Finite Group Databasebeta

← C3×#10605020 #14297281

Subgroups — C3×#10605020 #14297281

Order 1920.

1096 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66287C211yesmaximal, characteristic
2C3×#65795C212yesmaximal, complemented
3C3×#65794C212yesmaximal, complemented
4C3×#65787C211yesmaximal, characteristic, complemented
5C3×#65785C211yesmaximal, characteristic, complemented
6#10605020C311yesmaximal, characteristic, direct product factor, Hall
7C3×C5×2−1+4C411yescharacteristic, complemented
8C3×#4149—22no—
9C3×#4149—22no—
10C3×#4149C412yescomplemented
11C3×D5×#41—22no—
12C3×D5×#41—22no—
13C3×D5×#41C2212yes—
14C3×#4147—22no—
15C3×#4147C411yescharacteristic, complemented
16C2×C3×#713—22no—
17C2×C3×#713C2211yescharacteristic
18C3×#4130C2212yescomplemented
19C2×C3×#700C2211yescharacteristic, complemented
20C2×C3×D5⋊C8C2211yescharacteristic, complemented
21C3×#4010—22no—
22C3×#4009—22no—
23C3×#4008—22no—
24C3×#4007—22no—
25C3×#4003—22no—
26C3×#3999—22no—
27C3×#1794—51nomaximal, Hall
28#66287C2×C311yescharacteristic
29#65795C2×C312yescomplemented
30#65794C2×C312yescomplemented
31#65787C2×C311yescharacteristic, complemented
32#65785C2×C311yescharacteristic, complemented
33C3×C5×#41—22no—
34C3×C5×#41—22no—
35C3×C5×#41C2×C412yes—
36C2×C3×C5×Q8—22no—
37C2×C3×C5×Q8C2×C411yescharacteristic
38C3×#713—22no—
39C3×#713—24no—
40C3×#713—22no—
41C3×#713—22no—
42C3×#713—22no—
43C3×#713—22no—
44C3×#713D412yes—
45C3×Q8×D5—22no—
46C3×Q8×D5—22no—
47C3×Q8×D5D412yes—
48C3×#711—22no—
49C3×#711—22no—
50C3×#711D412yes—
51C3×D4×D5—24no—
52C3×D4×D5—22no—
53C3×D4×D5—24no—
54C3×D4×D5—22no—
55C3×D4×D5—22no—
56C3×D4×D5D412yes—
57C3×#709—24no—
58C3×#709—24no—
59C3×#709—22no—
60C3×#709—22no—
61C3×#709—22no—
62C3×#709C2×C412yes—
63C2×C3×D20—24no—
64C2×C3×D20—22no—
65C2×C3×D20C2×C411yescharacteristic
66C2×C3×D20D412yes—
67C2×C3×C4×D5—24no—
68C2×C3×C4×D5—22no—
69C2×C3×C4×D5D412yes—
70C2×C3×C4×D5C2311yescharacteristic
71C3×C22⋊AGL(1,𝔽5)—22no—
72C2×C3×#170—21no—
73C2×C3×#170—21no—
74C3×C4⋊AGL(1,𝔽5)—22no—
75C3×C4×AGL(1,𝔽5)—22no—
76C3×#700—22no—
77C3×D5⋊C8—22no—
78C3×#585—51no—
79C3×#421—52no—
80C3×#416—52no—
81C3×#410—51no—
82C3×#408—51no—
83C5×2−1+4C3×C411yescharacteristic, complemented
84#4149—22no—
85#4149—22no—
86#4149C3×C412yescomplemented
87D5×#41—22no—
88D5×#41—22no—
89D5×#41C22×C312yes—
90#4147—22no—
91#4147C3×C411yescharacteristic, complemented
92C2×#713—22no—
93C2×#713C22×C311yescharacteristic
