Finite Group Databasebeta

← C3×#10610423 #14425782

Subgroups — C3×#10610423 #14425782

Order 1920.

1432 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66283C211yesmaximal, characteristic
2C3×#66159C211yesmaximal, characteristic, complemented
3C3×#65860C212yesmaximal, complemented
4C3×C5×#410C211yesmaximal, characteristic, complemented
5C3×#65295C212yesmaximal, complemented
6C3×#65031C218yesmaximal, complemented
7#10610423C311yesmaximal, characteristic, direct product factor, Hall
8C2×C3×C5×#41C2211yescharacteristic
9C3×#4150—22no—
10C3×#4149—22no—
11C3×D5×#41—24no—
12C3×#4147—21no—
13C3×#4144—21no—
14C3×#4144C2214yes—
15C2×C3×#709—22no—
16C2×C3×#709C2212yes—
17C22×C3×Dic10C2211yescharacteristic, complemented
18C3×C5×#117C2212yescomplemented
19C3×#4090—21no—
20C3×#4082—21no—
21C3×#4081C2214yescomplemented
22C3×#4072C2211yescharacteristic, complemented
23C3×#4070—22no—
24C3×#4043—24no—
25C3×#4032C2218yescomplemented
26C3×#4021—22no—
27C3×#4015—22no—
28C3×C5×#99C2214yescomplemented
29C3×#3938C2218yescomplemented
30C3×#3028—51nomaximal, Hall
31#66283C2×C311yescharacteristic
32#66159C2×C311yescharacteristic, complemented
33#65860C2×C312yescomplemented
34C5×#410C2×C311yescharacteristic, complemented
35#65295C2×C312yescomplemented
36#65031C2×C318yescomplemented
37C3×C5×#41—22no—
38C3×C5×#41—22no—
39C2×C3×C5×Q8D411yescharacteristic
40C2×C3×C5×D4D411yescharacteristic
41C2×C3×C5×D4C2312yes—
42C22×C3×C4×C5D412yes—
43C22×C3×C4×C5C2311yescharacteristic
44C2×C3×C22⋊D5—24no—
45C2×C3×C22⋊D5C2314yes—
46C22×C3×Dic5C2312yes—
47C3×#713—24no—
48C3×Q8×D5—24no—
49C3×#711—24no—
50C3×#711—28no—
51C3×D4×D5—24no—
52C3×D4×D5—28no—
53C3×#709—24no—
54C3×#709—24no—
55C3×#709—22no—
56C3×#709—24no—
57C3×#709—22no—
58C3×#709—22no—
59C3×#709—22no—
60C2×C3×D20—22no—
61C2×C3×D20D412yes—
62C2×C3×C4×D5—24no—
63C2×C3×C4×D5D414yes—
64C2×C3×Dic10—22no—
65C2×C3×Dic10—21no—
66C2×C3×Dic10—22no—
67C2×C3×Dic10—22no—
68C2×C3×Dic10—21no—
69C2×C3×Dic10D412yes—
70C3×C5×C4⋊C4—22no—
71C3×C5×C22⋊C4C2314yes—
72C3×C42×C5—22no—
73C3×C22⋊Dic5—24no—
74C3×C22⋊Dic5C2312yes—
75C3×#685—28no—
76C3×C4⋊Dic5—21no—
77C3×C4⋊Dic5—28no—
78C3×#683—24no—
79C3×C4×Dic5—21no—
80C3×#585—51no—
81C3×#536—51no—
82C3×#536—52no—
83C3×#458—58no—
84C3×#410—52no—
85C3×#410D511yescharacteristic, complemented
86C2×C5×#41C22×C311yescharacteristic
87#4150—22no—
88#4149—22no—
89D5×#41—24no—
90#4147—21no—
91#4144—21no—
92#4144C22×C314yes—
93C2×#709—22no—
