Finite Group Databasebeta

← #10614317

Subgroups — #10614317

Order 640.

922 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#120C211yesmaximal, characteristic, complemented
2C2×#4099C211yesmaximal, characteristic, complemented
3C22×C22⋊Dic5C211yesmaximal, characteristic, complemented
4C2×#4085C211yesmaximal, characteristic, complemented
5C2×#4084C211yesmaximal, characteristic, complemented
6C2×#4027C211yesmaximal, characteristic, complemented
7#65512C211yesmaximal, characteristic, complemented
8#65511C211yesmaximal, characteristic, complemented
9#65510C211yesmaximal, characteristic, complemented
10#65509C211yesmaximal, characteristic, complemented
11#65250C211yesmaximal, characteristic, complemented
12#65245C211yesmaximal, characteristic, complemented
13#65239C211yesmaximal, characteristic, complemented
14#65237C211yesmaximal, characteristic, complemented
15#65233C211yesmaximal, characteristic, complemented
16C25×C5C2211yescharacteristic
17C22×C5×D4C2211yescharacteristic
18C22×C5×D4C2211yescharacteristic
19C22×C5×D4C2211yescharacteristic
20C22×C22⋊D5C2211yescharacteristic
21C22×C22⋊D5C2211yescharacteristic
22C22×C22⋊D5C2211yescharacteristic
23C22×C22⋊D5C2211yescharacteristic, complemented
24C23×Dic5C2211yescharacteristic
25C23×Dic5C2211yescharacteristic
26C22×C4×D5C2211yescharacteristic, complemented
27C5×#120—24no—
28C2×C5×C22⋊C4C2211yescharacteristic
29C2×C5×C22⋊C4C2211yescharacteristic
30C2×C5×C22⋊C4C2211yescharacteristic
31#4099—24no—
32C2×C22⋊Dic5—24no—
33C2×C22⋊Dic5C2211yescharacteristic
34C2×C22⋊Dic5C2211yescharacteristic
35C2×C22⋊Dic5C2211yescharacteristic
36C2×C22⋊Dic5C2211yescharacteristic
37C2×C22⋊Dic5C2211yescharacteristic
38C2×C22⋊Dic5C2211yescharacteristic
39C2×C22⋊Dic5C2211yescharacteristic
40C2×C22⋊Dic5C2211yescharacteristic, complemented
41C2×C22⋊Dic5C2211yescharacteristic, complemented
42C2×C22⋊Dic5C2211yescharacteristic, complemented
43#4085—24no—
44#4084—24no—
45C2×#685C2211yescharacteristic, complemented
46C2×#685C2211yescharacteristic, complemented
47C2×C4⋊Dic5C2211yescharacteristic, complemented
48C2×C4⋊Dic5C2211yescharacteristic, complemented
49C2×#683C2211yescharacteristic, complemented
50C2×#683C2211yescharacteristic, complemented
51C2×C4×Dic5C2211yescharacteristic, complemented
52#4027—24no—
53#3963C2211yescharacteristic, complemented
54#3963C2211yescharacteristic, complemented
55#3963C2211yescharacteristic, complemented
56#3963C2211yescharacteristic, complemented
57#2424—51nomaximal, Sylow
58C24×C5—22no—
59C24×C5—22no—
60C24×C5—22no—
61C24×C5—22no—
62C24×C5—22no—
63C24×C5—41no—
64C24×C5—41no—
65C24×C5C2311yescharacteristic
66C24×C5C2311yescharacteristic
67C24×C5C2311yescharacteristic
68C24×C5C2311yescharacteristic
69C23×D5C2311yescharacteristic
70C2×C5×D4—22no—
71C2×C5×D4—22no—
72C2×C5×D4—22no—
73C2×C5×D4—22no—
74C2×C5×D4—22no—
75C2×C5×D4—22no—
76C2×C5×D4—22no—
77C2×C5×D4—41no—
78C2×C5×D4—41no—
79C22×C4×C5C2311yescharacteristic
80C22×C4×C5C2311yescharacteristic
81C22×C4×C5C2311yescharacteristic
82C2×C22⋊D5—22no—
