Finite Group Databasebeta

← #10615159

Subgroups — #10615159

Order 640.

390 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#537C211yesmaximal, characteristic
2#66159C211yesmaximal, characteristic, complemented
3#65528C211yesmaximal, characteristic, complemented
4#65528C211yesmaximal, characteristic, complemented
5#65523C211yesmaximal, characteristic, complemented
6#65341C211yesmaximal, characteristic, complemented
7#65340C211yesmaximal, characteristic, complemented
8#65339C211yesmaximal, characteristic, complemented
9#65334C211yesmaximal, characteristic, complemented
10#65331C211yesmaximal, characteristic, complemented
11#65330C211yesmaximal, characteristic, complemented
12#65329C211yesmaximal, characteristic, complemented
13#65324C211yesmaximal, characteristic, complemented
14#65260C211yesmaximal, characteristic, complemented
15#65259C211yesmaximal, characteristic, complemented
16C2×C5×#41C2211yescharacteristic
17C2×C5×#41C2211yescharacteristic
18C22×Dic10C2211yescharacteristic
19C5×C22⋊Q8—21no—
20C5×C22⋊Q8C2211yescharacteristic
21C5×C22⋊Q8C2211yescharacteristic
22C5×#121—21no—
23C5×#121—21no—
24C5×#121—21no—
25C5×#121C2211yescharacteristic
26C5×#121C2211yescharacteristic
27C2×C5×C4⋊C4C2211yescharacteristic
28#4090—21no—
29C2×#689C2211yescharacteristic
30C2×#689C2211yescharacteristic
31#4082—21no—
32#4081—21no—
33#4081—21no—
34C2×#687C2211yescharacteristic
35C2×#687C2211yescharacteristic
36#4072C2211yescharacteristic
37#4070C2211yescharacteristic
38#4070C2211yescharacteristic
39#4070C2211yescharacteristic
40#4070C2211yescharacteristic
41C2×#681C2211yescharacteristic
42C22×#168C2211yescharacteristic
43#3967C2211yescharacteristic
44#3967C2211yescharacteristic
45#3964C2211yescharacteristic
46#3964C2211yescharacteristic
47C22⋊#168C2211yescharacteristic
48C22⋊#168C2211yescharacteristic
49C22⋊#168C2211yescharacteristic
50C22⋊#168C2211yescharacteristic
51#3942C2211yescharacteristic
52#3942C2211yescharacteristic
53#3942C2211yescharacteristic
54#3942C2211yescharacteristic
55#3940C2211yescharacteristic
56#3940C2211yescharacteristic
57#3939C2211yescharacteristic
58#3939C2211yescharacteristic
59#3201—51nomaximal, Sylow
60C5×#41—41no—
61C5×#41—41no—
62C5×#41—41no—
63C5×#41—41no—
64C2×C5×Q8D411yescharacteristic
65C2×C5×Q8C2311yescharacteristic
66C2×C5×D4—21no—
67C2×C5×D4—21no—
68C2×C5×D4—21no—
69C2×C5×D4D411yescharacteristic
70C2×C5×D4D411yescharacteristic
71C2×C5×D4D411yescharacteristic
72C2×C5×D4C2311yescharacteristic
73C22×C4×C5—21no—
74C22×C4×C5—21no—
75C22×C4×C5C2311yescharacteristic
76C22×C4×C5C2311yescharacteristic
77C22×C4×C5C2311yescharacteristic
78C22×Dic5—21no—
79C2×Dic10—21no—
80C2×Dic10—41no—
81C2×Dic10—41no—
82C2×Dic10C2311yescharacteristic
83C2×Dic10C2311yescharacteristic
84C5×C4⋊C4—21no—
85C5×C4⋊C4—21no—
86C5×C4⋊C4—21no—
87C5×C4⋊C4C2311yescharacteristic
88C5×C4⋊C4C2311yescharacteristic
89C5×C22⋊C4—21no—
90C5×C22⋊C4—21no—
