Finite Group Databasebeta

← C5×#36698 #11959418

Subgroups — C5×#36698 #11959418

Order 1280.

1484 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2844C211yesmaximal, characteristic, complemented
2C5×#2838C214yesmaximal, complemented
3C5×#2836C214yesmaximal, complemented
4C5×#2833C212yesmaximal, complemented
5C5×#2830C212yesmaximal
6C5×#2822C211yesmaximal, characteristic, complemented
7C5×#2820C212yesmaximal
8C5×#2401C214yesmaximal, complemented
9C5×#2399C212yesmaximal, complemented
10C5×#2396C214yesmaximal, complemented
11C5×#2395C212yesmaximal
12C4×C5×#125C212yesmaximal, complemented
13C4×C5×#124C211yesmaximal, characteristic
14C2×C5×#125C2212yes—
15C2×C5×#124C2211yescharacteristic
16C2×C4×C5×Q8C2211yescharacteristic
17C2×C4×C5×Q8C2212yes—
18C2×C4×C5×D4C2211yescharacteristic
19C2×C5×#117C2212yes—
20C5×#401C2214yes—
21C5×#400C2214yes—
22C5×#400C2214yes—
23C5×#400C2214yes—
24C5×#400C2212yes—
25C5×#400C2214yes—
26C5×#399C2214yes—
27C5×#399C2212yes—
28C5×#398C2212yes—
29C5×#398C2212yes—
30C5×#398C2214yes—
31C5×#398C2214yes—
32C5×#398C2212yes—
33C5×#398C2214yes—
34C5×#397C2214yes—
35C5×#397C2214yes—
36C5×#397C2214yes—
37C5×#397C2212yes—
38C5×#397C2214yes—
39C5×#396C2214yes—
40C5×#396C2214yes—
41C5×#396C2214yes—
42C5×#395C2214yes—
43C5×#395C2212yes—
44C5×#395C2212yes—
45C5×#395C2214yes—
46C5×#394C2214yes—
47C5×#393C2214yes—
48C5×#391C2211yescharacteristic
49C5×#390C2211yescharacteristic
50C5×#389C2212yes—
51C5×#389C2212yes—
52C5×#388C2214yes—
53C5×#387C2214yes—
54C5×#387C2214yes—
55C5×#382C2211yescharacteristic
56C5×#382C2212yes—
57C4×C5×C4⋊C4C2214yes—
58C4×C5×C4⋊C4C2212yes—
59C4×C5×C4⋊C4C2214yes—
60C4×C5×C4⋊C4C2212yes—
61C4×C5×C4⋊C4C2214yes—
62C4×C5×C22⋊C4C2214yes—
63C4×C5×C22⋊C4C2214yes—
64C4×C5×C22⋊C4C2212yes—
65C5×#376C2212yes—
66C43×C5C2211yescharacteristic
67#36698C511yesmaximal, characteristic, direct product factor, Sylow
68C22×C5×Q8C2311yescharacteristic
69C22×C5×Q8C2312yes—
70C22×C5×D4C2311yescharacteristic
71C23×C4×C5C2312yes—
72C5×#125—28no—
73C5×#124—24no—
74C4×C5×Q8—44no—
75C4×C5×Q8—42no—
76C4×C5×D4—42no—
77C5×#117—44no—
78C2×C5×C4⋊C4C2314yes—
79C2×C5×C4⋊C4C2314yes—
80C2×C5×C4⋊C4C2312yes—
81C2×C5×C4⋊C4C2312yes—
82C2×C5×C4⋊C4C2312yes—
83C2×C5×C4⋊C4C2312yes—
84C2×C5×C4⋊C4C2314yes—
85C2×C5×C4⋊C4C2314yes—
86C2×C5×C4⋊C4C2314yes—
87C2×C5×C4⋊C4C2314yes—
88C2×C5×C4⋊C4C2311yescharacteristic
89C2×C5×C4⋊C4C2311yescharacteristic
90C2×C5×C4⋊C4C2312yes—
91C2×C5×C4⋊C4C2312yes—
92C2×C5×C4⋊C4C2314yes—
93C2×C5×C4⋊C4C2312yes—
94C2×C5×C4⋊C4C2312yes—
95C2×C5×C22⋊C4C2314yes—
96C2×C5×C22⋊C4C2314yes—
97C2×C5×C22⋊C4C2314yes—
98C2×C5×C22⋊C4C2314yes—
99C2×C5×C22⋊C4C2314yes—
100C2×C5×C22⋊C4C2312yes—
101C2×C5×C22⋊C4C2314yes—
102C2×C42×C5C2312yes—
103C2×C42×C5C2311yescharacteristic
104C2×C42×C5C2314yes—
105C2×C42×C5C2312yes—
106C2×C42×C5C2314yes—
107C2×C42×C5C2311yescharacteristic
