Finite Group Databasebeta

← C3×#10599757 #14302984

Subgroups — C3×#10599757 #14302984

Order 1920.

760 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65827C211yesmaximal, characteristic
2C3×#65820C211yesmaximal, characteristic, complemented
3C2×C3×#3938C211yesmaximal, characteristic, complemented
4C2×C3×#3938C211yesmaximal, characteristic, complemented
5C2×C3×#3938C211yesmaximal, characteristic, complemented
6C3×C5×#342C211yesmaximal, characteristic, complemented
7C3×#64747C211yesmaximal, characteristic, complemented
8#10599757C311yesmaximal, characteristic, direct product factor, Hall
9C22×C3×C22⋊D5C2211yescharacteristic
10C22×C3×C22⋊D5C2211yescharacteristic
11C2×C3×C5×C22⋊C4C2211yescharacteristic
12C2×C3×C5×C22⋊C4C2211yescharacteristic
13C2×C3×C5×C22⋊C4C2211yescharacteristic
14C2×C3×C22⋊Dic5C2211yescharacteristic
15C2×C3×C22⋊Dic5C2211yescharacteristic
16C3×#4076—22no—
17C3×C4×C22⋊D5—22no—
18C3×#4032—21no—
19C3×#4032C412yescomplemented
20C3×#4028—21no—
21C3×#4028C412yescomplemented
22C3×#4027—22no—
23C3×D5×C22⋊C4—21no—
24C3×#4023—21no—
25C3×#3938—22no—
26C3×#3938—22no—
27C3×#3938—22no—
28C3×#1913—51nomaximal, Hall
29#65827C2×C311yescharacteristic
30#65820C2×C311yescharacteristic, complemented
31C2×#3938C2×C311yescharacteristic, complemented
32C2×#3938C2×C311yescharacteristic, complemented
33C2×#3938C2×C311yescharacteristic, complemented
34C5×#342C2×C311yescharacteristic, complemented
35#64747C2×C311yescharacteristic, complemented
36C24×C3×C5C2311yescharacteristic
37C23×C3×D5C2×C411yescharacteristic, complemented
38C23×C3×D5D411yescharacteristic
39C22×C3×C4×C5—21no—
40C22×C3×C4×C5—21no—
41C22×C3×C4×C5—21no—
42C2×C3×C22⋊D5—21no—
43C2×C3×C22⋊D5—21no—
44C2×C3×C22⋊D5—21no—
45C2×C3×C22⋊D5—21no—
46C2×C3×C22⋊D5—21no—
47C2×C3×C22⋊D5—21no—
48C2×C3×C22⋊D5—21no—
49C2×C3×C22⋊D5—21no—
50C2×C3×C22⋊D5—21no—
51C2×C3×C22⋊D5—21no—
52C2×C3×C22⋊D5C2×C412yes—
53C2×C3×C22⋊D5D412yes—
54C22×C3×Dic5—21no—
55C22×C3×Dic5—21no—
56C22×C3×Dic5C2×C411yescharacteristic, complemented
57C22×C3×Dic5D411yescharacteristic
58C2×C3×D20—22no—
59C2×C3×C4×D5—22no—
60C3×C5×C22⋊C4—21no—
61C3×C5×C22⋊C4—21no—
62C3×C5×C22⋊C4—21no—
63C3×C5×C22⋊C4—21no—
64C3×C5×C22⋊C4C2×C412yescomplemented
65C3×C5×C22⋊C4D412yescomplemented
66C3×C22⋊Dic5—21no—
67C3×C22⋊Dic5—21no—
68C3×C22⋊Dic5—21no—
69C3×C22⋊Dic5D412yescomplemented
70C3×#685—22no—
71C3×#685—22no—
72C3×#685—22no—
73C3×C4⋊Dic5—22no—
74C3×#683—22no—
