Finite Group Databasebeta

← #27466152

Subgroups — #27466152

Order 13568.

854 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133191C211yesmaximal, characteristic
2#20276277C211yesmaximal, characteristic, complemented
3#20276272C211yesmaximal, characteristic, complemented
4#20276249C211yesmaximal, characteristic, complemented
5#20276244C211yesmaximal, characteristic, complemented
6#20275481C212yesmaximal, complemented
7#20275037C212yesmaximal
8#20274914C212yesmaximal, complemented
9#20274722C212yesmaximal, complemented
10#20274695C211yesmaximal, characteristic, complemented
11C53×#2285C211yesmaximal, characteristic, complemented
12C2×#13133191C2211yescharacteristic
13C2×C4×D212C2211yescharacteristic
14C2×C4×C22⋊D53—21no—
15C2×#13133150C2212yes—
16C2×#13133148C2212yes—
17C2×C4⋊#88322C2212yes—
18C2×#13133137C2212yes—
19C2×#13133137C2212yes—
20C2×C22⋊#88322C2212yes—
21#16810525C2211yescharacteristic
22#16810521C2211yescharacteristic
23#16810514—21no—
24#16810502—21no—
25#16810483C2211yescharacteristic
26C4×C4⋊Dic53C2211yescharacteristic
27C4×#11719921—21no—
28#16810385C2212yes—
29#16810385C2212yes—
30#16810385C2212yes—
31#16810385C2212yes—
32#16810378C2211yescharacteristic
33#16810378C2211yescharacteristic
34C2×#117×C53C2212yes—
35C22×C42×C53C2211yescharacteristic
36C53×#384C2211yescharacteristic
37C53×#384C2211yescharacteristic
38C53×#382C2212yes—
39C53×#380—21no—
40C53×#380—21no—
41C4×C4⋊C4×C53—21no—
42C4×C4⋊C4×C53—21no—
43C22×D212C2311yescharacteristic
44C22×C22⋊D53—21no—
45C22×C4×D53—21no—
46C22×#88322C2312yes—
47C22×#88322C2312yes—
48#13133191—22no—
49#13133191Q814yes—
50C4×D212—24no—
51C4×C22⋊D53—44no—
52C2×C4⋊Dic53C2311yescharacteristic
53C2×C4⋊Dic53C2311yescharacteristic
54C2×C4⋊Dic53C2311yescharacteristic
55C2×#11719919—21no—
56C2×C22⋊Dic53—21no—
57C2×#11719921—21no—
58C2×#11719921—21no—
59C2×#11719921—21no—
60C2×#11719921—21no—
61C2×C4×Dic53—21no—
62#13133150—42no—
63#13133148—42no—
64C4⋊#88322—24no—
65#13133137—42no—
66#13133137—42no—
67C22⋊#88322—24no—
68#13133124—21no—
69#13133124—21no—
70#13133124—21no—
71C23×C4×C53D411yescharacteristic
72C23×C4×C53D411yescharacteristic
73C23×C4×C53C2311yescharacteristic
74#117×C53—42no—
75#117×C53—42no—
76C2×C4⋊C4×C53—21no—
77C2×C4⋊C4×C53—21no—
78C2×C4⋊C4×C53—21no—
79C2×C4⋊C4×C53—21no—
80C2×C4⋊C4×C53C2312yes—
81C2×C4⋊C4×C53C2312yes—
82C2×C22⋊C4×C53—22no—
83C2×C42×C53—24no—
84C2×C42×C53—22no—
85C2×C42×C53D411yescharacteristic
86C2×C42×C53D411yescharacteristic
87C2×C42×C53C2311yescharacteristic
88C2×C42×C53C2311yescharacteristic
89#95×C53—22no—
90#95×C53—22no—
91C24×C53C2×D411yescharacteristic
92C23×D53—21no—
93C22×C4×C53—22no—
94C22×C4×C53—22no—
95C22×C4×C53—22no—
96C22×C4×C53—22no—
97C22×C4×C53—22no—
98C22×C4×C53—22no—
99C22×C4×C53—22no—
100C22×C4×C53—22no—
101C22×C4×C53—41no—
