Finite Group Databasebeta

← C5×#29697 #11952417

Subgroups — C5×#29697 #11952417

Order 1280.

1012 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#542C211yesmaximal, characteristic
2C2×C5×#475C211yesmaximal, characteristic, complemented
3C2×C5×#476C211yesmaximal, characteristic, complemented
4C5×#2628C211yesmaximal, characteristic, complemented
5C5×#2058C212yesmaximal, complemented
6C5×#2042C212yesmaximal, complemented
7C5×#2035C212yesmaximal, complemented
8C5×#2034C212yesmaximal
9C5×#1805C211yesmaximal, characteristic, complemented
10C5×#1804C211yesmaximal, characteristic, complemented
11C5×#1798C211yesmaximal, characteristic, complemented
12C5×#542—24no—
13C5×#542—24no—
14C2×C5×C4⋊Q8C2211yescharacteristic
15C2×C5×C4⋊Q8C2211yescharacteristic
16C2×C5×#126—21no—
17C2×C5×#124—21no—
18C2×C5×#124C2211yescharacteristic
19C2×C5×#123—21no—
20C2×C5×C22⋊Q8—21no—
21C2×C5×C22⋊Q8C2211yescharacteristic
22C2×C5×#121C2211yescharacteristic
23C2×C4×C5×Q8C2211yescharacteristic
24C2×C4×C5×D4C2211yescharacteristic
25C2×C5×#117C2211yescharacteristic
26C5×#476—24no—
27C5×#475—24no—
28C2×C5×#106C2212yes—
29C2×C5×#106C2212yes—
30C2×C5×C4⋊C8C2211yescharacteristic
31C2×C5×C4⋊C8C2211yescharacteristic
32C2×C5×#103C2212yes—
33C2×C5×#103C2212yes—
34C2×C5×#102C2212yes—
35C2×C5×#102C2212yes—
36C2×C5×C22⋊C8C2212yes—
37C5×#400—21no—
38C5×#399—21no—
39C5×#395—21no—
40C5×#393—21no—
41C5×#389C2212yes—
42C5×#382C2211yescharacteristic
43C5×#380C2212yes—
44C5×#340C2212yes—
45C5×#340C2212yes—
46C5×#336C2212yes—
47#29697C511yesmaximal, characteristic, direct product factor, Sylow
48C22×C5×Q8—21no—
49C22×C5×Q8—21no—
50C22×C5×Q8C2311yescharacteristic
51C22×C5×D4—21no—
52C22×C5×D4C2311yescharacteristic
53C23×C4×C5—21no—
54C23×C4×C5C2311yescharacteristic
55C22×C5×C8C2312yes—
56C22×C5×C8C2312yes—
57C5×C4⋊Q8—22no—
58C5×C4⋊Q8—22no—
59C5×C4⋊Q8—22no—
60C5×C4⋊Q8—22no—
61C5×#126—28no—
62C5×#124—28no—
63C5×#124—22no—
64C5×#124—22no—
65C5×#123—28no—
66C5×C22⋊Q8—28no—
67C5×C22⋊Q8—22no—
68C5×C22⋊Q8—22no—
69C5×#121—22no—
70C5×#121—22no—
71C4×C5×Q8—22no—
72C4×C5×Q8D414yes—
73C4×C5×D4—22no—
74C4×C5×D4D414yes—
75C5×#117—22no—
76C5×#117—22no—
77C2×C5×C4⋊C4—21no—
78C2×C5×C4⋊C4—21no—
79C2×C5×C4⋊C4—21no—
80C2×C5×C4⋊C4—21no—
81C2×C5×C4⋊C4—22no—
82C2×C5×C4⋊C4D411yescharacteristic
83C2×C5×C4⋊C4D411yescharacteristic
84C2×C5×C4⋊C4C2312yes—
85C2×C5×C4⋊C4C2312yes—
86C2×C5×C4⋊C4C2311yescharacteristic
87C2×C5×C4⋊C4C2311yescharacteristic
88C2×C5×C22⋊C4—21no—
89C2×C5×C22⋊C4—21no—
90C2×C5×C22⋊C4—21no—
91C2×C5×C22⋊C4—21no—
92C2×C5×C22⋊C4—21no—
93C2×C5×C22⋊C4—22no—
94C2×C5×C22⋊C4D411yescharacteristic
95C2×C5×C22⋊C4D411yescharacteristic
96C2×C42×C5—21no—
97C2×C42×C5C2311yescharacteristic
98C2×C42×C5C2311yescharacteristic
99C5×#106—24no—
100C5×#106—24no—
101C5×C4⋊C8—22no—
102C5×C4⋊C8—22no—
103C5×#103—24no—
104C5×#103—42no—
105C5×#102—24no—
106C5×#102—42no—
107C5×C22⋊C8—42no—
108C5×#95—22no—
109C5×#95—21no—
110C5×#95—21no—
111C5×#95—22no—
112C2×#542C2×C511yescharacteristic
