Finite Group Databasebeta

← #12046809

Subgroups — #12046809

Order 1280.

1170 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#382C211yesmaximal, characteristic
2C2×#65275C211yesmaximal, characteristic
3#10602361C212yesmaximal
4#10602357C211yesmaximal, characteristic
5#10602329C212yesmaximal
6#10600213C212yesmaximal
7#10600206C212yesmaximal
8#10600206C212yesmaximal
9#10600197C212yesmaximal
10C22×C5×C4⋊C4C2211yescharacteristic
11C22×C5×C4⋊C4C2211yescharacteristic
12C22×C42×C5C2211yescharacteristic
13C22×C4⋊Dic5C2211yescharacteristic
14C22×#683C2212yes—
15C22×#683C2212yes—
16C22×#683C2212yes—
17C22×C4×Dic5C2211yescharacteristic
18C5×#382—24no—
19C2×C5×#95C2212yes—
20C2×C5×#95C2212yes—
21C2×#3963C2212yes—
22C2×#3963C2212yes—
23C2×#3963C2212yes—
24C2×#3963C2212yes—
25C2×#3963C2211yescharacteristic
26C2×#3963C2212yes—
27C2×#3963C2212yes—
28C2×#3963C2211yescharacteristic
29C2×#3963C2212yes—
30C2×#3963C2212yes—
31C2×#3963C2212yes—
32#65275—24no—
33#15946—51nomaximal, Sylow
34C23×C4×C5C2312yes—
35C23×C4×C5C2311yescharacteristic
36C23×C4×C5C2311yescharacteristic
37C23×C4×C5C2312yes—
38C23×C4×C5C2311yescharacteristic
39C23×Dic5C2312yes—
40C23×Dic5C2312yes—
41C23×Dic5C2312yes—
42C23×Dic5C2312yes—
43C2×C5×C4⋊C4—22no—
44C2×C5×C4⋊C4—41no—
45C2×C5×C4⋊C4—41no—
46C2×C5×C4⋊C4—41no—
47C2×C5×C4⋊C4—41no—
48C2×C5×C4⋊C4D414yes—
49C2×C42×C5—22no—
50C2×C42×C5—24no—
51C2×C4⋊Dic5—22no—
52C2×C4⋊Dic5—41no—
53C2×C4⋊Dic5—41no—
54C2×#683—24no—
55C2×#683—24no—
56C2×#683—24no—
57C2×#683—42no—
58C2×#683—42no—
59C2×#683—42no—
60C2×#683—42no—
61C2×#683—42no—
62C2×#683—42no—
63C2×C4×Dic5—24no—
64C2×C4×Dic5—41no—
65C5×#95—24no—
66C5×#95—24no—
67#3963—22no—
68#3963—24no—
69#3963—24no—
70#3963—24no—
71#3963—24no—
72#3963—24no—
73#3963—24no—
74#3963—41no—
75#3963—42no—
76#3963—42no—
77#3963Q818yes—
78C2×#400—51no—
79C2×#382D511yescharacteristic
80#1444—52no—
81#1444—52no—
82#1442—52no—
83#1442—52no—
84#1436—52no—
85#1436—51no—
86C4⋊#95—52no—
87C24×C5C2411yescharacteristic
88C22×C4×C5—22no—
89C22×C4×C5—21no—
90C22×C4×C5—21no—
91C22×C4×C5—22no—
92C22×C4×C5—22no—
93C22×C4×C5—22no—
94C22×C4×C5—22no—
95C22×C4×C5—22no—
96C22×C4×C5—22no—
97C22×C4×C5—21no—
98C22×C4×C5—21no—
99C22×C4×C5—21no—
100C22×C4×C5—21no—
101C22×C4×C5—22no—
102C22×C4×C5—22no—
103C22×C4×C5—22no—
104C22×C4×C5—22no—
105C22×C4×C5—22no—
106C22×C4×C5—22no—
107C22×C4×C5—22no—
108C22×C4×C5—22no—
109C22×C4×C5—22no—
110C22×C4×C5—22no—
111C22×C4×C5—22no—
112C22×C4×C5C2×D412yes—
113C22×C4×C5C2×D412yes—
114C22×C4×C5C2×D412yes—
115C22×C4×C5C2×Q814yes—
116C22×C4×C5#4114yes—
117C22×Dic5—22no—
118C22×Dic5—22no—
119C22×Dic5—22no—
120C22×Dic5—22no—
121C22×Dic5—22no—
122C22×Dic5—22no—
123C22×Dic5—22no—
124C22×Dic5—22no—
125C22×Dic5—22no—
126C22×Dic5—22no—
127C22×Dic5—22no—
128C22×Dic5—22no—
129C22×Dic5—22no—
130C22×Dic5—22no—
131C22×Dic5—22no—
132C22×Dic5—22no—
133C22×Dic5—22no—
134C22×Dic5—22no—
135C22×Dic5—22no—
136C22×Dic5—22no—
137C22×Dic5—22no—
138C22×Dic5—22no—
139C22×Dic5—22no—
140C22×Dic5—22no—
141C22×Dic5C2×Q814yes—
142C22×Dic5C2×Q814yes—
143C22×Dic5#4114yes—
144C22×Dic5#4114yes—
145C5×C4⋊C4—44no—
146C5×C4⋊C4—44no—
147C42×C5—28no—
