Finite Group Databasebeta

← C3×#10605012 #14297332

Subgroups — C3×#10605012 #14297332

Order 1920.

1176 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4149C211yesmaximal, characteristic
2C3×#65784C212yesmaximal, complemented
3C3×#65767C212yesmaximal, complemented
4C3×#65761C211yesmaximal, characteristic, complemented
5C3×#65745C211yesmaximal, characteristic, complemented
6#10605012C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×C5×#41C411yescharacteristic, complemented
8C3×#4149—24no—
9C3×#4149—24no—
10C2×C3×#713C411yescharacteristic, complemented
11C2×C3×#713C2211yescharacteristic
12C2×C3×D4×D5—21no—
13C2×C3×D4×D5—21no—
14C2×C3×D4×D5C411yescharacteristic, complemented
15C2×C3×D4×D5C2211yescharacteristic
16C2×C3×#709—21no—
17C2×C3×#709C2211yescharacteristic
18C22×C3×D20—21no—
19C22×C3×D20C411yescharacteristic, complemented
20C3×#4130C2212yescomplemented
21C3×D5⋊M16C2212yescomplemented
22C3×#4013—21no—
23C3×#4011—21no—
24C3×#4010—24no—
25C3×#4007—24no—
26C3×#4005—21no—
27C3×#3999—21no—
28C3×#1795—51nomaximal, Hall
29C2×#4149C2×C311yescharacteristic
30#65784C2×C312yescomplemented
31#65767C2×C312yescomplemented
32#65761C2×C311yescharacteristic, complemented
33#65745C2×C311yescharacteristic, complemented
34C23×C3×D5—21no—
35C23×C3×D5—21no—
36C23×C3×D5—21no—
37C3×C5×#41—22no—
38C3×C5×#41—22no—
39C2×C3×C5×Q8C2×C411yescharacteristic
40C2×C3×C5×D4—21no—
41C2×C3×C5×D4C2×C411yescharacteristic
42C22×C3×C4×C5—21no—
43C22×C3×C4×C5C2×C411yescharacteristic
44C2×C3×C22⋊D5—21no—
45C2×C3×C22⋊D5—21no—
46C2×C3×C22⋊D5D411yescharacteristic
47C2×C3×C22⋊D5D411yescharacteristic
48C3×#713—22no—
49C3×#713—22no—
50C3×#713—22no—
51C3×#713D414yes—
52C3×D4×D5—28no—
53C3×D4×D5—28no—
54C3×D4×D5—22no—
55C3×D4×D5—22no—
56C3×D4×D5—22no—
57C3×D4×D5D414yes—
58C3×#709—28no—
59C3×#709—22no—
60C3×#709—22no—
61C2×C3×D20—28no—
62C2×C3×D20—21no—
63C2×C3×D20—21no—
64C2×C3×D20—21no—
65C2×C3×D20—22no—
66C2×C3×D20—22no—
67C2×C3×D20C2×C411yescharacteristic
68C2×C3×D20C2×C411yescharacteristic
69C2×C3×D20D411yescharacteristic
70C2×C3×C4×D5—21no—
71C2×C3×C4×D5—21no—
72C2×C3×C4×D5C2×C411yescharacteristic
73C2×C3×C4×D5C2311yescharacteristic
74C2×C3×Dic10D411yescharacteristic
75C3×C22⋊AGL(1,𝔽5)—22no—
76C3×#704—22no—
77C3×C4⋊AGL(1,𝔽5)—22no—
78C3×C4×AGL(1,𝔽5)—22no—
79C3×#700—22no—
80C3×D5⋊C8—22no—
81C2×C3×H(5,𝔽2)—51no—
82C3×#421—52no—
83C3×#418—52no—
84C3×#413—51no—
85C3×#410—51no—
86C2×C5×#41C3×C411yescharacteristic, complemented
87#4149—24no—
88#4149—24no—
89C2×#713C3×C411yescharacteristic, complemented
90C2×#713C22×C311yescharacteristic
91C2×D4×D5—21no—
92C2×D4×D5—21no—
93C2×D4×D5C3×C411yescharacteristic, complemented
94C2×D4×D5C22×C311yescharacteristic
95C2×#709—21no—
96C2×#709C22×C311yescharacteristic
97C22×D20—21no—
98C22×D20C3×C411yescharacteristic, complemented
99#4130C22×C312yescomplemented
