Finite Group Databasebeta

← C2×C5×#1661 #11941701

Subgroups — C2×C5×#1661 #11941701

Order 1280.

1022 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#562C211yesmaximal, characteristic
2C2×C5×#501C211yesmaximal, characteristic, complemented
3C2×C5×#497C211yesmaximal, characteristic, complemented
4C2×C5×#489C211yesmaximal, characteristic, complemented
5C5×#1661C218yesmaximal, direct product factor
6C2×C5×#330C211yesmaximal, characteristic, complemented
7C2×C5×#329C211yesmaximal, characteristic, complemented
8C2×C5×#329C211yesmaximal, characteristic, complemented
9C22×C5×Q16—21no—
10C22×C5×SD16—21no—
11C5×#562—22no—
12C5×#562—22no—
13C5×#562—22no—
14C5×#562C2214yes—
15C2×C5×C4⋊Q8C2211yescharacteristic
16C2×C5×#125C2211yescharacteristic
17C2×C5×#124C2211yescharacteristic
18C2×C5×#124C2211yescharacteristic
19C2×C5×#123—21no—
20C2×C5×#123—21no—
21C2×C5×C22⋊Q8—21no—
22C2×C5×C22⋊Q8—21no—
23C2×C5×#121—21no—
24C2×C5×#121—21no—
25C5×#501—22no—
26C5×#501C2214yescomplemented
27C5×#497—22no—
28C5×#497C2214yescomplemented
29C5×#489—22no—
30C5×#489C2214yescomplemented
31C2×C5×#107—21no—
32C2×C5×#106—21no—
33C2×C5×#103—21no—
34C2×C5×#102—21no—
35C2×C5×#97C2211yescharacteristic
36C2×C5×#97C2211yescharacteristic
37C2×C5×#97C2211yescharacteristic
38C5×#330C2214yescomplemented
39C5×#329C2214yescomplemented
40C5×#329C2214yescomplemented
41C2×#1661C511yesmaximal, characteristic, direct product factor, Sylow
42C22×C5×Q8—21no—
43C22×C5×Q8D411yescharacteristic
44C22×C5×Q8D411yescharacteristic
45C22×C5×D4—21no—
46C22×C5×D4—21no—
47C22×C5×D4D411yescharacteristic
48C22×C5×D4D411yescharacteristic
49C23×C4×C5—21no—
50C23×C4×C5—21no—
51C2×C5×Q16—24no—
52C2×C5×Q16—44no—
53C2×C5×SD16—24no—
54C2×C5×SD16—44no—
55C22×C5×C8—21no—
56C22×C5×C8—21no—
57C22×C5×C8—21no—
58C5×C4⋊Q8—21no—
59C5×C4⋊Q8—21no—
60C5×C4⋊Q8—21no—
61C5×C4⋊Q8C2312yes—
62C5×#125—21no—
63C5×#125—21no—
64C5×#125D412yes—
65C5×#125C2312yes—
66C5×#124—21no—
67C5×#124—21no—
68C5×#124—21no—
69C5×#124—21no—
70C5×#124D412yes—
71C5×#124D412yes—
72C5×#124C2312yes—
73C5×#124C2312yes—
74C5×#123—24no—
75C5×#123—24no—
76C5×#123—24no—
77C5×#123—24no—
78C5×C22⋊Q8—24no—
79C5×C22⋊Q8—24no—
80C5×C22⋊Q8—24no—
81C5×C22⋊Q8—24no—
82C5×#121—24no—
83C5×#121—24no—
84C5×#121—24no—
85C5×#121—24no—
86C2×C5×C4⋊C4—21no—
87C2×C5×C4⋊C4—21no—
88C2×C5×C4⋊C4—21no—
89C2×C5×C4⋊C4—21no—
90C2×C5×C4⋊C4—21no—
91C2×C5×C4⋊C4—21no—
92C2×C5×C4⋊C4D411yescharacteristic
93C2×C5×C4⋊C4D411yescharacteristic
94C2×C5×C22⋊C4—21no—
95C2×C5×C22⋊C4—21no—
96C2×C5×C22⋊C4—21no—
97C2×C5×C22⋊C4—21no—
98C2×C5×C22⋊C4—21no—
99C2×C5×C22⋊C4—21no—
100C2×C42×C5C2311yescharacteristic
101C5×#107—24no—
102C5×#106—24no—
103C5×#103—24no—
104C5×#102—24no—
105C5×#97—21no—
106C5×#97—21no—
107C5×#97—21no—
108C5×#97C2312yes—
