Finite Group Databasebeta

← #10605325

Subgroups — #10605325

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
1C22×#685C211yesmaximal, characteristic, complemented
2C2×#4028C211yesmaximal, characteristic
3C2×#4024C211yesmaximal, characteristic, complemented
4C2×#4021C211yesmaximal, characteristic, complemented
5C4×C5×C22⋊C4C211yesmaximal, characteristic, complemented
6#65502C211yesmaximal, characteristic, complemented
7#65247C211yesmaximal, characteristic, complemented
8#65244C211yesmaximal, characteristic, complemented
9#65229C211yesmaximal, characteristic, complemented
10#65229C211yesmaximal, characteristic, complemented
11#65228C211yesmaximal, characteristic, complemented
12#64963C211yesmaximal, characteristic, complemented
13#64962C211yesmaximal, characteristic, complemented
14#64961C211yesmaximal, characteristic, complemented
15#64961C211yesmaximal, characteristic, complemented
16C24×D5C2211yescharacteristic
17C23×C4×C5C2211yescharacteristic
18C22×C22⋊D5C2211yescharacteristic
19C23×Dic5C2211yescharacteristic
20C22×D20C2211yescharacteristic
21C22×D20C2211yescharacteristic
22C22×Dic10C2211yescharacteristic, complemented
23C2×C5×C22⋊C4C2211yescharacteristic
24C2×C5×C22⋊C4C2211yescharacteristic
25C2×C42×C5C2211yescharacteristic, complemented
26C2×C42×C5C2211yescharacteristic, complemented
27C2×C22⋊Dic5C2211yescharacteristic
28C2×#685—24no—
29C2×#685C2211yescharacteristic
30C2×#685C2211yescharacteristic
31C2×#685C2211yescharacteristic
32C2×#685C2211yescharacteristic
33C2×#685C2211yescharacteristic
34C2×#685C2211yescharacteristic
35C2×#685C2211yescharacteristic
36C2×#685C2211yescharacteristic
37C2×#685C2211yescharacteristic
38C2×#685C2211yescharacteristic
39C2×#685C2211yescharacteristic, complemented
40C2×#685C2211yescharacteristic, complemented
41C2×#685C2211yescharacteristic, complemented
42C2×#685C2211yescharacteristic, complemented
43C2×C4⋊Dic5C2211yescharacteristic, complemented
44C2×#683C2211yescharacteristic, complemented
45C2×#683C2211yescharacteristic, complemented
46#4028—24no—
47#4024—24no—
48#4021—24no—
49C5×#95C2211yescharacteristic, complemented
50C5×#95C2211yescharacteristic, complemented
51#3963C2211yescharacteristic, complemented
52#3963C2211yescharacteristic, complemented
53#3963C2211yescharacteristic, complemented
54#3963C2211yescharacteristic, complemented
55#2472—51nomaximal, Sylow
56C24×C5C2311yescharacteristic
57C23×D5—22no—
58C23×D5—22no—
59C23×D5—22no—
60C23×D5—22no—
61C23×D5—22no—
62C23×D5—41no—
63C23×D5—41no—
64C23×D5C2311yescharacteristic
65C23×D5C2311yescharacteristic
66C23×D5C2311yescharacteristic
67C22×C4×C5—22no—
68C22×C4×C5—22no—
69C22×C4×C5—41no—
70C22×C4×C5C2311yescharacteristic
71C22×C4×C5C2311yescharacteristic
72C22×C4×C5C2311yescharacteristic
73C22×C4×C5C2311yescharacteristic
74C22×C4×C5C2311yescharacteristic
75C22×C4×C5C2311yescharacteristic
76C2×C22⋊D5—22no—
77C2×C22⋊D5—22no—
78C2×C22⋊D5—22no—
79C22×Dic5—22no—
80C22×Dic5—22no—
81C22×Dic5—41no—
82C22×Dic5C2311yescharacteristic
83C22×Dic5C2311yescharacteristic
84C22×Dic5C2311yescharacteristic
85C22×Dic5C2311yescharacteristic
86C22×Dic5C2311yescharacteristic
87C2×D20—22no—
88C2×D20—22no—
89C2×D20—22no—
90C2×D20—22no—
91C2×D20—41no—
92C2×D20—41no—
93C2×Dic10—22no—
94C2×Dic10—22no—
95C2×Dic10—41no—
96C5×C22⋊C4—22no—
97C5×C22⋊C4D414yes—
98C42×C5—22no—
99C42×C5—41no—
100C22⋊Dic5—22no—
101#685—22no—
102#685—22no—
