Finite Group Databasebeta

← C5×#29708 #11952428

Subgroups — C5×#29708 #11952428

Order 1280.

930 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3461C211yesmaximal, characteristic
2C5×#3080C212yesmaximal, complemented
3C5×#2565C211yesmaximal, characteristic, complemented
4C5×#2061C211yesmaximal, characteristic, complemented
5C5×#2060C211yesmaximal, characteristic, complemented
6C5×#2040C211yesmaximal, characteristic
7C5×#2039C211yesmaximal, characteristic
8C5×#2038C212yesmaximal, complemented
9C5×#2038C212yesmaximal, complemented
10C5×#1806C212yesmaximal, complemented
11C5×#1801C211yesmaximal, characteristic, complemented
12C5×#544C2211yescharacteristic
13C5×#542—22no—
14C5×#541—22no—
15C5×#539—22no—
16C5×#537C2211yescharacteristic
17C5×#536C2212yes—
18C5×#533—22no—
19C5×#529—22no—
20C5×#529—22no—
21C2×C5×#125C2211yescharacteristic
22C4×C5×#41C2212yes—
23C2×C4×C5×Q8C2211yescharacteristic
24C5×#479—22no—
25C5×#479—22no—
26C5×#478—22no—
27C5×#478—22no—
28C5×#428C2212yes—
29C5×#428C2212yes—
30C5×#423C2212yes—
31C2×C5×#104C2212yes—
32C2×C5×#104C2212yes—
33C5×#417C2212yes—
34C5×#417C2212yes—
35C2×C5×#101C2212yes—
36C5×#398—22no—
37C5×#391C2211yescharacteristic
38C5×#391C2211yescharacteristic
39C5×#387—22no—
40C5×#384C2211yescharacteristic
41C5×#382C2211yescharacteristic
42C4×C5×C4⋊C4C2211yescharacteristic
43C5×#343C2211yescharacteristic
44C5×#343C2211yescharacteristic
45C5×#339C2212yes—
46C5×#339C2212yes—
47#29708C511yesmaximal, characteristic, direct product factor, Sylow
48C2×C5×#41C2312yes—
49C22×C5×Q8C2311yescharacteristic
50C2×C5×M16C2312yes—
51C2×C5×M16C2312yes—
52C5×C4⋊Q8—22no—
53C5×C4⋊Q8—22no—
54C5×#126—24no—
55C5×#126—24no—
56C5×#125—22no—
57C5×#125—24no—
58C5×#125—22no—
59C5×#125—22no—
60C5×#125—22no—
61C5×#124—22no—
62C5×#124—22no—
63C5×#123—24no—
64C5×#123—24no—
65C5×C22⋊Q8—24no—
66C5×C22⋊Q8—22no—
67C5×C22⋊Q8—22no—
68C5×C22⋊Q8—22no—
69C5×C22⋊Q8—22no—
70C5×#121—22no—
71C5×#121—22no—
72C4×C5×Q8—22no—
73C4×C5×Q8—22no—
74C4×C5×Q8—22no—
75C4×C5×Q8D412yes—
76C4×C5×Q8D412yes—
77C4×C5×D4—22no—
78C4×C5×D4—22no—
79C4×C5×D4—22no—
80C4×C5×D4D412yes—
81C4×C5×D4D412yes—
82C5×#117—24no—
83C5×#117—22no—
84C5×#117C2312yes—
85C2×C5×C4⋊C4—21no—
86C2×C5×C4⋊C4—22no—
87C2×C5×C4⋊C4—21no—
88C2×C5×C4⋊C4—21no—
89C2×C5×C4⋊C4D412yes—
90C2×C5×C4⋊C4C2311yescharacteristic
91C2×C5×C4⋊C4C2311yescharacteristic
92C2×C5×C4⋊C4C2311yescharacteristic
93C2×C42×C5—22no—
94C2×C42×C5D412yes—
95C2×C42×C5C2311yescharacteristic
96C2×C42×C5C2311yescharacteristic
97C2×C42×C5C2311yescharacteristic
98C5×#107—22no—
99C5×#107—22no—
100C5×#106—22no—
101C5×#106—22no—
102C5×C4⋊C8—22no—
103C5×C4⋊C8—22no—
