Finite Group Databasebeta

← #27278061

Subgroups — #27278061

Order 13568.

532 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#20283741C211yesmaximal, characteristic
2#20275007C211yesmaximal, characteristic
3#20274893C211yesmaximal, characteristic, complemented
4C4×#13133131C211yesmaximal, characteristic
5#20272563C211yesmaximal, characteristic, complemented
6#20272562C211yesmaximal, characteristic, complemented
7C53×#1850C211yesmaximal, characteristic, complemented
8C2×C4×C22⋊D53C2211yescharacteristic
9#16810694—21no—
10#16810694—21no—
11#16810691—21no—
12#16810691—21no—
13#16810677—21no—
14C2×#13133146—21no—
15C2×#13133131C411yescharacteristic
16C2×#13133131C411yescharacteristic
17#16810585—21no—
18C2×C22⋊#88322C2211yescharacteristic
19C2×C22⋊#88322C2211yescharacteristic
20C2×#13133129—21no—
21C4×C22⋊Dic53C2211yescharacteristic
22#16810389C411yescharacteristic
23#16810389C411yescharacteristic
24#16810377—21no—
25#16810377—21no—
26C23×C8×C53C2211yescharacteristic
27C2×C4⋊C8×C53—21no—
28C2×C22⋊C8×C53C2211yescharacteristic
29C4×C22⋊C4×C53C2211yescharacteristic
30C53×#336—21no—
31C22×C22⋊D53C2×C411yescharacteristic
32C22×C4×D53D411yescharacteristic
33C2×#11719912—21no—
34C2×#11719912—21no—
35C22×#88322—21no—
36C22×#88322D411yescharacteristic
37C22×#88322D411yescharacteristic
38C4×C22⋊D53—21no—
39C4×C22⋊D53—21no—
40C4×C22⋊D53—21no—
41C4×C22⋊D53—21no—
42C2×#11719919Q811yescharacteristic
43C2×C22⋊Dic53C2×C411yescharacteristic
44C2×C22⋊Dic53C2×C411yescharacteristic
45C2×C22⋊Dic53C2×C411yescharacteristic
46C2×#11719921Q811yescharacteristic
47C2×C4×Dic53—21no—
48C2×C4×Dic53D411yescharacteristic
49#13133146—21no—
50#13133146—21no—
51#13133146—42no—
52#13133131—22no—
53#13133131—22no—
54C22⋊#88322—21no—
55C22⋊#88322—21no—
56C22⋊#88322—21no—
57C22⋊#88322—21no—
58#13133129—21no—
59#13133129—21no—
60#13133129—42no—
61#13133127—21no—
62#13133127—21no—
63#13133124—21no—
64C23×C4×C53C2311yescharacteristic
65C22×C8×C53—21no—
66C22×C8×C53—21no—
67C22×C8×C53—21no—
68C22×C8×C53—42no—
69C22×C8×C53D411yescharacteristic
70C22×C8×C53D411yescharacteristic
71C2×C22⋊C4×C53C2×C411yescharacteristic
72C2×C22⋊C4×C53C2×C411yescharacteristic
73C2×C42×C53—21no—
74C4⋊C8×C53—42no—
75C22⋊C8×C53—21no—
76C22⋊C8×C53—21no—
77#95×C53—21no—
78C24×C53C22×C411yescharacteristic
79C23×D53C4⋊C411yescharacteristic
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—41no—
83C22×C4×C53—41no—
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×Q811yescharacteristic
88C22×C4×C53#4111yescharacteristic
89C22×C4×C53#4111yescharacteristic
90C2×C22⋊D53—21no—
91C2×C22⋊D53—21no—
92C2×C22⋊D53—21no—
93C2×C22⋊D53—21no—
94C2×C22⋊D53—21no—
95C2×C22⋊D53—21no—
96C22×Dic53—21no—
