Finite Group Databasebeta

← #27278625

Subgroups — #27278625

Order 13568.

712 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810675C211yesmaximal, characteristic, complemented
2C4×#13133188C211yesmaximal, characteristic
3C4×#13133129C211yesmaximal, characteristic, complemented
4C2×D4⋊#88322C211yesmaximal, characteristic, complemented
5#20272555C211yesmaximal, characteristic, complemented
6#20272554C211yesmaximal, characteristic, complemented
7C2×C53×D4⋊C8C211yesmaximal, characteristic, complemented
8C2×#13133188C411yescharacteristic
9C2×C4×C22⋊D53—21no—
10C2×C42×D53C2211yescharacteristic
11#16810675—22no—
12#16810675—22no—
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133129—21no—
15C2×#13133129—21no—
16C2×C4×#88322C2211yescharacteristic
17C2×C8×Dic53—21no—
18#16810526C411yescharacteristic
19#16810519C411yescharacteristic
20#16810511—21no—
21#16810499—21no—
22#16810480C411yescharacteristic
23C4×C4⋊Dic53C2211yescharacteristic
24C4×C22⋊Dic53—21no—
25#16810377—21no—
26D4⋊#88322—22no—
27C2×C4×D4×C53C2211yescharacteristic
28C2×C4⋊C8×C53C2211yescharacteristic
29C2×C4×C8×C53C2211yescharacteristic
30C53×D4⋊C8—22no—
31C22×C22⋊D53—21no—
32C22×C4×D53C2×C411yescharacteristic
33C22×C4×D53D411yescharacteristic
34C22×C4×D53D411yescharacteristic
35C22×#88322—21no—
36C22×#88322—21no—
37#13133188—22no—
38#13133188C814yescomplemented
39C4×C22⋊D53—42no—
40C4×C22⋊D53—42no—
41C2×C4⋊Dic53C2×C411yescharacteristic
42C2×C4⋊Dic53C2×C411yescharacteristic
43C2×#11719919—21no—
44C2×C22⋊Dic53—21no—
45C2×C22⋊Dic53—21no—
46C2×C22⋊Dic53—21no—
47C2×#11719921—21no—
48C2×C4×Dic53—21no—
49C2×C4×Dic53C2×C411yescharacteristic
50C2×C4×Dic53D411yescharacteristic
51C2×C4×Dic53D411yescharacteristic
52C42×D53—21no—
53C42×D53—21no—
54C42×D53—21no—
55C42×D53—21no—
56C4⋊#88322—21no—
57C4⋊#88322D412yescomplemented
58#13133129—24no—
59#13133129—24no—
60C4×#88322—21no—
61C4×#88322—21no—
62C8×Dic53—24no—
63#13133124—21no—
64#13133124—21no—
65#13133124—21no—
66C22×D4×C53C2×C411yescharacteristic
67C23×C4×C53—21no—
68C22×C8×C53—21no—
69C22×C8×C53—21no—
70C4×D4×C53—21no—
71C4×D4×C53—21no—
72C4×D4×C53—21no—
73C4×D4×C53—21no—
74C2×C4⋊C4×C53C2×C411yescharacteristic
75C2×C22⋊C4×C53—21no—
76C2×C42×C53C2311yescharacteristic
77C4⋊C8×C53—21no—
78C4⋊C8×C53D412yescomplemented
79C4×C8×C53—21no—
80C4×C8×C53—21no—
81C24×C53—21no—
82C23×D53C22⋊C411yescharacteristic
83C2×D4×C53—21no—
84C2×D4×C53—21no—
85C2×D4×C53—21no—
86C2×D4×C53—21no—
87C2×D4×C53—21no—
88C2×D4×C53C2×C812yes—
89C22×C4×C53—21no—
90C22×C4×C53—41no—
91C22×C4×C53—41no—
92C22×C4×C53—41no—
93C22×C4×C53—41no—
94C22×C4×C53—41no—
95C22×C4×C53—41no—
96C22×C4×C53C22×C411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C2×C22⋊D53—24no—
100C2×C22⋊D53—42no—
101C2×C22⋊D53—42no—
102C22×Dic53—21no—
103C22×Dic53—21no—
104C22×Dic53C22⋊C411yescharacteristic
105C22×Dic53C22⋊C411yescharacteristic
106C22×Dic53C22⋊C411yescharacteristic
107C2×C4×D53—21no—
108C2×C4×D53—21no—
109C2×C4×D53—21no—
110C2×C4×D53—21no—
111C2×C4×D53—21no—
112C2×C4×D53—21no—
113C2×C4×D53—21no—
114C2×C4×D53—21no—
115C2×C4×D53—21no—
116C2×C4×D53—21no—
117C2×C4×D53—21no—
118C2×C4×D53—21no—
119C2×C4×D53—22no—
120C2×C4×D53—41no—
121C2×C4×D53C2×C812yes—
122C2×C4×D53M1612yes—
