Finite Group Databasebeta

← C3×C5×#2008 #14275042

Subgroups — C3×C5×#2008 #14275042

Order 1920.

760 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#575C211yesmaximal, characteristic, complemented
2C2×C3×C5×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
3C3×C5×#535C211yesmaximal, characteristic, complemented
4C3×C5×#421C211yesmaximal, characteristic, complemented
5C3×C5×#418C211yesmaximal, characteristic, complemented
6C3×C5×#410C211yesmaximal, characteristic, complemented
7C3×C5×#344C211yesmaximal, characteristic, complemented
8C5×#2008C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C2×C3×C5×#41C2211yescharacteristic
11C22×C3×C5×D4C2211yescharacteristic
12C3×C5×#137—21no—
13C3×C5×#137—21no—
14C3×C5×AGL(1,ℤ/8)—22no—
15C3×C5×AGL(1,ℤ/8)—21no—
16C3×C5×AGL(1,ℤ/8)—21no—
17C3×C5×AGL(1,ℤ/8)—22no—
18C3×C5×#135—22no—
19C2×C3×C5×SD16—21no—
20C2×C3×C5×D8—21no—
21C2×C3×C5×M16C2211yescharacteristic
22C2×C3×C5×M16C2211yescharacteristic
23C3×C5×#127—21no—
24C3×C5×#124—21no—
25C3×C5×#121—22no—
26C3×C5×#120—22no—
27C3×C5×#117C2211yescharacteristic
28C3×C5×#117C2211yescharacteristic
29C3×C5×#104—22no—
30C3×C5×#102—22no—
31C3×C5×#99—22no—
32C3×#2008C511yesmaximal, characteristic, direct product factor, Hall
33C5×#575C2×C311yescharacteristic, complemented
34C2×C5×AGL(1,ℤ/8)C2×C311yescharacteristic, complemented
35C5×#535C2×C311yescharacteristic, complemented
36C5×#421C2×C311yescharacteristic, complemented
37C5×#418C2×C311yescharacteristic, complemented
38C5×#410C2×C311yescharacteristic, complemented
39C5×#344C2×C311yescharacteristic, complemented
40C24×C3×C5—21no—
41C3×C5×#41—21no—
42C3×C5×#41—21no—
43C3×C5×#41—21no—
44C3×C5×#41—21no—
45C3×C5×#41—21no—
46C3×C5×#41—21no—
47C3×C5×#41—21no—
48C3×C5×#41—21no—
49C2×C3×C5×Q8D411yescharacteristic
50C2×C3×C5×Q8D411yescharacteristic, complemented
51C2×C3×C5×D4—21no—
52C2×C3×C5×D4—21no—
53C2×C3×C5×D4—24no—
54C2×C3×C5×D4—21no—
55C2×C3×C5×D4—21no—
56C2×C3×C5×D4—21no—
57C2×C3×C5×D4—21no—
58C2×C3×C5×D4—21no—
59C2×C3×C5×D4D412yes—
60C2×C3×C5×D4D411yescharacteristic
61C2×C3×C5×D4D411yescharacteristic, complemented
62C22×C3×C4×C5—21no—
63C22×C3×C4×C5—21no—
64C22×C3×C4×C5C2311yescharacteristic
65C3×C5×Q16—22no—
66C3×C5×SD16—24no—
67C3×C5×SD16—22no—
68C3×C5×SD16—22no—
69C3×C5×D8—24no—
70C3×C5×D8—22no—
71C3×C5×M16—21no—
72C3×C5×M16—21no—
73C3×C5×M16—21no—
74C3×C5×M16D412yescomplemented
75C2×C3×C5×C8—21no—
76C2×C3×C5×C8D412yescomplemented
77C3×C5×C4⋊C4—21no—
78C3×C5×C4⋊C4—21no—
79C3×C5×C22⋊C4—24no—