91C5×C22⋊C4—21no—
92C5×C22⋊C4—21no—
93C22⋊Dic5—21no—
94C22⋊Dic5—21no—
95C22⋊Dic5—21no—
96C22⋊Dic5—21no—
97C22⋊Dic5—21no—
98#689—41no—
99#689—41no—
100#687—41no—
101#687—41no—
102C4⋊Dic5C2311yescharacteristic
103C4⋊Dic5C2311yescharacteristic
104#683—21no—
105#683—21no—
106#683—21no—
107#683—21no—
108C4×Dic5—21no—
109#681—41no—
110C2×#168—41no—
111C2×#168C2311yescharacteristic
112C2×#168C2311yescharacteristic
113C2×#168C2311yescharacteristic
114C2×#168C2311yescharacteristic
115#537D511yescharacteristic
116#536—51no—
117#485—51no—
118#484—51no—
119#483—51no—
120#482—51no—
121#470—51no—
122#468—51no—
123#467—51no—
124#465—51no—
125#452—51no—
126#452—51no—
127#451—51no—
128#448—51no—
129#408—51no—
130C23×C5—21no—
131C23×C5C2×D411yescharacteristic
132C23×C5C2×D411yescharacteristic
133C23×C5C2×D411yescharacteristic
134C5×Q8—21no—
135C5×Q8—21no—
136C5×Q8—41no—
137C5×D4—21no—
138C5×D4—21no—
139C5×D4—41no—
140C5×D4—41no—
141C5×D4—41no—
142C5×D4—42no—
143C5×D4—81no—
144C5×D4—81no—
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—21no—
152C2×C4×C5—21no—
153C2×C4×C5—21no—
154C2×C4×C5—42no—
155C2×C4×C5—81no—
156C2×C4×C5C2×D411yescharacteristic
157C2×C4×C5C2×D411yescharacteristic
158C2×C4×C5C2×D411yescharacteristic
159C2×C4×C5C2411yescharacteristic
160C2×Dic5—21no—
161C2×Dic5—21no—
162C2×Dic5—21no—
163C2×Dic5—21no—
164C2×Dic5—42no—
165Dic10—21no—
166Dic10—21no—
167Dic10—21no—
168Dic10—21no—
169Dic10—42no—
170#168—21no—
171#168—21no—
172#168—21no—
173#168—21no—
174C2×#41C2×D511yescharacteristic
175C2×#41C2×D511yescharacteristic
176C22×Q8—51no—
177C2×Q16—51no—
178C2×Q16—51no—
179C2×SD16—51no—
180C2×SD16—51no—
181C2×M16—51no—
182C22×C8—51no—
183C4⋊Q8—101no—
184#124—101no—
185#123—101no—
186#123—101no—
187C22⋊Q8—21no—
188C22⋊Q8—51no—
189C22⋊Q8—51no—
190C22⋊Q8—51no—
191C22⋊Q8—51no—
192C22⋊Q8C2×D511yescharacteristic
193C22⋊Q8C2×D511yescharacteristic
194#121—21no—
195#121—21no—
196#121—21no—
197#121C2×D511yescharacteristic
198#121C2×D511yescharacteristic
199#117—51no—
200C2×C4⋊C4C2×D511yescharacteristic
201#107—51no—
202#107—51no—
203#106—51no—
204#106—51no—
205#103—51no—
206#103—51no—
207#103—51no—
208#103—51no—
209#103—51no—
210#103—51no—
211#102—51no—
212#102—51no—
213C22⋊C8—51no—
214C22⋊C8—51no—
215C22⋊C8—51no—
216C22⋊C8—51no—
217C22×C5—21no—
218C22×C5—21no—
219C22×C5—21no—
220C22×C5—21no—
221C22×C5—21no—
222C22×C5—21no—
223C22×C5—21no—
224C22×C5—21no—
225C22×C5—21no—
226C22×C5—41no—
227C22×C5—41no—
228C22×C5—41no—
229C22×C5C22×D411yescharacteristic
230C4×C5—21no—
231C4×C5—21no—
232C4×C5—21no—
233C4×C5—41no—
234C4×C5—41no—
235C4×C5—41no—
236C4×C5H(5,𝔽2)11yescharacteristic
237C4×C5H(5,𝔽2)11yescharacteristic
238Dic5—41no—
239Dic5—41no—
240Dic5—41no—
241Dic5—41no—
242#41—41no—
243#41—41no—
244#41—41no—
245#41—41no—
246C2×Q8—51no—
247C2×Q8—51no—
248C2×Q8—101no—
249C2×Q8—201no—
250C2×Q8—201no—
251C2×Q8C22⋊D511yescharacteristic
252C2×Q8C22×D511yescharacteristic
