Finite Group Databasebeta

← #10614607

Subgroups — #10614607

Order 640.

490 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×C5×#123C211yesmaximal, characteristic
2C2×#4036C211yesmaximal, characteristic, complemented
3#65511C211yesmaximal, characteristic
4#65510C211yesmaximal, characteristic
5#65501C211yesmaximal, characteristic, complemented
6#65499C211yesmaximal, characteristic, complemented
7#65272C211yesmaximal, characteristic, complemented
8#65270C211yesmaximal, characteristic, complemented
9#65261C211yesmaximal, characteristic, complemented
10#65242C211yesmaximal, characteristic, complemented
11#65238C211yesmaximal, characteristic, complemented
12#65238C211yesmaximal, characteristic, complemented
13#65237C211yesmaximal, characteristic, complemented
14#65233C211yesmaximal, characteristic, complemented
15#65233C211yesmaximal, characteristic, complemented
16C22×C5×D4C2211yescharacteristic
17C23×C4×C5C2211yescharacteristic
18C5×#123—24no—
19C2×C5×C4⋊C4C2211yescharacteristic
20C2×C5×C4⋊C4C2211yescharacteristic
21C2×C5×C22⋊C4C2211yescharacteristic
22C2×C5×C22⋊C4C2211yescharacteristic
23C2×C5×C22⋊C4C2211yescharacteristic
24C2×C22⋊Dic5C2211yescharacteristic
25C2×C22⋊Dic5C2211yescharacteristic
26C2×C22⋊Dic5C2211yescharacteristic
27C2×C22⋊Dic5C2211yescharacteristic
28C2×C22⋊Dic5C2211yescharacteristic
29C2×C22⋊Dic5C2211yescharacteristic
30C2×C22⋊Dic5C2211yescharacteristic
31C2×C22⋊Dic5C2211yescharacteristic
32C2×C4⋊Dic5C2211yescharacteristic
33C2×C4⋊Dic5C2211yescharacteristic
34C2×C4⋊Dic5C2211yescharacteristic
35C2×C4⋊Dic5C2211yescharacteristic
36C2×#683C2211yescharacteristic
37C2×#683C2211yescharacteristic
38C2×#683C2211yescharacteristic
39C2×#683C2211yescharacteristic
40C2×#683C2211yescharacteristic
41C2×#683C2211yescharacteristic
42C2×C4×Dic5C2211yescharacteristic
43C2×C4×Dic5C2211yescharacteristic
44#4036—24no—
45#3963C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#2694—51nomaximal, Sylow
54C24×C5C2311yescharacteristic
55C24×C5C2311yescharacteristic
56C2×C5×D4—22no—
57C2×C5×D4—22no—
58C2×C5×D4—41no—
59C22×C4×C5—22no—
60C22×C4×C5—22no—
61C22×C4×C5—41no—
62C22×C4×C5C2311yescharacteristic
63C22×C4×C5C2311yescharacteristic
64C22×C4×C5C2311yescharacteristic
65C22×C4×C5C2311yescharacteristic
66C22×C4×C5C2311yescharacteristic
67C22×Dic5C2311yescharacteristic
68C22×Dic5C2311yescharacteristic
69C22×Dic5C2311yescharacteristic
70C22×Dic5C2311yescharacteristic
71C22×Dic5C2311yescharacteristic
72C22×Dic5C2311yescharacteristic
73C22×Dic5C2311yescharacteristic
74C22×Dic5C2311yescharacteristic
75C5×C4⋊C4—22no—
76C5×C4⋊C4D414yes—
77C5×C22⋊C4—22no—
78C5×C22⋊C4—22no—
79C5×C22⋊C4—22no—
80C22⋊Dic5—22no—
81C22⋊Dic5—22no—
82C22⋊Dic5—22no—
83C22⋊Dic5—41no—
84C22⋊Dic5—41no—
85C22⋊Dic5—41no—
86C22⋊Dic5—41no—
87C22⋊Dic5—41no—
88C4⋊Dic5—22no—
89C4⋊Dic5—22no—
90C4⋊Dic5—22no—
91C4⋊Dic5—41no—
92#683—22no—
93#683—22no—
94#683—22no—
95#683—41no—
96#683—41no—
97#683—41no—
98C4×Dic5—22no—
99C4×Dic5—41no—
100C2×#125—51no—
101C2×#123D511yescharacteristic
102#399—51no—
103#399—51no—
104#398—51no—
105#397—51no—
106#396—51no—
107#394—51no—
108#390—51no—
109#389—51no—
110#388—51no—
111#388—51no—
112#387—51no—
113#385—51no—
114#380—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×C5—21no—
126C23×C5—21no—
127C23×C5—21no—
128C23×C5#4112yes—
129C23×C5C2411yescharacteristic
130C5×D4—44no—
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—44no—
144C2×C4×C5C2×D412yes—
145C2×C4×C5C2×D412yes—
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—21no—
170C2×Dic5#4112yes—
171C22×D4C2×D511yescharacteristic
172C23×C4C2×D511yescharacteristic
173#125—104no—
174#123—24no—
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⋊C4—51no—
182C2×C4⋊C4—51no—
183C2×C4⋊C4—51no—
184C2×C4⋊C4—51no—
185C2×C4⋊C4C2×D511yescharacteristic
186C2×C4⋊C4C2×D511yescharacteristic
187C2×C22⋊C4—51no—
188C2×C22⋊C4—51no—
