Finite Group Databasebeta

← #12065189

Subgroups — #12065189

Order 1280.

1214 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4033C211yesmaximal, characteristic
2#10610350C212yesmaximal
3C5×#1463C211yesmaximal, characteristic
4#10602345C211yesmaximal, characteristic
5#10600764C212yesmaximal
6#10598920C214yesmaximal, complemented
7#10598918C214yesmaximal, complemented
8C23×Dic10C411yescharacteristic
9C22×C5×C4⋊C4C2211yescharacteristic
10C22×#683C411yescharacteristic
11C22×#683C2211yescharacteristic
12C22×C4×Dic5C411yescharacteristic
13C22×C4×Dic5C411yescharacteristic
14C22×C4×Dic5C2211yescharacteristic
15C2×#4033—22no—
16C2×#4033—22no—
17C2×#4033—22no—
18C2×#4033—22no—
19C2×#4033C412yes—
20C2×#4033C412yes—
21C2×#4033C2212yes—
22C2×#4033C2212yes—
23C5×#407C2212yes—
24#65496C2212yes—
25#65407—22no—
26M16×Dic5—22no—
27#65008—22no—
28#65008C2214yes—
29#65007—22no—
30#65007C2214yes—
31C5×#324C2214yes—
32#64748C2214yes—
33#64689C2218yes—
34#17365—51nomaximal, Sylow
35C23×C4×C5C2×C411yescharacteristic
36C23×C4×C5C2×C411yescharacteristic
37C23×C4×C5C2311yescharacteristic
38C23×Dic5C2×C411yescharacteristic
39C23×Dic5C2×C411yescharacteristic
40C23×Dic5C2×C411yescharacteristic
41C23×Dic5C2×C411yescharacteristic
42C22×Dic10—22no—
43C22×Dic10—22no—
44C22×Dic10—28no—
45C22×Dic10C2×C412yes—
46C22×Dic10C2×C412yes—
47C2×C5×M16—22no—
48C2×C5×C4⋊C4—22no—
49C2×C5×C4⋊C4—22no—
50C2×C5×C4⋊C4C2×C412yes—
51C2×C5×C4⋊C4C2312yes—
52C2×#683—22no—
53C2×#683—22no—
54C2×#683—22no—
55C2×#683—22no—
56C2×#683C2×C412yes—
57C2×#683C2×C412yes—
58C2×#683C2×C412yes—
59C2×#683C2312yes—
60C2×C4×Dic5—21no—
61C2×C4×Dic5—21no—
62C2×C4×Dic5—21no—
63C2×C4×Dic5—21no—
64C2×C4×Dic5—21no—
65C2×C4×Dic5—21no—
66C2×C4×Dic5—21no—
67C2×C4×Dic5—21no—
68C2×C4×Dic5C2×C412yes—
69C2×C4×Dic5C2×C412yes—
70C2×C4×Dic5C2×C412yes—
71C2×C4×Dic5C2×C412yes—
72C2×C4×Dic5C2×C411yescharacteristic
73C2×C4×Dic5C2×C411yescharacteristic
74C2×C4×Dic5D412yes—
75C2×C4×Dic5D412yes—
76C2×C4×Dic5D412yes—
77C2×C4×Dic5D412yes—
78C2×C4×Dic5C2311yescharacteristic
79C2×C4×Dic5C2311yescharacteristic
80C2×#681—22no—
81#4033—24no—
82#4033—24no—
83#4033—24no—
84#4033—24no—
85#4033—24no—
86#4033—24no—
87#4033—24no—
88#4033—24no—
89C5×C22⋊C8—22no—
90C5×C22⋊C8C2314yes—
91C22⋊#168—22no—
92C22⋊#168C2314yes—
93#3947—24no—
94#3946—24no—
95#3946—24no—
96C8×Dic5—24no—
97C22×C4×Q8—51no—
98#2920—52no—
99#1467—54no—
100#1463—52no—
101#1463—51no—
102#1463D511yescharacteristic
103#1461—54no—
104C24×C5C22×C411yescharacteristic
105C22×C4×C5—21no—