100D5⋊M16C22×C312yescomplemented
101#4013—21no—
102#4011—21no—
103#4010—24no—
104#4007—24no—
105#4005—21no—
106#3999—21no—
107#1795—51noSylow
108C23×C3×C5—21no—
109C23×C3×C5C22⋊C411yescharacteristic
110C22×C3×D5—21no—
111C22×C3×D5—21no—
112C22×C3×D5—24no—
113C22×C3×D5—24no—
114C22×C3×D5—24no—
115C22×C3×D5—82no—
116C22×C3×D5—82no—
117C22×C3×D5—82no—
118C22×C3×D5C22⋊C411yescharacteristic
119C22×C3×D5C2×D411yescharacteristic
120C2×C3×AGL(1,𝔽5)—42no—
121C3×C5×Q8—21no—
122C3×C5×Q8C22⋊C412yes—
123C3×C5×D4—24no—
124C3×C5×D4—21no—
125C3×C5×D4C22⋊C412yes—
126C2×C3×C4×C5—24no—
127C2×C3×C4×C5—21no—
128C2×C3×C4×C5—21no—
129C2×C3×C4×C5—21no—
130C2×C3×C4×C5C22⋊C411yescharacteristic
131C2×C3×C4×C5C22×C411yescharacteristic
132C3×C22⋊D5—24no—
133C3×C22⋊D5—22no—
134C3×C22⋊D5—22no—
135C3×C22⋊D5—24no—
136C2×C3×Dic5C22⋊C411yescharacteristic
137C2×C3×Dic5C2×D411yescharacteristic
138C3×D20—24no—
139C3×D20—24no—
140C3×D20—22no—
141C3×D20—24no—
142C3×D20—21no—
143C3×D20—21no—
144C3×D20C22⋊C412yes—
145C3×D20C22⋊C412yes—
146C3×C4×D5—24no—
147C3×C4×D5—21no—
148C3×C4×D5—21no—
149C3×C4×D5—24no—
150C3×C4×D5C2×D412yes—
151C3×C4×D5C2×D412yes—
152C3×Dic10—22no—
153C3×#170—42no—
154C3×H(5,𝔽2)—104no—
155C3×H(5,𝔽2)—104no—
156C2×C3×#41—51no—
157C2×C3×#41—51no—
158C2×C3×#41—51no—
159C2×C3×#41—101no—
160C2×C3×#41AGL(1,𝔽5)11yescharacteristic, complemented
161C22×C3×D4—51no—
162C22×C3×D4—51no—
163C22×C3×D4—51no—
164C22×C3×D4—101no—
165C22×C3×D4—101no—
166C22×C3×D4—101no—
167C2×C3×M16—52no—
168C3×#117—52no—
169C3×#104—104no—
170C3×#102—104no—
171C3×#101—101no—
172C3×#100—101no—
173C3×#99—101no—
174C3×#99—101no—
175C23×D5—21no—
176C23×D5—21no—
177C23×D5—21no—
178C5×#41—22no—
179C5×#41—22no—
180C2×C5×Q8C2×C3×C411yescharacteristic
181C2×C5×D4—21no—
182C2×C5×D4C2×C3×C411yescharacteristic
183C22×C4×C5—21no—
184C22×C4×C5C2×C3×C411yescharacteristic
185C2×C22⋊D5—21no—
186C2×C22⋊D5—21no—
187C2×C22⋊D5C3×D411yescharacteristic
188C2×C22⋊D5C3×D411yescharacteristic
189#713—22no—
190#713—22no—
191#713—22no—
192#713C3×D414yes—
193D4×D5—28no—
194D4×D5—28no—
195D4×D5—22no—
196D4×D5—22no—
197D4×D5—22no—
198D4×D5C3×D414yes—
199#709—28no—
200#709—22no—
201#709—22no—
202C2×D20—28no—
203C2×D20—21no—
204C2×D20—21no—
205C2×D20—21no—
206C2×D20—22no—
207C2×D20—22no—
208C2×D20C2×C3×C411yescharacteristic
209C2×D20C2×C3×C411yescharacteristic
210C2×D20C3×D411yescharacteristic
211C2×C4×D5—21no—
212C2×C4×D5—21no—
213C2×C4×D5C2×C3×C411yescharacteristic
214C2×C4×D5C23×C311yescharacteristic
215C2×Dic10C3×D411yescharacteristic
216C22⋊AGL(1,𝔽5)—22no—
217#704—22no—
218C4⋊AGL(1,𝔽5)—22no—
219C4×AGL(1,𝔽5)—22no—
220#700—22no—
221D5⋊C8—22no—
222C2×H(5,𝔽2)—51no—
223#421—52no—
224#418—52no—
225#413—51no—
226#410—51no—
227C22×C3×C5—21no—
228C22×C3×C5—22no—
229C22×C3×C5—41no—
230C22×C3×C5—42no—
231C22×C3×C5C2×C22⋊C411yescharacteristic
232C2×C3×D5—21no—