109C5×#97C2312yes—
110C5×#97C2312yes—
111C2×#562C2×C511yescharacteristic
112C2×#501C2×C511yescharacteristic, complemented
113C2×#497C2×C511yescharacteristic, complemented
114C2×#489C2×C511yescharacteristic, complemented
115#1661C2×C518yesdirect product factor
116C2×#330C2×C511yescharacteristic, complemented
117C2×#329C2×C511yescharacteristic, complemented
118C2×#329C2×C511yescharacteristic, complemented
119C24×C5—21no—
120C24×C5—21no—
121C2×C5×Q8—21no—
122C2×C5×Q8—21no—
123C2×C5×Q8—22no—
124C2×C5×Q8—42no—
125C2×C5×Q8—42no—
126C2×C5×Q8—42no—
127C2×C5×Q8—42no—
128C2×C5×Q8—41no—
129C2×C5×Q8C2×D412yes—
130C2×C5×Q8C2×D412yes—
131C2×C5×D4—24no—
132C2×C5×D4—24no—
133C2×C5×D4—21no—
134C2×C5×D4—21no—
135C2×C5×D4—42no—
136C2×C5×D4—42no—
137C2×C5×D4—82no—
138C2×C5×D4—82no—
139C2×C5×D4C2×D412yes—
140C2×C5×D4C2×D412yes—
141C22×C4×C5—21no—
142C22×C4×C5—21no—
143C22×C4×C5—21no—
144C22×C4×C5—21no—
145C22×C4×C5—24no—
146C22×C4×C5—24no—
147C22×C4×C5—41no—
148C22×C4×C5—82no—
149C22×C4×C5—82no—
150C22×C4×C5C2×D411yescharacteristic
151C22×C4×C5C2×D411yescharacteristic
152C22×C4×C5C2×D411yescharacteristic
153C5×Q16—48no—
154C5×SD16—48no—
155C2×C5×C8—24no—
156C2×C5×C8—24no—
157C2×C5×C8—22no—
158C2×C5×C8—22no—
159C2×C5×C8—22no—
160C2×C5×C8—42no—
161C5×C4⋊C4—24no—
162C5×C4⋊C4—24no—
163C5×C4⋊C4—22no—
164C5×C4⋊C4—22no—
165C5×C4⋊C4—21no—
166C5×C4⋊C4—21no—
167C5×C4⋊C4—22no—
168C5×C4⋊C4—22no—
169C5×C4⋊C4—22no—
170C5×C4⋊C4—22no—
171C5×C4⋊C4—41no—
172C5×C4⋊C4—41no—
173C5×C4⋊C4C2×D412yes—
174C5×C4⋊C4C2×D412yes—
175C5×C22⋊C4—24no—
176C5×C22⋊C4—24no—
177C5×C22⋊C4—22no—
178C5×C22⋊C4—22no—
179C5×C22⋊C4—22no—
180C5×C22⋊C4—22no—
181C5×C22⋊C4—22no—
182C5×C22⋊C4—22no—
183C5×C22⋊C4—22no—
184C5×C22⋊C4—22no—
185C42×C5C2×D411yescharacteristic
186C42×C5C2×D411yescharacteristic
187C42×C5C2×D411yescharacteristic
188C42×C5C2411yescharacteristic
189C22×Q16—21no—
190C22×SD16—21no—
191#562—22no—
192#562—22no—
193#562—22no—
194#562C22×C514yes—
195C2×C4⋊Q8C22×C511yescharacteristic
196C2×#125C22×C511yescharacteristic
197C2×#124C22×C511yescharacteristic
198C2×#124C22×C511yescharacteristic
199C2×#123—21no—
200C2×#123—21no—
201C2×C22⋊Q8—21no—
202C2×C22⋊Q8—21no—
203C2×#121—21no—
204C2×#121—21no—
205#501—22no—
206#501C22×C514yescomplemented
207#497—22no—
208#497C22×C514yescomplemented
209#489—22no—
210#489C22×C514yescomplemented
211C2×#107—21no—
212C2×#106—21no—
213C2×#103—21no—
214C2×#102—21no—
215C2×#97C22×C511yescharacteristic
216C2×#97C22×C511yescharacteristic
217C2×#97C22×C511yescharacteristic
218#330C22×C514yescomplemented
219#329C22×C514yescomplemented
220#329C22×C514yescomplemented
221C23×C5—22no—
222C23×C5—22no—
223C23×C5—42no—
224C23×C5—42no—
225C23×C5—42no—
226C23×C5—42no—
227C23×C5—42no—
228C23×C5—42no—
229C23×C5#12011yescharacteristic
