Finite Group Databasebeta

← C3×#10601207 #14297322

Subgroups — C3×#10601207 #14297322

Order 1920.

1176 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4149C211yesmaximal, characteristic
2C3×C5×#421C211yesmaximal, characteristic, complemented
3C3×#65525C211yesmaximal, characteristic, complemented
4C3×#65421C211yesmaximal, characteristic, complemented
5C3×#65420C211yesmaximal, characteristic, complemented
6C3×#65295C211yesmaximal, characteristic, complemented
7C3×#65290C211yesmaximal, characteristic, complemented
8#10601207C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C3×#4149—22no—
11C3×#4149—22no—
12C3×#4149—22no—
13C3×#4149—22no—
14C2×C3×#713C412yescomplemented
15C2×C3×D4×D5—22no—
16C2×C3×D4×D5C412yescomplemented
17C2×C3×#709—21no—
18C2×C3×#709C2211yescharacteristic
19C22×C3×D20—21no—
20C22×C3×D20C2211yescharacteristic
21C2×C3×C5×M16C2211yescharacteristic, complemented
22C3×C5×#117C2211yescharacteristic, complemented
23C3×#4072C2211yescharacteristic, complemented
24C2×C3×#681C2211yescharacteristic, complemented
25C3×C5×#104—22no—
26C3×#3969—22no—
27C3×#3956—21no—
28C3×#3955—21no—
29C3×#3953—22no—
30C3×#3941—22no—
31C3×#3938—22no—
32C3×#1795—51nomaximal, Hall
33C2×#4149C2×C311yescharacteristic
34C5×#421C2×C311yescharacteristic, complemented
35#65525C2×C311yescharacteristic, complemented
36#65421C2×C311yescharacteristic, complemented
37#65420C2×C311yescharacteristic, complemented
38#65295C2×C311yescharacteristic, complemented
39#65290C2×C311yescharacteristic, complemented
40C23×C3×D5—21no—
41C23×C3×D5—22no—
42C3×C5×#41—21no—
43C3×C5×#41—21no—
44C3×C5×#41D412yes—
45C3×C5×#41D412yes—
46C2×C3×C5×Q8C2×C411yescharacteristic
47C2×C3×C5×D4—21no—
48C2×C3×C5×D4C2×C411yescharacteristic
49C22×C3×C4×C5—21no—
50C22×C3×C4×C5C2311yescharacteristic
51C2×C3×C22⋊D5—22no—
52C2×C3×C22⋊D5D412yes—
53C3×#713—24no—
54C3×#713—24no—
55C3×D4×D5—28no—
56C3×D4×D5—28no—
57C3×D4×D5—24no—
58C3×D4×D5—24no—
59C3×#709—24no—
60C3×#709—24no—
61C3×#709—21no—
62C3×#709—21no—
63C3×#709—21no—
64C3×#709—21no—
65C2×C3×D20—24no—
66C2×C3×D20—24no—
67C2×C3×D20—22no—
68C2×C3×D20—21no—
69C2×C3×D20—21no—
70C2×C3×D20—21no—
71C2×C3×D20C2×C412yes—
72C2×C3×D20D411yescharacteristic
73C2×C3×D20D412yes—
74C2×C3×D20D412yes—
75C2×C3×C4×D5—22no—
76C2×C3×C4×D5C2×C412yes—
77C2×C3×Dic10D411yescharacteristic
78C3×C5×M16—21no—
79C3×C5×M16—21no—
80C2×C3×C5×C8—21no—
81C3×C5×C4⋊C4—21no—
82C3×C5×C22⋊C4—21no—
83C3×C42×C5—21no—
84C3×C22⋊Dic5—21no—