94#4130C22×C312yescomplemented
95C2×#700C22×C311yescharacteristic, complemented
96C2×D5⋊C8C22×C311yescharacteristic, complemented
97#4010—22no—
98#4009—22no—
99#4008—22no—
100#4007—22no—
101#4003—22no—
102#3999—22no—
103#1794—51noSylow
104C22×C3×D5—24no—
105C22×C3×D5—22no—
106C22×C3×D5C22⋊C412yes—
107C22×C3×D5C2×D411yescharacteristic
108C2×C3×AGL(1,𝔽5)—42no—
109C3×C5×Q8—22no—
110C3×C5×Q8—22no—
111C3×C5×Q8C22⋊C412yes—
112C3×C5×D4—22no—
113C3×C5×D4—22no—
114C3×C5×D4C22⋊C412yes—
115C2×C3×C4×C5—24no—
116C2×C3×C4×C5—22no—
117C2×C3×C4×C5C22⋊C412yes—
118C2×C3×C4×C5C22×C411yescharacteristic
119C3×C22⋊D5—24no—
120C3×C22⋊D5—22no—
121C3×C22⋊D5—22no—
122C3×C22⋊D5—22no—
123C2×C3×Dic5—22no—
124C2×C3×Dic5C2×D411yescharacteristic
125C3×D20—24no—
126C3×D20—22no—
127C3×D20—22no—
128C3×D20—24no—
129C3×D20—22no—
130C3×D20C22⋊C412yes—
131C3×C4×D5—24no—
132C3×C4×D5—24no—
133C3×C4×D5—22no—
134C3×C4×D5—22no—
135C3×C4×D5—22no—
136C3×C4×D5—22no—
137C3×C4×D5—22no—
138C3×C4×D5C2×D412yes—
139C3×C4×D5C2×D412yes—
140C3×Dic10—22no—
141C3×Dic10—22no—
142C3×Dic10C22⋊C412yes—
143C3×#170—41no—
144C3×#170—41no—
145C3×2−1+4—51no—
146C3×2−1+4—102no—
147C3×2−1+4AGL(1,𝔽5)11yescharacteristic, complemented
148C3×H(5,𝔽2)—52no—
149C3×H(5,𝔽2)—102no—
150C3×H(5,𝔽2)—102no—
151C2×C3×#41—51no—
152C2×C3×#41—52no—
153C2×C3×#41—102no—
154C2×C3×#41—102no—
155C2×C3×#41—102no—
156C2×C3×M16—51no—
157C22×C3×C8—51no—
158C3×#117—52no—
159C3×#104—102no—
160C3×#104—102no—
161C3×#103—102no—
162C3×#102—102no—
163C3×#99—102no—
164C3×C22⋊C8—102no—
165C5×#41—22no—
166C5×#41—22no—
167C5×#41C2×C3×C412yes—
168C2×C5×Q8—22no—
169C2×C5×Q8C2×C3×C411yescharacteristic
170#713—22no—
171#713—24no—
172#713—22no—
173#713—22no—
174#713—22no—
175#713—22no—
176#713C3×D412yes—
177Q8×D5—22no—
178Q8×D5—22no—
179Q8×D5C3×D412yes—
180#711—22no—
181#711—22no—
182#711C3×D412yes—
183D4×D5—24no—
184D4×D5—22no—
185D4×D5—24no—
186D4×D5—22no—
187D4×D5—22no—
188D4×D5C3×D412yes—
189#709—24no—
190#709—24no—
191#709—22no—
192#709—22no—
193#709—22no—
194#709C2×C3×C412yes—
195C2×D20—24no—
196C2×D20—22no—
197C2×D20C2×C3×C411yescharacteristic
198C2×D20C3×D412yes—
199C2×C4×D5—24no—
200C2×C4×D5—22no—
201C2×C4×D5C3×D412yes—
202C2×C4×D5C23×C311yescharacteristic
203C22⋊AGL(1,𝔽5)—22no—
204C2×#170—21no—
205C2×#170—21no—
206C4⋊AGL(1,𝔽5)—22no—
207C4×AGL(1,𝔽5)—22no—
208#700—22no—
209D5⋊C8—22no—
210#585—51no—
211#421—52no—
212#416—52no—
213#410—51no—
214#408—51no—
