Finite Group Databasebeta

← #12070520

Subgroups — #12070520

Order 1280.

598 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10611033C211yesmaximal, characteristic
2#10611025C211yesmaximal, characteristic
3#10610998C211yesmaximal, characteristic, complemented
4C5×#1450C211yesmaximal, characteristic, complemented
5#10601546C211yesmaximal, characteristic, complemented
6#10600531C211yesmaximal, characteristic, complemented
7#10600527C211yesmaximal, characteristic
8#10598881C212yesmaximal, complemented
9#10598880C212yesmaximal, complemented
10#10598848C212yesmaximal, complemented
11C8×Dic20C212yesmaximal, complemented
12C4×#709C2211yescharacteristic
13#65804C2211yescharacteristic
14C5×#431C2211yescharacteristic
15C5×#431C2211yescharacteristic
16C2×C4×C5×C8C2211yescharacteristic
17#65214C2211yescharacteristic
18#65213C2211yescharacteristic
19#65140C411yescharacteristic
20#65138C411yescharacteristic
21#65135C411yescharacteristic
22#65130C411yescharacteristic
23#65126—21no—
24#65122—21no—
25#65120—21no—
26#65119—21no—
27#65022C412yes—
28#65018—22no—
29#65012C412yes—
30#65009—22no—
31C4×Dic20C2212yes—
32C4×#677C2212yes—
33#64977C2212yes—
34#64968C2212yes—
35C8×Dic10C2212yes—
36C8×Dic10C2212yes—
37C5×#322C2212yes—
38C5×C82C2212yes—
39#64679C2211yescharacteristic
40#64679C2211yescharacteristic
41#64678C2212yes—
42#64677C2212yes—
43#64676C2211yescharacteristic
44#64676C2211yescharacteristic
45#64676C2212yes—
46#64676C2212yes—
47#18019—51nomaximal, Sylow
48C2×#709C2×C411yescharacteristic
49C22×C5×C8—21no—
50C2×C42×C5C2311yescharacteristic
51C4×C22⋊D5—21no—
52C4×C22⋊D5—21no—
53#4072—21no—
54#4072C2×C411yescharacteristic
55#4070—21no—
56#4070C2×C412yes—
57C2×#681—21no—
58C2×Dic20C2×C412yes—
59C2×#677C2×C412yes—
60C2×#676—22no—
61C4×D20C2×C411yescharacteristic
62C4×D20C2311yescharacteristic
63#4018—21no—
64C42×D5—21no—
65#4016—21no—
66#4015C2×C411yescharacteristic
67#4015C2×C411yescharacteristic
68C4×Dic10—21no—
69C4×Dic10C2×C411yescharacteristic
70C4×Dic10C2312yes—
71C4×Dic10C2311yescharacteristic
72C5×C4⋊C8C2×C411yescharacteristic
73C5×C4⋊C8C2×C411yescharacteristic
74C5×C4⋊C8D411yescharacteristic
75C5×C4⋊C8D411yescharacteristic
76C5×C22⋊C8C2×C412yes—
77C5×C22⋊C8D412yes—
78C4×C5×C8—21no—
79C4×C5×C8C2311yescharacteristic
80C4×C5×C8C2311yescharacteristic
81C4×C5×C8C2312yes—
82C4×C5×C8C2312yes—
83C22⋊#168—21no—
84#3953C2×C412yes—
85#3952—22no—
86#3950C2×C412yes—
87#3949C2×C412yes—
88#3948C2×C412yes—
89#3948C2×C412yes—
90#3948C2×C412yes—
91#3947—22no—
92#3946—22no—
93#3946—22no—
94#3946—22no—
95C8×Dic5—22no—
96C8×Dic5—22no—
97C4⋊#168—21no—
98C4⋊#168C2312yes—
99C4⋊#168C2311yescharacteristic
100C4⋊#168C2311yescharacteristic
101C4×#168—21no—
102#3107—51no—
103#2991—51no—
104#2986—51no—
105#1590—52no—
106#1588—52no—
107#1587—52no—
108C8×Q16—52no—
109#1574—51no—
110#1493—51no—
111#1476—51no—
112#1450D511yescharacteristic, complemented
113C22×C4×C5C22×C411yescharacteristic
114C22×C4×C5C2×D411yescharacteristic
115C22×C4×C5#4111yescharacteristic
116C2×C22⋊D5—21no—
117#709—22no—
118#709—41no—
119C2×D20C22×C411yescharacteristic
120C2×C4×D5—21no—
121C2×C4×D5—21no—
122C2×C4×D5—21no—
123C2×Dic10—21no—
124C2×Dic10C22×C412yes—
125C2×Dic10C22×C411yescharacteristic
126C2×C5×C8—24no—
