Finite Group Databasebeta

← C3×C5×#2009 #14275043

Subgroups — C3×C5×#2009 #14275043

Order 1920.

696 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#575C211yesmaximal, characteristic
2C2×C3×C5×#137C211yesmaximal, characteristic, complemented
3C3×C5×#536C211yesmaximal, characteristic, complemented
4C3×C5×#421C211yesmaximal, characteristic, complemented
5C3×C5×#419C211yesmaximal, characteristic, complemented
6C3×C5×#410C211yesmaximal, characteristic, complemented
7C3×C5×#344C211yesmaximal, characteristic, complemented
8C5×#2009C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C2×C3×C5×#41C2211yescharacteristic
11C22×C3×C5×Q8C2211yescharacteristic
12C3×C5×#137—22no—
13C3×C5×#137—21no—
14C3×C5×#137—21no—
15C3×C5×#137—22no—
16C3×C5×AGL(1,ℤ/8)—21no—
17C3×C5×AGL(1,ℤ/8)—21no—
18C3×C5×#135—22no—
19C2×C3×C5×Q16—21no—
20C2×C3×C5×SD16—21no—
21C2×C3×C5×M16C2211yescharacteristic
22C2×C3×C5×M16C2211yescharacteristic
23C3×C5×C4⋊Q8—21no—
24C3×C5×#124—21no—
25C3×C5×#123—22no—
26C3×C5×C22⋊Q8—22no—
27C3×C5×#117C2211yescharacteristic
28C3×C5×#117C2211yescharacteristic
29C3×C5×#104—22no—
30C3×C5×#103—22no—
31C3×C5×#99—22no—
32C3×#2009C511yesmaximal, characteristic, direct product factor, Hall
33C5×#575C2×C311yescharacteristic
34C2×C5×#137C2×C311yescharacteristic, complemented
35C5×#536C2×C311yescharacteristic, complemented
36C5×#421C2×C311yescharacteristic, complemented
37C5×#419C2×C311yescharacteristic, complemented
38C5×#410C2×C311yescharacteristic, complemented
39C5×#344C2×C311yescharacteristic, complemented
40C3×C5×#41—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—
48C2×C3×C5×Q8—21no—
49C2×C3×C5×Q8—21no—
50C2×C3×C5×Q8—21no—
51C2×C3×C5×Q8—21no—
52C2×C3×C5×Q8—21no—
53C2×C3×C5×Q8D412yes—
54C2×C3×C5×Q8D411yescharacteristic
55C2×C3×C5×Q8D411yescharacteristic
56C2×C3×C5×D4—21no—
57C2×C3×C5×D4—21no—
58C2×C3×C5×D4D411yescharacteristic
59C2×C3×C5×D4D411yescharacteristic
60C22×C3×C4×C5—21no—
61C22×C3×C4×C5—21no—
62C22×C3×C4×C5—21no—
63C22×C3×C4×C5C2311yescharacteristic
64C3×C5×Q16—24no—
65C3×C5×Q16—22no—
66C3×C5×SD16—24no—
67C3×C5×SD16—22no—
68C3×C5×SD16—22no—
69C3×C5×D8—22no—
70C3×C5×M16—21no—
71C3×C5×M16—21no—
72C3×C5×M16—21no—
73C3×C5×M16D412yes—
74C2×C3×C5×C8—21no—
75C2×C3×C5×C8D412yes—
76C3×C5×C4⋊C4—24no—
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
85C2×C3×#137C2×C511yescharacteristic, complemented
86C3×#536C2×C511yescharacteristic, complemented
87C3×#421C2×C511yescharacteristic, complemented
88C3×#419C2×C511yescharacteristic, complemented
89C3×#410C2×C511yescharacteristic, complemented
90C3×#344C2×C511yescharacteristic, complemented
91C2×C5×#41C22×C311yescharacteristic
92C2×C5×#41C22×C311yescharacteristic
