Finite Group Databasebeta

← #10608840

Subgroups — #10608840

Order 640.

786 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4083C211yesmaximal, characteristic
2C2×#4038C211yesmaximal, characteristic, complemented
3C2×#4029C211yesmaximal, characteristic, complemented
4C2×D5×C22⋊C4C211yesmaximal, characteristic, complemented
5C5×#394C211yesmaximal, characteristic, complemented
6#65510C211yesmaximal, characteristic, complemented
7#65283C211yesmaximal, characteristic, complemented
8#65250C211yesmaximal, characteristic, complemented
9#65246C211yesmaximal, characteristic, complemented
10#65245C211yesmaximal, characteristic, complemented
11#65244C211yesmaximal, characteristic, complemented
12#65240C211yesmaximal, characteristic, complemented
13#65237C211yesmaximal, characteristic, complemented
14#64959C211yesmaximal, characteristic, complemented
15#64957C211yesmaximal, characteristic, complemented
16C24×D5C2211yescharacteristic
17C22×C5×D4C2211yescharacteristic
18C22×C22⋊D5C2211yescharacteristic
19C22×C22⋊D5C2211yescharacteristic
20C22×C4×D5C2211yescharacteristic
21C22×C4×D5C2211yescharacteristic
22C2×C5×C4⋊C4C2211yescharacteristic, complemented
23C2×C5×C22⋊C4C2211yescharacteristic
24C2×C5×C22⋊C4C2211yescharacteristic
25C2×C5×C22⋊C4C2211yescharacteristic, complemented
26C2×C5×C22⋊C4C2211yescharacteristic, complemented
27C2×C22⋊Dic5C2211yescharacteristic
28C2×C22⋊Dic5C2211yescharacteristic
29C2×C22⋊Dic5C2211yescharacteristic
30C2×C22⋊Dic5C2211yescharacteristic, complemented
31C2×C22⋊Dic5C2211yescharacteristic, complemented
32#4083—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×C4⋊Dic5C2211yescharacteristic, complemented
44C2×#683C2211yescharacteristic, complemented
45C2×#683C2211yescharacteristic, complemented
46C2×C4×Dic5C2211yescharacteristic, complemented
47C2×C4×Dic5C2211yescharacteristic, complemented
48#4038—24no—
49#4029—24no—
50D5×C22⋊C4—24no—
51C5×#95C2211yescharacteristic, complemented
52#3963C2211yescharacteristic, complemented
53#3963C2211yescharacteristic, complemented
54#3963C2211yescharacteristic, complemented
55#2411—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×C5C2311yescharacteristic
71C22×C4×C5C2311yescharacteristic
72C22×C4×C5C2311yescharacteristic
73C22×C4×C5C2311yescharacteristic
74C22×C4×C5C2311yescharacteristic
75C2×C22⋊D5—22no—
76C2×C22⋊D5—22no—
77C2×C22⋊D5—22no—
78C2×C22⋊D5—22no—
79C2×C22⋊D5—22no—
80C2×C22⋊D5—41no—
81C22×Dic5C2311yescharacteristic
82C22×Dic5C2311yescharacteristic
83C22×Dic5C2311yescharacteristic
84C22×Dic5C2311yescharacteristic
85C22×Dic5C2311yescharacteristic
86C22×Dic5C2311yescharacteristic
87C2×C4×D5—22no—
88C2×C4×D5—22no—
89C2×C4×D5—22no—
90C2×C4×D5—41no—
91C2×C4×D5—41no—
92C2×C4×D5D414yescomplemented
93C5×C4⋊C4—22no—
94C5×C22⋊C4—22no—
95C5×C22⋊C4—22no—
96C5×C22⋊C4—22no—
97C5×C22⋊C4—41no—
98C22⋊Dic5—22no—
99C22⋊Dic5—22no—
100C22⋊Dic5—22no—
101C22⋊Dic5—41no—
102C22⋊Dic5—41no—
103#685—22no—
104#685—22no—
105#685—22no—
106#685—22no—
107#685—22no—
108#685—22no—
109#685—41no—
110#685D414yes—
111C4⋊Dic5—22no—
112C4⋊Dic5—22no—
113C4⋊Dic5—41no—
114#683—22no—
