Finite Group Databasebeta

← C2×C3×#65051 #14426014

Subgroups — C2×C3×#65051 #14426014

Order 1920.

1168 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4086C211yesmaximal, characteristic
2C22×C3×#687C211yesmaximal, characteristic, complemented
3C22×C3×#677C211yesmaximal, characteristic, complemented
4C2×C3×#4033C211yesmaximal, characteristic, complemented
5C2×C3×C5×#102C211yesmaximal, characteristic, complemented
6C2×C3×#3946C211yesmaximal, characteristic, complemented
7C2×C3×#3941C211yesmaximal, characteristic, complemented
8C3×#65051C218yesmaximal, direct product factor
9C2×#65051C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C5×D4C2211yescharacteristic
11C22×C3×C22⋊D5—21no—
12C22×C3×C22⋊D5—21no—
13C22×C3×D20C2211yescharacteristic
14C22×C3×Dic10C2211yescharacteristic
15C22×C3×C5×C8C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C3×#4086—22no—
18C3×#4086C2214yes—
19C2×C3×#687—24no—
20C2×C3×#687C2214yescomplemented
21C2×C3×#683—21no—
22C2×C3×C4×Dic5—21no—
23C2×C3×C4×Dic5C2211yescharacteristic
24C22×C3×#168C2211yescharacteristic
25C2×C3×#677—24no—
26C2×C3×#677C2214yescomplemented
27C3×#4033—22no—
28C3×#4033C2214yescomplemented
29C3×C5×#102C2214yescomplemented
30C3×#3946C2214yescomplemented
31C3×#3941C2214yescomplemented
32C2×C3×#460—51nomaximal, Hall
33C2×#4086C2×C311yescharacteristic
34C22×#687C2×C311yescharacteristic, complemented
35C22×#677C2×C311yescharacteristic, complemented
36C2×#4033C2×C311yescharacteristic, complemented
37C2×C5×#102C2×C311yescharacteristic, complemented
38C2×#3946C2×C311yescharacteristic, complemented
39C2×#3941C2×C311yescharacteristic, complemented
40#65051C2×C318yesdirect product factor
41C24×C3×C5—21no—
42C23×C3×D5—21no—
43C2×C3×C5×D4—22no—
44C2×C3×C5×D4—21no—
45C2×C3×C5×D4—22no—
46C2×C3×C5×D4C2312yes—
47C22×C3×C4×C5—21no—
48C22×C3×C4×C5C2311yescharacteristic
49C2×C3×C22⋊D5—24no—
50C2×C3×C22⋊D5—24no—
51C2×C3×C22⋊D5—44no—
52C2×C3×C22⋊D5—44no—
53C22×C3×Dic5—21no—
54C22×C3×Dic5D411yescharacteristic
55C22×C3×Dic5D411yescharacteristic
56C2×C3×D20—22no—
57C2×C3×D20—22no—
58C2×C3×D20—21no—
59C2×C3×D20C2312yes—
60C2×C3×Dic10—21no—
61C2×C3×Dic10—22no—
62C2×C3×Dic10D414yescomplemented
63C2×C3×Dic10C2312yes—
64C2×C3×C5×C8—22no—
65C2×C3×C5×C8C2312yes—
66C3×C5×C4⋊C4—21no—
67C3×C5×C4⋊C4C2312yes—
68C3×#687—28no—
69C3×#683—24no—
70C3×C4×Dic5—24no—
71C3×C4×Dic5D412yes—
72C3×C4×Dic5C2312yes—
73C2×C3×#168—22no—
74C2×C3×#168C2312yes—
75C3×#677—28no—
76C22×C3×SD16—51no—
77C22×C3×SD16—51no—
78C2×C3×#127—51no—
79C2×C3×C4×Q8—51no—
80C3×#460—58no—
81C2×C3×C4⋊C8—51no—
82C2×C3×#102—51no—
83C2×C3×#102D511yescharacteristic, complemented
84C22×C5×D4C22×C311yescharacteristic
85C22×C22⋊D5—21no—