93C22×C5×Q8C22×C311yescharacteristic
94C5×#137—22no—
95C5×#137—21no—
96C5×#137—21no—
97C5×#137—22no—
98C5×AGL(1,ℤ/8)—21no—
99C5×AGL(1,ℤ/8)—21no—
100C5×#135—22no—
101C2×C5×Q16—21no—
102C2×C5×SD16—21no—
103C2×C5×M16C22×C311yescharacteristic
104C2×C5×M16C22×C311yescharacteristic
105C5×C4⋊Q8—21no—
106C5×#124—21no—
107C5×#123—22no—
108C5×C22⋊Q8—22no—
109C5×#117C22×C311yescharacteristic
110C5×#117C22×C311yescharacteristic
111C5×#104—22no—
112C5×#103—22no—
113C5×#99—22no—
114#2009C3×C511yescharacteristic, direct product factor, Sylow
115C23×C3×C5—21no—
116C23×C3×C5—21no—
117C23×C3×C5C2×D411yescharacteristic
118C3×C5×Q8—22no—
119C3×C5×Q8—21no—
120C3×C5×Q8—21no—
121C3×C5×Q8—21no—
122C3×C5×Q8—22no—
123C3×C5×Q8—21no—
124C3×C5×Q8—41no—
125C3×C5×Q8—41no—
126C3×C5×D4—21no—
127C3×C5×D4—21no—
128C3×C5×D4—21no—
129C3×C5×D4—21no—
130C3×C5×D4—41no—
131C3×C5×D4—41no—
132C3×C5×D4—41no—
133C3×C5×D4—41no—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5—22no—
136C2×C3×C4×C5—21no—
137C2×C3×C4×C5—21no—
138C2×C3×C4×C5—41no—
139C2×C3×C4×C5—41no—
140C2×C3×C4×C5—41no—
141C2×C3×C4×C5—41no—
142C2×C3×C4×C5—41no—
143C2×C3×C4×C5—41no—
144C2×C3×C4×C5—41no—
145C2×C3×C4×C5C2×D411yescharacteristic
146C2×C3×C4×C5C2×D411yescharacteristic
147C2×C3×C4×C5C2×D411yescharacteristic
148C2×C3×C4×C5C2×D411yescharacteristic
149C2×C3×C4×C5C2×D411yescharacteristic
150C2×C3×C4×C5#4111yescharacteristic
151C3×C5×C8—22no—
152C3×C5×C8—22no—
153C2×C3×#41C22×C511yescharacteristic
154C2×C3×#41C22×C511yescharacteristic
155C22×C3×Q8C22×C511yescharacteristic
156C3×#137—22no—
157C3×#137—21no—
158C3×#137—21no—
159C3×#137—22no—
160C3×AGL(1,ℤ/8)—21no—
161C3×AGL(1,ℤ/8)—21no—
162C3×#135—22no—
163C2×C3×Q16—21no—
164C2×C3×SD16—21no—
165C2×C3×M16C22×C511yescharacteristic
166C2×C3×M16C22×C511yescharacteristic
167C3×C4⋊Q8—21no—
168C3×#124—21no—
169C3×#123—22no—
170C3×C22⋊Q8—22no—
171C3×#117C22×C511yescharacteristic
172C3×#117C22×C511yescharacteristic
173C3×#104—22no—
174C3×#103—22no—
175C3×#99—22no—
176C5×#41—21no—
177C5×#41—21no—
178C5×#41—21no—
179C5×#41—21no—
180C5×#41—21no—
181C5×#41—21no—
182C5×#41—21no—
183C5×#41—21no—
184C2×C5×Q8—21no—
185C2×C5×Q8—21no—
186C2×C5×Q8—21no—
187C2×C5×Q8—21no—
188C2×C5×Q8—21no—
189C2×C5×Q8C3×D412yes—
190C2×C5×Q8C3×D411yescharacteristic
191C2×C5×Q8C3×D411yescharacteristic
192C2×C5×D4—21no—
193C2×C5×D4—21no—
194C2×C5×D4C3×D411yescharacteristic
195C2×C5×D4C3×D411yescharacteristic
196C22×C4×C5—21no—
197C22×C4×C5—21no—
198C22×C4×C5—21no—
199C22×C4×C5C23×C311yescharacteristic
200C5×Q16—24no—
201C5×Q16—22no—
202C5×SD16—24no—
203C5×SD16—22no—
204C5×SD16—22no—
205C5×D8—22no—
206C5×M16—21no—
207C5×M16—21no—
208C5×M16—21no—
209C5×M16C3×D412yes—
