Finite Group Databasebeta

← #10607667

Subgroups — #10607667

Order 640.

560 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#388C211yesmaximal, characteristic, complemented
2#65282C211yesmaximal, characteristic, complemented
3#65279C211yesmaximal, characteristic, complemented
4#65244C211yesmaximal, characteristic
5#65244C211yesmaximal, characteristic
6#65244C211yesmaximal, characteristic
7#65241C211yesmaximal, characteristic, complemented
8#65236C211yesmaximal, characteristic, complemented
9#65233C211yesmaximal, characteristic, complemented
10#65229C211yesmaximal, characteristic, complemented
11#65229C211yesmaximal, characteristic, complemented
12#64964C211yesmaximal, characteristic, complemented
13#64962C211yesmaximal, characteristic, complemented
14#64962C211yesmaximal, characteristic, complemented
15#64958C211yesmaximal, characteristic, complemented
16C24×D5C2211yescharacteristic
17C22×Dic10C2211yescharacteristic, complemented
18C2×C5×C4⋊C4C2211yescharacteristic, complemented
19C2×C5×C22⋊C4C2211yescharacteristic
20C2×C5×C22⋊C4C2211yescharacteristic
21C2×C5×C22⋊C4C2211yescharacteristic
22C2×C42×C5C2211yescharacteristic, complemented
23C2×C22⋊Dic5C2211yescharacteristic
24C2×C22⋊Dic5C2211yescharacteristic
25C2×C22⋊Dic5C2211yescharacteristic
26C2×#685C2211yescharacteristic
27C2×#685C2211yescharacteristic
28C2×#685C2211yescharacteristic
29C2×#685C2211yescharacteristic
30C2×#685C2211yescharacteristic
31C2×#685C2211yescharacteristic
32C2×#685C2211yescharacteristic
33C2×#685C2211yescharacteristic
34C2×#685C2211yescharacteristic
35C2×#685C2211yescharacteristic
36C2×#685C2211yescharacteristic
37C2×#685C2211yescharacteristic
38C2×#683C2211yescharacteristic, complemented
39C2×#683C2211yescharacteristic, complemented
40C2×C4×Dic5C2211yescharacteristic, complemented
41C5×#95C2211yescharacteristic, complemented
42C5×#95C2211yescharacteristic, complemented
43#3963C2211yescharacteristic, complemented
44#3963C2211yescharacteristic, complemented
45#3963C2211yescharacteristic, complemented
46#3963C2211yescharacteristic, complemented
47#3963C2211yescharacteristic, complemented
48#3963C2211yescharacteristic, complemented
49#3963C2211yescharacteristic, complemented
50#3963C2211yescharacteristic, complemented
51#2736—51nomaximal, Sylow
52C24×C5C2311yescharacteristic
53C23×D5—22no—
54C23×D5—22no—
55C23×D5—22no—
56C23×D5—41no—
57C23×D5—41no—
58C23×D5—41no—
59C23×D5—41no—
60C23×D5C2311yescharacteristic
61C23×D5C2311yescharacteristic
62C22×C4×C5C2311yescharacteristic
63C22×C4×C5C2311yescharacteristic
64C22×C4×C5C2311yescharacteristic
65C22×C4×C5C2311yescharacteristic
66C22×C4×C5C2311yescharacteristic
67C22×C4×C5C2311yescharacteristic
68C22×Dic5C2311yescharacteristic
69C22×Dic5C2311yescharacteristic
70C22×Dic5C2311yescharacteristic
71C22×Dic5C2311yescharacteristic
72C22×Dic5C2311yescharacteristic
73C22×Dic5C2311yescharacteristic
74C2×Dic10—41no—
75C2×Dic10—41no—
76C2×Dic10—41no—
77C5×C4⋊C4—41no—
78C5×C22⋊C4—22no—
79C5×C22⋊C4—41no—
80C5×C22⋊C4—41no—
81C42×C5—41no—
82C22⋊Dic5—22no—
83C22⋊Dic5—41no—
84C22⋊Dic5—41no—
85#685—22no—
86#685—22no—
87#685—22no—
88#685—22no—
89#685—41no—
90#685—41no—
91#685—41no—
92#685—41no—
93#685—41no—
94#685—41no—
95#685—41no—
96#685—41no—
97#683—41no—
98#683—41no—
99C4×Dic5—41no—
100#397—51no—
101#397—51no—
102#397—51no—
103#393—51no—
104#393—51no—
105#393—51no—
106#388—51no—
107#388—51no—
108#388—51no—
