Finite Group Databasebeta

← C5×#34967 #11957687

Subgroups — C5×#34967 #11957687

Order 1280.

2164 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2522C214yesmaximal, complemented
2C5×#2516C214yesmaximal, complemented
3C5×#2464C214yesmaximal, complemented
4C5×#2457C214yesmaximal, complemented
5C5×#2402C211yesmaximal, characteristic, complemented
6C5×#2397C211yesmaximal, characteristic, complemented
7C5×#2392C212yesmaximal, complemented
8C4×C5×#123C214yesmaximal, complemented
9C4×C5×#121C214yesmaximal, complemented
10C5×#2298C211yesmaximal, characteristic, complemented
11C42×C5×D4C212yesmaximal, complemented
12C2×C5×#123C2214yes—
13C2×C5×#121C2214yes—
14C2×C4×C5×D4C2212yes—
15C2×C4×C5×D4C2214yes—
16C2×C4×C5×D4C2214yes—
17C2×C4×C5×D4C2214yes—
18C2×C5×#117C2211yescharacteristic
19C22×C42×C5C2214yes—
20C5×#400C2214yes—
21C5×#399C2214yes—
22C5×#398C2212yes—
23C5×#398C2214yes—
24C5×#397C2212yes—
25C5×#396C2214yes—
26C5×#394C2214yes—
27C5×#394C2212yes—
28C5×#392C2212yes—
29C5×#392C2214yes—
30C5×#390C2214yes—
31C5×#389C2214yes—
32C5×#388C2214yes—
33C5×#388C2214yes—
34C5×#388C2214yes—
35C5×#388C2214yes—
36C5×#388C2214yes—
37C5×#387C2214yes—
38C5×#387C2214yes—
39C5×#387C2214yes—
40C5×#386C2214yes—
41C5×#386C2214yes—
42C5×#386C2214yes—
43C5×#385C2214yes—
44C5×#381C2214yes—
45C5×#380C2214yes—
46C4×C5×C4⋊C4C2214yes—
47C4×C5×C4⋊C4C2212yes—
48C4×C5×C4⋊C4C2212yes—
49C4×C5×C4⋊C4C2214yes—
50C4×C5×C22⋊C4C2214yes—
51C4×C5×C22⋊C4C2214yes—
52C4×C5×C22⋊C4C2214yes—
53C4×C5×C22⋊C4C2212yes—
54C4×C5×C22⋊C4C2214yes—
55C5×#376C2211yescharacteristic
56C43×C5C2211yescharacteristic
57#34967C511yesmaximal, characteristic, direct product factor, Sylow
58C22×C5×D4C2312yes—
59C22×C5×D4C2314yes—
60C22×C5×D4C2314yes—
61C23×C4×C5C2314yes—
62C23×C4×C5C2311yescharacteristic
63C23×C4×C5C2314yes—
64C23×C4×C5C2314yes—
65C5×#123—216no—
66C5×#121—216no—
67C4×C5×D4—216no—
68C4×C5×D4—28no—
69C4×C5×D4—48no—
70C4×C5×D4—48no—
71C5×#117—24no—
72C2×C5×C4⋊C4C2314yes—
73C2×C5×C4⋊C4C2312yes—
74C2×C5×C4⋊C4C2314yes—
75C2×C5×C4⋊C4C2314yes—
76C2×C5×C4⋊C4C2314yes—
77C2×C5×C4⋊C4C2312yes—
78C2×C5×C4⋊C4C2314yes—
79C2×C5×C4⋊C4C2314yes—
80C2×C5×C22⋊C4C2314yes—
81C2×C5×C22⋊C4C2314yes—
82C2×C5×C22⋊C4C2314yes—
83C2×C5×C22⋊C4C2314yes—
84C2×C5×C22⋊C4C2314yes—
85C2×C5×C22⋊C4C2312yes—
86C2×C5×C22⋊C4C2314yes—
87C2×C5×C22⋊C4C2314yes—
88C2×C5×C22⋊C4C2314yes—
89C2×C5×C22⋊C4C2314yes—
90C2×C5×C22⋊C4C2314yes—
91C2×C42×C5—48no—
92C2×C42×C5C2314yes—
93C2×C42×C5C2312yes—
94C2×C42×C5C2312yes—
95C2×C42×C5C2311yescharacteristic
96C2×C42×C5C2314yes—
97C2×C42×C5C2314yes—
98C2×C42×C5C2314yes—
99C2×C42×C5C2311yescharacteristic
100C5×#95C2314yes—
101C5×#95C2314yes—