94C2×#709C22×C312yes—
95C22×Dic10C22×C311yescharacteristic, complemented
96C5×#117C22×C312yescomplemented
97#4090—21no—
98#4082—21no—
99#4081C22×C314yescomplemented
100#4072C22×C311yescharacteristic, complemented
101#4070—22no—
102#4043—24no—
103#4032C22×C318yescomplemented
104#4021—22no—
105#4015—22no—
106C5×#99C22×C314yescomplemented
107#3938C22×C318yescomplemented
108#3028—51noSylow
109C23×C3×C5C2×D412yes—
110C23×C3×C5C2411yescharacteristic
111C22×C3×D5—24no—
112C22×C3×D5C2×D414yes—
113C3×C5×Q8—22no—
114C3×C5×D4—24no—
115C3×C5×D4—22no—
116C2×C3×C4×C5—22no—
117C2×C3×C4×C5—22no—
118C2×C3×C4×C5—24no—
119C2×C3×C4×C5C2×D412yes—
120C2×C3×C4×C5C2×D411yescharacteristic
121C2×C3×C4×C5C2×D411yescharacteristic
122C2×C3×C4×C5C2×D412yes—
123C2×C3×C4×C5C2×D412yes—
124C3×C22⋊D5—28no—
125C3×C22⋊D5—24no—
126C3×C22⋊D5—24no—
127C2×C3×Dic5—24no—
128C2×C3×Dic5—22no—
129C2×C3×Dic5—22no—
130C2×C3×Dic5—24no—
131C2×C3×Dic5C2×D414yes—
132C3×D20—24no—
133C3×D20—22no—
134C3×D20—22no—
135C3×C4×D5—28no—
136C3×C4×D5—24no—
137C3×C4×D5—24no—
138C3×Dic10—24no—
139C3×Dic10—22no—
140C3×Dic10—22no—
141C3×Dic10—41no—
142C3×Dic10—42no—
143C3×Dic10—41no—
144C3×2−1+4—102no—
145C3×2−1+4—101no—
146C3×H(5,𝔽2)—54no—
147C3×H(5,𝔽2)—102no—
148C3×H(5,𝔽2)—101no—
149C2×C3×#41—52no—
150C2×C3×#41—104no—
151C2×C3×#41—102no—
152C2×C3×#41C2×D511yescharacteristic
153C22×C3×Q8—51no—
154C3×C4⋊Q8—102no—
155C3×C4⋊Q8—101no—
156C3×#124—102no—
157C3×#124—101no—
158C3×#123—54no—
159C3×#123—58no—
160C3×C22⋊Q8—104no—
161C3×C22⋊Q8—102no—
162C3×#117—51no—
163C3×#117C2×D512yescomplemented
164C3×#99—58no—
165C3×#99C2×D514yescomplemented
166C5×#41—22no—
167C5×#41—22no—
168C2×C5×Q8C3×D411yescharacteristic
169C2×C5×D4C3×D411yescharacteristic
170C2×C5×D4C23×C312yes—
171C22×C4×C5C3×D412yes—
172C22×C4×C5C23×C311yescharacteristic
173C2×C22⋊D5—24no—
174C2×C22⋊D5C23×C314yes—
175C22×Dic5C23×C312yes—
176#713—24no—
177Q8×D5—24no—
178#711—24no—
179#711—28no—
180D4×D5—24no—
181D4×D5—28no—
182#709—24no—
183#709—24no—
184#709—22no—
185#709—24no—
186#709—22no—
187#709—22no—
188#709—22no—
189C2×D20—22no—
190C2×D20C3×D412yes—
191C2×C4×D5—24no—
192C2×C4×D5C3×D414yes—
193C2×Dic10—22no—
194C2×Dic10—22no—
195C2×Dic10—22no—
196C2×Dic10—21no—
197C2×Dic10—21no—
198C2×Dic10C3×D412yes—
199C5×C4⋊C4—22no—
200C5×C22⋊C4C23×C314yes—
201C42×C5—22no—
202C22⋊Dic5—24no—
203C22⋊Dic5C23×C312yes—
204#685—28no—
205C4⋊Dic5—21no—
206C4⋊Dic5—28no—
207#683—24no—
208C4×Dic5—21no—
