Finite Group Databasebeta

← #10614522

Subgroups — #10614522

Order 640.

544 conjugacy classes of subgroups.

The subgroup lattice has 388 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
3C2×#4070C211yesmaximal, characteristic
4C2×#4034C211yesmaximal, characteristic, complemented
5#65518C211yesmaximal, characteristic, complemented
6#65499C211yesmaximal, characteristic
7#65273C211yesmaximal, characteristic, complemented
8#65272C211yesmaximal, characteristic, complemented
9#65270C211yesmaximal, characteristic, complemented
10#65268C211yesmaximal, characteristic, complemented
11#65267C211yesmaximal, characteristic, complemented
12#65238C211yesmaximal, characteristic
13#65236C211yesmaximal, characteristic
14#65234C211yesmaximal, characteristic
15#65233C211yesmaximal, characteristic
16C22×C5×Q8C2211yescharacteristic
17C23×C4×C5C2211yescharacteristic
18C22×Dic10C2211yescharacteristic
19C22×Dic10C2211yescharacteristic
20C5×C22⋊Q8—24no—
21C2×C5×C4⋊C4C2211yescharacteristic
22C2×C5×C4⋊C4C2211yescharacteristic
23C2×C5×C4⋊C4C2211yescharacteristic
24C2×C5×C22⋊C4C2211yescharacteristic
25C2×C5×C22⋊C4C2211yescharacteristic
26C2×C22⋊Dic5C2211yescharacteristic
27C2×C22⋊Dic5C2211yescharacteristic
28C2×C22⋊Dic5C2211yescharacteristic
29C2×C22⋊Dic5C2211yescharacteristic
30#4090—24no—
31C2×C4⋊Dic5C2211yescharacteristic
32C2×C4⋊Dic5C2211yescharacteristic
33#4070—24no—
34C2×#683C2211yescharacteristic
35C2×#683C2211yescharacteristic
36C2×#683C2211yescharacteristic
37C2×#683C2211yescharacteristic
38C2×#683C2211yescharacteristic
39C2×#683C2211yescharacteristic
40C2×#683C2211yescharacteristic
41C2×#683C2211yescharacteristic
42C2×C4×Dic5C2211yescharacteristic
43C2×C4×Dic5C2211yescharacteristic
44#4034—24no—
45#3963C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#3963C2211yescharacteristic
54#3963C2211yescharacteristic
55#2677—51nomaximal, Sylow
56C24×C5C2311yescharacteristic
57C2×C5×Q8—22no—
58C2×C5×Q8—22no—
59C2×C5×Q8—41no—
60C22×C4×C5—22no—
61C22×C4×C5—22no—
62C22×C4×C5—41no—
63C22×C4×C5C2311yescharacteristic
64C22×C4×C5C2311yescharacteristic
65C22×C4×C5C2311yescharacteristic
66C22×C4×C5C2311yescharacteristic
67C22×C4×C5C2311yescharacteristic
68C22×C4×C5C2311yescharacteristic
69C22×Dic5C2311yescharacteristic
70C22×Dic5C2311yescharacteristic
71C22×Dic5C2311yescharacteristic
72C22×Dic5C2311yescharacteristic
73C22×Dic5C2311yescharacteristic
74C22×Dic5C2311yescharacteristic
75C22×Dic5C2311yescharacteristic
76C22×Dic5C2311yescharacteristic
77C2×Dic10—22no—
78C2×Dic10—22no—
79C2×Dic10—22no—
80C2×Dic10—22no—
81C2×Dic10—41no—
82C2×Dic10—41no—
83C5×C4⋊C4—22no—
84C5×C4⋊C4—22no—
85C5×C4⋊C4D414yes—
86C5×C22⋊C4—22no—
87C5×C22⋊C4—22no—
88C22⋊Dic5—22no—
89C22⋊Dic5—22no—
90C22⋊Dic5—41no—
91C22⋊Dic5Q814yescomplemented
92C4⋊Dic5—22no—
93C4⋊Dic5—22no—
