Finite Group Databasebeta

← #27389177

Subgroups — #27389177

Order 13568.

1450 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811170C211yesmaximal, characteristic
2C2×#16810674C211yesmaximal, characteristic, complemented
3C4×#13133175C211yesmaximal, characteristic, complemented
4#20275661C212yesmaximal, complemented
5#20275640C212yesmaximal, complemented
6#20275036C212yesmaximal, complemented
7#20275034C212yesmaximal, complemented
8C4×#13133137C211yesmaximal, characteristic, complemented
9C4×#13133139C211yesmaximal, characteristic, complemented
10C4×#13133138C211yesmaximal, characteristic, complemented
11C4×#102×C53C211yesmaximal, characteristic, complemented
12C22×#11719945—21no—
13#16811170—24no—
14#16811170C418yescomplemented
15C2×#13133216C2211yescharacteristic
16C2×#13133213C2211yescharacteristic
17C2×#13133203—21no—
18C2×#13133192C2211yescharacteristic
19C2×#13133190—21no—
20C2×#13133188C2211yescharacteristic
21C2×C4×D212C2211yescharacteristic
22C2×C4×C22⋊D53—21no—
23C2×#13133175C2212yes—
24C2×C42×D53C2211yescharacteristic
25#16810674—24no—
26C2×#13133146C2212yes—
27C2×#13133139C2212yes—
28C2×#13133138C2212yes—
29C2×#13133137C2212yes—
30C2×#13133129C2212yes—
31C2×C4×#88322C2211yescharacteristic
32C4⋊C4×Dic53C2212yes—
33C4×C4⋊Dic53C2211yescharacteristic
34C4×#11719921—21no—
35#16810470—21no—
36#16810468—21no—
37C42×Dic53—21no—
38#16810386C2212yes—
39#16810385C2212yes—
40#16810385C2212yes—
41C2×C4×D4×C53C2211yescharacteristic
42C2×#102×C53C2212yes—
43C2×C4×C8×C53C2211yescharacteristic
44C4×C4⋊C4×C53C2211yescharacteristic
45C53×#340C2212yes—
46C2×#11719945—28no—
47C2×#11719945—44no—
48C2×#11719945—44no—
49C22×Dic106—21no—
50C22×D212—21no—
51C22×D212C2311yescharacteristic
52C22×C22⋊D53—21no—
53C22×C22⋊D53—21no—
54C22×C22⋊D53—21no—
55C22×C4×D53—21no—
56C22×C4×D53D411yescharacteristic
57C22×C4×D53D411yescharacteristic
58C22×C4×D53C2311yescharacteristic
59C22×#88322C2312yes—
60#13133216—22no—
61#13133216C2×C414yes—
62#13133213—22no—
63#13133213C2×C414yes—
64#13133203—28no—
65#13133192—22no—
66#13133192C2×C414yes—
67#13133190—28no—
68#13133188—22no—
69#13133188C2×C414yes—
70C4×D212—22no—
71C4×D212C2×C414yes—
72C4×C22⋊D53—28no—
73#13133175—24no—
74#13133175—24no—
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2312yes—
77C2×#11719919—21no—
78C2×C22⋊Dic53—21no—
79C2×#11719921—22no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—22no—
84C2×C4×Dic53—22no—
85C2×C4×Dic53D411yescharacteristic
86C2×C4×Dic53D411yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88C42×D53C2×C414yes—
89C42×D53D414yes—
90#13133146—24no—
91#13133139—24no—
92#13133138—24no—
93#13133137—24no—
94#13133129—24no—
95C4×#88322—22no—
96#13133124—21no—
97#13133124—22no—
98C22×D4×C53C2311yescharacteristic
99C23×C4×C53—21no—
100C22×C8×C53C2312yes—
101C4×D4×C53—22no—
102C4×D4×C53C2×C414yes—
103C2×C4⋊C4×C53C2311yescharacteristic
104C2×C4⋊C4×C53C2312yes—
105C2×C22⋊C4×C53—21no—
106C2×C42×C53—21no—
107C2×C42×C53C2311yescharacteristic
108#102×C53—24no—
109C4×C8×C53—22no—
110#95×C53—21no—
111C24×C53—21no—
112C23×D53—21no—
113C23×D53C2×D411yescharacteristic
114C2×D4×C53—21no—
115C2×D4×C53—22no—
116C2×D4×C53—22no—
117C2×D4×C53C22×C412yes—
118C22×C4×C53—22no—
119C22×C4×C53—21no—
120C22×C4×C53—24no—
121C22×C4×C53—42no—
122C22×C4×C53—42no—
