Finite Group Databasebeta

← C2×C5×#1664 #11941706

Subgroups — C2×C5×#1664 #11941706

Order 1280.

894 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#563C211yesmaximal, characteristic
2C2×C5×#501C211yesmaximal, characteristic, complemented
3C2×C5×#491C211yesmaximal, characteristic, complemented
4C2×C5×#489C211yesmaximal, characteristic
5C5×#1664C218yesmaximal, direct product factor
6C2×C5×#330C211yesmaximal, characteristic, complemented
7C2×C5×#330C211yesmaximal, characteristic, complemented
8C2×C5×#329C211yesmaximal, characteristic
9C5×#563—22no—
10C5×#563—22no—
11C5×#563—22no—
12C5×#563C2214yes—
13C2×C5×C4⋊Q8—21no—
14C2×C5×C4⋊Q8C2211yescharacteristic
15C2×C5×#126—21no—
16C2×C5×#126—21no—
17C2×C5×#125C2211yescharacteristic
18C2×C5×#125C2211yescharacteristic
19C2×C5×#124C2211yescharacteristic
20C2×C5×#123—21no—
21C2×C5×C22⋊Q8—21no—
22C2×C5×C22⋊Q8—21no—
23C5×#501—22no—
24C5×#501C2214yescomplemented
25C5×#491—22no—
26C5×#491C2214yescomplemented
27C5×#489—22no—
28C5×#489C2214yes—
29C2×C5×#107—21no—
30C2×C5×#106—21no—
31C2×C5×#103—21no—
32C2×C5×#103—21no—
33C2×C5×#103—21no—
34C2×C5×#102—21no—
35C2×C5×#97C2211yescharacteristic
36C2×C5×#97C2211yescharacteristic
37C2×C5×#97C2211yescharacteristic
38C5×#330C2214yescomplemented
39C5×#330C2214yescomplemented
40C5×#329C2214yes—
41C2×#1664C511yesmaximal, characteristic, direct product factor, Sylow
42C22×C5×Q8—21no—
43C22×C5×Q8—21no—
44C22×C5×Q8D411yescharacteristic
45C22×C5×D4D411yescharacteristic
46C23×C4×C5—21no—
47C22×C5×C8—21no—
48C22×C5×C8—21no—
49C22×C5×C8—21no—
50C5×C4⋊Q8—24no—
51C5×C4⋊Q8—24no—
52C5×C4⋊Q8—21no—
53C5×C4⋊Q8—21no—
54C5×C4⋊Q8—21no—
55C5×C4⋊Q8C2312yes—
56C5×#126—24no—
57C5×#126—24no—
58C5×#126—24no—
59C5×#126—24no—
60C5×#125—21no—
61C5×#125—21no—
62C5×#125—21no—
63C5×#125—21no—
64C5×#125D412yes—
65C5×#125D412yes—
66C5×#125C2312yes—
67C5×#125C2312yes—
68C5×#124—21no—
69C5×#124—21no—
70C5×#124D412yes—
71C5×#124C2312yes—
72C5×#123—24no—
73C5×#123—24no—
74C5×C22⋊Q8—24no—
75C5×C22⋊Q8—24no—
76C5×C22⋊Q8—24no—
77C5×C22⋊Q8—24no—
78C2×C5×C4⋊C4—21no—
79C2×C5×C4⋊C4—21no—
80C2×C5×C4⋊C4—21no—
81C2×C5×C4⋊C4—21no—
82C2×C5×C4⋊C4—21no—
83C2×C5×C4⋊C4—21no—
84C2×C5×C4⋊C4—21no—
85C2×C5×C4⋊C4—21no—
86C2×C5×C4⋊C4D411yescharacteristic
87C2×C5×C4⋊C4D411yescharacteristic
88C2×C5×C4⋊C4D411yescharacteristic
89C2×C5×C4⋊C4D411yescharacteristic
90C2×C5×C22⋊C4—21no—
91C2×C5×C22⋊C4—21no—
92C2×C5×C22⋊C4—21no—
93C2×C5×C22⋊C4—21no—
94C2×C5×C22⋊C4—21no—
95C2×C42×C5—21no—
96C2×C42×C5C2311yescharacteristic
97C5×#107—24no—
98C5×#106—24no—
99C5×#103—24no—
