Finite Group Databasebeta

← C3×C5×#2012 #14275046

Subgroups — C3×C5×#2012 #14275046

Order 1920.

852 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C5×SD16C211yesmaximal, characteristic, complemented
2C22×C3×C5×D8C211yesmaximal, characteristic
3C2×C3×C5×#124C211yesmaximal, characteristic, complemented
4C2×C3×C5×#103C211yesmaximal, characteristic, complemented
5C2×C3×C5×#102C211yesmaximal, characteristic, complemented
6C3×C5×#390C211yesmaximal, characteristic, complemented
7C3×C5×#336C211yesmaximal, characteristic, complemented
8C5×#2012C311yesmaximal, characteristic, direct product factor, Hall
9C22×C3×C5×Q8C2211yescharacteristic
10C22×C3×C5×D4C2211yescharacteristic
11C22×C3×C5×D4C2211yescharacteristic
12C2×C3×C5×SD16—22no—
13C2×C3×C5×SD16—22no—
14C2×C3×C5×SD16—22no—
15C2×C3×C5×D8—22no—
16C2×C3×C5×D8—22no—
17C2×C3×C5×D8—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×C5×C22⋊C4—22no—
25C2×C3×C42×C5C2211yescharacteristic
26C3×C5×#103—22no—
27C3×C5×#102—22no—
28C3×#2012C511yesmaximal, characteristic, direct product factor, Hall
29C22×C5×SD16C2×C311yescharacteristic, complemented
30C22×C5×D8C2×C311yescharacteristic
31C2×C5×#124C2×C311yescharacteristic, complemented
32C2×C5×#103C2×C311yescharacteristic, complemented
33C2×C5×#102C2×C311yescharacteristic, complemented
34C5×#390C2×C311yescharacteristic, complemented
35C5×#336C2×C311yescharacteristic, complemented
36C24×C3×C5—21no—
37C24×C3×C5—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×Q8D412yes—
44C2×C3×C5×D4—21no—
45C2×C3×C5×D4—21no—
46C2×C3×C5×D4—21no—
47C2×C3×C5×D4—21no—
48C2×C3×C5×D4—21no—
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—
55C2×C3×C5×D4D412yes—
56C22×C3×C4×C5—21no—
57C22×C3×C4×C5—21no—
58C22×C3×C4×C5D412yes—
59C22×C3×C4×C5C2311yescharacteristic
60C3×C5×SD16—28no—
61C3×C5×D8—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×C5×C22⋊C4—44no—
72C3×C42×C5—21no—
73C3×C42×C5—21no—
74C22×C3×SD16C2×C511yescharacteristic, complemented
75C22×C3×D8C2×C511yescharacteristic
76C2×C3×#124C2×C511yescharacteristic, complemented
77C2×C3×#103C2×C511yescharacteristic, complemented
78C2×C3×#102C2×C511yescharacteristic, complemented
79C3×#390C2×C511yescharacteristic, complemented
80C3×#336C2×C511yescharacteristic, complemented
81C22×C5×Q8C22×C311yescharacteristic
82C22×C5×D4C22×C311yescharacteristic
83C22×C5×D4C22×C311yescharacteristic
84C2×C5×SD16—22no—
85C2×C5×SD16—22no—
86C2×C5×SD16—22no—
87C2×C5×D8—22no—
88C2×C5×D8—22no—
89C2×C5×D8—22no—
90C22×C5×C8C22×C311yescharacteristic
91C22×C5×C8C22×C311yescharacteristic
92C5×#124—22no—
93C5×#124—22no—
94C2×C5×C4⋊C4C22×C311yescharacteristic