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4C22⋊D511yescharacteristic
257C2×D4C22⋊D511yescharacteristic
258C2×D4C22⋊D511yescharacteristic
259C2×D4C22×D511yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—101no—
263C22×C4C22×D511yescharacteristic
264C22×C4C22×D511yescharacteristic
265C22×C4C22×D511yescharacteristic
266Q16—201no—
267Q16—201no—
268SD16—201no—
269SD16—201no—
270M16—201no—
271C2×C8—51no—
272C2×C8—51no—
273C2×C8—51no—
274C2×C8—51no—
275C2×C8—201no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—51no—
280C4⋊C4—51no—
281C4⋊C4—101no—
282C4⋊C4—101no—
283C4⋊C4—101no—
284C4⋊C4—101no—
285C4⋊C4C22×D511yescharacteristic
286C4⋊C4C22×D511yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—101no—
292C22⋊C4—101no—
293C22⋊C4—101no—
294C22⋊C4—101no—
295C22⋊C4—101no—
296C42—101no—
297C2×C5—41no—
298C2×C5—41no—
299C2×C5—41no—
300C2×C5—81no—
301C2×C5#53411yescharacteristic
302C2×C5#57711yescharacteristic
303C2×C5#57811yescharacteristic
304C23—21no—
305C23C2×C22⋊D511yescharacteristic
306C23C2×C22⋊D511yescharacteristic
307C23C2×C22⋊D511yescharacteristic
308Q8—21no—
309Q8—21no—
310Q8—41no—
311Q8—101no—
312Q8—101no—
313Q8—101no—
314Q8—101no—
315Q8—202no—
316D4—21no—
317D4—21no—
318D4—41no—
319D4—41no—
320D4—41no—
321D4—42no—
322D4—81no—
323D4—81no—
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—42no—
334C2×C4—81no—
335C2×C4—101no—
336C2×C4—101no—
337C2×C4—101no—
338C2×C4—101no—
339C2×C4—202no—
340C2×C4C2×C22⋊D511yescharacteristic
341C2×C4C2×C22⋊D511yescharacteristic
342C2×C4C2×C22⋊D511yescharacteristic
343C2×C4C23×D511yescharacteristic
344C8—101no—
345C8—101no—
346C8—101no—
347C8—101no—
348C5#320111yescharacteristic, complemented, Sylow
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22C22×C22⋊D511yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—201no—
369C4—201no—
370C4—201no—
371C4—201no—
372C4#414411yescharacteristic
373C4#414411yescharacteristic
374C2—41no—
375C2—41no—
376C2—41no—
377C2—81no—
378C2#6613811yescharacteristic
379C2#6617011yescharacteristic
380C2#6617211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
Fitting, prime radicalC5×#537C2
socleC22×C5C22×D4
hypercenter#537D5
hypocenterC5#3201
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×#41 ◁ #537

Complements

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

NH ≅ G/Ntype
2. #66159C2semidirect
3. #65528C2semidirect
4. #65528C2semidirect
5. #65523C2semidirect
6. #65341C2semidirect
7. #65340C2semidirect
8. #65339C2semidirect
9. #65334C2semidirect
10. #65331C2semidirect
11. #65330C2semidirect
12. #65329C2semidirect
13. #65324C2semidirect
14. #65260C2semidirect
15. #65259C2semidirect
348. C5#3201semidirect

Maximal subgroups

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

subgroupindex
C5×#5372
#661592
#655282
#655232
#653412
#653402
#653392
#653342
#653312
#653302
#653292
#653242
#652602
#652592
#32015