123C2×C8×C53—22no—
124C2×C8×C53—22no—
125C2×C8×C53—41no—
126C2×C8×C53—41no—
127C2×C8×C53—41no—
128C2×C8×C53—41no—
129C4⋊C4×C53—21no—
130C4⋊C4×C53—21no—
131C22⋊C4×C53—41no—
132C22⋊C4×C53—41no—
133C42×C53C2×D411yescharacteristic
134C42×C53C2×D411yescharacteristic
135C42×C53#4111yescharacteristic
136C42×C53#4111yescharacteristic
137C22⋊Dic53—24no—
138C22⋊Dic53—42no—
139C22⋊Dic53—42no—
140#11719921—42no—
141C4⋊Dic53—21no—
142C4⋊Dic53—21no—
143C4⋊Dic53—21no—
144C4⋊Dic53C2×C812yes—
145#11719919—42no—
146C4×Dic53—21no—
147C4×Dic53—21no—
148C4×Dic53—21no—
149C4×Dic53—21no—
150C4×Dic53—21no—
151C4×Dic53—21no—
152C4×Dic53—41no—
153C4×Dic53—41no—
154C2×#88322—22no—
155C2×#88322—22no—
156C2×#88322—41no—
157C2×#88322—41no—
158C2×#88322—41no—
159C2×#88322—41no—
160C23×C53—22no—
161C23×C53—41no—
162C23×C53—41no—
163C23×C53—41no—
164C23×C53—41no—
165C23×C53—41no—
166C23×C53—41no—
167C23×C53C2×C22⋊C411yescharacteristic
168C22×D53—21no—
169C22×D53—21no—
170C22×D53—21no—
171C22×D53—21no—
172C22×D53—21no—
173C22×D53—21no—
174C22×D53C22⋊C812yes—
175D4×C53—22no—
176D4×C53—22no—
177D4×C53—41no—
178D4×C53—41no—
179C2×C4×C53—21no—
180C2×C4×C53—21no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53—41no—
184C2×C4×C53—81no—
185C2×C4×C53—81no—
186C2×C4×C53—81no—
187C2×C4×C53—81no—
188C2×C4×C53C4×D411yescharacteristic
189C2×C4×C53C4×D411yescharacteristic
190C2×C4×C53#12011yescharacteristic
191C2×C4×C53#12111yescharacteristic
192C2×C4×C53#12111yescharacteristic
193C2×C4×C53#12311yescharacteristic
194C2×C4×C53C22×C811yescharacteristic
195C2×C4×C53C2×M1611yescharacteristic
196C2×C4×C53#13111yescharacteristic
197C2×C4×C53#13111yescharacteristic
198C2×C4×C53#13511yescharacteristic
199C2×C4×C53#13511yescharacteristic
200C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
201C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
202C22⋊D53—48no—
203C2×Dic53—21no—
204C2×Dic53—21no—
205C2×Dic53—21no—
206C2×Dic53—21no—
207C2×Dic53—21no—
208C2×Dic53—22no—
209C2×Dic53—21no—
210C2×Dic53—21no—
211C2×Dic53—41no—
212C2×Dic53—41no—
213C2×Dic53—41no—
214C2×Dic53—41no—
215C2×Dic53—41no—
216C2×Dic53C22⋊C812yes—
217C2×Dic53#10412yes—
218C4×D53—22no—
219C4×D53—22no—
220C4×D53—22no—
221C4×D53—22no—
222C4×D53—42no—
223C4×D53—42no—
224C4×D53—41no—
225C4×D53—41no—
226C4×D53—41no—
227C4×D53—41no—
228C8×C53—42no—
229C8×C53—42no—
230#88322—42no—
231#88322—42no—
232#8940—531nomaximal, Sylow
233C22×C53—42no—
234C22×C53—42no—
235C22×C53—42no—
236C22×C53—81no—
237C22×C53—81no—
238C22×C53—81no—
239C22×C53—81no—
240C22×C53#38611yescharacteristic
241C22×C53C2×C22⋊C811yescharacteristic
242C22×C53#40811yescharacteristic
243C22×C53#41611yescharacteristic
244C22×C53#41811yescharacteristic
245C22×C53C2×#10411yescharacteristic
246C22×C53#42111yescharacteristic
247C2×D53—22no—
248C2×D53—22no—
249C2×D53—22no—
250C2×D53—41no—
251C2×D53—41no—
252C2×D53—41no—
253C2×D53—41no—
254C4×C53—21no—
255C4×C53—21no—
256C4×C53—81no—
257C4×C53C8×D411yescharacteristic
258C4×C53C8×D411yescharacteristic
259C4×C53#43511yescharacteristic
260C4×C53#43511yescharacteristic
261C4×C53#44911yescharacteristic
262C4×C53#44911yescharacteristic
263C4×C53#48211yescharacteristic
264C4×C53#48211yescharacteristic
265Dic53—22no—
266Dic53—22no—