103#685—22no—
104#685—22no—
105#685—22no—
106#685—22no—
107#685—22no—
108#685—22no—
109#685—22no—
110#685—41no—
111#685—41no—
112#685—41no—
113#685—41no—
114#685D414yescomplemented
115C4⋊Dic5—22no—
116#683—22no—
117#683—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#390—51no—
127#388—51no—
128#388—51no—
129#386—51no—
130#386—51no—
131#379—51no—
132C4×C22⋊C4D511yescharacteristic, complemented
133C23×C5—21no—
134C23×C5—21no—
135C23×C5—21no—
136C23×C5—21no—
137C23×C5C2×D412yes—
138C23×C5#4112yes—
139C23×C5#4112yes—
140C23×C5C2411yescharacteristic
141C22×D5—28no—
142C22×D5—21no—
143C22×D5—21no—
144C22×D5—21no—
145C22×D5—21no—
146C22×D5—21no—
147C22×D5—21no—
148C22×D5—21no—
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—
169C2×C4×C5—21no—
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—
186C2×C4×C5C2×D412yes—
187C2×C4×C5C2×D412yes—
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—44no—
204C2×Dic5C2×D412yes—
205D20—44no—
206D20—44no—
207Dic10—44no—
208C25—51no—
209C22×Q8—51no—
210C22×D4—51no—
211C22×D4—51no—
212C22×D4—51no—
213C23×C4—51no—
214C23×C4C2×D511yescharacteristic
215#124—104no—
216C22⋊Q8—104no—
217#120—104no—
218C2×C4⋊C4—51no—
219C2×C4⋊C4—51no—
220C2×C4⋊C4—51no—
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—51no—
235C2×C22⋊C4—51no—
236C2×C22⋊C4—104no—
237C2×C22⋊C4C2×D511yescharacteristic
238C2×C22⋊C4C2×D511yescharacteristic
239C2×C42C2×D511yescharacteristic, complemented
240C2×C42C2×D511yescharacteristic, complemented
241#95—51no—
242#95—51no—
243#95—51no—
244#95—51no—
245#95C2×D511yescharacteristic, complemented
246#95C2×D511yescharacteristic, complemented
247C22×C5—22no—
248C22×C5—22no—
249C22×C5—22no—
250C22×C5—22no—
251C22×C5—22no—
252C22×C5—22no—
253C22×C5#12414yescomplemented
254C22×C5C22×D411yescharacteristic
255C22×C5C22×D411yescharacteristic
256C22×C5C2×#4111yescharacteristic
257C22×C5C2×#4111yescharacteristic
258C22×C5C2×#4111yescharacteristic
259C22×C5H(5,𝔽2)11yescharacteristic
260C22×C5H(5,𝔽2)11yescharacteristic
261C2×D5—22no—
262C2×D5—22no—
263C2×D5—22no—
264C2×D5—22no—
265C2×D5—22no—
266C2×D5—22no—
267C2×D5—416no—
268C2×D5—41no—
269C2×D5—41no—
270C2×D5—41no—
271C2×D5—41no—
272C2×D5—41no—
273C2×D5—41no—
274C2×D5—41no—
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—22no—
286C4×C5—41no—
287C4×C5—41no—
288C4×C5—41no—
289Dic5—22no—
290Dic5—41no—
291Dic5—41no—
292Dic5—41no—
293Dic5—41no—
294C24—51no—
295C24—51no—
296C24—51no—
297C24—102no—
298C24—102no—
299C24—102no—
300C24—102no—
301C24—102no—
302C24—201no—
303C24—201no—
304C24C22×D511yescharacteristic
305C2×Q8—102no—
306C2×Q8—102no—
307C2×Q8—201no—
308C2×D4—102no—
309C2×D4—102no—
310C2×D4—102no—
311C2×D4—102no—
312C2×D4—102no—
313C2×D4—102no—
314C2×D4—102no—
315C2×D4—201no—
316C2×D4—201no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—41no—
320C22×C4—51no—
321C22×C4—51no—
322C22×C4—51no—
323C22×C4—51no—
324C22×C4—51no—
325C22×C4—102no—
326C22×C4—102no—
327C22×C4—201no—
328C22×C4C22×D511yescharacteristic
329C22×C4C22×D511yescharacteristic
330C22×C4C22×D511yescharacteristic
331C22×C4C22×D511yescharacteristic
332C22×C4C22×D511yescharacteristic
333C22×C4C22×D511yescharacteristic
334C4⋊C4—102no—
335C4⋊C4—102no—
336C4⋊C4—102no—
337C22⋊C4—22no—
338C22⋊C4—54no—
339C22⋊C4—102no—
340C22⋊C4—102no—
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—201no—
350C22⋊C4—201no—
351C22⋊C4—201no—
