Finite Group Databasebeta

← C3×#10608209 #14321158

Subgroups — C3×#10608209 #14321158

Order 1920.

1872 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#713C211yesmaximal, characteristic
2C2×C3×#4038C211yesmaximal, characteristic, complemented
3C2×C3×C42×D5C211yesmaximal, characteristic, complemented
4C3×C5×#391C211yesmaximal, characteristic, complemented
5C3×#65519C211yesmaximal, characteristic, complemented
6C3×#65277C211yesmaximal, characteristic, complemented
7C3×#65228C211yesmaximal, characteristic, complemented
8C3×#64955C214yesmaximal, complemented
9C3×#64954C214yesmaximal, complemented
10#10608209C311yesmaximal, characteristic, direct product factor, Hall
11C22×C3×C5×Q8C2211yescharacteristic
12C2×C3×#713—24no—
13C2×C3×#713—24no—
14C2×C3×#713C418yescomplemented
15C22×C3×D20C2212yes—
16C22×C3×D20C2211yescharacteristic
17C22×C3×C4×D5C2212yes—
18C22×C3×C4×D5C2212yes—
19C22×C3×C4×D5C2211yescharacteristic
20C2×C3×C5×C4⋊C4C2211yescharacteristic
21C2×C3×C42×C5C2211yescharacteristic
22C2×C3×#685C2212yes—
23C2×C3×#685C2214yes—
24C2×C3×#685C2214yes—
25C2×C3×C4⋊Dic5C2211yescharacteristic
26C2×C3×C4×Dic5C2212yes—
27C2×C3×C4×Dic5C2212yes—
28C2×C3×C4×Dic5C2211yescharacteristic
29C3×#4038—24no—
30C3×C42×D5—24no—
31C3×C5×#95C2214yes—
32C3×#3963C2214yes—
33C3×#2297—51nomaximal, Hall
34C22×#713C2×C311yescharacteristic
35C2×#4038C2×C311yescharacteristic, complemented
36C2×C42×D5C2×C311yescharacteristic, complemented
37C5×#391C2×C311yescharacteristic, complemented
38#65519C2×C311yescharacteristic, complemented
39#65277C2×C311yescharacteristic, complemented
40#65228C2×C311yescharacteristic, complemented
41#64955C2×C314yescomplemented
42#64954C2×C314yescomplemented
43C23×C3×D5C2312yes—
44C23×C3×D5C2311yescharacteristic
45C2×C3×C5×Q8—22no—
46C2×C3×C5×Q8—22no—
47C2×C3×C5×Q8C2×C414yescomplemented
48C22×C3×C4×C5C2312yes—
49C22×C3×C4×C5C2312yes—
50C22×C3×C4×C5C2312yes—
51C22×C3×C4×C5C2311yescharacteristic
52C22×C3×Dic5C2312yes—
53C22×C3×Dic5C2312yes—
54C22×C3×Dic5C2311yescharacteristic
55C3×#713—232no—
56C2×C3×D20—24no—
57C2×C3×D20—24no—
58C2×C3×D20—22no—
59C2×C3×D20—22no—
60C2×C3×D20C2×C418yes—
61C2×C3×D20C2×C414yescomplemented
62C2×C3×C4×D5—24no—
63C2×C3×C4×D5—24no—
64C2×C3×C4×D5—24no—
65C2×C3×C4×D5—24no—
66C2×C3×C4×D5—24no—
67C2×C3×C4×D5C2×C414yes—
68C2×C3×C4×D5C2×C418yescomplemented
69C2×C3×C4×D5D414yes—
70C2×C3×C4×D5D414yes—
71C3×C5×C4⋊C4—22no—
72C3×C42×C5—22no—
73C3×#685—24no—
74C3×#685—28no—
75C3×#685—28no—
76C3×C4⋊Dic5—22no—
77C3×C4×Dic5—22no—
78C3×C4×Dic5—24no—
79C3×C4×Dic5—24no—
80C22×C3×#41—51no—
81C2×C3×#117—51no—
82C22×C3×C42—51no—
83C3×#391—51no—
84C3×#391D511yescharacteristic, complemented
85C3×#390—51no—
86C3×#390—51no—
87C3×#386—54no—
88C3×C4×C22⋊C4—54no—
89C22×C5×Q8C22×C311yescharacteristic