102C5×#95C2314yes—
103C5×#95C2312yes—
104C5×#95C2312yes—
105C5×#95C2314yes—
106C5×#95C2314yes—
107C5×#95C2314yes—
108C5×#95C2314yes—
109C5×#95C2314yes—
110C5×#95C2314yes—
111#2522C2×C514yescomplemented
112#2516C2×C514yescomplemented
113#2464C2×C514yescomplemented
114#2457C2×C514yescomplemented
115#2402C2×C511yescharacteristic, complemented
116#2397C2×C511yescharacteristic, complemented
117#2392C2×C512yescomplemented
118C4×#123C2×C514yescomplemented
119C4×#121C2×C514yescomplemented
120#2298C2×C511yescharacteristic, complemented
121C42×D4C2×C512yescomplemented
122C24×C5C2411yescharacteristic
123C24×C5C2414yes—
124C2×C5×D4—28no—
125C2×C5×D4—28no—
126C2×C5×D4—28no—
127C2×C5×D4—28no—
128C2×C5×D4—24no—
129C2×C5×D4—28no—
130C2×C5×D4—44no—
131C2×C5×D4—44no—
132C22×C4×C5—28no—
133C22×C4×C5—22no—
134C22×C4×C5—28no—
135C22×C4×C5—28no—
136C22×C4×C5—28no—
137C22×C4×C5—28no—
138C22×C4×C5—28no—
139C22×C4×C5—28no—
140C22×C4×C5—24no—
141C22×C4×C5—44no—
142C22×C4×C5—44no—
143C22×C4×C5C2414yes—
144C22×C4×C5C2414yes—
145C22×C4×C5C2412yes—
146C22×C4×C5C2414yes—
147C22×C4×C5C2411yescharacteristic
148C22×C4×C5C2414yes—
149C22×C4×C5C2412yes—
150C22×C4×C5C2414yes—
151C22×C4×C5C2411yescharacteristic
152C5×C4⋊C4—24no—
153C5×C4⋊C4—28no—
154C5×C4⋊C4—24no—
155C5×C4⋊C4—28no—
156C5×C4⋊C4—28no—
157C5×C4⋊C4—28no—
158C5×C4⋊C4—44no—
159C5×C4⋊C4—44no—
160C5×C22⋊C4—28no—
161C5×C22⋊C4—24no—
162C5×C22⋊C4—28no—
163C5×C22⋊C4—28no—
164C5×C22⋊C4—28no—
165C5×C22⋊C4—28no—
166C5×C22⋊C4—44no—
167C5×C22⋊C4—44no—
168C5×C22⋊C4—44no—
169C5×C22⋊C4—44no—
170C5×C22⋊C4#41116yes—
171C42×C5—22no—
172C42×C5—28no—
173C42×C5—24no—
174C42×C5—28no—
175C42×C5—24no—
176C42×C5—44no—
177C42×C5—44no—
178C42×C5#4114yes—
179C2×#123C22×C514yes—
180C2×#121C22×C514yes—
181C2×C4×D4C22×C512yes—
182C2×C4×D4C22×C514yes—
183C2×C4×D4C22×C514yes—
184C2×C4×D4C22×C514yes—
185C2×#117C22×C511yescharacteristic
186C22×C42C22×C514yes—
187#400C22×C514yes—
188#399C22×C514yes—
189#398C22×C512yes—
190#398C22×C514yes—
191#397C22×C512yes—
192#396C22×C514yes—
193#394C22×C512yes—
194#394C22×C514yes—
195#392C22×C512yes—
196#392C22×C514yes—
197#390C22×C514yes—
198#389C22×C514yes—
199#388C22×C514yes—
200#388C22×C514yes—
201#388C22×C514yes—
202#388C22×C514yes—
203#388C22×C514yes—
204#387C22×C514yes—
205#387C22×C514yes—
206#387C22×C514yes—
207#386C22×C514yes—
208#386C22×C514yes—
209#386C22×C514yes—
210#385C22×C514yes—
211#381C22×C514yes—
212#380C22×C514yes—
213C4×C4⋊C4C22×C512yes—
214C4×C4⋊C4C22×C514yes—
215C4×C4⋊C4C22×C514yes—
216C4×C4⋊C4C22×C512yes—
217C4×C22⋊C4C22×C514yes—
218C4×C22⋊C4C22×C514yes—
219C4×C22⋊C4C22×C514yes—
220C4×C22⋊C4C22×C514yes—
221C4×C22⋊C4C22×C512yes—