95C2×C5×C22⋊C4—22no—
96C2×C5×C22⋊C4—22no—
97C2×C42×C5C22×C311yescharacteristic
98C5×#103—22no—
99C5×#102—22no—
100#2012C3×C511yescharacteristic, direct product factor, Sylow
101C23×C3×C5—22no—
102C23×C3×C5—22no—
103C23×C3×C5—41no—
104C23×C3×C5—41no—
105C23×C3×C5—41no—
106C23×C3×C5—41no—
107C23×C3×C5—42no—
108C23×C3×C5—42no—
109C23×C3×C5—42no—
110C23×C3×C5—42no—
111C23×C3×C5C2×D411yescharacteristic
112C3×C5×Q8—22no—
113C3×C5×Q8—22no—
114C3×C5×Q8—41no—
115C3×C5×Q8—41no—
116C3×C5×D4—22no—
117C3×C5×D4—22no—
118C3×C5×D4—22no—
119C3×C5×D4—22no—
120C3×C5×D4—41no—
121C3×C5×D4—41no—
122C3×C5×D4—41no—
123C3×C5×D4—41no—
124C2×C3×C4×C5—22no—
125C2×C3×C4×C5—22no—
126C2×C3×C4×C5—22no—
127C2×C3×C4×C5—22no—
128C2×C3×C4×C5—41no—
129C2×C3×C4×C5—41no—
130C2×C3×C4×C5—41no—
131C2×C3×C4×C5—42no—
132C2×C3×C4×C5C2×D411yescharacteristic
133C2×C3×C4×C5C2×D411yescharacteristic
134C2×C3×C4×C5C2×D411yescharacteristic
135C2×C3×C4×C5C2×D411yescharacteristic
136C2×C3×C4×C5C2×D411yescharacteristic
137C2×C3×C4×C5#4111yescharacteristic
138C3×C5×C8—22no—
139C3×C5×C8—22no—
140C22×C3×Q8C22×C511yescharacteristic
141C22×C3×D4C22×C511yescharacteristic
142C22×C3×D4C22×C511yescharacteristic
143C2×C3×SD16—22no—
144C2×C3×SD16—22no—
145C2×C3×SD16—22no—
146C2×C3×D8—22no—
147C2×C3×D8—22no—
148C2×C3×D8—22no—
149C22×C3×C8C22×C511yescharacteristic
150C22×C3×C8C22×C511yescharacteristic
151C3×#124—22no—
152C3×#124—22no—
153C2×C3×C4⋊C4C22×C511yescharacteristic
154C2×C3×C22⋊C4—22no—
155C2×C3×C22⋊C4—22no—
156C2×C3×C42C22×C511yescharacteristic
157C3×#103—22no—
158C3×#102—22no—
159C24×C5—21no—
160C24×C5—21no—
161C2×C5×Q8—21no—
162C2×C5×Q8—21no—
163C2×C5×Q8—21no—
164C2×C5×Q8—21no—
165C2×C5×Q8—21no—
166C2×C5×Q8C3×D412yes—
167C2×C5×D4—21no—
168C2×C5×D4—21no—
169C2×C5×D4—21no—
170C2×C5×D4—21no—
171C2×C5×D4—21no—
172C2×C5×D4—21no—
173C2×C5×D4—21no—
174C2×C5×D4—21no—
175C2×C5×D4—21no—
176C2×C5×D4—21no—
177C2×C5×D4C3×D412yes—
178C2×C5×D4C3×D412yes—
179C22×C4×C5—21no—
180C22×C4×C5—21no—
181C22×C4×C5C3×D412yes—
182C22×C4×C5C23×C311yescharacteristic
183C5×SD16—28no—
184C5×D8—28no—
185C2×C5×C8—21no—
186C2×C5×C8—21no—
187C2×C5×C8—21no—
188C2×C5×C8—21no—
189C2×C5×C8C3×D412yes—
190C2×C5×C8C3×D412yes—
191C5×C4⋊C4—21no—
192C5×C4⋊C4—21no—
193C5×C22⋊C4—28no—
194C5×C22⋊C4—44no—
195C42×C5—21no—
196C42×C5—21no—
197C22×SD16C2×C3×C511yescharacteristic, complemented
198C22×D8C2×C3×C511yescharacteristic
199C2×#124C2×C3×C511yescharacteristic, complemented
200C2×#103C2×C3×C511yescharacteristic, complemented
201C2×#102C2×C3×C511yescharacteristic, complemented
202#390C2×C3×C511yescharacteristic, complemented
203#336C2×C3×C511yescharacteristic, complemented
