Finite Group Databasebeta

← C5×#33810 #11956530

Subgroups — C5×#33810 #11956530

Order 1280.

454 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3175C211yesmaximal, characteristic
2C5×#3164C211yesmaximal, characteristic, complemented
3C5×#2255C211yesmaximal, characteristic, complemented
4C5×#2254C211yesmaximal, characteristic, complemented
5C5×#2243C211yesmaximal, characteristic, complemented
6C5×#2242C211yesmaximal, characteristic, complemented
7C5×#2237C211yesmaximal, characteristic, complemented
8C5×#2235C211yesmaximal, characteristic, complemented
9C5×#2234C211yesmaximal, characteristic
10C5×#2233C211yesmaximal, characteristic
11C5×#2229C211yesmaximal, characteristic, complemented
12C5×#2228C211yesmaximal, characteristic, complemented
13C5×#2226C211yesmaximal, characteristic, complemented
14C5×#2225C211yesmaximal, characteristic, complemented
15C5×#2164C211yesmaximal, characteristic, complemented
16C5×#544—21no—
17C5×#542—21no—
18C5×#501—21no—
19C5×#501—21no—
20C5×#500C2211yescharacteristic
21C5×#500C2211yescharacteristic
22C5×#500C2211yescharacteristic
23C5×#499C2211yescharacteristic
24C5×#499C2211yescharacteristic
25C5×#495C2211yescharacteristic
26C5×#489—21no—
27C5×#468—21no—
28C5×#467C2211yescharacteristic
29C5×#466C2211yescharacteristic
30C5×#440—21no—
31C4×C5×Q16C2211yescharacteristic
32C4×C5×D8C2211yescharacteristic
33C5×#428—21no—
34C5×#427C2211yescharacteristic
35C2×C5×#107C2211yescharacteristic
36C5×#405C2211yescharacteristic
37C5×#367C2211yescharacteristic
38C5×#367C2211yescharacteristic
39C5×#367C2211yescharacteristic
40C5×#367C2211yescharacteristic
41C5×#367C2211yescharacteristic
42C5×#367C2211yescharacteristic
43C5×#366C2211yescharacteristic
44C5×#366C2211yescharacteristic
45C5×C4⋊C16C2211yescharacteristic
46C5×C4⋊C16C2211yescharacteristic
47C5×#358C2211yescharacteristic
48C5×#358C2211yescharacteristic
49C5×#358C2211yescharacteristic
50C5×#358C2211yescharacteristic
51C5×#357C2211yescharacteristic
52C5×#357C2211yescharacteristic
53C5×#357C2211yescharacteristic
54C5×#357C2211yescharacteristic
55C5×C22⋊C16C2211yescharacteristic
56C5×C22⋊C16C2211yescharacteristic
57C5×#346C2211yescharacteristic
58C4×C5×C16C2211yescharacteristic
59#33810C511yesmaximal, characteristic, direct product factor, Sylow
60C2×C5×Q16C2311yescharacteristic
61C2×C5×SD16—21no—
62C2×C5×D8C2311yescharacteristic
63C2×C5×M16D411yescharacteristic
64C22×C5×C8C2311yescharacteristic
65C5×C4⋊Q8—21no—
66C5×C4⋊Q8—21no—
67C5×C4⋊Q8—21no—
68C5×C4⋊Q8—41no—
69C5×#126—41no—
70C5×#125—21no—
71C5×#125—21no—
72C5×#125—41no—
73C5×#124—21no—
74C5×#124—41no—
75C5×#123—41no—
76C5×C22⋊Q8—21no—
77C5×C22⋊Q8—41no—
78C5×C22⋊Q8—41no—
79C5×C22⋊Q8—41no—
80C5×#121—21no—
81C4×C5×Q8—21no—
82C4×C5×D4—21no—
83C5×#117—21no—
84C5×#117D411yescharacteristic
85C2×C5×C4⋊C4—21no—
86C2×C5×C16C2311yescharacteristic
87C2×C5×C16C2311yescharacteristic
88C2×C5×C16C2311yescharacteristic
89C2×C5×C16C2311yescharacteristic
90C5×#107—21no—
91C5×#107—21no—
