Finite Group Databasebeta

← #27291094

Subgroups — #27291094

Order 13568.

970 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283279C211yesmaximal, characteristic, complemented
2C2×#16810627C211yesmaximal, characteristic
3C2×#16810627C211yesmaximal, characteristic
4C22×#13133133C211yesmaximal, characteristic
5C2×#16810393C211yesmaximal, characteristic, complemented
6C2×#16810381C211yesmaximal, characteristic, complemented
7C53×#1733C211yesmaximal, characteristic, complemented
8C23×D212C2211yescharacteristic
9C22×#11719917C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11C22×#11719917C2211yescharacteristic
12#16810650—22no—
13#16810650—22no—
14#16810647—22no—
15#16810627—22no—
16#16810627—22no—
17#16810627—22no—
18#16810627—22no—
19C2×#13133137—22no—
20C2×#13133137C414yescomplemented
21C2×#13133133—22no—
22C2×#13133133—22no—
23C2×#13133133—22no—
24#16810393—22no—
25#16810381—22no—
26C22×M16×C53C2211yescharacteristic
27C2×#117×C53C2211yescharacteristic
28C2×#117×C53C2211yescharacteristic
29C53×#339—22no—
30C24×D53—21no—
31C22×D212—21no—
32C22×D212—21no—
33C22×D212—21no—
34C22×D212—21no—
35C22×D212—21no—
36C22×D212—21no—
37C22×D212—21no—
38C22×D212—21no—
39C22×D212—21no—
40C22×D212—21no—
41C22×D212—21no—
42C22×D212C2×C412yes—
43C22×D212D412yes—
44C22×D212Q812yes—
45C2×#11719917—21no—
46C2×#11719917—21no—
47C2×#11719917—21no—
48C2×#11719917—21no—
49C2×#11719917—21no—
50C2×#11719917—21no—
51C2×#11719917—21no—
52C2×#11719917—21no—
53C2×#11719917—21no—
54C2×#11719917—21no—
55C2×#11719917—21no—
56C2×#11719917—21no—
57C2×#11719917—21no—
58C2×#11719917—21no—
59C2×#11719917—21no—
60C2×#11719917—21no—
61C2×#11719917—21no—
62C2×#11719917D412yes—
63C22×#88322—21no—
64C22×#88322C2×C412yes—
65C22×#88322D412yes—
66C4⋊#88322—28no—
67#13133137—28no—
68#13133137—28no—
69#13133133—28no—
70C23×C4×C53C2311yescharacteristic
71C2×M16×C53—21no—
72C2×M16×C53—21no—
73C2×M16×C53—21no—
74C2×M16×C53—21no—
75C2×M16×C53—21no—
76C2×M16×C53D412yes—
77C22×C8×C53—21no—
78#117×C53—21no—
79#117×C53—21no—
80#117×C53—21no—
81#117×C53—21no—
82#117×C53—21no—
83#117×C53D412yes—
84#117×C53D412yes—
85#117×C53D412yes—
86C2×C4⋊C4×C53—21no—
87C2×C4⋊C4×C53C2×C412yes—
88C2×C22⋊C4×C53—21no—
89C2×C22⋊C4×C53—21no—
90C2×C42×C53—21no—
91C2×C42×C53—21no—
92C24×C53#4111yescharacteristic
93C23×D53—22no—
94C23×D53—22no—
95C23×D53—22no—
96C23×D53—41no—
97C23×D53—41no—
98C23×D53—41no—
99C23×D53—41no—
100C23×D53—84no—
101C22×C4×C53—22no—
102C22×C4×C53—41no—
103C22×C4×C53C22×C411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53C2×Q811yescharacteristic
112C22×C4×C53#4111yescharacteristic
113C22×C4×C53#4111yescharacteristic
114C22×C4×C53#4111yescharacteristic
115C22×C4×C53#4111yescharacteristic
116C22×C4×C53#4111yescharacteristic
117C2×D212—22no—
118C2×D212—22no—
119C2×D212—22no—
120C2×D212—22no—
121C2×D212—22no—
122C2×D212—22no—
123C2×D212—22no—
124C2×D212—22no—
125C2×D212—22no—
126C2×D212—22no—
127C2×D212—22no—
128C2×D212—22no—
129C2×D212—22no—
130C2×D212—22no—
131C2×D212—22no—
132C2×D212—41no—
133C2×D212—41no—
134C2×D212—41no—
135C2×D212—41no—
136C2×D212—41no—
137C2×D212—41no—
138C2×D212—41no—
139C2×D212—41no—
140C2×D212—41no—
141C2×D212—41no—
142C2×D212—41no—
143C2×D212—41no—
144C2×D212C4⋊C414yes—
145M16×C53—22no—
146M16×C53—22no—
147M16×C53—41no—
148M16×C53—41no—
149C2×C8×C53—22no—
150C2×C8×C53—41no—
151C2×C8×C53—41no—
152C4⋊C4×C53—22no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—41no—
