Finite Group Databasebeta

← #11994677

Subgroups — #11994677

Order 1280.

736 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#388C211yesmaximal, characteristic
2C2×#65214C211yesmaximal, characteristic, complemented
3C2×#64745C212yesmaximal
4#10602324C211yesmaximal, characteristic, complemented
5#10598104C212yesmaximal, complemented
6C22×C5×C4⋊C4C411yescharacteristic
7C22×C5×C22⋊C4C411yescharacteristic
8C22×C5×C22⋊C4C411yescharacteristic
9C22×C5×C22⋊C4C2211yescharacteristic
10C22×C42×C5C2211yescharacteristic
11C5×#388—21no—
12C5×#388—21no—
13C5×#388—21no—
14C5×#388—21no—
15C2×C5×#95C411yescharacteristic
16C2×C5×#95C2211yescharacteristic
17C2×C22⋊#168C2212yes—
18C2×C22⋊#168C2212yes—
19#65214—24no—
20C2×C4⋊#168—21no—
21C2×C4×#168—21no—
22#64745—24no—
23#64744—21no—
24#64744—21no—
25#9080—51nomaximal, Sylow
26C25×C5C2×C411yescharacteristic
27C23×C4×C5C2×C411yescharacteristic
28C23×C4×C5C2×C411yescharacteristic
29C23×C4×C5C2×C411yescharacteristic
30C23×C4×C5C2×C411yescharacteristic
31C23×C4×C5C2×C411yescharacteristic
32C23×C4×C5C2311yescharacteristic
33C2×C5×C4⋊C4—21no—
34C2×C5×C4⋊C4—21no—
35C2×C5×C4⋊C4—42no—
36C2×C5×C22⋊C4—21no—
37C2×C5×C22⋊C4—21no—
38C2×C5×C22⋊C4—22no—
39C2×C5×C22⋊C4—21no—
40C2×C5×C22⋊C4—21no—
41C2×C5×C22⋊C4—21no—
42C2×C5×C22⋊C4—21no—
43C2×C5×C22⋊C4—22no—
44C2×C5×C22⋊C4—22no—
45C2×C5×C22⋊C4—42no—
46C2×C5×C22⋊C4C814yescomplemented
47C2×C42×C5—22no—
48C2×C42×C5—22no—
49C2×C42×C5D411yescharacteristic
50C2×C42×C5D411yescharacteristic
51C2×C42×C5D411yescharacteristic
52C2×C42×C5D411yescharacteristic
53C22×#168—22no—
54C22×#168—22no—
55C5×#95—21no—
56C5×#95—21no—
57C5×#95—21no—
58C5×#95—21no—
59C22⋊#168—22no—
60C22⋊#168—22no—
61C22⋊#168—22no—
62C22⋊#168D414yes—
63C4⋊#168—24no—
64C4×#168—24no—
65C2×#431—51no—
66C2×#388D511yescharacteristic
67#1817—51no—
68C2×#323—52no—
69#1280—52no—
70C24×C5—21no—
71C24×C5—21no—
72C24×C5—22no—
73C24×C5—22no—
74C24×C5—41no—
75C24×C5—41no—
76C24×C5—41no—
77C24×C5—41no—
78C24×C5C2×C812yes—
79C24×C5C22×C411yescharacteristic
80C22×C4×C5—22no—
81C22×C4×C5—22no—
82C22×C4×C5—21no—
83C22×C4×C5—22no—
84C22×C4×C5—21no—
85C22×C4×C5—22no—
86C22×C4×C5—21no—
87C22×C4×C5—21no—
88C22×C4×C5—22no—
89C22×C4×C5—22no—
90C22×C4×C5—41no—
91C22×C4×C5—41no—
92C22×C4×C5—41no—
93C22×C4×C5—41no—
94C22×C4×C5—41no—
95C22×C4×C5—41no—
96C22×C4×C5—41no—
97C22×C4×C5—41no—
98C22×C4×C5—41no—
99C22×C4×C5C22⋊C411yescharacteristic
100C22×C4×C5C22⋊C411yescharacteristic
101C22×C4×C5C22⋊C411yescharacteristic
102C22×C4×C5C22⋊C411yescharacteristic
103C22×C4×C5C2×C812yes—
104C22×C4×C5C2×C812yes—
105C22×C4×C5M1612yes—
106C22×C4×C5C2×D412yes—
107C22×C4×C5C2×D411yescharacteristic
108C22×C4×C5C2×D411yescharacteristic
109C22×C4×C5#4112yes—
110C5×C4⋊C4—42no—
111C5×C4⋊C4—42no—
112C5×C22⋊C4—24no—
113C5×C22⋊C4—24no—
114C5×C22⋊C4—42no—
115C5×C22⋊C4—42no—
116C5×C22⋊C4—41no—
117C5×C22⋊C4—42no—
118C5×C22⋊C4—41no—
119C42×C5—21no—
120C42×C5—21no—
121C42×C5—22no—
122C42×C5—22no—
123C42×C5—21no—
124C42×C5—21no—
125C2×#168—24no—
126C2×#168—42no—
127C2×#168—42no—
128C2×#168—42no—
129C2×#168—42no—
130C2×#168—42no—
131C22×C4⋊C4Dic511yescharacteristic
