Finite Group Databasebeta

← C2×C5×#1854 #11943562

Subgroups — C2×C5×#1854 #11943562

Order 1280.

1662 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C5×#117C211yesmaximal, characteristic
2C22×C5×#107C211yesmaximal, characteristic, complemented
3C22×C5×#106C211yesmaximal, characteristic, complemented
4C22×C5×C22⋊C8C211yesmaximal, characteristic
5C2×C5×#380C211yesmaximal, characteristic
6C5×#1854C218yesmaximal, direct product factor
7C2×C5×#340C211yesmaximal, characteristic, complemented
8C2×C5×#340C211yesmaximal, characteristic, complemented
9C24×C4×C5C2211yescharacteristic
10C23×C5×C8C2211yescharacteristic
11C23×C5×C8C2211yescharacteristic
12C2×C5×#117—22no—
13C2×C5×#117—24no—
14C2×C5×#117—24no—
15C2×C5×#117C2214yes—
16C22×C5×C4⋊C4C2211yescharacteristic
17C22×C5×C4⋊C4C2211yescharacteristic
18C22×C5×C4⋊C4C2211yescharacteristic
19C22×C5×C4⋊C4C2211yescharacteristic
20C22×C5×C22⋊C4—21no—
21C22×C5×C22⋊C4—21no—
22C22×C42×C5—21no—
23C2×C5×#107—24no—
24C2×C5×#107C2214yescomplemented
25C2×C5×#106—24no—
26C2×C5×#106C2214yescomplemented
27C2×C5×C22⋊C8C418yescomplemented
28C2×C5×C22⋊C8C2214yes—
29C5×#380—22no—
30C5×#380C2214yes—
31C2×C5×#95—21no—
32C5×#340C2214yescomplemented
33C5×#340C2214yescomplemented
34C2×#1854C511yesmaximal, characteristic, direct product factor, Sylow
35C25×C5D411yescharacteristic
36C23×C4×C5—22no—
37C23×C4×C5—22no—
38C23×C4×C5—22no—
39C23×C4×C5—22no—
40C23×C4×C5—21no—
41C23×C4×C5—21no—
42C23×C4×C5—21no—
43C23×C4×C5—21no—
44C23×C4×C5C2×C414yes—
45C23×C4×C5D412yes—
46C23×C4×C5D411yescharacteristic
47C23×C4×C5Q814yes—
48C23×C4×C5C2312yes—
49C23×C4×C5C2311yescharacteristic
50C22×C5×C8—22no—
51C22×C5×C8—22no—
52C22×C5×C8—22no—
53C22×C5×C8—22no—
54C22×C5×C8C2×C414yes—
55C22×C5×C8C2×C414yes—
56C22×C5×C8C2312yes—
57C22×C5×C8C2312yes—
58C5×#117—216no—
59C5×#117—28no—
60C5×#117—28no—
61C2×C5×C4⋊C4—21no—
62C2×C5×C4⋊C4—21no—
63C2×C5×C4⋊C4—22no—
64C2×C5×C4⋊C4—22no—
65C2×C5×C4⋊C4—22no—
66C2×C5×C4⋊C4—22no—
67C2×C5×C4⋊C4—22no—
68C2×C5×C4⋊C4—22no—
69C2×C5×C4⋊C4—21no—
70C2×C5×C4⋊C4—21no—
71C2×C5×C4⋊C4D414yes—
72C2×C5×C4⋊C4D414yes—
73C2×C5×C4⋊C4C2312yes—
74C2×C5×C4⋊C4C2312yes—
75C2×C5×C4⋊C4C2312yes—
76C2×C5×C4⋊C4C2312yes—
77C2×C5×C22⋊C4—28no—
78C2×C5×C22⋊C4—24no—
79C2×C5×C22⋊C4—24no—
80C2×C5×C22⋊C4—44no—
81C2×C42×C5—28no—
82C2×C42×C5—24no—
83C5×#107—28no—
84C5×#106—28no—
85C5×C22⋊C8C2×C4116yescomplemented
86C5×#95—24no—
87C22×#117C2×C511yescharacteristic
88C22×#107C2×C511yescharacteristic, complemented
89C22×#106C2×C511yescharacteristic, complemented
90C22×C22⋊C8C2×C511yescharacteristic
91C2×#380C2×C511yescharacteristic
92#1854C2×C518yesdirect product factor
93C2×#340C2×C511yescharacteristic, complemented
94C2×#340C2×C511yescharacteristic, complemented
95C24×C5—22no—
96C24×C5—22no—
97C24×C5—22no—
98C24×C5—22no—
99C24×C5—22no—
100C24×C5—22no—
101C24×C5C4⋊C414yes—
102C24×C5C2×D412yes—
103C24×C5C2×D411yescharacteristic
104C22×C4×C5—22no—
105C22×C4×C5—22no—
106C22×C4×C5—28no—
107C22×C4×C5—28no—
108C22×C4×C5—24no—
