Finite Group Databasebeta

← #10605590

Subgroups — #10605590

Order 640.

688 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4099C211yesmaximal, characteristic
2C2×#4031C211yesmaximal, characteristic, complemented
3C5×#379C211yesmaximal, characteristic, complemented
4#65536C211yesmaximal, characteristic, complemented
5#65250C211yesmaximal, characteristic, complemented
6#65248C211yesmaximal, characteristic, complemented
7#65245C211yesmaximal, characteristic, complemented
8#65245C211yesmaximal, characteristic, complemented
9#65245C211yesmaximal, characteristic, complemented
10#65242C211yesmaximal, characteristic, complemented
11#65241C211yesmaximal, characteristic, complemented
12#65240C211yesmaximal, characteristic, complemented
13#65236C211yesmaximal, characteristic, complemented
14#65236C211yesmaximal, characteristic, complemented
15#65234C211yesmaximal, characteristic, complemented
16C25×C5C2211yescharacteristic
17C22×C22⋊D5C2211yescharacteristic
18C22×C22⋊D5C2211yescharacteristic
19C22×C22⋊D5C2211yescharacteristic
20C22×C4×D5C2211yescharacteristic, complemented
21C22×Dic10C2211yescharacteristic, complemented
22C2×C5×C22⋊C4C2211yescharacteristic
23C2×C5×C22⋊C4C2211yescharacteristic
24C2×C5×C22⋊C4C2211yescharacteristic
25C2×C5×C22⋊C4C2211yescharacteristic
26C2×C5×C22⋊C4C2211yescharacteristic
27C2×C5×C22⋊C4C2211yescharacteristic
28#4099—24no—
29C2×C22⋊Dic5C2211yescharacteristic
30C2×C22⋊Dic5C2211yescharacteristic
31C2×C22⋊Dic5C2211yescharacteristic
32C2×C22⋊Dic5C2211yescharacteristic
33C2×C22⋊Dic5C2211yescharacteristic
34C2×C22⋊Dic5C2211yescharacteristic
35C2×C22⋊Dic5C2211yescharacteristic
36C2×C22⋊Dic5C2211yescharacteristic
37C2×C22⋊Dic5C2211yescharacteristic
38C2×#685C2211yescharacteristic, complemented
39C2×#685C2211yescharacteristic, complemented
40C2×#685C2211yescharacteristic, complemented
41C2×C4⋊Dic5C2211yescharacteristic, complemented
42C2×#683C2211yescharacteristic, complemented
43C2×C4×Dic5C2211yescharacteristic, complemented
44#4031—24no—
45#3963C2211yescharacteristic, complemented
46#3963C2211yescharacteristic, complemented
47#3963C2211yescharacteristic, complemented
48#3963C2211yescharacteristic, complemented
49#3963C2211yescharacteristic, complemented
50#3963C2211yescharacteristic, complemented
51#3963C2211yescharacteristic, complemented
52#3963C2211yescharacteristic, complemented
53#2684—51nomaximal, Sylow
54C24×C5—22no—
55C24×C5—22no—
56C24×C5—22no—
57C24×C5—41no—
58C24×C5—41no—
59C24×C5—41no—
60C24×C5—41no—
61C24×C5C2311yescharacteristic
62C24×C5C2311yescharacteristic
63C24×C5C2311yescharacteristic
64C23×D5C2311yescharacteristic
65C22×C4×C5C2311yescharacteristic
66C22×C4×C5C2311yescharacteristic
67C22×C4×C5C2311yescharacteristic
68C22×C4×C5C2311yescharacteristic
69C2×C22⋊D5—22no—
70C2×C22⋊D5—22no—
71C2×C22⋊D5—22no—
72C2×C22⋊D5—22no—
73C2×C22⋊D5—41no—
74C2×C22⋊D5—41no—
75C2×C22⋊D5—41no—
76C2×C22⋊D5—41no—
77C2×C22⋊D5D414yes—
78C22×Dic5C2311yescharacteristic
79C22×Dic5C2311yescharacteristic
80C22×Dic5C2311yescharacteristic
81C22×Dic5C2311yescharacteristic
82C22×Dic5C2311yescharacteristic
83C22×Dic5C2311yescharacteristic
84C22×Dic5C2311yescharacteristic
85C2×C4×D5—41no—
86C2×C4×D5—41no—
87C2×C4×D5—41no—
88C2×Dic10—22no—
89C2×Dic10—41no—
90C2×Dic10—41no—
91C5×C22⋊C4—22no—
92C5×C22⋊C4—22no—
93C5×C22⋊C4—41no—
94C5×C22⋊C4—41no—
95C5×C22⋊C4—41no—
96C5×C22⋊C4—41no—
97C22⋊Dic5—22no—
98C22⋊Dic5—22no—
99C22⋊Dic5—22no—
100C22⋊Dic5—22no—
101C22⋊Dic5—22no—
102C22⋊Dic5—41no—
