Finite Group Databasebeta

← C3×C5×#2004 #14275038

Subgroups — C3×C5×#2004 #14275038

Order 1920.

788 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#137C211yesmaximal, characteristic, complemented
2C3×C5×#535C211yesmaximal, characteristic
3C2×C3×C5×#124C211yesmaximal, characteristic, complemented
4C2×C3×C5×#104C211yesmaximal, characteristic, complemented
5C3×C5×#419C211yesmaximal, characteristic, complemented
6C2×C3×C5×#100C211yesmaximal, characteristic, complemented
7C3×C5×#339C211yesmaximal, characteristic, complemented
8C5×#2004C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C22×C3×C5×Q8C2211yescharacteristic
11C22×C3×C5×D4C2211yescharacteristic
12C3×C5×#137—22no—
13C3×C5×#137—22no—
14C2×C3×C5×Q16—21no—
15C2×C3×C5×SD16—21no—
16C2×C3×C5×M16C2211yescharacteristic
17C2×C3×C5×M16C2211yescharacteristic
18C3×C5×#127—21no—
19C3×C5×#124—21no—
20C3×C5×#124—22no—
21C3×C5×#124—22no—
22C3×C5×#121—22no—
23C3×C5×#120—22no—
24C3×C5×#117C2211yescharacteristic
25C2×C3×C5×C22⋊C4—22no—
26C2×C3×C42×C5C2211yescharacteristic
27C3×C5×#104—22no—
28C3×C5×#103—22no—
29C3×C5×#100—22no—
30C3×#2004C511yesmaximal, characteristic, direct product factor, Hall
31C2×C5×#137C2×C311yescharacteristic, complemented
32C5×#535C2×C311yescharacteristic
33C2×C5×#124C2×C311yescharacteristic, complemented
34C2×C5×#104C2×C311yescharacteristic, complemented
35C5×#419C2×C311yescharacteristic, complemented
36C2×C5×#100C2×C311yescharacteristic, complemented
37C5×#339C2×C311yescharacteristic, complemented
38C24×C3×C5D412yes—
39C3×C5×#41—21no—
40C3×C5×#41—21no—
41C3×C5×#41—21no—
42C3×C5×#41—21no—
43C2×C3×C5×Q8—21no—
44C2×C3×C5×Q8—21no—
45C2×C3×C5×Q8—21no—
46C2×C3×C5×Q8—21no—
47C2×C3×C5×Q8—21no—
48C2×C3×C5×Q8D411yescharacteristic
49C2×C3×C5×Q8D412yescomplemented
50C2×C3×C5×D4—21no—
51C2×C3×C5×D4—24no—
52C2×C3×C5×D4—21no—
53C2×C3×C5×D4—21no—
54C2×C3×C5×D4—21no—
55C2×C3×C5×D4—21no—
56C2×C3×C5×D4—21no—
57C2×C3×C5×D4—21no—
58C2×C3×C5×D4D411yescharacteristic
59C22×C3×C4×C5—21no—
60C22×C3×C4×C5—21no—
61C22×C3×C4×C5—21no—
62C22×C3×C4×C5C2311yescharacteristic
63C3×C5×Q16—24no—
64C3×C5×SD16—24no—
65C3×C5×M16—21no—
66C3×C5×M16—21no—
67C3×C5×M16—21no—
68C3×C5×M16D412yescomplemented
69C2×C3×C5×C8—21no—
70C2×C3×C5×C8—21no—
71C3×C5×C4⋊C4D412yes—
72C3×C5×C22⋊C4—28no—
73C3×C5×C22⋊C4—24no—
74C3×C5×C22⋊C4—21no—
75C3×C42×C5—21no—
76C3×C42×C5—21no—
77C3×C42×C5D412yescomplemented
78C2×C3×#137C2×C511yescharacteristic, complemented
79C3×#535C2×C511yescharacteristic
80C2×C3×#124C2×C511yescharacteristic, complemented
81C2×C3×#104C2×C511yescharacteristic, complemented
82C3×#419C2×C511yescharacteristic, complemented
83C2×C3×#100C2×C511yescharacteristic, complemented
84C3×#339C2×C511yescharacteristic, complemented
85C2×C5×#41C22×C311yescharacteristic
