Finite Group Databasebeta

← #27468200

Subgroups — #27468200

Order 13568.

438 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20276298C211yesmaximal, characteristic, complemented
2#20276294C211yesmaximal, characteristic, complemented
3#20276168C211yesmaximal, characteristic
4#20276165C211yesmaximal, characteristic
5#20275502C211yesmaximal, characteristic, complemented
6#20275467C211yesmaximal, characteristic, complemented
7#20275381C211yesmaximal, characteristic
8#20274864C211yesmaximal, characteristic, complemented
9#20274861C211yesmaximal, characteristic, complemented
10#20274836C211yesmaximal, characteristic, complemented
11#20274767C211yesmaximal, characteristic
12#20274763C211yesmaximal, characteristic
13#20274622C211yesmaximal, characteristic, complemented
14#20274560C211yesmaximal, characteristic
15C53×#1932C211yesmaximal, characteristic
16C2×#13133224C2211yescharacteristic
17#16810671C2211yescharacteristic
18C2×#13133152C2211yescharacteristic
19C2×#13133152C2211yescharacteristic
20C2×#13133152C2211yescharacteristic
21#16810659C2211yescharacteristic
22#16810651C2211yescharacteristic
23#16810650C2211yescharacteristic
24#16810640C2211yescharacteristic
25#16810619C2211yescharacteristic
26#16810607C2211yescharacteristic
27#16810581C2211yescharacteristic
28C4×#11719917C2211yescharacteristic
29C2×#13133128C2211yescharacteristic
30#16810395C2211yescharacteristic
31#16810394C2211yescharacteristic
32#16810394C2211yescharacteristic
33#16810394C2211yescharacteristic
34#16810393C2211yescharacteristic
35#16810387C2211yescharacteristic
36#16810387C2211yescharacteristic
37#16810387C2211yescharacteristic
38#16810383C2211yescharacteristic
39#16810383C2211yescharacteristic
40#16810381C2211yescharacteristic
41#16810381C2211yescharacteristic
42#16810380C2211yescharacteristic
43#16810379C2211yescharacteristic
44C53×#430C2211yescharacteristic
45C53×#421C2211yescharacteristic
46C2×#104×C53C2211yescharacteristic
47C53×#413C2211yescharacteristic
48C4×M16×C53C2211yescharacteristic
49C53×#344C2211yescharacteristic
50C53×#339C2211yescharacteristic
51#13133224—21no—
52#13133224—21no—
53#13133224—21no—
54#13133224—21no—
55C2×#11719917—21no—
56C2×#11719917C2311yescharacteristic
57C2×#11719917C2311yescharacteristic
58C2×#11719917C2311yescharacteristic
59C2×#11719917C2311yescharacteristic
60C22×#88322—21no—
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63#13133176C2311yescharacteristic
64#13133176C2311yescharacteristic
65#13133152—21no—
66#13133152—21no—
67#13133152—21no—
68#13133152—21no—
69#13133152—21no—
70#13133152—21no—
71#13133152—41no—
72#13133150—21no—
73#13133148—21no—
74C4⋊#88322—41no—
75#13133146—41no—
76#13133144—41no—
77#13133141—21no—
78#13133138—21no—
79#13133136—21no—
80#13133134—21no—
81#13133128—21no—
82#13133128—21no—
83#13133128—21no—
84#13133127—21no—
85C4×#88322—21no—
86C8×Dic53—21no—
87C2×#41×C53C2311yescharacteristic
88C2×M16×C53C2311yescharacteristic
89C2×M16×C53C2311yescharacteristic
90C2×M16×C53C2311yescharacteristic
91C2×M16×C53C2311yescharacteristic
92#117×C53C2311yescharacteristic
93C2×C42×C53C2311yescharacteristic
94#108×C53—41no—
95#104×C53—21no—
96#104×C53—21no—
97#104×C53—41no—
98#101×C53—21no—
99#100×C53—21no—
100#97×C53—21no—
101C4×C8×C53—21no—
102#41×C53—21no—
103#41×C53—21no—
104#41×C53—21no—
105#41×C53—21no—
106C2×Q8×C53#4111yescharacteristic
107C2×D4×C53—21no—
108C2×D4×C53#4111yescharacteristic
109C22×C4×C53—21no—
110C22×C4×C53—21no—
111C22×C4×C53C2411yescharacteristic
112M16×C53—21no—
113M16×C53—21no—
114M16×C53—21no—
115M16×C53—21no—