103C22⋊Dic5—41no—
104C22⋊Dic5—41no—
105C22⋊Dic5—41no—
106#685—22no—
107#685—22no—
108#685—41no—
109C4⋊Dic5—41no—
110#683—41no—
111C4×Dic5—22no—
112C2×#124—51no—
113C2×#120—51no—
114#397—51no—
115#397—51no—
116#394—51no—
117#394—51no—
118#393—51no—
119#393—51no—
120#388—51no—
121#388—51no—
122#386—51no—
123#386—51no—
124#386—51no—
125#379—51no—
126#379D511yescharacteristic, complemented
127C23×C5—21no—
128C23×C5—21no—
129C23×C5—21no—
130C23×C5—21no—
131C23×C5—21no—
132C23×C5—21no—
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—44no—
146C23×C5—44no—
147C23×C5—44no—
148C23×C5—44no—
149C23×C5—44no—
150C23×C5—44no—
151C23×C5—44no—
152C23×C5C2×D412yes—
153C23×C5#4112yes—
154C23×C5#4112yes—
155C23×C5C2411yescharacteristic
156C22×D5—21no—
157C22×D5—21no—
158C22×D5—21no—
159C22×D5—21no—
160C22×D5—21no—
161C22×D5—21no—
162C22×D5C2×D412yes—
163C2×C4×C5—21no—
164C2×C4×C5—21no—
165C2×C4×C5—21no—
166C2×C4×C5—21no—
167C2×C4×C5—21no—
168C2×C4×C5—21no—
169C2×C4×C5—21no—
170C2×C4×C5—21no—
171C2×C4×C5—21no—
172C2×C4×C5—21no—
173C2×C4×C5—21no—
174C2×C4×C5—21no—
175C22⋊D5—44no—
176C22⋊D5—44no—
177C22⋊D5—44no—
178C2×Dic5—21no—
179C2×Dic5—21no—
180C2×Dic5—21no—
181C2×Dic5—21no—
182C2×Dic5—21no—
183C2×Dic5—21no—
184C2×Dic5—21no—
185C2×Dic5—21no—
186C2×Dic5—21no—
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×Dic5C2×D412yes—
199C4×D5—44no—
200Dic10—44no—
201C25C2×D511yescharacteristic
202C22×Q8—51no—
203C22×D4—51no—
204C22×D4—51no—
205C22×D4—51no—
206C23×C4—51no—
207#124—104no—
208#120—104no—
209C2×C4⋊C4—51no—
210C2×C4⋊C4—51no—
211C2×C22⋊C4—51no—
212C2×C22⋊C4—51no—
213C2×C22⋊C4—51no—
214C2×C22⋊C4—51no—
215C2×C22⋊C4—51no—
216C2×C22⋊C4—51no—
217C2×C22⋊C4—51no—
218C2×C22⋊C4—51no—
219C2×C22⋊C4—51no—
220C2×C22⋊C4—51no—
221C2×C22⋊C4—51no—
222C2×C22⋊C4—51no—
223C2×C22⋊C4C2×D511yescharacteristic
224C2×C22⋊C4C2×D511yescharacteristic
225C2×C22⋊C4C2×D511yescharacteristic
226C2×C22⋊C4C2×D511yescharacteristic
227C2×C22⋊C4C2×D511yescharacteristic
228C2×C22⋊C4C2×D511yescharacteristic
229C2×C42—51no—
230#95—51no—
231#95—51no—
232#95—51no—
233#95—51no—
234#95—51no—
235#95—51no—
236#95—51no—
237#95—51no—
238C22×C5—22no—
239C22×C5—22no—
240C22×C5—22no—
241C22×C5—22no—
242C22×C5—22no—
243C22×C5—22no—
244C22×C5—22no—
245C22×C5—22no—
246C22×C5—22no—
247C22×C5—416no—
248C22×C5—41no—
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×C5C2×#4111yescharacteristic
262C22×C5C2×#4111yescharacteristic
263C22×C5H(5,𝔽2)11yescharacteristic
264C22×C5H(5,𝔽2)11yescharacteristic
265C22×C5H(5,𝔽2)11yescharacteristic
266C22×C5H(5,𝔽2)11yescharacteristic
267C2×D5—22no—
268C2×D5—22no—
269C2×D5—22no—
270C2×D5—41no—
271C2×D5—41no—
272C2×D5—41no—
273C2×D5—41no—
274C4×C5—41no—
275C4×C5—41no—
276C4×C5—41no—
277C4×C5—41no—
278Dic5—22no—
279Dic5—41no—
280Dic5—41no—
281Dic5—41no—
282Dic5—41no—
283Dic5—41no—
284Dic5—41no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—41no—
289C24—41no—
290C24—41no—
291C24—41no—
292C24—51no—
293C24C22×D511yescharacteristic
294C24C22×D511yescharacteristic
295C24C22×D511yescharacteristic
296C2×Q8—102no—
297C2×Q8—201no—
298C2×Q8—201no—
299C2×D4—54no—
300C2×D4—102no—
301C2×D4—102no—
302C2×D4—102no—
303C2×D4—102no—
304C2×D4—201no—
305C2×D4—201no—
306C2×D4—201no—
307C2×D4—201no—
308C22×C4—51no—
309C22×C4—51no—
310C22×C4—51no—
311C22×C4—51no—
312C22×C4—51no—
313C22×C4—51no—
314C22×C4—51no—
315C22×C4—201no—
316C22×C4—201no—
317C22×C4—201no—
318C22×C4C22×D511yescharacteristic
319C22×C4C22×D511yescharacteristic