86C22×C5×Q8C22×C311yescharacteristic
87C22×C5×D4C22×C311yescharacteristic
88C5×#137—22no—
89C5×#137—22no—
90C2×C5×Q16—21no—
91C2×C5×SD16—21no—
92C2×C5×M16C22×C311yescharacteristic
93C2×C5×M16C22×C311yescharacteristic
94C5×#127—21no—
95C5×#124—21no—
96C5×#124—22no—
97C5×#124—22no—
98C5×#121—22no—
99C5×#120—22no—
100C5×#117C22×C311yescharacteristic
101C2×C5×C22⋊C4—22no—
102C2×C42×C5C22×C311yescharacteristic
103C5×#104—22no—
104C5×#103—22no—
105C5×#100—22no—
106#2004C3×C511yescharacteristic, direct product factor, Sylow
107C23×C3×C5—22no—
108C23×C3×C5—22no—
109C23×C3×C5—21no—
110C23×C3×C5—22no—
111C23×C3×C5—44no—
112C23×C3×C5C2×D411yescharacteristic
113C3×C5×Q8—22no—
114C3×C5×Q8—21no—
115C3×C5×Q8—22no—
116C3×C5×Q8—21no—
117C3×C5×Q8—41no—
118C3×C5×Q8—41no—
119C3×C5×D4—21no—
120C3×C5×D4—21no—
121C3×C5×D4—41no—
122C3×C5×D4—41no—
123C3×C5×D4—41no—
124C3×C5×D4—41no—
125C3×C5×D4—41no—
126C3×C5×D4—41no—
127C3×C5×D4—84no—
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5—22no—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5—21no—
132C2×C3×C4×C5—42no—
133C2×C3×C4×C5—41no—
134C2×C3×C4×C5—41no—
135C2×C3×C4×C5—41no—
136C2×C3×C4×C5—41no—
137C2×C3×C4×C5C2×D411yescharacteristic
138C2×C3×C4×C5C2×D411yescharacteristic
139C2×C3×C4×C5C2×D411yescharacteristic
140C2×C3×C4×C5C2×D411yescharacteristic
141C2×C3×C4×C5C2×D411yescharacteristic
142C2×C3×C4×C5#4111yescharacteristic
143C3×C5×C8—22no—
144C3×C5×C8—41no—
145C2×C3×#41C22×C511yescharacteristic
146C22×C3×Q8C22×C511yescharacteristic
147C22×C3×D4C22×C511yescharacteristic
148C3×#137—22no—
149C3×#137—22no—
150C2×C3×Q16—21no—
151C2×C3×SD16—21no—
152C2×C3×M16C22×C511yescharacteristic
153C2×C3×M16C22×C511yescharacteristic
154C3×#127—21no—
155C3×#124—21no—
156C3×#124—22no—
157C3×#124—22no—
158C3×#121—22no—
159C3×#120—22no—
160C3×#117C22×C511yescharacteristic
161C2×C3×C22⋊C4—22no—
162C2×C3×C42C22×C511yescharacteristic
163C3×#104—22no—
164C3×#103—22no—
165C3×#100—22no—
166C24×C5C3×D412yes—
167C5×#41—21no—
168C5×#41—21no—
169C5×#41—21no—
170C5×#41—21no—
171C2×C5×Q8—21no—
172C2×C5×Q8—21no—
173C2×C5×Q8—21no—
174C2×C5×Q8—21no—
175C2×C5×Q8—21no—
176C2×C5×Q8C3×D411yescharacteristic
177C2×C5×Q8C3×D412yescomplemented
178C2×C5×D4—21no—
179C2×C5×D4—24no—
180C2×C5×D4—21no—
181C2×C5×D4—21no—
182C2×C5×D4—21no—
183C2×C5×D4—21no—
184C2×C5×D4—21no—
185C2×C5×D4—21no—
186C2×C5×D4C3×D411yescharacteristic
187C22×C4×C5—21no—
188C22×C4×C5—21no—
189C22×C4×C5—21no—
190C22×C4×C5C23×C311yescharacteristic
191C5×Q16—24no—
192C5×SD16—24no—
193C5×M16—21no—
194C5×M16—21no—
195C5×M16—21no—
196C5×M16C3×D412yescomplemented
197C2×C5×C8—21no—
198C2×C5×C8—21no—
199C5×C4⋊C4C3×D412yes—
200C5×C22⋊C4—28no—
201C5×C22⋊C4—24no—
202C5×C22⋊C4—21no—
203C42×C5—21no—
204C42×C5—21no—
205C42×C5C3×D412yescomplemented
