Finite Group Databasebeta

← #10614575

Subgroups — #10614575

Order 640.

560 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#123C211yesmaximal, characteristic
2C2×D4×Dic5C211yesmaximal, characteristic
3#65509C211yesmaximal, characteristic
4#65500C211yesmaximal, characteristic, complemented
5C2×#3963C211yesmaximal, characteristic, complemented
6#65275C211yesmaximal, characteristic, complemented
7#65272C211yesmaximal, characteristic, complemented
8#65267C211yesmaximal, characteristic, complemented
9#65267C211yesmaximal, characteristic, complemented
10#65241C211yesmaximal, characteristic, complemented
11#65241C211yesmaximal, characteristic, complemented
12#65235C211yesmaximal, characteristic, complemented
13#65235C211yesmaximal, characteristic, complemented
14#65232C211yesmaximal, characteristic, complemented
15#65231C211yesmaximal, characteristic, complemented
16C22×C5×D4C2211yescharacteristic
17C23×C4×C5C2211yescharacteristic
18C23×Dic5C2211yescharacteristic
19C23×Dic5C2211yescharacteristic
20C23×Dic5C2211yescharacteristic
21C5×#123—24no—
22C2×C5×C4⋊C4C2211yescharacteristic
23C2×C5×C4⋊C4C2211yescharacteristic
24C2×C5×C22⋊C4C2211yescharacteristic
25C2×C5×C22⋊C4C2211yescharacteristic
26C2×C5×C22⋊C4C2211yescharacteristic
27C2×C22⋊Dic5C2211yescharacteristic
28C2×C22⋊Dic5C2211yescharacteristic
29C2×C22⋊Dic5C2211yescharacteristic
30C2×C22⋊Dic5C2211yescharacteristic
31C2×C22⋊Dic5C2211yescharacteristic
32D4×Dic5—24no—
33C2×C4⋊Dic5C2211yescharacteristic
34C2×#683C2211yescharacteristic
35C2×#683C2211yescharacteristic
36C2×#683C2211yescharacteristic
37C2×#683C2211yescharacteristic
38C2×C4×Dic5C2211yescharacteristic
39#3963C2211yescharacteristic
40#3963C2211yescharacteristic
41#3963C2211yescharacteristic
42#3963C2211yescharacteristic
43#3963C2211yescharacteristic
44#3963C2211yescharacteristic
45#3963C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#2419—51nomaximal, Sylow
54C24×C5C2311yescharacteristic
55C24×C5C2311yescharacteristic
56C2×C5×D4—22no—
57C2×C5×D4—22no—
58C2×C5×D4D414yes—
59C22×C4×C5—22no—
60C22×C4×C5—22no—
61C22×C4×C5—22no—
62C22×C4×C5C2311yescharacteristic
63C22×C4×C5C2311yescharacteristic
64C22×C4×C5C2311yescharacteristic
65C22×C4×C5C2311yescharacteristic
66C22×C4×C5C2311yescharacteristic
67C22×Dic5—22no—
68C22×Dic5—22no—
69C22×Dic5—22no—
70C22×Dic5—22no—
71C22×Dic5—22no—
72C22×Dic5—22no—
73C22×Dic5—41no—
74C22×Dic5—41no—
75C22×Dic5—41no—
76C22×Dic5C2311yescharacteristic
77C22×Dic5C2311yescharacteristic
78C22×Dic5C2311yescharacteristic
79C22×Dic5C2311yescharacteristic
80C22×Dic5C2311yescharacteristic
81C22×Dic5C2311yescharacteristic
82C22×Dic5C2311yescharacteristic
83C22×Dic5C2311yescharacteristic
84C5×C4⋊C4—22no—
85C5×C4⋊C4—22no—
86C5×C22⋊C4—22no—
87C5×C22⋊C4—22no—
88C5×C22⋊C4—22no—
89C22⋊Dic5—22no—
90C22⋊Dic5—22no—
91C22⋊Dic5—22no—
92C22⋊Dic5—22no—
93C22⋊Dic5—41no—
94C4⋊Dic5—22no—
95#683—22no—
96#683—22no—
97#683—41no—
98#683—41no—
99C4×Dic5—22no—
100C2×#123D511yescharacteristic
101C2×C4×D4—51no—
102#400—51no—
103#398—51no—
104#397—51no—
105#397—51no—
106#387—51no—
107#387—51no—
108#386—51no—
109#385—51no—
110#385—51no—
111#381—51no—
112#381—51no—
113#380—51no—
114C2×#95—51no—
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×C5C2×D412yes—
126C23×C5C2×D412yes—
127C23×C5#4112yes—
128C23×C5#4112yes—
129C23×C5C2411yescharacteristic
130C5×D4—28no—
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—44no—
146C2×C4×C5C2×D412yes—
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—21no—
169C2×Dic5—44no—
170C2×Dic5—44no—
171C2×Dic5—44no—
172C2×Dic5#4112yes—
173C2×Dic5#4112yes—
174C22×D4C2×D511yescharacteristic
175C23×C4—51no—
176C23×C4—51no—
177C23×C4—51no—
178C23×C4C2×D511yescharacteristic
179#123—24no—
180C4×D4—104no—
181C2×C4⋊C4—51no—
182C2×C4⋊C4—51no—
183C2×C4⋊C4—51no—
184C2×C4⋊C4—51no—
185C2×C4⋊C4—51no—
186C2×C4⋊C4C2×D511yescharacteristic
187C2×C4⋊C4C2×D511yescharacteristic
188C2×C22⋊C4—51no—
189C2×C22⋊C4—51no—
190C2×C22⋊C4—51no—
