Finite Group Databasebeta

← C2×C3×#65076 #14425927

Subgroups — C2×C3×#65076 #14425927

Order 1920.

1128 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×#4082C211yesmaximal, characteristic, complemented
2C22×C3×#686C211yesmaximal, characteristic
3C22×C3×#677C211yesmaximal, characteristic, complemented
4C2×C3×#4039C211yesmaximal, characteristic, complemented
5C2×C3×C5×#102C211yesmaximal, characteristic, complemented
6C2×C3×#3946C211yesmaximal, characteristic, complemented
7C2×C3×#3942C211yesmaximal, characteristic, complemented
8C3×#65076C218yesmaximal, direct product factor
9C2×#65076C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C5×D4C2211yescharacteristic
11C22×C3×D20C2211yescharacteristic
12C22×C3×C4×D5—21no—
13C22×C3×Dic10C2211yescharacteristic
14C22×C3×C5×C8C2211yescharacteristic
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C3×C22⋊Dic5—21no—
17C2×C3×C22⋊Dic5—21no—
18C3×#4082—22no—
19C3×#4082C2214yescomplemented
20C2×C3×#686—24no—
21C2×C3×#686C2214yes—
22C2×C3×#685—21no—
23C2×C3×C4×Dic5C2211yescharacteristic
24C22×C3×#168C2211yescharacteristic
25C2×C3×#677—24no—
26C2×C3×#677C2214yescomplemented
27C3×#4039—22no—
28C3×#4039C2214yescomplemented
29C3×C5×#102C2214yescomplemented
30C3×#3946C2214yescomplemented
31C3×#3942C2214yescomplemented
32C2×C3×#463—51nomaximal, Hall
33C2×#4082C2×C311yescharacteristic, complemented
34C22×#686C2×C311yescharacteristic
35C22×#677C2×C311yescharacteristic, complemented
36C2×#4039C2×C311yescharacteristic, complemented
37C2×C5×#102C2×C311yescharacteristic, complemented
38C2×#3946C2×C311yescharacteristic, complemented
39C2×#3942C2×C311yescharacteristic, complemented
40#65076C2×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
49C22×C3×Dic5—21no—
50C22×C3×Dic5D411yescharacteristic
51C22×C3×Dic5D411yescharacteristic
52C2×C3×D20—21no—
53C2×C3×D20—22no—
54C2×C3×D20D414yes—
55C2×C3×D20C2312yes—
56C2×C3×C4×D5—24no—
57C2×C3×C4×D5—44no—
58C2×C3×Dic10—22no—
59C2×C3×Dic10—22no—
60C2×C3×Dic10—21no—
61C2×C3×Dic10C2312yes—
62C2×C3×C5×C8—22no—
63C2×C3×C5×C8C2312yes—
64C3×C5×C4⋊C4—21no—
65C3×C5×C4⋊C4C2312yes—
66C3×C22⋊Dic5—24no—
67C3×C22⋊Dic5—24no—
68C3×#686—28no—
69C3×#685—24no—
70C3×C4×Dic5D412yes—
71C3×C4×Dic5C2312yes—
72C2×C3×#168—22no—
73C2×C3×#168C2312yes—
74C3×#677—28no—
75C22×C3×SD16—51no—
76C22×C3×D8—51no—
77C2×C3×#124—51no—
78C2×C3×C4×D4—51no—
79C3×#463—58no—
80C2×C3×C4⋊C8—51no—
81C2×C3×#103—51no—
82C2×C3×#102D511yescharacteristic, complemented
83C22×C5×D4C22×C311yescharacteristic
84C22×D20C22×C311yescharacteristic
