Finite Group Databasebeta

← C5×#33808 #11956528

Subgroups — C5×#33808 #11956528

Order 1280.

442 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3177C211yesmaximal, characteristic
2C5×#3164C211yesmaximal, characteristic
3C5×#2255C211yesmaximal, characteristic, complemented
4C5×#2253C211yesmaximal, characteristic, complemented
5C5×#2245C211yesmaximal, characteristic, complemented
6C5×#2241C211yesmaximal, characteristic, complemented
7C5×#2237C211yesmaximal, characteristic
8C5×#2237C211yesmaximal, characteristic
9C5×#2236C211yesmaximal, characteristic
10C5×#2235C211yesmaximal, characteristic
11C5×#2229C211yesmaximal, characteristic, complemented
12C5×#2229C211yesmaximal, characteristic, complemented
13C5×#2227C211yesmaximal, characteristic, complemented
14C5×#2225C211yesmaximal, characteristic, complemented
15C5×#2164C211yesmaximal, characteristic
16C5×#544—21no—
17C5×#541—21no—
18C5×#501—21no—
19C5×#501—21no—
20C5×#500C2211yescharacteristic
21C5×#500C2211yescharacteristic
22C5×#499C2211yescharacteristic
23C5×#499C2211yescharacteristic
24C5×#499C2211yescharacteristic
25C5×#494C2211yescharacteristic
26C5×#491—21no—
27C5×#470—21no—
28C5×#467C2211yescharacteristic
29C5×#467C2211yescharacteristic
30C5×#441—21no—
31C4×C5×Q16C2211yescharacteristic
32C4×C5×Q16C2211yescharacteristic
33C5×#428—21no—
34C5×#427C2211yescharacteristic
35C2×C5×#107C2211yescharacteristic
36C5×#405C2211yescharacteristic
37C5×#367C2211yescharacteristic
38C5×#367C2211yescharacteristic
39C5×#367C2211yescharacteristic
40C5×#367C2211yescharacteristic
41C5×#366C2211yescharacteristic
42C5×#366C2211yescharacteristic
43C5×#366C2211yescharacteristic
44C5×#366C2211yescharacteristic
45C5×C4⋊C16C2211yescharacteristic
46C5×C4⋊C16C2211yescharacteristic
47C5×#358C2211yescharacteristic
48C5×#358C2211yescharacteristic
49C5×#358C2211yescharacteristic
50C5×#358C2211yescharacteristic
51C5×#358C2211yescharacteristic
52C5×#358C2211yescharacteristic
53C5×#358C2211yescharacteristic
54C5×#358C2211yescharacteristic
55C5×C22⋊C16C2211yescharacteristic
56C5×C22⋊C16C2211yescharacteristic
57C5×#346C2211yescharacteristic
58C4×C5×C16C2211yescharacteristic
59#33808C511yesmaximal, characteristic, direct product factor, Sylow
60C2×C5×Q16—21no—
61C2×C5×Q16C2311yescharacteristic
62C2×C5×Q16C2311yescharacteristic
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×#126—41no—
71C5×#125—21no—
72C5×#125—21no—
73C5×#125—21no—
74C5×#125—41no—
75C5×#125—41no—
76C5×#125—41no—
77C5×C22⋊Q8—21no—
78C5×C22⋊Q8—21no—
79C5×C22⋊Q8—41no—
80C5×C22⋊Q8—41no—
81C4×C5×Q8—21no—
82C4×C5×Q8—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×#107C2311yescharacteristic
94C5×#107C2311yescharacteristic
95C5×#107C2311yescharacteristic
96C5×#107C2311yescharacteristic
97C5×#107C2311yescharacteristic
98C5×#107C2311yescharacteristic
99C5×#106—21no—
100C5×#106—21no—
101C5×#106—21no—
102C5×#106—21no—
103C5×#103—21no—
104C5×#103—21no—
105C5×#103—21no—
106C5×#103—21no—
107C5×#97D411yescharacteristic
108C5×#97D411yescharacteristic
109C4×C5×C8C2311yescharacteristic
110C4×C5×C8C2311yescharacteristic
111#3177C2×C511yescharacteristic
112#3164C2×C511yescharacteristic
113#2255C2×C511yescharacteristic, complemented
114#2253C2×C511yescharacteristic, complemented
115#2245C2×C511yescharacteristic, complemented
116#2241C2×C511yescharacteristic, complemented
117#2237C2×C511yescharacteristic
118#2237C2×C511yescharacteristic
