Finite Group Databasebeta

← #10614627

Subgroups — #10614627

Order 640.

800 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#123C211yesmaximal, characteristic, complemented
2C2×#4083C211yesmaximal, characteristic
3C2×C4×C22⋊D5C211yesmaximal, characteristic, complemented
4C2×D5×C22⋊C4C211yesmaximal, characteristic, complemented
5#65511C211yesmaximal, characteristic, complemented
6#65502C211yesmaximal, characteristic, complemented
7#65281C211yesmaximal, characteristic, complemented
8#65279C211yesmaximal, characteristic, complemented
9#65278C211yesmaximal, characteristic, complemented
10#65276C211yesmaximal, characteristic, complemented
11#65250C211yesmaximal, characteristic, complemented
12#65245C211yesmaximal, characteristic, complemented
13#65244C211yesmaximal, characteristic, complemented
14#65241C211yesmaximal, characteristic, complemented
15#65240C211yesmaximal, characteristic, complemented
16C24×D5C2211yescharacteristic
17C22×C5×D4C2211yescharacteristic
18C23×C4×C5C2211yescharacteristic, complemented
19C22×C22⋊D5C2211yescharacteristic
20C22×C22⋊D5C2211yescharacteristic
21C22×C4×D5C2211yescharacteristic
22C22×C4×D5C2211yescharacteristic
23C5×#123—24no—
24C2×C5×C4⋊C4C2211yescharacteristic, complemented
25C2×C5×C4⋊C4C2211yescharacteristic, complemented
26C2×C5×C22⋊C4C2211yescharacteristic
27C2×C5×C22⋊C4C2211yescharacteristic
28C2×C5×C22⋊C4C2211yescharacteristic, complemented
29C2×C22⋊Dic5C2211yescharacteristic
30C2×C22⋊Dic5C2211yescharacteristic
31C2×C22⋊Dic5C2211yescharacteristic
32C2×C22⋊Dic5C2211yescharacteristic, complemented
33C2×C22⋊Dic5C2211yescharacteristic, complemented
34#4083—24no—
35C4×C22⋊D5—24no—
36C2×#685C2211yescharacteristic
37C2×#685C2211yescharacteristic
38C2×#685C2211yescharacteristic
39C2×#685C2211yescharacteristic
40C2×#685C2211yescharacteristic
41C2×#685C2211yescharacteristic
42C2×#685C2211yescharacteristic
43C2×#685C2211yescharacteristic
44C2×C4⋊Dic5C2211yescharacteristic, complemented
45C2×#683C2211yescharacteristic, complemented
46C2×#683C2211yescharacteristic, complemented
47C2×C4×Dic5C2211yescharacteristic, complemented
48D5×C22⋊C4—24no—
49#3963C2211yescharacteristic, complemented
50#3963C2211yescharacteristic, complemented
51#3963C2211yescharacteristic, complemented
52#3963C2211yescharacteristic, complemented
53#3963C2211yescharacteristic, complemented
54#3963C2211yescharacteristic, complemented
55#2418—51nomaximal, Sylow
56C24×C5C2311yescharacteristic
57C24×C5C2311yescharacteristic
58C23×D5—22no—
59C23×D5—22no—
60C23×D5—22no—
61C23×D5—22no—
62C23×D5—22no—
63C23×D5—41no—
64C23×D5—41no—
65C23×D5C2311yescharacteristic
66C23×D5C2311yescharacteristic
67C2×C5×D4—22no—
68C2×C5×D4—22no—
69C2×C5×D4—41no—
70C22×C4×C5—22no—
71C22×C4×C5—22no—
72C22×C4×C5—41no—
73C22×C4×C5C2311yescharacteristic
74C22×C4×C5C2311yescharacteristic
75C22×C4×C5C2311yescharacteristic
76C22×C4×C5C2311yescharacteristic
77C22×C4×C5C2311yescharacteristic
78C2×C22⋊D5—22no—
79C2×C22⋊D5—22no—
80C2×C22⋊D5—22no—
81C2×C22⋊D5—22no—
82C2×C22⋊D5—41no—
83C2×C22⋊D5D414yes—
84C22×Dic5C2311yescharacteristic
85C22×Dic5C2311yescharacteristic
86C22×Dic5C2311yescharacteristic
87C22×Dic5C2311yescharacteristic
88C22×Dic5C2311yescharacteristic
89C22×Dic5C2311yescharacteristic
90C2×C4×D5—22no—
91C2×C4×D5—22no—
92C2×C4×D5—22no—
93C2×C4×D5—22no—
94C2×C4×D5—41no—
95C2×C4×D5—41no—
96C5×C4⋊C4—22no—
97C5×C4⋊C4—22no—
98C5×C22⋊C4—22no—
99C5×C22⋊C4—22no—
100C5×C22⋊C4D414yescomplemented
101C22⋊Dic5—22no—
102C22⋊Dic5—22no—
103C22⋊Dic5—22no—
