Finite Group Databasebeta

← C3×#10610447 #14425207

Subgroups — C3×#10610447 #14425207

Order 1920.

1448 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66283C211yesmaximal, characteristic, complemented
2C3×#66166C211yesmaximal, characteristic, complemented
3C3×#66082C212yesmaximal, complemented
4C3×C5×#413C211yesmaximal, characteristic, complemented
5C2×C3×#3957C212yesmaximal, complemented
6C3×#65042C212yesmaximal, complemented
7C3×#65041C212yesmaximal, complemented
8C3×#65036C212yesmaximal, complemented
9C3×#65035C212yesmaximal, complemented
10#10610447C311yesmaximal, characteristic, direct product factor, Hall
11C2×C3×C5×#41C2211yescharacteristic
12C3×#4150—22no—
13C3×#4149—22no—
14C3×D5×#41—24no—
15C3×#4147C2212yes—
16C3×#4144—22no—
17C3×#4144C2212yes—
18C2×C3×#709—22no—
19C2×C3×#709C2212yes—
20C2×C3×#709C2211yescharacteristic, complemented
21C2×C3×C5×M16C2212yescomplemented
22C3×#4093C2211yescharacteristic, complemented
23C3×#4090C2211yescharacteristic, complemented
24C3×#4086C2211yescharacteristic, complemented
25C3×#4085—22no—
26C3×#4082C2211yescharacteristic, complemented
27C3×C4×C22⋊D5—22no—
28C2×C3×C4×Dic5C2211yescharacteristic, complemented
29C3×#4055—22no—
30C3×#4055C2214yescomplemented
31C3×#4054—22no—
32C3×#4054C2214yescomplemented
33C3×#4050—24no—
34C3×C5×#101C2212yescomplemented
35C3×C5×#100C2212yescomplemented
36C3×#3957C2214yescomplemented
37C3×#3957C2214yescomplemented
38C3×#3018—51nomaximal, Hall
39#66283C2×C311yescharacteristic, complemented
40#66166C2×C311yescharacteristic, complemented
41#66082C2×C312yescomplemented
42C5×#413C2×C311yescharacteristic, complemented
43C2×#3957C2×C312yescomplemented
44#65042C2×C312yescomplemented
45#65041C2×C312yescomplemented
46#65036C2×C312yescomplemented
47#65035C2×C312yescomplemented
48C3×C5×#41—22no—
49C3×C5×#41—22no—
50C2×C3×C5×Q8C2311yescharacteristic
51C2×C3×C5×D4D412yes—
52C2×C3×C5×D4C2311yescharacteristic
53C22×C3×C4×C5D412yes—
54C22×C3×C4×C5C2311yescharacteristic
55C2×C3×C22⋊D5—24no—
56C2×C3×C22⋊D5—22no—
57C2×C3×C22⋊D5—21no—
58C2×C3×C22⋊D5—22no—
59C22×C3×Dic5—21no—
60C3×#713—24no—
61C3×Q8×D5—24no—
62C3×#711—28no—
63C3×#711—24no—
64C3×D4×D5—28no—
65C3×D4×D5—24no—
66C3×#709—24no—
67C3×#709—24no—
68C3×#709—22no—
69C3×#709—24no—
70C3×#709—22no—
71C3×#709—22no—
72C3×#709—22no—
73C3×#709D414yescomplemented
74C3×#709C2314yes—
75C2×C3×D20—22no—
76C2×C3×D20D412yescomplemented
77C2×C3×D20C2311yescharacteristic
78C2×C3×C4×D5—24no—
79C2×C3×C4×D5—22no—
80C2×C3×C4×D5—21no—
81C2×C3×Dic10—22no—
