Finite Group Databasebeta

← #12118536

Subgroups — #12118536

Order 1280.

990 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10614068C211yesmaximal, characteristic, complemented
2C2×C4×#686C211yesmaximal, characteristic
3#10606091C211yesmaximal, characteristic, complemented
4C5×#1802C211yesmaximal, characteristic, complemented
5#10601272C211yesmaximal, characteristic, complemented
6#10601258C211yesmaximal, characteristic, complemented
7#10601001C212yesmaximal
8#10600979C212yesmaximal, complemented
9#10600367C211yesmaximal, characteristic, complemented
10#10599580C212yesmaximal, complemented
11#10599573C212yesmaximal, complemented
12C2×C4×C5×D4C2211yescharacteristic
13C22×#686C2211yescharacteristic
14C2×#4072—21no—
15C2×#4039C2212yes—
16C2×D5×C4⋊C4—21no—
17C2×C4×D20C2211yescharacteristic
18C2×C5×C4⋊C8C2211yescharacteristic
19C2×C5×#102C2212yes—
20C5×#384C2211yescharacteristic
21#65510C2212yes—
22C2×#3964C2211yescharacteristic
23#65501—21no—
24C2×#3953C2212yes—
25C2×C8×Dic5C2212yes—
26C4×#686C418yescomplemented
27#65278—22no—
28#65276—21no—
29C4⋊C4×Dic5C2212yes—
30#65262C2212yes—
31C2×#3941C2211yescharacteristic
32C2×#3939C2211yescharacteristic
33#65240—22no—
34#65224C2211yescharacteristic
35C4×C4⋊Dic5C2211yescharacteristic
36C2×C4×#168C2211yescharacteristic
37C5×#340C2212yes—
38#64771C2212yes—
39#64770C2212yes—
40#64749C2212yes—
41#64749C2212yes—
42#22766—51nomaximal, Sylow
43C22×C5×D4C2311yescharacteristic
44C23×C4×C5—21no—
45C22×D20C2311yescharacteristic
46C22×C4×D5—21no—
47C22×C4×D5—22no—
48C22×C5×C8C2312yes—
49C4×C5×D4—22no—
50C4×C5×D4C2×C414yescomplemented
51C2×C5×C4⋊C4—21no—
52C2×C5×C4⋊C4—21no—
53C2×C5×C4⋊C4C2312yes—
54C2×C5×C4⋊C4C2311yescharacteristic
55C2×C5×C22⋊C4—21no—
56C2×C42×C5C2311yescharacteristic
57C2×C22⋊Dic5—22no—
58C2×C22⋊Dic5—22no—
59C2×#686—22no—
60C2×#686—22no—
61C2×#686C2×C414yes—
62C2×#685—21no—
63C2×#685—22no—
64#4072—44no—
65C2×C4⋊Dic5—21no—
66C2×C4⋊Dic5—21no—
67C2×C4⋊Dic5D411yescharacteristic
68C2×C4⋊Dic5D411yescharacteristic
69C2×C4⋊Dic5C2312yes—
70C2×C4⋊Dic5C2311yescharacteristic
71C2×#683—22no—
72C2×C4×Dic5—22no—
73C2×C4×Dic5C2312yes—
74C22×#168C2311yescharacteristic
75C22×#168C2311yescharacteristic
76#4039—24no—
77#4039—42no—
78D5×C4⋊C4—44no—
79C4×D20—22no—
80C4×D20C2×C414yes—
81C5×C4⋊C8—22no—
82C5×#102—42no—
83#3964C2×C414yescomplemented
84#3963—22no—
85#3963—22no—
86#3963D412yes—
87#3953—24no—
88C8×Dic5—42no—
89#3941C2×C414yes—
90#3939C2×C414yescomplemented
91C4×#168C2×C414yescomplemented
92C2×C4×D8—51no—
93#2354—51no—
94D4×C4⋊C4—51no—
95#1934—52no—
96#1918—52no—
97#1879—52no—
98#1834—51no—
99#1802—51no—
100#1802D511yescharacteristic, complemented
101C8×C4⋊C4—51no—
102C4×#102—52no—
103C24×C5—21no—
104C23×D5—21no—
105C2×C5×D4—21no—
106C2×C5×D4—21no—
107C2×C5×D4—21no—
108C2×C5×D4—21no—
109C2×C5×D4—21no—
110C2×C5×D4C22×C412yes—
111C22×C4×C5—21no—
112C22×C4×C5—24no—