206C2×#137C2×C3×C511yescharacteristic, complemented
207#535C2×C3×C511yescharacteristic
208C2×#124C2×C3×C511yescharacteristic, complemented
209C2×#104C2×C3×C511yescharacteristic, complemented
210#419C2×C3×C511yescharacteristic, complemented
211C2×#100C2×C3×C511yescharacteristic, complemented
212#339C2×C3×C511yescharacteristic, complemented
213C22×C3×C5—22no—
214C22×C3×C5—44no—
215C22×C3×C5—41no—
216C22×C3×C5—42no—
217C22×C3×C5—42no—
218C22×C3×C5—44no—
219C22×C3×C5—82no—
220C22×C3×C5#12011yescharacteristic
221C22×C3×C5#12011yescharacteristic
222C22×C3×C5#12111yescharacteristic
223C3×C4×C5—42no—
224C3×C4×C5—41no—
225C3×C4×C5—42no—
226C3×C4×C5—42no—
227C3×C4×C5#12011yescharacteristic
228C3×C4×C5#12111yescharacteristic
229C3×C4×C5#12111yescharacteristic
230C3×C4×C5#12711yescharacteristic
231C24×C3C5×D412yes—
232C3×#41—21no—
233C3×#41—21no—
234C3×#41—21no—
235C3×#41—21no—
236C2×C3×Q8—21no—
237C2×C3×Q8—21no—
238C2×C3×Q8—21no—
239C2×C3×Q8—21no—
240C2×C3×Q8—21no—
241C2×C3×Q8C5×D411yescharacteristic
242C2×C3×Q8C5×D412yescomplemented
243C2×C3×D4—21no—
244C2×C3×D4—24no—
245C2×C3×D4—21no—
246C2×C3×D4—21no—
247C2×C3×D4—21no—
248C2×C3×D4—21no—
249C2×C3×D4—21no—
250C2×C3×D4—21no—
251C2×C3×D4C5×D411yescharacteristic
252C22×C3×C4—21no—
253C22×C3×C4—21no—
254C22×C3×C4—21no—
255C22×C3×C4C23×C511yescharacteristic
256C3×Q16—24no—
257C3×SD16—24no—
258C3×M16—21no—
259C3×M16—21no—
260C3×M16—21no—
261C3×M16C5×D412yescomplemented
262C2×C3×C8—21no—
263C2×C3×C8—21no—
264C3×C4⋊C4C5×D412yes—
265C3×C22⋊C4—28no—
266C3×C22⋊C4—24no—
267C3×C22⋊C4—21no—
268C3×C42—21no—
269C3×C42—21no—
270C3×C42C5×D412yescomplemented
271C23×C5—22no—
272C23×C5—22no—
273C23×C5—21no—
274C23×C5—22no—
275C23×C5—44no—
276C23×C5C2×C3×D411yescharacteristic
277C5×Q8—22no—
278C5×Q8—21no—
279C5×Q8—22no—
280C5×Q8—21no—
281C5×Q8—41no—
282C5×Q8—41no—
283C5×D4—21no—
284C5×D4—21no—
285C5×D4—41no—
286C5×D4—41no—
287C5×D4—41no—
288C5×D4—41no—
289C5×D4—41no—
290C5×D4—41no—
291C5×D4—84no—
292C2×C4×C5—22no—
293C2×C4×C5—22no—
294C2×C4×C5—21no—
295C2×C4×C5—22no—
296C2×C4×C5—42no—
297C2×C4×C5—41no—
298C2×C4×C5—41no—
299C2×C4×C5—41no—
300C2×C4×C5—41no—
301C2×C4×C5C2×C3×D411yescharacteristic
302C2×C4×C5C2×C3×D411yescharacteristic
303C2×C4×C5C2×C3×D411yescharacteristic
304C2×C4×C5C2×C3×D411yescharacteristic
305C2×C4×C5C2×C3×D411yescharacteristic
306C2×C4×C5C3×#4111yescharacteristic
307C5×C8—22no—
308C5×C8—41no—
309C2×#41C22×C3×C511yescharacteristic
310C22×Q8C22×C3×C511yescharacteristic
311C22×D4C22×C3×C511yescharacteristic
312#137—22no—
313#137—22no—
314C2×Q16—21no—
315C2×SD16—21no—
316C2×M16C22×C3×C511yescharacteristic
317C2×M16C22×C3×C511yescharacteristic
318#127—21no—
319#124—21no—
320#124—22no—
321#124—22no—
322#121—22no—
323#120—22no—
324#117C22×C3×C511yescharacteristic
