Finite Group Databasebeta

← #27368094

Subgroups — #27368094

Order 13568.

712 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285270C211yesmaximal, characteristic, complemented
2#20284512C211yesmaximal, characteristic, complemented
3C2×#16810701C211yesmaximal, characteristic, complemented
4#20274264C212yesmaximal, complemented
5#20274246C212yesmaximal, complemented
6#20274212C212yesmaximal, complemented
7#20274178C212yesmaximal, complemented
8#20273473C211yesmaximal, characteristic, complemented
9#20273406C211yesmaximal, characteristic, complemented
10#20273392C211yesmaximal, characteristic, complemented
11C53×#3036C211yesmaximal, characteristic
12C22×#11719917—21no—
13#16811180C2211yescharacteristic
14#16811019C2211yescharacteristic, complemented
15#16810995C2211yescharacteristic, complemented
16#16810975C2211yescharacteristic, complemented
17#16810969C2211yescharacteristic, complemented
18#16810923—22no—
19#16810913C2212yes—
20#16810903C2212yes—
21#16810850—21no—
22#16810850—21no—
23C4×#11719923—21no—
24C4×#11719923—21no—
25#16810701—21no—
26#16810701—21no—
27#16810701C2212yes—
28#16810701C2212yes—
29#16810663—21no—
30#16810647C2211yescharacteristic
31C2×C4⋊#88322C2211yescharacteristic
32#16810628—21no—
33C2×C22⋊#88322—21no—
34C2×C4×#88322C2211yescharacteristic
35#16810375C2212yescomplemented
36#16810375C2212yescomplemented
37#16810371C2212yescomplemented
38#16810369C2212yescomplemented
39#16810368C2212yescomplemented
40#16810367C2212yescomplemented
41C2×AGL(1,ℤ/8)×C53—21no—
42C53×#532C2211yescharacteristic
43C2×C4×D4×C53C2211yescharacteristic
44C53×#463—22no—
45C53×#461C2212yes—
46C53×#459C2212yes—
47C53×#423C2211yescharacteristic
48C53×#417—21no—
49C2×#11719923—21no—
50C2×#11719923—21no—
51C2×#11719917—22no—
52C2×#11719917—22no—
53C2×#11719917—21no—
54C2×#11719917—44no—
55C22×#88322—21no—
56C22×#88322D411yescharacteristic
57C22×#88322D411yescharacteristic
58#13133217C2311yescharacteristic
59#13133217C2311yescharacteristic
60#13133209—21no—
61#13133196—21no—
62#13133196—21no—
63C4×Dic106—21no—
64C4×Dic106—21no—
65#13133176—21no—
66#13133150—22no—
67#13133148—22no—
68#13133148—21no—
69#13133148—21no—
70C4⋊#88322—21no—
71C4⋊#88322—21no—
72C4⋊#88322C2312yes—
73C4⋊#88322C2312yes—
74#13133140—21no—
75#13133140—21no—
76#13133138—21no—
77#13133138—21no—
78#13133138—22no—
79#13133138—22no—
80C22⋊#88322—22no—
81C22⋊#88322—22no—
82C4×#88322D411yescharacteristic
83C4×#88322D411yescharacteristic
84C4×#88322C2311yescharacteristic
85C4×#88322C2311yescharacteristic
86C2×#41×C53—21no—
87C22×D4×C53D411yescharacteristic
88C23×C4×C53—21no—
89AGL(1,ℤ/8)×C53—44no—
90C2×SD16×C53—22no—
91C2×D8×C53—22no—
92C2×M16×C53—21no—
93C4⋊Q8×C53C2311yescharacteristic
94#127×C53C2311yescharacteristic
95#124×C53—21no—
96#121×C53—21no—
97#121×C53—21no—
98C4×D4×C53—21no—
99C4×D4×C53—21no—
100C4×D4×C53—21no—
101C4×D4×C53—21no—
