Finite Group Databasebeta

← #10613028

Subgroups — #10613028

Order 640.

360 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#65948C211yesmaximal, characteristic, complemented
2#65848C211yesmaximal, characteristic, complemented
3C5×#480C211yesmaximal, characteristic, complemented
4#65328C211yesmaximal, characteristic, complemented
5#65253C211yesmaximal, characteristic, complemented
6#65180C211yesmaximal, characteristic, complemented
7#65171C211yesmaximal, characteristic, complemented
8#65170C211yesmaximal, characteristic, complemented
9#65167C211yesmaximal, characteristic, complemented
10#65073C211yesmaximal, characteristic, complemented
11#65071C211yesmaximal, characteristic, complemented
12#65052C211yesmaximal, characteristic, complemented
13#65045C211yesmaximal, characteristic, complemented
14#65026C211yesmaximal, characteristic, complemented
15#65018C211yesmaximal, characteristic, complemented
16C5×#121C2211yescharacteristic
17C2×C5×C4⋊C4C2211yescharacteristic
18#4084—21no—
19#4084C2211yescharacteristic
20#4081—21no—
21D4×Dic5C2211yescharacteristic
22C2×#686C2211yescharacteristic
23#4076—21no—
24#4076C2211yescharacteristic
25C4×C22⋊D5—21no—
26C4×C22⋊D5C2211yescharacteristic
27#4072C2211yescharacteristic
28C2×#681C2211yescharacteristic
29C2×#676C2211yescharacteristic
30#4044—21no—
31#4041C2211yescharacteristic
32#4040—21no—
33#4039C2211yescharacteristic
34#4038C2211yescharacteristic
35D5×C4⋊C4C2211yescharacteristic
36#4036C2211yescharacteristic
37#4036C2211yescharacteristic
38#4027—21no—
39#4025—21no—
40C5×#107C2211yescharacteristic
41C5×#107C2211yescharacteristic
42C5×#102C2211yescharacteristic
43C5×#102C2211yescharacteristic
44C5×C22⋊C8C2211yescharacteristic
45#3964C2211yescharacteristic
46#3953C2211yescharacteristic
47#3953C2211yescharacteristic
48#3952C2211yescharacteristic
49#3950C2211yescharacteristic
50#3949C2211yescharacteristic
51#3946C2211yescharacteristic
52C8×Dic5C2211yescharacteristic
53#3941C2211yescharacteristic
54#3940C2211yescharacteristic
55#3940C2211yescharacteristic
56#3940C2211yescharacteristic
57#3939C2211yescharacteristic
58#3939C2211yescharacteristic
59#3355—51nomaximal, Sylow
60C2×C5×D4—21no—
61C2×C5×D4C2311yescharacteristic
62C22×C4×C5—21no—
63C22×C4×C5C2311yescharacteristic
64C2×C22⋊D5—21no—
65C2×C22⋊D5—21no—
66C2×C22⋊D5D411yescharacteristic
67C22×Dic5—21no—
68C2×D20—21no—
69C2×D20C2311yescharacteristic
70C2×C4×D5—21no—
71C2×C4×D5—21no—
72C2×C4×D5C2311yescharacteristic
73C2×C5×C8C2311yescharacteristic
74C2×C5×C8C2311yescharacteristic
75C5×C4⋊C4—21no—
76C5×C4⋊C4C2311yescharacteristic
77C5×C4⋊C4C2311yescharacteristic
78C5×C4⋊C4C2311yescharacteristic
79C5×C22⋊C4—21no—
80C22⋊Dic5—21no—
81C22⋊Dic5—21no—
82C22⋊Dic5—21no—
83C22⋊Dic5D411yescharacteristic
84#686—22no—
85#685—21no—
86#685—21no—
87#685—21no—
88#685—21no—
89#685D411yescharacteristic
90C4⋊Dic5—21no—
91C4⋊Dic5—21no—
92C4⋊Dic5C2311yescharacteristic
93C4⋊Dic5C2311yescharacteristic
94C4⋊Dic5C2311yescharacteristic
95#683—21no—
96#683—21no—
97#683—21no—
98#683D411yescharacteristic
99C4×Dic5—21no—
100C4×Dic5—21no—
101C4×Dic5C2311yescharacteristic
102#681—22no—
103C2×#168C2311yescharacteristic
104C2×#168C2311yescharacteristic
105#676—22no—
106#553—51no—
107#553—51no—
108#499—51no—
109#482—51no—
110#482—51no—
111#480—51no—
112#480D511yescharacteristic, complemented
113#478—51no—
114#478—51no—
115#469—51no—
116#469—51no—
117C4×D8—51no—
118#436—51no—
119#428—51no—
120#428—51no—
121C23×C5—21no—
122C23×C5C2×D411yescharacteristic
123C22×D5—21no—
124C22×D5C2×D411yescharacteristic
125C5×D4—21no—
126C5×D4—21no—
127C5×D4—81no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—21no—
133C2×C4×C5—81no—
134C2×C4×C5C2×D411yescharacteristic
135C2×C4×C5C2411yescharacteristic
136C22⋊D5—41no—
137C22⋊D5—42no—
138C22⋊D5—42no—
139C2×Dic5—21no—
140C2×Dic5—21no—
141C2×Dic5—21no—
142C2×Dic5—81no—
143C2×Dic5C2×D411yescharacteristic
144C2×Dic5C2×D411yescharacteristic
145C2×Dic5C2×D411yescharacteristic
146D20—21no—
147D20—21no—
148D20—81no—
149C4×D5—21no—
150C4×D5—21no—
151C4×D5—81no—
152C4×D5—81no—
153C5×C8—21no—
154C5×C8—21no—
155#168—21no—