85C22×C4×D5—21no—
86C22×Dic10C22×C311yescharacteristic
87C22×C5×C8C22×C311yescharacteristic
88C2×C5×C4⋊C4C22×C311yescharacteristic
89C2×C22⋊Dic5—21no—
90C2×C22⋊Dic5—21no—
91#4082—22no—
92#4082C22×C314yescomplemented
93C2×#686—24no—
94C2×#686C22×C314yes—
95C2×#685—21no—
96C2×C4×Dic5C22×C311yescharacteristic
97C22×#168C22×C311yescharacteristic
98C2×#677—24no—
99C2×#677C22×C314yescomplemented
100#4039—22no—
101#4039C22×C314yescomplemented
102C5×#102C22×C314yescomplemented
103#3946C22×C314yescomplemented
104#3942C22×C314yescomplemented
105C2×#463—51noSylow
106C23×C3×C5—22no—
107C23×C3×C5—42no—
108C23×C3×C5—42no—
109C23×C3×C5—42no—
110C23×C3×C5C2×D411yescharacteristic
111C22×C3×D5—24no—
112C22×C3×D5—22no—
113C22×C3×D5—42no—
114C22×C3×D5—42no—
115C3×C5×D4—22no—
116C3×C5×D4—22no—
117C3×C5×D4—42no—
118C2×C3×C4×C5—22no—
119C2×C3×C4×C5—42no—
120C2×C3×C4×C5C2×D411yescharacteristic
121C2×C3×C4×C5C2×D412yes—
122C2×C3×C4×C5#4112yes—
123C2×C3×C4×C5C2411yescharacteristic
124C2×C3×Dic5—22no—
125C2×C3×Dic5—22no—
126C2×C3×Dic5—22no—
127C2×C3×Dic5—42no—
128C2×C3×Dic5C2×D412yes—
129C2×C3×Dic5C2×D412yes—
130C3×D20—24no—
131C3×D20—22no—
132C3×D20C2×D414yes—
133C3×C4×D5—48no—
134C3×Dic10—22no—
135C3×Dic10—22no—
136C3×Dic10—42no—
137C3×C5×C8—22no—
138C3×#168—22no—
139C22×C3×Q8—51no—
140C22×C3×D4—51no—
141C22×C3×D4C2×D511yescharacteristic
142C23×C3×C4—101no—
143C2×C3×SD16—54no—
144C2×C3×SD16—104no—
145C2×C3×D8—54no—
146C2×C3×D8—104no—
147C22×C3×C8—51no—
148C22×C3×C8C2×D511yescharacteristic
149C3×#124—54no—
150C3×#124—102no—
151C3×C4×D4—54no—
152C3×C4×D4—102no—
153C2×C3×C4⋊C4C2×D511yescharacteristic
154C2×C3×C22⋊C4—101no—
155C2×C3×C22⋊C4—101no—
156C2×C3×C22⋊C4—101no—
157C2×C3×C42—51no—
158C3×C4⋊C8—54no—
159C3×#103—54no—
160C3×#102C2×D514yescomplemented
161C24×C5—21no—
162C23×D5—21no—
163C2×C5×D4—22no—
164C2×C5×D4—21no—
165C2×C5×D4—22no—
166C2×C5×D4C23×C312yes—
167C22×C4×C5—21no—
168C22×C4×C5C23×C311yescharacteristic
169C22×Dic5—21no—
170C22×Dic5C3×D411yescharacteristic
171C22×Dic5C3×D411yescharacteristic
172C2×D20—21no—
173C2×D20—22no—
174C2×D20C3×D414yes—
175C2×D20C23×C312yes—
176C2×C4×D5—24no—
177C2×C4×D5—44no—
178C2×Dic10—22no—
179C2×Dic10—22no—
180C2×Dic10—21no—
181C2×Dic10C23×C312yes—
182C2×C5×C8—22no—
183C2×C5×C8C23×C312yes—
184C5×C4⋊C4—21no—
185C5×C4⋊C4C23×C312yes—
186C22⋊Dic5—24no—
187C22⋊Dic5—24no—
188#686—28no—
189#685—24no—
190C4×Dic5C3×D412yes—
191C4×Dic5C23×C312yes—
192C2×#168—22no—
193C2×#168C23×C312yes—
194#677—28no—
195C22×SD16—51no—
196C22×D8—51no—
197C2×#124—51no—