119#2236C2×C511yescharacteristic
120#2235C2×C511yescharacteristic
121#2229C2×C511yescharacteristic, complemented
122#2229C2×C511yescharacteristic, complemented
123#2227C2×C511yescharacteristic, complemented
124#2225C2×C511yescharacteristic, complemented
125#2164C2×C511yescharacteristic
126C2×C5×Q8—21no—
127C2×C5×Q8—21no—
128C2×C5×Q8—41no—
129C2×C5×Q8—41no—
130C22×C4×C5—41no—
131C22×C4×C5C2×D411yescharacteristic
132C5×Q16—21no—
133C5×Q16—21no—
134C5×Q16—21no—
135C5×Q16—21no—
136C5×Q16—42no—
137C5×M16—22no—
138C2×C5×C8—21no—
139C2×C5×C8—21no—
140C2×C5×C8C2×D411yescharacteristic
141C2×C5×C8C2×D411yescharacteristic
142C2×C5×C8C2×D411yescharacteristic
143C2×C5×C8C2411yescharacteristic
144C5×C4⋊C4—21no—
145C5×C4⋊C4—21no—
146C5×C4⋊C4—21no—
147C5×C4⋊C4—21no—
148C5×C4⋊C4—21no—
149C5×C4⋊C4—21no—
150C5×C4⋊C4—41no—
151C5×C4⋊C4—41no—
152C5×C4⋊C4—41no—
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⋊C4—41no—
164C5×C4⋊C4—41no—
165C5×C4⋊C4D811yescharacteristic
166C5×C4⋊C4D811yescharacteristic
167C5×C22⋊C4—41no—
168C5×C22⋊C4—41no—
169C5×C22⋊C4—41no—
170C5×C22⋊C4D811yescharacteristic
171C5×C22⋊C4D811yescharacteristic
172C42×C5—41no—
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#541—21no—
183#501—21no—
184#501—21no—
185#500C22×C511yescharacteristic
186#500C22×C511yescharacteristic
187#499C22×C511yescharacteristic
188#499C22×C511yescharacteristic
189#499C22×C511yescharacteristic
190#494C22×C511yescharacteristic
191#491—21no—
192#470—21no—
193#467C22×C511yescharacteristic
194#467C22×C511yescharacteristic
195#441—21no—
196C4×Q16C22×C511yescharacteristic
197C4×Q16C22×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#366C22×C511yescharacteristic
207#366C22×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#358C22×C511yescharacteristic
217#358C22×C511yescharacteristic
218#358C22×C511yescharacteristic
219#358C22×C511yescharacteristic
220C22⋊C16C22×C511yescharacteristic
221C22⋊C16C22×C511yescharacteristic
222#346C22×C511yescharacteristic
223C4×C16C22×C511yescharacteristic
224C23×C5C2×D811yescharacteristic
225C5×Q8—41no—
226C5×Q8—41no—
227C5×Q8—41no—
228C5×Q8—41no—
229C5×Q8—161no—
230C5×Q8—161no—
231C2×C4×C5—21no—
232C2×C4×C5—21no—
233C2×C4×C5—41no—
234C2×C4×C5—41no—
235C2×C4×C5—41no—
236C2×C4×C5—41no—
237C2×C4×C5—41no—
238C2×C4×C5—41no—
239C2×C4×C5—41no—
240C2×C4×C5—41no—
241C2×C4×C5—161no—
242C2×C4×C5C2×D811yescharacteristic
243C2×C4×C5C2×D811yescharacteristic
244C2×C4×C5C2×D811yescharacteristic
245C2×C4×C5C2×D811yescharacteristic
246C2×C4×C5C2×D811yescharacteristic
247C2×C4×C5C22×D411yescharacteristic
248C5×C8—21no—
249C5×C8—21no—
250C5×C8#13712yes—
251C5×C82−1+411yescharacteristic
252C5×C82−1+411yescharacteristic
253C2×Q16—21no—
254C2×Q16C23×C511yescharacteristic
255C2×Q16C23×C511yescharacteristic
256C2×M16C5×D411yescharacteristic
257C22×C8C23×C511yescharacteristic
258C4⋊Q8—21no—
259C4⋊Q8—21no—
260C4⋊Q8—21no—
261C4⋊Q8—41no—
262#126—41no—
263#126—41no—
264#125—21no—
265#125—21no—
266#125—21no—
267#125—41no—
268#125—41no—
269#125—41no—
270C22⋊Q8—21no—
271C22⋊Q8—21no—
272C22⋊Q8—41no—
273C22⋊Q8—41no—
274C4×Q8—21no—
275C4×Q8—21no—
276#117—21no—
277#117C5×D411yescharacteristic
278C2×C4⋊C4—21no—
279C2×C16C23×C511yescharacteristic
280C2×C16C23×C511yescharacteristic
281C2×C16C23×C511yescharacteristic
282C2×C16C23×C511yescharacteristic
283#107—21no—
284#107—21no—
285#107—21no—
286#107C23×C511yescharacteristic
287#107C23×C511yescharacteristic
