Finite Group Databasebeta

← C2×C3×#65187 #14429296

Subgroups — C2×C3×#65187 #14429296

Order 1920.

936 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4055C211yesmaximal, characteristic, complemented
2C2×C3×#4054C211yesmaximal, characteristic
3C22×C3×#677C211yesmaximal, characteristic, complemented
4C2×C3×#4047C211yesmaximal, characteristic, complemented
5C2×C3×C5×#108C211yesmaximal, characteristic, complemented
6C2×C3×#3956C211yesmaximal, characteristic, complemented
7C2×C3×#3955C211yesmaximal, characteristic, complemented
8C3×#65187C218yesmaximal, direct product factor
9C2×#65187C311yesmaximal, characteristic, direct product factor, Hall
10C2×C3×#709C2211yescharacteristic
11C22×C3×D20C2211yescharacteristic
12C22×C3×Dic10C2211yescharacteristic
13C2×C3×C5×M16C2211yescharacteristic
14C2×C3×C5×M16C2211yescharacteristic
15C22×C3×C5×C8C2211yescharacteristic
16C2×C3×#681C2211yescharacteristic
17C3×#4055—22no—
18C3×#4055C2214yescomplemented
19C3×#4054—22no—
20C3×#4054C2214yes—
21C2×C3×Dic20—21no—
22C2×C3×D40—21no—
23C2×C3×#677—21no—
24C2×C3×#677—21no—
25C2×C3×#677—21no—
26C2×C3×#677—21no—
27C2×C3×#677—22no—
28C2×C3×#677C2214yescomplemented
29C3×#4047—22no—
30C3×#4047C2214yescomplemented
31C2×C3×#676—21no—
32C2×C3×C8×D5—21no—
33C3×C5×#108C2214yescomplemented
34C3×#3956C2214yescomplemented
35C3×#3955C2214yescomplemented
36C2×C3×#471—51nomaximal, Hall
37C2×#4055C2×C311yescharacteristic, complemented
38C2×#4054C2×C311yescharacteristic
39C22×#677C2×C311yescharacteristic, complemented
40C2×#4047C2×C311yescharacteristic, complemented
41C2×C5×#108C2×C311yescharacteristic, complemented
42C2×#3956C2×C311yescharacteristic, complemented
43C2×#3955C2×C311yescharacteristic, complemented
44#65187C2×C318yesdirect product factor
45C23×C3×D5—21no—
46C22×C3×C4×C5C2311yescharacteristic
47C2×C3×C22⋊D5—21no—
48C22×C3×Dic5—21no—
49C3×#709—21no—
50C3×#709—21no—
51C3×#709D412yes—
52C3×#709C2312yes—
53C2×C3×D20—21no—
54C2×C3×D20—21no—
55C2×C3×D20—21no—
56C2×C3×D20—21no—
57C2×C3×D20—21no—
58C2×C3×D20D411yescharacteristic
59C2×C3×D20C2312yes—
60C2×C3×C4×D5—21no—
61C2×C3×Dic10—21no—
62C2×C3×Dic10—21no—
63C2×C3×Dic10—21no—
64C2×C3×Dic10—21no—
65C2×C3×Dic10—21no—
66C2×C3×Dic10D411yescharacteristic
67C2×C3×Dic10C2312yes—
68C3×C5×M16—21no—
69C3×C5×M16—21no—
70C3×C5×M16C2312yes—
71C3×C5×M16C2312yes—
72C2×C3×C5×C8—21no—
73C2×C3×C5×C8—21no—
74C2×C3×C5×C8D411yescharacteristic
75C2×C3×C5×C8D411yescharacteristic
76C2×C3×C5×C8D412yes—
77C2×C3×C5×C8C2312yes—
78C3×#681—21no—
79C3×#681C2312yes—
80C2×C3×#168—21no—
81C3×Dic20—24no—
82C3×D40—24no—
83C3×#677—24no—
84C3×#677—24no—
85C3×#677—24no—
86C3×#677—24no—
87C3×#676—24no—
88C3×C8×D5—24no—
89C2×C3×#137—51no—
90C2×C3×AGL(1,ℤ/8)—51no—
91C22×C3×SD16—51no—
92C2×C3×#131—51no—
93C3×#471—58no—
94C2×C3×#108D511yescharacteristic, complemented
95C2×C3×#101—51no—
