Finite Group Databasebeta

← #10608061

Subgroups — #10608061

Order 640.

786 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×#4083C211yesmaximal, characteristic
2C2×#4042C211yesmaximal, characteristic, complemented
3C2×#4031C211yesmaximal, characteristic, complemented
4C2×D5×C22⋊C4C211yesmaximal, characteristic, complemented
5C5×#390C211yesmaximal, characteristic, complemented
6#65511C211yesmaximal, characteristic, complemented
7#65282C211yesmaximal, characteristic, complemented
8#65248C211yesmaximal, characteristic, complemented
9#65245C211yesmaximal, characteristic, complemented
10#65245C211yesmaximal, characteristic, complemented
11#65244C211yesmaximal, characteristic, complemented
12#65236C211yesmaximal, characteristic, complemented
13#65233C211yesmaximal, characteristic, complemented
14#65229C211yesmaximal, characteristic, complemented
15C4×#685C211yesmaximal, characteristic, complemented
16C24×D5C2211yescharacteristic
17C22×C5×D4C2211yescharacteristic
18C22×C22⋊D5C2211yescharacteristic
19C22×C22⋊D5C2211yescharacteristic
20C22×C4×D5C2211yescharacteristic
21C22×C4×D5C2211yescharacteristic
22C22×Dic10C2211yescharacteristic, complemented
23C2×C5×C4⋊C4C2211yescharacteristic, complemented
24C2×C5×C22⋊C4C2211yescharacteristic
25C2×C5×C22⋊C4C2211yescharacteristic
26C2×C5×C22⋊C4C2211yescharacteristic, complemented
27C2×C5×C22⋊C4C2211yescharacteristic, complemented
28C2×C42×C5C2211yescharacteristic, complemented
29C2×C22⋊Dic5C2211yescharacteristic
30C2×C22⋊Dic5C2211yescharacteristic
31C2×C22⋊Dic5C2211yescharacteristic
32C2×C22⋊Dic5C2211yescharacteristic, complemented
33C2×C22⋊Dic5C2211yescharacteristic, complemented
34#4083—24no—
35C2×#685C2211yescharacteristic
36C2×#685C2211yescharacteristic
37C2×#685C2211yescharacteristic
38C2×#685C2211yescharacteristic
39C2×#685C2211yescharacteristic
40C2×#685C2211yescharacteristic
41C2×#685C2211yescharacteristic
42C2×#685C2211yescharacteristic
43C2×#683C2211yescharacteristic, complemented
44C2×#683C2211yescharacteristic, complemented
45C2×C4×Dic5C2211yescharacteristic, complemented
46#4042—24no—
47#4031—24no—
48D5×C22⋊C4—24no—
49#3963C2211yescharacteristic, complemented
50#3963C2211yescharacteristic, complemented
51#3963C2211yescharacteristic, complemented
52#3963C2211yescharacteristic, complemented
53#3963C2211yescharacteristic, complemented
54#3963C2211yescharacteristic, complemented
55#2472—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—22no—
91C2×C4×D5—41no—
92C2×C4×D5—41no—
93C2×Dic10—22no—
94C2×Dic10—22no—
95C2×Dic10—41no—
96C5×C4⋊C4—22no—
97C5×C22⋊C4—22no—
98C5×C22⋊C4—22no—
99C5×C22⋊C4—41no—
100C5×C22⋊C4—41no—
101C42×C5—41no—
102C22⋊Dic5—22no—
103C22⋊Dic5—22no—
104C22⋊Dic5—22no—
105C22⋊Dic5—22no—
106C22⋊Dic5—41no—
107#685—22no—
108#685—22no—
109#685—22no—
110#685—22no—
111#685—22no—
112#685—41no—
113#685D414yes—
114#685D414yescomplemented
115#683—22no—
116#683—22no—
117C4×Dic5—22no—
118C2×#124—51no—
119C2×C22⋊Q8—51no—
120C2×#120—51no—
121C22×C22⋊C4—51no—
122#397—51no—
123#394—51no—
124#394—51no—
125#393—51no—
126#390D511yescharacteristic, complemented
127#388—51no—
128#388—51no—
129#386—51no—
130#386—51no—
131#379—51no—
132C4×C22⋊C4—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—44no—
206C4×D5—44no—
207Dic10—44no—
208C25—51no—
209C22×Q8—51no—
210C22×D4—51no—
211C22×D4—51no—
212C22×D4C2×D511yescharacteristic
213C23×C4—51no—
214C23×C4—51no—
215#124—104no—
216C22⋊Q8—104no—
217#120—104no—
218C2×C4⋊C4—51no—
219C2×C4⋊C4—51no—
220C2×C4⋊C4C2×D511yescharacteristic, complemented
221C2×C22⋊C4—51no—
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—104no—
235C2×C22⋊C4C2×D511yescharacteristic
236C2×C22⋊C4C2×D511yescharacteristic
237C2×C22⋊C4C2×D511yescharacteristic, complemented