222#376C22×C511yescharacteristic
223C43C22×C511yescharacteristic
224C23×C5—24no—
225C23×C5—24no—
226C23×C5—24no—
227C23×C5—24no—
228C23×C5—24no—
229C23×C5—24no—
230C23×C5—21no—
231C23×C5—24no—
232C23×C5—22no—
233C23×C5C2×#4118yes—
234C23×C5C2511yescharacteristic
235C5×D4—216no—
236C5×D4—416no—
237C5×D4—416no—
238C2×C4×C5—24no—
239C2×C4×C5—24no—
240C2×C4×C5—24no—
241C2×C4×C5—24no—
242C2×C4×C5—24no—
243C2×C4×C5—22no—
244C2×C4×C5—24no—
245C2×C4×C5—24no—
246C2×C4×C5—24no—
247C2×C4×C5—24no—
248C2×C4×C5—24no—
249C2×C4×C5—21no—
250C2×C4×C5—24no—
251C2×C4×C5—24no—
252C2×C4×C5—24no—
253C2×C4×C5—24no—
254C2×C4×C5—24no—
255C2×C4×C5—22no—
256C2×C4×C5—416no—
257C2×C4×C5—416no—
258C2×C4×C5—416no—
259C2×C4×C5—44no—
260C2×C4×C5C2×#4118yes—
261C2×C4×C5C2×#4114yes—
262C2×C4×C5C2×#4118yes—
263C2×C4×C5C2×#4112yes—
264C2×C4×C5H(5,𝔽2)14yes—
265C22×D4C23×C512yes—
266C22×D4C23×C514yes—
267C22×D4C23×C514yes—
268C23×C4C23×C511yescharacteristic
269C23×C4C23×C514yes—
270C23×C4C23×C514yes—
271C23×C4C23×C514yes—
272#123—216no—
273#121—216no—
274C4×D4—216no—
275C4×D4—28no—
276C4×D4—48no—
277C4×D4—48no—
278#117—24no—
279C2×C4⋊C4C23×C512yes—
280C2×C4⋊C4C23×C514yes—
281C2×C4⋊C4C23×C512yes—
282C2×C4⋊C4C23×C514yes—
283C2×C4⋊C4C23×C514yes—
284C2×C4⋊C4C23×C514yes—
285C2×C4⋊C4C23×C514yes—
286C2×C4⋊C4C23×C514yes—
287C2×C22⋊C4C23×C512yes—
288C2×C22⋊C4C23×C514yes—
289C2×C22⋊C4C23×C514yes—
290C2×C22⋊C4C23×C514yes—
291C2×C22⋊C4C23×C514yes—
292C2×C22⋊C4C23×C514yes—
293C2×C22⋊C4C23×C514yes—
294C2×C22⋊C4C23×C514yes—
295C2×C22⋊C4C23×C514yes—
296C2×C22⋊C4C23×C514yes—
297C2×C22⋊C4C23×C514yes—
298C2×C42—48no—
299C2×C42C23×C511yescharacteristic
300C2×C42C23×C512yes—
301C2×C42C23×C514yes—
302C2×C42C23×C514yes—
303C2×C42C23×C512yes—
304C2×C42C23×C514yes—
305C2×C42C23×C514yes—
306C2×C42C23×C511yescharacteristic
307#95C23×C514yes—
308#95C23×C514yes—
309#95C23×C514yes—
310#95C23×C512yes—
311#95C23×C512yes—
312#95C23×C514yes—
313#95C23×C514yes—
314#95C23×C514yes—
315#95C23×C514yes—
316#95C23×C514yes—
317#95C23×C514yes—
318C22×C5—28no—
319C22×C5—28no—
320C22×C5—28no—
321C22×C5—44no—
322C22×C5—44no—
323C22×C5—44no—
324C22×C5—44no—
325C22×C5—42no—
326C22×C5—41no—
327C22×C5—44no—
328C22×C5C22×#4111yescharacteristic
329C22×C5C22×#4114yes—
330C22×C5C2×H(5,𝔽2)12yes—
331C4×C5—24no—
332C4×C5—28no—
333C4×C5—28no—
334C4×C5—42no—
335C4×C5—44no—
336C4×C5—41no—
337C4×C5—44no—
338C4×C5—44no—
339C4×C5#54014yes—
340C24C24×C511yescharacteristic
341C24C24×C514yes—
342C2×D4—28no—
343C2×D4—28no—
344C2×D4—28no—
345C2×D4—28no—
346C2×D4—24no—
347C2×D4—28no—
348C2×D4—44no—
349C2×D4—44no—
350C22×C4—28no—
