Finite Group Databasebeta

← #27281164

Subgroups — #27281164

Order 13568.

470 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283373C211yesmaximal, characteristic, complemented
2#20274810C211yesmaximal, characteristic, complemented
3#20273283C211yesmaximal, characteristic
4#20273075C211yesmaximal, characteristic
5#20272620C211yesmaximal, characteristic, complemented
6#20272602C211yesmaximal, characteristic, complemented
7C53×#3094C211yesmaximal, characteristic
8#16810684—22no—
9#16810680—21no—
10#16810676—21no—
11#16810650—21no—
12#16810587C2211yescharacteristic
13C2×C22⋊#88322C2211yescharacteristic
14C2×C22⋊#88322C2211yescharacteristic
15#16810579—21no—
16#16810483C2211yescharacteristic
17#16810385—21no—
18#16810385—21no—
19#16810357—21no—
20#16810357—21no—
21#16810347—22no—
22C2×#121×C53C2211yescharacteristic
23C2×#117×C53C2211yescharacteristic
24C53×#482—21no—
25C53×#482—21no—
26C53×#481—21no—
27C53×#480—21no—
28C53×#428C411yescharacteristic, complemented
29C53×#428C411yescharacteristic, complemented
30C53×#418C411yescharacteristic, complemented
31C53×#418C411yescharacteristic, complemented
32C53×#407C2211yescharacteristic
33C2×#11719917—21no—
34C22×#88322—21no—
35C22×#88322—21no—
36C2×C4⋊Dic53—21no—
37C2×C22⋊Dic53—21no—
38C4⋊#88322—22no—
39C4⋊#88322—22no—
40C22⋊#88322—21no—
41C22⋊#88322—21no—
42C22⋊#88322—21no—
43C22⋊#88322—21no—
44C22⋊#88322D411yescharacteristic
45C22⋊#88322D411yescharacteristic
46C22⋊#88322D412yes—
47#13133128—22no—
48C4×#88322—22no—
49#13133124—21no—
50C22×D4×C53—21no—
51C22×D4×C53C2×C411yescharacteristic
52C23×C4×C53C2311yescharacteristic
53C2×M16×C53C2×C411yescharacteristic
54C2×M16×C53C2×C411yescharacteristic
55#121×C53—21no—
56#121×C53—21no—
57#121×C53—21no—
58#121×C53—21no—
59#117×C53—21no—
60#117×C53—21no—
61#117×C53C2×C411yescharacteristic
62#117×C53C2×C411yescharacteristic
63#117×C53D411yescharacteristic
64#117×C53D411yescharacteristic
65C2×C4⋊C4×C53C2×C411yescharacteristic
66C2×C4⋊C4×C53D411yescharacteristic
67C2×C4⋊C4×C53D411yescharacteristic
68C2×C22⋊C4×C53—21no—
69C2×C22⋊C4×C53—21no—
70C2×C42×C53—21no—
71#107×C53—21no—
72#107×C53—21no—
73#106×C53—21no—
74#106×C53—21no—
75#102×C53—21no—
76#102×C53—21no—
77#102×C53—21no—
78#102×C53—21no—
79C22⋊C8×C53—21no—
80C22⋊C8×C53—21no—
81C24×C53—21no—
82C24×C53#4111yescharacteristic
83C2×D4×C53—21no—
84C2×D4×C53—21no—
85C2×D4×C53—41no—
86C2×D4×C53—41no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—81no—
90C2×D4×C53—81no—
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53—41no—
95C22×C4×C53—41no—
96C22×C4×C53C22×C411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53#4111yescharacteristic
102C22×Dic53—41no—
103M16×C53—41no—
104M16×C53—41no—
105C2×C8×C53—21no—
106C2×C8×C53—21no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53C22⋊C411yescharacteristic
112C4⋊C4×C53C22⋊C411yescharacteristic
113C4⋊C4×C53C22⋊C411yescharacteristic
114C4⋊C4×C53C22⋊C411yescharacteristic
115C22⋊C4×C53—21no—
116C22⋊C4×C53—21no—
117C22⋊C4×C53—21no—
118C22⋊C4×C53—21no—
119C22⋊C4×C53—41no—
120C22⋊C4×C53—41no—
121C42×C53—21no—
122C42×C53—21no—
123C42×C53—21no—
124C42×C53—21no—
125C22⋊Dic53—81no—
126C4⋊Dic53—41no—
127C4⋊Dic53—41no—
128#11719917—41no—
129#11719917—41no—
