Finite Group Databasebeta

← C3×C5×#2015 #14275049

Subgroups — C3×C5×#2015 #14275049

Order 1920.

780 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C5×Q16C211yesmaximal, characteristic, complemented
2C22×C3×C5×SD16C211yesmaximal, characteristic
3C2×C3×C5×#124C211yesmaximal, characteristic
4C2×C3×C5×#103C211yesmaximal, characteristic, complemented
5C2×C3×C5×#102C211yesmaximal, characteristic
6C3×C5×#391C211yesmaximal, characteristic, complemented
7C3×C5×#336C211yesmaximal, characteristic, complemented
8C5×#2015C311yesmaximal, characteristic, direct product factor, Hall
9C22×C3×C5×Q8C2211yescharacteristic
10C22×C3×C5×Q8C2211yescharacteristic
11C22×C3×C5×D4C2211yescharacteristic
12C2×C3×C5×Q16—22no—
13C2×C3×C5×Q16—22no—
14C2×C3×C5×Q16—22no—
15C2×C3×C5×SD16—22no—
16C2×C3×C5×SD16—22no—
17C2×C3×C5×SD16—22no—
18C22×C3×C5×C8C2211yescharacteristic
19C22×C3×C5×C8C2211yescharacteristic
20C3×C5×#124—22no—
21C3×C5×#124—22no—
22C2×C3×C5×C4⋊C4C2211yescharacteristic
23C2×C3×C5×C22⋊C4—22no—
24C2×C3×C42×C5C2211yescharacteristic
25C3×C5×#103—22no—
26C3×C5×#102—22no—
27C3×C5×#95—22no—
28C3×#2015C511yesmaximal, characteristic, direct product factor, Hall
29C22×C5×Q16C2×C311yescharacteristic, complemented
30C22×C5×SD16C2×C311yescharacteristic
31C2×C5×#124C2×C311yescharacteristic
32C2×C5×#103C2×C311yescharacteristic, complemented
33C2×C5×#102C2×C311yescharacteristic
34C5×#391C2×C311yescharacteristic, complemented
35C5×#336C2×C311yescharacteristic, complemented
36C24×C3×C5—21no—
37C2×C3×C5×Q8—21no—
38C2×C3×C5×Q8—21no—
39C2×C3×C5×Q8—21no—
40C2×C3×C5×Q8—21no—
41C2×C3×C5×Q8—21no—
42C2×C3×C5×Q8—21no—
43C2×C3×C5×Q8—21no—
44C2×C3×C5×Q8—21no—
45C2×C3×C5×Q8—21no—
46C2×C3×C5×Q8—21no—
47C2×C3×C5×Q8D412yes—
48C2×C3×C5×Q8D412yes—
49C2×C3×C5×D4—21no—
50C2×C3×C5×D4—21no—
51C2×C3×C5×D4—21no—
52C2×C3×C5×D4—21no—
53C2×C3×C5×D4—21no—
54C2×C3×C5×D4D412yes—
55C22×C3×C4×C5—21no—
56C22×C3×C4×C5—21no—
57C22×C3×C4×C5—21no—
58C22×C3×C4×C5D412yes—
59C22×C3×C4×C5C2311yescharacteristic
60C3×C5×Q16—28no—
61C3×C5×SD16—28no—
62C2×C3×C5×C8—21no—
63C2×C3×C5×C8—21no—
64C2×C3×C5×C8—21no—
65C2×C3×C5×C8—21no—
66C2×C3×C5×C8D412yes—
67C2×C3×C5×C8D412yes—
68C3×C5×C4⋊C4—21no—
69C3×C5×C4⋊C4—21no—
70C3×C5×C22⋊C4—28no—
71C3×C42×C5—21no—
72C3×C42×C5—21no—
73C22×C3×Q16C2×C511yescharacteristic, complemented
74C22×C3×SD16C2×C511yescharacteristic
75C2×C3×#124C2×C511yescharacteristic
76C2×C3×#103C2×C511yescharacteristic, complemented
77C2×C3×#102C2×C511yescharacteristic
78C3×#391C2×C511yescharacteristic, complemented
