteristic
29C5×#428C2×C311yescharacteristic, complemented
30C2×#3955C2×C311yescharacteristic
31#65277C2×C311yescharacteristic
32#65253C2×C311yescharacteristic, complemented
33#64775C2×C311yescharacteristic, complemented
34#64752C2×C311yescharacteristic, complemented
35C23×C3×D5D412yes—
36C22×C3×C4×C5—21no—
37C22×C3×C4×C5C2311yescharacteristic
38C22×C3×Dic5—21no—
39C2×C3×D20—21no—
40C2×C3×D20—21no—
41C2×C3×D20—21no—
42C2×C3×D20—21no—
43C2×C3×D20C2×C412yes—
44C2×C3×D20Q812yes—
45C2×C3×C4×D5—22no—
46C2×C3×C4×D5—22no—
47C3×C5×M16—21no—
48C3×C5×M16C2×C412yescomplemented
49C2×C3×C5×C8—21no—
50C3×C5×C4⋊C4—21no—
51C3×C5×C4⋊C4—21no—
52C3×C5×C4⋊C4D411yescharacteristic
53C3×C5×C4⋊C4D411yescharacteristic
54C3×C5×C22⋊C4—21no—
55C3×C42×C5—21no—
56C3×C22⋊Dic5—21no—
57C3×#685—22no—
58C3×#685—22no—
59C3×#685—44no—
60C3×C4⋊Dic5D411yescharacteristic
61C3×C4⋊Dic5D411yescharacteristic
62C3×C4×Dic5—21no—
63C3×C4×Dic5—21no—
64C3×C4×Dic5—21no—
65C3×#681—21no—
66C3×#681C2×C412yescomplemented
67C2×C3×#168—21no—
68C3×#518—51no—
69C3×#428—51no—
70C3×#428D511yescharacteristic, complemented
71C2×C3×#100—51no—
72C3×#390—51no—
73C3×#339—51no—
74C3×#339—51no—
75C22×D20C22×C311yescharacteristic
76C2×C5×M16C22×C311yescharacteristic
77C5×#117C22×C311yescharacteristic
78C2×C5×C4⋊C4C22×C311yescharacteristic
79C2×#685—22no—
80#4072C22×C311yescharacteristic
81C2×C4×Dic5C22×C311yescharacteristic
82C2×#681C22×C311yescharacteristic
83#4039—21no—
84#4039—21no—
85D5×C22⋊C4—22no—
86C4×D20—21no—
87C4×D20—21no—
88C5×#107—21no—
89C5×#106—21no—
90#3955C3×C414yescomplemented
91#3940—21no—
92#3939—21no—
93#1856—51noSylow
94C23×C3×C5C2×D411yescharacteristic
95C22×C3×D5—22no—
96C22×C3×D5—22no—
97C22×C3×D5—42no—
98C22×C3×D5—42no—
99C22×C3×D5C4⋊C414yes—
100C2×C3×C4×C5—21no—
101C2×C3×C4×C5—21no—
102C2×C3×C4×C5—42no—
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—22no—
111C2×C3×Dic5—21no—
112C2×C3×Dic5—21no—
113C2×C3×Dic5—41no—
114C2×C3×Dic5—41no—
115C3×D20—22no—
116C3×D20—22no—
117C3×D20—22no—
118C3×D20—22no—
119C3×C4×D5—82no—
120C3×C4×D5—82no—
121C3×C5×C8—22no—
122C3×#168—22no—
123C22×C3×D4—51no—
124C2×C3×M16—51no—
125C2×C3×M16C2×D511yescharacteristic
126C3×C4×D4—101no—
127C3×C4×D4—101no—
128C3×C4×D4—101no—
129C3×C4×D4—101no—
130C3×#117—51no—
131C3×#117C2×D511yescharacteristic
132C2×C3×C4⋊C4C2×D511yescharacteristic
133C2×C3×C22⋊C4—102no—
134C2×C3×C22⋊C4—102no—
135C2×C3×C42—51no—
136C3×#107—21no—
137C3×#107—101no—
138C3×#106—21no—
139C3×#106—101no—
140C3×#100—54no—
141C23×D5C3×D412yes—
142C22×C4×C5—21no—
143C22×C4×C5C23×C311yescharacteristic
144C22×Dic5—21no—
145C2×D20—21no—
146C2×D20—21no—
147C2×D20—21no—
