Finite Group Databasebeta

← C2×C5×#2035 #11946196

Subgroups — C2×C5×#2035 #11946196

Order 1280.

1662 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C5×C22⋊Q8C211yesmaximal, characteristic
2C22×C5×#106C211yesmaximal, characteristic, complemented
3C22×C5×#103C211yesmaximal, characteristic, complemented
4C22×C5×#103C211yesmaximal, characteristic, complemented
5C22×C5×C22⋊C8C211yesmaximal, characteristic
6C2×C5×#380C211yesmaximal, characteristic
7C5×#2035C218yesmaximal, direct product factor
8C2×C5×#340C211yesmaximal, characteristic, complemented
9C23×C5×Q8C2211yescharacteristic
10C24×C4×C5C2211yescharacteristic
11C23×C5×C8C2211yescharacteristic
12C23×C5×C8C2211yescharacteristic
13C2×C5×C22⋊Q8—24no—
14C2×C5×C22⋊Q8—22no—
15C2×C5×C22⋊Q8—24no—
16C2×C5×C22⋊Q8C2214yes—
17C22×C5×C4⋊C4—21no—
18C22×C5×C4⋊C4C2211yescharacteristic
19C22×C5×C4⋊C4C2211yescharacteristic
20C22×C5×C4⋊C4C2211yescharacteristic
21C22×C5×C22⋊C4—21no—
22C22×C5×C22⋊C4—21no—
23C2×C5×#106—24no—
24C2×C5×#106C2214yescomplemented
25C2×C5×#103—24no—
26C2×C5×#103—24no—
27C2×C5×#103C2214yescomplemented
28C2×C5×#103C2214yescomplemented
29C2×C5×C22⋊C8—24no—
30C2×C5×C22⋊C8C2214yes—
31C5×#380—22no—
32C5×#380C2214yes—
33C2×C5×#95—21no—
34C5×#340C2214yescomplemented
35C2×#2035C511yesmaximal, characteristic, direct product factor, Sylow
36C25×C5D411yescharacteristic
37C22×C5×Q8—22no—
38C22×C5×Q8—21no—
39C22×C5×Q8—22no—
40C22×C5×Q8—22no—
41C22×C5×Q8—22no—
42C22×C5×Q8—22no—
43C22×C5×Q8D414yes—
44C22×C5×Q8C2312yes—
45C23×C4×C5—21no—
46C23×C4×C5—21no—
47C23×C4×C5—22no—
48C23×C4×C5—22no—
49C23×C4×C5—22no—
50C23×C4×C5—22no—
51C23×C4×C5—22no—
52C23×C4×C5—22no—
53C23×C4×C5—21no—
54C23×C4×C5—21no—
55C23×C4×C5D412yes—
56C23×C4×C5D411yescharacteristic
57C23×C4×C5C2312yes—
58C23×C4×C5C2311yescharacteristic
59C22×C5×C8—22no—
60C22×C5×C8—22no—
61C22×C5×C8—22no—
62C22×C5×C8—22no—
63C22×C5×C8—22no—
64C22×C5×C8D414yes—
65C22×C5×C8C2312yes—
66C22×C5×C8C2312yes—
67C5×C22⋊Q8—216no—
68C5×C22⋊Q8—28no—
69C5×C22⋊Q8—28no—
70C2×C5×C4⋊C4—28no—
71C2×C5×C4⋊C4—24no—
72C2×C5×C4⋊C4—21no—
73C2×C5×C4⋊C4—22no—
74C2×C5×C4⋊C4—22no—
75C2×C5×C4⋊C4—22no—
76C2×C5×C4⋊C4—22no—
77C2×C5×C4⋊C4—22no—
78C2×C5×C4⋊C4—21no—
79C2×C5×C4⋊C4—21no—
80C2×C5×C4⋊C4D414yes—
81C2×C5×C4⋊C4C2312yes—
82C2×C5×C4⋊C4C2312yes—
83C2×C5×C4⋊C4C2312yes—
84C2×C5×C22⋊C4—28no—
85C2×C5×C22⋊C4—24no—
86C2×C5×C22⋊C4—24no—
87C2×C5×C22⋊C4—44no—
88C5×#106—28no—
89C5×#103—28no—
90C5×#103—28no—
91C5×C22⋊C8—28no—
92C5×#95—24no—
93C22×C22⋊Q8C2×C511yescharacteristic
94C22×#106C2×C511yescharacteristic, complemented
95C22×#103C2×C511yescharacteristic, complemented
96C22×#103C2×C511yescharacteristic, complemented
97C22×C22⋊C8C2×C511yescharacteristic
98C2×#380C2×C511yescharacteristic
99#2035C2×C518yesdirect product factor
100C2×#340C2×C511yescharacteristic, complemented
101C24×C5—22no—
102C24×C5—22no—
103C24×C5—22no—
104C24×C5—22no—
105C24×C5—22no—
106C24×C5—22no—
107C24×C5SD1614yes—
108C24×C5C2×D412yes—
109C24×C5C2×D411yescharacteristic
110C2×C5×Q8—24no—
111C2×C5×Q8—22no—
112C2×C5×Q8—28no—
113C2×C5×Q8—28no—
114C2×C5×Q8—22no—
115C2×C5×Q8—22no—