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2411yescharacteristic
126C2×C22⋊D53—24no—
127C2×C22⋊D53—24no—
128C2×C22⋊D53—24no—
129C2×C22⋊D53—42no—
130C2×C22⋊D53—42no—
131C2×C22⋊D53—42no—
132C2×C22⋊D53—42no—
133C2×C22⋊D53—44no—
134C22×Dic53—22no—
135C22×Dic53—21no—
136C22×Dic53C2×D411yescharacteristic
137C22×Dic53C2×D411yescharacteristic
138C22×Dic53C2×D411yescharacteristic
139#11719945—432no—
140C2×D212—22no—
141C2×D212—22no—
142C2×D212—24no—
143C2×D212—21no—
144C2×D212—42no—
145C2×D212—42no—
146C2×D212C22×C412yes—
147C2×C4×D53—22no—
148C2×C4×D53—22no—
149C2×C4×D53—22no—
150C2×C4×D53—22no—
151C2×C4×D53—24no—
152C2×C4×D53—24no—
153C2×C4×D53—42no—
154C2×C4×D53—42no—
155C2×C4×D53C22⋊C414yes—
156C2×C4×D53C22⋊C414yes—
157C2×C4×D53C22×C412yes—
158C2×C4×D53C2×D412yes—
159C2×Dic106—24no—
160C2×Dic106—42no—
161C2×Dic106—42no—
162C2×C8×C53—22no—
163C2×C8×C53—22no—
164C2×C8×C53—22no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—22no—
167C4⋊C4×C53C22×C412yes—
168C4⋊C4×C53#4114yescomplemented
169C22⋊C4×C53—24no—
170C42×C53—42no—
171C42×C53C22×C412yes—
172C42×C53C2×D412yes—
173C22⋊Dic53—24no—
174#11719921—28no—
175#11719921—24no—
176#11719921—24no—
177C4⋊Dic53—21no—
178C4⋊Dic53—22no—
179C4⋊Dic53C22×C412yes—
180C4⋊Dic53#4114yescomplemented
181#11719919—24no—
182C4×Dic53—28no—
183C4×Dic53—42no—
184C4×Dic53—44no—
185C4×Dic53C22⋊C414yes—
186C4×Dic53C22⋊C414yes—
187C4×Dic53C22×C412yes—
188C4×Dic53C2×D412yes—
189C2×#88322—22no—
190C2×#88322—22no—
191C2×#88322—22no—
192C23×C53—22no—
193C23×C53—42no—
194C23×C53—42no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53C22×D411yescharacteristic
198C22×D53—22no—
199C22×D53—22no—
200C22×D53—21no—
201C22×D53—21no—
202C22×D53—22no—
203C22×D53—42no—
204C22×D53—42no—
205C22×D53—41no—
206C22×D53—41no—
207C22×D53C2×C22⋊C412yes—
208D4×C53—22no—
209D4×C53—22no—
210D4×C53—42no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—24no—
214C2×C4×C53—42no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—42no—
218C2×C4×C53—48no—
219C2×C4×C53C2×C22⋊C412yes—
220C2×C4×C53C2×C22⋊C412yes—
221C2×C4×C53AGL(1,ℤ/8)12yes—
222C2×C4×C53AGL(1,ℤ/8)12yes—
223C2×C4×C53C23×C411yescharacteristic
224C2×C4×C53C22×D411yescharacteristic
225C2×C4×C53C2×#4112yes—
226C2×C4×C53C2×#4112yes—
227C22⋊D53—48no—
228C22⋊D53—48no—
229C22⋊D53—48no—
230C2×Dic53—22no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—22no—
234C2×Dic53—24no—
235C2×Dic53—42no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—42no—
239C2×Dic53C2×C22⋊C412yes—
240C2×Dic53C2×C22⋊C412yes—
241C2×Dic53C2×C22⋊C412yes—
242C2×Dic53#13514yes—
243D212—22no—
244D212—22no—
245D212—48no—
246D212—42no—
247C4×D53—22no—
248C4×D53—22no—
249C4×D53—24no—
250C4×D53—28no—
251C4×D53—28no—
252C4×D53—48no—
253Dic106—48no—
254C8×C53—42no—
255#88322—42no—
256#21390—531nomaximal, Sylow
257C22×C53—42no—
258C22×C53—42no—
259C22×C53—42no—
260C22×C53—82no—
261C22×C53—82no—
262C22×C53C22×C22⋊C411yescharacteristic
263C22×C53#52512yes—
264C22×C53C2×#13512yes—
265C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
267C2×D53—22no—
268C2×D53—22no—
269C2×D53—22no—
270C2×D53—42no—
271C2×D53—42no—
272C2×D53—42no—
273C2×D53—42no—
274C2×D53—42no—
275C2×D53—82no—
276C2×D53—82no—
277C4×C53—22no—
278C4×C53—44no—
279C4×C53—42no—
280C4×C53#41814yes—
281C4×C53C4×#4111yescharacteristic
282C4×C53C4×#4111yescharacteristic