90C2×#713—24no—
91C2×#713—24no—
92C2×#713C3×C418yescomplemented
93C22×D20C22×C312yes—
94C22×D20C22×C311yescharacteristic
95C22×C4×D5C22×C312yes—
96C22×C4×D5C22×C312yes—
97C22×C4×D5C22×C311yescharacteristic
98C2×C5×C4⋊C4C22×C311yescharacteristic
99C2×C42×C5C22×C311yescharacteristic
100C2×#685C22×C312yes—
101C2×#685C22×C314yes—
102C2×#685C22×C314yes—
103C2×C4⋊Dic5C22×C311yescharacteristic
104C2×C4×Dic5C22×C312yes—
105C2×C4×Dic5C22×C311yescharacteristic
106C2×C4×Dic5C22×C312yes—
107#4038—24no—
108C42×D5—24no—
109C5×#95C22×C314yes—
110#3963C22×C314yes—
111#2297—51noSylow
112C23×C3×C5C2411yescharacteristic
113C22×C3×D5—22no—
114C22×C3×D5—22no—
115C22×C3×D5—22no—
116C22×C3×D5—22no—
117C22×C3×D5—24no—
118C22×C3×D5—24no—
119C22×C3×D5C22×C414yes—
120C22×C3×D5C22×C412yes—
121C22×C3×D5C2×D412yes—
122C22×C3×D5C2×D412yes—
123C3×C5×Q8—28no—
124C2×C3×C4×C5—22no—
125C2×C3×C4×C5—22no—
126C2×C3×C4×C5—22no—
127C2×C3×C4×C5—22no—
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5—24no—
130C2×C3×C4×C5C22×C414yes—
131C2×C3×C4×C5C22×C412yes—
132C2×C3×C4×C5C2×D412yes—
133C2×C3×C4×C5C2×D412yes—
134C2×C3×C4×C5#4114yescomplemented
135C2×C3×Dic5—22no—
136C2×C3×Dic5—22no—
137C2×C3×Dic5—22no—
138C2×C3×Dic5—24no—
139C2×C3×Dic5C22×C412yes—
140C2×C3×Dic5C2×D412yes—
141C2×C3×Dic5C2×D412yes—
142C2×C3×Dic5#4114yescomplemented
143C3×D20—216no—
144C3×D20—28no—
145C3×C4×D5—216no—
146C3×C4×D5—216no—
147C3×C4×D5C22⋊C4116yescomplemented
148C2×C3×#41—58no—
149C2×C3×#41—104no—
150C2×C3×#41—104no—
151C22×C3×Q8C2×D511yescharacteristic
152C22×C3×D4—52no—
153C22×C3×D4—51no—
154C23×C3×C4—51no—
155C23×C3×C4—52no—
156C23×C3×C4—52no—
157C3×#117—104no—
158C2×C3×C4⋊C4—51no—
159C2×C3×C4⋊C4C2×D511yescharacteristic
160C2×C3×C22⋊C4—54no—
161C2×C3×C22⋊C4—54no—
162C2×C3×C22⋊C4—52no—
163C2×C3×C42—52no—
164C2×C3×C42—52no—
165C2×C3×C42—51no—
166C2×C3×C42—104no—
167C2×C3×C42C2×D511yescharacteristic
168C3×#95—54no—
169C3×#95C2×D514yes—
170C23×D5C23×C312yes—
171C23×D5C23×C311yescharacteristic
172C2×C5×Q8—22no—
173C2×C5×Q8—22no—
174C2×C5×Q8C2×C3×C414yescomplemented
175C22×C4×C5C23×C312yes—
176C22×C4×C5C23×C312yes—
177C22×C4×C5C23×C312yes—
178C22×C4×C5C23×C311yescharacteristic
179C22×Dic5C23×C312yes—
180C22×Dic5C23×C312yes—
181C22×Dic5C23×C311yescharacteristic
182#713—232no—
183C2×D20—24no—
184C2×D20—24no—
185C2×D20—22no—
186C2×D20—22no—
187C2×D20C2×C3×C418yes—
188C2×D20C2×C3×C414yescomplemented
189C2×C4×D5—24no—
190C2×C4×D5—24no—
191C2×C4×D5—24no—
192C2×C4×D5—24no—
193C2×C4×D5—24no—
194C2×C4×D5C2×C3×C414yes—
195C2×C4×D5C2×C3×C418yescomplemented
196C2×C4×D5C3×D414yes—
197C2×C4×D5C3×D414yes—
198C5×C4⋊C4—22no—
199C42×C5—22no—
200#685—24no—
201#685—28no—
202#685—28no—
203C4⋊Dic5—22no—
204C4×Dic5—22no—
205C4×Dic5—24no—
206C4×Dic5—24no—
207C22×#41—51no—
208C2×#117—51no—
