Finite Group Databasebeta

← #10607201

Subgroups — #10607201

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×#4029C211yesmaximal, characteristic
2C5×#388C211yesmaximal, characteristic, complemented
3#65276C211yesmaximal, characteristic, complemented
4#65261C211yesmaximal, characteristic, complemented
5#65247C211yesmaximal, characteristic
6#65246C211yesmaximal, characteristic
7#65239C211yesmaximal, characteristic, complemented
8#65233C211yesmaximal, characteristic, complemented
9C22⋊C4×Dic5C211yesmaximal, characteristic, complemented
10#65229C211yesmaximal, characteristic, complemented
11C4×C4⋊Dic5C211yesmaximal, characteristic, complemented
12#64959C211yesmaximal, characteristic, complemented
13#64955C211yesmaximal, characteristic, complemented
14#64944C211yesmaximal, characteristic, complemented
15#64943C211yesmaximal, characteristic, complemented
16C22×C22⋊D5C2211yescharacteristic
17C23×Dic5C2211yescharacteristic
18C22×C4×D5C2211yescharacteristic
19C2×C5×C4⋊C4C2211yescharacteristic
20C2×C5×C22⋊C4C2211yescharacteristic
21C2×C5×C22⋊C4C2211yescharacteristic
22C2×C5×C22⋊C4C2211yescharacteristic
23C2×C42×C5C2211yescharacteristic
24C2×C22⋊Dic5C2211yescharacteristic
25C2×C22⋊Dic5C2211yescharacteristic
26C2×#685C2211yescharacteristic
27C2×#685C2211yescharacteristic
28C2×#685C2211yescharacteristic
29C2×#685C2211yescharacteristic
30C2×#685C2211yescharacteristic
31C2×C4⋊Dic5C2211yescharacteristic
32C2×C4⋊Dic5C2211yescharacteristic
33C2×C4⋊Dic5C2211yescharacteristic
34C2×#683C2211yescharacteristic
35C2×#683C2211yescharacteristic
36C2×#683C2211yescharacteristic
37C2×C4×Dic5C2211yescharacteristic
38C2×C4×Dic5C2211yescharacteristic
39C2×C4×Dic5C2211yescharacteristic
40C2×C4×Dic5C2211yescharacteristic
41#4029C418yescomplemented
42C5×#95C2211yescharacteristic
43C5×#95C2211yescharacteristic
44#3963C2211yescharacteristic
45#3963C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#2335—51nomaximal, Sylow
53C24×C5C2311yescharacteristic
54C23×D5C2311yescharacteristic
55C22×C4×C5C2311yescharacteristic
56C22×C4×C5C2311yescharacteristic
57C22×C4×C5C2311yescharacteristic
58C22×C4×C5C2311yescharacteristic
59C22×C4×C5C2311yescharacteristic
60C22×C4×C5C2311yescharacteristic
61C2×C22⋊D5—41no—
62C2×C22⋊D5—41no—
63C2×C22⋊D5C2×C414yes—
64C22×Dic5—22no—
65C22×Dic5—41no—
66C22×Dic5—41no—
67C22×Dic5C2311yescharacteristic
68C22×Dic5C2311yescharacteristic
69C22×Dic5C2311yescharacteristic
70C22×Dic5C2311yescharacteristic
71C22×Dic5C2311yescharacteristic
72C22×Dic5C2311yescharacteristic
73C22×Dic5C2311yescharacteristic
74C2×C4×D5—41no—
75C2×C4×D5—41no—
76C2×C4×D5C2×C414yescomplemented
77C5×C4⋊C4—41no—
78C5×C22⋊C4—22no—
79C5×C22⋊C4—41no—
80C5×C22⋊C4C2×C414yes—
81C42×C5—22no—
82C22⋊Dic5—41no—
83C22⋊Dic5C2×C414yes—
84#685—22no—
85#685—22no—
86#685—41no—
87#685—41no—
88#685C2×C414yescomplemented
89C4⋊Dic5—22no—
90C4⋊Dic5—41no—
91C4⋊Dic5C2×C414yescomplemented
92#683—41no—
93#683—41no—
94#683C2×C414yescomplemented
95C4×Dic5—41no—
96C4×Dic5—41no—
97C4×Dic5—41no—
98C4×Dic5—41no—
99C2×#123—51no—
100#390—51no—
101#389—51no—
102#388—51no—
103#388—51no—
104#388—51no—
105#388D511yescharacteristic, complemented
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×C5C22×C412yes—
121C23×C5C2411yescharacteristic
122C22×D5—21no—
123C22×D5—21no—
124C22×D5—21no—
125C22×D5—21no—
126C22×D5—21no—
127C22×D5—21no—
128C22×D5C22×C412yes—
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—21no—
140C2×C4×C5—21no—
141C2×C4×C5—21no—
142C2×C4×C5—21no—
143C2×C4×C5—21no—
144C2×C4×C5C22×C412yes—
145C2×C4×C5C22×C412yes—
146C2×C4×C5#4112yes—
147C22⋊D5—44no—
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—44no—
166C2×Dic5C22×C412yes—
167C2×Dic5C22×C412yes—
168C2×Dic5C22×C412yes—
169C2×Dic5#4112yes—
170C4×D5—44no—
171C22×D4—51no—
172C23×C4—51no—
173C23×C4—51no—
174#123—58no—
175C2×C4⋊C4—51no—
176C2×C4⋊C4—51no—
177C2×C4⋊C4—51no—
178C2×C4⋊C4—51no—
179C2×C4⋊C4—51no—
180C2×C4⋊C4—51no—
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×C42C2×D511yescharacteristic
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#95C2×D511yescharacteristic