102C22×C4×C53—41no—
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×Q812yes—
109C22×C4×C53#4112yes—
110C22×C4×C53C2411yescharacteristic
111C2×C22⋊D53—24no—
112C2×C22⋊D53—42no—
113C2×C22⋊D53—42no—
114C22×Dic53—21no—
115C22×Dic53—21no—
116C22×Dic53—21no—
117C2×D212—22no—
118C2×D212—22no—
119C2×D212—21no—
120C2×D212C2×Q812yes—
121C2×C4×D53—42no—
122C2×C4×D53—42no—
123C2×C4×D53—42no—
124C4⋊C4×C53—22no—
125C4⋊C4×C53—22no—
126C4⋊C4×C53—22no—
127C4⋊C4×C53—22no—
128C4⋊C4×C53—42no—
129C4⋊C4×C53—42no—
130C4⋊C4×C53—42no—
131C4⋊C4×C53—42no—
132C22⋊C4×C53—44no—
133C42×C53—22no—
134C42×C53—22no—
135C42×C53—22no—
136C42×C53—44no—
137C42×C53#4114yes—
138C22⋊Dic53—42no—
139#11719921—24no—
140#11719921—42no—
141#11719921—42no—
142#11719921—42no—
143C4⋊Dic53—21no—
144C4⋊Dic53—22no—
145C4⋊Dic53—22no—
146C4⋊Dic53C2×Q812yes—
147#11719919—42no—
148C4×Dic53—42no—
149C2×#88322—22no—
150C2×#88322—22no—
151C2×#88322—22no—
152C2×#88322—22no—
153C2×#88322—22no—
154C2×#88322—22no—
155C23×C53—21no—
156C23×C53—21no—
157C23×C53—21no—
158C23×C53—21no—
159C23×C53—21no—
160C23×C53—21no—
161C23×C53C22⋊Q812yes—
162C23×C53C22×D411yescharacteristic
163C22×D53—22no—
164C22×D53—41no—
165C22×D53—41no—
166C22×D53—41no—
167C22×D53—41no—
168C22×D53—41no—
169C22×D53—41no—
170C2×C4×C53—21no—
171C2×C4×C53—21no—
172C2×C4×C53—21no—
173C2×C4×C53—21no—
174C2×C4×C53—21no—
175C2×C4×C53—21no—
176C2×C4×C53—21no—
177C2×C4×C53—21no—
178C2×C4×C53—21no—
179C2×C4×C53—21no—
180C2×C4×C53—24no—
181C2×C4×C53—24no—
182C2×C4×C53—44no—
183C2×C4×C53—44no—
184C2×C4×C53—42no—
185C2×C4×C53—42no—
186C2×C4×C53—42no—
187C2×C4×C53—42no—
188C2×C4×C53—42no—
189C2×C4×C53—42no—
190C2×C4×C53C22⋊Q812yes—
191C2×C4×C53#12312yes—
192C2×C4×C53#12312yes—
193C2×C4×C53#13512yes—
194C2×C4×C53AGL(1,ℤ/8)12yes—
195C2×C4×C53C22×Q811yescharacteristic
196C2×C4×C53C2×#4112yes—
197C2×C4×C53C2×#4111yescharacteristic
198C2×C4×C53H(5,𝔽2)12yes—
199C22⋊D53—48no—
200C2×Dic53—22no—
201C2×Dic53—41no—
202C2×Dic53—41no—
203C2×Dic53—41no—
204C2×Dic53—41no—
205C2×Dic53—41no—
206C2×Dic53—41no—
207C2×Dic53—41no—
208C2×Dic53—41no—
209D212—24no—
210D212—42no—
211C4×D53—84no—
212#88322—42no—
213#88322—42no—
214#27683—531nomaximal, Sylow
215C22×C53—22no—
216C22×C53—22no—
217C22×C53—22no—
218C22×C53—41no—
219C22×C53—41no—
220C22×C53—41no—
221C22×C53—41no—
222C22×C53C2×C22⋊Q811yescharacteristic
223C22×C53C2×#12311yescharacteristic
224C22×C53#53511yescharacteristic
225C22×C53C2×#13511yescharacteristic
226C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
227C22×C53#57511yescharacteristic
228C22×C53#57511yescharacteristic
229C2×D53—42no—
230C2×D53—42no—
231C2×D53—42no—
232C2×D53—81no—
233C2×D53—81no—
234C2×D53—81no—
235C2×D53—81no—
236C4×C53—22no—
237C4×C53—22no—
238C4×C53—22no—
239C4×C53—22no—
240C4×C53—22no—
241C4×C53—82no—
