Finite Group Databasebeta

← #10609717

Subgroups — #10609717

Order 640.

528 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4043C211yesmaximal, characteristic, complemented
2C2×#4042C211yesmaximal, characteristic, complemented
3C5×#398C211yesmaximal, characteristic, complemented
4#65283C211yesmaximal, characteristic, complemented
5#65281C211yesmaximal, characteristic, complemented
6#65281C211yesmaximal, characteristic, complemented
7#65278C211yesmaximal, characteristic, complemented
8#65277C211yesmaximal, characteristic
9#65273C211yesmaximal, characteristic, complemented
10#64965C211yesmaximal, characteristic, complemented
11#64957C211yesmaximal, characteristic, complemented
12#64954C211yesmaximal, characteristic
13#64954C211yesmaximal, characteristic
14#64936C211yesmaximal, characteristic, complemented
15#64935C211yesmaximal, characteristic, complemented
16C22×D20C2211yescharacteristic
17C22×C4×D5C2211yescharacteristic
18C22×C4×D5C2211yescharacteristic
19C22×Dic10C2211yescharacteristic
20C2×C5×C4⋊C4C2211yescharacteristic
21C2×C5×C4⋊C4C2211yescharacteristic
22C2×C5×C4⋊C4C2211yescharacteristic
23C2×C5×C4⋊C4C2211yescharacteristic
24C2×#685C2211yescharacteristic
25C2×#685C2211yescharacteristic
26C2×#685C2211yescharacteristic
27C2×#685C2211yescharacteristic
28C2×#685C2211yescharacteristic
29C2×#685C2211yescharacteristic
30C2×#685C2211yescharacteristic
31C2×#685C2211yescharacteristic
32C2×#685C2211yescharacteristic
33C2×#685C2211yescharacteristic
34C2×C4⋊Dic5C2211yescharacteristic
35C2×C4⋊Dic5C2211yescharacteristic
36C2×#683C2211yescharacteristic
37C2×#683C2211yescharacteristic
38C2×#683C2211yescharacteristic
39C2×#683C2211yescharacteristic
40C2×#683C2211yescharacteristic
41C2×#683C2211yescharacteristic
42C2×C4×Dic5C2211yescharacteristic
43#4043—24no—
44#4042—24no—
45C5×#95C2211yescharacteristic
46C5×#95C2211yescharacteristic
47C5×#95C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#2732—51nomaximal, Sylow
54C23×D5C2311yescharacteristic
55C23×D5C2311yescharacteristic
56C22×C4×C5C2311yescharacteristic
57C22×C4×C5C2311yescharacteristic
58C22×C4×C5C2311yescharacteristic
59C22×C4×C5C2311yescharacteristic
60C22×C4×C5C2311yescharacteristic
61C22×C4×C5C2311yescharacteristic
62C22×C4×C5C2311yescharacteristic
63C22×Dic5C2311yescharacteristic
64C22×Dic5C2311yescharacteristic
65C22×Dic5C2311yescharacteristic
66C22×Dic5C2311yescharacteristic
67C22×Dic5C2311yescharacteristic
68C22×Dic5C2311yescharacteristic
69C2×D20—41no—
70C2×D20—41no—
71C2×D20Q814yescomplemented
72C2×C4×D5—22no—
73C2×C4×D5—22no—
74C2×C4×D5—41no—
75C2×C4×D5—41no—
76C2×C4×D5—41no—
77C2×C4×D5—41no—
78C2×Dic10—41no—
79C2×Dic10—41no—
80C2×Dic10D414yescomplemented
81C5×C4⋊C4—22no—
82C5×C4⋊C4—22no—
83C5×C4⋊C4—41no—
84C5×C4⋊C4—41no—
85#685—22no—
86#685—22no—
87#685—22no—
88#685—22no—
89#685—22no—
90#685—22no—
91#685—41no—
92#685—41no—
93#685—41no—
