Finite Group Databasebeta

← #27496230

Subgroups — #27496230

Order 13568.

446 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#20276183C211yesmaximal, characteristic, complemented
2#20276179C211yesmaximal, characteristic, complemented
3#20276178C211yesmaximal, characteristic
4#20276177C211yesmaximal, characteristic
5#20276174C211yesmaximal, characteristic
6#20276116C211yesmaximal, characteristic, complemented
7#20276112C211yesmaximal, characteristic
8#20275619C211yesmaximal, characteristic
9#20275617C211yesmaximal, characteristic
10#20275617C211yesmaximal, characteristic
11#20275504C211yesmaximal, characteristic, complemented
12#20275504C211yesmaximal, characteristic, complemented
13#20275502C211yesmaximal, characteristic, complemented
14#20274469C211yesmaximal, characteristic, complemented
15C53×#2082C211yesmaximal, characteristic, complemented
16C22×#11719924C2211yescharacteristic
17#16810672C2211yescharacteristic
18#16810672C2211yescharacteristic
19#16810671C2211yescharacteristic
20#16810671C2211yescharacteristic
21#16810671C2211yescharacteristic
22C2×#13133152C2211yescharacteristic
23C2×#13133152C2211yescharacteristic
24C2×#13133152C2211yescharacteristic
25#16810632C2211yescharacteristic
26#16810629C2211yescharacteristic
27#16810627C2211yescharacteristic
28#16810627C2211yescharacteristic
29C2×#13133136C2211yescharacteristic
30C2×#13133136C2211yescharacteristic
31C2×#13133133C2211yescharacteristic
32C2×#13133133C2211yescharacteristic
33C2×#13133133C2211yescharacteristic
34#16810581C2211yescharacteristic
35#16810579C2211yescharacteristic
36#16810579C2211yescharacteristic
37#16810395C2211yescharacteristic
38#16810394C2211yescharacteristic
39#16810393C2211yescharacteristic
40#16810393C2211yescharacteristic
41#16810381C2211yescharacteristic
42#16810379C2211yescharacteristic
43#16810379C2211yescharacteristic
44C53×#430C2211yescharacteristic
45C53×#419C2211yescharacteristic
46C53×#419C2211yescharacteristic
47C2×#101×C53C2211yescharacteristic
48C53×#344C2211yescharacteristic
49C53×#344C2211yescharacteristic
50C53×H(3,ℤ/4)C2211yescharacteristic
51C22×D212C2311yescharacteristic
52C2×#11719924—21no—
53C2×#11719924—21no—
54C2×#11719924—21no—
55C2×#11719924—21no—
56C2×#11719924—21no—
57C2×#11719924—21no—
58C2×#11719917C2311yescharacteristic
59C2×#11719917C2311yescharacteristic
60C2×#11719917C2311yescharacteristic
61C2×#11719917C2311yescharacteristic
62C2×#11719917C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133176C2311yescharacteristic
65#13133153—41no—
66#13133153—41no—
67#13133152—21no—
68#13133152—21no—
69#13133152—21no—
70#13133152—21no—
71#13133152—21no—
72#13133152—21no—
73#13133152—41no—
74#13133152—41no—
75#13133152—41no—
76#13133141—41no—
77#13133139—41no—
78#13133137—41no—
79#13133137—41no—
80#13133136—41no—
81#13133136—41no—
82#13133133—41no—
83#13133133—41no—
84#13133133—41no—
85#13133128—21no—
86#13133128—21no—
87#13133127—21no—
88C4×#88322—21no—
89C4×#88322—21no—
90C8×Dic53—21no—
91C22×Q8×C53C2311yescharacteristic
92C2×M16×C53C2311yescharacteristic
93C2×M16×C53C2311yescharacteristic
94C2×M16×C53C2311yescharacteristic
95C2×M16×C53C2311yescharacteristic
96#117×C53C2311yescharacteristic
97#117×C53C2311yescharacteristic
98#108×C53—41no—
99#103×C53—41no—
100#103×C53—41no—
101#101×C53—41no—
102C23×D53—21no—
103C2×Q8×C53—21no—
104C2×Q8×C53—21no—
105C2×Q8×C53—21no—
106C2×Q8×C53—21no—
107C2×Q8×C53—21no—
108C2×Q8×C53—21no—
109C22×C4×C53—21no—
110C22×C4×C53C2411yescharacteristic
111C2×D212—21no—
112C2×D212—21no—
113C2×D212—21no—
114C2×D212—21no—
