Finite Group Databasebeta

← #10614568

Subgroups — #10614568

Order 640.

532 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#123C211yesmaximal, characteristic
2#65510C211yesmaximal, characteristic
3#65509C211yesmaximal, characteristic
4#65501C211yesmaximal, characteristic, complemented
5C4×C22⋊Dic5C211yesmaximal, characteristic, complemented
6#65274C211yesmaximal, characteristic, complemented
7#65270C211yesmaximal, characteristic, complemented
8#65270C211yesmaximal, characteristic, complemented
9C4⋊C4×Dic5C211yesmaximal, characteristic, complemented
10#65240C211yesmaximal, characteristic, complemented
11#65240C211yesmaximal, characteristic, complemented
12#65240C211yesmaximal, characteristic, complemented
13#65240C211yesmaximal, characteristic, complemented
14#65239C211yesmaximal, characteristic, complemented
15C22⋊C4×Dic5C211yesmaximal, characteristic, complemented
16C22×C5×D4C2211yescharacteristic
17C23×C4×C5C2211yescharacteristic
18C23×Dic5C2211yescharacteristic
19C5×#123C418yescomplemented
20C2×C5×C4⋊C4C2211yescharacteristic
21C2×C5×C4⋊C4C2211yescharacteristic
22C2×C5×C22⋊C4C2211yescharacteristic
23C2×C5×C22⋊C4C2211yescharacteristic
24C2×C5×C22⋊C4C2211yescharacteristic
25C2×C22⋊Dic5C2211yescharacteristic
26C2×C22⋊Dic5C2211yescharacteristic
27C2×C22⋊Dic5C2211yescharacteristic
28C2×C22⋊Dic5C2211yescharacteristic
29C2×C22⋊Dic5C2211yescharacteristic
30C2×C22⋊Dic5C2211yescharacteristic
31C2×C22⋊Dic5C2211yescharacteristic
32C2×C4⋊Dic5C2211yescharacteristic
33C2×C4⋊Dic5C2211yescharacteristic
34C2×C4⋊Dic5C2211yescharacteristic
35C2×C4⋊Dic5C2211yescharacteristic
36C2×C4⋊Dic5C2211yescharacteristic
37C2×C4×Dic5C2211yescharacteristic
38C2×C4×Dic5C2211yescharacteristic
39C2×C4×Dic5C2211yescharacteristic
40C2×C4×Dic5C2211yescharacteristic
41C2×C4×Dic5C2211yescharacteristic
42#3963C2211yescharacteristic
43#3963C2211yescharacteristic
44#3963C2211yescharacteristic
45#3963C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#2335—51nomaximal, Sylow
53C24×C5C2311yescharacteristic
54C24×C5C2311yescharacteristic
55C2×C5×D4—41no—
56C2×C5×D4—41no—
57C2×C5×D4C2×C414yes—
58C22×C4×C5—41no—
59C22×C4×C5—41no—
60C22×C4×C5C2×C414yescomplemented
61C22×C4×C5C2311yescharacteristic
62C22×C4×C5C2311yescharacteristic
63C22×C4×C5C2311yescharacteristic
64C22×C4×C5C2311yescharacteristic
65C22×C4×C5C2311yescharacteristic
66C22×Dic5—22no—
67C22×Dic5—41no—
68C22×Dic5—41no—
69C22×Dic5C2311yescharacteristic
70C22×Dic5C2311yescharacteristic
71C22×Dic5C2311yescharacteristic
72C22×Dic5C2311yescharacteristic
73C22×Dic5C2311yescharacteristic
74C22×Dic5C2311yescharacteristic
75C22×Dic5C2311yescharacteristic
76C22×Dic5C2311yescharacteristic
77C5×C4⋊C4C2×C414yescomplemented
78C5×C4⋊C4C2×C414yescomplemented
79C5×C22⋊C4C2×C414yes—
80C5×C22⋊C4C2×C414yes—
81C5×C22⋊C4C2×C414yescomplemented
82C22⋊Dic5—22no—
83C22⋊Dic5—22no—
84C22⋊Dic5—22no—
85C22⋊Dic5—41no—
86C22⋊Dic5—41no—
87C22⋊Dic5—41no—
88C22⋊Dic5—41no—