80C3×C5×C22⋊C4—21no—
81C3×C5×C22⋊C4D412yes—
82C3×C42×C5—21no—
83C3×C42×C5—21no—
84C3×#575C2×C511yescharacteristic, complemented
85C2×C3×AGL(1,ℤ/8)C2×C511yescharacteristic, complemented
86C3×#535C2×C511yescharacteristic, complemented
87C3×#421C2×C511yescharacteristic, complemented
88C3×#418C2×C511yescharacteristic, complemented
89C3×#410C2×C511yescharacteristic, complemented
90C3×#344C2×C511yescharacteristic, complemented
91C2×C5×#41C22×C311yescharacteristic
92C2×C5×#41C22×C311yescharacteristic
93C22×C5×D4C22×C311yescharacteristic
94C5×#137—21no—
95C5×#137—21no—
96C5×AGL(1,ℤ/8)—22no—
97C5×AGL(1,ℤ/8)—21no—
98C5×AGL(1,ℤ/8)—21no—
99C5×AGL(1,ℤ/8)—22no—
100C5×#135—22no—
101C2×C5×SD16—21no—
102C2×C5×D8—21no—
103C2×C5×M16C22×C311yescharacteristic
104C2×C5×M16C22×C311yescharacteristic
105C5×#127—21no—
106C5×#124—21no—
107C5×#121—22no—
108C5×#120—22no—
109C5×#117C22×C311yescharacteristic
110C5×#117C22×C311yescharacteristic
111C5×#104—22no—
112C5×#102—22no—
113C5×#99—22no—
114#2008C3×C511yescharacteristic, direct product factor, Sylow
115C23×C3×C5—22no—
116C23×C3×C5—21no—
117C23×C3×C5—21no—
118C23×C3×C5—41no—
119C23×C3×C5—41no—
120C23×C3×C5—82no—
121C23×C3×C5C2×D411yescharacteristic
122C3×C5×Q8—21no—
123C3×C5×Q8—21no—
124C3×C5×Q8—21no—
125C3×C5×Q8—21no—
126C3×C5×D4—21no—
127C3×C5×D4—22no—
128C3×C5×D4—21no—
129C3×C5×D4—21no—
130C3×C5×D4—22no—
131C3×C5×D4—21no—
132C3×C5×D4—41no—
133C3×C5×D4—41no—
134C3×C5×D4—41no—
135C3×C5×D4—41no—
136C3×C5×D4—41no—
137C3×C5×D4—41no—
138C3×C5×D4—84no—
139C2×C3×C4×C5—22no—
140C2×C3×C4×C5—21no—
141C2×C3×C4×C5—21no—
142C2×C3×C4×C5—41no—
143C2×C3×C4×C5—41no—
144C2×C3×C4×C5—41no—
145C2×C3×C4×C5—41no—
146C2×C3×C4×C5—41no—
147C2×C3×C4×C5C2×D411yescharacteristic
148C2×C3×C4×C5C2×D411yescharacteristic
149C2×C3×C4×C5C2×D411yescharacteristic
150C2×C3×C4×C5C2×D411yescharacteristic
151C2×C3×C4×C5C2×D411yescharacteristic
152C2×C3×C4×C5#4111yescharacteristic
153C3×C5×C8—22no—
154C3×C5×C8—22no—
155C2×C3×#41C22×C511yescharacteristic
156C2×C3×#41C22×C511yescharacteristic
157C22×C3×D4C22×C511yescharacteristic
158C3×#137—21no—
159C3×#137—21no—
160C3×AGL(1,ℤ/8)—22no—
161C3×AGL(1,ℤ/8)—21no—
162C3×AGL(1,ℤ/8)—21no—
163C3×AGL(1,ℤ/8)—22no—
164C3×#135—22no—
165C2×C3×SD16—21no—
166C2×C3×D8—21no—
167C2×C3×M16C22×C511yescharacteristic
168C2×C3×M16C22×C511yescharacteristic
169C3×#127—21no—
170C3×#124—21no—
171C3×#121—22no—
172C3×#120—22no—
173C3×#117C22×C511yescharacteristic
174C3×#117C22×C511yescharacteristic
175C3×#104—22no—
176C3×#102—22no—
