Finite Group Databasebeta

← #10618926

Subgroups — #10618926

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#66065C212yesmaximal, complemented
3#66062C211yesmaximal, characteristic, complemented
4#66061C211yesmaximal, characteristic, complemented
5#66061C212yesmaximal, complemented
6#66030C214yesmaximal, complemented
7#66027C214yesmaximal, complemented
8#66025C212yesmaximal
9#66019C212yesmaximal, complemented
10#65918C211yesmaximal, characteristic, complemented
11#65915C212yesmaximal, complemented
12#65914C211yesmaximal, characteristic
13#65912C212yesmaximal, complemented
14#65906C212yesmaximal, complemented
15C4×Q8×D5C211yesmaximal, characteristic, complemented
16#65902C212yesmaximal, complemented
17#65901C211yesmaximal, characteristic, complemented
18C2×Q8×D5C2211yescharacteristic
19C2×Q8×D5C2212yes—
20C5×C4⋊Q8C2211yescharacteristic
21C5×C4⋊Q8C2212yes—
22C5×#125C2212yes—
23C5×#125C2214yes—
24C4×C5×Q8C2211yescharacteristic
25C4×C5×Q8C2212yes—
26C4×C5×Q8C2211yescharacteristic
27C4×C5×Q8C2212yes—
28#4093C2211yescharacteristic
29#4092C2212yes—
30#4092C2212yes—
31#4092C2212yes—
32#4092C2211yescharacteristic
33Q8×Dic5C2212yes—
34Q8×Dic5C2211yescharacteristic
35#4090C2212yes—
36#4044C2214yes—
37#4044C2214yes—
38#4044C2212yes—
39#4044C2214yes—
40#4043C2212yes—
41#4042C2214yes—
42#4042C2214yes—
43#4042C2214yes—
44#4041C2211yescharacteristic
45#4040C2212yes—
46#4040C2214yes—
47#4039C2212yes—
48#4039C2214yes—
49#4038C2212yes—
50#4038C2212yes—
51#4038C2214yes—
52D5×C4⋊C4C2211yescharacteristic
53D5×C4⋊C4C2212yes—
54#4036C2212yes—
55#4036C2211yescharacteristic
56#4035C2214yes—
57#4035C2214yes—
58#4035C2214yes—
59#4035C2214yes—
60#4034C2212yes—
61#4034C2214yes—
62#4033C2212yes—
63#4033C2212yes—
64#4033C2211yescharacteristic
65#4033C2212yes—
66#4033C2214yes—
67#4022C2214yes—
68#4021C2212yes—
69C4×D20C2212yes—
70C4×D20C2211yescharacteristic
71#4018C2212yes—
72#4018C2214yes—
73#4018C2214yes—
74C42×D5C2211yescharacteristic
75C42×D5C2212yes—
76#4016C2212yes—
77#4015C2211yescharacteristic
78C4×Dic10C2211yescharacteristic
79C4×Dic10C2212yes—
80C4×Dic10C2212yes—
81C4×Dic10C2214yes—
82C4×Dic10C2212yes—
83#3523—51nomaximal, Sylow
84C2×C5×Q8C2311yescharacteristic
85C2×C5×Q8C2311yescharacteristic
86C2×C5×Q8C2312yes—
87Q8×D5—24no—
88Q8×D5—28no—
89C2×D20C2311yescharacteristic
90C2×D20C2312yes—
91C2×C4×D5C2311yescharacteristic
92C2×C4×D5C2312yes—
93C2×C4×D5C2312yes—
94C2×C4×D5C2314yes—
95C2×C4×D5C2311yescharacteristic
96C2×C4×D5C2312yes—
97C2×C4×D5C2311yescharacteristic
98C2×Dic10C2312yes—
99C2×Dic10C2314yes—
100C2×Dic10C2312yes—
101C2×Dic10C2311yescharacteristic
102C5×C4⋊C4C2311yescharacteristic
103C5×C4⋊C4C2311yescharacteristic
104C5×C4⋊C4C2314yes—
105C5×C4⋊C4C2312yes—
106C5×C4⋊C4C2312yes—
107C5×C4⋊C4C2312yes—
