Finite Group Databasebeta

← C5×#70270 #14301750

Subgroups — C5×#70270 #14301750

Order 1920.

760 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×C5×#946C211yesmaximal, characteristic
2C22×C5×D3×C8C211yesmaximal, characteristic
3C2×C3×C5×#103C211yesmaximal, characteristic, complemented
4C2×C5×#835C211yesmaximal, characteristic, complemented
5C5×#4987C211yesmaximal, characteristic
6C5×#4551C211yesmaximal, characteristic, complemented
7C5×#4529C211yesmaximal, characteristic, complemented
8C5×#1892—31nomaximal, Hall
9C22×C3×C5×Q8C2211yescharacteristic
10C22×C4×C5×D3C2211yescharacteristic
11C22×C3×C5×C8C2211yescharacteristic
12C2×C3×C5×C4⋊C4C2211yescharacteristic
13C5×#946—22no—
14C5×#946C414yescomplemented
15C2×C5×(C2×D3)⋊C4—21no—
16C2×C5×(C2×D3)⋊C4—21no—
17C2×C5×C4⋊Dic3C2211yescharacteristic
18C2×C5×C4⋊Dic3C2211yescharacteristic
19C2×C5×Dic3⋊C4—21no—
20C22×C5×C3⋊C8C2211yescharacteristic
21C2×C5×D3×C8—22no—
22C2×C5×D3×C8—22no—
23C2×C5×D3×C8—22no—
24C3×C5×#103—22no—
25C5×#835—22no—
26C5×C3⋊#95—21no—
27#70270C511yesmaximal, characteristic, direct product factor, Hall
28C23×C5×C8—31no—
29C2×C5×C22⋊Q8—31no—
30C2×C5×#103—31no—
31C2×C5×#103D311yescharacteristic, complemented
32C5×#380—31no—
33C5×#340—31no—
34C5×#340—31no—
35C23×C5×D3D411yescharacteristic
36C2×C3×C5×Q8—21no—
37C2×C3×C5×Q8—21no—
38C2×C3×C5×Q8—21no—
39C2×C3×C5×Q8—21no—
40C2×C3×C5×Q8—21no—
41C2×C3×C5×Q8C2×C412yes—
42C22×C3×C4×C5—21no—
43C22×C3×C4×C5—21no—
44C22×C3×C4×C5C2311yescharacteristic
45C22×C5×Dic3—21no—
46C22×C5×Dic3—21no—
47C22×C5×Dic3D411yescharacteristic
48C2×C4×C5×D3—21no—
49C2×C4×C5×D3—21no—
50C2×C4×C5×D3—21no—
51C2×C4×C5×D3—21no—
52C2×C4×C5×D3C2×C412yes—
53C2×C4×C5×D3D412yes—
54C2×C3×C5×C8—21no—
55C2×C3×C5×C8—21no—
56C2×C3×C5×C8D412yes—
57C3×C5×C4⋊C4—21no—
58C3×C5×C4⋊C4—21no—
59C5×(C2×D3)⋊C4—24no—
60C5×(C2×D3)⋊C4—42no—
61C5×C4⋊Dic3—21no—
62C5×C4⋊Dic3—21no—
63C5×C4⋊Dic3—21no—
64C5×C4⋊Dic3C2×C412yes—
65C5×Dic3⋊C4—24no—
66C2×C5×C3⋊C8—21no—
67C2×C5×C3⋊C8—21no—
68C2×C5×C3⋊C8D412yes—
69C5×D3×C8—28no—
70C2×#946C2×C511yescharacteristic
71C22×D3×C8C2×C511yescharacteristic
72C2×C3×#103C2×C511yescharacteristic, complemented
73C2×#835C2×C511yescharacteristic, complemented
74#4987C2×C511yescharacteristic
75#4551C2×C511yescharacteristic, complemented
76#4529C2×C511yescharacteristic, complemented
77C22×C5×Q8C2×D311yescharacteristic
78C23×C4×C5—31no—
79C22×C5×C8—31no—
80C22×C5×C8—62no—
81C22×C5×C8—62no—
82C22×C5×C8—62no—
83C22×C5×C8C2×D311yescharacteristic
84C5×C22⋊Q8—34no—
85C5×C22⋊Q8—62no—
86C2×C5×C4⋊C4—31no—
87C2×C5×C4⋊C4—31no—
88C2×C5×C4⋊C4—61no—
89C2×C5×C4⋊C4C2×D311yescharacteristic
90C2×C5×C22⋊C4—61no—
91C2×C5×C22⋊C4—61no—
92C5×#103—22no—
93C5×#103—62no—