177C3×#99—22no—
178C24×C5—21no—
179C5×#41—21no—
180C5×#41—21no—
181C5×#41—21no—
182C5×#41—21no—
183C5×#41—21no—
184C5×#41—21no—
185C5×#41—21no—
186C5×#41—21no—
187C2×C5×Q8C3×D411yescharacteristic
188C2×C5×Q8C3×D411yescharacteristic, complemented
189C2×C5×D4—21no—
190C2×C5×D4—21no—
191C2×C5×D4—24no—
192C2×C5×D4—21no—
193C2×C5×D4—21no—
194C2×C5×D4—21no—
195C2×C5×D4—21no—
196C2×C5×D4—21no—
197C2×C5×D4C3×D412yes—
198C2×C5×D4C3×D411yescharacteristic
199C2×C5×D4C3×D411yescharacteristic, complemented
200C22×C4×C5—21no—
201C22×C4×C5—21no—
202C22×C4×C5C23×C311yescharacteristic
203C5×Q16—22no—
204C5×SD16—24no—
205C5×SD16—22no—
206C5×SD16—22no—
207C5×D8—24no—
208C5×D8—22no—
209C5×M16—21no—
210C5×M16—21no—
211C5×M16—21no—
212C5×M16C3×D412yescomplemented
213C2×C5×C8—21no—
214C2×C5×C8C3×D412yescomplemented
215C5×C4⋊C4—21no—
216C5×C4⋊C4—21no—
217C5×C22⋊C4—24no—
218C5×C22⋊C4—21no—
219C5×C22⋊C4C3×D412yes—
220C42×C5—21no—
221C42×C5—21no—
222#575C2×C3×C511yescharacteristic, complemented
223C2×AGL(1,ℤ/8)C2×C3×C511yescharacteristic, complemented
224#535C2×C3×C511yescharacteristic, complemented
225#421C2×C3×C511yescharacteristic, complemented
226#418C2×C3×C511yescharacteristic, complemented
227#410C2×C3×C511yescharacteristic, complemented
228#344C2×C3×C511yescharacteristic, complemented
229C22×C3×C5—41no—
230C22×C3×C5—42no—
231C22×C3×C5—44no—
232C22×C3×C5—41no—
233C22×C3×C5—41no—
234C22×C3×C5—42no—
235C22×C3×C5—81no—
236C22×C3×C5—82no—
237C22×C3×C5—82no—
238C22×C3×C5#12011yescharacteristic
239C22×C3×C5#12111yescharacteristic
240C22×C3×C5#12711yescharacteristic
241C3×C4×C5—42no—
242C3×C4×C5—41no—
243C3×C4×C5—41no—
244C3×C4×C5—81no—
245C3×C4×C5#12011yescharacteristic
246C3×C4×C5#12011yescharacteristic
247C3×C4×C5#12111yescharacteristic
248C3×C4×C5#12111yescharacteristic
249C24×C3—21no—
250C3×#41—21no—
251C3×#41—21no—
252C3×#41—21no—
253C3×#41—21no—
254C3×#41—21no—
255C3×#41—21no—
256C3×#41—21no—
257C3×#41—21no—
258C2×C3×Q8C5×D411yescharacteristic
259C2×C3×Q8C5×D411yescharacteristic, complemented
260C2×C3×D4—21no—
261C2×C3×D4—21no—
262C2×C3×D4—24no—
263C2×C3×D4—21no—
264C2×C3×D4—21no—
265C2×C3×D4—21no—
266C2×C3×D4—21no—
267C2×C3×D4—21no—
268C2×C3×D4C5×D412yes—
269C2×C3×D4C5×D411yescharacteristic
270C2×C3×D4C5×D411yescharacteristic, complemented
271C22×C3×C4—21no—
272C22×C3×C4—21no—
273C22×C3×C4C23×C511yescharacteristic
274C3×Q16—22no—
275C3×SD16—24no—
276C3×SD16—22no—
277C3×SD16—22no—
278C3×D8—24no—
279C3×D8—22no—
280C3×M16—21no—
281C3×M16—21no—
