Finite Group Databasebeta

← C3×#10615121 #14424424

Subgroups — C3×#10615121 #14424424

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
2C3×C5×#535C211yesmaximal, characteristic, complemented
3C22×C3×#687C211yesmaximal, characteristic, complemented
4C2×C3×#4078C211yesmaximal, characteristic, complemented
5C3×#65523C211yesmaximal, characteristic, complemented
6C3×#65329C214yesmaximal, complemented
7C3×#65326C214yesmaximal, complemented
8C3×#65293C211yesmaximal, characteristic, complemented
9C3×#65291C211yesmaximal, characteristic, complemented
10#10615121C311yesmaximal, characteristic, direct product factor, Hall
11C2×C3×C5×#41C2211yescharacteristic
12C22×C3×C5×D4C2211yescharacteristic
13C3×#4150—24no—
14C3×#4150—24no—
15C2×C3×Q8×D5—21no—
16C2×C3×#711—21no—
17C2×C3×#711—22no—
18C2×C3×#709C2211yescharacteristic
19C2×C3×#709C2212yescomplemented
20C22×C3×Dic10C2211yescharacteristic
21C22×C3×Dic10C2212yescomplemented
22C3×C5×#127—21no—
23C3×C5×#124—21no—
24C3×C5×#121C2214yes—
25C3×C5×#120—22no—
26C3×C5×#117C2211yescharacteristic
27C2×C3×#687—24no—
28C2×C3×#687C2212yes—
29C2×C3×#687C2214yes—
30C3×#4078—24no—
31C2×C3×#686C2212yes—
32C2×C3×#681C2211yescharacteristic
33C22×C3×#168C2211yescharacteristic
34C3×C22⋊#168C2214yescomplemented
35C3×#3942C2212yescomplemented
36C3×#3942C2214yescomplemented
37C3×#3941C2212yescomplemented
38C3×#3010—51nomaximal, Hall
39C2×#4150C2×C311yescharacteristic, complemented
40C5×#535C2×C311yescharacteristic, complemented
41C22×#687C2×C311yescharacteristic, complemented
42C2×#4078C2×C311yescharacteristic, complemented
43#65523C2×C311yescharacteristic, complemented
44#65329C2×C314yescomplemented
45#65326C2×C314yescomplemented
46#65293C2×C311yescharacteristic, complemented
47#65291C2×C311yescharacteristic, complemented
48C24×C3×C5—21no—
49C3×C5×#41—22no—
50C3×C5×#41—22no—
51C2×C3×C5×Q8D411yescharacteristic
52C2×C3×C5×D4—24no—
53C2×C3×C5×D4—22no—
54C2×C3×C5×D4—22no—
55C2×C3×C5×D4—21no—
56C2×C3×C5×D4D412yes—
57C2×C3×C5×D4D411yescharacteristic
58C2×C3×C5×D4C2312yes—
59C22×C3×C4×C5C2312yes—
60C22×C3×C4×C5C2311yescharacteristic
61C2×C3×C22⋊D5—21no—
62C2×C3×C22⋊D5—22no—
63C22×C3×Dic5—21no—
64C22×C3×Dic5—22no—
65C3×Q8×D5—28no—
66C3×#711—216no—
67C3×#711—28no—
68C3×#709—22no—
69C3×#709—24no—
70C3×#709—24no—
71C3×#709D414yescomplemented
72C2×C3×D20C2311yescharacteristic
73C2×C3×C4×D5—21no—
74C2×C3×C4×D5—22no—
75C2×C3×Dic10—21no—
76C2×C3×Dic10—22no—
77C2×C3×Dic10—22no—
78C2×C3×Dic10—24no—
79C2×C3×Dic10—24no—
80C2×C3×Dic10D414yescomplemented
81C2×C3×Dic10C2312yes—
82C2×C3×Dic10C2311yescharacteristic
83C3×C5×C4⋊C4C2312yes—
84C3×C5×C22⋊C4—21no—
85C3×C5×C22⋊C4—24no—
86C3×C42×C5—21no—
87C3×#687—24no—
88C3×#687—28no—
89C3×#686—24no—
90C3×#681—22no—
91C2×C3×#168—22no—
92C2×C3×#168C2312yes—
93C2×C3×#168C2312yes—
94C2×C3×2−1+4—51no—