325C2×C22⋊C4—22no—
326C2×C42C22×C3×C511yescharacteristic
327#104—22no—
328#103—22no—
329#100—22no—
330C2×C3×C5—21no—
331C2×C3×C5—21no—
332C2×C3×C5—81no—
333C2×C3×C5—82no—
334C2×C3×C5#39211yescharacteristic
335C2×C3×C5#45512yes—
336C23×C3—22no—
337C23×C3—22no—
338C23×C3—21no—
339C23×C3—22no—
340C23×C3—44no—
341C23×C3C2×C5×D411yescharacteristic
342C3×Q8—22no—
343C3×Q8—21no—
344C3×Q8—22no—
345C3×Q8—21no—
346C3×Q8—41no—
347C3×Q8—41no—
348C3×D4—21no—
349C3×D4—21no—
350C3×D4—41no—
351C3×D4—41no—
352C3×D4—41no—
353C3×D4—41no—
354C3×D4—41no—
355C3×D4—41no—
356C3×D4—84no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—42no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4C2×C5×D411yescharacteristic
367C2×C3×C4C2×C5×D411yescharacteristic
368C2×C3×C4C2×C5×D411yescharacteristic
369C2×C3×C4C2×C5×D411yescharacteristic
370C2×C3×C4C2×C5×D411yescharacteristic
371C2×C3×C4C5×#4111yescharacteristic
372C3×C8—22no—
373C3×C8—41no—
374C22×C5—22no—
375C22×C5—44no—
376C22×C5—41no—
377C22×C5—42no—
378C22×C5—42no—
379C22×C5—44no—
380C22×C5—82no—
381C22×C5C3×#12011yescharacteristic
382C22×C5C3×#12011yescharacteristic
383C22×C5C3×#12111yescharacteristic
384C4×C5—42no—
385C4×C5—41no—
386C4×C5—42no—
387C4×C5—42no—
388C4×C5C3×#12011yescharacteristic
389C4×C5C3×#12111yescharacteristic
390C4×C5C3×#12111yescharacteristic
391C4×C5C3×#12711yescharacteristic
392C24C3×C5×D412yes—
393#41—21no—
394#41—21no—
395#41—21no—
396#41—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8C3×C5×D411yescharacteristic
403C2×Q8C3×C5×D412yescomplemented
404C2×D4—21no—
405C2×D4—24no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4C3×C5×D411yescharacteristic
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4C23×C3×C511yescharacteristic
417Q16—24no—
418SD16—24no—
419M16—21no—
420M16—21no—
421M16—21no—
422M16C3×C5×D412yescomplemented
423C2×C8—21no—
424C2×C8—21no—
425C4⋊C4C3×C5×D412yes—
426C22⋊C4—28no—
427C22⋊C4—24no—
428C22⋊C4—21no—
429C42—21no—
430C42—21no—
431C42C3×C5×D412yescomplemented
432C3×C5#200411yescharacteristic, direct product factor, Hall
433C22×C3—22no—
434C22×C3—44no—
435C22×C3—41no—
436C22×C3—42no—
437C22×C3—42no—
438C22×C3—44no—
439C22×C3—82no—
440C22×C3C5×#12011yescharacteristic
441C22×C3C5×#12011yescharacteristic
442C22×C3C5×#12111yescharacteristic
443C3×C4—42no—
444C3×C4—41no—
445C3×C4—42no—
446C3×C4—42no—
447C3×C4C5×#12011yescharacteristic
448C3×C4C5×#12111yescharacteristic
449C3×C4C5×#12111yescharacteristic
450C3×C4C5×#12711yescharacteristic
451C2×C5—21no—
452C2×C5—21no—
453C2×C5—82no—
454C2×C5—81no—
455C2×C5C3×#39211yescharacteristic
456C2×C5C3×#45512yes—
457C23—22no—
458C23—22no—
459C23—21no—
460C23—22no—
461C23—44no—
462C23C2×C3×C5×D411yescharacteristic
463Q8—22no—
464Q8—21no—
465Q8—22no—
466Q8—21no—
467Q8—41no—