85C3×C4⋊Dic5—21no—
86C3×C4×Dic5—21no—
87C3×#681—21no—
88C3×#681—21no—
89C2×C3×#168—21no—
90C2×C3×H(5,𝔽2)—51no—
91C3×#421—51no—
92C3×#421D511yescharacteristic, complemented
93C3×#418—51no—
94C3×#418—51no—
95C3×#413—51no—
96C3×#410—51no—
97C2×C5×#41C22×C311yescharacteristic
98#4149—22no—
99#4149—22no—
100#4149—22no—
101#4149—22no—
102C2×#713C3×C412yescomplemented
103C2×D4×D5—22no—
104C2×D4×D5C3×C412yescomplemented
105C2×#709—21no—
106C2×#709C22×C311yescharacteristic
107C22×D20—21no—
108C22×D20C22×C311yescharacteristic
109C2×C5×M16C22×C311yescharacteristic, complemented
110C5×#117C22×C311yescharacteristic, complemented
111#4072C22×C311yescharacteristic, complemented
112C2×#681C22×C311yescharacteristic, complemented
113C5×#104—22no—
114#3969—22no—
115#3956—21no—
116#3955—21no—
117#3953—22no—
118#3941—22no—
119#3938—22no—
120#1795—51noSylow
121C23×C3×C5—21no—
122C23×C3×C5C2×D411yescharacteristic
123C22×C3×D5—22no—
124C22×C3×D5—28no—
125C22×C3×D5—22no—
126C22×C3×D5—22no—
127C22×C3×D5—82no—
128C22×C3×D5—84no—
129C22×C3×D5C22⋊C412yes—
130C3×C5×Q8—21no—
131C3×C5×Q8C22⋊C412yes—
132C3×C5×D4—22no—
133C3×C5×D4—22no—
134C3×C5×D4—21no—
135C3×C5×D4C22⋊C412yes—
136C2×C3×C4×C5—21no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5—22no—
139C2×C3×C4×C5—41no—
140C2×C3×C4×C5C22×C411yescharacteristic
141C2×C3×C4×C5C2×D411yescharacteristic
142C2×C3×C4×C5C2×D411yescharacteristic
143C2×C3×C4×C5C2×D411yescharacteristic
144C2×C3×C4×C5C2×D411yescharacteristic
145C2×C3×C4×C5C2×D411yescharacteristic
146C3×C22⋊D5—28no—
147C3×C22⋊D5—22no—
148C3×C22⋊D5—22no—
149C2×C3×Dic5—41no—
150C2×C3×Dic5C22⋊C412yes—
151C3×D20—28no—
152C3×D20—22no—
153C3×D20—21no—
154C3×D20—22no—
155C3×D20—21no—
156C3×D20—22no—
157C3×D20C22⋊C414yes—
158C3×C4×D5—28no—
159C3×C4×D5—22no—
160C3×C4×D5—22no—
161C3×Dic10—21no—
162C3×Dic10—21no—
163C3×C5×C8—41no—
164C3×#168—41no—
165C3×H(5,𝔽2)—102no—
166C3×H(5,𝔽2)—102no—
167C3×H(5,𝔽2)—102no—
168C3×H(5,𝔽2)—102no—
169C2×C3×#41—51no—
170C2×C3×#41—52no—
171C2×C3×#41—101no—
172C2×C3×#41C2×D511yescharacteristic
173C22×C3×D4—52no—
174C22×C3×D4—51no—
175C22×C3×D4—101no—
176C22×C3×D4—102no—
177C2×C3×M16—51no—
178C2×C3×M16C2×D511yescharacteristic, complemented
179C3×#117—51no—
180C3×#117C2×D511yescharacteristic, complemented
181C3×#104—22no—
182C3×#104—102no—
183C3×#102—102no—
184C3×#102—102no—
185C3×#101—101no—
186C3×#100—101no—
187C3×#99—102no—
188C23×D5—21no—
189C23×D5—22no—
190C5×#41—21no—
191C5×#41—21no—
