Finite Group Databasebeta

← C3×#10600450 #14301022

Subgroups — C3×#10600450 #14301022

Order 1920.

876 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#686C211yesmaximal, characteristic
2C2×C3×C4×D20C211yesmaximal, characteristic, complemented
3C3×C5×#390C211yesmaximal, characteristic, complemented
4C2×C3×#3964C211yesmaximal, characteristic, complemented
5C2×C3×#3941C211yesmaximal, characteristic, complemented
6C3×#64749C211yesmaximal, characteristic, complemented
7C3×#64744C211yesmaximal, characteristic, complemented
8#10600450C311yesmaximal, characteristic, direct product factor, Hall
9C22×C3×C5×D4C2211yescharacteristic
10C22×C3×D20C2211yescharacteristic
11C22×C3×C4×D5—21no—
12C2×C3×C5×C4⋊C4C2211yescharacteristic
13C2×C3×C5×C22⋊C4—21no—
14C2×C3×C5×C22⋊C4—21no—
15C2×C3×C42×C5C2211yescharacteristic
16C2×C3×#686—22no—
17C2×C3×#686—22no—
18C2×C3×#686C414yescomplemented
19C2×C3×#685—21no—
20C2×C3×C4⋊Dic5C2211yescharacteristic
21C22×C3×#168C2211yescharacteristic
22C22×C3×#168C2211yescharacteristic
23C3×C4×D20—22no—
24C3×C4×D20—22no—
25C3×#3964—22no—
26C3×#3941—22no—
27C3×#1879—51nomaximal, Hall
28C22×#686C2×C311yescharacteristic
29C2×C4×D20C2×C311yescharacteristic, complemented
30C5×#390C2×C311yescharacteristic, complemented
31C2×#3964C2×C311yescharacteristic, complemented
32C2×#3941C2×C311yescharacteristic, complemented
33#64749C2×C311yescharacteristic, complemented
34#64744C2×C311yescharacteristic, complemented
35C24×C3×C5—21no—
36C23×C3×D5—21no—
37C2×C3×C5×D4—21no—
38C2×C3×C5×D4—21no—
39C2×C3×C5×D4—21no—
40C2×C3×C5×D4—21no—
41C2×C3×C5×D4—21no—
42C2×C3×C5×D4C2×C412yescomplemented
43C22×C3×C4×C5—21no—
44C22×C3×C4×C5D411yescharacteristic
45C22×C3×C4×C5D411yescharacteristic
46C22×C3×C4×C5C2311yescharacteristic
47C22×C3×Dic5—21no—
48C2×C3×D20—21no—
49C2×C3×D20—21no—
50C2×C3×D20—21no—
51C2×C3×D20C2×C412yes—
52C2×C3×D20D412yes—
53C2×C3×D20D412yes—
54C2×C3×C4×D5—24no—
55C2×C3×C4×D5—42no—
56C2×C3×C4×D5—42no—
57C3×C5×C4⋊C4—21no—
58C3×C5×C4⋊C4—21no—
59C3×C5×C22⋊C4—42no—
60C3×C5×C22⋊C4—42no—
61C3×C42×C5—21no—
62C3×C42×C5—21no—
63C3×#686—28no—
64C3×#685—24no—
65C3×C4⋊Dic5—21no—
66C3×C4⋊Dic5D412yes—
67C2×C3×#168—21no—
68C2×C3×#168—21no—
69C2×C3×#168—21no—
70C2×C3×#168—21no—
71C2×C3×#168—21no—
72C2×C3×#168C2×C412yescomplemented
73C22×C3×D8—51no—
74C2×C3×C4×D4—51no—
75C2×C3×#102—51no—
76C2×C3×#102—51no—
77C3×#390D511yescharacteristic, complemented
78C3×#340—51no—
79C3×#336—51no—
80C22×C5×D4C22×C311yescharacteristic
81C22×D20C22×C311yescharacteristic
82C22×C4×D5—21no—
83C2×C5×C4⋊C4C22×C311yescharacteristic
84C2×C5×C22⋊C4—21no—
85C2×C5×C22⋊C4—21no—
86C2×C42×C5C22×C311yescharacteristic
87C2×#686—22no—
88C2×#686—22no—
89C2×#686C3×C414yescomplemented
90C2×#685—21no—
91C2×C4⋊Dic5C22×C311yescharacteristic
92C22×#168C22×C311yescharacteristic
93C22×#168C22×C311yescharacteristic
94C4×D20—22no—
95C4×D20—22no—
96#3964—22no—
97#3941—22no—
98#1879—51noSylow
99C23×C3×C5—22no—
100C23×C3×C5—41no—
101C23×C3×C5—41no—
