Finite Group Databasebeta

← #10614512

Subgroups — #10614512

Order 640.

544 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×C22⋊Q8C211yesmaximal, characteristic
2C2×#4024C211yesmaximal, characteristic
3#65519C211yesmaximal, characteristic, complemented
4#65518C211yesmaximal, characteristic, complemented
5#65499C211yesmaximal, characteristic
6C2×#3963C211yesmaximal, characteristic
7#65272C211yesmaximal, characteristic, complemented
8#65269C211yesmaximal, characteristic, complemented
9#65268C211yesmaximal, characteristic, complemented
10#65263C211yesmaximal, characteristic, complemented
11#65261C211yesmaximal, characteristic, complemented
12#65261C211yesmaximal, characteristic, complemented
13#65235C211yesmaximal, characteristic
14#65231C211yesmaximal, characteristic
15C22⋊C4×Dic5C211yesmaximal, characteristic
16C22×C5×Q8C2211yescharacteristic
17C23×C4×C5C2211yescharacteristic
18C23×Dic5C2211yescharacteristic
19C23×Dic5C2211yescharacteristic
20C22×Dic10C2211yescharacteristic
21C5×C22⋊Q8—24no—
22C2×C5×C4⋊C4C2211yescharacteristic
23C2×C5×C4⋊C4C2211yescharacteristic
24C2×C5×C4⋊C4C2211yescharacteristic
25C2×C5×C22⋊C4C2211yescharacteristic
26C2×C5×C22⋊C4C2211yescharacteristic
27C2×C22⋊Dic5C2211yescharacteristic
28C2×C22⋊Dic5C2211yescharacteristic
29C2×C4⋊Dic5C2211yescharacteristic
30C2×#683C2211yescharacteristic
31C2×#683C2211yescharacteristic
32C2×#683C2211yescharacteristic
33C2×#683C2211yescharacteristic
34C2×#683C2211yescharacteristic
35C2×#683C2211yescharacteristic
36C2×#683C2211yescharacteristic
37C2×#683C2211yescharacteristic
38C2×C4×Dic5C2211yescharacteristic
39C2×C4×Dic5C2211yescharacteristic
40#4024—24no—
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#2549—51nomaximal, Sylow
54C24×C5C2311yescharacteristic
55C2×C5×Q8—22no—
56C2×C5×Q8—22no—
57C2×C5×Q8—41no—
58C22×C4×C5—22no—
59C22×C4×C5—22no—
60C22×C4×C5—41no—
61C22×C4×C5C2311yescharacteristic
62C22×C4×C5C2311yescharacteristic
63C22×C4×C5C2311yescharacteristic
64C22×C4×C5C2311yescharacteristic
65C22×C4×C5C2311yescharacteristic
66C22×C4×C5C2311yescharacteristic
67C22×Dic5—22no—
68C22×Dic5—22no—
69C22×Dic5—41no—
70C22×Dic5—41no—
71C22×Dic5—41no—
72C22×Dic5Q814yescomplemented
73C22×Dic5C2311yescharacteristic
74C22×Dic5C2311yescharacteristic
75C22×Dic5C2311yescharacteristic
76C22×Dic5C2311yescharacteristic
77C22×Dic5C2311yescharacteristic
78C22×Dic5C2311yescharacteristic
79C22×Dic5C2311yescharacteristic
80C22×Dic5C2311yescharacteristic
81C2×Dic10—22no—
82C2×Dic10—22no—
83C2×Dic10—41no—
84C5×C4⋊C4—22no—
85C5×C4⋊C4—22no—
86C5×C4⋊C4—22no—
87C5×C22⋊C4—22no—
88C5×C22⋊C4D414yes—
89C22⋊Dic5—22no—
90C22⋊Dic5Q814yescomplemented
91C4⋊Dic5—22no—
92#683—22no—
93#683—22no—
94#683—22no—
95#683—22no—
96#683—41no—
97#683—41no—
98#683—41no—
99#683—41no—
100C4×Dic5—41no—
101C4×Dic5—41no—
102C2×C22⋊Q8—51no—
103C2×C22⋊Q8D511yescharacteristic
104#398—51no—
105#398—51no—
106#395—51no—
107#395—51no—
108#391—51no—
109#391—51no—
110#389—51no—
111#389—51no—
112#385—51no—
113#385—51no—
114#380—51no—
115C4×C22⋊C4—51no—
116C2×#95—51no—
117C23×C5—21no—
118C23×C5—21no—
119C23×C5—21no—
120C23×C5—21no—
121C23×C5C2×D412yes—
122C23×C5C2×Q812yes—
123C23×C5C2×Q812yes—
124C23×C5C2411yescharacteristic
125C5×Q8—44no—
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—21no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
140C2×C4×C5—21no—
141C2×C4×C5—44no—
142C2×C4×C5C2×D412yes—
143C2×C4×C5C2×D412yes—
144C2×C4×C5#4112yes—
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×Dic5—21no—
163C2×Dic5—21no—
164C2×Dic5—44no—
165C2×Dic5—44no—
166C2×Dic5C2×Q812yes—
167C2×Dic5C2×Q812yes—
168C2×Dic5C2×Q812yes—
169C2×Dic5C2×Q812yes—
170C2×Dic5#4112yes—
171Dic10—44no—
172C22×Q8—51no—
173C22×Q8C2×D511yescharacteristic
174C23×C4—51no—
175C23×C4—51no—
176C23×C4C2×D511yescharacteristic
177C22⋊Q8—24no—
178C22⋊Q8—104no—
179C2×C4⋊C4—51no—
180C2×C4⋊C4—51no—