83C2×C22⋊D5—22no—
84C2×C22⋊D5—22no—
85C2×C22⋊D5—22no—
86C2×C22⋊D5—22no—
87C2×C22⋊D5—22no—
88C2×C22⋊D5—22no—
89C2×C22⋊D5—22no—
90C2×C22⋊D5—41no—
91C2×C22⋊D5—41no—
92C2×C22⋊D5—41no—
93C2×C22⋊D5D414yescomplemented
94C22×Dic5—22no—
95C22×Dic5—22no—
96C22×Dic5—22no—
97C22×Dic5—22no—
98C22×Dic5—41no—
99C22×Dic5—41no—
100C22×Dic5C2311yescharacteristic
101C22×Dic5C2311yescharacteristic
102C22×Dic5C2311yescharacteristic
103C22×Dic5C2311yescharacteristic
104C22×Dic5C2311yescharacteristic
105C22×Dic5C2311yescharacteristic
106C22×Dic5C2311yescharacteristic
107C2×C4×D5—22no—
108C2×C4×D5—22no—
109C2×C4×D5—41no—
110C5×C22⋊C4—22no—
111C5×C22⋊C4—22no—
112C5×C22⋊C4D414yescomplemented
113C22⋊Dic5—22no—
114C22⋊Dic5—22no—
115C22⋊Dic5—22no—
116C22⋊Dic5—22no—
117C22⋊Dic5—22no—
118C22⋊Dic5—22no—
119C22⋊Dic5—22no—
120C22⋊Dic5—22no—
121C22⋊Dic5—41no—
122C22⋊Dic5D414yescomplemented
123#685—22no—
124#685—22no—
125C4⋊Dic5—22no—
126C4⋊Dic5—41no—
127#683—22no—
128#683—22no—
129C4×Dic5—22no—
130C2×#121—51no—
131C2×#121—51no—
132C2×#120—51no—
133C2×#120D511yescharacteristic, complemented
134C2×C4×D4—51no—
135C22×C22⋊C4—51no—
136#396—51no—
137#394—51no—
138#394—51no—
139#392—51no—
140#390—51no—
141#388—51no—
142#386—51no—
143#386—51no—
144#385—51no—
145C23×C5—21no—
146C23×C5—21no—
147C23×C5—21no—
148C23×C5—21no—
149C23×C5—28no—
150C23×C5—21no—
151C23×C5—21no—
152C23×C5—21no—
153C23×C5—21no—
154C23×C5—21no—
155C23×C5—21no—
156C23×C5—21no—
157C23×C5—21no—
158C23×C5—21no—
159C23×C5—21no—
160C23×C5—21no—
161C23×C5—21no—
162C23×C5—21no—
163C23×C5—21no—
164C23×C5—21no—
165C23×C5—21no—
166C23×C5—21no—
167C23×C5—21no—
168C23×C5—21no—
169C23×C5—44no—
170C23×C5—44no—
171C23×C5—44no—
172C23×C5—44no—
173C23×C5—44no—
174C23×C5—44no—
175C23×C5C2×D412yes—
176C23×C5C2×D412yes—
177C23×C5C2×D412yes—
178C23×C5#4112yes—
179C23×C5#4112yes—
180C23×C5C2411yescharacteristic
181C22×D5—21no—
182C22×D5—21no—
183C22×D5—21no—
184C22×D5—21no—
185C22×D5—21no—
186C22×D5—21no—
187C22×D5C2×D412yes—
188C5×D4—44no—
189C5×D4—44no—
190C5×D4—44no—
191C2×C4×C5—21no—
192C2×C4×C5—21no—
193C2×C4×C5—21no—
194C2×C4×C5—21no—
195C2×C4×C5—21no—
196C2×C4×C5—21no—
197C2×C4×C5—21no—
198C2×C4×C5C2×D412yes—
199C2×C4×C5C2×D412yes—
200C22⋊D5—28no—
201C22⋊D5—44no—
202C22⋊D5—44no—
203C22⋊D5—44no—
204C2×Dic5—21no—
205C2×Dic5—21no—
206C2×Dic5—21no—
207C2×Dic5—21no—
208C2×Dic5—21no—
209C2×Dic5—21no—
210C2×Dic5—21no—
211C2×Dic5—21no—
212C2×Dic5—21no—
213C2×Dic5—21no—
214C2×Dic5—21no—
215C2×Dic5—21no—
216C2×Dic5—21no—
217C2×Dic5—21no—
218C2×Dic5—21no—
219C2×Dic5—21no—
220C2×Dic5—21no—
221C2×Dic5—21no—
222C2×Dic5—44no—
223C2×Dic5—44no—
224C2×Dic5C2×D412yes—
225C2×Dic5C2×D412yes—
226C2×Dic5C2×D412yes—
227C4×D5—44no—
228C25C2×D511yescharacteristic
229C22×D4—51no—
230C22×D4—51no—
231C22×D4—51no—
232C22×D4—51no—
233C22×D4C2×D511yescharacteristic
234C22×D4C2×D511yescharacteristic
235C22×D4C2×D511yescharacteristic
236C23×C4—51no—
237C23×C4—51no—
238C23×C4—51no—
239#121—104no—
240#121—104no—
241#120—24no—
242#120—104no—
243C4×D4—104no—
244C2×C4⋊C4—51no—
245C2×C4⋊C4—51no—
