Finite Group Databasebeta

← C3×#10610440 #14425387

Subgroups — C3×#10610440 #14425387

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×#66150C211yesmaximal, characteristic, complemented
3C3×#66082C212yesmaximal, complemented
4C2×C3×C5×#101C211yesmaximal, characteristic, complemented
5C3×#65428C212yesmaximal, complemented
6C3×#65042C214yesmaximal, complemented
7C3×#65041C214yesmaximal, complemented
8#10610440C311yesmaximal, characteristic, direct product factor, Hall
9C22×C3×C5×Q8C2211yescharacteristic
10C3×#4147—22no—
11C3×#4147—24no—
12C3×#4147C2214yes—
13C2×C3×#713—22no—
14C2×C3×Q8×D5—22no—
15C2×C3×#709—22no—
16C2×C3×#709C2212yes—
17C2×C3×#709C2211yescharacteristic
18C2×C3×C5×M16C2212yes—
19C3×#4093C2212yescomplemented
20C3×#4092—22no—
21C3×#4090C2212yescomplemented
22C3×#4075—22no—
23C3×#4072C2211yescharacteristic
24C3×#4055—22no—
25C3×#4055C2214yescomplemented
26C3×#4054—22no—
27C3×#4054C2214yescomplemented
28C3×#4050—24no—
29C3×C5×#101C2214yescomplemented
30C3×#3957C2218yescomplemented
31C3×#3020—51nomaximal, Hall
32C2×#4147C2×C311yescharacteristic, complemented
33#66150C2×C311yescharacteristic, complemented
34#66082C2×C312yescomplemented
35C2×C5×#101C2×C311yescharacteristic, complemented
36#65428C2×C312yescomplemented
37#65042C2×C314yescomplemented
38#65041C2×C314yescomplemented
39C2×C3×C5×Q8—22no—
40C2×C3×C5×Q8—22no—
41C2×C3×C5×Q8D412yes—
42C2×C3×C5×Q8C2312yes—
43C22×C3×C4×C5D412yes—
44C22×C3×C4×C5C2311yescharacteristic
45C2×C3×C22⋊D5—22no—
46C2×C3×C22⋊D5—21no—
47C3×#713—28no—
48C3×#713—28no—
49C3×Q8×D5—28no—
50C3×Q8×D5—28no—
51C3×#709—28no—
52C3×#709—28no—
53C3×#709—22no—
54C3×#709—22no—
55C3×#709—22no—
56C3×#709—22no—
57C3×#709D414yes—
58C3×#709C2314yes—
59C2×C3×D20—22no—
60C2×C3×D20D412yes—
61C2×C3×D20C2311yescharacteristic
62C2×C3×C4×D5—24no—
63C2×C3×C4×D5—22no—
64C2×C3×C4×D5—21no—
65C2×C3×Dic10—22no—
66C2×C3×Dic10D412yes—
67C2×C3×Dic10C2311yescharacteristic
68C3×C5×M16—22no—
69C3×C5×M16C2314yes—
70C2×C3×C5×C8—22no—
71C3×C22⋊Dic5—21no—
72C3×#685—24no—
73C3×C4⋊Dic5—21no—
74C3×#683—24no—
75C3×C4×Dic5C2312yes—
76C3×Dic20—24no—
77C3×D40—24no—
78C3×#677—24no—
79C3×#677—24no—
80C2×C3×2−1+4—51no—
81C3×#575—52no—
82C3×#536—51no—
83C3×#456—54no—
84C3×#454—54no—
85C3×#421—52no—
86C2×C3×#101D511yescharacteristic, complemented
87C22×C5×Q8C22×C311yescharacteristic
88#4147—22no—
89#4147—24no—
90#4147C22×C314yes—
91C2×#713—22no—
92C2×Q8×D5—22no—
93C2×#709—22no—
94C2×#709C22×C312yes—
95C2×#709C22×C311yescharacteristic
96C2×C5×M16C22×C312yes—
97#4093C22×C312yescomplemented
98#4092—22no—
99#4090C22×C312yescomplemented
100#4075—22no—
101#4072C22×C311yescharacteristic
102#4055—22no—
103#4055C22×C314yescomplemented
104#4054—22no—
105#4054C22×C314yescomplemented
106#4050—24no—
107C5×#101C22×C314yescomplemented
108#3957C22×C318yescomplemented
109#3020—51noSylow
110C23×C3×C5C2×D411yescharacteristic