109C22×C4×C5—28no—
110C22×C4×C5—28no—
111C22×C4×C5—24no—
112C22×C4×C5—22no—
113C22×C4×C5—22no—
114C22×C4×C5—22no—
115C22×C4×C5—22no—
116C22×C4×C5—24no—
117C22×C4×C5—24no—
118C22×C4×C5—22no—
119C22×C4×C5—22no—
120C22×C4×C5—22no—
121C22×C4×C5—22no—
122C22×C4×C5—42no—
123C22×C4×C5—42no—
124C22×C4×C5—42no—
125C22×C4×C5—42no—
126C22×C4×C5—42no—
127C22×C4×C5—42no—
128C22×C4×C5—42no—
129C22×C4×C5—42no—
130C22×C4×C5—42no—
131C22×C4×C5—42no—
132C22×C4×C5C4⋊C414yes—
133C22×C4×C5C4⋊C418yes—
134C22×C4×C5C22×C414yes—
135C22×C4×C5C22×C412yes—
136C22×C4×C5C2×D412yes—
137C22×C4×C5C2×D411yescharacteristic
138C22×C4×C5C2×D412yes—
139C22×C4×C5C2×D412yes—
140C22×C4×C5C2×Q814yes—
141C22×C4×C5C2×Q812yes—
142C22×C4×C5#4112yes—
143C22×C4×C5#4112yes—
144C22×C4×C5C2411yescharacteristic
145C2×C5×C8—28no—
146C2×C5×C8—28no—
147C2×C5×C8—22no—
148C2×C5×C8—22no—
149C2×C5×C8—22no—
150C2×C5×C8—22no—
151C2×C5×C8C22×C414yes—
152C2×C5×C8C22×C414yes—
153C5×C4⋊C4—24no—
154C5×C4⋊C4—24no—
155C5×C4⋊C4—22no—
156C5×C4⋊C4—22no—
157C5×C4⋊C4—22no—
158C5×C4⋊C4—22no—
159C5×C4⋊C4—22no—
160C5×C4⋊C4—22no—
161C5×C4⋊C4—42no—
162C5×C4⋊C4—42no—
163C5×C4⋊C4C2×D414yes—
164C5×C4⋊C4C2×D414yes—
165C5×C22⋊C4—216no—
166C5×C22⋊C4—48no—
167C42×C5—216no—
168C24×C4C22×C511yescharacteristic
169C23×C8C22×C511yescharacteristic
170C23×C8C22×C511yescharacteristic
171C2×#117—22no—
172C2×#117—24no—
173C2×#117—24no—
174C2×#117C22×C514yes—
175C22×C4⋊C4C22×C511yescharacteristic
176C22×C4⋊C4C22×C511yescharacteristic
177C22×C4⋊C4C22×C511yescharacteristic
178C22×C4⋊C4C22×C511yescharacteristic
179C22×C22⋊C4—21no—
180C22×C22⋊C4—21no—
181C22×C42—21no—
182C2×#107—24no—
183C2×#107C22×C514yescomplemented
184C2×#106—24no—
185C2×#106C22×C514yescomplemented
186C2×C22⋊C8C4×C518yescomplemented
187C2×C22⋊C8C22×C514yes—
188#380—22no—
189#380C22×C514yes—
190C2×#95—21no—
191#340C22×C514yescomplemented
192#340C22×C514yescomplemented
193C23×C5—22no—
194C23×C5—22no—
195C23×C5—22no—
196C23×C5—22no—
197C23×C5—22no—
198C23×C5—22no—
199C23×C5—28no—
200C23×C5—28no—
201C23×C5—28no—
202C23×C5—44no—
203C23×C5—44no—
204C23×C5—44no—
205C23×C5—44no—
206C23×C5C2×C4⋊C414yes—
207C23×C5C2×C4⋊C412yes—
208C23×C5#12012yes—
209C23×C5#12312yes—
210C23×C5#13512yes—
211C23×C5#13512yes—
212C23×C5AGL(1,ℤ/8)12yes—
213C23×C5#13712yes—
214C23×C5C22×D411yescharacteristic
215C2×C4×C5—22no—
216C2×C4×C5—22no—
217C2×C4×C5—28no—
218C2×C4×C5—24no—
219C2×C4×C5—24no—
220C2×C4×C5—24no—
221C2×C4×C5—24no—
222C2×C4×C5—216no—
223C2×C4×C5—24no—
224C2×C4×C5—24no—
225C2×C4×C5—42no—
226C2×C4×C5—42no—
227C2×C4×C5—42no—
228C2×C4×C5—42no—
229C2×C4×C5—48no—
230C2×C4×C5—48no—
231C2×C4×C5—42no—
232C2×C4×C5—42no—
233C2×C4×C5—42no—
234C2×C4×C5—42no—
235C2×C4×C5—42no—
236C2×C4×C5—42no—
237C2×C4×C5—84no—
238C2×C4×C5—84no—
239C2×C4×C5C2×C4⋊C414yes—
240C2×C4×C5C2×C4⋊C412yes—
241C2×C4×C5C4×D414yes—
242C2×C4×C5C4×D414yes—
243C2×C4×C5#12012yes—
244C2×C4×C5C22⋊Q814yes—
245C2×C4×C5C22⋊Q814yes—
246C2×C4×C5#12312yes—
247C2×C4×C5C23×C411yescharacteristic
248C2×C4×C5C22×D411yescharacteristic