148C2×D20—21no—
149C2×D20C2×C3×C412yes—
150C2×D20C3×Q812yes—
151C2×C4×D5—22no—
152C2×C4×D5—22no—
153C5×M16—21no—
154C5×M16C2×C3×C412yescomplemented
155C2×C5×C8—21no—
156C5×C4⋊C4—21no—
157C5×C4⋊C4—21no—
158C5×C4⋊C4C3×D411yescharacteristic
159C5×C4⋊C4C3×D411yescharacteristic
160C5×C22⋊C4—21no—
161C42×C5—21no—
162C22⋊Dic5—21no—
163#685—22no—
164#685—22no—
165#685—44no—
166C4⋊Dic5C3×D411yescharacteristic
167C4⋊Dic5C3×D411yescharacteristic
168C4×Dic5—21no—
169C4×Dic5—21no—
170C4×Dic5—21no—
171#681—21no—
172#681C2×C3×C412yescomplemented
173C2×#168—21no—
174#518—51no—
175#428—51no—
176#428C3×D511yescharacteristic, complemented
177C2×#100—51no—
178#390—51no—
179#339—51no—
180#339—51no—
181C22×C3×C5—21no—
182C22×C3×C5—21no—
183C22×C3×C5C2×C4⋊C411yescharacteristic
184C22×C3×C5#12011yescharacteristic
185C22×C3×C5#12311yescharacteristic
186C2×C3×D5—24no—
187C2×C3×D5—42no—
188C2×C3×D5—42no—
189C2×C3×D5—44no—
190C2×C3×D5—82no—
191C3×C4×C5—41no—
192C3×C4×C5—41no—
193C3×C4×C5—81no—
194C3×C4×C5C4×D411yescharacteristic
195C3×C4×C5C4×D411yescharacteristic
196C3×C4×C5C22⋊Q811yescharacteristic
197C3×C4×C5C22⋊Q811yescharacteristic
198C3×Dic5—42no—
199C3×Dic5—41no—
200C3×Dic5—41no—
201C24×C3—52no—
202C2×C3×D4—52no—
203C2×C3×D4—52no—
204C2×C3×D4—101no—
205C2×C3×D4—101no—
206C2×C3×D4—101no—
207C2×C3×D4—101no—
208C22×C3×C4—21no—
209C22×C3×C4—102no—
210C22×C3×C4—102no—
211C22×C3×C4—101no—
212C22×C3×C4C22×D511yescharacteristic
213C3×M16—21no—
214C3×M16—52no—
215C3×M16—101no—
216C3×M16C4×D512yescomplemented
217C2×C3×C8—21no—
218C2×C3×C8—101no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—51no—
222C3×C4⋊C4—51no—
223C3×C4⋊C4C22⋊D511yescharacteristic
224C3×C4⋊C4C22⋊D511yescharacteristic
225C3×C22⋊C4—21no—
226C3×C22⋊C4—102no—
227C3×C22⋊C4—102no—
228C3×C22⋊C4—101no—
229C3×C22⋊C4—204no—
230C3×C42—21no—
231C3×C42—101no—
232C3×C42—101no—
233C3×C42—101no—
234C23×C5C2×C3×D411yescharacteristic
235C22×D5—22no—
236C22×D5—22no—
237C22×D5—42no—
238C22×D5—42no—
239C22×D5C3×C4⋊C414yes—
240C2×C4×C5—21no—
241C2×C4×C5—21no—
242C2×C4×C5—42no—
243C2×C4×C5—41no—
244C2×C4×C5C22×C3×C411yescharacteristic
245C2×C4×C5C2×C3×D411yescharacteristic
246C2×C4×C5C2×C3×D411yescharacteristic
247C2×C4×C5C2×C3×Q811yescharacteristic
248C2×C4×C5C3×#4111yescharacteristic
249C2×C4×C5C3×#4111yescharacteristic
250C2×Dic5—22no—
251C2×Dic5—21no—
252C2×Dic5—21no—
253C2×Dic5—41no—
254C2×Dic5—41no—
255D20—22no—
256D20—22no—
257D20—22no—
258D20—22no—
259C4×D5—82no—
260C4×D5—82no—
261C5×C8—22no—
262#168—22no—
263C22×D4—51no—
264C2×M16—51no—
265C2×M16C2×C3×D511yescharacteristic
266C4×D4—101no—
267C4×D4—101no—
268C4×D4—101no—