209C22×C42—51no—
210#391—51no—
211#391C3×D511yescharacteristic, complemented
212#390—51no—
213#390—51no—
214#386—54no—
215C4×C22⋊C4—54no—
216C22×C3×C5C23×C411yescharacteristic
217C22×C3×C5C22×D411yescharacteristic
218C22×C3×C5C22×D411yescharacteristic
219C22×C3×C5C2×#4112yes—
220C22×C3×C5C2×#4112yes—
221C2×C3×D5—24no—
222C2×C3×D5—24no—
223C2×C3×D5—22no—
224C2×C3×D5—22no—
225C2×C3×D5—24no—
226C2×C3×D5—28no—
227C2×C3×D5—44no—
228C2×C3×D5—42no—
229C2×C3×D5—42no—
230C2×C3×D5C2×C22⋊C414yes—
231C3×C4×C5—24no—
232C3×C4×C5—24no—
233C3×C4×C5—42no—
234C3×C4×C5C2×C22⋊C414yes—
235C3×Dic5—24no—
236C3×Dic5—42no—
237C3×Dic5C2×C22⋊C414yes—
238C24×C3—51no—
239C24×C3—52no—
240C3×#41—1032no—
241C2×C3×Q8—22no—
242C2×C3×Q8—22no—
243C2×C3×Q8C4×D514yescomplemented
244C2×C3×D4—58no—
245C2×C3×D4—54no—
246C2×C3×D4—104no—
247C2×C3×D4—104no—
248C2×C3×D4—102no—
249C2×C3×D4—102no—
250C22×C3×C4—51no—
251C22×C3×C4—54no—
252C22×C3×C4—54no—
253C22×C3×C4—54no—
254C22×C3×C4—52no—
255C22×C3×C4—58no—
256C22×C3×C4—52no—
257C22×C3×C4—104no—
258C22×C3×C4—104no—
259C22×C3×C4—104no—
260C22×C3×C4—104no—
261C22×C3×C4—104no—
262C22×C3×C4C22×D512yes—
263C22×C3×C4C22×D512yes—
264C22×C3×C4C22×D512yes—
265C22×C3×C4C22×D511yescharacteristic
266C3×C4⋊C4—22no—
267C3×C4⋊C4—102no—
268C3×C22⋊C4—108no—
269C3×C22⋊C4—104no—
270C3×C22⋊C4—108no—
271C3×C42—22no—
272C3×C42—104no—
273C3×C42—104no—
274C3×C42—102no—
275C23×C5C24×C311yescharacteristic
276C22×D5—22no—
277C22×D5—22no—
278C22×D5—22no—
279C22×D5—22no—
280C22×D5—24no—
281C22×D5—24no—
282C22×D5C22×C3×C414yes—
283C22×D5C22×C3×C412yes—
284C22×D5C2×C3×D412yes—
285C22×D5C2×C3×D412yes—
286C5×Q8—28no—
287C2×C4×C5—22no—
288C2×C4×C5—22no—
289C2×C4×C5—22no—
290C2×C4×C5—22no—
291C2×C4×C5—22no—
292C2×C4×C5—24no—
293C2×C4×C5C22×C3×C414yes—
294C2×C4×C5C22×C3×C412yes—
295C2×C4×C5C2×C3×D412yes—
296C2×C4×C5C2×C3×D412yes—
297C2×C4×C5C3×#4114yescomplemented
298C2×Dic5—22no—
299C2×Dic5—22no—
300C2×Dic5—22no—
301C2×Dic5—24no—
302C2×Dic5C22×C3×C412yes—
303C2×Dic5C2×C3×D412yes—
304C2×Dic5C2×C3×D412yes—
305C2×Dic5C3×#4114yescomplemented
306D20—216no—
307D20—28no—
308C4×D5—216no—
309C4×D5—216no—
310C4×D5C3×C22⋊C4116yescomplemented
311C2×#41—58no—
312C2×#41—104no—
313C2×#41—104no—
314C22×Q8C2×C3×D511yescharacteristic
315C22×D4—52no—
316C22×D4—51no—
317C23×C4—51no—
318C23×C4—52no—
319C23×C4—52no—
320#117—104no—
321C2×C4⋊C4—51no—
322C2×C4⋊C4C2×C3×D511yescharacteristic
323C2×C22⋊C4—52no—
324C2×C22⋊C4—54no—
325C2×C22⋊C4—54no—
326C2×C42—52no—
327C2×C42—52no—
328C2×C42—51no—
329C2×C42—104no—
330C2×C42C2×C3×D511yescharacteristic
331#95—54no—
332#95C2×C3×D514yes—
333C2×C3×C5C22×C22⋊C411yescharacteristic
334C2×C3×C5C4×#4112yes—
335C2×C3×C5#52512yes—
336C2×C3×C5#53211yescharacteristic
337C2×C3×C5#53211yescharacteristic
338C3×D5—24no—
