Finite Group Databasebeta

← #10606474

Subgroups — #10606474

Order 640.

482 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#386C211yesmaximal, characteristic
2#65509C211yesmaximal, characteristic
3#65509C211yesmaximal, characteristic
4#65501C211yesmaximal, characteristic, complemented
5C2×#3963C211yesmaximal, characteristic, complemented
6#65242C211yesmaximal, characteristic, complemented
7#65241C211yesmaximal, characteristic, complemented
8#65240C211yesmaximal, characteristic, complemented
9#65235C211yesmaximal, characteristic, complemented
10#65234C211yesmaximal, characteristic, complemented
11C22⋊C4×Dic5C211yesmaximal, characteristic, complemented
12C5⋊#401C211yesmaximal, characteristic, complemented
13#64949C211yesmaximal, characteristic, complemented
14#64942C211yesmaximal, characteristic, complemented
15#64940C211yesmaximal, characteristic, complemented
16C22×C5×D4C2211yescharacteristic
17C23×C4×C5C2211yescharacteristic
18C23×Dic5C2211yescharacteristic
19C23×Dic5C2211yescharacteristic
20C2×C5×C22⋊C4C2211yescharacteristic
21C2×C5×C22⋊C4C2211yescharacteristic
22C2×C5×C22⋊C4C2211yescharacteristic
23C2×C22⋊Dic5C2211yescharacteristic
24C2×C22⋊Dic5C2211yescharacteristic
25C2×C22⋊Dic5C2211yescharacteristic
26C2×C22⋊Dic5C2211yescharacteristic
27C2×C22⋊Dic5C2211yescharacteristic
28C2×C22⋊Dic5C2211yescharacteristic
29C2×C4⋊Dic5C2211yescharacteristic
30C2×C4⋊Dic5C2211yescharacteristic
31C2×#683C2211yescharacteristic
32C2×#683C2211yescharacteristic
33C2×C4×Dic5C2211yescharacteristic
34C2×C4×Dic5C2211yescharacteristic
35C5×#95C2211yescharacteristic
36C5×#95C2211yescharacteristic
37#3963C2211yescharacteristic
38#3963C2211yescharacteristic
39#3963C2211yescharacteristic
40#3963C2211yescharacteristic
41#3963C2211yescharacteristic
42#3963C2211yescharacteristic
43#3963C2211yescharacteristic
44#3963C2211yescharacteristic
45#3963C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#2510—51nomaximal, Sylow
52C24×C5C2311yescharacteristic
53C24×C5C2311yescharacteristic
54C2×C5×D4—22no—
55C2×C5×D4—22no—
56C2×C5×D4—22no—
57C22×C4×C5—22no—
58C22×C4×C5—22no—
59C22×C4×C5—41no—
60C22×C4×C5C2311yescharacteristic
61C22×C4×C5C2311yescharacteristic
62C22×C4×C5C2311yescharacteristic
63C22×C4×C5C2311yescharacteristic
64C22×C4×C5C2311yescharacteristic
65C22×Dic5—22no—
66C22×Dic5—22no—
67C22×Dic5—22no—
68C22×Dic5—41no—
69C22×Dic5—41no—
70C22×Dic5—41no—
71C22×Dic5C2311yescharacteristic
72C22×Dic5C2311yescharacteristic
73C22×Dic5C2311yescharacteristic
74C22×Dic5C2311yescharacteristic
75C22×Dic5C2311yescharacteristic
76C22×Dic5C2311yescharacteristic
77C22×Dic5C2311yescharacteristic
78C22×Dic5C2311yescharacteristic
79C5×C22⋊C4—22no—
80C5×C22⋊C4—22no—
81C5×C22⋊C4—41no—
82C22⋊Dic5—22no—
83C22⋊Dic5—22no—
84C22⋊Dic5—22no—
85C22⋊Dic5—41no—
86C22⋊Dic5—41no—
87C22⋊Dic5—41no—
88C4⋊Dic5—22no—
89C4⋊Dic5—41no—
90#683—41no—