116M16×C53—21no—
117M16×C53—21no—
118M16×C53—21no—
119M16×C53—21no—
120C2×C8×C53—21no—
121C2×C8×C53—21no—
122C2×C8×C53—21no—
123C2×C8×C53—21no—
124C4⋊C4×C53—21no—
125C22⋊C4×C53—21no—
126C42×C53—21no—
127C42×C53—21no—
128C42×C53#4111yescharacteristic
129C42×C53#4111yescharacteristic
130C22⋊Dic53—21no—
131C22⋊Dic53—21no—
132C4⋊Dic53—21no—
133C4⋊Dic53—21no—
134C4×Dic53—21no—
135C4×Dic53—21no—
136#11719917—21no—
137#11719917—21no—
138#11719917—21no—
139#11719917—21no—
140#11719917—21no—
141#11719917—21no—
142#11719917—21no—
143#11719917—21no—
144#11719917—41no—
145#11719917—41no—
146C2×#88322—21no—
147C2×#88322—21no—
148C2×#88322—21no—
149C2×#88322—21no—
150C2×#88322—21no—
151C2×#88322—21no—
152C2×#88322—21no—
153C2×#88322—41no—
154C2×#88322—41no—
155C2×#88322#4111yescharacteristic
156C2×#88322#4111yescharacteristic
157C2×#88322#4111yescharacteristic
158C2×#88322#4111yescharacteristic
159C2×#88322#4111yescharacteristic
160C2×#88322#4111yescharacteristic
161C23×C53—21no—
162C23×C532−1+411yescharacteristic
163Q8×C53—21no—
164Q8×C53—21no—
165D4×C53—21no—
166D4×C53—21no—
167D4×C53—41no—
168D4×C53—41no—
169C2×C4×C53—21no—
170C2×C4×C53—21no—
171C2×C4×C53—21no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53—81no—
176C2×C4×C53C2×#4111yescharacteristic
177C2×C4×C53C2×#4111yescharacteristic
178C2×C4×C53C2×#4111yescharacteristic
179C2×C4×C53C2×#4111yescharacteristic
180C2×C4×C53C2×#4111yescharacteristic
181C2×C4×C53H(5,𝔽2)11yescharacteristic
182C2×Dic53—41no—
183C2×Dic53—41no—
184C8×C53—41no—
185C8×C53—41no—
186C8×C53—41no—
187C8×C53—41no—
188#88322—41no—
189#88322—41no—
190#88322—41no—
191#88322—41no—
192#88322—41no—
193#88322#12611yescharacteristic
194#88322#12611yescharacteristic
195#88322#12611yescharacteristic
196#88322#12611yescharacteristic
197#27834—531nomaximal, Sylow
198C22×C53—41no—
199C22×C53—41no—
200C22×C53—81no—
201C22×C53#54211yescharacteristic
202C22×C53#55211yescharacteristic
203C22×C53#55211yescharacteristic
204C4×C53—41no—
205C4×C53—41no—
206C4×C53—41no—
207C4×C53—81no—
208C4×C53C2×#12611yescharacteristic
209C4×C53#52911yescharacteristic
210C4×C53#55311yescharacteristic
211C4×C53#55511yescharacteristic
212Dic53—81no—
213Dic53—81no—
214#2103—531no—
215#2103—531no—
216#2080—531no—
217#2080—531no—
218#1953—531no—
219#1946—531no—
220#1932—531no—
221#1932D5311yescharacteristic
222#1841—531no—
223#1841—531no—
224#1833—531no—
225#1814—531no—
226#1813—531no—
227#1778—531no—
228#1765—531no—
229C2×C53—21no—
230C2×C53—21no—
231C2×C53—21no—
232C2×C53—81no—
233C2×C53#249511yescharacteristic
234C2×#131—531no—
235#430—531no—
236#430C2×D5311yescharacteristic
237C2×#108—531no—
238C2×#108—531no—
239C2×#108—531no—
240#427—531no—
241#424—531no—
242#424—531no—
243#421—531no—
244#421C2×D5311yescharacteristic
245C2×#104C2×D5311yescharacteristic
246#416—531no—
247#413—531no—
248#413C2×D5311yescharacteristic
249#405—531no—
250C4×M16—531no—
251C4×M16C2×D5311yescharacteristic
252C2×#97—531no—
253#344—531no—
254#344—531no—
255#344—531no—
256#344—531no—
257#344—531no—
258#344C2×D5311yescharacteristic
259#341—531no—
260#341—531no—
261#341—531no—
262#339—531no—
263#339—531no—
264#339C2×D5311yescharacteristic
265#338—531no—
266#338—531no—
267H(3,ℤ/4)—531no—
268H(3,ℤ/4)—531no—
269C53#2783411yescharacteristic, complemented, Sylow
270C2×#41C22×D5311yescharacteristic
271#131—1061no—
272#131—1061no—
273#131—1061no—
274#131—1061no—
275C2×M16—531no—
276C2×M16—531no—
277C2×M16—531no—
278C2×M16—531no—
279C2×M16—1061no—
280C2×M16C22×D5311yescharacteristic
281C2×M16C22×D5311yescharacteristic