102C4×D4×C53D412yes—
103C4×D4×C53C2312yes—
104C2×C4⋊C4×C53D411yescharacteristic
105C2×C22⋊C4×C53—21no—
106C2×C42×C53C2311yescharacteristic
107C4⋊C8×C53—21no—
108C4⋊C8×C53C2312yes—
109#103×C53—22no—
110#102×C53—22no—
111C24×C53—21no—
112#41×C53—42no—
113#41×C53—42no—
114C2×Q8×C53—21no—
115C2×D4×C53—21no—
116C2×D4×C53—22no—
117C2×D4×C53—22no—
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—41no—
122C2×D4×C53C2×D412yes—
123C22×C4×C53—21no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—42no—
127C22×C4×C53—42no—
128C22×C4×C53—41no—
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53#4111yescharacteristic
132C2×Dic106—21no—
133C2×Dic106—21no—
134SD16×C53—44no—
135D8×C53—44no—
136M16×C53—42no—
137C2×C8×C53—22no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53C2×D412yes—
142C22⋊C4×C53—22no—
143C22⋊C4×C53—22no—
144C22⋊C4×C53—41no—
145C22⋊C4×C53—41no—
146C42×C53C2×D411yescharacteristic
147C42×C53C2×D411yescharacteristic
148C42×C53#4111yescharacteristic
149C42×C53C2411yescharacteristic
150C22⋊Dic53—41no—
151#11719923—42no—
152#11719923—42no—
153C4⋊Dic53—21no—
154C4⋊Dic53—21no—
155C4⋊Dic53—21no—
156C4⋊Dic53—21no—
157#11719919—41no—
158#11719919—41no—
159C4×Dic53—41no—
160#11719917—44no—
161#11719917—42no—
162#11719917—44no—
163C2×#88322—22no—
164C2×#88322—22no—
165C2×#88322—22no—
166C2×#88322—41no—
167C2×#88322—42no—
168C2×#88322C2×D411yescharacteristic
169C2×#88322C2×D411yescharacteristic
170C2×#88322C2×D411yescharacteristic
171C2×#88322C2×D411yescharacteristic
172C23×C53—22no—
173C23×C53—42no—
174C23×C53—42no—
175C23×C53—42no—
176C23×C53—41no—
177C23×C53#12111yescharacteristic
178Q8×C53—41no—
179Q8×C53—41no—
180D4×C53—22no—
181D4×C53—22no—
182D4×C53—41no—
183D4×C53—41no—
184D4×C53—41no—
185D4×C53—41no—
186D4×C53—42no—
187D4×C53—81no—
188D4×C53—82no—
189C2×C4×C53—21no—
190C2×C4×C53—21no—
191C2×C4×C53—22no—
192C2×C4×C53—41no—
193C2×C4×C53—44no—
194C2×C4×C53—42no—
195C2×C4×C53—42no—
196C2×C4×C53—42no—
197C2×C4×C53—41no—
198C2×C4×C53—82no—
199C2×C4×C53#12111yescharacteristic
200C2×C4×C53#12111yescharacteristic
201C2×C4×C53#12111yescharacteristic
202C2×C4×C53AGL(1,ℤ/8)12yes—
203C2×C4×C53#13712yes—
204C2×C4×C53C22×D411yescharacteristic
205C2×C4×C53C22×D411yescharacteristic
206C2×C4×C53C2×#4111yescharacteristic
207C2×Dic53—41no—
208C2×Dic53—41no—
209Dic106—41no—
210Dic106—41no—
211Dic106—81no—
212C8×C53—42no—
213#88322—21no—
214#88322—21no—
215#88322—21no—
216#88322—42no—
217#88322—42no—
218#88322AGL(1,ℤ/8)12yescomplemented
219#19279—531nomaximal, Sylow
220C22×C53—21no—
221C22×C53—21no—
222C22×C53—42no—
223C22×C53—42no—
224C22×C53—42no—
225C22×C53—84no—
226C22×C53—81no—
227C22×C53—81no—
228C22×C53—81no—
229C22×C53#46812yes—
230C22×C53C2×#12111yescharacteristic
231C4×C53—21no—
232C4×C53—21no—
233C4×C53—21no—
234C4×C53—42no—
235C4×C53—81no—
236C4×C53#46812yes—
237C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
