Finite Group Databasebeta

← #27276412

Subgroups — #27276412

Order 13568.

686 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#20275709C211yesmaximal, characteristic, complemented
2#20273437C211yesmaximal, characteristic, complemented
3#20273322C211yesmaximal, characteristic, complemented
4#20273180C211yesmaximal, characteristic, complemented
5#20273132C211yesmaximal, characteristic, complemented
6#20273071C211yesmaximal, characteristic, complemented
7C53×#3559C211yesmaximal, characteristic
8#16810680C2211yescharacteristic
9#16810676C2211yescharacteristic
10#16810647C2211yescharacteristic
11C4×#11719917C2211yescharacteristic
12#16810392—21no—
13#16810392—21no—
14#16810369—21no—
15#16810367—21no—
16#16810365—21no—
17#16810363—21no—
18Q8⋊#88322—21no—
19D4⋊#88322—21no—
20D4⋊#88322—22no—
21#16810349—22no—
22C53×#563—21no—
23C53×#562—21no—
24C53×#554—21no—
25C53×#553—21no—
26C53×#553—21no—
27C53×#547—22no—
28C53×#547—21no—
29C53×#544—21no—
30C53×#540—21no—
31C53×#539—22no—
32C53×#539C412yes—
33C53×#537C412yes—
34C53×#532C2211yescharacteristic
35C53×#529C2211yescharacteristic
36C2×#125×C53C2211yescharacteristic
37C2×#11719917—21no—
38C2×#11719917—21no—
39C4⋊#88322—21no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C22⋊#88322—21no—
44C22⋊#88322—21no—
45#13133128—21no—
46#13133128—21no—
47C4×#88322—21no—
48C4×#88322—21no—
49C2×#41×C53—22no—
50C2×#41×C53—21no—
51C4⋊Q8×C53—21no—
52C4⋊Q8×C53—22no—
53C4⋊Q8×C53D411yescharacteristic
54#127×C53D411yescharacteristic
55#126×C53—22no—
56#126×C53—22no—
57#126×C53—21no—
58#125×C53—21no—
59#125×C53—22no—
60#125×C53—21no—
61#125×C53—21no—
62#125×C53—21no—
63#125×C53—21no—
64#125×C53D411yescharacteristic
65#124×C53—21no—
66#124×C53D411yescharacteristic
67#123×C53—22no—
68#123×C53—22no—
69#123×C53—22no—
70#123×C53—22no—
71#123×C53—22no—
72#123×C53—22no—
73#123×C53—21no—
74C22⋊Q8×C53—22no—
75C22⋊Q8×C53—22no—
76C22⋊Q8×C53—22no—
77C22⋊Q8×C53—22no—
78C22⋊Q8×C53—22no—
79C22⋊Q8×C53—22no—
80C22⋊Q8×C53C2×C411yescharacteristic
81#121×C53—21no—
82#121×C53—22no—
83#121×C53—21no—
84#121×C53—22no—
85#121×C53—22no—
86#121×C53—22no—
87#121×C53—21no—
88#121×C53C2×C411yescharacteristic
89#121×C53C2×C412yes—
90C4×Q8×C53D411yescharacteristic
91C4×D4×C53—22no—
92C4×D4×C53—21no—
93C4×D4×C53—21no—
94C4×D4×C53—21no—
95C4×D4×C53—22no—
96C4×D4×C53—21no—
97C4×D4×C53D411yescharacteristic
98C4×D4×C53D412yes—
99#117×C53—21no—
100#117×C53—21no—
101C2×C4⋊C4×C53—22no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53C2×C412yes—
104C2×C4⋊C4×C53D412yes—
105C2×C4⋊C4×C53D411yescharacteristic
106C2×C4⋊C4×C53D411yescharacteristic
107C2×C42×C53C2311yescharacteristic
108#41×C53—82no—
109#41×C53—84no—
110C2×Q8×C53—22no—
111C2×Q8×C53—21no—
112C2×Q8×C53C22⋊C411yescharacteristic
113C2×D4×C53—22no—
114C2×D4×C53—21no—
115C2×D4×C53—22no—
116C2×D4×C53—22no—
117C2×D4×C53—21no—
118C2×D4×C53—21no—
119C2×D4×C53C22⋊C411yescharacteristic
120C2×D4×C53C22⋊C412yes—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—21no—
125C22×C4×C53—22no—
126C22×C4×C53C22⋊C412yes—
127C22×C4×C53C22⋊C412yes—
128C22×C4×C53C22×C411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C4⋊C4×C53—22no—
132C4⋊C4×C53—22no—
133C4⋊C4×C53—22no—
134C4⋊C4×C53—22no—
135C4⋊C4×C53—22no—
136C4⋊C4×C53—22no—
137C4⋊C4×C53—21no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—22no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—22no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53C22⋊C412yes—