113C22×C4×C5—22no—
114C22×C4×C5—42no—
115C22×C4×C5—42no—
116C22×C4×C5C2×D411yescharacteristic
117C22×C4×C5C2×D411yescharacteristic
118C22×C4×C5C2411yescharacteristic
119C22×Dic5—21no—
120C22×Dic5—22no—
121C22×Dic5C2×D412yes—
122C22×Dic5C2×D412yes—
123C2×D20—21no—
124C2×D20—21no—
125C2×D20—21no—
126C2×D20C22×C412yes—
127C2×D20#4112yes—
128C2×D20#4112yes—
129C2×C4×D5—24no—
130C2×C4×D5—42no—
131C2×C4×D5—42no—
132C2×C4×D5—44no—
133C2×C4×D5—44no—
134C2×C4×D5—82no—
135C2×C5×C8—22no—
136C2×C5×C8—22no—
137C2×C5×C8—22no—
138C5×C4⋊C4—21no—
139C5×C4⋊C4—22no—
140C5×C4⋊C4—22no—
141C5×C4⋊C4—42no—
142C5×C4⋊C4—81no—
143C5×C4⋊C4C22×C412yes—
144C5×C22⋊C4—24no—
145C42×C5C22×C412yes—
146C42×C5#4112yes—
147C22⋊Dic5—44no—
148C22⋊Dic5—82no—
149#686—28no—
150#685—24no—
151#685—44no—
152C4⋊Dic5—22no—
153C4⋊Dic5—21no—
154C4⋊Dic5—22no—
155C4⋊Dic5—22no—
156C4⋊Dic5—42no—
157C4⋊Dic5—81no—
158C4⋊Dic5D814yescomplemented
159C4⋊Dic5C22×C412yes—
160#683—44no—
161C4×Dic5—22no—
162C4×Dic5—22no—
163C4×Dic5—44no—
164C2×#168—21no—
165C2×#168—21no—
166C2×#168—21no—
167C2×#168—21no—
168C2×#168C22×C412yes—
169C2×#168C22×C412yes—
170C22×D8—51no—
171C2×C4×D4—52no—
172C2×C4×D4—51no—
173C2×C4×D4C2×D511yescharacteristic
174C2×#117—101no—
175C22×C4⋊C4—101no—
176C4×D8—58no—
177C2×#107—51no—
178C2×C4⋊C8C2×D511yescharacteristic
179C2×#102—51no—
180C2×#102—51no—
181C2×#102—52no—
182C2×#102C2×D512yes—
183C2×C4×C8—51no—
184C2×C4×C8—52no—
185#390—52no—
186#389—52no—
187#388—102no—
188#385—102no—
189#384—51no—
190#384C2×D511yescharacteristic
191#380—101no—
192#380—101no—
193C4×C4⋊C4—51no—
194C4×C4⋊C4—52no—
195#340—52no—
196#340—52no—
197#340—52no—
198#340C2×D512yes—
199#336—52no—
200C23×C5—22no—
201C23×C5—41no—
202C23×C5—41no—
203C23×C5—41no—
204C23×C5—41no—
205C23×C5—41no—
206C23×C5—41no—
207C23×C5C22×D411yescharacteristic
208C22×D5—22no—
209C22×D5—22no—
210C22×D5—22no—
211C22×D5—41no—
212C22×D5—41no—
213C22×D5—41no—
214C22×D5—41no—
215C5×D4—22no—
216C5×D4—22no—
217C5×D4—41no—
218C5×D4—41no—
219C2×C4×C5—21no—
220C2×C4×C5—21no—
221C2×C4×C5—22no—
222C2×C4×C5—41no—
223C2×C4×C5—41no—
224C2×C4×C5—48no—
225C2×C4×C5—42no—
226C2×C4×C5—42no—
227C2×C4×C5—42no—
228C2×C4×C5C4×D412yes—
229C2×C4×C5C4×D412yes—
230C2×C4×C5C2×D812yes—
231C2×C4×C5#13512yes—
232C2×C4×C5C23×C411yescharacteristic
233C2×C4×C5C2×#4111yescharacteristic
234C2×C4×C5C2×#4111yescharacteristic
235C2×C4×C5C2×#4111yescharacteristic
236C2×C4×C5H(5,𝔽2)11yescharacteristic
237C2×C4×C52−1+411yescharacteristic
238C2×Dic5—22no—
239C2×Dic5—22no—
240C2×Dic5—22no—
241C2×Dic5—22no—
242C2×Dic5—22no—
243C2×Dic5—22no—
244C2×Dic5—41no—
245C2×Dic5—41no—
246C2×Dic5—42no—
247C2×Dic5—42no—
248C2×Dic5—42no—
249C2×Dic5C2×D814yes—
250D20—22no—
251D20—22no—
252D20—22no—
253D20#11714yescomplemented
254C4×D5—48no—
255C4×D5—88no—
256C5×C8—42no—
257#168—22no—
258#168—22no—
259C22×D4—51no—
260C22×D4C22×D511yescharacteristic