111C22×C3×D5—22no—
112C22×C3×D5—21no—
113C3×C5×Q8—24no—
114C3×C5×Q8—24no—
115C2×C3×C4×C5—22no—
116C2×C3×C4×C5—22no—
117C2×C3×C4×C5C2×D414yes—
118C2×C3×C4×C5C2×D411yescharacteristic
119C2×C3×C4×C5C2×D412yes—
120C2×C3×C4×C5C2×D412yes—
121C2×C3×C4×C5C2411yescharacteristic
122C3×C22⋊D5—24no—
123C3×C22⋊D5—24no—
124C3×C22⋊D5—41no—
125C3×C22⋊D5—41no—
126C2×C3×Dic5—22no—
127C2×C3×Dic5—22no—
128C2×C3×Dic5—21no—
129C3×D20—28no—
130C3×D20—22no—
131C3×D20—22no—
132C3×D20C2×D414yes—
133C3×C4×D5—216no—
134C3×C4×D5—24no—
135C3×C4×D5—24no—
136C3×C4×D5—42no—
137C3×Dic10—28no—
138C3×Dic10—22no—
139C3×Dic10—22no—
140C3×Dic10C2×D414yes—
141C3×C5×C8—24no—
142C3×2−1+4—54no—
143C3×2−1+4—104no—
144C3×2−1+4—102no—
145C2×C3×#41—52no—
146C2×C3×#41—51no—
147C2×C3×#41—102no—
148C2×C3×#41—102no—
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×M16C2×D512yes—
157C3×C4⋊Q8—52no—
158C3×#124—52no—
159C3×#123—102no—
160C3×C22⋊Q8—102no—
161C3×#117—51no—
162C3×#104—58no—
163C3×#101C2×D514yescomplemented
164C2×C5×Q8—22no—
165C2×C5×Q8—22no—
166C2×C5×Q8C3×D412yes—
167C2×C5×Q8C23×C312yes—
168C22×C4×C5C3×D412yes—
169C22×C4×C5C23×C311yescharacteristic
170C2×C22⋊D5—22no—
171C2×C22⋊D5—21no—
172#713—28no—
173#713—28no—
174Q8×D5—28no—
175Q8×D5—28no—
176#709—28no—
177#709—28no—
178#709—22no—
179#709—22no—
180#709—22no—
181#709—22no—
182#709C3×D414yes—
183#709C23×C314yes—
184C2×D20—22no—
185C2×D20C3×D412yes—
186C2×D20C23×C311yescharacteristic
187C2×C4×D5—24no—
188C2×C4×D5—22no—
189C2×C4×D5—21no—
190C2×Dic10—22no—
191C2×Dic10C3×D412yes—
192C2×Dic10C23×C311yescharacteristic
193C5×M16—22no—
194C5×M16C23×C314yes—
195C2×C5×C8—22no—
196C22⋊Dic5—21no—
197#685—24no—
198C4⋊Dic5—21no—
199#683—24no—
200C4×Dic5C23×C312yes—
201Dic20—24no—
202D40—24no—
203#677—24no—
204#677—24no—
205C2×2−1+4—51no—
206#575—52no—
207#536—51no—
208#456—54no—
209#454—54no—
210#421—52no—
211C2×#101C3×D511yescharacteristic, complemented
212C22×C3×C5—41no—
213C22×C3×C5#12011yescharacteristic
214C22×C3×C5#12011yescharacteristic
215C22×C3×C5C22×D411yescharacteristic
216C2×C3×D5—24no—
217C2×C3×D5—44no—
218C2×C3×D5—42no—
219C2×C3×D5—41no—
220C3×C4×C5—24no—
221C3×C4×C5#12012yes—
222C3×C4×C5C22×D412yes—
223C3×Dic5—24no—
224C3×Dic5—42no—
225C3×Dic5—41no—
226C3×#41—54no—
227C3×#41—54no—
228C3×#41—108no—
229C3×#41—108no—
230C3×#41—108no—
231C3×#41—102no—
232C3×#41—108no—
233C3×#41—102no—
234C3×#41—102no—
235C3×#41—102no—
236C2×C3×Q8—22no—
237C2×C3×Q8—22no—
238C2×C3×Q8—52no—
239C2×C3×Q8—51no—
240C2×C3×Q8—102no—
241C2×C3×Q8—108no—
242C2×C3×Q8—108no—
243C2×C3×Q8D2012yes—
244C2×C3×Q8C22×D512yes—
245C2×C3×D4—52no—
246C2×C3×D4—51no—
247C2×C3×D4—102no—
248C2×C3×D4—102no—
249C2×C3×D4—101no—
250C22×C3×C4—104no—
251C22×C3×C4—102no—
252C22×C3×C4—101no—
253C22×C3×C4D2012yes—
254C22×C3×C4C22×D511yescharacteristic
255C3×Q16—104no—
256C3×SD16—104no—
257C3×SD16—104no—
258C3×D8—104no—
259C3×M16—22no—