192C5×#41C3×D412yes—
193C5×#41C3×D412yes—
194C2×C5×Q8C2×C3×C411yescharacteristic
195C2×C5×D4—21no—
196C2×C5×D4C2×C3×C411yescharacteristic
197C22×C4×C5—21no—
198C22×C4×C5C23×C311yescharacteristic
199C2×C22⋊D5—22no—
200C2×C22⋊D5C3×D412yes—
201#713—24no—
202#713—24no—
203D4×D5—28no—
204D4×D5—28no—
205D4×D5—24no—
206D4×D5—24no—
207#709—24no—
208#709—24no—
209#709—21no—
210#709—21no—
211#709—21no—
212#709—21no—
213C2×D20—24no—
214C2×D20—24no—
215C2×D20—22no—
216C2×D20—21no—
217C2×D20—21no—
218C2×D20—21no—
219C2×D20C2×C3×C412yes—
220C2×D20C3×D411yescharacteristic
221C2×D20C3×D412yes—
222C2×D20C3×D412yes—
223C2×C4×D5—22no—
224C2×C4×D5C2×C3×C412yes—
225C2×Dic10C3×D411yescharacteristic
226C5×M16—21no—
227C5×M16—21no—
228C2×C5×C8—21no—
229C5×C4⋊C4—21no—
230C5×C22⋊C4—21no—
231C42×C5—21no—
232C22⋊Dic5—21no—
233C4⋊Dic5—21no—
234C4×Dic5—21no—
235#681—21no—
236#681—21no—
237C2×#168—21no—
238C2×H(5,𝔽2)—51no—
239#421—51no—
240#421C3×D511yescharacteristic, complemented
241#418—51no—
242#418—51no—
243#413—51no—
244#410—51no—
245C22×C3×C5—22no—
246C22×C3×C5—41no—
247C22×C3×C5—42no—
248C22×C3×C5C2×C22⋊C411yescharacteristic
249C22×C3×C5#12011yescharacteristic
250C22×C3×C5#12011yescharacteristic
251C2×C3×D5—22no—
252C2×C3×D5—24no—
253C2×C3×D5—44no—
254C2×C3×D5—42no—
255C2×C3×D5—82no—
256C2×C3×D5—82no—
257C2×C3×D5—88no—
258C3×C4×C5—22no—
259C3×C4×C5—81no—
260C3×C4×C5C2×C22⋊C411yescharacteristic
261C3×C4×C5C2×C22⋊C411yescharacteristic
262C3×C4×C5#12011yescharacteristic
263C3×C4×C5#12011yescharacteristic
264C3×Dic5—22no—
265C3×Dic5—81no—
266C24×C3—102no—
267C24×C3—101no—
268C3×#41—21no—
269C3×#41—21no—
270C3×#41—104no—
271C3×#41—104no—
272C3×#41—104no—
273C3×#41—104no—
274C3×#41—101no—
275C3×#41—101no—
276C3×#41—101no—
277C3×#41—101no—
278C3×#41D2012yes—
279C3×#41C22⋊D512yes—
280C2×C3×Q8—51no—
281C2×C3×Q8C4×D511yescharacteristic
282C2×C3×D4—21no—
283C2×C3×D4—52no—
284C2×C3×D4—52no—
285C2×C3×D4—52no—
286C2×C3×D4—52no—
287C2×C3×D4—51no—
288C2×C3×D4—102no—
289C2×C3×D4—104no—
290C2×C3×D4—104no—
291C2×C3×D4—102no—
292C2×C3×D4—104no—
293C2×C3×D4—108no—
294C2×C3×D4—108no—
295C2×C3×D4—104no—
296C2×C3×D4—101no—
297C2×C3×D4—101no—
298C2×C3×D4—101no—
299C2×C3×D4C4×D511yescharacteristic
300C22×C3×C4—21no—
301C22×C3×C4—52no—
302C22×C3×C4—102no—
303C22×C3×C4C22×D511yescharacteristic
304C3×M16—21no—
305C3×M16—21no—
306C3×M16—101no—
307C3×M16—101no—
308C2×C3×C8—21no—
309C2×C3×C8—101no—
310C3×C4⋊C4—21no—
311C3×C4⋊C4—101no—
312C3×C22⋊C4—21no—