92C5×#107—21no—
93C5×#107—21no—
94C5×#107C2311yescharacteristic
95C5×#107C2311yescharacteristic
96C5×#107C2311yescharacteristic
97C5×#107C2311yescharacteristic
98C5×#107C2311yescharacteristic
99C5×#107C2311yescharacteristic
100C5×#106—21no—
101C5×#106—21no—
102C5×#106—21no—
103C5×#103—21no—
104C5×#103—21no—
105C5×#102—21no—
106C5×#102—21no—
107C5×#97D411yescharacteristic
108C5×#97D411yescharacteristic
109C4×C5×C8C2311yescharacteristic
110C4×C5×C8C2311yescharacteristic
111#3175C2×C511yescharacteristic
112#3164C2×C511yescharacteristic, complemented
113#2255C2×C511yescharacteristic, complemented
114#2254C2×C511yescharacteristic, complemented
115#2243C2×C511yescharacteristic, complemented
116#2242C2×C511yescharacteristic, complemented
117#2237C2×C511yescharacteristic, complemented
118#2235C2×C511yescharacteristic, complemented
119#2234C2×C511yescharacteristic
120#2233C2×C511yescharacteristic
121#2229C2×C511yescharacteristic, complemented
122#2228C2×C511yescharacteristic, complemented
123#2226C2×C511yescharacteristic, complemented
124#2225C2×C511yescharacteristic, complemented
125#2164C2×C511yescharacteristic, complemented
126C2×C5×Q8—21no—
127C2×C5×Q8—41no—
128C2×C5×Q8—41no—
129C2×C5×Q8—41no—
130C2×C5×D4—21no—
131C2×C5×D4—41no—
132C22×C4×C5—41no—
133C22×C4×C5—41no—
134C22×C4×C5C2×D411yescharacteristic
135C5×Q16—21no—
136C5×Q16—21no—
137C5×SD16—42no—
138C5×D8—21no—
139C5×D8—21no—
140C5×M16—22no—
141C2×C5×C8—21no—
142C2×C5×C8—21no—
143C2×C5×C8C2×D411yescharacteristic
144C2×C5×C8C2×D411yescharacteristic
145C2×C5×C8C2×D411yescharacteristic
146C2×C5×C8C2411yescharacteristic
147C5×C4⋊C4—21no—
148C5×C4⋊C4—21no—
149C5×C4⋊C4—21no—
150C5×C4⋊C4—21no—
151C5×C4⋊C4—21no—
152C5×C4⋊C4—21no—
153C5×C4⋊C4—41no—
154C5×C4⋊C4—41no—
155C5×C4⋊C4—41no—
156C5×C4⋊C4—41no—
157C5×C4⋊C4—41no—
158C5×C4⋊C4—41no—
159C5×C4⋊C4—41no—
160C5×C4⋊C4—41no—
161C5×C4⋊C4—41no—
162C5×C4⋊C4—41no—
163C5×C4⋊C4D811yescharacteristic
164C5×C4⋊C4D811yescharacteristic
165C5×C22⋊C4—41no—
166C5×C22⋊C4—41no—
167C5×C22⋊C4—41no—
168C5×C22⋊C4—41no—
169C5×C22⋊C4—41no—
170C5×C22⋊C4—41no—
171C5×C22⋊C4D811yescharacteristic
172C5×C22⋊C4D811yescharacteristic
173C42×C5—41no—
174C42×C5—41no—
175C42×C5C2×D411yescharacteristic
176C42×C5C2×D411yescharacteristic
177C5×C16—21no—
178C5×C16—21no—
179C5×C16—21no—
180C5×C16—21no—
181#544—21no—
182#542—21no—
183#501—21no—
184#501—21no—
185#500C22×C511yescharacteristic
186#500C22×C511yescharacteristic
187#500C22×C511yescharacteristic
188#499C22×C511yescharacteristic
189#499C22×C511yescharacteristic
190#495C22×C511yescharacteristic
191#489—21no—
192#468—21no—
193#467C22×C511yescharacteristic
194#466C22×C511yescharacteristic
195#440—21no—
196C4×Q16C22×C511yescharacteristic
197C4×D8C22×C511yescharacteristic
198#428—21no—
199#427C22×C511yescharacteristic
200C2×#107C22×C511yescharacteristic
201#405C22×C511yescharacteristic
202#367C22×C511yescharacteristic
203#367C22×C511yescharacteristic
204#367C22×C511yescharacteristic
