Finite Group Databasebeta

← #10618039

Subgroups — #10618039

Order 640.

910 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#529C211yesmaximal, characteristic
2#66035C214yesmaximal, complemented
3#66019C212yesmaximal, complemented
4#66005C212yesmaximal, complemented
5#65985C214yesmaximal, complemented
6#65959C212yesmaximal, complemented
7#65937C212yesmaximal
8#65901C211yesmaximal, characteristic, complemented
9Q8×Dic10C211yesmaximal, characteristic, complemented
10#65877C211yesmaximal, characteristic, complemented
11#65876C212yesmaximal, complemented
12#65876C211yesmaximal, characteristic, complemented
13#65875C212yesmaximal, complemented
14#65853C212yesmaximal, complemented
15#65852C212yesmaximal, complemented
16#65804C211yesmaximal, characteristic, complemented
17#65804C211yesmaximal, characteristic, complemented
18C5×#126C2212yes—
19C5×#125C2211yescharacteristic
20C5×#124C2211yescharacteristic
21C5×#123C2212yes—
22C5×C22⋊Q8C2211yescharacteristic
23C5×#121C2211yescharacteristic
24C4×C5×Q8C2211yescharacteristic
25C4×C5×D4C2212yes—
26C4×C5×D4C2211yescharacteristic
27C5×#117C2212yes—
28C2×C42×C5C2211yescharacteristic
29Q8×Dic5C2212yes—
30#4090C2212yes—
31#4082C2214yes—
32#4082C2212yes—
33#4081C2214yes—
34D4×Dic5C2212yes—
35#4072C2212yes—
36#4072C2212yes—
37#4072C2212yes—
38C2×C4⋊Dic5C2212yes—
39#4070C2211yescharacteristic
40#4070C2211yescharacteristic
41#4070C2212yes—
42#4070C2212yes—
43#4070C2212yes—
44#4070C2212yes—
45#4070C2212yes—
46#4036C2214yes—
47#4036C2212yes—
48#4035C2214yes—
49#4035C2214yes—
50#4035C2214yes—
51#4035C2214yes—
52#4034C2214yes—
53#4034C2212yes—
54#4034C2214yes—
55#4033C2212yes—
56#4033C2214yes—
57#4033C2212yes—
58#4025C2214yes—
59#4025C2214yes—
60#4025C2214yes—
61#4025C2214yes—
62#4025C2214yes—
63#4025C2214yes—
64#4024C2214yes—
65#4024C2214yes—
66#4024C2214yes—
67#4023C2214yes—
68#4016C2212yes—
69#4016C2212yes—
70#4016C2212yes—
71#4015C2211yescharacteristic
72#4015C2211yescharacteristic
73#4015C2211yescharacteristic
74#4015C2211yescharacteristic
75#4015C2212yes—
76C4×Dic10C2211yescharacteristic
77C4×Dic10C2211yescharacteristic
78C4×Dic10C2212yes—
79C4×Dic10C2212yes—
80C4×Dic10C2212yes—
81C4×Dic10C2212yes—
82C4×Dic10C2212yes—
83#3555—51nomaximal, Sylow
84C2×C5×Q8C2311yescharacteristic
85C2×C5×D4C2311yescharacteristic
86C2×C5×D4C2312yes—
87C22×C4×C5C2311yescharacteristic
88C22×C4×C5C2311yescharacteristic
89C22×C4×C5C2312yes—
90C22×C4×C5C2311yescharacteristic
91C22×Dic5C2314yes—
92C2×Dic10C2311yescharacteristic
93C2×Dic10C2311yescharacteristic
94C2×Dic10C2312yes—
95C2×Dic10C2312yes—
96C2×Dic10C2312yes—
97C2×Dic10C2314yes—
98C5×C4⋊C4C2312yes—
99C5×C4⋊C4C2312yes—
100C5×C4⋊C4C2311yescharacteristic
101C5×C4⋊C4C2312yes—
102C5×C4⋊C4C2311yescharacteristic
103C5×C4⋊C4C2312yes—
104C5×C22⋊C4C2312yes—
105C5×C22⋊C4C2312yes—
106C5×C22⋊C4C2312yes—
107C5×C22⋊C4C2312yes—
108C5×C22⋊C4C2312yes—
109C42×C5C2311yescharacteristic
110C42×C5C2311yescharacteristic
111C42×C5C2311yescharacteristic
112C42×C5C2311yescharacteristic
113C42×C5C2312yes—
114C22⋊Dic5C2314yes—
115C22⋊Dic5C2314yes—
116C22⋊Dic5C2314yes—
117C22⋊Dic5C2312yes—
118C22⋊Dic5C2314yes—
119C22⋊Dic5C2312yes—