91#683—41no—
92C4×Dic5—22no—
93C4×Dic5—41no—
94#401—51no—
95#400—51no—
96#397—51no—
97#397—51no—
98#388—51no—
99#388—51no—
100#387—51no—
101#386—51no—
102#386—51no—
103#386D511yescharacteristic
104#385—51no—
105#383—51no—
106#380—51no—
107C4×C22⋊C4—51no—
108C2×#95—51no—
109C23×C5—21no—
110C23×C5—21no—
111C23×C5—21no—
112C23×C5—21no—
113C23×C5—21no—
114C23×C5—21no—
115C23×C5—21no—
116C23×C5—21no—
117C23×C5—21no—
118C23×C5—21no—
119C23×C5—21no—
120C23×C5#4112yes—
121C23×C5#4112yes—
122C23×C5#4112yes—
123C23×C5C2411yescharacteristic
124C5×D4—44no—
125C2×C4×C5—21no—
126C2×C4×C5—21no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—21no—
133C2×C4×C5—21no—
134C2×C4×C5—21no—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
137C2×C4×C5—21no—
138C2×C4×C5—21no—
139C2×C4×C5—44no—
140C2×C4×C5#4112yes—
141C2×Dic5—21no—
142C2×Dic5—21no—
143C2×Dic5—21no—
144C2×Dic5—21no—
145C2×Dic5—21no—
146C2×Dic5—21no—
147C2×Dic5—21no—
148C2×Dic5—21no—
149C2×Dic5—21no—
150C2×Dic5—21no—
151C2×Dic5—21no—
152C2×Dic5—21no—
153C2×Dic5—21no—
154C2×Dic5—21no—
155C2×Dic5—21no—
156C2×Dic5—21no—
157C2×Dic5—21no—
158C2×Dic5—21no—
159C2×Dic5—21no—
160C2×Dic5—21no—
161C2×Dic5—21no—
162C2×Dic5—21no—
163C2×Dic5—21no—
164C2×Dic5—44no—
165C2×Dic5—44no—
166C2×Dic5#4112yes—
167C22×D4C2×D511yescharacteristic
168C23×C4—51no—
169C23×C4—51no—
170C23×C4C2×D511yescharacteristic
171C2×C4⋊C4—51no—
172C2×C4⋊C4—51no—
173C2×C4⋊C4—51no—
174C2×C4⋊C4—51no—
175C2×C22⋊C4—51no—
176C2×C22⋊C4—51no—
177C2×C22⋊C4—51no—
178C2×C22⋊C4—51no—
179C2×C22⋊C4—51no—
180C2×C22⋊C4—51no—
181C2×C22⋊C4C2×D511yescharacteristic
182C2×C22⋊C4C2×D511yescharacteristic
183C2×C22⋊C4C2×D511yescharacteristic
184C2×C42—51no—
185C2×C42—51no—
186#95—51no—
187#95—51no—
188#95—51no—
189#95—51no—
190#95—51no—
191#95—51no—
192#95—51no—
193#95—51no—
194#95—51no—
195#95—51no—
196#95—51no—
197#95—51no—
198#95—51no—
199#95—51no—
200#95C2×D511yescharacteristic
201#95C2×D511yescharacteristic
202C22×C5—22no—
203C22×C5—22no—
204C22×C5—22no—
205C22×C5—22no—
206C22×C5—22no—
207C22×C5—22no—
208C22×C5—41no—
209C22×C5—41no—
210C22×C5—41no—
211C22×C5—41no—
212C22×C5—41no—
213C22×C5—41no—
214C22×C5—41no—
215C22×C5#12614yes—
216C22×C5C2×#4111yescharacteristic
217C22×C5C2×#4111yescharacteristic
218C22×C5C2×#4111yescharacteristic
219C22×C5C2×#4111yescharacteristic
220C22×C5C2×#4111yescharacteristic
221C22×C5H(5,𝔽2)11yescharacteristic
222C22×C5H(5,𝔽2)11yescharacteristic
223C4×C5—22no—
224C4×C5—41no—
225C4×C5—41no—
226C4×C5—41no—
227C4×C5—41no—
228Dic5—22no—
229Dic5—41no—
230Dic5—41no—
231Dic5—41no—
232Dic5—41no—
233Dic5—41no—
234Dic5—41no—
235Dic5—41no—
236C24C22×D511yescharacteristic
237C24C22×D511yescharacteristic
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—41no—
244C22×C4—51no—
245C22×C4—51no—
246C22×C4—51no—
247C22×C4—51no—
248C22×C4—51no—
249C22×C4—51no—
250C22×C4—51no—
251C22×C4—51no—
