Finite Group Databasebeta

← #10608531

Subgroups — #10608531

Order 640.

476 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#393C211yesmaximal, characteristic, complemented
2#65520C211yesmaximal, characteristic, complemented
3#65519C211yesmaximal, characteristic, complemented
4#65250C211yesmaximal, characteristic
5#65247C211yesmaximal, characteristic
6#65245C211yesmaximal, characteristic
7#65234C211yesmaximal, characteristic, complemented
8#65232C211yesmaximal, characteristic, complemented
9C22⋊C4×Dic5C211yesmaximal, characteristic, complemented
10#64958C211yesmaximal, characteristic, complemented
11#64955C211yesmaximal, characteristic, complemented
12#64953C211yesmaximal, characteristic, complemented
13#64949C211yesmaximal, characteristic, complemented
14#64940C211yesmaximal, characteristic, complemented
15#64940C211yesmaximal, characteristic, complemented
16C22×C5×Q8C2211yescharacteristic
17C22×C22⋊D5C2211yescharacteristic
18C23×Dic5C2211yescharacteristic
19C22×C4×D5C2211yescharacteristic
20C2×C5×C22⋊C4C2211yescharacteristic
21C2×C5×C22⋊C4C2211yescharacteristic
22C2×C5×C22⋊C4C2211yescharacteristic
23C2×C22⋊Dic5C2211yescharacteristic
24C2×C22⋊Dic5C2211yescharacteristic
25C2×#685C2211yescharacteristic
26C2×#685C2211yescharacteristic
27C2×#685C2211yescharacteristic
28C2×#685C2211yescharacteristic
29C2×#685C2211yescharacteristic
30C2×C4⋊Dic5C2211yescharacteristic
31C2×#683C2211yescharacteristic
32C2×C4×Dic5C2211yescharacteristic
33C2×C4×Dic5C2211yescharacteristic
34C2×C4×Dic5C2211yescharacteristic
35C5×#95C2211yescharacteristic
36C5×#95C2211yescharacteristic
37C5×#95C2211yescharacteristic
38#3963C2211yescharacteristic
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#2572—51nomaximal, Sylow
52C24×C5C2311yescharacteristic
53C23×D5C2311yescharacteristic
54C2×C5×Q8—22no—
55C2×C5×Q8—41no—
56C2×C5×Q8—41no—
57C22×C4×C5C2311yescharacteristic
58C22×C4×C5C2311yescharacteristic
59C22×C4×C5C2311yescharacteristic
60C22×C4×C5C2311yescharacteristic
61C22×C4×C5C2311yescharacteristic
62C22×C4×C5C2311yescharacteristic
63C2×C22⋊D5—22no—
64C2×C22⋊D5—22no—
65C2×C22⋊D5—41no—
66C22×Dic5—22no—
67C22×Dic5—22no—
68C22×Dic5—41no—
69C22×Dic5C2311yescharacteristic
70C22×Dic5C2311yescharacteristic
71C22×Dic5C2311yescharacteristic
72C22×Dic5C2311yescharacteristic
73C22×Dic5C2311yescharacteristic
74C22×Dic5C2311yescharacteristic
75C22×Dic5C2311yescharacteristic
76C2×C4×D5—22no—
77C2×C4×D5—22no—
78C2×C4×D5—41no—
79C5×C22⋊C4—22no—
80C5×C22⋊C4—41no—
81C5×C22⋊C4—41no—
82C22⋊Dic5—22no—
83C22⋊Dic5—22no—
84#685—22no—
85#685—22no—
86#685—22no—
87#685—41no—
88#685—41no—
89C4⋊Dic5—41no—
90#683—41no—
91C4×Dic5—22no—
92C4×Dic5—22no—
93C4×Dic5—41no—
94#400—51no—
95#394—51no—
96#393—51no—
97#393D511yescharacteristic, complemented
98#391—51no—
99#388—51no—
100#388—51no—
101#386—51no—
102#386—51no—
103#383—51no—
104#383—51no—
105#381—51no—
106#381—51no—
107C4×C22⋊C4—51no—
108C4×C22⋊C4—51no—
109C23×C5—21no—
110C23×C5—21no—
111C23×C5—21no—
112C23×C5—21no—
113C23×C5—21no—
114C23×C5—21no—
115C23×C5#4112yes—
116C23×C5C2411yescharacteristic
117C22×D5—21no—
118C22×D5—21no—
119C22×D5—21no—
120C22×D5—21no—
121C22×D5—21no—
122C22×D5—21no—
123C22×D5#4112yes—
124C5×Q8—44no—
125C2×C4×C5—21no—
126C2×C4×C5—21no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
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#4112yes—
143C22⋊D5—44no—
144C2×Dic5—21no—
145C2×Dic5—21no—
146C2×Dic5—21no—
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—44no—
164C2×Dic5#4112yes—
165C2×Dic5#4112yes—
166C4×D5—44no—
167C22×Q8C2×D511yescharacteristic
168C22×D4—51no—
169C23×C4—51no—
170C23×C4—51no—
171C2×C4⋊C4—51no—
172C2×C4⋊C4—51no—
173C2×C22⋊C4—51no—
174C2×C22⋊C4—51no—
175C2×C22⋊C4—51no—
176C2×C22⋊C4—51no—
177C2×C22⋊C4—51no—
178C2×C22⋊C4—51no—
179C2×C22⋊C4—51no—
180C2×C22⋊C4C2×D511yescharacteristic
181C2×C22⋊C4C2×D511yescharacteristic
182C2×C22⋊C4C2×D511yescharacteristic
183C2×C42—51no—
184C2×C42—51no—
185C2×C42—51no—
186#95—51no—
187#95—51no—
188#95—51no—
189#95—51no—