210C2×C5×C8—21no—
211C2×C5×C8C3×D412yes—
212C5×C4⋊C4—24no—
213C5×C4⋊C4—21no—
214C5×C4⋊C4—21no—
215C5×C22⋊C4—24no—
216C5×C22⋊C4—21no—
217C5×C22⋊C4C3×D412yes—
218C42×C5—21no—
219C42×C5—21no—
220#575C2×C3×C511yescharacteristic
221C2×#137C2×C3×C511yescharacteristic, complemented
222#536C2×C3×C511yescharacteristic, complemented
223#421C2×C3×C511yescharacteristic, complemented
224#419C2×C3×C511yescharacteristic, complemented
225#410C2×C3×C511yescharacteristic, complemented
226#344C2×C3×C511yescharacteristic, complemented
227C22×C3×C5—41no—
228C22×C3×C5—42no—
229C22×C3×C5—41no—
230C22×C3×C5—41no—
231C22×C3×C5—81no—
232C22×C3×C5#12011yescharacteristic
233C22×C3×C5#12111yescharacteristic
234C22×C3×C5#12711yescharacteristic
235C3×C4×C5—42no—
236C3×C4×C5—41no—
237C3×C4×C5—41no—
238C3×C4×C5—42no—
239C3×C4×C5—81no—
240C3×C4×C5#12011yescharacteristic
241C3×C4×C5#12011yescharacteristic
242C3×C4×C5#12111yescharacteristic
243C3×C4×C5#12111yescharacteristic
244C3×#41—21no—
245C3×#41—21no—
246C3×#41—21no—
247C3×#41—21no—
248C3×#41—21no—
249C3×#41—21no—
250C3×#41—21no—
251C3×#41—21no—
252C2×C3×Q8—21no—
253C2×C3×Q8—21no—
254C2×C3×Q8—21no—
255C2×C3×Q8—21no—
256C2×C3×Q8—21no—
257C2×C3×Q8C5×D412yes—
258C2×C3×Q8C5×D411yescharacteristic
259C2×C3×Q8C5×D411yescharacteristic
260C2×C3×D4—21no—
261C2×C3×D4—21no—
262C2×C3×D4C5×D411yescharacteristic
263C2×C3×D4C5×D411yescharacteristic
264C22×C3×C4—21no—
265C22×C3×C4—21no—
266C22×C3×C4—21no—
267C22×C3×C4C23×C511yescharacteristic
268C3×Q16—24no—
269C3×Q16—22no—
270C3×SD16—24no—
271C3×SD16—22no—
272C3×SD16—22no—
273C3×D8—22no—
274C3×M16—21no—
275C3×M16—21no—
276C3×M16—21no—
277C3×M16C5×D412yes—
278C2×C3×C8—21no—
279C2×C3×C8C5×D412yes—
280C3×C4⋊C4—24no—
281C3×C4⋊C4—21no—
282C3×C4⋊C4—21no—
283C3×C22⋊C4—24no—
284C3×C22⋊C4—21no—
285C3×C22⋊C4C5×D412yes—
286C3×C42—21no—
287C3×C42—21no—
288C23×C5—21no—
289C23×C5—21no—
290C23×C5C2×C3×D411yescharacteristic
291C5×Q8—22no—
292C5×Q8—21no—
293C5×Q8—21no—
294C5×Q8—21no—
295C5×Q8—22no—
296C5×Q8—21no—
297C5×Q8—41no—
298C5×Q8—41no—
299C5×D4—21no—
300C5×D4—21no—
301C5×D4—21no—
302C5×D4—21no—
303C5×D4—41no—
304C5×D4—41no—
305C5×D4—41no—
306C5×D4—41no—
307C2×C4×C5—22no—
308C2×C4×C5—22no—
309C2×C4×C5—21no—
310C2×C4×C5—21no—
311C2×C4×C5—41no—
312C2×C4×C5—41no—
313C2×C4×C5—41no—
314C2×C4×C5—41no—
315C2×C4×C5—41no—
316C2×C4×C5—41no—
317C2×C4×C5—41no—
318C2×C4×C5C2×C3×D411yescharacteristic
319C2×C4×C5C2×C3×D411yescharacteristic
320C2×C4×C5C2×C3×D411yescharacteristic
321C2×C4×C5C2×C3×D411yescharacteristic
322C2×C4×C5C2×C3×D411yescharacteristic
323C2×C4×C5C3×#4111yescharacteristic
324C5×C8—22no—
325C5×C8—22no—
326C2×#41C22×C3×C511yescharacteristic
