Finite Group Databasebeta

← C3×#10610567 #14424439

Subgroups — C3×#10610567 #14424439

Order 1920.

1464 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4150C211yesmaximal, characteristic, complemented
2C2×C3×#4095C211yesmaximal, characteristic, complemented
3C22×C3×#677C211yesmaximal, characteristic, complemented
4C3×#65859C211yesmaximal, characteristic, complemented
5C3×C5×#416C211yesmaximal, characteristic, complemented
6C3×#65399C211yesmaximal, characteristic, complemented
7C3×#65293C211yesmaximal, characteristic, complemented
8C3×#65095C214yesmaximal, complemented
9C3×#65070C214yesmaximal, complemented
10#10610567C311yesmaximal, characteristic, direct product factor, Hall
11C2×C3×C5×#41C2211yescharacteristic
12C3×#4150—24no—
13C3×#4150—24no—
14C2×C3×Q8×D5C2212yescomplemented
15C2×C3×#711—22no—
16C2×C3×#711C2212yescomplemented
17C2×C3×#709—21no—
18C2×C3×#709C2211yescharacteristic
19C22×C3×D20C2211yescharacteristic
20C22×C3×Dic10—21no—
21C22×C3×Dic10C2211yescharacteristic
22C22×C3×C5×C8C2211yescharacteristic
23C3×C5×#117C2211yescharacteristic
24C3×#4095—24no—
25C2×C3×#688C2212yes—
26C2×C3×#686C2212yes—
27C2×C3×#681C2211yescharacteristic
28C2×C3×#677—24no—
29C2×C3×#677C2214yes—
30C3×#4041C2214yes—
31C3×#4028—22no—
32C3×#4021—21no—
33C3×#4020—21no—
34C3×C5×#103C2212yescomplemented
35C3×C5×#102C2212yescomplemented
36C3×#3952C2214yescomplemented
37C3×#3942C2214yescomplemented
38C3×#3010—51nomaximal, Hall
39C2×#4150C2×C311yescharacteristic, complemented
40C2×#4095C2×C311yescharacteristic, complemented
41C22×#677C2×C311yescharacteristic, complemented
42#65859C2×C311yescharacteristic, complemented
43C5×#416C2×C311yescharacteristic, complemented
44#65399C2×C311yescharacteristic, complemented
45#65293C2×C311yescharacteristic, complemented
46#65095C2×C314yescomplemented
47#65070C2×C314yescomplemented
48C23×C3×D5—21no—
49C3×C5×#41—22no—
50C3×C5×#41D414yescomplemented
51C2×C3×C5×Q8C2311yescharacteristic
52C2×C3×C5×D4—21no—
53C2×C3×C5×D4C2311yescharacteristic
54C22×C3×C4×C5—21no—
55C22×C3×C4×C5C2311yescharacteristic
56C2×C3×C22⋊D5—22no—
57C2×C3×C22⋊D5D412yes—
58C22×C3×Dic5—21no—
59C22×C3×Dic5—22no—
60C3×Q8×D5—24no—
61C3×Q8×D5—24no—
62C3×#711—216no—
63C3×#711—24no—
64C3×#711—24no—
65C3×#709—28no—
66C3×#709—22no—
67C3×#709—22no—
68C2×C3×D20—22no—
69C2×C3×D20—22no—
70C2×C3×D20—21no—
71C2×C3×D20—24no—
72C2×C3×D20D411yescharacteristic
73C2×C3×D20C2312yes—
74C2×C3×C4×D5—22no—
75C2×C3×C4×D5C2312yes—
76C2×C3×Dic10—28no—
77C2×C3×Dic10—21no—
78C2×C3×Dic10—22no—
79C2×C3×Dic10—22no—
80C2×C3×Dic10D411yescharacteristic
81C2×C3×Dic10D414yescomplemented
82C2×C3×Dic10C2312yes—
83C2×C3×C5×C8—22no—
84C2×C3×C5×C8C2312yes—
85C3×C5×C4⋊C4C2312yes—
86C3×C5×C22⋊C4—21no—
87C3×C42×C5—21no—
88C3×#688—24no—
89C3×#686—24no—
90C3×#685—24no—
91C3×#681—22no—
92C2×C3×#168C2312yes—
93C3×#677—28no—
94C2×C3×2−1+4—51no—
