Finite Group Databasebeta

← #10607708

Subgroups — #10607708

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×#4040C211yesmaximal, characteristic
2C5×#389C211yesmaximal, characteristic, complemented
3#65279C211yesmaximal, characteristic, complemented
4#65279C211yesmaximal, characteristic, complemented
5#65279C211yesmaximal, characteristic, complemented
6#65277C211yesmaximal, characteristic
7#65276C211yesmaximal, characteristic, complemented
8#65270C211yesmaximal, characteristic, complemented
9#65267C211yesmaximal, characteristic, complemented
10#65229C211yesmaximal, characteristic, complemented
11C4×#685C211yesmaximal, characteristic, complemented
12#64957C211yesmaximal, characteristic, complemented
13#64955C211yesmaximal, characteristic, complemented
14#64954C211yesmaximal, characteristic
15#64938C211yesmaximal, characteristic, complemented
16C22×D20C2211yescharacteristic
17C22×C4×D5C2211yescharacteristic
18C22×C4×D5C2211yescharacteristic
19C2×C5×C4⋊C4C2211yescharacteristic
20C2×C5×C4⋊C4C2211yescharacteristic
21C2×C5×C4⋊C4C2211yescharacteristic
22C2×C5×C4⋊C4C2211yescharacteristic
23C2×C42×C5C2211yescharacteristic
24C2×#685C2211yescharacteristic
25C2×#685C2211yescharacteristic
26C2×#685C2211yescharacteristic
27C2×#685C2211yescharacteristic
28C2×#685C2211yescharacteristic
29C2×#685C2211yescharacteristic
30C2×#685C2211yescharacteristic
31C2×#685C2211yescharacteristic
32C2×#685C2211yescharacteristic
33C2×#685C2211yescharacteristic
34C2×C4⋊Dic5C2211yescharacteristic
35C2×#683C2211yescharacteristic
36C2×#683C2211yescharacteristic
37C2×C4×Dic5C2211yescharacteristic
38C2×C4×Dic5C2211yescharacteristic
39C2×C4×Dic5C2211yescharacteristic
40C2×C4×Dic5C2211yescharacteristic
41#4040C418yescomplemented
42C5×#95C2211yescharacteristic
43C5×#95C2211yescharacteristic
44#3963C2211yescharacteristic
45#3963C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#2335—51nomaximal, Sylow
53C23×D5C2311yescharacteristic
54C23×D5C2311yescharacteristic
55C22×C4×C5C2311yescharacteristic
56C22×C4×C5C2311yescharacteristic
57C22×C4×C5C2311yescharacteristic
58C22×C4×C5C2311yescharacteristic
59C22×C4×C5C2311yescharacteristic
60C22×C4×C5C2311yescharacteristic
61C22×C4×C5C2311yescharacteristic
62C22×Dic5C2311yescharacteristic
63C22×Dic5C2311yescharacteristic
64C22×Dic5C2311yescharacteristic
65C22×Dic5C2311yescharacteristic
66C22×Dic5C2311yescharacteristic
67C22×Dic5C2311yescharacteristic
68C2×D20—41no—
69C2×D20—41no—
70C2×D20C2×C414yes—
71C2×C4×D5—22no—
72C2×C4×D5—41no—
73C2×C4×D5—41no—
74C2×C4×D5—41no—
75C2×C4×D5—41no—
76C2×C4×D5C2×C414yescomplemented
77C5×C4⋊C4—41no—
78C5×C4⋊C4—41no—
79C5×C4⋊C4—41no—
80C5×C4⋊C4C2×C414yescomplemented
81C42×C5—41no—
82#685—22no—
83#685—22no—
84#685—22no—
85#685—41no—
86#685—41no—
87#685—41no—
88#685—41no—
89#685C2×C414yes—
90#685C2×C414yes—
91#685C2×C414yescomplemented
92C4⋊Dic5—41no—
93#683—22no—
94#683C2×C414yescomplemented
95C4×Dic5—22no—
96C4×Dic5—41no—
97C4×Dic5—41no—
98C4×Dic5—41no—
99C2×#123—51no—
100#390—51no—
101#389D511yescharacteristic, complemented
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×C5C2411yescharacteristic
115C22×D5—21no—
116C22×D5—21no—
117C22×D5—21no—
118C22×D5—21no—
119C22×D5—21no—
120C22×D5—21no—
121C22×D5—21no—
122C22×D5—21no—
123C22×D5—21no—
124C22×D5—21no—
125C22×D5—21no—
126C22×D5—21no—
127C22×D5C22×C412yes—
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×C5—21no—
145C2×C4×C5—21no—
146C2×C4×C5C22×C412yes—
147C2×C4×C5C22×C412yes—
148C2×C4×C5C22×C412yes—
149C2×C4×C5#4112yes—
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×Dic5C22×C412yes—
166C2×Dic5C22×C412yes—
167C2×Dic5#4112yes—
168D20—44no—
169C4×D5—44no—
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⋊C4C2×D511yescharacteristic
179C2×C4⋊C4C2×D511yescharacteristic
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⋊C4—51no—
190C2×C22⋊C4—51no—
191C2×C22⋊C4—51no—
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×C5C23×C411yescharacteristic
