Finite Group Databasebeta

← #10605749

Subgroups — #10605749

Order 640.

814 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
1C22×#685C211yesmaximal, characteristic, complemented
2C2×#4040C211yesmaximal, characteristic, complemented
3C2×#4028C211yesmaximal, characteristic
4C2×D5×C22⋊C4C211yesmaximal, characteristic, complemented
5C5×#380C211yesmaximal, characteristic, complemented
6#65502C211yesmaximal, characteristic, complemented
7#65282C211yesmaximal, characteristic, complemented
8#65280C211yesmaximal, characteristic, complemented
9#65278C211yesmaximal, characteristic, complemented
10#65247C211yesmaximal, characteristic, complemented
11#65231C211yesmaximal, characteristic, complemented
12#64964C211yesmaximal, characteristic, complemented
13#64961C211yesmaximal, characteristic, complemented
14#64954C211yesmaximal, characteristic, complemented
15#64953C211yesmaximal, characteristic, complemented
16C24×D5C2211yescharacteristic
17C23×C4×C5C2211yescharacteristic
18C22×C22⋊D5C2211yescharacteristic
19C23×Dic5C2211yescharacteristic
20C22×D20C2211yescharacteristic
21C22×D20C2211yescharacteristic
22C22×C4×D5C2211yescharacteristic
23C22×C4×D5C2211yescharacteristic
24C2×C5×C4⋊C4C2211yescharacteristic, complemented
25C2×C5×C4⋊C4C2211yescharacteristic, complemented
26C2×C5×C22⋊C4C2211yescharacteristic
27C2×C5×C22⋊C4C2211yescharacteristic
28C2×C22⋊Dic5C2211yescharacteristic
29C2×#685—24no—
30C2×#685C2211yescharacteristic
31C2×#685C2211yescharacteristic
32C2×#685C2211yescharacteristic
33C2×#685C2211yescharacteristic
34C2×#685C2211yescharacteristic
35C2×#685C2211yescharacteristic
36C2×#685C2211yescharacteristic
37C2×#685C2211yescharacteristic
38C2×#685C2211yescharacteristic, complemented
39C2×#685C2211yescharacteristic, complemented
40C2×#685C2211yescharacteristic, complemented
41C2×#685C2211yescharacteristic, complemented
42C2×#683C2211yescharacteristic, complemented
43C2×#683C2211yescharacteristic, complemented
44#4040—24no—
45#4028—24no—
46D5×C22⋊C4—24no—
47C5×#95C2211yescharacteristic, complemented
48C5×#95C2211yescharacteristic, complemented
49#3963C2211yescharacteristic, complemented
50#3963C2211yescharacteristic, complemented
51#3963C2211yescharacteristic, complemented
52#3963C2211yescharacteristic, complemented
53#3963C2211yescharacteristic, complemented
54#3963C2211yescharacteristic, complemented
55#2412—51nomaximal, Sylow
56C24×C5C2311yescharacteristic
57C23×D5—22no—
58C23×D5—22no—
59C23×D5—22no—
60C23×D5—22no—
61C23×D5—41no—
62C23×D5—41no—
63C23×D5D414yes—
64C23×D5C2311yescharacteristic
65C23×D5C2311yescharacteristic
66C23×D5C2311yescharacteristic
67C22×C4×C5—22no—
68C22×C4×C5—22no—
69C22×C4×C5—41no—
70C22×C4×C5C2311yescharacteristic
71C22×C4×C5C2311yescharacteristic
72C22×C4×C5C2311yescharacteristic
73C22×C4×C5C2311yescharacteristic
74C22×C4×C5C2311yescharacteristic
75C22×C4×C5C2311yescharacteristic
76C2×C22⋊D5—22no—
77C2×C22⋊D5—22no—
78C2×C22⋊D5—22no—