238C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
239C4×C53C2×#13711yescharacteristic
240C4×C53#57511yescharacteristic
241Dic53—81no—
242Dic53—81no—
243#3167—531no—
244#3077—531no—
245#3036D5311yescharacteristic
246C2×#436—531no—
247#1699—532no—
248#1697—532no—
249#1691—532no—
250#1685—532no—
251#1567—531no—
252#1550—531no—
253#1547—531no—
254C2×C53—22no—
255C2×C53—82no—
256C2×C53—161no—
257C2×C53#303611yescharacteristic
258C2×C53C2×#46811yescharacteristic
259C2×C53#306711yescharacteristic
260C2×AGL(1,ℤ/8)—21no—
261C22×M16—1061no—
262#533—531no—
263#532C2×D5311yescharacteristic
264C2×C4×D4C2×D5311yescharacteristic
265#498—531no—
266#492—531no—
267#487—531no—
268#486—531no—
269#478—1062no—
270#476—532no—
271#474—532no—
272#465—1061no—
273#465—1061no—
274#463—22no—
275#461C2×D5312yes—
276#459C2×D5312yes—
277C4×SD16—1061no—
278C4×SD16—1061no—
279#436—532no—
280#436—532no—
281#436—1061no—
282#436—1061no—
283#428—1061no—
284#423—531no—
285#423C2×D5311yescharacteristic
286C2×C4⋊C8—531no—
287#418—1061no—
288#417—21no—
289C2×C22⋊C8—1061no—
290C2×C4×C8—531no—
291#335—532no—
292#335—532no—
293#333—532no—
294#332—532no—
295#332—532no—
296#331—532no—
297C53#1927911yescharacteristic, complemented, Sylow
298C2×#41—21no—
299C22×D4C22⋊D5311yescharacteristic
300C23×C4—21no—
301AGL(1,ℤ/8)—44no—
302C2×SD16—22no—
303C2×SD16—1061no—
304C2×SD16—1061no—
305C2×D8—22no—
306C2×M16—21no—
307C2×M16—1062no—
308C2×M16—1062no—
309C2×M16—1061no—
310C2×M16—2124no—
311C22×C8—531no—
312C22×C8—531no—
313C22×C8—1061no—
314C4⋊Q8—531no—
315C4⋊Q8—531no—
316C4⋊Q8C22×D5311yescharacteristic
317#127C22×D5311yescharacteristic
318#125—1061no—
319#124—21no—
320C22⋊Q8—1061no—
321C22⋊Q8—1061no—
322#121—21no—
323#121—21no—
324C4×Q8—1061no—
325C4×Q8—1061no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4C22⋊D5312yes—
331C4×D4C22×D5312yes—
332#117—1061no—
333C2×C4⋊C4C22⋊D5311yescharacteristic
334C2×C22⋊C4—21no—
335C2×C42C22×D5311yescharacteristic
336#107—1062no—
337#106—1061no—
338#106—1061no—
339#106—1062no—
340C4⋊C8—21no—
341C4⋊C8—532no—
342C4⋊C8—532no—
343C4⋊C8—1061no—
344C4⋊C8—1061no—
345C4⋊C8C22×D5312yes—
346#103—22no—
347#103—1061no—
348#103—1061no—
349#102—22no—
350#102—1061no—
351#102—1061no—
352#102—1062no—
353#102—1062no—
354C22⋊C8—1062no—
355C22⋊C8—1062no—
356C4×C8—531no—
357C4×C8—531no—
358C4×C8—531no—
359C4×C8—531no—
360C24—21no—
361#41—42no—
362#41—42no—
363C2×Q8—21no—
364C2×Q8—1061no—
365C2×Q8—1061no—
366C2×D4—21no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—41no—
373C2×D4C2×C22⋊D5312yes—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4D4×D5311yescharacteristic
381C22×C4#1171994711yescharacteristic
382C22×C4C2×C22⋊D5311yescharacteristic
383SD16—44no—
384SD16—2122no—
385SD16—2122no—
386D8—44no—
387M16—42no—
388M16—2124no—
389M16—2124no—
390M16—2122no—
391C2×C8—22no—
392C2×C8—531no—