282C2×M16C22×D5311yescharacteristic
283C2×M16C22×D5311yescharacteristic
284C22×C8—531no—
285C22×C8—531no—
286C22×C8—1061no—
287#117—531no—
288#117—531no—
289#117C22×D5311yescharacteristic
290C2×C42C22×D5311yescharacteristic
291#108—41no—
292#108—1061no—
293#108—1061no—
294#108—1061no—
295#108—1061no—
296#108—1061no—
297#108—1061no—
298#108—2121no—
299#107—1061no—
300#106—1061no—
301C4⋊C8—2121no—
302C4⋊C8—2121no—
303#104—21no—
304#104—21no—
305#104—41no—
306#104—2121no—
307#103—1061no—
308#102—1061no—
309#101—21no—
310#101—1061no—
311#100—21no—
312#100—1061no—
313#97—21no—
314#97—1061no—
315#97—1061no—
316#97—1061no—
317#97—1061no—
318C4×C8—21no—
319C4×C8—1061no—
320C4×C8—1061no—
321#41—21no—
322#41—21no—
323#41—21no—
324#41—21no—
325C2×Q8#1171994511yescharacteristic
326C2×D4—21no—
327C2×D4#1171994511yescharacteristic
328C22×C4—21no—
329C22×C4—21no—
330C22×C4C23×D5311yescharacteristic
331M16—21no—
332M16—21no—
333M16—21no—
334M16—21no—
335M16—21no—
336M16—21no—
337M16—21no—
338M16—21no—
339M16—1061no—
340M16—1061no—
341M16—1061no—
342M16—1061no—
343M16—1061no—
344M16—1061no—
345M16—1061no—
346M16—1061no—
347M16—2121no—
348M16—2121no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—531no—
354C2×C8—531no—
355C2×C8—531no—
356C2×C8—531no—
357C2×C8—531no—
358C2×C8—531no—
359C2×C8—1061no—
360C2×C8—1061no—
361C2×C8—1061no—
362C2×C8—1061no—
363C2×C8—1061no—
364C2×C8—1061no—
365C2×C8—1061no—
366C2×C8—2121no—
367C2×C8—2121no—
368C4⋊C4—21no—
369C4⋊C4—1061no—
370C4⋊C4—1061no—
371C22⋊C4—21no—
372C22⋊C4—1061no—
373C22⋊C4—1061no—
374C42—21no—
375C42—21no—
376C42—1061no—
377C42—1061no—
378C42#1171994711yescharacteristic
379C42#1171994711yescharacteristic
380C23—21no—
381C23#1313326411yescharacteristic
382Q8—21no—
383Q8—21no—
384D4—21no—
385D4—21no—
386D4—41no—
387D4—41no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—81no—
395C2×C4—2121no—
396C2×C4—2121no—
397C2×C4#41×D5311yescharacteristic
398C2×C4#41×D5311yescharacteristic
399C2×C4#41×D5311yescharacteristic
400C2×C4C2×#1171994711yescharacteristic
401C2×C4C2×#1171994511yescharacteristic
402C2×C4#1313326211yescharacteristic
403C8—41no—
404C8—41no—
405C8—41no—
406C8—41no—
407C8—531no—
408C8—531no—
409C8—531no—
410C8—531no—
411C8—2121no—
412C8—2121no—
413C8—2121no—
414C8—2121no—
415C8—2121no—
416C22—41no—
417C22—41no—
418C22—81no—
419C22#1681123711yescharacteristic
420C22#1681130611yescharacteristic
421C22#1681131011yescharacteristic
422C4—41no—
423C4—41no—
424C4—41no—
425C4—81no—
426C4—4241no—
427C4—4241no—
428C4#126×D5311yescharacteristic
429C4#1681115311yescharacteristic
430C4#1681132211yescharacteristic
431C4#1681133711yescharacteristic
432C2—21no—
433C2—21no—
434C2—21no—
435C2—81no—
436C2#2027912311yescharacteristic
subgroupG/H
centerC4#16811337
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1932C2
socleC2×C53#2495
prime radicalC2×#41×C53C23
hypercenter#1932D53
hypocenterC53#27834
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1932

Complements

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

NH ≅ G/Ntype
1. #20276298C2semidirect
2. #20276294C2semidirect
5. #20275502C2semidirect
6. #20275467C2semidirect
8. #20274864C2semidirect
9. #20274861C2semidirect
10. #20274836C2semidirect
13. #20274622C2semidirect
269. C53#27834semidirect

Maximal subgroups

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

subgroupindex
#202762982
#202762942
#202761682
#202761652
#202755022
#202754672
#202753812
#202748642
#202748612
#202748362
#202747672
#202747632
#202746222
#202745602
C53×#19322
#2783453