82C2×C3×Dic10D412yescomplemented
83C2×C3×Dic10C2311yescharacteristic
84C3×C5×M16—22no—
85C3×C5×M16C2314yes—
86C2×C3×C5×C8—22no—
87C3×C22⋊Dic5—22no—
88C3×#685—22no—
89C3×#683—22no—
90C3×C4×Dic5—21no—
91C3×C4×Dic5C2311yescharacteristic
92C3×C4×Dic5C2311yescharacteristic
93C3×Dic20—24no—
94C3×D40—24no—
95C3×#677—24no—
96C3×#677—24no—
97C3×#585—51no—
98C3×#575—52no—
99C3×#532—51no—
100C3×#456—52no—
101C3×#455—52no—
102C3×#454—52no—
103C3×#453—52no—
104C2×C3×#104—52no—
105C3×#413D511yescharacteristic, complemented
106C2×C5×#41C22×C311yescharacteristic
107#4150—22no—
108#4149—22no—
109D5×#41—24no—
110#4147C22×C312yes—
111#4144—22no—
112#4144C22×C312yes—
113C2×#709—22no—
114C2×#709C22×C312yes—
115C2×#709C22×C311yescharacteristic, complemented
116C2×C5×M16C22×C312yescomplemented
117#4093C22×C311yescharacteristic, complemented
118#4090C22×C311yescharacteristic, complemented
119#4086C22×C311yescharacteristic, complemented
120#4085—22no—
121#4082C22×C311yescharacteristic, complemented
122C4×C22⋊D5—22no—
123C2×C4×Dic5C22×C311yescharacteristic, complemented
124#4055—22no—
125#4055C22×C314yescomplemented
126#4054—22no—
127#4054C22×C314yescomplemented
128#4050—24no—
129C5×#101C22×C312yescomplemented
130C5×#100C22×C312yescomplemented
131#3957C22×C314yescomplemented
132#3957C22×C314yescomplemented
133#3018—51noSylow
134C23×C3×C5C2×D412yes—
135C23×C3×C5C2×D411yescharacteristic
136C22×C3×D5—24no—
137C22×C3×D5—22no—
138C22×C3×D5—21no—
139C3×C5×Q8—22no—
140C3×C5×D4—24no—
141C3×C5×D4—22no—
142C2×C3×C4×C5—22no—
143C2×C3×C4×C5—22no—
144C2×C3×C4×C5C2×D412yes—
145C2×C3×C4×C5C2×D411yescharacteristic
146C2×C3×C4×C5C2×D412yes—
147C2×C3×C4×C5C2×D412yes—
148C2×C3×C4×C5C2411yescharacteristic
149C3×C22⋊D5—28no—
150C3×C22⋊D5—24no—
151C3×C22⋊D5—24no—
152C3×C22⋊D5—44no—
153C3×C22⋊D5—42no—
154C2×C3×Dic5—24no—
155C2×C3×Dic5—22no—
156C2×C3×Dic5—21no—
157C2×C3×Dic5—21no—
158C2×C3×Dic5—21no—
159C2×C3×Dic5—42no—
160C3×D20—24no—
161C3×D20—22no—
162C3×D20—22no—
163C3×D20C2×D414yes—
164C3×C4×D5—28no—
165C3×C4×D5—24no—
166C3×C4×D5—24no—
167C3×C4×D5—41no—
168C3×C4×D5—41no—
169C3×Dic10—24no—
170C3×Dic10—22no—
171C3×Dic10—22no—
172C3×Dic10C2×D414yes—
173C3×C5×C8—24no—
174C3×2−1+4—52no—
175C3×2−1+4—102no—
176C3×H(5,𝔽2)—52no—
177C3×H(5,𝔽2)—102no—
178C3×H(5,𝔽2)—102no—
179C2×C3×#41—52no—
180C2×C3×#41—51no—
181C2×C3×#41—104no—
182C2×C3×#41—102no—
183C2×C3×#41C2×D511yescharacteristic
184C3×#137—54no—
185C3×#137—102no—
186C3×AGL(1,ℤ/8)—54no—
187C3×AGL(1,ℤ/8)—102no—
188C3×#135—104no—
189C2×C3×M16C2×D512yescomplemented
