Finite Group Databasebeta

← C3×#10605123 #14326795

Subgroups — C3×#10605123 #14326795

Order 1920.

1008 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4018C211yesmaximal, characteristic
2C3×C5×#376C211yesmaximal, characteristic, complemented
3C3×#65229C212yesmaximal
4C3×#65225C211yesmaximal, characteristic, complemented
5C3×C4×#683C212yesmaximal, complemented
6C3×#64959C212yesmaximal
7C3×#64956C211yesmaximal, characteristic
8C3×#64955C211yesmaximal, characteristic
9C3×#64939C211yesmaximal, characteristic, complemented
10C3×#64937C211yesmaximal, characteristic, complemented
11C3×#64936C212yesmaximal, complemented
12#10605123C311yesmaximal, characteristic, direct product factor, Hall
13C22×C3×C4×D5C2211yescharacteristic
14C2×C3×C42×C5C2212yes—
15C2×C3×C42×C5C2211yescharacteristic
16C2×C3×#685C2212yes—
17C2×C3×#685C2212yes—
18C2×C3×#685C2211yescharacteristic
19C2×C3×#685C2211yescharacteristic
20C2×C3×C4⋊Dic5C2212yes—
21C2×C3×#683C2212yes—
22C2×C3×#683C2212yes—
23C2×C3×#683C2211yescharacteristic
24C2×C3×#683C2211yescharacteristic
25C2×C3×C4×Dic5C2212yes—
26C2×C3×C4×Dic5C2212yes—
27C2×C3×C4×Dic5C2211yescharacteristic
28C3×#4018C418yescomplemented
29C3×C5×#95C2212yes—
30C3×C5×#95C2211yescharacteristic
31C3×C5×#95C2211yescharacteristic
32C3×#3963C2212yes—
33C3×#3963C2211yescharacteristic
34C3×#3963C2211yescharacteristic
35C3×#3963C2212yes—
36C3×#3963C2212yes—
37C3×#2346—51nomaximal, Hall
38C2×#4018C2×C311yescharacteristic
39C5×#376C2×C311yescharacteristic, complemented
40#65229C2×C312yes—
41#65225C2×C311yescharacteristic, complemented
42C4×#683C2×C312yescomplemented
43#64959C2×C312yes—
44#64956C2×C311yescharacteristic
45#64955C2×C311yescharacteristic
46#64939C2×C311yescharacteristic, complemented
47#64937C2×C311yescharacteristic, complemented
48#64936C2×C312yescomplemented
49C23×C3×D5C2311yescharacteristic
50C22×C3×C4×C5C2312yes—
51C22×C3×C4×C5C2311yescharacteristic
52C22×C3×C4×C5C2311yescharacteristic
53C22×C3×C4×C5C2311yescharacteristic
54C22×C3×C4×C5C2312yes—
55C22×C3×Dic5C2312yes—
56C22×C3×Dic5C2312yes—
57C22×C3×Dic5C2311yescharacteristic
58C22×C3×Dic5C2311yescharacteristic
59C22×C3×Dic5C2311yescharacteristic
60C2×C3×C4×D5—24no—
61C2×C3×C4×D5C2×C414yes—
62C3×C42×C5—24no—
63C3×C42×C5C2×C414yes—
64C3×#685—24no—
65C3×#685—42no—
66C3×#685C2×C414yes—
67C3×#685C2×C414yes—
68C3×C4⋊Dic5—42no—
69C3×#683—24no—
70C3×#683—42no—
71C3×#683C2×C414yes—
72C3×#683C2×C414yes—
73C3×C4×Dic5—24no—