267Dic53—41no—
268Dic53—42no—
269Dic53—81no—
270C2×#431—531no—
271C4×#121—531no—
272C4×C22⋊C8—531no—
273C2×D4⋊C8—531no—
274C2×D4⋊C8D5311yescharacteristic, complemented
275#1279—531no—
276#1279—531no—
277C2×C53—82no—
278C2×C53#147611yescharacteristic
279C2×C53#147811yescharacteristic
280C2×C53#148211yescharacteristic
281C2×C53#148411yescharacteristic
282C2×C53#186911yescharacteristic
283C2×C53#187111yescharacteristic
284C2×C53#187511yescharacteristic
285D53—42no—
286C2×#121—531no—
287C2×C4×D4—1061no—
288C2×C4×D4C2×D5311yescharacteristic
289C22×C42—531no—
290#431—1062no—
291#431—1062no—
292C2×C4⋊C8—531no—
293C2×C4⋊C8C2×D5311yescharacteristic
294C2×C22⋊C8—1061no—
295C2×C22⋊C8—1061no—
296C2×C4×C8—531no—
297C2×C4×C8—1061no—
298C2×C4×C8C2×D5311yescharacteristic
299#390—531no—
300#389—531no—
301#388—1061no—
302#386—1061no—
303#380—531no—
304C4×C4⋊C4—531no—
305C4×C22⋊C4—1061no—
306#336—1061no—
307D4⋊C8—22no—
308D4⋊C8—1062no—
309C53#894011yescharacteristic, complemented, Sylow
310C22×D4—1061no—
311C22×D4C4×D5311yescharacteristic
312C23×C4—21no—
313C23×C4—531no—
314C23×C4—531no—
315C23×C4—531no—
316C22×C8—21no—
317C22×C8—21no—
318C22×C8—1061no—
319C22×C8—1061no—
320#121—534no—
321#121—1062no—
322C4×D4—21no—
323C4×D4—21no—
324C4×D4—21no—
325C4×D4—21no—
326C4×D4—2122no—
327C4×D4—2122no—
328C2×C4⋊C4—531no—
329C2×C4⋊C4—531no—
330C2×C4⋊C4—1061no—
331C2×C4⋊C4C4×D5311yescharacteristic
332C2×C22⋊C4—21no—
333C2×C22⋊C4—1061no—
334C2×C22⋊C4—1061no—
335C2×C22⋊C4—1061no—
336C2×C22⋊C4—1061no—
337C2×C42—531no—
338C2×C42—531no—
339C2×C42—531no—
340C2×C42—1061no—
341C2×C42—1061no—
342C2×C42—1061no—
343C2×C42—1061no—
344C2×C42—1061no—
345C2×C42C22×D5311yescharacteristic
346C4⋊C8—21no—
347C4⋊C8—532no—
348C4⋊C8—1061no—
349C4⋊C8C22⋊D5312yescomplemented
350C22⋊C8—1064no—
351C22⋊C8—1064no—
352C4×C8—21no—
353C4×C8—21no—
354C4×C8—1064no—
355C4×C8—1061no—
356C4×C8—1061no—
357#95—1061no—
358#95—1061no—
359#95—1061no—
360C24—21no—
361C24—531no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—1064no—
368C2×D4—2122no—
369C2×D4—2122no—
370C2×D4C8×D5312yes—
371C22×C4—21no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—531no—
379C22×C4—531no—
380C22×C4—532no—
381C22×C4—532no—
382C22×C4—531no—
383C22×C4—1062no—
384C22×C4—1061no—
385C22×C4—1061no—
386C22×C4—1061no—
387C22×C4—1061no—
388C22×C4—1061no—
389C22×C4—1061no—
390C22×C4—1061no—
391C22×C4—1061no—
392C22×C4—1061no—
393C22×C4—1061no—
394C22×C4—1061no—
395C22×C4—1061no—
396C22×C4—1061no—
397C22×C4—1061no—
398C22×C4—2121no—
399C22×C4C2×C4×D5311yescharacteristic
400C22×C4D4×D5311yescharacteristic
401C22×C4D4×D5311yescharacteristic
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—1062no—
409C2×C8—1062no—
410C2×C8—2121no—
411C2×C8—2121no—
412C2×C8—2121no—
413C2×C8—2121no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—532no—
417C4⋊C4—1061no—
418C4⋊C4—1061no—
419C4⋊C4—1061no—
420C4⋊C4—2122no—
421C22⋊C4—41no—
422C22⋊C4—41no—
423C22⋊C4—1064no—
424C22⋊C4—2122no—
425C22⋊C4—2122no—
426C22⋊C4—2122no—
427C42—1061no—
428C42—1061no—
429C42—1061no—
430C42—1061no—
431C42—1061no—
432C42—1061no—
433C42—2121no—
434C42—2121no—
435C42#1171994511yescharacteristic
436C42D4×D5311yescharacteristic
437C42#1171994711yescharacteristic
