Finite Group Databasebeta

← C5×#69499 #14301609

Subgroups — C5×#69499 #14301609

Order 1920.

1156 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#953C211yesmaximal, characteristic
2C5×#5486C211yesmaximal, characteristic, complemented
3C3×C5×#379C211yesmaximal, characteristic, complemented
4C5×#5247C211yesmaximal, characteristic, complemented
5C2×C5×#806C211yesmaximal, characteristic, complemented
6C5×#4527C211yesmaximal, characteristic, complemented
7C5×#4526C211yesmaximal, characteristic, complemented
8C5×#1889—31nomaximal, Hall
9C25×C3×C5C2211yescharacteristic
10C22×C5×#67—21no—
11C22×C5×#67C2211yescharacteristic
12C2×C3×C5×C22⋊C4—22no—
13C2×C3×C5×C22⋊C4C2211yescharacteristic
14C2×C3×C5×C22⋊C4C2211yescharacteristic
15C5×#953—22no—
16C5×#953C414yescomplemented
17C2×C5×#217—21no—
18C2×C5×#217—22no—
19C2×C5×#217C2211yescharacteristic
20C2×C5×#217C2211yescharacteristic
21C2×C5×#217C2211yescharacteristic
22C4×C5×#67—22no—
23C5×#67⋊C4—22no—
24C5×D3×C22⋊C4—21no—
25C5×#877—21no—
26C5×#806—22no—
27#69499C511yesmaximal, characteristic, direct product factor, Hall
28C2×C5×#120—31no—
29C5×#518—31no—
30C2×C5×#99—31no—
31C5×#379—31no—
32C5×#379D311yescharacteristic, complemented
33C5×#342—31no—
34C5×#342—31no—
35C24×C3×C5—21no—
36C24×C3×C5—21no—
37C24×C3×C5—21no—
38C24×C3×C5—21no—
39C24×C3×C5—21no—
40C24×C3×C5—21no—
41C24×C3×C5—44no—
42C24×C3×C5C2×C412yescomplemented
43C24×C3×C5C2311yescharacteristic
44C23×C5×D3D411yescharacteristic
45C22×C3×C4×C5—21no—
46C22×C3×C4×C5D412yescomplemented
47C2×C5×#67—21no—
48C2×C5×#67—21no—
49C2×C5×#67—21no—
50C2×C5×#67—21no—
51C2×C5×#67—24no—
52C2×C5×#67—44no—
53C2×C5×#67C2×C412yes—
54C2×C5×#67D412yes—
55C22×C5×Dic3—21no—
56C22×C5×Dic3—21no—
57C22×C5×Dic3D411yescharacteristic
58C22×C5×Dic3D412yescomplemented
59C2×C4×C5×D3—22no—
60C3×C5×C22⋊C4—21no—
61C3×C5×C22⋊C4—21no—
62C3×C5×C22⋊C4—21no—
63C3×C5×C22⋊C4—21no—
64C3×C5×C22⋊C4—44no—
65C5×#217—21no—
66C5×#217—21no—
67C5×#217—21no—
68C5×#217—24no—
69C5×#217—21no—
70C5×#217—21no—
71C5×#217—44no—
72C5×#217C2×C412yescomplemented
73C5×(C2×D3)⋊C4—22no—
74C5×Dic3⋊C4—22no—
75C4×C5×Dic3—22no—
76C2×#953C2×C511yescharacteristic
77#5486C2×C511yescharacteristic, complemented
78C3×#379C2×C511yescharacteristic, complemented
79#5247C2×C511yescharacteristic, complemented
80C2×#806C2×C511yescharacteristic, complemented
81#4527C2×C511yescharacteristic, complemented
82#4526C2×C511yescharacteristic, complemented
83C25×C5C2×D311yescharacteristic
84C22×C5×D4—31no—