204C22×C3×C5—42no—
205C22×C3×C5—42no—
206C22×C3×C5—44no—
207C22×C3×C5—44no—
208C22×C3×C5—82no—
209C22×C3×C5—82no—
210C22×C3×C5—81no—
211C22×C3×C5—81no—
212C22×C3×C5—81no—
213C22×C3×C5—81no—
214C22×C3×C5#12011yescharacteristic
215C22×C3×C5#12111yescharacteristic
216C22×C3×C5#12711yescharacteristic
217C22×C3×C5#13512yes—
218C22×C3×C5AGL(1,ℤ/8)12yes—
219C3×C4×C5—42no—
220C3×C4×C5—42no—
221C3×C4×C5—81no—
222C3×C4×C5#12011yescharacteristic
223C3×C4×C5#12011yescharacteristic
224C3×C4×C5#12111yescharacteristic
225C3×C4×C5#12111yescharacteristic
226C24×C3—21no—
227C24×C3—21no—
228C2×C3×Q8—21no—
229C2×C3×Q8—21no—
230C2×C3×Q8—21no—
231C2×C3×Q8—21no—
232C2×C3×Q8—21no—
233C2×C3×Q8C5×D412yes—
234C2×C3×D4—21no—
235C2×C3×D4—21no—
236C2×C3×D4—21no—
237C2×C3×D4—21no—
238C2×C3×D4—21no—
239C2×C3×D4—21no—
240C2×C3×D4—21no—
241C2×C3×D4—21no—
242C2×C3×D4—21no—
243C2×C3×D4—21no—
244C2×C3×D4C5×D412yes—
245C2×C3×D4C5×D412yes—
246C22×C3×C4—21no—
247C22×C3×C4—21no—
248C22×C3×C4C5×D412yes—
249C22×C3×C4C23×C511yescharacteristic
250C3×SD16—28no—
251C3×D8—28no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8C5×D412yes—
257C2×C3×C8C5×D412yes—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—21no—
260C3×C22⋊C4—28no—
261C3×C22⋊C4—44no—
262C3×C42—21no—
263C3×C42—21no—
264C23×C5—22no—
265C23×C5—22no—
266C23×C5—41no—
267C23×C5—41no—
268C23×C5—41no—
269C23×C5—41no—
270C23×C5—42no—
271C23×C5—42no—
272C23×C5—42no—
273C23×C5—42no—
274C23×C5C2×C3×D411yescharacteristic
275C5×Q8—22no—
276C5×Q8—22no—
277C5×Q8—41no—
278C5×Q8—41no—
279C5×D4—22no—
280C5×D4—22no—
281C5×D4—22no—
282C5×D4—22no—
283C5×D4—41no—
284C5×D4—41no—
285C5×D4—41no—
286C5×D4—41no—
287C2×C4×C5—22no—
288C2×C4×C5—22no—
289C2×C4×C5—22no—
290C2×C4×C5—22no—
291C2×C4×C5—41no—
292C2×C4×C5—41no—
293C2×C4×C5—41no—
294C2×C4×C5—42no—
295C2×C4×C5C2×C3×D411yescharacteristic
296C2×C4×C5C2×C3×D411yescharacteristic
297C2×C4×C5C2×C3×D411yescharacteristic
298C2×C4×C5C2×C3×D411yescharacteristic
299C2×C4×C5C2×C3×D411yescharacteristic
300C2×C4×C5C3×#4111yescharacteristic
301C5×C8—22no—
302C5×C8—22no—
303C22×Q8C22×C3×C511yescharacteristic
304C22×D4C22×C3×C511yescharacteristic
305C22×D4C22×C3×C511yescharacteristic
306C2×SD16—22no—
307C2×SD16—22no—
308C2×SD16—22no—
309C2×D8—22no—
310C2×D8—22no—
311C2×D8—22no—
312C22×C8C22×C3×C511yescharacteristic
313C22×C8C22×C3×C511yescharacteristic
314#124—22no—
315#124—22no—
316C2×C4⋊C4C22×C3×C511yescharacteristic
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C42C22×C3×C511yescharacteristic
320#103—22no—
321#102—22no—
322C2×C3×C5—82no—
323C2×C3×C5—82no—
324C2×C3×C5#39211yescharacteristic
325C2×C3×C5#44912yes—