94#683—22no—
95#683—22no—
96#683—22no—
97#683—22no—
98#683—22no—
99#683—22no—
100#683—41no—
101#683D414yes—
102C4×Dic5—22no—
103C4×Dic5—22no—
104C2×C4⋊Q8—51no—
105C2×C4⋊Q8—51no—
106C2×C22⋊Q8—51no—
107C2×C22⋊Q8D511yescharacteristic
108#399—51no—
109#398—51no—
110#398—51no—
111#395—51no—
112#395—51no—
113#393—51no—
114#388—51no—
115#388—51no—
116#387—51no—
117#387—51no—
118#385—51no—
119C23×C5—21no—
120C23×C5—21no—
121C23×C5—21no—
122C23×C5—21no—
123C23×C5—21no—
124C23×C5—21no—
125C23×C5C2×Q812yes—
126C23×C5C2411yescharacteristic
127C5×Q8—44no—
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—21no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
140C2×C4×C5—21no—
141C2×C4×C5—21no—
142C2×C4×C5—44no—
143C2×C4×C5C2×D412yes—
144C2×C4×C5C2×D412yes—
145C2×C4×C5C2×D412yes—
146C2×C4×C5C2×D412yes—
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×Dic5—21no—
163C2×Dic5—21no—
164C2×Dic5—21no—
165C2×Dic5—21no—
166C2×Dic5—21no—
167C2×Dic5C2×D412yes—
168C2×Dic5C2×D412yes—
169C2×Dic5C2×Q812yes—
170C2×Dic5C2×Q812yes—
171Dic10—44no—
172Dic10—44no—
173C22×Q8—51no—
174C22×Q8—51no—
175C22×Q8C2×D511yescharacteristic
176C23×C4C2×D511yescharacteristic
177C4⋊Q8—104no—
178C4⋊Q8—104no—
179C22⋊Q8—24no—
180C22⋊Q8—104no—
181C2×C4⋊C4—51no—
182C2×C4⋊C4—51no—
183C2×C4⋊C4—51no—
184C2×C4⋊C4—51no—
185C2×C4⋊C4—51no—
186C2×C4⋊C4—51no—
187C2×C4⋊C4—51no—
188C2×C4⋊C4—51no—
189C2×C4⋊C4—51no—
190C2×C4⋊C4—51no—
191C2×C4⋊C4C2×D511yescharacteristic
192C2×C4⋊C4C2×D511yescharacteristic
193C2×C4⋊C4C2×D511yescharacteristic
194C2×C22⋊C4—51no—
195C2×C22⋊C4—51no—
196C2×C22⋊C4—51no—
197C2×C22⋊C4—51no—
198C2×C22⋊C4C2×D511yescharacteristic
199C2×C22⋊C4C2×D511yescharacteristic
200C2×C42—51no—
201C2×C42—51no—
202#95—51no—
203#95—51no—
204#95—51no—
205#95—51no—
206#95—51no—
207#95—51no—
208#95—51no—
209#95—51no—
210#95—51no—
211#95—51no—
212C22×C5—22no—
213C22×C5—22no—
214C22×C5—22no—
215C22×C5—41no—
216C22×C5—41no—
217C22×C5—41no—
218C22×C5—41no—
219C22×C5C22×D411yescharacteristic
220C22×C5C22×D411yescharacteristic
221C22×C5C22×Q811yescharacteristic
222C22×C5H(5,𝔽2)11yescharacteristic
223C22×C52−1+411yescharacteristic
224C22×C52−1+411yescharacteristic
225C22×C52−1+411yescharacteristic
226C4×C5—22no—
227C4×C5—22no—
228C4×C5—22no—
229C4×C5—22no—
230C4×C5—41no—
231C4×C5—41no—
232Dic5—22no—
233Dic5—22no—
234Dic5—22no—
235Dic5—22no—
236Dic5—41no—
237Dic5—41no—
238Dic5—41no—
239Dic5—41no—
240C24C22×D511yescharacteristic
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—41no—
244C2×Q8—102no—
245C2×Q8—102no—
246C2×Q8—102no—
247C2×Q8—102no—
248C2×Q8—201no—
249C2×Q8—201no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—41no—
253C22×C4—51no—
254C22×C4—51no—
255C22×C4—51no—
256C22×C4—51no—
257C22×C4—51no—
258C22×C4—51no—
259C22×C4—51no—
260C22×C4—51no—
