Finite Group Databasebeta

← #27496231

Subgroups — #27496231

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#20276182C211yesmaximal, characteristic, complemented
3#20276179C211yesmaximal, characteristic, complemented
4#20276176C211yesmaximal, characteristic
5#20276175C211yesmaximal, characteristic
6#20276172C211yesmaximal, characteristic
7#20276116C211yesmaximal, characteristic, complemented
8#20276113C211yesmaximal, characteristic
9#20275620C211yesmaximal, characteristic
10#20275620C211yesmaximal, characteristic
11#20275506C211yesmaximal, characteristic, complemented
12#20275505C211yesmaximal, characteristic, complemented
13#20275503C211yesmaximal, characteristic, complemented
14#20274470C211yesmaximal, characteristic, complemented
15C53×#1977C211yesmaximal, characteristic
16#16810672C2211yescharacteristic
17#16810672C2211yescharacteristic
18#16810671C2211yescharacteristic
19#16810671C2211yescharacteristic
20#16810671C2211yescharacteristic
21#16810671C2211yescharacteristic
22C2×#13133153C2211yescharacteristic
23C2×#13133153C2211yescharacteristic
24#16810632C2211yescharacteristic
25#16810632C2211yescharacteristic
26#16810628C2211yescharacteristic
27#16810628C2211yescharacteristic
28C2×#13133136C2211yescharacteristic
29C2×#13133136C2211yescharacteristic
30C2×#13133134C2211yescharacteristic
31C2×#13133134C2211yescharacteristic
32#16810578C2211yescharacteristic
33#16810578C2211yescharacteristic
34#16810395C2211yescharacteristic
35#16810395C2211yescharacteristic
36#16810394C2211yescharacteristic
37#16810394C2211yescharacteristic
38#16810393C2211yescharacteristic
39#16810393C2211yescharacteristic
40#16810381C2211yescharacteristic
41#16810380C2211yescharacteristic
42#16810379C2211yescharacteristic
43#16810379C2211yescharacteristic
44C22×SD16×C53C2211yescharacteristic
45C2×#108×C53C2211yescharacteristic
46C53×#419C2211yescharacteristic
47C53×#418C2211yescharacteristic
48C2×#101×C53C2211yescharacteristic
49C2×#100×C53C2211yescharacteristic
50C53×#405C2211yescharacteristic
51C2×#11719917C2311yescharacteristic
52C2×#11719917C2311yescharacteristic
53C2×#11719917C2311yescharacteristic
54C2×#11719917C2311yescharacteristic
55C2×#11719917C2311yescharacteristic
56C2×#11719917C2311yescharacteristic
57#13133176C2311yescharacteristic
58#13133176C2311yescharacteristic
59#13133153—21no—
60#13133153—21no—
61#13133153—21no—
62#13133153—21no—
63#13133153—41no—
64#13133153—41no—
65#13133152—41no—
66#13133152—41no—
67#13133152—41no—
68#13133152—41no—
69#13133141—41no—
70#13133141—41no—
71#13133138—41no—
72#13133138—41no—
73#13133136—41no—
74#13133136—41no—
75#13133134—41no—
76#13133134—41no—
77#13133127—21no—
78#13133127—21no—
79C8×Dic53—21no—
80C8×Dic53—21no—
81C22×Q8×C53C2311yescharacteristic
82C22×D4×C53C2311yescharacteristic
83C2×SD16×C53—21no—
84C2×SD16×C53—21no—
85C2×SD16×C53—21no—
86C2×SD16×C53—21no—
87C2×SD16×C53—21no—
88C2×SD16×C53—21no—
89C2×M16×C53C2311yescharacteristic
90C2×M16×C53C2311yescharacteristic
91C2×M16×C53C2311yescharacteristic
92C22×C8×C53C2311yescharacteristic
93#117×C53C2311yescharacteristic
94#108×C53—21no—
95#108×C53—21no—
96#103×C53—41no—
97#102×C53—41no—
98#101×C53—41no—
99#100×C53—41no—
100#97×C53—21no—
101C4×C8×C53—21no—
102C24×C53—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—
109C2×D4×C53—21no—
110C2×D4×C53—21no—
111C2×D4×C53—21no—
112C2×D4×C53—21no—
113C2×D4×C53—21no—
114C2×D4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53C2411yescharacteristic
117SD16×C53—41no—
118SD16×C53—41no—
119SD16×C53—41no—