313C3×C22⋊C4—101no—
314C3×C42—21no—
315C3×C42—101no—
316C23×C5—21no—
317C23×C5C2×C3×D411yescharacteristic
318C22×D5—22no—
319C22×D5—28no—
320C22×D5—22no—
321C22×D5—22no—
322C22×D5—82no—
323C22×D5—84no—
324C22×D5C3×C22⋊C412yes—
325C5×Q8—21no—
326C5×Q8C3×C22⋊C412yes—
327C5×D4—22no—
328C5×D4—22no—
329C5×D4—21no—
330C5×D4C3×C22⋊C412yes—
331C2×C4×C5—21no—
332C2×C4×C5—22no—
333C2×C4×C5—22no—
334C2×C4×C5—41no—
335C2×C4×C5C22×C3×C411yescharacteristic
336C2×C4×C5C2×C3×D411yescharacteristic
337C2×C4×C5C2×C3×D411yescharacteristic
338C2×C4×C5C2×C3×D411yescharacteristic
339C2×C4×C5C2×C3×D411yescharacteristic
340C2×C4×C5C2×C3×D411yescharacteristic
341C22⋊D5—28no—
342C22⋊D5—22no—
343C22⋊D5—22no—
344C2×Dic5—41no—
345C2×Dic5C3×C22⋊C412yes—
346D20—28no—
347D20—22no—
348D20—21no—
349D20—22no—
350D20—21no—
351D20—22no—
352D20C3×C22⋊C414yes—
353C4×D5—28no—
354C4×D5—22no—
355C4×D5—22no—
356Dic10—21no—
357Dic10—21no—
358C5×C8—41no—
359#168—41no—
360H(5,𝔽2)—102no—
361H(5,𝔽2)—102no—
362H(5,𝔽2)—102no—
363H(5,𝔽2)—102no—
364C2×#41—51no—
365C2×#41—52no—
366C2×#41—101no—
367C2×#41C2×C3×D511yescharacteristic
368C22×D4—52no—
369C22×D4—51no—
370C22×D4—101no—
371C22×D4—102no—
372C2×M16—51no—
373C2×M16C2×C3×D511yescharacteristic, complemented
374#117—51no—
375#117C2×C3×D511yescharacteristic, complemented
376#104—22no—
377#104—102no—
378#102—102no—
379#102—102no—
380#101—101no—
381#100—101no—
382#99—102no—
383C2×C3×C5—21no—
384C2×C3×C5—21no—
385C2×C3×C5—21no—
386C2×C3×C5—42no—
387C2×C3×C5#37911yescharacteristic
388C3×D5—44no—
389C3×D5—42no—
390C23×C3—21no—
391C23×C3—52no—
392C23×C3—108no—
393C23×C3—102no—
394C23×C3—102no—
395C23×C3—102no—
396C23×C3—402no—
397C23×C3—404no—
398C23×C3D4×D511yescharacteristic
399C3×Q8—21no—
400C3×Q8—101no—
401C3×Q8—101no—
402C3×Q8#68512yes—
403C3×D4—22no—
404C3×D4—22no—
405C3×D4—21no—
406C3×D4—54no—
407C3×D4—108no—
408C3×D4—108no—
409C3×D4—102no—
410C3×D4—102no—
411C3×D4—102no—
412C3×D4—101no—
413C3×D4—102no—
414C3×D4—102no—
415C3×D4—101no—
416C3×D4#68512yes—
417C2×C3×C4—21no—
418C2×C3×C4—22no—
419C2×C3×C4—22no—
420C2×C3×C4—41no—
421C2×C3×C4—52no—
422C2×C3×C4—108no—
423C2×C3×C4—102no—
424C2×C3×C4—102no—
425C2×C3×C4—201no—
426C2×C3×C4C2×C4×D511yescharacteristic
427C2×C3×C4C2×D2011yescharacteristic
428C2×C3×C4D4×D511yescharacteristic
429C2×C3×C4D4×D511yescharacteristic
430C2×C3×C4D4×D511yescharacteristic
431C2×C3×C4C2×C22⋊D511yescharacteristic
432C3×C8—41no—
433C3×C8—201no—
434C22×C5—22no—
435C22×C5—41no—