233C2×C3×D5—22no—
234C2×C3×D5—22no—
235C2×C3×D5—41no—
236C2×C3×D5—42no—
237C2×C3×D5—42no—
238C2×C3×D5—41no—
239C2×C3×D5—84no—
240C2×C3×D5—84no—
241C2×C3×D5—84no—
242C2×C3×D5#12012yes—
243C3×AGL(1,𝔽5)—82no—
244C3×C4×C5—21no—
245C3×C4×C5—22no—
246C3×C4×C5C2×C22⋊C411yescharacteristic
247C3×C4×C5C2×C22⋊C411yescharacteristic
248C3×Dic5—21no—
249C3×Dic5#12012yes—
250C24×C3—101no—
251C24×C3—101no—
252C24×C3—101no—
253C3×#41—22no—
254C3×#41—22no—
255C3×#41—54no—
256C3×#41—102no—
257C3×#41—102no—
258C3×#41—108no—
259C3×#41—102no—
260C3×#41—102no—
261C3×#41—102no—
262C2×C3×Q8—51no—
263C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
264C2×C3×D4—21no—
265C2×C3×D4—54no—
266C2×C3×D4—51no—
267C2×C3×D4—51no—
268C2×C3×D4—51no—
269C2×C3×D4—51no—
270C2×C3×D4—51no—
271C2×C3×D4—102no—
272C2×C3×D4—102no—
273C2×C3×D4—101no—
274C2×C3×D4—101no—
275C2×C3×D4—108no—
276C2×C3×D4—102no—
277C2×C3×D4—101no—
278C2×C3×D4—101no—
279C2×C3×D4—102no—
280C2×C3×D4—108no—
281C2×C3×D4—108no—
282C2×C3×D4—102no—
283C2×C3×D4—101no—
284C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
285C22×C3×C4—21no—
286C22×C3×C4—51no—
287C22×C3×C4—51no—
288C22×C3×C4—101no—
289C22×C3×C4—101no—
290C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
291C3×M16—102no—
292C3×M16—102no—
293C2×C3×C8—102no—
294C3×C4⋊C4—102no—
295C3×C22⋊C4—102no—
296C3×C42—102no—
297C23×C5—21no—
298C23×C5C3×C22⋊C411yescharacteristic
299C22×D5—21no—
300C22×D5—21no—
301C22×D5—24no—
302C22×D5—24no—
303C22×D5—24no—
304C22×D5—82no—
305C22×D5—82no—
306C22×D5—82no—
307C22×D5C3×C22⋊C411yescharacteristic
308C22×D5C2×C3×D411yescharacteristic
309C2×AGL(1,𝔽5)—42no—
310C5×Q8—21no—
311C5×Q8C3×C22⋊C412yes—
312C5×D4—24no—
313C5×D4—21no—
314C5×D4C3×C22⋊C412yes—
315C2×C4×C5—24no—
316C2×C4×C5—21no—
317C2×C4×C5—21no—
318C2×C4×C5—21no—
319C2×C4×C5C3×C22⋊C411yescharacteristic
320C2×C4×C5C22×C3×C411yescharacteristic
321C22⋊D5—24no—
322C22⋊D5—22no—
323C22⋊D5—22no—
324C22⋊D5—24no—
325C2×Dic5C3×C22⋊C411yescharacteristic
326C2×Dic5C2×C3×D411yescharacteristic
327D20—24no—
328D20—24no—
329D20—22no—
330D20—24no—
331D20—21no—
332D20—21no—
333D20C3×C22⋊C412yes—
334D20C3×C22⋊C412yes—
335C4×D5—24no—
336C4×D5—21no—
337C4×D5—21no—
338C4×D5—24no—
339C4×D5C2×C3×D412yes—
340C4×D5C2×C3×D412yes—
341Dic10—22no—
342#170—42no—
343H(5,𝔽2)—104no—
344H(5,𝔽2)—104no—
345C2×#41—51no—
346C2×#41—51no—
347C2×#41—51no—
348C2×#41—101no—
349C2×#41C3×AGL(1,𝔽5)11yescharacteristic, complemented
350C22×D4—51no—
351C22×D4—51no—
352C22×D4—51no—
353C22×D4—101no—
354C22×D4—101no—
355C22×D4—101no—
356C2×M16—52no—
357#117—52no—
358#104—104no—
359#102—104no—
360#101—101no—
361#100—101no—
362#99—101no—
363#99—101no—
364C2×C3×C5—21no—
365C2×C3×C5—41no—
366C2×C3×C5—42no—
367C2×C3×C5#37911yescharacteristic
368C3×D5—22no—
369C3×D5—42no—
370C3×D5—42no—
371C3×D5—41no—
372C23×C3—21no—
373C23×C3—51no—
374C23×C3—51no—