102C23×C3×C5—41no—
103C23×C3×C5—41no—
104C23×C3×C5—41no—
105C23×C3×C5—41no—
106C23×C3×C5C2×D411yescharacteristic
107C22×C3×D5—22no—
108C22×C3×D5—22no—
109C22×C3×D5—22no—
110C22×C3×D5—41no—
111C22×C3×D5—41no—
112C22×C3×D5—41no—
113C22×C3×D5—41no—
114C3×C5×D4—22no—
115C3×C5×D4—22no—
116C3×C5×D4—41no—
117C3×C5×D4—41no—
118C2×C3×C4×C5—21no—
119C2×C3×C4×C5—21no—
120C2×C3×C4×C5—21no—
121C2×C3×C4×C5—21no—
122C2×C3×C4×C5—21no—
123C2×C3×C4×C5—41no—
124C2×C3×C4×C5—41no—
125C2×C3×C4×C5—41no—
126C2×C3×C4×C5D812yescomplemented
127C2×C3×C4×C5C22×C411yescharacteristic
128C2×C3×C4×C5C2×D411yescharacteristic
129C2×C3×C4×C5C2×D411yescharacteristic
130C2×C3×C4×C5C2×D411yescharacteristic
131C2×C3×C4×C5#4111yescharacteristic
132C2×C3×C4×C5#4111yescharacteristic
133C2×C3×Dic5—22no—
134C2×C3×Dic5—41no—
135C2×C3×Dic5—41no—
136C3×D20—22no—
137C3×D20—22no—
138C3×D20—22no—
139C3×D20C22⋊C414yescomplemented
140C3×C4×D5—48no—
141C3×#168—22no—
142C3×#168—41no—
143C22×C3×D4—51no—
144C22×C3×D4C2×D511yescharacteristic
145C23×C3×C4—101no—
146C2×C3×D8—54no—
147C2×C3×D8—102no—
148C2×C3×D8—102no—
149C22×C3×C8—51no—
150C22×C3×C8—51no—
151C3×C4×D4—102no—
152C3×C4×D4—102no—
153C2×C3×C4⋊C4—51no—
154C2×C3×C4⋊C4C2×D511yescharacteristic
155C2×C3×C22⋊C4—21no—
156C2×C3×C22⋊C4—21no—
157C2×C3×C22⋊C4—101no—
158C2×C3×C42C2×D511yescharacteristic
159C3×#102—102no—
160C3×#102—102no—
161C24×C5—21no—
162C23×D5—21no—
163C2×C5×D4—21no—
164C2×C5×D4—21no—
165C2×C5×D4—21no—
166C2×C5×D4—21no—
167C2×C5×D4—21no—
168C2×C5×D4C2×C3×C412yescomplemented
169C22×C4×C5—21no—
170C22×C4×C5C3×D411yescharacteristic
171C22×C4×C5C3×D411yescharacteristic
172C22×C4×C5C23×C311yescharacteristic
173C22×Dic5—21no—
174C2×D20—21no—
175C2×D20—21no—
176C2×D20—21no—
177C2×D20C2×C3×C412yes—
178C2×D20C3×D412yes—
179C2×D20C3×D412yes—
180C2×C4×D5—24no—
181C2×C4×D5—42no—
182C2×C4×D5—42no—
183C5×C4⋊C4—21no—
184C5×C4⋊C4—21no—
185C5×C22⋊C4—42no—
186C5×C22⋊C4—42no—
187C42×C5—21no—
188C42×C5—21no—
189#686—28no—
190#685—24no—
191C4⋊Dic5—21no—
192C4⋊Dic5C3×D412yes—
193C2×#168—21no—
194C2×#168—21no—
195C2×#168—21no—
196C2×#168—21no—
197C2×#168—21no—
198C2×#168C2×C3×C412yescomplemented
199C22×D8—51no—
200C2×C4×D4—51no—
201C2×#102—51no—
202C2×#102—51no—
203#390C3×D511yescharacteristic, complemented
204#340—51no—
205#336—51no—
206C22×C3×C5—42no—
207C22×C3×C5—42no—
208C22×C3×C5—42no—
209C22×C3×C5—81no—
210C22×C3×C5—81no—
211C22×C3×C5—81no—
212C22×C3×C5—81no—
213C22×C3×C5C4×D411yescharacteristic
214C22×C3×C5#12011yescharacteristic
215C22×C3×C5#12111yescharacteristic
216C22×C3×C5C2×D811yescharacteristic
217C22×C3×C5#13511yescharacteristic
218C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
219C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
220C2×C3×D5—24no—
221C2×C3×D5—42no—
222C2×C3×D5—42no—
223C2×C3×D5—42no—
224C2×C3×D5—42no—
225C2×C3×D5—42no—
226C2×C3×D5—42no—
227C3×C4×C5—22no—
228C3×C4×C5—41no—
229C3×C4×C5—81no—
230C3×C4×C5C2×C22⋊C411yescharacteristic
231C3×C4×C5C4×D411yescharacteristic