95C2×C3×AGL(1,ℤ/8)—51no—
96C22×C3×SD16—51no—
97C3×#535D511yescharacteristic, complemented
98C3×#449—54no—
99C3×#448—54no—
100C3×#419—51no—
101C3×#416—51no—
102C3×#408—51no—
103C2×C5×#41C22×C311yescharacteristic
104C22×C5×D4C22×C311yescharacteristic
105#4150—24no—
106#4150—24no—
107C2×Q8×D5—21no—
108C2×#711—21no—
109C2×#711—22no—
110C2×#709C22×C311yescharacteristic
111C2×#709C22×C312yescomplemented
112C22×Dic10C22×C311yescharacteristic
113C22×Dic10C22×C312yescomplemented
114C5×#127—21no—
115C5×#124—21no—
116C5×#121C22×C314yes—
117C5×#120—22no—
118C5×#117C22×C311yescharacteristic
119C2×#687—24no—
120C2×#687C22×C314yes—
121C2×#687C22×C312yes—
122#4078—24no—
123C2×#686C22×C312yes—
124C2×#681C22×C311yescharacteristic
125C22×#168C22×C311yescharacteristic
126C22⋊#168C22×C314yescomplemented
127#3942C22×C312yescomplemented
128#3942C22×C314yescomplemented
129#3941C22×C312yescomplemented
130#3010—51noSylow
131C23×C3×C5—22no—
132C23×C3×C5—42no—
133C23×C3×C5—82no—
134C23×C3×C5C2×D412yes—
135C23×C3×C5C2×D411yescharacteristic
136C22×C3×D5—21no—
137C3×C5×Q8—22no—
138C3×C5×D4—24no—
139C3×C5×D4—22no—
140C3×C5×D4—22no—
141C3×C5×D4—22no—
142C3×C5×D4—42no—
143C3×C5×D4—84no—
144C2×C3×C4×C5—22no—
145C2×C3×C4×C5—22no—
146C2×C3×C4×C5—22no—
147C2×C3×C4×C5C2×D412yes—
148C2×C3×C4×C5C2×D411yescharacteristic
149C2×C3×C4×C5C2×D412yes—
150C2×C3×C4×C5C2×D412yes—
151C2×C3×C4×C5C2411yescharacteristic
152C3×C22⋊D5—28no—
153C3×C22⋊D5—24no—
154C2×C3×Dic5—28no—
155C2×C3×Dic5—24no—
156C2×C3×Dic5—22no—
157C2×C3×Dic5—21no—
158C3×D20—21no—
159C3×D20C2×D412yes—
160C3×C4×D5—28no—
161C3×C4×D5—24no—
162C3×Dic10—28no—
163C3×Dic10—24no—
164C3×Dic10—22no—
165C3×Dic10—21no—
166C3×Dic10C2×D412yes—
167C3×Dic10C2×D414yes—
168C3×#168—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—102no—
175C2×C3×#41—101no—
176C2×C3×#41C2×D511yescharacteristic
177C22×C3×Q8—51no—
178C22×C3×Q8—52no—
179C22×C3×Q8—101no—
180C22×C3×D4C2×D511yescharacteristic
181C3×AGL(1,ℤ/8)—104no—
182C2×C3×SD16—54no—
183C2×C3×SD16—52no—
184C2×C3×SD16—104no—
185C2×C3×D8—52no—
186C2×C3×M16—51no—
187C22×C3×C8—51no—
188C3×#127—21no—
189C3×#124—21no—
190C3×#121C2×D514yes—
191C3×#120—22no—
192C3×#117C2×D511yescharacteristic
193C3×#103—52no—
194C3×#103—54no—
195C3×#102—52no—
196C3×C22⋊C8—54no—
197C24×C5—21no—
198C5×#41—22no—
199C5×#41—22no—
200C2×C5×Q8C3×D411yescharacteristic
201C2×C5×D4—24no—
202C2×C5×D4—22no—
203C2×C5×D4—22no—
204C2×C5×D4—21no—
205C2×C5×D4C3×D412yes—
206C2×C5×D4C3×D411yescharacteristic
207C2×C5×D4C23×C312yes—
208C22×C4×C5C23×C312yes—
209C22×C4×C5C23×C311yescharacteristic
210C2×C22⋊D5—21no—
211C2×C22⋊D5—22no—
212C22×Dic5—21no—
213C22×Dic5—22no—
214Q8×D5—28no—
215#711—216no—
216#711—28no—
217#709—22no—
218#709—24no—
219#709—24no—
220#709C3×D414yescomplemented
221C2×D20C23×C311yescharacteristic
222C2×C4×D5—21no—
