Finite Group Databasebeta

← #10618921

Subgroups — #10618921

Order 640.

992 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#557C211yesmaximal, characteristic, complemented
2#66066C211yesmaximal, characteristic, complemented
3#66058C212yesmaximal
4D5×C4⋊Q8C211yesmaximal, characteristic, complemented
5#66055C212yesmaximal, complemented
6#66030C214yesmaximal, complemented
7#66028C212yesmaximal, complemented
8#66027C214yesmaximal, complemented
9#66022C212yesmaximal, complemented
10#65918C211yesmaximal, characteristic, complemented
11#65915C212yesmaximal, complemented
12#65912C212yesmaximal, complemented
13#65912C211yesmaximal, characteristic, complemented
14C4×#713C211yesmaximal, characteristic
15#65906C212yesmaximal, complemented
16#65902C212yesmaximal, complemented
17#65901C211yesmaximal, characteristic, complemented
18C2×#713C2211yescharacteristic
19C2×Q8×D5C2212yes—
20C5×C4⋊Q8C2212yes—
21C5×C4⋊Q8C2211yescharacteristic
22C5×#125C2212yes—
23C5×#125C2214yes—
24C4×C5×Q8C2211yescharacteristic
25C4×C5×Q8C2212yes—
26C4×C5×Q8C2212yes—
27C4×C5×Q8C2211yescharacteristic
28#4093C2212yes—
29#4092C2211yescharacteristic
30#4092C2212yes—
31Q8×Dic5C2211yescharacteristic
32Q8×Dic5C2212yes—
33#4090C2212yes—
34#4090C2212yes—
35#4090C2211yescharacteristic
36#4044C2214yes—
37#4044C2214yes—
38#4044C2214yes—
39#4043C2212yes—
40#4043C2211yescharacteristic
41#4042C2214yes—
42#4042C2214yes—
43#4042C2214yes—
44#4042C2212yes—
45#4040C2214yes—
46#4040C2214yes—
47#4039C2211yescharacteristic
48#4039C2212yes—
49#4038C2212yes—
50#4038C2211yescharacteristic
51#4038C2214yes—
52#4038C2212yes—
53#4038C2212yes—
54D5×C4⋊C4C2212yes—
55D5×C4⋊C4C2212yes—
56#4036C2214yes—
57#4035C2214yes—
58#4035C2214yes—
59#4035C2214yes—
60#4035C2212yes—
61#4034C2212yes—
62#4034C2212yes—
63#4034C2211yescharacteristic
64#4033C2214yes—
65#4033C2212yes—
66#4033C2214yes—
67#4022C2214yes—
68#4021C2212yes—
69C4×D20C2211yescharacteristic
70C4×D20C2212yes—
71#4018C2212yes—
72#4018C2214yes—
73#4018C2214yes—
74C42×D5C2211yescharacteristic
75C42×D5C2212yes—
76#4016C2212yes—
77#4015C2211yescharacteristic
78C4×Dic10C2212yes—
79C4×Dic10C2212yes—
80C4×Dic10C2211yescharacteristic
81C4×Dic10C2212yes—
82C4×Dic10C2214yes—
83#3509—51nomaximal, Sylow
84C2×C5×Q8C2311yescharacteristic
85C2×C5×Q8C2312yes—
86C2×C5×Q8C2311yescharacteristic
87#713—24no—
88Q8×D5—28no—
89C2×D20C2311yescharacteristic
90C2×D20C2312yes—
91C2×C4×D5C2314yes—
92C2×C4×D5C2312yes—
93C2×C4×D5C2312yes—
94C2×C4×D5C2311yescharacteristic
95C2×C4×D5C2312yes—
96C2×C4×D5C2311yescharacteristic
97C2×C4×D5C2311yescharacteristic
98C2×Dic10C2312yes—
99C2×Dic10C2314yes—
100C2×Dic10C2311yescharacteristic
101C2×Dic10C2312yes—
102C5×C4⋊C4C2311yescharacteristic
103C5×C4⋊C4C2314yes—
104C5×C4⋊C4C2311yescharacteristic
105C5×C4⋊C4C2312yes—
106C5×C4⋊C4C2312yes—
107C5×C4⋊C4C2312yes—
108C5×C4⋊C4C2314yes—
109C5×C4⋊C4C2314yes—
110C5×C4⋊C4C2312yes—
111C42×C5C2312yes—
112C42×C5C2314yes—
113C42×C5C2312yes—
114C42×C5C2311yescharacteristic
115#685C2312yes—
116#685C2312yes—
117#685C2314yes—
118#685C2312yes—
119#685C2314yes—
120#685C2312yes—