113C2×#475C2×C511yescharacteristic, complemented
114C2×#476C2×C511yescharacteristic, complemented
115#2628C2×C511yescharacteristic, complemented
116#2058C2×C512yescomplemented
117#2042C2×C512yescomplemented
118#2035C2×C512yescomplemented
119#2034C2×C512yes—
120#1805C2×C511yescharacteristic, complemented
121#1804C2×C511yescharacteristic, complemented
122#1798C2×C511yescharacteristic, complemented
123C24×C5—21no—
124C24×C5C2×D411yescharacteristic
125C2×C5×Q8—24no—
126C2×C5×Q8—21no—
127C2×C5×Q8—21no—
128C2×C5×Q8—21no—
129C2×C5×Q8—21no—
130C2×C5×Q8—21no—
131C2×C5×Q8—42no—
132C2×C5×Q8—82no—
133C2×C5×Q8—82no—
134C2×C5×Q8C2×D412yes—
135C2×C5×D4—24no—
136C2×C5×D4—21no—
137C2×C5×D4—21no—
138C2×C5×D4—21no—
139C2×C5×D4—21no—
140C2×C5×D4—21no—
141C2×C5×D4—82no—
142C2×C5×D4C2×D412yes—
143C22×C4×C5—21no—
144C22×C4×C5—21no—
145C22×C4×C5—21no—
146C22×C4×C5—21no—
147C22×C4×C5—21no—
148C22×C4×C5—21no—
149C22×C4×C5—21no—
150C22×C4×C5—21no—
151C22×C4×C5—21no—
152C22×C4×C5—24no—
153C22×C4×C5—22no—
154C22×C4×C5—22no—
155C22×C4×C5—82no—
156C22×C4×C5C2×D411yescharacteristic
157C22×C4×C5C2×D411yescharacteristic
158C22×C4×C5C2×D411yescharacteristic
159C22×C4×C5C2×D411yescharacteristic
160C22×C4×C5C2×D411yescharacteristic
161C22×C4×C5C2411yescharacteristic
162C2×C5×C8—22no—
163C2×C5×C8—22no—
164C2×C5×C8—22no—
165C2×C5×C8—22no—
166C2×C5×C8—22no—
167C2×C5×C8—22no—
168C5×C4⋊C4—24no—
169C5×C4⋊C4—24no—
170C5×C4⋊C4—24no—
171C5×C4⋊C4—22no—
172C5×C4⋊C4—22no—
173C5×C4⋊C4—21no—
174C5×C4⋊C4—21no—
175C5×C4⋊C4—22no—
176C5×C4⋊C4—22no—
177C5×C4⋊C4—22no—
178C5×C4⋊C4—22no—
179C5×C4⋊C4—42no—
180C5×C4⋊C4—82no—
181C5×C4⋊C4C2×D412yes—
182C5×C4⋊C4C2×D412yes—
183C5×C22⋊C4—24no—
184C5×C22⋊C4—24no—
185C5×C22⋊C4—24no—
186C5×C22⋊C4—24no—
187C5×C22⋊C4—24no—
188C5×C22⋊C4—22no—
189C5×C22⋊C4—22no—
190C5×C22⋊C4—82no—
191C42×C5—24no—
192C42×C5—21no—
193C42×C5—21no—
194C42×C5C2×D412yes—
195C42×C5C2×D412yes—
196#542—24no—
197#542—24no—
198C2×C4⋊Q8C22×C511yescharacteristic
199C2×C4⋊Q8C22×C511yescharacteristic
200C2×#126—21no—
201C2×#124—21no—
202C2×#124C22×C511yescharacteristic
203C2×#123—21no—
204C2×C22⋊Q8—21no—
205C2×C22⋊Q8C22×C511yescharacteristic
206C2×#121C22×C511yescharacteristic
207C2×C4×Q8C22×C511yescharacteristic
208C2×C4×D4C22×C511yescharacteristic
209C2×#117C22×C511yescharacteristic
210#476—24no—
211#475—24no—
212C2×#106C22×C512yes—
213C2×#106C22×C512yes—
214C2×C4⋊C8C22×C511yescharacteristic
215C2×C4⋊C8C22×C511yescharacteristic
216C2×#103C22×C512yes—
217C2×#103C22×C512yes—
218C2×#102C22×C512yes—
219C2×#102C22×C512yes—
220C2×C22⋊C8C22×C512yes—
221#400—21no—
222#399—21no—
223#395—21no—
224#393—21no—
225#389C22×C512yes—
226#382C22×C511yescharacteristic
227#380C22×C512yes—
228#340C22×C512yes—
229#340C22×C512yes—
230#336C22×C512yes—
231C23×C5—21no—
232C23×C5—22no—
233C23×C5—21no—
234C23×C5—21no—
235C23×C5—22no—
236C23×C5—22no—
237C23×C5—41no—
238C23×C5—41no—
239C23×C5—42no—
240C23×C5—42no—
241C23×C5C22×D411yescharacteristic
242C5×Q8—22no—