436C22×C5—42no—
437C22×C5C2×C3×C22⋊C411yescharacteristic
438C22×C5C3×#12011yescharacteristic
439C22×C5C3×#12011yescharacteristic
440C2×D5—22no—
441C2×D5—24no—
442C2×D5—44no—
443C2×D5—42no—
444C2×D5—82no—
445C2×D5—82no—
446C2×D5—88no—
447C4×C5—22no—
448C4×C5—81no—
449C4×C5C2×C3×C22⋊C411yescharacteristic
450C4×C5C2×C3×C22⋊C411yescharacteristic
451C4×C5C3×#12011yescharacteristic
452C4×C5C3×#12011yescharacteristic
453Dic5—22no—
454Dic5—81no—
455C24—102no—
456C24—101no—
457#41—21no—
458#41—21no—
459#41—104no—
460#41—104no—
461#41—104no—
462#41—104no—
463#41—101no—
464#41—101no—
465#41—101no—
466#41—101no—
467#41C3×D2012yes—
468#41C3×C22⋊D512yes—
469C2×Q8—51no—
470C2×Q8C3×C4×D511yescharacteristic
471C2×D4—21no—
472C2×D4—52no—
473C2×D4—52no—
474C2×D4—52no—
475C2×D4—52no—
476C2×D4—51no—
477C2×D4—102no—
478C2×D4—104no—
479C2×D4—104no—
480C2×D4—102no—
481C2×D4—104no—
482C2×D4—108no—
483C2×D4—108no—
484C2×D4—104no—
485C2×D4—101no—
486C2×D4—101no—
487C2×D4—101no—
488C2×D4C3×C4×D511yescharacteristic
489C22×C4—21no—
490C22×C4—52no—
491C22×C4—102no—
492C22×C4C22×C3×D511yescharacteristic
493M16—21no—
494M16—21no—
495M16—101no—
496M16—101no—
497C2×C8—21no—
498C2×C8—101no—
499C4⋊C4—21no—
500C4⋊C4—101no—
501C22⋊C4—21no—
502C22⋊C4—101no—
503C42—21no—
504C42—101no—
505C3×C5#179511yescharacteristic, complemented, Hall
506C22×C3—22no—
507C22×C3—41no—
508C22×C3—42no—
509C22×C3—104no—
510C22×C3—102no—
511C22×C3—202no—
512C22×C3—204no—
513C22×C3—402no—
514C22×C3—402no—
515C22×C3—408no—
516C22×C3D5×C22⋊C411yescharacteristic
517C22×C3#402811yescharacteristic
518C22×C3#408311yescharacteristic
519C3×C4—22no—
520C3×C4—81no—
521C3×C4—102no—
522C3×C4—401no—
523C3×C4D5×C22⋊C411yescharacteristic
524C3×C4#402811yescharacteristic
525C3×C4C2×#68511yescharacteristic
526C3×C4#408311yescharacteristic
527C2×C5—21no—
528C2×C5—21no—
529C2×C5—21no—
530C2×C5—42no—
531C2×C5C3×#37911yescharacteristic
532D5—44no—
533D5—42no—
534C23—21no—
535C23—52no—
536C23—108no—
537C23—102no—
538C23—102no—
539C23—102no—
540C23—402no—
541C23—404no—
542C23C3×D4×D511yescharacteristic
543Q8—21no—
544Q8—101no—
545Q8—101no—
546Q8C3×#68512yes—
547D4—22no—
548D4—22no—
549D4—21no—
550D4—54no—
551D4—108no—
552D4—108no—
553D4—102no—
554D4—102no—
555D4—102no—
556D4—101no—
557D4—102no—
558D4—101no—
559D4—102no—
560D4C3×#68512yes—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—41no—
565C2×C4—52no—
566C2×C4—108no—
567C2×C4—102no—
568C2×C4—102no—
569C2×C4—201no—
