Finite Group Databasebeta

← #10614135

Subgroups — #10614135

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×C4×C5×D4C211yesmaximal, characteristic, complemented
2C2×#4083C211yesmaximal, characteristic
3C2×#4075C211yesmaximal, characteristic, complemented
4C22×#685C211yesmaximal, characteristic, complemented
5#65509C211yesmaximal, characteristic, complemented
6#65502C211yesmaximal, characteristic, complemented
7#65499C211yesmaximal, characteristic, complemented
8#65282C211yesmaximal, characteristic, complemented
9#65281C211yesmaximal, characteristic, complemented
10#65248C211yesmaximal, characteristic, complemented
11#65248C211yesmaximal, characteristic, complemented
12#65244C211yesmaximal, characteristic, complemented
13#65231C211yesmaximal, characteristic, complemented
14#65229C211yesmaximal, characteristic, complemented
15#65229C211yesmaximal, characteristic, complemented
16C24×D5C2211yescharacteristic
17C22×C5×D4C2211yescharacteristic
18C23×C4×C5C2211yescharacteristic
19C23×C4×C5C2211yescharacteristic, complemented
20C22×C22⋊D5C2211yescharacteristic
21C22×C22⋊D5C2211yescharacteristic
22C23×Dic5C2211yescharacteristic
23C4×C5×D4—24no—
24C2×C5×C4⋊C4C2211yescharacteristic, complemented
25C2×C5×C22⋊C4C2211yescharacteristic
26C2×C5×C22⋊C4C2211yescharacteristic, complemented
27C2×C42×C5C2211yescharacteristic, complemented
28C2×C22⋊Dic5C2211yescharacteristic
29C2×C22⋊Dic5C2211yescharacteristic
30C2×C22⋊Dic5C2211yescharacteristic, complemented
31C2×C22⋊Dic5C2211yescharacteristic, complemented
32#4083—24no—
33#4075—24no—
34C2×#685—24no—
35C2×#685C2211yescharacteristic
36C2×#685C2211yescharacteristic
37C2×#685C2211yescharacteristic
38C2×#685C2211yescharacteristic
39C2×#685C2211yescharacteristic
40C2×#685C2211yescharacteristic
41C2×#685C2211yescharacteristic
42C2×#685C2211yescharacteristic
43C2×#685C2211yescharacteristic
44C2×#685C2211yescharacteristic
45C2×#683C2211yescharacteristic, complemented
46C2×#683C2211yescharacteristic, complemented
47C2×#683C2211yescharacteristic, complemented
48C2×#683C2211yescharacteristic, complemented
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×D4D414yes—
70C22×C4×C5—22no—
71C22×C4×C5—22no—
72C22×C4×C5—22no—
73C22×C4×C5—22no—
74C22×C4×C5—41no—
75C22×C4×C5—41no—
76C22×C4×C5C2311yescharacteristic
77C22×C4×C5C2311yescharacteristic
78C22×C4×C5C2311yescharacteristic
79C22×C4×C5C2311yescharacteristic
80C22×C4×C5C2311yescharacteristic
81C2×C22⋊D5—22no—
82C2×C22⋊D5—22no—
83C2×C22⋊D5—22no—
84C2×C22⋊D5—22no—
85C2×C22⋊D5—41no—
86C2×C22⋊D5—41no—
87C22×Dic5—22no—
88C22×Dic5—22no—
89C22×Dic5—41no—
90C22×Dic5C2311yescharacteristic
91C22×Dic5C2311yescharacteristic
92C22×Dic5C2311yescharacteristic
93C22×Dic5C2311yescharacteristic
94C22×Dic5C2311yescharacteristic
95C22×Dic5C2311yescharacteristic
96C5×C4⋊C4—22no—
97C5×C22⋊C4—22no—
98C5×C22⋊C4—22no—
99C42×C5—22no—
100C22⋊Dic5—22no—
101C22⋊Dic5—22no—
102C22⋊Dic5—22no—
103C22⋊Dic5—41no—
104#685—22no—
105#685—22no—
106#685—22no—
107#685—22no—
108#685—22no—
109#685—22no—
110#685—22no—
111#685—41no—
112#685—41no—
113#685D414yescomplemented
114#683—22no—
115#683—22no—
116#683—41no—
117#683—41no—
118C2×#123—51no—
119C2×#120—51no—
120C2×C4×D4D511yescharacteristic, complemented
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×C5C2×D412yes—
144C23×C5C2×D412yes—
145C23×C5#4112yes—
146C23×C5#4112yes—
147C23×C5C2411yescharacteristic
148C22×D5—28no—
149C22×D5—21no—
150C22×D5—21no—
151C22×D5—21no—
152C22×D5—21no—
153C22×D5—21no—
154C22×D5—21no—
155C22×D5—21no—
156C22×D5—21no—
157C22×D5—21no—
158C22×D5—21no—
159C22×D5—21no—
160C22×D5—21no—
161C22×D5—44no—
162C22×D5—44no—
163C22×D5—44no—
164C22×D5—44no—
165C22×D5—44no—
166C22×D5—44no—
167C22×D5C2×D412yes—
168C22×D5#4112yes—
169C5×D4—28no—
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×C5—44no—
185C2×C4×C5C2×D412yes—
186C2×C4×C5C2×D412yes—
187C22⋊D5—44no—
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—21no—
204C2×Dic5—21no—
205C2×Dic5—21no—
206C2×Dic5—44no—
207C2×Dic5C2×D412yes—
208C25—51no—
209C22×D4—51no—
210C22×D4—51no—
211C22×D4C2×D511yescharacteristic
212C23×C4—51no—
213C23×C4C2×D511yescharacteristic
214C23×C4C2×D511yescharacteristic, complemented
215#123—104no—
216#120—104no—
217C4×D4—24no—
218C2×C4⋊C4—51no—
219C2×C4⋊C4—51no—
220C2×C4⋊C4—51no—
221C2×C4⋊C4—51no—
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—51no—