106C22×C4×C5—21no—
107C22×C4×C5—24no—
108C22×C4×C5—21no—
109C22×C4×C5—21no—
110C22×C4×C5—42no—
111C22×C4×C5—42no—
112C22×C4×C5C22×C412yes—
113C22×C4×C5C22×C412yes—
114C22×C4×C5C22×C411yescharacteristic
115C22×C4×C5C2×D412yes—
116C22×C4×C5C2×D412yes—
117C22×C4×C5C2411yescharacteristic
118C22×Dic5—21no—
119C22×Dic5—21no—
120C22×Dic5—24no—
121C22×Dic5—24no—
122C22×Dic5—24no—
123C22×Dic5—21no—
124C22×Dic5—21no—
125C22×Dic5—42no—
126C22×Dic5C22⋊C412yes—
127C22×Dic5C22⋊C412yes—
128C22×Dic5C22⋊C411yescharacteristic
129C22×Dic5C22⋊C411yescharacteristic
130C22×Dic5C22⋊C411yescharacteristic
131C22×Dic5C22⋊C411yescharacteristic
132C22×Dic5C22×C411yescharacteristic
133C22×Dic5C22×C411yescharacteristic
134C22×Dic5C22×C411yescharacteristic
135C22×Dic5C22×C412yes—
136C22×Dic5C22×C412yes—
137C22×Dic5C22×C411yescharacteristic
138C2×Dic10—216no—
139C2×Dic10—216no—
140C2×Dic10—24no—
141C2×Dic10—24no—
142C2×Dic10—24no—
143C2×Dic10—24no—
144C2×Dic10—24no—
145C2×Dic10M1618yescomplemented
146C5×M16—44no—
147C2×C5×C8—24no—
148C5×C4⋊C4—24no—
149C5×C4⋊C4—24no—
150#683—24no—
151#683—24no—
152#683—24no—
153#683—24no—
154C4×Dic5—22no—
155C4×Dic5—22no—
156C4×Dic5—22no—
157C4×Dic5—22no—
158C4×Dic5—22no—
159C4×Dic5—22no—
160C4×Dic5—22no—
161C4×Dic5—22no—
162C4×Dic5C22⋊C412yes—
163C4×Dic5C22⋊C412yes—
164C4×Dic5C22⋊C412yes—
165C4×Dic5C22⋊C412yes—
166C4×Dic5C2×D412yes—
167C4×Dic5C2×D412yes—
168C4×Dic5C2×D412yes—
169C4×Dic5C2×D412yes—
170#681—44no—
171C2×#168—24no—
172C23×Q8—51no—
173C2×C4×Q8—52no—
174C2×C4×Q8—52no—
175C2×C4×Q8—52no—
176C2×C4×Q8—52no—
177C2×C4×Q8—102no—
178C2×C4×Q8—102no—
179C2×C4×Q8—102no—
180C2×C4×Q8—102no—
181C22×C4⋊C4—51no—
182C22×C4⋊C4—51no—
183C22×C4⋊C4C2×D511yescharacteristic
184C22×C42—51no—
185C22×C42—51no—
186C22×C42—51no—
187#432—54no—
188#432—102no—
189#431—54no—
190#431—102no—
191#423—102no—
192#407—52no—
193#407C2×D512yes—
194C4×M16—102no—
195#324—54no—
196#324—58no—
197#324C2×D514yes—
198C23×C5—22no—
199C23×C5—22no—
200C23×C5—22no—
201C23×C5C2×C22⋊C411yescharacteristic
202C23×C5C2×C22⋊C411yescharacteristic
203C23×C5C23×C411yescharacteristic
204C2×C4×C5—24no—
205C2×C4×C5—24no—
206C2×C4×C5—22no—
207C2×C4×C5—22no—
208C2×C4×C5—24no—
209C2×C4×C5—22no—
210C2×C4×C5—41no—
211C2×C4×C5—42no—
212C2×C4×C5—41no—
213C2×C4×C5—42no—
214C2×C4×C5—42no—
215C2×C4×C5—42no—
216C2×C4×C5C2×C22⋊C412yes—
217C2×C4×C5C2×M1614yes—
218C2×C4×C5C22×D412yes—
219C2×Dic5—24no—
220C2×Dic5—24no—
221C2×Dic5—22no—
222C2×Dic5—22no—
223C2×Dic5—22no—
224C2×Dic5—22no—
225C2×Dic5—24no—
226C2×Dic5—24no—