89C4⋊Dic5—22no—
90C4⋊Dic5—41no—
91C4⋊Dic5—41no—
92C4⋊Dic5—41no—
93C4⋊Dic5—41no—
94C4×Dic5—22no—
95C4×Dic5—41no—
96C4×Dic5—41no—
97C4×Dic5—41no—
98C4×Dic5—41no—
99C2×#123D511yescharacteristic
100#390—51no—
101#389—51no—
102#388—51no—
103#388—51no—
104#388—51no—
105#388—51no—
106#387—51no—
107#387—51no—
108#386—51no—
109#385—51no—
110#380—51no—
111C4×C4⋊C4—51no—
112C4×C22⋊C4—51no—
113C4×C22⋊C4—51no—
114C23×C5—21no—
115C23×C5—21no—
116C23×C5—21no—
117C23×C5—21no—
118C23×C5—21no—
119C23×C5—21no—
120C23×C5—21no—
121C23×C5—21no—
122C23×C5—21no—
123C23×C5—21no—
124C23×C5—21no—
125C23×C5—21no—
126C23×C5C22×C412yes—
127C23×C5C22×C412yes—
128C23×C5C2411yescharacteristic
129C5×D4—44no—
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—21no—
140C2×C4×C5—44no—
141C2×C4×C5C22×C412yes—
142C2×C4×C5C22×C412yes—
143C2×C4×C5C22×C412yes—
144C2×C4×C5C22×C412yes—
145C2×C4×C5C22×C412yes—
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—21no—
165C2×Dic5—21no—
166C2×Dic5—21no—
167C2×Dic5—21no—
168C2×Dic5—44no—
169C2×Dic5#4112yes—
170C2×Dic5#4112yes—
171C22×D4C2×D511yescharacteristic
172C23×C4—51no—
173C23×C4C2×D511yescharacteristic
174#123Dic518yescomplemented
175C2×C4⋊C4—51no—
176C2×C4⋊C4—51no—
177C2×C4⋊C4—51no—
178C2×C4⋊C4—51no—
179C2×C4⋊C4—51no—
180C2×C4⋊C4C2×D511yescharacteristic
181C2×C4⋊C4C2×D511yescharacteristic
182C2×C22⋊C4—51no—
183C2×C22⋊C4—51no—
184C2×C22⋊C4—51no—
185C2×C22⋊C4—51no—
186C2×C22⋊C4—51no—
187C2×C22⋊C4—51no—
188C2×C22⋊C4—51no—
189C2×C22⋊C4C2×D511yescharacteristic
190C2×C22⋊C4C2×D511yescharacteristic
191C2×C22⋊C4C2×D511yescharacteristic
192C2×C42—51no—
193C2×C42—51no—
194C2×C42—51no—
195C2×C42—51no—
196C2×C42—51no—
197#95—51no—
198#95—51no—
199#95—51no—
200#95—51no—
201#95—51no—
202#95—51no—
203#95—51no—
204#95—51no—
205#95—51no—
206#95—51no—
207C22×C5—22no—
208C22×C5—22no—
209C22×C5—22no—
210C22×C5—22no—
211C22×C5—22no—
212C22×C5—22no—
213C22×C5—41no—
214C22×C5—41no—
215C22×C5—41no—
216C22×C5—41no—
217C22×C5—41no—
218C22×C5—41no—
219C22×C5—41no—
220C22×C5—41no—
221C22×C5C23×C411yescharacteristic
222C22×C5C2×#4111yescharacteristic
223C22×C5C2×#4111yescharacteristic
224C22×C5H(5,𝔽2)11yescharacteristic
225C22×C5H(5,𝔽2)11yescharacteristic
226C22×C5H(5,𝔽2)11yescharacteristic
227C22×C52−1+411yescharacteristic
228C4×C5—22no—
229C4×C5—22no—
230C4×C5—22no—
231C4×C5—22no—
232C4×C5—22no—
233Dic5—22no—
234Dic5—22no—
235Dic5—41no—
236Dic5—41no—
237Dic5—41no—
238Dic5—41no—
239Dic5—41no—
240Dic5—41no—
241C24C22×D511yescharacteristic
242C24C22×D511yescharacteristic
243C2×D4—41no—
244C2×D4—41no—
245C2×D4C2×Dic514yes—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—51no—
249C22×C4—51no—
250C22×C4—51no—
251C22×C4—51no—
252C22×C4—51no—
253C22×C4—51no—
254C22×C4—51no—
255C22×C4—51no—
256C22×C4—102no—
257C22×C4—201no—
258C22×C4—201no—
259C22×C4C2×Dic514yescomplemented
260C22×C4C22×D511yescharacteristic
261C22×C4C22×D511yescharacteristic
262C22×C4C22×D511yescharacteristic