260C3×M16C22×D514yes—
261C2×C3×C8—22no—
262C3×C4⋊C4—104no—
263C3×C4⋊C4—101no—
264C3×C22⋊C4—101no—
265C3×C22⋊C4—104no—
266C3×C42—52no—
267C23×C5C2×C3×D411yescharacteristic
268C22×D5—22no—
269C22×D5—21no—
270C5×Q8—24no—
271C5×Q8—24no—
272C2×C4×C5—22no—
273C2×C4×C5—22no—
274C2×C4×C5C2×C3×D414yes—
275C2×C4×C5C2×C3×D411yescharacteristic
276C2×C4×C5C2×C3×D412yes—
277C2×C4×C5C2×C3×D412yes—
278C2×C4×C5C24×C311yescharacteristic
279C22⋊D5—24no—
280C22⋊D5—24no—
281C22⋊D5—41no—
282C22⋊D5—41no—
283C2×Dic5—22no—
284C2×Dic5—22no—
285C2×Dic5—21no—
286D20—28no—
287D20—22no—
288D20—22no—
289D20C2×C3×D414yes—
290C4×D5—216no—
291C4×D5—24no—
292C4×D5—24no—
293C4×D5—42no—
294Dic10—28no—
295Dic10—22no—
296Dic10—22no—
297Dic10C2×C3×D414yes—
298C5×C8—24no—
2992−1+4—54no—
3002−1+4—104no—
3012−1+4—102no—
302C2×#41—52no—
303C2×#41—51no—
304C2×#41—102no—
305C2×#41—102no—
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×M16C2×C3×D512yes—
314C4⋊Q8—52no—
315#124—52no—
316#123—102no—
317C22⋊Q8—102no—
318#117—51no—
319#104—58no—
320#101C2×C3×D514yescomplemented
321C2×C3×C5—21no—
322C2×C3×C5—21no—
323C2×C3×C5—21no—
324C2×C3×C5C2×#12011yescharacteristic
325C3×D5—44no—
326C3×D5—81no—
327C23×C3—102no—
328C23×C3—101no—
329C23×C3C2×D2011yescharacteristic
330C3×Q8—24no—
331C3×Q8—24no—
332C3×Q8—54no—
333C3×Q8—108no—
334C3×Q8—102no—
335C3×Q8—102no—
336C3×D4—54no—
337C3×D4—108no—
338C3×D4—104no—
339C3×D4—104no—
340C3×D4—102no—
341C3×D4—102no—
342C3×D4—201no—
343C3×D4—201no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—1016no—
347C2×C3×C4—102no—
348C2×C3×C4—104no—
349C2×C3×C4—104no—
350C2×C3×C4—102no—
351C2×C3×C4—101no—
352C2×C3×C4—202no—
353C2×C3×C4C2×D2014yes—
354C2×C3×C4C2×D2011yescharacteristic
355C2×C3×C4D4×D512yes—
356C2×C3×C4D4×D512yes—
357C2×C3×C4C23×D511yescharacteristic
358C3×C8—24no—
359C22×C5—41no—
360C22×C5C3×#12011yescharacteristic
361C22×C5C3×#12011yescharacteristic
362C22×C5C22×C3×D411yescharacteristic
363C2×D5—24no—
364C2×D5—44no—
365C2×D5—42no—
366C2×D5—41no—
367C4×C5—24no—
368C4×C5C3×#12012yes—
369C4×C5C22×C3×D412yes—
370Dic5—24no—
371Dic5—42no—
372Dic5—41no—
373#41—54no—
374#41—54no—
375#41—108no—
376#41—108no—
377#41—108no—
378#41—102no—
379#41—108no—
380#41—102no—
381#41—102no—
382#41—102no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—52no—
386C2×Q8—51no—
387C2×Q8—102no—
388C2×Q8—108no—
389C2×Q8—108no—
390C2×Q8C3×D2012yes—
391C2×Q8C22×C3×D512yes—
392C2×D4—52no—
393C2×D4—51no—
394C2×D4—102no—
395C2×D4—102no—
396C2×D4—101no—
397C22×C4—104no—
398C22×C4—102no—
399C22×C4—101no—
400C22×C4C3×D2012yes—
401C22×C4C22×C3×D511yescharacteristic
402Q16—104no—
403SD16—104no—
404SD16—104no—
405D8—104no—
406M16—22no—
407M16C22×C3×D514yes—
408C2×C8—22no—
409C4⋊C4—101no—
410C4⋊C4—104no—
411C22⋊C4—101no—
412C22⋊C4—104no—
413C42—52no—
414C3×C5#302011yescharacteristic, complemented, Hall
415C22×C3—41no—
416C22×C3—104no—
417C22×C3—204no—