339C3×D5—44no—
340C23×C3—52no—
341C23×C3—52no—
342C23×C3—52no—
343C23×C3—54no—
344C23×C3—102no—
345C23×C3—102no—
346C23×C3—102no—
347C23×C3—102no—
348C23×C3—104no—
349C23×C3—104no—
350C23×C3C23×D511yescharacteristic
351C3×Q8—28no—
352C3×D4—1016no—
353C3×D4—108no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—24no—
360C2×C3×C4—52no—
361C2×C3×C4—52no—
362C2×C3×C4—52no—
363C2×C3×C4—516no—
364C2×C3×C4—54no—
365C2×C3×C4—1016no—
366C2×C3×C4—1016no—
367C2×C3×C4—102no—
368C2×C3×C4—102no—
369C2×C3×C4—102no—
370C2×C3×C4—104no—
371C2×C3×C4C2×C4×D514yes—
372C2×C3×C4C2×C4×D512yes—
373C2×C3×C4D4×D512yes—
374C2×C3×C4D4×D512yes—
375C2×C3×C4#71314yescomplemented
376C22×C5C23×C3×C411yescharacteristic
377C22×C5C22×C3×D411yescharacteristic
378C22×C5C22×C3×D411yescharacteristic
379C22×C5C2×C3×#4112yes—
380C22×C5C2×C3×#4112yes—
381C2×D5—24no—
382C2×D5—24no—
383C2×D5—22no—
384C2×D5—22no—
385C2×D5—24no—
386C2×D5—28no—
387C2×D5—44no—
388C2×D5—42no—
389C2×D5—42no—
390C2×D5C2×C3×C22⋊C414yes—
391C4×C5—24no—
392C4×C5—24no—
393C4×C5—42no—
394C4×C5C2×C3×C22⋊C414yes—
395Dic5—24no—
396Dic5—42no—
397Dic5C2×C3×C22⋊C414yes—
398C24—51no—
399C24—52no—
400#41—1032no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8C3×C4×D514yescomplemented
404C2×D4—58no—
405C2×D4—54no—
406C2×D4—104no—
407C2×D4—104no—
408C2×D4—102no—
409C2×D4—102no—
410C22×C4—51no—
411C22×C4—54no—
412C22×C4—54no—
413C22×C4—54no—
414C22×C4—52no—
415C22×C4—58no—
416C22×C4—52no—
417C22×C4—104no—
418C22×C4—104no—
419C22×C4—104no—
420C22×C4—104no—
421C22×C4—104no—
422C22×C4C22×C3×D512yes—
423C22×C4C22×C3×D512yes—
424C22×C4C22×C3×D512yes—
425C22×C4C22×C3×D511yescharacteristic
426C4⋊C4—22no—
427C4⋊C4—102no—
428C22⋊C4—108no—
429C22⋊C4—104no—
430C22⋊C4—108no—
431C42—22no—
432C42—104no—
433C42—104no—
434C42—102no—
435C3×C5#229711yescharacteristic, complemented, Hall
436C22×C3—54no—
437C22×C3—102no—
438C22×C3—104no—
439C22×C3—104no—
440C22×C3—102no—
441C22×C3—104no—
442C22×C3—108no—
443C22×C3—202no—
444C22×C3—204no—
445C22×C3—202no—
446C22×C3C22×C4×D511yescharacteristic
447C22×C3C2×D4×D511yescharacteristic
448C22×C3C2×D4×D511yescharacteristic
449C22×C3C2×#71312yes—
450C22×C3D5×#4112yes—
451C3×C4—24no—
452C3×C4—24no—
453C3×C4—42no—
454C3×C4—54no—
455C3×C4—104no—
456C3×C4—202no—
457C3×C4D5×C22⋊C414yes—
458C2×C5C22×C3×C22⋊C411yescharacteristic
459C2×C5C3×C4×#4112yes—
460C2×C5C3×#52512yes—
461C2×C5C3×#53211yescharacteristic
462C2×C5C3×#53211yescharacteristic
463D5—24no—
464D5—44no—
465C23—52no—
466C23—52no—
467C23—52no—
468C23—54no—
469C23—102no—
470C23—102no—
471C23—102no—
472C23—102no—
473C23—104no—
474C23—104no—
475C23C23×C3×D511yescharacteristic
476Q8—28no—
477D4—1016no—
478D4—108no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—52no—
486C2×C4—52no—
487C2×C4—52no—