237C2×C22⋊C4—104no—
238C2×C22⋊C4C2×D511yescharacteristic
239C2×C22⋊C4C2×D511yescharacteristic, complemented
240C2×C42C2×D511yescharacteristic, complemented
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—22no—
254C22×C5—22no—
255C22×C5—22no—
256C22×C5—41no—
257C22×C5—41no—
258C22×C5—41no—
259C22×C5—41no—
260C22×C5#12314yescomplemented
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—416no—
275C2×D5—41no—
276C2×D5—41no—
277C2×D5—41no—
278C2×D5—41no—
279C2×D5—41no—
280C2×D5—41no—
281C2×D5—41no—
282C2×D5—41no—
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—102no—
308C2×D4—102no—
309C2×D4—102no—
310C2×D4—102no—
311C2×D4—201no—
312C2×D4—201no—
313C2×D4C22⋊D514yes—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—51no—
321C22×C4—51no—
322C22×C4—51no—
323C22×C4—51no—
324C22×C4—51no—
325C22×C4—51no—
326C22×C4—102no—
327C22×C4—102no—
328C22×C4—201no—
329C22×C4C22×D511yescharacteristic
330C22×C4C22×D511yescharacteristic
331C22×C4C22×D511yescharacteristic
332C22×C4C22×D511yescharacteristic
333C22×C4C22×D511yescharacteristic
334C4⋊C4—22no—
335C4⋊C4—102no—
336C4⋊C4—102no—
337C4⋊C4—201no—
338C4⋊C4—201no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—54no—
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—102no—
352C22⋊C4—201no—
353C22⋊C4—201no—
354C22⋊C4—201no—
355C42—22no—
356C2×C5—24no—
357C2×C5—42no—
358C2×C5C2×#12311yescharacteristic
359C2×C5#52511yescharacteristic
360C2×C5#53811yescharacteristic
361C2×C5D4211yescharacteristic
362C2×C5#54611yescharacteristic
363C2×C5#54611yescharacteristic
364C2×C5#55111yescharacteristic
365D5—42no—
366D5—42no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—52no—
378C23—52no—
379C23—108no—
380C23—101no—
381C23—101no—
382C23—101no—
383C23—101no—
384C23—101no—
385C23—101no—
386C23—101no—
387C23—101no—
388C23—101no—
389C23—101no—
390C23—101no—
391C23—101no—
392C23—204no—
393C23—204no—
394C23—204no—
395C23—204no—
396C23—204no—
397C23—204no—
398C23#70912yes—
399C23#70912yes—
400C23C2×C22⋊D512yes—
401C23C2×C22⋊D512yes—
402C23C23×D511yescharacteristic
403D4—28no—
404D4—204no—
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—44no—
421C2×C4—52no—
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—101no—
439C2×C4—204no—
440C2×C4D4×D512yes—
441C2×C4C2×C22⋊D512yes—
442C5#241811yescharacteristic, complemented, Sylow
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—102no—
457C22—102no—
458C22—102no—
459C22—102no—
460C22—102no—
461C22—102no—
462C22—2016no—
463C22—201no—
464C22—201no—
465C22—201no—
466C22—201no—
467C22—201no—
468C22—201no—
469C22—201no—
470C22—201no—
471C22#407514yescomplemented
472C22C2×#70911yescharacteristic
473C22C2×#70911yescharacteristic
474C22C2×D4×D511yescharacteristic
475C22#414411yescharacteristic
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—24no—
491C2—42no—
492C2—202no—
493C2—202no—
494C2#6588111yescharacteristic
495C2#6588411yescharacteristic
496C2#6589011yescharacteristic
497C2#6589411yescharacteristic
498C2C2×#407511yescharacteristic
499C2D4×C22⋊D511yescharacteristic
500C2#6616311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C4×C5×D4C2
prime radicalC2×#4083C2
hypercenterC2×C4×D4D5
hypocenterC5#2418
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C4×C5×D4C2semidirect
3. C2×#4075C2semidirect
4. C22×#685C2semidirect
5. #65509C2semidirect
6. #65502C2semidirect
7. #65499C2semidirect
8. #65282C2semidirect
9. #65281C2semidirect
10. #65248C2semidirect
11. #65248C2semidirect
12. #65244C2semidirect
13. #65231C2semidirect
14. #65229C2semidirect
15. #65229C2semidirect
19. C23×C4×C5C22semidirect
24. C2×C5×C4⋊C4C22semidirect
26. C2×C5×C22⋊C4C22semidirect
27. C2×C42×C5C22semidirect
30. C2×C22⋊Dic5C22semidirect
31. C2×C22⋊Dic5C22semidirect
45. C2×#683C22semidirect
46. C2×#683C22semidirect
47. C2×#683C22semidirect
48. C2×#683C22semidirect
49. #3963C22semidirect
50. #3963C22semidirect
51. #3963C22semidirect
52. #3963C22semidirect
53. #3963C22semidirect
54. #3963C22semidirect
113. #685D4semidirect
120. C2×C4×D4D5semidirect
214. C23×C4C2×D5semidirect
222. C2×C4⋊C4C2×D5semidirect
239. C2×C22⋊C4C2×D5semidirect
240. C2×C42C2×D5semidirect
260. C22×C5#123semidirect
442. C5#2418semidirect
471. C22#4075semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×D42
C2×#40832
C2×#40752
C22×#6852
#655092
#655022
#654992
#652822
#652812
#652482
#652442
#652312
#652292
#24185