104C5×#104—42no—
105C5×#104—42no—
106C5×#103—22no—
107C5×#103—22no—
108C5×#102—22no—
109C5×#102—22no—
110C5×#101—42no—
111C5×#95—21no—
112C5×#95—22no—
113C5×#95—22no—
114C5×#95—22no—
115#3461C2×C511yescharacteristic
116#3080C2×C512yescomplemented
117#2565C2×C511yescharacteristic, complemented
118#2061C2×C511yescharacteristic, complemented
119#2060C2×C511yescharacteristic, complemented
120#2040C2×C511yescharacteristic
121#2039C2×C511yescharacteristic
122#2038C2×C512yescomplemented
123#2038C2×C512yescomplemented
124#1806C2×C512yescomplemented
125#1801C2×C511yescharacteristic, complemented
126C5×#41—22no—
127C5×#41—22no—
128C5×#41—22no—
129C5×#41—22no—
130C2×C5×Q8—22no—
131C2×C5×Q8—22no—
132C2×C5×Q8—21no—
133C2×C5×Q8—22no—
134C2×C5×Q8—21no—
135C2×C5×Q8C2×D412yes—
136C2×C5×D4—22no—
137C2×C5×D4—22no—
138C2×C5×D4C2×D412yes—
139C22×C4×C5—21no—
140C22×C4×C5—22no—
141C22×C4×C5—22no—
142C22×C4×C5—22no—
143C22×C4×C5—21no—
144C22×C4×C5—21no—
145C22×C4×C5—21no—
146C22×C4×C5C2×D412yes—
147C22×C4×C5C2×D412yes—
148C22×C4×C5C2×D411yescharacteristic
149C22×C4×C5C2×D411yescharacteristic
150C22×C4×C5C2411yescharacteristic
151C5×M16—22no—
152C5×M16—22no—
153C5×M16—22no—
154C5×M16—22no—
155C2×C5×C8—22no—
156C2×C5×C8—22no—
157C5×C4⋊C4—24no—
158C5×C4⋊C4—24no—
159C5×C4⋊C4—22no—
160C5×C4⋊C4—22no—
161C5×C4⋊C4—24no—
162C5×C4⋊C4—24no—
163C5×C4⋊C4—22no—
164C5×C4⋊C4—22no—
165C5×C4⋊C4—22no—
166C5×C4⋊C4—22no—
167C5×C4⋊C4—22no—
168C5×C4⋊C4—42no—
169C5×C4⋊C4—44no—
170C5×C4⋊C4—41no—
171C5×C4⋊C4—41no—
172C5×C4⋊C4—42no—
173C5×C4⋊C4C2×D412yes—
174C5×C4⋊C4C2×D412yes—
175C5×C22⋊C4—24no—
176C5×C22⋊C4—24no—
177C5×C22⋊C4—24no—
178C5×C22⋊C4—22no—
179C5×C22⋊C4—22no—
180C5×C22⋊C4—22no—
181C42×C5—22no—
182C42×C5—22no—
183C42×C5—22no—
184C42×C5—22no—
185C42×C5—22no—
186C42×C5—21no—
187C42×C5—21no—
188C42×C5—21no—
189C42×C5—21no—
190C42×C5C2×D412yes—
191C42×C5C2×D412yes—
192#544C22×C511yescharacteristic
193#542—22no—
194#541—22no—
195#539—22no—
196#537C22×C511yescharacteristic
197#536C22×C512yes—
198#533—22no—
199#529—22no—
200#529—22no—
201C2×#125C22×C511yescharacteristic
202C4×#41C22×C512yes—
203C2×C4×Q8C22×C511yescharacteristic
204#479—22no—
205#479—22no—
206#478—22no—
207#478—22no—
208#428C22×C512yes—
209#428C22×C512yes—
210#423C22×C512yes—
211C2×#104C22×C512yes—
212C2×#104C22×C512yes—
213#417C22×C512yes—
214#417C22×C512yes—
215C2×#101C22×C512yes—
216#398—22no—
217#391C22×C511yescharacteristic
218#391C22×C511yescharacteristic
219#387—22no—
220#384C22×C511yescharacteristic
221#382C22×C511yescharacteristic
222C4×C4⋊C4C22×C511yescharacteristic
223#343C22×C511yescharacteristic
224#343C22×C511yescharacteristic
225#339C22×C512yes—