215C22×C3×C5—22no—
216C22×C3×C5C2×C22⋊C411yescharacteristic
217C2×C3×D5—22no—
218C2×C3×D5—22no—
219C2×C3×D5—41no—
220C2×C3×D5—42no—
221C2×C3×D5—82no—
222C2×C3×D5—84no—
223C2×C3×D5#12012yes—
224C3×AGL(1,𝔽5)—82no—
225C3×C4×C5—22no—
226C3×C4×C5—22no—
227C3×C4×C5C2×C22⋊C412yes—
228C3×Dic5—22no—
229C3×Dic5#12011yescharacteristic
230C3×Dic5#12011yescharacteristic
231C3×#41—22no—
232C3×#41—22no—
233C3×#41—52no—
234C3×#41—52no—
235C3×#41—52no—
236C3×#41—102no—
237C3×#41—102no—
238C3×#41—104no—
239C3×#41—104no—
240C3×#41—102no—
241C3×#41—102no—
242C3×#41—102no—
243C3×#41—102no—
244C3×#41—102no—
245C3×#41—102no—
246C3×#41—104no—
247C3×#41—102no—
248C3×#41—102no—
249C3×#41C2×AGL(1,𝔽5)12yes—
250C2×C3×Q8—22no—
251C2×C3×Q8—52no—
252C2×C3×Q8—102no—
253C2×C3×Q8—102no—
254C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
255C2×C3×D4—52no—
256C2×C3×D4—52no—
257C2×C3×D4—51no—
258C2×C3×D4—102no—
259C2×C3×D4—104no—
260C2×C3×D4—102no—
261C2×C3×D4—104no—
262C2×C3×D4—102no—
263C2×C3×D4—104no—
264C2×C3×D4—102no—
265C22×C3×C4—51no—
266C22×C3×C4—52no—
267C22×C3×C4—102no—
268C22×C3×C4—104no—
269C3×M16—102no—
270C2×C3×C8—101no—
271C2×C3×C8—101no—
272C2×C3×C8—102no—
273C3×C4⋊C4—102no—
274C3×C22⋊C4—102no—
275C3×C42—102no—
276C22×D5—24no—
277C22×D5—22no—
278C22×D5C3×C22⋊C412yes—
279C22×D5C2×C3×D411yescharacteristic
280C2×AGL(1,𝔽5)—42no—
281C5×Q8—22no—
282C5×Q8—22no—
283C5×Q8C3×C22⋊C412yes—
284C5×D4—22no—
285C5×D4—22no—
286C5×D4C3×C22⋊C412yes—
287C2×C4×C5—24no—
288C2×C4×C5—22no—
289C2×C4×C5C3×C22⋊C412yes—
290C2×C4×C5C22×C3×C411yescharacteristic
291C22⋊D5—24no—
292C22⋊D5—22no—
293C22⋊D5—22no—
294C22⋊D5—22no—
295C2×Dic5—22no—
296C2×Dic5C2×C3×D411yescharacteristic
297D20—24no—
298D20—22no—
299D20—22no—
300D20—24no—
301D20—22no—
302D20C3×C22⋊C412yes—
303C4×D5—24no—
304C4×D5—24no—
305C4×D5—22no—
306C4×D5—22no—
307C4×D5—22no—
308C4×D5—22no—
309C4×D5—22no—
310C4×D5C2×C3×D412yes—
311C4×D5C2×C3×D412yes—
312Dic10—22no—
313Dic10—22no—
314Dic10C3×C22⋊C412yes—
315#170—41no—
316#170—41no—
3172−1+4—51no—
3182−1+4—102no—
3192−1+4C3×AGL(1,𝔽5)11yescharacteristic, complemented
320H(5,𝔽2)—52no—
321H(5,𝔽2)—102no—
322H(5,𝔽2)—102no—
323C2×#41—51no—
324C2×#41—52no—
325C2×#41—102no—
326C2×#41—102no—
327C2×#41—102no—
328C2×M16—51no—
329C22×C8—51no—
330#117—52no—
331#104—102no—
332#104—102no—
333#103—102no—
334#102—102no—
335#99—102no—
336C22⋊C8—102no—
337C2×C3×C5—21no—
338C2×C3×C5—42no—
339C2×C3×C5#37911yescharacteristic