79C22×Dic5—22no—
80C22×Dic5—22no—
81C22×Dic5—41no—
82C22×Dic5C2311yescharacteristic
83C22×Dic5C2311yescharacteristic
84C22×Dic5C2311yescharacteristic
85C22×Dic5C2311yescharacteristic
86C22×Dic5C2311yescharacteristic
87C2×D20—22no—
88C2×D20—22no—
89C2×D20—22no—
90C2×D20—22no—
91C2×D20—41no—
92C2×D20—41no—
93C2×C4×D5—22no—
94C2×C4×D5—22no—
95C2×C4×D5—22no—
96C2×C4×D5—22no—
97C2×C4×D5—41no—
98C2×C4×D5—41no—
99C5×C4⋊C4—22no—
100C5×C4⋊C4—41no—
101C5×C22⋊C4—22no—
102C5×C22⋊C4—22no—
103C22⋊Dic5—22no—
104#685—22no—
105#685—22no—
106#685—22no—
107#685—22no—
108#685—22no—
109#685—22no—
110#685—22no—
111#685—22no—
112#685—22no—
113#685—41no—
114#685—41no—
115#685D414yescomplemented
116#683—22no—
117#683—41no—
118C2×#123—51no—
119C2×#120—51no—
120C22×C22⋊C4—51no—
121C22×C22⋊C4—51no—
122#397—51no—
123#397—51no—
124#394—51no—
125#394—51no—
126#386—51no—
127#386—51no—
128#386—51no—
129#385—51no—
130#381—51no—
131#380—51no—
132#380D511yescharacteristic, complemented
133C23×C5—21no—
134C23×C5—21no—
135C23×C5—21no—
136C23×C5—21no—
137C23×C5C2×D412yes—
138C23×C5#4112yes—
139C23×C5#4112yes—
140C23×C5C2411yescharacteristic
141C22×D5—21no—
142C22×D5—21no—
143C22×D5—21no—
144C22×D5—21no—
145C22×D5—28no—
146C22×D5—21no—
147C22×D5—21no—
148C22×D5—21no—
149C22×D5—21no—
150C22×D5—21no—
151C22×D5—21no—
152C22×D5—28no—
153C22×D5—21no—
154C22×D5—21no—
155C22×D5—21no—
156C22×D5—21no—
157C22×D5—21no—
158C22×D5—21no—
159C22×D5—21no—
160C22×D5—44no—
161C22×D5—44no—
162C22×D5—44no—
163C22×D5—44no—
164C22×D5—44no—
165C22×D5C2×D412yes—
166C22×D5C2×D412yes—
167C22×D5C2×D412yes—
168C22×D5#4112yes—
169C2×C4×C5—21no—
170C2×C4×C5—21no—
171C2×C4×C5—21no—
172C2×C4×C5—21no—
173C2×C4×C5—21no—
174C2×C4×C5—21no—
175C2×C4×C5—21no—
176C2×C4×C5—21no—
177C2×C4×C5—21no—
178C2×C4×C5—21no—
179C2×C4×C5—21no—
180C2×C4×C5—21no—
181C2×C4×C5—21no—
182C2×C4×C5—21no—
183C2×C4×C5—21no—
184C2×C4×C5—21no—
185C2×C4×C5—21no—
186C2×C4×C5—44no—
187C2×C4×C5C2×D412yes—
188C22⋊D5—44no—
189C2×Dic5—21no—
190C2×Dic5—21no—
191C2×Dic5—21no—
192C2×Dic5—21no—
193C2×Dic5—21no—
194C2×Dic5—21no—
195C2×Dic5—21no—
196C2×Dic5—21no—
197C2×Dic5—21no—
198C2×Dic5—21no—
199C2×Dic5—21no—
200C2×Dic5—21no—
201C2×Dic5—21no—
202C2×Dic5—21no—
203C2×Dic5—44no—
204C2×Dic5C2×D412yes—
205D20—44no—
206D20—44no—
207C4×D5—44no—
208C4×D5—44no—
209C25—51no—
210C22×D4—51no—
211C22×D4—51no—
212C22×D4—51no—
213C23×C4—51no—
214C23×C4—51no—
215C23×C4—51no—
216C23×C4C2×D511yescharacteristic
217#123—104no—
218#120—104no—
219C2×C4⋊C4—51no—
220C2×C4⋊C4—51no—
221C2×C4⋊C4C2×D511yescharacteristic, complemented
222C2×C4⋊C4C2×D511yescharacteristic, complemented
223C2×C22⋊C4—51no—
224C2×C22⋊C4—51no—
225C2×C22⋊C4—51no—
226C2×C22⋊C4—51no—
227C2×C22⋊C4—51no—
228C2×C22⋊C4—51no—
229C2×C22⋊C4—51no—
230C2×C22⋊C4—51no—
231C2×C22⋊C4—51no—
232C2×C22⋊C4—51no—
233C2×C22⋊C4—51no—
234C2×C22⋊C4—51no—
235C2×C22⋊C4—51no—
236C2×C22⋊C4—104no—
237C2×C22⋊C4—104no—
238C2×C22⋊C4C2×D511yescharacteristic