94#685—41no—
95C4⋊Dic5—22no—
96C4⋊Dic5—22no—
97#683—22no—
98#683—22no—
99#683—22no—
100#683—41no—
101#683—41no—
102#683—41no—
103C4×Dic5—41no—
104C2×C22⋊Q8—51no—
105C2×C22⋊Q8—51no—
106#399—51no—
107#399—51no—
108#398—51no—
109#398D511yescharacteristic, complemented
110#396—51no—
111#396—51no—
112#391—51no—
113#390—51no—
114#389—51no—
115#386—51no—
116#386—51no—
117#385—51no—
118#385—51no—
119C23×C5C2411yescharacteristic
120C22×D5—21no—
121C22×D5—21no—
122C22×D5—21no—
123C22×D5—21no—
124C22×D5—21no—
125C22×D5—21no—
126C22×D5—21no—
127C22×D5—21no—
128C22×D5—21no—
129C22×D5—21no—
130C22×D5—21no—
131C22×D5—21no—
132C22×D5C2×Q812yes—
133C22×D5C2×Q812yes—
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×C5—21no—
147C2×C4×C5—21no—
148C2×C4×C5—21no—
149C2×C4×C5—21no—
150C2×C4×C5—21no—
151C2×C4×C5—21no—
152C2×C4×C5—21no—
153C2×C4×C5C2×D412yes—
154C2×C4×C5C2×Q812yes—
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—21no—
169C2×Dic5—21no—
170C2×Dic5C2×D412yes—
171C2×Dic5C2×D412yes—
172C2×Dic5#4112yes—
173D20—44no—
174C4×D5—44no—
175C4×D5—44no—
176Dic10—44no—
177C22×Q8—51no—
178C22×D4—51no—
179C23×C4—51no—
180C23×C4—51no—
181C22⋊Q8—104no—
182C22⋊Q8—104no—
183C2×C4⋊C4—51no—
184C2×C4⋊C4—51no—
185C2×C4⋊C4—51no—
186C2×C4⋊C4—51no—
187C2×C4⋊C4—51no—
188C2×C4⋊C4—51no—
189C2×C4⋊C4—51no—
190C2×C4⋊C4—51no—
191C2×C4⋊C4C2×D511yescharacteristic
192C2×C4⋊C4C2×D511yescharacteristic
193C2×C4⋊C4C2×D511yescharacteristic
194C2×C4⋊C4C2×D511yescharacteristic
195C2×C22⋊C4—51no—
196C2×C22⋊C4—51no—
197C2×C22⋊C4—51no—
198C2×C22⋊C4—51no—
199C2×C22⋊C4—51no—
200C2×C22⋊C4—51no—
201C2×C22⋊C4—51no—
202C2×C22⋊C4—51no—
203C2×C22⋊C4—51no—
204C2×C22⋊C4—51no—
205C2×C42—51no—
206#95—51no—
207#95—51no—
208#95—51no—
209#95—51no—
210#95—51no—
211#95C2×D511yescharacteristic
212#95C2×D511yescharacteristic
213#95C2×D511yescharacteristic
214C22×C5C22×D411yescharacteristic
215C22×C5C22×Q811yescharacteristic
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—22no—
225C2×D5—22no—
226C2×D5—22no—
227C2×D5—41no—
228C2×D5—41no—
229C2×D5—41no—
230C2×D5—41no—
231C2×D5—41no—
232C2×D5—41no—
233C2×D5—41no—
234C2×D5—41no—
235C4×C5—22no—
236C4×C5—22no—
237C4×C5—41no—
238C4×C5—41no—
239C4×C5—41no—
240C4×C5—41no—
241C4×C5—41no—
242Dic5—22no—
243Dic5—22no—
244Dic5—22no—
245Dic5—41no—
246Dic5—41no—
247Dic5—41no—
248C24—51no—
249C24—51no—
250C2×Q8—54no—
251C2×Q8—201no—
252C2×Q8—201no—
253C2×D4—54no—
254C2×D4—201no—
255C2×D4—201no—
256C22×C4—51no—
257C22×C4—51no—
258C22×C4—51no—
259C22×C4—51no—
260C22×C4—51no—
261C22×C4—51no—
262C22×C4—102no—
263C22×C4—102no—
264C22×C4—201no—
265C22×C4—201no—
266C22×C4—201no—
267C22×C4—201no—
268C22×C4C22×D511yescharacteristic
269C22×C4C22×D511yescharacteristic
270C22×C4C22×D511yescharacteristic
271C22×C4C22×D511yescharacteristic
272C22×C4C22×D511yescharacteristic
273C22×C4C22×D511yescharacteristic
274C22×C4C22×D511yescharacteristic
