Finite Group Databasebeta

← #27538617

Subgroups — #27538617

Order 13568.

674 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811053C211yesmaximal, characteristic, complemented
2C2×#16811040C211yesmaximal, characteristic, complemented
3#20284385C211yesmaximal, characteristic, complemented
4#20276821C212yesmaximal, complemented
5#20276816C212yesmaximal, complemented
6#20276813C212yesmaximal, complemented
7#20276809C212yesmaximal, complemented
8C2×#16810450C211yesmaximal, characteristic, complemented
9#20276427C211yesmaximal, characteristic, complemented
10#20276425C211yesmaximal, characteristic, complemented
11C53×#3061C211yesmaximal, characteristic, complemented
12C2×#13133239C2211yescharacteristic
13#16811201—21no—
14#16811053—24no—
15C2×#13133163C2211yescharacteristic
16C2×#13133161C2211yescharacteristic
17C2×#13133160C2211yescharacteristic
18C2×#13133158C2211yescharacteristic
19#16811040—24no—
20C2×#13133157C2212yes—
21C2×#13133157C2211yescharacteristic
22C22×#11719908C2212yes—
23C22×#11719908C2211yescharacteristic
24#16810869—22no—
25#16810864C2212yes—
26#16810859C2212yes—
27C2×#13133151C2211yescharacteristic
28#16810625—21no—
29#16810450C2212yes—
30#16810450C2212yes—
31#16810427C2212yes—
32#16810426C2212yes—
33#16810424C2212yes—
34#16810423C2212yes—
35C2×#135×C53C2211yescharacteristic
36C2×#131×C53C2211yescharacteristic
37C53×#537—21no—
38C53×#468—22no—
39C53×#467C2212yes—
40C53×#466C2212yes—
41C2×#107×C53C2211yescharacteristic
42C53×#417—21no—
43C2×#11719945—21no—
44#13133239—22no—
45#13133239—22no—
46C2×Dic212C2311yescharacteristic
47C2×#11719913—21no—
48C2×D424C2311yescharacteristic
49#13133196—22no—
50#13133193—41no—
51#13133191—22no—
52#13133189—42no—
53#13133188—41no—
54C2×C4⋊Dic53—21no—
55#13133163—22no—
56#13133161—22no—
57#13133160—22no—
58#13133158—22no—
59#13133157—22no—
60#13133157—24no—
61C2×#11719908—24no—
62C2×#11719908D414yescomplemented
63C2×#11719908C2312yes—
64C2×#11719908C2311yescharacteristic
65C2×#11719908C2311yescharacteristic
66#13133151—21no—
67#13133151C2312yes—
68#13133142—22no—
69#13133139—22no—
70C2×#41×C53—21no—
71C2×#41×C53D411yescharacteristic
72#135×C53—22no—
73#135×C53—22no—
74C2×Q16×C53C2311yescharacteristic
75C2×SD16×C53—21no—
76C2×D8×C53C2311yescharacteristic
77#131×C53—22no—
78#131×C53—22no—
79C2×M16×C53D412yes—
80C2×M16×C53D411yescharacteristic
81C22×C8×C53C2312yes—
82C22×C8×C53C2311yescharacteristic
83C22⋊Q8×C53—22no—
84C22⋊Q8×C53—41no—
85#121×C53—22no—
86#121×C53—42no—
87#121×C53—41no—
88C2×C4⋊C4×C53—21no—
89#107×C53—21no—
90#107×C53C2312yes—
91#103×C53—22no—
92#102×C53—22no—
93#41×C53—22no—
94#41×C53—22no—
95#41×C53—42no—
96#41×C53—42no—
97C2×Q8×C53—21no—
98C2×Q8×C53D811yescharacteristic
99C2×D4×C53—21no—
100C2×D4×C53—41no—
101C2×D4×C53—42no—
102C2×D4×C53D811yescharacteristic
103C2×D4×C53D812yes—
104C22×C4×C53—41no—
105C22×C4×C53—41no—
106C22×C4×C53C2×D412yes—
107C22×C4×C53C2×D411yescharacteristic
108C2×C22⋊D53—41no—
109C2×C22⋊D53—42no—
110C22×Dic53—41no—
111#11719945—42no—
112#11719945—42no—
113C2×D212—21no—
114C2×C4×D53—41no—
115C2×Dic106—21no—
116Q16×C53—21no—
117Q16×C53—21no—
118SD16×C53—42no—
119D8×C53—21no—
120D8×C53—21no—
121M16×C53—22no—
122M16×C53—24no—
123C2×C8×C53—22no—
124C2×C8×C53—22no—
125C2×C8×C53C2×D412yes—
126C2×C8×C53C2×D411yescharacteristic
127C2×C8×C53C2×D412yes—