198C2×C4×D4—51no—
199#463—58no—
200C2×C4⋊C8—51no—
201C2×#103—51no—
202C2×#102C3×D511yescharacteristic, complemented
203C22×C3×C5—42no—
204C22×C3×C5—42no—
205C22×C3×C5—42no—
206C22×C3×C5—84no—
207C22×C3×C5#12112yes—
208C22×C3×C5#13512yes—
209C22×C3×C5AGL(1,ℤ/8)12yes—
210C22×C3×C5C22×D411yescharacteristic
211C2×C3×D5—24no—
212C2×C3×D5—42no—
213C2×C3×D5—42no—
214C2×C3×D5—48no—
215C3×C4×C5—42no—
216C3×C4×C5#12112yes—
217C3×C4×C5C22×D411yescharacteristic
218C3×C4×C5C2×#4111yescharacteristic
219C3×Dic5—22no—
220C3×Dic5—22no—
221C3×Dic5—42no—
222C24×C3—21no—
223C24×C3—101no—
224C2×C3×Q8—52no—
225C2×C3×Q8—101no—
226C2×C3×Q8—102no—
227C2×C3×Q8—102no—
228C2×C3×D4—22no—
229C2×C3×D4—22no—
230C2×C3×D4—21no—
231C2×C3×D4—54no—
232C2×C3×D4—52no—
233C2×C3×D4—102no—
234C2×C3×D4—101no—
235C2×C3×D4C22×D512yes—
236C22×C3×C4—21no—
237C22×C3×C4—51no—
238C22×C3×C4—51no—
239C22×C3×C4—104no—
240C22×C3×C4—101no—
241C22×C3×C4—204no—
242C22×C3×C4C22×D511yescharacteristic
243C3×SD16—108no—
244C3×D8—108no—
245C2×C3×C8—22no—
246C2×C3×C8—52no—
247C2×C3×C8—102no—
248C2×C3×C8C22×D512yes—
249C3×C4⋊C4—21no—
250C3×C4⋊C4C22×D512yes—
251C3×C22⋊C4—104no—
252C3×C22⋊C4—104no—
253C3×C22⋊C4—104no—
254C3×C42—52no—
255C3×C42—52no—
256C23×C5—22no—
257C23×C5—42no—
258C23×C5—42no—
259C23×C5—42no—
260C23×C5C2×C3×D411yescharacteristic
261C22×D5—24no—
262C22×D5—22no—
263C22×D5—42no—
264C22×D5—42no—
265C5×D4—22no—
266C5×D4—22no—
267C5×D4—42no—
268C2×C4×C5—22no—
269C2×C4×C5—42no—
270C2×C4×C5C2×C3×D411yescharacteristic
271C2×C4×C5C2×C3×D412yes—
272C2×C4×C5C3×#4112yes—
273C2×C4×C5C24×C311yescharacteristic
274C2×Dic5—22no—
275C2×Dic5—22no—
276C2×Dic5—22no—
277C2×Dic5—42no—
278C2×Dic5C2×C3×D412yes—
279C2×Dic5C2×C3×D412yes—
280D20—24no—
281D20—22no—
282D20C2×C3×D414yes—
283C4×D5—48no—
284Dic10—22no—
285Dic10—22no—
286Dic10—42no—
287C5×C8—22no—
288#168—22no—
289C22×Q8—51no—
290C22×D4—51no—
291C22×D4C2×C3×D511yescharacteristic
292C23×C4—101no—
293C2×SD16—54no—
294C2×SD16—104no—
295C2×D8—54no—
296C2×D8—104no—
297C22×C8—51no—
298C22×C8C2×C3×D511yescharacteristic
299#124—54no—
300#124—102no—
301C4×D4—54no—
302C4×D4—102no—
303C2×C4⋊C4C2×C3×D511yescharacteristic
304C2×C22⋊C4—101no—
305C2×C22⋊C4—101no—
306C2×C22⋊C4—101no—
307C2×C42—51no—
308C4⋊C8—54no—
309#103—54no—
310#102C2×C3×D514yescomplemented
311C2×C3×C5—82no—
312C2×C3×C5#46314yescomplemented
313C2×C3×C5C2×#12111yescharacteristic
314C2×C3×C5C2×#13511yescharacteristic
315C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
316C3×D5—44no—