282C3×M16—21no—
283C3×M16C5×D412yescomplemented
284C2×C3×C8—21no—
285C2×C3×C8C5×D412yescomplemented
286C3×C4⋊C4—21no—
287C3×C4⋊C4—21no—
288C3×C22⋊C4—24no—
289C3×C22⋊C4—21no—
290C3×C22⋊C4C5×D412yes—
291C3×C42—21no—
292C3×C42—21no—
293C23×C5—22no—
294C23×C5—21no—
295C23×C5—21no—
296C23×C5—41no—
297C23×C5—41no—
298C23×C5—82no—
299C23×C5C2×C3×D411yescharacteristic
300C5×Q8—21no—
301C5×Q8—21no—
302C5×Q8—21no—
303C5×Q8—21no—
304C5×D4—21no—
305C5×D4—22no—
306C5×D4—21no—
307C5×D4—21no—
308C5×D4—22no—
309C5×D4—21no—
310C5×D4—41no—
311C5×D4—41no—
312C5×D4—41no—
313C5×D4—41no—
314C5×D4—41no—
315C5×D4—41no—
316C5×D4—84no—
317C2×C4×C5—22no—
318C2×C4×C5—21no—
319C2×C4×C5—21no—
320C2×C4×C5—41no—
321C2×C4×C5—41no—
322C2×C4×C5—41no—
323C2×C4×C5—41no—
324C2×C4×C5—41no—
325C2×C4×C5C2×C3×D411yescharacteristic
326C2×C4×C5C2×C3×D411yescharacteristic
327C2×C4×C5C2×C3×D411yescharacteristic
328C2×C4×C5C2×C3×D411yescharacteristic
329C2×C4×C5C2×C3×D411yescharacteristic
330C2×C4×C5C3×#4111yescharacteristic
331C5×C8—22no—
332C5×C8—22no—
333C2×#41C22×C3×C511yescharacteristic
334C2×#41C22×C3×C511yescharacteristic
335C22×D4C22×C3×C511yescharacteristic
336#137—21no—
337#137—21no—
338AGL(1,ℤ/8)—22no—
339AGL(1,ℤ/8)—21no—
340AGL(1,ℤ/8)—21no—
341AGL(1,ℤ/8)—22no—
342#135—22no—
343C2×SD16—21no—
344C2×D8—21no—
345C2×M16C22×C3×C511yescharacteristic
346C2×M16C22×C3×C511yescharacteristic
347#127—21no—
348#124—21no—
349#121—22no—
350#120—22no—
351#117C22×C3×C511yescharacteristic
352#117C22×C3×C511yescharacteristic
353#104—22no—
354#102—22no—
355#99—22no—
356C2×C3×C5—21no—
357C2×C3×C5—21no—
358C2×C3×C5—21no—
359C2×C3×C5—81no—
360C2×C3×C5—82no—
361C2×C3×C5—81no—
362C2×C3×C5#39211yescharacteristic
363C23×C3—22no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3—82no—
369C23×C3C2×C5×D411yescharacteristic
370C3×Q8—21no—
371C3×Q8—21no—
372C3×Q8—21no—
373C3×Q8—21no—
374C3×D4—21no—
375C3×D4—22no—
376C3×D4—21no—
377C3×D4—21no—
378C3×D4—22no—
379C3×D4—21no—
380C3×D4—41no—
381C3×D4—41no—
382C3×D4—41no—
383C3×D4—41no—
384C3×D4—41no—
385C3×D4—41no—
386C3×D4—84no—
387C2×C3×C4—22no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4C2×C5×D411yescharacteristic
396C2×C3×C4C2×C5×D411yescharacteristic
397C2×C3×C4C2×C5×D411yescharacteristic
398C2×C3×C4C2×C5×D411yescharacteristic
399C2×C3×C4C2×C5×D411yescharacteristic
400C2×C3×C4C5×#4111yescharacteristic
401C3×C8—22no—
402C3×C8—22no—
403C22×C5—41no—
404C22×C5—42no—
405C22×C5—44no—
406C22×C5—41no—
407C22×C5—41no—
408C22×C5—42no—
409C22×C5—81no—