230C5×Q8—42no—
231C5×Q8—42no—
232C5×Q8—42no—
233C5×Q8—42no—
234C5×Q8—81no—
235C5×Q8—81no—
236C5×Q8—82no—
237C5×D4—42no—
238C5×D4—42no—
239C5×D4—81no—
240C5×D4—81no—
241C5×D4—84no—
242C5×D4—84no—
243C2×C4×C5—22no—
244C2×C4×C5—22no—
245C2×C4×C5—22no—
246C2×C4×C5—22no—
247C2×C4×C5—22no—
248C2×C4×C5—22no—
249C2×C4×C5—22no—
250C2×C4×C5—42no—
251C2×C4×C5—42no—
252C2×C4×C5—42no—
253C2×C4×C5—42no—
254C2×C4×C5—42no—
255C2×C4×C5—84no—
256C2×C4×C5—84no—
257C2×C4×C5—82no—
258C2×C4×C5#12011yescharacteristic
259C2×C4×C5#12011yescharacteristic
260C2×C4×C5#12011yescharacteristic
261C2×C4×C5C22×D411yescharacteristic
262C2×C4×C5C22×D411yescharacteristic
263C2×C4×C5C22×D411yescharacteristic
264C5×C8—24no—
265C5×C8—24no—
266C5×C8—42no—
267C22×Q8—21no—
268C22×Q8C5×D411yescharacteristic
269C22×Q8C5×D411yescharacteristic
270C22×D4—21no—
271C22×D4—21no—
272C22×D4C5×D411yescharacteristic
273C22×D4C5×D411yescharacteristic
274C23×C4—21no—
275C23×C4—21no—
276C2×Q16—24no—
277C2×Q16—44no—
278C2×SD16—24no—
279C2×SD16—44no—
280C22×C8—21no—
281C22×C8—21no—
282C22×C8—21no—
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8C23×C512yes—
287#125—21no—
288#125—21no—
289#125C5×D412yes—
290#125C23×C512yes—
291#124—21no—
292#124—21no—
293#124—21no—
294#124—21no—
295#124C5×D412yes—
296#124C5×D412yes—
297#124C23×C512yes—
298#124C23×C512yes—
299#123—24no—
300#123—24no—
301#123—24no—
302#123—24no—
303C22⋊Q8—24no—
304C22⋊Q8—24no—
305C22⋊Q8—24no—
306C22⋊Q8—24no—
307#121—24no—
308#121—24no—
309#121—24no—
310#121—24no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4C5×D411yescharacteristic
318C2×C4⋊C4C5×D411yescharacteristic
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C42C23×C511yescharacteristic
326#107—24no—
327#106—24no—
328#103—24no—
329#102—24no—
330#97—21no—
331#97—21no—
332#97—21no—
333#97C23×C512yes—
334#97C23×C512yes—
335#97C23×C512yes—
336C22×C5—42no—
337C22×C5—42no—
338C22×C5—42no—
339C22×C5—42no—
340C22×C5—42no—
341C22×C5—42no—
342C22×C5—84no—
343C22×C5—84no—
344C22×C5#45312yes—
345C22×C5#45512yes—
346C22×C5#45612yes—
347C22×C5C2×#12011yescharacteristic
348C4×C5—21no—
349C4×C5—21no—
350C4×C5—21no—
351C4×C5—21no—
352C4×C5—21no—
353C4×C5—21no—
354C4×C5—42no—
355C4×C5—42no—
356C4×C5—42no—
357C4×C5—42no—
358C4×C5—82no—
359C24—21no—
360C24—21no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—22no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8—41no—
369C2×Q8C2×C5×D412yes—
370C2×Q8C2×C5×D412yes—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—82no—
378C2×D4—82no—
379C2×D4C2×C5×D412yes—
380C2×D4C2×C5×D412yes—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—41no—
388C22×C4—82no—
389C22×C4—82no—
390C22×C4C2×C5×D411yescharacteristic
391C22×C4C2×C5×D411yescharacteristic
392C22×C4C2×C5×D411yescharacteristic
