Finite Group Databasebeta

← #10615270

Subgroups — #10615270

Order 640.

392 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#548C211yesmaximal, characteristic, complemented
2#65897C211yesmaximal, characteristic, complemented
3#65527C211yesmaximal, characteristic, complemented
4#65522C211yesmaximal, characteristic, complemented
5#65516C211yesmaximal, characteristic, complemented
6#65514C211yesmaximal, characteristic, complemented
7#65346C211yesmaximal, characteristic, complemented
8#65344C211yesmaximal, characteristic, complemented
9#65337C211yesmaximal, characteristic, complemented
10#65334C211yesmaximal, characteristic, complemented
11#65328C211yesmaximal, characteristic, complemented
12#65324C211yesmaximal, characteristic, complemented
13#65315C211yesmaximal, characteristic, complemented
14#65309C211yesmaximal, characteristic, complemented
15#65300C211yesmaximal, characteristic, complemented
16C2×C5×#41C2211yescharacteristic
17C22×C5×Q8C2211yescharacteristic
18C5×#124—21no—
19C5×#124C2211yescharacteristic
20C5×C22⋊Q8—21no—
21C5×C22⋊Q8C2211yescharacteristic
22C5×#121—21no—
23C5×#121C2211yescharacteristic
24C4×C5×Q8C2211yescharacteristic
25C4×C5×D4—21no—
26C4×C5×D4C2211yescharacteristic
27C2×#688C2211yescharacteristic
28C2×#688C2211yescharacteristic
29#4084—21no—
30C2×#686C2211yescharacteristic
31#4076C2211yescharacteristic
32#4076C2211yescharacteristic
33#4072C2211yescharacteristic
34#4072C2211yescharacteristic
35C2×#681C2211yescharacteristic
36C22×#168C2211yescharacteristic
37#4038—21no—
38#4031—21no—
39#4031—21no—
40C4×D20C2211yescharacteristic
41#4015C2211yescharacteristic
42#3967C2211yescharacteristic
43#3967C2211yescharacteristic
44#3967C2211yescharacteristic
45#3967C2211yescharacteristic
46#3967C2211yescharacteristic
47#3967C2211yescharacteristic
48#3964C2211yescharacteristic
49#3964C2211yescharacteristic
50#3964C2211yescharacteristic
51C22⋊#168C2211yescharacteristic
52C22⋊#168C2211yescharacteristic
53#3941C2211yescharacteristic
54#3940C2211yescharacteristic
55#3940C2211yescharacteristic
56#3939C2211yescharacteristic
57C4⋊#168C2211yescharacteristic
58#3935C2211yescharacteristic
59#3321—51nomaximal, Sylow
60C5×#41—22no—
61C5×#41—41no—
62C2×C5×Q8—21no—
63C2×C5×Q8—22no—
64C2×C5×Q8—41no—
65C2×C5×Q8C2311yescharacteristic
66C2×C5×Q8C2311yescharacteristic
67C2×C5×D4—21no—
68C2×C5×D4—21no—
69C2×C5×D4D411yescharacteristic
70C2×C5×D4C2311yescharacteristic
71C22×C4×C5—21no—
72C22×C4×C5—21no—
73C22×C4×C5C2311yescharacteristic
74C22×C4×C5C2311yescharacteristic
75C2×C22⋊D5—21no—
76C2×C22⋊D5—21no—
77C2×D20C2311yescharacteristic
78C2×C4×D5—21no—
79C2×Dic10—21no—
80C5×C4⋊C4—21no—
81C5×C4⋊C4—21no—
82C5×C4⋊C4D411yescharacteristic
83C5×C4⋊C4C2311yescharacteristic
84C5×C22⋊C4—21no—
85C5×C22⋊C4—21no—
86C5×C22⋊C4—21no—
87C5×C22⋊C4—21no—
88C5×C22⋊C4D411yescharacteristic
89C5×C22⋊C4D411yescharacteristic