75C3×C4×Dic5—22no—
76C3×#534—51no—
77C3×#518—51no—
78C2×C3×#99—51no—
79C2×C3×#99—51no—
80C2×C3×#99—51no—
81C3×#342—51no—
82C3×#342D511yescharacteristic, complemented
83C22×C22⋊D5C22×C311yescharacteristic
84C22×C22⋊D5C22×C311yescharacteristic
85C2×C5×C22⋊C4C22×C311yescharacteristic
86C2×C5×C22⋊C4C22×C311yescharacteristic
87C2×C5×C22⋊C4C22×C311yescharacteristic
88C2×C22⋊Dic5C22×C311yescharacteristic
89C2×C22⋊Dic5C22×C311yescharacteristic
90#4076—22no—
91C4×C22⋊D5—22no—
92#4032—21no—
93#4032C3×C412yescomplemented
94#4028—21no—
95#4028C3×C412yescomplemented
96#4027—22no—
97D5×C22⋊C4—21no—
98#4023—21no—
99#3938—22no—
100#3938—22no—
101#3938—22no—
102#1913—51noSylow
103C23×C3×C5—81no—
104C23×C3×C5C22×C411yescharacteristic
105C23×C3×C5C2×D411yescharacteristic
106C23×C3×C5C2×D411yescharacteristic
107C23×C3×C5C2×D411yescharacteristic
108C23×C3×C5C2×D411yescharacteristic
109C23×C3×C5#4111yescharacteristic
110C23×C3×C5#4111yescharacteristic
111C22×C3×D5—21no—
112C22×C3×D5—21no—
113C22×C3×D5—21no—
114C22×C3×D5—21no—
115C22×C3×D5—21no—
116C22×C3×D5—81no—
117C22×C3×D5—81no—
118C22×C3×D5C22⋊C412yes—
119C2×C3×C4×C5—22no—
120C2×C3×C4×C5—22no—
121C2×C3×C4×C5—41no—
122C2×C3×C4×C5—81no—
123C2×C3×C4×C5—81no—
124C2×C3×C4×C5—81no—
125C3×C22⋊D5—22no—
126C3×C22⋊D5—22no—
127C3×C22⋊D5—22no—
128C3×C22⋊D5—22no—
129C3×C22⋊D5—41no—
130C3×C22⋊D5—41no—
131C3×C22⋊D5—41no—
132C3×C22⋊D5—41no—
133C2×C3×Dic5—21no—
134C2×C3×Dic5—21no—
135C2×C3×Dic5—21no—
136C2×C3×Dic5—21no—
137C2×C3×Dic5—21no—
138C2×C3×Dic5—22no—
139C2×C3×Dic5—41no—
140C2×C3×Dic5—81no—
141C2×C3×Dic5—81no—
142C2×C3×Dic5C22⋊C412yes—
143C3×D20—82no—
144C3×C4×D5—82no—
145C22×C3×D4—51no—
146C22×C3×D4—51no—
147C3×#123—52no—
148C3×#123—101no—
149C3×#121—102no—
150C3×#120—52no—
151C3×#120—101no—
152C3×C4×D4—102no—
153C3×C4×D4—102no—
154C3×#117—101no—
155C2×C3×C22⋊C4—51no—
156C2×C3×C22⋊C4—51no—
157C2×C3×C22⋊C4—101no—
158C2×C3×C22⋊C4C2×D511yescharacteristic
159C2×C3×C22⋊C4C2×D511yescharacteristic
160C2×C3×C22⋊C4C2×D511yescharacteristic
161C3×#99—102no—
162C3×#99—102no—
163C3×#99—102no—
164C24×C5C23×C311yescharacteristic
165C23×D5C2×C3×C411yescharacteristic, complemented
166C23×D5C3×D411yescharacteristic
167C22×C4×C5—21no—
168C22×C4×C5—21no—
169C22×C4×C5—21no—
170C2×C22⋊D5—21no—
171C2×C22⋊D5—21no—
172C2×C22⋊D5—21no—
173C2×C22⋊D5—21no—
174C2×C22⋊D5—21no—
175C2×C22⋊D5—21no—
176C2×C22⋊D5—21no—
177C2×C22⋊D5—21no—
178C2×C22⋊D5—21no—
179C2×C22⋊D5—21no—
