Finite Group Databasebeta

← C3×#10608227 #14331131

Subgroups — C3×#10608227 #14331131

Order 1920.

1040 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×#4093C211yesmaximal, characteristic
2C3×C5×#391C211yesmaximal, characteristic, complemented
3C3×#65519C211yesmaximal, characteristic, complemented
4C3×#65278C212yesmaximal, complemented
5C3×#65229C212yesmaximal, complemented
6C3×#64958C212yesmaximal, complemented
7C3×#64957C212yesmaximal, complemented
8C3×#64954C212yesmaximal
9C3×#64939C212yesmaximal, complemented
10#10608227C311yesmaximal, characteristic, direct product factor, Hall
11C22×C3×C5×Q8C2211yescharacteristic
12C22×C3×D20C2211yescharacteristic
13C22×C3×C4×D5C2212yes—
14C2×C3×C5×C4⋊C4C2211yescharacteristic
15C2×C3×C42×C5C2211yescharacteristic
16C3×#4093C418yescomplemented
17C2×C3×#685C2212yes—
18C2×C3×#685C2212yes—
19C2×C3×#685C2212yes—
20C2×C3×#685C2212yes—
21C2×C3×#685C2212yes—
22C2×C3×C4⋊Dic5C2211yescharacteristic
23C2×C3×#683C2212yes—
24C2×C3×#683C2212yes—
25C2×C3×C4×Dic5C2211yescharacteristic
26C2×C3×C4×Dic5C2211yescharacteristic
27C3×C5×#95C2212yes—
28C3×C5×#95C2212yes—
29C3×#3963C2212yes—
30C3×#3963C2212yes—
31C3×#3963C2212yes—
32C3×#3963C2212yes—
33C3×#2379—51nomaximal, Hall
34C2×#4093C2×C311yescharacteristic
35C5×#391C2×C311yescharacteristic, complemented
36#65519C2×C311yescharacteristic, complemented
37#65278C2×C312yescomplemented
38#65229C2×C312yescomplemented
39#64958C2×C312yescomplemented
40#64957C2×C312yescomplemented
41#64954C2×C312yes—
42#64939C2×C312yescomplemented
43C23×C3×D5C2312yes—
44C2×C3×C5×Q8—41no—
45C2×C3×C5×Q8—41no—
46C2×C3×C5×Q8C2×C414yescomplemented
47C22×C3×C4×C5C2312yes—
48C22×C3×C4×C5C2312yes—
49C22×C3×C4×C5C2311yescharacteristic
50C22×C3×C4×C5C2311yescharacteristic
51C22×C3×C4×C5C2311yescharacteristic
52C22×C3×Dic5C2312yes—
53C22×C3×Dic5C2312yes—
54C22×C3×Dic5C2312yes—
55C2×C3×D20—41no—
56C2×C3×D20—41no—
57C2×C3×D20C2×C414yes—
58C2×C3×C4×D5—42no—
59C2×C3×C4×D5—42no—
60C2×C3×C4×D5—42no—
61C3×C5×C4⋊C4—41no—
62C3×C42×C5—41no—
63C3×#685—42no—
64C3×#685—42no—
65C3×#685—42no—
66C3×#685C2×C418yescomplemented
67C3×#685C2×C418yescomplemented
68C3×C4⋊Dic5—41no—
69C3×#683—42no—
70C3×#683—42no—
71C3×C4×Dic5—41no—
72C3×C4×Dic5C2×C414yescomplemented
73C2×C3×#124—51no—
74C3×#391—51no—
75C3×#391D511yescharacteristic, complemented
76C3×#388—52no—
77C3×#388—52no—
78C3×#386—52no—
79C3×#385—52no—
80C3×#385—52no—
81C3×#382—52no—