85C22×C5×D4—61no—
86C5×#120—34no—
87C5×#120—62no—
88C4×C5×D4—62no—
89C4×C5×D4—62no—
90C5×#117—61no—
91C2×C5×C22⋊C4—22no—
92C2×C5×C22⋊C4—31no—
93C2×C5×C22⋊C4—31no—
94C2×C5×C22⋊C4—31no—
95C2×C5×C22⋊C4—61no—
96C2×C5×C22⋊C4—61no—
97C2×C5×C22⋊C4—62no—
98C2×C5×C22⋊C4C2×D311yescharacteristic
99C2×C5×C22⋊C4C2×D311yescharacteristic
100C5×#99—62no—
101#1889—31noSylow
102C23×C3×C5—24no—
103C23×C3×C5—24no—
104C23×C3×C5—24no—
105C23×C3×C5—22no—
106C23×C3×C5—22no—
107C23×C3×C5—22no—
108C23×C3×C5—44no—
109C23×C3×C5—42no—
110C23×C3×C5—42no—
111C23×C3×C5—42no—
112C23×C3×C5—42no—
113C23×C3×C5—44no—
114C23×C3×C5—44no—
115C23×C3×C5—44no—
116C23×C3×C5—41no—
117C23×C3×C5—41no—
118C23×C3×C5—41no—
119C23×C3×C5—41no—
120C23×C3×C5C22×C411yescharacteristic
121C23×C3×C5C2×D411yescharacteristic
122C23×C3×C5C2×D411yescharacteristic
123C23×C3×C5C2×D411yescharacteristic
124C23×C3×C5C2×D411yescharacteristic
125C23×C3×C5#4111yescharacteristic
126C23×C3×C5#4111yescharacteristic
127C22×C5×D3—21no—
128C22×C5×D3—21no—
129C22×C5×D3—42no—
130C22×C5×D3C22⋊C412yescomplemented
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—42no—
133C2×C3×C4×C5—42no—
134C2×C3×C4×C5—41no—
135C5×#67—22no—
136C5×#67—22no—
137C5×#67—22no—
138C5×#67—22no—
139C5×#67—48no—
140C2×C5×Dic3—21no—
141C2×C5×Dic3—21no—
142C2×C5×Dic3—22no—
143C2×C5×Dic3—22no—
144C2×C5×Dic3—42no—
145C2×C5×Dic3—42no—
146C2×C5×Dic3—42no—
147C2×C5×Dic3—41no—
148C2×C5×Dic3C22⋊C412yescomplemented
149C4×C5×D3—82no—
150C25×C3C22×C511yescharacteristic
151C22×#67—21no—
152C22×#67C22×C511yescharacteristic
153C2×C3×C22⋊C4—22no—
154C2×C3×C22⋊C4C22×C511yescharacteristic
155C2×C3×C22⋊C4C22×C511yescharacteristic
156#953—22no—
157#953C4×C514yescomplemented
158C2×#217—21no—
159C2×#217—22no—
160C2×#217C22×C511yescharacteristic
161C2×#217C22×C511yescharacteristic
162C2×#217C22×C511yescharacteristic
163C4×#67—22no—
164#67⋊C4—22no—
165D3×C22⋊C4—21no—
166#877—21no—
167#806—22no—
168C24×C5—21no—
169C24×C5—21no—
170C24×C5—21no—
171C24×C5—21no—
172C24×C5—21no—
173C24×C5—21no—
174C24×C5—31no—
175C24×C5—44no—
176C24×C5C4×D312yescomplemented
177C24×C5C22×D311yescharacteristic
178C2×C5×D4—32no—
179C2×C5×D4—32no—
180C2×C5×D4—61no—
181C2×C5×D4—61no—
182C2×C5×D4—61no—
183C2×C5×D4—61no—
184C2×C5×D4—64no—
185C2×C5×D4—124no—
186C22×C4×C5—21no—
187C22×C4×C5—31no—
188C22×C4×C5—32no—
189C22×C4×C5—62no—
190C22×C4×C5—61no—
191C22×C4×C5—61no—
192C22×C4×C5#6712yescomplemented
193C5×C4⋊C4—62no—
194C5×C22⋊C4—21no—
195C5×C22⋊C4—21no—
196C5×C22⋊C4—21no—