90C42×C5—21no—
91C42×C5C2311yescharacteristic
92C22⋊Dic5—21no—
93C22⋊Dic5—21no—
94#688—22no—
95#688—41no—
96#686—41no—
97#685—21no—
98#685—21no—
99#685—21no—
100C4⋊Dic5—21no—
101C4⋊Dic5C2311yescharacteristic
102C4⋊Dic5C2311yescharacteristic
103C4⋊Dic5C2311yescharacteristic
104C4×Dic5—21no—
105C4×Dic5—21no—
106#681—41no—
107C2×#168—41no—
108C2×#168C2311yescharacteristic
109C2×#168C2311yescharacteristic
110C2×#168C2311yescharacteristic
111C2×#168C2311yescharacteristic
112#555—51no—
113#548D511yescharacteristic, complemented
114#489—51no—
115#485—51no—
116#482—51no—
117#475—51no—
118#469—51no—
119#465—51no—
120#463—51no—
121#449—51no—
122#448—51no—
123#440—51no—
124#435—51no—
125#419—51no—
126#416—51no—
127C23×C5—21no—
128C23×C5C2×D411yescharacteristic
129C23×C5C2×D411yescharacteristic
130C22×D5—21no—
131C5×Q8—21no—
132C5×Q8—21no—
133C5×Q8—21no—
134C5×Q8—42no—
135C5×Q8#4112yes—
136C5×D4—21no—
137C5×D4—21no—
138C5×D4—41no—
139C5×D4—42no—
140C5×D4—42no—
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—42no—
149C2×C4×C5—42no—
150C2×C4×C5C2×D411yescharacteristic
151C2×C4×C5C2×D411yescharacteristic
152C2×C4×C5C2×D411yescharacteristic
153C2×C4×C5C2×D411yescharacteristic
154C2×C4×C5C2411yescharacteristic
155C22⋊D5—81no—
156C22⋊D5—81no—
157C2×Dic5—21no—
158C2×Dic5—21no—
159C2×Dic5—21no—
160D20—21no—
161D20—21no—
162C4×D5—42no—
163Dic10—81no—
164#168—21no—
165#168—21no—
166#168—21no—
167#168—21no—
168C2×#41C2×D511yescharacteristic
169C22×Q8C2×D511yescharacteristic
170C2×SD16—51no—
171C2×SD16—51no—
172C2×D8—51no—
173C2×M16—51no—
174C22×C8—51no—
175C4⋊Q8—51no—
176#124—21no—
177#124—101no—
178#124—101no—
179#124C2×D511yescharacteristic
180C22⋊Q8—21no—
181C22⋊Q8C2×D511yescharacteristic
182#121—21no—
183#121—51no—
184#121—51no—
185#121—101no—
186#121C2×D511yescharacteristic
187C4×Q8C2×D511yescharacteristic
188C4×D4—21no—
189C4×D4—51no—
190C4×D4C2×D511yescharacteristic
191#117—51no—
192#117—51no—
193#117—101no—
194#107—51no—
195#106—51no—
196#106—51no—
197C4⋊C8—51no—
198#103—51no—
199#103—51no—
200#103—51no—
201#103—51no—
202#103—51no—
203#103—51no—
204#102—51no—
205#102—51no—
206#102—51no—
207#102—51no—
208C22⋊C8—51no—
209C22⋊C8—51no—
210#97—51no—
211C22×C5—21no—
212C22×C5—21no—
213C22×C5—21no—
214C22×C5—21no—
215C22×C5—21no—
216C22×C5—21no—
217C22×C5—41no—
218C22×C5—41no—
219C22×C5—41no—
220C22×C5C22×D411yescharacteristic
221C2×D5—41no—
222C2×D5—41no—
223C2×D5—41no—
224C4×C5—21no—
225C4×C5—21no—
226C4×C5—22no—
227C4×C5—21no—
228C4×C5—21no—
229C4×C5—41no—
230C4×C5—41no—
231C4×C5C2×#4111yescharacteristic
232C4×C5H(5,𝔽2)11yescharacteristic
233Dic5—41no—
234Dic5—41no—
235Dic5—41no—
236#41—22no—
237#41—41no—
238C2×Q8—21no—
239C2×Q8—22no—
240C2×Q8—41no—
241C2×Q8—101no—
242C2×Q8C22×D511yescharacteristic
243C2×Q8C22×D511yescharacteristic
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—51no—