190#95—51no—
191#95—51no—
192#95—51no—
193#95—51no—
194#95—51no—
195#95—51no—
196#95—51no—
197#95—51no—
198#95—51no—
199#95C2×D511yescharacteristic
200#95C2×D511yescharacteristic
201#95C2×D511yescharacteristic
202C22×C5—22no—
203C22×C5—22no—
204C22×C5—22no—
205C22×C5—41no—
206C22×C5—41no—
207C22×C5—41no—
208C22×C5—41no—
209C22×C5C2×#4111yescharacteristic
210C22×C5C2×#4111yescharacteristic
211C22×C5C2×#4111yescharacteristic
212C22×C5C2×#4111yescharacteristic
213C22×C5C2×#4111yescharacteristic
214C22×C5H(5,𝔽2)11yescharacteristic
215C22×C5H(5,𝔽2)11yescharacteristic
216C2×D5—22no—
217C2×D5—22no—
218C2×D5—22no—
219C2×D5—41no—
220C2×D5—41no—
221C2×D5—41no—
222C2×D5—41no—
223C4×C5—22no—
224C4×C5—41no—
225C4×C5—41no—
226C4×C5—41no—
227C4×C5—41no—
228C4×C5—41no—
229Dic5—22no—
230Dic5—22no—
231Dic5—41no—
232Dic5—41no—
233Dic5—41no—
234Dic5—41no—
235Dic5—41no—
236C24—51no—
237C24C22×D511yescharacteristic
238C2×Q8—22no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×D4—102no—
242C2×D4—102no—
243C2×D4—201no—
244C22×C4—51no—
245C22×C4—51no—
246C22×C4—51no—
247C22×C4—51no—
248C22×C4—51no—
249C22×C4—51no—
250C22×C4—51no—
251C22×C4—102no—
252C22×C4—102no—
253C22×C4—102no—
254C22×C4—102no—
255C22×C4—201no—
256C22×C4—201no—
257C22×C4C22×D511yescharacteristic
258C22×C4C22×D511yescharacteristic
259C22×C4C22×D511yescharacteristic
260C22×C4C22×D511yescharacteristic
261C22×C4C22×D511yescharacteristic
262C22×C4C22×D511yescharacteristic
263C4⋊C4—201no—
264C4⋊C4—201no—
265C22⋊C4—22no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—102no—
269C22⋊C4—102no—
270C22⋊C4—102no—
271C22⋊C4—102no—
272C22⋊C4—102no—
273C22⋊C4—201no—
274C22⋊C4—201no—
275C42—102no—
276C42—102no—
277C42—201no—
278C2×C5—42no—
279C2×C5#52911yescharacteristic
280C2×C5#52911yescharacteristic
281C2×C5#53811yescharacteristic
282C2×C5#55111yescharacteristic
283C2×C5#55111yescharacteristic
284C2×C5#55111yescharacteristic
285C2×C5#55511yescharacteristic
286D5—42no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—52no—
294C23—101no—
295C23—101no—
296C23—101no—
297C23—101no—
298C23—101no—
299C23—101no—
300C23#71312yes—
301C23C23×D511yescharacteristic
302Q8—44no—
303D4—204no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
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—52no—
322C2×C4—52no—
323C2×C4—101no—
324C2×C4—101no—
325C2×C4—101no—
326C2×C4—101no—
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—204no—
343C2×C4—204no—
344C2×C4#71112yes—
345C5#257211yescharacteristic, complemented, Sylow
346C22—22no—
347C22—22no—
348C22—22no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—102no—
354C22—102no—
355C22—102no—
356C22—201no—
357C22—201no—
358C22—201no—
359C22—201no—
360C22C2×#71111yescharacteristic
361C22#414411yescharacteristic
362C22#414411yescharacteristic
363C22C2×#71311yescharacteristic
364C22D5×#4111yescharacteristic
365C22D5×#4111yescharacteristic
366C22D5×#4111yescharacteristic
367C4—22no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—102no—
374C4—102no—
375C4—201no—
376C4—201no—
377C4—201no—
378C4—201no—
379C4—201no—
380C2—42no—
381C2—202no—
382C2#6592711yescharacteristic
383C2#6595811yescharacteristic
384C2#6597111yescharacteristic
385C2#6597111yescharacteristic
386C2#6600411yescharacteristic
387C2#6601311yescharacteristic
388C2#6601411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#393C2
prime radicalC22×C22⋊D5C22
hypercenter#393D5
hypocenterC5#2572
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C5×#393C2semidirect
2. #65520C2semidirect
3. #65519C2semidirect
7. #65234C2semidirect
8. #65232C2semidirect
9. C22⋊C4×Dic5C2semidirect
10. #64958C2semidirect
11. #64955C2semidirect
12. #64953C2semidirect
13. #64949C2semidirect
14. #64940C2semidirect
15. #64940C2semidirect
97. #393D5semidirect
345. C5#2572semidirect

Maximal subgroups

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

subgroupindex
C5×#3932
#655202
#655192
#652502
#652472
#652452
#652342
#652322
C22⋊C4×Dic52
#649582
#649552
#649532
#649492
#649402
#25725