Finite Group Databasebeta

← C5×#11757 #11934477

Subgroups — C5×#11757 #11934477

Order 1280.

700 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2539C211yesmaximal, characteristic
2C2×C5×#342C211yesmaximal, characteristic, complemented
3C5×#1462C211yesmaximal, characteristic, complemented
4C5×#1352C212yesmaximal, complemented
5C5×#1343C212yesmaximal, complemented
6C2×C5×#123—21no—
7C2×C5×#121C411yescharacteristic, complemented
8C2×C5×#120—21no—
9C22×C5×C22⋊C4—21no—
10C22×C5×C22⋊C4C2211yescharacteristic
11C5×#407—21no—
12C5×#399C411yescharacteristic, complemented
13C5×#399C2211yescharacteristic
14C5×#392C411yescharacteristic, complemented
15C5×#392C2211yescharacteristic
16C5×#390—21no—
17C5×#390—21no—
18C4×C5×C22⋊C4C411yescharacteristic, complemented
19C5×#342—21no—
20C5×#342C2212yes—
21C5×#323—21no—
22C5×#323C2212yes—
23#11757C511yesmaximal, characteristic, direct product factor, Sylow
24C25×C5D411yescharacteristic
25C22×C5×D4—21no—
26C22×C5×D4—21no—
27C22×C5×D4—21no—
28C22×C5×D4C2×C411yescharacteristic, complemented
29C23×C4×C5—41no—
30C23×C4×C5C2×C411yescharacteristic
31C23×C4×C5D411yescharacteristic
32C2×C5×M16—41no—
33C5×#123—44no—
34C5×#121—41no—
35C5×#121—41no—
36C5×#120—44no—
37C2×C5×C4⋊C4—21no—
38C2×C5×C4⋊C4—21no—
39C2×C5×C4⋊C4C2×C411yescharacteristic, complemented
40C2×C5×C22⋊C4—21no—
41C2×C5×C22⋊C4—24no—
42C2×C5×C22⋊C4—22no—
43C2×C5×C22⋊C4—21no—
44C2×C5×C22⋊C4—21no—
45C2×C5×C22⋊C4—21no—
46C2×C5×C22⋊C4—21no—
47C2×C5×C22⋊C4—21no—
48C2×C5×C22⋊C4—41no—
49C2×C5×C22⋊C4—41no—
50C2×C5×C22⋊C4—41no—
51C2×C5×C22⋊C4—42no—
52C2×C5×C22⋊C4C2×C411yescharacteristic
53C2×C5×C22⋊C4D411yescharacteristic
54C2×C5×C22⋊C4D411yescharacteristic
55C2×C5×C22⋊C4C2311yescharacteristic
56C2×C42×C5—21no—
57C5×C22⋊C8—22no—
58C5×C22⋊C8—41no—
59C5×#95C2×C411yescharacteristic, complemented
60C5×#95C2×C411yescharacteristic, complemented
61#2539C2×C511yescharacteristic
62C2×#342C2×C511yescharacteristic, complemented
63#1462C2×C511yescharacteristic, complemented
64#1352C2×C512yescomplemented
65#1343C2×C512yescomplemented
66C24×C5—21no—
67C24×C5—21no—
68C24×C5—22no—
69C24×C5—21no—
70C24×C5—21no—
71C24×C5—21no—
72C24×C5—42no—
73C24×C5—81no—
74C24×C5C22⋊C411yescharacteristic
75C24×C5C22⋊C411yescharacteristic
76C24×C5C2×D411yescharacteristic
77C2×C5×D4—41no—
78C2×C5×D4—42no—
79C2×C5×D4—42no—
80C2×C5×D4—42no—
81C2×C5×D4—42no—
82C2×C5×D4—41no—
83C2×C5×D4—41no—
84C2×C5×D4—42no—
85C2×C5×D4—41no—
86C2×C5×D4—41no—
87C2×C5×D4—81no—
88C2×C5×D4—81no—
89C2×C5×D4—81no—
90C22×C4×C5—21no—
91C22×C4×C5—21no—
92C22×C4×C5—21no—
93C22×C4×C5—21no—
94C22×C4×C5—21no—
95C22×C4×C5—21no—
96C22×C4×C5—21no—
97C22×C4×C5—42no—
98C22×C4×C5—44no—
99C22×C4×C5—42no—
100C22×C4×C5—81no—
101C22×C4×C5—82no—
102C22×C4×C5—82no—
103C22×C4×C5C22⋊C411yescharacteristic
104C22×C4×C5C22⋊C411yescharacteristic
105C22×C4×C5C22×C411yescharacteristic
106C22×C4×C5C2×D411yescharacteristic
107C5×M16—82no—
108C2×C5×C8—42no—
109C5×C4⋊C4—41no—
110C5×C4⋊C4—42no—
111C5×C4⋊C4—42no—
112C5×C22⋊C4—24no—
113C5×C22⋊C4—24no—
114C5×C22⋊C4—41no—
115C5×C22⋊C4—41no—
116C5×C22⋊C4—41no—
117C5×C22⋊C4—41no—
118C5×C22⋊C4—42no—
119C5×C22⋊C4—42no—
120C5×C22⋊C4—42no—
121C5×C22⋊C4—42no—
122C5×C22⋊C4—42no—
123C5×C22⋊C4—44no—
124C5×C22⋊C4—42no—
125C42×C5—81no—
126C2×#123—21no—
127C2×#121C4×C511yescharacteristic, complemented
128C2×#120—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4C22×C511yescharacteristic