96C2×C3×#100—51no—
97C2×#709C22×C311yescharacteristic
98C22×D20C22×C311yescharacteristic
99C22×Dic10C22×C311yescharacteristic
100C2×C5×M16C22×C311yescharacteristic
101C2×C5×M16C22×C311yescharacteristic
102C22×C5×C8C22×C311yescharacteristic
103C2×#681C22×C311yescharacteristic
104#4055—22no—
105#4055C22×C314yescomplemented
106#4054—22no—
107#4054C22×C314yes—
108C2×Dic20—21no—
109C2×D40—21no—
110C2×#677—21no—
111C2×#677—21no—
112C2×#677—21no—
113C2×#677—21no—
114C2×#677—22no—
115C2×#677C22×C314yescomplemented
116#4047—22no—
117#4047C22×C314yescomplemented
118C2×#676—21no—
119C2×C8×D5—21no—
120C5×#108C22×C314yescomplemented
121#3956C22×C314yescomplemented
122#3955C22×C314yescomplemented
123C2×#471—51noSylow
124C23×C3×C5#4111yescharacteristic
125C22×C3×D5—21no—
126C22×C3×D5—22no—
127C22×C3×D5—41no—
128C22×C3×D5—41no—
129C22×C3×D5—82no—
130C2×C3×C4×C5C2×D411yescharacteristic
131C2×C3×C4×C5C2×D411yescharacteristic
132C2×C3×C4×C5C2×D411yescharacteristic
133C2×C3×C4×C5C2×D411yescharacteristic
134C2×C3×C4×C5#4111yescharacteristic
135C2×C3×C4×C5C2411yescharacteristic
136C3×C22⋊D5—22no—
137C3×C22⋊D5—22no—
138C2×C3×Dic5—21no—
139C2×C3×Dic5—22no—
140C2×C3×Dic5—41no—
141C2×C3×Dic5—41no—
142C3×D20—21no—
143C3×D20—22no—
144C3×D20—22no—
145C3×D20—41no—
146C3×D20—41no—
147C3×D20C2×D412yes—
148C3×C4×D5—22no—
149C3×C4×D5—22no—
150C3×Dic10—21no—
151C3×Dic10—22no—
152C3×Dic10—22no—
153C3×Dic10—41no—
154C3×Dic10—41no—
155C3×Dic10C2×D412yes—
156C3×C5×C8—22no—
157C3×C5×C8—22no—
158C3×C5×C8C2×D412yes—
159C3×C5×C8C2×D412yes—
160C3×#168—22no—
161C2×C3×#41—51no—
162C22×C3×Q8—51no—
163C22×C3×D4—51no—
164C3×#137—54no—
165C3×#137—102no—
166C3×AGL(1,ℤ/8)—54no—
167C3×AGL(1,ℤ/8)—102no—
168C2×C3×Q16—101no—
169C2×C3×SD16—54no—
170C2×C3×SD16—101no—
171C2×C3×SD16—101no—
172C2×C3×SD16—101no—
173C2×C3×SD16—101no—
174C2×C3×SD16—102no—
175C2×C3×D8—101no—
176C3×#131—54no—
177C3×#131—102no—
178C2×C3×M16—51no—
179C2×C3×M16—101no—
180C2×C3×M16C2×D511yescharacteristic
181C2×C3×M16C2×D511yescharacteristic
182C22×C3×C8—101no—
183C22×C3×C8C2×D511yescharacteristic
184C3×#108C2×D514yescomplemented
185C3×#101—54no—
186C3×#100—54no—
187C23×D5—21no—
188C22×C4×C5C23×C311yescharacteristic
189C2×C22⋊D5—21no—
190C22×Dic5—21no—
191#709—21no—
192#709—21no—
193#709C3×D412yes—
194#709C23×C312yes—
195C2×D20—21no—
196C2×D20—21no—
197C2×D20—21no—
198C2×D20—21no—
199C2×D20—21no—
200C2×D20C3×D411yescharacteristic
201C2×D20C23×C312yes—
202C2×C4×D5—21no—
203C2×Dic10—21no—
204C2×Dic10—21no—
205C2×Dic10—21no—
206C2×Dic10—21no—
207C2×Dic10—21no—
208C2×Dic10C3×D411yescharacteristic
209C2×Dic10C23×C312yes—
210C5×M16—21no—
211C5×M16—21no—
212C5×M16C23×C312yes—
213C5×M16C23×C312yes—
214C2×C5×C8—21no—
215C2×C5×C8—21no—
216C2×C5×C8C3×D411yescharacteristic
217C2×C5×C8C3×D411yescharacteristic
218C2×C5×C8C3×D412yes—
219C2×C5×C8C23×C312yes—
220#681—21no—
