Finite Group Databasebeta

← #10608151

Subgroups — #10608151

Order 640.

470 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#390C211yesmaximal, characteristic, complemented
2#65512C211yesmaximal, characteristic
3#65511C211yesmaximal, characteristic, complemented
4#65279C211yesmaximal, characteristic, complemented
5#65267C211yesmaximal, characteristic, complemented
6#65250C211yesmaximal, characteristic, complemented
7#65250C211yesmaximal, characteristic, complemented
8#65248C211yesmaximal, characteristic, complemented
9#65246C211yesmaximal, characteristic, complemented
10#65241C211yesmaximal, characteristic, complemented
11#65240C211yesmaximal, characteristic, complemented
12#65240C211yesmaximal, characteristic, complemented
13#65233C211yesmaximal, characteristic, complemented
14#65229C211yesmaximal, characteristic, complemented
15#65217C211yesmaximal, characteristic, complemented
16C22×C5×D4C2211yescharacteristic
17C22×C22⋊D5C2211yescharacteristic
18C22×C22⋊D5C2211yescharacteristic
19C2×C5×C4⋊C4C2211yescharacteristic
20C2×C5×C22⋊C4C2211yescharacteristic
21C2×C5×C22⋊C4C2211yescharacteristic
22C2×C5×C22⋊C4C2211yescharacteristic
23C2×C5×C22⋊C4C2211yescharacteristic
24C2×C42×C5C2211yescharacteristic
25C2×C22⋊Dic5C2211yescharacteristic
26C2×C22⋊Dic5C2211yescharacteristic
27C2×C22⋊Dic5C2211yescharacteristic
28C2×C22⋊Dic5C2211yescharacteristic
29C2×C22⋊Dic5C2211yescharacteristic
30C2×C22⋊Dic5C2211yescharacteristic
31C2×#685C2211yescharacteristic
32C2×#685C2211yescharacteristic
33C2×#685C2211yescharacteristic
34C2×#685C2211yescharacteristic
35C2×#685C2211yescharacteristic
36C2×C4⋊Dic5C2211yescharacteristic
37C2×C4⋊Dic5C2211yescharacteristic
38C2×#683C2211yescharacteristic
39C2×#683C2211yescharacteristic
40C2×#683C2211yescharacteristic
41C2×C4×Dic5C2211yescharacteristic
42C2×C4×Dic5C2211yescharacteristic
43#3963C2211yescharacteristic
44#3963C2211yescharacteristic
45#3963C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#2803—51nomaximal, Sylow
52C24×C5C2311yescharacteristic
53C24×C5C2311yescharacteristic
54C23×D5C2311yescharacteristic
55C2×C5×D4—22no—
56C2×C5×D4—41no—
57C2×C5×D4—41no—
58C22×C4×C5C2311yescharacteristic
59C22×C4×C5C2311yescharacteristic
60C22×C4×C5C2311yescharacteristic
61C22×C4×C5C2311yescharacteristic
62C22×C4×C5C2311yescharacteristic
63C2×C22⋊D5—22no—
64C2×C22⋊D5—41no—
65C2×C22⋊D5—41no—
66C2×C22⋊D5—41no—
67C2×C22⋊D5—41no—
68C2×C22⋊D5—41no—
69C22×Dic5C2311yescharacteristic
70C22×Dic5C2311yescharacteristic
71C22×Dic5C2311yescharacteristic
72C22×Dic5C2311yescharacteristic
73C22×Dic5C2311yescharacteristic
74C22×Dic5C2311yescharacteristic
75C22×Dic5C2311yescharacteristic
76C5×C4⋊C4—41no—
77C5×C22⋊C4—22no—
78C5×C22⋊C4—22no—
79C5×C22⋊C4—41no—
80C5×C22⋊C4—41no—
81C42×C5—41no—
82C22⋊Dic5—22no—
83C22⋊Dic5—41no—
84C22⋊Dic5—41no—
85C22⋊Dic5—41no—
86C22⋊Dic5—41no—
87C22⋊Dic5—41no—
88#685—22no—
89#685—22no—
90#685—41no—
91#685—41no—
92#685—41no—
93C4⋊Dic5—22no—
94C4⋊Dic5—41no—
95#683—22no—
96#683—41no—
97#683—41no—
98C4×Dic5—41no—
99C4×Dic5—41no—
100#397—51no—
101#394—51no—
102#394—51no—
103#394—51no—
104#394—51no—
105#392—51no—
106#390—51no—
107#390D511yescharacteristic, complemented
108#389—51no—
109#388—51no—
110#388—51no—
111#388—51no—
112#388—51no—
113#388—51no—
114#387—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—21no—
129C23×C5C2411yescharacteristic
130C22×D5—21no—
131C22×D5—21no—
132C22×D5—21no—
133C22×D5—21no—
134C22×D5—21no—
135C22×D5—21no—
136C22×D5—21no—
137C5×D4—44no—
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#4112yes—
152C2×C4×C5#4112yes—
153C22⋊D5—44no—
154C22⋊D5—44no—
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—21no—
171C2×Dic5—21no—
172C2×Dic5—21no—
173C2×Dic5—21no—
174C2×Dic5—21no—
175C2×Dic5#4112yes—
176C22×D4—51no—
177C22×D4—51no—
178C22×D4C2×D511yescharacteristic
179C2×C4⋊C4—51no—
180C2×C4⋊C4—51no—
181C2×C4⋊C4—51no—
182C2×C4⋊C4—51no—
183C2×C4⋊C4—51no—
184C2×C4⋊C4C2×D511yescharacteristic
185C2×C22⋊C4—51no—
186C2×C22⋊C4—51no—
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⋊C4—51no—
196C2×C22⋊C4C2×D511yescharacteristic