340C3×D5—22no—
341C3×D5—42no—
342C3×D5—42no—
343C23×C3—51no—
344C23×C3—52no—
345C23×C3—102no—
346C23×C3—104no—
347C3×Q8—22no—
348C3×Q8—22no—
349C3×Q8—52no—
350C3×Q8—102no—
351C3×Q8—102no—
352C3×Q8C22⋊AGL(1,𝔽5)12yes—
353C3×D4—22no—
354C3×D4—22no—
355C3×D4—52no—
356C3×D4—104no—
357C3×D4—102no—
358C3×D4—104no—
359C3×D4—102no—
360C3×D4—102no—
361C3×D4—102no—
362C3×D4—104no—
363C3×D4—102no—
364C3×D4—102no—
365C3×D4C22⋊AGL(1,𝔽5)12yes—
366C2×C3×C4—24no—
367C2×C3×C4—22no—
368C2×C3×C4—51no—
369C2×C3×C4—52no—
370C2×C3×C4—52no—
371C2×C3×C4—102no—
372C2×C3×C4—102no—
373C2×C3×C4—102no—
374C2×C3×C4—102no—
375C2×C3×C4—104no—
376C2×C3×C4—104no—
377C2×C3×C4—102no—
378C2×C3×C4—102no—
379C2×C3×C4—202no—
380C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
381C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
382C3×C8—201no—
383C3×C8—201no—
384C22×C5—22no—
385C22×C5C2×C3×C22⋊C411yescharacteristic
386C2×D5—22no—
387C2×D5—22no—
388C2×D5—41no—
389C2×D5—42no—
390C2×D5—82no—
391C2×D5—84no—
392C2×D5C3×#12012yes—
393AGL(1,𝔽5)—82no—
394C4×C5—22no—
395C4×C5—22no—
396C4×C5C2×C3×C22⋊C412yes—
397Dic5—22no—
398Dic5C3×#12011yescharacteristic
399Dic5C3×#12011yescharacteristic
400#41—22no—
401#41—22no—
402#41—52no—
403#41—52no—
404#41—52no—
405#41—102no—
406#41—102no—
407#41—104no—
408#41—104no—
409#41—102no—
410#41—102no—
411#41—102no—
412#41—102no—
413#41—102no—
414#41—102no—
415#41—104no—
416#41—102no—
417#41—102no—
418#41C2×C3×AGL(1,𝔽5)12yes—
419C2×Q8—22no—
420C2×Q8—52no—
421C2×Q8—102no—
422C2×Q8—102no—
423C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
424C2×D4—52no—
425C2×D4—52no—
426C2×D4—51no—
427C2×D4—102no—
428C2×D4—104no—
429C2×D4—102no—
430C2×D4—104no—
431C2×D4—102no—
432C2×D4—104no—
433C2×D4—102no—
434C22×C4—51no—
435C22×C4—52no—
436C22×C4—102no—
437C22×C4—104no—
438M16—102no—
439C2×C8—101no—
440C2×C8—101no—
441C2×C8—102no—
442C4⋊C4—102no—
443C22⋊C4—102no—
444C42—102no—
445C3×C5#179411yescharacteristic, complemented, Hall
446C22×C3—22no—
447C22×C3—52no—
448C22×C3—102no—
449C22×C3—102no—
450C22×C3—201no—
451C22×C3—202no—
452C22×C3—402no—
453C22×C3—404no—
454C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
455C3×C4—22no—
456C3×C4—22no—
457C3×C4—51no—
458C3×C4—51no—
459C3×C4—102no—
460C3×C4—402no—
461C3×C4C2×C22⋊AGL(1,𝔽5)12yes—
462C2×C5—21no—
463C2×C5—42no—
464C2×C5C3×#37911yescharacteristic
465D5—22no—
466D5—42no—
467D5—42no—
468C23—51no—
469C23—52no—
470C23—102no—