97C22×Dic53—21no—
98C22×Dic53C4⋊C411yescharacteristic
99C22×Dic53C4⋊C411yescharacteristic
100C22×Dic53C4⋊C411yescharacteristic
101C2×C4×D53—21no—
102C2×C4×D53—21no—
103C2×C4×D53—41no—
104C2×C4×D53—41no—
105C2×C8×C53—22no—
106C2×C8×C53—22no—
107C2×C8×C53—22no—
108C2×C8×C53—21no—
109C2×C8×C53—21no—
110C2×C8×C53—41no—
111C2×C8×C53—41no—
112C2×C8×C53—41no—
113C2×C8×C53—41no—
114C2×C8×C53—41no—
115C2×C8×C53—41no—
116C22⋊C4×C53—21no—
117C22⋊C4×C53—21no—
118C22⋊C4×C53—21no—
119C22⋊C4×C53M1612yescomplemented
120C42×C53—41no—
121C42×C53—41no—
122C22⋊Dic53—21no—
123C22⋊Dic53—21no—
124C22⋊Dic53—21no—
125C22⋊Dic53—21no—
126C22⋊Dic53—21no—
127C22⋊Dic53M1612yescomplemented
128#11719921—21no—
129#11719921—21no—
130#11719919—21no—
131#11719919—21no—
132C4×Dic53—21no—
133C4×Dic53—21no—
134C4×Dic53—41no—
135C4×Dic53—41no—
136C2×#88322—21no—
137C2×#88322—21no—
138C2×#88322—41no—
139C2×#88322—41no—
140C2×#88322—41no—
141C2×#88322—41no—
142C2×#88322—41no—
143#11719912—81no—
144#11719912—81no—
145C23×C53—41no—
146C23×C53—41no—
147C23×C53C2×C4⋊C411yescharacteristic
148C23×C53C4×D411yescharacteristic
149C23×C53C4×D411yescharacteristic
150C23×C53C2×M1611yescharacteristic
151C23×C53C2×M1611yescharacteristic
152C23×C53#13111yescharacteristic
153C23×C53#13111yescharacteristic
154C22×D53—21no—
155C22×D53—21no—
156C22×D53—21no—
157C22×D53—41no—
158C22×D53—41no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—22no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53#12011yescharacteristic
175C2×C4×C53C22⋊Q811yescharacteristic
176C2×C4×C53C22⋊Q811yescharacteristic
177C2×C4×C53#12311yescharacteristic
178C22⋊D53—41no—
179C22⋊D53—41no—
180C22⋊D53—41no—
181C22⋊D53—41no—
182C2×Dic53—21no—
183C2×Dic53—21no—
184C2×Dic53—21no—
185C2×Dic53—21no—
186C2×Dic53—21no—
187C2×Dic53—22no—
188C2×Dic53—41no—
189C2×Dic53—41no—
190C2×Dic53—41no—
191C2×Dic53—41no—
192C2×Dic53—41no—
193C2×Dic53—41no—
194C2×Dic53—41no—
195C4×D53—41no—
196C4×D53—41no—
197C4×D53—81no—
198C8×C53—41no—
199C8×C53—42no—
200C8×C53—41no—
201#88322—41no—
202#88322—41no—
203#88322—81no—
204#8852—531nomaximal, Sylow
205C22×C53—21no—
206C22×C53—21no—
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—21no—
210C22×C53—21no—
211C22×C53—41no—
212C22×C53—41no—
213C22×C53—41no—
214C22×C53—41no—
215C22×C53#38511yescharacteristic
216C22×C53#42311yescharacteristic
217C22×C53#42411yescharacteristic
218C22×C53#43511yescharacteristic
219C22×C53#43511yescharacteristic
220C22×C53#43611yescharacteristic
221C22×C53#43611yescharacteristic
222C2×D53—42no—
223C2×D53—41no—
224C2×D53—41no—
225C2×D53—41no—
226C2×D53—81no—
227C4×C53—21no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—42no—
231C4×C53—81no—