221#681C23×C312yes—
222C2×#168—21no—
223Dic20—24no—
224D40—24no—
225#677—24no—
226#677—24no—
227#677—24no—
228#677—24no—
229#676—24no—
230C8×D5—24no—
231C2×#137—51no—
232C2×AGL(1,ℤ/8)—51no—
233C22×SD16—51no—
234C2×#131—51no—
235#471—58no—
236C2×#108C3×D511yescharacteristic, complemented
237C2×#101—51no—
238C2×#100—51no—
239C22×C3×C5—22no—
240C22×C3×C5#12111yescharacteristic
241C22×C3×C5#12111yescharacteristic
242C22×C3×C5C2×#4111yescharacteristic
243C2×C3×D5—22no—
244C2×C3×D5—42no—
245C2×C3×D5—42no—
246C2×C3×D5—82no—
247C2×C3×D5—82no—
248C2×C3×D5—82no—
249C3×C4×C5#12111yescharacteristic
250C3×C4×C5#12111yescharacteristic
251C3×C4×C5C22×D411yescharacteristic
252C3×C4×C5C22×D411yescharacteristic
253C3×Dic5—22no—
254C3×Dic5—42no—
255C24×C3—101no—
256C3×#41—52no—
257C3×#41—52no—
258C3×#41—101no—
259C3×#41—101no—
260C2×C3×Q8—51no—
261C2×C3×Q8—52no—
262C2×C3×Q8—101no—
263C2×C3×Q8—101no—
264C2×C3×Q8—101no—
265C2×C3×Q8—101no—
266C2×C3×Q8—101no—
267C2×C3×D4—51no—
268C2×C3×D4—52no—
269C2×C3×D4—101no—
270C2×C3×D4—101no—
271C2×C3×D4—101no—
272C2×C3×D4—101no—
273C2×C3×D4—101no—
274C2×C3×D4—101no—
275C22×C3×C4—101no—
276C22×C3×C4—101no—
277C22×C3×C4C22×D511yescharacteristic
278C3×Q16—104no—
279C3×SD16—104no—
280C3×SD16—104no—
281C3×SD16—104no—
282C3×SD16—104no—
283C3×D8—104no—
284C3×M16—21no—
285C3×M16—21no—
286C3×M16—52no—
287C3×M16—101no—
288C3×M16—104no—
289C3×M16C22×D512yes—
290C3×M16C22×D512yes—
291C2×C3×C8—21no—
292C2×C3×C8—21no—
293C2×C3×C8—101no—
294C2×C3×C8—104no—
295C2×C3×C8D2011yescharacteristic
296C2×C3×C8D2011yescharacteristic
297C2×C3×C8D2012yes—
298C2×C3×C8C22×D512yes—
299C23×C5C3×#4111yescharacteristic
300C22×D5—21no—
301C22×D5—22no—
302C22×D5—41no—
303C22×D5—41no—
304C22×D5—82no—
305C2×C4×C5C2×C3×D411yescharacteristic
306C2×C4×C5C2×C3×D411yescharacteristic
307C2×C4×C5C2×C3×D411yescharacteristic
308C2×C4×C5C2×C3×D411yescharacteristic
309C2×C4×C5C3×#4111yescharacteristic
310C2×C4×C5C24×C311yescharacteristic
311C22⋊D5—22no—
312C22⋊D5—22no—
313C2×Dic5—21no—
314C2×Dic5—22no—
315C2×Dic5—41no—
316C2×Dic5—41no—
317D20—21no—
318D20—22no—
319D20—22no—
320D20—41no—
321D20—41no—
322D20C2×C3×D412yes—
323C4×D5—22no—
324C4×D5—22no—
325Dic10—21no—
326Dic10—22no—
327Dic10—22no—
328Dic10—41no—
329Dic10—41no—
330Dic10C2×C3×D412yes—
331C5×C8—22no—
332C5×C8—22no—
333C5×C8C2×C3×D412yes—
334C5×C8C2×C3×D412yes—
335#168—22no—
336C2×#41—51no—
337C22×Q8—51no—
338C22×D4—51no—
339#137—54no—
340#137—102no—
341AGL(1,ℤ/8)—54no—
342AGL(1,ℤ/8)—102no—
343C2×Q16—101no—
344C2×SD16—54no—
345C2×SD16—101no—
346C2×SD16—101no—
347C2×SD16—101no—
348C2×SD16—101no—
349C2×SD16—102no—
350C2×D8—101no—
351#131—54no—
352#131—102no—
353C2×M16—51no—
354C2×M16—101no—
355C2×M16C2×C3×D511yescharacteristic
356C2×M16C2×C3×D511yescharacteristic
357C22×C8—101no—
358C22×C8C2×C3×D511yescharacteristic
359#108C2×C3×D514yescomplemented