132C22×C22⋊C4Dic511yescharacteristic
133C22×C22⋊C4Dic511yescharacteristic
134C22×C22⋊C4C2×D511yescharacteristic
135C22×C42C2×D511yescharacteristic
136#431—104no—
137C2×C4⋊C8—101no—
138C2×C22⋊C8—52no—
139C2×C22⋊C8—52no—
140C2×C4×C8—101no—
141#388—21no—
142#388—21no—
143#388—21no—
144#388—21no—
145C2×#95Dic511yescharacteristic
146C2×#95C2×D511yescharacteristic
147#336—101no—
148#336—101no—
149#323—104no—
150C23×C5—21no—
151C23×C5—21no—
152C23×C5—21no—
153C23×C5—21no—
154C23×C5—22no—
155C23×C5—24no—
156C23×C5—42no—
157C23×C5—42no—
158C23×C5—42no—
159C23×C5—42no—
160C23×C5—41no—
161C23×C5—41no—
162C23×C5—41no—
163C23×C5—41no—
164C23×C5—41no—
165C23×C5—41no—
166C23×C5—42no—
167C23×C5—82no—
168C23×C5—82no—
169C23×C5—82no—
170C23×C5—82no—
171C23×C5C2×C22⋊C411yescharacteristic
172C23×C5C4×D412yes—
173C23×C5C22×C811yescharacteristic
174C23×C5C2×M1611yescharacteristic
175C23×C5#13112yes—
176C2×C4×C5—21no—
177C2×C4×C5—21no—
178C2×C4×C5—21no—
179C2×C4×C5—21no—
180C2×C4×C5—21no—
181C2×C4×C5—21no—
182C2×C4×C5—21no—
183C2×C4×C5—21no—
184C2×C4×C5—21no—
185C2×C4×C5—21no—
186C2×C4×C5—24no—
187C2×C4×C5—24no—
188C2×C4×C5—21no—
189C2×C4×C5—22no—
190C2×C4×C5—21no—
191C2×C4×C5—21no—
192C2×C4×C5—42no—
193C2×C4×C5—42no—
194C2×C4×C5—44no—
195C2×C4×C5—42no—
196C2×C4×C5—41no—
197C2×C4×C5—41no—
198C2×C4×C5—42no—
199C2×C4×C5—41no—
200C2×C4×C5—41no—
201C2×C4×C5—41no—
202C2×C4×C5—41no—
203C2×C4×C5—41no—
204C2×C4×C5—41no—
205C2×C4×C5—82no—
206C2×C4×C5—82no—
207C2×C4×C5C22⋊C812yes—
208C2×C4×C5C22⋊C812yes—
209C2×C4×C5#10012yes—
210C2×C4×C5#12011yescharacteristic
211C2×C4×C5#12112yes—
212C2×C4×C5#12311yescharacteristic
213#168—44no—
214#168—82no—
215C25C2×Dic511yescharacteristic
216C23×C4C2×Dic511yescharacteristic
217C23×C4C2×Dic511yescharacteristic
218C23×C4C2×Dic511yescharacteristic
219C23×C4C2×Dic511yescharacteristic
220C23×C4C2×Dic511yescharacteristic
221C23×C4C22×D511yescharacteristic
222C22×C8—102no—
223C22×C8—102no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—42no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4#16814yescomplemented
238C2×C42—22no—
239C2×C42—22no—
240C2×C42C22⋊D511yescharacteristic
241C2×C42C22⋊D511yescharacteristic
242C2×C42C22⋊D511yescharacteristic
243C2×C42C22⋊D511yescharacteristic
244C4⋊C8—104no—
245C22⋊C8—54no—
246C22⋊C8—102no—
247C22⋊C8—102no—
248C22⋊C8—102no—
249C4×C8—104no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254C22×C5—22no—
255C22×C5—22no—
256C22×C5—21no—
257C22×C5—21no—
258C22×C5—21no—
259C22×C5—21no—
260C22×C5—22no—
261C22×C5—22no—
262C22×C5—42no—
263C22×C5—42no—
264C22×C5—42no—
265C22×C5—84no—
266C22×C5—84no—
267C22×C5—81no—
268C22×C5—81no—
269C22×C5—81no—
270C22×C5—81no—
271C22×C5#38611yescharacteristic
272C22×C5C2×C22⋊C811yescharacteristic
273C22×C5#40811yescharacteristic
274C22×C5#41011yescharacteristic
275C22×C5#41011yescharacteristic
276C22×C5C2×#10011yescharacteristic
277C22×C5#41311yescharacteristic
278C22×C5C8×D412yes—
279C22×C5#43512yes—
280C4×C5—22no—
281C4×C5—22no—
282C4×C5—21no—
283C4×C5—22no—
284C4×C5—21no—
285C4×C5—22no—
286C4×C5—41no—
287C4×C5—42no—
288C4×C5—81no—
289C4×C5—81no—
290C24—21no—
291C24—21no—
292C24—22no—