352C22⋊C4—201no—
353C22⋊C4D2014yes—
354C42—22no—
355C42—41no—
356C2×C5—24no—
357C2×C5#52511yescharacteristic
358C2×C5C2×#12411yescharacteristic
359C2×C5#53811yescharacteristic
360C2×C5D4211yescharacteristic
361C2×C5#54611yescharacteristic
362C2×C5#54611yescharacteristic
363C2×C5#55111yescharacteristic
364D5—42no—
365D5—42no—
366D5—81no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—52no—
372C23—52no—
373C23—108no—
374C23—101no—
375C23—101no—
376C23—101no—
377C23—101no—
378C23—101no—
379C23—101no—
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—101no—
393C23—204no—
394C23—204no—
395C23—204no—
396C23—204no—
397C23—204no—
398C23—204no—
399C23C2×D2012yes—
400C23#70912yes—
401C23#70912yes—
402C23C23×D511yescharacteristic
403Q8—204no—
404D4—204no—
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—21no—
422C2×C4—44no—
423C2×C4—52no—
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×C4C2×D2012yes—
440C2×C4C2×D2012yes—
441C2×C4D4×D512yes—
442C5#247211yescharacteristic, complemented, Sylow
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—102no—
450C22—102no—
451C22—102no—
452C22—102no—
453C22—102no—
454C22—102no—
455C22—2016no—
456C22—201no—
457C22—201no—
458C22—201no—
459C22—201no—
460C22—201no—
461C22—201no—
462C22—201no—
463C22—201no—
464C22—201no—
465C22—201no—
466C22—201no—
467C22—201no—
468C22—201no—
469C22—201no—
470C22—201no—
471C22#402114yescomplemented
472C22C22×D2011yescharacteristic
473C22C2×#70911yescharacteristic
474C22C2×#70911yescharacteristic
475C22C2×D4×D511yescharacteristic
476C22#414411yescharacteristic
477C22D5×#4111yescharacteristic
478C22#414911yescharacteristic
479C4—22no—
480C4—22no—
481C4—22no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—102no—
486C4—201no—
487C4—201no—
488C4—201no—
489C4—201no—
490C2—24no—
491C2—202no—
492C2—202no—
493C2—401no—
494C2C2×#402111yescharacteristic
495C2#6582411yescharacteristic
496C2#6586111yescharacteristic
497C2#6588111yescharacteristic
498C2D4×D2011yescharacteristic
499C2#6588811yescharacteristic
500C2#6589411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC4×C5×C22⋊C4C2
prime radicalC2×#4028C2
hypercenterC4×C22⋊C4D5
hypocenterC5#2472
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#685C2semidirect
3. C2×#4024C2semidirect
4. C2×#4021C2semidirect
5. C4×C5×C22⋊C4C2semidirect
6. #65502C2semidirect
7. #65247C2semidirect
8. #65244C2semidirect
9. #65229C2semidirect
10. #65229C2semidirect
11. #65228C2semidirect
12. #64963C2semidirect
13. #64962C2semidirect
14. #64961C2semidirect
15. #64961C2semidirect
22. C22×Dic10C22semidirect
25. C2×C42×C5C22semidirect
26. C2×C42×C5C22semidirect
39. C2×#685C22semidirect
40. C2×#685C22semidirect
41. C2×#685C22semidirect
42. C2×#685C22semidirect
43. C2×C4⋊Dic5C22semidirect
44. C2×#683C22semidirect
45. C2×#683C22semidirect
49. C5×#95C22semidirect
50. C5×#95C22semidirect
51. #3963C22semidirect
52. #3963C22semidirect
53. #3963C22semidirect
54. #3963C22semidirect
114. #685D4semidirect
132. C4×C22⋊C4D5semidirect
239. C2×C42C2×D5semidirect
240. C2×C42C2×D5semidirect
245. #95C2×D5semidirect
246. #95C2×D5semidirect
253. C22×C5#124semidirect
442. C5#2472semidirect
471. C22#4021semidirect

Maximal subgroups

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

subgroupindex
C22×#6852
C2×#40282
C2×#40242
C2×#40212
C4×C5×C22⋊C42
#655022
#652472
#652442
#652292
#652282
#649632
#649622
#649612
#24725