Finite Group Databasebeta

← #10609145

Subgroups — #10609145

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×#4040C211yesmaximal, characteristic, complemented
2C2×#4038C211yesmaximal, characteristic, complemented
3C2×#4028C211yesmaximal, characteristic
4C2×D5×C22⋊C4C211yesmaximal, characteristic, complemented
5C5×#396C211yesmaximal, characteristic, complemented
6#65283C211yesmaximal, characteristic, complemented
7#65280C211yesmaximal, characteristic, complemented
8#65278C211yesmaximal, characteristic, complemented
9#65277C211yesmaximal, characteristic, complemented
10#65250C211yesmaximal, characteristic, complemented
11#65246C211yesmaximal, characteristic, complemented
12#65244C211yesmaximal, characteristic, complemented
13#65233C211yesmaximal, characteristic, complemented
14#64958C211yesmaximal, characteristic, complemented
15#64954C211yesmaximal, characteristic, complemented
16C24×D5C2211yescharacteristic
17C22×C22⋊D5C2211yescharacteristic
18C22×D20C2211yescharacteristic
19C22×D20C2211yescharacteristic
20C22×C4×D5C2211yescharacteristic
21C22×C4×D5C2211yescharacteristic
22C2×C5×C4⋊C4C2211yescharacteristic, complemented
23C2×C5×C4⋊C4C2211yescharacteristic, complemented
24C2×C5×C4⋊C4C2211yescharacteristic, complemented
25C2×C5×C22⋊C4C2211yescharacteristic
26C2×C5×C22⋊C4C2211yescharacteristic
27C2×C5×C22⋊C4C2211yescharacteristic
28C2×C22⋊Dic5C2211yescharacteristic
29C2×C22⋊Dic5C2211yescharacteristic
30C2×#685C2211yescharacteristic
31C2×#685C2211yescharacteristic
32C2×#685C2211yescharacteristic
33C2×#685C2211yescharacteristic
34C2×#685C2211yescharacteristic
35C2×#685C2211yescharacteristic
36C2×#685C2211yescharacteristic
37C2×#685C2211yescharacteristic
38C2×#685C2211yescharacteristic, complemented
39C2×#685C2211yescharacteristic, complemented
40C2×#685C2211yescharacteristic, complemented
41C2×#685C2211yescharacteristic, complemented
42C2×C4⋊Dic5C2211yescharacteristic, complemented
43C2×#683C2211yescharacteristic, complemented
44C2×#683C2211yescharacteristic, complemented
45C2×C4×Dic5C2211yescharacteristic, complemented
46C2×C4×Dic5C2211yescharacteristic, complemented
47#4040—24no—
48#4038—24no—
49#4028—24no—
50D5×C22⋊C4—24no—
51C5×#95C2211yescharacteristic, complemented
52#3963C2211yescharacteristic, complemented
53#3963C2211yescharacteristic, complemented
54#3963C2211yescharacteristic, complemented
55#2411—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×C5C2311yescharacteristic
68C22×C4×C5C2311yescharacteristic
69C22×C4×C5C2311yescharacteristic
70C22×C4×C5C2311yescharacteristic
71C22×C4×C5C2311yescharacteristic
72C22×C4×C5C2311yescharacteristic
73C2×C22⋊D5—22no—
74C2×C22⋊D5—22no—
75C2×C22⋊D5—41no—
76C22×Dic5C2311yescharacteristic
77C22×Dic5C2311yescharacteristic
78C22×Dic5C2311yescharacteristic
79C22×Dic5C2311yescharacteristic
80C22×Dic5C2311yescharacteristic
81C2×D20—22no—
82C2×D20—22no—
83C2×D20—22no—
84C2×D20—22no—
85C2×D20—22no—
86C2×D20—41no—
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×C4⋊C4—22no—
95C5×C4⋊C4—41no—
96C5×C22⋊C4—22no—
97C5×C22⋊C4—22no—
98C5×C22⋊C4—41no—
99C22⋊Dic5—22no—
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—41no—
110#685—41no—
111#685—41no—
112#685D414yes—
113C4⋊Dic5—22no—
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#396D511yescharacteristic, complemented
124#394—51no—
125#394—51no—
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#4112yes—
140C23×C5C2411yescharacteristic
141C22×D5—21no—
142C22×D5—21no—
143C22×D5—21no—
144C22×D5—21no—
145C22×D5—28no—
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—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—
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—21no—
185C2×C4×C5C2×D412yes—
186C2×C4×C5C2×D412yes—
187C22⋊D5—44no—
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×Dic5C2×D412yes—
202C2×Dic5C2×D412yes—
203D20—44no—
204D20—44no—
205C4×D5—28no—
206C4×D5—44no—
207C25—51no—
208C22×D4—51no—
209C22×D4—51no—
210C22×D4—51no—
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⋊C4C2×D511yescharacteristic, complemented
220C2×C4⋊C4C2×D511yescharacteristic, complemented
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—51no—
236C2×C22⋊C4—104no—
237C2×C22⋊C4C2×D511yescharacteristic
238C2×C22⋊C4C2×D511yescharacteristic
239C2×C22⋊C4C2×D511yescharacteristic
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×C5C22×D411yescharacteristic
254C22×C5C22×D411yescharacteristic
255C22×C5C2×#4111yescharacteristic
256C22×C5C2×#4111yescharacteristic