120C4⋊Dic5C2311yescharacteristic
121C4⋊Dic5C2312yes—
122C4⋊Dic5C2311yescharacteristic
123C4⋊Dic5C2312yes—
124C4⋊Dic5C2312yes—
125C4⋊Dic5C2312yes—
126C4⋊Dic5C2312yes—
127C4⋊Dic5C2312yes—
128C4⋊Dic5C2312yes—
129C4⋊Dic5C2314yes—
130C4⋊Dic5C2314yes—
131#683C2312yes—
132#683C2312yes—
133#683C2314yes—
134#683C2314yes—
135#683C2312yes—
136#683C2312yes—
137#683C2314yes—
138#683C2314yes—
139#683C2314yes—
140#683C2314yes—
141#683C2314yes—
142#683C2314yes—
143C4×Dic5C2314yes—
144C4×Dic5C2312yes—
145C4×Dic5C2312yes—
146C4×Dic5C2314yes—
147C4×Dic5C2314yes—
148C4×Dic5C2314yes—
149#563—54no—
150Q82—51no—
151#557—52no—
152#557—51no—
153#556—52no—
154#556—52no—
155#556—51no—
156#554—51no—
157#552—52no—
158#544—52no—
159#542—52no—
160#541—54no—
161#541—52no—
162#541—52no—
163#533—51no—
164#533—51no—
165#529D511yescharacteristic
166C23×C5C2411yescharacteristic
167C23×C5C2412yes—
168C5×Q8—22no—
169C5×D4—22no—
170C5×D4—42no—
171C2×C4×C5—22no—
172C2×C4×C5—24no—
173C2×C4×C5—22no—
174C2×C4×C5—41no—
175C2×C4×C5C2411yescharacteristic
176C2×C4×C5C2412yes—
177C2×C4×C5C2412yes—
178C2×C4×C5C2412yes—
179C2×C4×C5C2411yescharacteristic
180C2×C4×C5C2411yescharacteristic
181C2×C4×C5C2411yescharacteristic
182C2×C4×C5C2411yescharacteristic
183C2×C4×C5C2411yescharacteristic
184C2×Dic5—44no—
185C2×Dic5C2414yes—
186C2×Dic5C2412yes—
187C2×Dic5C2412yes—
188C2×Dic5C2412yes—
189C2×Dic5C2414yes—
190C2×Dic5C2412yes—
191Dic10—22no—
192Dic10—22no—
193Dic10—24no—
194Dic10—44no—
195Dic10—42no—
196Dic10—42no—
197C4⋊Q8—52no—
198C4⋊Q8—54no—
199C4⋊Q8—52no—
200C4⋊Q8—54no—
201C4⋊Q8—51no—
202C4⋊Q8—51no—
203C4⋊Q8—51no—
204C4⋊Q8—51no—
205C4⋊Q8—52no—
206#126—54no—
207#126—54no—
208#126—54no—
209#126—54no—
210#126—54no—
211#126—54no—
212#126C2×D512yes—
213#125—54no—
214#125—54no—
215#125—54no—
216#125—52no—
217#125—54no—
218#125—54no—
219#125—52no—
220#125—52no—
221#125—52no—
222#125C2×D511yescharacteristic
223#124—52no—
224#124—54no—
225#124C2×D511yescharacteristic
226#123—54no—
227#123C2×D512yes—
228C22⋊Q8—52no—
229C22⋊Q8—51no—
230C22⋊Q8—51no—
231C22⋊Q8—52no—
232C22⋊Q8—52no—
233C22⋊Q8—52no—
234C22⋊Q8—52no—
235C22⋊Q8—54no—
236C22⋊Q8—54no—
237C22⋊Q8—54no—
238C22⋊Q8C2×D511yescharacteristic
239#121C2×D511yescharacteristic
240C4×Q8—52no—
241C4×Q8—54no—
242C4×Q8—52no—
243C4×Q8—52no—
244C4×Q8—52no—
245C4×Q8—51no—
246C4×Q8—52no—
247C4×Q8—52no—
248C4×Q8—51no—
249C4×Q8—52no—
250C4×Q8—52no—
251C4×Q8C2×D511yescharacteristic
252C4×D4—52no—
253C4×D4C2×D512yes—
254C4×D4C2×D511yescharacteristic
255#117—52no—
256#117—52no—
257#117—52no—
258#117—54no—
259#117C2×D512yes—
260C2×C4⋊C4—52no—
261C2×C42C2×D511yescharacteristic
262C22×C5—21no—
263C22×C5—21no—
264C22×C5—22no—
265C22×C5—22no—
266C22×C5—22no—
267C22×C5—21no—
268C22×C5C2511yescharacteristic
269C4×C5—21no—
270C4×C5—22no—
271C4×C5—21no—
272C4×C5—22no—
273C4×C5—21no—
274C4×C5—21no—
275C4×C5—22no—
276C4×C52−1+412yes—
277C4×C52−1+412yes—
278Dic5—22no—
279Dic5—22no—
280Dic5—24no—
281Dic5—22no—
282Dic5—24no—
283Dic5—22no—
284C2×Q8—51no—
285C2×Q8—51no—
286C2×Q8—52no—
287C2×Q8—52no—
288C2×Q8—52no—
289C2×Q8—54no—
290C2×Q8C22×D511yescharacteristic
291C2×D4C22×D511yescharacteristic
292C2×D4C22×D512yes—