488C2×C4—516no—
489C2×C4—54no—
490C2×C4—1016no—
491C2×C4—1016no—
492C2×C4—102no—
493C2×C4—102no—
494C2×C4—102no—
495C2×C4—104no—
496C2×C4C2×C3×C4×D514yes—
497C2×C4C2×C3×C4×D512yes—
498C2×C4C3×D4×D512yes—
499C2×C4C3×D4×D512yes—
500C2×C4C3×#71314yescomplemented
501C2×C3—104no—
502C2×C3—204no—
503C2×C3C2×D5×C22⋊C411yescharacteristic
504C2×C3C4×#71312yes—
505C2×C3#6596112yes—
506C2×C3#6600211yescharacteristic
507C2×C3#6605911yescharacteristic
508C5C3×#229711yescharacteristic, complemented, Sylow
509C22—54no—
510C22—104no—
511C22—104no—
512C22—102no—
513C22—102no—
514C22—108no—
515C22—104no—
516C22—204no—
517C22—202no—
518C22—202no—
519C22C22×C3×C4×D511yescharacteristic
520C22C2×C3×D4×D511yescharacteristic
521C22C2×C3×D4×D511yescharacteristic
522C22C2×C3×#71312yes—
523C22C3×D5×#4112yes—
524C4—24no—
525C4—24no—
526C4—42no—
527C4—54no—
528C4—104no—
529C4—202no—
530C4C3×D5×C22⋊C414yes—
531C3#1060820911yescharacteristic, direct product factor, Sylow
532C2—104no—
533C2—204no—
534C2C2×C3×D5×C22⋊C411yescharacteristic
535C2C3×C4×#71312yes—
536C2C3×#6596112yes—
537C2C3×#6600211yescharacteristic
538C2C3×#6605911yescharacteristic
subgroupG/H
centerC23×C3C23×D5
commutatorC22×C5C23×C3×C4
FrattiniC23C23×C3×D5
FittingC3×C5×#391C2
socleC23×C3×C5C24
prime radicalC22×C3×#713C2
cosocleC23×C5C24×C3
hypercenterC3×#391D5
hypocenterC5C3×#2297
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C3×#391

Complements

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

NH ≅ G/Ntype
2. C2×C3×#4038C2semidirect
3. C2×C3×C42×D5C2semidirect
4. C3×C5×#391C2semidirect
5. C3×#65519C2semidirect
6. C3×#65277C2semidirect
7. C3×#65228C2semidirect
8. C3×#64955C2semidirect
9. C3×#64954C2semidirect
10. #10608209C3direct (N × H)
14. C2×C3×#713C4semidirect
35. C2×#4038C2×C3semidirect
36. C2×C42×D5C2×C3semidirect
37. C5×#391C2×C3semidirect
38. #65519C2×C3semidirect
39. #65277C2×C3semidirect
40. #65228C2×C3semidirect
41. #64955C2×C3semidirect
42. #64954C2×C3semidirect
47. C2×C3×C5×Q8C2×C4semidirect
61. C2×C3×D20C2×C4semidirect
68. C2×C3×C4×D5C2×C4semidirect
84. C3×#391D5semidirect
92. C2×#713C3×C4semidirect
134. C2×C3×C4×C5#41semidirect
142. C2×C3×Dic5#41semidirect
147. C3×C4×D5C22⋊C4semidirect
174. C2×C5×Q8C2×C3×C4semidirect
188. C2×D20C2×C3×C4semidirect
195. C2×C4×D5C2×C3×C4semidirect
211. #391C3×D5semidirect
243. C2×C3×Q8C4×D5semidirect
297. C2×C4×C5C3×#41semidirect
305. C2×Dic5C3×#41semidirect
310. C4×D5C3×C22⋊C4semidirect
375. C2×C3×C4#713semidirect
403. C2×Q8C3×C4×D5semidirect
435. C3×C5#2297semidirect
500. C2×C4C3×#713semidirect
508. C5C3×#2297semidirect
531. C3#10608209direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#7132
C2×C3×#40382
C2×C3×C42×D52
C3×C5×#3912
C3×#655192
C3×#652772
C3×#652282
C3×#649552
C3×#649542
#106082093
C3×#22975