Finite Group Databasebeta

← C5×#34976 #11957696

Subgroups — C5×#34976 #11957696

Order 1280.

2810 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C5×#41C212yesmaximal, complemented
2C5×#2609C214yesmaximal, complemented
3C5×#2582C211yesmaximal, characteristic, complemented
4C5×#2538C212yesmaximal, complemented
5C5×#2537C211yesmaximal, characteristic, complemented
6C5×#2485C214yesmaximal, complemented
7C5×#2407C214yesmaximal, complemented
8C5×#2392C214yesmaximal, complemented
9C4×C5×C22⋊Q8C214yesmaximal, complemented
10C4×C5×#121C214yesmaximal, complemented
11C5×#2285C211yesmaximal, characteristic, complemented
12C22×C5×#41C2212yes—
13C2×C5×C22⋊Q8C2214yes—
14C2×C5×#121C2214yes—
15C4×C5×#41—216no—
16C2×C4×C5×Q8C2212yes—
17C2×C4×C5×Q8C2212yes—
18C2×C4×C5×D4C2214yes—
19C2×C4×C5×D4C2212yes—
20C2×C4×C5×D4C2212yes—
21C2×C4×C5×D4C2214yes—
22C2×C4×C5×D4C2214yes—
23C2×C5×#117C2212yes—
24C2×C5×#117C2214yes—
25C2×C5×#117C2212yes—
26C22×C42×C5C2214yes—
27C22×C42×C5C2211yescharacteristic
28C5×#398C2214yes—
29C5×#397C2214yes—
30C5×#396C2218yes—
31C5×#394C2218yes—
32C5×#394C2214yes—
33C5×#392C2214yes—
34C5×#391C2214yes—
35C5×#390C2214yes—
36C5×#389C2214yes—
37C5×#389C2214yes—
38C5×#388C2218yes—
39C5×#387C2218yes—
40C5×#386C2218yes—
41C5×#385C2218yes—
42C5×#384C2211yescharacteristic
43C5×#384C2211yescharacteristic
44C5×#382C2212yes—
45C5×#380C2214yes—
46C5×#380C2214yes—
47C5×#380C2214yes—
48C4×C5×C4⋊C4C2214yes—
49C4×C5×C4⋊C4C2214yes—
50C4×C5×C22⋊C4C2218yes—
51#34976C511yesmaximal, characteristic, direct product factor, Sylow
52C2×C5×#41—28no—
53C2×C5×#41—28no—
54C2×C5×#41—28no—
55C22×C5×Q8C2312yes—
56C22×C5×D4C2314yes—
57C22×C5×D4C2314yes—
58C22×C5×D4C2312yes—
59C23×C4×C5C2314yes—
60C23×C4×C5C2314yes—
61C23×C4×C5C2314yes—
62C23×C4×C5C2312yes—
63C23×C4×C5C2311yescharacteristic
64C5×C22⋊Q8—216no—
65C5×#121—216no—
66C4×C5×Q8—28no—
67C4×C5×Q8—28no—
68C4×C5×D4—216no—
69C4×C5×D4—216no—
70C4×C5×D4—28no—
71C4×C5×D4—28no—
72C4×C5×D4—48no—
73C5×#117—28no—
74C5×#117—216no—
75C5×#117—44no—
76C2×C5×C4⋊C4C2314yes—
77C2×C5×C4⋊C4C2314yes—
78C2×C5×C4⋊C4C2318yes—
79C2×C5×C4⋊C4C2312yes—
80C2×C5×C4⋊C4C2312yes—
81C2×C5×C4⋊C4C2314yes—
82C2×C5×C4⋊C4C2314yes—
83C2×C5×C4⋊C4C2312yes—
84C2×C5×C4⋊C4C2312yes—
85C2×C5×C4⋊C4C2314yes—
86C2×C5×C22⋊C4C2318yes—
87C2×C5×C22⋊C4C2318yes—
88C2×C5×C22⋊C4C2314yes—
89C2×C5×C22⋊C4C2314yes—
90C2×C5×C22⋊C4C2314yes—
91C2×C5×C22⋊C4C2314yes—
92C2×C5×C22⋊C4C2318yes—
93C2×C42×C5—216no—
94C2×C42×C5D418yescomplemented
95C2×C42×C5C2314yes—
96C2×C42×C5C2314yes—
97C2×C42×C5C2314yes—
98C2×C42×C5C2314yes—
99C2×C42×C5C2311yescharacteristic
100C2×C42×C5C2311yescharacteristic
101C2×C42×C5C2312yes—
102C5×#95C2318yes—
103C5×#95C2318yes—
104C5×#95C2314yes—
105C5×#95C2318yes—
106C5×#95C2314yes—
107C2×C4×#41C2×C512yescomplemented
108#2609C2×C514yescomplemented
109#2582C2×C511yescharacteristic, complemented
110#2538C2×C512yescomplemented
