Finite Group Databasebeta

← C2×C5×#5053 #14425627

Subgroups — C2×C5×#5053 #14425627

Order 1920.

1588 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#1004C211yesmaximal, characteristic
2C2×C3×C5×#123C211yesmaximal, characteristic, complemented
3C2×C5×#929C212yesmaximal, complemented
4C2×C5×#834C211yesmaximal, characteristic, complemented
5C5×#5053C218yesmaximal, direct product factor
6C2×C5×#806C212yesmaximal, complemented
7C2×C5×#458—31nomaximal, Hall
8C22×C3×C5×D4C2211yescharacteristic
9C23×C3×C4×C5C2211yescharacteristic, complemented
10C22×C5×#67—21no—
11C22×C5×#67C2212yes—
12C5×#1004—22no—
13C5×#1004—24no—
14C5×#1004C2214yes—
15C2×C5×#237—22no—
16C2×C5×D3×D4—22no—
17C2×C5×#235—21no—
18C3×C5×#123—22no—
19C3×C5×#123C2214yescomplemented
20C2×C3×C5×C4⋊C4—21no—
21C2×C3×C5×C22⋊C4—21no—
22C2×C3×C5×C22⋊C4C2211yescharacteristic, complemented
23C2×C5×#217C2212yescomplemented
24C5×#929—24no—
25C5×#929C2218yescomplemented
26C2×C5×(C2×D3)⋊C4—22no—
27C2×C5×Dic3⋊C4—22no—
28C5×#834C2214yescomplemented
29C5×#806C2218yescomplemented
30C2×#5053C511yesmaximal, characteristic, direct product factor, Hall
31C2×C5×H(5,𝔽2)—31no—
32C2×C5×#123—32no—
33C2×C5×#123D311yescharacteristic, complemented
34C5×#458—38no—
35C2×C5×#99—31no—
36C2×C5×#99—32no—
37C24×C3×C5D411yescharacteristic
38C24×C3×C5C2311yescharacteristic
39C23×C5×D3—21no—
40C23×C5×D3D412yes—
41C2×C3×C5×D4—22no—
42C2×C3×C5×D4—22no—
43C2×C3×C5×D4D412yes—
44C2×C3×C5×D4C2312yes—
45C22×C3×C4×C5—22no—
46C22×C3×C4×C5—22no—
47C22×C3×C4×C5—21no—
48C22×C3×C4×C5—21no—
49C22×C3×C4×C5—21no—
50C22×C3×C4×C5D411yescharacteristic
51C22×C3×C4×C5C2312yescomplemented
52C2×C5×#67—22no—
53C2×C5×#67—24no—
54C2×C5×#67—22no—
55C2×C5×#67—22no—
56C2×C5×#67—22no—
57C2×C5×#67—22no—
58C2×C5×#67—24no—
59C2×C5×#67D414yes—
60C2×C5×#67C2314yes—
61C22×C5×Dic3—21no—
62C22×C5×Dic3—22no—
63C22×C5×Dic3D412yes—
64C5×#237—28no—
65C5×#237—28no—
66C5×D3×D4—28no—
67C5×D3×D4—28no—
68C5×#235—24no—
69C5×#235—24no—
70C2×C5×D12—21no—
71C2×C4×C5×D3—22no—
72C2×C5×Dic6—21no—
73C3×C5×C4⋊C4—24no—
74C3×C5×C22⋊C4—24no—
75C3×C5×C22⋊C4—21no—
76C3×C5×C22⋊C4C2312yescomplemented
77C5×#217—22no—
78C5×#217C2314yescomplemented
79C5×(C2×D3)⋊C4—28no—
80C5×Dic3⋊C4—28no—
81C2×#1004C2×C511yescharacteristic
82C2×C3×#123C2×C511yescharacteristic, complemented
83C2×#929C2×C512yescomplemented
84C2×#834C2×C511yescharacteristic, complemented
85#5053C2×C518yesdirect product factor
86C2×#806C2×C512yescomplemented
87C5×H(5,𝔽2)—34no—
88C5×H(5,𝔽2)—64no—
89C5×H(5,𝔽2)—62no—
90C2×C5×#41—62no—
91C2×C5×#41—61no—
92C22×C5×D4—32no—
93C22×C5×D4—62no—
94C22×C5×D4—61no—
95C22×C5×D4C2×D311yescharacteristic
96C23×C4×C5C2×D311yescharacteristic, complemented
97C5×#123—22no—
98C5×#123—38no—
99C5×#123—64no—
100C5×#123C2×D314yescomplemented
101C2×C5×C4⋊C4—21no—
102C2×C5×C4⋊C4—62no—
103C2×C5×C22⋊C4—21no—
104C2×C5×C22⋊C4—32no—
105C2×C5×C22⋊C4—62no—