410C22×C5—82no—
411C22×C5—82no—
412C22×C5C3×#12011yescharacteristic
413C22×C5C3×#12111yescharacteristic
414C22×C5C3×#12711yescharacteristic
415C4×C5—42no—
416C4×C5—41no—
417C4×C5—41no—
418C4×C5—81no—
419C4×C5C3×#12011yescharacteristic
420C4×C5C3×#12011yescharacteristic
421C4×C5C3×#12111yescharacteristic
422C4×C5C3×#12111yescharacteristic
423C24—21no—
424#41—21no—
425#41—21no—
426#41—21no—
427#41—21no—
428#41—21no—
429#41—21no—
430#41—21no—
431#41—21no—
432C2×Q8C3×C5×D411yescharacteristic
433C2×Q8C3×C5×D411yescharacteristic, complemented
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—24no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4C3×C5×D412yes—
443C2×D4C3×C5×D411yescharacteristic
444C2×D4C3×C5×D411yescharacteristic, complemented
445C22×C4—21no—
446C22×C4—21no—
447C22×C4C23×C3×C511yescharacteristic
448Q16—22no—
449SD16—24no—
450SD16—22no—
451SD16—22no—
452D8—24no—
453D8—22no—
454M16—21no—
455M16—21no—
456M16—21no—
457M16C3×C5×D412yescomplemented
458C2×C8—21no—
459C2×C8C3×C5×D412yescomplemented
460C4⋊C4—21no—
461C4⋊C4—21no—
462C22⋊C4—24no—
463C22⋊C4—21no—
464C22⋊C4C3×C5×D412yes—
465C42—21no—
466C42—21no—
467C3×C5#200811yescharacteristic, direct product factor, Hall
468C22×C3—41no—
469C22×C3—42no—
470C22×C3—44no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—42no—
474C22×C3—81no—
475C22×C3—82no—
476C22×C3—82no—
477C22×C3C5×#12011yescharacteristic
478C22×C3C5×#12111yescharacteristic
479C22×C3C5×#12711yescharacteristic
480C3×C4—42no—
481C3×C4—41no—
482C3×C4—41no—
483C3×C4—81no—
484C3×C4C5×#12011yescharacteristic
485C3×C4C5×#12011yescharacteristic
486C3×C4C5×#12111yescharacteristic
487C3×C4C5×#12111yescharacteristic
488C2×C5—21no—
489C2×C5—21no—
490C2×C5—21no—
491C2×C5—82no—
492C2×C5—81no—
493C2×C5—81no—
494C2×C5C3×#39211yescharacteristic
495C23—22no—
496C23—21no—
497C23—21no—
498C23—41no—
499C23—41no—
500C23—82no—
501C23C2×C3×C5×D411yescharacteristic
502Q8—21no—
503Q8—21no—
504Q8—21no—
505Q8—21no—
506D4—21no—
507D4—22no—
508D4—21no—
509D4—21no—
510D4—22no—
511D4—21no—
512D4—41no—
513D4—41no—
514D4—41no—
515D4—41no—
516D4—41no—
517D4—41no—
518D4—84no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4C2×C3×C5×D411yescharacteristic
528C2×C4C2×C3×C5×D411yescharacteristic
529C2×C4C2×C3×C5×D411yescharacteristic
530C2×C4C2×C3×C5×D411yescharacteristic
531C2×C4C2×C3×C5×D411yescharacteristic
532C2×C4C3×C5×#4111yescharacteristic
533C8—22no—
534C8—22no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—82no—