111#2537C2×C511yescharacteristic, complemented
112#2485C2×C514yescomplemented
113#2407C2×C514yescomplemented
114#2392C2×C514yescomplemented
115C4×C22⋊Q8C2×C514yescomplemented
116C4×#121C2×C514yescomplemented
117#2285C2×C511yescharacteristic, complemented
118C24×C5C2411yescharacteristic
119C24×C5C2414yes—
120C5×#41—264no—
121C2×C5×Q8—24no—
122C2×C5×Q8—24no—
123C2×C5×Q8#4118yescomplemented
124C2×C5×D4—28no—
125C2×C5×D4—28no—
126C2×C5×D4—216no—
127C2×C5×D4—24no—
128C2×C5×D4—28no—
129C2×C5×D4—24no—
130C2×C5×D4—44no—
131C2×C5×D4#4118yescomplemented
132C22×C4×C5—28no—
133C22×C4×C5—28no—
134C22×C4×C5—28no—
135C22×C4×C5—24no—
136C22×C4×C5—24no—
137C22×C4×C5—22no—
138C22×C4×C5—22no—
139C22×C4×C5—28no—
140C22×C4×C5—28no—
141C22×C4×C5—28no—
142C22×C4×C5—28no—
143C22×C4×C5—28no—
144C22×C4×C5—28no—
145C22×C4×C5C2×D418yes—
146C22×C4×C5C2×D414yes—
147C22×C4×C5C2414yes—
148C22×C4×C5C2414yes—
149C22×C4×C5C2414yes—
150C22×C4×C5C2414yes—
151C22×C4×C5C2412yes—
152C22×C4×C5C2411yescharacteristic
153C22×C4×C5C2414yes—
154C22×C4×C5C2412yes—
155C22×C4×C5C2411yescharacteristic
156C5×C4⋊C4—28no—
157C5×C4⋊C4—28no—
158C5×C4⋊C4—24no—
159C5×C4⋊C4—24no—
160C5×C4⋊C4—28no—
161C5×C4⋊C4—28no—
162C5×C4⋊C4—216no—
163C5×C4⋊C4—42no—
164C5×C4⋊C4—42no—
165C5×C4⋊C4—44no—
166C5×C22⋊C4—28no—
167C5×C22⋊C4—216no—
168C5×C22⋊C4—28no—
169C5×C22⋊C4—28no—
170C5×C22⋊C4—216no—
171C5×C22⋊C4—44no—
172C5×C22⋊C4—48no—
173C42×C5—28no—
174C42×C5—28no—
175C42×C5—28no—
176C42×C5—28no—
177C42×C5C2×D418yes—
178C42×C5C2×D414yes—
179C42×C5C2×D414yes—
180C22×#41C22×C512yes—
181C2×C22⋊Q8C22×C514yes—
182C2×#121C22×C514yes—
183C4×#41—216no—
184C2×C4×Q8C22×C512yes—
185C2×C4×Q8C22×C512yes—
186C2×C4×D4C22×C514yes—
187C2×C4×D4C22×C514yes—
188C2×C4×D4C22×C512yes—
189C2×C4×D4C22×C512yes—
190C2×C4×D4C22×C514yes—
191C2×#117C22×C514yes—
192C2×#117C22×C512yes—
193C2×#117C22×C512yes—
194C22×C42C22×C514yes—
195C22×C42C22×C511yescharacteristic
196#398C22×C514yes—
197#397C22×C514yes—
198#396C22×C518yes—
199#394C22×C514yes—
200#394C22×C518yes—
201#392C22×C514yes—
202#391C22×C514yes—
203#390C22×C514yes—
204#389C22×C514yes—
205#389C22×C514yes—
206#388C22×C518yes—
207#387C22×C518yes—
208#386C22×C518yes—
209#385C22×C518yes—
210#384C22×C511yescharacteristic
211#384C22×C511yescharacteristic
212#382C22×C512yes—
213#380C22×C514yes—
214#380C22×C514yes—
215#380C22×C514yes—
216C4×C4⋊C4C22×C514yes—
217C4×C4⋊C4C22×C514yes—
218C4×C22⋊C4C22×C518yes—
219C23×C5—24no—
220C23×C5—24no—
221C23×C5—22no—
222C23×C5—22no—
223C23×C5—24no—
224C23×C5—24no—
225C23×C5—21no—
226C23×C5—24no—
227C23×C5—21no—
228C23×C5—24no—
229C23×C5C22×D412yes—
230C23×C5C2×#4118yes—
231C23×C5C2511yescharacteristic
232C5×Q8—216no—
233C5×D4—232no—
234C5×D4—216no—
235C5×D4—416no—
236C2×C4×C5—232no—
237C2×C4×C5—24no—
238C2×C4×C5—22no—
239C2×C4×C5—22no—
240C2×C4×C5—24no—
241C2×C4×C5—21no—
242C2×C4×C5—21no—
243C2×C4×C5—216no—
244C2×C4×C5—28no—