393Q16—48no—
394SD16—48no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—42no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4C2×C5×D412yes—
414C4⋊C4C2×C5×D412yes—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C42C2×C5×D411yescharacteristic
426C42C2×C5×D411yescharacteristic
427C42C2×C5×D411yescharacteristic
428C42C24×C511yescharacteristic
429C2×C5—82no—
430C2×C5—82no—
431C2×C5#166114yesdirect product factor
432C2×C5C2×#45311yescharacteristic
433C2×C5C2×#45511yescharacteristic
434C2×C5C2×#45611yescharacteristic
435C23—22no—
436C23—22no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23C5×#12011yescharacteristic
444Q8—42no—
445Q8—42no—
446Q8—42no—
447Q8—42no—
448Q8—81no—
449Q8—81no—
450Q8—82no—
451D4—42no—
452D4—42no—
453D4—81no—
454D4—81no—
455D4—84no—
456D4—84no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—84no—
470C2×C4—84no—
471C2×C4—82no—
472C2×C4C5×#12011yescharacteristic
473C2×C4C5×#12011yescharacteristic
474C2×C4C5×#12011yescharacteristic
475C2×C4C22×C5×D411yescharacteristic
476C2×C4C22×C5×D411yescharacteristic
477C2×C4C22×C5×D411yescharacteristic
478C8—24no—
479C8—24no—
480C8—42no—
481C5C2×#166111yescharacteristic, direct product factor, Sylow
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—84no—
489C22—84no—
490C22C5×#45312yes—
491C22C5×#45512yes—
492C22C5×#45612yes—
493C22C2×C5×#12011yescharacteristic
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—82no—
505C2—82no—
506C2—82no—
507C2C5×#166114yesdirect product factor
508C2C2×C5×#45311yescharacteristic
509C2C2×C5×#45511yescharacteristic
510C2C2×C5×#45611yescharacteristic
subgroupG/H
center, socleC23×C5#120
commutator, Frattini, cosocleC42C24×C5
prime radicalC2×C5×#562C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C5 ◁ C2×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C5×#501C2semidirect
3. C2×C5×#497C2semidirect
4. C2×C5×#489C2semidirect
5. C5×#1661C2direct (N × H)
6. C2×C5×#330C2semidirect
7. C2×C5×#329C2semidirect
8. C2×C5×#329C2semidirect
26. C5×#501C22semidirect
28. C5×#497C22semidirect
30. C5×#489C22semidirect
38. C5×#330C22semidirect
39. C5×#329C22semidirect
40. C5×#329C22semidirect
41. C2×#1661C5direct (N × H)
112. C2×#501C2×C5semidirect
113. C2×#497C2×C5semidirect
114. C2×#489C2×C5semidirect
115. #1661C2×C5direct (N × H)
116. C2×#330C2×C5semidirect
117. C2×#329C2×C5semidirect
118. C2×#329C2×C5semidirect
206. #501C22×C5semidirect
208. #497C22×C5semidirect
210. #489C22×C5semidirect
218. #330C22×C5semidirect
219. #329C22×C5semidirect
220. #329C22×C5semidirect
431. C2×C5#1661direct (N × H)
481. C5C2×#1661direct (N × H)
507. C2C5×#1661direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5622
C2×C5×#5012
C2×C5×#4972
C2×C5×#4892
C5×#16612
C2×C5×#3302
C2×C5×#3292
C2×#16615