148C4⋊Dic5—44no—
149#683—48no—
150#683—48no—
151#683—48no—
152C4×Dic5—44no—
153C22×C4⋊C4—52no—
154C22×C4⋊C4—52no—
155C22×C4⋊C4—51no—
156C22×C4⋊C4—52no—
157C22×C4⋊C4C2×D511yescharacteristic
158C22×C4⋊C4C2×D511yescharacteristic
159C22×C42—51no—
160C22×C42C2×D511yescharacteristic
161#400—104no—
162#382—24no—
163C2×#95—52no—
164C2×#95—52no—
165C2×#95—52no—
166C2×#95—51no—
167C2×#95—52no—
168C2×#95—52no—
169C2×#95—52no—
170C2×#95—52no—
171C2×#95—51no—
172C2×#95—52no—
173C2×#95—52no—
174C2×#95C2×D512yes—
175C2×#95C2×D512yes—
176C23×C5C22×D411yescharacteristic
177C23×C5C22×Q812yes—
178C23×C5C2×#4112yes—
179C23×C5C2×#4112yes—
180C23×C5C2×#4112yes—
181C23×C5H(5,𝔽2)11yescharacteristic
182C23×C5H(5,𝔽2)11yescharacteristic
183C23×C5H(5,𝔽2)12yes—
184C23×C52−1+412yes—
185C2×C4×C5—22no—
186C2×C4×C5—22no—
187C2×C4×C5—24no—
188C2×C4×C5—24no—
189C2×C4×C5—24no—
190C2×C4×C5—22no—
191C2×C4×C5—22no—
192C2×C4×C5—24no—
193C2×C4×C5—24no—
194C2×C4×C5—24no—
195C2×C4×C5—24no—
196C2×C4×C5—42no—
197C2×C4×C5—42no—
198C2×C4×C5—42no—
199C2×C4×C5—42no—
200C2×C4×C5—42no—
201C2×C4×C5—42no—
202C2×C4×C5—41no—
203C2×C4×C5—41no—
204C2×C4×C5—41no—
205C2×C4×C5—41no—
206C2×C4×C5—42no—
207C2×C4×C5—42no—
208C2×C4×C5—42no—
209C2×C4×C5—42no—
210C2×C4×C5—42no—
211C2×C4×C5—42no—
212C2×C4×C5—42no—
213C2×C4×C5C4⋊Q814yes—
214C2×Dic5—24no—
215C2×Dic5—24no—
216C2×Dic5—24no—
217C2×Dic5—24no—
218C2×Dic5—24no—
219C2×Dic5—24no—
220C2×Dic5—24no—
221C2×Dic5—24no—
222C2×Dic5—24no—
223C2×Dic5—42no—
224C2×Dic5—42no—
225C2×Dic5—42no—
226C2×Dic5—42no—
227C2×Dic5—42no—
228C2×Dic5—42no—
229C2×Dic5—42no—
230C2×Dic5—42no—
231C2×Dic5—42no—
232C2×Dic5—42no—
233C2×Dic5—42no—
234C2×Dic5—42no—
235C2×Dic5—42no—
236C2×Dic5—42no—
237C2×Dic5—42no—
238C2×Dic5—42no—
239C2×Dic5—42no—
240C2×Dic5—42no—
241C2×Dic5#12518yes—
242C23×C4—52no—
243C23×C4—52no—
244C23×C4—52no—
245C23×C4—52no—
246C23×C4C22×D512yes—
247C23×C4C22×D512yes—
248C23×C4C22×D511yescharacteristic
249C23×C4C22×D511yescharacteristic
250C23×C4C22×D511yescharacteristic
251C2×C4⋊C4—22no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—104no—
257C2×C4⋊C4—104no—
258C2×C4⋊C4—104no—
259C2×C4⋊C4—102no—
260C2×C4⋊C4—202no—
261C2×C4⋊C4—202no—
262C2×C4⋊C4—202no—
263C2×C4⋊C4—202no—
264C2×C4⋊C4—202no—
265C2×C4⋊C4—202no—
266C2×C4⋊C4—201no—
267C2×C4⋊C4—201no—
268C2×C4⋊C4C22⋊D514yes—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—104no—
272C2×C42—201no—
273#95—24no—
274#95—24no—
275#95—58no—
276#95—104no—
277#95—104no—
278#95—104no—
279#95—104no—
280#95—102no—
281#95—104no—
282#95—104no—
283#95—201no—
284#95—202no—
285#95—202no—
286C22×C5#52511yescharacteristic
287C22×C5C2×#12512yes—
288C22×C5#52912yes—
289C22×C5C2×C4⋊Q811yescharacteristic
290C22×C5#53411yescharacteristic
291C22×C5#53812yes—
292C22×C5#54212yes—
293C22×C5#54212yes—
294C22×C5#54412yes—
295C22×C5#54712yes—
296C22×C5#55012yes—
297C22×C5#55112yes—
298C22×C5#55111yescharacteristic
299C22×C5#55212yes—
300C22×C5#55412yes—
301C22×C5#55412yes—
302C22×C5#55412yes—
303C22×C5#55511yescharacteristic
304C22×C5#55712yes—
305C22×C5H(3,𝔽4)11yescharacteristic
306C22×C5#56311yescharacteristic
307C4×C5—24no—
308C4×C5—42no—