232C4×C53B(4,𝔽2)11yescharacteristic
233C4×C53#45811yescharacteristic
234Dic53—42no—
235Dic53—41no—
236Dic53—41no—
237Dic53—41no—
238Dic53—81no—
239#2973—531no—
240#1869—531no—
241#1850—531no—
242#1850D5311yescharacteristic, complemented
243C4×#99—531no—
244#1282—531no—
245#1282—531no—
246C2×C53—21no—
247C2×C53—21no—
248C2×C53—21no—
249C2×C53—21no—
250C2×C53—21no—
251C2×C53—21no—
252C2×C53#185111yescharacteristic
253C2×C53#186711yescharacteristic
254C2×C53#186811yescharacteristic
255D53—81no—
256C23×C8C2×D5311yescharacteristic
257C2×C4×D4—531no—
258#435—1061no—
259#435—1061no—
260#435—1061no—
261#435—1061no—
262#432—1061no—
263C2×C4⋊C8—21no—
264C2×C4⋊C8—1061no—
265C2×#99—531no—
266C2×#99—531no—
267#407—1061no—
268C2×C22⋊C8—531no—
269C2×C22⋊C8—531no—
270C2×C22⋊C8—1061no—
271C2×C22⋊C8C2×D5311yescharacteristic
272C4×C22⋊C4—531no—
273C4×C22⋊C4C2×D5311yescharacteristic
274#342—531no—
275#342—531no—
276#336—21no—
277#336—1061no—
278#336—1061no—
279C53#885211yescharacteristic, complemented, Sylow
280C22×D4—531no—
281C23×C4—531no—
282C23×C4C22×D5311yescharacteristic
283C2×M16—1061no—
284C2×M16—1061no—
285C22×C8—21no—
286C22×C8—21no—
287C22×C8—21no—
288C22×C8—42no—
289C22×C8—531no—
290C22×C8—531no—
291C22×C8—1061no—
292C22×C8C22⋊D5311yescharacteristic
293C22×C8C22⋊D5311yescharacteristic
294C4×D4—1061no—
295C4×D4—1061no—
296C4×D4—1061no—
297C4×D4—1061no—
298C2×C4⋊C4—531no—
299C2×C22⋊C4—531no—
300C2×C22⋊C4—531no—
301C2×C22⋊C4—531no—
302C2×C22⋊C4—531no—
303C2×C22⋊C4C4×D5311yescharacteristic
304C2×C22⋊C4C4×D5311yescharacteristic
305C2×C42—21no—
306C2×C42—531no—
307C2×C42—1061no—
308C4⋊C8—42no—
309C4⋊C8—1061no—
310C4⋊C8—1061no—
311C4⋊C8—2122no—
312#99—1062no—
313#99—1062no—
314C22⋊C8—21no—
315C22⋊C8—21no—
316C22⋊C8—1061no—
317C22⋊C8—1061no—
318C22⋊C8—1061no—
319C22⋊C8—1061no—
320C22⋊C8—1061no—
321C22⋊C8—1061no—
322C22⋊C8—2122no—
323#97—1061no—
324#97—1061no—
325#95—21no—
326#95—1061no—
327C24—531no—
328C24C2×C4×D5311yescharacteristic
329C2×D4—1061no—
330C2×D4—1061no—
331C2×D4—1061no—
332C2×D4—1061no—
333C2×D4—1061no—
334C2×D4—1061no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—531no—
340C22×C4—531no—
341C22×C4—531no—
342C22×C4—1061no—
343C22×C4—1061no—
344C22×C4—1061no—
345C22×C4—1061no—
346C22×C4—2121no—
347C22×C4—2121no—
348C22×C4#1171994511yescharacteristic
349C22×C4D4×D5311yescharacteristic
350C22×C4D4×D5311yescharacteristic
351C22×C4Q8×D5311yescharacteristic
352C22×C4#1171994911yescharacteristic
353C22×C4C2×C22⋊D5311yescharacteristic
354M16—4241no—
355M16—4241no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—1061no—
368C2×C8—1061no—
369C2×C8—2121no—
370C2×C8—2121no—
371C2×C8—2121no—
372C2×C8—2121no—