86C22×C22⋊D5—21no—
87C22×D20C22×C311yescharacteristic
88C22×Dic10C22×C311yescharacteristic
89C22×C5×C8C22×C311yescharacteristic
90C2×C5×C4⋊C4C22×C311yescharacteristic
91#4086—22no—
92#4086C22×C314yes—
93C2×#687—24no—
94C2×#687C22×C314yescomplemented
95C2×#683—21no—
96C2×C4×Dic5—21no—
97C2×C4×Dic5C22×C311yescharacteristic
98C22×#168C22×C311yescharacteristic
99C2×#677—24no—
100C2×#677C22×C314yescomplemented
101#4033—22no—
102#4033C22×C314yescomplemented
103C5×#102C22×C314yescomplemented
104#3946C22×C314yescomplemented
105#3941C22×C314yescomplemented
106C2×#460—51noSylow
107C23×C3×C5—22no—
108C23×C3×C5—42no—
109C23×C3×C5—42no—
110C23×C3×C5—42no—
111C23×C3×C5C2×D411yescharacteristic
112C22×C3×D5—22no—
113C22×C3×D5—42no—
114C22×C3×D5—42no—
115C22×C3×D5—42no—
116C3×C5×D4—22no—
117C3×C5×D4—22no—
118C3×C5×D4—42no—
119C2×C3×C4×C5—22no—
120C2×C3×C4×C5—42no—
121C2×C3×C4×C5C2×D411yescharacteristic
122C2×C3×C4×C5C2×D412yes—
123C2×C3×C4×C5#4112yes—
124C2×C3×C4×C5C2411yescharacteristic
125C3×C22⋊D5—48no—
126C3×C22⋊D5—48no—
127C2×C3×Dic5—24no—
128C2×C3×Dic5—22no—
129C2×C3×Dic5—22no—
130C2×C3×Dic5SD1614yescomplemented
131C2×C3×Dic5C2×D412yes—
132C2×C3×Dic5C2×D412yes—
133C3×D20—22no—
134C3×D20—22no—
135C3×D20—42no—
136C3×Dic10—24no—
137C3×Dic10—22no—
138C3×Dic10C2×D414yescomplemented
139C3×C5×C8—22no—
140C3×#168—22no—
141C22×C3×Q8—51no—
142C22×C3×D4—51no—
143C22×C3×D4—101no—
144C22×C3×D4—101no—
145C22×C3×D4C2×D511yescharacteristic
146C2×C3×SD16—54no—
147C2×C3×SD16—54no—
148C2×C3×SD16—104no—
149C2×C3×SD16—104no—
150C22×C3×C8—51no—
151C22×C3×C8C2×D511yescharacteristic
152C3×#127—54no—
153C3×#127—102no—
154C3×C4×Q8—54no—
155C3×C4×Q8—102no—
156C2×C3×C4⋊C4—101no—
157C2×C3×C4⋊C4C2×D511yescharacteristic
158C2×C3×C42—51no—
159C2×C3×C42—101no—
160C3×C4⋊C8—54no—
161C3×#102—54no—
162C3×#102C2×D514yescomplemented
163C24×C5—21no—
164C23×D5—21no—
165C2×C5×D4—22no—
166C2×C5×D4—21no—
167C2×C5×D4—22no—
168C2×C5×D4C23×C312yes—
169C22×C4×C5—21no—
170C22×C4×C5C23×C311yescharacteristic
171C2×C22⋊D5—24no—
172C2×C22⋊D5—24no—
173C2×C22⋊D5—44no—
174C2×C22⋊D5—44no—
175C22×Dic5—21no—
176C22×Dic5C3×D411yescharacteristic
177C22×Dic5C3×D411yescharacteristic
178C2×D20—22no—
179C2×D20—22no—
180C2×D20—21no—
181C2×D20C23×C312yes—
182C2×Dic10—21no—
183C2×Dic10—22no—
184C2×Dic10C3×D414yescomplemented
185C2×Dic10C23×C312yes—
186C2×C5×C8—22no—
187C2×C5×C8C23×C312yes—
188C5×C4⋊C4—21no—
189C5×C4⋊C4C23×C312yes—
190#687—28no—
191#683—24no—
192C4×Dic5—24no—
193C4×Dic5C3×D412yes—
194C4×Dic5C23×C312yes—
195C2×#168—22no—
196C2×#168C23×C312yes—
197#677—28no—
198C22×SD16—51no—
199C22×SD16—51no—
200C2×#127—51no—
201C2×C4×Q8—51no—
202#460—58no—
203C2×C4⋊C8—51no—
204C2×#102—51no—