79C3×#336C2×C511yescharacteristic, complemented
80C22×C5×Q8C22×C311yescharacteristic
81C22×C5×Q8C22×C311yescharacteristic
82C22×C5×D4C22×C311yescharacteristic
83C2×C5×Q16—22no—
84C2×C5×Q16—22no—
85C2×C5×Q16—22no—
86C2×C5×SD16—22no—
87C2×C5×SD16—22no—
88C2×C5×SD16—22no—
89C22×C5×C8C22×C311yescharacteristic
90C22×C5×C8C22×C311yescharacteristic
91C5×#124—22no—
92C5×#124—22no—
93C2×C5×C4⋊C4C22×C311yescharacteristic
94C2×C5×C22⋊C4—22no—
95C2×C42×C5C22×C311yescharacteristic
96C5×#103—22no—
97C5×#102—22no—
98C5×#95—22no—
99#2015C3×C511yescharacteristic, direct product factor, Sylow
100C23×C3×C5—22no—
101C23×C3×C5—41no—
102C23×C3×C5—41no—
103C23×C3×C5—42no—
104C23×C3×C5—42no—
105C23×C3×C5C2×D411yescharacteristic
106C3×C5×Q8—22no—
107C3×C5×Q8—22no—
108C3×C5×Q8—22no—
109C3×C5×Q8—22no—
110C3×C5×Q8—41no—
111C3×C5×Q8—41no—
112C3×C5×Q8—41no—
113C3×C5×Q8—41no—
114C3×C5×D4—22no—
115C3×C5×D4—22no—
116C3×C5×D4—41no—
117C3×C5×D4—41no—
118C2×C3×C4×C5—22no—
119C2×C3×C4×C5—22no—
120C2×C3×C4×C5—22no—
121C2×C3×C4×C5—22no—
122C2×C3×C4×C5—22no—
123C2×C3×C4×C5—41no—
124C2×C3×C4×C5—41no—
125C2×C3×C4×C5—41no—
126C2×C3×C4×C5—41no—
127C2×C3×C4×C5—41no—
128C2×C3×C4×C5—42no—
129C2×C3×C4×C5C2×D411yescharacteristic
130C2×C3×C4×C5C2×D411yescharacteristic
131C2×C3×C4×C5C2×D411yescharacteristic
132C2×C3×C4×C5C2×D411yescharacteristic
133C2×C3×C4×C5C2×D411yescharacteristic
134C2×C3×C4×C5#4111yescharacteristic
135C3×C5×C8—22no—
136C3×C5×C8—22no—
137C22×C3×Q8C22×C511yescharacteristic
138C22×C3×Q8C22×C511yescharacteristic
139C22×C3×D4C22×C511yescharacteristic
140C2×C3×Q16—22no—
141C2×C3×Q16—22no—
142C2×C3×Q16—22no—
143C2×C3×SD16—22no—
144C2×C3×SD16—22no—
145C2×C3×SD16—22no—
146C22×C3×C8C22×C511yescharacteristic
147C22×C3×C8C22×C511yescharacteristic
148C3×#124—22no—
149C3×#124—22no—
150C2×C3×C4⋊C4C22×C511yescharacteristic
151C2×C3×C22⋊C4—22no—
152C2×C3×C42C22×C511yescharacteristic
153C3×#103—22no—
154C3×#102—22no—
155C3×#95—22no—
156C24×C5—21no—
157C2×C5×Q8—21no—
158C2×C5×Q8—21no—
159C2×C5×Q8—21no—
160C2×C5×Q8—21no—
161C2×C5×Q8—21no—
162C2×C5×Q8—21no—
163C2×C5×Q8—21no—
164C2×C5×Q8—21no—
165C2×C5×Q8—21no—
166C2×C5×Q8—21no—
167C2×C5×Q8C3×D412yes—
168C2×C5×Q8C3×D412yes—
169C2×C5×D4—21no—
170C2×C5×D4—21no—
171C2×C5×D4—21no—
172C2×C5×D4—21no—
173C2×C5×D4—21no—
174C2×C5×D4C3×D412yes—
175C22×C4×C5—21no—
176C22×C4×C5—21no—
177C22×C4×C5—21no—
178C22×C4×C5C3×D412yes—
179C22×C4×C5C23×C311yescharacteristic
180C5×Q16—28no—
181C5×SD16—28no—
182C2×C5×C8—21no—
183C2×C5×C8—21no—
184C2×C5×C8—21no—
185C2×C5×C8—21no—
186C2×C5×C8C3×D412yes—