74C3×C4×Dic5—24no—
75C3×C4×Dic5C2×C414yes—
76C2×C3×#117—51no—
77C3×#389—52no—
78C3×#388—52no—
79C3×#388—52no—
80C3×#382—51no—
81C3×#382—51no—
82C3×#380—51no—
83C3×C4×C4⋊C4—52no—
84C3×C4×C22⋊C4—51no—
85C3×#376—51no—
86C3×#376D511yescharacteristic, complemented
87C22×C4×D5C22×C311yescharacteristic
88C2×C42×C5C22×C312yes—
89C2×C42×C5C22×C311yescharacteristic
90C2×#685C22×C312yes—
91C2×#685C22×C312yes—
92C2×#685C22×C311yescharacteristic
93C2×#685C22×C311yescharacteristic
94C2×C4⋊Dic5C22×C312yes—
95C2×#683C22×C312yes—
96C2×#683C22×C312yes—
97C2×#683C22×C311yescharacteristic
98C2×#683C22×C311yescharacteristic
99C2×C4×Dic5C22×C312yes—
100C2×C4×Dic5C22×C312yes—
101C2×C4×Dic5C22×C311yescharacteristic
102#4018C3×C418yescomplemented
103C5×#95C22×C312yes—
104C5×#95C22×C311yescharacteristic
105C5×#95C22×C311yescharacteristic
106#3963C22×C312yes—
107#3963C22×C311yescharacteristic
108#3963C22×C311yescharacteristic
109#3963C22×C312yes—
110#3963C22×C312yes—
111#2346—51noSylow
112C23×C3×C5C2411yescharacteristic
113C22×C3×D5—22no—
114C22×C3×D5—22no—
115C22×C3×D5—21no—
116C22×C3×D5—21no—
117C22×C3×D5C22×C412yes—
118C2×C3×C4×C5—22no—
119C2×C3×C4×C5—22no—
120C2×C3×C4×C5—22no—
121C2×C3×C4×C5—21no—
122C2×C3×C4×C5—21no—
123C2×C3×C4×C5—22no—
124C2×C3×C4×C5—22no—
125C2×C3×C4×C5—22no—
126C2×C3×C4×C5—22no—
127C2×C3×C4×C5C22×C412yes—
128C2×C3×C4×C5C22×C412yes—
129C2×C3×C4×C5C22×C412yes—
130C2×C3×C4×C5#4114yes—
131C2×C3×Dic5—22no—
132C2×C3×Dic5—21no—
133C2×C3×Dic5—21no—
134C2×C3×Dic5—22no—
135C2×C3×Dic5—22no—
136C2×C3×Dic5—22no—
137C2×C3×Dic5—22no—
138C2×C3×Dic5—22no—
139C2×C3×Dic5—22no—
140C2×C3×Dic5C22×C412yes—
141C2×C3×Dic5C22×C412yes—
142C2×C3×Dic5C22×C412yes—
143C2×C3×Dic5#4114yes—
144C3×C4×D5—28no—
145C23×C3×C4—51no—
146C3×#117—58no—
147C2×C3×C4⋊C4—51no—
148C2×C3×C4⋊C4—51no—
149C2×C3×C4⋊C4—52no—
150C2×C3×C4⋊C4—52no—
151C2×C3×C4⋊C4—52no—
152C2×C3×C22⋊C4—51no—
153C2×C3×C22⋊C4—51no—
154C2×C3×C22⋊C4—52no—
155C2×C3×C22⋊C4—52no—
156C2×C3×C42—51no—
157C2×C3×C42—52no—
158C2×C3×C42—52no—
159C2×C3×C42C2×D512yes—
160C2×C3×C42C2×D511yescharacteristic
161C3×#95—51no—
162C3×#95—52no—
163C3×#95—51no—
164C3×#95—52no—
165C3×#95—52no—
166C3×#95C2×D511yescharacteristic
167C3×#95C2×D511yescharacteristic
168C3×#95C2×D512yes—
169C23×D5C23×C311yescharacteristic
170C22×C4×C5C23×C312yes—
171C22×C4×C5C23×C311yescharacteristic
172C22×C4×C5C23×C311yescharacteristic
173C22×C4×C5C23×C311yescharacteristic
174C22×C4×C5C23×C312yes—