269C4×D4—101no—
270#117—51no—
271#117C2×C3×D511yescharacteristic
272C2×C4⋊C4C2×C3×D511yescharacteristic
273C2×C22⋊C4—102no—
274C2×C22⋊C4—102no—
275C2×C42—51no—
276#107—21no—
277#107—101no—
278#106—21no—
279#106—101no—
280#100—54no—
281C2×C3×C5—21no—
282C2×C3×C5—21no—
283C2×C3×C5#38511yescharacteristic
284C2×C3×C5#45311yescharacteristic
285C2×C3×C5#45511yescharacteristic
286C3×D5—44no—
287C23×C3—54no—
288C23×C3—102no—
289C23×C3—102no—
290C23×C3—202no—
291C23×C3—202no—
292C23×C3D4×D511yescharacteristic
293C3×D4—102no—
294C3×D4—102no—
295C3×D4—102no—
296C3×D4—102no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—42no—
300C2×C3×C4—41no—
301C2×C3×C4—102no—
302C2×C3×C4—101no—
303C2×C3×C4—101no—
304C2×C3×C4—201no—
305C2×C3×C4—201no—
306C2×C3×C4—402no—
307C2×C3×C4—402no—
308C2×C3×C4C2×C4×D511yescharacteristic
309C2×C3×C4#70911yescharacteristic
310C2×C3×C4D4×D511yescharacteristic
311C2×C3×C4Q8×D511yescharacteristic
312C2×C3×C4#71311yescharacteristic
313C2×C3×C4C2×C22⋊D511yescharacteristic
314C3×C8—22no—
315C3×C8—102no—
316C22×C5—21no—
317C22×C5—21no—
318C22×C5C2×C3×C4⋊C411yescharacteristic
319C22×C5C3×#12011yescharacteristic
320C22×C5C3×#12311yescharacteristic
321C2×D5—24no—
322C2×D5—42no—
323C2×D5—42no—
324C2×D5—44no—
325C2×D5—82no—
326C4×C5—41no—
327C4×C5—41no—
328C4×C5—81no—
329C4×C5C3×C4×D411yescharacteristic
330C4×C5C3×C4×D411yescharacteristic
331C4×C5C3×C22⋊Q811yescharacteristic
332C4×C5C3×C22⋊Q811yescharacteristic
333Dic5—42no—
334Dic5—41no—
335Dic5—41no—
336C24—52no—
337C2×D4—52no—
338C2×D4—52no—
339C2×D4—101no—
340C2×D4—101no—
341C2×D4—101no—
342C2×D4—101no—
343C22×C4—21no—
344C22×C4—102no—
345C22×C4—102no—
346C22×C4—101no—
347C22×C4C22×C3×D511yescharacteristic
348M16—21no—
349M16—52no—
350M16—101no—
351M16C3×C4×D512yescomplemented
352C2×C8—21no—
353C2×C8—101no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—51no—
357C4⋊C4—51no—
358C4⋊C4C3×C22⋊D511yescharacteristic
359C4⋊C4C3×C22⋊D511yescharacteristic
360C22⋊C4—21no—
361C22⋊C4—102no—
362C22⋊C4—102no—
363C22⋊C4—101no—
364C22⋊C4—204no—
365C42—21no—
366C42—101no—
367C42—101no—
368C42—101no—
369C3×C5#185611yescharacteristic, complemented, Hall
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—104no—
373C22×C3—204no—
374C22×C3—202no—
375C22×C3—202no—
376C22×C3—402no—
377C22×C3D5×C4⋊C411yescharacteristic
378C22×C3#404011yescharacteristic
379C22×C3#408311yescharacteristic
380C3×C4—41no—
381C3×C4—41no—
382C3×C4—81no—
383C3×C4—202no—
384C3×C4—201no—
385C3×C4—201no—
386C3×C4#403911yescharacteristic
387C3×C4#404211yescharacteristic
388C3×C4C4×C22⋊D511yescharacteristic
389C3×C4#409211yescharacteristic