115#683—41no—
116C4×Dic5—22no—
117C4×Dic5—22no—
118C2×#123—51no—
119C2×#120—51no—
120C2×#117—51no—
121C22×C22⋊C4—51no—
122#396—51no—
123#396—51no—
124#394—51no—
125#394D511yescharacteristic, complemented
126#390—51no—
127#390—51no—
128#388—51no—
129#388—51no—
130#386—51no—
131#385—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×C5—21no—
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—44no—
160C22×D5—44no—
161C22×D5—44no—
162C22×D5—44no—
163C22×D5—44no—
164C22×D5—44no—
165C22×D5C2×D412yes—
166C22×D5C2×D412yes—
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×C5C2×D412yes—
184C2×C4×C5C2×D412yes—
185C22⋊D5—44no—
186C22⋊D5—44no—
187C2×Dic5—21no—
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×Dic5C2×D412yes—
204C2×Dic5C2×D412yes—
205C4×D5—28no—
206C4×D5—44no—
207C25—51no—
208C22×D4—51no—
209C22×D4—51no—
210C22×D4C2×D511yescharacteristic
211C23×C4—51no—
212C23×C4—51no—
213#123—104no—
214#120—104no—
215#117—104no—
216C2×C4⋊C4—51no—
217C2×C4⋊C4—51no—
218C2×C4⋊C4—51no—
219C2×C4⋊C4—51no—
220C2×C4⋊C4—51no—
221C2×C4⋊C4C2×D511yescharacteristic, complemented
222C2×C22⋊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⋊C4C2×D511yescharacteristic
237C2×C22⋊C4C2×D511yescharacteristic
238C2×C22⋊C4C2×D511yescharacteristic, complemented
239C2×C22⋊C4C2×D511yescharacteristic, complemented
240C2×C42—51no—
241C2×C42—51no—
242#95—51no—
243#95—51no—
244#95—51no—
245#95C2×D511yescharacteristic, complemented
246C22×C5—22no—
247C22×C5—22no—
248C22×C5—22no—
249C22×C5—41no—
250C22×C5—41no—
251C22×C5—41no—
252C22×C5—41no—
253C22×C5—41no—
254C22×C5—41no—
255C22×C5—41no—
256C22×C5—41no—
257C22×C5—41no—
258C22×C5—41no—
259C22×C5—41no—
260C22×C5C22×D411yescharacteristic
261C22×C5C22×D411yescharacteristic
262C22×C5C2×#4111yescharacteristic
263C22×C5C2×#4111yescharacteristic
264C22×C5C2×#4111yescharacteristic
265C22×C5H(5,𝔽2)11yescharacteristic
266C22×C5H(5,𝔽2)11yescharacteristic
267C2×D5—22no—
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—416no—
277C2×D5—41no—
278C2×D5—41no—
279C2×D5—41no—
280C2×D5—41no—
281C2×D5#12114yes—
282C4×C5—22no—
283C4×C5—22no—
284C4×C5—41no—
285C4×C5—41no—
286C4×C5—41no—
287Dic5—22no—
288Dic5—22no—
289Dic5—41no—
290Dic5—41no—
291Dic5—41no—
292Dic5—41no—
293C24—51no—
294C24—51no—
295C24—102no—
296C24—102no—
297C24—102no—
298C24—102no—
299C24—102no—
300C24—201no—
301C24—201no—
302C24C22×D511yescharacteristic
303C24C22×D511yescharacteristic
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—41no—
307C2×D4—102no—
308C2×D4—102no—
309C2×D4—102no—
310C2×D4—102no—
311C2×D4—102no—
312C2×D4—201no—
313C22×C4—51no—
314C22×C4—51no—
315C22×C4—54no—
316C22×C4—51no—
317C22×C4—51no—
318C22×C4—51no—
319C22×C4—51no—
320C22×C4—102no—
321C22×C4—102no—
322C22×C4—102no—
323C22×C4—201no—
324C22×C4—201no—
325C22×C4C22×D511yescharacteristic
326C22×C4C22×D511yescharacteristic
327C22×C4C22×D511yescharacteristic
328C22×C4C22×D511yescharacteristic
329C22×C4C22×D511yescharacteristic
330C4⋊C4—22no—
331C4⋊C4—102no—
332C4⋊C4—102no—
333C4⋊C4—102no—
334C4⋊C4—201no—
335C4⋊C4—201no—
336C22⋊C4—22no—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C22⋊C4—41no—
