Finite Group Databasebeta

← C3×#10615146 #14425375

Subgroups — C3×#10615146 #14425375

Order 1920.

1396 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×#4147C211yesmaximal, characteristic, complemented
2C3×C5×#536C211yesmaximal, characteristic, complemented
3C3×#66156C212yesmaximal, complemented
4C2×C3×#3968C211yesmaximal, characteristic, complemented
5C3×#65384C214yesmaximal, complemented
6C3×#65351C214yesmaximal, complemented
7C3×#65294C212yesmaximal, complemented
8#10615146C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C22×C3×C5×Q8C2211yescharacteristic
11C3×#4147—22no—
12C3×#4147—24no—
13C3×#4147C2214yes—
14C2×C3×#713—22no—
15C2×C3×Q8×D5—22no—
16C2×C3×#709—22no—
17C2×C3×#709C2212yes—
18C3×C5×C4⋊Q8C2212yescomplemented
19C3×C5×#124C2212yescomplemented
20C3×C5×#123—22no—
21C3×C5×C22⋊Q8—22no—
22C3×C5×#117C2211yescharacteristic
23C3×#4097—22no—
24C3×#4096—24no—
25C3×#4095—22no—
26C3×#4088C2214yescomplemented
27C3×#4078C2214yescomplemented
28C2×C3×#681C2212yes—
29C3×#3968C2214yescomplemented
30C3×#3937C2218yescomplemented
31C3×#3020—51nomaximal, Hall
32C2×#4147C2×C311yescharacteristic, complemented
33C5×#536C2×C311yescharacteristic, complemented
34#66156C2×C312yescomplemented
35C2×#3968C2×C311yescharacteristic, complemented
36#65384C2×C314yescomplemented
37#65351C2×C314yescomplemented
38#65294C2×C312yescomplemented
39C3×C5×#41—22no—
40C3×C5×#41—22no—
41C2×C3×C5×Q8—22no—
42C2×C3×C5×Q8—22no—
43C2×C3×C5×Q8D412yes—
44C2×C3×C5×Q8C2311yescharacteristic
45C2×C3×C5×Q8C2312yes—
46C2×C3×C5×D4—21no—
47C2×C3×C5×D4C2311yescharacteristic
48C22×C3×C4×C5—21no—
49C22×C3×C4×C5D412yes—
50C22×C3×C4×C5C2311yescharacteristic
51C2×C3×C22⋊D5—22no—
52C3×#713—28no—
53C3×#713—28no—
54C3×Q8×D5—28no—
55C3×Q8×D5—28no—
56C3×#709—28no—
57C3×#709—28no—
58C3×#709—22no—
59C3×#709—22no—
60C3×#709D414yes—
61C3×#709C2314yes—
62C2×C3×D20—22no—
63C2×C3×D20D412yes—
64C2×C3×C4×D5—24no—
65C2×C3×C4×D5—22no—
66C2×C3×Dic10—22no—
67C2×C3×Dic10D412yes—
68C3×C5×C4⋊C4—21no—
69C3×C5×C4⋊C4—24no—
70C3×C5×C22⋊C4—21no—
71C3×C5×C22⋊C4—24no—
72C3×C42×C5C2312yes—
73C3×#689—24no—
74C3×#688—24no—
75C3×#687—24no—
76C3×#686—24no—
77C3×#681—22no—
78C3×#681C2314yes—
79C2×C3×#168—22no—
80C2×C3×2−1+4—51no—
81C3×#575—52no—
82C3×#536D511yescharacteristic, complemented
83C3×#456—54no—
84C3×#454—54no—
85C3×#421—52no—
86C2×C3×#101—51no—
87C2×C5×#41C22×C311yescharacteristic
88C22×C5×Q8C22×C311yescharacteristic
89#4147—22no—
90#4147—24no—
91#4147C22×C314yes—
92C2×#713—22no—
93C2×Q8×D5—22no—
94C2×#709—22no—
95C2×#709C22×C312yes—
96C5×C4⋊Q8C22×C312yescomplemented
97C5×#124C22×C312yescomplemented
98C5×#123—22no—
99C5×C22⋊Q8—22no—
100C5×#117C22×C311yescharacteristic
101#4097—22no—
102#4096—24no—
103#4095—22no—
104#4088C22×C314yescomplemented
105#4078C22×C314yescomplemented
106C2×#681C22×C312yes—
107#3968C22×C314yescomplemented
108#3937C22×C318yescomplemented
109#3020—51noSylow
110C23×C3×C5—21no—
111C23×C3×C5C2×D411yescharacteristic
112C22×C3×D5—22no—
113C3×C5×Q8—24no—
114C3×C5×Q8—24no—
115C3×C5×Q8—22no—
116C3×C5×D4—22no—
117C3×C5×D4—41no—
118C3×C5×D4—41no—
119C2×C3×C4×C5—22no—
120C2×C3×C4×C5—22no—
121C2×C3×C4×C5—22no—
122C2×C3×C4×C5—21no—
123C2×C3×C4×C5—42no—
124C2×C3×C4×C5C2×D414yes—
125C2×C3×C4×C5C2×D411yescharacteristic
126C2×C3×C4×C5C2×D412yes—
127C2×C3×C4×C5C2×D412yes—
128C2×C3×C4×C5C2411yescharacteristic
129C3×C22⋊D5—24no—
130C3×C22⋊D5—24no—
131C2×C3×Dic5—22no—
132C3×D20—28no—
133C3×D20—22no—
134C3×D20C2×D414yes—
135C3×C4×D5—216no—
136C3×C4×D5—24no—
137C3×C4×D5—24no—
138C3×Dic10—28no—
139C3×Dic10—22no—
140C3×Dic10C2×D414yes—
141C3×#168—24no—