108C5×C4⋊C4C2314yes—
109C5×C4⋊C4C2314yes—
110C5×C4⋊C4C2312yes—
111C42×C5C2312yes—
112C42×C5C2314yes—
113C42×C5C2311yescharacteristic
114C42×C5C2312yes—
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#683C2312yes—
132#683C2314yes—
133#683C2312yes—
134#683C2312yes—
135#683C2312yes—
136#683C2314yes—
137#683C2314yes—
138#683C2314yes—
139#683C2312yes—
140#683C2314yes—
141#683C2314yes—
142C4×Dic5C2311yescharacteristic
143C4×Dic5C2312yes—
144C4×Dic5C2312yes—
145C4×Dic5C2314yes—
146C4×Dic5C2314yes—
147C4×Dic5C2311yescharacteristic
148C4×Dic5C2311yescharacteristic
149C4×Dic5C2312yes—
150C4×Dic5C2314yes—
151#557—52no—
152#557—51no—
153#557D511yescharacteristic, complemented
154#556—52no—
155#556—51no—
156#552—54no—
157#552—52no—
158D4×Q8—52no—
159D4×Q8—51no—
160#541—52no—
161#541—54no—
162#533—51no—
163#533—52no—
164#529—52no—
165#529—51no—
166#519—52no—
167C2×C4×Q8—51no—
168C22×D5C2412yes—
169C22×D5C2411yescharacteristic
170C5×Q8—22no—
171C5×Q8—24no—
172C5×Q8—22no—
173C2×C4×C5C2412yes—
174C2×C4×C5C2412yes—
175C2×C4×C5C2411yescharacteristic
176C2×C4×C5C2412yes—
177C2×C4×C5C2414yes—
178C2×C4×C5C2412yes—
179C2×C4×C5C2411yescharacteristic
180C2×C4×C5C2411yescharacteristic
181C2×Dic5C2411yescharacteristic
182C2×Dic5C2411yescharacteristic
183C2×Dic5C2414yes—
184C2×Dic5C2412yes—
185C2×Dic5C2411yescharacteristic
186C2×Dic5C2412yes—
187C2×Dic5C2412yes—
188D20—24no—
189D20—22no—
190C4×D5—24no—
191C4×D5—22no—
192C4×D5—22no—
193C4×D5—24no—
194C4×D5—22no—
195C4×D5—24no—
196C4×D5—28no—
197Dic10—24no—
198Dic10—24no—
199Dic10—28no—
200Dic10#4114yes—
201C22×Q8—51no—
202C22×Q8—52no—
203C4⋊Q8—52no—
204C4⋊Q8—52no—
205C4⋊Q8—54no—
206C4⋊Q8—51no—
207C4⋊Q8C2×D511yescharacteristic
208C4⋊Q8C2×D512yes—
209#126—52no—
210#126—54no—
211#126—54no—
212#126—54no—
213#126—54no—
214#125—51no—
215#125—54no—
216#125—52no—
217#125—54no—
218#125—54no—
219#125—54no—
220#125—52no—
221#125C2×D512yes—
222#125C2×D514yes—
223#124—51no—
224#124—52no—
225#123—52no—
226#123—54no—
227C22⋊Q8—51no—
228C22⋊Q8—52no—
229C22⋊Q8—52no—
230C22⋊Q8—52no—
231C22⋊Q8—52no—
232C22⋊Q8—54no—
233C22⋊Q8—54no—
234C22⋊Q8—54no—
235#121—51no—
236C4×Q8—51no—
237C4×Q8—52no—
238C4×Q8—51no—
239C4×Q8—52no—
240C4×Q8—54no—
241C4×Q8—52no—
242C4×Q8—52no—
243C4×Q8—51no—
244C4×Q8—52no—
245C4×Q8—54no—
246C4×Q8—52no—
247C4×Q8—52no—
248C4×Q8C2×D511yescharacteristic
249C4×Q8C2×D512yes—
250C4×Q8C2×D511yescharacteristic
251C4×Q8C2×D512yes—
252C4×D4—54no—
253C4×D4—52no—
254C4×D4—52no—
255C4×D4—51no—
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—51no—
264C2×C42—52no—
265C2×C42—51no—
266C22×C5C2511yescharacteristic
267C2×D5—21no—
268C2×D5—21no—
269C2×D5—22no—
270C2×D5—22no—
271C2×D5—22no—
272C2×D52−1+412yes—
273C4×C5—22no—
274C4×C5—24no—
275C4×C5—22no—
276C4×C5—21no—
277C4×C5—22no—