180C2×C22⋊D5C2×C3×C412yes—
181C2×C22⋊D5C3×D412yes—
182C22×Dic5—21no—
183C22×Dic5—21no—
184C22×Dic5C2×C3×C411yescharacteristic, complemented
185C22×Dic5C3×D411yescharacteristic
186C2×D20—22no—
187C2×C4×D5—22no—
188C5×C22⋊C4—21no—
189C5×C22⋊C4—21no—
190C5×C22⋊C4—21no—
191C5×C22⋊C4—21no—
192C5×C22⋊C4C2×C3×C412yescomplemented
193C5×C22⋊C4C3×D412yescomplemented
194C22⋊Dic5—21no—
195C22⋊Dic5—21no—
196C22⋊Dic5—21no—
197C22⋊Dic5C3×D412yescomplemented
198#685—22no—
199#685—22no—
200#685—22no—
201C4⋊Dic5—22no—
202#683—22no—
203C4×Dic5—22no—
204#534—51no—
205#518—51no—
206C2×#99—51no—
207C2×#99—51no—
208C2×#99—51no—
209#342—51no—
210#342C3×D511yescharacteristic, complemented
211C22×C3×C5—41no—
212C22×C3×C5—41no—
213C22×C3×C5—41no—
214C22×C3×C5—41no—
215C22×C3×C5—41no—
216C22×C3×C5—41no—
217C22×C3×C5—41no—
218C22×C3×C5C2×C22⋊C411yescharacteristic
219C22×C3×C5C4×D411yescharacteristic
220C22×C3×C5C4×D411yescharacteristic
221C22×C3×C5#12011yescharacteristic
222C22×C3×C5#12111yescharacteristic
223C22×C3×C5#12111yescharacteristic
224C22×C3×C5#12311yescharacteristic
225C2×C3×D5—22no—
226C2×C3×D5—42no—
227C2×C3×D5—42no—
228C2×C3×D5—42no—
229C2×C3×D5—41no—
230C2×C3×D5—81no—
231C2×C3×D5—81no—
232C2×C3×D5—81no—
233C3×C4×C5—42no—
234C3×C4×C5—42no—
235C3×C4×C5—81no—
236C3×Dic5—22no—
237C3×Dic5—42no—
238C3×Dic5—41no—
239C3×Dic5—81no—
240C24×C3—51no—
241C24×C3—51no—
242C24×C3C22×D511yescharacteristic
243C2×C3×D4—52no—
244C2×C3×D4—52no—
245C2×C3×D4—102no—
246C2×C3×D4—101no—
247C2×C3×D4—101no—
248C2×C3×D4—101no—
249C2×C3×D4—101no—
250C2×C3×D4—101no—
251C2×C3×D4—101no—
252C2×C3×D4—101no—
253C2×C3×D4—101no—
254C2×C3×D4—101no—
255C2×C3×D4—101no—
256C22×C3×C4—21no—
257C22×C3×C4—21no—
258C22×C3×C4—21no—
259C22×C3×C4—51no—
260C22×C3×C4—51no—
261C22×C3×C4—102no—
262C22×C3×C4—101no—
263C22×C3×C4—101no—
264C3×C4⋊C4—102no—
265C3×C4⋊C4—102no—
266C3×C22⋊C4—21no—
267C3×C22⋊C4—21no—
268C3×C22⋊C4—21no—
269C3×C22⋊C4—21no—
270C3×C22⋊C4—52no—
271C3×C22⋊C4—102no—
272C3×C22⋊C4—102no—
273C3×C22⋊C4—102no—
274C3×C22⋊C4—101no—
275C3×C22⋊C4—101no—
276C3×C22⋊C4—101no—
277C3×C22⋊C4C4×D512yescomplemented
278C3×C22⋊C4D2012yescomplemented
279C3×C42—102no—
280C23×C5—81no—
281C23×C5C22×C3×C411yescharacteristic
282C23×C5C2×C3×D411yescharacteristic
283C23×C5C2×C3×D411yescharacteristic
284C23×C5C2×C3×D411yescharacteristic
285C23×C5C2×C3×D411yescharacteristic
286C23×C5C3×#4111yescharacteristic
287C23×C5C3×#4111yescharacteristic