82C22×C5×Q8C22×C311yescharacteristic
83C22×D20C22×C311yescharacteristic
84C22×C4×D5C22×C312yes—
85C2×C5×C4⋊C4C22×C311yescharacteristic
86C2×C42×C5C22×C311yescharacteristic
87#4093C3×C418yescomplemented
88C2×#685C22×C312yes—
89C2×#685C22×C312yes—
90C2×#685C22×C312yes—
91C2×#685C22×C312yes—
92C2×#685C22×C312yes—
93C2×C4⋊Dic5C22×C311yescharacteristic
94C2×#683C22×C312yes—
95C2×#683C22×C312yes—
96C2×C4×Dic5C22×C311yescharacteristic
97C2×C4×Dic5C22×C311yescharacteristic
98C5×#95C22×C312yes—
99C5×#95C22×C312yes—
100#3963C22×C312yes—
101#3963C22×C312yes—
102#3963C22×C312yes—
103#3963C22×C312yes—
104#2379—51noSylow
105C23×C3×C5C2411yescharacteristic
106C22×C3×D5—22no—
107C22×C3×D5—22no—
108C22×C3×D5—22no—
109C22×C3×D5—22no—
110C22×C3×D5—22no—
111C22×C3×D5—22no—
112C22×C3×D5C22×C414yes—
113C3×C5×Q8—44no—
114C2×C3×C4×C5—21no—
115C2×C3×C4×C5—21no—
116C2×C3×C4×C5—21no—
117C2×C3×C4×C5—21no—
118C2×C3×C4×C5—21no—
119C2×C3×C4×C5—21no—
120C2×C3×C4×C5—22no—
121C2×C3×C4×C5—22no—
122C2×C3×C4×C5—22no—
123C2×C3×C4×C5—22no—
124C2×C3×C4×C5—22no—
125C2×C3×C4×C5—22no—
126C2×C3×C4×C5C22×C412yes—
127C2×C3×C4×C5C22×C412yes—
128C2×C3×C4×C5C22×C412yes—
129C2×C3×Dic5—22no—
130C2×C3×Dic5—22no—
131C2×C3×Dic5—22no—
132C2×C3×Dic5—22no—
133C2×C3×Dic5—22no—
134C2×C3×Dic5—22no—
135C2×C3×Dic5—22no—
136C2×C3×Dic5—22no—
137C2×C3×Dic5C22×C414yes—
138C3×D20—44no—
139C3×C4×D5—48no—
140C22×C3×Q8C2×D511yescharacteristic
141C22×C3×D4—51no—
142C23×C3×C4—52no—
143C3×#124—58no—
144C2×C3×C4⋊C4—52no—
145C2×C3×C4⋊C4—52no—
146C2×C3×C4⋊C4—51no—
147C2×C3×C4⋊C4C2×D511yescharacteristic
148C2×C3×C22⋊C4—52no—
149C2×C3×C22⋊C4—52no—
150C2×C3×C22⋊C4—52no—
151C2×C3×C22⋊C4—52no—
152C2×C3×C22⋊C4—52no—
153C2×C3×C42—51no—
154C2×C3×C42—51no—
155C2×C3×C42C2×D511yescharacteristic
156C3×#95—52no—
157C3×#95—52no—
158C3×#95—52no—
159C3×#95—52no—
160C3×#95C2×D512yes—
161C3×#95C2×D512yes—
162C23×D5C23×C312yes—
163C2×C5×Q8—41no—
164C2×C5×Q8—41no—
165C2×C5×Q8C2×C3×C414yescomplemented
166C22×C4×C5C23×C312yes—
167C22×C4×C5C23×C312yes—
168C22×C4×C5C23×C311yescharacteristic
169C22×C4×C5C23×C311yescharacteristic
170C22×C4×C5C23×C311yescharacteristic
171C22×Dic5C23×C312yes—
172C22×Dic5C23×C312yes—
173C22×Dic5C23×C312yes—
174C2×D20—41no—
175C2×D20—41no—
176C2×D20C2×C3×C414yes—
177C2×C4×D5—42no—
178C2×C4×D5—42no—
179C2×C4×D5—42no—
180C5×C4⋊C4—41no—
181C42×C5—41no—
182#685—42no—
183#685—42no—
184#685—42no—
185#685C2×C3×C418yescomplemented
186#685C2×C3×C418yescomplemented