249C2×C4×C5C22×D411yescharacteristic
250C2×C4×C5C22×Q811yescharacteristic
251C2×C4×C5C2×#4111yescharacteristic
252C2×C4×C5C2×#4111yescharacteristic
253C5×C8—24no—
254C5×C8—24no—
255C25C5×D411yescharacteristic
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—21no—
264C23×C4C2×C4×C514yes—
265C23×C4C5×D412yes—
266C23×C4C5×D411yescharacteristic
267C23×C4C5×Q814yes—
268C23×C4C23×C512yes—
269C23×C4C23×C511yescharacteristic
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8C2×C4×C514yes—
275C22×C8C2×C4×C514yes—
276C22×C8C23×C512yes—
277C22×C8C23×C512yes—
278#117—216no—
279#117—28no—
280#117—28no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4C5×D414yes—
292C2×C4⋊C4C5×D414yes—
293C2×C4⋊C4C23×C512yes—
294C2×C4⋊C4C23×C512yes—
295C2×C4⋊C4C23×C512yes—
296C2×C4⋊C4C23×C512yes—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—44no—
301C2×C42—28no—
302C2×C42—24no—
303#107—28no—
304#106—28no—
305C22⋊C8C2×C4×C5116yescomplemented
306#95—24no—
307C22×C5—24no—
308C22×C5—24no—
309C22×C5—24no—
310C22×C5—42no—
311C22×C5—42no—
312C22×C5—416no—
313C22×C5—42no—
314C22×C5—42no—
315C22×C5#38514yes—
316C22×C5#42714yes—
317C22×C5#42814yes—
318C22×C5#44914yes—
319C22×C5#45214yes—
320C22×C5#48214yes—
321C22×C5#48514yes—
322C22×C5C22×C4⋊C411yescharacteristic
323C22×C5C2×#12011yescharacteristic
324C22×C5C2×#12311yescharacteristic
325C22×C5C2×#13511yescharacteristic
326C22×C5C2×#13511yescharacteristic
327C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
328C22×C5C2×#13711yescharacteristic
329C4×C5—24no—
330C4×C5—44no—
331C4×C5—44no—
332C4×C5—82no—
333C4×C5—82no—
334C4×C5#38514yes—
335C4×C5C2×C4×D411yescharacteristic
336C4×C5C2×C4×D411yescharacteristic
337C4×C5C2×C22⋊Q811yescharacteristic
338C4×C5C2×C22⋊Q811yescharacteristic
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24C5×C4⋊C414yes—
346C24C2×C5×D412yes—
347C24C2×C5×D411yescharacteristic
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—24no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4C5×C4⋊C414yes—
377C22×C4C5×C4⋊C418yes—
378C22×C4C22×C4×C514yes—
379C22×C4C22×C4×C512yes—
380C22×C4C2×C5×D412yes—
381C22×C4C2×C5×D411yescharacteristic
382C22×C4C2×C5×D412yes—
383C22×C4C2×C5×D412yes—
384C22×C4C2×C5×Q814yes—
385C22×C4C2×C5×Q812yes—
386C22×C4C5×#4112yes—
387C22×C4C5×#4112yes—
388C22×C4C24×C511yescharacteristic
389C2×C8—28no—
390C2×C8—28no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8C22×C4×C514yes—
396C2×C8C22×C4×C514yes—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4C2×C5×D414yes—
408C4⋊C4C2×C5×D414yes—
409C22⋊C4—216no—
410C22⋊C4—48no—
411C42—216no—
412C2×C5—44no—
413C2×C5#185418yesdirect product factor
414C2×C5C2×#38511yescharacteristic
415C2×C5C2×#42711yescharacteristic
416C2×C5C2×#42811yescharacteristic
417C2×C5C2×#44911yescharacteristic
418C2×C5C2×#45211yescharacteristic
419C2×C5C2×#48211yescharacteristic
420C2×C5C2×#48511yescharacteristic
421C23—28no—
422C23—28no—
423C23—28no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23C2×C5×C4⋊C414yes—