205C2×#102C3×D511yescharacteristic, complemented
206C22×C3×C5—42no—
207C22×C3×C5—42no—
208C22×C3×C5—42no—
209C22×C3×C5—84no—
210C22×C3×C5#12112yes—
211C22×C3×C5C2×SD1612yes—
212C22×C3×C5AGL(1,ℤ/8)12yes—
213C22×C3×C5C22×D411yescharacteristic
214C2×C3×D5—42no—
215C2×C3×D5—42no—
216C2×C3×D5—42no—
217C2×C3×D5—84no—
218C3×C4×C5—42no—
219C3×C4×C5#12112yes—
220C3×C4×C5C22×D411yescharacteristic
221C3×C4×C5C2×#4111yescharacteristic
222C3×Dic5—24no—
223C3×Dic5—22no—
224C3×Dic5C2×SD1614yescomplemented
225C24×C3—21no—
226C24×C3—101no—
227C2×C3×Q8—54no—
228C2×C3×Q8—52no—
229C2×C3×Q8—102no—
230C2×C3×Q8—101no—
231C2×C3×D4—22no—
232C2×C3×D4—22no—
233C2×C3×D4—21no—
234C2×C3×D4—52no—
235C2×C3×D4—104no—
236C2×C3×D4—104no—
237C2×C3×D4—102no—
238C2×C3×D4—102no—
239C2×C3×D4—101no—
240C2×C3×D4—204no—
241C2×C3×D4—204no—
242C2×C3×D4C22×D512yes—
243C22×C3×C4—21no—
244C22×C3×C4—51no—
245C22×C3×C4—51no—
246C22×C3×C4—101no—
247C22×C3×C4C22×D511yescharacteristic
248C3×SD16—108no—
249C3×SD16—108no—
250C2×C3×C8—22no—
251C2×C3×C8—52no—
252C2×C3×C8—102no—
253C2×C3×C8C22×D512yes—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—104no—
256C3×C4⋊C4C22×D512yes—
257C3×C42—52no—
258C3×C42—52no—
259C3×C42—104no—
260C23×C5—22no—
261C23×C5—42no—
262C23×C5—42no—
263C23×C5—42no—
264C23×C5C2×C3×D411yescharacteristic
265C22×D5—22no—
266C22×D5—42no—
267C22×D5—42no—
268C22×D5—42no—
269C5×D4—22no—
270C5×D4—22no—
271C5×D4—42no—
272C2×C4×C5—22no—
273C2×C4×C5—42no—
274C2×C4×C5C2×C3×D411yescharacteristic
275C2×C4×C5C2×C3×D412yes—
276C2×C4×C5C3×#4112yes—
277C2×C4×C5C24×C311yescharacteristic
278C22⋊D5—48no—
279C22⋊D5—48no—
280C2×Dic5—24no—
281C2×Dic5—22no—
282C2×Dic5—22no—
283C2×Dic5C3×SD1614yescomplemented
284C2×Dic5C2×C3×D412yes—
285C2×Dic5C2×C3×D412yes—
286D20—22no—
287D20—22no—
288D20—42no—
289Dic10—24no—
290Dic10—22no—
291Dic10C2×C3×D414yescomplemented
292C5×C8—22no—
293#168—22no—
294C22×Q8—51no—
295C22×D4—51no—
296C22×D4—101no—
297C22×D4—101no—
298C22×D4C2×C3×D511yescharacteristic
299C2×SD16—54no—
300C2×SD16—54no—
301C2×SD16—104no—
302C2×SD16—104no—
303C22×C8—51no—
304C22×C8C2×C3×D511yescharacteristic
305#127—54no—
306#127—102no—
307C4×Q8—54no—
308C4×Q8—102no—
309C2×C4⋊C4—101no—
310C2×C4⋊C4C2×C3×D511yescharacteristic
311C2×C42—51no—
312C2×C42—101no—
313C4⋊C8—54no—
314#102—54no—
315#102C2×C3×D514yescomplemented
316C2×C3×C5—82no—
317C2×C3×C5#46014yescomplemented
318C2×C3×C5C2×#12111yescharacteristic
319C2×C3×C5C22×SD1611yescharacteristic
320C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
321C3×D5—82no—
322C23×C3—22no—
323C23×C3—42no—
324C23×C3—42no—
325C23×C3—42no—
326C23×C3—102no—
327C23×C3—202no—
328C23×C3—202no—
329C23×C3—202no—