326C2×C3×C5#46312yes—
327C2×C3×C5#49511yescharacteristic
328C2×C3×C5#49611yescharacteristic
329C23×C3—22no—
330C23×C3—22no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3—42no—
336C23×C3—42no—
337C23×C3—42no—
338C23×C3—42no—
339C23×C3C2×C5×D411yescharacteristic
340C3×Q8—22no—
341C3×Q8—22no—
342C3×Q8—41no—
343C3×Q8—41no—
344C3×D4—22no—
345C3×D4—22no—
346C3×D4—22no—
347C3×D4—22no—
348C3×D4—41no—
349C3×D4—41no—
350C3×D4—41no—
351C3×D4—41no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—42no—
360C2×C3×C4C2×C5×D411yescharacteristic
361C2×C3×C4C2×C5×D411yescharacteristic
362C2×C3×C4C2×C5×D411yescharacteristic
363C2×C3×C4C2×C5×D411yescharacteristic
364C2×C3×C4C2×C5×D411yescharacteristic
365C2×C3×C4C5×#4111yescharacteristic
366C3×C8—22no—
367C3×C8—22no—
368C22×C5—42no—
369C22×C5—42no—
370C22×C5—44no—
371C22×C5—44no—
372C22×C5—82no—
373C22×C5—82no—
374C22×C5—81no—
375C22×C5—81no—
376C22×C5—81no—
377C22×C5—81no—
378C22×C5C3×#12011yescharacteristic
379C22×C5C3×#12111yescharacteristic
380C22×C5C3×#12711yescharacteristic
381C22×C5C3×#13512yes—
382C22×C5C3×AGL(1,ℤ/8)12yes—
383C4×C5—42no—
384C4×C5—42no—
385C4×C5—81no—
386C4×C5C3×#12011yescharacteristic
387C4×C5C3×#12011yescharacteristic
388C4×C5C3×#12111yescharacteristic
389C4×C5C3×#12111yescharacteristic
390C24—21no—
391C24—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8C3×C5×D412yes—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4C3×C5×D412yes—
409C2×D4C3×C5×D412yes—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4C3×C5×D412yes—
413C22×C4C23×C3×C511yescharacteristic
414SD16—28no—
415D8—28no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8C3×C5×D412yes—
421C2×C8C3×C5×D412yes—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C22⋊C4—28no—
425C22⋊C4—44no—
426C42—21no—
427C42—21no—
428C3×C5#201211yescharacteristic, direct product factor, Hall
429C22×C3—42no—
430C22×C3—42no—
431C22×C3—44no—
432C22×C3—44no—
433C22×C3—82no—
434C22×C3—82no—
435C22×C3—81no—
436C22×C3—81no—
437C22×C3—81no—
438C22×C3—81no—
439C22×C3C5×#12011yescharacteristic
440C22×C3C5×#12111yescharacteristic
441C22×C3C5×#12711yescharacteristic
442C22×C3C5×#13512yes—
443C22×C3C5×AGL(1,ℤ/8)12yes—
444C3×C4—42no—
445C3×C4—42no—
446C3×C4—81no—
447C3×C4C5×#12011yescharacteristic
448C3×C4C5×#12011yescharacteristic
449C3×C4C5×#12111yescharacteristic
450C3×C4C5×#12111yescharacteristic
451C2×C5—82no—
452C2×C5—82no—
453C2×C5C3×#39211yescharacteristic
454C2×C5C3×#44912yes—
455C2×C5C3×#46312yes—
456C2×C5C3×#49511yescharacteristic
457C2×C5C3×#49611yescharacteristic
458C23—22no—
459C23—22no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23C2×C3×C5×D411yescharacteristic