438C42C2×C22⋊D5311yescharacteristic
439C23—22no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—532no—
447C23—1061no—
448C23—1061no—
449C23—1061no—
450C23—1061no—
451C23—1061no—
452C23—1061no—
453C23C22⋊C4×D5311yescharacteristic
454D4—22no—
455D4—22no—
456D4—41no—
457D4—41no—
458D4—2128no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—532no—
469C2×C4—532no—
470C2×C4—1061no—
471C2×C4—1061no—
472C2×C4—1061no—
473C2×C4—1061no—
474C2×C4—1062no—
475C2×C4—1062no—
476C2×C4—1062no—
477C2×C4—1062no—
478C2×C4—1061no—
479C2×C4—1062no—
480C2×C4—1061no—
481C2×C4—1061no—
482C2×C4—2122no—
483C2×C4—2122no—
484C2×C4—2121no—
485C2×C4—2121no—
486C2×C4—2121no—
487C2×C4—2121no—
488C2×C4—2121no—
489C2×C4—2121no—
490C2×C4—2121no—
491C2×C4—2121no—
492C2×C4—2121no—
493C2×C4#1313317711yescharacteristic
494C2×C4C4×C22⋊D5311yescharacteristic
495C2×C4#1313318511yescharacteristic
496C2×C4#1313318811yescharacteristic
497C2×C4#1313319011yescharacteristic
498C2×C4#1313319911yescharacteristic
499C2×C4C2×C8×D5311yescharacteristic
500C2×C4M16×D5311yescharacteristic
501C2×C4#1313322311yescharacteristic
502C2×C4#1313322511yescharacteristic
503C2×C4#1313323611yescharacteristic
504C2×C4#1313323811yescharacteristic
505C2×C4#1313324111yescharacteristic
506C2×C4#1313324411yescharacteristic
507C8—42no—
508C8—42no—
509C8—2122no—
510C8—2122no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22—81no—
518C22—1062no—
519C22—1062no—
520C22—1062no—
521C22—2121no—
522C22—2121no—
523C22—2121no—
524C22—2121no—
525C22#1681049911yescharacteristic
526C22C22⋊C8×D5311yescharacteristic
527C22#1681058811yescharacteristic
528C22#1681061611yescharacteristic
529C22#1681062611yescharacteristic
530C22#104×D5311yescharacteristic
531C22#1681063811yescharacteristic
532C4—21no—
533C4—21no—
534C4—81no—
535C4—1062no—
536C4—1062no—
537C4—4241no—
538C4—2121no—
539C4—2122no—
540C4#1681068611yescharacteristic
541C4C8×C22⋊D5311yescharacteristic
542C4#1681069211yescharacteristic
543C4#1681069711yescharacteristic
544C4#1681076511yescharacteristic
545C4#1681076711yescharacteristic
546C4#1681094511yescharacteristic
547C4#1681094611yescharacteristic
548C2—82no—
549C2—2122no—
550C2#2027309411yescharacteristic
551C2#2027310411yescharacteristic
552C2#2027312011yescharacteristic
553C2#2027313011yescharacteristic
554C2#2027500411yescharacteristic
555C2#2027501411yescharacteristic
556C2#2027503411yescharacteristic
subgroupG/H
centerC22×C4D4×D53
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC2×C53×D4⋊C8C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133188C4
cosocleC2×C42×C53C23
hypercenterC2×D4⋊C8D53
hypocenterC53#8940
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×D4⋊C8

Complements

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

NH ≅ G/Ntype
1. C2×#16810675C2semidirect
3. C4×#13133129C2semidirect
4. C2×D4⋊#88322C2semidirect
5. #20272555C2semidirect
6. #20272554C2semidirect
7. C2×C53×D4⋊C8C2semidirect
38. #13133188C8semidirect
57. C4⋊#88322D4semidirect
78. C4⋊C8×C53D4semidirect
274. C2×D4⋊C8D53semidirect
309. C53#8940semidirect
349. C4⋊C8C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168106752
C4×#131331882
C4×#131331292
C2×D4⋊#883222
#202725552
#202725542
C2×C53×D4⋊C82
#894053