330C23×C3D4×D511yescharacteristic
331C3×Q8—54no—
332C3×Q8—102no—
333C3×Q8—104no—
334C3×D4—22no—
335C3×D4—22no—
336C3×D4—42no—
337C3×D4—102no—
338C3×D4—102no—
339C3×D4—208no—
340C3×D4—208no—
341C3×D4—202no—
342C2×C3×C4—22no—
343C2×C3×C4—42no—
344C2×C3×C4—52no—
345C2×C3×C4—54no—
346C2×C3×C4—52no—
347C2×C3×C4—102no—
348C2×C3×C4—104no—
349C2×C3×C4—102no—
350C2×C3×C4#70912yes—
351C2×C3×C4D4×D512yes—
352C2×C3×C4D4×D511yescharacteristic
353C2×C3×C4C23×D511yescharacteristic
354C3×C8—22no—
355C3×C8—102no—
356C22×C5—42no—
357C22×C5—42no—
358C22×C5—42no—
359C22×C5—84no—
360C22×C5C3×#12112yes—
361C22×C5C2×C3×SD1612yes—
362C22×C5C3×AGL(1,ℤ/8)12yes—
363C22×C5C22×C3×D411yescharacteristic
364C2×D5—42no—
365C2×D5—42no—
366C2×D5—42no—
367C2×D5—84no—
368C4×C5—42no—
369C4×C5C3×#12112yes—
370C4×C5C22×C3×D411yescharacteristic
371C4×C5C2×C3×#4111yescharacteristic
372Dic5—22no—
373Dic5—24no—
374Dic5C2×C3×SD1614yescomplemented
375C24—21no—
376C24—101no—
377C2×Q8—54no—
378C2×Q8—52no—
379C2×Q8—102no—
380C2×Q8—101no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—21no—
384C2×D4—52no—
385C2×D4—104no—
386C2×D4—104no—
387C2×D4—102no—
388C2×D4—102no—
389C2×D4—101no—
390C2×D4—204no—
391C2×D4—204no—
392C2×D4C22×C3×D512yes—
393C22×C4—21no—
394C22×C4—51no—
395C22×C4—51no—
396C22×C4—101no—
397C22×C4C22×C3×D511yescharacteristic
398SD16—108no—
399SD16—108no—
400C2×C8—22no—
401C2×C8—52no—
402C2×C8—102no—
403C2×C8C22×C3×D512yes—
404C4⋊C4—21no—
405C4⋊C4—104no—
406C4⋊C4C22×C3×D512yes—
407C42—52no—
408C42—52no—
409C42—104no—
410C3×C5C2×#46011yescharacteristic, complemented, Hall
411C22×C3—42no—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3—84no—
415C22×C3—202no—
416C22×C3—202no—
417C22×C3—202no—
418C22×C3—404no—
419C22×C3#403012yes—
420C22×C3#405712yes—
421C22×C3D5×SD1612yes—
422C22×C3C2×D4×D511yescharacteristic
423C3×C4—42no—
424C3×C4—54no—
425C3×C4—104no—
426C3×C4—102no—
427C3×C4#403012yes—
428C3×C4C2×#70911yescharacteristic
429C3×C4C2×D4×D511yescharacteristic
430C2×C5—82no—
431C2×C5C3×#46014yescomplemented
432C2×C5C2×C3×#12111yescharacteristic
433C2×C5C22×C3×SD1611yescharacteristic
434C2×C5C2×C3×AGL(1,ℤ/8)11yescharacteristic
435D5—82no—
436C23—22no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—102no—
441C23—202no—
442C23—202no—
443C23—202no—
444C23C3×D4×D511yescharacteristic
445Q8—54no—
446Q8—102no—
447Q8—104no—
448D4—22no—
449D4—22no—
450D4—42no—
451D4—102no—
452D4—102no—
453D4—208no—
454D4—208no—
455D4—202no—
456C2×C4—22no—
457C2×C4—42no—
458C2×C4—52no—
459C2×C4—54no—
460C2×C4—52no—
461C2×C4—102no—
462C2×C4—104no—
463C2×C4—102no—
464C2×C4C3×#70912yes—
465C2×C4C3×D4×D512yes—
466C2×C4C3×D4×D511yescharacteristic
467C2×C4C23×C3×D511yescharacteristic