104C22⋊Dic5—41no—
105C22⋊Dic5—41no—
106#685—22no—
107#685—22no—
108#685—22no—
109#685—22no—
110#685—22no—
111#685—22no—
112#685—22no—
113#685—41no—
114C4⋊Dic5—41no—
115#683—22no—
116#683—41no—
117C4×Dic5—22no—
118C2×#123D511yescharacteristic, complemented
119C2×#120—51no—
120C2×C4×D4—51no—
121C22×C22⋊C4—51no—
122#399—51no—
123#397—51no—
124#394—51no—
125#394—51no—
126#388—51no—
127#388—51no—
128#386—51no—
129#386—51no—
130#385—51no—
131#380—51no—
132#379—51no—
133C23×C5—21no—
134C23×C5—21no—
135C23×C5—21no—
136C23×C5—21no—
137C23×C5—21no—
138C23×C5—21no—
139C23×C5—21no—
140C23×C5—21no—
141C23×C5—21no—
142C23×C5—21no—
143C23×C5—21no—
144C23×C5—21no—
145C23×C5C2×D412yes—
146C23×C5C2×D412yes—
147C23×C5C2411yescharacteristic
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—44no—
160C22×D5—44no—
161C22×D5—44no—
162C22×D5—44no—
163C22×D5—44no—
164C22×D5—44no—
165C22×D5C2×D412yes—
166C22×D5#4112yes—
167C22×D5#4112yes—
168C22×D5#4112yes—
169C5×D4—44no—
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—44no—
184C2×C4×C5C2×D412yes—
185C2×C4×C5C2×D412yes—
186C22⋊D5—28no—
187C22⋊D5—44no—
188C2×Dic5—21no—
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—21no—
204C2×Dic5—21no—
205C2×Dic5C2×D412yes—
206C4×D5—44no—
207C4×D5—44no—
208C25—51no—
209C22×D4—51no—
210C22×D4—51no—
211C22×D4C2×D511yescharacteristic
212C23×C4—51no—
213C23×C4—51no—
214C23×C4C2×D511yescharacteristic, complemented
215#123—24no—
216#120—104no—
217C4×D4—104no—
218C2×C4⋊C4—51no—
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⋊C4C2×D511yescharacteristic
238C2×C22⋊C4C2×D511yescharacteristic
239C2×C22⋊C4C2×D511yescharacteristic, complemented
240C2×C42—51no—
241#95—51no—
242#95—51no—
243#95—51no—
244#95—51no—
245#95—51no—
246#95—51no—
247C22×C5—22no—
248C22×C5—22no—
249C22×C5—22no—
250C22×C5—22no—
251C22×C5—22no—
252C22×C5—22no—
253C22×C5—41no—
254C22×C5—41no—
255C22×C5—41no—
256C22×C5—41no—
257C22×C5—41no—
258C22×C5—41no—
259C22×C5—41no—
260C22×C5—41no—
261C22×C5C22×D411yescharacteristic
262C22×C5C22×D411yescharacteristic
263C22×C5C2×#4111yescharacteristic
264C22×C5C2×#4111yescharacteristic
265C22×C5C2×#4111yescharacteristic
266C22×C5H(5,𝔽2)11yescharacteristic
267C22×C5H(5,𝔽2)11yescharacteristic
268C2×D5—22no—
269C2×D5—22no—
270C2×D5—22no—
271C2×D5—22no—
272C2×D5—22no—
273C2×D5—22no—
274C2×D5—22no—
275C2×D5—22no—
276C2×D5—22no—
277C2×D5—416no—
278C2×D5—41no—
279C2×D5—41no—
280C2×D5—41no—
281C2×D5—41no—
282C2×D5#12314yescomplemented
283C4×C5—22no—
284C4×C5—22no—
285C4×C5—41no—
286C4×C5—41no—
287C4×C5—41no—
288Dic5—22no—
289Dic5—41no—
290Dic5—41no—
291Dic5—41no—
292Dic5—41no—
293Dic5—41no—
294C24—51no—
295C24—51no—
296C24—102no—
297C24—102no—
298C24—102no—
299C24—102no—
300C24—102no—
301C24—201no—
302C24—201no—
303C24C22×D511yescharacteristic
304C24C22×D511yescharacteristic
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—41no—
308C2×D4—54no—
309C2×D4—102no—
310C2×D4—102no—
311C2×D4—102no—
312C2×D4—102no—
313C2×D4—201no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—41no—
317C22×C4—51no—
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—201no—
328C22×C4—201no—
329C22×C4C22×D511yescharacteristic
330C22×C4C22×D511yescharacteristic
331C22×C4C22×D511yescharacteristic
332C22×C4C22×D511yescharacteristic
333C22×C4C22×D511yescharacteristic
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—102no—
337C4⋊C4—201no—
338C4⋊C4—201no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—102no—
342C22⋊C4—102no—
343C22⋊C4—102no—
344C22⋊C4—102no—
345C22⋊C4—102no—