155C4⋊C4×C53C22⋊C414yes—
156C22⋊C4×C53—22no—
157C22⋊C4×C53—22no—
158C22⋊C4×C53—22no—
159C22⋊C4×C53—41no—
160C42×C53—22no—
161C42×C53—22no—
162C42×C53—22no—
163C42×C53—41no—
164#11719917—22no—
165#11719917—22no—
166#11719917—41no—
167#11719917—41no—
168#11719917—41no—
169#11719917—41no—
170#11719917—41no—
171#11719917—41no—
172#11719917—41no—
173#11719917—41no—
174#11719917—41no—
175#11719917—41no—
176C2×#88322—22no—
177C2×#88322—22no—
178C2×#88322—22no—
179C2×#88322—22no—
180C2×#88322—22no—
181C2×#88322—22no—
182C2×#88322—22no—
183C2×#88322—41no—
184C2×#88322—41no—
185C23×C53—42no—
186C23×C53C4×D411yescharacteristic
187C23×C53#12111yescharacteristic
188C23×C53C22⋊Q811yescharacteristic
189C23×C53#12311yescharacteristic
190C23×C53#12311yescharacteristic
191C23×C53#12311yescharacteristic
192C23×C53#12611yescharacteristic
193C22×D53—24no—
194C22×D53—42no—
195C22×D53—42no—
196C22×D53—42no—
197C22×D53—42no—
198C22×D53—42no—
199C22×D53—42no—
200C22×D53—84no—
201C22×D53—84no—
202C22×D53—84no—
203C22×D53—84no—
204C22×D53—84no—
205C22×D53—84no—
206C22×D53—84no—
207C2×C4×C53—24no—
208C2×C4×C53—42no—
209C2×C4×C53—82no—
210C2×C4×C53—82no—
211C2×C4×C53C2×C22⋊C411yescharacteristic
212C2×C4×C53C2×C4⋊C411yescharacteristic
213C2×C4×C53#11711yescharacteristic
214C2×C4×C53C4×D411yescharacteristic
215C2×C4×C53C4×D411yescharacteristic
216C2×C4×C53C4×D411yescharacteristic
217C2×C4×C53#12011yescharacteristic
218C2×C4×C53#12011yescharacteristic
219C2×C4×C53#12011yescharacteristic
220C2×C4×C53#12111yescharacteristic
221C2×C4×C53#12111yescharacteristic
222C2×C4×C53#12111yescharacteristic
223C2×C4×C53#12111yescharacteristic
224C2×C4×C53#12111yescharacteristic
225C2×C4×C53#12111yescharacteristic
226C2×C4×C53#12111yescharacteristic
227C2×C4×C53C22⋊Q811yescharacteristic
228C2×C4×C53C22⋊Q811yescharacteristic
229C2×C4×C53C22⋊Q811yescharacteristic
230C2×C4×C53C22⋊Q811yescharacteristic
231C2×C4×C53C22⋊Q811yescharacteristic
232C2×C4×C53#12311yescharacteristic
233C2×C4×C53#12311yescharacteristic
234C2×C4×C53#12411yescharacteristic
235C2×C4×C53#12411yescharacteristic
236C2×C4×C53#12411yescharacteristic
237C2×C4×C53#12611yescharacteristic
238C2×C4×C53#12711yescharacteristic
239D212—24no—
240D212—24no—
241D212—42no—
242D212—42no—
243D212—42no—
244D212—42no—
245D212—42no—
246D212—42no—
247C8×C53—42no—
248#88322—42no—
249#88322—42no—
250#88322—42no—
251#11058—531nomaximal, Sylow
252C22×C53—22no—
253C22×C53—22no—
254C22×C53—22no—
255C22×C53—41no—
256C22×C53—41no—
257C22×C53—41no—
258C22×C53—41no—
259C22×C53#38511yescharacteristic
260C22×C53#38611yescharacteristic
261C22×C53#38811yescharacteristic
262C22×C53#39411yescharacteristic
263C22×C53#39611yescharacteristic
264C22×C53#39711yescharacteristic
265C22×C53#39911yescharacteristic
266C2×D53—44no—
267C2×D53—84no—
268C2×D53—84no—
269C2×D53—84no—
270C2×D53—84no—
271C2×D53—84no—
272C2×D53—84no—
273C2×D53—84no—
274C4×C53—44no—
275C4×C53—82no—
276C4×C53#38011yescharacteristic
277C4×C53#38511yescharacteristic
278C4×C53#38811yescharacteristic
279C4×C53#39011yescharacteristic
280C4×C53#39211yescharacteristic
281C4×C53#39311yescharacteristic
282C4×C53#39411yescharacteristic
283C4×C53#39611yescharacteristic
284#2905—531no—
285C2×#418—531no—
286C2×#418—531no—
287C22×#100—531no—
288C2×#344—531no—
289#1733D5311yescharacteristic, complemented
290C2×#338—531no—
291C2×C53—21no—
292C2×C53—21no—
293C2×C53—21no—
294C2×C53—21no—