373C2×C8—2121no—
374C4⋊C4—1061no—
375C4⋊C4—1061no—
376C22⋊C4—21no—
377C22⋊C4—21no—
378C22⋊C4—21no—
379C22⋊C4—532no—
380C22⋊C4—1061no—
381C22⋊C4—1061no—
382C22⋊C4—1061no—
383C22⋊C4—1061no—
384C22⋊C4—1061no—
385C22⋊C4—1061no—
386C22⋊C4—1061no—
387C22⋊C4#1171991212yescomplemented
388C42—41no—
389C42—41no—
390C42—1061no—
391C42—1061no—
392C42—2121no—
393C42—2121no—
394C23—41no—
395C23—41no—
396C23—1061no—
397C23—1061no—
398C23—1061no—
399C23—2121no—
400C23—2121no—
401C23C4⋊C4×D5311yescharacteristic
402C23C4×C22⋊D5311yescharacteristic
403C23#1313317911yescharacteristic
404C23M16×D5311yescharacteristic
405C23C2×#1171991211yescharacteristic
406C23#1313322311yescharacteristic
407C23#1313322511yescharacteristic
408D4—2121no—
409D4—2121no—
410D4—2121no—
411D4—2121no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—1061no—
428C2×C4—1061no—
429C2×C4—1062no—
430C2×C4—1061no—
431C2×C4—1061no—
432C2×C4—1061no—
433C2×C4—4241no—
434C2×C4—2121no—
435C2×C4—2121no—
436C2×C4—2121no—
437C2×C4—2121no—
438C2×C4—2121no—
439C2×C4—2121no—
440C2×C4—2121no—
441C2×C4—2121no—
442C2×C4—2121no—
443C2×C4#1313318511yescharacteristic
444C2×C4#1313319311yescharacteristic
445C2×C4#1313319511yescharacteristic
446C2×C4#1313320211yescharacteristic
447C8—42no—
448C8—41no—
449C8—41no—
450C8—4241no—
451C8—2121no—
452C8—2121no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—4241no—
464C22—2122no—
465C22—2121no—
466C22—2121no—
467C22—2121no—
468C22#1681049411yescharacteristic
469C22#1681064511yescharacteristic
470C22#1681064811yescharacteristic
471C22#1681069311yescharacteristic
472C22#1681069411yescharacteristic
473C22#1681069911yescharacteristic
474C22#1681070011yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—42no—
479C4—81no—
480C4—4241no—
481C4—2122no—
482C4—2121no—
483C4—2121no—
484C4—2121no—
485C4#1681080911yescharacteristic
486C4#1681081211yescharacteristic
487C2—21no—
488C2—21no—
489C2—21no—
490C2—21no—
491C2—21no—
492C2—21no—
493C2—4241no—
494C2#2027489611yescharacteristic
495C2#2027499411yescharacteristic
496C2#2027499911yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1850C2
socleC22×C53#385
prime radicalC22×C22⋊D53C2×C4
cosocleC23×C4×C53C23
hypercenter#1850D53
hypocenterC53#8852
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
3. #20274893C2semidirect
5. #20272563C2semidirect
6. #20272562C2semidirect
7. C53×#1850C2semidirect
119. C22⋊C4×C53M16semidirect
127. C22⋊Dic53M16semidirect
242. #1850D53semidirect
279. C53#8852semidirect
387. C22⋊C4#11719912semidirect

Maximal subgroups

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

subgroupindex
#202837412
#202750072
#202748932
C4×#131331312
#202725632
#202725622
C53×#18502
#885253