127C2×C5×C8—21no—
128C2×C5×C8—21no—
129C2×C5×C8C22×C412yes—
130C2×C5×C8C22×C412yes—
131C2×C5×C8C2×D412yes—
132C2×C5×C8C2×D412yes—
133C42×C5C22×C411yescharacteristic
134C42×C5C2×D411yescharacteristic
135C42×C5#4111yescharacteristic
136C42×C5C2411yescharacteristic
137C22⋊Dic5—22no—
138C22⋊Dic5—21no—
139#685—21no—
140#685—21no—
141C4⋊Dic5—21no—
142C4⋊Dic5—21no—
143C4⋊Dic5—21no—
144C4⋊Dic5C22×C411yescharacteristic
145C4⋊Dic5C22×C411yescharacteristic
146C4⋊Dic5C22×C412yes—
147#683—22no—
148#683—21no—
149#683—22no—
150#683—21no—
151C4×Dic5—21no—
152C4×Dic5—21no—
153C4×Dic5—21no—
154C4×Dic5—22no—
155#681—81no—
156C2×#168—21no—
157C2×#168—21no—
158C2×#168—22no—
159Dic20—24no—
160#677—42no—
161#676—82no—
162#533—51no—
163C4×#41—51no—
164#485—52no—
165#475—51no—
166#474—51no—
167#462—51no—
168#461—51no—
169#452—52no—
170#446—52no—
171#446—101no—
172C8×Q8—52no—
173C8×Q8—52no—
174C8×Q8—101no—
175C4×Q16—52no—
176C4×SD16—52no—
177#436—52no—
178#436—102no—
179#433—102no—
180#431—51no—
181#431—101no—
182#431C2×D511yescharacteristic
183#431C2×D511yescharacteristic
184#423—51no—
185#423—101no—
186C2×C4×C8C2×D511yescharacteristic
187#335—52no—
188#334—52no—
189Q8⋊C8—51no—
190Q8⋊C8—52no—
191Q8⋊C8—51no—
192Q8⋊C8—52no—
193D4⋊C8—51no—
194D4⋊C8—51no—
195#322C2×D512yes—
196C82C2×D512yes—
197C23×C5C4×D411yescharacteristic
198C22×D5—21no—
199C2×C4×C5—21no—
200C2×C4×C5—21no—
201C2×C4×C5—21no—
202C2×C4×C5—21no—
203C2×C4×C5—22no—
204C2×C4×C5C4×D411yescharacteristic
205C2×C4×C5C4×D411yescharacteristic
206C2×C4×C5C4×D411yescharacteristic
207C2×C4×C5C23×C411yescharacteristic
208C2×C4×C5C22×D411yescharacteristic
209C2×C4×C5C2×#4111yescharacteristic
210C22⋊D5—42no—
211C2×Dic5—21no—
212C2×Dic5—21no—
213C2×Dic5—21no—
214C2×Dic5—22no—
215C2×Dic5—22no—
216D20—21no—
217D20—21no—
218C4×D5—42no—
219C4×D5—41no—
220C4×D5—41no—
221C4×D5—42no—
222Dic10—21no—
223Dic10—22no—
224Dic10—22no—
225Dic10—42no—
226Dic10#13112yes—
227C5×C8—22no—
228C5×C8—22no—
229C5×C8—22no—
230C5×C8—22no—
231C5×C8—22no—
232C5×C8—22no—
233#168—42no—
234#168—41no—
235#168—41no—
236C2×#41—51no—
237C2×Q16—52no—
238C2×SD16—52no—
239C2×M16—101no—
240C2×M16—102no—
241C22×C8—21no—
242C4⋊Q8—51no—
243C4⋊Q8—51no—
244#125—101no—
245C22⋊Q8—52no—
246C22⋊Q8—101no—
247C4×Q8—51no—
248C4×Q8—51no—
249C4×Q8—52no—
250C4×Q8—101no—
251C4×D4—51no—
252C4×D4—51no—
253C4×D4—101no—
254C4×D4—101no—
255#117—51no—
256#117—101no—
257#117—101no—
258C2×C42—101no—
259C2×C42C22×D511yescharacteristic
260#107—52no—
261#106—52no—
262C4⋊C8—51no—
263C4⋊C8—51no—
264C4⋊C8—52no—
265C4⋊C8—102no—
266C4⋊C8—102no—
267C4⋊C8—102no—
268C4⋊C8—101no—
269C4⋊C8C4×D511yescharacteristic
270C4⋊C8C4×D511yescharacteristic
271C4⋊C8D2011yescharacteristic
272C4⋊C8D2011yescharacteristic
273#103—52no—
274#103—52no—
275#103—52no—
276#102—52no—
277C22⋊C8—101no—
278C22⋊C8—102no—
279C22⋊C8C4×D512yes—
280C22⋊C8D2012yes—
281#97—102no—
282C4×C8—21no—
283C4×C8—102no—
284C4×C8—102no—
285C4×C8—101no—
286C4×C8C22×D511yescharacteristic
287C4×C8C22×D511yescharacteristic
288C4×C8C22×D512yes—
289C4×C8C22×D512yes—
290C22×C5—21no—
291C22×C5—21no—
292C22×C5—21no—
293C22×C5C2×C4×D411yescharacteristic
294C2×D5—41no—
295C2×D5—41no—
296C2×D5—41no—
297C4×C5—21no—
298C4×C5—21no—
299C4×C5—21no—
300C4×C5#44312yes—