539C2×C3—81no—
540C2×C3—81no—
541C2×C3C5×#39211yescharacteristic
542C5C3×#200811yescharacteristic, direct product factor, Sylow
543C22—41no—
544C22—42no—
545C22—44no—
546C22—41no—
547C22—41no—
548C22—42no—
549C22—81no—
550C22—82no—
551C22—82no—
552C22C3×C5×#12011yescharacteristic
553C22C3×C5×#12111yescharacteristic
554C22C3×C5×#12711yescharacteristic
555C4—42no—
556C4—41no—
557C4—41no—
558C4—81no—
559C4C3×C5×#12011yescharacteristic
560C4C3×C5×#12011yescharacteristic
561C4C3×C5×#12111yescharacteristic
562C4C3×C5×#12111yescharacteristic
563C3C5×#200811yescharacteristic, direct product factor, Sylow
564C2—21no—
565C2—21no—
566C2—21no—
567C2—82no—
568C2—81no—
569C2—81no—
570C2C3×C5×#39211yescharacteristic
subgroupG/H
center, socleC2×C3×C5#392
commutator, Frattini, cosocleC22×C4C23×C3×C5
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C5 ◁ C22×C3×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×C5×#575C2semidirect
2. C2×C3×C5×AGL(1,ℤ/8)C2semidirect
3. C3×C5×#535C2semidirect
4. C3×C5×#421C2semidirect
5. C3×C5×#418C2semidirect
6. C3×C5×#410C2semidirect
7. C3×C5×#344C2semidirect
8. C5×#2008C3direct (N × H)
32. C3×#2008C5direct (N × H)
33. C5×#575C2×C3semidirect
34. C2×C5×AGL(1,ℤ/8)C2×C3semidirect
35. C5×#535C2×C3semidirect
36. C5×#421C2×C3semidirect
37. C5×#418C2×C3semidirect
38. C5×#410C2×C3semidirect
39. C5×#344C2×C3semidirect
50. C2×C3×C5×Q8D4semidirect
61. C2×C3×C5×D4D4semidirect
74. C3×C5×M16D4semidirect
76. C2×C3×C5×C8D4semidirect
84. C3×#575C2×C5semidirect
85. C2×C3×AGL(1,ℤ/8)C2×C5semidirect
86. C3×#535C2×C5semidirect
87. C3×#421C2×C5semidirect
88. C3×#418C2×C5semidirect
89. C3×#410C2×C5semidirect
90. C3×#344C2×C5semidirect
114. #2008C3×C5direct (N × H)
188. C2×C5×Q8C3×D4semidirect
199. C2×C5×D4C3×D4semidirect
212. C5×M16C3×D4semidirect
214. C2×C5×C8C3×D4semidirect
222. #575C2×C3×C5semidirect
223. C2×AGL(1,ℤ/8)C2×C3×C5semidirect
224. #535C2×C3×C5semidirect
225. #421C2×C3×C5semidirect
226. #418C2×C3×C5semidirect
227. #410C2×C3×C5semidirect
228. #344C2×C3×C5semidirect
259. C2×C3×Q8C5×D4semidirect
270. C2×C3×D4C5×D4semidirect
283. C3×M16C5×D4semidirect
285. C2×C3×C8C5×D4semidirect
433. C2×Q8C3×C5×D4semidirect
444. C2×D4C3×C5×D4semidirect
457. M16C3×C5×D4semidirect
459. C2×C8C3×C5×D4semidirect
467. C3×C5#2008direct (N × H)
542. C5C3×#2008direct (N × H)
563. C3C5×#2008direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5752
C2×C3×C5×AGL(1,ℤ/8)2
C3×C5×#5352
C3×C5×#4212
C3×C5×#4182
C3×C5×#4102
C3×C5×#3442
C5×#20083
C3×#20085