375C23×C3—104no—
376C23×C3—104no—
377C23×C3—104no—
378C23×C3—101no—
379C23×C3—101no—
380C23×C3—402no—
381C23×C3—402no—
382C23×C3—402no—
383C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
384C3×Q8—21no—
385C3×Q8—102no—
386C3×Q8C22⋊AGL(1,𝔽5)12yes—
387C3×D4—24no—
388C3×D4—21no—
389C3×D4—52no—
390C3×D4—52no—
391C3×D4—102no—
392C3×D4—104no—
393C3×D4—104no—
394C3×D4—104no—
395C3×D4—102no—
396C3×D4—104no—
397C3×D4—102no—
398C3×D4—104no—
399C3×D4—101no—
400C3×D4—101no—
401C3×D4C22⋊AGL(1,𝔽5)12yes—
402C2×C3×C4—24no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—51no—
407C2×C3×C4—52no—
408C2×C3×C4—52no—
409C2×C3×C4—51no—
410C2×C3×C4—104no—
411C2×C3×C4—104no—
412C2×C3×C4—101no—
413C2×C3×C4—101no—
414C2×C3×C4—202no—
415C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
416C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
417C3×C8—202no—
418C22×C5—21no—
419C22×C5—22no—
420C22×C5—41no—
421C22×C5—42no—
422C22×C5C2×C3×C22⋊C411yescharacteristic
423C2×D5—21no—
424C2×D5—22no—
425C2×D5—22no—
426C2×D5—41no—
427C2×D5—42no—
428C2×D5—42no—
429C2×D5—41no—
430C2×D5—84no—
431C2×D5—84no—
432C2×D5—84no—
433C2×D5C3×#12012yes—
434AGL(1,𝔽5)—82no—
435C4×C5—21no—
436C4×C5—22no—
437C4×C5C2×C3×C22⋊C411yescharacteristic
438C4×C5C2×C3×C22⋊C411yescharacteristic
439Dic5—21no—
440Dic5C3×#12012yes—
441C24—101no—
442C24—101no—
443C24—101no—
444#41—22no—
445#41—22no—
446#41—54no—
447#41—102no—
448#41—102no—
449#41—108no—
450#41—102no—
451#41—102no—
452#41—102no—
453C2×Q8—51no—
454C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
455C2×D4—21no—
456C2×D4—54no—
457C2×D4—51no—
458C2×D4—51no—
459C2×D4—51no—
460C2×D4—51no—
461C2×D4—51no—
462C2×D4—102no—
463C2×D4—102no—
464C2×D4—101no—
465C2×D4—101no—
466C2×D4—108no—
467C2×D4—102no—
468C2×D4—101no—
469C2×D4—101no—
470C2×D4—102no—
471C2×D4—108no—
472C2×D4—108no—
473C2×D4—102no—
474C2×D4—101no—
475C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
476C22×C4—21no—
477C22×C4—51no—
478C22×C4—51no—
479C22×C4—101no—
480C22×C4—101no—
481C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
482M16—102no—
483M16—102no—
484C2×C8—102no—
485C4⋊C4—102no—
486C22⋊C4—102no—
487C42—102no—
488C3×C5#179511yescharacteristic, complemented, Hall
489C22×C3—21no—
490C22×C3—22no—
491C22×C3—41no—
492C22×C3—42no—
493C22×C3—52no—
494C22×C3—102no—
495C22×C3—102no—
496C22×C3—101no—
497C22×C3—201no—
498C22×C3—201no—
499C22×C3—202no—
500C22×C3—202no—
501C22×C3—404no—
502C22×C3—404no—
503C22×C3—404no—
504C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
505C3×C4—21no—
506C3×C4—22no—
507C3×C4—52no—
508C3×C4—101no—
509C3×C4—402no—
510C3×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
511C3×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
512C2×C5—21no—
513C2×C5—42no—
514C2×C5—41no—
515C2×C5C3×#37911yescharacteristic
516D5—22no—
517D5—42no—
518D5—42no—