95C2×C3×AGL(1,ℤ/8)—51no—
96C22×C3×SD16—51no—
97C3×#535—51no—
98C3×#449—54no—
99C3×#448—54no—
100C3×#419—51no—
101C3×#416D511yescharacteristic, complemented
102C3×#408—51no—
103C2×C5×#41C22×C311yescharacteristic
104#4150—24no—
105#4150—24no—
106C2×Q8×D5C22×C312yescomplemented
107C2×#711—22no—
108C2×#711C22×C312yescomplemented
109C2×#709—21no—
110C2×#709C22×C311yescharacteristic
111C22×D20C22×C311yescharacteristic
112C22×Dic10—21no—
113C22×Dic10C22×C311yescharacteristic
114C22×C5×C8C22×C311yescharacteristic
115C5×#117C22×C311yescharacteristic
116#4095—24no—
117C2×#688C22×C312yes—
118C2×#686C22×C312yes—
119C2×#681C22×C311yescharacteristic
120C2×#677—24no—
121C2×#677C22×C314yes—
122#4041C22×C314yes—
123#4028—22no—
124#4021—21no—
125#4020—21no—
126C5×#103C22×C312yescomplemented
127C5×#102C22×C312yescomplemented
128#3952C22×C314yescomplemented
129#3942C22×C314yescomplemented
130#3010—51noSylow
131C23×C3×C5—21no—
132C23×C3×C5C2×D411yescharacteristic
133C22×C3×D5—22no—
134C22×C3×D5—42no—
135C22×C3×D5—82no—
136C22×C3×D5C2×D412yes—
137C3×C5×Q8—21no—
138C3×C5×Q8C2×D412yes—
139C3×C5×D4—24no—
140C3×C5×D4—21no—
141C3×C5×D4C2×D412yes—
142C2×C3×C4×C5—21no—
143C2×C3×C4×C5—24no—
144C2×C3×C4×C5—22no—
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—
151C2×C3×Dic5—28no—
152C2×C3×Dic5—24no—
153C2×C3×Dic5—22no—
154C2×C3×Dic5C2×D412yes—
155C3×D20—22no—
156C3×D20—22no—
157C3×D20—22no—
158C3×D20—42no—
159C3×D20—84no—
160C3×C4×D5—28no—
161C3×C4×D5—22no—
162C3×C4×D5—22no—
163C3×Dic10—28no—
164C3×Dic10—22no—
165C3×Dic10—24no—
166C3×Dic10—22no—
167C3×Dic10C2×D414yes—
168C3×C5×C8—22no—
169C3×#168—22no—
170C3×2−1+4—104no—
171C3×2−1+4—104no—
172C2×C3×#41—51no—
173C2×C3×#41—52no—
174C2×C3×#41—101no—
175C2×C3×#41—102no—
176C2×C3×#41C2×D511yescharacteristic
177C22×C3×Q8—52no—
178C22×C3×Q8—51no—
179C22×C3×Q8—101no—
180C22×C3×D4—51no—
181C3×AGL(1,ℤ/8)—104no—
182C2×C3×SD16—52no—
183C2×C3×SD16—54no—
184C2×C3×SD16—104no—
185C2×C3×D8—52no—
186C2×C3×M16—51no—
187C22×C3×C8C2×D511yescharacteristic
188C3×#127—101no—
189C3×#124—101no—
190C3×#121—54no—
191C3×#120—102no—
192C3×#117C2×D511yescharacteristic
193C3×#103—54no—
194C3×#103C2×D512yescomplemented
195C3×#102C2×D512yescomplemented
196C3×C22⋊C8—54no—
197C23×D5—21no—
198C5×#41—22no—
199C5×#41C3×D414yescomplemented
200C2×C5×Q8C23×C311yescharacteristic
201C2×C5×D4—21no—
202C2×C5×D4C23×C311yescharacteristic
203C22×C4×C5—21no—
204C22×C4×C5C23×C311yescharacteristic
205C2×C22⋊D5—22no—
206C2×C22⋊D5C3×D412yes—
207C22×Dic5—21no—
208C22×Dic5—22no—
209Q8×D5—24no—
210Q8×D5—24no—
211#711—216no—
212#711—24no—
213#711—24no—
214#709—28no—
215#709—22no—
216#709—22no—
217C2×D20—22no—
218C2×D20—22no—
219C2×D20—21no—
220C2×D20—24no—
221C2×D20C3×D411yescharacteristic
222C2×D20C23×C312yes—
223C2×C4×D5—22no—
224C2×C4×D5C23×C312yes—
225C2×Dic10—28no—
226C2×Dic10—21no—
227C2×Dic10—22no—
228C2×Dic10—22no—