327C2×#41C22×C3×C511yescharacteristic
328C22×Q8C22×C3×C511yescharacteristic
329#137—22no—
330#137—21no—
331#137—21no—
332#137—22no—
333AGL(1,ℤ/8)—21no—
334AGL(1,ℤ/8)—21no—
335#135—22no—
336C2×Q16—21no—
337C2×SD16—21no—
338C2×M16C22×C3×C511yescharacteristic
339C2×M16C22×C3×C511yescharacteristic
340C4⋊Q8—21no—
341#124—21no—
342#123—22no—
343C22⋊Q8—22no—
344#117C22×C3×C511yescharacteristic
345#117C22×C3×C511yescharacteristic
346#104—22no—
347#103—22no—
348#99—22no—
349C2×C3×C5—21no—
350C2×C3×C5—21no—
351C2×C3×C5—21no—
352C2×C3×C5—81no—
353C2×C3×C5—81no—
354C2×C3×C5#39211yescharacteristic
355C23×C3—21no—
356C23×C3—21no—
357C23×C3C2×C5×D411yescharacteristic
358C3×Q8—22no—
359C3×Q8—21no—
360C3×Q8—21no—
361C3×Q8—21no—
362C3×Q8—22no—
363C3×Q8—21no—
364C3×Q8—41no—
365C3×Q8—41no—
366C3×D4—21no—
367C3×D4—21no—
368C3×D4—21no—
369C3×D4—21no—
370C3×D4—41no—
371C3×D4—41no—
372C3×D4—41no—
373C3×D4—41no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4C2×C5×D411yescharacteristic
386C2×C3×C4C2×C5×D411yescharacteristic
387C2×C3×C4C2×C5×D411yescharacteristic
388C2×C3×C4C2×C5×D411yescharacteristic
389C2×C3×C4C2×C5×D411yescharacteristic
390C2×C3×C4C5×#4111yescharacteristic
391C3×C8—22no—
392C3×C8—22no—
393C22×C5—41no—
394C22×C5—42no—
395C22×C5—41no—
396C22×C5—41no—
397C22×C5—81no—
398C22×C5C3×#12011yescharacteristic
399C22×C5C3×#12111yescharacteristic
400C22×C5C3×#12711yescharacteristic
401C4×C5—42no—
402C4×C5—41no—
403C4×C5—41no—
404C4×C5—42no—
405C4×C5—81no—
406C4×C5C3×#12011yescharacteristic
407C4×C5C3×#12011yescharacteristic
408C4×C5C3×#12111yescharacteristic
409C4×C5C3×#12111yescharacteristic
410#41—21no—
411#41—21no—
412#41—21no—
413#41—21no—
414#41—21no—
415#41—21no—
416#41—21no—
417#41—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8C3×C5×D412yes—
424C2×Q8C3×C5×D411yescharacteristic
425C2×Q8C3×C5×D411yescharacteristic
426C2×D4—21no—
427C2×D4—21no—
428C2×D4C3×C5×D411yescharacteristic
429C2×D4C3×C5×D411yescharacteristic
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4C23×C3×C511yescharacteristic
434Q16—24no—
435Q16—22no—
436SD16—24no—
437SD16—22no—
438SD16—22no—
439D8—22no—
440M16—21no—
441M16—21no—
442M16—21no—
443M16C3×C5×D412yes—
444C2×C8—21no—
445C2×C8C3×C5×D412yes—
446C4⋊C4—24no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C22⋊C4—24no—
450C22⋊C4—21no—
451C22⋊C4C3×C5×D412yes—
452C42—21no—
453C42—21no—
454C3×C5#200911yescharacteristic, direct product factor, Hall
455C22×C3—41no—
456C22×C3—42no—
457C22×C3—41no—
458C22×C3—41no—
459C22×C3—81no—
460C22×C3C5×#12011yescharacteristic
461C22×C3C5×#12111yescharacteristic
462C22×C3C5×#12711yescharacteristic
463C3×C4—42no—
464C3×C4—41no—
465C3×C4—41no—
466C3×C4—42no—