288C22×D5—21no—
289C22×D5—21no—
290C22×D5—21no—
291C22×D5—21no—
292C22×D5—21no—
293C22×D5—81no—
294C22×D5—81no—
295C22×D5C3×C22⋊C412yes—
296C2×C4×C5—22no—
297C2×C4×C5—22no—
298C2×C4×C5—41no—
299C2×C4×C5—81no—
300C2×C4×C5—81no—
301C2×C4×C5—81no—
302C22⋊D5—22no—
303C22⋊D5—22no—
304C22⋊D5—22no—
305C22⋊D5—22no—
306C22⋊D5—41no—
307C22⋊D5—41no—
308C22⋊D5—41no—
309C22⋊D5—41no—
310C2×Dic5—21no—
311C2×Dic5—21no—
312C2×Dic5—21no—
313C2×Dic5—21no—
314C2×Dic5—21no—
315C2×Dic5—22no—
316C2×Dic5—41no—
317C2×Dic5—81no—
318C2×Dic5—81no—
319C2×Dic5C3×C22⋊C412yes—
320D20—82no—
321C4×D5—82no—
322C22×D4—51no—
323C22×D4—51no—
324#123—52no—
325#123—101no—
326#121—102no—
327#120—52no—
328#120—101no—
329C4×D4—102no—
330C4×D4—102no—
331#117—101no—
332C2×C22⋊C4—51no—
333C2×C22⋊C4—51no—
334C2×C22⋊C4—101no—
335C2×C22⋊C4C2×C3×D511yescharacteristic
336C2×C22⋊C4C2×C3×D511yescharacteristic
337C2×C22⋊C4C2×C3×D511yescharacteristic
338#99—102no—
339#99—102no—
340#99—102no—
341C2×C3×C5—21no—
342C2×C3×C5—21no—
343C2×C3×C5—21no—
344C2×C3×C5—21no—
345C2×C3×C5—21no—
346C2×C3×C5—21no—
347C2×C3×C5—21no—
348C2×C3×C5#38611yescharacteristic
349C3×D5—42no—
350C3×D5—81no—
351C23×C3—52no—
352C23×C3—81no—
353C23×C3—101no—
354C23×C3—101no—
355C23×C3—101no—
356C23×C3—101no—
357C23×C3—101no—
358C23×C3—401no—
359C23×C3—401no—
360C23×C3C2×C4×D511yescharacteristic
361C23×C3C2×D2011yescharacteristic
362C23×C3#70911yescharacteristic
363C23×C3D4×D511yescharacteristic
364C23×C3D4×D511yescharacteristic
365C23×C3D4×D511yescharacteristic
366C23×C3#71311yescharacteristic
367C3×D4—102no—
368C3×D4—102no—
369C3×D4—102no—
370C3×D4—102no—
371C3×D4—201no—
372C3×D4—201no—
373C3×D4—201no—
374C3×D4—201no—
375C3×D4—402no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—41no—
379C2×C3×C4—52no—
380C2×C3×C4—81no—
381C2×C3×C4—81no—
382C2×C3×C4—81no—
383C2×C3×C4—101no—
384C2×C3×C4—101no—
385C2×C3×C4—101no—
386C2×C3×C4—101no—
387C2×C3×C4—102no—
388C2×C3×C4—101no—
389C2×C3×C4—201no—
390C2×C3×C4—402no—
391C2×C3×C4—401no—
392C2×C3×C4—401no—
393C22×C5—41no—
394C22×C5—41no—
395C22×C5—41no—
396C22×C5—41no—
397C22×C5—41no—
398C22×C5—41no—
399C22×C5—41no—
400C22×C5C2×C3×C22⋊C411yescharacteristic
401C22×C5C3×C4×D411yescharacteristic
402C22×C5C3×C4×D411yescharacteristic
403C22×C5C3×#12011yescharacteristic
404C22×C5C3×#12111yescharacteristic
405C22×C5C3×#12111yescharacteristic
406C22×C5C3×#12311yescharacteristic
407C2×D5—22no—