149C4⋊C4×C53C22⋊C411yescharacteristic
150C4⋊C4×C53C22⋊C411yescharacteristic
151C22⋊C4×C53—22no—
152C22⋊C4×C53—22no—
153C22⋊C4×C53—22no—
154C22⋊C4×C53—22no—
155C22⋊C4×C53—22no—
156C22⋊C4×C53—21no—
157C22⋊C4×C53—21no—
158C22⋊C4×C53—22no—
159C22⋊C4×C53—21no—
160C22⋊C4×C53—21no—
161C22⋊C4×C53—21no—
162C22⋊C4×C53—22no—
163C42×C53—22no—
164C42×C53—21no—
165C42×C53C2×D411yescharacteristic
166C42×C53C2×D411yescharacteristic
167C42×C53C2×D411yescharacteristic
168C42×C53C2×D411yescharacteristic
169#11719917—22no—
170#11719917—41no—
171#11719917—42no—
172C2×#88322—41no—
173C2×#88322—41no—
174C2×#88322—41no—
175C2×#88322—41no—
176C23×C53—21no—
177C23×C53—22no—
178C23×C53C2×C22⋊C411yescharacteristic
179Q8×C53—41no—
180Q8×C53—81no—
181Q8×C53—82no—
182D4×C53—42no—
183D4×C53—41no—
184D4×C53—44no—
185D4×C53—42no—
186D4×C53—41no—
187D4×C53—41no—
188D4×C53—81no—
189D4×C53—82no—
190D4×C53—82no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—22no—
194C2×C4×C53—21no—
195C2×C4×C53—22no—
196C2×C4×C53—21no—
197C2×C4×C53—22no—
198C2×C4×C53—21no—
199C2×C4×C53—41no—
200C2×C4×C53—41no—
201C2×C4×C53—41no—
202C2×C4×C53—41no—
203C2×C4×C53—42no—
204C2×C4×C53—42no—
205C2×C4×C53—82no—
206C2×C4×C53—82no—
207C2×C4×C53—82no—
208C2×C4×C53—82no—
209C2×C4×C53C2×C22⋊C411yescharacteristic
210C2×C4×C53C2×C22⋊C411yescharacteristic
211C2×C4×C53#12011yescharacteristic
212C2×C4×C53#12011yescharacteristic
213C2×C4×C53#12011yescharacteristic
214C2×C4×C53#12011yescharacteristic
215#88322—42no—
216#88322—81no—
217#88322—81no—
218#88322—81no—
219#8486—531nomaximal, Sylow
220C22×C53—21no—
221C22×C53—21no—
222C22×C53—21no—
223C22×C53—41no—
224C22×C53—42no—
225C22×C53—42no—
226C22×C53—41no—
227C22×C53—42no—
228C22×C53—41no—
229C22×C53#37911yescharacteristic
230C4×C53—21no—
231C4×C53—21no—
232C4×C53—21no—
233C4×C53—21no—
234C4×C53—21no—
235C4×C53—41no—
236C4×C53—42no—
237C4×C53—42no—
238C4×C53—41no—
239C4×C53—42no—
240C4×C53—41no—
241C4×C53#45411yescharacteristic
242C4×C53#45611yescharacteristic
243#3559D5311yescharacteristic
244#1998—531no—
245#1557—531no—
246#1529—531no—
247#1495—531no—
248#1484—531no—
249#1470—531no—
250C2×C53—41no—
251C2×C53—81no—
252C2×C53—82no—
253C2×C53#179111yescharacteristic
254C2×C53#179311yescharacteristic
255C2×C53#179511yescharacteristic
256#563—21no—
257#562—21no—
258#554—21no—
259#553—21no—
260#553—21no—
261#547—22no—
262#547—21no—
263#544—21no—
264#540—21no—
265#539—22no—
266#539Dic5312yes—
267#537Dic5312yes—
268#532C2×D5311yescharacteristic
269#529C2×D5311yescharacteristic
270C2×#125C2×D5311yescharacteristic
271#432—531no—
272#431—531no—
273#423—531no—
274C4×M16—531no—
275#343—1061no—
276#343—1061no—
277#332—1061no—
278#331—1061no—
279#330—1061no—
280#329—1061no—
281Q8⋊C8—1061no—
282D4⋊C8—1062no—
283D4⋊C8—1061no—
284#324—1062no—
285C53#848611yescharacteristic, complemented, Sylow
286C2×#41—21no—
287C2×#41—22no—
288C2×M16—1061no—
289C2×M16—1061no—
290C4⋊Q8—21no—
291C4⋊Q8—22no—
292C4⋊Q8C22⋊D5311yescharacteristic
293#127C22⋊D5311yescharacteristic
294#126—22no—
295#126—22no—
296#126—21no—
297#125—21no—
298#125—22no—
299#125—21no—
300#125—21no—
301#125—21no—
302#125—21no—
303#125C22⋊D5311yescharacteristic
304#124—21no—
305#124C22⋊D5311yescharacteristic
306#123—22no—
307#123—22no—
308#123—22no—
309#123—22no—
310#123—22no—
311#123—21no—
312#123—22no—
313C22⋊Q8—22no—
314C22⋊Q8—22no—
315C22⋊Q8—22no—
316C22⋊Q8—22no—
317C22⋊Q8—22no—
318C22⋊Q8—22no—
319C22⋊Q8C2×Dic5311yescharacteristic