226#339C22×C512yes—
227C23×C5—22no—
228C23×C5C22×D411yescharacteristic
229C5×Q8—22no—
230C5×Q8—22no—
231C5×Q8—42no—
232C5×Q8—42no—
233C5×Q8—82no—
234C5×D4—22no—
235C5×D4—22no—
236C5×D4—42no—
237C5×D4—42no—
238C5×D4—82no—
239C2×C4×C5—22no—
240C2×C4×C5—22no—
241C2×C4×C5—22no—
242C2×C4×C5—22no—
243C2×C4×C5—22no—
244C2×C4×C5—22no—
245C2×C4×C5—22no—
246C2×C4×C5—22no—
247C2×C4×C5—41no—
248C2×C4×C5—41no—
249C2×C4×C5—44no—
250C2×C4×C5—41no—
251C2×C4×C5—41no—
252C2×C4×C5—42no—
253C2×C4×C5—42no—
254C2×C4×C5—42no—
255C2×C4×C5—41no—
256C2×C4×C5—42no—
257C2×C4×C5—41no—
258C2×C4×C5—42no—
259C2×C4×C5—41no—
260C2×C4×C5—41no—
261C2×C4×C5—41no—
262C2×C4×C5—82no—
263C2×C4×C5#12011yescharacteristic
264C2×C4×C5#12011yescharacteristic
265C2×C4×C5#12012yes—
266C2×C4×C5C22×D412yes—
267C2×C4×C5H(5,𝔽2)11yescharacteristic
268C2×C4×C52−1+411yescharacteristic
269C2×C4×C52−1+412yes—
270C5×C8—42no—
271C5×C8—42no—
272C2×#41C23×C512yes—
273C22×Q8C23×C511yescharacteristic
274C2×M16C23×C512yes—
275C2×M16C23×C512yes—
276C4⋊Q8—22no—
277C4⋊Q8—22no—
278#126—24no—
279#126—24no—
280#125—22no—
281#125—24no—
282#125—22no—
283#125—22no—
284#125—22no—
285#124—22no—
286#124—22no—
287#123—24no—
288#123—24no—
289C22⋊Q8—24no—
290C22⋊Q8—22no—
291C22⋊Q8—22no—
292C22⋊Q8—22no—
293C22⋊Q8—22no—
294#121—22no—
295#121—22no—
296C4×Q8—22no—
297C4×Q8—22no—
298C4×Q8—22no—
299C4×Q8C5×D412yes—
300C4×Q8C5×D412yes—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4—22no—
304C4×D4C5×D412yes—
305C4×D4C5×D412yes—
306#117—24no—
307#117—22no—
308#117C23×C512yes—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4C5×D412yes—
314C2×C4⋊C4C23×C511yescharacteristic
315C2×C4⋊C4C23×C511yescharacteristic
316C2×C4⋊C4C23×C511yescharacteristic
317C2×C42—22no—
318C2×C42C5×D412yes—
319C2×C42C23×C511yescharacteristic
320C2×C42C23×C511yescharacteristic
321C2×C42C23×C511yescharacteristic
322#107—22no—
323#107—22no—
324#106—22no—
325#106—22no—
326C4⋊C8—22no—
327C4⋊C8—22no—
328#104—42no—
329#104—42no—
330#103—22no—
331#103—22no—
332#102—22no—
333#102—22no—
334#101—42no—
335#95—21no—
336#95—22no—
337#95—22no—
338#95—22no—
339C22×C5—21no—
340C22×C5—21no—
341C22×C5—42no—
342C22×C5—42no—
343C22×C5—42no—
344C22×C5C2×#12011yescharacteristic
345C22×C5#53611yescharacteristic
346C22×C5#53711yescharacteristic
347C4×C5—22no—
348C4×C5—21no—
349C4×C5—21no—
350C4×C5—42no—
351C4×C5—42no—
352C4×C5—42no—
353C4×C5—42no—
354C4×C5—42no—
355C4×C5—42no—
356C4×C5—81no—
357C4×C5—81no—
358C4×C5—81no—
359C4×C5#53612yes—
360C4×C5#53712yes—
361#41—22no—
362#41—22no—
363#41—22no—
364#41—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—21no—
368C2×Q8—22no—
369C2×Q8—21no—