209#585—51no—
210#536—51no—
211#536—52no—
212#458—58no—
213#410—52no—
214#410C3×D511yescharacteristic, complemented
215C22×C3×C5—22no—
216C22×C3×C5—41no—
217C22×C3×C5—42no—
218C22×C3×C5C22×D411yescharacteristic
219C22×C3×C5C22×D412yes—
220C2×C3×D5—24no—
221C2×C3×D5—44no—
222C2×C3×D5—84no—
223C3×C4×C5—22no—
224C3×C4×C5—44no—
225C3×C4×C5#12011yescharacteristic
226C3×C4×C5#12012yes—
227C3×C4×C5#12011yescharacteristic
228C3×Dic5—24no—
229C3×Dic5—42no—
230C3×Dic5—42no—
231C3×#41—22no—
232C3×#41—22no—
233C3×#41—104no—
234C3×#41—104no—
235C3×#41—108no—
236C3×#41—104no—
237C3×#41—104no—
238C3×#41—102no—
239C3×#41—102no—
240C3×#41—104no—
241C3×#41—102no—
242C3×#41—102no—
243C2×C3×Q8—52no—
244C2×C3×Q8—102no—
245C2×C3×Q8—104no—
246C2×C3×Q8—101no—
247C2×C3×Q8—101no—
248C2×C3×Q8—102no—
249C2×C3×Q8—102no—
250C2×C3×Q8D2011yescharacteristic
251C2×C3×D4—54no—
252C2×C3×D4—52no—
253C2×C3×D4—102no—
254C2×C3×D4—108no—
255C2×C3×D4—104no—
256C2×C3×D4—104no—
257C2×C3×D4D2011yescharacteristic
258C2×C3×D4C22×D512yes—
259C22×C3×C4—54no—
260C22×C3×C4—52no—
261C22×C3×C4—104no—
262C22×C3×C4D2012yes—
263C22×C3×C4C22×D511yescharacteristic
264C3×C4⋊C4—22no—
265C3×C4⋊C4—108no—
266C3×C4⋊C4—104no—
267C3×C4⋊C4—101no—
268C3×C22⋊C4—52no—
269C3×C22⋊C4—108no—
270C3×C22⋊C4—104no—
271C3×C22⋊C4C22×D514yes—
272C3×C42—22no—
273C3×C42—101no—
274C23×C5C2×C3×D412yes—
275C23×C5C24×C311yescharacteristic
276C22×D5—24no—
277C22×D5C2×C3×D414yes—
278C5×Q8—22no—
279C5×D4—24no—
280C5×D4—22no—
281C2×C4×C5—22no—
282C2×C4×C5—22no—
283C2×C4×C5—24no—
284C2×C4×C5C2×C3×D412yes—
285C2×C4×C5C2×C3×D411yescharacteristic
286C2×C4×C5C2×C3×D411yescharacteristic
287C2×C4×C5C2×C3×D412yes—
288C2×C4×C5C2×C3×D412yes—
289C22⋊D5—28no—
290C22⋊D5—24no—
291C22⋊D5—24no—
292C2×Dic5—24no—
293C2×Dic5—22no—
294C2×Dic5—22no—
295C2×Dic5—24no—
296C2×Dic5C2×C3×D414yes—
297D20—24no—
298D20—22no—
299D20—22no—
300C4×D5—28no—
301C4×D5—24no—
302C4×D5—24no—
303Dic10—24no—
304Dic10—22no—
305Dic10—22no—
306Dic10—41no—
307Dic10—42no—
308Dic10—41no—
3092−1+4—102no—
3102−1+4—101no—
311H(5,𝔽2)—54no—
312H(5,𝔽2)—102no—
313H(5,𝔽2)—101no—
314C2×#41—52no—
315C2×#41—104no—
316C2×#41—102no—
317C2×#41C2×C3×D511yescharacteristic
318C22×Q8—51no—
319C4⋊Q8—102no—
320C4⋊Q8—101no—
321#124—102no—
322#124—101no—
323#123—54no—
324#123—58no—
325C22⋊Q8—104no—
326C22⋊Q8—102no—
327#117—51no—
328#117C2×C3×D512yescomplemented
329#99—58no—
330#99C2×C3×D514yescomplemented
331C2×C3×C5—21no—
332C2×C3×C5—22no—