229C2×Dic10C3×D411yescharacteristic
230C2×Dic10C3×D414yescomplemented
231C2×Dic10C23×C312yes—
232C2×C5×C8—22no—
233C2×C5×C8C23×C312yes—
234C5×C4⋊C4C23×C312yes—
235C5×C22⋊C4—21no—
236C42×C5—21no—
237#688—24no—
238#686—24no—
239#685—24no—
240#681—22no—
241C2×#168C23×C312yes—
242#677—28no—
243C2×2−1+4—51no—
244C2×AGL(1,ℤ/8)—51no—
245C22×SD16—51no—
246#535—51no—
247#449—54no—
248#448—54no—
249#419—51no—
250#416C3×D511yescharacteristic, complemented
251#408—51no—
252C22×C3×C5—22no—
253C22×C3×C5—22no—
254C22×C3×C5—42no—
255C22×C3×C5#12012yes—
256C22×C3×C5C22×D411yescharacteristic
257C2×C3×D5—22no—
258C2×C3×D5—24no—
259C2×C3×D5—42no—
260C2×C3×D5—44no—
261C2×C3×D5—84no—
262C3×C4×C5—22no—
263C3×C4×C5—42no—
264C3×C4×C5#12012yes—
265C3×C4×C5C22×D411yescharacteristic
266C3×C4×C5C22×D411yescharacteristic
267C3×Dic5—22no—
268C3×Dic5—24no—
269C24×C3—101no—
270C3×#41—22no—
271C3×#41—108no—
272C3×#41—104no—
273C3×#41—1016no—
274C3×#41—104no—
275C3×#41—102no—
276C3×#41—102no—
277C3×#41D2014yescomplemented
278C2×C3×Q8—51no—
279C2×C3×Q8—54no—
280C2×C3×Q8—52no—
281C2×C3×Q8—102no—
282C2×C3×Q8—108no—
283C2×C3×Q8—104no—
284C2×C3×Q8—104no—
285C2×C3×Q8—102no—
286C2×C3×Q8—101no—
287C2×C3×Q8C22×D511yescharacteristic
288C2×C3×D4—21no—
289C2×C3×D4—52no—
290C2×C3×D4—51no—
291C2×C3×D4—52no—
292C2×C3×D4—104no—
293C2×C3×D4—102no—
294C2×C3×D4—102no—
295C2×C3×D4—102no—
296C2×C3×D4—101no—
297C2×C3×D4C22×D511yescharacteristic
298C22×C3×C4—21no—
299C22×C3×C4—52no—
300C22×C3×C4—102no—
301C22×C3×C4—102no—
302C22×C3×C4—101no—
303C22×C3×C4C22×D511yescharacteristic
304C3×SD16—108no—
305C3×SD16—104no—
306C3×D8—104no—
307C3×M16—102no—
308C2×C3×C8—22no—
309C2×C3×C8—52no—
310C2×C3×C8C22×D512yes—
311C3×C4⋊C4C22×D512yes—
312C3×C22⋊C4—21no—
313C3×C22⋊C4—104no—
314C3×C42—21no—
315C23×C5—21no—
316C23×C5C2×C3×D411yescharacteristic
317C22×D5—22no—
318C22×D5—42no—
319C22×D5—82no—
320C22×D5C2×C3×D412yes—
321C5×Q8—21no—
322C5×Q8C2×C3×D412yes—
323C5×D4—24no—
324C5×D4—21no—
325C5×D4C2×C3×D412yes—
326C2×C4×C5—21no—
327C2×C4×C5—24no—
328C2×C4×C5—22no—
329C2×C4×C5C2×C3×D412yes—
330C2×C4×C5C2×C3×D411yescharacteristic
331C2×C4×C5C2×C3×D412yes—
332C2×C4×C5C24×C311yescharacteristic
333C22⋊D5—28no—
334C22⋊D5—24no—
335C2×Dic5—28no—
336C2×Dic5—24no—
337C2×Dic5—22no—
338C2×Dic5C2×C3×D412yes—
339D20—22no—
340D20—22no—
341D20—22no—
342D20—42no—
343D20—84no—
344C4×D5—28no—
345C4×D5—22no—
346C4×D5—22no—
347Dic10—28no—
348Dic10—22no—
349Dic10—24no—
350Dic10—22no—
351Dic10C2×C3×D414yes—
352C5×C8—22no—
353#168—22no—
3542−1+4—104no—
3552−1+4—104no—
356C2×#41—51no—
357C2×#41—52no—
358C2×#41—101no—
359C2×#41—102no—
360C2×#41C2×C3×D511yescharacteristic
361C22×Q8—52no—
362C22×Q8—51no—
363C22×Q8—101no—
364C22×D4—51no—
365AGL(1,ℤ/8)—104no—
366C2×SD16—52no—
367C2×SD16—54no—
368C2×SD16—104no—
369C2×D8—52no—
370C2×M16—51no—
371C22×C8C2×C3×D511yescharacteristic
372#127—101no—