227C2×Dic5—22no—
228C2×Dic5—22no—
229C2×Dic5—22no—
230C2×Dic5—22no—
231C2×Dic5—22no—
232C2×Dic5—42no—
233C2×Dic5—42no—
234C2×Dic5—42no—
235C2×Dic5#10114yes—
236C2×Dic5C2×C22⋊C411yescharacteristic
237C2×Dic5C2×C22⋊C411yescharacteristic
238C2×Dic5C2×C22⋊C412yes—
239C2×Dic5C2×C22⋊C412yes—
240C2×Dic5C2×C22⋊C411yescharacteristic
241C2×Dic5C2×C22⋊C411yescharacteristic
242C2×Dic5C2×M1614yes—
243C2×Dic5C2×M1614yes—
244Dic10—216no—
245Dic10—216no—
246C5×C8—44no—
247#168—44no—
248C22×Q8—52no—
249C22×Q8—52no—
250C22×Q8—102no—
251C22×Q8—102no—
252C22×Q8—108no—
253C23×C4—51no—
254C23×C4—51no—
255C23×C4—51no—
256C23×C4—51no—
257C23×C4C4×D511yescharacteristic
258C23×C4C4×D511yescharacteristic
259C23×C4C22×D511yescharacteristic
260C2×M16—22no—
261C2×M16—102no—
262C4×Q8—104no—
263C4×Q8—104no—
264C4×Q8—104no—
265C4×Q8—104no—
266C4×Q8—104no—
267C4×Q8—104no—
268C4×Q8—104no—
269C4×Q8—104no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—52no—
273C2×C4⋊C4—52no—
274C2×C4⋊C4—52no—
275C2×C4⋊C4—52no—
276C2×C4⋊C4—102no—
277C2×C4⋊C4—102no—
278C2×C4⋊C4—102no—
279C2×C4⋊C4—102no—
280C2×C4⋊C4C4×D512yes—
281C2×C4⋊C4C22×D512yes—
282C2×C42—51no—
283C2×C42—51no—
284C2×C42—52no—
285C2×C42—52no—
286C2×C42—52no—
287C2×C42—52no—
288C2×C42—51no—
289C2×C42—51no—
290C2×C42—52no—
291C2×C42—52no—
292C2×C42—52no—
293C2×C42—52no—
294C2×C42—101no—
295C2×C42—101no—
296C2×C42—101no—
297C2×C42—101no—
298C2×C42—101no—
299C2×C42—101no—
300C2×C42—101no—
301C2×C42—101no—
302C4⋊C8—104no—
303C4⋊C8—104no—
304C22⋊C8—22no—
305C22⋊C8—54no—
306C22⋊C8—102no—
307C22⋊C8C22×D514yes—
308#97—104no—
309C4×C8—104no—
310C22×C5—28no—
311C22×C5—21no—
312C22×C5—21no—
313C22×C5—21no—
314C22×C5—21no—
315C22×C5—21no—
316C22×C5—21no—
317C22×C5#41012yes—
318C22×C5C2×#10112yes—
319C22×C5C22×C22⋊C411yescharacteristic
320C22×C5C22×M1612yes—
321C4×C5—24no—
322C4×C5—22no—
323C4×C5—22no—
324C4×C5—42no—
325Dic5—22no—
326Dic5—24no—
327Dic5—21no—
328Dic5—21no—
329Dic5—42no—
330Dic5#40714yes—
331Dic5C2×#10112yes—
332Dic5C2×#10112yes—
333C24C2×C4×D511yescharacteristic
334C2×Q8—58no—
335C2×Q8—104no—
336C2×Q8—104no—
337C2×Q8—104no—
338C2×Q8—104no—
339C2×Q8—1016no—
340C2×Q8—1016no—
341C2×Q8—104no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—24no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—51no—
350C22×C4—51no—
351C22×C4—51no—
352C22×C4—51no—
353C22×C4—51no—
354C22×C4—52no—
355C22×C4—51no—
356C22×C4—51no—
357C22×C4—51no—
358C22×C4—52no—
359C22×C4—52no—
360C22×C4—52no—
361C22×C4—104no—
362C22×C4—104no—
363C22×C4—104no—
364C22×C4—101no—