295C2×C53—21no—
296C2×C53—21no—
297C2×C53#143911yescharacteristic
298C2×C53#190312yes—
299D53—84no—
300C23×D4—531no—
301C22×M16—531no—
302C22×M16—531no—
303C22×M16—531no—
304C22×M16C2×D5311yescharacteristic
305C2×#117C2×D5311yescharacteristic
306C2×#117C2×D5311yescharacteristic
307#424—1062no—
308#424—1062no—
309#423—1062no—
310#418—1062no—
311#418—1062no—
312#418—1062no—
313#418—1062no—
314C2×#102—534no—
315C2×#102—1062no—
316C2×#100—1062no—
317C2×#100—1062no—
318C2×#100—1062no—
319#344—1062no—
320#339—22no—
321#338—1062no—
322C53#1105811yescharacteristic, complemented, Sylow
323C25—1061no—
324C22×D4—532no—
325C22×D4—532no—
326C22×D4—532no—
327C22×D4—1061no—
328C22×D4—1061no—
329C22×D4—1061no—
330C22×D4—1061no—
331C22×D4—1061no—
332C22×D4—1061no—
333C22×D4—1061no—
334C22×D4—1061no—
335C22×D4—1061no—
336C22×D4—1061no—
337C22×D4—1061no—
338C23×C4C22×D5311yescharacteristic
339C2×M16—21no—
340C2×M16—21no—
341C2×M16—21no—
342C2×M16—21no—
343C2×M16—21no—
344C2×M16—532no—
345C2×M16—1061no—
346C2×M16—1061no—
347C2×M16—1061no—
348C2×M16—1061no—
349C2×M16—1061no—
350C2×M16—1061no—
351C2×M16—1061no—
352C2×M16—1061no—
353C2×M16—1061no—
354C2×M16—1061no—
355C2×M16—1061no—
356C2×M16—1061no—
357C2×M16—1061no—
358C2×M16—1061no—
359C2×M16—1061no—
360C2×M16—1061no—
361C2×M16—1061no—
362C2×M16D21212yes—
363C22×C8—21no—
364C22×C8—532no—
365C22×C8—532no—
366C22×C8—1061no—
367#117—21no—
368#117—21no—
369#117—21no—
370#117—21no—
371#117—21no—
372#117D21212yes—
373#117D21212yes—
374#117C22⋊D5312yes—
375C2×C4⋊C4—21no—
376C2×C4⋊C4C4×D5312yes—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C42—21no—
380C2×C42—21no—
381C4⋊C8—1068no—
382#102—1068no—
383#102—1068no—
384#100—1068no—
385C24—1062no—
386C24—1062no—
387C24—1062no—
388C24—4244no—
389C24—2121no—
390C24—2121no—
391C24—2121no—
392C24—2121no—
393C24#1171994711yescharacteristic
394C2×D4—534no—
395C2×D4—1062no—
396C2×D4—1062no—
397C2×D4—1062no—
398C2×D4—1062no—
399C2×D4—1062no—
400C2×D4—1062no—
401C2×D4—1062no—
402C2×D4—1062no—
403C2×D4—1062no—
404C2×D4—1062no—
405C2×D4—1062no—
406C2×D4—1062no—
407C2×D4—1062no—
408C2×D4—1062no—
409C2×D4—1062no—
410C2×D4—2121no—
411C2×D4—2121no—
412C2×D4—2121no—
413C2×D4—2121no—
414C2×D4—2121no—
415C2×D4—2121no—
416C2×D4—2121no—
417C2×D4—2121no—
418C2×D4—2121no—
419C2×D4—2121no—
420C2×D4—2121no—
421C2×D4—2121no—
422C22×C4—22no—
423C22×C4—41no—
424C22×C4C2×C4×D5311yescharacteristic
425C22×C4C2×D21211yescharacteristic
426C22×C4C2×D21211yescharacteristic
427C22×C4C2×D21211yescharacteristic
428C22×C4#1171994511yescharacteristic
429C22×C4#1171994511yescharacteristic
430C22×C4D4×D5311yescharacteristic
431C22×C4D4×D5311yescharacteristic
432C22×C4D4×D5311yescharacteristic
433C22×C4#1171994711yescharacteristic
434C22×C4#1171994711yescharacteristic
435C22×C4Q8×D5311yescharacteristic
436C22×C4#1171994911yescharacteristic
437C22×C4C2×C22⋊D5311yescharacteristic
438M16—22no—
439M16—22no—
440M16—41no—
441M16—41no—
442M16—1062no—
443M16—1062no—
444M16—2121no—
445M16—2121no—
446M16—2121no—
447M16—2121no—
448M16—2121no—
449M16—2121no—
450M16—2121no—
451M16—2121no—
452M16—2121no—
453M16—2121no—
454C2×C8—22no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—1062no—
458C2×C8—1062no—
459C2×C8—1062no—
460C2×C8—1062no—
461C2×C8—1062no—
462C2×C8—1062no—
463C2×C8—1062no—
464C2×C8—2121no—
465C2×C8—2121no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—41no—