232C3×C4×C5#12111yescharacteristic
233C3×C4×C5#12311yescharacteristic
234C3×Dic5—42no—
235C24×C3—21no—
236C24×C3—101no—
237C2×C3×D4—21no—
238C2×C3×D4—21no—
239C2×C3×D4—21no—
240C2×C3×D4—21no—
241C2×C3×D4—21no—
242C2×C3×D4—52no—
243C2×C3×D4—52no—
244C2×C3×D4—52no—
245C2×C3×D4—101no—
246C2×C3×D4—101no—
247C2×C3×D4—101no—
248C2×C3×D4C4×D512yescomplemented
249C22×C3×C4—21no—
250C22×C3×C4—104no—
251C22×C3×C4—101no—
252C22×C3×C4—202no—
253C22×C3×C4—202no—
254C22×C3×C4C22⋊D511yescharacteristic
255C22×C3×C4C22⋊D511yescharacteristic
256C22×C3×C4C22×D511yescharacteristic
257C3×D8—108no—
258C2×C3×C8—52no—
259C2×C3×C8—101no—
260C2×C3×C8—101no—
261C2×C3×C8—101no—
262C2×C3×C8—101no—
263C2×C3×C8—101no—
264C3×C4⋊C4—21no—
265C3×C4⋊C4—21no—
266C3×C4⋊C4—52no—
267C3×C4⋊C4—101no—
268C3×C22⋊C4—42no—
269C3×C22⋊C4—42no—
270C3×C22⋊C4—104no—
271C3×C42—21no—
272C3×C42—21no—
273C23×C5—22no—
274C23×C5—41no—
275C23×C5—41no—
276C23×C5—41no—
277C23×C5—41no—
278C23×C5—41no—
279C23×C5—41no—
280C23×C5C2×C3×D411yescharacteristic
281C22×D5—22no—
282C22×D5—22no—
283C22×D5—22no—
284C22×D5—41no—
285C22×D5—41no—
286C22×D5—41no—
287C22×D5—41no—
288C5×D4—22no—
289C5×D4—22no—
290C5×D4—41no—
291C5×D4—41no—
292C2×C4×C5—21no—
293C2×C4×C5—21no—
294C2×C4×C5—21no—
295C2×C4×C5—21no—
296C2×C4×C5—21no—
297C2×C4×C5—41no—
298C2×C4×C5—41no—
299C2×C4×C5—41no—
300C2×C4×C5C3×D812yescomplemented
301C2×C4×C5C22×C3×C411yescharacteristic
302C2×C4×C5C2×C3×D411yescharacteristic
303C2×C4×C5C2×C3×D411yescharacteristic
304C2×C4×C5C2×C3×D411yescharacteristic
305C2×C4×C5C3×#4111yescharacteristic
306C2×C4×C5C3×#4111yescharacteristic
307C2×Dic5—22no—
308C2×Dic5—41no—
309C2×Dic5—41no—
310D20—22no—
311D20—22no—
312D20—22no—
313D20C3×C22⋊C414yescomplemented
314C4×D5—48no—
315#168—22no—
316#168—41no—
317C22×D4—51no—
318C22×D4C2×C3×D511yescharacteristic
319C23×C4—101no—
320C2×D8—54no—
321C2×D8—102no—
322C2×D8—102no—
323C22×C8—51no—
324C22×C8—51no—
325C4×D4—102no—
326C4×D4—102no—
327C2×C4⋊C4—51no—
328C2×C4⋊C4C2×C3×D511yescharacteristic
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—101no—
332C2×C42C2×C3×D511yescharacteristic
333#102—102no—
334#102—102no—
335C2×C3×C5—82no—
336C2×C3×C5#38611yescharacteristic
337C2×C3×C5C4×D811yescharacteristic
338C2×C3×C5#44211yescharacteristic
339C2×C3×C5#44711yescharacteristic
340C2×C3×C5#44911yescharacteristic
341C2×C3×C5#45911yescharacteristic
342C2×C3×C5#46311yescharacteristic
343C3×D5—44no—
344C23×C3—22no—
345C23×C3—41no—
346C23×C3—41no—
347C23×C3—41no—
348C23×C3—41no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3—102no—
352C23×C3—102no—
353C23×C3—102no—
354C23×C3—201no—
355C23×C3—201no—
356C23×C3—201no—
357C23×C3—201no—
358C23×C3C2×C22⋊D511yescharacteristic
359C3×D4—22no—
360C3×D4—22no—
361C3×D4—41no—
362C3×D4—41no—
363C3×D4—54no—
364C3×D4—102no—
365C3×D4—102no—
366C3×D4—102no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—102no—
376C2×C3×C4—208no—
377C2×C3×C4—201no—
378C2×C3×C4—201no—
379C2×C3×C4#68612yescomplemented