365C22×C4—101no—
366C22×C4—101no—
367C22×C4—101no—
368C22×C4—202no—
369C22×C4C2×C4×D512yes—
370C22×C4C2×C4×D512yes—
371C22×C4C2×C4×D511yescharacteristic
372C22×C4D4×D512yes—
373C22×C4D4×D512yes—
374C22×C4C23×D511yescharacteristic
375M16—44no—
376M16—204no—
377C2×C8—24no—
378C2×C8—104no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—104no—
382C4⋊C4—104no—
383C4⋊C4—104no—
384C4⋊C4—104no—
385C42—52no—
386C42—52no—
387C42—52no—
388C42—52no—
389C42—52no—
390C42—52no—
391C42—52no—
392C42—52no—
393C42—102no—
394C42—102no—
395C42—102no—
396C42—102no—
397C42—102no—
398C42—102no—
399C42—102no—
400C42—102no—
401C2×C5—22no—
402C2×C5—22no—
403C2×C5—22no—
404C2×C5C2×#40711yescharacteristic
405C2×C5C2×#41011yescharacteristic
406C2×C5C22×#10111yescharacteristic
407C23—22no—
408C23—22no—
409C23—22no—
410C23D5×C22⋊C411yescharacteristic
411C23D5×C22⋊C411yescharacteristic
412C23C22×C4×D511yescharacteristic
413Q8—1016no—
414Q8—1016no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—54no—
428C2×C4—51no—
429C2×C4—51no—
430C2×C4—52no—
431C2×C4—54no—
432C2×C4—51no—
433C2×C4—51no—
434C2×C4—52no—
435C2×C4—54no—
436C2×C4—104no—
437C2×C4—104no—
438C2×C4—102no—
439C2×C4—102no—
440C2×C4—102no—
441C2×C4—102no—
442C2×C4—104no—
443C2×C4—104no—
444C2×C4—102no—
445C2×C4—102no—
446C2×C4—102no—
447C2×C4—102no—
448C2×C4—102no—
449C2×C4—202no—
450C2×C4—202no—
451C2×C4—202no—
452C2×C4D5×C22⋊C412yes—
453C2×C4D5×M1614yes—
454C2×C4C2×D4×D512yes—
455C8—44no—
456C8—204no—
457C5#1736511yescharacteristic, complemented, Sylow
458C22—28no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22#6502912yes—
466C22D5×#10112yes—
467C22C2×D5×C22⋊C411yescharacteristic
468C22C2×D5×M1612yes—
469C4—24no—
470C4—22no—
471C4—22no—
472C4—42no—
473C4—52no—
474C4—52no—
475C4—54no—
476C4—101no—
477C4—101no—
478C4—102no—
479C4—104no—
480C4—202no—
481C2—22no—
482C2—22no—
483C2—22no—
484C2D5×#40711yescharacteristic
485C2C2×#6502911yescharacteristic
486C2C2×D5×#10111yescharacteristic
subgroupG/H
centerC22C2×D5×C22⋊C4
commutatorC23×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1463C2
socleC22×C5C22×C22⋊C4
prime radicalC24×C5C22×C4
cosocleC22×C4×C5C24
hypercenter#1463D5
hypocenterC5#17365
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
6. #10598920C2semidirect
7. #10598918C2semidirect
145. C2×Dic10M16semidirect
457. C5#17365semidirect

Maximal subgroups

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

subgroupindex
C22×#40332
#106103502
C5×#14632
#106023452
#106007642
#105989202
#105989182
#173655