Finite Group Databasebeta

← #12014244

Subgroups — #12014244

Order 1280.

838 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#413C211yesmaximal, characteristic
2#10615817C211yesmaximal, characteristic
3C2×#65529C211yesmaximal, characteristic
4#10598189C214yesmaximal, complemented
5C22×C5×#41C2211yescharacteristic
6C22×C5×M16—21no—
7C2×#4072—21no—
8C2×#4072C411yescharacteristic, complemented
9C22×C4×Dic5C411yescharacteristic, complemented
10C5×#413—22no—
11C5×#413C414yescomplemented
12C2×C5×#101—21no—
13C2×C5×#100C2212yes—
14#65529—22no—
15#65529C414yescomplemented
16#65519—21no—
17#65510—21no—
18#65509C412yescomplemented
19#65509C2212yes—
20C2×#3966—21no—
21C2×#3966C2212yes—
22C4×C22⋊Dic5—21no—
23C4×C22⋊Dic5—21no—
24#12573—51nomaximal, Sylow
25C2×C5×#41—41no—
26C2×C5×#41—41no—
27C2×C5×#41—42no—
28C2×C5×#41C2×C412yes—
29C2×C5×#41C2×C412yes—
30C2×C5×#41D412yes—
31C2×C5×#41Q812yes—
32C22×C5×Q8D411yescharacteristic
33C22×C5×D4C2×C411yescharacteristic
34C22×C5×D4D411yescharacteristic
35C22×C5×D4C2311yescharacteristic
36C23×C4×C5C2×C411yescharacteristic
37C23×C4×C5D411yescharacteristic
38C23×C4×C5D411yescharacteristic
39C23×Dic5C2×C412yes—
40C2×C5×M16—24no—
41C2×C5×M16—22no—
42C2×C5×M16—41no—
43C2×C5×M16—42no—
44C22×C5×C8—41no—
45C2×C22⋊Dic5—22no—
46C2×C22⋊Dic5—22no—
47C2×C22⋊Dic5—22no—
48C2×C22⋊Dic5C2×C412yes—
49#4072—24no—
50#4072—42no—
51#4072—42no—
52C2×C4⋊Dic5—21no—
53C2×C4⋊Dic5—41no—
54C2×C4×Dic5—21no—
55C2×C4×Dic5—21no—
56C2×C4×Dic5—21no—
57C2×C4×Dic5—41no—
58C2×C4×Dic5—41no—
59C2×C4×Dic5—41no—
60C5×#101—24no—
61C5×#100—22no—
62C5×#100C2×C414yes—
63#3966—24no—
64#3966—22no—
65#3966C2×C414yes—
66#3963—22no—
67#3963—22no—
68C2×#413D511yescharacteristic
69C2×#410—51no—
70#2297—51no—
71#1390—54no—
72C24×C5C22⋊C411yescharacteristic
73C24×C5C22×C411yescharacteristic
74C24×C5C2×D411yescharacteristic
75C5×#41—42no—
76C5×#41—42no—
77C5×#41—42no—
78C5×#41—42no—
79C5×#41—42no—
80C5×#41—42no—
81C5×#41—42no—
82C5×#41—42no—
83C2×C5×Q8—41no—
84C2×C5×Q8—41no—
85C2×C5×Q8C22⋊C412yes—
86C2×C5×Q8C4⋊C412yes—
87C2×C5×D4—41no—
88C2×C5×D4—41no—
89C2×C5×D4—41no—
90C2×C5×D4—41no—
91C2×C5×D4—41no—
92C2×C5×D4—41no—
93C2×C5×D4C4212yes—
94C2×C5×D4C22⋊C412yes—
95C2×C5×D4C4⋊C412yes—
96C2×C5×D4C4⋊C412yes—
97C2×C5×D4C22×C411yescharacteristic
98C2×C5×D4C22×C411yescharacteristic
99C2×C5×D4C2×D411yescharacteristic
100C2×C5×D4C2×Q811yescharacteristic
101C22×C4×C5—21no—
102C22×C4×C5—21no—
103C22×C4×C5—22no—
104C22×C4×C5—21no—
105C22×C4×C5—21no—
106C22×C4×C5—22no—
107C22×C4×C5—21no—
108C22×C4×C5—21no—
109C22×C4×C5—22no—
110C22×C4×C5C4212yes—
111C22×C4×C5C22⋊C411yescharacteristic
112C22×C4×C5C22⋊C411yescharacteristic
113C22×C4×C5C22⋊C412yes—
114C22×C4×C5C22⋊C411yescharacteristic
115C22×C4×C5C22⋊C412yes—
116C22×C4×C5C4⋊C412yes—
117C22×C4×C5C4⋊C412yes—
118C22×C4×C5C4⋊C412yes—
119C22×C4×C5C2×D411yescharacteristic
120C22×Dic5—22no—
121C22×Dic5—22no—
122C22×Dic5—44no—
123C22×Dic5—42no—
124C22×Dic5—42no—
125C5×M16—24no—
126C5×M16—42no—
127C5×M16—44no—
128C2×C5×C8—42no—
129C2×C5×C8—44no—
130C22⋊Dic5—24no—
131C22⋊Dic5—44no—
132C22⋊Dic5—44no—
133C22⋊Dic5—42no—
134C22⋊Dic5—42no—
135C4⋊Dic5—42no—
136C4⋊Dic5—44no—
137C4×Dic5—42no—
138C4×Dic5—44no—
139C4×Dic5—41no—
