Finite Group Databasebeta

← #10612313

Subgroups — #10612313

Order 640.

338 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#66005C211yesmaximal, characteristic
2Q8×Dic10C211yesmaximal, characteristic, complemented
3C5×#464C211yesmaximal, characteristic
4#65473C211yesmaximal, characteristic, complemented
5#65471C211yesmaximal, characteristic, complemented
6#65455C211yesmaximal, characteristic
7SD16×Dic5C211yesmaximal, characteristic
8#65344C211yesmaximal, characteristic
9#65319C211yesmaximal, characteristic, complemented
10#65140C211yesmaximal, characteristic, complemented
11#65117C211yesmaximal, characteristic, complemented
12#65081C211yesmaximal, characteristic, complemented
13#65080C211yesmaximal, characteristic, complemented
14#65054C211yesmaximal, characteristic
15#65050C211yesmaximal, characteristic
16C2×C5×Q16C2211yescharacteristic
17C2×C5×SD16C2211yescharacteristic
18C5×#124C2211yescharacteristic
19C4×C5×Q8C2211yescharacteristic
20Q8×Dic5C2211yescharacteristic
21Q8×Dic5C2211yescharacteristic
22Q8×Dic5C2211yescharacteristic
23#4090C2211yescharacteristic
24C2×#689C2211yescharacteristic
25#4082C2211yescharacteristic
26D4×Dic5C2211yescharacteristic
27#4034—21no—
28#4034—21no—
29#4034C2211yescharacteristic
30#4034C2211yescharacteristic
31#4025—21no—
32#4025—21no—
33#4024—21no—
34#4023—21no—
35#4015—21no—
36#4015C2211yescharacteristic
37C4×Dic10—21no—
38C4×Dic10C2211yescharacteristic
39C4×Dic10C2211yescharacteristic
40C5×C4⋊C8C2211yescharacteristic
41C5×#103C2211yescharacteristic
42C5×#102C2211yescharacteristic
43#3967C2211yescharacteristic
44#3967C2211yescharacteristic
45#3967C2211yescharacteristic
46#3964C2211yescharacteristic
47#3964C2211yescharacteristic
48#3949C2211yescharacteristic
49#3949C2211yescharacteristic
50#3948C2211yescharacteristic
51#3948C2211yescharacteristic
52#3947C2211yescharacteristic
53#3946C2211yescharacteristic
54#3946C2211yescharacteristic
55C8×Dic5C2211yescharacteristic
56#3942C2211yescharacteristic
57#3940C2211yescharacteristic
58#3935C2211yescharacteristic
59#3369—51nomaximal, Sylow
60C2×C5×Q8C2311yescharacteristic
61C2×C5×Q8C2311yescharacteristic
62C2×C5×D4C2311yescharacteristic
63C22×Dic5—21no—
64C2×Dic10—21no—
65C2×Dic10—21no—
66C2×Dic10—21no—
67C2×Dic10D411yescharacteristic
68C2×Dic10C2311yescharacteristic
69C5×Q16—22no—
70C5×SD16—22no—
71C2×C5×C8C2311yescharacteristic
72C2×C5×C8C2311yescharacteristic
73C5×C4⋊C4—21no—
74C5×C4⋊C4C2311yescharacteristic
75C5×C22⋊C4—21no—
76C5×C22⋊C4—21no—
77C42×C5—21no—
78C42×C5C2311yescharacteristic
79C22⋊Dic5—21no—
80C22⋊Dic5—21no—
81C22⋊Dic5—21no—
82#689—22no—
83C4⋊Dic5—21no—
84C4⋊Dic5—21no—
85C4⋊Dic5—21no—
86C4⋊Dic5—21no—
87C4⋊Dic5D411yescharacteristic
88C4⋊Dic5C2311yescharacteristic
89C4⋊Dic5C2311yescharacteristic
90C4⋊Dic5C2311yescharacteristic
91#683—21no—
92#683—21no—
93#683—21no—
94#683—21no—
95#683—21no—
96#683D411yescharacteristic
97#683D411yescharacteristic
98C4×Dic5—21no—
99C4×Dic5—21no—
100C4×Dic5—21no—
101C4×Dic5—21no—
102C4×Dic5C2311yescharacteristic
103C4×Dic5C2311yescharacteristic
104C2×#168C2311yescharacteristic
105C2×#168C2311yescharacteristic
106Q82—51no—
107#556—51no—
108#489—51no—
109#489—51no—
110#487—51no—
111#476—51no—
112#475—51no—
113#475—51no—
114#464—51no—
115#464D511yescharacteristic
116#462—51no—
117#446—51no—
118#441—51no—
119#441—51no—
120C4×SD16—51no—
121C23×C5—21no—
122C5×Q8—21no—
123C5×Q8—21no—
124C5×Q8—21no—
125C5×Q8#4112yes—
126C5×D4—21no—
127C5×D4—21no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5C2×D411yescharacteristic
132C2×C4×C5C2×D411yescharacteristic
133C2×C4×C5C2411yescharacteristic
134C2×Dic5—21no—
135C2×Dic5—21no—
136C2×Dic5—21no—
137C2×Dic5—21no—
138C2×Dic5—42no—
139C2×Dic5C2×D411yescharacteristic
140C2×Dic5C2×D411yescharacteristic
141C2×Dic5C2×D411yescharacteristic