156#168#4112yes—
157C2×D8—51no—
158C2×M16—51no—
159C2×M16—51no—
160#126—101no—
161#126—101no—
162#125—51no—
163#125—51no—
164#123—101no—
165#123—101no—
166#121—51no—
167#121—51no—
168#121—51no—
169#121—101no—
170#121—101no—
171#121C2×D511yescharacteristic
172C4×D4—51no—
173C4×D4—51no—
174C4×D4—51no—
175C4×D4—101no—
176C4×D4—101no—
177#117—51no—
178#117—51no—
179C2×C4⋊C4—51no—
180C2×C4⋊C4C2×D511yescharacteristic
181#107—51no—
182#107—51no—
183#107—51no—
184#107C2×D511yescharacteristic
185#107C2×D511yescharacteristic
186#106—51no—
187#106—51no—
188#106—51no—
189#106—51no—
190C4⋊C8—51no—
191#102—51no—
192#102—51no—
193#102—51no—
194#102—51no—
195#102C2×D511yescharacteristic
196#102C2×D511yescharacteristic
197C22⋊C8—51no—
198C22⋊C8C2×D511yescharacteristic
199C4×C8—51no—
200C22×C5—21no—
201C22×C5—21no—
202C22×C5—21no—
203C22×C5—41no—
204C22×C5—41no—
205C22×C5—41no—
206C22×C5C22×D411yescharacteristic
207C2×D5—21no—
208C2×D5—21no—
209C2×D5—21no—
210C2×D5—41no—
211C2×D5—41no—
212C2×D5—41no—
213C4×C5—21no—
214C4×C5—41no—
215C4×C5—41no—
216C4×C5—41no—
217C4×C5C2×#4111yescharacteristic
218C4×C52−1+411yescharacteristic
219Dic5—21no—
220Dic5—21no—
221Dic5—21no—
222Dic5—41no—
223Dic5—41no—
224Dic5—41no—
225C2×D4—21no—
226C2×D4—51no—
227C2×D4—51no—
228C2×D4—101no—
229C2×D4—101no—
230C2×D4—101no—
231C2×D4C22×D511yescharacteristic
232C22×C4—21no—
233C22×C4—51no—
234C22×C4—101no—
235C22×C4—101no—
236C22×C4—101no—
237C22×C4C22×D511yescharacteristic
238D8—102no—
239M16—102no—
240M16—102no—
241C2×C8—51no—
242C2×C8—51no—
243C2×C8C22×D511yescharacteristic
244C2×C8C22×D511yescharacteristic
245C4⋊C4—21no—
246C4⋊C4—51no—
247C4⋊C4—51no—
248C4⋊C4—51no—
249C4⋊C4—51no—
250C4⋊C4—101no—
251C4⋊C4—101no—
252C4⋊C4—101no—
253C4⋊C4—101no—
254C4⋊C4—101no—
255C4⋊C4C22×D511yescharacteristic
256C4⋊C4C22×D511yescharacteristic
257C4⋊C4C22×D511yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—51no—
260C22⋊C4—51no—
261C22⋊C4—101no—
262C22⋊C4—101no—
263C22⋊C4—101no—
264C22⋊C4—101no—
265C22⋊C4—101no—
266C22⋊C4—101no—
267C22⋊C4—101no—
268C42—51no—
269C42—101no—
270C42—101no—
271C2×C5—41no—
272C2×C5—81no—
273C2×C5#54711yescharacteristic
274C2×C5#57511yescharacteristic
275C2×C5#57611yescharacteristic
276D5—41no—
277D5—81no—
278C23—21no—
279C23—51no—
280C23—101no—
281C23D4×D511yescharacteristic
282D4—21no—
283D4—21no—
284D4—81no—
285D4—101no—
286D4—101no—
287D4—201no—
288D4—202no—
289D4—202no—
290D4—401no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—51no—
297C2×C4—51no—
298C2×C4—51no—
299C2×C4—81no—
300C2×C4—101no—
301C2×C4—101no—
302C2×C4—101no—
303C2×C4—101no—
304C2×C4—101no—
305C2×C4—401no—
306C2×C4—401no—
307C2×C4—401no—
308C2×C4D4×D511yescharacteristic
309C2×C4C23×D511yescharacteristic
310C8—21no—
311C8—21no—
312C8—52no—
313C8—101no—
314C5#335511yescharacteristic, complemented, Sylow
315C22—21no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—101no—
322C22—101no—
323C22—101no—
324C22—201no—
325C22—201no—
326C22—201no—
327C22C2×D4×D511yescharacteristic
328C4—21no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—101no—
333C4—101no—
334C4—101no—
335C4—201no—
336C4—201no—
337C4—201no—
338C4D5×#4111yescharacteristic
339C4#415011yescharacteristic
340C2—41no—
341C2—81no—
342C2—201no—
343C2—401no—
344C2#6598911yescharacteristic
345C2#6609111yescharacteristic
346C2#6611111yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#480C2
socleC22×C5C22×D4
hypercenter#480D5
hypocenterC5#3355
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #65948C2semidirect
2. #65848C2semidirect
3. C5×#480C2semidirect
4. #65328C2semidirect
5. #65253C2semidirect
6. #65180C2semidirect
7. #65171C2semidirect
8. #65170C2semidirect
9. #65167C2semidirect
10. #65073C2semidirect
11. #65071C2semidirect
12. #65052C2semidirect
13. #65045C2semidirect
14. #65026C2semidirect
15. #65018C2semidirect
112. #480D5semidirect
314. C5#3355semidirect

Maximal subgroups

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

subgroupindex
#659482
#658482
C5×#4802
#653282
#652532
#651802
#651712
#651702
#651672
#650732
#650712
#650522
#650452
#650262
#650182
#33555