106C2×C5×C22⋊C4C2×D311yescharacteristic, complemented
107C5×#99—34no—
108C5×#99—38no—
109C2×#458—31noSylow
110C23×C3×C5—21no—
111C23×C3×C5—21no—
112C23×C3×C5—42no—
113C23×C3×C5—42no—
114C23×C3×C5C2×D412yes—
115C23×C3×C5C2×D411yescharacteristic
116C23×C3×C5C2×D412yes—
117C23×C3×C5C2×D412yes—
118C23×C3×C5C2×D411yescharacteristic
119C23×C3×C5C2411yescharacteristic
120C22×C5×D3—22no—
121C22×C5×D3—24no—
122C22×C5×D3—22no—
123C22×C5×D3—44no—
124C22×C5×D3—82no—
125C22×C5×D3C2×D414yes—
126C3×C5×D4—24no—
127C3×C5×D4—24no—
128C2×C3×C4×C5—21no—
129C2×C3×C4×C5—21no—
130C2×C3×C4×C5—24no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—42no—
134C2×C3×C4×C5—42no—
135C2×C3×C4×C5—41no—
136C2×C3×C4×C5—41no—
137C2×C3×C4×C5C2×D412yes—
138C5×#67—28no—
139C5×#67—24no—
140C5×#67—24no—
141C5×#67—24no—
142C5×#67—24no—
143C2×C5×Dic3—24no—
144C2×C5×Dic3—22no—
145C2×C5×Dic3—22no—
146C2×C5×Dic3—24no—
147C2×C5×Dic3—44no—
148C2×C5×Dic3C2×D414yes—
149C5×D12—24no—
150C4×C5×D3—28no—
151C5×Dic6—24no—
152C22×C3×D4C22×C511yescharacteristic
153C23×C3×C4C22×C511yescharacteristic, complemented
154C22×#67—21no—
155C22×#67C22×C512yes—
156#1004—22no—
157#1004—24no—
158#1004C22×C514yes—
159C2×#237—22no—
160C2×D3×D4—22no—
161C2×#235—21no—
162C3×#123—22no—
163C3×#123C22×C514yescomplemented
164C2×C3×C4⋊C4—21no—
165C2×C3×C22⋊C4—21no—
166C2×C3×C22⋊C4C22×C511yescharacteristic, complemented
167C2×#217C22×C512yescomplemented
168#929—24no—
169#929C22×C518yescomplemented
170C2×(C2×D3)⋊C4—22no—
171C2×Dic3⋊C4—22no—
172#834C22×C514yescomplemented
173#806C22×C518yescomplemented
174C24×C5—32no—
175C24×C5—61no—
176C24×C5#6711yescharacteristic
177C24×C5C22×D311yescharacteristic
178C5×#41—64no—
179C5×#41—68no—
180C5×#41—68no—
181C5×#41—64no—
182C2×C5×Q8—61no—
183C2×C5×D4—22no—
184C2×C5×D4—22no—
185C2×C5×D4—34no—
186C2×C5×D4—34no—
187C2×C5×D4—61no—
188C2×C5×D4—68no—
189C2×C5×D4—68no—
190C2×C5×D4—62no—
191C2×C5×D4—64no—
192C2×C5×D4—62no—
193C2×C5×D4—62no—
194C2×C5×D4—62no—
195C2×C5×D4—62no—
196C2×C5×D4—64no—
197C2×C5×D4#6712yes—
198C2×C5×D4C22×D312yes—
199C22×C4×C5—22no—
200C22×C4×C5—22no—
201C22×C4×C5—21no—
202C22×C4×C5—21no—
203C22×C4×C5—21no—
204C22×C4×C5—32no—
205C22×C4×C5—62no—
206C22×C4×C5—61no—
207C22×C4×C5—62no—
208C22×C4×C5#6711yescharacteristic
209C22×C4×C5C22×D312yescomplemented
210C5×C4⋊C4—24no—
211C5×C4⋊C4—68no—
212C5×C22⋊C4—24no—
213C5×C22⋊C4—21no—
214C5×C22⋊C4—34no—
215C5×C22⋊C4—68no—
216C5×C22⋊C4—62no—
217C5×C22⋊C4C22×D312yescomplemented
218C2×H(5,𝔽2)—31no—
219C2×#123—32no—
220C2×#123C5×D311yescharacteristic, complemented
221#458—38no—
222C2×#99—31no—
223C2×#99—32no—
224C22×C3×C5—24no—
225C22×C3×C5—22no—
226C22×C3×C5—22no—
227C22×C3×C5—42no—
228C22×C3×C5—41no—
229C22×C3×C5—41no—
230C22×C3×C5—42no—
231C22×C3×C5—42no—
232C22×C3×C5—81no—
233C22×C3×C5#12012yes—
234C22×C3×C5#12011yescharacteristic
235C22×C3×C5#12011yescharacteristic
236C22×C3×C5C22×D412yes—