519D5—41no—
520C23—21no—
521C23—51no—
522C23—51no—
523C23—104no—
524C23—104no—
525C23—104no—
526C23—101no—
527C23—101no—
528C23—402no—
529C23—402no—
530C23—402no—
531C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
532Q8—21no—
533Q8—102no—
534Q8C3×C22⋊AGL(1,𝔽5)12yes—
535D4—24no—
536D4—21no—
537D4—52no—
538D4—52no—
539D4—102no—
540D4—104no—
541D4—104no—
542D4—104no—
543D4—102no—
544D4—104no—
545D4—102no—
546D4—104no—
547D4—101no—
548D4—101no—
549D4C3×C22⋊AGL(1,𝔽5)12yes—
550C2×C4—24no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—51no—
555C2×C4—52no—
556C2×C4—52no—
557C2×C4—51no—
558C2×C4—104no—
559C2×C4—104no—
560C2×C4—101no—
561C2×C4—101no—
562C2×C4—202no—
563C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
564C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
565C8—202no—
566C2×C3—21no—
567C2×C3—42no—
568C2×C3—41no—
569C2×C3—102no—
570C2×C3—202no—
571C2×C3—202no—
572C2×C3—201no—
573C2×C3#6580011yescharacteristic
574C5C3×#179511yescharacteristic, complemented, Sylow
575C22—21no—
576C22—22no—
577C22—41no—
578C22—42no—
579C22—52no—
580C22—102no—
581C22—102no—
582C22—101no—
583C22—201no—
584C22—201no—
585C22—202no—
586C22—202no—
587C22—404no—
588C22—404no—
589C22—404no—
590C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
591C4—21no—
592C4—22no—
593C4—52no—
594C4—101no—
595C4—402no—
596C4C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
597C4C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
598C3#1060501211yescharacteristic, direct product factor, Sylow
599C2—21no—
600C2—42no—
601C2—41no—
602C2—102no—
603C2—202no—
604C2—202no—
605C2—201no—
606C2C3×#6580011yescharacteristic
subgroupG/H
centerC2×C3#65800
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC2×C3×C5×#41C4
socleC2×C3×C5#379
prime radicalC2×C3×#4149C2
cosocleC2×C4×D5C23×C3
hypercenterC2×C3×#41AGL(1,𝔽5)
hypocenterC5C3×#1795
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×#41

Complements

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

NH ≅ G/Ntype
2. C3×#65784C2semidirect
3. C3×#65767C2semidirect
4. C3×#65761C2semidirect
5. C3×#65745C2semidirect
6. #10605012C3direct (N × H)
7. C2×C3×C5×#41C4semidirect
10. C2×C3×#713C4semidirect
14. C2×C3×D4×D5C4semidirect
19. C22×C3×D20C4semidirect
20. C3×#4130C22semidirect
21. C3×D5⋊M16C22semidirect
30. #65784C2×C3semidirect
31. #65767C2×C3semidirect
32. #65761C2×C3semidirect
33. #65745C2×C3semidirect
86. C2×C5×#41C3×C4semidirect
89. C2×#713C3×C4semidirect
93. C2×D4×D5C3×C4semidirect
98. C22×D20C3×C4semidirect
99. #4130C22×C3semidirect
100. D5⋊M16C22×C3semidirect
160. C2×C3×#41AGL(1,𝔽5)semidirect
349. C2×#41C3×AGL(1,𝔽5)semidirect
488. C3×C5#1795semidirect
574. C5C3×#1795semidirect
598. C3#10605012direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#41492
C3×#657842
C3×#657672
C3×#657612
C3×#657452
#106050123
C3×#17955