261C23×C4—21no—
262C23×C4—102no—
263C23×C4—101no—
264C2×D8—54no—
265C2×D8—102no—
266C2×D8—102no—
267C22×C8—51no—
268C22×C8—51no—
269C22×C8C22×D512yes—
270C4×D4—22no—
271C4×D4—54no—
272C4×D4—104no—
273C4×D4—102no—
274C4×D4—202no—
275C4×D4C4×D514yescomplemented
276#117—204no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—51no—
280C2×C4⋊C4—51no—
281C2×C4⋊C4—51no—
282C2×C4⋊C4—52no—
283C2×C4⋊C4—102no—
284C2×C4⋊C4—101no—
285C2×C4⋊C4—101no—
286C2×C4⋊C4—204no—
287C2×C4⋊C4C22×D512yes—
288C2×C4⋊C4C22×D511yescharacteristic
289C2×C22⋊C4—21no—
290C2×C22⋊C4—101no—
291C2×C22⋊C4—102no—
292C2×C22⋊C4—102no—
293C2×C22⋊C4—102no—
294C2×C42—52no—
295C2×C42—102no—
296C2×C42C22×D511yescharacteristic
297#107—54no—
298C4⋊C8—22no—
299#102—42no—
300#102—54no—
301#102—54no—
302#102—104no—
303C4×C8—54no—
304C4×C8—202no—
305#95—52no—
306#95—102no—
307#95—102no—
308C22×C5—42no—
309C22×C5—42no—
310C22×C5—42no—
311C22×C5—81no—
312C22×C5—81no—
313C22×C5—81no—
314C22×C5—81no—
315C22×C5C2×C4×D411yescharacteristic
316C22×C5#54611yescharacteristic
317C22×C5#54811yescharacteristic
318C22×C5C22×D811yescharacteristic
319C22×C5C2×#13511yescharacteristic
320C22×C5#57711yescharacteristic
321C22×C5#57711yescharacteristic
322C2×D5—24no—
323C2×D5—42no—
324C2×D5—42no—
325C2×D5—42no—
326C2×D5—42no—
327C2×D5—42no—
328C2×D5—42no—
329C4×C5—22no—
330C4×C5—42no—
331C4×C5—82no—
332C4×C5C4×D814yes—
333C4×C5C2×#11711yescharacteristic
334C4×C5#52011yescharacteristic
335C4×C5#55211yescharacteristic
336C4×C5#55311yescharacteristic
337Dic5—24no—
338Dic5—42no—
339Dic5—42no—
340Dic5—82no—
341C24—21no—
342C24—101no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—52no—
349C2×D4—52no—
350C2×D4—52no—
351C2×D4—101no—
352C2×D4—101no—
353C2×D4—101no—
354C2×D4C2×C4×D512yes—
355C22×C4—21no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—52no—
361C22×C4—52no—
362C22×C4—101no—
363C22×C4—104no—
364C22×C4—102no—
365C22×C4—204no—
366C22×C4—204no—
367C22×C4—202no—
368C22×C4—202no—
369C22×C4—402no—
370C22×C4D4×D511yescharacteristic
371C22×C4D4×D511yescharacteristic
372C22×C4C23×D511yescharacteristic
373D8—108no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—52no—
378C2×C8—52no—
379C2×C8—101no—
380C2×C8—101no—
381C2×C8—101no—
382C2×C8—101no—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—42no—
387C4⋊C4—54no—
388C4⋊C4—52no—
389C4⋊C4—81no—
390C4⋊C4—101no—
391C4⋊C4—102no—
392C4⋊C4—102no—
393C4⋊C4—102no—
394C4⋊C4—204no—
395C4⋊C4—202no—
396C4⋊C4—401no—
397C4⋊C4C2×C4×D512yes—
398C22⋊C4—24no—
399C22⋊C4—104no—
400C22⋊C4—204no—
401C22⋊C4—204no—
402C22⋊C4—402no—
403C42—102no—
404C42—102no—
405C42—204no—
406C42C2×C4×D512yes—
407C42#71112yes—
408C2×C5—82no—
409C2×C5#235211yescharacteristic
410C2×C5C2×C4×D811yescharacteristic
411C2×C5#295211yescharacteristic
412C2×C5#329411yescharacteristic
413C2×C5#330311yescharacteristic
414C2×C5#335811yescharacteristic