142C2×Dic5C2×D411yescharacteristic
143Dic10—22no—
144Dic10—21no—
145Dic10—21no—
146Dic10—81no—
147Dic10—81no—
148Dic10—81no—
149C5×C8—21no—
150C5×C8—21no—
151#168—21no—
152#168—21no—
153C2×Q16—51no—
154C2×Q16C2×D511yescharacteristic
155C2×SD16C2×D511yescharacteristic
156C4⋊Q8—51no—
157C4⋊Q8—51no—
158C4⋊Q8—51no—
159C4⋊Q8—51no—
160C4⋊Q8—101no—
161C4⋊Q8—101no—
162C4⋊Q8—101no—
163#126—101no—
164#126—101no—
165#124—51no—
166#124C2×D511yescharacteristic
167C22⋊Q8—101no—
168C4×Q8—51no—
169C4×Q8—51no—
170C4×Q8—51no—
171C4×Q8—51no—
172C4×Q8—51no—
173C4×Q8—101no—
174C4×Q8C2×D511yescharacteristic
175C4×D4—51no—
176#117—101no—
177#106—51no—
178#106—51no—
179#106—51no—
180C4⋊C8—51no—
181C4⋊C8—51no—
182C4⋊C8C2×D511yescharacteristic
183#103—51no—
184#103—51no—
185#103—51no—
186#103—51no—
187#103—51no—
188#103—51no—
189#103C2×D511yescharacteristic
190#102—51no—
191#102—51no—
192#102C2×D511yescharacteristic
193#97—51no—
194#97—51no—
195C4×C8—51no—
196C22×C5—41no—
197C22×C5—41no—
198C22×C5—41no—
199C22×C5C22×D411yescharacteristic
200C4×C5—22no—
201C4×C5—21no—
202C4×C5—21no—
203C4×C5—41no—
204C4×C5—41no—
205C4×C5C2×#4111yescharacteristic
206C4×C52−1+411yescharacteristic
207Dic5—21no—
208Dic5—21no—
209Dic5—21no—
210Dic5—41no—
211Dic5—41no—
212Dic5—41no—
213Dic5—41no—
214Dic5#13712yes—
215C2×Q8—51no—
216C2×Q8—51no—
217C2×Q8—101no—
218C2×Q8—101no—
219C2×Q8—101no—
220C2×Q8C22×D511yescharacteristic
221C2×Q8C22×D511yescharacteristic
222C2×D4C22×D511yescharacteristic
223C22×C4—101no—
224Q16—22no—
225Q16—102no—
226SD16—22no—
227C2×C8—51no—
228C2×C8—51no—
229C2×C8C22×D511yescharacteristic
230C2×C8C22×D511yescharacteristic
231C4⋊C4—21no—
232C4⋊C4—51no—
233C4⋊C4—51no—
234C4⋊C4—51no—
235C4⋊C4—51no—
236C4⋊C4—51no—
237C4⋊C4—51no—
238C4⋊C4—101no—
239C4⋊C4—101no—
240C4⋊C4—101no—
241C4⋊C4—101no—
242C4⋊C4—101no—
243C4⋊C4—101no—
244C4⋊C4—101no—
245C4⋊C4—101no—
246C4⋊C4—101no—
247C4⋊C4C22×D511yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—21no—
250C22⋊C4—101no—
251C22⋊C4—101no—
252C22⋊C4—101no—
253C42—21no—
254C42—51no—
255C42—51no—
256C42—101no—
257C42—101no—
258C42—101no—
259C42—101no—
260C42C22×D511yescharacteristic
261C2×C5—81no—
262C2×C5#54811yescharacteristic
263C2×C5C2×#13711yescharacteristic
264C2×C5#57711yescharacteristic
265C23—21no—
266Q8—21no—
267Q8—21no—
268Q8—21no—
269Q8—102no—
270Q8—101no—
271Q8—101no—
272Q8—401no—
273Q8—401no—
274Q8—401no—
275Q8#71112yes—
276D4—21no—
277D4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—51no—
282C2×C4—51no—
283C2×C4—51no—
284C2×C4—51no—
285C2×C4—101no—
286C2×C4—101no—
287C2×C4—101no—
288C2×C4—101no—
289C2×C4—202no—
290C2×C4D4×D511yescharacteristic
291C2×C4D4×D511yescharacteristic
292C2×C4C23×D511yescharacteristic
293C8—21no—
294C8—21no—
295C8—101no—
296C8—101no—
297C5#336911yescharacteristic, complemented, Sylow
298C22—41no—
299C22—41no—
300C22—41no—
301C22C2×D4×D511yescharacteristic
302C4—21no—
303C4—21no—
304C4—22no—
305C4—41no—
306C4—41no—
307C4—52no—
308C4—101no—
309C4—101no—
310C4—101no—
311C4—201no—
312C4—201no—
313C4—201no—
314C4—201no—
315C4C2×#71111yescharacteristic
316C4#415011yescharacteristic
317C2—81no—
318C2#6593211yescharacteristic
319C2#6610411yescharacteristic
320C2D5×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#464C2
socleC22×C5C22×D4
prime radicalC2×C5×D4C23
hypercenter#464D5
hypocenterC5#3369
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
2. Q8×Dic10C2semidirect
4. #65473C2semidirect
5. #65471C2semidirect
9. #65319C2semidirect
10. #65140C2semidirect
11. #65117C2semidirect
12. #65081C2semidirect
13. #65080C2semidirect
297. C5#3369semidirect

Maximal subgroups

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

subgroupindex
#660052
Q8×Dic102
C5×#4642
#654732
#654712
#654552
SD16×Dic52
#653442
#653192
#651402
#651172
#650812
#650802
#650542
#650502
#33695