140C4×Dic5—41no—
141C4×Dic5—42no—
142C22×#41C2×D511yescharacteristic
143C22×M16—21no—
144C2×#117—51no—
145C2×#117—101no—
146C22×C42—51no—
147#413—22no—
148#413Dic514yescomplemented
149C2×#101—21no—
150C2×#100C2×D512yes—
151#410—54no—
152#410—102no—
153C2×#99—52no—
154C2×#99—101no—
155#391—101no—
156#390—101no—
157#386—52no—
158#386—52no—
159C4×C22⋊C4—101no—
160C4×C22⋊C4—101no—
161C23×C5—21no—
162C23×C5—21no—
163C23×C5—21no—
164C23×C5—21no—
165C23×C5—21no—
166C23×C5—21no—
167C23×C5—42no—
168C23×C5—42no—
169C23×C5—42no—
170C23×C5#9512yes—
171C23×C5C2×C4211yescharacteristic
172C23×C5C2×C22⋊C411yescharacteristic
173C23×C5C2×C22⋊C411yescharacteristic
174C23×C5C2×C4⋊C411yescharacteristic
175C23×C5C2×C4⋊C411yescharacteristic
176C5×Q8—42no—
177C5×Q8—42no—
178C5×D4—42no—
179C5×D4—42no—
180C5×D4—42no—
181C5×D4—42no—
182C5×D4—41no—
183C5×D4—41no—
184C5×D4—41no—
185C5×D4—41no—
186C2×C4×C5—21no—
187C2×C4×C5—21no—
188C2×C4×C5—21no—
189C2×C4×C5—21no—
190C2×C4×C5—21no—
191C2×C4×C5—21no—
192C2×C4×C5—22no—
193C2×C4×C5—22no—
194C2×C4×C5—22no—
195C2×C4×C5—22no—
196C2×C4×C5—22no—
197C2×C4×C5—22no—
198C2×C4×C5—22no—
199C2×C4×C5—22no—
200C2×C4×C5—22no—
201C2×C4×C5—22no—
202C2×C4×C5—22no—
203C2×C4×C5—22no—
204C2×C4×C5#9512yes—
205C2×C4×C5#9512yes—
206C2×C4×C5#9512yes—
207C2×C4×C5#9912yes—
208C2×C4×C5#9911yescharacteristic
209C2×C4×C5#9911yescharacteristic
210C2×C4×C5C2×C22⋊C411yescharacteristic
211C2×C4×C5C2×C4⋊C411yescharacteristic
212C2×Dic5—24no—
213C2×Dic5—42no—
214C2×Dic5—44no—
215C2×Dic5—42no—
216C2×Dic5—44no—
217C2×Dic5—44no—
218C2×Dic5—42no—
219C2×Dic5—42no—
220C2×Dic5—42no—
221C2×Dic5—84no—
222C5×C8—44no—
223C2×#41—41no—
224C2×#41—42no—
225C2×#41—41no—
226C2×#41Dic1012yes—
227C2×#41C4×D512yes—
228C2×#41C2×Dic512yes—
229C2×#41C22⋊D512yes—
230C22×Q8C22⋊D511yescharacteristic
231C22×D4C4×D511yescharacteristic
232C22×D4D2011yescharacteristic
233C22×D4C22×D511yescharacteristic
234C23×C4—52no—
235C23×C4C4×D511yescharacteristic
236C23×C4D2011yescharacteristic
237C23×C4C22⋊D511yescharacteristic
238C2×M16—24no—
239C2×M16—22no—
240C2×M16—41no—
241C2×M16—42no—
242C22×C8—41no—
243#117—104no—
244#117—202no—
245#117—202no—
246C2×C4⋊C4—101no—
247C2×C4⋊C4—201no—
248C2×C22⋊C4—52no—
249C2×C22⋊C4—102no—
250C2×C22⋊C4—102no—
251C2×C22⋊C4—102no—
252C2×C42—101no—
253C2×C42—101no—
254C2×C42—101no—
255C2×C42—201no—
256C2×C42—201no—
257C2×C42—201no—
258#101—24no—
259#100—22no—
260#100C2×Dic514yes—
261#99—54no—
262#99—104no—
263#99—102no—
264#95—102no—
265#95—102no—
266C22×C5—22no—
267C22×C5—21no—
268C22×C5—22no—
269C22×C5—21no—
270C22×C5—21no—
271C22×C5—21no—
272C22×C5—42no—
273C22×C5—41no—
274C22×C5—41no—
275C22×C5—41no—
276C22×C5—41no—
277C22×C5—42no—
278C22×C5—42no—
279C22×C5—42no—
280C22×C5—42no—
281C22×C5—81no—
282C22×C5—81no—
283C22×C5C2×#9511yescharacteristic
284C22×C5C2×#9911yescharacteristic
285C22×C5C2×#9911yescharacteristic
286C4×C5—21no—
287C4×C5—22no—
288C4×C5—22no—
289C4×C5—21no—
290C4×C5#34212yes—
291C4×C5#34212yes—
292Dic5—44no—
293Dic5—82no—
294Dic5—84no—
295C24#68511yescharacteristic
296C24C2×C4×D511yescharacteristic
297C24C2×D2011yescharacteristic
298#41—42no—
299#41—42no—
300#41—42no—
301#41—42no—
302#41—42no—
303#41—42no—
304#41—42no—