293C22×C4—54no—
294C22×C4C22×D511yescharacteristic
295C22×C4C22×D511yescharacteristic
296C22×C4C22×D512yes—
297C22×C4C22×D511yescharacteristic
298C4⋊C4—51no—
299C4⋊C4—52no—
300C4⋊C4—54no—
301C4⋊C4—52no—
302C4⋊C4—52no—
303C4⋊C4—51no—
304C4⋊C4—52no—
305C4⋊C4—54no—
306C4⋊C4—52no—
307C4⋊C4—52no—
308C4⋊C4—52no—
309C4⋊C4—52no—
310C4⋊C4—54no—
311C4⋊C4—54no—
312C4⋊C4—54no—
313C4⋊C4—54no—
314C4⋊C4—54no—
315C4⋊C4—52no—
316C4⋊C4—52no—
317C4⋊C4—52no—
318C4⋊C4—54no—
319C4⋊C4—54no—
320C4⋊C4—54no—
321C4⋊C4C22×D512yes—
322C4⋊C4C22×D512yes—
323C4⋊C4C22×D511yescharacteristic
324C4⋊C4C22×D512yes—
325C4⋊C4C22×D511yescharacteristic
326C4⋊C4C22×D512yes—
327C22⋊C4—52no—
328C22⋊C4—54no—
329C22⋊C4—52no—
330C22⋊C4—54no—
331C22⋊C4—54no—
332C22⋊C4—54no—
333C22⋊C4C22×D512yes—
334C22⋊C4C22×D512yes—
335C22⋊C4C22×D512yes—
336C22⋊C4C22×D512yes—
337C22⋊C4C22×D512yes—
338C42—54no—
339C42—52no—
340C42—52no—
341C42—54no—
342C42—54no—
343C42—54no—
344C42C22×D511yescharacteristic
345C42C22×D511yescharacteristic
346C42C22×D511yescharacteristic
347C42C22×D511yescharacteristic
348C42C22×D512yes—
349C2×C5—41no—
350C2×C5—42no—
351C2×C5C2×2−1+411yescharacteristic
352C2×C5C2×2−1+411yescharacteristic
353C2×C5#58511yescharacteristic
354C23C23×D511yescharacteristic
355C23C23×D512yes—
356Q8—22no—
357Q8—102no—
358Q8—102no—
359Q8—104no—
360Q8—202no—
361Q8—202no—
362Q8—204no—
363D4—22no—
364D4—42no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—54no—
370C2×C4—52no—
371C2×C4—52no—
372C2×C4—52no—
373C2×C4—54no—
374C2×C4—52no—
375C2×C4—204no—
376C2×C4C23×D511yescharacteristic
377C2×C4C23×D512yes—
378C2×C4C23×D512yes—
379C2×C4C23×D512yes—
380C2×C4C23×D511yescharacteristic
381C2×C4C23×D511yescharacteristic
382C2×C4C23×D511yescharacteristic
383C2×C4C23×D511yescharacteristic
384C2×C4C23×D511yescharacteristic
385C5#355511yescharacteristic, complemented, Sylow
386C22—21no—
387C22—21no—
388C22—22no—
389C22—22no—
390C22—21no—
391C22—22no—
392C22C24×D511yescharacteristic
393C4—21no—
394C4—22no—
395C4—21no—
396C4—22no—
397C4—21no—
398C4—21no—
399C4—22no—
400C4—102no—
401C4—102no—
402C4—104no—
403C4—102no—
404C4—104no—
405C4—102no—
406C4#415012yes—
407C4#415012yes—
408C2—41no—
409C2—42no—
410C2C2×#415011yescharacteristic
411C2C2×#415011yescharacteristic
412C2#6628311yescharacteristic
subgroupG/H
center, FrattiniC22C24×D5
commutator, socle, cosocleC22×C5C25
FittingC5×#529C2
prime radicalC5×#121C22
hypercenter#529D5
hypocenterC5#3555
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ #529

Complements

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

NH ≅ G/Ntype
2. #66035C2semidirect
3. #66019C2semidirect
4. #66005C2semidirect
5. #65985C2semidirect
6. #65959C2semidirect
8. #65901C2semidirect
9. Q8×Dic10C2semidirect
10. #65877C2semidirect
11. #65876C2semidirect
12. #65876C2semidirect
13. #65875C2semidirect
14. #65853C2semidirect
15. #65852C2semidirect
16. #65804C2semidirect
17. #65804C2semidirect
385. C5#3555semidirect

Maximal subgroups

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

subgroupindex
C5×#5292
#660352
#660192
#660052
#659852
#659592
#659372
#659012
Q8×Dic102
#658772
#658762
#658752
#658532
#658522
#658042
#35555