Finite Group Databasebeta

← #10606218

Subgroups — #10606218

Order 640.

564 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×C4×C22⋊D5C211yesmaximal, characteristic
2C2×#4033C211yesmaximal, characteristic, complemented
3C5×#385C211yesmaximal, characteristic, complemented
4#65499C211yesmaximal, characteristic, complemented
5#65281C211yesmaximal, characteristic, complemented
6#65278C211yesmaximal, characteristic, complemented
7#65261C211yesmaximal, characteristic, complemented
8#65246C211yesmaximal, characteristic
9#65246C211yesmaximal, characteristic
10#65238C211yesmaximal, characteristic, complemented
11#65236C211yesmaximal, characteristic, complemented
12#64962C211yesmaximal, characteristic, complemented
13#64957C211yesmaximal, characteristic, complemented
14#64939C211yesmaximal, characteristic, complemented
15#64936C211yesmaximal, characteristic, complemented
16C23×C4×C5C2211yescharacteristic
17C22×C22⋊D5C2211yescharacteristic
18C22×C4×D5C2211yescharacteristic
19C22×Dic10C2211yescharacteristic
20C2×C5×C4⋊C4C2211yescharacteristic
21C2×C5×C4⋊C4C2211yescharacteristic
22C2×C5×C22⋊C4C2211yescharacteristic
23C2×C5×C22⋊C4C2211yescharacteristic
24C2×C22⋊Dic5C2211yescharacteristic
25C2×C22⋊Dic5C2211yescharacteristic
26C2×C22⋊Dic5C2211yescharacteristic
27C4×C22⋊D5—24no—
28C2×#685C2211yescharacteristic
29C2×#685C2211yescharacteristic
30C2×#685C2211yescharacteristic
31C2×#685C2211yescharacteristic
32C2×#685C2211yescharacteristic
33C2×C4⋊Dic5C2211yescharacteristic
34C2×#683C2211yescharacteristic
35C2×#683C2211yescharacteristic
36C2×#683C2211yescharacteristic
37C2×#683C2211yescharacteristic
38C2×#683C2211yescharacteristic
39C2×#683C2211yescharacteristic
40C2×#683C2211yescharacteristic
41C2×C4×Dic5C2211yescharacteristic
42C2×C4×Dic5C2211yescharacteristic
43C2×C4×Dic5C2211yescharacteristic
44#4033—24no—
45C5×#95C2211yescharacteristic
46C5×#95C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#2580—51nomaximal, Sylow
54C24×C5C2311yescharacteristic
55C23×D5C2311yescharacteristic
56C22×C4×C5—22no—
57C22×C4×C5—41no—
58C22×C4×C5—41no—
59C22×C4×C5C2311yescharacteristic
60C22×C4×C5C2311yescharacteristic
61C22×C4×C5C2311yescharacteristic
62C22×C4×C5C2311yescharacteristic
63C22×C4×C5C2311yescharacteristic
64C22×C4×C5C2311yescharacteristic
65C2×C22⋊D5—22no—
66C2×C22⋊D5—22no—
67C2×C22⋊D5Q814yescomplemented
68C22×Dic5C2311yescharacteristic
69C22×Dic5C2311yescharacteristic
70C22×Dic5C2311yescharacteristic
71C22×Dic5C2311yescharacteristic
72C22×Dic5C2311yescharacteristic
73C22×Dic5C2311yescharacteristic
74C22×Dic5C2311yescharacteristic
75C2×C4×D5—22no—
76C2×C4×D5—41no—
77C2×C4×D5—41no—
78C2×Dic10—22no—
79C2×Dic10—22no—
80C2×Dic10—22no—
81C5×C4⋊C4—22no—
82C5×C4⋊C4—41no—
83C5×C22⋊C4—22no—
84C5×C22⋊C4—41no—
85C22⋊Dic5—22no—
86C22⋊Dic5—22no—
87C22⋊Dic5—41no—
88#685—22no—
89#685—22no—
90#685—22no—
91#685—41no—
92#685—41no—
93C4⋊Dic5—41no—
94#683—22no—
95#683—22no—
96#683—22no—
97#683—22no—
98#683—22no—
99#683—41no—
100#683—41no—
101C4×Dic5—22no—
102C4×Dic5—22no—
103C4×Dic5D414yes—
104C2×C4×Q8—51no—
105C2×C4×D4—51no—
106#399—51no—
107#399—51no—
108#393—51no—
109#393—51no—
110#390—51no—
111#390—51no—
112#389—51no—
113#389—51no—
114#385—51no—
115#385—51no—
116#385—51no—
117#385D511yescharacteristic, complemented
118#382—51no—
119C23×C5—21no—
120C23×C5—21no—
121C23×C5—21no—
122C23×C5—21no—
123C23×C5—21no—
124C23×C5—21no—
125C23×C5C2×Q812yes—
126C23×C5C2411yescharacteristic
127C22×D5—21no—
128C22×D5—21no—
129C22×D5—21no—
130C22×D5—21no—
131C22×D5—21no—
132C22×D5—21no—
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—44no—
151C2×C4×C5C2×D412yes—
152C2×C4×C5#4112yes—
153C22⋊D5—28no—
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—21no—
169C2×Dic5—21no—
170C2×Dic5C2×D412yes—
171C2×Dic5C2×D412yes—
172C2×Dic5C2×Q812yes—
173C2×Dic5#4112yes—
174C2×Dic5#4112yes—
175C4×D5—44no—
176Dic10—28no—
177C22×Q8—51no—
178C22×D4—51no—
179C23×C4—51no—
180C23×C4C2×D511yescharacteristic
181C4×Q8—104no—
182C4×D4—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×C22⋊C4—51no—