109#388—51no—
110#388—51no—
111#388D511yescharacteristic, complemented
112#379—51no—
113#379—51no—
114#379—51no—
115C23×C5—21no—
116C23×C5—21no—
117C23×C5—21no—
118C23×C5—21no—
119C23×C5—21no—
120C23×C5—21no—
121C23×C5#4112yes—
122C23×C5C2411yescharacteristic
123C22×D5—21no—
124C22×D5—21no—
125C22×D5—21no—
126C22×D5—21no—
127C22×D5—21no—
128C22×D5—21no—
129C22×D5—21no—
130C22×D5—21no—
131C22×D5—21no—
132C22×D5—21no—
133C22×D5—21no—
134C22×D5—21no—
135C22×D5—44no—
136C22×D5—44no—
137C22×D5—44no—
138C22×D5—44no—
139C22×D5—44no—
140C22×D5—44no—
141C22×D5—44no—
142C22×D5#4112yes—
143C22×D5#4112yes—
144C2×C4×C5—21no—
145C2×C4×C5—21no—
146C2×C4×C5—21no—
147C2×C4×C5—21no—
148C2×C4×C5—21no—
149C2×C4×C5—21no—
150C2×C4×C5—21no—
151C2×C4×C5—21no—
152C2×C4×C5—21no—
153C2×C4×C5—21no—
154C2×C4×C5—21no—
155C2×C4×C5—21no—
156C2×C4×C5—21no—
157C2×C4×C5—21no—
158C2×C4×C5—21no—
159C2×C4×C5—21no—
160C2×C4×C5—21no—
161C2×C4×C5—21no—
162C2×Dic5—21no—
163C2×Dic5—21no—
164C2×Dic5—21no—
165C2×Dic5—21no—
166C2×Dic5—21no—
167C2×Dic5—21no—
168C2×Dic5—21no—
169C2×Dic5—21no—
170C2×Dic5—21no—
171C2×Dic5—21no—
172C2×Dic5—21no—
173C2×Dic5—21no—
174C2×Dic5—21no—
175C2×Dic5—21no—
176C2×Dic5—21no—
177C2×Dic5—21no—
178C2×Dic5—21no—
179C2×Dic5—21no—
180Dic10—44no—
181C25—51no—
182C22×Q8—51no—
183C2×C4⋊C4—51no—
184C2×C4⋊C4—51no—
185C2×C4⋊C4C2×D511yescharacteristic, complemented
186C2×C22⋊C4—51no—
187C2×C22⋊C4—51no—
188C2×C22⋊C4—51no—
189C2×C22⋊C4—51no—
190C2×C22⋊C4—51no—
191C2×C22⋊C4—51no—
192C2×C22⋊C4—51no—
193C2×C22⋊C4—51no—
194C2×C22⋊C4—51no—
195C2×C22⋊C4—51no—
196C2×C22⋊C4—51no—
197C2×C22⋊C4—51no—
198C2×C22⋊C4—51no—
199C2×C22⋊C4—51no—
200C2×C22⋊C4—51no—
201C2×C22⋊C4C2×D511yescharacteristic
202C2×C22⋊C4C2×D511yescharacteristic
203C2×C22⋊C4C2×D511yescharacteristic
204C2×C42—51no—
205C2×C42C2×D511yescharacteristic, complemented
206#95—51no—
207#95—51no—
208#95—51no—
209#95—51no—
210#95—51no—
211#95—51no—
212#95—51no—
213#95—51no—
214#95C2×D511yescharacteristic, complemented
215#95C2×D511yescharacteristic, complemented
216C22×C5—22no—
217C22×C5—22no—
218C22×C5—22no—
219C22×C5—41no—
220C22×C5—41no—
221C22×C5—41no—
222C22×C5—41no—
223C22×C5C2×#4111yescharacteristic
224C22×C5C2×#4111yescharacteristic
225C22×C5C2×#4111yescharacteristic
226C22×C5H(5,𝔽2)11yescharacteristic
227C22×C5H(5,𝔽2)11yescharacteristic
228C22×C5H(5,𝔽2)11yescharacteristic
229C22×C5H(5,𝔽2)11yescharacteristic
230C2×D5—22no—
231C2×D5—22no—
232C2×D5—22no—
233C2×D5—22no—
234C2×D5—22no—
235C2×D5—22no—
236C2×D5—416no—
237C2×D5—41no—
238C2×D5—41no—
239C2×D5—41no—
240C2×D5—41no—
241C2×D5—41no—
242C2×D5—41no—
243C2×D5—41no—
244C2×D5—41no—
245C4×C5—41no—
246C4×C5—41no—
247C4×C5—41no—
248C4×C5—41no—
249C4×C5—41no—
250C4×C5—41no—
251Dic5—41no—
252Dic5—41no—
253Dic5—41no—
254Dic5—41no—
255Dic5—41no—
256Dic5—41no—
257C24—51no—
258C24—51no—
259C24—102no—
260C24—102no—
261C24—102no—
262C24—201no—
263C24—201no—
264C24—201no—
265C24—201no—
266C24C22×D511yescharacteristic
267C2×Q8—201no—
268C2×Q8—201no—
269C2×Q8—201no—
270C22×C4—51no—
271C22×C4—51no—
272C22×C4—51no—
273C22×C4—51no—
274C22×C4—51no—
275C22×C4—51no—
276C22×C4C22×D511yescharacteristic
277C22×C4C22×D511yescharacteristic
278C22×C4C22×D511yescharacteristic
279C22×C4C22×D511yescharacteristic
280C22×C4C22×D511yescharacteristic
281C22×C4C22×D511yescharacteristic
282C4⋊C4—41no—
283C4⋊C4—201no—
284C4⋊C4—201no—
285C22⋊C4—22no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—102no—