108C2×C42×C5C2312yes—
109C2×C42×C5C2311yescharacteristic
110C5×#95C2314yes—
111C5×#95C2312yes—
112C5×#95C2312yes—
113C5×#95C2314yes—
114C5×#95C2314yes—
115C5×#95C2314yes—
116C5×#95C2314yes—
117C5×#95C2312yes—
118C5×#95C2312yes—
119C5×#95C2314yes—
120C5×#95C2314yes—
121C5×#95C2314yes—
122C5×#95C2314yes—
123C5×#95C2314yes—
124C5×#95C2314yes—
125C5×#95C2314yes—
126C5×#95C2314yes—
127#2844C2×C511yescharacteristic, complemented
128#2838C2×C514yescomplemented
129#2836C2×C514yescomplemented
130#2833C2×C512yescomplemented
131#2830C2×C512yes—
132#2822C2×C511yescharacteristic, complemented
133#2820C2×C512yes—
134#2401C2×C514yescomplemented
135#2399C2×C512yescomplemented
136#2396C2×C514yescomplemented
137#2395C2×C512yes—
138C4×#125C2×C512yescomplemented
139C4×#124C2×C511yescharacteristic
140C24×C5C2412yes—
141C2×C5×Q8—22no—
142C2×C5×Q8—41no—
143C2×C5×Q8—42no—
144C2×C5×Q8—41no—
145C2×C5×Q8—44no—
146C2×C5×D4—22no—
147C2×C5×D4—41no—
148C2×C5×D4—41no—
149C22×C4×C5—24no—
150C22×C4×C5—28no—
151C22×C4×C5C2411yescharacteristic
152C22×C4×C5C2412yes—
153C22×C4×C5C2414yes—
154C22×C4×C5C2412yes—
155C22×C4×C5C2414yes—
156C22×C4×C5C2412yes—
157C22×C4×C5C2414yes—
158C22×C4×C5C2414yes—
159C22×C4×C5C2412yes—
160C22×C4×C5C2412yes—
161C22×C4×C5C2411yescharacteristic
162C22×C4×C5C2411yescharacteristic
163C5×C4⋊C4—28no—
164C5×C4⋊C4—24no—
165C5×C4⋊C4—24no—
166C5×C4⋊C4—28no—
167C5×C4⋊C4—42no—
168C5×C4⋊C4—42no—
169C5×C4⋊C4—44no—
170C5×C4⋊C4—44no—
171C5×C4⋊C4—44no—
172C5×C4⋊C4—44no—
173C5×C4⋊C4—42no—
174C5×C4⋊C4—42no—
175C5×C4⋊C4—41no—
176C5×C4⋊C4—41no—
177C5×C4⋊C4—42no—
178C5×C4⋊C4—42no—
179C5×C4⋊C4—44no—
180C5×C22⋊C4—28no—
181C5×C22⋊C4—44no—
182C5×C22⋊C4—44no—
183C5×C22⋊C4—44no—
184C5×C22⋊C4—44no—
185C5×C22⋊C4—44no—
186C5×C22⋊C4—42no—
187C42×C5—28no—
188C42×C5—22no—
189C42×C5—24no—
190C42×C5—24no—
191C42×C5—28no—
192C42×C5—42no—
193C42×C5—41no—
194C42×C5#4114yes—
195C2×#125C22×C512yes—
196C2×#124C22×C511yescharacteristic
197C2×C4×Q8C22×C512yes—
198C2×C4×Q8C22×C511yescharacteristic
199C2×C4×D4C22×C511yescharacteristic
200C2×#117C22×C512yes—
201#401C22×C514yes—
202#400C22×C514yes—
203#400C22×C514yes—
204#400C22×C514yes—
205#400C22×C512yes—
206#400C22×C514yes—
207#399C22×C514yes—
208#399C22×C512yes—
209#398C22×C512yes—
210#398C22×C512yes—
211#398C22×C514yes—
212#398C22×C514yes—
213#398C22×C512yes—
214#398C22×C514yes—
215#397C22×C514yes—
216#397C22×C514yes—
217#397C22×C514yes—
218#397C22×C512yes—
219#397C22×C514yes—
220#396C22×C514yes—
221#396C22×C514yes—
222#396C22×C514yes—
223#395C22×C514yes—
224#395C22×C512yes—
225#395C22×C512yes—
226#395C22×C514yes—
227#394C22×C514yes—
228#393C22×C514yes—
229#391C22×C511yescharacteristic
230#390C22×C511yescharacteristic
231#389C22×C512yes—
232#389C22×C512yes—
233#388C22×C514yes—
234#387C22×C514yes—
235#387C22×C514yes—
236#382C22×C512yes—