187C4⋊Dic5—41no—
188#683—42no—
189#683—42no—
190C4×Dic5—41no—
191C4×Dic5C2×C3×C414yescomplemented
192C2×#124—51no—
193#391—51no—
194#391C3×D511yescharacteristic, complemented
195#388—52no—
196#388—52no—
197#386—52no—
198#385—52no—
199#385—52no—
200#382—52no—
201C22×C3×C5C23×C411yescharacteristic
202C22×C3×C5H(5,𝔽2)11yescharacteristic
203C22×C3×C5H(5,𝔽2)11yescharacteristic
204C22×C3×C5H(5,𝔽2)12yes—
205C22×C3×C52−1+412yes—
206C2×C3×D5—24no—
207C2×C3×D5—24no—
208C2×C3×D5—24no—
209C2×C3×D5—42no—
210C2×C3×D5—42no—
211C2×C3×D5—42no—
212C2×C3×D5—42no—
213C3×C4×C5—22no—
214C3×C4×C5—22no—
215C3×C4×C5—22no—
216C3×C4×C5—42no—
217C3×C4×C5—42no—
218C3×Dic5—24no—
219C3×Dic5—42no—
220C3×Dic5—42no—
221C24×C3—52no—
222C2×C3×Q8—41no—
223C2×C3×Q8—41no—
224C2×C3×Q8C4×D514yescomplemented
225C2×C3×D4—54no—
226C2×C3×D4—201no—
227C2×C3×D4—201no—
228C22×C3×C4—52no—
229C22×C3×C4—52no—
230C22×C3×C4—52no—
231C22×C3×C4—202no—
232C22×C3×C4—202no—
233C22×C3×C4—202no—
234C22×C3×C4C22×D512yes—
235C22×C3×C4C22×D512yes—
236C22×C3×C4C22×D511yescharacteristic
237C22×C3×C4C22×D511yescharacteristic
238C22×C3×C4C22×D511yescharacteristic
239C3×C4⋊C4—41no—
240C3×C4⋊C4—202no—
241C3×C4⋊C4—202no—
242C3×C4⋊C4—201no—
243C3×C22⋊C4—58no—
244C3×C22⋊C4—58no—
245C3×C22⋊C4—202no—
246C3×C22⋊C4—202no—
247C3×C22⋊C4—202no—
248C3×C42—41no—
249C3×C42—54no—
250C3×C42—201no—
251C23×C5C24×C311yescharacteristic
252C22×D5—22no—
253C22×D5—22no—
254C22×D5—22no—
255C22×D5—22no—
256C22×D5—22no—
257C22×D5—22no—
258C22×D5C22×C3×C414yes—
259C5×Q8—44no—
260C2×C4×C5—21no—
261C2×C4×C5—21no—
262C2×C4×C5—21no—
263C2×C4×C5—21no—
264C2×C4×C5—21no—
265C2×C4×C5—21no—
266C2×C4×C5—22no—
267C2×C4×C5—22no—
268C2×C4×C5—22no—
269C2×C4×C5—22no—
270C2×C4×C5—22no—
271C2×C4×C5—22no—
272C2×C4×C5C22×C3×C412yes—
273C2×C4×C5C22×C3×C412yes—
274C2×C4×C5C22×C3×C412yes—
275C2×Dic5—22no—
276C2×Dic5—22no—
277C2×Dic5—22no—
278C2×Dic5—22no—
279C2×Dic5—22no—
280C2×Dic5—22no—
281C2×Dic5—22no—
282C2×Dic5—22no—
283C2×Dic5C22×C3×C414yes—
284D20—44no—
285C4×D5—48no—
286C22×Q8C2×C3×D511yescharacteristic
287C22×D4—51no—
288C23×C4—52no—
289#124—58no—
290C2×C4⋊C4—52no—
291C2×C4⋊C4—52no—
292C2×C4⋊C4—51no—
293C2×C4⋊C4C2×C3×D511yescharacteristic
294C2×C22⋊C4—52no—
295C2×C22⋊C4—52no—
296C2×C22⋊C4—52no—
297C2×C22⋊C4—52no—
298C2×C22⋊C4—52no—
299C2×C42—51no—
300C2×C42—51no—
301C2×C42C2×C3×D511yescharacteristic
302#95—52no—
303#95—52no—
304#95—52no—
305#95—52no—
306#95C2×C3×D512yes—