320#121—21no—
321#121—22no—
322#121—21no—
323#121—22no—
324#121—22no—
325#121—21no—
326#121—22no—
327#121C2×Dic5311yescharacteristic
328#121C2×Dic5312yes—
329C4×Q8C22⋊D5311yescharacteristic
330C4×D4—22no—
331C4×D4—21no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—22no—
335C4×D4—21no—
336C4×D4C22⋊D5311yescharacteristic
337C4×D4C22⋊D5312yes—
338#117—21no—
339#117—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4C2×Dic5312yes—
343C2×C4⋊C4C22⋊D5312yes—
344C2×C4⋊C4C22⋊D5311yescharacteristic
345C2×C4⋊C4C22⋊D5311yescharacteristic
346C2×C42C22×D5311yescharacteristic
347C4⋊C8—1061no—
348C4⋊C8—1061no—
349C4⋊C8—1061no—
350C4⋊C8—1061no—
351C22⋊C8—1061no—
352C22⋊C8—1061no—
353#97—1061no—
354#97—1061no—
355C4×C8—1061no—
356C4×C8—1061no—
357#41—82no—
358#41—84no—
359C2×Q8—22no—
360C2×Q8—21no—
361C2×Q8C22⋊Dic5311yescharacteristic
362C2×D4—21no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4C22⋊Dic5311yescharacteristic
369C2×D4C22⋊Dic5312yes—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4C22⋊Dic5312yes—
376C22×C4C22⋊Dic5312yes—
377C22×C4C22×Dic5311yescharacteristic
378C22×C4C2×C22⋊D5311yescharacteristic
379C22×C4C2×C22⋊D5311yescharacteristic
380M16—1062no—
381M16—2122no—
382M16—2121no—
383C2×C8—2121no—
384C2×C8—2121no—
385C2×C8—2121no—
386C2×C8—2121no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4C22⋊Dic5312yes—
405C4⋊C4C22⋊Dic5311yescharacteristic
406C4⋊C4C22⋊Dic5311yescharacteristic
407C22⋊C4—22no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—21no—
413C22⋊C4—21no—
414C22⋊C4—22no—
415C22⋊C4—21no—
416C22⋊C4—21no—
417C22⋊C4—21no—
418C22⋊C4—22no—
419C42—22no—
420C42—21no—
421C42C2×C22⋊D5311yescharacteristic
422C42C2×C22⋊D5311yescharacteristic
423C42C2×C22⋊D5311yescharacteristic
424C42C2×C22⋊D5311yescharacteristic
425C23—22no—
426C23—21no—
427C23C2×C22⋊Dic5311yescharacteristic
428Q8—41no—
429Q8—81no—
430Q8—82no—
431D4—42no—
432D4—41no—
433D4—44no—
434D4—42no—
435D4—41no—
436D4—41no—
437D4—81no—
438D4—82no—
439D4—82no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4C2×C22⋊Dic5311yescharacteristic
459C2×C4C2×C22⋊Dic5311yescharacteristic
460C2×C4#1313318611yescharacteristic
461C2×C4#1313318611yescharacteristic
462C2×C4#1313318611yescharacteristic
463C2×C4#1313318611yescharacteristic
464C8—4241no—
465C8—4241no—
466C8—4241no—
467C8—2122no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—42no—
472C22—41no—
473C22—41no—
474C22—42no—
475C22—42no—
476C22—41no—
477C22#1681047911yescharacteristic
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—42no—
484C4—41no—
485C4—41no—
486C4—42no—
487C4—42no—
488C4—41no—
489C4#1681079611yescharacteristic
490C4#1681080611yescharacteristic
491C2—41no—
492C2—82no—
493C2—81no—
494C2#2027466611yescharacteristic
495C2#2027467411yescharacteristic
496C2#2027468511yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3559C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenter#3559D53
hypocenterC53#8486
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3559

Complements

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

NH ≅ G/Ntype
1. #20275709C2semidirect
2. #20273437C2semidirect
3. #20273322C2semidirect
4. #20273180C2semidirect
5. #20273132C2semidirect
6. #20273071C2semidirect
285. C53#8486semidirect

Maximal subgroups

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

subgroupindex
#202757092
#202734372
#202733222
#202731802
#202731322
#202730712
C53×#35592
#848653