309C4×C5—42no—
310C4×C5—44no—
311C4×C5—82no—
312Dic5—28no—
313Dic5—44no—
314Dic5—82no—
315Dic5—82no—
316C24C23×D511yescharacteristic
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—54no—
342C22×C4—54no—
343C22×C4—54no—
344C22×C4—54no—
345C22×C4—102no—
346C22×C4—102no—
347C22×C4—102no—
348C22×C4—102no—
349C22×C4—102no—
350C22×C4—102no—
351C22×C4—102no—
352C22×C4—102no—
353C22×C4—102no—
354C22×C4—102no—
355C22×C4—102no—
356C22×C4—102no—
357C22×C4—102no—
358C22×C4—102no—
359C22×C4—102no—
360C22×C4—102no—
361C22×C4—102no—
362C22×C4—102no—
363C22×C4—102no—
364C22×C4—102no—
365C22×C4—102no—
366C22×C4—102no—
367C22×C4—102no—
368C22×C4—102no—
369C22×C4#70914yes—
370C22×C4Q8×D514yes—
371C22×C4C2×C22⋊D512yes—
372C22×C4C2×C22⋊D512yes—
373C22×C4C2×C22⋊D512yes—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—208no—
377C4⋊C4—204no—
378C4⋊C4—208no—
379C4⋊C4—208no—
380C42—28no—
381C42—204no—
382C2×C5#247912yes—
383C2×C5#250512yes—
384C2×C5#250812yes—
385C2×C5#255212yes—
386C2×C5#258311yescharacteristic
387C2×C5#271711yescharacteristic
388C2×C5#273311yescharacteristic
389C2×C5#276312yes—
390C2×C5#277412yes—
391C23C2×#70912yes—
392C23#414411yescharacteristic
393C23#414411yescharacteristic
394C23C2×Q8×D512yes—
395C23D5×#4112yes—
396C23D5×#4112yes—
397C23#414912yes—
398C23#415012yes—
399C23C22×C22⋊D511yescharacteristic
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—58no—
429C2×C4—104no—
430C2×C4—104no—
431C2×C4—104no—
432C2×C4—104no—
433C2×C4—104no—
434C2×C4—104no—
435C2×C4—104no—
436C2×C4—104no—
437C2×C4—104no—
438C2×C4—202no—
439C2×C4—202no—
440C2×C4—202no—
441C2×C4—202no—
442C2×C4—202no—
443C2×C4—202no—
444C2×C4—202no—
445C2×C4—202no—
446C2×C4—202no—
447C2×C4—202no—
448C2×C4—202no—
449C2×C4—202no—
450C2×C4—202no—
451C2×C4—202no—
452C2×C4—202no—
453C2×C4—202no—
454C2×C4—202no—
455C2×C4—202no—
456C2×C4#409014yes—
457C5#1594611yescharacteristic, complemented, Sylow
458C22#6584512yes—
459C22#6584512yes—
460C22#6586112yes—
461C22#6586212yes—
462C22#6586512yes—
463C22#6586812yes—
464C22#6600312yes—
465C22#6600411yescharacteristic
466C22#6600611yescharacteristic
467C22#6601212yes—
468C22#6601311yescharacteristic
469C22#6601711yescharacteristic
470C22#6601912yes—
471C22D5×#12512yes—
472C22#6602312yes—
473C22#6602412yes—
474C22#6612511yescharacteristic
475C22#6613811yescharacteristic
476C22C2×#409011yescharacteristic
477C22#6615812yes—
478C22#6616812yes—
479C4—24no—
480C4—42no—
481C4—42no—
482C4—44no—
483C4—82no—
484C4—108no—
485C4—204no—
486C4—402no—
487C4—402no—
488C2#1060590112yes—
489C2#1060593512yes—
490C2#1060594312yes—
491C2#1060596712yes—
492C2#1061391211yescharacteristic
493C2#1061398412yes—
494C2#1061472011yescharacteristic
495C2#1061472211yescharacteristic
496C2#1061476812yes—
subgroupG/H
center, FrattiniC24C23×D5
commutator, socle, prime radical, cosocleC24×C5C24
FittingC2×C5×#382C2
hypercenterC2×#382D5
hypocenterC5#15946
LCSG ▷ C24×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
457. C5#15946semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#3822
C2×#652752
#106023612
#106023572
#106023292
#106002132
#106002062
#106001972
#159465