415C2×C5#336011yescharacteristic
416D5—44no—
417C23—22no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—102no—
425C23—102no—
426C23—102no—
427C23—201no—
428C23—201no—
429C23—201no—
430C23—201no—
431C23C2×D4×D511yescharacteristic
432D4—22no—
433D4—22no—
434D4—41no—
435D4—41no—
436D4—54no—
437D4—102no—
438D4—102no—
439D4—102no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—48no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—54no—
450C2×C4—102no—
451C2×C4—102no—
452C2×C4—102no—
453C2×C4—102no—
454C2×C4—102no—
455C2×C4—102no—
456C2×C4—201no—
457C2×C4—201no—
458C2×C4—208no—
459C2×C4—202no—
460C2×C4—202no—
461C2×C4—202no—
462C2×C4—408no—
463C2×C4#402712yes—
464C2×C4#402712yes—
465C2×C4D5×D812yes—
466C2×C4#406212yes—
467C2×C4C22×C4×D511yescharacteristic
468C2×C4C2×#71111yescharacteristic
469C2×C4#414411yescharacteristic
470C2×C4#414711yescharacteristic
471C2×C4D5×#4111yescharacteristic
472C2×C4D5×#4111yescharacteristic
473C8—42no—
474C8—102no—
475C8—102no—
476C5#2276611yescharacteristic, complemented, Sylow
477C22—42no—
478C22—42no—
479C22—42no—
480C22—81no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22—104no—
485C22—202no—
486C22—202no—
487C22—202no—
488C22—202no—
489C22—202no—
490C22—202no—
491C22C2×#402711yescharacteristic
492C22#6594611yescharacteristic
493C22#6596711yescharacteristic
494C22C2×D5×D811yescharacteristic
495C22C2×#406211yescharacteristic
496C22#6610911yescharacteristic
497C22#6611211yescharacteristic
498C4—22no—
499C4—42no—
500C4—82no—
501C4—104no—
502C4—202no—
503C4—202no—
504C4—402no—
505C4#6504514yes—
506C4#6583811yescharacteristic
507C4D5×#11711yescharacteristic
508C4#6594811yescharacteristic
509C4#6597311yescharacteristic
510C2—82no—
511C2—204no—
512C2#1060574311yescharacteristic
513C2C2×#6504511yescharacteristic
514C2#1061060011yescharacteristic
515C2#1061208311yescharacteristic
516C2#1061218211yescharacteristic
517C2#1061272311yescharacteristic
518C2#1061281811yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1802C2
socleC23×C5C22×D4
prime radicalC22×#686C22
cosocleC22×C4×C5C24
hypercenter#1802D5
hypocenterC5#22766
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
1. #10614068C2semidirect
3. #10606091C2semidirect
4. C5×#1802C2semidirect
5. #10601272C2semidirect
6. #10601258C2semidirect
8. #10600979C2semidirect
9. #10600367C2semidirect
10. #10599580C2semidirect
11. #10599573C2semidirect
26. C4×#686C4semidirect
50. C4×C5×D4C2×C4semidirect
83. #3964C2×C4semidirect
90. #3939C2×C4semidirect
91. C4×#168C2×C4semidirect
100. #1802D5semidirect
158. C4⋊Dic5D8semidirect
253. D20#117semidirect
275. C4×D4C4×D5semidirect
476. C5#22766semidirect

Maximal subgroups

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

subgroupindex
#106140682
C2×C4×#6862
#106060912
C5×#18022
#106012722
#106012582
#106010012
#106009792
#106003672
#105995802
#105995732
#227665