175C22×Dic5C23×C312yes—
176C22×Dic5C23×C312yes—
177C22×Dic5C23×C311yescharacteristic
178C22×Dic5C23×C311yescharacteristic
179C22×Dic5C23×C311yescharacteristic
180C2×C4×D5—24no—
181C2×C4×D5C2×C3×C414yes—
182C42×C5—24no—
183C42×C5C2×C3×C414yes—
184#685—24no—
185#685—42no—
186#685C2×C3×C414yes—
187#685C2×C3×C414yes—
188C4⋊Dic5—42no—
189#683—24no—
190#683—42no—
191#683C2×C3×C414yes—
192#683C2×C3×C414yes—
193C4×Dic5—24no—
194C4×Dic5—24no—
195C4×Dic5C2×C3×C414yes—
196C2×#117—51no—
197#389—52no—
198#388—52no—
199#388—52no—
200#382—51no—
201#382—51no—
202#380—51no—
203C4×C4⋊C4—52no—
204C4×C22⋊C4—51no—
205#376—51no—
206#376C3×D511yescharacteristic, complemented
207C22×C3×C5C23×C411yescharacteristic
208C22×C3×C5C2×#4112yes—
209C22×C3×C5C2×#4112yes—
210C22×C3×C5H(5,𝔽2)11yescharacteristic
211C22×C3×C52−1+411yescharacteristic
212C2×C3×D5—22no—
213C2×C3×D5—22no—
214C2×C3×D5—22no—
215C2×C3×D5—42no—
216C2×C3×D5—42no—
217C3×C4×C5—22no—
218C3×C4×C5—22no—
219C3×C4×C5—22no—
220C3×C4×C5—24no—
221C3×C4×C5—42no—
222C3×Dic5—22no—
223C3×Dic5—22no—
224C3×Dic5—42no—
225C3×Dic5—42no—
226C3×Dic5#11714yes—
227C24×C3—51no—
228C22×C3×C4—51no—
229C22×C3×C4—51no—
230C22×C3×C4—51no—
231C22×C3×C4—54no—
232C22×C3×C4—52no—
233C22×C3×C4—52no—
234C22×C3×C4—104no—
235C22×C3×C4C22×D512yes—
236C22×C3×C4C22×D511yescharacteristic
237C22×C3×C4C22×D511yescharacteristic
238C22×C3×C4C22×D511yescharacteristic
239C22×C3×C4C22×D512yes—
240C3×C4⋊C4—54no—
241C3×C4⋊C4—54no—
242C3×C4⋊C4—104no—
243C3×C4⋊C4—202no—
244C3×C4⋊C4—202no—
245C3×C22⋊C4—54no—
246C3×C22⋊C4—54no—
247C3×C22⋊C4—104no—
248C3×C22⋊C4—202no—
249C3×C42—24no—
250C3×C42—54no—
251C3×C42—104no—
252C3×C42—104no—
253C3×C42C4×D514yes—
254C23×C5C24×C311yescharacteristic
255C22×D5—22no—
256C22×D5—22no—
257C22×D5—21no—
258C22×D5—21no—
259C22×D5C22×C3×C412yes—
260C2×C4×C5—22no—
261C2×C4×C5—22no—
262C2×C4×C5—22no—
263C2×C4×C5—21no—
264C2×C4×C5—21no—
265C2×C4×C5—22no—
266C2×C4×C5—22no—
267C2×C4×C5—22no—
268C2×C4×C5—22no—
269C2×C4×C5C22×C3×C412yes—
270C2×C4×C5C22×C3×C412yes—
271C2×C4×C5C22×C3×C412yes—
272C2×C4×C5C3×#4114yes—
273C2×Dic5—22no—
274C2×Dic5—21no—
275C2×Dic5—21no—
276C2×Dic5—22no—
277C2×Dic5—22no—
278C2×Dic5—22no—
279C2×Dic5—22no—
280C2×Dic5—22no—
281C2×Dic5—22no—
282C2×Dic5C22×C3×C412yes—
283C2×Dic5C22×C3×C412yes—
284C2×Dic5C22×C3×C412yes—
285C2×Dic5C3×#4114yes—
286C4×D5—28no—
287C23×C4—51no—
288#117—58no—
289C2×C4⋊C4—51no—
290C2×C4⋊C4—51no—
291C2×C4⋊C4—52no—
292C2×C4⋊C4—52no—
293C2×C4⋊C4—52no—