418C22×C3—202no—
419C22×C3—201no—
420C22×C3#402811yescharacteristic
421C22×C3#402811yescharacteristic
422C22×C3C22×D2011yescharacteristic
423C3×C4—24no—
424C3×C4—104no—
425C3×C4—202no—
426C3×C4—201no—
427C3×C4#402812yes—
428C3×C4C2×D4×D512yes—
429C2×C5—21no—
430C2×C5—21no—
431C2×C5—21no—
432C2×C5C2×C3×#12011yescharacteristic
433D5—44no—
434D5—81no—
435C23—102no—
436C23—101no—
437C23C2×C3×D2011yescharacteristic
438Q8—24no—
439Q8—24no—
440Q8—54no—
441Q8—108no—
442Q8—102no—
443Q8—102no—
444D4—54no—
445D4—108no—
446D4—104no—
447D4—104no—
448D4—102no—
449D4—102no—
450D4—201no—
451D4—201no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—1016no—
455C2×C4—102no—
456C2×C4—104no—
457C2×C4—104no—
458C2×C4—102no—
459C2×C4—101no—
460C2×C4—202no—
461C2×C4C2×C3×D2014yes—
462C2×C4C2×C3×D2011yescharacteristic
463C2×C4C3×D4×D512yes—
464C2×C4C3×D4×D512yes—
465C2×C4C23×C3×D511yescharacteristic
466C8—24no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—204no—
471C2×C3—401no—
472C2×C3C2×#402811yescharacteristic
473C5C3×#302011yescharacteristic, complemented, Sylow
474C22—41no—
475C22—104no—
476C22—204no—
477C22—202no—
478C22—201no—
479C22C3×#402811yescharacteristic
480C22C3×#402811yescharacteristic
481C22C22×C3×D2011yescharacteristic
482C4—24no—
483C4—104no—
484C4—202no—
485C4—201no—
486C4C3×#402812yes—
487C4C2×C3×D4×D512yes—
488C3#1061044011yescharacteristic, direct product factor, Sylow
489C2—21no—
490C2—21no—
491C2—21no—
492C2—204no—
493C2—401no—
494C2C2×C3×#402811yescharacteristic
subgroupG/H
centerC2×C3C2×#4028
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×#101C2
socleC2×C3×C5C2×#120
hypercenterC2×C3×#101D5
hypocenterC5C3×#3020
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#101

Complements

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

NH ≅ G/Ntype
1. C2×C3×#4147C2semidirect
2. C3×#66150C2semidirect
3. C3×#66082C2semidirect
4. C2×C3×C5×#101C2semidirect
5. C3×#65428C2semidirect
6. C3×#65042C2semidirect
7. C3×#65041C2semidirect
8. #10610440C3direct (N × H)
19. C3×#4093C22semidirect
21. C3×#4090C22semidirect
25. C3×#4055C22semidirect
27. C3×#4054C22semidirect
29. C3×C5×#101C22semidirect
30. C3×#3957C22semidirect
32. C2×#4147C2×C3semidirect
33. #66150C2×C3semidirect
34. #66082C2×C3semidirect
35. C2×C5×#101C2×C3semidirect
36. #65428C2×C3semidirect
37. #65042C2×C3semidirect
38. #65041C2×C3semidirect
86. C2×C3×#101D5semidirect
97. #4093C22×C3semidirect
99. #4090C22×C3semidirect
103. #4055C22×C3semidirect
105. #4054C22×C3semidirect
107. C5×#101C22×C3semidirect
108. #3957C22×C3semidirect
163. C3×#101C2×D5semidirect
211. C2×#101C3×D5semidirect
320. #101C2×C3×D5semidirect
414. C3×C5#3020semidirect
473. C5C3×#3020semidirect
488. C3#10610440direct (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×#661502
C3×#660822
C2×C3×C5×#1012
C3×#654282
C3×#650422
C3×#650412
#106104403
C3×#30205