467C3×C4—81no—
468C3×C4C5×#12011yescharacteristic
469C3×C4C5×#12011yescharacteristic
470C3×C4C5×#12111yescharacteristic
471C3×C4C5×#12111yescharacteristic
472C2×C5—21no—
473C2×C5—21no—
474C2×C5—21no—
475C2×C5—81no—
476C2×C5—81no—
477C2×C5C3×#39211yescharacteristic
478C23—21no—
479C23—21no—
480C23C2×C3×C5×D411yescharacteristic
481Q8—22no—
482Q8—21no—
483Q8—21no—
484Q8—21no—
485Q8—22no—
486Q8—21no—
487Q8—41no—
488Q8—41no—
489D4—21no—
490D4—21no—
491D4—21no—
492D4—21no—
493D4—41no—
494D4—41no—
495D4—41no—
496D4—41no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4C2×C3×C5×D411yescharacteristic
509C2×C4C2×C3×C5×D411yescharacteristic
510C2×C4C2×C3×C5×D411yescharacteristic
511C2×C4C2×C3×C5×D411yescharacteristic
512C2×C4C2×C3×C5×D411yescharacteristic
513C2×C4C3×C5×#4111yescharacteristic
514C8—22no—
515C8—22no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—81no—
520C2×C3—81no—
521C2×C3C5×#39211yescharacteristic
522C5C3×#200911yescharacteristic, direct product factor, Sylow
523C22—41no—
524C22—42no—
525C22—41no—
526C22—41no—
527C22—81no—
528C22C3×C5×#12011yescharacteristic
529C22C3×C5×#12111yescharacteristic
530C22C3×C5×#12711yescharacteristic
531C4—42no—
532C4—41no—
533C4—41no—
534C4—42no—
535C4—81no—
536C4C3×C5×#12011yescharacteristic
537C4C3×C5×#12011yescharacteristic
538C4C3×C5×#12111yescharacteristic
539C4C3×C5×#12111yescharacteristic
540C3C5×#200911yescharacteristic, direct product factor, Sylow
541C2—21no—
542C2—21no—
543C2—21no—
544C2—81no—
545C2—81no—
546C2C3×C5×#39211yescharacteristic
subgroupG/H
center, socleC2×C3×C5#392
commutator, Frattini, cosocleC22×C4C23×C3×C5
prime radicalC3×C5×#575C2
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
2. C2×C3×C5×#137C2semidirect
3. C3×C5×#536C2semidirect
4. C3×C5×#421C2semidirect
5. C3×C5×#419C2semidirect
6. C3×C5×#410C2semidirect
7. C3×C5×#344C2semidirect
8. C5×#2009C3direct (N × H)
32. C3×#2009C5direct (N × H)
34. C2×C5×#137C2×C3semidirect
35. C5×#536C2×C3semidirect
36. C5×#421C2×C3semidirect
37. C5×#419C2×C3semidirect
38. C5×#410C2×C3semidirect
39. C5×#344C2×C3semidirect
85. C2×C3×#137C2×C5semidirect
86. C3×#536C2×C5semidirect
87. C3×#421C2×C5semidirect
88. C3×#419C2×C5semidirect
89. C3×#410C2×C5semidirect
90. C3×#344C2×C5semidirect
114. #2009C3×C5direct (N × H)
221. C2×#137C2×C3×C5semidirect
222. #536C2×C3×C5semidirect
223. #421C2×C3×C5semidirect
224. #419C2×C3×C5semidirect
225. #410C2×C3×C5semidirect
226. #344C2×C3×C5semidirect
454. C3×C5#2009direct (N × H)
522. C5C3×#2009direct (N × H)
540. C3C5×#2009direct (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×#1372
C3×C5×#5362
C3×C5×#4212
C3×C5×#4192
C3×C5×#4102
C3×C5×#3442
C5×#20093
C3×#20095