94C5×#95—61no—
95#1892—31noSylow
96C23×C3×C5C2×D411yescharacteristic
97C22×C5×D3—21no—
98C22×C5×D3—21no—
99C22×C5×D3—21no—
100C22×C5×D3—21no—
101C22×C5×D3C22⋊C412yes—
102C22×C5×D3SD1612yes—
103C22×C5×D3Q1612yes—
104C3×C5×Q8—22no—
105C3×C5×Q8—22no—
106C3×C5×Q8—41no—
107C3×C5×Q8—41no—
108C2×C3×C4×C5—22no—
109C2×C3×C4×C5—41no—
110C2×C3×C4×C5—41no—
111C2×C3×C4×C5—41no—
112C2×C3×C4×C5—41no—
113C2×C3×C4×C5—41no—
114C2×C3×C4×C5C22×C411yescharacteristic
115C2×C3×C4×C5C2×D411yescharacteristic
116C2×C3×C4×C5C2×D411yescharacteristic
117C2×C3×C4×C5C2×D411yescharacteristic
118C2×C3×C4×C5#4111yescharacteristic
119C2×C3×C4×C5#4111yescharacteristic
120C2×C5×Dic3—21no—
121C2×C5×Dic3—21no—
122C2×C5×Dic3—22no—
123C2×C5×Dic3—41no—
124C2×C5×Dic3—41no—
125C2×C5×Dic3—41no—
126C2×C5×Dic3—41no—
127C2×C5×Dic3—41no—
128C2×C5×Dic3C22⋊C412yes—
129C4×C5×D3—22no—
130C4×C5×D3—22no—
131C4×C5×D3—22no—
132C4×C5×D3—22no—
133C3×C5×C8—22no—
134C5×C3⋊C8—22no—
135C22×C3×Q8C22×C511yescharacteristic
136C22×C4×D3C22×C511yescharacteristic
137C22×C3×C8C22×C511yescharacteristic
138C2×C3×C4⋊C4C22×C511yescharacteristic
139#946—22no—
140#946C4×C514yescomplemented
141C2×(C2×D3)⋊C4—21no—
142C2×(C2×D3)⋊C4—21no—
143C2×C4⋊Dic3C22×C511yescharacteristic
144C2×C4⋊Dic3C22×C511yescharacteristic
145C2×Dic3⋊C4—21no—
146C22×C3⋊C8C22×C511yescharacteristic
147C2×D3×C8—22no—
148C2×D3×C8—22no—
149C2×D3×C8—22no—
150C3×#103—22no—
151#835—22no—
152C3⋊#95—21no—
153C24×C5—31no—
154C2×C5×Q8—21no—
155C2×C5×Q8—21no—
156C2×C5×Q8—21no—
157C2×C5×Q8—21no—
158C2×C5×Q8—21no—
159C2×C5×Q8C4×D312yes—
160C22×C4×C5—21no—
161C22×C4×C5—21no—
162C22×C4×C5—31no—
163C22×C4×C5—32no—
164C22×C4×C5—32no—
165C22×C4×C5—61no—
166C22×C4×C5—61no—
167C22×C4×C5—61no—
168C22×C4×C5—61no—
169C22×C4×C5—61no—
170C22×C4×C5—61no—
171C22×C4×C5C22×D311yescharacteristic
172C2×C5×C8—21no—
173C2×C5×C8—21no—
174C2×C5×C8—32no—
175C2×C5×C8—68no—
176C2×C5×C8—61no—
177C2×C5×C8—61no—
178C2×C5×C8#6712yes—
179C5×C4⋊C4—21no—
180C5×C4⋊C4—21no—
181C5×C4⋊C4—32no—
182C5×C4⋊C4—64no—
183C5×C4⋊C4—61no—
184C5×C4⋊C4—61no—
185C5×C4⋊C4—61no—
186C5×C22⋊C4—64no—
187C5×C22⋊C4—122no—
188C23×C8—31no—
189C2×C22⋊Q8—31no—
190C2×#103—31no—
191C2×#103C5×D311yescharacteristic, complemented
192#380—31no—
193#340—31no—
194#340—31no—
195C22×C3×C5C2×C22⋊C411yescharacteristic
196C22×C3×C5#12111yescharacteristic
197C22×C3×C5#12111yescharacteristic
198C22×C3×C5C2×SD1611yescharacteristic
199C22×C3×C5C2×Q1611yescharacteristic
200C22×C3×C5#13511yescharacteristic
201C22×C3×C5#13511yescharacteristic
202C2×C5×D3—22no—
203C2×C5×D3—22no—
204C2×C5×D3—22no—
205C2×C5×D3—22no—
206C2×C5×D3—22no—
207C2×C5×D3—22no—
208C2×C5×D3#10314yescomplemented
209C3×C4×C5—42no—