468Q8—41no—
469D4—21no—
470D4—21no—
471D4—41no—
472D4—41no—
473D4—41no—
474D4—41no—
475D4—41no—
476D4—41no—
477D4—84no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4C2×C3×C5×D411yescharacteristic
488C2×C4C2×C3×C5×D411yescharacteristic
489C2×C4C2×C3×C5×D411yescharacteristic
490C2×C4C2×C3×C5×D411yescharacteristic
491C2×C4C2×C3×C5×D411yescharacteristic
492C2×C4C3×C5×#4111yescharacteristic
493C8—22no—
494C8—41no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—82no—
498C2×C3—81no—
499C2×C3C5×#39211yescharacteristic
500C2×C3C5×#45512yes—
501C5C3×#200411yescharacteristic, direct product factor, Sylow
502C22—22no—
503C22—44no—
504C22—42no—
505C22—44no—
506C22—41no—
507C22—42no—
508C22—82no—
509C22C3×C5×#12011yescharacteristic
510C22C3×C5×#12011yescharacteristic
511C22C3×C5×#12111yescharacteristic
512C4—42no—
513C4—42no—
514C4—41no—
515C4—42no—
516C4C3×C5×#12011yescharacteristic
517C4C3×C5×#12111yescharacteristic
518C4C3×C5×#12111yescharacteristic
519C4C3×C5×#12711yescharacteristic
520C3C5×#200411yescharacteristic, direct product factor, Sylow
521C2—21no—
522C2—21no—
523C2—82no—
524C2—81no—
525C2C3×C5×#39211yescharacteristic
526C2C3×C5×#45512yes—
subgroupG/H
center, socleC22×C3×C5#120
commutator, Frattini, cosocleC22×C4C23×C3×C5
prime radicalC3×C5×#535C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3×C5 ◁ C22×C3×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×#137C2semidirect
3. C2×C3×C5×#124C2semidirect
4. C2×C3×C5×#104C2semidirect
5. C3×C5×#419C2semidirect
6. C2×C3×C5×#100C2semidirect
7. C3×C5×#339C2semidirect
8. C5×#2004C3direct (N × H)
30. C3×#2004C5direct (N × H)
31. C2×C5×#137C2×C3semidirect
33. C2×C5×#124C2×C3semidirect
34. C2×C5×#104C2×C3semidirect
35. C5×#419C2×C3semidirect
36. C2×C5×#100C2×C3semidirect
37. C5×#339C2×C3semidirect
49. C2×C3×C5×Q8D4semidirect
68. C3×C5×M16D4semidirect
77. C3×C42×C5D4semidirect
78. C2×C3×#137C2×C5semidirect
80. C2×C3×#124C2×C5semidirect
81. C2×C3×#104C2×C5semidirect
82. C3×#419C2×C5semidirect
83. C2×C3×#100C2×C5semidirect
84. C3×#339C2×C5semidirect
106. #2004C3×C5direct (N × H)
177. C2×C5×Q8C3×D4semidirect
196. C5×M16C3×D4semidirect
205. C42×C5C3×D4semidirect
206. C2×#137C2×C3×C5semidirect
208. C2×#124C2×C3×C5semidirect
209. C2×#104C2×C3×C5semidirect
210. #419C2×C3×C5semidirect
211. C2×#100C2×C3×C5semidirect
212. #339C2×C3×C5semidirect
242. C2×C3×Q8C5×D4semidirect
261. C3×M16C5×D4semidirect
270. C3×C42C5×D4semidirect
403. C2×Q8C3×C5×D4semidirect
422. M16C3×C5×D4semidirect
431. C42C3×C5×D4semidirect
432. C3×C5#2004direct (N × H)
501. C5C3×#2004direct (N × H)
520. C3C5×#2004direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×#1372
C3×C5×#5352
C2×C3×C5×#1242
C2×C3×C5×#1042
C3×C5×#4192
C2×C3×C5×#1002
C3×C5×#3392
C5×#20043
C3×#20045