130C2×#88322—21no—
131C2×#88322—22no—
132C2×#88322—21no—
133C2×#88322—41no—
134C2×#88322—41no—
135C2×#88322—41no—
136C2×#88322—41no—
137C2×#88322—41no—
138C23×C53—21no—
139C23×C53—21no—
140C23×C53—41no—
141C23×C53—41no—
142C23×C53—42no—
143C23×C53—41no—
144C23×C53—81no—
145C23×C53—81no—
146C23×C53C4×D411yescharacteristic
147C23×C53#12111yescharacteristic
148C23×C53#12311yescharacteristic
149D4×C53—41no—
150D4×C53—41no—
151D4×C53—81no—
152D4×C53—82no—
153D4×C53—81no—
154D4×C53—81no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
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—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—81no—
172C2×C4×C53—81no—
173C2×C4×C53C2×C22⋊C411yescharacteristic
174C2×C4×C53C4×D411yescharacteristic
175C2×C4×C53#12011yescharacteristic
176C2×C4×C53#12111yescharacteristic
177C2×Dic53—81no—
178C2×Dic53—81no—
179C2×Dic53—81no—
180C8×C53—41no—
181C8×C53—41no—
182#88322—41no—
183#88322—41no—
184#88322—42no—
185#88322—81no—
186#9348—531nomaximal, Sylow
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—21no—
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—81no—
196C22×C53—81no—
197C22×C53—81no—
198C22×C53—81no—
199C22×C53—81no—
200C22×C53—162no—
201C22×C53#38611yescharacteristic
202C22×C53#47111yescharacteristic
203C22×C53#47211yescharacteristic
204C4×C53—21no—
205C4×C53—21no—
206C4×C53—21no—
207C4×C53—41no—
208C4×C53—41no—
209C4×C53—41no—
210C4×C53—41no—
211C4×C53—81no—
212C4×C53#41011yescharacteristic
213C4×C53#41011yescharacteristic
214Dic53—161no—
215#3094D5311yescharacteristic
216#2924—531no—
217#1825—531no—
218#1519—531no—
219#1471—531no—
220#1305—531no—
221#1298—531no—
222C2×C53—21no—
223C2×C53—21no—
224C2×C53—41no—
225C2×C53—41no—
226C2×C53—161no—
227C2×C53#188111yescharacteristic
228C2×C53#189911yescharacteristic
229C2×C53#190711yescharacteristic
230C2×#121C2×D5311yescharacteristic
231C2×#117C2×D5311yescharacteristic
232#482—21no—
233#482—21no—
234#481—21no—
235#480—21no—
236#433—1062no—
237#432—1061no—
238#431—1061no—
239#428Dic5311yescharacteristic, complemented
240#428Dic5311yescharacteristic, complemented
241#424—1061no—
242#418Dic5311yescharacteristic, complemented
243#418Dic5311yescharacteristic, complemented
244#407—531no—
245#407C2×D5311yescharacteristic
246C2×C22⋊C8—531no—
247C2×C22⋊C8—531no—
248#405—1061no—
249#380—531no—
250#340—1061no—
251#340—1061no—
252#327—1061no—
253#327—1061no—
254#323—1062no—
255C53#934811yescharacteristic, complemented, Sylow
256C22×D4—21no—
257C22×D4C2×Dic5311yescharacteristic
258C23×C4C22×D5311yescharacteristic
259C2×M16—1061no—
260C2×M16C2×Dic5311yescharacteristic
261C2×M16C2×Dic5311yescharacteristic
262C22×C8—1061no—
263C22×C8—1061no—
264#121—21no—
265#121—21no—
266#121—21no—
267#121—21no—
268#117—21no—
269#117—21no—
270#117C2×Dic5311yescharacteristic
271#117C2×Dic5311yescharacteristic
272#117C22⋊D5311yescharacteristic
273#117C22⋊D5311yescharacteristic
274C2×C4⋊C4—1061no—
275C2×C4⋊C4C2×Dic5311yescharacteristic
276C2×C4⋊C4C22⋊D5311yescharacteristic
277C2×C4⋊C4C22⋊D5311yescharacteristic
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—1061no—
281C2×C42—21no—
282#107—21no—
283#107—21no—
284#106—21no—
285#106—21no—
286C4⋊C8—1062no—
287C4⋊C8—1062no—
