Finite Group Databasebeta

← #10614536

Subgroups — #10614536

Order 640.

480 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×C22⋊Q8C211yesmaximal, characteristic
2C2×#4090C211yesmaximal, characteristic, complemented
3#65519C211yesmaximal, characteristic, complemented
4#65499C211yesmaximal, characteristic
5#65499C211yesmaximal, characteristic
6#65273C211yesmaximal, characteristic, complemented
7#65271C211yesmaximal, characteristic, complemented
8#65269C211yesmaximal, characteristic, complemented
9#65268C211yesmaximal, characteristic, complemented
10#65267C211yesmaximal, characteristic, complemented
11#65261C211yesmaximal, characteristic, complemented
12#65237C211yesmaximal, characteristic
13#65236C211yesmaximal, characteristic
14#65234C211yesmaximal, characteristic
15#65233C211yesmaximal, characteristic
16C22×C5×Q8C2211yescharacteristic
17C23×C4×C5C2211yescharacteristic
18C22×Dic10C2211yescharacteristic
19C5×C22⋊Q8—24no—
20C2×C5×C4⋊C4C2211yescharacteristic
21C2×C5×C4⋊C4C2211yescharacteristic
22C2×C5×C4⋊C4C2211yescharacteristic
23C2×C5×C22⋊C4C2211yescharacteristic
24C2×C5×C22⋊C4C2211yescharacteristic
25C2×C22⋊Dic5C2211yescharacteristic
26C2×C22⋊Dic5C2211yescharacteristic
27C2×C22⋊Dic5C2211yescharacteristic
28C2×C22⋊Dic5C2211yescharacteristic
29#4090—24no—
30C2×C4⋊Dic5C2211yescharacteristic
31C2×#683C2211yescharacteristic
32C2×#683C2211yescharacteristic
33C2×#683C2211yescharacteristic
34C2×#683C2211yescharacteristic
35C2×#683C2211yescharacteristic
36C2×#683C2211yescharacteristic
37C2×#683C2211yescharacteristic
38C2×#683C2211yescharacteristic
39C2×C4×Dic5C2211yescharacteristic
40C2×C4×Dic5C2211yescharacteristic
41#3963C2211yescharacteristic
42#3963C2211yescharacteristic
43#3963C2211yescharacteristic
44#3963C2211yescharacteristic
45#3963C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#2729—51nomaximal, Sylow
54C24×C5C2311yescharacteristic
55C2×C5×Q8—41no—
56C2×C5×Q8—41no—
57C2×C5×Q8D414yes—
58C22×C4×C5—22no—
59C22×C4×C5—41no—
60C22×C4×C5—41no—
61C22×C4×C5C2311yescharacteristic
62C22×C4×C5C2311yescharacteristic
63C22×C4×C5C2311yescharacteristic
64C22×C4×C5C2311yescharacteristic
65C22×C4×C5C2311yescharacteristic
66C22×C4×C5C2311yescharacteristic
67C22×Dic5C2311yescharacteristic
68C22×Dic5C2311yescharacteristic
69C22×Dic5C2311yescharacteristic
70C22×Dic5C2311yescharacteristic
71C22×Dic5C2311yescharacteristic
72C22×Dic5C2311yescharacteristic
73C22×Dic5C2311yescharacteristic
74C22×Dic5C2311yescharacteristic
75C2×Dic10—22no—
76C2×Dic10—41no—
77C2×Dic10—41no—
78C5×C4⋊C4—22no—
79C5×C4⋊C4—22no—
80C5×C4⋊C4—22no—
81C5×C22⋊C4—22no—
82C5×C22⋊C4—22no—
83C22⋊Dic5—22no—
84C22⋊Dic5—41no—
85C22⋊Dic5—41no—
86C22⋊Dic5Q814yescomplemented
87C4⋊Dic5—41no—
88#683—22no—
89#683—22no—
90#683—22no—
91#683—22no—
92#683—22no—
93#683—22no—