245C2×C4×C5—24no—
246C2×C4×C5—24no—
247C2×C4×C5—24no—
248C2×C4×C5—28no—
249C2×C4×C5—28no—
250C2×C4×C5—24no—
251C2×C4×C5—24no—
252C2×C4×C5—24no—
253C2×C4×C5—24no—
254C2×C4×C5—24no—
255C2×C4×C5—416no—
256C2×C4×C5—416no—
257C2×C4×C5C22×D414yes—
258C2×C4×C5C22×D412yes—
259C2×C4×C5C22×D414yes—
260C2×C4×C5C22×D412yes—
261C2×C4×C5C2×#4118yes—
262C2×C4×C5C2×#4114yes—
263C2×C4×C5C2×#4114yes—
264C2×C4×C5H(5,𝔽2)12yes—
265C2×C4×C52−1+412yes—
266C2×#41—28no—
267C2×#41—28no—
268C2×#41—28no—
269C22×Q8C23×C512yes—
270C22×D4C23×C514yes—
271C22×D4C23×C514yes—
272C22×D4C23×C512yes—
273C23×C4C23×C514yes—
274C23×C4C23×C514yes—
275C23×C4C23×C514yes—
276C23×C4C23×C511yescharacteristic
277C23×C4C23×C512yes—
278C22⋊Q8—216no—
279#121—216no—
280C4×Q8—28no—
281C4×Q8—28no—
282C4×D4—216no—
283C4×D4—216no—
284C4×D4—28no—
285C4×D4—28no—
286C4×D4—48no—
287#117—28no—
288#117—216no—
289#117—44no—
290C2×C4⋊C4C23×C512yes—
291C2×C4⋊C4C23×C512yes—
292C2×C4⋊C4C23×C514yes—
293C2×C4⋊C4C23×C514yes—
294C2×C4⋊C4C23×C514yes—
295C2×C4⋊C4C23×C514yes—
296C2×C4⋊C4C23×C518yes—
297C2×C4⋊C4C23×C512yes—
298C2×C4⋊C4C23×C512yes—
299C2×C4⋊C4C23×C514yes—
300C2×C22⋊C4C23×C514yes—
301C2×C22⋊C4C23×C514yes—
302C2×C22⋊C4C23×C514yes—
303C2×C22⋊C4C23×C518yes—
304C2×C22⋊C4C23×C518yes—
305C2×C22⋊C4C23×C514yes—
306C2×C22⋊C4C23×C518yes—
307C2×C42—216no—
308C2×C42C5×D418yescomplemented
309C2×C42C23×C514yes—
310C2×C42C23×C514yes—
311C2×C42C23×C514yes—
312C2×C42C23×C514yes—
313C2×C42C23×C511yescharacteristic
314C2×C42C23×C511yescharacteristic
315C2×C42C23×C512yes—
316#95C23×C518yes—
317#95C23×C518yes—
318#95C23×C514yes—
319#95C23×C518yes—
320#95C23×C514yes—
321C22×C5—24no—
322C22×C5—28no—
323C22×C5—22no—
324C22×C5—28no—
325C22×C5—28no—
326C22×C5—44no—
327C22×C5—44no—
328C22×C5—41no—
329C22×C5—42no—
330C22×C5—44no—
331C22×C5—44no—
332C22×C5—41no—
333C22×C5C23×D411yescharacteristic
334C22×C5C22×#4112yes—
335C22×C5C22×#4112yes—
336C22×C5C2×H(5,𝔽2)11yescharacteristic
337C22×C5C2×2−1+411yescharacteristic
338C4×C5—24no—
339C4×C5—28no—
340C4×C5—22no—
341C4×C5—44no—
342C4×C5—44no—
343C4×C5—44no—
344C4×C5—44no—
345C4×C5#52518yes—
346C4×C5#53714yes—
347C24C24×C511yescharacteristic
348C24C24×C514yes—
349#41—264no—
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8C5×#4118yescomplemented
353C2×D4—28no—
354C2×D4—28no—
355C2×D4—216no—
356C2×D4—24no—
357C2×D4—28no—
358C2×D4—24no—
359C2×D4—44no—
360C2×D4C5×#4118yescomplemented
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4C2×C5×D418yes—
375C22×C4C2×C5×D414yes—
376C22×C4C24×C514yes—
377C22×C4C24×C514yes—
378C22×C4C24×C514yes—
379C22×C4C24×C514yes—
380C22×C4C24×C512yes—
381C22×C4C24×C511yescharacteristic
382C22×C4C24×C514yes—
383C22×C4C24×C512yes—
384C22×C4C24×C511yescharacteristic
385C4⋊C4—28no—
386C4⋊C4—28no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—28no—
390C4⋊C4—28no—
391C4⋊C4—216no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—44no—