142C3×2−1+4—54no—
143C3×2−1+4—104no—
144C3×2−1+4—102no—
145C2×C3×#41—52no—
146C2×C3×#41—102no—
147C2×C3×#41—102no—
148C2×C3×#41C2×D511yescharacteristic
149C22×C3×Q8—102no—
150C22×C3×Q8C2×D511yescharacteristic
151C3×#137—54no—
152C3×#137—102no—
153C3×AGL(1,ℤ/8)—54no—
154C3×AGL(1,ℤ/8)—102no—
155C3×#135—104no—
156C2×C3×M16—52no—
157C3×C4⋊Q8C2×D512yescomplemented
158C3×#124C2×D512yescomplemented
159C3×#123—22no—
160C3×C22⋊Q8—22no—
161C3×#117C2×D511yescharacteristic
162C3×#104—58no—
163C3×#101—54no—
164C5×#41—22no—
165C5×#41—22no—
166C2×C5×Q8—22no—
167C2×C5×Q8—22no—
168C2×C5×Q8C3×D412yes—
169C2×C5×Q8C23×C311yescharacteristic
170C2×C5×Q8C23×C312yes—
171C2×C5×D4—21no—
172C2×C5×D4C23×C311yescharacteristic
173C22×C4×C5—21no—
174C22×C4×C5C3×D412yes—
175C22×C4×C5C23×C311yescharacteristic
176C2×C22⋊D5—22no—
177#713—28no—
178#713—28no—
179Q8×D5—28no—
180Q8×D5—28no—
181#709—28no—
182#709—28no—
183#709—22no—
184#709—22no—
185#709C3×D414yes—
186#709C23×C314yes—
187C2×D20—22no—
188C2×D20C3×D412yes—
189C2×C4×D5—24no—
190C2×C4×D5—22no—
191C2×Dic10—22no—
192C2×Dic10C3×D412yes—
193C5×C4⋊C4—21no—
194C5×C4⋊C4—24no—
195C5×C22⋊C4—21no—
196C5×C22⋊C4—24no—
197C42×C5C23×C312yes—
198#689—24no—
199#688—24no—
200#687—24no—
201#686—24no—
202#681—22no—
203#681C23×C314yes—
204C2×#168—22no—
205C2×2−1+4—51no—
206#575—52no—
207#536C3×D511yescharacteristic, complemented
208#456—54no—
209#454—54no—
210#421—52no—
211C2×#101—51no—
212C22×C3×C5—41no—
213C22×C3×C5—42no—
214C22×C3×C5—41no—
215C22×C3×C5#12011yescharacteristic
216C22×C3×C5#12011yescharacteristic
217C22×C3×C5C22×D411yescharacteristic
218C2×C3×D5—24no—
219C2×C3×D5—44no—
220C3×C4×C5—24no—
221C3×C4×C5—42no—
222C3×C4×C5—41no—
223C3×C4×C5#12012yes—
224C3×C4×C5C22×D412yes—
225C3×Dic5—24no—
226C3×#41—22no—
227C3×#41—22no—
228C3×#41—54no—
229C3×#41—54no—
230C3×#41—108no—
231C3×#41—108no—
232C3×#41—108no—
233C3×#41—102no—
234C3×#41—102no—
235C3×#41—108no—
236C2×C3×Q8—22no—
237C2×C3×Q8—22no—
238C2×C3×Q8—52no—
239C2×C3×Q8—102no—
240C2×C3×Q8—108no—
241C2×C3×Q8—108no—
242C2×C3×Q8C22⋊D512yes—
243C2×C3×Q8C22×D511yescharacteristic
244C2×C3×Q8C22×D512yes—
245C2×C3×D4—21no—
246C2×C3×D4—52no—
247C2×C3×D4—102no—
248C2×C3×D4—102no—
249C2×C3×D4C22×D511yescharacteristic
250C22×C3×C4—21no—
251C22×C3×C4—104no—
252C22×C3×C4—102no—
253C22×C3×C4C22⋊D512yes—
254C22×C3×C4C22×D511yescharacteristic
255C3×Q16—104no—
256C3×SD16—104no—
257C3×SD16—104no—
258C3×D8—104no—
259C3×M16—54no—
260C3×M16—102no—
261C2×C3×C8—102no—
262C3×C4⋊C4—21no—
263C3×C4⋊C4—24no—
264C3×C22⋊C4—21no—
265C3×C22⋊C4—24no—
266C3×C42C22×D512yes—
267C23×C5—21no—
268C23×C5C2×C3×D411yescharacteristic
269C22×D5—22no—
270C5×Q8—24no—
271C5×Q8—24no—
272C5×Q8—22no—
273C5×D4—22no—
274C5×D4—41no—
275C5×D4—41no—
276C2×C4×C5—22no—
277C2×C4×C5—22no—
278C2×C4×C5—22no—
279C2×C4×C5—21no—
280C2×C4×C5—42no—
281C2×C4×C5C2×C3×D412yes—
282C2×C4×C5C2×C3×D412yes—
283C2×C4×C5C2×C3×D414yes—
284C2×C4×C5C2×C3×D411yescharacteristic
285C2×C4×C5C24×C311yescharacteristic
286C22⋊D5—24no—
287C22⋊D5—24no—
288C2×Dic5—22no—
289D20—28no—
290D20—22no—
291D20C2×C3×D414yes—
292C4×D5—216no—
293C4×D5—24no—
294C4×D5—24no—
295Dic10—28no—
296Dic10—22no—
297Dic10C2×C3×D414yes—
298#168—24no—
2992−1+4—54no—
3002−1+4—104no—
3012−1+4—102no—
302C2×#41—52no—
303C2×#41—102no—
304C2×#41—102no—
305C2×#41C2×C3×D511yescharacteristic
306C22×Q8—102no—
307C22×Q8C2×C3×D511yescharacteristic
308#137—54no—
309#137—102no—
310AGL(1,ℤ/8)—54no—
311AGL(1,ℤ/8)—102no—
312#135—104no—
313C2×M16—52no—
314C4⋊Q8C2×C3×D512yescomplemented
315#124C2×C3×D512yescomplemented
316#123—22no—
317C22⋊Q8—22no—
318#117C2×C3×D511yescharacteristic