94#683—41no—
95#683—41no—
96C4×Dic5—22no—
97C4×Dic5—41no—
98C2×C4⋊Q8—51no—
99C2×C22⋊Q8D511yescharacteristic
100#400—51no—
101#398—51no—
102#398—51no—
103#396—51no—
104#395—51no—
105#393—51no—
106#391—51no—
107#389—51no—
108#388—51no—
109#388—51no—
110#387—51no—
111#385—51no—
112#385—51no—
113C23×C5—21no—
114C23×C5—21no—
115C23×C5—21no—
116C23×C5—21no—
117C23×C5—21no—
118C23×C5—21no—
119C23×C5C2×Q812yes—
120C23×C5C2411yescharacteristic
121C5×Q8—44no—
122C2×C4×C5—21no—
123C2×C4×C5—21no—
124C2×C4×C5—21no—
125C2×C4×C5—21no—
126C2×C4×C5—21no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—21no—
133C2×C4×C5—21no—
134C2×C4×C5—21no—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
137C2×C4×C5—44no—
138C2×C4×C5C2×D412yes—
139C2×C4×C5C2×D412yes—
140C2×C4×C5C2×D412yes—
141C2×Dic5—21no—
142C2×Dic5—21no—
143C2×Dic5—21no—
144C2×Dic5—21no—
145C2×Dic5—21no—
146C2×Dic5—21no—
147C2×Dic5—21no—
148C2×Dic5—21no—
149C2×Dic5—21no—
150C2×Dic5—21no—
151C2×Dic5—21no—
152C2×Dic5—21no—
153C2×Dic5—21no—
154C2×Dic5—21no—
155C2×Dic5—21no—
156C2×Dic5—21no—
157C2×Dic5—21no—
158C2×Dic5—21no—
159C2×Dic5—21no—
160C2×Dic5—21no—
161C2×Dic5—21no—
162C2×Dic5C2×Q812yes—
163C2×Dic5C2×Q812yes—
164C2×Dic5#4112yes—
165Dic10—44no—
166C22×Q8—51no—
167C22×Q8C2×D511yescharacteristic
168C23×C4C2×D511yescharacteristic
169C4⋊Q8—104no—
170C22⋊Q8—24no—
171C2×C4⋊C4—51no—
172C2×C4⋊C4—51no—
173C2×C4⋊C4—51no—
174C2×C4⋊C4—51no—
175C2×C4⋊C4—51no—
176C2×C4⋊C4—51no—
177C2×C4⋊C4—51no—
178C2×C4⋊C4—51no—
179C2×C4⋊C4—51no—
180C2×C4⋊C4C2×D511yescharacteristic
181C2×C4⋊C4C2×D511yescharacteristic
182C2×C4⋊C4C2×D511yescharacteristic
183C2×C22⋊C4—51no—
184C2×C22⋊C4—51no—
185C2×C22⋊C4—51no—
186C2×C22⋊C4—51no—
187C2×C22⋊C4C2×D511yescharacteristic
188C2×C22⋊C4C2×D511yescharacteristic
189C2×C42—51no—
190C2×C42—51no—
191#95—51no—
192#95—51no—
193#95—51no—
194#95—51no—
195#95—51no—
196#95—51no—
197#95—51no—
198#95—51no—
199#95—51no—
200#95—51no—
201#95—51no—
202#95—51no—
203C22×C5—22no—
204C22×C5—22no—
205C22×C5—22no—
206C22×C5—41no—
207C22×C5—41no—
208C22×C5—41no—
209C22×C5—41no—
210C22×C5C22×D411yescharacteristic
211C22×C5C22×Q811yescharacteristic
212C22×C5C2×#4111yescharacteristic
213C22×C5H(5,𝔽2)11yescharacteristic
214C22×C5H(5,𝔽2)11yescharacteristic
215C22×C52−1+411yescharacteristic
216C22×C52−1+411yescharacteristic
217C4×C5—22no—
218C4×C5—22no—
219C4×C5—22no—
220C4×C5—41no—
221C4×C5—41no—
222C4×C5—41no—
223Dic5—22no—
224Dic5—22no—
225Dic5—22no—
226Dic5—41no—
227Dic5—41no—
228Dic5—41no—
229Dic5—41no—
230Dic5—41no—
231C24C22×D511yescharacteristic
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—102no—
235C2×Q8—201no—
236C2×Q8—201no—
237C2×Q8C22⋊D514yes—
238C22×C4—22no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—51no—
242C22×C4—51no—
243C22×C4—51no—
244C22×C4—51no—
245C22×C4—51no—