237#382C22×C511yescharacteristic
238C4×C4⋊C4C22×C514yes—
239C4×C4⋊C4C22×C512yes—
240C4×C4⋊C4C22×C514yes—
241C4×C4⋊C4C22×C512yes—
242C4×C4⋊C4C22×C514yes—
243C4×C22⋊C4C22×C514yes—
244C4×C22⋊C4C22×C512yes—
245C4×C22⋊C4C22×C514yes—
246#376C22×C512yes—
247C43C22×C511yescharacteristic
248C23×C5—22no—
249C23×C5—22no—
250C23×C5—24no—
251C23×C5—24no—
252C23×C5—22no—
253C23×C5C2511yescharacteristic
254C5×Q8—44no—
255C5×Q8—48no—
256C5×D4—44no—
257C2×C4×C5—21no—
258C2×C4×C5—22no—
259C2×C4×C5—22no—
260C2×C4×C5—21no—
261C2×C4×C5—22no—
262C2×C4×C5—24no—
263C2×C4×C5—24no—
264C2×C4×C5—21no—
265C2×C4×C5—24no—
266C2×C4×C5—24no—
267C2×C4×C5—22no—
268C2×C4×C5—22no—
269C2×C4×C5—22no—
270C2×C4×C5—24no—
271C2×C4×C5—22no—
272C2×C4×C5—24no—
273C2×C4×C5—24no—
274C2×C4×C5—22no—
275C2×C4×C5—22no—
276C2×C4×C5—22no—
277C2×C4×C5—24no—
278C2×C4×C5—24no—
279C2×C4×C5—24no—
280C2×C4×C5—24no—
281C2×C4×C5—24no—
282C2×C4×C5—24no—
283C2×C4×C5—22no—
284C2×C4×C5—24no—
285C2×C4×C5—48no—
286C2×C4×C5C2×#4114yes—
287C2×C4×C5C2×#4112yes—
288C2×C4×C5H(5,𝔽2)12yes—
289C2×C4×C52−1+414yes—
290C22×Q8C23×C511yescharacteristic
291C22×Q8C23×C512yes—
292C22×D4C23×C511yescharacteristic
293C23×C4C23×C512yes—
294#125—28no—
295#124—24no—
296C4×Q8—44no—
297C4×Q8—42no—
298C4×D4—42no—
299#117—44no—
300C2×C4⋊C4C23×C514yes—
301C2×C4⋊C4C23×C514yes—
302C2×C4⋊C4C23×C512yes—
303C2×C4⋊C4C23×C512yes—
304C2×C4⋊C4C23×C512yes—
305C2×C4⋊C4C23×C512yes—
306C2×C4⋊C4C23×C511yescharacteristic
307C2×C4⋊C4C23×C511yescharacteristic
308C2×C4⋊C4C23×C512yes—
309C2×C4⋊C4C23×C514yes—
310C2×C4⋊C4C23×C512yes—
311C2×C4⋊C4C23×C512yes—
312C2×C4⋊C4C23×C512yes—
313C2×C4⋊C4C23×C514yes—
314C2×C4⋊C4C23×C514yes—
315C2×C4⋊C4C23×C514yes—
316C2×C4⋊C4C23×C514yes—
317C2×C22⋊C4C23×C514yes—
318C2×C22⋊C4C23×C514yes—
319C2×C22⋊C4C23×C512yes—
320C2×C22⋊C4C23×C514yes—
321C2×C22⋊C4C23×C514yes—
322C2×C22⋊C4C23×C514yes—
323C2×C22⋊C4C23×C514yes—
324C2×C42C23×C514yes—
325C2×C42C23×C511yescharacteristic
326C2×C42C23×C512yes—
327C2×C42C23×C512yes—
328C2×C42C23×C514yes—
329C2×C42C23×C511yescharacteristic
330C2×C42C23×C512yes—
331C2×C42C23×C511yescharacteristic
332#95C23×C514yes—
333#95C23×C512yes—
334#95C23×C512yes—
335#95C23×C514yes—
336#95C23×C514yes—
337#95C23×C514yes—
338#95C23×C514yes—
339#95C23×C512yes—
340#95C23×C512yes—
341#95C23×C514yes—
342#95C23×C514yes—
343#95C23×C514yes—
344#95C23×C514yes—
345#95C23×C514yes—
346#95C23×C514yes—
347#95C23×C514yes—
348#95C23×C514yes—
349C22×C5—42no—
350C22×C5—44no—
351C22×C5—42no—
352C22×C5—42no—
353C22×C5—44no—
354C22×C5C22×#4111yescharacteristic
355C22×C5C2×H(5,𝔽2)11yescharacteristic
356C22×C5C2×2−1+412yes—
357C22×C5#58511yescharacteristic
358C22×C5#58512yes—
359C4×C5—22no—
360C4×C5—24no—
361C4×C5—44no—
362C4×C5—42no—
363C4×C5—44no—
364C4×C5—42no—
365C4×C5—41no—
366C4×C5—44no—
367C4×C5—42no—
368C4×C5—42no—
369C4×C5—44no—