191C2×C22⋊C4—51no—
192C2×C22⋊C4—51no—
193C2×C22⋊C4C2×D511yescharacteristic
194C2×C22⋊C4C2×D511yescharacteristic
195C2×C22⋊C4C2×D511yescharacteristic
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—
207#95—51no—
208#95—51no—
209#95—51no—
210#95—51no—
211C22×C5—22no—
212C22×C5—22no—
213C22×C5—22no—
214C22×C5—22no—
215C22×C5—22no—
216C22×C5—22no—
217C22×C5—22no—
218C22×C5—22no—
219C22×C5—22no—
220C22×C5—41no—
221C22×C5—41no—
222C22×C5—41no—
223C22×C5—41no—
224C22×C5#12314yes—
225C22×C5C22×D411yescharacteristic
226C22×C5C2×#4111yescharacteristic
227C22×C5C2×#4111yescharacteristic
228C22×C5C2×#4111yescharacteristic
229C22×C5C2×#4111yescharacteristic
230C22×C5H(5,𝔽2)11yescharacteristic
231C22×C52−1+411yescharacteristic
232C4×C5—22no—
233C4×C5—41no—
234C4×C5—41no—
235C4×C5—41no—
236C4×C5—41no—
237Dic5—22no—
238Dic5—22no—
239Dic5—41no—
240Dic5—41no—
241Dic5—41no—
242Dic5—41no—
243Dic5—41no—
244Dic5—41no—
245C24C22×D511yescharacteristic
246C24C22×D511yescharacteristic
247C2×D4—22no—
248C2×D4—22no—
249C2×D4C22⋊D514yes—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—51no—
254C22×C4—51no—
255C22×C4—51no—
256C22×C4—51no—
257C22×C4—51no—
258C22×C4—51no—
259C22×C4—51no—
260C22×C4—51no—
261C22×C4—102no—
262C22×C4—102no—
263C22×C4—102no—
264C22×C4—102no—
265C22×C4—102no—
266C22×C4—102no—
267C22×C4—201no—
268C22×C4—201no—
269C22×C4—201no—
270C22×C4C22×D511yescharacteristic
271C22×C4C22×D511yescharacteristic
272C22×C4C22×D511yescharacteristic
273C22×C4C22×D511yescharacteristic
274C22×C4C22×D511yescharacteristic
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—102no—
278C4⋊C4—102no—
279C4⋊C4—102no—
280C4⋊C4—201no—
281C4⋊C4—201no—
282C22⋊C4—22no—
283C22⋊C4—22no—
284C22⋊C4—22no—
285C22⋊C4—102no—
286C22⋊C4—102no—
287C22⋊C4—102no—
288C22⋊C4—102no—
289C22⋊C4—201no—
290C42—102no—
291C2×C5—24no—
292C2×C5—42no—
293C2×C5C2×#12311yescharacteristic
294C2×C5#52911yescharacteristic
295C2×C5#53911yescharacteristic
296C2×C5#54611yescharacteristic
297C2×C5#54711yescharacteristic
298C2×C5#55111yescharacteristic
299C2×C5#55211yescharacteristic
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23#71112yes—
311C23#71112yes—
312C23C2×C22⋊D512yes—
313C23C2×C22⋊D512yes—
314C23C23×D511yescharacteristic
315D4—28no—
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—21no—
330C2×C4—44no—
331C2×C4—52no—
332C2×C4—52no—
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—101no—
349C2×C4—101no—
350C2×C4—101no—
351C2×C4—101no—
352C2×C4—101no—
353C2×C4—101no—
354C2×C4—101no—
355C2×C4—204no—
356C2×C4—204no—
357C2×C4—204no—
358C2×C4C2×C22⋊D512yes—
359C5#241911yescharacteristic, complemented, Sylow
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22#408114yes—
374C22C2×#71111yescharacteristic
375C22C2×#71111yescharacteristic
376C22D5×#4111yescharacteristic
377C22D5×#4111yescharacteristic
378C22#414911yescharacteristic
379C22#415011yescharacteristic
380C22C22×C22⋊D511yescharacteristic
381C4—22no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—102no—
387C4—102no—
388C4—201no—
389C4—201no—
390C4—201no—
391C4—201no—
392C4—201no—
393C4—201no—
394C2—24no—
395C2—42no—
396C2#6598311yescharacteristic
397C2#6598411yescharacteristic
398C2#6599211yescharacteristic
399C2#6599911yescharacteristic
400C2C2×#408111yescharacteristic
401C2#6615811yescharacteristic
402C2#6616311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×#123C2
prime radicalC22×C5×D4C22
hypercenterC2×#123D5
hypocenterC5#2419
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
4. #65500C2semidirect
5. C2×#3963C2semidirect
6. #65275C2semidirect
7. #65272C2semidirect
8. #65267C2semidirect
9. #65267C2semidirect
10. #65241C2semidirect
11. #65241C2semidirect
12. #65235C2semidirect
13. #65235C2semidirect
14. #65232C2semidirect
15. #65231C2semidirect
359. C5#2419semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1232
C2×D4×Dic52
#655092
#655002
C2×#39632
#652752
#652722
#652672
#652412
#652352
#652322
#652312
#24195