390C2×C5—21no—
391C2×C5—21no—
392C2×C5C3×#38511yescharacteristic
393C2×C5C3×#45311yescharacteristic
394C2×C5C3×#45511yescharacteristic
395D5—44no—
396C23—54no—
397C23—102no—
398C23—102no—
399C23—202no—
400C23—202no—
401C23C3×D4×D511yescharacteristic
402D4—102no—
403D4—102no—
404D4—102no—
405D4—102no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—102no—
411C2×C4—101no—
412C2×C4—101no—
413C2×C4—201no—
414C2×C4—201no—
415C2×C4—402no—
416C2×C4—402no—
417C2×C4C2×C3×C4×D511yescharacteristic
418C2×C4C3×#70911yescharacteristic
419C2×C4C3×D4×D511yescharacteristic
420C2×C4C3×Q8×D511yescharacteristic
421C2×C4C3×#71311yescharacteristic
422C2×C4C2×C3×C22⋊D511yescharacteristic
423C8—22no—
424C8—102no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—204no—
428C2×C3#6527811yescharacteristic
429C2×C3#6548711yescharacteristic
430C2×C3#6549111yescharacteristic
431C5C3×#185611yescharacteristic, complemented, Sylow
432C22—21no—
433C22—21no—
434C22—104no—
435C22—204no—
436C22—202no—
437C22—202no—
438C22—402no—
439C22C3×D5×C4⋊C411yescharacteristic
440C22C3×#404011yescharacteristic
441C22C3×#408311yescharacteristic
442C4—41no—
443C4—41no—
444C4—81no—
445C4—202no—
446C4—201no—
447C4—201no—
448C4C3×#403911yescharacteristic
449C4C3×#404211yescharacteristic
450C4C3×C4×C22⋊D511yescharacteristic
451C4C3×#409211yescharacteristic
452C3#1060146711yescharacteristic, direct product factor, Sylow
453C2—21no—
454C2—21no—
455C2—204no—
456C2C3×#6527811yescharacteristic
457C2C3×#6548711yescharacteristic
458C2C3×#6549111yescharacteristic
subgroupG/H
centerC22×C3#4083
commutatorC2×C4×C5C22×C3×C4
FrattiniC22×C4C22×C3×D5
FittingC3×C5×#428C2
socleC22×C3×C5#120
prime radicalC22×C3×D20C22
cosocleC22×C4×C5C23×C3
hypercenterC3×#428D5
hypocenterC5C3×#1856
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#428

Complements

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

NH ≅ G/Ntype
2. C3×C5×#428C2semidirect
5. C3×#65253C2semidirect
6. C3×#64775C2semidirect
7. C3×#64752C2semidirect
8. #10601467C3direct (N × H)
24. C3×#3955C4semidirect
29. C5×#428C2×C3semidirect
32. #65253C2×C3semidirect
33. #64775C2×C3semidirect
34. #64752C2×C3semidirect
48. C3×C5×M16C2×C4semidirect
66. C3×#681C2×C4semidirect
70. C3×#428D5semidirect
90. #3955C3×C4semidirect
154. C5×M16C2×C3×C4semidirect
172. #681C2×C3×C4semidirect
176. #428C3×D5semidirect
216. C3×M16C4×D5semidirect
351. M16C3×C4×D5semidirect
369. C3×C5#1856semidirect
431. C5C3×#1856semidirect
452. C3#10601467direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#658552
C3×C5×#4282
C2×C3×#39552
C3×#652772
C3×#652532
C3×#647752
C3×#647522
#106014673
C3×#18565