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—102no—
280C4⋊C4—102no—
281C4⋊C4—102no—
282C4⋊C4—102no—
283C4⋊C4—102no—
284C4⋊C4—201no—
285C4⋊C4—201no—
286C4⋊C4—201no—
287C22⋊C4—102no—
288C22⋊C4—102no—
289C22⋊C4—102no—
290C22⋊C4—102no—
291C22⋊C4—102no—
292C22⋊C4—102no—
293C22⋊C4—201no—
294C22⋊C4—201no—
295C22⋊C4—201no—
296C22⋊C4—201no—
297C42—201no—
298C2×C5#53411yescharacteristic
299C2×C5#53911yescharacteristic
300C2×C5#54311yescharacteristic
301C2×C5D4×Q811yescharacteristic
302C2×C5#55011yescharacteristic
303C2×C5#55411yescharacteristic
304C2×C5#56211yescharacteristic
305D5—42no—
306D5—42no—
307C23—52no—
308C23—52no—
309C23—101no—
310C23—101no—
311C23—101no—
312C23—101no—
313C23—101no—
314C23—101no—
315C23—101no—
316C23—101no—
317C23—101no—
318C23—101no—
319C23—101no—
320C23—101no—
321C23C23×D511yescharacteristic
322Q8—204no—
323D4—204no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—52no—
344C2×C4—52no—
345C2×C4—52no—
346C2×C4—101no—
347C2×C4—101no—
348C2×C4—101no—
349C2×C4—101no—
350C2×C4—101no—
351C2×C4—101no—
352C2×C4—101no—
353C2×C4—101no—
354C2×C4—101no—
355C2×C4—101no—
356C2×C4—101no—
357C2×C4—101no—
358C2×C4—101no—
359C2×C4—101no—
360C2×C4—101no—
361C2×C4—204no—
362C2×C4—204no—
363C2×C4D4×D512yes—
364C2×C4Q8×D512yes—
365C5#273211yescharacteristic, complemented, Sylow
366C22—102no—
367C22—102no—
368C22—102no—
369C22—102no—
370C22—102no—
371C22—102no—
372C22—201no—
373C22—201no—
374C22—201no—
375C22—201no—
376C22—201no—
377C22—201no—
378C22—201no—
379C22—201no—
380C22C2×D4×D511yescharacteristic
381C22#414411yescharacteristic
382C22C2×Q8×D511yescharacteristic
383C22#414711yescharacteristic
384C22D5×#4111yescharacteristic
385C22#414911yescharacteristic
386C22#414911yescharacteristic
387C4—22no—
388C4—22no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—102no—
395C4—102no—
396C4—102no—
397C4—201no—
398C4—201no—
399C4—201no—
400C2—202no—
401C2—202no—
402C2#6593311yescharacteristic
403C2#6596811yescharacteristic
404C2#6598111yescharacteristic
405C2#6598711yescharacteristic
406C2#6602411yescharacteristic
407C2#6602611yescharacteristic
408C2#6606111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#398C2
prime radicalC22×D20C22
hypercenter#398D5
hypocenterC5#2732
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#4043C2semidirect
2. C2×#4042C2semidirect
3. C5×#398C2semidirect
4. #65283C2semidirect
5. #65281C2semidirect
6. #65281C2semidirect
7. #65278C2semidirect
9. #65273C2semidirect
10. #64965C2semidirect
11. #64957C2semidirect
14. #64936C2semidirect
15. #64935C2semidirect
71. C2×D20Q8semidirect
80. C2×Dic10D4semidirect
109. #398D5semidirect
365. C5#2732semidirect

Maximal subgroups

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

subgroupindex
C2×#40432
C2×#40422
C5×#3982
#652832
#652812
#652782
#652772
#652732
#649652
#649572
#649542
#649362
#649352
#27325