346C22⋊C4—102no—
347C22⋊C4—102no—
348C22⋊C4—102no—
349C22⋊C4—102no—
350C22⋊C4—102no—
351C22⋊C4—201no—
352C22⋊C4—201no—
353C22⋊C4—201no—
354C22⋊C4C22⋊D514yescomplemented
355C42—102no—
356C2×C5—42no—
357C2×C5—42no—
358C2×C5C2×#12311yescharacteristic
359C2×C5#52511yescharacteristic
360C2×C5#53811yescharacteristic
361C2×C5D4211yescharacteristic
362C2×C5#54611yescharacteristic
363C2×C5#54611yescharacteristic
364C2×C5#55111yescharacteristic
365D5—24no—
366D5—42no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—52no—
380C23—52no—
381C23—52no—
382C23—52no—
383C23—108no—
384C23—101no—
385C23—101no—
386C23—101no—
387C23—101no—
388C23—101no—
389C23—101no—
390C23—101no—
391C23—101no—
392C23—101no—
393C23—101no—
394C23—204no—
395C23—204no—
396C23—204no—
397C23—204no—
398C23—204no—
399C23—204no—
400C23D4×D512yes—
401C23C2×C22⋊D512yes—
402C23C23×D511yescharacteristic
403D4—44no—
404D4—108no—
405D4—204no—
406C2×C4—21no—
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—44no—
420C2×C4—52no—
421C2×C4—101no—
422C2×C4—101no—
423C2×C4—101no—
424C2×C4—101no—
425C2×C4—101no—
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—204no—
439C2×C4—204no—
440C2×C4C2×C22⋊D512yes—
441C2×C4C2×C22⋊D512yes—
442C5#241811yescharacteristic, complemented, Sylow
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—54no—
458C22—102no—
459C22—102no—
460C22—102no—
461C22—102no—
462C22—102no—
463C22—102no—
464C22—102no—
465C22—102no—
466C22—102no—
467C22—2016no—
468C22—201no—
469C22—201no—
470C22—201no—
471C22—201no—
472C22C2×D4×D511yescharacteristic
473C22#414411yescharacteristic
474C22D5×#4111yescharacteristic
475C22D5×#4111yescharacteristic
476C22D5×#4111yescharacteristic
477C22#414911yescharacteristic
478C22C22×C22⋊D511yescharacteristic
479C4—22no—
480C4—22no—
481C4—41no—
482C4—41no—
483C4—41no—
484C4—102no—
485C4—201no—
486C4—201no—
487C4—201no—
488C4—201no—
489C4—201no—
490C2—42no—
491C2—42no—
492C2—104no—
493C2—202no—
494C2D5×#12311yescharacteristic
495C2#6598811yescharacteristic
496C2#6599111yescharacteristic
497C2#6599311yescharacteristic
498C2D4×C22⋊D511yescharacteristic
499C2#6616211yescharacteristic
500C2#6616311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×#123C2
prime radicalC2×#4083C2
hypercenterC2×#123D5
hypocenterC5#2418
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×C5×#123C2semidirect
3. C2×C4×C22⋊D5C2semidirect
4. C2×D5×C22⋊C4C2semidirect
5. #65511C2semidirect
6. #65502C2semidirect
7. #65281C2semidirect
8. #65279C2semidirect
9. #65278C2semidirect
10. #65276C2semidirect
11. #65250C2semidirect
12. #65245C2semidirect
13. #65244C2semidirect
14. #65241C2semidirect
15. #65240C2semidirect
18. C23×C4×C5C22semidirect
24. C2×C5×C4⋊C4C22semidirect
25. C2×C5×C4⋊C4C22semidirect
28. C2×C5×C22⋊C4C22semidirect
32. C2×C22⋊Dic5C22semidirect
33. C2×C22⋊Dic5C22semidirect
44. C2×C4⋊Dic5C22semidirect
45. C2×#683C22semidirect
46. C2×#683C22semidirect
47. C2×C4×Dic5C22semidirect
49. #3963C22semidirect
50. #3963C22semidirect
51. #3963C22semidirect
52. #3963C22semidirect
53. #3963C22semidirect
54. #3963C22semidirect
100. C5×C22⋊C4D4semidirect
118. C2×#123D5semidirect
214. C23×C4C2×D5semidirect
221. C2×C4⋊C4C2×D5semidirect
222. C2×C4⋊C4C2×D5semidirect
239. C2×C22⋊C4C2×D5semidirect
282. C2×D5#123semidirect
354. C22⋊C4C22⋊D5semidirect
442. C5#2418semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1232
C2×#40832
C2×C4×C22⋊D52
C2×D5×C22⋊C42
#655112
#655022
#652812
#652792
#652782
#652762
#652502
#652452
#652442
#652412
#652402
#24185