288#107C23×C511yescharacteristic
289#107C23×C511yescharacteristic
290#107C23×C511yescharacteristic
291#107C23×C511yescharacteristic
292#106—21no—
293#106—21no—
294#106—21no—
295#106—21no—
296#103—21no—
297#103—21no—
298#103—21no—
299#103—21no—
300#97C5×D411yescharacteristic
301#97C5×D411yescharacteristic
302C4×C8C23×C511yescharacteristic
303C4×C8C23×C511yescharacteristic
304C22×C5—21no—
305C22×C5—21no—
306C22×C5—21no—
307C22×C5C22×D811yescharacteristic
308C4×C5—21no—
309C4×C5—21no—
310C4×C5—21no—
311C4×C5—21no—
312C4×C5—21no—
313C4×C5—81no—
314C4×C5—81no—
315C4×C5—81no—
316C4×C5—81no—
317C4×C5—81no—
318C4×C5—81no—
319C4×C5—81no—
320C4×C5—81no—
321C4×C5#53611yescharacteristic
322C4×C5C2×#13711yescharacteristic
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—41no—
326C2×Q8—41no—
327C22×C4—41no—
328C22×C4C2×C5×D411yescharacteristic
329Q16—21no—
330Q16—21no—
331Q16—21no—
332Q16—21no—
333Q16—42no—
334M16—22no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8C2×C5×D411yescharacteristic
338C2×C8C2×C5×D411yescharacteristic
339C2×C8C2×C5×D411yescharacteristic
340C2×C8C24×C511yescharacteristic
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4C5×D811yescharacteristic
363C4⋊C4C5×D811yescharacteristic
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4C5×D811yescharacteristic
368C22⋊C4C5×D811yescharacteristic
369C42—41no—
370C42—41no—
371C42—41no—
372C42C2×C5×D411yescharacteristic
373C42C2×C5×D411yescharacteristic
374C16—21no—
375C16—21no—
376C16—21no—
377C16—21no—
378C2×C5—41no—
379C2×C5#322611yescharacteristic
380C2×C5#341611yescharacteristic
381C2×C5#341711yescharacteristic
382C23C2×C5×D811yescharacteristic
383Q8—41no—
384Q8—41no—
385Q8—41no—
386Q8—41no—
387Q8—161no—
388Q8—161no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—161no—
400C2×C4C2×C5×D811yescharacteristic
401C2×C4C2×C5×D811yescharacteristic
402C2×C4C2×C5×D811yescharacteristic
403C2×C4C2×C5×D811yescharacteristic
404C2×C4C2×C5×D811yescharacteristic
405C2×C4C22×C5×D411yescharacteristic
406C8—21no—
407C8—21no—
408C8C5×#13712yes—
409C8C5×2−1+411yescharacteristic
410C8C5×2−1+411yescharacteristic
411C5#3380811yescharacteristic, direct product factor, Sylow
412C22—21no—
413C22—21no—
414C22—21no—
415C22C22×C5×D811yescharacteristic
416C4—21no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—81no—
422C4—81no—
423C4—81no—
424C4—81no—
425C4—81no—
426C4—81no—
427C4—81no—
428C4—81no—
429C4C5×#53611yescharacteristic
430C4C2×C5×#13711yescharacteristic
431C2—41no—
432C2C5×#322611yescharacteristic
433C2C5×#341611yescharacteristic
434C2C5×#341711yescharacteristic
subgroupG/H
center, socleC22×C5C22×D8
commutator, Frattini, cosocleC2×C8C24×C5
prime radicalC23×C5C2×D8
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
3. C5×#2255C2semidirect
4. C5×#2253C2semidirect
5. C5×#2245C2semidirect
6. C5×#2241C2semidirect
11. C5×#2229C2semidirect
12. C5×#2229C2semidirect
13. C5×#2227C2semidirect
14. C5×#2225C2semidirect
59. #33808C5direct (N × H)
113. #2255C2×C5semidirect
114. #2253C2×C5semidirect
115. #2245C2×C5semidirect
116. #2241C2×C5semidirect
121. #2229C2×C5semidirect
122. #2229C2×C5semidirect
123. #2227C2×C5semidirect
124. #2225C2×C5semidirect
411. C5#33808direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#31772
C5×#31642
C5×#22552
C5×#22532
C5×#22452
C5×#22412
C5×#22372
C5×#22362
C5×#22352
C5×#22292
C5×#22272
C5×#22252
C5×#21642
#338085