307#95C2×C3×D512yes—
308C2×C3×C5#51811yescharacteristic
309C2×C3×C5#52012yes—
310C2×C3×C5H(3,𝔽4)11yescharacteristic
311C2×C3×C5#56212yes—
312C2×C3×C5#56311yescharacteristic
313C3×D5—44no—
314C23×C3—54no—
315C23×C3—102no—
316C23×C3—102no—
317C23×C3—102no—
318C23×C3—102no—
319C23×C3—102no—
320C23×C3—102no—
321C23×C3C23×D511yescharacteristic
322C3×Q8—44no—
323C3×D4—204no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—54no—
337C2×C3×C4—102no—
338C2×C3×C4—102no—
339C2×C3×C4—102no—
340C2×C3×C4—102no—
341C2×C3×C4—102no—
342C2×C3×C4—102no—
343C2×C3×C4—102no—
344C2×C3×C4—102no—
345C2×C3×C4—208no—
346C2×C3×C4C2×C4×D512yes—
347C2×C3×C4C2×C4×D512yes—
348C2×C3×C4C2×C4×D512yes—
349C22×C5C23×C3×C411yescharacteristic
350C22×C5C3×H(5,𝔽2)11yescharacteristic
351C22×C5C3×H(5,𝔽2)11yescharacteristic
352C22×C5C3×H(5,𝔽2)12yes—
353C22×C5C3×2−1+412yes—
354C2×D5—24no—
355C2×D5—24no—
356C2×D5—24no—
357C2×D5—42no—
358C2×D5—42no—
359C2×D5—42no—
360C2×D5—42no—
361C4×C5—22no—
362C4×C5—22no—
363C4×C5—22no—
364C4×C5—42no—
365C4×C5—42no—
366Dic5—24no—
367Dic5—42no—
368Dic5—42no—
369C24—52no—
370C2×Q8—41no—
371C2×Q8—41no—
372C2×Q8C3×C4×D514yescomplemented
373C2×D4—54no—
374C2×D4—201no—
375C2×D4—201no—
376C22×C4—52no—
377C22×C4—52no—
378C22×C4—52no—
379C22×C4—202no—
380C22×C4—202no—
381C22×C4—202no—
382C22×C4C22×C3×D512yes—
383C22×C4C22×C3×D512yes—
384C22×C4C22×C3×D511yescharacteristic
385C22×C4C22×C3×D511yescharacteristic
386C22×C4C22×C3×D511yescharacteristic
387C4⋊C4—41no—
388C4⋊C4—202no—
389C4⋊C4—202no—
390C4⋊C4—201no—
391C22⋊C4—58no—
392C22⋊C4—58no—
393C22⋊C4—202no—
394C22⋊C4—202no—
395C22⋊C4—202no—
396C42—41no—
397C42—54no—
398C42—201no—
399C3×C5#237911yescharacteristic, complemented, Hall
400C22×C3—104no—
401C22×C3—104no—
402C22×C3—104no—
403C22×C3—202no—
404C22×C3—202no—
405C22×C3—202no—
406C22×C3—202no—
407C22×C3C22×C4×D511yescharacteristic
408C22×C3#414411yescharacteristic
409C22×C3#414411yescharacteristic
410C22×C3#414712yes—
411C22×C3#414912yes—
412C3×C4—22no—
413C3×C4—22no—
414C3×C4—22no—
415C3×C4—42no—
416C3×C4—42no—
417C3×C4—104no—
418C3×C4—202no—
419C3×C4—202no—
420C2×C5C3×#51811yescharacteristic
421C2×C5C3×#52012yes—
422C2×C5C3×H(3,𝔽4)11yescharacteristic
423C2×C5C3×#56212yes—
424C2×C5C3×#56311yescharacteristic
425D5—44no—
426C23—54no—
427C23—102no—
428C23—102no—
429C23—102no—
430C23—102no—
431C23—102no—
432C23—102no—
433C23C23×C3×D511yescharacteristic
434Q8—44no—