238C2×C22⋊C4C2×D511yescharacteristic, complemented
239C2×C42—51no—
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—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×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#12414yescomplemented
283C4×C5—22no—
284C4×C5—22no—
285C4×C5—41no—
286C4×C5—41no—
287C4×C5—41no—
288Dic5—22no—
289Dic5—22no—
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×Q8—102no—
306C2×Q8—102no—
307C2×Q8—201no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—41no—
311C2×D4—102no—
312C2×D4—102no—
313C2×D4—102no—
314C2×D4—102no—
315C2×D4—102no—
316C2×D4—201no—
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—102no—
336C4⋊C4—102no—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—54no—
342C22⋊C4—54no—
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—
354C42—41no—
355C42—102no—
356C2×C5—42no—
357C2×C5—81no—
358C2×C5#52511yescharacteristic
359C2×C5C2×#12411yescharacteristic
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—21no—
380C23—52no—
381C23—52no—
382C23—52no—
383C23—52no—
384C23—108no—
385C23—101no—
386C23—101no—
387C23—101no—
388C23—101no—
389C23—101no—
390C23—101no—
391C23—101no—
392C23—101no—
393C23—101no—
394C23—101no—
395C23—204no—
396C23—204no—
397C23—204no—
398C23—204no—
399C23—204no—
400C23—204no—
401C23#70912yes—
402C23C23×D511yescharacteristic
403Q8—204no—
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—52no—
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—204no—
439C2×C4—204no—
440C2×C4D4×D512yes—
441C2×C4D4×D512yes—
442C5#247211yescharacteristic, complemented, Sylow
443C22—22no—
444C22—22no—
445C22—22no—
446C22—41no—
447C22—41no—
448C22—41no—
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×#70911yescharacteristic
473C22C2×D4×D511yescharacteristic
474C22C2×D4×D511yescharacteristic
475C22#414411yescharacteristic
476C22#414411yescharacteristic
477C22D5×#4111yescharacteristic
478C22D5×#4111yescharacteristic
479C4—22no—
480C4—22no—
481C4—41no—
482C4—41no—
483C4—41no—
484C4—102no—
485C4—102no—
486C4—201no—
487C4—201no—
488C4—201no—
489C4—201no—
490C2—42no—
491C2—81no—
492C2—104no—
493C2—202no—
494C2#6582411yescharacteristic
495C2#6588111yescharacteristic
496C2#6589011yescharacteristic
497C2#6594411yescharacteristic
498C2#6594711yescharacteristic
499C2D5×#12411yescharacteristic
500C2#6605111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#390C2
prime radicalC2×#4083C2
hypercenter#390D5
hypocenterC5#2472
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#4042C2semidirect
3. C2×#4031C2semidirect
4. C2×D5×C22⋊C4C2semidirect
5. C5×#390C2semidirect
6. #65511C2semidirect
7. #65282C2semidirect
8. #65248C2semidirect
9. #65245C2semidirect
10. #65245C2semidirect
11. #65244C2semidirect
12. #65236C2semidirect
13. #65233C2semidirect
14. #65229C2semidirect
15. C4×#685C2semidirect
22. C22×Dic10C22semidirect
23. C2×C5×C4⋊C4C22semidirect
26. C2×C5×C22⋊C4C22semidirect
27. C2×C5×C22⋊C4C22semidirect
28. C2×C42×C5C22semidirect
32. C2×C22⋊Dic5C22semidirect
33. C2×C22⋊Dic5C22semidirect
43. C2×#683C22semidirect
44. C2×#683C22semidirect
45. C2×C4×Dic5C22semidirect
49. #3963C22semidirect
50. #3963C22semidirect
51. #3963C22semidirect
52. #3963C22semidirect
53. #3963C22semidirect
54. #3963C22semidirect
114. #685D4semidirect
126. #390D5semidirect
220. C2×C4⋊C4C2×D5semidirect
237. C2×C22⋊C4C2×D5semidirect
238. C2×C22⋊C4C2×D5semidirect
240. C2×C42C2×D5semidirect
282. C2×D5#124semidirect
442. C5#2472semidirect

Maximal subgroups

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

subgroupindex
C2×#40832
C2×#40422
C2×#40312
C2×D5×C22⋊C42
C5×#3902
#655112
#652822
#652482
#652452
#652442
#652362
#652332
#652292
C4×#6852
#24725