197C2×C22⋊C4C2×D511yescharacteristic
198C2×C22⋊C4C2×D511yescharacteristic
199C2×C22⋊C4C2×D511yescharacteristic
200C2×C42—51no—
201C2×C42—51no—
202C2×C42C2×D511yescharacteristic
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—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×C5—41no—
223C22×C5—41no—
224C22×C5—41no—
225C22×C5C2×#4111yescharacteristic
226C22×C5C2×#4111yescharacteristic
227C22×C5C2×#4111yescharacteristic
228C22×C5H(5,𝔽2)11yescharacteristic
229C22×C5H(5,𝔽2)11yescharacteristic
230C22×C5H(5,𝔽2)11yescharacteristic
231C22×C5H(5,𝔽2)11yescharacteristic
232C2×D5—41no—
233C2×D5—41no—
234C2×D5—41no—
235C2×D5—41no—
236C2×D5—41no—
237C2×D5—41no—
238C2×D5—41no—
239C4×C5—22no—
240C4×C5—22no—
241C4×C5—41no—
242C4×C5—41no—
243C4×C5—41no—
244Dic5—22no—
245Dic5—41no—
246Dic5—41no—
247Dic5—41no—
248Dic5—41no—
249Dic5—41no—
250Dic5—41no—
251C24—51no—
252C24C22×D511yescharacteristic
253C24C22×D511yescharacteristic
254C2×D4—22no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—102no—
258C2×D4—201no—
259C2×D4—201no—
260C2×D4—201no—
261C2×D4—201no—
262C2×D4—201no—
263C22×C4—51no—
264C22×C4—51no—
265C22×C4—51no—
266C22×C4—51no—
267C22×C4—51no—
268C22×C4—51no—
269C22×C4—51no—
270C22×C4C22×D511yescharacteristic
271C22×C4C22×D511yescharacteristic
272C22×C4C22×D511yescharacteristic
273C22×C4C22×D511yescharacteristic
274C22×C4C22×D511yescharacteristic
275C4⋊C4—41no—
276C4⋊C4—102no—
277C4⋊C4—102no—
278C4⋊C4—201no—
279C4⋊C4—201no—
280C4⋊C4—201no—
281C22⋊C4—22no—
282C22⋊C4—22no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—102no—
286C22⋊C4—102no—
287C22⋊C4—102no—
288C22⋊C4—201no—
289C22⋊C4—201no—
290C22⋊C4—201no—
291C22⋊C4—201no—
292C22⋊C4—201no—
293C22⋊C4—201no—
294C22⋊C4—201no—
295C22⋊C4—201no—
296C42—41no—
297C42—201no—
298C42—201no—
299C2×C5—81no—
300C2×C5—81no—
301C2×C5#53811yescharacteristic
302C2×C5#53811yescharacteristic
303C2×C5#54011yescharacteristic
304C2×C5#55311yescharacteristic
305C2×C5#55511yescharacteristic
306C2×C5#55911yescharacteristic
307C2×C5#56011yescharacteristic
308D5—81no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—101no—
324C23—101no—
325C23—101no—
326C23—101no—
327C23—101no—
328C23—101no—
329C23—101no—
330C23C23×D511yescharacteristic
331D4—44no—
332D4—204no—
333D4—204no—
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—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—52no—
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—101no—
363C2×C4—101no—
364C2×C4—101no—
365C2×C4—101no—
366C2×C4—101no—
367C2×C4—101no—
368C2×C4#70912yes—
369C2×C4#71112yes—
370C5#280311yescharacteristic, complemented, Sylow
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—201no—
386C22—201no—
387C22—201no—
388C22—201no—
389C22—201no—
390C22—201no—
391C22—201no—
392C22C2×#70911yescharacteristic
393C22C2×#71111yescharacteristic
394C22#414411yescharacteristic
395C22#414411yescharacteristic
396C22#414411yescharacteristic
397C22D5×#4111yescharacteristic
398C22#414911yescharacteristic
399C4—22no—
400C4—22no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—102no—
405C4—201no—
406C4—201no—
407C4—201no—
408C4—201no—
409C4—201no—
410C4—201no—
411C2—81no—
412C2—81no—
413C2—401no—
414C2#6582811yescharacteristic
415C2#6589311yescharacteristic
416C2#6589711yescharacteristic
417C2#6593511yescharacteristic
418C2#6595411yescharacteristic
419C2#6601211yescharacteristic
420C2#6605311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#390C2
prime radical#65512C2
hypercenter#390D5
hypocenterC5#2803
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C5×#390C2semidirect
3. #65511C2semidirect
4. #65279C2semidirect
5. #65267C2semidirect
6. #65250C2semidirect
7. #65250C2semidirect
8. #65248C2semidirect
9. #65246C2semidirect
10. #65241C2semidirect
11. #65240C2semidirect
12. #65240C2semidirect
13. #65233C2semidirect
14. #65229C2semidirect
15. #65217C2semidirect
107. #390D5semidirect
370. C5#2803semidirect

Maximal subgroups

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

subgroupindex
C5×#3902
#655122
#655112
#652792
#652672
#652502
#652482
#652462
#652412
#652402
#652332
#652292
#652172
#28035