116C2×C5×Q8—22no—
117C2×C5×Q8—22no—
118C2×C5×Q8—42no—
119C2×C5×Q8—44no—
120C2×C5×Q8—42no—
121C2×C5×Q8—44no—
122C2×C5×Q8C2×D414yes—
123C22×C4×C5—24no—
124C22×C4×C5—24no—
125C22×C4×C5—22no—
126C22×C4×C5—22no—
127C22×C4×C5—22no—
128C22×C4×C5—22no—
129C22×C4×C5—22no—
130C22×C4×C5—24no—
131C22×C4×C5—24no—
132C22×C4×C5—24no—
133C22×C4×C5—22no—
134C22×C4×C5—22no—
135C22×C4×C5—22no—
136C22×C4×C5—22no—
137C22×C4×C5—22no—
138C22×C4×C5—22no—
139C22×C4×C5—22no—
140C22×C4×C5—22no—
141C22×C4×C5—42no—
142C22×C4×C5—42no—
143C22×C4×C5—42no—
144C22×C4×C5—42no—
145C22×C4×C5—44no—
146C22×C4×C5—44no—
147C22×C4×C5—44no—
148C22×C4×C5—44no—
149C22×C4×C5—42no—
150C22×C4×C5—42no—
151C22×C4×C5—42no—
152C22×C4×C5—42no—
153C22×C4×C5—42no—
154C22×C4×C5—42no—
155C22×C4×C5C2×D412yes—
156C22×C4×C5C2×D411yescharacteristic
157C22×C4×C5C2×D412yes—
158C22×C4×C5C2×D412yes—
159C22×C4×C5C2×D412yes—
160C22×C4×C5#4112yes—
161C22×C4×C5#4112yes—
162C22×C4×C5#4112yes—
163C22×C4×C5C2411yescharacteristic
164C2×C5×C8—28no—
165C2×C5×C8—22no—
166C2×C5×C8—22no—
167C2×C5×C8—22no—
168C2×C5×C8—22no—
169C2×C5×C8—22no—
170C2×C5×C8—44no—
171C2×C5×C8C2×D414yes—
172C5×C4⋊C4—216no—
173C5×C4⋊C4—24no—
174C5×C4⋊C4—22no—
175C5×C4⋊C4—22no—
176C5×C4⋊C4—22no—
177C5×C4⋊C4—22no—
178C5×C4⋊C4—22no—
179C5×C4⋊C4—42no—
180C5×C4⋊C4—42no—
181C5×C4⋊C4C2×D414yes—
182C5×C22⋊C4—216no—
183C5×C22⋊C4—48no—
184C23×Q8C22×C511yescharacteristic
185C24×C4C22×C511yescharacteristic
186C23×C8C22×C511yescharacteristic
187C23×C8C22×C511yescharacteristic
188C2×C22⋊Q8—24no—
189C2×C22⋊Q8—22no—
190C2×C22⋊Q8—24no—
191C2×C22⋊Q8C22×C514yes—
192C22×C4⋊C4—21no—
193C22×C4⋊C4C22×C511yescharacteristic
194C22×C4⋊C4C22×C511yescharacteristic
195C22×C4⋊C4C22×C511yescharacteristic
196C22×C22⋊C4—21no—
197C22×C22⋊C4—21no—
198C2×#106—24no—
199C2×#106C22×C514yescomplemented
200C2×#103—24no—
201C2×#103—24no—
202C2×#103C22×C514yescomplemented
203C2×#103C22×C514yescomplemented
204C2×C22⋊C8—24no—
205C2×C22⋊C8C22×C514yes—
206#380—22no—
207#380C22×C514yes—
208C2×#95—21no—
209#340C22×C514yescomplemented
210C23×C5—22no—
211C23×C5—22no—
212C23×C5—22no—
213C23×C5—22no—
214C23×C5—22no—
215C23×C5—22no—
216C23×C5—28no—
217C23×C5—28no—
218C23×C5—28no—
219C23×C5—44no—
220C23×C5—44no—
221C23×C5—44no—
222C23×C5—44no—
223C23×C5#12012yes—
224C23×C5#12112yes—
225C23×C5#12312yes—
226C23×C5C2×SD1614yes—
227C23×C5C2×SD1612yes—
228C23×C5#13512yes—
229C23×C5#13712yes—
230C23×C5#13712yes—
231C23×C5C22×D411yescharacteristic
232C5×Q8—24no—
233C5×Q8—24no—
234C5×Q8—42no—
235C5×Q8—42no—
236C5×Q8—48no—
237C2×C4×C5—24no—
238C2×C4×C5—24no—
239C2×C4×C5—22no—
240C2×C4×C5—22no—
241C2×C4×C5—22no—
242C2×C4×C5—42no—
243C2×C4×C5—42no—
244C2×C4×C5—42no—
245C2×C4×C5—42no—
246C2×C4×C5—44no—
247C2×C4×C5—48no—
248C2×C4×C5—48no—
249C2×C4×C5—42no—
250C2×C4×C5—42no—
251C2×C4×C5—42no—
252C2×C4×C5—42no—
253C2×C4×C5—48no—
254C2×C4×C5—42no—
255C2×C4×C5—42no—
256C2×C4×C5—42no—
257C2×C4×C5—42no—
258C2×C4×C5—42no—
259C2×C4×C5—42no—
260C2×C4×C5—84no—
261C2×C4×C5—84no—
262C2×C4×C5#12012yes—
263C2×C4×C5#12114yes—
264C2×C4×C5#12114yes—
265C2×C4×C5#12112yes—