294C2×C22⋊C4—51no—
295C2×C22⋊C4—51no—
296C2×C22⋊C4—52no—
297C2×C22⋊C4—52no—
298C2×C42—51no—
299C2×C42—52no—
300C2×C42—52no—
301C2×C42C2×C3×D512yes—
302C2×C42C2×C3×D511yescharacteristic
303#95—51no—
304#95—52no—
305#95—51no—
306#95—52no—
307#95—52no—
308#95C2×C3×D511yescharacteristic
309#95C2×C3×D511yescharacteristic
310#95C2×C3×D512yes—
311C2×C3×C5C2×#11711yescharacteristic
312C2×C3×C5C4×#4111yescharacteristic
313C2×C3×C5#52011yescharacteristic
314C2×C3×C5#55212yes—
315C2×C3×C5#55512yes—
316C3×D5—42no—
317C23×C3—52no—
318C23×C3—102no—
319C23×C3—102no—
320C23×C3—101no—
321C23×C3—101no—
322C23×C3C23×D511yescharacteristic
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—52no—
333C2×C3×C4—54no—
334C2×C3×C4—52no—
335C2×C3×C4—52no—
336C2×C3×C4—108no—
337C2×C3×C4—102no—
338C2×C3×C4—101no—
339C2×C3×C4—101no—
340C2×C3×C4—102no—
341C2×C3×C4—102no—
342C2×C3×C4—102no—
343C2×C3×C4—102no—
344C2×C3×C4—102no—
345C2×C3×C4—102no—
346C2×C3×C4C2×C4×D512yes—
347C2×C3×C4C2×C4×D512yes—
348C2×C3×C4C2×C4×D512yes—
349C2×C3×C4#70914yes—
350C22×C5C23×C3×C411yescharacteristic
351C22×C5C2×C3×#4112yes—
352C22×C5C2×C3×#4112yes—
353C22×C5C3×H(5,𝔽2)11yescharacteristic
354C22×C5C3×2−1+411yescharacteristic
355C2×D5—22no—
356C2×D5—22no—
357C2×D5—22no—
358C2×D5—42no—
359C2×D5—42no—
360C4×C5—22no—
361C4×C5—22no—
362C4×C5—22no—
363C4×C5—24no—
364C4×C5—42no—
365Dic5—22no—
366Dic5—22no—
367Dic5—42no—
368Dic5—42no—
369Dic5C3×#11714yes—
370C24—51no—
371C22×C4—51no—
372C22×C4—51no—
373C22×C4—51no—
374C22×C4—54no—
375C22×C4—52no—
376C22×C4—52no—
377C22×C4—104no—
378C22×C4C22×C3×D512yes—
379C22×C4C22×C3×D511yescharacteristic
380C22×C4C22×C3×D511yescharacteristic
381C22×C4C22×C3×D511yescharacteristic
382C22×C4C22×C3×D512yes—
383C4⋊C4—54no—
384C4⋊C4—54no—
385C4⋊C4—104no—
386C4⋊C4—202no—
387C4⋊C4—202no—
388C22⋊C4—54no—
389C22⋊C4—54no—
390C22⋊C4—104no—
391C22⋊C4—202no—
392C42—24no—
393C42—54no—
394C42—104no—
395C42—104no—
396C42C3×C4×D514yes—
397C3×C5#234611yescharacteristic, complemented, Hall
398C22×C3—102no—
399C22×C3—102no—
400C22×C3—102no—
401C22×C3—202no—
402C22×C3—202no—
403C22×C3C22×C4×D511yescharacteristic
404C22×C3C2×#70912yes—
405C22×C3D5×#4112yes—
406C22×C3#414911yescharacteristic
407C22×C3#415011yescharacteristic
408C3×C4—22no—
409C3×C4—22no—
410C3×C4—22no—
411C3×C4—24no—
412C3×C4—42no—
413C3×C4—54no—
414C3×C4—102no—
415C3×C4—102no—
416C3×C4—202no—
417C3×C4—202no—
418C2×C5C2×C3×#11711yescharacteristic