252C22×C4—102no—
253C22×C4—102no—
254C22×C4—102no—
255C22×C4—201no—
256C22×C4—201no—
257C22×C4—201no—
258C22×C4C22×D511yescharacteristic
259C22×C4C22×D511yescharacteristic
260C22×C4C22×D511yescharacteristic
261C22×C4C22×D511yescharacteristic
262C22×C4C22×D511yescharacteristic
263C4⋊C4—102no—
264C4⋊C4—201no—
265C4⋊C4—201no—
266C4⋊C4—201no—
267C22⋊C4—22no—
268C22⋊C4—22no—
269C22⋊C4—41no—
270C22⋊C4—102no—
271C22⋊C4—102no—
272C22⋊C4—102no—
273C22⋊C4—201no—
274C22⋊C4—201no—
275C22⋊C4—201no—
276C42—102no—
277C42—201no—
278C2×C5—24no—
279C2×C5—81no—
280C2×C5C2×#12611yescharacteristic
281C2×C5#52911yescharacteristic
282C2×C5#53811yescharacteristic
283C2×C5#55111yescharacteristic
284C2×C5#55111yescharacteristic
285C2×C5#55311yescharacteristic
286C2×C5#55311yescharacteristic
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23#70912yes—
299C23#71112yes—
300C23#71112yes—
301C23C23×D511yescharacteristic
302D4—44no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—44no—
318C2×C4—52no—
319C2×C4—101no—
320C2×C4—101no—
321C2×C4—101no—
322C2×C4—101no—
323C2×C4—101no—
324C2×C4—101no—
325C2×C4—101no—
326C2×C4—101no—
327C2×C4—101no—
328C2×C4—101no—
329C2×C4—101no—
330C2×C4—101no—
331C2×C4—101no—
332C2×C4—101no—
333C2×C4—101no—
334C2×C4—101no—
335C2×C4—101no—
336C2×C4—101no—
337C2×C4—101no—
338C2×C4—101no—
339C2×C4—101no—
340C2×C4—101no—
341C2×C4—101no—
342C2×C4—204no—
343C2×C4—204no—
344C2×C4#71112yes—
345C5#251011yescharacteristic, complemented, Sylow
346C22—22no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22#402514yes—
360C22C2×#70911yescharacteristic
361C22C2×#71111yescharacteristic
362C22C2×#71111yescharacteristic
363C22C2×#71111yescharacteristic
364C22#414411yescharacteristic
365C22D5×#4111yescharacteristic
366C22#414911yescharacteristic
367C4—22no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—102no—
373C4—201no—
374C4—201no—
375C4—201no—
376C4—201no—
377C4—201no—
378C4—201no—
379C4—201no—
380C2—24no—
381C2—81no—
382C2C2×#402511yescharacteristic
383C2#6587411yescharacteristic
384C2#6589911yescharacteristic
385C2#6592111yescharacteristic
386C2#6593411yescharacteristic
387C2#6595111yescharacteristic
388C2#6599211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#386C2
prime radicalC22×C5×D4C22
hypercenter#386D5
hypocenterC5#2510
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
4. #65501C2semidirect
5. C2×#3963C2semidirect
6. #65242C2semidirect
7. #65241C2semidirect
8. #65240C2semidirect
9. #65235C2semidirect
10. #65234C2semidirect
11. C22⋊C4×Dic5C2semidirect
12. C5⋊#401C2semidirect
13. #64949C2semidirect
14. #64942C2semidirect
15. #64940C2semidirect
345. C5#2510semidirect

Maximal subgroups

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

subgroupindex
C5×#3862
#655092
#655012
C2×#39632
#652422
#652412
#652402
#652352
#652342
C22⋊C4×Dic52
C5⋊#4012
#649492
#649422
#649402
#25105