223C2×C4×D5—22no—
224C2×Dic10—21no—
225C2×Dic10—22no—
226C2×Dic10—22no—
227C2×Dic10—24no—
228C2×Dic10—24no—
229C2×Dic10C3×D414yescomplemented
230C2×Dic10C23×C312yes—
231C2×Dic10C23×C311yescharacteristic
232C5×C4⋊C4C23×C312yes—
233C5×C22⋊C4—21no—
234C5×C22⋊C4—24no—
235C42×C5—21no—
236#687—24no—
237#687—28no—
238#686—24no—
239#681—22no—
240C2×#168—22no—
241C2×#168C23×C312yes—
242C2×#168C23×C312yes—
243C2×2−1+4—51no—
244C2×AGL(1,ℤ/8)—51no—
245C22×SD16—51no—
246#535C3×D511yescharacteristic, complemented
247#449—54no—
248#448—54no—
249#419—51no—
250#416—51no—
251#408—51no—
252C22×C3×C5—22no—
253C22×C3×C5—22no—
254C22×C3×C5—24no—
255C22×C3×C5—42no—
256C22×C3×C5—44no—
257C22×C3×C5—84no—
258C22×C3×C5#12012yes—
259C22×C3×C5C22×D411yescharacteristic
260C2×C3×D5—22no—
261C2×C3×D5—42no—
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—21no—
270C3×#41—22no—
271C3×#41—22no—
272C3×#41—54no—
273C3×#41—104no—
274C3×#41—104no—
275C3×#41—1016no—
276C3×#41—108no—
277C3×#41—102no—
278C2×C3×Q8—54no—
279C2×C3×Q8—52no—
280C2×C3×Q8—51no—
281C2×C3×Q8—104no—
282C2×C3×Q8—104no—
283C2×C3×Q8—108no—
284C2×C3×Q8—102no—
285C2×C3×Q8—102no—
286C2×C3×Q8—101no—
287C2×C3×Q8C22⋊D511yescharacteristic
288C2×C3×D4—24no—
289C2×C3×D4—22no—
290C2×C3×D4—22no—
291C2×C3×D4—21no—
292C2×C3×D4—51no—
293C2×C3×D4—102no—
294C2×C3×D4—101no—
295C2×C3×D4C22⋊D512yes—
296C2×C3×D4C22⋊D511yescharacteristic
297C2×C3×D4C22×D512yes—
298C22×C3×C4—102no—
299C22×C3×C4—101no—
300C22×C3×C4—102no—
301C22×C3×C4—101no—
302C22×C3×C4C22×D512yes—
303C22×C3×C4C22×D511yescharacteristic
304C3×SD16—108no—
305C3×SD16—104no—
306C3×D8—104no—
307C3×M16—102no—
308C2×C3×C8—52no—
309C2×C3×C8—52no—
310C2×C3×C8—102no—
311C3×C4⋊C4C22×D512yes—
312C3×C22⋊C4—21no—
313C3×C22⋊C4—24no—
314C3×C42—21no—
315C23×C5—22no—
316C23×C5—42no—
317C23×C5—82no—
318C23×C5C2×C3×D412yes—
319C23×C5C2×C3×D411yescharacteristic
320C22×D5—21no—
321C5×Q8—22no—
322C5×D4—24no—
323C5×D4—22no—
324C5×D4—22no—
325C5×D4—22no—
326C5×D4—42no—
327C5×D4—84no—
328C2×C4×C5—22no—
329C2×C4×C5—22no—
330C2×C4×C5—22no—
331C2×C4×C5C2×C3×D412yes—
332C2×C4×C5C2×C3×D412yes—
333C2×C4×C5C2×C3×D412yes—
334C2×C4×C5C2×C3×D411yescharacteristic
335C2×C4×C5C24×C311yescharacteristic
336C22⋊D5—28no—
337C22⋊D5—24no—
338C2×Dic5—28no—
339C2×Dic5—24no—
340C2×Dic5—22no—
341C2×Dic5—21no—
342D20—21no—
343D20C2×C3×D412yes—
344C4×D5—28no—
345C4×D5—24no—
346Dic10—28no—
347Dic10—24no—
348Dic10—22no—
349Dic10—21no—
350Dic10C2×C3×D412yes—
351Dic10C2×C3×D414yes—
352#168—22no—
353#168—22no—
3542−1+4—104no—
3552−1+4—104no—
356C2×#41—51no—
357C2×#41—52no—
358C2×#41—102no—
359C2×#41—101no—
360C2×#41C2×C3×D511yescharacteristic
361C22×Q8—51no—
362C22×Q8—52no—
363C22×Q8—101no—
364C22×D4C2×C3×D511yescharacteristic
365AGL(1,ℤ/8)—104no—
366C2×SD16—54no—
367C2×SD16—52no—
368C2×SD16—104no—
369C2×D8—52no—