319#104—58no—
320#101—54no—
321C2×C3×C5—21no—
322C2×C3×C5—21no—
323C2×C3×C5—21no—
324C2×C3×C5—81no—
325C2×C3×C5C2×#12011yescharacteristic
326C3×D5—44no—
327C23×C3—21no—
328C23×C3—102no—
329C23×C3C2×C22⋊D511yescharacteristic
330C3×Q8—24no—
331C3×Q8—24no—
332C3×Q8—22no—
333C3×Q8—54no—
334C3×Q8—108no—
335C3×Q8—102no—
336C3×D4—22no—
337C3×D4—41no—
338C3×D4—41no—
339C3×D4—54no—
340C3×D4—108no—
341C3×D4—104no—
342C3×D4—104no—
343C3×D4—102no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—42no—
349C2×C3×C4—1016no—
350C2×C3×C4—102no—
351C2×C3×C4—104no—
352C2×C3×C4—104no—
353C2×C3×C4D4×D512yes—
354C2×C3×C4D4×D512yes—
355C2×C3×C4C2×C22⋊D514yes—
356C2×C3×C4C2×C22⋊D511yescharacteristic
357C2×C3×C4C23×D511yescharacteristic
358C3×C8—104no—
359C22×C5—41no—
360C22×C5—42no—
361C22×C5—41no—
362C22×C5C3×#12011yescharacteristic
363C22×C5C3×#12011yescharacteristic
364C22×C5C22×C3×D411yescharacteristic
365C2×D5—24no—
366C2×D5—44no—
367C4×C5—24no—
368C4×C5—42no—
369C4×C5—41no—
370C4×C5C3×#12012yes—
371C4×C5C22×C3×D412yes—
372Dic5—24no—
373#41—22no—
374#41—22no—
375#41—54no—
376#41—54no—
377#41—108no—
378#41—108no—
379#41—108no—
380#41—102no—
381#41—102no—
382#41—108no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—52no—
386C2×Q8—102no—
387C2×Q8—108no—
388C2×Q8—108no—
389C2×Q8C3×C22⋊D512yes—
390C2×Q8C22×C3×D511yescharacteristic
391C2×Q8C22×C3×D512yes—
392C2×D4—21no—
393C2×D4—52no—
394C2×D4—102no—
395C2×D4—102no—
396C2×D4C22×C3×D511yescharacteristic
397C22×C4—21no—
398C22×C4—104no—
399C22×C4—102no—
400C22×C4C3×C22⋊D512yes—
401C22×C4C22×C3×D511yescharacteristic
402Q16—104no—
403SD16—104no—
404SD16—104no—
405D8—104no—
406M16—54no—
407M16—102no—
408C2×C8—102no—
409C4⋊C4—21no—
410C4⋊C4—24no—
411C22⋊C4—21no—
412C22⋊C4—24no—
413C42C22×C3×D512yes—
414C3×C5#302011yescharacteristic, complemented, Hall
415C22×C3—41no—
416C22×C3—42no—
417C22×C3—41no—
418C22×C3—104no—
419C22×C3—204no—
420C22×C3#408311yescharacteristic
421C22×C3#408311yescharacteristic
422C22×C3C22×C22⋊D511yescharacteristic
423C3×C4—24no—
424C3×C4—42no—
425C3×C4—41no—
426C3×C4—104no—
427C3×C4#408312yes—
428C3×C4C2×D4×D512yes—
429C2×C5—21no—
430C2×C5—21no—
431C2×C5—21no—
432C2×C5—81no—
433C2×C5C2×C3×#12011yescharacteristic
434D5—44no—
435C23—21no—
436C23—102no—
437C23C2×C3×C22⋊D511yescharacteristic
438Q8—24no—
439Q8—24no—
440Q8—22no—
441Q8—54no—
442Q8—108no—
443Q8—102no—
444D4—22no—
445D4—41no—
446D4—41no—
447D4—54no—
448D4—108no—
449D4—104no—
450D4—104no—
451D4—102no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—42no—
457C2×C4—1016no—
458C2×C4—102no—
459C2×C4—104no—
460C2×C4—104no—
461C2×C4C3×D4×D512yes—
462C2×C4C3×D4×D512yes—
463C2×C4C2×C3×C22⋊D514yes—
464C2×C4C2×C3×C22⋊D511yescharacteristic
465C2×C4C23×C3×D511yescharacteristic
466C8—104no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—81no—
471C2×C3—204no—
472C2×C3C2×#408311yescharacteristic
473C5C3×#302011yescharacteristic, complemented, Sylow
474C22—41no—
475C22—42no—
476C22—41no—
477C22—104no—
478C22—204no—
479C22C3×#408311yescharacteristic
480C22C3×#408311yescharacteristic
481C22C22×C3×C22⋊D511yescharacteristic
482C4—24no—
483C4—42no—
484C4—41no—
485C4—104no—
486C4C3×#408312yes—
487C4C2×C3×D4×D512yes—
488C3#1061514611yescharacteristic, direct product factor, Sylow
489C2—21no—
490C2—21no—
491C2—21no—
492C2—81no—
493C2—204no—
494C2C2×C3×#408311yescharacteristic
subgroupG/H
centerC2×C3C2×#4083
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC3×C5×#536C2
socleC2×C3×C5C2×#120
hypercenterC3×#536D5
hypocenterC5C3×#3020
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#536