419C2×C5C3×C4×#4111yescharacteristic
420C2×C5C3×#52011yescharacteristic
421C2×C5C3×#55212yes—
422C2×C5C3×#55512yes—
423D5—42no—
424C23—52no—
425C23—102no—
426C23—102no—
427C23—101no—
428C23—101no—
429C23C23×C3×D511yescharacteristic
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—52no—
440C2×C4—54no—
441C2×C4—52no—
442C2×C4—52no—
443C2×C4—108no—
444C2×C4—102no—
445C2×C4—101no—
446C2×C4—101no—
447C2×C4—102no—
448C2×C4—102no—
449C2×C4—102no—
450C2×C4—102no—
451C2×C4—102no—
452C2×C4—102no—
453C2×C4C2×C3×C4×D512yes—
454C2×C4C2×C3×C4×D512yes—
455C2×C4C2×C3×C4×D512yes—
456C2×C4C3×#70914yes—
457C2×C3—202no—
458C2×C3C4×#70911yescharacteristic
459C2×C3D5×#11711yescharacteristic
460C2×C3#6585711yescharacteristic
461C2×C3#6586512yes—
462C2×C3#6586612yes—
463C5C3×#234611yescharacteristic, complemented, Sylow
464C22—102no—
465C22—102no—
466C22—102no—
467C22—202no—
468C22—202no—
469C22C22×C3×C4×D511yescharacteristic
470C22C2×C3×#70912yes—
471C22C3×D5×#4112yes—
472C22C3×#414911yescharacteristic
473C22C3×#415011yescharacteristic
474C4—22no—
475C4—22no—
476C4—24no—
477C4—22no—
478C4—42no—
479C4—54no—
480C4—102no—
481C4—102no—
482C4—202no—
483C4—202no—
484C3#1060512311yescharacteristic, direct product factor, Sylow
485C2—202no—
486C2C3×C4×#70911yescharacteristic
487C2C3×D5×#11711yescharacteristic
488C2C3×#6585711yescharacteristic
489C2C3×#6586512yes—
490C2C3×#6586612yes—
subgroupG/H
centerC23×C3C23×D5
commutatorC22×C5C23×C3×C4
FrattiniC23C23×C3×D5
FittingC3×C5×#376C2
socleC23×C3×C5C24
prime radicalC23×C3×D5C23
cosocleC23×C5C24×C3
hypercenterC3×#376D5
hypocenterC5C3×#2346
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C3×#376

Complements

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

NH ≅ G/Ntype
2. C3×C5×#376C2semidirect
4. C3×#65225C2semidirect
5. C3×C4×#683C2semidirect
9. C3×#64939C2semidirect
10. C3×#64937C2semidirect
11. C3×#64936C2semidirect
12. #10605123C3direct (N × H)
28. C3×#4018C4semidirect
39. C5×#376C2×C3semidirect
41. #65225C2×C3semidirect
42. C4×#683C2×C3semidirect
46. #64939C2×C3semidirect
47. #64937C2×C3semidirect
48. #64936C2×C3semidirect
86. C3×#376D5semidirect
102. #4018C3×C4semidirect
206. #376C3×D5semidirect
397. C3×C5#2346semidirect
463. C5C3×#2346semidirect
484. C3#10605123direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40182
C3×C5×#3762
C3×#652292
C3×#652252
C3×C4×#6832
C3×#649592
C3×#649562
C3×#649552
C3×#649392
C3×#649372
C3×#649362
#106051233
C3×#23465