370C2×Q8C2×C5×D412yes—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4C2×C5×D412yes—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4C2×C5×D412yes—
382C22×C4C2×C5×D412yes—
383C22×C4C2×C5×D411yescharacteristic
384C22×C4C2×C5×D411yescharacteristic
385C22×C4C24×C511yescharacteristic
386M16—22no—
387M16—22no—
388M16—22no—
389M16—22no—
390C2×C8—22no—
391C2×C8—22no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—42no—
404C4⋊C4—44no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—42no—
408C4⋊C4C2×C5×D412yes—
409C4⋊C4C2×C5×D412yes—
410C22⋊C4—24no—
411C22⋊C4—24no—
412C22⋊C4—24no—
413C22⋊C4—22no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42C2×C5×D412yes—
426C42C2×C5×D412yes—
427C2×C5—21no—
428C2×C5—21no—
429C2×C5—82no—
430C2×C5#261611yescharacteristic
431C2×C5#301811yescharacteristic
432C2×C5#302011yescharacteristic
433C23—22no—
434C23C22×C5×D411yescharacteristic
435Q8—22no—
436Q8—22no—
437Q8—42no—
438Q8—42no—
439Q8—82no—
440D4—22no—
441D4—22no—
442D4—42no—
443D4—42no—
444D4—82no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—44no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—82no—
469C2×C4C5×#12011yescharacteristic
470C2×C4C5×#12011yescharacteristic
471C2×C4C5×#12012yes—
472C2×C4C22×C5×D412yes—
473C2×C4C5×H(5,𝔽2)11yescharacteristic
474C2×C4C5×2−1+411yescharacteristic
475C2×C4C5×2−1+412yes—
476C8—42no—
477C8—42no—
478C5#2970811yescharacteristic, direct product factor, Sylow
479C22—21no—
480C22—21no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22C2×C5×#12011yescharacteristic
485C22C5×#53611yescharacteristic
486C22C5×#53711yescharacteristic
487C4—22no—
488C4—21no—
489C4—21no—
490C4—42no—
491C4—42no—
492C4—42no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—81no—
497C4—81no—
498C4—81no—
499C4C5×#53612yes—
500C4C5×#53712yes—
501C2—21no—
502C2—21no—
503C2—82no—
504C2C5×#261611yescharacteristic
505C2C5×#301811yescharacteristic
506C2C5×#302011yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC5×#537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#3080C2semidirect
3. C5×#2565C2semidirect
4. C5×#2061C2semidirect
5. C5×#2060C2semidirect
8. C5×#2038C2semidirect
9. C5×#2038C2semidirect
10. C5×#1806C2semidirect
11. C5×#1801C2semidirect
47. #29708C5direct (N × H)
116. #3080C2×C5semidirect
117. #2565C2×C5semidirect
118. #2061C2×C5semidirect
119. #2060C2×C5semidirect
122. #2038C2×C5semidirect
123. #2038C2×C5semidirect
124. #1806C2×C5semidirect
125. #1801C2×C5semidirect
478. C5#29708direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#34612
C5×#30802
C5×#25652
C5×#20612
C5×#20602
C5×#20402
C5×#20392
C5×#20382
C5×#18062
C5×#18012
#297085