360#101—54no—
361#100—54no—
362C2×C3×C5—21no—
363C2×C3×C5—21no—
364C2×C3×C5#47112yescomplemented
365C2×C3×C5C2×#12111yescharacteristic
366C3×D5—42no—
367C3×D5—82no—
368C23×C3—101no—
369C23×C3—102no—
370C23×C3—201no—
371C23×C3—201no—
372C23×C3—402no—
373C23×C3#71311yescharacteristic
374C3×Q8—52no—
375C3×Q8—101no—
376C3×Q8—102no—
377C3×Q8—102no—
378C3×Q8—201no—
379C3×Q8—201no—
380C3×D4—52no—
381C3×D4—102no—
382C3×D4—102no—
383C3×D4—102no—
384C3×D4—102no—
385C3×D4—101no—
386C3×D4—201no—
387C3×D4—201no—
388C2×C3×C4—101no—
389C2×C3×C4—102no—
390C2×C3×C4—102no—
391C2×C3×C4—102no—
392C2×C3×C4—201no—
393C2×C3×C4—201no—
394C2×C3×C4C2×D2011yescharacteristic
395C2×C3×C4C2×D2011yescharacteristic
396C2×C3×C4D4×D511yescharacteristic
397C2×C3×C4D4×D511yescharacteristic
398C2×C3×C4#71311yescharacteristic
399C2×C3×C4C23×D511yescharacteristic
400C3×C8—22no—
401C3×C8—22no—
402C3×C8—102no—
403C3×C8C2×D2012yes—
404C3×C8C2×D2012yes—
405C22×C5—22no—
406C22×C5C3×#12111yescharacteristic
407C22×C5C3×#12111yescharacteristic
408C22×C5C2×C3×#4111yescharacteristic
409C2×D5—22no—
410C2×D5—42no—
411C2×D5—42no—
412C2×D5—82no—
413C2×D5—82no—
414C2×D5—82no—
415C4×C5C3×#12111yescharacteristic
416C4×C5C3×#12111yescharacteristic
417C4×C5C22×C3×D411yescharacteristic
418C4×C5C22×C3×D411yescharacteristic
419Dic5—22no—
420Dic5—42no—
421C24—101no—
422#41—52no—
423#41—52no—
424#41—101no—
425#41—101no—
426C2×Q8—51no—
427C2×Q8—52no—
428C2×Q8—101no—
429C2×Q8—101no—
430C2×Q8—101no—
431C2×Q8—101no—
432C2×Q8—101no—
433C2×D4—51no—
434C2×D4—52no—
435C2×D4—101no—
436C2×D4—101no—
437C2×D4—101no—
438C2×D4—101no—
439C2×D4—101no—
440C2×D4—101no—
441C22×C4—101no—
442C22×C4—101no—
443C22×C4C22×C3×D511yescharacteristic
444Q16—104no—
445SD16—104no—
446SD16—104no—
447SD16—104no—
448SD16—104no—
449D8—104no—
450M16—21no—
451M16—21no—
452M16—52no—
453M16—101no—
454M16—104no—
455M16C22×C3×D512yes—
456M16C22×C3×D512yes—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—101no—
460C2×C8—104no—
461C2×C8C3×D2011yescharacteristic
462C2×C8C3×D2011yescharacteristic
463C2×C8C3×D2012yes—
464C2×C8C22×C3×D512yes—
465C3×C5C2×#47111yescharacteristic, complemented, Hall
466C22×C3—22no—
467C22×C3—102no—
468C22×C3—202no—
469C22×C3—202no—
470C22×C3—402no—
471C22×C3—402no—
472C22×C3—402no—
473C22×C3#404111yescharacteristic
474C22×C3#404111yescharacteristic
475C22×C3C2×#71311yescharacteristic
476C3×C4—102no—
477C3×C4—202no—
478C3×C4#404111yescharacteristic
479C3×C4#404111yescharacteristic
480C3×C4C22×D2011yescharacteristic
481C3×C4C2×D4×D511yescharacteristic
482C2×C5—21no—
483C2×C5—21no—
484C2×C5C3×#47112yescomplemented
485C2×C5C2×C3×#12111yescharacteristic
486D5—42no—
487D5—82no—
488C23—101no—
489C23—102no—
490C23—201no—
491C23—201no—
492C23—402no—
493C23C3×#71311yescharacteristic
494Q8—52no—
495Q8—101no—
496Q8—102no—
497Q8—102no—
498Q8—201no—