121#685C2312yes—
122#685C2314yes—
123#685C2314yes—
124C4⋊Dic5C2311yescharacteristic
125C4⋊Dic5C2312yes—
126C4⋊Dic5C2311yescharacteristic
127C4⋊Dic5C2312yes—
128C4⋊Dic5C2314yes—
129C4⋊Dic5C2312yes—
130C4⋊Dic5C2312yes—
131#683C2314yes—
132#683C2312yes—
133#683C2312yes—
134#683C2312yes—
135#683C2312yes—
136#683C2314yes—
137#683C2314yes—
138#683C2314yes—
139#683C2312yes—
140#683C2314yes—
141#683C2314yes—
142C4×Dic5C2312yes—
143C4×Dic5C2311yescharacteristic
144C4×Dic5C2314yes—
145C4×Dic5C2312yes—
146C4×Dic5C2314yes—
147C4×Dic5C2311yescharacteristic
148C4×Dic5C2312yes—
149C4×Dic5C2311yescharacteristic
150C4×Dic5C2314yes—
151#557—52no—
152#557—51no—
153#557D511yescharacteristic, complemented
154#556—51no—
155#556—52no—
156#556—51no—
157#552—54no—
158#552—52no—
159#552—52no—
160#541—54no—
161#536—52no—
162#533—52no—
163#533—51no—
164#533—52no—
165C2×C4⋊Q8—51no—
166#519—52no—
167C4×#41—51no—
168C22×D5C2412yes—
169C22×D5C2411yescharacteristic
170C5×Q8—24no—
171C5×Q8—22no—
172C5×Q8—22no—
173C2×C4×C5C2412yes—
174C2×C4×C5C2412yes—
175C2×C4×C5C2412yes—
176C2×C4×C5C2411yescharacteristic
177C2×C4×C5C2414yes—
178C2×C4×C5C2412yes—
179C2×C4×C5C2411yescharacteristic
180C2×C4×C5C2411yescharacteristic
181C2×Dic5C2412yes—
182C2×Dic5C2412yes—
183C2×Dic5C2412yes—
184C2×Dic5C2411yescharacteristic
185C2×Dic5C2414yes—
186C2×Dic5C2411yescharacteristic
187C2×Dic5C2411yescharacteristic
188D20—24no—
189D20—22no—
190C4×D5—28no—
191C4×D5—22no—
192C4×D5—24no—
193C4×D5—24no—
194C4×D5—22no—
195C4×D5—24no—
196C4×D5#4114yes—
197Dic10—24no—
198Dic10—22no—
199Dic10—28no—
200Dic10—24no—
201C2×#41—51no—
202C22×Q8—52no—
203C4⋊Q8—52no—
204C4⋊Q8—51no—
205C4⋊Q8—52no—
206C4⋊Q8—52no—
207C4⋊Q8—52no—
208C4⋊Q8—51no—
209C4⋊Q8—51no—
210C4⋊Q8C2×D511yescharacteristic
211C4⋊Q8C2×D512yes—
212#126—54no—
213#126—54no—
214#126—54no—
215#126—54no—
216#125—52no—
217#125—54no—
218#125—54no—
219#125—54no—
220#125—54no—
221#125—52no—
222#125C2×D512yes—
223#125C2×D514yes—
224#124—52no—
225#124—52no—
226#123—54no—
227#123—54no—
228C22⋊Q8—51no—
229C22⋊Q8—52no—
230C22⋊Q8—54no—
231C22⋊Q8—54no—
232C22⋊Q8—52no—
233C22⋊Q8—51no—
234C22⋊Q8—54no—
235C22⋊Q8—52no—
236C4×Q8—51no—
237C4×Q8—52no—
238C4×Q8—54no—
239C4×Q8—52no—
240C4×Q8—54no—
241C4×Q8—54no—
242C4×Q8—52no—
243C4×Q8—52no—
244C4×Q8—52no—
245C4×Q8—51no—
246C4×Q8C2×D511yescharacteristic
247C4×Q8C2×D512yes—
248C4×Q8C2×D512yes—
249C4×Q8C2×D511yescharacteristic
250C4×D4—52no—
251C4×D4—51no—
252C4×D4—52no—
253C4×D4—51no—
254#117—51no—
255#117—52no—
256#117—54no—
257#117—52no—
258#117—52no—
259#117—52no—
260#117—54no—
261#117—54no—
262C2×C4⋊C4—52no—
263C2×C4⋊C4—52no—
264C2×C42—51no—
265C2×C42—52no—
266C22×C5C2511yescharacteristic
267C2×D5—21no—
268C2×D5—21no—
269C2×D5—22no—
270C2×D5—22no—
271C2×D5—22no—
272C2×D5C2×#4112yes—
273C4×C5—24no—
274C4×C5—22no—
275C4×C5—22no—
276C4×C5—22no—
277C4×C5—21no—
278C4×C5—22no—
279C4×C5C2×#4112yes—
280C4×C52−1+412yes—
281Dic5—24no—
282Dic5—22no—
283Dic5—22no—
284Dic5—22no—
285Dic5—21no—
286Dic5C2×#4112yes—
287Dic52−1+412yes—
288#41—104no—
289C2×Q8—52no—
290C2×Q8—51no—
291C2×Q8—54no—
292C2×Q8—52no—