333C2×C3×C5—42no—
334C2×C3×C5C2×#12011yescharacteristic
335C3×D5—44no—
336C23×C3—54no—
337C23×C3—104no—
338C23×C3C2×D2012yes—
339C23×C3C23×D511yescharacteristic
340C3×Q8—22no—
341C3×Q8—104no—
342C3×Q8—102no—
343C3×Q8—102no—
344C3×Q8—201no—
345C3×Q8—201no—
346C3×Q8—202no—
347C3×D4—24no—
348C3×D4—22no—
349C3×D4—104no—
350C3×D4—108no—
351C3×D4—104no—
352C3×D4—104no—
353C3×D4—102no—
354C3×D4—102no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—24no—
358C2×C3×C4—54no—
359C2×C3×C4—108no—
360C2×C3×C4—104no—
361C2×C3×C4—104no—
362C2×C3×C4—104no—
363C2×C3×C4—102no—
364C2×C3×C4—102no—
365C2×C3×C4—104no—
366C2×C3×C4C2×D2012yes—
367C2×C3×C4C2×D2011yescharacteristic
368C2×C3×C4C2×D2011yescharacteristic
369C2×C3×C4D4×D512yes—
370C2×C3×C4D4×D512yes—
371C22×C5—22no—
372C22×C5—41no—
373C22×C5—42no—
374C22×C5C22×C3×D411yescharacteristic
375C22×C5C22×C3×D412yes—
376C2×D5—24no—
377C2×D5—44no—
378C2×D5—84no—
379C4×C5—22no—
380C4×C5—44no—
381C4×C5C3×#12011yescharacteristic
382C4×C5C3×#12012yes—
383C4×C5C3×#12011yescharacteristic
384Dic5—24no—
385Dic5—42no—
386Dic5—42no—
387#41—22no—
388#41—22no—
389#41—104no—
390#41—104no—
391#41—108no—
392#41—104no—
393#41—104no—
394#41—102no—
395#41—102no—
396#41—104no—
397#41—102no—
398#41—102no—
399C2×Q8—52no—
400C2×Q8—102no—
401C2×Q8—104no—
402C2×Q8—101no—
403C2×Q8—101no—
404C2×Q8—102no—
405C2×Q8—102no—
406C2×Q8C3×D2011yescharacteristic
407C2×D4—54no—
408C2×D4—52no—
409C2×D4—102no—
410C2×D4—108no—
411C2×D4—104no—
412C2×D4—104no—
413C2×D4C3×D2011yescharacteristic
414C2×D4C22×C3×D512yes—
415C22×C4—54no—
416C22×C4—52no—
417C22×C4—104no—
418C22×C4C3×D2012yes—
419C22×C4C22×C3×D511yescharacteristic
420C4⋊C4—22no—
421C4⋊C4—108no—
422C4⋊C4—101no—
423C4⋊C4—104no—
424C22⋊C4—52no—
425C22⋊C4—108no—
426C22⋊C4—104no—
427C22⋊C4C22×C3×D514yes—
428C42—22no—
429C42—101no—
430C3×C5#302811yescharacteristic, complemented, Hall
431C22×C3—22no—
432C22×C3—41no—
433C22×C3—42no—
434C22×C3—104no—
435C22×C3—204no—
436C22×C3—404no—
437C22×C3C22×D2011yescharacteristic
438C22×C3C2×D4×D512yes—
439C3×C4—22no—
440C3×C4—44no—
441C3×C4—104no—
442C3×C4—202no—
443C3×C4—202no—
444C3×C4#402811yescharacteristic
445C3×C4#402811yescharacteristic
446C3×C4#402812yes—
447C2×C5—21no—
448C2×C5—22no—
449C2×C5—42no—
450C2×C5C2×C3×#12011yescharacteristic
451D5—44no—
452C23—54no—
453C23—104no—
454C23C2×C3×D2012yes—
455C23C23×C3×D511yescharacteristic
456Q8—22no—
457Q8—104no—
458Q8—102no—
459Q8—102no—