289C22⋊C4—102no—
290C22⋊C4—102no—
291C22⋊C4—102no—
292C22⋊C4—102no—
293C22⋊C4—201no—
294C22⋊C4—201no—
295C22⋊C4—201no—
296C22⋊C4—201no—
297C22⋊C4—201no—
298C22⋊C4—201no—
299C22⋊C4—201no—
300C22⋊C4—201no—
301C22⋊C4—201no—
302C22⋊C4—201no—
303C42—41no—
304C42—201no—
305C2×C5—42no—
306C2×C5#53811yescharacteristic
307C2×C5#53811yescharacteristic
308C2×C5#53811yescharacteristic
309C2×C5#55111yescharacteristic
310C2×C5#55111yescharacteristic
311C2×C5#55111yescharacteristic
312C2×C5H(3,𝔽4)11yescharacteristic
313D5—42no—
314D5—42no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—52no—
322C23—52no—
323C23—101no—
324C23—101no—
325C23—101no—
326C23—101no—
327C23—101no—
328C23—101no—
329C23—101no—
330C23—101no—
331C23—101no—
332C23—101no—
333C23—101no—
334C23—101no—
335C23—204no—
336C23—204no—
337C23—204no—
338C23—204no—
339C23—204no—
340C23—204no—
341C23—204no—
342C23#70912yes—
343C23C23×D511yescharacteristic
344Q8—204no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—101no—
364C2×C4—101no—
365C2×C4—101no—
366C2×C4—101no—
367C2×C4—101no—
368C2×C4—101no—
369C2×C4—101no—
370C2×C4—101no—
371C2×C4—101no—
372C2×C4—101no—
373C2×C4—101no—
374C2×C4—101no—
375C2×C4—101no—
376C2×C4—101no—
377C2×C4—101no—
378C2×C4—101no—
379C2×C4—101no—
380C2×C4—101no—
381C5#273611yescharacteristic, complemented, Sylow
382C22—22no—
383C22—22no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—102no—
390C22—102no—
391C22—102no—
392C22—102no—
393C22—102no—
394C22—102no—
395C22—2016no—
396C22—201no—
397C22—201no—
398C22—201no—
399C22—201no—
400C22—201no—
401C22—201no—
402C22—201no—
403C22—201no—
404C22C2×#70911yescharacteristic
405C22#414411yescharacteristic
406C22#414411yescharacteristic
407C22D5×#4111yescharacteristic
408C22D5×#4111yescharacteristic
409C22#414911yescharacteristic
410C22#414911yescharacteristic
411C4—41no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—201no—
418C4—201no—
419C4—201no—
420C4—201no—
421C4—201no—
422C4—201no—
423C2—42no—
424C2—202no—
425C2—202no—
426C2#6586111yescharacteristic
427C2#6589011yescharacteristic
428C2#6589411yescharacteristic
429C2#6594711yescharacteristic
430C2#6599111yescharacteristic
431C2#6601711yescharacteristic
432C2#6603911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#388C2
prime radicalC24×D5C22
hypercenter#388D5
hypocenterC5#2736
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C5×#388C2semidirect
2. #65282C2semidirect
3. #65279C2semidirect
7. #65241C2semidirect
8. #65236C2semidirect
9. #65233C2semidirect
10. #65229C2semidirect
11. #65229C2semidirect
12. #64964C2semidirect
13. #64962C2semidirect
14. #64962C2semidirect
15. #64958C2semidirect
17. C22×Dic10C22semidirect
18. C2×C5×C4⋊C4C22semidirect
22. C2×C42×C5C22semidirect
38. C2×#683C22semidirect
39. C2×#683C22semidirect
40. C2×C4×Dic5C22semidirect
41. C5×#95C22semidirect
42. C5×#95C22semidirect
43. #3963C22semidirect
44. #3963C22semidirect
45. #3963C22semidirect
46. #3963C22semidirect
47. #3963C22semidirect
48. #3963C22semidirect
49. #3963C22semidirect
50. #3963C22semidirect
111. #388D5semidirect
185. C2×C4⋊C4C2×D5semidirect
205. C2×C42C2×D5semidirect
214. #95C2×D5semidirect
215. #95C2×D5semidirect
381. C5#2736semidirect

Maximal subgroups

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

subgroupindex
C5×#3882
#652822
#652792
#652442
#652412
#652362
#652332
#652292
#649642
#649622
#649582
#27365