351C22×C4—22no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—24no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4C24×C514yes—
362C22×C4C24×C512yes—
363C22×C4C24×C514yes—
364C22×C4C24×C514yes—
365C22×C4C24×C511yescharacteristic
366C22×C4C24×C514yes—
367C22×C4C24×C512yes—
368C22×C4C24×C514yes—
369C22×C4C24×C511yescharacteristic
370C4⋊C4—24no—
371C4⋊C4—28no—
372C4⋊C4—24no—
373C4⋊C4—28no—
374C4⋊C4—28no—
375C4⋊C4—28no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C22⋊C4—28no—
379C22⋊C4—24no—
380C22⋊C4—28no—
381C22⋊C4—28no—
382C22⋊C4—28no—
383C22⋊C4—28no—
384C22⋊C4—44no—
385C22⋊C4—44no—
386C22⋊C4—44no—
387C22⋊C4—44no—
388C22⋊C4C5×#41116yes—
389C42—22no—
390C42—28no—
391C42—24no—
392C42—28no—
393C42—24no—
394C42—44no—
395C42—44no—
396C42C5×#4114yes—
397C2×C5—48no—
398C2×C5—81no—
399C2×C5#344412yes—
400C2×C5C2×#54011yescharacteristic
401C2×C5#348114yes—
402C23—24no—
403C23—24no—
404C23—24no—
405C23—24no—
406C23—24no—
407C23—24no—
408C23—21no—
409C23—24no—
410C23—22no—
411C23C2×C5×#4118yes—
412C23C25×C511yescharacteristic
413D4—216no—
414D4—416no—
415D4—416no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—21no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—416no—
435C2×C4—416no—
436C2×C4—416no—
437C2×C4—44no—
438C2×C4C2×C5×#4118yes—
439C2×C4C2×C5×#4114yes—
440C2×C4C2×C5×#4118yes—
441C2×C4C2×C5×#4112yes—
442C2×C4C5×H(5,𝔽2)14yes—
443C5#3496711yescharacteristic, direct product factor, Sylow
444C22—28no—
445C22—28no—
446C22—28no—
447C22—44no—
448C22—44no—
449C22—44no—
450C22—44no—
451C22—42no—
452C22—41no—
453C22—44no—
454C22C22×C5×#4111yescharacteristic
455C22C22×C5×#4114yes—
456C22C2×C5×H(5,𝔽2)12yes—
457C4—24no—
458C4—28no—
459C4—28no—
460C4—42no—
461C4—44no—
462C4—41no—
463C4—44no—
464C4—44no—
465C4C5×#54014yes—
466C2—48no—
467C2—81no—
468C2C5×#344412yes—
469C2C2×C5×#54011yescharacteristic
470C2C5×#348114yes—
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. C5×#2522C2semidirect
2. C5×#2516C2semidirect
3. C5×#2464C2semidirect
4. C5×#2457C2semidirect
5. C5×#2402C2semidirect
6. C5×#2397C2semidirect
7. C5×#2392C2semidirect
8. C4×C5×#123C2semidirect
9. C4×C5×#121C2semidirect
10. C5×#2298C2semidirect
11. C42×C5×D4C2semidirect
57. #34967C5direct (N × H)
111. #2522C2×C5semidirect
112. #2516C2×C5semidirect
113. #2464C2×C5semidirect
114. #2457C2×C5semidirect
115. #2402C2×C5semidirect
116. #2397C2×C5semidirect
117. #2392C2×C5semidirect
118. C4×#123C2×C5semidirect
119. C4×#121C2×C5semidirect
120. #2298C2×C5semidirect
121. C42×D4C2×C5semidirect
443. C5#34967direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#25222
C5×#25162
C5×#24642
C5×#24572
C5×#24022
C5×#23972
C5×#23922
C4×C5×#1232
C4×C5×#1212
C5×#22982
C42×C5×D42
#349675