246C2×C4⋊C4—51no—
247C2×C4⋊C4—51no—
248C2×C22⋊C4—51no—
249C2×C22⋊C4—51no—
250C2×C22⋊C4—51no—
251C2×C22⋊C4—51no—
252C2×C22⋊C4—51no—
253C2×C22⋊C4—51no—
254C2×C22⋊C4—51no—
255C2×C22⋊C4—51no—
256C2×C22⋊C4—51no—
257C2×C22⋊C4—51no—
258C2×C22⋊C4—51no—
259C2×C22⋊C4—51no—
260C2×C22⋊C4—104no—
261C2×C22⋊C4C2×D511yescharacteristic
262C2×C22⋊C4C2×D511yescharacteristic
263C2×C22⋊C4C2×D511yescharacteristic
264C2×C42—51no—
265#95—51no—
266#95—51no—
267#95—51no—
268#95—51no—
269C22×C5—22no—
270C22×C5—22no—
271C22×C5—22no—
272C22×C5—22no—
273C22×C5—22no—
274C22×C5—22no—
275C22×C5—22no—
276C22×C5—22no—
277C22×C5—22no—
278C22×C5—22no—
279C22×C5—22no—
280C22×C5—22no—
281C22×C5—416no—
282C22×C5—41no—
283C22×C5—41no—
284C22×C5—41no—
285C22×C5—41no—
286C22×C5—41no—
287C22×C5—41no—
288C22×C5—41no—
289C22×C5—41no—
290C22×C5—41no—
291C22×C5—41no—
292C22×C5—41no—
293C22×C5—41no—
294C22×C5—41no—
295C22×C5—41no—
296C22×C5—41no—
297C22×C5#12114yescomplemented
298C22×C5C22×D411yescharacteristic
299C22×C5C22×D411yescharacteristic
300C22×C5C22×D411yescharacteristic
301C22×C5C2×#4111yescharacteristic
302C22×C5C2×#4111yescharacteristic
303C22×C5H(5,𝔽2)11yescharacteristic
304C22×C5H(5,𝔽2)11yescharacteristic
305C2×D5—22no—
306C2×D5—22no—
307C2×D5—22no—
308C2×D5—41no—
309C2×D5—41no—
310C2×D5—41no—
311C2×D5—41no—
312C4×C5—22no—
313C4×C5—22no—
314C4×C5—41no—
315Dic5—22no—
316Dic5—22no—
317Dic5—22no—
318Dic5—41no—
319Dic5—41no—
320Dic5—41no—
321Dic5—41no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—41no—
328C24—41no—
329C24—51no—
330C24C22×D511yescharacteristic
331C24C22×D511yescharacteristic
332C24C22×D511yescharacteristic
333C24C22×D511yescharacteristic
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—54no—
344C2×D4—102no—
345C2×D4—102no—
346C2×D4—102no—
347C2×D4—102no—
348C2×D4—102no—
349C2×D4—102no—
350C2×D4—102no—
351C2×D4—102no—
352C2×D4—201no—
353C2×D4—201no—
354C2×D4—201no—
355C22×C4—51no—
356C22×C4—51no—
357C22×C4—51no—
358C22×C4—51no—
359C22×C4—51no—
360C22×C4—51no—
361C22×C4—51no—
362C22×C4—102no—
363C22×C4—102no—
364C22×C4—102no—
365C22×C4—102no—
366C22×C4—102no—
367C22×C4—102no—
368C22×C4—201no—
369C22×C4—201no—
370C22×C4—201no—
371C22×C4C22×D511yescharacteristic
372C22×C4C22×D511yescharacteristic
373C22×C4C22×D511yescharacteristic
374C4⋊C4—102no—
375C4⋊C4—102no—
376C4⋊C4—102no—
377C4⋊C4—201no—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—54no—
381C22⋊C4—102no—
382C22⋊C4—102no—
383C22⋊C4—102no—
384C22⋊C4—102no—
385C22⋊C4—102no—
386C22⋊C4—102no—
387C22⋊C4—102no—
388C22⋊C4—102no—
389C22⋊C4—102no—
390C22⋊C4—102no—
391C22⋊C4—201no—
392C22⋊C4C22⋊D514yescomplemented
393C42—102no—
394C2×C5—24no—
395C2×C5—42no—
396C2×C5—42no—
397C2×C5—81no—
398C2×C5C2×#12111yescharacteristic
399C2×C5#52511yescharacteristic
400C2×C5#53811yescharacteristic
401C2×C5D4211yescharacteristic
402C2×C5D4211yescharacteristic
403C2×C5#54611yescharacteristic
404C2×C5#54611yescharacteristic
405D5—42no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—28no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—52no—