189C2×C22⋊C4—51no—
190C2×C22⋊C4—51no—
191C2×C22⋊C4—51no—
192C2×C22⋊C4—51no—
193C2×C22⋊C4—51no—
194C2×C22⋊C4—51no—
195C2×C22⋊C4C2×D511yescharacteristic
196C2×C22⋊C4C2×D511yescharacteristic
197C2×C22⋊C4C2×D511yescharacteristic
198C2×C42—51no—
199C2×C42—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—
208C22×C5—22no—
209C22×C5—22no—
210C22×C5—22no—
211C22×C5—41no—
212C22×C5—41no—
213C22×C5—41no—
214C22×C5—41no—
215C22×C5—41no—
216C22×C5—41no—
217C22×C5—41no—
218C22×C5—41no—
219C22×C5—41no—
220C22×C5—41no—
221C22×C5—41no—
222C22×C5C22×D411yescharacteristic
223C22×C5C2×#4111yescharacteristic
224C22×C5C2×#4111yescharacteristic
225C22×C5H(5,𝔽2)11yescharacteristic
226C22×C5H(5,𝔽2)11yescharacteristic
227C22×C52−1+411yescharacteristic
228C22×C52−1+411yescharacteristic
229C4×C5—22no—
230C4×C5—22no—
231C4×C5—22no—
232C4×C5—41no—
233C4×C5—41no—
234Dic5—22no—
235Dic5—41no—
236Dic5—41no—
237Dic5—41no—
238Dic5—41no—
239Dic5—41no—
240Dic5—41no—
241Dic5—41no—
242C24C22×D511yescharacteristic
243C24C22×D511yescharacteristic
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—41no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—41no—
250C22×C4—51no—
251C22×C4—51no—
252C22×C4—51no—
253C22×C4—51no—
254C22×C4—51no—
255C22×C4—51no—
256C22×C4—51no—
257C22×C4—51no—
258C22×C4C22×D511yescharacteristic
259C22×C4C22×D511yescharacteristic
260C22×C4C22×D511yescharacteristic
261C22×C4C22×D511yescharacteristic
262C22×C4C22×D511yescharacteristic
263C4⋊C4—22no—
264C4⋊C4—102no—
265C4⋊C4—102no—
266C4⋊C4—102no—
267C4⋊C4—102no—
268C4⋊C4—102no—
269C4⋊C4—102no—
270C4⋊C4—201no—
271C4⋊C4—201no—
272C4⋊C4—201no—
273C4⋊C4—201no—
274C4⋊C4C22⋊D514yes—
275C22⋊C4—22no—
276C22⋊C4—22no—
277C22⋊C4—22no—
278C22⋊C4—102no—
279C22⋊C4—102no—
280C22⋊C4—102no—
281C22⋊C4—201no—
282C22⋊C4—201no—
283C22⋊C4—201no—
284C22⋊C4—201no—
285C22⋊C4—201no—
286C42—102no—
287C42—201no—
288C2×C5—42no—
289C2×C5—81no—
290C2×C5#53611yescharacteristic
291C2×C5#53911yescharacteristic
292C2×C5#53911yescharacteristic
293C2×C5#54611yescharacteristic
294C2×C5#55011yescharacteristic
295C2×C5#55211yescharacteristic
296C2×C5#56211yescharacteristic
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23#71112yes—
311C23C23×D511yescharacteristic
312D4—44no—
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—44no—
326C2×C4—52no—
327C2×C4—101no—
328C2×C4—101no—
329C2×C4—101no—
330C2×C4—101no—
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—101no—
349C2×C4—101no—
350C2×C4C2×C22⋊D512yes—
351C2×C4C2×C22⋊D512yes—
352C2×C4C2×C22⋊D512yes—
353C5#269411yescharacteristic, complemented, Sylow
354C22—22no—
355C22—22no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22C2×#71111yescharacteristic
369C22#414411yescharacteristic
370C22D5×#4111yescharacteristic
371C22#414911yescharacteristic
372C22#415011yescharacteristic
373C22#415011yescharacteristic
374C22C22×C22⋊D511yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—102no—
381C4—201no—
382C4—201no—
383C4—201no—
384C4—201no—
385C4—201no—
386C4—201no—
387C4—201no—
388C2—42no—
389C2—81no—
390C2#6598411yescharacteristic
391C2#6599411yescharacteristic
392C2#6599911yescharacteristic
393C2#6600111yescharacteristic
394C2#6614011yescharacteristic
395C2#6615911yescharacteristic
396C2#6616811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×#123C2
prime radicalC22×C5×D4C22
hypercenterC2×#123D5
hypocenterC5#2694
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#4036C2semidirect
5. #65501C2semidirect
6. #65499C2semidirect
7. #65272C2semidirect
8. #65270C2semidirect
9. #65261C2semidirect
10. #65242C2semidirect
11. #65238C2semidirect
12. #65238C2semidirect
13. #65237C2semidirect
14. #65233C2semidirect
15. #65233C2semidirect
353. C5#2694semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1232
C2×#40362
#655112
#655102
#655012
#654992
#652722
#652702
#652612
#652422
#652382
#652372
#652332
#26945