237C22×C3×C5C22×D411yescharacteristic
238C2×C5×D3—24no—
239C2×C5×D3—44no—
240C2×C5×D3—44no—
241C2×C5×D3—82no—
242C2×C5×D3—84no—
243C3×C4×C5—22no—
244C3×C4×C5—42no—
245C3×C4×C5—42no—
246C5×Dic3—24no—
247C5×Dic3—44no—
248C24×C3C5×D411yescharacteristic
249C24×C3C23×C511yescharacteristic
250C23×D3—21no—
251C23×D3C5×D412yes—
252C2×C3×D4—22no—
253C2×C3×D4—22no—
254C2×C3×D4C5×D412yes—
255C2×C3×D4C23×C512yes—
256C22×C3×C4—22no—
257C22×C3×C4—22no—
258C22×C3×C4—21no—
259C22×C3×C4—21no—
260C22×C3×C4—21no—
261C22×C3×C4C5×D411yescharacteristic
262C22×C3×C4C23×C512yescomplemented
263C2×#67—22no—
264C2×#67—24no—
265C2×#67—22no—
266C2×#67—22no—
267C2×#67—22no—
268C2×#67—22no—
269C2×#67—24no—
270C2×#67C5×D414yes—
271C2×#67C23×C514yes—
272C22×Dic3—21no—
273C22×Dic3—22no—
274C22×Dic3C5×D412yes—
275#237—28no—
276#237—28no—
277D3×D4—28no—
278D3×D4—28no—
279#235—24no—
280#235—24no—
281C2×D12—21no—
282C2×C4×D3—22no—
283C2×Dic6—21no—
284C3×C4⋊C4—24no—
285C3×C22⋊C4—24no—
286C3×C22⋊C4—21no—
287C3×C22⋊C4C23×C512yescomplemented
288#217—22no—
289#217C23×C514yescomplemented
290(C2×D3)⋊C4—28no—
291Dic3⋊C4—28no—
292C23×C5—21no—
293C23×C5—21no—
294C23×C5—34no—
295C23×C5—42no—
296C23×C5—42no—
297C23×C5—64no—
298C23×C5—62no—
299C23×C5—62no—
300C23×C5—124no—
301C23×C5—242no—
302C23×C5D3×D412yes—
303C23×C5D3×D412yes—
304C23×C5C2×#6711yescharacteristic
305C23×C5C2×#6712yes—
306C23×C5C2×#6711yescharacteristic
307C23×C5C23×D311yescharacteristic
308C5×Q8—64no—
309C5×D4—24no—
310C5×D4—24no—
311C5×D4—64no—
312C5×D4—68no—
313C5×D4—64no—
314C5×D4—64no—
315C5×D4—64no—
316C5×D4—64no—
317C2×C4×C5—21no—
318C2×C4×C5—21no—
319C2×C4×C5—24no—
320C2×C4×C5—22no—
321C2×C4×C5—22no—
322C2×C4×C5—34no—
323C2×C4×C5—42no—
324C2×C4×C5—41no—
325C2×C4×C5—42no—
326C2×C4×C5—41no—
327C2×C4×C5—68no—
328C2×C4×C5—64no—
329C2×C4×C5—62no—
330C2×C4×C5—62no—
331C2×C4×C5—64no—
332C2×C4×C5—124no—
333C2×C4×C5C2×#6712yes—
334H(5,𝔽2)—34no—
335H(5,𝔽2)—64no—
336H(5,𝔽2)—62no—
337C2×#41—62no—
338C2×#41—61no—
339C22×D4—32no—
340C22×D4—62no—
341C22×D4—61no—
342C22×D4C2×C5×D311yescharacteristic
343C23×C4C2×C5×D311yescharacteristic, complemented
344#123—22no—
345#123—38no—
346#123—64no—
347#123C2×C5×D314yescomplemented
348C2×C4⋊C4—21no—
349C2×C4⋊C4—62no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—32no—
352C2×C22⋊C4—62no—
353C2×C22⋊C4C2×C5×D311yescharacteristic, complemented
354#99—34no—
355#99—38no—
356C2×C3×C5—22no—
357C2×C3×C5—21no—
358C2×C3×C5—22no—
359C2×C3×C5—21no—
360C2×C3×C5—42no—
361C2×C3×C5#45812yescomplemented
362C2×C3×C5C2×#12011yescharacteristic
363C5×D3—44no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—42no—
367C23×C3—42no—
368C23×C3C2×C5×D412yes—
369C23×C3C2×C5×D412yes—
370C23×C3C2×C5×D411yescharacteristic
371C23×C3C2×C5×D412yes—
372C23×C3C2×C5×D411yescharacteristic
373C23×C3C24×C511yescharacteristic
374C22×D3—22no—
375C22×D3—24no—
376C22×D3—22no—
377C22×D3—44no—