197C5×C22⋊C4—21no—
198C5×C22⋊C4—32no—
199C5×C22⋊C4—44no—
200C5×C22⋊C4—62no—
201C5×C22⋊C4—64no—
202C5×C22⋊C4—61no—
203C5×C22⋊C4—61no—
204C5×C22⋊C4—61no—
205C5×C22⋊C4—61no—
206C5×C22⋊C4—61no—
207C5×C22⋊C4—124no—
208C42×C5—62no—
209C2×#120—31no—
210#518—31no—
211C2×#99—31no—
212#379—31no—
213#379C5×D311yescharacteristic, complemented
214#342—31no—
215#342—31no—
216C22×C3×C5—22no—
217C22×C3×C5—22no—
218C22×C3×C5—22no—
219C22×C3×C5—24no—
220C22×C3×C5—24no—
221C22×C3×C5—24no—
222C22×C3×C5—41no—
223C22×C3×C5—41no—
224C22×C3×C5—41no—
225C22×C3×C5—41no—
226C22×C3×C5—42no—
227C22×C3×C5—44no—
228C22×C3×C5—44no—
229C22×C3×C5—44no—
230C22×C3×C5—42no—
231C22×C3×C5—42no—
232C22×C3×C5—42no—
233C22×C3×C5—44no—
234C22×C3×C5C2×C22⋊C411yescharacteristic
235C22×C3×C5C4×D411yescharacteristic
236C22×C3×C5C4×D411yescharacteristic
237C22×C3×C5#12011yescharacteristic
238C22×C3×C5#12111yescharacteristic
239C22×C3×C5#12111yescharacteristic
240C22×C3×C5#12311yescharacteristic
241C2×C5×D3—22no—
242C2×C5×D3—42no—
243C2×C5×D3—42no—
244C2×C5×D3—81no—
245C3×C4×C5—42no—
246C3×C4×C5—81no—
247C5×Dic3—22no—
248C5×Dic3—42no—
249C5×Dic3—42no—
250C5×Dic3—81no—
251C24×C3—21no—
252C24×C3—21no—
253C24×C3—21no—
254C24×C3—21no—
255C24×C3—21no—
256C24×C3—21no—
257C24×C3—44no—
258C24×C3C2×C4×C512yescomplemented
259C24×C3C23×C511yescharacteristic
260C23×D3C5×D411yescharacteristic
261C22×C3×C4—21no—
262C22×C3×C4C5×D412yescomplemented
263C2×#67—21no—
264C2×#67—21no—
265C2×#67—21no—
266C2×#67—21no—
267C2×#67—24no—
268C2×#67—44no—
269C2×#67C2×C4×C512yes—
270C2×#67C5×D412yes—
271C22×Dic3—21no—
272C22×Dic3—21no—
273C22×Dic3C5×D411yescharacteristic
274C22×Dic3C5×D412yescomplemented
275C2×C4×D3—22no—
276C3×C22⋊C4—21no—
277C3×C22⋊C4—21no—
278C3×C22⋊C4—21no—
279C3×C22⋊C4—21no—
280C3×C22⋊C4—44no—
281#217—21no—
282#217—21no—
283#217—21no—
284#217—24no—
285#217—21no—
286#217—21no—
287#217—44no—
288#217C2×C4×C512yescomplemented
289(C2×D3)⋊C4—22no—
290Dic3⋊C4—22no—
291C4×Dic3—22no—
292C23×C5—24no—
293C23×C5—24no—
294C23×C5—24no—
295C23×C5—22no—
296C23×C5—22no—
297C23×C5—22no—
298C23×C5—32no—
299C23×C5—44no—
300C23×C5—42no—
301C23×C5—42no—
302C23×C5—42no—
303C23×C5—42no—
304C23×C5—44no—
305C23×C5—44no—
306C23×C5—44no—
307C23×C5—41no—
308C23×C5—41no—
309C23×C5—41no—
310C23×C5—41no—
311C23×C5—61no—
312C23×C5—61no—
313C23×C5—122no—
314C23×C5C2×C4×D311yescharacteristic
315C23×C5#23511yescharacteristic
316C23×C5D3×D411yescharacteristic
317C23×C5D3×D411yescharacteristic