242C4×C53—82no—
243C4×C53#55312yes—
244C4×C53#55412yes—
245Dic53—42no—
246Dic53—81no—
247Dic53—81no—
248C4×#121—531no—
249#2285D5311yescharacteristic, complemented
250#2100—531no—
251#2099—531no—
252#2094—531no—
253#2093—531no—
254#1949—532no—
255#1875—532no—
256#1854—532no—
257#1804—532no—
258#1797—531no—
259C2×C53—42no—
260C2×C53#248711yescharacteristic
261C2×C53#307611yescharacteristic
262C2×C53#307711yescharacteristic
263C2×C53#309111yescharacteristic
264C2×C53#309411yescharacteristic
265C2×C53#320711yescharacteristic
266C2×C53#321911yescharacteristic
267D53—82no—
268C2×#121—531no—
269C2×C4×D4—531no—
270C2×C4×D4—1061no—
271C2×#117C2×D5312yes—
272C22×C42C2×D5311yescharacteristic
273C2×#107—532no—
274C2×#106—532no—
275C2×C4⋊C8—532no—
276C2×#102—532no—
277C2×#102—532no—
278C2×C22⋊C8—532no—
279#390—531no—
280#389—531no—
281#388—1061no—
282#386—1061no—
283#384C2×D5311yescharacteristic
284#384C2×D5311yescharacteristic
285#382C2×D5312yes—
286#380—21no—
287#380—21no—
288#380—531no—
289C4×C4⋊C4—21no—
290C4×C4⋊C4—21no—
291C4×C4⋊C4—531no—
292C4×C22⋊C4—1061no—
293#340—532no—
294#340—532no—
295#340—532no—
296#340—532no—
297#336—531no—
298#336—531no—
299C53#2768311yescharacteristic, complemented, Sylow
300C22×D4—531no—
301C22×D4—1061no—
302C23×C4—1061no—
303C23×C4C22⋊D5311yescharacteristic
304C23×C4C22⋊D5311yescharacteristic
305C23×C4C22×D5311yescharacteristic
306C22×C8—532no—
307C22×C8—532no—
308#121—534no—
309#121—1062no—
310C4×D4—1064no—
311C4×D4—2124no—
312#117—42no—
313#117—42no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—531no—
319C2×C4⋊C4—531no—
320C2×C4⋊C4—531no—
321C2×C4⋊C4—1061no—
322C2×C4⋊C4C22×D5312yes—
323C2×C4⋊C4C22×D5312yes—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—1061no—
326C2×C22⋊C4—1061no—
327C2×C22⋊C4—1061no—
328C2×C22⋊C4—1061no—
329C2×C22⋊C4—1061no—
330C2×C42—24no—
331C2×C42—22no—
332C2×C42—1061no—
333C2×C42C22⋊D5311yescharacteristic
334C2×C42C22⋊D5311yescharacteristic
335C2×C42C22×D5311yescharacteristic
336C2×C42C22×D5311yescharacteristic
337#107—2122no—
338#106—2122no—
339C4⋊C8—1064no—
340#102—2122no—
341#102—2122no—
342C22⋊C8—1064no—
343#95—22no—
344#95—22no—
345#95—1061no—
346#95—1061no—
347#95—1061no—
348C24—1061no—
349C24C2×C22⋊D5311yescharacteristic
350C2×D4—532no—
351C2×D4—1064no—
352C2×D4—1062no—
353C2×D4—1061no—
354C2×D4—1062no—
355C2×D4—2122no—
356C2×D4—2122no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—1061no—
368C22×C4—1061no—
369C22×C4—1061no—
370C22×C4—2122no—
371C22×C4—2122no—
372C22×C4—2122no—
373C22×C4Q8×D5312yes—
374C22×C4#1171994912yes—
375C22×C4C2×C22⋊D5311yescharacteristic
376C22×C4C2×C22⋊D5311yescharacteristic
377C22×C4C2×C22⋊D5311yescharacteristic
378C22×C4C2×C22⋊D5311yescharacteristic
379C22×C4C2×C22⋊D5311yescharacteristic
380C22×C4C23×D5311yescharacteristic