393C2×C8—531no—
394C2×C8—531no—
395C2×C8—531no—
396C2×C8—1062no—
397C2×C8—1062no—
398C2×C8—1062no—
399C2×C8—2121no—
400C2×C8—2122no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—1061no—
405C4⋊C4—1061no—
406C4⋊C4—1061no—
407C4⋊C4—1061no—
408C4⋊C4—2121no—
409C4⋊C4—2121no—
410C4⋊C4C2×C22⋊D5312yes—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—2121no—
416C42—2121no—
417C42D4×D5311yescharacteristic
418C42#1171994711yescharacteristic
419C42C2×C22⋊D5311yescharacteristic
420C42C23×D5311yescharacteristic
421C23—22no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—41no—
426C23#1313318911yescharacteristic
427Q8—41no—
428Q8—41no—
429Q8—4241no—
430Q8—2121no—
431Q8—2121no—
432D4—22no—
433D4—22no—
434D4—42no—
435D4—41no—
436D4—41no—
437D4—41no—
438D4—41no—
439D4—81no—
440D4—82no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—41no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—82no—
451C2×C4—2121no—
452C2×C4—2121no—
453C2×C4#1313318911yescharacteristic
454C2×C4#1313318911yescharacteristic
455C2×C4#1313318911yescharacteristic
456C2×C4#1313324312yes—
457C2×C4#1313324712yes—
458C2×C4C2×D4×D5311yescharacteristic
459C2×C4C22×C22⋊D5311yescharacteristic
460C2×C4C2×#1171994711yescharacteristic
461C8—42no—
462C8—532no—
463C8—1061no—
464C8—1061no—
465C8—1061no—
466C8—2122no—
467C8—2122no—
468C22—21no—
469C22—21no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—84no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22#1681086712yes—
478C22C2×#1313318911yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—42no—
483C4—81no—
484C4—4241no—
485C4—4241no—
486C4#1681086712yes—
487C4AGL(1,ℤ/8)×D5311yescharacteristic
488C4C2×#1313324311yescharacteristic
489C4C2×#1313324711yescharacteristic
490C4#1681141911yescharacteristic
491C2—22no—
492C2—82no—
493C2—161no—
494C2#2028417611yescharacteristic
495C2C2×#1681086711yescharacteristic
496C2#2028443211yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3036C2
socleC22×C53C2×#121
hypercenter#3036D53
hypocenterC53#19279
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
1. #20285270C2semidirect
2. #20284512C2semidirect
3. C2×#16810701C2semidirect
4. #20274264C2semidirect
5. #20274246C2semidirect
6. #20274212C2semidirect
7. #20274178C2semidirect
8. #20273473C2semidirect
9. #20273406C2semidirect
10. #20273392C2semidirect
14. #16811019C22semidirect
15. #16810995C22semidirect
16. #16810975C22semidirect
17. #16810969C22semidirect
35. #16810375C22semidirect
36. #16810375C22semidirect
37. #16810371C22semidirect
38. #16810369C22semidirect
39. #16810368C22semidirect
40. #16810367C22semidirect
218. #88322AGL(1,ℤ/8)semidirect
297. C53#19279semidirect

Maximal subgroups

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

subgroupindex
#202852702
#202845122
C2×#168107012
#202742642
#202742462
#202742122
#202741782
#202734732
#202734062
#202733922
C53×#30362
#1927953