318C23×C5D3×D411yescharacteristic
319C23×C5#23711yescharacteristic
320C23×C5C2×#6711yescharacteristic
321C5×D4—62no—
322C5×D4—62no—
323C5×D4—62no—
324C5×D4—62no—
325C5×D4—128no—
326C2×C4×C5—22no—
327C2×C4×C5—32no—
328C2×C4×C5—42no—
329C2×C4×C5—42no—
330C2×C4×C5—41no—
331C2×C4×C5—61no—
332C2×C4×C5—61no—
333C2×C4×C5—62no—
334C2×C4×C5—62no—
335C2×C4×C5—122no—
336C2×C4×C5—122no—
337C2×C4×C5—122no—
338C2×C4×C5—121no—
339C2×C4×C5—242no—
340C25C2×C5×D311yescharacteristic
341C22×D4—31no—
342C22×D4—61no—
343#120—34no—
344#120—62no—
345C4×D4—62no—
346C4×D4—62no—
347#117—61no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—31no—
350C2×C22⋊C4—31no—
351C2×C22⋊C4—31no—
352C2×C22⋊C4—61no—
353C2×C22⋊C4—61no—
354C2×C22⋊C4—62no—
355C2×C22⋊C4C2×C5×D311yescharacteristic
356C2×C22⋊C4C2×C5×D311yescharacteristic
357#99—62no—
358C2×C3×C5—21no—
359C2×C3×C5—21no—
360C2×C3×C5—21no—
361C2×C3×C5—21no—
362C2×C3×C5—21no—
363C2×C3×C5—21no—
364C2×C3×C5—44no—
365C2×C3×C5#38611yescharacteristic
366C2×C3×C5B(4,𝔽2)12yes—
367C5×D3—42no—
368C23×C3—24no—
369C23×C3—24no—
370C23×C3—24no—
371C23×C3—22no—
372C23×C3—22no—
373C23×C3—22no—
374C23×C3—44no—
375C23×C3—42no—
376C23×C3—42no—
377C23×C3—42no—
378C23×C3—42no—
379C23×C3—44no—
380C23×C3—44no—
381C23×C3—44no—
382C23×C3—41no—
383C23×C3—41no—
384C23×C3—41no—
385C23×C3—41no—
386C23×C3C22×C4×C511yescharacteristic
387C23×C3C2×C5×D411yescharacteristic
388C23×C3C2×C5×D411yescharacteristic
389C23×C3C2×C5×D411yescharacteristic
390C23×C3C2×C5×D411yescharacteristic
391C23×C3C5×#4111yescharacteristic
392C23×C3C5×#4111yescharacteristic
393C22×D3—21no—
394C22×D3—21no—
395C22×D3—42no—
396C22×D3C5×C22⋊C412yescomplemented
397C2×C3×C4—22no—
398C2×C3×C4—42no—
399C2×C3×C4—42no—
400C2×C3×C4—41no—
401#67—22no—
402#67—22no—
403#67—22no—
404#67—22no—
405#67—48no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—22no—
409C2×Dic3—22no—
410C2×Dic3—42no—
411C2×Dic3—42no—
412C2×Dic3—42no—
413C2×Dic3—41no—
414C2×Dic3C5×C22⋊C412yescomplemented
415C4×D3—82no—
416C22×C5—22no—
417C22×C5—22no—
418C22×C5—22no—
419C22×C5—24no—
420C22×C5—24no—
421C22×C5—24no—
422C22×C5—41no—
423C22×C5—41no—
424C22×C5—41no—
425C22×C5—41no—
426C22×C5—44no—
427C22×C5—42no—
428C22×C5—44no—
429C22×C5—42no—
430C22×C5—42no—
431C22×C5—44no—
432C22×C5—42no—
433C22×C5—44no—
434C22×C5—62no—
435C22×C5—122no—
436C22×C5—122no—
437C22×C5—241no—
438C22×C5D3×C22⋊C411yescharacteristic
439C22×C5#67⋊C411yescharacteristic
440C22×C5#88311yescharacteristic
441C22×C5#88411yescharacteristic