128C2×C8×C53#4112yes—
129C2×C8×C53C2411yescharacteristic
130C4⋊C4×C53—22no—
131C4⋊C4×C53—41no—
132C4⋊C4×C53—42no—
133C22⋊C4×C53—42no—
134C22⋊C4×C53—42no—
135C22⋊Dic53—42no—
136#11719921—42no—
137C4⋊Dic53—22no—
138C4⋊Dic53—41no—
139#11719919—42no—
140Dic212—21no—
141Dic212—21no—
142D424—21no—
143D424—21no—
144#11719913—42no—
145#11719908—22no—
146#11719908C2×D412yes—
147#11719908C2×D412yes—
148C23×C53—41no—
149C23×C53C2×D812yes—
150C23×C53C2×D811yescharacteristic
151C22×D53—41no—
152Q8×C53—22no—
153Q8×C53—41no—
154Q8×C53—41no—
155D4×C53—24no—
156D4×C53—22no—
157D4×C53—41no—
158D4×C53—41no—
159D4×C53—82no—
160D4×C53—162no—
161C2×C4×C53—22no—
162C2×C4×C53—22no—
163C2×C4×C53—41no—
164C2×C4×C53—42no—
165C2×C4×C53—82no—
166C2×C4×C53—161no—
167C2×C4×C53#12112yes—
168C2×C4×C53C2×D812yes—
169C2×C4×C53C2×D811yescharacteristic
170C2×C4×C53#13512yes—
171C2×C4×C53C22×D411yescharacteristic
172C22⋊D53—82no—
173C22⋊D53—162no—
174C2×Dic53—41no—
175C2×Dic53—42no—
176C2×Dic53—161no—
177D212—41no—
178D212—41no—
179C4×D53—82no—
180Dic106—41no—
181Dic106—41no—
182C8×C53—22no—
183C8×C53#12112yes—
184C8×C53C22×D411yescharacteristic
185C8×C53C2×#4111yescharacteristic
186#33611—531nomaximal, Sylow
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—22no—
190C22×C53—22no—
191C22×C53—22no—
192C22×C53—81no—
193C22×C53—81no—
194C22×C53—81no—
195C22×C53#46612yes—
196C22×C53C22×D811yescharacteristic
197C2×D53—81no—
198C2×D53—81no—
199C2×D53—81no—
200C4×C53—22no—
201C4×C53—82no—
202C4×C53—81no—
203C4×C53#46612yes—
204C4×C53C2×#12111yescharacteristic
205C4×C53C2×#13511yescharacteristic
206Dic53—82no—
207Dic53—81no—
208C2×#508—531no—
209C2×#504—531no—
210#3061—531no—
211#3061D5311yescharacteristic, complemented
212#2219—532no—
213#2218—532no—
214#2217—532no—
215#2217—532no—
216C2×#367—531no—
217#2139—531no—
218#2139—531no—
219C2×C53—22no—
220C2×C53—42no—
221C2×C53—161no—
222C2×C53C2×#46611yescharacteristic
223C2×C53#341611yescharacteristic
224C2×C53#341611yescharacteristic
225D53—161no—
226C2×#135—531no—
227C2×#135C2×D5311yescharacteristic
228C2×#131C2×D5311yescharacteristic
229#537—21no—
230#537—1061no—
231#508—1064no—
232C2×Q32—531no—
233C2×SD32—531no—
234C2×SD32—531no—
235C2×D16—531no—
236#504—1064no—
237C2×M32—531no—
238C2×M32—532no—
239C22×C16—531no—
240C22×C16—532no—
241#468—22no—
242#468—1062no—
243#467—532no—
244#467C2×D5312yes—
245#466—532no—
246#466C2×D5312yes—
247C2×#107—531no—
248C2×#107C2×D5311yescharacteristic
249#417—21no—
250#417—1061no—
251#367—532no—
252#367—532no—
253#358—532no—
254#358—532no—
255#357—532no—
256#357—532no—
257C53#3361111yescharacteristic, complemented, Sylow
258C2×#41—21no—
259C2×#41—1061no—
260C2×#41C22⋊D5311yescharacteristic
261#135—22no—
262#135—22no—
263#135—1062no—
264#135—1062no—
265C2×Q16—531no—
266C2×Q16C22×D5311yescharacteristic
267C2×SD16—21no—
268C2×SD16—1061no—
269C2×D8—531no—
270C2×D8C22×D5311yescharacteristic
271#131—22no—
272#131—22no—
273C2×M16C22⋊D5311yescharacteristic
274C2×M16C22⋊D5312yes—
275C22×C8C22×D5312yes—
276C22×C8C22×D5311yescharacteristic
277C22⋊Q8—22no—
278C22⋊Q8—41no—
279C22⋊Q8—1062no—
280C22⋊Q8—2121no—
281#121—22no—
282#121—41no—
283#121—42no—
284#121—1062no—
285#121—2121no—
286#121—2122no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—1061no—
289Q32—1062no—
290SD32—1062no—
291SD32—1062no—