370C2×M16—51no—
371C22×C8—51no—
372#127—21no—
373#124—21no—
374#121C2×C3×D514yes—
375#120—22no—
376#117C2×C3×D511yescharacteristic
377#103—52no—
378#103—54no—
379#102—52no—
380C22⋊C8—54no—
381C2×C3×C5—22no—
382C2×C3×C5—42no—
383C2×C3×C5—82no—
384C2×C3×C5C2×#12011yescharacteristic
385C2×C3×C5#57712yes—
386C3×D5—42no—
387C23×C3—22no—
388C23×C3—42no—
389C23×C3—82no—
390C23×C3—101no—
391C23×C3C2×C22⋊D512yes—
392C23×C3C2×C22⋊D511yescharacteristic
393C3×Q8—22no—
394C3×Q8—54no—
395C3×Q8—52no—
396C3×Q8—108no—
397C3×Q8—102no—
398C3×Q8—101no—
399C3×Q8—104no—
400C3×D4—24no—
401C3×D4—22no—
402C3×D4—22no—
403C3×D4—22no—
404C3×D4—42no—
405C3×D4—52no—
406C3×D4—84no—
407C3×D4—108no—
408C3×D4—104no—
409C3×D4—101no—
410C2×C3×C4—22no—
411C2×C3×C4—22no—
412C2×C3×C4—22no—
413C2×C3×C4—108no—
414C2×C3×C4—108no—
415C2×C3×C4—102no—
416C2×C3×C4—101no—
417C2×C3×C4—104no—
418C2×C3×C4—104no—
419C2×C3×C4D4×D512yes—
420C2×C3×C4D4×D512yes—
421C2×C3×C4C2×C22⋊D512yes—
422C2×C3×C4C2×C22⋊D511yescharacteristic
423C2×C3×C4C23×D511yescharacteristic
424C3×C8—102no—
425C3×C8—102no—
426C22×C5—22no—
427C22×C5—24no—
428C22×C5—22no—
429C22×C5—44no—
430C22×C5—42no—
431C22×C5—84no—
432C22×C5C3×#12012yes—
433C22×C5C22×C3×D411yescharacteristic
434C2×D5—22no—
435C2×D5—42no—
436C4×C5—22no—
437C4×C5—42no—
438C4×C5C3×#12012yes—
439C4×C5C22×C3×D411yescharacteristic
440C4×C5C22×C3×D411yescharacteristic
441Dic5—22no—
442Dic5—24no—
443C24—21no—
444#41—22no—
445#41—22no—
446#41—54no—
447#41—104no—
448#41—104no—
449#41—1016no—
450#41—108no—
451#41—102no—
452C2×Q8—54no—
453C2×Q8—52no—
454C2×Q8—51no—
455C2×Q8—104no—
456C2×Q8—104no—
457C2×Q8—108no—
458C2×Q8—102no—
459C2×Q8—102no—
460C2×Q8—101no—
461C2×Q8C3×C22⋊D511yescharacteristic
462C2×D4—24no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—21no—
466C2×D4—51no—
467C2×D4—102no—
468C2×D4—101no—
469C2×D4C3×C22⋊D512yes—
470C2×D4C3×C22⋊D511yescharacteristic
471C2×D4C22×C3×D512yes—
472C22×C4—102no—
473C22×C4—101no—
474C22×C4—102no—
475C22×C4—101no—
476C22×C4C22×C3×D512yes—
477C22×C4C22×C3×D511yescharacteristic
478SD16—108no—
479SD16—104no—
480D8—104no—
481M16—102no—
482C2×C8—52no—
483C2×C8—52no—
484C2×C8—102no—
485C4⋊C4C22×C3×D512yes—
486C22⋊C4—21no—
487C22⋊C4—24no—
488C42—21no—
489C3×C5#301011yescharacteristic, complemented, Hall
490C22×C3—22no—
491C22×C3—22no—
492C22×C3—24no—
493C22×C3—42no—
494C22×C3—44no—
495C22×C3—84no—
496C22×C3—102no—
497C22×C3—202no—
498C22×C3#408312yes—
499C22×C3C22×C22⋊D511yescharacteristic
500C3×C4—22no—
501C3×C4—42no—
502C3×C4—104no—
503C3×C4—102no—
504C3×C4#408312yes—
505C3×C4C2×D4×D511yescharacteristic
506C3×C4C2×D4×D511yescharacteristic
507C2×C5—22no—
508C2×C5—42no—
509C2×C5—82no—
510C2×C5C2×C3×#12011yescharacteristic
511C2×C5C3×#57712yes—
512D5—42no—
513C23—22no—
514C23—42no—
515C23—82no—
516C23—101no—
