Finite Group Databasebeta

← #10606621

Subgroups — #10606621

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
1C2×#4083C211yesmaximal, characteristic
2C22×#685C211yesmaximal, characteristic, complemented
3C2×#4029C211yesmaximal, characteristic, complemented
4C2×D5×C22⋊C4C211yesmaximal, characteristic, complemented
5C5×#386C211yesmaximal, characteristic, complemented
6#65509C211yesmaximal, characteristic, complemented
7#65501C211yesmaximal, characteristic, complemented
8#65250C211yesmaximal, characteristic, complemented
9#65248C211yesmaximal, characteristic, complemented
10#65245C211yesmaximal, characteristic, complemented
11#65231C211yesmaximal, characteristic, complemented
12#64964C211yesmaximal, characteristic, complemented
13#64963C211yesmaximal, characteristic, complemented
14#64957C211yesmaximal, characteristic, complemented
15#64953C211yesmaximal, characteristic, complemented
16C24×D5C2211yescharacteristic
17C22×C5×D4C2211yescharacteristic
18C23×C4×C5C2211yescharacteristic
19C22×C22⋊D5C2211yescharacteristic
20C22×C22⋊D5C2211yescharacteristic
21C23×Dic5C2211yescharacteristic
22C22×C4×D5C2211yescharacteristic
23C22×C4×D5C2211yescharacteristic
24C2×C5×C22⋊C4C2211yescharacteristic
25C2×C5×C22⋊C4C2211yescharacteristic, complemented
26C2×C5×C22⋊C4C2211yescharacteristic, complemented
27C2×C22⋊Dic5C2211yescharacteristic
28C2×C22⋊Dic5C2211yescharacteristic
29C2×C22⋊Dic5C2211yescharacteristic, complemented
30C2×C22⋊Dic5C2211yescharacteristic, complemented
31#4083—24no—
32C2×#685—24no—
33C2×#685C2211yescharacteristic
34C2×#685C2211yescharacteristic
35C2×#685C2211yescharacteristic
36C2×#685C2211yescharacteristic
37C2×#685C2211yescharacteristic
38C2×#685C2211yescharacteristic
39C2×#685C2211yescharacteristic
40C2×#685C2211yescharacteristic
41C2×C4⋊Dic5C2211yescharacteristic, complemented
42C2×C4⋊Dic5C2211yescharacteristic, complemented
43C2×#683C2211yescharacteristic, complemented
44C2×#683C2211yescharacteristic, complemented
45#4029—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
57C24×C5C2311yescharacteristic
58C23×D5—22no—
59C23×D5—22no—
60C23×D5—22no—
61C23×D5—22no—
62C23×D5—41no—
63C23×D5—41no—
64C23×D5D414yes—
65C23×D5C2311yescharacteristic
66C23×D5C2311yescharacteristic
67C2×C5×D4—22no—
68C2×C5×D4—22no—
69C2×C5×D4—22no—
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⋊D5—41no—
84C22×Dic5—22no—
85C22×Dic5—22no—
86C22×Dic5—41no—
87C22×Dic5C2311yescharacteristic
88C22×Dic5C2311yescharacteristic
89C22×Dic5C2311yescharacteristic
90C22×Dic5C2311yescharacteristic
91C22×Dic5C2311yescharacteristic
92C22×Dic5C2311yescharacteristic
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×C22⋊C4—22no—
100C5×C22⋊C4—22no—
101C5×C22⋊C4—41no—
102C22⋊Dic5—22no—
103C22⋊Dic5—22no—
104C22⋊Dic5—22no—
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#685D414yescomplemented
114C4⋊Dic5—22no—
115C4⋊Dic5—41no—
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#386D511yescharacteristic, complemented
129#385—51no—
130#381—51no—
131#380—51no—
132#380—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×C5C2×D412yes—
145C23×C5#4112yes—
146C23×C5#4112yes—
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—28no—
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—
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—21no—
184C2×C4×C5—44no—
185C2×C4×C5C2×D412yes—
186C22⋊D5—44no—
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×Dic5—44no—
206C2×Dic5C2×D412yes—
207C4×D5—44no—
208C4×D5—44no—
209C25—51no—
210C22×D4—51no—
211C22×D4—51no—
212C22×D4C2×D511yescharacteristic
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⋊C4—51no—
222C2×C4⋊C4—51no—
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—104no—
236C2×C22⋊C4—104no—
237C2×C22⋊C4C2×D511yescharacteristic
238C2×C22⋊C4C2×D511yescharacteristic, complemented
239C2×C22⋊C4C2×D511yescharacteristic, complemented
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—41no—
255C22×C5—41no—
256C22×C5—41no—
257C22×C5—41no—
258C22×C5—41no—
259C22×C5—41no—
260C22×C5—41no—
261C22×C5#12314yescomplemented
262C22×C5C22×D411yescharacteristic
263C22×C5C22×D411yescharacteristic
264C22×C5C2×#4111yescharacteristic