292D16—1062no—
293M32—1064no—
294M32—1062no—
295C2×C16—534no—
296C2×C16—531no—
297C2×C16—531no—
298C2×C16—532no—
299C2×C16—1064no—
300#107—21no—
301#107—532no—
302#107—1061no—
303#107C22×D5312yes—
304#103—22no—
305#103—1062no—
306#102—22no—
307#102—1062no—
308#41—22no—
309#41—22no—
310#41—42no—
311#41—42no—
312#41—2122no—
313#41—2122no—
314C2×Q8—21no—
315C2×Q8—1061no—
316C2×Q8#1171992211yescharacteristic
317C2×D4—21no—
318C2×D4—41no—
319C2×D4—42no—
320C2×D4—1061no—
321C2×D4—2122no—
322C2×D4—2121no—
323C2×D4#1171992212yes—
324C2×D4#1171992211yescharacteristic
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—2121no—
328C22×C4—2121no—
329C22×C4C2×C22⋊D5312yes—
330C22×C4C2×C22⋊D5311yescharacteristic
331Q16—21no—
332Q16—21no—
333Q16—1061no—
334Q16—1061no—
335SD16—42no—
336SD16—2122no—
337D8—21no—
338D8—21no—
339D8—1061no—
340D8—1061no—
341M16—22no—
342M16—24no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8D4×D5312yes—
346C2×C8#1171994712yes—
347C2×C8C2×C22⋊D5312yes—
348C2×C8C2×C22⋊D5311yescharacteristic
349C2×C8C23×D5311yescharacteristic
350C4⋊C4—22no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—1062no—
354C4⋊C4—2121no—
355C4⋊C4—2122no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—2122no—
359C22⋊C4—2122no—
360C16—532no—
361C16—532no—
362C16—1062no—
363C23—41no—
364C23—2121no—
365C23C2×#1171992212yes—
366C23C2×#1171992211yescharacteristic
367Q8—22no—
368Q8—41no—
369Q8—41no—
370Q8—2121no—
371Q8—2121no—
372D4—24no—
373D4—22no—
374D4—41no—
375D4—41no—
376D4—82no—
377D4—162no—
378D4—8482no—
379D4—4242no—
380D4—2121no—
381D4—2121no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—82no—
387C2×C4—161no—
388C2×C4—8481no—
389C2×C4—4242no—
390C2×C4—2122no—
391C2×C4—2121no—
392C2×C4#1313318912yes—
393C2×C4C2×#1171992212yes—
394C2×C4C2×#1171992211yescharacteristic
395C2×C4#1313323712yes—
396C2×C4C22×C22⋊D5311yescharacteristic
397C8—22no—
398C8#1313318912yes—
399C8C2×D4×D5311yescharacteristic
400C8C2×#1171994711yescharacteristic
401C22—21no—
402C22—21no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—4241no—
410C22—4241no—
411C22—4241no—
412C22#1681085712yes—
413C22C22×#1171992211yescharacteristic
414C4—22no—
415C4—82no—
416C4—81no—
417C4—4242no—
418C4—4241no—
419C4#1681085712yes—
420C4C2×#1313318911yescharacteristic
421C4C2×#1313323711yescharacteristic
422C2—22no—
423C2—42no—
424C2—161no—
425C2—8481no—
426C2C2×#1681085711yescharacteristic
427C2#2028785911yescharacteristic
428C2#2028786211yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3061C2
socleC22×C53C22×D8
hypercenter#3061D53
hypocenterC53#33611
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ #3061

Complements

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

NH ≅ G/Ntype
1. C2×#16811053C2semidirect
2. C2×#16811040C2semidirect
3. #20284385C2semidirect
4. #20276821C2semidirect
5. #20276816C2semidirect
6. #20276813C2semidirect
7. #20276809C2semidirect
8. C2×#16810450C2semidirect
9. #20276427C2semidirect
10. #20276425C2semidirect
11. C53×#3061C2semidirect
62. C2×#11719908D4semidirect
211. #3061D53semidirect
257. C53#33611semidirect

Maximal subgroups

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

subgroupindex
C2×#168110532
C2×#168110402
#202843852
#202768212
#202768162
#202768132
#202768092
C2×#168104502
#202764272
#202764252
C53×#30612
#3361153