381C2×C8—1062no—
382C2×C8—1062no—
383C2×C8—1062no—
384C2×C8—1062no—
385C2×C8—1062no—
386C2×C8—1062no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—532no—
396C4⋊C4—1061no—
397C4⋊C4—1062no—
398C4⋊C4—1062no—
399C4⋊C4—2122no—
400C22⋊C4—44no—
401C22⋊C4—1064no—
402C22⋊C4—2122no—
403C22⋊C4—2122no—
404C22⋊C4—2122no—
405C22⋊C4—2122no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—44no—
410C42—2122no—
411C42#1171994514yes—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—1062no—
419C23—2121no—
420C23—2121no—
421C23—2121no—
422C23—2121no—
423C23—2121no—
424C23—2121no—
425C23#1313319312yes—
426C23C22×C22⋊D5311yescharacteristic
427D4—1064no—
428D4—2128no—
429D4—2122no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—1062no—
451C2×C4—4244no—
452C2×C4—2121no—
453C2×C4—2121no—
454C2×C4—2121no—
455C2×C4—2121no—
456C2×C4—2121no—
457C2×C4—2121no—
458C2×C4—2121no—
459C2×C4—2121no—
460C2×C4#1313319312yes—
461C2×C4#1313320012yes—
462C2×C4#1313320012yes—
463C2×C4#1313323712yes—
464C2×C4#1313324212yes—
465C2×C4C2×Q8×D5311yescharacteristic
466C2×C4C2×#1171994512yes—
467C2×C4C2×#1171994911yescharacteristic
468C2×C4#1313326212yes—
469C8—2122no—
470C8—2122no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—4241no—
479C22—4241no—
480C22—4241no—
481C22—4241no—
482C22—2122no—
483C22—2122no—
484C22—2122no—
485C22C2×#1313319311yescharacteristic
486C22C2×#1313320011yescharacteristic
487C22#1681119011yescharacteristic
488C22C2×#1313323711yescharacteristic
489C22C2×#1313324211yescharacteristic
490C22#1681142111yescharacteristic
491C22#1681142111yescharacteristic
492C4—22no—
493C4—22no—
494C4—22no—
495C4—22no—
496C4—22no—
497C4—82no—
498C4—82no—
499C4—4241no—
500C4—4241no—
501C4—2122no—
502C4#1681132112yes—
503C4#1681132812yes—
504C2—42no—
505C2—4242no—
506C2#2027902611yescharacteristic
507C2#2028449711yescharacteristic
508C2#2028450811yescharacteristic
509C2#2028462811yescharacteristic
510C2#2028465511yescharacteristic
511C2#2028564811yescharacteristic
512C2#2028575211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2285C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#2285D53
hypocenterC53#27683
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2285

Complements

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

NH ≅ G/Ntype
2. #20276277C2semidirect
3. #20276272C2semidirect
4. #20276249C2semidirect
5. #20276244C2semidirect
6. #20275481C2semidirect
8. #20274914C2semidirect
9. #20274722C2semidirect
10. #20274695C2semidirect
11. C53×#2285C2semidirect
249. #2285D53semidirect
299. C53#27683semidirect

Maximal subgroups

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

subgroupindex
C4×#131331912
#202762772
#202762722
#202762492
#202762442
#202754812
#202750372
#202749142
#202747222
#202746952
C53×#22852
#2768353