442C22×C5C4×#6711yescharacteristic
443C22×C5#93711yescharacteristic
444C22×C5#93811yescharacteristic
445C4×C5—42no—
446C4×C5—62no—
447C4×C5—81no—
448C4×C5—122no—
449C4×C5—122no—
450C4×C5—241no—
451C24—21no—
452C24—21no—
453C24—21no—
454C24—21no—
455C24—21no—
456C24—21no—
457C24—31no—
458C24—44no—
459C24C4×C5×D312yescomplemented
460C24C22×C5×D311yescharacteristic
461C2×D4—32no—
462C2×D4—32no—
463C2×D4—61no—
464C2×D4—61no—
465C2×D4—61no—
466C2×D4—61no—
467C2×D4—64no—
468C2×D4—124no—
469C22×C4—21no—
470C22×C4—31no—
471C22×C4—32no—
472C22×C4—62no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4C5×#6712yescomplemented
476C4⋊C4—62no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—32no—
482C22⋊C4—44no—
483C22⋊C4—62no—
484C22⋊C4—64no—
485C22⋊C4—61no—
486C22⋊C4—61no—
487C22⋊C4—61no—
488C22⋊C4—61no—
489C22⋊C4—61no—
490C22⋊C4—124no—
491C42—62no—
492C3×C5#188911yescharacteristic, complemented, Hall
493C22×C3—22no—
494C22×C3—22no—
495C22×C3—22no—
496C22×C3—24no—
497C22×C3—24no—
498C22×C3—24no—
499C22×C3—41no—
500C22×C3—41no—
501C22×C3—41no—
502C22×C3—41no—
503C22×C3—42no—
504C22×C3—44no—
505C22×C3—44no—
506C22×C3—44no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3—42no—
510C22×C3—44no—
511C22×C3C2×C5×C22⋊C411yescharacteristic
512C22×C3C4×C5×D411yescharacteristic
513C22×C3C4×C5×D411yescharacteristic
514C22×C3C5×#12011yescharacteristic
515C22×C3C5×#12111yescharacteristic
516C22×C3C5×#12111yescharacteristic
517C22×C3C5×#12311yescharacteristic
518C2×D3—22no—
519C2×D3—42no—
520C2×D3—42no—
521C2×D3—81no—
522C3×C4—42no—
523C3×C4—81no—
524Dic3—22no—
525Dic3—42no—
526Dic3—42no—
527Dic3—81no—
528C2×C5—21no—
529C2×C5—21no—
530C2×C5—21no—
531C2×C5—21no—
532C2×C5—21no—
533C2×C5—21no—
534C2×C5—44no—
535C2×C5—122no—
536C2×C5#495611yescharacteristic
537C2×C5#503212yes—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—32no—
545C23—44no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—61no—
558C23—61no—
559C23—122no—
560C23C2×C4×C5×D311yescharacteristic
561C23C5×#23511yescharacteristic
562C23C5×D3×D411yescharacteristic
563C23C5×D3×D411yescharacteristic
564C23C5×D3×D411yescharacteristic
565C23C5×#23711yescharacteristic
566C23C2×C5×#6711yescharacteristic
567D4—62no—
568D4—62no—
569D4—62no—
570D4—62no—
571D4—128no—
572C2×C4—22no—
573C2×C4—32no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—62no—
580C2×C4—62no—
581C2×C4—122no—
582C2×C4—122no—
583C2×C4—122no—
584C2×C4—121no—
585C2×C4—242no—