190C3×C4⋊Q8—51no—
191C3×#127—51no—
192C3×#124—51no—
193C3×#124—51no—
194C3×#121—102no—
195C3×C4×D4—102no—
196C2×C3×C42—51no—
197C3×#104—54no—
198C3×#104—54no—
199C3×#101C2×D512yescomplemented
200C3×#100C2×D512yescomplemented
201C5×#41—22no—
202C5×#41—22no—
203C2×C5×Q8C23×C311yescharacteristic
204C2×C5×D4C3×D412yes—
205C2×C5×D4C23×C311yescharacteristic
206C22×C4×C5C3×D412yes—
207C22×C4×C5C23×C311yescharacteristic
208C2×C22⋊D5—24no—
209C2×C22⋊D5—22no—
210C2×C22⋊D5—21no—
211C2×C22⋊D5—22no—
212C22×Dic5—21no—
213#713—24no—
214Q8×D5—24no—
215#711—28no—
216#711—24no—
217D4×D5—28no—
218D4×D5—24no—
219#709—24no—
220#709—24no—
221#709—22no—
222#709—24no—
223#709—22no—
224#709—22no—
225#709—22no—
226#709C3×D414yescomplemented
227#709C23×C314yes—
228C2×D20—22no—
229C2×D20C3×D412yescomplemented
230C2×D20C23×C311yescharacteristic
231C2×C4×D5—24no—
232C2×C4×D5—22no—
233C2×C4×D5—21no—
234C2×Dic10—22no—
235C2×Dic10C3×D412yescomplemented
236C2×Dic10C23×C311yescharacteristic
237C5×M16—22no—
238C5×M16C23×C314yes—
239C2×C5×C8—22no—
240C22⋊Dic5—22no—
241#685—22no—
242#683—22no—
243C4×Dic5—21no—
244C4×Dic5C23×C311yescharacteristic
245C4×Dic5C23×C311yescharacteristic
246Dic20—24no—
247D40—24no—
248#677—24no—
249#677—24no—
250#585—51no—
251#575—52no—
252#532—51no—
253#456—52no—
254#455—52no—
255#454—52no—
256#453—52no—
257C2×#104—52no—
258#413C3×D511yescharacteristic, complemented
259C22×C3×C5—22no—
260C22×C3×C5—41no—
261C22×C3×C5—42no—
262C22×C3×C5#12012yes—
263C22×C3×C5C22×D411yescharacteristic
264C2×C3×D5—24no—
265C2×C3×D5—42no—
266C2×C3×D5—41no—
267C2×C3×D5—82no—
268C2×C3×D5—84no—
269C3×C4×C5—22no—
270C3×C4×C5#12011yescharacteristic
271C3×C4×C5#12011yescharacteristic
272C3×C4×C5C22×D412yes—
273C3×Dic5—22no—
274C3×Dic5—24no—
275C3×Dic5—41no—
276C3×Dic5—41no—
277C3×#41—22no—
278C3×#41—22no—
279C3×#41—54no—
280C3×#41—54no—
281C3×#41—104no—
282C3×#41—104no—
283C3×#41—104no—
284C3×#41—108no—
285C3×#41—104no—
286C3×#41—102no—
287C3×#41—104no—
288C3×#41—102no—
289C3×#41—102no—
290C3×#41—102no—
291C2×C3×Q8—52no—
292C2×C3×Q8—51no—
293C2×C3×Q8—102no—
294C2×C3×Q8—104no—
295C2×C3×Q8C22×D511yescharacteristic
296C2×C3×D4—52no—
297C2×C3×D4—51no—
298C2×C3×D4—102no—
299C2×C3×D4—104no—
300C2×C3×D4—108no—
301C2×C3×D4—102no—
302C2×C3×D4—104no—
303C2×C3×D4—102no—
304C2×C3×D4—101no—
305C2×C3×D4D2012yes—
306C2×C3×D4C22×D511yescharacteristic
307C22×C3×C4—104no—
308C22×C3×C4—102no—
309C22×C3×C4—101no—
310C22×C3×C4—101no—
311C22×C3×C4D2012yes—
312C22×C3×C4C22×D511yescharacteristic
313C3×Q16—104no—