100C5×#103—24no—
101C5×#103—24no—
102C5×#102—24no—
103C5×#97—21no—
104C5×#97—21no—
105C5×#97—21no—
106C5×#97C2312yes—
107C5×#97C2312yes—
108C5×#97C2312yes—
109C2×#563C2×C511yescharacteristic
110C2×#501C2×C511yescharacteristic, complemented
111C2×#491C2×C511yescharacteristic, complemented
112C2×#489C2×C511yescharacteristic
113#1664C2×C518yesdirect product factor
114C2×#330C2×C511yescharacteristic, complemented
115C2×#330C2×C511yescharacteristic, complemented
116C2×#329C2×C511yescharacteristic
117C24×C5—21no—
118C2×C5×Q8—24no—
119C2×C5×Q8—21no—
120C2×C5×Q8—22no—
121C2×C5×Q8—42no—
122C2×C5×Q8—44no—
123C2×C5×Q8—41no—
124C2×C5×Q8—82no—
125C2×C5×Q8C2×D412yes—
126C2×C5×D4—21no—
127C2×C5×D4—42no—
128C2×C5×D4C2×D412yes—
129C22×C4×C5—21no—
130C22×C4×C5—21no—
131C22×C4×C5—21no—
132C22×C4×C5—21no—
133C22×C4×C5—21no—
134C22×C4×C5—24no—
135C22×C4×C5—41no—
136C22×C4×C5—82no—
137C22×C4×C5C2×D411yescharacteristic
138C22×C4×C5C2×D411yescharacteristic
139C22×C4×C5C2×D411yescharacteristic
140C2×C5×C8—24no—
141C2×C5×C8—22no—
142C2×C5×C8—22no—
143C2×C5×C8—22no—
144C2×C5×C8—42no—
145C2×C5×C8—42no—
146C5×C4⋊C4—24no—
147C5×C4⋊C4—24no—
148C5×C4⋊C4—22no—
149C5×C4⋊C4—22no—
150C5×C4⋊C4—22no—
151C5×C4⋊C4—22no—
152C5×C4⋊C4—21no—
153C5×C4⋊C4—21no—
154C5×C4⋊C4—21no—
155C5×C4⋊C4—21no—
156C5×C4⋊C4—22no—
157C5×C4⋊C4—22no—
158C5×C4⋊C4—22no—
159C5×C4⋊C4—22no—
160C5×C4⋊C4—22no—
161C5×C4⋊C4—22no—
162C5×C4⋊C4—41no—
163C5×C4⋊C4—41no—
164C5×C4⋊C4C2×D412yes—
165C5×C4⋊C4C2×D412yes—
166C5×C4⋊C4C2×D412yes—
167C5×C4⋊C4C2×D412yes—
168C5×C22⋊C4—24no—
169C5×C22⋊C4—24no—
170C5×C22⋊C4—24no—
171C5×C22⋊C4—22no—
172C5×C22⋊C4—22no—
173C5×C22⋊C4—22no—
174C5×C22⋊C4—22no—
175C42×C5—24no—
176C42×C5C2×D411yescharacteristic
177C42×C5C2×D411yescharacteristic
178C42×C5C2×D411yescharacteristic
179C42×C5C2411yescharacteristic
180#563—22no—
181#563—22no—
182#563—22no—
183#563C22×C514yes—
184C2×C4⋊Q8—21no—
185C2×C4⋊Q8C22×C511yescharacteristic
186C2×#126—21no—
187C2×#126—21no—
188C2×#125C22×C511yescharacteristic
189C2×#125C22×C511yescharacteristic
190C2×#124C22×C511yescharacteristic
191C2×#123—21no—
192C2×C22⋊Q8—21no—
193C2×C22⋊Q8—21no—
194#501—22no—
195#501C22×C514yescomplemented
196#491—22no—
197#491C22×C514yescomplemented
198#489—22no—
199#489C22×C514yes—
200C2×#107—21no—
201C2×#106—21no—
202C2×#103—21no—
203C2×#103—21no—
204C2×#103—21no—
205C2×#102—21no—
206C2×#97C22×C511yescharacteristic
207C2×#97C22×C511yescharacteristic
208C2×#97C22×C511yescharacteristic
209#330C22×C514yescomplemented
210#330C22×C514yescomplemented
211#329C22×C514yes—