469Q8—22no—
470Q8—22no—
471Q8—41no—
472Q8—41no—
473D4—22no—
474D4—22no—
475D4—22no—
476D4—22no—
477D4—41no—
478D4—41no—
479D4—41no—
480D4—41no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4C2×C3×C5×D411yescharacteristic
490C2×C4C2×C3×C5×D411yescharacteristic
491C2×C4C2×C3×C5×D411yescharacteristic
492C2×C4C2×C3×C5×D411yescharacteristic
493C2×C4C2×C3×C5×D411yescharacteristic
494C2×C4C3×C5×#4111yescharacteristic
495C8—22no—
496C8—22no—
497C2×C3—82no—
498C2×C3—82no—
499C2×C3C5×#39211yescharacteristic
500C2×C3C5×#44912yes—
501C2×C3C5×#46312yes—
502C2×C3C5×#49511yescharacteristic
503C2×C3C5×#49611yescharacteristic
504C5C3×#201211yescharacteristic, direct product factor, Sylow
505C22—44no—
506C22—44no—
507C22—42no—
508C22—42no—
509C22—82no—
510C22—82no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22C3×C5×#12011yescharacteristic
516C22C3×C5×#12111yescharacteristic
517C22C3×C5×#12711yescharacteristic
518C22C3×C5×#13512yes—
519C22C3×C5×AGL(1,ℤ/8)12yes—
520C4—42no—
521C4—42no—
522C4—81no—
523C4C3×C5×#12011yescharacteristic
524C4C3×C5×#12011yescharacteristic
525C4C3×C5×#12111yescharacteristic
526C4C3×C5×#12111yescharacteristic
527C3C5×#201211yescharacteristic, direct product factor, Sylow
528C2—82no—
529C2—82no—
530C2C3×C5×#39211yescharacteristic
531C2C3×C5×#44912yes—
532C2C3×C5×#46312yes—
533C2C3×C5×#49511yescharacteristic
534C2C3×C5×#49611yescharacteristic
subgroupG/H
center, socleC23×C3×C5C2×D4
commutator, Frattini, cosocleC22×C4C23×C3×C5
prime radicalC22×C3×C5×D8C2
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×SD16C2semidirect
3. C2×C3×C5×#124C2semidirect
4. C2×C3×C5×#103C2semidirect
5. C2×C3×C5×#102C2semidirect
6. C3×C5×#390C2semidirect
7. C3×C5×#336C2semidirect
8. C5×#2012C3direct (N × H)
28. C3×#2012C5direct (N × H)
29. C22×C5×SD16C2×C3semidirect
31. C2×C5×#124C2×C3semidirect
32. C2×C5×#103C2×C3semidirect
33. C2×C5×#102C2×C3semidirect
34. C5×#390C2×C3semidirect
35. C5×#336C2×C3semidirect
74. C22×C3×SD16C2×C5semidirect
76. C2×C3×#124C2×C5semidirect
77. C2×C3×#103C2×C5semidirect
78. C2×C3×#102C2×C5semidirect
79. C3×#390C2×C5semidirect
80. C3×#336C2×C5semidirect
100. #2012C3×C5direct (N × H)
197. C22×SD16C2×C3×C5semidirect
199. C2×#124C2×C3×C5semidirect
200. C2×#103C2×C3×C5semidirect
201. C2×#102C2×C3×C5semidirect
202. #390C2×C3×C5semidirect
203. #336C2×C3×C5semidirect
428. C3×C5#2012direct (N × H)
504. C5C3×#2012direct (N × H)
527. C3C5×#2012direct (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×SD162
C22×C3×C5×D82
C2×C3×C5×#1242
C2×C3×C5×#1032
C2×C3×C5×#1022
C3×C5×#3902
C3×C5×#3362
C5×#20123
C3×#20125