320C22×C4C22×D511yescharacteristic
321C22×C4C22×D511yescharacteristic
322C4⋊C4—201no—
323C4⋊C4—201no—
324C22⋊C4—22no—
325C22⋊C4—22no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—102no—
331C22⋊C4—102no—
332C22⋊C4—102no—
333C22⋊C4—102no—
334C22⋊C4—102no—
335C22⋊C4—102no—
336C22⋊C4—102no—
337C22⋊C4—201no—
338C22⋊C4—201no—
339C22⋊C4—201no—
340C22⋊C4—201no—
341C22⋊C4—201no—
342C42—102no—
343C2×C5—42no—
344C2×C5—42no—
345C2×C5—42no—
346C2×C5#53411yescharacteristic
347C2×C5#53811yescharacteristic
348C2×C5#53811yescharacteristic
349C2×C5#54611yescharacteristic
350C2×C5#54611yescharacteristic
351C2×C5#55111yescharacteristic
352C2×C5H(3,𝔽4)11yescharacteristic
353D5—42no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—44no—
373C23—44no—
374C23—44no—
375C23—44no—
376C23—44no—
377C23—44no—
378C23—44no—
379C23—52no—
380C23—101no—
381C23—101no—
382C23—101no—
383C23—101no—
384C23—101no—
385C23—101no—
386C23#70912yes—
387C23D4×D512yes—
388C23#71112yes—
389C23C23×D511yescharacteristic
390Q8—204no—
391D4—204no—
392D4—204no—
393D4—204no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—52no—
407C2×C4—101no—
408C2×C4—101no—
409C2×C4—101no—
410C2×C4—101no—
411C2×C4—101no—
412C2×C4—101no—
413C2×C4—101no—
414C2×C4—101no—
415C2×C4—101no—
416C2×C4—101no—
417C2×C4—101no—
418C2×C4—101no—
419C2×C4—101no—
420C2×C4—101no—
421C2×C4—101no—
422C2×C4—101no—
423C2×C4—101no—
424C2×C4—101no—
425C2×C4—101no—
426C2×C4—101no—
427C2×C4—204no—
428C5#268411yescharacteristic, complemented, Sylow
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—416no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—102no—
452C22—102no—
453C22—102no—
454C22—201no—
455C22—201no—
456C22—201no—
457C22—201no—
458C22C2×#70911yescharacteristic
459C22C2×D4×D511yescharacteristic
460C22C2×#71111yescharacteristic
461C22#414411yescharacteristic
462C22#414411yescharacteristic
463C22#414411yescharacteristic
464C22#414411yescharacteristic
465C4—41no—
466C4—41no—
467C4—41no—
468C4—41no—
469C4—102no—
470C4—201no—
471C4—201no—
472C4—201no—
473C4—201no—
474C4—201no—
475C4—201no—
476C2—42no—
477C2—42no—
478C2—42no—
479C2—202no—
480C2#6582411yescharacteristic
481C2#6582811yescharacteristic
482C2#6582911yescharacteristic
483C2#6592311yescharacteristic
484C2#6592511yescharacteristic
485C2#6592711yescharacteristic
486C2#6592811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#379C2
prime radicalC2×#4099C2
hypercenter#379D5
hypocenterC5#2684
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×#4031C2semidirect
3. C5×#379C2semidirect
4. #65536C2semidirect
5. #65250C2semidirect
6. #65248C2semidirect
7. #65245C2semidirect
8. #65245C2semidirect
9. #65245C2semidirect
10. #65242C2semidirect
11. #65241C2semidirect
12. #65240C2semidirect
13. #65236C2semidirect
14. #65236C2semidirect
15. #65234C2semidirect
20. C22×C4×D5C22semidirect
21. C22×Dic10C22semidirect
38. C2×#685C22semidirect
39. C2×#685C22semidirect
40. C2×#685C22semidirect
41. C2×C4⋊Dic5C22semidirect
42. C2×#683C22semidirect
43. C2×C4×Dic5C22semidirect
45. #3963C22semidirect
46. #3963C22semidirect
47. #3963C22semidirect
48. #3963C22semidirect
49. #3963C22semidirect
50. #3963C22semidirect
51. #3963C22semidirect
52. #3963C22semidirect
126. #379D5semidirect
428. C5#2684semidirect

Maximal subgroups

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

subgroupindex
C2×#40992
C2×#40312
C5×#3792
#655362
#652502
#652482
#652452
#652422
#652412
#652402
#652362
#652342
#26845