115C2×D212—21no—
116C2×D212—21no—
117M16×C53—21no—
118M16×C53—21no—
119M16×C53—21no—
120M16×C53—21no—
121M16×C53—21no—
122M16×C53—21no—
123M16×C53—21no—
124M16×C53—21no—
125C2×C8×C53—21no—
126C2×C8×C53—21no—
127C2×C8×C53—21no—
128C2×C8×C53—21no—
129C4⋊C4×C53—21no—
130C4⋊C4×C53—21no—
131C22⋊C4×C53—21no—
132C22⋊C4×C53—21no—
133C42×C53—21no—
134C42×C53—21no—
135C22⋊Dic53—21no—
136#11719924—41no—
137#11719924—41no—
138#11719924—41no—
139#11719924—41no—
140C4⋊Dic53—21no—
141C4×Dic53—21no—
142#11719917—21no—
143#11719917—21no—
144#11719917—21no—
145#11719917—21no—
146#11719917—21no—
147#11719917—21no—
148#11719917—21no—
149#11719917—21no—
150#11719917—21no—
151#11719917—21no—
152C2×#88322—21no—
153C2×#88322—21no—
154C2×#88322—21no—
155C2×#88322—21no—
156C2×#88322—21no—
157C2×#88322#4111yescharacteristic
158C2×#88322#4111yescharacteristic
159C2×#88322#4111yescharacteristic
160C2×#88322#4111yescharacteristic
161C2×#88322#4111yescharacteristic
162C2×#88322#4111yescharacteristic
163C23×C532−1+411yescharacteristic
164C22×D53—41no—
165C22×D53—41no—
166C22×D53—41no—
167C22×D53—161no—
168Q8×C53—41no—
169Q8×C53—41no—
170Q8×C53—41no—
171Q8×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53C2×#4111yescharacteristic
178C2×C4×C53C2×#4111yescharacteristic
179C2×C4×C53C2×#4111yescharacteristic
180C2×C4×C53H(5,𝔽2)11yescharacteristic
181C2×C4×C53H(5,𝔽2)11yescharacteristic
182C2×C4×C53H(5,𝔽2)11yescharacteristic
183C2×Dic53—41no—
184D212—41no—
185D212—41no—
186D212—41no—
187D212—41no—
188C8×C53—41no—
189C8×C53—41no—
190C8×C53—41no—
191C8×C53—41no—
192#88322—41no—
193#88322—41no—
194#88322—41no—
195#88322—41no—
196#88322—41no—
197#88322#12611yescharacteristic
198#88322#12611yescharacteristic
199#88322#12611yescharacteristic
200#88322#12611yescharacteristic
201#30236—531nomaximal, Sylow
202C22×C53—41no—
203C22×C53#54211yescharacteristic
204C22×C53#54211yescharacteristic
205C22×C53#54211yescharacteristic
206C2×D53—81no—
207C2×D53—161no—
208C2×D53—161no—
209C2×D53—161no—
210C4×C53—81no—
211C4×C53—81no—
212C4×C53—81no—
213C4×C53C2×#12611yescharacteristic
214C4×C53#54011yescharacteristic
215C4×C53#54011yescharacteristic
216C4×C53#54011yescharacteristic
217Dic53—81no—
218#2082—531no—
219#2082—531no—
220#2082D5311yescharacteristic, complemented
221#2081—531no—
222#2081—531no—
223#2081—531no—
224#2070—531no—
225#2069—531no—
226#1977—531no—
227#1977—531no—
228#1977—531no—
229#1953—531no—
230#1953—531no—
231#1953—531no—
232#1747—531no—
233C2×C53—21no—
234C2×C53—21no—
235C2×C53—21no—
236C2×C53#264411yescharacteristic
237D53—161no—
238C22×SD16—531no—
239#430—531no—
240#430—531no—
241#430—531no—
242#430—531no—
243#430—531no—
244#430C2×D5311yescharacteristic
245C2×#108—531no—
246C2×#108—531no—
247C2×#108—531no—
248#419—531no—
249#419C2×D5311yescharacteristic
250#419C2×D5311yescharacteristic
251#418—531no—
252#418—531no—
253#418—531no—
254C2×#101—531no—
255C2×#101—531no—
256C2×#101C2×D5311yescharacteristic
257C2×#100—531no—
258C2×#100—531no—
259C2×#100—531no—
260#405—531no—
261#405—531no—
262#405—531no—
263#344—531no—
264#344—531no—
265#344—531no—
266#344—531no—
267#344C2×D5311yescharacteristic
268#344C2×D5311yescharacteristic
269#338—531no—
270H(3,ℤ/4)—531no—
271H(3,ℤ/4)—531no—
272H(3,ℤ/4)C2×D5311yescharacteristic
273C53#3023611yescharacteristic, complemented, Sylow
274C22×Q8C22×D5311yescharacteristic
275C22×D4—531no—
276C2×SD16—1061no—
277C2×SD16—1061no—
278C2×SD16—1061no—
279C2×SD16—1061no—
280C2×SD16—1061no—
281C2×SD16—1061no—
282C2×M16—531no—
283C2×M16—531no—
284C2×M16—531no—
285C2×M16—531no—
286C2×M16—531no—