408C2×D5—42no—
409C2×D5—42no—
410C2×D5—42no—
411C2×D5—41no—
412C2×D5—81no—
413C2×D5—81no—
414C2×D5—81no—
415C4×C5—42no—
416C4×C5—42no—
417C4×C5—81no—
418Dic5—22no—
419Dic5—42no—
420Dic5—41no—
421Dic5—81no—
422C24—51no—
423C24—51no—
424C24C22×C3×D511yescharacteristic
425C2×D4—52no—
426C2×D4—52no—
427C2×D4—102no—
428C2×D4—101no—
429C2×D4—101no—
430C2×D4—101no—
431C2×D4—101no—
432C2×D4—101no—
433C2×D4—101no—
434C2×D4—101no—
435C2×D4—101no—
436C2×D4—101no—
437C2×D4—101no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—51no—
442C22×C4—51no—
443C22×C4—102no—
444C22×C4—101no—
445C22×C4—101no—
446C4⋊C4—102no—
447C4⋊C4—102no—
448C22⋊C4—21no—
449C22⋊C4—21no—
450C22⋊C4—21no—
451C22⋊C4—21no—
452C22⋊C4—52no—
453C22⋊C4—102no—
454C22⋊C4—102no—
455C22⋊C4—102no—
456C22⋊C4—101no—
457C22⋊C4—101no—
458C22⋊C4—101no—
459C22⋊C4C3×C4×D512yescomplemented
460C22⋊C4C3×D2012yescomplemented
461C42—102no—
462C3×C5#191311yescharacteristic, complemented, Hall
463C22×C3—41no—
464C22×C3—41no—
465C22×C3—41no—
466C22×C3—41no—
467C22×C3—41no—
468C22×C3—41no—
469C22×C3—41no—
470C22×C3—102no—
471C22×C3—202no—
472C22×C3—202no—
473C22×C3—202no—
474C22×C3—201no—
475C22×C3—401no—
476C22×C3—401no—
477C22×C3—401no—
478C22×C3C4×D2011yescharacteristic
479C22×C3D5×C22⋊C411yescharacteristic
480C22×C3#402811yescharacteristic
481C22×C3#403011yescharacteristic
482C22×C3#403911yescharacteristic
483C22×C3#404011yescharacteristic
484C22×C3#404111yescharacteristic
485C3×C4—42no—
486C3×C4—42no—
487C3×C4—81no—
488C3×C4—102no—
489C3×C4—202no—
490C3×C4—201no—
491C3×C4—401no—
492C2×C5—21no—
493C2×C5—21no—
494C2×C5—21no—
495C2×C5—21no—
496C2×C5—21no—
497C2×C5—21no—
498C2×C5—21no—
499C2×C5C3×#38611yescharacteristic
500D5—42no—
501D5—81no—
502C23—52no—
503C23—81no—
504C23—101no—
505C23—101no—
506C23—101no—
507C23—101no—
508C23—101no—
509C23—401no—
510C23—401no—
511C23C2×C3×C4×D511yescharacteristic
512C23C2×C3×D2011yescharacteristic
513C23C3×#70911yescharacteristic
514C23C3×D4×D511yescharacteristic
515C23C3×D4×D511yescharacteristic
516C23C3×D4×D511yescharacteristic
517C23C3×#71311yescharacteristic
518D4—102no—
519D4—102no—
520D4—102no—
521D4—102no—
522D4—201no—
523D4—201no—
524D4—201no—
525D4—201no—
526D4—402no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—41no—
530C2×C4—52no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—101no—
535C2×C4—101no—
536C2×C4—101no—
537C2×C4—101no—
538C2×C4—102no—
539C2×C4—101no—