395C22⋊C4—28no—
396C22⋊C4—216no—
397C22⋊C4—28no—
398C22⋊C4—28no—
399C22⋊C4—216no—
400C22⋊C4—44no—
401C22⋊C4—48no—
402C42—28no—
403C42—28no—
404C42—28no—
405C42—28no—
406C42C2×C5×D418yes—
407C42C2×C5×D414yes—
408C42C2×C5×D414yes—
409C2×C5—42no—
410C2×C5—48no—
411C2×C5C2×#52512yes—
412C2×C5C2×#53711yescharacteristic
413C2×C5#348111yescharacteristic
414C2×C5#348212yes—
415C2×C5#348311yescharacteristic
416C23—24no—
417C23—24no—
418C23—22no—
419C23—22no—
420C23—24no—
421C23—24no—
422C23—21no—
423C23—24no—
424C23—21no—
425C23—24no—
426C23C22×C5×D412yes—
427C23C2×C5×#4118yes—
428C23C25×C511yescharacteristic
429Q8—216no—
430D4—232no—
431D4—216no—
432D4—416no—
433C2×C4—232no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—216no—
441C2×C4—28no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—28no—
446C2×C4—28no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—416no—
453C2×C4—416no—
454C2×C4C22×C5×D414yes—
455C2×C4C22×C5×D412yes—
456C2×C4C22×C5×D414yes—
457C2×C4C22×C5×D412yes—
458C2×C4C2×C5×#4118yes—
459C2×C4C2×C5×#4114yes—
460C2×C4C2×C5×#4114yes—
461C2×C4C5×H(5,𝔽2)12yes—
462C2×C4C5×2−1+412yes—
463C5#3497611yescharacteristic, direct product factor, Sylow
464C22—24no—
465C22—28no—
466C22—22no—
467C22—28no—
468C22—28no—
469C22—44no—
470C22—44no—
471C22—41no—
472C22—42no—
473C22—44no—
474C22—44no—
475C22—41no—
476C22C23×C5×D411yescharacteristic
477C22C22×C5×#4112yes—
478C22C22×C5×#4112yes—
479C22C2×C5×H(5,𝔽2)11yescharacteristic
480C22C2×C5×2−1+411yescharacteristic
481C4—24no—
482C4—28no—
483C4—22no—
484C4—44no—
485C4—44no—
486C4—44no—
487C4—44no—
488C4C5×#52518yes—
489C4C5×#53714yes—
490C2—42no—
491C2—48no—
492C2C2×C5×#52512yes—
493C2C2×C5×#53711yescharacteristic
494C2C5×#348111yescharacteristic
495C2C5×#348212yes—
496C2C5×#348311yescharacteristic
subgroupG/H
centerC22×C4×C5C24
commutator, Frattini, cosocleC23C25×C5
socleC23×C5C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C4×C5×#41C2semidirect
2. C5×#2609C2semidirect
3. C5×#2582C2semidirect
4. C5×#2538C2semidirect
5. C5×#2537C2semidirect
6. C5×#2485C2semidirect
7. C5×#2407C2semidirect
8. C5×#2392C2semidirect
9. C4×C5×C22⋊Q8C2semidirect
10. C4×C5×#121C2semidirect
11. C5×#2285C2semidirect
51. #34976C5direct (N × H)
94. C2×C42×C5D4semidirect
107. C2×C4×#41C2×C5semidirect
108. #2609C2×C5semidirect
109. #2582C2×C5semidirect
110. #2538C2×C5semidirect
111. #2537C2×C5semidirect
112. #2485C2×C5semidirect
113. #2407C2×C5semidirect
114. #2392C2×C5semidirect
115. C4×C22⋊Q8C2×C5semidirect
116. C4×#121C2×C5semidirect
117. #2285C2×C5semidirect
123. C2×C5×Q8#41semidirect
131. C2×C5×D4#41semidirect
308. C2×C42C5×D4semidirect
352. C2×Q8C5×#41semidirect
360. C2×D4C5×#41semidirect
463. C5#34976direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×C5×#412
C5×#26092
C5×#25822
C5×#25382
C5×#25372
C5×#24852
C5×#24072
C5×#23922
C4×C5×C22⋊Q82
C4×C5×#1212
C5×#22852
#349765