263C22×C4C22×D511yescharacteristic
264C22×C4C22×D511yescharacteristic
265C4⋊C4—102no—
266C4⋊C4—201no—
267C4⋊C4—201no—
268C4⋊C4—201no—
269C4⋊C4—201no—
270C4⋊C4C2×Dic514yescomplemented
271C4⋊C4C2×Dic514yescomplemented
272C22⋊C4—102no—
273C22⋊C4—102no—
274C22⋊C4—102no—
275C22⋊C4—201no—
276C22⋊C4—201no—
277C22⋊C4—201no—
278C22⋊C4—201no—
279C22⋊C4C2×Dic514yes—
280C22⋊C4C2×Dic514yes—
281C22⋊C4C2×Dic514yescomplemented
282C42—102no—
283C42—201no—
284C42—201no—
285C42—201no—
286C42—201no—
287C2×C5—42no—
288C2×C5—42no—
289C2×C5C4×#4111yescharacteristic
290C2×C5#51811yescharacteristic
291C2×C5#52011yescharacteristic
292C2×C5#53811yescharacteristic
293C2×C5#53911yescharacteristic
294C2×C5#54011yescharacteristic
295C2×C5#54211yescharacteristic
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23C22×Dic512yes—
309C23C22×Dic512yes—
310C23C23×D511yescharacteristic
311D4—44no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—44no—
323C2×C4—52no—
324C2×C4—52no—
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—101no—
343C2×C4—101no—
344C2×C4—101no—
345C2×C4—101no—
346C2×C4—101no—
347C2×C4—204no—
348C2×C4C22×Dic512yes—
349C2×C4C22×Dic512yes—
350C2×C4C22×Dic512yes—
351C2×C4C22×Dic512yes—
352C2×C4C22×Dic512yes—
353C5#233511yescharacteristic, complemented, Sylow
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22#414411yescharacteristic
369C22#414411yescharacteristic
370C22D5×#4111yescharacteristic
371C22D5×#4111yescharacteristic
372C22#414911yescharacteristic
373C22#415011yescharacteristic
374C22C23×Dic511yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—22no—
379C4—22no—
380C4—102no—
381C4—102no—
382C4—201no—
383C4—201no—
384C4—201no—
385C4—201no—
386C4—201no—
387C4—201no—
388C2—42no—
389C2—42no—
390C2#6599111yescharacteristic
391C2#6599411yescharacteristic
392C2#6599611yescharacteristic
393C2#6599711yescharacteristic
394C2#6613211yescharacteristic
395C2#41×Dic511yescharacteristic
396C2#6616111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutatorC22×C5C23×C4
FittingC2×C5×#123C2
socle, cosocleC23×C5C24
prime radicalC22×C5×D4C22
hypercenterC2×#123D5
hypocenterC5#2335
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
4. #65501C2semidirect
5. C4×C22⋊Dic5C2semidirect
6. #65274C2semidirect
7. #65270C2semidirect
8. #65270C2semidirect
9. C4⋊C4×Dic5C2semidirect
10. #65240C2semidirect
11. #65240C2semidirect
12. #65240C2semidirect
13. #65240C2semidirect
14. #65239C2semidirect
15. C22⋊C4×Dic5C2semidirect
19. C5×#123C4semidirect
60. C22×C4×C5C2×C4semidirect
77. C5×C4⋊C4C2×C4semidirect
78. C5×C4⋊C4C2×C4semidirect
81. C5×C22⋊C4C2×C4semidirect
174. #123Dic5semidirect
259. C22×C4C2×Dic5semidirect
270. C4⋊C4C2×Dic5semidirect
271. C4⋊C4C2×Dic5semidirect
281. C22⋊C4C2×Dic5semidirect
353. C5#2335semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1232
#655102
#655092
#655012
C4×C22⋊Dic52
#652742
#652702
C4⋊C4×Dic52
#652402
#652392
C22⋊C4×Dic52
#23355