468C8—22no—
469C8—102no—
470C2×C3—82no—
471C2×C3—402no—
472C2×C3#6505114yesdirect product factor
473C2×C3C2×#403011yescharacteristic
474C2×C3C2×#405711yescharacteristic
475C2×C3C2×D5×SD1611yescharacteristic
476C5C2×C3×#46011yescharacteristic, complemented, Sylow
477C22—42no—
478C22—42no—
479C22—42no—
480C22—84no—
481C22—202no—
482C22—202no—
483C22—202no—
484C22—404no—
485C22C3×#403012yes—
486C22C3×#405712yes—
487C22C3×D5×SD1612yes—
488C22C2×C3×D4×D511yescharacteristic
489C4—42no—
490C4—54no—
491C4—104no—
492C4—102no—
493C4C3×#403012yes—
494C4C2×C3×#70911yescharacteristic
495C4C2×C3×D4×D511yescharacteristic
496C3C2×#6505111yescharacteristic, direct product factor, Sylow
497C2—82no—
498C2—402no—
499C2C3×#6505114yesdirect product factor
500C2C2×C3×#403011yescharacteristic
501C2C2×C3×#405711yescharacteristic
502C2C2×C3×D5×SD1611yescharacteristic
subgroupG/H
centerC23×C3D4×D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×#102C2
socleC23×C3×C5C2×D4
prime radicalC2×C3×#4086C2
hypercenterC2×C3×#102D5
hypocenterC5C2×C3×#460
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C2×C3×#102

Complements

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

NH ≅ G/Ntype
2. C22×C3×#687C2semidirect
3. C22×C3×#677C2semidirect
4. C2×C3×#4033C2semidirect
5. C2×C3×C5×#102C2semidirect
6. C2×C3×#3946C2semidirect
7. C2×C3×#3941C2semidirect
8. C3×#65051C2direct (N × H)
9. C2×#65051C3direct (N × H)
20. C2×C3×#687C22semidirect
26. C2×C3×#677C22semidirect
28. C3×#4033C22semidirect
29. C3×C5×#102C22semidirect
30. C3×#3946C22semidirect
31. C3×#3941C22semidirect
34. C22×#687C2×C3semidirect
35. C22×#677C2×C3semidirect
36. C2×#4033C2×C3semidirect
37. C2×C5×#102C2×C3semidirect
38. C2×#3946C2×C3semidirect
39. C2×#3941C2×C3semidirect
40. #65051C2×C3direct (N × H)
62. C2×C3×Dic10D4semidirect
83. C2×C3×#102D5semidirect
94. C2×#687C22×C3semidirect
100. C2×#677C22×C3semidirect
102. #4033C22×C3semidirect
103. C5×#102C22×C3semidirect
104. #3946C22×C3semidirect
105. #3941C22×C3semidirect
130. C2×C3×Dic5SD16semidirect
138. C3×Dic10C2×D4semidirect
162. C3×#102C2×D5semidirect
184. C2×Dic10C3×D4semidirect
205. C2×#102C3×D5semidirect
224. C3×Dic5C2×SD16semidirect
283. C2×Dic5C3×SD16semidirect
291. Dic10C2×C3×D4semidirect
315. #102C2×C3×D5semidirect
317. C2×C3×C5#460semidirect
374. Dic5C2×C3×SD16semidirect
410. C3×C5C2×#460semidirect
431. C2×C5C3×#460semidirect
472. C2×C3#65051direct (N × H)
476. C5C2×C3×#460semidirect
496. C3C2×#65051direct (N × H)
499. C2C3×#65051direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40862
C22×C3×#6872
C22×C3×#6772
C2×C3×#40332
C2×C3×C5×#1022
C2×C3×#39462
C2×C3×#39412
C3×#650512
C2×#650513
C2×C3×#4605