Finite Group Databasebeta

← #12106402

Subgroups — #12106402

Order 1280.

1060 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×Dic5C211yesmaximal, characteristic, complemented
2C2×#65256C211yesmaximal, characteristic
3#10605596C211yesmaximal, characteristic
4C5×#1823C211yesmaximal, characteristic, complemented
5#10600725C211yesmaximal, characteristic
6Dic5×C22⋊C8C211yesmaximal, characteristic, complemented
7#10600297C211yesmaximal, characteristic, complemented
8#10599584C212yesmaximal, complemented
9#10599580C212yesmaximal, complemented
10#10599578C212yesmaximal, complemented
11#10599573C212yesmaximal, complemented
12C22×C5×C4⋊C4C2211yescharacteristic
13C2×#4076C2211yescharacteristic
14C22×C4⋊Dic5C2211yescharacteristic
15C22×C4×Dic5—21no—
16C22×C4×Dic5C2211yescharacteristic
17C2×#4039C2211yescharacteristic
18C2×#4027—21no—
19C2×C5×C22⋊C8C2211yescharacteristic
20C5×#380C2211yescharacteristic
21C2×#3963—21no—
22C2×C22⋊#168C2211yescharacteristic
23C2×#3953C2212yes—
24C2×C8×Dic5C2212yes—
25#65277C2211yescharacteristic
26C4⋊C4×Dic5—21no—
27C4⋊C4×Dic5—21no—
28C4⋊C4×Dic5C2211yescharacteristic
29C4⋊C4×Dic5C2212yes—
30C4⋊C4×Dic5C2212yes—
31#65262C2211yescharacteristic
32#65256C418yescomplemented
33C2×#3941C2212yes—
34C2×#3939C2212yes—
35C2×#3939C2212yes—
36#65247—21no—
37#64958—21no—
38#64955—21no—
39C5×#340C2212yes—
40C5×#340C2212yes—
41#64771C2212yes—
42#64770C2212yes—
43#64749C2212yes—
44C4×#480—51nomaximal, Sylow
45C23×C4×C5—21no—
46C23×C4×C5C2311yescharacteristic
47C22×C22⋊D5—21no—
48C23×Dic5—21no—
49C23×Dic5D411yescharacteristic
50C23×Dic5D411yescharacteristic
51C22×D20C2311yescharacteristic
52C22×C4×D5—21no—
53C22×C5×C8C2312yes—
54C2×C5×C4⋊C4—21no—
55C2×C5×C4⋊C4—22no—
56C2×C5×C4⋊C4C2×C414yescomplemented
57C2×C5×C4⋊C4C2311yescharacteristic
58C2×C5×C4⋊C4C2311yescharacteristic
59C2×C5×C4⋊C4C2312yes—
60C2×C5×C22⋊C4—21no—
61#4076—22no—
62#4076C2×C414yes—
63C2×#685—21no—
64C2×#685—21no—
65C2×#685—21no—
66C2×#685—21no—
67C2×C4⋊Dic5—21no—
68C2×C4⋊Dic5—24no—
69C2×C4⋊Dic5C2311yescharacteristic
70C2×C4⋊Dic5C2312yes—
71C2×#683—21no—
72C2×C4×Dic5—24no—
73C2×C4×Dic5—21no—
74C2×C4×Dic5—22no—
75C2×C4×Dic5—22no—
76C2×C4×Dic5—44no—
77C2×C4×Dic5D411yescharacteristic
78C2×C4×Dic5D411yescharacteristic
79C2×C4×Dic5C2311yescharacteristic
80C2×C4×Dic5C2311yescharacteristic
81C22×#168C2312yes—
82#4039—24no—
83#4027—44no—
84C5×C22⋊C8—22no—
85C5×#95—21no—
86#3963—21no—
87#3963—21no—
88#3963—22no—
89#3963—22no—
90C22⋊#168C2×C414yescomplemented
91#3953—24no—
92C8×Dic5—24no—
93#3941C2×C418yescomplemented
94#3939—24no—
95#3939C2×C418yescomplemented
96C2×#480—51no—
97C4×#121—51no—
98C2×C4×C4⋊C4—51no—
99#1925—52no—
100#1918—52no—
101#1874—51no—
102#1852—51no—
103#1823D511yescharacteristic, complemented
104C4×#107—52no—
105C4×#102—52no—
106C4×C22⋊C8—51no—
107C24×C5C2×D411yescharacteristic
108C23×D5—21no—
109C22×C4×C5—22no—