Complements

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

NH ≅ G/Ntype
1. C2×C3×#4147C2semidirect
2. C3×C5×#536C2semidirect
3. C3×#66156C2semidirect
4. C2×C3×#3968C2semidirect
5. C3×#65384C2semidirect
6. C3×#65351C2semidirect
7. C3×#65294C2semidirect
8. #10615146C3direct (N × H)
18. C3×C5×C4⋊Q8C22semidirect
19. C3×C5×#124C22semidirect
26. C3×#4088C22semidirect
27. C3×#4078C22semidirect
29. C3×#3968C22semidirect
30. C3×#3937C22semidirect
32. C2×#4147C2×C3semidirect
33. C5×#536C2×C3semidirect
34. #66156C2×C3semidirect
35. C2×#3968C2×C3semidirect
36. #65384C2×C3semidirect
37. #65351C2×C3semidirect
38. #65294C2×C3semidirect
82. C3×#536D5semidirect
96. C5×C4⋊Q8C22×C3semidirect
97. C5×#124C22×C3semidirect
104. #4088C22×C3semidirect
105. #4078C22×C3semidirect
107. #3968C22×C3semidirect
108. #3937C22×C3semidirect
157. C3×C4⋊Q8C2×D5semidirect
158. C3×#124C2×D5semidirect
207. #536C3×D5semidirect
314. C4⋊Q8C2×C3×D5semidirect
315. #124C2×C3×D5semidirect
414. C3×C5#3020semidirect
473. C5C3×#3020semidirect
488. C3#10615146direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#41472
C3×C5×#5362
C3×#661562
C2×C3×#39682
C3×#653842
C3×#653512
C3×#652942
#106151463
C3×#30205