293C2×Q8—108no—
294C2×Q8C22×D512yes—
295C2×Q8C22×D511yescharacteristic
296C2×Q8C22×D511yescharacteristic
297C2×D4—52no—
298C2×D4—51no—
299C22×C4—51no—
300C22×C4—51no—
301C22×C4—51no—
302C22×C4—52no—
303C22×C4—52no—
304C22×C4—52no—
305C22×C4—54no—
306C4⋊C4—54no—
307C4⋊C4—52no—
308C4⋊C4—54no—
309C4⋊C4—52no—
310C4⋊C4—52no—
311C4⋊C4—51no—
312C4⋊C4—51no—
313C4⋊C4—52no—
314C4⋊C4—52no—
315C4⋊C4—52no—
316C4⋊C4—54no—
317C4⋊C4—54no—
318C4⋊C4—54no—
319C4⋊C4—54no—
320C4⋊C4—54no—
321C4⋊C4—52no—
322C4⋊C4—52no—
323C4⋊C4—52no—
324C4⋊C4C22×D512yes—
325C4⋊C4C22×D512yes—
326C4⋊C4C22×D511yescharacteristic
327C4⋊C4C22×D511yescharacteristic
328C4⋊C4C22×D512yes—
329C4⋊C4C22×D514yes—
330C4⋊C4C22×D514yes—
331C4⋊C4C22×D514yes—
332C4⋊C4C22×D512yes—
333C22⋊C4—52no—
334C22⋊C4—52no—
335C22⋊C4—52no—
336C22⋊C4—52no—
337C22⋊C4—52no—
338C22⋊C4—54no—
339C22⋊C4—54no—
340C22⋊C4—54no—
341C22⋊C4—54no—
342C42—51no—
343C42—52no—
344C42—51no—
345C42—52no—
346C42—51no—
347C42—54no—
348C42—54no—
349C42—54no—
350C42—52no—
351C42C22×D512yes—
352C42C22×D511yescharacteristic
353C42C22×D514yes—
354C42C22×D512yes—
355C2×C5C22×#4111yescharacteristic
356C2×C5C2×2−1+411yescharacteristic
357C2×C5C2×2−1+411yescharacteristic
358D5—22no—
359D5—42no—
360C23—51no—
361C23—52no—
362Q8—22no—
363Q8—24no—
364Q8—22no—
365Q8—108no—
366Q8—104no—
367Q8—102no—
368Q8—104no—
369D4—104no—
370D4—102no—
371C2×C4—51no—
372C2×C4—51no—
373C2×C4—54no—
374C2×C4—52no—
375C2×C4—52no—
376C2×C4—52no—
377C2×C4—51no—
378C2×C4—54no—
379C2×C4—104no—
380C2×C4—102no—
381C2×C4—104no—
382C2×C4—108no—
383C2×C4—102no—
384C2×C4—104no—
385C2×C4C23×D512yes—
386C2×C4C23×D512yes—
387C2×C4C23×D511yescharacteristic
388C2×C4C23×D514yes—
389C2×C4C23×D512yes—
390C2×C4C23×D512yes—
391C2×C4C23×D511yescharacteristic
392C2×C4C23×D511yescharacteristic
393C5#350911yescharacteristic, complemented, Sylow
394C22—52no—
395C22—101no—
396C22—101no—
397C22—102no—
398C22—102no—
399C22—102no—
400C22C24×D511yescharacteristic
401C4—22no—
402C4—22no—
403C4—24no—
404C4—22no—
405C4—21no—
406C4—22no—
407C4—52no—
408C4—52no—
409C4—102no—
410C4—101no—
411C4—102no—
412C4—104no—
413C4—102no—
414C4#414712yes—
415C4D5×#4112yes—
416C2—102no—
417C2—202no—
418C2C2×#414711yescharacteristic
419C2C2×D5×#4111yescharacteristic
420C2D5×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D5
commutator, socle, cosocleC22×C5C25
FittingC5×#557C2
prime radicalC2×#713C22
hypercenter#557D5
hypocenterC5#3509
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. C5×#557C2semidirect
2. #66066C2semidirect
4. D5×C4⋊Q8C2semidirect
5. #66055C2semidirect
6. #66030C2semidirect
7. #66028C2semidirect
8. #66027C2semidirect
9. #66022C2semidirect
10. #65918C2semidirect
11. #65915C2semidirect
12. #65912C2semidirect
13. #65912C2semidirect
15. #65906C2semidirect
16. #65902C2semidirect
17. #65901C2semidirect
153. #557D5semidirect
393. C5#3509semidirect

Maximal subgroups

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

subgroupindex
C5×#5572
#660662
#660582
D5×C4⋊Q82
#660552
#660302
#660282
#660272
#660222
#659182
#659152
#659122
C4×#7132
#659062
#659022
#659012
#35095