210C3×C4×C5—81no—
211C3×C4×C5C4×D411yescharacteristic
212C3×C4×C5C4×D411yescharacteristic
213C3×C4×C5#12011yescharacteristic
214C3×C4×C5#12311yescharacteristic
215C5×Dic3—22no—
216C5×Dic3—42no—
217C5×Dic3—81no—
218C23×D3C5×D411yescharacteristic
219C2×C3×Q8—21no—
220C2×C3×Q8—21no—
221C2×C3×Q8—21no—
222C2×C3×Q8—21no—
223C2×C3×Q8—21no—
224C2×C3×Q8C2×C4×C512yes—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4C23×C511yescharacteristic
228C22×Dic3—21no—
229C22×Dic3—21no—
230C22×Dic3C5×D411yescharacteristic
231C2×C4×D3—21no—
232C2×C4×D3—21no—
233C2×C4×D3—21no—
234C2×C4×D3—21no—
235C2×C4×D3C2×C4×C512yes—
236C2×C4×D3C5×D412yes—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8C5×D412yes—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242(C2×D3)⋊C4—24no—
243(C2×D3)⋊C4—42no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3C2×C4×C512yes—
248Dic3⋊C4—24no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8C5×D412yes—
252D3×C8—28no—
253C23×C5—32no—
254C23×C5—32no—
255C23×C5—32no—
256C23×C5—61no—
257C23×C5—61no—
258C23×C5—61no—
259C23×C5—61no—
260C23×C5D3×D411yescharacteristic
261C5×Q8—22no—
262C5×Q8—22no—
263C5×Q8—41no—
264C5×Q8—41no—
265C2×C4×C5—22no—
266C2×C4×C5—32no—
267C2×C4×C5—41no—
268C2×C4×C5—41no—
269C2×C4×C5—41no—
270C2×C4×C5—41no—
271C2×C4×C5—41no—
272C2×C4×C5—61no—
273C2×C4×C5—61no—
274C2×C4×C5—62no—
275C2×C4×C5—62no—
276C2×C4×C5—62no—
277C2×C4×C5—62no—
278C2×C4×C5—62no—
279C2×C4×C5—121no—
280C2×C4×C5—121no—
281C2×C4×C5—121no—
282C2×C4×C5—121no—
283C2×C4×C5—121no—
284C2×C4×C5C2×C4×D311yescharacteristic
285C2×C4×C5#23511yescharacteristic
286C2×C4×C5D3×D411yescharacteristic
287C2×C4×C5D3×D411yescharacteristic
288C2×C4×C5#23711yescharacteristic
289C2×C4×C5C2×#6711yescharacteristic
290C5×C8—22no—
291C5×C8—62no—
292C22×Q8C2×C5×D311yescharacteristic
293C23×C4—31no—
294C22×C8—31no—
295C22×C8—62no—
296C22×C8—62no—
297C22×C8—62no—
298C22×C8C2×C5×D311yescharacteristic
299C22⋊Q8—34no—
300C22⋊Q8—62no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4C2×C5×D311yescharacteristic
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307#103—22no—
308#103—62no—
309#95—61no—
310C2×C3×C5#38611yescharacteristic
311C2×C3×C5C2×#10311yescharacteristic
312C2×C3×C5#41611yescharacteristic
313C2×C3×C5#46511yescharacteristic
314C2×C3×C5#46511yescharacteristic
315C2×C3×C5#46711yescharacteristic
316C2×C3×C5#46711yescharacteristic
317C5×D3—24no—
318C23×C3C2×C5×D411yescharacteristic
319C22×D3—21no—
320C22×D3—21no—
321C22×D3—21no—
322C22×D3—21no—
323C22×D3C5×C22⋊C412yes—
324C22×D3C5×SD1612yes—
325C22×D3C5×Q1612yes—
326C3×Q8—22no—
327C3×Q8—22no—
328C3×Q8—41no—
329C3×Q8—41no—
330C2×C3×C4—22no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4C22×C4×C511yescharacteristic