110C22×C4×C5—22no—
111C22×C4×C5—24no—
112C22×C4×C5—22no—
113C22×C4×C5—21no—
114C22×C4×C5—21no—
115C22×C4×C5—42no—
116C22×C4×C5—42no—
117C22×C4×C5C22×C412yes—
118C22×C4×C5C2×D411yescharacteristic
119C22×C4×C5#4112yes—
120C22×C4×C5C2411yescharacteristic
121C2×C22⋊D5—24no—
122C2×C22⋊D5—42no—
123C2×C22⋊D5—42no—
124C22×Dic5—22no—
125C22×Dic5—22no—
126C22×Dic5—24no—
127C22×Dic5—22no—
128C22×Dic5—22no—
129C22×Dic5—21no—
130C22×Dic5—22no—
131C22×Dic5—41no—
132C22×Dic5—42no—
133C22×Dic5—42no—
134C22×Dic5D814yes—
135C22×Dic5C2×D411yescharacteristic
136C22×Dic5C2×D411yescharacteristic
137C22×Dic5C2×D411yescharacteristic
138C22×Dic5C2×D411yescharacteristic
139C2×D20—22no—
140C2×D20—22no—
141C2×D20—21no—
142C2×D20C22×C412yes—
143C2×C4×D5—42no—
144C2×C4×D5—42no—
145C2×C4×D5—42no—
146C2×C5×C8—22no—
147C2×C5×C8—22no—
148C2×C5×C8—22no—
149C5×C4⋊C4—24no—
150C5×C4⋊C4—22no—
151C5×C4⋊C4—22no—
152C5×C4⋊C4—22no—
153C5×C4⋊C4C22×C414yes—
154C5×C22⋊C4—42no—
155#685—24no—
156#685—42no—
157#685—42no—
158#685—42no—
159C4⋊Dic5—24no—
160C4⋊Dic5—22no—
161C4⋊Dic5—21no—
162C4⋊Dic5C22×C412yes—
163C4⋊Dic5#4114yes—
164#683—42no—
165C4×Dic5—22no—
166C4×Dic5—22no—
167C4×Dic5—22no—
168C4×Dic5—42no—
169C4×Dic5—44no—
170C4×Dic5—44no—
171C4×Dic5#4114yes—
172C2×#168—22no—
173C2×#168—22no—
174C2×#168C22×C414yes—
175C2×#121—51no—
176C2×C4×D4—51no—
177C2×C4×D4—101no—
178C22×C4⋊C4—51no—
179C22×C4⋊C4C2×D511yescharacteristic
180C22×C42—51no—
181C22×C42—101no—
182#480—58no—
183C2×#107—52no—
184C2×#107—52no—
185C2×#102—52no—
186C2×#102—52no—
187C2×C22⋊C8—51no—
188C2×C22⋊C8C2×D511yescharacteristic
189C2×C4×C8—52no—
190#390—51no—
191#389—51no—
192#388—101no—
193#386—101no—
194#380C2×D511yescharacteristic
195C4×C4⋊C4—52no—
196C4×C4⋊C4—52no—
197C4×C4⋊C4—51no—
198C4×C4⋊C4—101no—
199C4×C4⋊C4—101no—
200C4×C22⋊C4—101no—
201C2×#95—101no—
202#340—52no—
203#340—52no—
204#340C2×D512yes—
205#340C2×D512yes—
206#336—52no—
207C23×C5—21no—
208C23×C5—21no—
209C23×C5—21no—
210C23×C5—21no—
211C23×C5C4×D412yes—
212C23×C5C2×D812yes—
213C23×C5#13512yes—
214C23×C5C22×D411yescharacteristic
215C22×D5—22no—
216C22×D5—41no—
217C22×D5—41no—
218C22×D5—41no—
219C22×D5—41no—
220C22×D5—41no—
221C22×D5—41no—
222C2×C4×C5—21no—
223C2×C4×C5—21no—
224C2×C4×C5—24no—
225C2×C4×C5—24no—
226C2×C4×C5—24no—
227C2×C4×C5—42no—
228C2×C4×C5—42no—
229C2×C4×C5—48no—
230C2×C4×C5—41no—
231C2×C4×C5—41no—
232C2×C4×C5—41no—
233C2×C4×C5—41no—
234C2×C4×C5—41no—
235C2×C4×C5—41no—
236C2×C4×C5C4×D412yes—
237C2×C4×C5C23×C411yescharacteristic
238C2×C4×C5C2×#4111yescharacteristic
239C2×C4×C5C2×#4112yes—
240C2×C4×C5C2×#4112yes—
241C22⋊D5—48no—
242C2×Dic5—21no—
243C2×Dic5—21no—
244C2×Dic5—21no—
245C2×Dic5—21no—
246C2×Dic5—28no—
247C2×Dic5—22no—
248C2×Dic5—21no—
249C2×Dic5—24no—
250C2×Dic5—21no—
251C2×Dic5—21no—
252C2×Dic5—44no—
253C2×Dic5—48no—
254C2×Dic5—42no—
255C2×Dic5—41no—
256C2×Dic5—41no—