287C2×M16C22×D5311yescharacteristic
288C2×M16C22×D5311yescharacteristic
289C2×M16C22×D5311yescharacteristic
290C2×M16C22×D5311yescharacteristic
291C22×C8—531no—
292#117—531no—
293#117C22×D5311yescharacteristic
294#117C22×D5311yescharacteristic
295#108—41no—
296#108—1061no—
297#108—1061no—
298#108—1061no—
299#108—1061no—
300#108—1061no—
301#108—1061no—
302#108—2121no—
303#108—2121no—
304#108—2121no—
305#108—2121no—
306#108—2121no—
307#103—41no—
308#103—41no—
309#103—2121no—
310#102—2121no—
311#102—2121no—
312#102—2121no—
313#101—41no—
314#101—2121no—
315#101—2121no—
316#100—2121no—
317#100—2121no—
318#100—2121no—
319#97—1061no—
320#97—1061no—
321#97—1061no—
322C4×C8—1061no—
323C4×C8—1061no—
324C4×C8—1061no—
325C24—1061no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×D4—1061no—
333C2×D4—1061no—
334C2×D4—1061no—
335C2×D4—1061no—
336C2×D4—1061no—
337C2×D4—1061no—
338C22×C4—21no—
339C22×C4C23×D5311yescharacteristic
340SD16—2121no—
341SD16—2121no—
342SD16—2121no—
343SD16—2121no—
344M16—21no—
345M16—21no—
346M16—21no—
347M16—21no—
348M16—21no—
349M16—21no—
350M16—21no—
351M16—21no—
352M16—1061no—
353M16—1061no—
354M16—1061no—
355M16—1061no—
356M16—1061no—
357M16—1061no—
358M16—1061no—
359M16—1061no—
360M16—1061no—
361M16—1061no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—531no—
367C2×C8—531no—
368C2×C8—531no—
369C2×C8—531no—
370C2×C8—531no—
371C2×C8—531no—
372C2×C8—1061no—
373C2×C8—1061no—
374C2×C8—1061no—
375C2×C8—1061no—
376C2×C8—1061no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—1061no—
380C22⋊C4—21no—
381C22⋊C4—21no—
382C22⋊C4—1061no—
383C42—21no—
384C42—21no—
385C42—1061no—
386C23—8481no—
387C23—2121no—
388C23—2121no—
389C23—2121no—
390C23#1313326411yescharacteristic
391Q8—41no—
392Q8—41no—
393Q8—41no—
394Q8—41no—
395D4—2121no—
396D4—2121no—
397D4—2121no—
398D4—2121no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—2121no—
405C2×C4#41×D5311yescharacteristic
406C2×C4#41×D5311yescharacteristic
407C2×C4#41×D5311yescharacteristic
408C2×C4#1313326211yescharacteristic
409C2×C4#1313326211yescharacteristic
410C2×C4#1313326311yescharacteristic
411C8—41no—
412C8—41no—
413C8—41no—
414C8—41no—
415C8—531no—
416C8—531no—
417C8—531no—
418C8—531no—
419C8—2121no—
420C8—2121no—
421C8—2121no—
422C8—2121no—
423C8—2121no—
424C22—41no—
425C22—8481no—
426C22—8481no—
427C22—8481no—
428C22—4241no—
429C22#1681123711yescharacteristic
430C22#1681123911yescharacteristic
431C22#1681124111yescharacteristic
432C4—81no—
433C4—81no—
434C4—81no—
435C4—4241no—
436C4#126×D5311yescharacteristic
437C4#1681122311yescharacteristic
438C4#1681122511yescharacteristic
439C4#1681122511yescharacteristic
440C2—21no—
441C2—21no—
442C2—21no—
443C2—8481no—
444C2#2028067411yescharacteristic
subgroupG/H
centerC2#20280674
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2082C2
socleC2×C53#2644
prime radicalC22×D212C23
hypercenter#2082D53
hypocenterC53#30236
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #2082

Complements

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

NH ≅ G/Ntype
1. #20276183C2semidirect
2. #20276179C2semidirect
6. #20276116C2semidirect
11. #20275504C2semidirect
12. #20275504C2semidirect
13. #20275502C2semidirect
14. #20274469C2semidirect
15. C53×#2082C2semidirect
220. #2082D53semidirect
273. C53#30236semidirect

Maximal subgroups

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

subgroupindex
#202761832
#202761792
#202761782
#202761772
#202761742
#202761162
#202761122
#202756192
#202756172
#202755042
#202755022
#202744692
C53×#20822
#3023653