278C4×C5—22no—
279C4×C5C2×#4112yes—
280C4×C52−1+412yes—
281Dic5—21no—
282Dic5—22no—
283Dic5—24no—
284Dic5—21no—
285Dic5—21no—
286Dic5—22no—
287Dic5C2×#4114yes—
288C2×Q8—51no—
289C2×Q8—52no—
290C2×Q8—52no—
291C2×Q8—54no—
292C2×Q8—104no—
293C2×Q8—108no—
294C2×Q8C22×D511yescharacteristic
295C2×Q8C22×D512yes—
296C2×Q8C22×D511yescharacteristic
297C2×D4—52no—
298C2×D4—51no—
299C22×C4—51no—
300C22×C4—51no—
301C22×C4—52no—
302C22×C4—52no—
303C22×C4—51no—
304C22×C4—54no—
305C22×C4—52no—
306C4⋊C4—52no—
307C4⋊C4—52no—
308C4⋊C4—51no—
309C4⋊C4—52no—
310C4⋊C4—52no—
311C4⋊C4—54no—
312C4⋊C4—52no—
313C4⋊C4—52no—
314C4⋊C4—54no—
315C4⋊C4—52no—
316C4⋊C4—54no—
317C4⋊C4—54no—
318C4⋊C4—52no—
319C4⋊C4—51no—
320C4⋊C4—52no—
321C4⋊C4—54no—
322C4⋊C4—54no—
323C4⋊C4—54no—
324C4⋊C4C22×D511yescharacteristic
325C4⋊C4C22×D512yes—
326C4⋊C4C22×D514yes—
327C4⋊C4C22×D512yes—
328C4⋊C4C22×D512yes—
329C4⋊C4C22×D514yes—
330C4⋊C4C22×D511yescharacteristic
331C4⋊C4C22×D514yes—
332C4⋊C4C22×D512yes—
333C22⋊C4—52no—
334C22⋊C4—52no—
335C22⋊C4—54no—
336C22⋊C4—54no—
337C22⋊C4—52no—
338C22⋊C4—54no—
339C22⋊C4—52no—
340C22⋊C4—54no—
341C22⋊C4—52no—
342C42—52no—
343C42—51no—
344C42—52no—
345C42—51no—
346C42—52no—
347C42—51no—
348C42—54no—
349C42—54no—
350C42—54no—
351C42C22×D511yescharacteristic
352C42C22×D512yes—
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—22no—
364Q8—24no—
365Q8—54no—
366Q8—104no—
367Q8—108no—
368Q8—104no—
369D4—104no—
370D4—102no—
371C2×C4—52no—
372C2×C4—51no—
373C2×C4—51no—
374C2×C4—54no—
375C2×C4—52no—
376C2×C4—51no—
377C2×C4—52no—
378C2×C4—104no—
379C2×C4—108no—
380C2×C4—104no—
381C2×C4—102no—
382C2×C4—102no—
383C2×C4—102no—
384C2×C4—104no—
385C2×C4C23×D512yes—
386C2×C4C23×D512yes—
387C2×C4C23×D511yescharacteristic
388C2×C4C23×D512yes—
389C2×C4C23×D514yes—
390C2×C4C23×D512yes—
391C2×C4C23×D511yescharacteristic
392C2×C4C23×D511yescharacteristic
393C5#352311yescharacteristic, 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—54no—
408C4—101no—
409C4—104no—
410C4—101no—
411C4—102no—
412C4—101no—
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 radical#4041C22
hypercenter#557D5
hypocenterC5#3523
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. #66065C2semidirect
3. #66062C2semidirect
4. #66061C2semidirect
5. #66061C2semidirect
6. #66030C2semidirect
7. #66027C2semidirect
9. #66019C2semidirect
10. #65918C2semidirect
11. #65915C2semidirect
13. #65912C2semidirect
14. #65906C2semidirect
15. C4×Q8×D5C2semidirect
16. #65902C2semidirect
17. #65901C2semidirect
153. #557D5semidirect
393. C5#3523semidirect

Maximal subgroups

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

subgroupindex
C5×#5572
#660652
#660622
#660612
#660302
#660272
#660252
#660192
#659182
#659152
#659142
#659122
#659062
C4×Q8×D52
#659022
#659012
#35235