243C5×Q8—22no—
244C5×Q8—41no—
245C5×Q8—41no—
246C5×Q8—84no—
247C5×Q8—84no—
248C5×D4—22no—
249C5×D4—22no—
250C5×D4—41no—
251C5×D4—41no—
252C5×D4—84no—
253C2×C4×C5—22no—
254C2×C4×C5—22no—
255C2×C4×C5—21no—
256C2×C4×C5—22no—
257C2×C4×C5—21no—
258C2×C4×C5—22no—
259C2×C4×C5—21no—
260C2×C4×C5—22no—
261C2×C4×C5—21no—
262C2×C4×C5—21no—
263C2×C4×C5—22no—
264C2×C4×C5—22no—
265C2×C4×C5—41no—
266C2×C4×C5—41no—
267C2×C4×C5—42no—
268C2×C4×C5—42no—
269C2×C4×C5—41no—
270C2×C4×C5—41no—
271C2×C4×C5—41no—
272C2×C4×C5—41no—
273C2×C4×C5—42no—
274C2×C4×C5—42no—
275C2×C4×C5—42no—
276C2×C4×C5—42no—
277C2×C4×C5—42no—
278C2×C4×C5—42no—
279C2×C4×C5—84no—
280C2×C4×C5#12012yes—
281C2×C4×C5#12012yes—
282C2×C4×C5C22×D411yescharacteristic
283C2×C4×C5C22×D411yescharacteristic
284C2×C4×C5H(5,𝔽2)11yescharacteristic
285C2×C4×C52−1+411yescharacteristic
286C2×C4×C52−1+411yescharacteristic
287C2×C4×C52−1+411yescharacteristic
288C5×C8—42no—
289C5×C8—42no—
290C22×Q8—21no—
291C22×Q8—21no—
292C22×Q8C23×C511yescharacteristic
293C22×D4—21no—
294C22×D4C23×C511yescharacteristic
295C23×C4—21no—
296C23×C4C23×C511yescharacteristic
297C22×C8C23×C512yes—
298C22×C8C23×C512yes—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303#126—28no—
304#124—28no—
305#124—22no—
306#124—22no—
307#123—28no—
308C22⋊Q8—28no—
309C22⋊Q8—22no—
310C22⋊Q8—22no—
311#121—22no—
312#121—22no—
313C4×Q8—22no—
314C4×Q8C5×D414yes—
315C4×D4—22no—
316C4×D4C5×D414yes—
317#117—22no—
318#117—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4C5×D411yescharacteristic
325C2×C4⋊C4C5×D411yescharacteristic
326C2×C4⋊C4C23×C512yes—
327C2×C4⋊C4C23×C512yes—
328C2×C4⋊C4C23×C511yescharacteristic
329C2×C4⋊C4C23×C511yescharacteristic
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4C5×D411yescharacteristic
337C2×C22⋊C4C5×D411yescharacteristic
338C2×C42—21no—
339C2×C42C23×C511yescharacteristic
340C2×C42C23×C511yescharacteristic
341#106—24no—
342#106—24no—
343C4⋊C8—22no—
344C4⋊C8—22no—
345#103—24no—
346#103—42no—
347#102—24no—
348#102—42no—
349C22⋊C8—42no—
350#95—22no—
351#95—21no—
352#95—21no—
353#95—22no—
354C22×C5—41no—
355C22×C5—42no—
356C22×C5—42no—
357C22×C5—42no—
358C22×C5—42no—
359C22×C5—41no—
360C22×C5—41no—
361C22×C5—42no—
362C22×C5—82no—
363C22×C5—82no—
364C22×C5C2×#12011yescharacteristic
365C22×C5#53611yescharacteristic
366C22×C5#53711yescharacteristic
367C22×C5#57712yes—
368C22×C5#57712yes—
369C4×C5—22no—
370C4×C5—22no—
371C4×C5—41no—
372C4×C5—42no—
373C4×C5—41no—
374C4×C5—42no—
375C4×C5—42no—
376C4×C5—41no—
377C4×C5—82no—
378C4×C5—82no—
379C4×C5#53611yescharacteristic
380C4×C5#53611yescharacteristic
381C4×C5#53711yescharacteristic
382C4×C5#53711yescharacteristic
383C24—21no—
384C24C2×C5×D411yescharacteristic
385C2×Q8—24no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—42no—
392C2×Q8—82no—
393C2×Q8—82no—
394C2×Q8C2×C5×D412yes—
395C2×D4—24no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—82no—
402C2×D4C2×C5×D412yes—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—82no—