246C22×C4—51no—
247C22×C4—51no—
248C22×C4—51no—
249C22×C4C22×D511yescharacteristic
250C22×C4C22×D511yescharacteristic
251C22×C4C22×D511yescharacteristic
252C22×C4C22×D511yescharacteristic
253C22×C4C22×D511yescharacteristic
254C22×C4C22×D511yescharacteristic
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—102no—
259C4⋊C4—102no—
260C4⋊C4—102no—
261C4⋊C4—102no—
262C4⋊C4—102no—
263C4⋊C4—102no—
264C4⋊C4—201no—
265C4⋊C4—201no—
266C4⋊C4—201no—
267C22⋊C4—22no—
268C22⋊C4—22no—
269C22⋊C4—54no—
270C22⋊C4—102no—
271C22⋊C4—201no—
272C22⋊C4—201no—
273C42—102no—
274C42—201no—
275C2×C5—42no—
276C2×C5#53411yescharacteristic
277C2×C5#54211yescharacteristic
278C2×C5#54411yescharacteristic
279C2×C5#54811yescharacteristic
280C2×C5D4×Q811yescharacteristic
281C2×C5#55411yescharacteristic
282C2×C5#56311yescharacteristic
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23Q8×D512yes—
290C23C23×D511yescharacteristic
291Q8—44no—
292Q8—204no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—44no—
309C2×C4—52no—
310C2×C4—52no—
311C2×C4—52no—
312C2×C4—101no—
313C2×C4—101no—
314C2×C4—101no—
315C2×C4—101no—
316C2×C4—101no—
317C2×C4—101no—
318C2×C4—101no—
319C2×C4—101no—
320C2×C4—101no—
321C2×C4—101no—
322C2×C4—101no—
323C2×C4—101no—
324C2×C4—101no—
325C2×C4—101no—
326C2×C4—101no—
327C2×C4—101no—
328C2×C4—101no—
329C2×C4—101no—
330C2×C4—101no—
331C2×C4—101no—
332C2×C4—101no—
333C2×C4C2×C22⋊D512yes—
334C2×C4C2×C22⋊D512yes—
335C2×C4C2×C22⋊D512yes—
336C5#272911yescharacteristic, complemented, Sylow
337C22—22no—
338C22—22no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22#414411yescharacteristic
345C22#414411yescharacteristic
346C22C2×Q8×D511yescharacteristic
347C22#414711yescharacteristic
348C22D5×#4111yescharacteristic
349C22#415011yescharacteristic
350C22C22×C22⋊D511yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—102no—
358C4—102no—
359C4—102no—
360C4—201no—
361C4—201no—
362C4—201no—
363C4—201no—
364C4—201no—
365C2—42no—
366C2#6595711yescharacteristic
367C2#6597011yescharacteristic
368C2#6597411yescharacteristic
369C2#6598011yescharacteristic
370C2#6613811yescharacteristic
371C2Q8×C22⋊D511yescharacteristic
372C2#6616511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×C22⋊Q8C2
prime radicalC24×C5C23
hypercenterC2×C22⋊Q8D5
hypocenterC5#2729
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#4090C2semidirect
3. #65519C2semidirect
6. #65273C2semidirect
7. #65271C2semidirect
8. #65269C2semidirect
9. #65268C2semidirect
10. #65267C2semidirect
11. #65261C2semidirect
86. C22⋊Dic5Q8semidirect
336. C5#2729semidirect

Maximal subgroups

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

subgroupindex
C2×C5×C22⋊Q82
C2×#40902
#655192
#654992
#652732
#652712
#652692
#652682
#652672
#652612
#652372
#652362
#652342
#652332
#27295