314C3×SD16—104no—
315C3×SD16—104no—
316C3×D8—104no—
317C3×M16—22no—
318C3×M16C22×D514yes—
319C2×C3×C8—22no—
320C3×C4⋊C4—102no—
321C3×C22⋊C4—102no—
322C3×C22⋊C4—102no—
323C3×C42—51no—
324C3×C42—51no—
325C3×C42—101no—
326C23×C5C2×C3×D412yes—
327C23×C5C2×C3×D411yescharacteristic
328C22×D5—24no—
329C22×D5—22no—
330C22×D5—21no—
331C5×Q8—22no—
332C5×D4—24no—
333C5×D4—22no—
334C2×C4×C5—22no—
335C2×C4×C5—22no—
336C2×C4×C5C2×C3×D412yes—
337C2×C4×C5C2×C3×D411yescharacteristic
338C2×C4×C5C2×C3×D412yes—
339C2×C4×C5C2×C3×D412yes—
340C2×C4×C5C24×C311yescharacteristic
341C22⋊D5—28no—
342C22⋊D5—24no—
343C22⋊D5—24no—
344C22⋊D5—44no—
345C22⋊D5—42no—
346C2×Dic5—24no—
347C2×Dic5—22no—
348C2×Dic5—21no—
349C2×Dic5—21no—
350C2×Dic5—21no—
351C2×Dic5—42no—
352D20—24no—
353D20—22no—
354D20—22no—
355D20C2×C3×D414yes—
356C4×D5—28no—
357C4×D5—24no—
358C4×D5—24no—
359C4×D5—41no—
360C4×D5—41no—
361Dic10—24no—
362Dic10—22no—
363Dic10—22no—
364Dic10C2×C3×D414yes—
365C5×C8—24no—
3662−1+4—52no—
3672−1+4—102no—
368H(5,𝔽2)—52no—
369H(5,𝔽2)—102no—
370H(5,𝔽2)—102no—
371C2×#41—52no—
372C2×#41—51no—
373C2×#41—104no—
374C2×#41—102no—
375C2×#41C2×C3×D511yescharacteristic
376#137—54no—
377#137—102no—
378AGL(1,ℤ/8)—54no—
379AGL(1,ℤ/8)—102no—
380#135—104no—
381C2×M16C2×C3×D512yescomplemented
382C4⋊Q8—51no—
383#127—51no—
384#124—51no—
385#124—51no—
386#121—102no—
387C4×D4—102no—
388C2×C42—51no—
389#104—54no—
390#104—54no—
391#101C2×C3×D512yescomplemented
392#100C2×C3×D512yescomplemented
393C2×C3×C5—21no—
394C2×C3×C5—22no—
395C2×C3×C5—42no—
396C2×C3×C5C2×#12011yescharacteristic
397C3×D5—44no—
398C3×D5—81no—
399C23×C3—104no—
400C23×C3—102no—
401C23×C3—101no—
402C23×C3C2×D2012yes—
403C23×C3C2×D2011yescharacteristic
404C3×Q8—22no—
405C3×Q8—54no—
406C3×Q8—104no—
407C3×Q8—102no—
408C3×Q8—102no—
409C3×D4—24no—
410C3×D4—22no—
411C3×D4—54no—
412C3×D4—104no—
413C3×D4—108no—
414C3×D4—104no—
415C3×D4—104no—
416C3×D4—102no—
417C3×D4—102no—
418C3×D4—204no—
419C3×D4—202no—
420C2×C3×C4—22no—
421C2×C3×C4—22no—
422C2×C3×C4—108no—
423C2×C3×C4—104no—
424C2×C3×C4—101no—
425C2×C3×C4—102no—
426C2×C3×C4—104no—
427C2×C3×C4—104no—
428C2×C3×C4—101no—
429C2×C3×C4—101no—
430C2×C3×C4—202no—
431C2×C3×C4—201no—
432C2×C3×C4—201no—
433C2×C3×C4C2×D2012yes—
434C2×C3×C4C2×D2011yescharacteristic
435C2×C3×C4D4×D512yes—
436C2×C3×C4D4×D512yes—
437C2×C3×C4C23×D511yescharacteristic
438C3×C8—24no—
439C22×C5—22no—
440C22×C5—41no—
441C22×C5—42no—
442C22×C5C3×#12012yes—
443C22×C5C22×C3×D411yescharacteristic
444C2×D5—24no—
445C2×D5—42no—