340C22⋊C4—54no—
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—201no—
351C22⋊C4—201no—
352C22⋊C4—201no—
353C42—102no—
354C42—102no—
355C2×C5—42no—
356C2×C5—81no—
357C2×C5C2×#12111yescharacteristic
358C2×C5#52511yescharacteristic
359C2×C5#53511yescharacteristic
360C2×C5#54611yescharacteristic
361C2×C5#54611yescharacteristic
362C2×C5#55111yescharacteristic
363C2×C5#55111yescharacteristic
364D5—24no—
365D5—42no—
366C23—21no—
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—
400C23#71112yes—
401C23C23×D511yescharacteristic
402D4—44no—
403D4—204no—
404D4—204no—
405C2×C4—21no—
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—52no—
419C2×C4—52no—
420C2×C4—108no—
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—204no—
438C2×C4D4×D512yes—
439C2×C4D4×D512yes—
440C5#241111yescharacteristic, complemented, Sylow
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—54no—
456C22—102no—
457C22—102no—
458C22—102no—
459C22—102no—
460C22—102no—
461C22—102no—
462C22—102no—
463C22—102no—
464C22—102no—
465C22—2016no—
466C22—201no—
467C22—201no—
468C22—201no—
469C22—201no—
470C22C2×D4×D511yescharacteristic
471C22C2×D4×D511yescharacteristic
472C22C2×#71111yescharacteristic
473C22#414411yescharacteristic
474C22D5×#4111yescharacteristic
475C22D5×#4111yescharacteristic
476C22#414911yescharacteristic
477C4—22no—
478C4—22no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—102no—
483C4—102no—
484C4—201no—
485C4—201no—
486C4—201no—
487C4—201no—
488C2—42no—
489C2—81no—
490C2—104no—
491C2—202no—
492C2#6592511yescharacteristic
493C2D5×#12111yescharacteristic
494C2#6594011yescharacteristic
495C2#6594111yescharacteristic
496C2#6599211yescharacteristic
497C2#6599311yescharacteristic
498C2#6601011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#394C2
prime radicalC2×#4083C2
hypercenter#394D5
hypocenterC5#2411
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#4038C2semidirect
3. C2×#4029C2semidirect
4. C2×D5×C22⋊C4C2semidirect
5. C5×#394C2semidirect
6. #65510C2semidirect
7. #65283C2semidirect
8. #65250C2semidirect
9. #65246C2semidirect
10. #65245C2semidirect
11. #65244C2semidirect
12. #65240C2semidirect
13. #65237C2semidirect
14. #64959C2semidirect
15. #64957C2semidirect
22. C2×C5×C4⋊C4C22semidirect
25. C2×C5×C22⋊C4C22semidirect
26. C2×C5×C22⋊C4C22semidirect
30. C2×C22⋊Dic5C22semidirect
31. C2×C22⋊Dic5C22semidirect
41. C2×C4⋊Dic5C22semidirect
42. C2×C4⋊Dic5C22semidirect
43. C2×C4⋊Dic5C22semidirect
44. C2×#683C22semidirect
45. C2×#683C22semidirect
46. C2×C4×Dic5C22semidirect
47. C2×C4×Dic5C22semidirect
51. C5×#95C22semidirect
52. #3963C22semidirect
53. #3963C22semidirect
54. #3963C22semidirect
92. C2×C4×D5D4semidirect
125. #394D5semidirect
221. C2×C4⋊C4C2×D5semidirect
238. C2×C22⋊C4C2×D5semidirect
239. C2×C22⋊C4C2×D5semidirect
245. #95C2×D5semidirect
440. C5#2411semidirect

Maximal subgroups

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

subgroupindex
C2×#40832
C2×#40382
C2×#40292
C2×D5×C22⋊C42
C5×#3942
#655102
#652832
#652502
#652462
#652452
#652442
#652402
#652372
#649592
#649572
#24115