317C23×C3—22no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—102no—
322C23×C3—104no—
323C23×C3—202no—
324C23×C3—202no—
325C23×C3D4×D511yescharacteristic
326C3×Q8—102no—
327C3×Q8—102no—
328C3×Q8—202no—
329C3×D4—22no—
330C3×D4—22no—
331C3×D4—42no—
332C3×D4—54no—
333C3×D4—102no—
334C3×D4—104no—
335C2×C3×C4—22no—
336C2×C3×C4—42no—
337C2×C3×C4—52no—
338C2×C3×C4—52no—
339C2×C3×C4—102no—
340C2×C3×C4—102no—
341C2×C3×C4—102no—
342C2×C3×C4—208no—
343C2×C3×C4—202no—
344C2×C3×C4#70912yes—
345C2×C3×C4D4×D512yes—
346C2×C3×C4D4×D511yescharacteristic
347C2×C3×C4C23×D511yescharacteristic
348C3×C8—22no—
349C3×C8—102no—
350C22×C5—42no—
351C22×C5—42no—
352C22×C5—42no—
353C22×C5—84no—
354C22×C5C3×#12112yes—
355C22×C5C3×#13512yes—
356C22×C5C3×AGL(1,ℤ/8)12yes—
357C22×C5C22×C3×D411yescharacteristic
358C2×D5—24no—
359C2×D5—42no—
360C2×D5—42no—
361C2×D5—48no—
362C4×C5—42no—
363C4×C5C3×#12112yes—
364C4×C5C22×C3×D411yescharacteristic
365C4×C5C2×C3×#4111yescharacteristic
366Dic5—22no—
367Dic5—22no—
368Dic5—42no—
369C24—21no—
370C24—101no—
371C2×Q8—52no—
372C2×Q8—101no—
373C2×Q8—102no—
374C2×Q8—102no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—21no—
378C2×D4—54no—
379C2×D4—52no—
380C2×D4—102no—
381C2×D4—101no—
382C2×D4C22×C3×D512yes—
383C22×C4—21no—
384C22×C4—51no—
385C22×C4—51no—
386C22×C4—104no—
387C22×C4—101no—
388C22×C4—204no—
389C22×C4C22×C3×D511yescharacteristic
390SD16—108no—
391D8—108no—
392C2×C8—22no—
393C2×C8—52no—
394C2×C8—102no—
395C2×C8C22×C3×D512yes—
396C4⋊C4—21no—
397C4⋊C4C22×C3×D512yes—
398C22⋊C4—104no—
399C22⋊C4—104no—
400C22⋊C4—104no—
401C42—52no—
402C42—52no—
403C3×C5C2×#46311yescharacteristic, complemented, Hall
404C22×C3—42no—
405C22×C3—42no—
406C22×C3—42no—
407C22×C3—84no—
408C22×C3—104no—
409C22×C3—202no—
410C22×C3—202no—
411C22×C3—208no—
412C22×C3#403012yes—
413C22×C3#405712yes—
414C22×C3#406212yes—
415C22×C3C2×D4×D511yescharacteristic
416C3×C4—42no—
417C3×C4—102no—
418C3×C4—102no—
419C3×C4—202no—
420C3×C4#403012yes—
421C3×C4C2×#70911yescharacteristic
422C3×C4C2×D4×D511yescharacteristic
423C2×C5—82no—
424C2×C5C3×#46314yescomplemented
425C2×C5C2×C3×#12111yescharacteristic
426C2×C5C2×C3×#13511yescharacteristic
427C2×C5C2×C3×AGL(1,ℤ/8)11yescharacteristic
428D5—44no—
429C23—22no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—102no—
434C23—104no—
435C23—202no—
436C23—202no—
437C23C3×D4×D511yescharacteristic
438Q8—102no—
439Q8—102no—
440Q8—202no—
441D4—22no—
442D4—22no—
443D4—42no—
444D4—54no—
445D4—102no—
446D4—104no—
447C2×C4—22no—
448C2×C4—42no—
449C2×C4—52no—
450C2×C4—52no—