469C4⋊C4#1171992114yes—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—41no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—41no—
478C23—42no—
479C23—1064no—
480C23—4244no—
481C23—4244no—
482C23—4244no—
483C23—4244no—
484C23—4244no—
485C23—4244no—
486C23—4244no—
487C23—2122no—
488C23—2122no—
489C23—2122no—
490C23—2122no—
491C23—2122no—
492C23—2122no—
493C23#1313317711yescharacteristic
494C23#1313318911yescharacteristic
495C23#1313319711yescharacteristic
496C23#1313319911yescharacteristic
497C23#1313320111yescharacteristic
498C23#1313320111yescharacteristic
499C23#1313321211yescharacteristic
500D4—1064no—
501D4—1064no—
502D4—2122no—
503D4—2122no—
504D4—2122no—
505D4—2122no—
506D4—2122no—
507D4—2122no—
508C2×C4—24no—
509C2×C4—42no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4C2×#1171992111yescharacteristic
513C2×C4C4⋊C4×D5311yescharacteristic
514C2×C4#1313317511yescharacteristic
515C2×C4#1313317711yescharacteristic
516C2×C4C4×D21211yescharacteristic
517C2×C4C4×D21211yescharacteristic
518C2×C4#1313318711yescharacteristic
519C2×C4#1313318711yescharacteristic
520C2×C4#1313318711yescharacteristic
521C2×C4#1313318811yescharacteristic
522C2×C4#1313318911yescharacteristic
523C2×C4#1313319111yescharacteristic
524C2×C4#1313319111yescharacteristic
525C2×C4#1313319211yescharacteristic
526C2×C4#1313319211yescharacteristic
527C2×C4#1313319211yescharacteristic
528C2×C4#1313319311yescharacteristic
529C2×C4#1313319511yescharacteristic
530C2×C4#1313319511yescharacteristic
531C2×C4#1313319711yescharacteristic
532C2×C4#1313319711yescharacteristic
533C2×C4#1313319911yescharacteristic
534C2×C4#1313320111yescharacteristic
535C2×C4#1313320311yescharacteristic
536C2×C4#1313320311yescharacteristic
537C2×C4#1313320511yescharacteristic
538C2×C4#1313321211yescharacteristic
539C2×C4#1313321411yescharacteristic
540C8—42no—
541C8—2122no—
542C8—2122no—
543C8—2122no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—4244no—
552C22—4244no—
553C22—4244no—
554C22—4244no—
555C22—4244no—
556C22—4244no—
557C22—4244no—
558C22—2124no—
559C22#1681049711yescharacteristic
560C22#1681050011yescharacteristic
561C22#1681051511yescharacteristic
562C22#1681053911yescharacteristic
563C22#1681054511yescharacteristic
564C22#1681055011yescharacteristic
565C22#1681056011yescharacteristic
566C4—44no—
567C4—82no—
568C4#1681048011yescharacteristic
569C4#1681049711yescharacteristic
570C4#1681051511yescharacteristic
571C4#1681052511yescharacteristic
572C4#1681053311yescharacteristic
573C4#1681053611yescharacteristic
574C4#1681053911yescharacteristic
575C4#1681054511yescharacteristic
576C2—21no—
577C2—21no—
578C2—21no—
579C2—21no—
580C2—21no—
581C2—21no—
582C2—4244no—
583C2#2027296111yescharacteristic
584C2#2027519312yes—
subgroupG/H
centerC22#16810500
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1733C2
socleC22×C53#386
prime radicalC23×D212C22
cosocleC23×C4×C53C23
hypercenter#1733D53
hypocenterC53#11058
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1733

Complements

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

NH ≅ G/Ntype
1. #20283279C2semidirect
5. C2×#16810393C2semidirect
6. C2×#16810381C2semidirect
7. C53×#1733C2semidirect
20. C2×#13133137C4semidirect
289. #1733D53semidirect
322. C53#11058semidirect

Maximal subgroups

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

subgroupindex
#202832792
C2×#168106272
C22×#131331332
C2×#168103932
C2×#168103812
C53×#17332
#1105853