266C2×C4×C5#12314yes—
267C2×C4×C5#12312yes—
268C2×C4×C5#12414yes—
269C2×C4×C5C22×D411yescharacteristic
270C2×C4×C5C22×D411yescharacteristic
271C2×C4×C5C22×D411yescharacteristic
272C2×C4×C5C2×#4111yescharacteristic
273C2×C4×C5C2×#4111yescharacteristic
274C2×C4×C5C2×#4111yescharacteristic
275C5×C8—24no—
276C5×C8—42no—
277C25C5×D411yescharacteristic
278C22×Q8—22no—
279C22×Q8—21no—
280C22×Q8—22no—
281C22×Q8—22no—
282C22×Q8—22no—
283C22×Q8—22no—
284C22×Q8C5×D414yes—
285C22×Q8C23×C512yes—
286C23×C4—21no—
287C23×C4—21no—
288C23×C4—22no—
289C23×C4—22no—
290C23×C4—22no—
291C23×C4—22no—
292C23×C4—22no—
293C23×C4—22no—
294C23×C4—21no—
295C23×C4—21no—
296C23×C4C5×D411yescharacteristic
297C23×C4C5×D412yes—
298C23×C4C23×C512yes—
299C23×C4C23×C511yescharacteristic
300C22×C8—22no—
301C22×C8—22no—
302C22×C8—22no—
303C22×C8—22no—
304C22×C8—22no—
305C22×C8C5×D414yes—
306C22×C8C23×C512yes—
307C22×C8C23×C512yes—
308C22⋊Q8—216no—
309C22⋊Q8—28no—
310C22⋊Q8—28no—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4C5×D414yes—
322C2×C4⋊C4C23×C512yes—
323C2×C4⋊C4C23×C512yes—
324C2×C4⋊C4C23×C512yes—
325C2×C22⋊C4—28no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—44no—
329#106—28no—
330#103—28no—
331#103—28no—
332C22⋊C8—28no—
333#95—24no—
334C22×C5—24no—
335C22×C5—24no—
336C22×C5—24no—
337C22×C5—42no—
338C22×C5—42no—
339C22×C5—42no—
340C22×C5—42no—
341C22×C5—416no—
342C22×C5#39414yes—
343C22×C5#44814yes—
344C22×C5#45214yes—
345C22×C5#46514yes—
346C22×C5#47014yes—
347C22×C5#48314yes—
348C22×C5#48514yes—
349C22×C5C2×#12011yescharacteristic
350C22×C5C2×#12111yescharacteristic
351C22×C5C2×#12311yescharacteristic
352C22×C5C22×SD1611yescharacteristic
353C22×C5C2×#13511yescharacteristic
354C22×C5C2×#13711yescharacteristic
355C22×C5C2×#13711yescharacteristic
356C4×C5—42no—
357C4×C5—44no—
358C4×C5—44no—
359C4×C5—82no—
360C4×C5—82no—
361C4×C5#39414yes—
362C4×C5C2×#12111yescharacteristic
363C4×C5C2×#12111yescharacteristic
364C4×C5C2×#12311yescharacteristic
365C4×C5C2×#12411yescharacteristic
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24C5×SD1614yes—
373C24C2×C5×D412yes—
374C24C2×C5×D411yescharacteristic
375C2×Q8—24no—
376C2×Q8—22no—
377C2×Q8—28no—
378C2×Q8—28no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—44no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—44no—
387C2×Q8C2×C5×D414yes—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4C2×C5×D412yes—
421C22×C4C2×C5×D411yescharacteristic
422C22×C4C2×C5×D412yes—
423C22×C4C2×C5×D412yes—
424C22×C4C2×C5×D412yes—
425C22×C4C5×#4112yes—
426C22×C4C5×#4112yes—
427C22×C4C5×#4112yes—
428C22×C4C24×C511yescharacteristic
429C2×C8—28no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—44no—
436C2×C8C2×C5×D414yes—
437C4⋊C4—216no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4C2×C5×D414yes—
447C22⋊C4—216no—
448C22⋊C4—48no—
449C2×C5—44no—
450C2×C5#203518yesdirect product factor
451C2×C5C2×#39411yescharacteristic
452C2×C5C2×#44811yescharacteristic
453C2×C5C2×#45211yescharacteristic
454C2×C5C2×#46511yescharacteristic
455C2×C5C2×#47011yescharacteristic
456C2×C5C2×#48311yescharacteristic