378C22×D3—82no—
379C22×D3C2×C5×D414yes—
380C3×D4—24no—
381C3×D4—24no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—24no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—42no—
388C2×C3×C4—42no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4C2×C5×D412yes—
392#67—28no—
393#67—24no—
394#67—24no—
395#67—24no—
396#67—24no—
397C2×Dic3—24no—
398C2×Dic3—22no—
399C2×Dic3—22no—
400C2×Dic3—24no—
401C2×Dic3—44no—
402C2×Dic3C2×C5×D414yes—
403D12—24no—
404C4×D3—28no—
405Dic6—24no—
406C22×C5—22no—
407C22×C5—24no—
408C22×C5—22no—
409C22×C5—42no—
410C22×C5—41no—
411C22×C5—41no—
412C22×C5—42no—
413C22×C5—42no—
414C22×C5—64no—
415C22×C5—81no—
416C22×C5—124no—
417C22×C5—124no—
418C22×C5—242no—
419C22×C5—244no—
420C22×C5#93711yescharacteristic
421C22×C5#93712yes—
422C22×C5#93711yescharacteristic
423C22×C5C2×D3×D412yes—
424C22×C5C22×#6711yescharacteristic
425C4×C5—22no—
426C4×C5—42no—
427C4×C5—42no—
428C4×C5—64no—
429C4×C5—124no—
430C24—32no—
431C24—61no—
432C24C5×#6711yescharacteristic
433C24C22×C5×D311yescharacteristic
434#41—64no—
435#41—68no—
436#41—68no—
437#41—64no—
438C2×Q8—61no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—34no—
442C2×D4—34no—
443C2×D4—61no—
444C2×D4—68no—
445C2×D4—68no—
446C2×D4—62no—
447C2×D4—64no—
448C2×D4—62no—
449C2×D4—62no—
450C2×D4—62no—
451C2×D4—62no—
452C2×D4—64no—
453C2×D4C5×#6712yes—
454C2×D4C22×C5×D312yes—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—32no—
461C22×C4—62no—
462C22×C4—61no—
463C22×C4—62no—
464C22×C4C5×#6711yescharacteristic
465C22×C4C22×C5×D312yescomplemented
466C4⋊C4—24no—
467C4⋊C4—68no—
468C22⋊C4—24no—
469C22⋊C4—21no—
470C22⋊C4—34no—
471C22⋊C4—68no—
472C22⋊C4—62no—
473C22⋊C4C22×C5×D312yescomplemented
474C3×C5C2×#45811yescharacteristic, complemented, Hall
475C22×C3—24no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—42no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3—81no—
484C22×C3C5×#12012yes—
485C22×C3C5×#12011yescharacteristic
486C22×C3C5×#12011yescharacteristic
487C22×C3C22×C5×D412yes—
488C22×C3C22×C5×D411yescharacteristic
489C2×D3—24no—
490C2×D3—44no—
491C2×D3—44no—
492C2×D3—82no—
493C2×D3—84no—
494C3×C4—22no—
495C3×C4—42no—
496C3×C4—42no—
497Dic3—24no—
498Dic3—44no—
499C2×C5—21no—
500C2×C5—22no—
501C2×C5—21no—
502C2×C5—22no—
503C2×C5—42no—
504C2×C5—124no—
505C2×C5#505312yesdirect product factor
506C2×C5C2×#93711yescharacteristic
507C23—21no—
508C23—21no—
509C23—34no—
510C23—42no—
511C23—42no—
512C23—64no—
513C23—62no—
514C23—62no—
515C23—124no—
516C23—242no—
517C23C5×D3×D412yes—
518C23C5×D3×D412yes—
519C23C2×C5×#6711yescharacteristic
520C23C2×C5×#6712yes—
521C23C2×C5×#6711yescharacteristic
522C23C23×C5×D311yescharacteristic
523Q8—64no—
524D4—24no—
525D4—24no—
526D4—64no—
527D4—68no—
528D4—64no—
529D4—64no—
530D4—64no—
531D4—64no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—34no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—68no—