373#124—101no—
374#121—54no—
375#120—102no—
376#117C2×C3×D511yescharacteristic
377#103—54no—
378#103C2×C3×D512yescomplemented
379#102C2×C3×D512yescomplemented
380C22⋊C8—54no—
381C2×C3×C5—22no—
382C2×C3×C5—42no—
383C2×C3×C5C2×#12011yescharacteristic
384C2×C3×C5#57712yes—
385C3×D5—42no—
386C3×D5—82no—
387C23×C3—21no—
388C23×C3—52no—
389C23×C3—102no—
390C23×C3—202no—
391C23×C3—402no—
392C23×C3D4×D511yescharacteristic
393C3×Q8—21no—
394C3×Q8—54no—
395C3×Q8—108no—
396C3×Q8—102no—
397C3×Q8—102no—
398C3×Q8—104no—
399C3×Q8C2×D2012yes—
400C3×D4—24no—
401C3×D4—21no—
402C3×D4—108no—
403C3×D4—104no—
404C3×D4—102no—
405C3×D4—102no—
406C3×D4—102no—
407C3×D4—202no—
408C3×D4—404no—
409C3×D4C2×D2012yes—
410C2×C3×C4—21no—
411C2×C3×C4—24no—
412C2×C3×C4—22no—
413C2×C3×C4—52no—
414C2×C3×C4—108no—
415C2×C3×C4—108no—
416C2×C3×C4—102no—
417C2×C3×C4—104no—
418C2×C3×C4—102no—
419C2×C3×C4—102no—
420C2×C3×C4C2×D2012yes—
421C2×C3×C4D4×D512yes—
422C2×C3×C4D4×D511yescharacteristic
423C2×C3×C4C23×D511yescharacteristic
424C3×C8—22no—
425C3×C8—102no—
426C22×C5—22no—
427C22×C5—22no—
428C22×C5—42no—
429C22×C5C3×#12012yes—
430C22×C5C22×C3×D411yescharacteristic
431C2×D5—24no—
432C2×D5—22no—
433C2×D5—44no—
434C2×D5—42no—
435C2×D5—84no—
436C4×C5—22no—
437C4×C5—42no—
438C4×C5C3×#12012yes—
439C4×C5C22×C3×D411yescharacteristic
440C4×C5C22×C3×D411yescharacteristic
441Dic5—22no—
442Dic5—24no—
443C24—101no—
444#41—22no—
445#41—108no—
446#41—104no—
447#41—1016no—
448#41—104no—
449#41—102no—
450#41—102no—
451#41C3×D2014yescomplemented
452C2×Q8—51no—
453C2×Q8—54no—
454C2×Q8—52no—
455C2×Q8—102no—
456C2×Q8—108no—
457C2×Q8—104no—
458C2×Q8—104no—
459C2×Q8—102no—
460C2×Q8—101no—
461C2×Q8C22×C3×D511yescharacteristic
462C2×D4—21no—
463C2×D4—52no—
464C2×D4—51no—
465C2×D4—52no—
466C2×D4—104no—
467C2×D4—102no—
468C2×D4—102no—
469C2×D4—102no—
470C2×D4—101no—
471C2×D4C22×C3×D511yescharacteristic
472C22×C4—21no—
473C22×C4—52no—
474C22×C4—102no—
475C22×C4—102no—
476C22×C4—101no—
477C22×C4C22×C3×D511yescharacteristic
478SD16—108no—
479SD16—104no—
480D8—104no—
481M16—102no—
482C2×C8—22no—
483C2×C8—52no—
484C2×C8C22×C3×D512yes—
485C4⋊C4C22×C3×D512yes—
486C22⋊C4—21no—
487C22⋊C4—104no—
488C42—21no—
489C3×C5#301011yescharacteristic, complemented, Hall
490C22×C3—22no—
491C22×C3—22no—
492C22×C3—42no—
493C22×C3—102no—
494C22×C3—104no—
495C22×C3—202no—
496C22×C3—204no—
497C22×C3—404no—
498C22×C3#402812yes—
499C22×C3C2×D4×D511yescharacteristic
500C3×C4—22no—
501C3×C4—42no—
502C3×C4—102no—
503C3×C4—104no—
504C3×C4#402812yes—
505C3×C4C22×D2011yescharacteristic
506C3×C4C2×D4×D511yescharacteristic
507C2×C5—22no—
508C2×C5—42no—
509C2×C5C2×C3×#12011yescharacteristic
510C2×C5C3×#57712yes—
511D5—42no—
512D5—82no—
513C23—21no—
514C23—52no—
515C23—102no—
516C23—202no—
517C23—402no—
518C23C3×D4×D511yescharacteristic
519Q8—21no—
520Q8—54no—
521Q8—108no—
522Q8—102no—
523Q8—102no—