293C24—22no—
294C24—41no—
295C24—41no—
296C24—41no—
297C24—41no—
298C24C2×#16812yes—
299C24C22×Dic511yescharacteristic
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4C2×#16812yes—
320C22×C4C2×#16812yes—
321C22×C4#68112yes—
322C22×C4C22⋊Dic511yescharacteristic
323C22×C4C22⋊Dic511yescharacteristic
324C22×C4C22⋊Dic511yescharacteristic
325C22×C4C22⋊Dic511yescharacteristic
326C22×C4D4×D512yes—
327C22×C4#71112yes—
328C22×C4C2×C22⋊D511yescharacteristic
329C22×C4C2×C22⋊D511yescharacteristic
330C2×C8—104no—
331C2×C8—202no—
332C2×C8—202no—
333C2×C8—202no—
334C2×C8—202no—
335C2×C8—202no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C22⋊C4—24no—
339C22⋊C4—24no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—42no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—22no—
350C42—22no—
351C2×C5—22no—
352C2×C5—22no—
353C2×C5—82no—
354C2×C5#145811yescharacteristic
355C2×C5#146011yescharacteristic
356C2×C5#146911yescharacteristic
357C2×C5#147311yescharacteristic
358C2×C5#186911yescharacteristic
359C2×C5#189911yescharacteristic
360C2×C5#190011yescharacteristic
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—22no—
366C23—24no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23—82no—
382C23C22×#16811yescharacteristic
383C23C2×#68111yescharacteristic
384C23D4×Dic512yes—
385C23#409412yes—
386C23C2×C22⋊Dic511yescharacteristic
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4C22⋊#16812yes—
419C2×C4C22⋊#16812yes—
420C2×C4#396512yes—
421C2×C4#408111yescharacteristic
422C2×C4#408311yescharacteristic
423C2×C4#408512yes—
424C8—204no—
425C8—402no—
426C5#908011yescharacteristic, complemented, Sylow
427C22—22no—
428C22—22no—
429C22—22no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—22no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—84no—
439C22—84no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22D4×#16812yes—
445C22#6530012yes—
446C22C2×C22⋊#16811yescharacteristic
447C22C2×#396511yescharacteristic
448C22#6550911yescharacteristic
449C22#6552311yescharacteristic
450C22#6552611yescharacteristic
451C22#6552911yescharacteristic
452C22#6552911yescharacteristic
453C4—21no—
454C4—22no—
455C4—21no—
456C4—22no—
457C4—22no—
458C4—22no—
459C4—42no—
460C4—41no—
461C4—81no—
462C4—81no—
463C2—22no—
464C2—22no—
465C2—82no—
466C2#1060236411yescharacteristic
467C2#1060239411yescharacteristic
468C2#1060240311yescharacteristic
469C2#1060247611yescharacteristic
470C2#1060249811yescharacteristic
471C2#1060251311yescharacteristic
472C2#1060253811yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic5
commutatorC23×C5C22×C8
FrattiniC23×C4C22×D5
FittingC2×C5×#388C2
socleC23×C5C2×C22⋊C4
prime radicalC25×C5C2×C4
cosocleC23×C4×C5C23
hypercenterC2×#388D5
hypocenterC5#9080
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#388

Complements

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

NH ≅ G/Ntype
2. C2×#65214C2semidirect
4. #10602324C2semidirect
5. #10598104C2semidirect
46. C2×C5×C22⋊C4C8semidirect
237. C2×C22⋊C4#168semidirect
426. C5#9080semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#3882
C2×#652142
C2×#647452
#106023242
#105981042
#90805