208C22×C5C2×#4111yescharacteristic
209C22×C5C2×#4111yescharacteristic
210C22×C5H(5,𝔽2)11yescharacteristic
211C22×C5H(5,𝔽2)11yescharacteristic
212C22×C5H(5,𝔽2)11yescharacteristic
213C22×C52−1+411yescharacteristic
214C2×D5—22no—
215C2×D5—22no—
216C2×D5—22no—
217C2×D5—22no—
218C2×D5—22no—
219C2×D5—22no—
220C2×D5—41no—
221C2×D5—41no—
222C2×D5—41no—
223C2×D5—41no—
224C2×D5—41no—
225C2×D5—41no—
226C2×D5—41no—
227C2×D5—41no—
228C4×C5—22no—
229C4×C5—22no—
230C4×C5—22no—
231C4×C5—22no—
232C4×C5—41no—
233C4×C5—41no—
234C4×C5—41no—
235Dic5—22no—
236Dic5—22no—
237Dic5—22no—
238Dic5—41no—
239Dic5—41no—
240Dic5—41no—
241C24—51no—
242C24—51no—
243C2×D4—54no—
244C2×D4—201no—
245C2×D4—201no—
246C22×C4—51no—
247C22×C4—51no—
248C22×C4—51no—
249C22×C4—54no—
250C22×C4—51no—
251C22×C4—51no—
252C22×C4—51no—
253C22×C4—102no—
254C22×C4—201no—
255C22×C4—201no—
256C22×C4—201no—
257C22×C4—201no—
258C22×C4C22×D511yescharacteristic
259C22×C4C22×D511yescharacteristic
260C22×C4C22×D511yescharacteristic
261C22×C4C22×D511yescharacteristic
262C22×C4C22×D511yescharacteristic
263C22×C4C22×D511yescharacteristic
264C22×C4C22×D511yescharacteristic
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—54no—
269C4⋊C4—102no—
270C4⋊C4—201no—
271C4⋊C4C4×D514yescomplemented
272C22⋊C4—54no—
273C22⋊C4—54no—
274C22⋊C4—54no—
275C22⋊C4—102no—
276C22⋊C4—102no—
277C22⋊C4—102no—
278C22⋊C4—201no—
279C22⋊C4—201no—
280C22⋊C4—201no—
281C22⋊C4—201no—
282C42—41no—
283C42—102no—
284C42—201no—
285C42—201no—
286C42—201no—
287C2×C5C4×#4111yescharacteristic
288C2×C5#51811yescharacteristic
289C2×C5#52011yescharacteristic
290C2×C5#53811yescharacteristic
291C2×C5#53911yescharacteristic
292C2×C5#54011yescharacteristic
293C2×C5#54211yescharacteristic
294D5—42no—
295D5—42no—
296C23—52no—
297C23—52no—
298C23—101no—
299C23—101no—
300C23—101no—
301C23—101no—
302C23—101no—
303C23—101no—
304C23—101no—
305C23—101no—
306C23—101no—
307C23—101no—
308C23—101no—
309C23—101no—
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—21no—
328C2×C4—21no—
329C2×C4—52no—
330C2×C4—52no—
331C2×C4—52no—
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×C4—204no—
349C2×C4C2×C4×D512yes—
350C2×C4C2×C4×D512yes—
351C2×C4C2×C4×D512yes—
352C2×C4#71312yes—
353C5#233511yescharacteristic, complemented, Sylow
354C22—102no—
355C22—102no—
356C22—102no—
357C22—102no—
358C22—102no—
359C22—102no—
360C22—201no—
361C22—201no—
362C22—201no—
363C22—201no—
364C22—201no—
365C22—201no—
366C22—201no—
367C22—201no—
368C22C22×C4×D511yescharacteristic
369C22#414411yescharacteristic
370C22#414411yescharacteristic
371C22C2×#71311yescharacteristic
372C22#414711yescharacteristic
373C22D5×#4111yescharacteristic
374C22#414911yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—22no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—102no—
383C4—102no—
384C4—102no—
385C4—201no—
386C4—201no—
387C4—201no—
388C2—202no—
389C2—202no—
390C2#6583811yescharacteristic
391C2#6587911yescharacteristic
392C2C4×#71311yescharacteristic
393C2#6594911yescharacteristic
394C2#6597811yescharacteristic
395C2#6602611yescharacteristic
396C2#6606711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutatorC22×C5C23×C4
FittingC5×#389C2
socle, cosocleC23×C5C24
prime radicalC22×D20C22
hypercenter#389D5
hypocenterC5#2335
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
2. C5×#389C2semidirect
3. #65279C2semidirect
4. #65279C2semidirect
5. #65279C2semidirect
7. #65276C2semidirect
8. #65270C2semidirect
9. #65267C2semidirect
10. #65229C2semidirect
11. C4×#685C2semidirect
12. #64957C2semidirect
13. #64955C2semidirect
15. #64938C2semidirect
41. #4040C4semidirect
76. C2×C4×D5C2×C4semidirect
80. C5×C4⋊C4C2×C4semidirect
91. #685C2×C4semidirect
94. #683C2×C4semidirect
101. #389D5semidirect
271. C4⋊C4C4×D5semidirect
353. C5#2335semidirect

Maximal subgroups

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

subgroupindex
C2×#40402
C5×#3892
#652792
#652772
#652762
#652702
#652672
#652292
C4×#6852
#649572
#649552
#649542
#649382
#23355