239C2×C22⋊C4C2×D511yescharacteristic
240#95—51no—
241#95—51no—
242#95—51no—
243#95—51no—
244#95—51no—
245#95—51no—
246#95C2×D511yescharacteristic, complemented
247#95C2×D511yescharacteristic, complemented
248C22×C5—22no—
249C22×C5—22no—
250C22×C5—22no—
251C22×C5—22no—
252C22×C5—22no—
253C22×C5—22no—
254C22×C5#12314yescomplemented
255C22×C5C22×D411yescharacteristic
256C22×C5C22×D411yescharacteristic
257C22×C5C2×#4111yescharacteristic
258C22×C5C2×#4111yescharacteristic
259C22×C5C2×#4111yescharacteristic
260C22×C5H(5,𝔽2)11yescharacteristic
261C22×C5H(5,𝔽2)11yescharacteristic
262C2×D5—22no—
263C2×D5—22no—
264C2×D5—22no—
265C2×D5—22no—
266C2×D5—22no—
267C2×D5—22no—
268C2×D5—22no—
269C2×D5—22no—
270C2×D5—22no—
271C2×D5—416no—
272C2×D5—41no—
273C2×D5—41no—
274C2×D5—41no—
275C2×D5—41no—
276C2×D5—41no—
277C2×D5—41no—
278C2×D5—41no—
279C2×D5—41no—
280C2×D5—41no—
281C2×D5—41no—
282C2×D5—41no—
283C2×D5#12114yes—
284C4×C5—22no—
285C4×C5—41no—
286C4×C5—41no—
287C4×C5—41no—
288C4×C5—41no—
289C4×C5—41no—
290Dic5—22no—
291Dic5—41no—
292Dic5—41no—
293Dic5—41no—
294Dic5—41no—
295C24—51no—
296C24—51no—
297C24—54no—
298C24—51no—
299C24—102no—
300C24—102no—
301C24—102no—
302C24—102no—
303C24—201no—
304C24—201no—
305C24C22×D511yescharacteristic
306C2×D4—102no—
307C2×D4—102no—
308C2×D4—102no—
309C2×D4—102no—
310C2×D4—102no—
311C2×D4—102no—
312C2×D4—102no—
313C2×D4—201no—
314C2×D4—201no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—41no—
318C22×C4—51no—
319C22×C4—51no—
320C22×C4—51no—
321C22×C4—51no—
322C22×C4—51no—
323C22×C4—102no—
324C22×C4—102no—
325C22×C4—102no—
326C22×C4—102no—
327C22×C4—102no—
328C22×C4—102no—
329C22×C4—201no—
330C22×C4—201no—
331C22×C4—201no—
332C22×C4C22×D511yescharacteristic
333C22×C4C22×D511yescharacteristic
334C22×C4C22×D511yescharacteristic
335C22×C4C22×D511yescharacteristic
336C22×C4C22×D511yescharacteristic
337C22×C4C22×D511yescharacteristic
338C4⋊C4—22no—
339C4⋊C4—41no—
340C4⋊C4—102no—
341C4⋊C4—201no—
342C22⋊C4—22no—
343C22⋊C4—22no—
344C22⋊C4—54no—
345C22⋊C4—102no—
346C22⋊C4—102no—
347C22⋊C4—102no—
348C22⋊C4—102no—
349C22⋊C4—102no—
350C22⋊C4—102no—
351C22⋊C4—102no—
352C22⋊C4—102no—
353C22⋊C4—102no—
354C22⋊C4—102no—
355C22⋊C4—201no—
356C22⋊C4—201no—
357C2×C5—24no—
358C2×C5C2×#12111yescharacteristic
359C2×C5C2×#12311yescharacteristic
360C2×C5#53411yescharacteristic
361C2×C5#54611yescharacteristic
362C2×C5#54611yescharacteristic
363C2×C5#55111yescharacteristic
364C2×C5#55111yescharacteristic
365D5—24no—
366D5—42no—
367D5—81no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—52no—
373C23—52no—
374C23—52no—
375C23—52no—
376C23—108no—
377C23—101no—
378C23—101no—
379C23—101no—
380C23—101no—
381C23—101no—
382C23—101no—
383C23—101no—
384C23—101no—
385C23—101no—
386C23—101no—
387C23—108no—
388C23—101no—
389C23—101no—
390C23—101no—
391C23—101no—
392C23—101no—
393C23—101no—
394C23—101no—
395C23—204no—
396C23—204no—
397C23—204no—