120SD16×C53—41no—
121M16×C53—21no—
122M16×C53—21no—
123M16×C53—21no—
124M16×C53—21no—
125M16×C53—21no—
126M16×C53—21no—
127C2×C8×C53—21no—
128C2×C8×C53—21no—
129C2×C8×C53—21no—
130C2×C8×C53#4111yescharacteristic
131C2×C8×C53#4111yescharacteristic
132C2×C8×C53#4111yescharacteristic
133C2×C8×C53#4111yescharacteristic
134C2×C8×C53#4111yescharacteristic
135C2×C8×C53#4111yescharacteristic
136C4⋊C4×C53—21no—
137C22⋊C4×C53—21no—
138C42×C53—21no—
139C22⋊Dic53—21no—
140C22⋊Dic53—21no—
141C4⋊Dic53—21no—
142C4⋊Dic53—21no—
143C4×Dic53—21no—
144C4×Dic53—21no—
145#11719917—21no—
146#11719917—21no—
147#11719917—21no—
148#11719917—21no—
149#11719917—21no—
150#11719917—21no—
151#11719917—21no—
152#11719917—21no—
153#11719917—21no—
154#11719917—21no—
155#11719917—21no—
156#11719917—21no—
157C2×#88322—21no—
158C2×#88322—21no—
159C2×#88322—21no—
160C2×#88322—21no—
161C2×#88322—21no—
162C2×#88322—21no—
163C23×C53—41no—
164C23×C53—41no—
165C23×C53—41no—
166C23×C53—161no—
167C23×C532−1+411yescharacteristic
168Q8×C53—41no—
169Q8×C53—41no—
170Q8×C53—41no—
171Q8×C53—41no—
172D4×C53—41no—
173D4×C53—41no—
174D4×C53—41no—
175D4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—41no—
180C2×C4×C53C2×#4111yescharacteristic
181C2×C4×C53C2×#4111yescharacteristic
182C2×C4×C53C2×#4111yescharacteristic
183C2×C4×C53H(5,𝔽2)11yescharacteristic
184C2×C4×C53H(5,𝔽2)11yescharacteristic
185C2×C4×C53H(5,𝔽2)11yescharacteristic
186C2×Dic53—41no—
187C2×Dic53—41no—
188C8×C53—41no—
189C8×C53—41no—
190C8×C53—41no—
191C8×C53#12611yescharacteristic
192C8×C53#12611yescharacteristic
193C8×C53#12611yescharacteristic
194C8×C53#12611yescharacteristic
195#88322—41no—
196#88322—41no—
197#88322—41no—
198#88322—41no—
199#88322—41no—
200#88322—41no—
201#30236—531nomaximal, Sylow
202C22×C53—41no—
203C22×C53—81no—
204C22×C53—161no—
205C22×C53—161no—
206C22×C53—161no—
207C22×C53#54211yescharacteristic
208C22×C53#54211yescharacteristic
209C22×C53#54211yescharacteristic
210C4×C53—81no—
211C4×C53—81no—
212C4×C53C2×#12611yescharacteristic
213C4×C53#54011yescharacteristic
214C4×C53#54011yescharacteristic
215C4×C53#54011yescharacteristic
216Dic53—81no—
217Dic53—81no—
218#2082—531no—
219#2082—531no—
220#2082—531no—
221#2081—531no—
222#2081—531no—
223#2081—531no—
224#2070—531no—
225#2069—531no—
226#1977—531no—
227#1977—531no—
228#1977D5311yescharacteristic
229#1953—531no—
230#1953—531no—
231#1953—531no—
232#1747—531no—
233C2×C53—21no—
234C2×C53—21no—
235C2×C53—21no—
236C2×C53—161no—
237C2×C53#264411yescharacteristic
238C22×SD16C2×D5311yescharacteristic
239#430—531no—
240#430—531no—
241#430—531no—
242#430—531no—
243#430—531no—
244#430—531no—
245C2×#108—531no—
246C2×#108—531no—
247C2×#108C2×D5311yescharacteristic
248#419—531no—
249#419—531no—
250#419C2×D5311yescharacteristic
251#418—531no—
252#418—531no—
253#418C2×D5311yescharacteristic
254C2×#101—531no—
255C2×#101—531no—
256C2×#101C2×D5311yescharacteristic
257C2×#100—531no—
258C2×#100—531no—
259C2×#100C2×D5311yescharacteristic
260#405—531no—
261#405—531no—
262#405C2×D5311yescharacteristic
263#344—531no—
264#344—531no—
265#344—531no—
266#344—531no—
267#344—531no—
268#344—531no—
269#338—531no—
270H(3,ℤ/4)—531no—
271H(3,ℤ/4)—531no—
272H(3,ℤ/4)—531no—
273C53#3023611yescharacteristic, complemented, Sylow
274C22×Q8C22×D5311yescharacteristic
275C22×D4C22×D5311yescharacteristic
276C2×SD16—21no—
277C2×SD16—21no—
278C2×SD16—21no—
279C2×SD16—21no—
280C2×SD16—21no—
281C2×SD16—21no—
282C2×M16—531no—
283C2×M16—531no—
284C2×M16—531no—
285C2×M16—531no—