416C22×C4C2×C5×D411yescharacteristic
417C22×C4C2×C5×D411yescharacteristic
418C22×C4C2×C5×D411yescharacteristic
419C22×C4C2×C5×D411yescharacteristic
420C22×C4C2×C5×D411yescharacteristic
421C22×C4C24×C511yescharacteristic
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—42no—
440C4⋊C4—82no—
441C4⋊C4C2×C5×D412yes—
442C4⋊C4C2×C5×D412yes—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—82no—
451C42—24no—
452C42—21no—
453C42—21no—
454C42C2×C5×D412yes—
455C42C2×C5×D412yes—
456C2×C5—81no—
457C2×C5—82no—
458C2×C5#261611yescharacteristic
459C2×C5#300911yescharacteristic
460C2×C5#301011yescharacteristic
461C2×C5#324412yes—
462C2×C5#326312yes—
463C23—21no—
464C23—22no—
465C23—21no—
466C23—21no—
467C23—22no—
468C23—22no—
469C23—41no—
470C23—41no—
471C23—42no—
472C23—42no—
473C23C22×C5×D411yescharacteristic
474Q8—22no—
475Q8—22no—
476Q8—41no—
477Q8—41no—
478Q8—84no—
479Q8—84no—
480D4—22no—
481D4—22no—
482D4—41no—
483D4—41no—
484D4—84no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—84no—
512C2×C4C5×#12012yes—
513C2×C4C5×#12012yes—
514C2×C4C22×C5×D411yescharacteristic
515C2×C4C22×C5×D411yescharacteristic
516C2×C4C5×H(5,𝔽2)11yescharacteristic
517C2×C4C5×2−1+411yescharacteristic
518C2×C4C5×2−1+411yescharacteristic
519C2×C4C5×2−1+411yescharacteristic
520C8—42no—
521C8—42no—
522C5#2969711yescharacteristic, direct product factor, Sylow
523C22—42no—
524C22—42no—
525C22—42no—
526C22—41no—
527C22—41no—
528C22—42no—
529C22—41no—
530C22—42no—
531C22—82no—
532C22—82no—
533C22C2×C5×#12011yescharacteristic
534C22C5×#53611yescharacteristic
535C22C5×#53711yescharacteristic
536C22C5×#57712yes—
537C22C5×#57712yes—
538C4—22no—
539C4—22no—
540C4—41no—
541C4—42no—
542C4—42no—
543C4—41no—
544C4—41no—
545C4—42no—
546C4—82no—
547C4—82no—
548C4C5×#53611yescharacteristic
549C4C5×#53611yescharacteristic
550C4C5×#53711yescharacteristic
551C4C5×#53711yescharacteristic
552C2—81no—
553C2—82no—
554C2C5×#261611yescharacteristic
555C2C5×#300911yescharacteristic
556C2C5×#301011yescharacteristic
557C2C5×#324412yes—
558C2C5×#326312yes—
subgroupG/H
center, socleC23×C5C22×D4
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC2×C5×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C5 ◁ C2×C5×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C5×#475C2semidirect
3. C2×C5×#476C2semidirect
4. C5×#2628C2semidirect
5. C5×#2058C2semidirect
6. C5×#2042C2semidirect
7. C5×#2035C2semidirect
9. C5×#1805C2semidirect
10. C5×#1804C2semidirect
11. C5×#1798C2semidirect
47. #29697C5direct (N × H)
113. C2×#475C2×C5semidirect
114. C2×#476C2×C5semidirect
115. #2628C2×C5semidirect
116. #2058C2×C5semidirect
117. #2042C2×C5semidirect
118. #2035C2×C5semidirect
120. #1805C2×C5semidirect
121. #1804C2×C5semidirect
122. #1798C2×C5semidirect
522. C5#29697direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5422
C2×C5×#4752
C2×C5×#4762
C5×#26282
C5×#20582
C5×#20422
C5×#20352
C5×#20342
C5×#18052
C5×#18042
C5×#17982
#296975