540C2×C4—201no—
541C2×C4—402no—
542C2×C4—401no—
543C2×C4—401no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—202no—
552C2×C3—401no—
553C2×C3#6495411yescharacteristic
554C5C3×#191311yescharacteristic, complemented, Sylow
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—102no—
563C22—202no—
564C22—202no—
565C22—202no—
566C22—201no—
567C22—401no—
568C22—401no—
569C22—401no—
570C22C3×C4×D2011yescharacteristic
571C22C3×D5×C22⋊C411yescharacteristic
572C22C3×#402811yescharacteristic
573C22C3×#403011yescharacteristic
574C22C3×#403911yescharacteristic
575C22C3×#404011yescharacteristic
576C22C3×#404111yescharacteristic
577C4—42no—
578C4—42no—
579C4—81no—
580C4—102no—
581C4—202no—
582C4—201no—
583C4—401no—
584C3#1059975711yescharacteristic, direct product factor, Sylow
585C2—21no—
586C2—21no—
587C2—21no—
588C2—21no—
589C2—21no—
590C2—21no—
591C2—21no—
592C2—202no—
593C2—401no—
594C2C3×#6495411yescharacteristic
subgroupG/H
centerC2×C3#64954
commutatorC23×C5C22×C3×C4
FrattiniC24C22×C3×D5
FittingC3×C5×#342C2
socleC2×C3×C5#386
prime radicalC3×#65827C2
cosocleC24×C5C23×C3
hypercenterC3×#342D5
hypocenterC5C3×#1913
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C24×C3 ◁ C3×#342

Complements

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

NH ≅ G/Ntype
2. C3×#65820C2semidirect
3. C2×C3×#3938C2semidirect
4. C2×C3×#3938C2semidirect
5. C2×C3×#3938C2semidirect
6. C3×C5×#342C2semidirect
7. C3×#64747C2semidirect
8. #10599757C3direct (N × H)
19. C3×#4032C4semidirect
21. C3×#4028C4semidirect
30. #65820C2×C3semidirect
31. C2×#3938C2×C3semidirect
32. C2×#3938C2×C3semidirect
33. C2×#3938C2×C3semidirect
34. C5×#342C2×C3semidirect
35. #64747C2×C3semidirect
37. C23×C3×D5C2×C4semidirect
56. C22×C3×Dic5C2×C4semidirect
64. C3×C5×C22⋊C4C2×C4semidirect
65. C3×C5×C22⋊C4D4semidirect
69. C3×C22⋊Dic5D4semidirect
82. C3×#342D5semidirect
93. #4032C3×C4semidirect
95. #4028C3×C4semidirect
165. C23×D5C2×C3×C4semidirect
184. C22×Dic5C2×C3×C4semidirect
192. C5×C22⋊C4C2×C3×C4semidirect
193. C5×C22⋊C4C3×D4semidirect
197. C22⋊Dic5C3×D4semidirect
210. #342C3×D5semidirect
277. C3×C22⋊C4C4×D5semidirect
278. C3×C22⋊C4D20semidirect
459. C22⋊C4C3×C4×D5semidirect
460. C22⋊C4C3×D20semidirect
462. C3×C5#1913semidirect
554. C5C3×#1913semidirect
584. C3#10599757direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#658272
C3×#658202
C2×C3×#39382
C3×C5×#3422
C3×#647472
#105997573
C3×#19135