435C23C2×C5×C4⋊C412yes—
436C23C5×#12012yes—
437C23C5×#12312yes—
438C23C5×#13512yes—
439C23C5×#13512yes—
440C23C5×AGL(1,ℤ/8)12yes—
441C23C5×#13712yes—
442C23C22×C5×D411yescharacteristic
443C2×C4—28no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—216no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—48no—
454C2×C4—48no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—84no—
466C2×C4—84no—
467C2×C4C2×C5×C4⋊C414yes—
468C2×C4C2×C5×C4⋊C412yes—
469C2×C4C4×C5×D414yes—
470C2×C4C4×C5×D414yes—
471C2×C4C5×#12012yes—
472C2×C4C5×C22⋊Q814yes—
473C2×C4C5×C22⋊Q814yes—
474C2×C4C5×#12312yes—
475C2×C4C23×C4×C511yescharacteristic
476C2×C4C22×C5×D411yescharacteristic
477C2×C4C22×C5×D411yescharacteristic
478C2×C4C22×C5×Q811yescharacteristic
479C2×C4C2×C5×#4111yescharacteristic
480C2×C4C2×C5×#4111yescharacteristic
481C8—24no—
482C8—24no—
483C5C2×#185411yescharacteristic, direct product factor, Sylow
484C22—24no—
485C22—24no—
486C22—24no—
487C22—42no—
488C22—42no—
489C22—416no—
490C22—42no—
491C22—42no—
492C22C5×#38514yes—
493C22C5×#42714yes—
494C22C5×#42814yes—
495C22C5×#44914yes—
496C22C5×#45214yes—
497C22C5×#48214yes—
498C22C5×#48514yes—
499C22C22×C5×C4⋊C411yescharacteristic
500C22C2×C5×#12011yescharacteristic
501C22C2×C5×#12311yescharacteristic
502C22C2×C5×#13511yescharacteristic
503C22C2×C5×#13511yescharacteristic
504C22C2×C5×AGL(1,ℤ/8)11yescharacteristic
505C22C2×C5×#13711yescharacteristic
506C4—24no—
507C4—44no—
508C4—44no—
509C4—82no—
510C4—82no—
511C4C5×#38514yes—
512C4C2×C4×C5×D411yescharacteristic
513C4C2×C4×C5×D411yescharacteristic
514C4C2×C5×C22⋊Q811yescharacteristic
515C4C2×C5×C22⋊Q811yescharacteristic
516C2—44no—
517C2C5×#185418yesdirect product factor
518C2C2×C5×#38511yescharacteristic
519C2C2×C5×#42711yescharacteristic
520C2C2×C5×#42811yescharacteristic
521C2C2×C5×#44911yescharacteristic
522C2C2×C5×#45211yescharacteristic
523C2C2×C5×#48211yescharacteristic
524C2C2×C5×#48511yescharacteristic
subgroupG/H
center, socleC24×C5C2×D4
commutatorC2×C4C23×C4×C5
Frattini, cosocleC22×C4C24×C5
prime radicalC25×C5D4
LCSG ▷ C2×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×#107C2semidirect
3. C22×C5×#106C2semidirect
6. C5×#1854C2direct (N × H)
7. C2×C5×#340C2semidirect
8. C2×C5×#340C2semidirect
24. C2×C5×#107C22semidirect
26. C2×C5×#106C22semidirect
27. C2×C5×C22⋊C8C4semidirect
32. C5×#340C22semidirect
33. C5×#340C22semidirect
34. C2×#1854C5direct (N × H)
85. C5×C22⋊C8C2×C4semidirect
88. C22×#107C2×C5semidirect
89. C22×#106C2×C5semidirect
92. #1854C2×C5direct (N × H)
93. C2×#340C2×C5semidirect
94. C2×#340C2×C5semidirect
183. C2×#107C22×C5semidirect
185. C2×#106C22×C5semidirect
186. C2×C22⋊C8C4×C5semidirect
191. #340C22×C5semidirect
192. #340C22×C5semidirect
305. C22⋊C8C2×C4×C5semidirect
413. C2×C5#1854direct (N × H)
483. C5C2×#1854direct (N × H)
517. C2C5×#1854direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C5×#1172
C22×C5×#1072
C22×C5×#1062
C22×C5×C22⋊C82
C2×C5×#3802
C5×#18542
C2×C5×#3402
C2×#18545