261C22×C4C22×D511yescharacteristic
262C22×C4C22×D511yescharacteristic
263C22×C4C22×D511yescharacteristic
264C22×C4C22×D511yescharacteristic
265C22×C4C22×D511yescharacteristic
266C22×C4C22×D511yescharacteristic
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—54no—
270C4⋊C4—102no—
271C4⋊C4—102no—
272C4⋊C4—102no—
273C4⋊C4—102no—
274C4⋊C4—102no—
275C4⋊C4—102no—
276C4⋊C4—102no—
277C4⋊C4—102no—
278C4⋊C4—201no—
279C4⋊C4C22⋊D514yes—
280C22⋊C4—22no—
281C22⋊C4—22no—
282C22⋊C4—54no—
283C22⋊C4—102no—
284C22⋊C4—102no—
285C22⋊C4—201no—
286C42—102no—
287C42—102no—
288C2×C5—42no—
289C2×C5#53611yescharacteristic
290C2×C5#53611yescharacteristic
291C2×C5#54411yescharacteristic
292C2×C5D4211yescharacteristic
293C2×C5D4×Q811yescharacteristic
294C2×C5D4×Q811yescharacteristic
295C2×C5#56311yescharacteristic
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23Q8×D512yes—
303C23C23×D511yescharacteristic
304Q8—44no—
305Q8—204no—
306Q8—204no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—44no—
322C2×C4—52no—
323C2×C4—52no—
324C2×C4—52no—
325C2×C4—52no—
326C2×C4—101no—
327C2×C4—101no—
328C2×C4—101no—
329C2×C4—101no—
330C2×C4—101no—
331C2×C4—101no—
332C2×C4—101no—
333C2×C4—101no—
334C2×C4—101no—
335C2×C4—101no—
336C2×C4—101no—
337C2×C4—101no—
338C2×C4—101no—
339C2×C4—101no—
340C2×C4—101no—
341C2×C4—101no—
342C2×C4—101no—
343C2×C4—101no—
344C2×C4—101no—
345C2×C4—101no—
346C2×C4D4×D512yes—
347C2×C4C2×C22⋊D512yes—
348C2×C4C2×C22⋊D512yes—
349C2×C4C2×C22⋊D512yes—
350C5#267711yescharacteristic, complemented, Sylow
351C22—22no—
352C22—22no—
353C22—22no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22C2×D4×D511yescharacteristic
359C22#414411yescharacteristic
360C22C2×Q8×D511yescharacteristic
361C22#414711yescharacteristic
362C22#415011yescharacteristic
363C22#415011yescharacteristic
364C22C22×C22⋊D511yescharacteristic
365C4—22no—
366C4—22no—
367C4—22no—
368C4—22no—
369C4—41no—
370C4—41no—
371C4—102no—
372C4—102no—
373C4—102no—
374C4—102no—
375C4—201no—
376C4—201no—
377C4—201no—
378C4—201no—
379C2—42no—
380C2#6595711yescharacteristic
381C2#6596211yescharacteristic
382C2#6596611yescharacteristic
383C2#6598011yescharacteristic
384C2D4×C22⋊D511yescharacteristic
385C2Q8×C22⋊D511yescharacteristic
386C2#6615911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×C22⋊Q8C2
prime radicalC24×C5C23
hypercenterC2×C22⋊Q8D5
hypocenterC5#2677
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
4. C2×#4034C2semidirect
5. #65518C2semidirect
7. #65273C2semidirect
8. #65272C2semidirect
9. #65270C2semidirect
10. #65268C2semidirect
11. #65267C2semidirect
91. C22⋊Dic5Q8semidirect
350. C5#2677semidirect

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
C2×#40702
C2×#40342
#655182
#654992
#652732
#652722
#652702
#652682
#652672
#652382
#652362
#652342
#652332
#26775