212C23×C5—22no—
213C23×C5—42no—
214C23×C5—42no—
215C23×C5—42no—
216C23×C5#12011yescharacteristic
217C5×Q8—42no—
218C5×Q8—44no—
219C5×Q8—81no—
220C5×Q8—82no—
221C5×Q8—84no—
222C5×D4—42no—
223C5×D4—81no—
224C2×C4×C5—22no—
225C2×C4×C5—22no—
226C2×C4×C5—22no—
227C2×C4×C5—22no—
228C2×C4×C5—22no—
229C2×C4×C5—22no—
230C2×C4×C5—22no—
231C2×C4×C5—22no—
232C2×C4×C5—42no—
233C2×C4×C5—42no—
234C2×C4×C5—42no—
235C2×C4×C5—42no—
236C2×C4×C5—42no—
237C2×C4×C5—42no—
238C2×C4×C5—84no—
239C2×C4×C5—82no—
240C2×C4×C5#12011yescharacteristic
241C2×C4×C5#12011yescharacteristic
242C2×C4×C5#12011yescharacteristic
243C2×C4×C5C22×D411yescharacteristic
244C2×C4×C5C22×D411yescharacteristic
245C2×C4×C5C22×D411yescharacteristic
246C5×C8—24no—
247C5×C8—42no—
248C5×C8—42no—
249C22×Q8—21no—
250C22×Q8—21no—
251C22×Q8C5×D411yescharacteristic
252C22×D4C5×D411yescharacteristic
253C23×C4—21no—
254C22×C8—21no—
255C22×C8—21no—
256C22×C8—21no—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8—21no—
260C4⋊Q8—21no—
261C4⋊Q8—21no—
262C4⋊Q8C23×C512yes—
263#126—24no—
264#126—24no—
265#126—24no—
266#126—24no—
267#125—21no—
268#125—21no—
269#125—21no—
270#125—21no—
271#125C5×D412yes—
272#125C5×D412yes—
273#125C23×C512yes—
274#125C23×C512yes—
275#124—21no—
276#124—21no—
277#124C5×D412yes—
278#124C23×C512yes—
279#123—24no—
280#123—24no—
281C22⋊Q8—24no—
282C22⋊Q8—24no—
283C22⋊Q8—24no—
284C22⋊Q8—24no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4C5×D411yescharacteristic
294C2×C4⋊C4C5×D411yescharacteristic
295C2×C4⋊C4C5×D411yescharacteristic
296C2×C4⋊C4C5×D411yescharacteristic
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C42—21no—
303C2×C42C23×C511yescharacteristic
304#107—24no—
305#106—24no—
306#103—24no—
307#103—24no—
308#103—24no—
309#102—24no—
310#97—21no—
311#97—21no—
312#97—21no—
313#97C23×C512yes—
314#97C23×C512yes—
315#97C23×C512yes—
316C22×C5—42no—
317C22×C5—42no—
318C22×C5—42no—
319C22×C5—84no—
320C22×C5#45412yes—
321C22×C5#45512yes—
322C22×C5#45612yes—
323C22×C5C2×#12011yescharacteristic
324C4×C5—21no—
325C4×C5—21no—
326C4×C5—21no—
327C4×C5—21no—
328C4×C5—21no—
329C4×C5—21no—
330C4×C5—42no—
331C4×C5—42no—
332C4×C5—42no—
333C4×C5—42no—
334C4×C5—42no—
335C4×C5—82no—
336C24—21no—
337C2×Q8—24no—
338C2×Q8—21no—
339C2×Q8—22no—
340C2×Q8—42no—
341C2×Q8—44no—
342C2×Q8—41no—
343C2×Q8—82no—
344C2×Q8C2×C5×D412yes—
345C2×D4—21no—
346C2×D4—42no—
347C2×D4C2×C5×D412yes—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—24no—
354C22×C4—41no—
355C22×C4—82no—
356C22×C4C2×C5×D411yescharacteristic