257C2×Dic5—42no—
258C2×Dic5#12312yes—
259C2×Dic5#12312yes—
260C2×Dic5C2×D812yes—
261C2×Dic5C2×D812yes—
262C2×Dic5#13712yes—
263D20—24no—
264D20—42no—
265C4×D5—84no—
266C5×C8—42no—
267#168—24no—
268C22×D4—51no—
269C22×D4—101no—
270C23×C4—21no—
271C23×C4—51no—
272C23×C4—51no—
273C23×C4—101no—
274C23×C4—101no—
275C23×C4C22×D511yescharacteristic
276C22×C8—52no—
277C22×C8C22×D512yes—
278#121—54no—
279#121—102no—
280C4×D4—104no—
281C4×D4—204no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—52no—
285C2×C4⋊C4—51no—
286C2×C4⋊C4—104no—
287C2×C4⋊C4—101no—
288C2×C4⋊C4—101no—
289C2×C4⋊C4C4×D514yescomplemented
290C2×C4⋊C4C22×D511yescharacteristic
291C2×C4⋊C4C22×D511yescharacteristic
292C2×C4⋊C4C22×D512yes—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—101no—
295C2×C22⋊C4—101no—
296C2×C22⋊C4—101no—
297C2×C22⋊C4—101no—
298C2×C42—51no—
299C2×C42—51no—
300C2×C42—51no—
301C2×C42—51no—
302C2×C42—104no—
303C2×C42—101no—
304C2×C42—102no—
305C2×C42—102no—
306C2×C42—204no—
307#107—58no—
308#107—104no—
309#102—58no—
310#102—104no—
311C22⋊C8—22no—
312C22⋊C8—54no—
313C4×C8—104no—
314#95—21no—
315#95—102no—
316#95—102no—
317#95—101no—
318#95—101no—
319C22×C5—22no—
320C22×C5—22no—
321C22×C5—22no—
322C22×C5—22no—
323C22×C5—22no—
324C22×C5—22no—
325C22×C5C4×D814yes—
326C22×C5C2×C4×D411yescharacteristic
327C22×C5C2×#12311yescharacteristic
328C22×C5#52511yescharacteristic
329C22×C5C22×D811yescharacteristic
330C22×C5C2×#13511yescharacteristic
331C22×C5C2×#13711yescharacteristic
332C22×C5#57511yescharacteristic
333C2×D5—42no—
334C2×D5—42no—
335C2×D5—42no—
336C2×D5—81no—
337C2×D5—81no—
338C2×D5—81no—
339C2×D5—81no—
340C4×C5—22no—
341C4×C5—44no—
342C4×C5—81no—
343C4×C5—81no—
344C4×C5C4×#4112yes—
345C4×C5#52912yes—
346Dic5—22no—
347Dic5—22no—
348Dic5—41no—
349Dic5—44no—
350Dic5—42no—
351Dic5#48014yescomplemented
352C24—101no—
353C24D4×D511yescharacteristic
354C2×D4—52no—
355C2×D4—104no—
356C2×D4—101no—
357C2×D4—102no—
358C2×D4—102no—
359C2×D4—202no—
360C2×D4—202no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—51no—
370C22×C4—51no—
371C22×C4—51no—
372C22×C4—54no—
373C22×C4—51no—
374C22×C4—102no—
375C22×C4—102no—
376C22×C4—102no—
377C22×C4—104no—
378C22×C4—102no—
379C22×C4—101no—
380C22×C4—102no—
381C22×C4—202no—
382C22×C4—202no—
383C22×C4—202no—
384C22×C4—201no—
385C22×C4—202no—
386C22×C4—202no—
387C22×C4C2×C4×D512yes—
388C22×C4D4×D511yescharacteristic
389C22×C4#71112yes—
390C22×C4C23×D511yescharacteristic
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—54no—
395C2×C8—102no—
396C2×C8—102no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—52no—
402C4⋊C4—54no—
403C4⋊C4—104no—
404C4⋊C4—101no—
405C4⋊C4—102no—
406C4⋊C4—202no—
407C4⋊C4C2×C4×D514yes—
408C22⋊C4—42no—
409C22⋊C4—104no—
410C22⋊C4—202no—
411C22⋊C4—202no—
412C22⋊C4—202no—
413C42—54no—
414C42—102no—
415C42—102no—
416C42—102no—
417C42—204no—