286C2×M16—531no—
287C2×M16—531no—
288C2×M16C22×D5311yescharacteristic
289C2×M16C22×D5311yescharacteristic
290C2×M16C22×D5311yescharacteristic
291C22×C8C22×D5311yescharacteristic
292#117—531no—
293#117—531no—
294#117C22×D5311yescharacteristic
295#108—21no—
296#108—21no—
297#108—1061no—
298#108—1061no—
299#108—1061no—
300#108—1061no—
301#108—2121no—
302#108—2121no—
303#108—2121no—
304#108—2121no—
305#108—2121no—
306#108—2121no—
307#103—41no—
308#103—2121no—
309#103—2121no—
310#102—41no—
311#102—2121no—
312#102—2121no—
313#101—41no—
314#101—2121no—
315#101—2121no—
316#100—41no—
317#100—2121no—
318#100—2121no—
319#97—21no—
320#97—1061no—
321#97—1061no—
322C4×C8—21no—
323C4×C8—1061no—
324C4×C8—1061no—
325C24—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—21no—
338C22×C4—21no—
339C22×C4C23×D5311yescharacteristic
340SD16—41no—
341SD16—41no—
342SD16—41no—
343SD16—41no—
344M16—21no—
345M16—21no—
346M16—21no—
347M16—21no—
348M16—21no—
349M16—21no—
350M16—1061no—
351M16—1061no—
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—1061no—
366C2×C8—1061no—
367C2×C8—1061no—
368C2×C8—1061no—
369C2×C8—1061no—
370C2×C8—1061no—
371C2×C8#1171994511yescharacteristic
372C2×C8#1171994511yescharacteristic
373C2×C8#1171994711yescharacteristic
374C2×C8#1171994711yescharacteristic
375C2×C8#1171994711yescharacteristic
376C2×C8#1171994711yescharacteristic
377C4⋊C4—21no—
378C4⋊C4—1061no—
379C4⋊C4—1061no—
380C22⋊C4—21no—
381C22⋊C4—1061no—
382C22⋊C4—1061no—
383C42—21no—
384C42—1061no—
385C42—1061no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—161no—
390C23#1313326411yescharacteristic
391Q8—41no—
392Q8—41no—
393Q8—41no—
394Q8—41no—
395D4—41no—
396D4—41no—
397D4—41no—
398D4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—2121no—
404C2×C4—2121no—
405C2×C4C2×#1171994711yescharacteristic
406C2×C4C2×#1171994711yescharacteristic
407C2×C4C2×#1171994511yescharacteristic
408C2×C4#1313326211yescharacteristic
409C2×C4#1313326211yescharacteristic
410C2×C4#1313326311yescharacteristic
411C8—41no—
412C8—41no—
413C8—41no—
414C8—2121no—
415C8—2121no—
416C8—2121no—
417C8—2121no—
418C8—2121no—
419C8—2121no—
420C8#1313321011yescharacteristic
421C8#1313321011yescharacteristic
422C8#1313321011yescharacteristic
423C8#1313321011yescharacteristic
424C22—41no—
425C22—81no—
426C22—161no—
427C22—161no—
428C22—161no—
429C22#1681123811yescharacteristic
430C22#1681124011yescharacteristic
431C22#1681124211yescharacteristic
432C4—81no—
433C4—81no—
434C4—4241no—
435C4—4241no—
436C4C2×#1313321011yescharacteristic
437C4#1681122411yescharacteristic
438C4#1681122611yescharacteristic
439C4#1681122711yescharacteristic
440C2—21no—
441C2—21no—
442C2—21no—
443C2—161no—
444C2#2028067611yescharacteristic
subgroupG/H
centerC2#20280676
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1977C2
socleC2×C53#2644
prime radicalC22×D4×C53C23
hypercenter#1977D53
hypocenterC53#30236
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1977

Complements

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

NH ≅ G/Ntype
1. #20276183C2semidirect
2. #20276182C2semidirect
3. #20276179C2semidirect
7. #20276116C2semidirect
11. #20275506C2semidirect
12. #20275505C2semidirect
13. #20275503C2semidirect
14. #20274470C2semidirect
273. C53#30236semidirect

Maximal subgroups

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

subgroupindex
#202761832
#202761822
#202761792
#202761762
#202761752
#202761722
#202761162
#202761132
#202756202
#202755062
#202755052
#202755032
#202744702
C53×#19772
#3023653