337C2×C3×C4C2×C5×D411yescharacteristic
338C2×C3×C4C2×C5×D411yescharacteristic
339C2×C3×C4C2×C5×D411yescharacteristic
340C2×C3×C4C5×#4111yescharacteristic
341C2×C3×C4C5×#4111yescharacteristic
342C2×Dic3—21no—
343C2×Dic3—21no—
344C2×Dic3—22no—
345C2×Dic3—41no—
346C2×Dic3—41no—
347C2×Dic3—41no—
348C2×Dic3—41no—
349C2×Dic3—41no—
350C2×Dic3C5×C22⋊C412yes—
351C4×D3—22no—
352C4×D3—22no—
353C4×D3—22no—
354C4×D3—22no—
355C3×C8—22no—
356C3⋊C8—22no—
357C22×C5—34no—
358C22×C5—62no—
359C22×C5—62no—
360C22×C5—62no—
361C22×C5—62no—
362C22×C5—62no—
363C22×C5—62no—
364C22×C5D3×C22⋊C411yescharacteristic
365C22×C5#88411yescharacteristic
366C22×C5D3×SD1611yescharacteristic
367C22×C5#91611yescharacteristic
368C22×C5D3×Q1611yescharacteristic
369C22×C5#91911yescharacteristic
370C22×C5#93811yescharacteristic
371C4×C5—42no—
372C4×C5—62no—
373C4×C5—81no—
374C4×C5—122no—
375C4×C5—241no—
376C4×C5#67⋊C411yescharacteristic
377C4×C5#88311yescharacteristic
378C4×C5C4×#6711yescharacteristic
379C4×C5#93711yescharacteristic
380C24—31no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8C4×C5×D312yes—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—31no—
390C22×C4—32no—
391C22×C4—32no—
392C22×C4—61no—
393C22×C4—61no—
394C22×C4—61no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4C22×C5×D311yescharacteristic
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—32no—
402C2×C8—68no—
403C2×C8—61no—
404C2×C8—61no—
405C2×C8C5×#6712yes—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—32no—
409C4⋊C4—64no—
410C4⋊C4—61no—
411C4⋊C4—61no—
412C4⋊C4—61no—
413C22⋊C4—64no—
414C22⋊C4—122no—
415C3×C5#189211yescharacteristic, complemented, Hall
416C22×C3C2×C5×C22⋊C411yescharacteristic
417C22×C3C5×#12111yescharacteristic
418C22×C3C5×#12111yescharacteristic
419C22×C3C2×C5×SD1611yescharacteristic
420C22×C3C2×C5×Q1611yescharacteristic
421C22×C3C5×#13511yescharacteristic
422C22×C3C5×#13511yescharacteristic
423C2×D3—22no—
424C2×D3—22no—
425C2×D3—22no—
426C2×D3—22no—
427C2×D3—22no—
428C2×D3—22no—
429C2×D3C5×#10314yescomplemented
430C3×C4—42no—
431C3×C4—81no—
432C3×C4C4×C5×D411yescharacteristic
433C3×C4C4×C5×D411yescharacteristic
434C3×C4C5×#12011yescharacteristic
435C3×C4C5×#12311yescharacteristic
436Dic3—22no—
437Dic3—42no—
438Dic3—81no—
439C2×C5—64no—
440C2×C5D3×#10311yescharacteristic
441C2×C5#480411yescharacteristic
442C2×C5#480711yescharacteristic
443C2×C5#481311yescharacteristic
444C2×C5#495611yescharacteristic
445C2×C5#517111yescharacteristic
446C2×C5#518811yescharacteristic
447C23—32no—
448C23—32no—
449C23—32no—
450C23—61no—
451C23—61no—
452C23—61no—
453C23—61no—
454C23C5×D3×D411yescharacteristic
455Q8—22no—