398C23—204no—
399C23—204no—
400C23#70912yes—
401C23D4×D512yes—
402C23#71312yes—
403C23C23×D511yescharacteristic
404D4—204no—
405D4—204no—
406D4—204no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—44no—
425C2×C4—52no—
426C2×C4—101no—
427C2×C4—101no—
428C2×C4—101no—
429C2×C4—101no—
430C2×C4—101no—
431C2×C4—101no—
432C2×C4—101no—
433C2×C4—101no—
434C2×C4—101no—
435C2×C4—101no—
436C2×C4—101no—
437C2×C4—101no—
438C2×C4—101no—
439C2×C4—101no—
440C2×C4—204no—
441C2×C4—204no—
442C2×C4—204no—
443C2×C4D4×D512yes—
444C5#241211yescharacteristic, complemented, Sylow
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—54no—
452C22—102no—
453C22—102no—
454C22—102no—
455C22—102no—
456C22—102no—
457C22—102no—
458C22—102no—
459C22—102no—
460C22—102no—
461C22—2016no—
462C22—201no—
463C22—201no—
464C22—201no—
465C22—201no—
466C22—201no—
467C22—201no—
468C22—201no—
469C22—201no—
470C22—201no—
471C22—201no—
472C22—201no—
473C22#404014yescomplemented
474C22C2×#70911yescharacteristic
475C22C2×D4×D511yescharacteristic
476C22C2×D4×D511yescharacteristic
477C22#414411yescharacteristic
478C22C2×#71311yescharacteristic
479C22D5×#4111yescharacteristic
480C22#414911yescharacteristic
481C4—22no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—41no—
487C4—102no—
488C4—201no—
489C4—201no—
490C4—201no—
491C4—201no—
492C2—24no—
493C2—104no—
494C2—202no—
495C2—401no—
496C2#6582411yescharacteristic
497C2C2×#404011yescharacteristic
498C2#6586111yescharacteristic
499C2D5×#12111yescharacteristic
500C2#6593911yescharacteristic
501C2#6596411yescharacteristic
502C2#6597111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#380C2
prime radicalC2×#4028C2
hypercenter#380D5
hypocenterC5#2412
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C22×#685C2semidirect
2. C2×#4040C2semidirect
4. C2×D5×C22⋊C4C2semidirect
5. C5×#380C2semidirect
6. #65502C2semidirect
7. #65282C2semidirect
8. #65280C2semidirect
9. #65278C2semidirect
10. #65247C2semidirect
11. #65231C2semidirect
12. #64964C2semidirect
13. #64961C2semidirect
14. #64954C2semidirect
15. #64953C2semidirect
24. C2×C5×C4⋊C4C22semidirect
25. C2×C5×C4⋊C4C22semidirect
38. C2×#685C22semidirect
39. C2×#685C22semidirect
40. C2×#685C22semidirect
41. C2×#685C22semidirect
42. C2×#683C22semidirect
43. C2×#683C22semidirect
47. C5×#95C22semidirect
48. C5×#95C22semidirect
49. #3963C22semidirect
50. #3963C22semidirect
51. #3963C22semidirect
52. #3963C22semidirect
53. #3963C22semidirect
54. #3963C22semidirect
115. #685D4semidirect
132. #380D5semidirect
221. C2×C4⋊C4C2×D5semidirect
222. C2×C4⋊C4C2×D5semidirect
246. #95C2×D5semidirect
247. #95C2×D5semidirect
254. C22×C5#123semidirect
444. C5#2412semidirect
473. C22#4040semidirect

Maximal subgroups

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

subgroupindex
C22×#6852
C2×#40402
C2×#40282
C2×D5×C22⋊C42
C5×#3802
#655022
#652822
#652802
#652782
#652472
#652312
#649642
#649612
#649542
#649532
#24125