131#407—21no—
132#399C4×C511yescharacteristic, complemented
133#399C22×C511yescharacteristic
134#392C4×C511yescharacteristic, complemented
135#392C22×C511yescharacteristic
136#390—21no—
137#390—21no—
138C4×C22⋊C4C4×C511yescharacteristic, complemented
139#342—21no—
140#342C22×C512yes—
141#323—21no—
142#323C22×C512yes—
143C23×C5—22no—
144C23×C5—22no—
145C23×C5—22no—
146C23×C5—22no—
147C23×C5—22no—
148C23×C5—21no—
149C23×C5—21no—
150C23×C5—21no—
151C23×C5—21no—
152C23×C5—21no—
153C23×C5—21no—
154C23×C5—41no—
155C23×C5—42no—
156C23×C5—41no—
157C23×C5—41no—
158C23×C5—41no—
159C23×C5—41no—
160C23×C5—41no—
161C23×C5—41no—
162C23×C5—41no—
163C23×C5—41no—
164C23×C5—42no—
165C23×C5—41no—
166C23×C5—41no—
167C23×C5—41no—
168C23×C5—81no—
169C23×C5—81no—
170C23×C5—84no—
171C23×C5—82no—
172C23×C5—82no—
173C23×C5#9911yescharacteristic
174C23×C5#9911yescharacteristic
175C23×C5#10412yes—
176C23×C5C2×C22⋊C411yescharacteristic
177C23×C5AGL(1,ℤ/8)11yescharacteristic
178C23×C5#13711yescharacteristic
179C5×D4—84no—
180C5×D4—82no—
181C5×D4—82no—
182C5×D4—82no—
183C5×D4—82no—
184C5×D4—82no—
185C2×C4×C5—22no—
186C2×C4×C5—22no—
187C2×C4×C5—21no—
188C2×C4×C5—21no—
189C2×C4×C5—21no—
190C2×C4×C5—21no—
191C2×C4×C5—41no—
192C2×C4×C5—41no—
193C2×C4×C5—42no—
194C2×C4×C5—41no—
195C2×C4×C5—41no—
196C2×C4×C5—41no—
197C2×C4×C5—41no—
198C2×C4×C5—41no—
199C2×C4×C5—41no—
200C2×C4×C5—44no—
201C2×C4×C5—41no—
202C2×C4×C5—41no—
203C2×C4×C5—41no—
204C2×C4×C5—81no—
205C2×C4×C5—81no—
206C2×C4×C5—81no—
207C2×C4×C5—88no—
208C2×C4×C5—82no—
209C2×C4×C5—82no—
210C5×C8—82no—
211C25C5×D411yescharacteristic
212C22×D4—21no—
213C22×D4—21no—
214C22×D4—21no—
215C22×D4C2×C4×C511yescharacteristic, complemented
216C23×C4—41no—
217C23×C4C2×C4×C511yescharacteristic
218C23×C4C5×D411yescharacteristic
219C2×M16—41no—
220#123—44no—
221#121—41no—
222#121—41no—
223#120—44no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4C2×C4×C511yescharacteristic, complemented
227C2×C22⋊C4—21no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4C2×C4×C511yescharacteristic
240C2×C22⋊C4C5×D411yescharacteristic
241C2×C22⋊C4C5×D411yescharacteristic
242C2×C22⋊C4C23×C511yescharacteristic
243C2×C42—21no—
244C22⋊C8—22no—
245C22⋊C8—41no—
246#95C2×C4×C511yescharacteristic, complemented
247#95C2×C4×C511yescharacteristic, complemented
248C22×C5—21no—
249C22×C5—21no—
250C22×C5—21no—
251C22×C5—21no—
252C22×C5—21no—
253C22×C5—21no—
254C22×C5—21no—
255C22×C5—21no—
256C22×C5—22no—
257C22×C5—21no—
258C22×C5—42no—
259C22×C5—42no—
260C22×C5—42no—
261C22×C5—41no—
262C22×C5—41no—
263C22×C5—44no—
264C22×C5—44no—
265C22×C5—41no—
266C22×C5—41no—
267C22×C5—41no—
268C22×C5—41no—
269C22×C5—81no—
270C22×C5—81no—
271C22×C5—81no—
272C22×C5—81no—
273C22×C5—81no—
274C22×C5—81no—
275C22×C5—82no—
276C22×C5—82no—
277C22×C5—81no—
278C22×C5—81no—
279C22×C5—81no—
280C22×C5#35312yes—
281C22×C5C2×#9911yescharacteristic
282C22×C5#41711yescharacteristic
283C22×C5C2×#10411yescharacteristic
284C4×C5—42no—
285C4×C5—42no—
286C4×C5—41no—
287C4×C5—41no—
288C4×C5—41no—
289C4×C5—41no—
290C4×C5—84no—
291C4×C5—81no—
292C24—21no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—22no—
297C24—21no—
298C24—42no—
299C24—81no—
300C24C5×C22⋊C411yescharacteristic
301C24C5×C22⋊C411yescharacteristic
302C24C2×C5×D411yescharacteristic
303C2×D4—41no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—42no—