283C4×C53#53212yes—
284Dic53—22no—
285Dic53—22no—
286Dic53—44no—
287Dic53—42no—
288Dic53#41614yes—
289C2×#532—531no—
290C2×#431—531no—
291C4×#117—531no—
292#1987—532no—
293#1982—532no—
294#1875—532no—
295#1875—532no—
296C4×#102—531no—
297C4×#102—531no—
298C4×#102—531no—
299C4×#102D5311yescharacteristic, complemented
300C2×C53—82no—
301C2×C53#229711yescharacteristic
302C2×C53C2×#41611yescharacteristic
303C2×C53C2×#41811yescharacteristic
304C2×C53#310612yes—
305C2×C53#310612yes—
306D53—42no—
307D53—82no—
308C22×#41—1061no—
309#532—538no—
310#532—1064no—
311C2×C4⋊Q8—531no—
312C2×#127—531no—
313C2×#124—1061no—
314C2×#121—531no—
315C2×#121—531no—
316C2×#121—1061no—
317C2×C4×D4—531no—
318C2×C4×D4—1061no—
319C2×C4×D4C2×D5311yescharacteristic
320C2×#117—532no—
321C22×C42—531no—
322#431—1064no—
323C2×C4⋊C8—532no—
324C2×#102—532no—
325C2×#102—532no—
326C2×#102—532no—
327C2×#102C2×D5312yes—
328C2×C22⋊C8—532no—
329C2×C4×C8—531no—
330C2×C4×C8C2×D5311yescharacteristic
331C4×C4⋊C4—532no—
332C4×C4⋊C4—531no—
333C4×C4⋊C4C2×D5311yescharacteristic
334C4×C22⋊C4—1061no—
335C4×C22⋊C4—1061no—
336#376—1061no—
337C43—1061no—
338#340—532no—
339#340—532no—
340#340—532no—
341#340C2×D5312yes—
342C53#2139011yescharacteristic, complemented, Sylow
343C2×#41—1068no—
344C2×#41—2124no—
345C2×#41—2124no—
346C22×Q8—1061no—
347C22×D4—531no—
348C22×D4—1061no—
349C22×D4—1061no—
350C22×D4—1061no—
351C22×D4—1061no—
352C22×D4C22×D5311yescharacteristic
353C23×C4—21no—
354C23×C4—531no—
355C23×C4—531no—
356C23×C4—531no—
357C23×C4—1061no—
358C22×C8—532no—
359C22×C8C22×D5312yes—
360C4⋊Q8—534no—
361C4⋊Q8—1062no—
362#127—534no—
363#127—1062no—
364#124—1068no—
365#121—534no—
366#121—534no—
367#121—1068no—
368#121—1062no—
369#121—1062no—
370C4×D4—22no—
371C4×D4—534no—
372C4×D4—1068no—
373C4×D4—1062no—
374C4×D4C4×D5314yes—
375#117—1064no—
376#117—1064no—
377C2×C4⋊C4—532no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4C22×D5311yescharacteristic
381C2×C4⋊C4C22×D5312yes—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—1061no—
384C2×C22⋊C4—1062no—
385C2×C22⋊C4—1061no—
386C2×C22⋊C4—1061no—
387C2×C42—21no—
388C2×C42—531no—
389C2×C42—531no—
390C2×C42—531no—
391C2×C42—534no—
392C2×C42—534no—
393C2×C42—1062no—
394C2×C42—1062no—
395C2×C42—1061no—
396C2×C42C22×D5311yescharacteristic
397C4⋊C8—1064no—
398#102—24no—
399#102—1064no—
400#102—1064no—
401#102—1064no—
402C22⋊C8—1064no—
403C4×C8—22no—
404C4×C8—1062no—
405#95—21no—
406#95—1062no—
407#95—1061no—
408C24—21no—
409C24—531no—
410C24—1061no—
411#41—21232no—
412C2×Q8—1064no—
413C2×Q8—2122no—
414C2×Q8—2122no—
415C2×D4—21no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—532no—
419C2×D4—1064no—
420C2×D4—1064no—
421C2×D4—1064no—
422C2×D4—1064no—
423C2×D4—1061no—
424C2×D4—1062no—
425C2×D4—1062no—
426C2×D4—2122no—
427C2×D4—2122no—
428C2×D4—2124no—
429C2×D4—2122no—
430C2×D4—2122no—
431C2×D4—2122no—
432C2×D4—2122no—
433C2×D4C2×C4×D5312yes—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—531no—
440C22×C4—531no—
441C22×C4—531no—
442C22×C4—534no—
443C22×C4—534no—
444C22×C4—532no—
445C22×C4—532no—
446C22×C4—1064no—
447C22×C4—1062no—
448C22×C4—1062no—
449C22×C4—1062no—
450C22×C4—1062no—
451C22×C4—1062no—
452C22×C4—1061no—
453C22×C4—1064no—
454C22×C4—2122no—
455C22×C4—2122no—
456C22×C4D4×D5311yescharacteristic
457C22×C4D4×D5311yescharacteristic