206#95C2×D511yescharacteristic
207C22×C5—22no—
208C22×C5—22no—
209C22×C5—22no—
210C22×C5—41no—
211C22×C5—41no—
212C22×C5—41no—
213C22×C5—41no—
214C22×C5C23×C411yescharacteristic
215C22×C5C2×#4111yescharacteristic
216C22×C5C2×#4111yescharacteristic
217C22×C5H(5,𝔽2)11yescharacteristic
218C22×C5H(5,𝔽2)11yescharacteristic
219C22×C5H(5,𝔽2)11yescharacteristic
220C22×C52−1+411yescharacteristic
221C2×D5—22no—
222C2×D5—22no—
223C2×D5—22no—
224C2×D5—41no—
225C2×D5—41no—
226C2×D5—41no—
227C2×D5—41no—
228C4×C5—22no—
229C4×C5—22no—
230C4×C5—22no—
231C4×C5—41no—
232C4×C5—41no—
233C4×C5—41no—
234Dic5—22no—
235Dic5—22no—
236Dic5—22no—
237Dic5—22no—
238Dic5—41no—
239Dic5—41no—
240Dic5—41no—
241C24—51no—
242C24C22×D511yescharacteristic
243C2×D4—54no—
244C2×D4—201no—
245C2×D4—201no—
246C22×C4—51no—
247C22×C4—51no—
248C22×C4—51no—
249C22×C4—51no—
250C22×C4—54no—
251C22×C4—51no—
252C22×C4—51no—
253C22×C4—51no—
254C22×C4—102no—
255C22×C4—201no—
256C22×C4—201no—
257C22×C4—201no—
258C22×C4—201no—
259C22×C4C22×D511yescharacteristic
260C22×C4C22×D511yescharacteristic
261C22×C4C22×D511yescharacteristic
262C22×C4C22×D511yescharacteristic
263C22×C4C22×D511yescharacteristic
264C22×C4C22×D511yescharacteristic
265C4⋊C4—41no—
266C4⋊C4—54no—
267C4⋊C4—54no—
268C4⋊C4—102no—
269C4⋊C4—201no—
270C4⋊C4—201no—
271C4⋊C4—201no—
272C22⋊C4—22no—
273C22⋊C4—41no—
274C22⋊C4—54no—
275C22⋊C4—54no—
276C22⋊C4—102no—
277C22⋊C4—102no—
278C22⋊C4—201no—
279C22⋊C4—201no—
280C22⋊C4—201no—
281C22⋊C4C4×D514yes—
282C42—22no—
283C42—201no—
284C42—201no—
285C42—201no—
286C42—201no—
287C2×C5—42no—
288C2×C5C4×#4111yescharacteristic
289C2×C5#51811yescharacteristic
290C2×C5#52011yescharacteristic
291C2×C5#53811yescharacteristic
292C2×C5#53911yescharacteristic
293C2×C5#54011yescharacteristic
294C2×C5#54211yescharacteristic
295D5—42no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—52no—
303C23—101no—
304C23—101no—
305C23—101no—
306C23—101no—
307C23—101no—
308C23—101no—
309C23C2×C4×D512yes—
310C23C23×D511yescharacteristic
311D4—204no—
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—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—52no—
328C2×C4—52no—
329C2×C4—52no—
330C2×C4—52no—
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—101no—
348C2×C4—204no—
349C2×C4—204no—
350C2×C4C2×C4×D512yes—
351C2×C4C2×C4×D512yes—
352C2×C4#71112yes—
353C5#233511yescharacteristic, complemented, Sylow
354C22—22no—
355C22—22no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—102no—
362C22—102no—
363C22—102no—
364C22—201no—
365C22—201no—
366C22—201no—
367C22—201no—
368C22C22×C4×D511yescharacteristic
369C22C2×#71111yescharacteristic
370C22#414411yescharacteristic
371C22D5×#4111yescharacteristic
372C22#414911yescharacteristic
373C22#414911yescharacteristic
374C22#415011yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—102no—
382C4—102no—
383C4—102no—
384C4—102no—
385C4—201no—
386C4—201no—
387C4—201no—
388C2—42no—
389C2—202no—
390C2#6585711yescharacteristic
391C2C4×#71111yescharacteristic
392C2#6587911yescharacteristic
393C2#6595311yescharacteristic
394C2#6599911yescharacteristic
395C2#6600311yescharacteristic
396C2#6603811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutatorC22×C5C23×C4
FittingC5×#388C2
socle, cosocleC23×C5C24
prime radicalC22×C22⋊D5C22
hypercenter#388D5
hypocenterC5#2335
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C5×#388C2semidirect
3. #65276C2semidirect
4. #65261C2semidirect
7. #65239C2semidirect
8. #65233C2semidirect
9. C22⋊C4×Dic5C2semidirect
10. #65229C2semidirect
11. C4×C4⋊Dic5C2semidirect
12. #64959C2semidirect
13. #64955C2semidirect
14. #64944C2semidirect
15. #64943C2semidirect
41. #4029C4semidirect
76. C2×C4×D5C2×C4semidirect
88. #685C2×C4semidirect
91. C4⋊Dic5C2×C4semidirect
94. #683C2×C4semidirect
105. #388D5semidirect
353. C5#2335semidirect

Maximal subgroups

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

subgroupindex
C2×#40292
C5×#3882
#652762
#652612
#652472
#652462
#652392
#652332
C22⋊C4×Dic52
#652292
C4×C4⋊Dic52
#649592
#649552
#649442
#649432
#23355