457C2×C5C2×#48511yescharacteristic
458C23—28no—
459C23—28no—
460C23—28no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23C5×#12012yes—
472C23C5×#12112yes—
473C23C5×#12312yes—
474C23C2×C5×SD1614yes—
475C23C2×C5×SD1612yes—
476C23C5×#13512yes—
477C23C5×#13712yes—
478C23C5×#13712yes—
479C23C22×C5×D411yescharacteristic
480Q8—24no—
481Q8—24no—
482Q8—48no—
483Q8—42no—
484Q8—42no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—44no—
491C2×C4—48no—
492C2×C4—48no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—48no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—84no—
509C2×C4—84no—
510C2×C4C5×#12012yes—
511C2×C4C5×#12112yes—
512C2×C4C5×#12114yes—
513C2×C4C5×#12114yes—
514C2×C4C5×#12312yes—
515C2×C4C5×#12314yes—
516C2×C4C5×#12414yes—
517C2×C4C22×C5×D411yescharacteristic
518C2×C4C22×C5×D411yescharacteristic
519C2×C4C22×C5×D411yescharacteristic
520C2×C4C2×C5×#4111yescharacteristic
521C2×C4C2×C5×#4111yescharacteristic
522C2×C4C2×C5×#4111yescharacteristic
523C8—24no—
524C8—42no—
525C5C2×#203511yescharacteristic, direct product factor, Sylow
526C22—24no—
527C22—24no—
528C22—24no—
529C22—42no—
530C22—416no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22C5×#39414yes—
535C22C5×#44814yes—
536C22C5×#45214yes—
537C22C5×#46514yes—
538C22C5×#47014yes—
539C22C5×#48314yes—
540C22C5×#48514yes—
541C22C2×C5×#12011yescharacteristic
542C22C2×C5×#12111yescharacteristic
543C22C2×C5×#12311yescharacteristic
544C22C22×C5×SD1611yescharacteristic
545C22C2×C5×#13511yescharacteristic
546C22C2×C5×#13711yescharacteristic
547C22C2×C5×#13711yescharacteristic
548C4—44no—
549C4—42no—
550C4—44no—
551C4—82no—
552C4—82no—
553C4C5×#39414yes—
554C4C2×C5×#12111yescharacteristic
555C4C2×C5×#12111yescharacteristic
556C4C2×C5×#12311yescharacteristic
557C4C2×C5×#12411yescharacteristic
558C2—44no—
559C2C5×#203518yesdirect product factor
560C2C2×C5×#39411yescharacteristic
561C2C2×C5×#44811yescharacteristic
562C2C2×C5×#45211yescharacteristic
563C2C2×C5×#46511yescharacteristic
564C2C2×C5×#47011yescharacteristic
565C2C2×C5×#48311yescharacteristic
566C2C2×C5×#48511yescharacteristic
subgroupG/H
center, socleC24×C5C2×D4
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC25×C5D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24×C5 ◁ C24×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C5×#106C2semidirect
3. C22×C5×#103C2semidirect
4. C22×C5×#103C2semidirect
7. C5×#2035C2direct (N × H)
8. C2×C5×#340C2semidirect
24. C2×C5×#106C22semidirect
27. C2×C5×#103C22semidirect
28. C2×C5×#103C22semidirect
34. C5×#340C22semidirect
35. C2×#2035C5direct (N × H)
94. C22×#106C2×C5semidirect
95. C22×#103C2×C5semidirect
96. C22×#103C2×C5semidirect
99. #2035C2×C5direct (N × H)
100. C2×#340C2×C5semidirect
199. C2×#106C22×C5semidirect
202. C2×#103C22×C5semidirect
203. C2×#103C22×C5semidirect
209. #340C22×C5semidirect
450. C2×C5#2035direct (N × H)
525. C5C2×#2035direct (N × H)
559. C2C5×#2035direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C5×C22⋊Q82
C22×C5×#1062
C22×C5×#1032
C22×C5×C22⋊C82
C2×C5×#3802
C5×#20352
C2×C5×#3402
C2×#20355