446C2×D5—41no—
447C2×D5—82no—
448C2×D5—84no—
449C4×C5—22no—
450C4×C5C3×#12011yescharacteristic
451C4×C5C3×#12011yescharacteristic
452C4×C5C22×C3×D412yes—
453Dic5—22no—
454Dic5—24no—
455Dic5—41no—
456Dic5—41no—
457#41—22no—
458#41—22no—
459#41—54no—
460#41—54no—
461#41—104no—
462#41—104no—
463#41—104no—
464#41—108no—
465#41—104no—
466#41—102no—
467#41—104no—
468#41—102no—
469#41—102no—
470#41—102no—
471C2×Q8—52no—
472C2×Q8—51no—
473C2×Q8—102no—
474C2×Q8—104no—
475C2×Q8C22×C3×D511yescharacteristic
476C2×D4—52no—
477C2×D4—51no—
478C2×D4—102no—
479C2×D4—104no—
480C2×D4—108no—
481C2×D4—102no—
482C2×D4—104no—
483C2×D4—102no—
484C2×D4—101no—
485C2×D4C3×D2012yes—
486C2×D4C22×C3×D511yescharacteristic
487C22×C4—104no—
488C22×C4—102no—
489C22×C4—101no—
490C22×C4—101no—
491C22×C4C3×D2012yes—
492C22×C4C22×C3×D511yescharacteristic
493Q16—104no—
494SD16—104no—
495SD16—104no—
496D8—104no—
497M16—22no—
498M16C22×C3×D514yes—
499C2×C8—22no—
500C4⋊C4—102no—
501C22⋊C4—102no—
502C22⋊C4—102no—
503C42—51no—
504C42—51no—
505C42—101no—
506C3×C5#301811yescharacteristic, complemented, Hall
507C22×C3—22no—
508C22×C3—41no—
509C22×C3—42no—
510C22×C3—104no—
511C22×C3—202no—
512C22×C3—201no—
513C22×C3—402no—
514C22×C3—404no—
515C22×C3#402812yes—
516C22×C3C22×D2011yescharacteristic
517C3×C4—22no—
518C3×C4—102no—
519C3×C4—104no—
520C3×C4—201no—
521C3×C4—201no—
522C3×C4#402811yescharacteristic
523C3×C4#402811yescharacteristic
524C3×C4C2×D4×D512yes—
525C2×C5—21no—
526C2×C5—22no—
527C2×C5—42no—
528C2×C5C2×C3×#12011yescharacteristic
529D5—44no—
530D5—81no—
531C23—104no—
532C23—102no—
533C23—101no—
534C23C2×C3×D2012yes—
535C23C2×C3×D2011yescharacteristic
536Q8—22no—
537Q8—54no—
538Q8—104no—
539Q8—102no—
540Q8—102no—
541D4—24no—
542D4—22no—
543D4—54no—
544D4—104no—
545D4—108no—
546D4—104no—
547D4—104no—
548D4—102no—
549D4—102no—
550D4—204no—
551D4—202no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—108no—
555C2×C4—104no—
556C2×C4—101no—
557C2×C4—102no—
558C2×C4—104no—
559C2×C4—104no—
560C2×C4—101no—
561C2×C4—101no—
562C2×C4—202no—
563C2×C4—201no—
564C2×C4—201no—
565C2×C4C2×C3×D2012yes—
566C2×C4C2×C3×D2011yescharacteristic
567C2×C4C3×D4×D512yes—
568C2×C4C3×D4×D512yes—
569C2×C4C23×C3×D511yescharacteristic
570C8—24no—
571C2×C3—21no—
572C2×C3—22no—
573C2×C3—42no—
574C2×C3—204no—
575C2×C3—401no—
576C2×C3C2×#402811yescharacteristic
577C5C3×#301811yescharacteristic, complemented, Sylow
578C22—22no—
579C22—41no—
580C22—42no—
581C22—104no—
582C22—202no—
583C22—201no—
584C22—402no—
585C22—404no—