357C22×C4C2×C5×D411yescharacteristic
358C22×C4C2×C5×D411yescharacteristic
359C2×C8—24no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—42no—
364C2×C8—42no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4C2×C5×D412yes—
384C4⋊C4C2×C5×D412yes—
385C4⋊C4C2×C5×D412yes—
386C4⋊C4C2×C5×D412yes—
387C22⋊C4—24no—
388C22⋊C4—24no—
389C22⋊C4—24no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C42—24no—
395C42C2×C5×D411yescharacteristic
396C42C2×C5×D411yescharacteristic
397C42C2×C5×D411yescharacteristic
398C42C24×C511yescharacteristic
399C2×C5—82no—
400C2×C5#166414yesdirect product factor
401C2×C5C2×#45511yescharacteristic
402C2×C5C2×#45411yescharacteristic
403C2×C5C2×#45611yescharacteristic
404C23—22no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23C5×#12011yescharacteristic
409Q8—42no—
410Q8—44no—
411Q8—81no—
412Q8—82no—
413Q8—84no—
414D4—42no—
415D4—81no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—84no—
431C2×C4—82no—
432C2×C4C5×#12011yescharacteristic
433C2×C4C5×#12011yescharacteristic
434C2×C4C5×#12011yescharacteristic
435C2×C4C22×C5×D411yescharacteristic
436C2×C4C22×C5×D411yescharacteristic
437C2×C4C22×C5×D411yescharacteristic
438C8—24no—
439C8—42no—
440C8—42no—
441C5C2×#166411yescharacteristic, direct product factor, Sylow
442C22—42no—
443C22—42no—
444C22—42no—
445C22—84no—
446C22C5×#45412yes—
447C22C5×#45512yes—
448C22C5×#45612yes—
449C22C2×C5×#12011yescharacteristic
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—42no—
457C4—42no—
458C4—42no—
459C4—42no—
460C4—42no—
461C4—82no—
462C2—82no—
463C2C5×#166414yesdirect product factor
464C2C2×C5×#45511yescharacteristic
465C2C2×C5×#45411yescharacteristic
466C2C2×C5×#45611yescharacteristic
subgroupG/H
center, socleC23×C5#120
commutator, Frattini, cosocleC42C24×C5
prime radicalC22×C5×D4D4
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×#491C2semidirect
5. C5×#1664C2direct (N × H)
6. C2×C5×#330C2semidirect
7. C2×C5×#330C2semidirect
24. C5×#501C22semidirect
26. C5×#491C22semidirect
38. C5×#330C22semidirect
39. C5×#330C22semidirect
41. C2×#1664C5direct (N × H)
110. C2×#501C2×C5semidirect
111. C2×#491C2×C5semidirect
113. #1664C2×C5direct (N × H)
114. C2×#330C2×C5semidirect
115. C2×#330C2×C5semidirect
195. #501C22×C5semidirect
197. #491C22×C5semidirect
209. #330C22×C5semidirect
210. #330C22×C5semidirect
400. C2×C5#1664direct (N × H)
441. C5C2×#1664direct (N × H)
463. C2C5×#1664direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5632
C2×C5×#5012
C2×C5×#4912
C2×C5×#4892
C5×#16642
C2×C5×#3302
C2×C5×#3292
C2×#16645