456Q8—22no—
457Q8—41no—
458Q8—41no—
459C2×C4—22no—
460C2×C4—32no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—62no—
469C2×C4—62no—
470C2×C4—62no—
471C2×C4—62no—
472C2×C4—62no—
473C2×C4—121no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4—121no—
477C2×C4—121no—
478C2×C4C2×C4×C5×D311yescharacteristic
479C2×C4C5×#23511yescharacteristic
480C2×C4C5×D3×D411yescharacteristic
481C2×C4C5×D3×D411yescharacteristic
482C2×C4C5×#23711yescharacteristic
483C2×C4C2×C5×#6711yescharacteristic
484C8—22no—
485C8—62no—
486C2×C3C5×#38611yescharacteristic
487C2×C3C2×C5×#10311yescharacteristic
488C2×C3C5×#41611yescharacteristic
489C2×C3C5×#46511yescharacteristic
490C2×C3C5×#46511yescharacteristic
491C2×C3C5×#46711yescharacteristic
492C2×C3C5×#46711yescharacteristic
493D3—24no—
494C5#7027011yescharacteristic, direct product factor, Sylow
495C22—34no—
496C22—62no—
497C22—62no—
498C22—62no—
499C22—62no—
500C22—62no—
501C22—62no—
502C22C5×D3×C22⋊C411yescharacteristic
503C22C5×#88411yescharacteristic
504C22C5×D3×SD1611yescharacteristic
505C22C5×#91611yescharacteristic
506C22C5×D3×Q1611yescharacteristic
507C22C5×#91911yescharacteristic
508C22C5×#93811yescharacteristic
509C4—42no—
510C4—62no—
511C4—81no—
512C4—122no—
513C4—241no—
514C4C5×#67⋊C411yescharacteristic
515C4C5×#88311yescharacteristic
516C4C4×C5×#6711yescharacteristic
517C4C5×#93711yescharacteristic
518C3C5×#189211yescharacteristic, complemented, Sylow
519C2—64no—
520C2C5×D3×#10311yescharacteristic
521C2C5×#480411yescharacteristic
522C2C5×#480711yescharacteristic
523C2C5×#481311yescharacteristic
524C2C5×#495611yescharacteristic
525C2C5×#517111yescharacteristic
526C2C5×#518811yescharacteristic
subgroupG/H
centerC23×C5D3×D4
commutatorC2×C3×C4C22×C4×C5
FrattiniC22×C4C22×C5×D3
FittingC2×C3×C5×#103C2
socleC23×C3×C5C2×D4
prime radicalC23×C5×D3D4
cosocleC22×C3×C4C23×C5
hypercenterC2×C5×#103D3
hypocenterC3C5×#1892
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C5 ◁ C22×C4×C5 ◁ C2×C5×#103

Complements

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

NH ≅ G/Ntype
3. C2×C3×C5×#103C2semidirect
4. C2×C5×#835C2semidirect
6. C5×#4551C2semidirect
7. C5×#4529C2semidirect
14. C5×#946C4semidirect
27. #70270C5direct (N × H)
31. C2×C5×#103D3semidirect
72. C2×C3×#103C2×C5semidirect
73. C2×#835C2×C5semidirect
75. #4551C2×C5semidirect
76. #4529C2×C5semidirect
140. #946C4×C5semidirect
191. C2×#103C5×D3semidirect
208. C2×C5×D3#103semidirect
415. C3×C5#1892semidirect
429. C2×D3C5×#103semidirect
494. C5#70270direct (N × H)
518. C3C5×#1892semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#9462
C22×C5×D3×C82
C2×C3×C5×#1032
C2×C5×#8352
C5×#49872
C5×#45512
C5×#45292
C5×#18923
#702705