418C42—204no—
419C42—202no—
420C2×C5—24no—
421C2×C5C4×#12311yescharacteristic
422C2×C5C2×C4×D811yescharacteristic
423C2×C5C4×#13711yescharacteristic
424C2×C5C2×#48011yescharacteristic
425C2×C5#309111yescharacteristic
426C2×C5#310011yescharacteristic
427C2×C5#310811yescharacteristic
428D5—82no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—102no—
434C23—201no—
435C23—201no—
436C23—201no—
437C23—201no—
438C23—201no—
439C23—201no—
440C23#402712yes—
441C23D5×D812yes—
442C23#406212yes—
443C23C2×D4×D511yescharacteristic
444D4—104no—
445D4—208no—
446D4—202no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—48no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—52no—
462C2×C4—52no—
463C2×C4—52no—
464C2×C4—52no—
465C2×C4—52no—
466C2×C4—108no—
467C2×C4—101no—
468C2×C4—101no—
469C2×C4—101no—
470C2×C4—101no—
471C2×C4—102no—
472C2×C4—101no—
473C2×C4—101no—
474C2×C4—101no—
475C2×C4—104no—
476C2×C4—204no—
477C2×C4—208no—
478C2×C4—202no—
479C2×C4—201no—
480C2×C4—201no—
481C2×C4—202no—
482C2×C4—404no—
483C2×C4#402712yes—
484C2×C4C22×C4×D511yescharacteristic
485C2×C4C2×#71111yescharacteristic
486C2×C4D5×#4112yes—
487C2×C4D5×#4112yes—
488C8—42no—
489C8—104no—
490C5C4×#48011yescharacteristic, complemented, Sylow
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—202no—
498C22—202no—
499C22—202no—
500C22—401no—
501C22—401no—
502C22—401no—
503C22—401no—
504C22#6504514yes—
505C22C2×#402711yescharacteristic
506C22D5×#12311yescharacteristic
507C22#6599011yescharacteristic
508C22C2×D5×D811yescharacteristic
509C22C2×#406211yescharacteristic
510C22D5×#13711yescharacteristic
511C22#6611111yescharacteristic
512C4—22no—
513C4—44no—
514C4—54no—
515C4—81no—
516C4—81no—
517C4—102no—
518C4—102no—
519C4—204no—
520C4—202no—
521C4—201no—
522C4#6585612yes—
523C4#6598312yes—
524C2—24no—
525C2—402no—
526C2#1060578311yescharacteristic
527C2C2×#6504511yescharacteristic
528C2#1061067711yescharacteristic
529C2#1061299911yescharacteristic
530C2D5×#48011yescharacteristic
531C2#1061314411yescharacteristic
532C2#1061315711yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1823C2
socleC23×C5C22×D4
prime radicalC2×#4076C22
cosocleC22×C4×C5C24
hypercenter#1823D5
hypocenterC5C4×#480
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×Dic5C2semidirect
4. C5×#1823C2semidirect
6. Dic5×C22⋊C8C2semidirect
7. #10600297C2semidirect
8. #10599584C2semidirect
9. #10599580C2semidirect
10. #10599578C2semidirect
11. #10599573C2semidirect
32. #65256C4semidirect
56. C2×C5×C4⋊C4C2×C4semidirect
90. C22⋊#168C2×C4semidirect
93. #3941C2×C4semidirect
95. #3939C2×C4semidirect
103. #1823D5semidirect
289. C2×C4⋊C4C4×D5semidirect
351. Dic5#480semidirect
490. C5C4×#480semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×Dic52
C2×#652562
#106055962
C5×#18232
#106007252
Dic5×C22⋊C82
#106002972
#105995842
#105995802
#105995782
#105995732
C4×#4805