570C2×C4C2×C3×C4×D511yescharacteristic
571C2×C4C2×C3×D2011yescharacteristic
572C2×C4C3×D4×D511yescharacteristic
573C2×C4C3×D4×D511yescharacteristic
574C2×C4C3×D4×D511yescharacteristic
575C2×C4C2×C3×C22⋊D511yescharacteristic
576C8—41no—
577C8—201no—
578C2×C3—21no—
579C2×C3—21no—
580C2×C3—21no—
581C2×C3—42no—
582C2×C3—204no—
583C2×C3—202no—
584C2×C3#6524411yescharacteristic
585C5C3×#179511yescharacteristic, complemented, Sylow
586C22—22no—
587C22—41no—
588C22—42no—
589C22—104no—
590C22—102no—
591C22—202no—
592C22—204no—
593C22—402no—
594C22—402no—
595C22—408no—
596C22C3×D5×C22⋊C411yescharacteristic
597C22C3×#402811yescharacteristic
598C22C3×#408311yescharacteristic
599C4—22no—
600C4—81no—
601C4—102no—
602C4—401no—
603C4C3×D5×C22⋊C411yescharacteristic
604C4C3×#402811yescharacteristic
605C4C2×C3×#68511yescharacteristic
606C4C3×#408311yescharacteristic
607C3#1060120711yescharacteristic, direct product factor, Sylow
608C2—21no—
609C2—21no—
610C2—21no—
611C2—42no—
612C2—204no—
613C2—202no—
614C2C3×#6524411yescharacteristic
subgroupG/H
centerC2×C3#65244
commutatorC2×C4×C5C22×C3×C4
FrattiniC22×C4C22×C3×D5
FittingC3×C5×#421C2
socleC2×C3×C5#379
prime radicalC2×C3×#4149C2
cosocleC22×C4×C5C23×C3
hypercenterC3×#421D5
hypocenterC5C3×#1795
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#421

Complements

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

NH ≅ G/Ntype
2. C3×C5×#421C2semidirect
3. C3×#65525C2semidirect
4. C3×#65421C2semidirect
5. C3×#65420C2semidirect
6. C3×#65295C2semidirect
7. C3×#65290C2semidirect
8. #10601207C3direct (N × H)
14. C2×C3×#713C4semidirect
16. C2×C3×D4×D5C4semidirect
21. C2×C3×C5×M16C22semidirect
22. C3×C5×#117C22semidirect
23. C3×#4072C22semidirect
24. C2×C3×#681C22semidirect
34. C5×#421C2×C3semidirect
35. #65525C2×C3semidirect
36. #65421C2×C3semidirect
37. #65420C2×C3semidirect
38. #65295C2×C3semidirect
39. #65290C2×C3semidirect
92. C3×#421D5semidirect
102. C2×#713C3×C4semidirect
104. C2×D4×D5C3×C4semidirect
109. C2×C5×M16C22×C3semidirect
110. C5×#117C22×C3semidirect
111. #4072C22×C3semidirect
112. C2×#681C22×C3semidirect
178. C2×C3×M16C2×D5semidirect
180. C3×#117C2×D5semidirect
240. #421C3×D5semidirect
373. C2×M16C2×C3×D5semidirect
375. #117C2×C3×D5semidirect
505. C3×C5#1795semidirect
585. C5C3×#1795semidirect
607. C3#10601207direct (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×C5×#4212
C3×#655252
C3×#654212
C3×#654202
C3×#652952
C3×#652902
#106012073
C3×#17955