380C2×C3×C4C2×C4×D511yescharacteristic
381C2×C3×C4#70911yescharacteristic
382C2×C3×C4D4×D511yescharacteristic
383C2×C3×C4D4×D511yescharacteristic
384C2×C3×C4D4×D511yescharacteristic
385C2×C3×C4#71111yescharacteristic
386C3×C8—102no—
387C3×C8—201no—
388C22×C5—42no—
389C22×C5—42no—
390C22×C5—42no—
391C22×C5—81no—
392C22×C5—81no—
393C22×C5—81no—
394C22×C5—81no—
395C22×C5C3×C4×D411yescharacteristic
396C22×C5C3×#12011yescharacteristic
397C22×C5C3×#12111yescharacteristic
398C22×C5C2×C3×D811yescharacteristic
399C22×C5C3×#13511yescharacteristic
400C22×C5C3×AGL(1,ℤ/8)11yescharacteristic
401C22×C5C3×AGL(1,ℤ/8)11yescharacteristic
402C2×D5—24no—
403C2×D5—42no—
404C2×D5—42no—
405C2×D5—42no—
406C2×D5—42no—
407C2×D5—42no—
408C2×D5—42no—
409C4×C5—22no—
410C4×C5—41no—
411C4×C5—81no—
412C4×C5C2×C3×C22⋊C411yescharacteristic
413C4×C5C3×C4×D411yescharacteristic
414C4×C5C3×#12111yescharacteristic
415C4×C5C3×#12311yescharacteristic
416Dic5—42no—
417C24—21no—
418C24—101no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—52no—
425C2×D4—52no—
426C2×D4—52no—
427C2×D4—101no—
428C2×D4—101no—
429C2×D4—101no—
430C2×D4C3×C4×D512yescomplemented
431C22×C4—21no—
432C22×C4—104no—
433C22×C4—101no—
434C22×C4—202no—
435C22×C4—202no—
436C22×C4C3×C22⋊D511yescharacteristic
437C22×C4C3×C22⋊D511yescharacteristic
438C22×C4C22×C3×D511yescharacteristic
439D8—108no—
440C2×C8—52no—
441C2×C8—101no—
442C2×C8—101no—
443C2×C8—101no—
444C2×C8—101no—
445C2×C8—101no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—52no—
449C4⋊C4—101no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—104no—
453C42—21no—
454C42—21no—
455C3×C5#187911yescharacteristic, complemented, Hall
456C22×C3—42no—
457C22×C3—42no—
458C22×C3—42no—
459C22×C3—81no—
460C22×C3—81no—
461C22×C3—81no—
462C22×C3—81no—
463C22×C3—104no—
464C22×C3—202no—
465C22×C3—202no—
466C22×C3—202no—
467C22×C3—202no—
468C22×C3—202no—
469C22×C3—202no—
470C22×C3C4×C22⋊D511yescharacteristic
471C22×C3C2×#68611yescharacteristic
472C22×C3#407811yescharacteristic
473C22×C3#408311yescharacteristic
474C22×C3#408411yescharacteristic
475C22×C3#409511yescharacteristic
476C22×C3#409611yescharacteristic
477C3×C4—22no—
478C3×C4—41no—
479C3×C4—81no—
480C3×C4—202no—
481C3×C4D5×C22⋊C411yescharacteristic
482C3×C4#402711yescharacteristic
483C3×C4#402911yescharacteristic
484C3×C4#403011yescharacteristic
485C2×C5—82no—
486C2×C5C3×#38611yescharacteristic
487C2×C5C3×C4×D811yescharacteristic
488C2×C5C3×#44211yescharacteristic
489C2×C5C3×#44711yescharacteristic
490C2×C5C3×#44911yescharacteristic
491C2×C5C3×#45911yescharacteristic
492C2×C5C3×#46311yescharacteristic
493D5—44no—
494C23—22no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—102no—
502C23—102no—
503C23—102no—
504C23—201no—
505C23—201no—
506C23—201no—
507C23—201no—
508C23C2×C3×C22⋊D511yescharacteristic
509D4—22no—
510D4—22no—
511D4—41no—
512D4—41no—
513D4—54no—
514D4—102no—
515D4—102no—
516D4—102no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—102no—