205#367C22×C511yescharacteristic
206#367C22×C511yescharacteristic
207#367C22×C511yescharacteristic
208#366C22×C511yescharacteristic
209#366C22×C511yescharacteristic
210C4⋊C16C22×C511yescharacteristic
211C4⋊C16C22×C511yescharacteristic
212#358C22×C511yescharacteristic
213#358C22×C511yescharacteristic
214#358C22×C511yescharacteristic
215#358C22×C511yescharacteristic
216#357C22×C511yescharacteristic
217#357C22×C511yescharacteristic
218#357C22×C511yescharacteristic
219#357C22×C511yescharacteristic
220C22⋊C16C22×C511yescharacteristic
221C22⋊C16C22×C511yescharacteristic
222#346C22×C511yescharacteristic
223C4×C16C22×C511yescharacteristic
224C23×C5—41no—
225C23×C5C2×D811yescharacteristic
226C5×Q8—41no—
227C5×Q8—41no—
228C5×Q8—161no—
229C5×Q8—161no—
230C5×Q8—161no—
231C5×D4—41no—
232C5×D4—41no—
233C5×D4—161no—
234C2×C4×C5—21no—
235C2×C4×C5—21no—
236C2×C4×C5—41no—
237C2×C4×C5—41no—
238C2×C4×C5—41no—
239C2×C4×C5—41no—
240C2×C4×C5—41no—
241C2×C4×C5—41no—
242C2×C4×C5—41no—
243C2×C4×C5—161no—
244C2×C4×C5—161no—
245C2×C4×C5C2×D811yescharacteristic
246C2×C4×C5C2×D811yescharacteristic
247C2×C4×C5C2×D811yescharacteristic
248C2×C4×C5C2×D811yescharacteristic
249C2×C4×C5C2×D811yescharacteristic
250C2×C4×C5C22×D411yescharacteristic
251C5×C8—21no—
252C5×C8—21no—
253C5×C8#13712yes—
254C5×C82−1+411yescharacteristic
255C5×C82−1+411yescharacteristic
256C2×Q16C23×C511yescharacteristic
257C2×SD16—21no—
258C2×D8C23×C511yescharacteristic
259C2×M16C5×D411yescharacteristic
260C22×C8C23×C511yescharacteristic
261C4⋊Q8—21no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4⋊Q8—41no—
265#126—41no—
266#125—21no—
267#125—21no—
268#125—41no—
269#124—21no—
270#124—41no—
271#123—41no—
272C22⋊Q8—21no—
273C22⋊Q8—41no—
274C22⋊Q8—41no—
275C22⋊Q8—41no—
276#121—21no—
277C4×Q8—21no—
278C4×D4—21no—
279#117—21no—
280#117C5×D411yescharacteristic
281C2×C4⋊C4—21no—
282C2×C16C23×C511yescharacteristic
283C2×C16C23×C511yescharacteristic
284C2×C16C23×C511yescharacteristic
285C2×C16C23×C511yescharacteristic
286#107—21no—
287#107—21no—
288#107—21no—
289#107—21no—
290#107C23×C511yescharacteristic
291#107C23×C511yescharacteristic
292#107C23×C511yescharacteristic
293#107C23×C511yescharacteristic
294#107C23×C511yescharacteristic
295#107C23×C511yescharacteristic
296#106—21no—
297#106—21no—
298#106—21no—
299#103—21no—
300#103—21no—
301#102—21no—
302#102—21no—
303#97C5×D411yescharacteristic
304#97C5×D411yescharacteristic
305C4×C8C23×C511yescharacteristic
306C4×C8C23×C511yescharacteristic
307C22×C5—21no—
308C22×C5—21no—
309C22×C5—21no—
310C22×C5—81no—
311C22×C5—81no—
312C22×C5—81no—
313C22×C5C22×D811yescharacteristic
314C4×C5—21no—
315C4×C5—21no—
316C4×C5—21no—
317C4×C5—21no—
318C4×C5—21no—
319C4×C5—81no—
320C4×C5—81no—
321C4×C5—81no—
322C4×C5—81no—
323C4×C5—81no—
324C4×C5—81no—
325C4×C5—81no—
326C4×C5#53611yescharacteristic
327C4×C5C2×#13711yescharacteristic
328C2×Q8—21no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×D4—21no—