471C23—104no—
472Q8—22no—
473Q8—22no—
474Q8—52no—
475Q8—102no—
476Q8—102no—
477Q8C3×C22⋊AGL(1,𝔽5)12yes—
478D4—22no—
479D4—22no—
480D4—52no—
481D4—104no—
482D4—102no—
483D4—104no—
484D4—102no—
485D4—102no—
486D4—102no—
487D4—104no—
488D4—102no—
489D4—102no—
490D4C3×C22⋊AGL(1,𝔽5)12yes—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—51no—
494C2×C4—52no—
495C2×C4—52no—
496C2×C4—102no—
497C2×C4—102no—
498C2×C4—102no—
499C2×C4—102no—
500C2×C4—104no—
501C2×C4—104no—
502C2×C4—102no—
503C2×C4—102no—
504C2×C4—202no—
505C2×C4C3×C22⋊AGL(1,𝔽5)12yes—
506C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
507C8—201no—
508C8—201no—
509C2×C3—21no—
510C2×C3—42no—
511C2×C3—102no—
512C2×C3—202no—
513C2×C3—202no—
514C2×C3#6580011yescharacteristic
515C5C3×#179411yescharacteristic, complemented, Sylow
516C22—22no—
517C22—52no—
518C22—102no—
519C22—102no—
520C22—201no—
521C22—202no—
522C22—402no—
523C22—404no—
524C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
525C4—22no—
526C4—22no—
527C4—51no—
528C4—51no—
529C4—102no—
530C4—402no—
531C4C2×C3×C22⋊AGL(1,𝔽5)12yes—
532C3#1060502011yescharacteristic, direct product factor, Sylow
533C2—21no—
534C2—42no—
535C2—102no—
536C2—202no—
537C2—202no—
538C2C3×#6580011yescharacteristic
subgroupG/H
centerC2×C3#65800
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×2−1+4C4
socleC2×C3×C5#379
prime radicalC3×#66287C2
cosocleC2×C4×D5C23×C3
hypercenterC3×2−1+4AGL(1,𝔽5)
hypocenterC5C3×#1794
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×2−1+4

Complements

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

NH ≅ G/Ntype
2. C3×#65795C2semidirect
3. C3×#65794C2semidirect
4. C3×#65787C2semidirect
5. C3×#65785C2semidirect
6. #10605020C3direct (N × H)
7. C3×C5×2−1+4C4semidirect
10. C3×#4149C4semidirect
15. C3×#4147C4semidirect
18. C3×#4130C22semidirect
19. C2×C3×#700C22semidirect
20. C2×C3×D5⋊C8C22semidirect
29. #65795C2×C3semidirect
30. #65794C2×C3semidirect
31. #65787C2×C3semidirect
32. #65785C2×C3semidirect
83. C5×2−1+4C3×C4semidirect
86. #4149C3×C4semidirect
91. #4147C3×C4semidirect
94. #4130C22×C3semidirect
95. C2×#700C22×C3semidirect
96. C2×D5⋊C8C22×C3semidirect
147. C3×2−1+4AGL(1,𝔽5)semidirect
319. 2−1+4C3×AGL(1,𝔽5)semidirect
445. C3×C5#1794semidirect
515. C5C3×#1794semidirect
532. C3#10605020direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#662872
C3×#657952
C3×#657942
C3×#657872
C3×#657852
#106050203
C3×#17945