435D4—204no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—54no—
449C2×C4—102no—
450C2×C4—102no—
451C2×C4—102no—
452C2×C4—102no—
453C2×C4—102no—
454C2×C4—102no—
455C2×C4—102no—
456C2×C4—102no—
457C2×C4—208no—
458C2×C4C2×C3×C4×D512yes—
459C2×C4C2×C3×C4×D512yes—
460C2×C4C2×C3×C4×D512yes—
461C2×C3—204no—
462C2×C3#6582011yescharacteristic
463C2×C3#6590812yes—
464C2×C3#6598112yes—
465C2×C3#6601711yescharacteristic
466C2×C3#6606911yescharacteristic
467C5C3×#237911yescharacteristic, complemented, Sylow
468C22—104no—
469C22—104no—
470C22—104no—
471C22—202no—
472C22—202no—
473C22—202no—
474C22—202no—
475C22C22×C3×C4×D511yescharacteristic
476C22C3×#414411yescharacteristic
477C22C3×#414411yescharacteristic
478C22C3×#414712yes—
479C22C3×#414912yes—
480C4—22no—
481C4—22no—
482C4—22no—
483C4—42no—
484C4—42no—
485C4—104no—
486C4—202no—
487C4—202no—
488C3#1060822711yescharacteristic, direct product factor, Sylow
489C2—204no—
490C2C3×#6582011yescharacteristic
491C2C3×#6590812yes—
492C2C3×#6598112yes—
493C2C3×#6601711yescharacteristic
494C2C3×#6606911yescharacteristic
subgroupG/H
centerC23×C3C23×D5
commutatorC22×C5C23×C3×C4
FrattiniC23C23×C3×D5
FittingC3×C5×#391C2
socleC23×C3×C5C24
prime radicalC22×C3×D20C22
cosocleC23×C5C24×C3
hypercenterC3×#391D5
hypocenterC5C3×#2379
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C3×#391

Complements

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

NH ≅ G/Ntype
2. C3×C5×#391C2semidirect
3. C3×#65519C2semidirect
4. C3×#65278C2semidirect
5. C3×#65229C2semidirect
6. C3×#64958C2semidirect
7. C3×#64957C2semidirect
9. C3×#64939C2semidirect
10. #10608227C3direct (N × H)
16. C3×#4093C4semidirect
35. C5×#391C2×C3semidirect
36. #65519C2×C3semidirect
37. #65278C2×C3semidirect
38. #65229C2×C3semidirect
39. #64958C2×C3semidirect
40. #64957C2×C3semidirect
42. #64939C2×C3semidirect
46. C2×C3×C5×Q8C2×C4semidirect
66. C3×#685C2×C4semidirect
67. C3×#685C2×C4semidirect
72. C3×C4×Dic5C2×C4semidirect
75. C3×#391D5semidirect
87. #4093C3×C4semidirect
165. C2×C5×Q8C2×C3×C4semidirect
185. #685C2×C3×C4semidirect
186. #685C2×C3×C4semidirect
191. C4×Dic5C2×C3×C4semidirect
194. #391C3×D5semidirect
224. C2×C3×Q8C4×D5semidirect
372. C2×Q8C3×C4×D5semidirect
399. C3×C5#2379semidirect
467. C5C3×#2379semidirect
488. C3#10608227direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40932
C3×C5×#3912
C3×#655192
C3×#652782
C3×#652292
C3×#649582
C3×#649572
C3×#649542
C3×#649392
#106082273
C3×#23795