Finite Group Databasebeta

← #10610631

Subgroups — #10610631

Order 640.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#66165C211yesmaximal, characteristic
2#65845C211yesmaximal, characteristic, complemented
3C5×#417C211yesmaximal, characteristic, complemented
4#65528C211yesmaximal, characteristic, complemented
5#65426C211yesmaximal, characteristic, complemented
6#65416C211yesmaximal, characteristic, complemented
7#65260C211yesmaximal, characteristic, complemented
8#65097C211yesmaximal, characteristic, complemented
9D5×#103C211yesmaximal, characteristic, complemented
10#65080C211yesmaximal, characteristic, complemented
11#65078C211yesmaximal, characteristic, complemented
12#65067C211yesmaximal, characteristic, complemented
13#65066C211yesmaximal, characteristic, complemented
14#65056C211yesmaximal, characteristic, complemented
15#65054C211yesmaximal, characteristic, complemented
16C2×C5×#41C2211yescharacteristic
17C2×Q8×D5C2211yescharacteristic
18C2×C5×M16C2211yescharacteristic
19C2×C5×C4⋊C4C2211yescharacteristic
20Q8×Dic5C2211yescharacteristic
21C2×#689C2211yescharacteristic
22#4084—21no—
23#4084C2211yescharacteristic
24#4082—21no—
25#4082C2211yescharacteristic
26C2×#687C2211yescharacteristic
27C4×C22⋊D5—21no—
28C4×C22⋊D5—21no—
29C4×C22⋊D5C2211yescharacteristic
30#4070—21no—
31#4070—21no—
32#4070C2211yescharacteristic
33#4070C2211yescharacteristic
34C2×Dic20C2211yescharacteristic
35C2×C8×D5C2211yescharacteristic
36#4043C2211yescharacteristic
37#4042—21no—
38D5×C4⋊C4C2211yescharacteristic
39#4035—21no—
40#4034C2211yescharacteristic
41#4033C2211yescharacteristic
42C5×#103C2211yescharacteristic
43C5×#103C2211yescharacteristic
44C5×#102C2211yescharacteristic
45C5×#102C2211yescharacteristic
46#3967C2211yescharacteristic
47#3964C2211yescharacteristic
48C22⋊#168C2211yescharacteristic
49#3952C2211yescharacteristic
50#3949C2211yescharacteristic
51#3948C2211yescharacteristic
52#3948C2211yescharacteristic
53#3947C2211yescharacteristic
54#3946C2211yescharacteristic
55#3942C2211yescharacteristic
56#3942C2211yescharacteristic
57#3940C2211yescharacteristic
58#3940C2211yescharacteristic
59#3316—51nomaximal, Sylow
60C5×#41—22no—
61C5×#41—41no—
62C2×C5×Q8C2311yescharacteristic
63C2×C5×D4—21no—
64C2×C5×D4C2311yescharacteristic
65C22×C4×C5—21no—
66C22×C4×C5—21no—
67C22×C4×C5C2311yescharacteristic
68C2×C22⋊D5—21no—
69C2×C22⋊D5D411yescharacteristic
70Q8×D5—22no—
71Q8×D5—41no—
72C2×C4×D5—21no—
73C2×C4×D5—21no—
74C2×C4×D5C2311yescharacteristic
75C2×Dic10—21no—
76C2×Dic10—21no—
77C2×Dic10C2311yescharacteristic
78C2×Dic10C2311yescharacteristic
79C5×M16—41no—
80C2×C5×C8C2311yescharacteristic
81C2×C5×C8C2311yescharacteristic
82C5×C4⋊C4—21no—
83C5×C4⋊C4C2311yescharacteristic
84C5×C4⋊C4C2311yescharacteristic
85C22⋊Dic5—21no—
86C22⋊Dic5—21no—
87C22⋊Dic5—21no—
88C22⋊Dic5—21no—
89C22⋊Dic5—21no—