187C2×C5×C8C3×D412yes—
188C5×C4⋊C4—21no—
189C5×C4⋊C4—21no—
190C5×C22⋊C4—28no—
191C42×C5—21no—
192C42×C5—21no—
193C22×Q16C2×C3×C511yescharacteristic, complemented
194C22×SD16C2×C3×C511yescharacteristic
195C2×#124C2×C3×C511yescharacteristic
196C2×#103C2×C3×C511yescharacteristic, complemented
197C2×#102C2×C3×C511yescharacteristic
198#391C2×C3×C511yescharacteristic, complemented
199#336C2×C3×C511yescharacteristic, complemented
200C22×C3×C5—42no—
201C22×C3×C5—44no—
202C22×C3×C5—82no—
203C22×C3×C5—81no—
204C22×C3×C5—81no—
205C22×C3×C5#12011yescharacteristic
206C22×C3×C5#12111yescharacteristic
207C22×C3×C5#12711yescharacteristic
208C22×C3×C5#13512yes—
209C22×C3×C5#13712yes—
210C3×C4×C5—42no—
211C3×C4×C5—42no—
212C3×C4×C5—42no—
213C3×C4×C5—81no—
214C3×C4×C5#12011yescharacteristic
215C3×C4×C5#12011yescharacteristic
216C3×C4×C5#12111yescharacteristic
217C3×C4×C5#12111yescharacteristic
218C24×C3—21no—
219C2×C3×Q8—21no—
220C2×C3×Q8—21no—
221C2×C3×Q8—21no—
222C2×C3×Q8—21no—
223C2×C3×Q8—21no—
224C2×C3×Q8—21no—
225C2×C3×Q8—21no—
226C2×C3×Q8—21no—
227C2×C3×Q8—21no—
228C2×C3×Q8—21no—
229C2×C3×Q8C5×D412yes—
230C2×C3×Q8C5×D412yes—
231C2×C3×D4—21no—
232C2×C3×D4—21no—
233C2×C3×D4—21no—
234C2×C3×D4—21no—
235C2×C3×D4—21no—
236C2×C3×D4C5×D412yes—
237C22×C3×C4—21no—
238C22×C3×C4—21no—
239C22×C3×C4—21no—
240C22×C3×C4C5×D412yes—
241C22×C3×C4C23×C511yescharacteristic
242C3×Q16—28no—
243C3×SD16—28no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8C5×D412yes—
249C2×C3×C8C5×D412yes—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C22⋊C4—28no—
253C3×C42—21no—
254C3×C42—21no—
255C23×C5—22no—
256C23×C5—41no—
257C23×C5—41no—
258C23×C5—42no—
259C23×C5—42no—
260C23×C5C2×C3×D411yescharacteristic
261C5×Q8—22no—
262C5×Q8—22no—
263C5×Q8—22no—
264C5×Q8—22no—
265C5×Q8—41no—
266C5×Q8—41no—
267C5×Q8—41no—
268C5×Q8—41no—
269C5×D4—22no—
270C5×D4—22no—
271C5×D4—41no—
272C5×D4—41no—
273C2×C4×C5—22no—
274C2×C4×C5—22no—
275C2×C4×C5—22no—
276C2×C4×C5—22no—
277C2×C4×C5—22no—
278C2×C4×C5—41no—
279C2×C4×C5—41no—
280C2×C4×C5—41no—
281C2×C4×C5—41no—
282C2×C4×C5—41no—
283C2×C4×C5—42no—
284C2×C4×C5C2×C3×D411yescharacteristic
285C2×C4×C5C2×C3×D411yescharacteristic
286C2×C4×C5C2×C3×D411yescharacteristic
287C2×C4×C5C2×C3×D411yescharacteristic
288C2×C4×C5C2×C3×D411yescharacteristic
289C2×C4×C5C3×#4111yescharacteristic
290C5×C8—22no—
291C5×C8—22no—
292C22×Q8C22×C3×C511yescharacteristic
293C22×Q8C22×C3×C511yescharacteristic
294C22×D4C22×C3×C511yescharacteristic
295C2×Q16—22no—
296C2×Q16—22no—
297C2×Q16—22no—
298C2×SD16—22no—
299C2×SD16—22no—