543C2×C4—64no—
544C2×C4—62no—
545C2×C4—62no—
546C2×C4—64no—
547C2×C4—124no—
548C2×C4C2×C5×#6712yes—
549C2×C3—22no—
550C2×C3—21no—
551C2×C3—22no—
552C2×C3—21no—
553C2×C3—42no—
554C2×C3C5×#45812yescomplemented
555C2×C3C2×C5×#12011yescharacteristic
556D3—44no—
557C5C2×#505311yescharacteristic, direct product factor, Sylow
558C22—22no—
559C22—24no—
560C22—22no—
561C22—42no—
562C22—41no—
563C22—41no—
564C22—42no—
565C22—42no—
566C22—64no—
567C22—81no—
568C22—124no—
569C22—124no—
570C22—242no—
571C22—244no—
572C22C5×#93711yescharacteristic
573C22C5×#93712yes—
574C22C5×#93711yescharacteristic
575C22C2×C5×D3×D412yes—
576C22C22×C5×#6711yescharacteristic
577C4—22no—
578C4—42no—
579C4—42no—
580C4—64no—
581C4—124no—
582C3C2×C5×#45811yescharacteristic, complemented, Sylow
583C2—21no—
584C2—22no—
585C2—21no—
586C2—22no—
587C2—42no—
588C2—124no—
589C2C5×#505312yesdirect product factor
590C2C2×C5×#93711yescharacteristic
subgroupG/H
centerC22×C5#937
commutator, cosocleC23×C3C24×C5
FrattiniC23C23×C5×D3
FittingC2×C3×C5×#123C2
socleC22×C3×C5#120
prime radicalC2×C5×#1004C2
hypercenterC2×C5×#123D3
hypocenterC3C2×C5×#458
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C24×C5 ◁ C2×C5×#123

Complements

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

NH ≅ G/Ntype
2. C2×C3×C5×#123C2semidirect
3. C2×C5×#929C2semidirect
4. C2×C5×#834C2semidirect
5. C5×#5053C2direct (N × H)
6. C2×C5×#806C2semidirect
9. C23×C3×C4×C5C22semidirect
19. C3×C5×#123C22semidirect
22. C2×C3×C5×C22⋊C4C22semidirect
23. C2×C5×#217C22semidirect
25. C5×#929C22semidirect
28. C5×#834C22semidirect
29. C5×#806C22semidirect
30. C2×#5053C5direct (N × H)
33. C2×C5×#123D3semidirect
51. C22×C3×C4×C5C23semidirect
76. C3×C5×C22⋊C4C23semidirect
78. C5×#217C23semidirect
82. C2×C3×#123C2×C5semidirect
83. C2×#929C2×C5semidirect
84. C2×#834C2×C5semidirect
85. #5053C2×C5direct (N × H)
86. C2×#806C2×C5semidirect
96. C23×C4×C5C2×D3semidirect
100. C5×#123C2×D3semidirect
106. C2×C5×C22⋊C4C2×D3semidirect
153. C23×C3×C4C22×C5semidirect
163. C3×#123C22×C5semidirect
166. C2×C3×C22⋊C4C22×C5semidirect
167. C2×#217C22×C5semidirect
169. #929C22×C5semidirect
172. #834C22×C5semidirect
173. #806C22×C5semidirect
209. C22×C4×C5C22×D3semidirect
217. C5×C22⋊C4C22×D3semidirect
220. C2×#123C5×D3semidirect
262. C22×C3×C4C23×C5semidirect
287. C3×C22⋊C4C23×C5semidirect
289. #217C23×C5semidirect
343. C23×C4C2×C5×D3semidirect
347. #123C2×C5×D3semidirect
353. C2×C22⋊C4C2×C5×D3semidirect
361. C2×C3×C5#458semidirect
465. C22×C4C22×C5×D3semidirect
473. C22⋊C4C22×C5×D3semidirect
474. C3×C5C2×#458semidirect
505. C2×C5#5053direct (N × H)
554. C2×C3C5×#458semidirect
557. C5C2×#5053direct (N × H)
582. C3C2×C5×#458semidirect
589. C2C5×#5053direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#10042
C2×C3×C5×#1232
C2×C5×#9292
C2×C5×#8342
C5×#50532
C2×C5×#8062
C2×C5×#4583
C2×#50535