194C2×C22⋊C4—51no—
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⋊C4C2×D511yescharacteristic
202C2×C22⋊C4C2×D511yescharacteristic
203C2×C42—51no—
204C2×C42—51no—
205C2×C42—51no—
206#95—51no—
207#95—51no—
208#95—51no—
209#95—51no—
210#95—51no—
211#95—51no—
212#95C2×D511yescharacteristic
213#95C2×D511yescharacteristic
214C22×C5—22no—
215C22×C5—22no—
216C22×C5—22no—
217C22×C5—41no—
218C22×C5—41no—
219C22×C5—41no—
220C22×C5—41no—
221C22×C5C22×D411yescharacteristic
222C22×C5C22×Q811yescharacteristic
223C22×C5C2×#4111yescharacteristic
224C22×C5C2×#4111yescharacteristic
225C22×C5C2×#4111yescharacteristic
226C22×C5H(5,𝔽2)11yescharacteristic
227C22×C5H(5,𝔽2)11yescharacteristic
228C2×D5—22no—
229C2×D5—22no—
230C2×D5—22no—
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—
241Dic5—22no—
242Dic5—22no—
243Dic5—41no—
244Dic5—41no—
245Dic5—41no—
246Dic5—41no—
247Dic5C22⋊Q814yes—
248C24—51no—
249C24C22×D511yescharacteristic
250C2×Q8—102no—
251C2×Q8—102no—
252C2×Q8—102no—
253C2×D4—54no—
254C2×D4—102no—
255C2×D4—102no—
256C22×C4—22no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—51no—
260C22×C4—51no—
261C22×C4—51no—
262C22×C4—51no—
263C22×C4—51no—
264C22×C4—51no—
265C22×C4—51no—
266C22×C4—102no—
267C22×C4—201no—
268C22×C4—201no—
269C22×C4C22×D511yescharacteristic
270C22×C4C22×D511yescharacteristic
271C22×C4C22×D511yescharacteristic
272C22×C4C22×D511yescharacteristic
273C22×C4C22×D511yescharacteristic
274C22×C4C22×D511yescharacteristic
275C4⋊C4—22no—
276C4⋊C4—41no—
277C4⋊C4—102no—
278C4⋊C4—102no—
279C4⋊C4—102no—
280C4⋊C4—102no—
281C4⋊C4—102no—
282C4⋊C4—201no—
283C4⋊C4—201no—
284C4⋊C4—201no—
285C22⋊C4—22no—
286C22⋊C4—41no—
287C22⋊C4—102no—
288C22⋊C4—102no—
289C22⋊C4—102no—
290C22⋊C4—102no—
291C22⋊C4—102no—
292C22⋊C4—201no—
293C22⋊C4—201no—
294C22⋊C4—201no—
295C42—54no—
296C42—102no—
297C42—102no—
298C2×C5—42no—
299C2×C5C2×C22⋊Q811yescharacteristic
300C2×C5#52511yescharacteristic
301C2×C5#53511yescharacteristic
302C2×C5#55411yescharacteristic
303C2×C5#55411yescharacteristic
304C2×C5#55911yescharacteristic
305C2×C5#55911yescharacteristic
306D5—42no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—52no—
314C23—101no—
315C23—101no—
316C23—101no—
317C23—101no—
318C23—101no—
319C23—101no—
320C23Q8×D512yes—
321C23C23×D511yescharacteristic
322Q8—108no—
323D4—108no—
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—44no—
341C2×C4—52no—
342C2×C4—52no—
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—101no—
362C2×C4—204no—
363C2×C4#70912yes—
364C2×C4D4×D512yes—
365C5#258011yescharacteristic, complemented, Sylow
366C22—22no—
367C22—22no—
368C22—22no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—102no—
374C22—102no—
375C22—102no—
376C22—201no—
377C22—201no—
378C22—201no—
379C22—201no—
380C22C2×#70911yescharacteristic
381C22C2×D4×D511yescharacteristic
382C22#414411yescharacteristic
383C22#414411yescharacteristic
384C22C2×Q8×D511yescharacteristic
385C22D5×#4111yescharacteristic
386C22D5×#4111yescharacteristic
387C4—22no—
388C4—22no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—54no—
394C4—102no—
395C4—102no—
396C4—201no—
397C4—201no—
398C4—201no—
399C4—201no—
400C2—42no—
401C2—202no—
402C2#6584511yescharacteristic
403C2#6588111yescharacteristic
404C2#6589311yescharacteristic
405C2#6592611yescharacteristic
406C2D5×C22⋊Q811yescharacteristic
407C2#6597011yescharacteristic
408C2#6599811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#385C2
prime radicalC22×C22⋊D5C22
hypercenter#385D5
hypocenterC5#2580
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#4033C2semidirect
3. C5×#385C2semidirect
4. #65499C2semidirect
5. #65281C2semidirect
6. #65278C2semidirect
7. #65261C2semidirect
10. #65238C2semidirect
11. #65236C2semidirect
12. #64962C2semidirect
13. #64957C2semidirect
14. #64939C2semidirect
15. #64936C2semidirect
67. C2×C22⋊D5Q8semidirect
117. #385D5semidirect
365. C5#2580semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D52
C2×#40332
C5×#3852
#654992
#652812
#652782
#652612
#652462
#652382
#652362
#649622
#649572
#649392
#649362
#25805