288#102—21no—
289#102—21no—
290#102—21no—
291#102—21no—
292C22⋊C8—21no—
293C22⋊C8—21no—
294C22⋊C8—532no—
295C22⋊C8—531no—
296C22⋊C8—531no—
297C22⋊C8—1061no—
298C22⋊C8—1061no—
299C22⋊C8—1061no—
300C22⋊C8—1061no—
301#97—1062no—
302C4×C8—1062no—
303#95—1061no—
304C24—21no—
305C24#1171994711yescharacteristic
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—81no—
313C2×D4—81no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—2121no—
320C22×C4D4×D5311yescharacteristic
321C22×C4D4×D5311yescharacteristic
322C22×C4#1171994711yescharacteristic
323C22×C4C22×Dic5311yescharacteristic
324C22×C4C2×C22⋊D5311yescharacteristic
325C22×C4C2×C22⋊D5311yescharacteristic
326M16—41no—
327M16—41no—
328M16—2121no—
329M16—2121no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—1061no—
333C2×C8—1062no—
334C2×C8—1061no—
335C2×C8—2121no—
336C2×C8—2121no—
337C2×C8—2121no—
338C2×C8—2121no—
339C2×C8—2121no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—2121no—
345C4⋊C4—2121no—
346C4⋊C4C22⋊Dic5311yescharacteristic
347C4⋊C4C22⋊Dic5311yescharacteristic
348C4⋊C4C22⋊Dic5311yescharacteristic
349C4⋊C4C22⋊Dic5311yescharacteristic
350C22⋊C4—21no—
351C22⋊C4—21no—
352C22⋊C4—21no—
353C22⋊C4—21no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—4241no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—21no—
361C23—21no—
362C23—21no—
363C23—41no—
364C23—41no—
365C23—42no—
366C23—41no—
367C23—81no—
368C23—81no—
369C23D4×Dic5311yescharacteristic
370C23#1313318911yescharacteristic
371C23#1313319811yescharacteristic
372D4—41no—
373D4—41no—
374D4—81no—
375D4—82no—
376D4—81no—
377D4—81no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—4241no—
397C2×C4—4241no—
398C2×C4—4241no—
399C2×C4C2×C22⋊Dic5311yescharacteristic
400C2×C4D4×Dic5311yescharacteristic
401C2×C4#1313318511yescharacteristic
402C2×C4#1313318911yescharacteristic
403C8—41no—
404C8—41no—
405C8—4241no—
406C8—2121no—
407C8—2121no—
408C8—2122no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—162no—
423C22#1681050311yescharacteristic
424C22#1681088411yescharacteristic
425C22#1681089111yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—41no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—81no—
434C4—8481no—
435C4#1681059711yescharacteristic
436C4#1681059711yescharacteristic
437C2—21no—
438C2—21no—
439C2—41no—
440C2—41no—
441C2—161no—
442C2#2027507911yescharacteristic
443C2#2027517411yescharacteristic
444C2#2027522511yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#3094C2
socleC22×C53#386
prime radicalC2×#121×C53C22
cosocleC23×C4×C53C23
hypercenter#3094D53
hypocenterC53#9348
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3094

Complements

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

NH ≅ G/Ntype
1. #20283373C2semidirect
2. #20274810C2semidirect
5. #20272620C2semidirect
6. #20272602C2semidirect
28. C53×#428C4semidirect
29. C53×#428C4semidirect
30. C53×#418C4semidirect
31. C53×#418C4semidirect
239. #428Dic53semidirect
240. #428Dic53semidirect
242. #418Dic53semidirect
243. #418Dic53semidirect
255. C53#9348semidirect

Maximal subgroups

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

subgroupindex
#202833732
#202748102
#202732832
#202730752
#202726202
#202726022
C53×#30942
#934853