526C2×C4—208no—
527C2×C4—201no—
528C2×C4—201no—
529C2×C4C3×#68612yescomplemented
530C2×C4C2×C3×C4×D511yescharacteristic
531C2×C4C3×#70911yescharacteristic
532C2×C4C3×D4×D511yescharacteristic
533C2×C4C3×D4×D511yescharacteristic
534C2×C4C3×D4×D511yescharacteristic
535C2×C4C3×#71111yescharacteristic
536C8—102no—
537C8—201no—
538C2×C3—82no—
539C2×C3—204no—
540C2×C3#6524511yescharacteristic
541C2×C3C4×#68611yescharacteristic
542C2×C3#6530311yescharacteristic
543C2×C3#6532511yescharacteristic
544C2×C3#6532611yescharacteristic
545C2×C3#6534611yescharacteristic
546C2×C3#6536111yescharacteristic
547C5C3×#187911yescharacteristic, complemented, Sylow
548C22—42no—
549C22—42no—
550C22—42no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—104no—
556C22—202no—
557C22—202no—
558C22—202no—
559C22—202no—
560C22—202no—
561C22—202no—
562C22C3×C4×C22⋊D511yescharacteristic
563C22C2×C3×#68611yescharacteristic
564C22C3×#407811yescharacteristic
565C22C3×#408311yescharacteristic
566C22C3×#408411yescharacteristic
567C22C3×#409511yescharacteristic
568C22C3×#409611yescharacteristic
569C4—22no—
570C4—41no—
571C4—81no—
572C4—202no—
573C4C3×D5×C22⋊C411yescharacteristic
574C4C3×#402711yescharacteristic
575C4C3×#402911yescharacteristic
576C4C3×#403011yescharacteristic
577C3#1060045011yescharacteristic, direct product factor, Sylow
578C2—82no—
579C2—204no—
580C2C3×#6524511yescharacteristic
581C2C3×C4×#68611yescharacteristic
582C2C3×#6530311yescharacteristic
583C2C3×#6532511yescharacteristic
584C2C3×#6532611yescharacteristic
585C2C3×#6534611yescharacteristic
586C2C3×#6536111yescharacteristic
subgroupG/H
centerC23×C3C2×C22⋊D5
commutatorC2×C4×C5C22×C3×C4
FrattiniC22×C4C22×C3×D5
FittingC3×C5×#390C2
socleC23×C3×C5C2×D4
prime radicalC22×C3×#686C2
cosocleC22×C4×C5C23×C3
hypercenterC3×#390D5
hypocenterC5C3×#1879
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C3×#390

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×D20C2semidirect
3. C3×C5×#390C2semidirect
4. C2×C3×#3964C2semidirect
5. C2×C3×#3941C2semidirect
6. C3×#64749C2semidirect
7. C3×#64744C2semidirect
8. #10600450C3direct (N × H)
18. C2×C3×#686C4semidirect
29. C2×C4×D20C2×C3semidirect
30. C5×#390C2×C3semidirect
31. C2×#3964C2×C3semidirect
32. C2×#3941C2×C3semidirect
33. #64749C2×C3semidirect
34. #64744C2×C3semidirect
42. C2×C3×C5×D4C2×C4semidirect
72. C2×C3×#168C2×C4semidirect
77. C3×#390D5semidirect
89. C2×#686C3×C4semidirect
126. C2×C3×C4×C5D8semidirect
139. C3×D20C22⋊C4semidirect
168. C2×C5×D4C2×C3×C4semidirect
198. C2×#168C2×C3×C4semidirect
203. #390C3×D5semidirect
248. C2×C3×D4C4×D5semidirect
300. C2×C4×C5C3×D8semidirect
313. D20C3×C22⋊C4semidirect
379. C2×C3×C4#686semidirect
430. C2×D4C3×C4×D5semidirect
455. C3×C5#1879semidirect
529. C2×C4C3×#686semidirect
547. C5C3×#1879semidirect
577. C3#10600450direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6862
C2×C3×C4×D202
C3×C5×#3902
C2×C3×#39642
C2×C3×#39412
C3×#647492
C3×#647442
#106004503
C3×#18795