437C23—101no—
438C23—101no—
439C23—101no—
440C23—101no—
441C23—101no—
442C23—101no—
443C23D4×D512yes—
444C23D4×D512yes—
445C23#71112yes—
446C23#71112yes—
447C23C2×C22⋊D512yes—
448C23C23×D511yescharacteristic
449D4—44no—
450D4—44no—
451D4—44no—
452D4—108no—
453D4—204no—
454D4—204no—
455D4—204no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—52no—
464C2×C4—52no—
465C2×C4—52no—
466C2×C4—101no—
467C2×C4—101no—
468C2×C4—101no—
469C2×C4—101no—
470C2×C4—101no—
471C2×C4—101no—
472C2×C4—101no—
473C2×C4—101no—
474C2×C4—101no—
475C2×C4—101no—
476C2×C4—101no—
477C2×C4—101no—
478C2×C4—101no—
479C2×C4—101no—
480C2×C4—101no—
481C2×C4—101no—
482C2×C4—101no—
483C2×C4—101no—
484C2×C4—204no—
485C2×C4—204no—
486C2×C4—204no—
487C2×C4C2×C22⋊D512yes—
488C2×C4C2×C22⋊D512yes—
489C5#242411yescharacteristic, complemented, Sylow
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—416no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—102no—
519C22—102no—
520C22—102no—
521C22—201no—
522C22—201no—
523C22—201no—
524C22—201no—
525C22#408414yescomplemented
526C22C2×D4×D511yescharacteristic
527C22C2×D4×D511yescharacteristic
528C22C2×#71111yescharacteristic
529C22C2×#71111yescharacteristic
530C22#414411yescharacteristic
531C22#414411yescharacteristic
532C22C22×C22⋊D511yescharacteristic
533C4—22no—
534C4—22no—
535C4—41no—
536C4—102no—
537C4—102no—
538C4—102no—
539C4—201no—
540C4—201no—
541C4—201no—
542C4—201no—
543C2—24no—
544C2—42no—
545C2—42no—
546C2—81no—
547C2—202no—
548C2#6592011yescharacteristic
549C2#6592411yescharacteristic
550C2#6592511yescharacteristic
551C2#6592711yescharacteristic
552C2C2×#408411yescharacteristic
553C2D4×C22⋊D511yescharacteristic
554C2#6614011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×#120C2
hypercenterC2×#120D5
hypocenterC5#2424
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×C5×#120C2semidirect
2. C2×#4099C2semidirect
3. C22×C22⋊Dic5C2semidirect
4. C2×#4085C2semidirect
5. C2×#4084C2semidirect
6. C2×#4027C2semidirect
7. #65512C2semidirect
8. #65511C2semidirect
9. #65510C2semidirect
10. #65509C2semidirect
11. #65250C2semidirect
12. #65245C2semidirect
13. #65239C2semidirect
14. #65237C2semidirect
15. #65233C2semidirect
23. C22×C22⋊D5C22semidirect
26. C22×C4×D5C22semidirect
40. C2×C22⋊Dic5C22semidirect
41. C2×C22⋊Dic5C22semidirect
42. C2×C22⋊Dic5C22semidirect
45. C2×#685C22semidirect
46. C2×#685C22semidirect
47. C2×C4⋊Dic5C22semidirect
48. C2×C4⋊Dic5C22semidirect
49. C2×#683C22semidirect
50. C2×#683C22semidirect
51. C2×C4×Dic5C22semidirect
53. #3963C22semidirect
54. #3963C22semidirect
55. #3963C22semidirect
56. #3963C22semidirect
93. C2×C22⋊D5D4semidirect
112. C5×C22⋊C4D4semidirect
122. C22⋊Dic5D4semidirect
133. C2×#120D5semidirect
297. C22×C5#121semidirect
392. C22⋊C4C22⋊D5semidirect
489. C5#2424semidirect
525. C22#4084semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1202
C2×#40992
C22×C22⋊Dic52
C2×#40852
C2×#40842
C2×#40272
#655122
#655112
#655102
#655092
#652502
#652452
#652392
#652372
#652332
#24245