265C22×C5C2×#4111yescharacteristic
266C22×C5C2×#4111yescharacteristic
267C22×C5H(5,𝔽2)11yescharacteristic
268C22×C5H(5,𝔽2)11yescharacteristic
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—22no—
278C2×D5—416no—
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—
289Dic5—22no—
290Dic5—41no—
291Dic5—41no—
292Dic5—41no—
293Dic5—41no—
294Dic5—41no—
295C24—51no—
296C24—51no—
297C24—54no—
298C24—102no—
299C24—102no—
300C24—102no—
301C24—102no—
302C24—201no—
303C24—201no—
304C24C22×D511yescharacteristic
305C24C22×D511yescharacteristic
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
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—51no—
324C22×C4—102no—
325C22×C4—102no—
326C22×C4—102no—
327C22×C4—102no—
328C22×C4—102no—
329C22×C4—102no—
330C22×C4—201no—
331C22×C4—201no—
332C22×C4—201no—
333C22×C4C22×D511yescharacteristic
334C22×C4C22×D511yescharacteristic
335C22×C4C22×D511yescharacteristic
336C22×C4C22×D511yescharacteristic
337C22×C4C22×D511yescharacteristic
338C4⋊C4—102no—
339C4⋊C4—102no—
340C4⋊C4—201no—
341C4⋊C4—201no—
342C22⋊C4—22no—
343C22⋊C4—22no—
344C22⋊C4—41no—
345C22⋊C4—54no—
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—102no—
356C22⋊C4—201no—
357C2×C5—24no—
358C2×C5—81no—
359C2×C5C2×#12111yescharacteristic
360C2×C5C2×#12311yescharacteristic
361C2×C5#53411yescharacteristic
362C2×C5#54611yescharacteristic
363C2×C5#54611yescharacteristic
364C2×C5#55111yescharacteristic
365C2×C5#55111yescharacteristic
366D5—24no—
367D5—42no—
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—108no—
395C23—204no—
396C23—204no—
397C23—204no—
398C23—204no—
399C23—204no—
400C23#70912yes—
401C23D4×D512yes—
402C23#71112yes—
403C23C23×D511yescharacteristic
404D4—44no—
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—44no—
422C2×C4—52no—
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—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—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—54no—
459C22—102no—
460C22—102no—
461C22—102no—
462C22—102no—
463C22—102no—
464C22—102no—
465C22—102no—
466C22—102no—
467C22—102no—
468C22—2016no—
469C22—201no—
470C22—201no—
471C22—201no—
472C22—201no—
473C22#402914yescomplemented
474C22C2×#70911yescharacteristic
475C22C2×D4×D511yescharacteristic
476C22C2×D4×D511yescharacteristic
477C22C2×#71111yescharacteristic
478C22#414411yescharacteristic
479C22D5×#4111yescharacteristic
480C22#414911yescharacteristic
481C4—22no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—102no—
487C4—201no—
488C4—201no—
489C4—201no—
490C4—201no—
491C4—201no—
492C2—24no—
493C2—81no—
494C2—104no—
495C2—202no—
496C2C2×#402911yescharacteristic
497C2#6588411yescharacteristic
498C2#6589011yescharacteristic
499C2#6592511yescharacteristic
500C2D5×#12111yescharacteristic
501C2#6593911yescharacteristic
502C2#6599211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#386C2
prime radicalC2×#4083C2
hypercenter#386D5
hypocenterC5#2412
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×#685C2semidirect
3. C2×#4029C2semidirect
4. C2×D5×C22⋊C4C2semidirect
5. C5×#386C2semidirect
6. #65509C2semidirect
7. #65501C2semidirect
8. #65250C2semidirect
9. #65248C2semidirect
10. #65245C2semidirect
11. #65231C2semidirect
12. #64964C2semidirect
13. #64963C2semidirect
14. #64957C2semidirect
15. #64953C2semidirect
25. C2×C5×C22⋊C4C22semidirect
26. C2×C5×C22⋊C4C22semidirect
29. C2×C22⋊Dic5C22semidirect
30. C2×C22⋊Dic5C22semidirect
41. C2×C4⋊Dic5C22semidirect
42. C2×C4⋊Dic5C22semidirect
43. C2×#683C22semidirect
44. C2×#683C22semidirect
47. C5×#95C22semidirect
48. C5×#95C22semidirect
49. #3963C22semidirect
50. #3963C22semidirect
51. #3963C22semidirect
52. #3963C22semidirect
53. #3963C22semidirect
54. #3963C22semidirect
113. #685D4semidirect
128. #386D5semidirect
238. C2×C22⋊C4C2×D5semidirect
239. C2×C22⋊C4C2×D5semidirect
246. #95C2×D5semidirect
247. #95C2×D5semidirect
261. C22×C5#123semidirect
444. C5#2412semidirect
473. C22#4029semidirect

Maximal subgroups

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

subgroupindex
C2×#40832
C22×#6852
C2×#40292
C2×D5×C22⋊C42
C5×#3862
#655092
#655012
#652502
#652482
#652452
#652312
#649642
#649632
#649572
#649532
#24125