451C2×C4—102no—
452C2×C4—102no—
453C2×C4—102no—
454C2×C4—208no—
455C2×C4—202no—
456C2×C4C3×#70912yes—
457C2×C4C3×D4×D512yes—
458C2×C4C3×D4×D511yescharacteristic
459C2×C4C23×C3×D511yescharacteristic
460C8—22no—
461C8—102no—
462C2×C3—82no—
463C2×C3—204no—
464C2×C3#6507614yesdirect product factor
465C2×C3C2×#403011yescharacteristic
466C2×C3C2×#405711yescharacteristic
467C2×C3C2×#406211yescharacteristic
468C5C2×C3×#46311yescharacteristic, complemented, Sylow
469C22—42no—
470C22—42no—
471C22—42no—
472C22—84no—
473C22—104no—
474C22—208no—
475C22—202no—
476C22—202no—
477C22C3×#403012yes—
478C22C3×#405712yes—
479C22C3×#406212yes—
480C22C2×C3×D4×D511yescharacteristic
481C4—42no—
482C4—102no—
483C4—102no—
484C4—202no—
485C4C3×#403012yes—
486C4C2×C3×#70911yescharacteristic
487C4C2×C3×D4×D511yescharacteristic
488C3C2×#6507611yescharacteristic, direct product factor, Sylow
489C2—82no—
490C2—204no—
491C2C3×#6507614yesdirect product factor
492C2C2×C3×#403011yescharacteristic
493C2C2×C3×#405711yescharacteristic
494C2C2×C3×#406211yescharacteristic
subgroupG/H
centerC23×C3D4×D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×#102C2
socleC23×C3×C5C2×D4
prime radicalC22×C3×#686C2
hypercenterC2×C3×#102D5
hypocenterC5C2×C3×#463
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
1. C2×C3×#4082C2semidirect
3. C22×C3×#677C2semidirect
4. C2×C3×#4039C2semidirect
5. C2×C3×C5×#102C2semidirect
6. C2×C3×#3946C2semidirect
7. C2×C3×#3942C2semidirect
8. C3×#65076C2direct (N × H)
9. C2×#65076C3direct (N × H)
19. C3×#4082C22semidirect
26. C2×C3×#677C22semidirect
28. C3×#4039C22semidirect
29. C3×C5×#102C22semidirect
30. C3×#3946C22semidirect
31. C3×#3942C22semidirect
33. C2×#4082C2×C3semidirect
35. C22×#677C2×C3semidirect
36. C2×#4039C2×C3semidirect
37. C2×C5×#102C2×C3semidirect
38. C2×#3946C2×C3semidirect
39. C2×#3942C2×C3semidirect
40. #65076C2×C3direct (N × H)
82. C2×C3×#102D5semidirect
92. #4082C22×C3semidirect
99. C2×#677C22×C3semidirect
101. #4039C22×C3semidirect
102. C5×#102C22×C3semidirect
103. #3946C22×C3semidirect
104. #3942C22×C3semidirect
160. C3×#102C2×D5semidirect
202. C2×#102C3×D5semidirect
310. #102C2×C3×D5semidirect
312. C2×C3×C5#463semidirect
403. C3×C5C2×#463semidirect
424. C2×C5C3×#463semidirect
464. C2×C3#65076direct (N × H)
468. C5C2×C3×#463semidirect
488. C3C2×#65076direct (N × H)
491. C2C3×#65076direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40822
C22×C3×#6862
C22×C3×#6772
C2×C3×#40392
C2×C3×C5×#1022
C2×C3×#39462
C2×C3×#39422
C3×#650762
C2×#650763
C2×C3×#4635