305#41—42no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8#68312yes—
309C2×Q8C22⋊Dic512yes—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4C4×Dic512yes—
317C2×D4#68312yes—
318C2×D4C4⋊Dic512yes—
319C2×D4#68512yes—
320C2×D4C2×Dic1011yescharacteristic
321C2×D4C2×C4×D511yescharacteristic
322C2×D4C22×Dic511yescharacteristic
323C2×D4C2×C22⋊D511yescharacteristic
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—102no—
334C22×C4—102no—
335C22×C4—204no—
336C22×C4—202no—
337C22×C4—202no—
338C22×C4C4×Dic512yes—
339C22×C4#68312yes—
340C22×C4#68312yes—
341C22×C4C4⋊Dic512yes—
342C22×C4#68511yescharacteristic
343C22×C4#68511yescharacteristic
344C22×C4#68512yes—
345C22×C4#68511yescharacteristic
346C22×C4C22⋊Dic512yes—
347C22×C4C2×C22⋊D511yescharacteristic
348M16—24no—
349M16—42no—
350M16—44no—
351C2×C8—42no—
352C2×C8—44no—
353C4⋊C4—202no—
354C4⋊C4—204no—
355C22⋊C4—104no—
356C22⋊C4—204no—
357C22⋊C4—204no—
358C22⋊C4—202no—
359C22⋊C4—202no—
360C42—202no—
361C42—204no—
362C42—201no—
363C42—202no—
364C42—201no—
365C2×C5—21no—
366C2×C5—21no—
367C2×C5—41no—
368C2×C5—41no—
369C2×C5—41no—
370C2×C5—42no—
371C2×C5—41no—
372C2×C5C2×#34211yescharacteristic
373C2×C5#212012yes—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23#396312yes—
384C23C2×C4×Dic511yescharacteristic
385C23C2×#68311yescharacteristic
386C23C2×C4⋊Dic511yescharacteristic
387C23C2×#68511yescharacteristic
388C23C2×#68511yescharacteristic
389Q8—42no—
390Q8—42no—
391D4—42no—
392D4—42no—
393D4—42no—
394D4—42no—
395D4—41no—
396D4—41no—
397D4—41no—
398D4—41no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—104no—
418C2×C4—202no—
419C2×C4—204no—
420C2×C4—202no—
421C2×C4—204no—
422C2×C4—204no—
423C2×C4—202no—
424C2×C4—202no—
425C2×C4—202no—
426C2×C4—404no—
427C2×C4#393811yescharacteristic
428C2×C4#393812yes—
429C2×C4#393811yescharacteristic
430C2×C4#396312yes—
431C2×C4#396312yes—
432C2×C4#396312yes—
433C2×C4C2×#68311yescharacteristic
434C2×C4C2×C22⋊Dic511yescharacteristic
435C8—44no—
436C5#1257311yescharacteristic, complemented, Sylow
437C22—22no—
438C22—21no—
439C22—22no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—42no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—81no—
453C22—81no—
454C22C2×#393811yescharacteristic
455C22C2×#393811yescharacteristic
456C22C2×#396311yescharacteristic
457C4—21no—
458C4—22no—
459C4—22no—
460C4—21no—
461C4—204no—
462C4—402no—
463C4—404no—
464C4#6474712yes—
465C4#6474712yes—
466C2—21no—
467C2—21no—
468C2—41no—
469C2—41no—
470C2—41no—
471C2—42no—
472C2—41no—
473C2#1060093812yes—
474C2C2×#6474711yescharacteristic
subgroupG/H
centerC2×C4#3938
commutatorC23×C5C2×C42
FrattiniC22×D4C22×D5
FittingC2×C5×#413C2
socleC22×C5C2×#99
prime radicalC22×C5×#41C22
cosocleC22×C5×D4C23
hypercenterC2×#413D5
hypocenterC5#12573
LCSG ▷ C23×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ C2×#413

Complements

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

NH ≅ G/Ntype
4. #10598189C2semidirect
8. C2×#4072C4semidirect
9. C22×C4×Dic5C4semidirect
11. C5×#413C4semidirect
15. #65529C4semidirect
18. #65509C4semidirect
148. #413Dic5semidirect
436. C5#12573semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#4132
#106158172
C2×#655292
#105981892
#125735