460Q8—201no—
461Q8—201no—
462Q8—202no—
463D4—24no—
464D4—22no—
465D4—104no—
466D4—108no—
467D4—104no—
468D4—104no—
469D4—102no—
470D4—102no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—54no—
475C2×C4—108no—
476C2×C4—104no—
477C2×C4—104no—
478C2×C4—104no—
479C2×C4—102no—
480C2×C4—102no—
481C2×C4—104no—
482C2×C4C2×C3×D2012yes—
483C2×C4C2×C3×D2011yescharacteristic
484C2×C4C2×C3×D2011yescharacteristic
485C2×C4C3×D4×D512yes—
486C2×C4C3×D4×D512yes—
487C2×C3—22no—
488C2×C3—21no—
489C2×C3—42no—
490C2×C3—204no—
491C2×C3C2×#402811yescharacteristic
492C5C3×#302811yescharacteristic, complemented, Sylow
493C22—22no—
494C22—41no—
495C22—42no—
496C22—104no—
497C22—204no—
498C22—404no—
499C22C22×C3×D2011yescharacteristic
500C22C2×C3×D4×D512yes—
501C4—22no—
502C4—44no—
503C4—104no—
504C4—202no—
505C4—202no—
506C4C3×#402811yescharacteristic
507C4C3×#402811yescharacteristic
508C4C3×#402812yes—
509C3#1061042311yescharacteristic, direct product factor, Sylow
510C2—22no—
511C2—21no—
512C2—42no—
513C2—204no—
514C2C2×C3×#402811yescharacteristic
subgroupG/H
centerC2×C3C2×#4028
commutator, cosocleC23×C5C24×C3
FrattiniC23C23×C3×D5
FittingC3×C5×#410C2
socleC2×C3×C5C2×#120
prime radicalC3×#66283C2
hypercenterC3×#410D5
hypocenterC5C3×#3028
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#410

Complements

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

NH ≅ G/Ntype
2. C3×#66159C2semidirect
3. C3×#65860C2semidirect
4. C3×C5×#410C2semidirect
5. C3×#65295C2semidirect
6. C3×#65031C2semidirect
7. #10610423C3direct (N × H)
17. C22×C3×Dic10C22semidirect
18. C3×C5×#117C22semidirect
21. C3×#4081C22semidirect
22. C3×#4072C22semidirect
25. C3×#4032C22semidirect
28. C3×C5×#99C22semidirect
29. C3×#3938C22semidirect
32. #66159C2×C3semidirect
33. #65860C2×C3semidirect
34. C5×#410C2×C3semidirect
35. #65295C2×C3semidirect
36. #65031C2×C3semidirect
85. C3×#410D5semidirect
95. C22×Dic10C22×C3semidirect
96. C5×#117C22×C3semidirect
99. #4081C22×C3semidirect
100. #4072C22×C3semidirect
103. #4032C22×C3semidirect
106. C5×#99C22×C3semidirect
107. #3938C22×C3semidirect
163. C3×#117C2×D5semidirect
165. C3×#99C2×D5semidirect
214. #410C3×D5semidirect
328. #117C2×C3×D5semidirect
330. #99C2×C3×D5semidirect
430. C3×C5#3028semidirect
492. C5C3×#3028semidirect
509. C3#10610423direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#662832
C3×#661592
C3×#658602
C3×C5×#4102
C3×#652952
C3×#650312
#106104233
C3×#30285