257C22×C5C2×#4111yescharacteristic
258C22×C5H(5,𝔽2)11yescharacteristic
259C22×C5H(5,𝔽2)11yescharacteristic
260C2×D5—22no—
261C2×D5—22no—
262C2×D5—22no—
263C2×D5—22no—
264C2×D5—22no—
265C2×D5—22no—
266C2×D5—22no—
267C2×D5—22no—
268C2×D5—22no—
269C2×D5—416no—
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#12114yes—
282C4×C5—22no—
283C4×C5—22no—
284C4×C5—41no—
285C4×C5—41no—
286C4×C5—41no—
287C4×C5—41no—
288Dic5—22no—
289Dic5—22no—
290Dic5—41no—
291Dic5—41no—
292Dic5—41no—
293C24—51no—
294C24—51no—
295C24—51no—
296C24—102no—
297C24—102no—
298C24—102no—
299C24—102no—
300C24—102no—
301C24—201no—
302C24—201no—
303C24C22×D511yescharacteristic
304C2×D4—102no—
305C2×D4—102no—
306C2×D4—102no—
307C2×D4—102no—
308C2×D4—102no—
309C2×D4—102no—
310C2×D4—102no—
311C2×D4—201no—
312C2×D4—201no—
313C22×C4—51no—
314C22×C4—51no—
315C22×C4—54no—
316C22×C4—51no—
317C22×C4—51no—
318C22×C4—51no—
319C22×C4—102no—
320C22×C4—102no—
321C22×C4—102no—
322C22×C4—201no—
323C22×C4—201no—
324C22×C4C22×D511yescharacteristic
325C22×C4C22×D511yescharacteristic
326C22×C4C22×D511yescharacteristic
327C22×C4C22×D511yescharacteristic
328C22×C4C22×D511yescharacteristic
329C22×C4C22×D511yescharacteristic
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—41no—
333C4⋊C4—102no—
334C4⋊C4—102no—
335C4⋊C4—201no—
336C22⋊C4—22no—
337C22⋊C4—22no—
338C22⋊C4—41no—
339C22⋊C4—54no—
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—102no—
350C22⋊C4—201no—
351C22⋊C4—201no—
352C22⋊C4—201no—
353C42—102no—
354C42—102no—
355C2×C5—42no—
356C2×C5C2×#12111yescharacteristic
357C2×C5#52511yescharacteristic
358C2×C5#53511yescharacteristic
359C2×C5#54611yescharacteristic
360C2×C5#54611yescharacteristic
361C2×C5#55111yescharacteristic
362C2×C5#55111yescharacteristic
363D5—24no—
364D5—42no—
365D5—81no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—52no—
373C23—52no—
374C23—52no—
375C23—52no—
376C23—108no—
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—101no—
394C23—204no—
395C23—204no—
396C23—204no—
397C23—204no—
398C23—204no—
399C23—204no—
400C23#71312yes—
401C23C23×D511yescharacteristic
402D4—204no—
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—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—52no—
422C2×C4—52no—
423C2×C4—108no—
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—54no—
449C22—102no—
450C22—102no—
451C22—102no—
452C22—102no—
453C22—102no—
454C22—102no—
455C22—102no—
456C22—102no—
457C22—102no—
458C22—2016no—
459C22—201no—
460C22—201no—
461C22—201no—
462C22—201no—
463C22—201no—
464C22—201no—
465C22—201no—
466C22—201no—
467C22—201no—
468C22—201no—
469C22—201no—
470C22C2×D4×D511yescharacteristic
471C22C2×D4×D511yescharacteristic
472C22#414411yescharacteristic
473C22C2×#71311yescharacteristic
474C22D5×#4111yescharacteristic
475C22D5×#4111yescharacteristic
476C22#414911yescharacteristic
477C4—22no—
478C4—22no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—102no—
484C4—102no—
485C4—201no—
486C4—201no—
487C4—201no—
488C2—42no—
489C2—104no—
490C2—202no—
491C2—401no—
492C2D5×#12111yescharacteristic
493C2#6594111yescharacteristic
494C2#6594411yescharacteristic
495C2#6596111yescharacteristic
496C2#6596411yescharacteristic
497C2#6597111yescharacteristic
498C2#6603911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#396C2
prime radicalC2×#4028C2
hypercenter#396D5
hypocenterC5#2411
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#4040C2semidirect
2. C2×#4038C2semidirect
4. C2×D5×C22⋊C4C2semidirect
5. C5×#396C2semidirect
6. #65283C2semidirect
7. #65280C2semidirect
8. #65278C2semidirect
9. #65277C2semidirect
10. #65250C2semidirect
11. #65246C2semidirect
12. #65244C2semidirect
13. #65233C2semidirect
14. #64958C2semidirect
15. #64954C2semidirect
22. C2×C5×C4⋊C4C22semidirect
23. C2×C5×C4⋊C4C22semidirect
24. C2×C5×C4⋊C4C22semidirect
38. C2×#685C22semidirect
39. C2×#685C22semidirect
40. C2×#685C22semidirect
41. C2×#685C22semidirect
42. C2×C4⋊Dic5C22semidirect
43. C2×#683C22semidirect
44. C2×#683C22semidirect
45. C2×C4×Dic5C22semidirect
46. C2×C4×Dic5C22semidirect
51. C5×#95C22semidirect
52. #3963C22semidirect
53. #3963C22semidirect
54. #3963C22semidirect
92. C2×C4×D5D4semidirect
123. #396D5semidirect
219. C2×C4⋊C4C2×D5semidirect
220. C2×C4⋊C4C2×D5semidirect
221. C2×C4⋊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×#40402
C2×#40382
C2×#40282
C2×D5×C22⋊C42
C5×#3962
#652832
#652802
#652782
#652772
#652502
#652462
#652442
#652332
#649582
#649542
#24115