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⋊C4C2×D511yescharacteristic
189C2×C4⋊C4C2×D511yescharacteristic
190C2×C4⋊C4C2×D511yescharacteristic
191C2×C22⋊C4—51no—
192C2×C22⋊C4—51no—
193C2×C22⋊C4C2×D511yescharacteristic
194C2×C22⋊C4C2×D511yescharacteristic
195C2×C42—51no—
196C2×C42—51no—
197#95—51no—
198#95—51no—
199#95—51no—
200#95—51no—
201#95—51no—
202#95—51no—
203#95—51no—
204#95—51no—
205#95—51no—
206#95—51no—
207#95—51no—
208#95—51no—
209C22×C5—22no—
210C22×C5—22no—
211C22×C5—22no—
212C22×C5—22no—
213C22×C5—22no—
214C22×C5—22no—
215C22×C5C4⋊Q814yes—
216C22×C5C22×D411yescharacteristic
217C22×C5C22×Q811yescharacteristic
218C22×C5C22×Q811yescharacteristic
219C22×C5C2×#4111yescharacteristic
220C22×C5C2×#4111yescharacteristic
221C22×C5H(5,𝔽2)11yescharacteristic
222C22×C5H(5,𝔽2)11yescharacteristic
223C4×C5—22no—
224C4×C5—22no—
225C4×C5—22no—
226C4×C5—41no—
227C4×C5—41no—
228C4×C5—41no—
229Dic5—22no—
230Dic5—22no—
231Dic5—22no—
232Dic5—22no—
233Dic5—22no—
234Dic5—41no—
235Dic5—41no—
236Dic5—41no—
237C24C22×D511yescharacteristic
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—41no—
241C2×Q8—102no—
242C2×Q8—102no—
243C2×Q8—201no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—41no—
247C22×C4—51no—
248C22×C4—51no—
249C22×C4—51no—
250C22×C4—51no—
251C22×C4—51no—
252C22×C4—54no—
253C22×C4—51no—
254C22×C4—51no—
255C22×C4—51no—
256C22×C4—102no—
257C22×C4—102no—
258C22×C4—201no—
259C22×C4—201no—
260C22×C4—201no—
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—22no—
270C4⋊C4—102no—
271C4⋊C4—102no—
272C4⋊C4—102no—
273C4⋊C4—102no—
274C4⋊C4—102no—
275C4⋊C4—201no—
276C4⋊C4—201no—
277C4⋊C4—201no—
278C4⋊C4—201no—
279C22⋊C4—22no—
280C22⋊C4—54no—
281C22⋊C4—102no—
282C22⋊C4C22⋊D514yes—
283C42—201no—
284C42—201no—
285C2×C5—24no—
286C2×C5C2×C4⋊Q811yescharacteristic
287C2×C5#53311yescharacteristic
288C2×C5#54311yescharacteristic
289C2×C5#54611yescharacteristic
290C2×C5#54611yescharacteristic
291C2×C5#55411yescharacteristic
292C2×C5#55411yescharacteristic
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23Q8×D512yes—
298C23Q8×D512yes—
299C23C2×C22⋊D512yes—
300C23C23×D511yescharacteristic
301Q8—44no—
302Q8—204no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
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—44no—
319C2×C4—52no—
320C2×C4—52no—
321C2×C4—52no—
322C2×C4—52no—
323C2×C4—52no—
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×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—204no—
344C2×C4—204no—
345C2×C4#71112yes—
346C2×C4C2×C22⋊D512yes—
347C2×C4C2×C22⋊D512yes—
348C5#254911yescharacteristic, complemented, Sylow
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22#409014yes—
356C22C2×#71111yescharacteristic
357C22#414411yescharacteristic
358C22C2×Q8×D511yescharacteristic
359C22C2×Q8×D511yescharacteristic
360C22D5×#4111yescharacteristic
361C22#414911yescharacteristic
362C22C22×C22⋊D511yescharacteristic
363C4—22no—
364C4—22no—
365C4—22no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—102no—
370C4—102no—
371C4—102no—
372C4—102no—
373C4—102no—
374C4—201no—
375C4—201no—
376C4—201no—
377C2—24no—
378C2#6595611yescharacteristic
379C2#6596811yescharacteristic
380C2#6596911yescharacteristic
381C2#6597011yescharacteristic
382C2#6614011yescharacteristic
383C2C2×#409011yescharacteristic
384C2#6616311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×C22⋊Q8C2
prime radicalC24×C5C23
hypercenterC2×C22⋊Q8D5
hypocenterC5#2549
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #65519C2semidirect
4. #65518C2semidirect
7. #65272C2semidirect
8. #65269C2semidirect
9. #65268C2semidirect
10. #65263C2semidirect
11. #65261C2semidirect
12. #65261C2semidirect
72. C22×Dic5Q8semidirect
90. C22⋊Dic5Q8semidirect
348. C5#2549semidirect

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×#40242
#655192
#655182
#654992
C2×#39632
#652722
#652692
#652682
#652632
#652612
#652352
#652312
C22⋊C4×Dic52
#25495