301C4×C5C2×#13111yescharacteristic
302C4×C5#56811yescharacteristic
303C4×C5#57711yescharacteristic
304C4×C5#57811yescharacteristic
305Dic5—22no—
306Dic5—42no—
307Dic5—41no—
308Dic5—42no—
309Dic5—41no—
310#41—102no—
311#41—201no—
312C2×Q8—51no—
313C2×Q8—52no—
314C2×Q8—101no—
315C2×D4—51no—
316C2×D4—101no—
317C22×C4—101no—
318C22×C4—101no—
319C22×C4—101no—
320C22×C4C2×C4×D511yescharacteristic
321C22×C4C2×D2011yescharacteristic
322C22×C4#70911yescharacteristic
323Q16—104no—
324SD16—202no—
325M16—402no—
326M16—401no—
327C2×C8—24no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—101no—
331C2×C8—101no—
332C2×C8—102no—
333C2×C8C2×C4×D512yes—
334C2×C8C2×C4×D512yes—
335C2×C8C2×D2012yes—
336C2×C8C2×D2012yes—
337C4⋊C4—52no—
338C4⋊C4—51no—
339C4⋊C4—51no—
340C4⋊C4—101no—
341C4⋊C4—102no—
342C4⋊C4—101no—
343C4⋊C4—101no—
344C4⋊C4—101no—
345C4⋊C4—102no—
346C4⋊C4—101no—
347C22⋊C4—101no—
348C22⋊C4—102no—
349C22⋊C4—101no—
350C22⋊C4—101no—
351C42—101no—
352C42—102no—
353C42—101no—
354C42—101no—
355C42C2×C4×D511yescharacteristic
356C42C2×D2011yescharacteristic
357C42#70911yescharacteristic
358C42C23×D511yescharacteristic
359C2×C5—41no—
360C2×C5#293311yescharacteristic
361C2×C5#294711yescharacteristic
362C2×C5C2×#44311yescharacteristic
363D5—81no—
364C23—101no—
365C23C4×D2011yescharacteristic
366Q8—52no—
367Q8—101no—
368Q8—102no—
369Q8—102no—
370Q8—202no—
371D4—101no—
372D4—101no—
373D4—202no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—101no—
380C2×C4—101no—
381C2×C4—101no—
382C2×C4—102no—
383C2×C4—102no—
384C2×C4—202no—
385C2×C4—201no—
386C2×C4—201no—
387C2×C4—202no—
388C2×C4C4×D2011yescharacteristic
389C2×C4C4×D2011yescharacteristic
390C2×C4C4×D2011yescharacteristic
391C2×C4C22×C4×D511yescharacteristic
392C2×C4C22×D2011yescharacteristic
393C2×C4C2×#70911yescharacteristic
394C8—22no—
395C8—22no—
396C8—22no—
397C8—22no—
398C8—22no—
399C8—22no—
400C8—202no—
401C8—201no—
402C8—201no—
403C5#1801911yescharacteristic, complemented, Sylow
404C22—21no—
405C22—21no—
406C22—21no—
407C22—201no—
408C22—201no—
409C22—201no—
410C22C2×C4×D2011yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—102no—
415C4—201no—
416C4—202no—
417C4—202no—
418C4—201no—
419C4#6498912yes—
420C4D5×#13111yescharacteristic
421C4#6608411yescharacteristic
422C4#6608511yescharacteristic
423C4#6608711yescharacteristic
424C2—41no—
425C2—401no—
426C2C2×#6498911yescharacteristic
427C2#1061028611yescharacteristic
428C2#1061030211yescharacteristic
subgroupG/H
centerC2×C4C22×D20
commutatorC2×C4×C5C23×C4
FrattiniC42C23×D5
FittingC5×#1450C2
socleC22×C5C2×C4×D4
prime radicalC2×#709C2×C4
cosocleC42×C5C24
hypercenter#1450D5
hypocenterC5#18019
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1450

Complements

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

NH ≅ G/Ntype
3. #10610998C2semidirect
4. C5×#1450C2semidirect
5. #10601546C2semidirect
6. #10600531C2semidirect
8. #10598881C2semidirect
9. #10598880C2semidirect
10. #10598848C2semidirect
11. C8×Dic20C2semidirect
112. #1450D5semidirect
403. C5#18019semidirect

Maximal subgroups

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

subgroupindex
#106110332
#106110252
#106109982
C5×#14502
#106015462
#106005312
#106005272
#105988812
#105988802
#105988482
C8×Dic202
#180195