524Q8—104no—
525Q8C2×C3×D2012yes—
526D4—24no—
527D4—21no—
528D4—108no—
529D4—104no—
530D4—102no—
531D4—102no—
532D4—102no—
533D4—202no—
534D4—404no—
535D4C2×C3×D2012yes—
536C2×C4—21no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—52no—
540C2×C4—108no—
541C2×C4—108no—
542C2×C4—102no—
543C2×C4—104no—
544C2×C4—102no—
545C2×C4—102no—
546C2×C4C2×C3×D2012yes—
547C2×C4C3×D4×D512yes—
548C2×C4C3×D4×D511yescharacteristic
549C2×C4C23×C3×D511yescharacteristic
550C8—22no—
551C8—102no—
552C2×C3—22no—
553C2×C3—42no—
554C2×C3—202no—
555C2×C3—402no—
556C2×C3C2×#402811yescharacteristic
557C2×C3#6610412yes—
558C5C3×#301011yescharacteristic, complemented, Sylow
559C22—22no—
560C22—22no—
561C22—42no—
562C22—104no—
563C22—102no—
564C22—204no—
565C22—202no—
566C22—404no—
567C22C3×#402812yes—
568C22C2×C3×D4×D511yescharacteristic
569C4—22no—
570C4—42no—
571C4—102no—
572C4—104no—
573C4C3×#402812yes—
574C4C22×C3×D2011yescharacteristic
575C4C2×C3×D4×D511yescharacteristic
576C3#1061056711yescharacteristic, direct product factor, Sylow
577C2—22no—
578C2—42no—
579C2—202no—
580C2—402no—
581C2C2×C3×#402811yescharacteristic
582C2C3×#6610412yes—
subgroupG/H
centerC22×C3C2×D4×D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC3×C5×#416C2
socleC22×C3×C5C22×D4
hypercenterC3×#416D5
hypocenterC5C3×#3010
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#416

Complements

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

NH ≅ G/Ntype
1. C2×C3×#4150C2semidirect
2. C2×C3×#4095C2semidirect
3. C22×C3×#677C2semidirect
4. C3×#65859C2semidirect
5. C3×C5×#416C2semidirect
6. C3×#65399C2semidirect
7. C3×#65293C2semidirect
8. C3×#65095C2semidirect
9. C3×#65070C2semidirect
10. #10610567C3direct (N × H)
14. C2×C3×Q8×D5C22semidirect
16. C2×C3×#711C22semidirect
34. C3×C5×#103C22semidirect
35. C3×C5×#102C22semidirect
36. C3×#3952C22semidirect
37. C3×#3942C22semidirect
39. C2×#4150C2×C3semidirect
40. C2×#4095C2×C3semidirect
41. C22×#677C2×C3semidirect
42. #65859C2×C3semidirect
43. C5×#416C2×C3semidirect
44. #65399C2×C3semidirect
45. #65293C2×C3semidirect
46. #65095C2×C3semidirect
47. #65070C2×C3semidirect
50. C3×C5×#41D4semidirect
81. C2×C3×Dic10D4semidirect
101. C3×#416D5semidirect
106. C2×Q8×D5C22×C3semidirect
108. C2×#711C22×C3semidirect
126. C5×#103C22×C3semidirect
127. C5×#102C22×C3semidirect
128. #3952C22×C3semidirect
129. #3942C22×C3semidirect
194. C3×#103C2×D5semidirect
195. C3×#102C2×D5semidirect
199. C5×#41C3×D4semidirect
230. C2×Dic10C3×D4semidirect
250. #416C3×D5semidirect
277. C3×#41D20semidirect
378. #103C2×C3×D5semidirect
379. #102C2×C3×D5semidirect
451. #41C3×D20semidirect
489. C3×C5#3010semidirect
558. C5C3×#3010semidirect
576. C3#10610567direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#41502
C2×C3×#40952
C22×C3×#6772
C3×#658592
C3×C5×#4162
C3×#653992
C3×#652932
C3×#650952
C3×#650702
#106105673
C3×#30105