517C23C2×C3×C22⋊D512yes—
518C23C2×C3×C22⋊D511yescharacteristic
519Q8—22no—
520Q8—54no—
521Q8—52no—
522Q8—108no—
523Q8—104no—
524Q8—102no—
525Q8—101no—
526D4—24no—
527D4—22no—
528D4—22no—
529D4—22no—
530D4—42no—
531D4—52no—
532D4—84no—
533D4—108no—
534D4—104no—
535D4—101no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—108no—
540C2×C4—108no—
541C2×C4—104no—
542C2×C4—102no—
543C2×C4—101no—
544C2×C4—104no—
545C2×C4C3×D4×D512yes—
546C2×C4C3×D4×D512yes—
547C2×C4C2×C3×C22⋊D512yes—
548C2×C4C2×C3×C22⋊D511yescharacteristic
549C2×C4C23×C3×D511yescharacteristic
550C8—102no—
551C8—102no—
552C2×C3—22no—
553C2×C3—42no—
554C2×C3—82no—
555C2×C3—202no—
556C2×C3C2×#408311yescharacteristic
557C2×C3#6617012yes—
558C5C3×#301011yescharacteristic, complemented, Sylow
559C22—22no—
560C22—22no—
561C22—24no—
562C22—42no—
563C22—44no—
564C22—84no—
565C22—102no—
566C22—202no—
567C22C3×#408312yes—
568C22C22×C3×C22⋊D511yescharacteristic
569C4—22no—
570C4—42no—
571C4—104no—
572C4—102no—
573C4C3×#408312yes—
574C4C2×C3×D4×D511yescharacteristic
575C4C2×C3×D4×D511yescharacteristic
576C3#1061512111yescharacteristic, direct product factor, Sylow
577C2—22no—
578C2—42no—
579C2—82no—
580C2—202no—
581C2C2×C3×#408311yescharacteristic
582C2C3×#6617012yes—
subgroupG/H
centerC22×C3C22×C22⋊D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC3×C5×#535C2
socleC22×C3×C5C22×D4
hypercenterC3×#535D5
hypocenterC5C3×#3010
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C2×C3×#41 ◁ C3×#535

Complements

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

NH ≅ G/Ntype
1. C2×C3×#4150C2semidirect
2. C3×C5×#535C2semidirect
3. C22×C3×#687C2semidirect
4. C2×C3×#4078C2semidirect
5. C3×#65523C2semidirect
6. C3×#65329C2semidirect
7. C3×#65326C2semidirect
8. C3×#65293C2semidirect
9. C3×#65291C2semidirect
10. #10615121C3direct (N × H)
19. C2×C3×#709C22semidirect
21. C22×C3×Dic10C22semidirect
34. C3×C22⋊#168C22semidirect
35. C3×#3942C22semidirect
36. C3×#3942C22semidirect
37. C3×#3941C22semidirect
39. C2×#4150C2×C3semidirect
40. C5×#535C2×C3semidirect
41. C22×#687C2×C3semidirect
42. C2×#4078C2×C3semidirect
43. #65523C2×C3semidirect
44. #65329C2×C3semidirect
45. #65326C2×C3semidirect
46. #65293C2×C3semidirect
47. #65291C2×C3semidirect
71. C3×#709D4semidirect
80. C2×C3×Dic10D4semidirect
97. C3×#535D5semidirect
111. C2×#709C22×C3semidirect
113. C22×Dic10C22×C3semidirect
126. C22⋊#168C22×C3semidirect
127. #3942C22×C3semidirect
128. #3942C22×C3semidirect
129. #3941C22×C3semidirect
220. #709C3×D4semidirect
229. C2×Dic10C3×D4semidirect
246. #535C3×D5semidirect
489. C3×C5#3010semidirect
558. C5C3×#3010semidirect
576. C3#10615121direct (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
C3×C5×#5352
C22×C3×#6872
C2×C3×#40782
C3×#655232
C3×#653292
C3×#653262
C3×#652932
C3×#652912
#106151213
C3×#30105