586C22C3×#402812yes—
587C22C22×C3×D2011yescharacteristic
588C4—22no—
589C4—102no—
590C4—104no—
591C4—201no—
592C4—201no—
593C4C3×#402811yescharacteristic
594C4C3×#402811yescharacteristic
595C4C2×C3×D4×D512yes—
596C3#1061044711yescharacteristic, direct product factor, Sylow
597C2—21no—
598C2—22no—
599C2—42no—
600C2—204no—
601C2—401no—
602C2C2×C3×#402811yescharacteristic
subgroupG/H
centerC3×C4#4028
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC3×C5×#413C2
socleC2×C3×C5C2×#120
hypercenterC3×#413D5
hypocenterC5C3×#3018
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C3×#413

Complements

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

NH ≅ G/Ntype
1. C3×#66283C2semidirect
2. C3×#66166C2semidirect
3. C3×#66082C2semidirect
4. C3×C5×#413C2semidirect
5. C2×C3×#3957C2semidirect
6. C3×#65042C2semidirect
7. C3×#65041C2semidirect
8. C3×#65036C2semidirect
9. C3×#65035C2semidirect
10. #10610447C3direct (N × H)
20. C2×C3×#709C22semidirect
21. C2×C3×C5×M16C22semidirect
22. C3×#4093C22semidirect
23. C3×#4090C22semidirect
24. C3×#4086C22semidirect
26. C3×#4082C22semidirect
28. C2×C3×C4×Dic5C22semidirect
30. C3×#4055C22semidirect
32. C3×#4054C22semidirect
34. C3×C5×#101C22semidirect
35. C3×C5×#100C22semidirect
36. C3×#3957C22semidirect
37. C3×#3957C22semidirect
39. #66283C2×C3semidirect
40. #66166C2×C3semidirect
41. #66082C2×C3semidirect
42. C5×#413C2×C3semidirect
43. C2×#3957C2×C3semidirect
44. #65042C2×C3semidirect
45. #65041C2×C3semidirect
46. #65036C2×C3semidirect
47. #65035C2×C3semidirect
73. C3×#709D4semidirect
76. C2×C3×D20D4semidirect
82. C2×C3×Dic10D4semidirect
105. C3×#413D5semidirect
115. C2×#709C22×C3semidirect
116. C2×C5×M16C22×C3semidirect
117. #4093C22×C3semidirect
118. #4090C22×C3semidirect
119. #4086C22×C3semidirect
121. #4082C22×C3semidirect
123. C2×C4×Dic5C22×C3semidirect
125. #4055C22×C3semidirect
127. #4054C22×C3semidirect
129. C5×#101C22×C3semidirect
130. C5×#100C22×C3semidirect
131. #3957C22×C3semidirect
132. #3957C22×C3semidirect
189. C2×C3×M16C2×D5semidirect
199. C3×#101C2×D5semidirect
200. C3×#100C2×D5semidirect
226. #709C3×D4semidirect
229. C2×D20C3×D4semidirect
235. C2×Dic10C3×D4semidirect
258. #413C3×D5semidirect
381. C2×M16C2×C3×D5semidirect
391. #101C2×C3×D5semidirect
392. #100C2×C3×D5semidirect
506. C3×C5#3018semidirect
577. C5C3×#3018semidirect
596. C3#10610447direct (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×#661662
C3×#660822
C3×C5×#4132
C2×C3×#39572
C3×#650422
C3×#650412
C3×#650362
C3×#650352
#106104473
C3×#30185