313C2×D4—81no—
314C2×D4—81no—
315C2×D4—81no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—42no—
324C22×C4—44no—
325C22×C4—42no—
326C22×C4—81no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4C5×C22⋊C411yescharacteristic
330C22×C4C5×C22⋊C411yescharacteristic
331C22×C4C22×C4×C511yescharacteristic
332C22×C4C2×C5×D411yescharacteristic
333M16—82no—
334C2×C8—42no—
335C4⋊C4—41no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C22⋊C4—24no—
339C22⋊C4—24no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C22⋊C4—42no—
349C22⋊C4—44no—
350C22⋊C4—42no—
351C42—81no—
352C2×C5—21no—
353C2×C5—21no—
354C2×C5—22no—
355C2×C5—41no—
356C2×C5—41no—
357C2×C5—42no—
358C2×C5—41no—
359C2×C5—161no—
360C2×C5#150411yescharacteristic
361C2×C5#212111yescharacteristic
362C2×C5C2×#35311yescharacteristic
363C23—22no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—22no—
372C23—21no—
373C23—21no—
374C23—41no—
375C23—42no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—42no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—81no—
389C23—81no—
390C23—82no—
391C23—82no—
392C23—84no—
393C23C5×#9911yescharacteristic
394C23C5×#9911yescharacteristic
395C23C5×#10412yes—
396C23C2×C5×C22⋊C411yescharacteristic
397C23C5×AGL(1,ℤ/8)11yescharacteristic
398C23C5×#13711yescharacteristic
399D4—82no—
400D4—82no—
401D4—84no—
402D4—82no—
403D4—82no—
404D4—82no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—44no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—88no—
428C2×C4—82no—
429C2×C4—82no—
430C8—82no—
431C5#1175711yescharacteristic, direct product factor, Sylow
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—22no—
440C22—21no—
441C22—21no—
442C22—42no—
443C22—42no—
444C22—41no—
445C22—41no—
446C22—42no—
447C22—44no—
448C22—44no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—82no—
460C22—82no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22C5×#35312yes—
465C22C2×C5×#9911yescharacteristic
466C22C5×#41711yescharacteristic
467C22C2×C5×#10411yescharacteristic
468C4—42no—
469C4—42no—
470C4—41no—
471C4—41no—
472C4—41no—
473C4—41no—
474C4—84no—
475C4—81no—
476C2—21no—
477C2—21no—
478C2—22no—
479C2—41no—
480C2—41no—
481C2—41no—
482C2—42no—
483C2—161no—
484C2C5×#150411yescharacteristic
485C2C5×#212111yescharacteristic
486C2C2×C5×#35311yescharacteristic
subgroupG/H
center, socleC22×C5C2×#99
commutatorC22×C4C22×C4×C5
Frattini, cosocleC2×C22⋊C4C23×C5
prime radicalC5×#2539C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C24×C5 ◁ C22×C5×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C5×#342C2semidirect
3. C5×#1462C2semidirect
4. C5×#1352C2semidirect
5. C5×#1343C2semidirect
7. C2×C5×#121C4semidirect
12. C5×#399C4semidirect
14. C5×#392C4semidirect
18. C4×C5×C22⋊C4C4semidirect
23. #11757C5direct (N × H)
28. C22×C5×D4C2×C4semidirect
39. C2×C5×C4⋊C4C2×C4semidirect
59. C5×#95C2×C4semidirect
60. C5×#95C2×C4semidirect
62. C2×#342C2×C5semidirect
63. #1462C2×C5semidirect
64. #1352C2×C5semidirect
65. #1343C2×C5semidirect
127. C2×#121C4×C5semidirect
132. #399C4×C5semidirect
134. #392C4×C5semidirect
138. C4×C22⋊C4C4×C5semidirect
215. C22×D4C2×C4×C5semidirect
226. C2×C4⋊C4C2×C4×C5semidirect
246. #95C2×C4×C5semidirect
247. #95C2×C4×C5semidirect
431. C5#11757direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#25392
C2×C5×#3422
C5×#14622
C5×#13522
C5×#13432
#117575