458C22×C4C23×D5311yescharacteristic
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—1062no—
463C2×C8—1062no—
464C2×C8—1062no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—532no—
468C4⋊C4—534no—
469C4⋊C4—1064no—
470C4⋊C4—1061no—
471C4⋊C4—1062no—
472C4⋊C4C2×C4×D5312yes—
473C4⋊C4#1171994514yescomplemented
474C22⋊C4—24no—
475C22⋊C4—1064no—
476C22⋊C4—1068no—
477C22⋊C4—1064no—
478C22⋊C4—1064no—
479C42—42no—
480C42—534no—
481C42—534no—
482C42—532no—
483C42—532no—
484C42—1068no—
485C42—2124no—
486C42—2122no—
487C42C2×C4×D5312yes—
488C42D4×D5312yes—
489C23—22no—
490C23—42no—
491C23—42no—
492C23—41no—
493C23—41no—
494C23—532no—
495C23—1062no—
496C23—1062no—
497C23—1061no—
498C23—1061no—
499C23—1062no—
500C23—2122no—
501C23—2122no—
502C23—2121no—
503C23—2121no—
504C23C2×D4×D5311yescharacteristic
505Q8—2128no—
506D4—22no—
507D4—22no—
508D4—42no—
509D4—1062no—
510D4—1062no—
511D4—2128no—
512D4—2128no—
513D4—2128no—
514D4—2128no—
515D4—2122no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—48no—
524C2×C4—532no—
525C2×C4—534no—
526C2×C4—532no—
527C2×C4—532no—
528C2×C4—1068no—
529C2×C4—1062no—
530C2×C4—1061no—
531C2×C4—1061no—
532C2×C4—1062no—
533C2×C4—1062no—
534C2×C4—1068no—
535C2×C4—1064no—
536C2×C4—1062no—
537C2×C4—1064no—
538C2×C4—2122no—
539C2×C4—2121no—
540C2×C4—2122no—
541C2×C4—2121no—
542C2×C4—2128no—
543C2×C4C22⋊C4×D5312yes—
544C2×C4C22⋊C4×D5312yes—
545C2×C4#1313324112yes—
546C2×C4#1313324412yes—
547C2×C4C22×C4×D5311yescharacteristic
548C2×C4C2×D4×D5311yescharacteristic
549C2×C4#41×D5312yes—
550C2×C4C2×#1171994512yes—
551C8—42no—
552C8—2122no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—82no—
557C22—82no—
558C22—1062no—
559C22—1062no—
560C22—1062no—
561C22—4242no—
562C22—4242no—
563C22—2122no—
564C22—2122no—
565C22—2122no—
566C22—2122no—
567C22—2122no—
568C22C2×C22⋊C4×D5311yescharacteristic
569C22#1681113212yes—
570C22#135×D5312yes—
571C22C2×#1313324111yescharacteristic
572C22C2×#1313324411yescharacteristic
573C4—22no—
574C4—42no—
575C4—44no—
576C4—534no—
577C4—1062no—
578C4—1062no—
579C4—2122no—
580C4—2124no—
581C4#1681062614yes—
582C4#1681109111yescharacteristic
583C4C4×#1171994511yescharacteristic
584C4#1681117012yes—
585C2—82no—
586C2—4242no—
587C2—2122no—
588C2#2027720111yescharacteristic
589C2#416×D5311yescharacteristic
590C2C2×#1681062611yescharacteristic
591C2#2028477112yes—
592C2#2028477312yes—
subgroupG/H
centerC22×C4D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC4×#102×C53C2
socleC23×C53C22×D4
prime radicalC2×#16811170C2
cosocleC22×C4×C53C24
hypercenterC4×#102D53
hypocenterC53#21390
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. C2×#16810674C2semidirect
3. C4×#13133175C2semidirect
4. #20275661C2semidirect
5. #20275640C2semidirect
6. #20275036C2semidirect
7. #20275034C2semidirect
8. C4×#13133137C2semidirect
9. C4×#13133139C2semidirect
10. C4×#13133138C2semidirect
11. C4×#102×C53C2semidirect
14. #16811170C4semidirect
168. C4⋊C4×C53#41semidirect
180. C4⋊Dic53#41semidirect
299. C4×#102D53semidirect
342. C53#21390semidirect
473. C4⋊C4#11719945semidirect

Maximal subgroups

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

subgroupindex
C2×#168111702
C2×#168106742
C4×#131331752
#202756612
#202756402
#202750362
#202750342
C4×#131331372
C4×#131331392
C4×#131331382
C4×#102×C532
#2139053