247C2×D4—101no—
248C2×D4—101no—
249C2×D4C22⋊D511yescharacteristic
250C2×D4C22×D511yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—101no—
254C22×C4C22×D511yescharacteristic
255C22×C4C22×D511yescharacteristic
256SD16—102no—
257SD16—201no—
258D8—201no—
259M16—201no—
260C2×C8—51no—
261C2×C8—51no—
262C2×C8—51no—
263C2×C8—51no—
264C2×C8—201no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—51no—
268C4⋊C4—51no—
269C4⋊C4—51no—
270C4⋊C4—101no—
271C4⋊C4C22⋊D511yescharacteristic
272C4⋊C4C22×D511yescharacteristic
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—101no—
278C22⋊C4—101no—
279C22⋊C4—101no—
280C22⋊C4—101no—
281C22⋊C4—101no—
282C22⋊C4C22⋊D511yescharacteristic
283C22⋊C4C22⋊D511yescharacteristic
284C42—21no—
285C42—101no—
286C42—101no—
287C42C22×D511yescharacteristic
288C2×C5—41no—
289C2×C5—41no—
290C2×C5—81no—
291C2×C5#54611yescharacteristic
292C2×C5#57511yescharacteristic
293C2×C5#57711yescharacteristic
294D5—81no—
295C23—21no—
296C23—101no—
297C23C2×C22⋊D511yescharacteristic
298C23C2×C22⋊D511yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—21no—
302Q8—42no—
303Q8—401no—
304Q8#71112yes—
305D4—21no—
306D4—21no—
307D4—41no—
308D4—42no—
309D4—42no—
310D4—101no—
311D4—101no—
312D4—401no—
313D4—401no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—101no—
324C2×C4—101no—
325C2×C4—101no—
326C2×C4—202no—
327C2×C4C2×C22⋊D511yescharacteristic
328C2×C4C2×C22⋊D511yescharacteristic
329C2×C4C2×C22⋊D511yescharacteristic
330C2×C4C2×C22⋊D511yescharacteristic
331C2×C4C23×D511yescharacteristic
332C8—101no—
333C8—101no—
334C8—101no—
335C8—101no—
336C5#332111yescharacteristic, complemented, Sylow
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—201no—
347C22—201no—
348C22—201no—
349C22C22×C22⋊D511yescharacteristic
350C4—21no—
351C4—21no—
352C4—22no—
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—201no—
358C4—201no—
359C4—201no—
360C4C2×#71111yescharacteristic
361C4#414411yescharacteristic
362C2—41no—
363C2—41no—
364C2—81no—
365C2—401no—
366C2#6614011yescharacteristic
367C2#6615611yescharacteristic
368C2#6617111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#548C2
socleC22×C5C22×D4
hypercenter#548D5
hypocenterC5#3321
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C5×#548C2semidirect
2. #65897C2semidirect
3. #65527C2semidirect
4. #65522C2semidirect
5. #65516C2semidirect
6. #65514C2semidirect
7. #65346C2semidirect
8. #65344C2semidirect
9. #65337C2semidirect
10. #65334C2semidirect
11. #65328C2semidirect
12. #65324C2semidirect
13. #65315C2semidirect
14. #65309C2semidirect
15. #65300C2semidirect
113. #548D5semidirect
336. C5#3321semidirect

Maximal subgroups

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

subgroupindex
C5×#5482
#658972
#655272
#655222
#655162
#655142
#653462
#653442
#653372
#653342
#653282
#653242
#653152
#653092
#653002
#33215