333C2×D4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4C2×C5×D411yescharacteristic
337Q16—21no—
338Q16—21no—
339SD16—42no—
340D8—21no—
341D8—21no—
342M16—22no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8C2×C5×D411yescharacteristic
346C2×C8C2×C5×D411yescharacteristic
347C2×C8C2×C5×D411yescharacteristic
348C2×C8C24×C511yescharacteristic
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4C5×D811yescharacteristic
366C4⋊C4C5×D811yescharacteristic
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4C5×D811yescharacteristic
374C22⋊C4C5×D811yescharacteristic
375C42—41no—
376C42—41no—
377C42C2×C5×D411yescharacteristic
378C42C2×C5×D411yescharacteristic
379C16—21no—
380C16—21no—
381C16—21no—
382C16—21no—
383C2×C5—41no—
384C2×C5—161no—
385C2×C5#322611yescharacteristic
386C2×C5#341611yescharacteristic
387C2×C5#341611yescharacteristic
388C23—41no—
389C23C2×C5×D811yescharacteristic
390Q8—41no—
391Q8—41no—
392Q8—161no—
393Q8—161no—
394Q8—161no—
395D4—41no—
396D4—41no—
397D4—161no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—161no—
408C2×C4—161no—
409C2×C4C2×C5×D811yescharacteristic
410C2×C4C2×C5×D811yescharacteristic
411C2×C4C2×C5×D811yescharacteristic
412C2×C4C2×C5×D811yescharacteristic
413C2×C4C2×C5×D811yescharacteristic
414C2×C4C22×C5×D411yescharacteristic
415C8—21no—
416C8—21no—
417C8C5×#13712yes—
418C8C5×2−1+411yescharacteristic
419C8C5×2−1+411yescharacteristic
420C5#3381011yescharacteristic, direct product factor, Sylow
421C22—21no—
422C22—21no—
423C22—21no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22C22×C5×D811yescharacteristic
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—81no—
434C4—81no—
435C4—81no—
436C4—81no—
437C4—81no—
438C4—81no—
439C4—81no—
440C4C5×#53611yescharacteristic
441C4C2×C5×#13711yescharacteristic
442C2—41no—
443C2—161no—
444C2C5×#322611yescharacteristic
445C2C5×#341611yescharacteristic
446C2C5×#341611yescharacteristic
subgroupG/H
center, socleC22×C5C22×D8
commutator, Frattini, cosocleC2×C8C24×C5
prime radicalC5×#466C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C5×#117 ◁ C5×#405 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#3164C2semidirect
3. C5×#2255C2semidirect
4. C5×#2254C2semidirect
5. C5×#2243C2semidirect
6. C5×#2242C2semidirect
7. C5×#2237C2semidirect
8. C5×#2235C2semidirect
11. C5×#2229C2semidirect
12. C5×#2228C2semidirect
13. C5×#2226C2semidirect
14. C5×#2225C2semidirect
15. C5×#2164C2semidirect
59. #33810C5direct (N × H)
112. #3164C2×C5semidirect
113. #2255C2×C5semidirect
114. #2254C2×C5semidirect
115. #2243C2×C5semidirect
116. #2242C2×C5semidirect
117. #2237C2×C5semidirect
118. #2235C2×C5semidirect
121. #2229C2×C5semidirect
122. #2228C2×C5semidirect
123. #2226C2×C5semidirect
124. #2225C2×C5semidirect
125. #2164C2×C5semidirect
420. C5#33810direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#31752
C5×#31642
C5×#22552
C5×#22542
C5×#22432
C5×#22422
C5×#22372
C5×#22352
C5×#22342
C5×#22332
C5×#22292
C5×#22282
C5×#22262
C5×#22252
C5×#21642
#338105