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3—21no—
591C2×C3—21no—
592C2×C3—44no—
593C2×C3C5×#38611yescharacteristic
594C2×C3C5×B(4,𝔽2)12yes—
595D3—42no—
596C5#6949911yescharacteristic, direct product factor, Sylow
597C22—22no—
598C22—22no—
599C22—22no—
600C22—24no—
601C22—24no—
602C22—24no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—44no—
608C22—42no—
609C22—44no—
610C22—42no—
611C22—42no—
612C22—44no—
613C22—42no—
614C22—44no—
615C22—62no—
616C22—122no—
617C22—122no—
618C22—241no—
619C22C5×D3×C22⋊C411yescharacteristic
620C22C5×#67⋊C411yescharacteristic
621C22C5×#88311yescharacteristic
622C22C5×#88411yescharacteristic
623C22C4×C5×#6711yescharacteristic
624C22C5×#93711yescharacteristic
625C22C5×#93811yescharacteristic
626C4—42no—
627C4—62no—
628C4—81no—
629C4—122no—
630C4—122no—
631C4—241no—
632C3C5×#188911yescharacteristic, complemented, Sylow
633C2—21no—
634C2—21no—
635C2—21no—
636C2—21no—
637C2—21no—
638C2—21no—
639C2—44no—
640C2—122no—
641C2C5×#495611yescharacteristic
642C2C5×#503212yes—
subgroupG/H
centerC22×C5#937
commutatorC23×C3C22×C4×C5
FrattiniC24C22×C5×D3
FittingC3×C5×#379C2
socleC22×C3×C5#120
prime radicalC2×C5×#953C2
cosocleC24×C3C23×C5
hypercenterC5×#379D3
hypocenterC3C5×#1889
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C24×C5 ◁ C5×#379

Complements

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

NH ≅ G/Ntype
2. C5×#5486C2semidirect
3. C3×C5×#379C2semidirect
4. C5×#5247C2semidirect
5. C2×C5×#806C2semidirect
6. C5×#4527C2semidirect
7. C5×#4526C2semidirect
16. C5×#953C4semidirect
27. #69499C5direct (N × H)
32. C5×#379D3semidirect
42. C24×C3×C5C2×C4semidirect
46. C22×C3×C4×C5D4semidirect
58. C22×C5×Dic3D4semidirect
72. C5×#217C2×C4semidirect
77. #5486C2×C5semidirect
78. C3×#379C2×C5semidirect
79. #5247C2×C5semidirect
80. C2×#806C2×C5semidirect
81. #4527C2×C5semidirect
82. #4526C2×C5semidirect
130. C22×C5×D3C22⋊C4semidirect
148. C2×C5×Dic3C22⋊C4semidirect
157. #953C4×C5semidirect
176. C24×C5C4×D3semidirect
192. C22×C4×C5#67semidirect
213. #379C5×D3semidirect
258. C24×C3C2×C4×C5semidirect
262. C22×C3×C4C5×D4semidirect
274. C22×Dic3C5×D4semidirect
288. #217C2×C4×C5semidirect
396. C22×D3C5×C22⋊C4semidirect
414. C2×Dic3C5×C22⋊C4semidirect
459. C24C4×C5×D3semidirect
475. C22×C4C5×#67semidirect
492. C3×C5#1889semidirect
596. C5#69499direct (N × H)
632. C3C5×#1889semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#9532
C5×#54862
C3×C5×#3792
C5×#52472
C2×C5×#8062
C5×#45272
C5×#45262
C5×#18893
#694995