300C2×SD16—22no—
301C22×C8C22×C3×C511yescharacteristic
302C22×C8C22×C3×C511yescharacteristic
303#124—22no—
304#124—22no—
305C2×C4⋊C4C22×C3×C511yescharacteristic
306C2×C22⋊C4—22no—
307C2×C42C22×C3×C511yescharacteristic
308#103—22no—
309#102—22no—
310#95—22no—
311C2×C3×C5—82no—
312C2×C3×C5#39211yescharacteristic
313C2×C3×C5#45212yes—
314C2×C3×C5#46412yes—
315C2×C3×C5#49511yescharacteristic
316C2×C3×C5#49711yescharacteristic
317C23×C3—22no—
318C23×C3—41no—
319C23×C3—41no—
320C23×C3—42no—
321C23×C3—42no—
322C23×C3C2×C5×D411yescharacteristic
323C3×Q8—22no—
324C3×Q8—22no—
325C3×Q8—22no—
326C3×Q8—22no—
327C3×Q8—41no—
328C3×Q8—41no—
329C3×Q8—41no—
330C3×Q8—41no—
331C3×D4—22no—
332C3×D4—22no—
333C3×D4—41no—
334C3×D4—41no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—42no—
346C2×C3×C4C2×C5×D411yescharacteristic
347C2×C3×C4C2×C5×D411yescharacteristic
348C2×C3×C4C2×C5×D411yescharacteristic
349C2×C3×C4C2×C5×D411yescharacteristic
350C2×C3×C4C2×C5×D411yescharacteristic
351C2×C3×C4C5×#4111yescharacteristic
352C3×C8—22no—
353C3×C8—22no—
354C22×C5—42no—
355C22×C5—44no—
356C22×C5—82no—
357C22×C5—81no—
358C22×C5—81no—
359C22×C5C3×#12011yescharacteristic
360C22×C5C3×#12111yescharacteristic
361C22×C5C3×#12711yescharacteristic
362C22×C5C3×#13512yes—
363C22×C5C3×#13712yes—
364C4×C5—42no—
365C4×C5—42no—
366C4×C5—42no—
367C4×C5—81no—
368C4×C5C3×#12011yescharacteristic
369C4×C5C3×#12011yescharacteristic
370C4×C5C3×#12111yescharacteristic
371C4×C5C3×#12111yescharacteristic
372C24—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8C3×C5×D412yes—
384C2×Q8C3×C5×D412yes—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4C3×C5×D412yes—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4C3×C5×D412yes—
395C22×C4C23×C3×C511yescharacteristic
396Q16—28no—
397SD16—28no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8C3×C5×D412yes—
403C2×C8C3×C5×D412yes—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C22⋊C4—28no—
407C42—21no—
408C42—21no—
409C3×C5#201511yescharacteristic, direct product factor, Hall
410C22×C3—42no—
411C22×C3—44no—
412C22×C3—82no—
413C22×C3—81no—
414C22×C3—81no—
415C22×C3C5×#12011yescharacteristic
416C22×C3C5×#12111yescharacteristic
417C22×C3C5×#12711yescharacteristic
418C22×C3C5×#13512yes—
419C22×C3C5×#13712yes—
420C3×C4—42no—
421C3×C4—42no—
422C3×C4—42no—
423C3×C4—81no—
424C3×C4C5×#12011yescharacteristic
425C3×C4C5×#12011yescharacteristic
426C3×C4C5×#12111yescharacteristic
427C3×C4C5×#12111yescharacteristic
428C2×C5—82no—
429C2×C5C3×#39211yescharacteristic