499Q8—201no—
500D4—52no—
501D4—102no—
502D4—102no—
503D4—102no—
504D4—102no—
505D4—101no—
506D4—201no—
507D4—201no—
508C2×C4—101no—
509C2×C4—102no—
510C2×C4—102no—
511C2×C4—102no—
512C2×C4—201no—
513C2×C4—201no—
514C2×C4C2×C3×D2011yescharacteristic
515C2×C4C2×C3×D2011yescharacteristic
516C2×C4C3×D4×D511yescharacteristic
517C2×C4C3×D4×D511yescharacteristic
518C2×C4C3×#71311yescharacteristic
519C2×C4C23×C3×D511yescharacteristic
520C8—22no—
521C8—22no—
522C8—102no—
523C8C2×C3×D2012yes—
524C8C2×C3×D2012yes—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—202no—
528C2×C3—402no—
529C2×C3#6518712yesdirect product factor
530C2×C3C2×#404111yescharacteristic
531C5C2×C3×#47111yescharacteristic, complemented, Sylow
532C22—22no—
533C22—102no—
534C22—202no—
535C22—202no—
536C22—402no—
537C22—402no—
538C22—402no—
539C22C3×#404111yescharacteristic
540C22C3×#404111yescharacteristic
541C22C2×C3×#71311yescharacteristic
542C4—102no—
543C4—202no—
544C4C3×#404111yescharacteristic
545C4C3×#404111yescharacteristic
546C4C22×C3×D2011yescharacteristic
547C4C2×C3×D4×D511yescharacteristic
548C3C2×#6518711yescharacteristic, direct product factor, Sylow
549C2—21no—
550C2—21no—
551C2—202no—
552C2—402no—
553C2C3×#6518712yesdirect product factor
554C2C2×C3×#404111yescharacteristic
subgroupG/H
centerC22×C3#4041
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×#108C2
socleC22×C3×C5#121
prime radicalC2×C3×#4054C2
hypercenterC2×C3×#108D5
hypocenterC5C2×C3×#471
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#108

Complements

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

NH ≅ G/Ntype
1. C2×C3×#4055C2semidirect
3. C22×C3×#677C2semidirect
4. C2×C3×#4047C2semidirect
5. C2×C3×C5×#108C2semidirect
6. C2×C3×#3956C2semidirect
7. C2×C3×#3955C2semidirect
8. C3×#65187C2direct (N × H)
9. C2×#65187C3direct (N × H)
18. C3×#4055C22semidirect
28. C2×C3×#677C22semidirect
30. C3×#4047C22semidirect
33. C3×C5×#108C22semidirect
34. C3×#3956C22semidirect
35. C3×#3955C22semidirect
37. C2×#4055C2×C3semidirect
39. C22×#677C2×C3semidirect
40. C2×#4047C2×C3semidirect
41. C2×C5×#108C2×C3semidirect
42. C2×#3956C2×C3semidirect
43. C2×#3955C2×C3semidirect
44. #65187C2×C3direct (N × H)
94. C2×C3×#108D5semidirect
105. #4055C22×C3semidirect
115. C2×#677C22×C3semidirect
117. #4047C22×C3semidirect
120. C5×#108C22×C3semidirect
121. #3956C22×C3semidirect
122. #3955C22×C3semidirect
184. C3×#108C2×D5semidirect
236. C2×#108C3×D5semidirect
359. #108C2×C3×D5semidirect
364. C2×C3×C5#471semidirect
465. C3×C5C2×#471semidirect
484. C2×C5C3×#471semidirect
529. C2×C3#65187direct (N × H)
531. C5C2×C3×#471semidirect
548. C3C2×#65187direct (N × H)
553. C2C3×#65187direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40552
C2×C3×#40542
C22×C3×#6772
C2×C3×#40472
C2×C3×C5×#1082
C2×C3×#39562
C2×C3×#39552
C3×#651872
C2×#651873
C2×C3×#4715