90C22⋊Dic5D411yescharacteristic
91#689—22no—
92#687—41no—
93#685—21no—
94#685D411yescharacteristic
95C4⋊Dic5—21no—
96C4⋊Dic5—21no—
97C4⋊Dic5C2311yescharacteristic
98C4⋊Dic5C2311yescharacteristic
99#683—21no—
100#683—21no—
101#683—21no—
102#683—21no—
103#683—21no—
104#683D411yescharacteristic
105C4×Dic5—21no—
106C4×Dic5—21no—
107C4×Dic5C2311yescharacteristic
108C2×#168C2311yescharacteristic
109C2×#168C2311yescharacteristic
110Dic20—41no—
111C8×D5—41no—
112#554—51no—
113#548—51no—
114#489—51no—
115#483—51no—
116#481—51no—
117#475—51no—
118#470—51no—
119#465—51no—
120#464—51no—
121#452—51no—
122#451—51no—
123#441—51no—
124#435—51no—
125#417D511yescharacteristic, complemented
126C2×#103—51no—
127C23×C5—21no—
128C23×C5C2×D411yescharacteristic
129C22×D5C2×D411yescharacteristic
130C5×Q8—21no—
131C5×Q8#4112yes—
132C5×D4—21no—
133C5×D4—21no—
134C5×D4—42no—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
137C2×C4×C5—21no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
140C2×C4×C5—42no—
141C2×C4×C5—81no—
142C2×C4×C5C2×D411yescharacteristic
143C2×C4×C5C2411yescharacteristic
144C22⋊D5—22no—
145C22⋊D5—42no—
146C2×Dic5—21no—
147C2×Dic5—21no—
148C2×Dic5—21no—
149C2×Dic5—21no—
150C2×Dic5C2×D411yescharacteristic
151C2×Dic5C2×D411yescharacteristic
152C2×Dic5C2×D411yescharacteristic
153C4×D5—21no—
154C4×D5—21no—
155C4×D5—42no—
156C4×D5—81no—
157Dic10—21no—
158Dic10—21no—
159Dic10—21no—
160Dic10—21no—
161Dic10—42no—
162Dic10—81no—
163C5×C8—21no—
164C5×C8—21no—
165#168—21no—
166#168—21no—
167C2×#41C2×D511yescharacteristic
168C22×Q8—51no—
169C2×Q16—51no—
170C2×Q16—51no—
171C2×SD16—51no—
172C2×M16C2×D511yescharacteristic
173C22×C8—51no—
174C4⋊Q8—51no—
175#125—101no—
176#124—51no—
177#124—101no—
178C22⋊Q8—51no—
179C22⋊Q8—51no—
180C22⋊Q8—51no—
181C22⋊Q8—101no—
182C22⋊Q8—101no—
183C22⋊Q8—101no—
184#121—51no—
185#121—101no—
186C4×Q8—51no—
187C4×Q8—51no—
188C4×D4—51no—
189C4×D4—101no—
190C4×D4—101no—
191C2×C4⋊C4—51no—
192C2×C4⋊C4C2×D511yescharacteristic
193#106—51no—
194#106—51no—
195#106—51no—
196C4⋊C8—51no—
197#103—51no—
198#103—51no—
199#103—51no—
200#103—51no—
201#103—51no—
202#103C2×D511yescharacteristic
203#103C2×D511yescharacteristic
204#102—51no—
205#102C2×D511yescharacteristic
206#102C2×D511yescharacteristic
207C22⋊C8—51no—
208C22⋊C8—51no—
209#97—51no—
210C22×C5—21no—
211C22×C5—21no—
212C22×C5—21no—
213C22×C5—41no—
214C22×C5—41no—
215C22×C5—41no—
216C22×C5C22×D411yescharacteristic
217C2×D5—21no—
218C2×D5—21no—
219C2×D5#13712yes—
220C4×C5—22no—
221C4×C5—21no—
222C4×C5—41no—
223C4×C5—41no—
224C4×C5C2×#4111yescharacteristic
225C4×C5H(5,𝔽2)11yescharacteristic
226Dic5—21no—
227Dic5—21no—
228Dic5—21no—
229Dic5—41no—
230Dic5—41no—
231Dic5—41no—
232Dic5—41no—
233#41—22no—
234#41—41no—
235C2×Q8—51no—
236C2×Q8—51no—
237C2×Q8—101no—
238C2×Q8—101no—
239C2×Q8—102no—
240C2×Q8—201no—