370C4×C5#56314yes—
371C24C24×C512yes—
372C2×Q8—22no—
373C2×Q8—41no—
374C2×Q8—42no—
375C2×Q8—41no—
376C2×Q8—44no—
377C2×D4—22no—
378C2×D4—41no—
379C2×D4—41no—
380C22×C4—24no—
381C22×C4—28no—
382C22×C4C24×C512yes—
383C22×C4C24×C512yes—
384C22×C4C24×C514yes—
385C22×C4C24×C514yes—
386C22×C4C24×C512yes—
387C22×C4C24×C514yes—
388C22×C4C24×C512yes—
389C22×C4C24×C511yescharacteristic
390C22×C4C24×C514yes—
391C22×C4C24×C512yes—
392C22×C4C24×C511yescharacteristic
393C22×C4C24×C511yescharacteristic
394C4⋊C4—28no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—28no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—44no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C22⋊C4—28no—
412C22⋊C4—44no—
413C22⋊C4—42no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C42—28no—
419C42—22no—
420C42—24no—
421C42—24no—
422C42—28no—
423C42—42no—
424C42—41no—
425C42C5×#4114yes—
426C2×C5—82no—
427C2×C5#346011yescharacteristic
428C2×C5#346112yes—
429C2×C5C2×#56311yescharacteristic
430C2×C5#352911yescharacteristic
431C2×C5#353112yes—
432C23—22no—
433C23—22no—
434C23—24no—
435C23—24no—
436C23—22no—
437C23C25×C511yescharacteristic
438Q8—44no—
439Q8—48no—
440D4—44no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—21no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—48no—
470C2×C4C2×C5×#4114yes—
471C2×C4C2×C5×#4112yes—
472C2×C4C5×H(5,𝔽2)12yes—
473C2×C4C5×2−1+414yes—
474C5#3669811yescharacteristic, direct product factor, Sylow
475C22—42no—
476C22—44no—
477C22—42no—
478C22—42no—
479C22—44no—
480C22C22×C5×#4111yescharacteristic
481C22C2×C5×H(5,𝔽2)11yescharacteristic
482C22C2×C5×2−1+412yes—
483C22C5×#58511yescharacteristic
484C22C5×#58512yes—
485C4—22no—
486C4—24no—
487C4—42no—
488C4—44no—
489C4—44no—
490C4—42no—
491C4—44no—
492C4—42no—
493C4—41no—
494C4—42no—
495C4—44no—
496C4C5×#56314yes—
497C2—82no—
498C2C5×#346011yescharacteristic
499C2C5×#346112yes—
500C2C2×C5×#56311yescharacteristic
501C2C5×#352911yescharacteristic
502C2C5×#353112yes—
subgroupG/H
centerC22×C4×C5C24
commutator, Frattini, cosocleC23C25×C5
socleC23×C5C25
prime radicalC22×C5×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C5×#2844C2semidirect
2. C5×#2838C2semidirect
3. C5×#2836C2semidirect
4. C5×#2833C2semidirect
6. C5×#2822C2semidirect
8. C5×#2401C2semidirect
9. C5×#2399C2semidirect
10. C5×#2396C2semidirect
12. C4×C5×#125C2semidirect
67. #36698C5direct (N × H)
127. #2844C2×C5semidirect
128. #2838C2×C5semidirect
129. #2836C2×C5semidirect
130. #2833C2×C5semidirect
132. #2822C2×C5semidirect
134. #2401C2×C5semidirect
135. #2399C2×C5semidirect
136. #2396C2×C5semidirect
138. C4×#125C2×C5semidirect
474. C5#36698direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#28442
C5×#28382
C5×#28362
C5×#28332
C5×#28302
C5×#28222
C5×#28202
C5×#24012
C5×#23992
C5×#23962
C5×#23952
C4×C5×#1252
C4×C5×#1242
#366985