430C2×C5C3×#45212yes—
431C2×C5C3×#46412yes—
432C2×C5C3×#49511yescharacteristic
433C2×C5C3×#49711yescharacteristic
434C23—22no—
435C23—41no—
436C23—41no—
437C23—42no—
438C23—42no—
439C23C2×C3×C5×D411yescharacteristic
440Q8—22no—
441Q8—22no—
442Q8—22no—
443Q8—22no—
444Q8—41no—
445Q8—41no—
446Q8—41no—
447Q8—41no—
448D4—22no—
449D4—22no—
450D4—41no—
451D4—41no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4C2×C3×C5×D411yescharacteristic
464C2×C4C2×C3×C5×D411yescharacteristic
465C2×C4C2×C3×C5×D411yescharacteristic
466C2×C4C2×C3×C5×D411yescharacteristic
467C2×C4C2×C3×C5×D411yescharacteristic
468C2×C4C3×C5×#4111yescharacteristic
469C8—22no—
470C8—22no—
471C2×C3—82no—
472C2×C3C5×#39211yescharacteristic
473C2×C3C5×#45212yes—
474C2×C3C5×#46412yes—
475C2×C3C5×#49511yescharacteristic
476C2×C3C5×#49711yescharacteristic
477C5C3×#201511yescharacteristic, direct product factor, Sylow
478C22—44no—
479C22—42no—
480C22—82no—
481C22—81no—
482C22—81no—
483C22C3×C5×#12011yescharacteristic
484C22C3×C5×#12111yescharacteristic
485C22C3×C5×#12711yescharacteristic
486C22C3×C5×#13512yes—
487C22C3×C5×#13712yes—
488C4—42no—
489C4—42no—
490C4—42no—
491C4—81no—
492C4C3×C5×#12011yescharacteristic
493C4C3×C5×#12011yescharacteristic
494C4C3×C5×#12111yescharacteristic
495C4C3×C5×#12111yescharacteristic
496C3C5×#201511yescharacteristic, direct product factor, Sylow
497C2—82no—
498C2C3×C5×#39211yescharacteristic
499C2C3×C5×#45212yes—
500C2C3×C5×#46412yes—
501C2C3×C5×#49511yescharacteristic
502C2C3×C5×#49711yescharacteristic
subgroupG/H
center, socleC23×C3×C5C2×D4
commutator, Frattini, cosocleC22×C4C23×C3×C5
prime radicalC22×C3×C5×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3×C5 ◁ C2×C3×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×C5×Q16C2semidirect
4. C2×C3×C5×#103C2semidirect
6. C3×C5×#391C2semidirect
7. C3×C5×#336C2semidirect
8. C5×#2015C3direct (N × H)
28. C3×#2015C5direct (N × H)
29. C22×C5×Q16C2×C3semidirect
32. C2×C5×#103C2×C3semidirect
34. C5×#391C2×C3semidirect
35. C5×#336C2×C3semidirect
73. C22×C3×Q16C2×C5semidirect
76. C2×C3×#103C2×C5semidirect
78. C3×#391C2×C5semidirect
79. C3×#336C2×C5semidirect
99. #2015C3×C5direct (N × H)
193. C22×Q16C2×C3×C5semidirect
196. C2×#103C2×C3×C5semidirect
198. #391C2×C3×C5semidirect
199. #336C2×C3×C5semidirect
409. C3×C5#2015direct (N × H)
477. C5C3×#2015direct (N × H)
496. C3C5×#2015direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C5×Q162
C22×C3×C5×SD162
C2×C3×C5×#1242
C2×C3×C5×#1032
C2×C3×C5×#1022
C3×C5×#3912
C3×C5×#3362
C5×#20153
C3×#20155