241C2×Q8C22×D511yescharacteristic
242C2×D4—21no—
243C2×D4—51no—
244C2×D4—101no—
245C2×D4C22×D511yescharacteristic
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—51no—
249C22×C4—101no—
250C22×C4—101no—
251C22×C4C22×D511yescharacteristic
252Q16—102no—
253Q16—201no—
254SD16—201no—
255M16—41no—
256C2×C8—51no—
257C2×C8—51no—
258C2×C8—201no—
259C2×C8C22×D511yescharacteristic
260C2×C8C22×D511yescharacteristic
261C4⋊C4—21no—
262C4⋊C4—51no—
263C4⋊C4—51no—
264C4⋊C4—51no—
265C4⋊C4—101no—
266C4⋊C4—101no—
267C4⋊C4—101no—
268C4⋊C4—101no—
269C4⋊C4—101no—
270C4⋊C4—101no—
271C4⋊C4—101no—
272C4⋊C4C22×D511yescharacteristic
273C4⋊C4C22×D511yescharacteristic
274C22⋊C4—51no—
275C22⋊C4—51no—
276C22⋊C4—101no—
277C22⋊C4—101no—
278C22⋊C4—101no—
279C22⋊C4—101no—
280C22⋊C4—101no—
281C22⋊C4—101no—
282C42—51no—
283C42—101no—
284C42—101no—
285C2×C5—41no—
286C2×C5—81no—
287C2×C5#54611yescharacteristic
288C2×C5C2×#13711yescharacteristic
289C2×C5#57711yescharacteristic
290D5—22no—
291C23—21no—
292C23—51no—
293C23D4×D511yescharacteristic
294Q8—21no—
295Q8—101no—
296Q8—101no—
297Q8—101no—
298Q8—101no—
299Q8—202no—
300Q8—401no—
301Q8#70912yes—
302D4—21no—
303D4—21no—
304D4—42no—
305D4—102no—
306D4—202no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—51no—
314C2×C4—51no—
315C2×C4—51no—
316C2×C4—81no—
317C2×C4—101no—
318C2×C4—101no—
319C2×C4—101no—
320C2×C4—101no—
321C2×C4—101no—
322C2×C4—101no—
323C2×C4—202no—
324C2×C4—401no—
325C2×C4D4×D511yescharacteristic
326C2×C4C23×D511yescharacteristic
327C8—21no—
328C8—21no—
329C8—101no—
330C8—101no—
331C5#331611yescharacteristic, complemented, Sylow
332C22—21no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—52no—
339C22—101no—
340C22—101no—
341C22C2×D4×D511yescharacteristic
342C4—21no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—101no—
347C4—101no—
348C4—101no—
349C4—201no—
350C4—201no—
351C4—201no—
352C4—201no—
353C4C2×#70911yescharacteristic
354C4#414411yescharacteristic
355C2—41no—
356C2—81no—
357C2—102no—
358C2#6582411yescharacteristic
359C2#6610911yescharacteristic
360C2D5×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#417C2
socleC22×C5C22×D4
prime radical#66165C2
hypercenter#417D5
hypocenterC5#3316
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. #65845C2semidirect
3. C5×#417C2semidirect
4. #65528C2semidirect
5. #65426C2semidirect
6. #65416C2semidirect
7. #65260C2semidirect
8. #65097C2semidirect
9. D5×#103C2semidirect
10. #65080C2semidirect
11. #65078C2semidirect
12. #65067C2semidirect
13. #65066C2semidirect
14. #65056C2semidirect
15. #65054C2semidirect
125. #417D5semidirect
331. C5#3316semidirect

Maximal subgroups

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

subgroupindex
#661652
#658452
C5×#4172
#655282
#654262
#654162
#652602
#650972
D5×#1032
#650802
#650782
#650672
#650662
#650562
#650542
#33165