Finite Group Databasebeta

← #27538565

Subgroups — #27538565

Order 13568.

738 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811059C211yesmaximal, characteristic, complemented
2C22×#13133157C211yesmaximal, characteristic, complemented
3#20284296C211yesmaximal, characteristic, complemented
4#20276826C212yesmaximal, complemented
5#20276816C212yesmaximal, complemented
6#20276809C212yesmaximal, complemented
7#20276799C212yesmaximal, complemented
8#20276501C211yesmaximal, characteristic, complemented
9#20276431C211yesmaximal, characteristic, complemented
10#20276425C211yesmaximal, characteristic, complemented
11C2×C53×#466C211yesmaximal, characteristic, complemented
12C2×#13133239C2211yescharacteristic
13#16811134—21no—
14#16811059—24no—
15C2×#13133161C2212yescomplemented
16C2×#13133158C2212yescomplemented
17C2×#13133157—24no—
18C2×#13133157C2211yescharacteristic
19C2×#13133157C2211yescharacteristic
20C2×#13133157C2212yescomplemented
21C22×#11719908C2212yescomplemented
22#16810864C2211yescharacteristic, complemented
23#16810864C2211yescharacteristic, complemented
24#16810859C2211yescharacteristic, complemented
25#16810859C2211yescharacteristic, complemented
26#16810852—21no—
27#16810852—21no—
28#16810661C2211yescharacteristic
29#16810620—21no—
30#16810450C2212yescomplemented
31#16810448C2212yescomplemented
32#16810426C2212yescomplemented
33#16810424C2212yescomplemented
34#16810424C2212yescomplemented
35#16810423C2212yescomplemented
36C22×D8×C53C2211yescharacteristic
37C23×C8×C53C2211yescharacteristic
38C2×#121×C53—21no—
39C53×#466—21no—
40C53×#466—21no—
41C53×#466C2212yes—
42C53×#466C2212yes—
43C2×#107×C53C2211yescharacteristic
44C2×#102×C53—21no—
45C2×#11719945—21no—
46#13133239—22no—
47#13133239—22no—
48C2×Dic212C2311yescharacteristic
49C2×#11719913—21no—
50C2×D424C2311yescharacteristic
51#13133200—41no—
52#13133196—21no—
53#13133196—21no—
54#13133191—21no—
55#13133191—21no—
56#13133186—41no—
57C4×C22⋊D53—41no—
58C4×C22⋊D53—41no—
59#13133176—21no—
60#13133161—24no—
61#13133158—24no—
62#13133157—22no—
63#13133157—24no—
64#13133157D414yescomplemented
65C2×#11719908—24no—
66C2×#11719908C2312yes—
67C2×#11719908C2312yes—
68#13133151C2311yescharacteristic
69#13133151C2311yescharacteristic
70#13133149—21no—
71#13133142—21no—
72#13133142—21no—
73#13133139—21no—
74#13133139—21no—
75C22×D4×C53—21no—
76C22×D4×C53—41no—
77C23×C4×C53D411yescharacteristic
78C2×D8×C53—21no—
79C2×D8×C53—22no—
80C2×D8×C53—22no—
81C2×D8×C53C2312yes—
82C22×C8×C53—22no—
83C22×C8×C53—22no—
84C22×C8×C53D411yescharacteristic
85C22×C8×C53D411yescharacteristic
86C22×C8×C53D411yescharacteristic
87C22×C8×C53C2311yescharacteristic
88C22×C8×C53C2311yescharacteristic
89C22×C8×C53C2311yescharacteristic
90#121×C53—22no—
91#121×C53—22no—
92#121×C53—41no—
93#121×C53—41no—
94C2×C4⋊C4×C53—21no—
95C2×C22⋊C4×C53—41no—
96#107×C53—21no—
97#107×C53C2312yes—
98#102×C53—22no—
99#102×C53—22no—
100C24×C53—41no—
101C24×C53D811yescharacteristic
102C2×D4×C53—22no—
103C2×D4×C53—42no—
104C2×D4×C53—42no—
105C2×D4×C53—42no—
106C2×D4×C53—41no—
107C2×D4×C53—42no—
108C2×D4×C53—84no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—41no—
112C22×C4×C53D811yescharacteristic
113C22×C4×C53D811yescharacteristic
114C22×C4×C53D811yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D411yescharacteristic
118C2×C22⋊D53—41no—
119C2×C22⋊D53—41no—
120C2×C22⋊D53—41no—
121#11719945—42no—
122#11719945—42no—
123C2×D212—21no—
124C2×C4×D53—41no—
125C2×Dic106—21no—
126D8×C53—22no—
127D8×C53—22no—
128D8×C53—42no—
129C2×C8×C53—22no—
130C2×C8×C53—22no—
131C2×C8×C53—22no—
132C2×C8×C53—21no—
133C2×C8×C53—21no—
134C2×C8×C53—21no—
135C2×C8×C53—21no—
136C2×C8×C53C2×D411yescharacteristic
137C2×C8×C53C2×D411yescharacteristic
138C2×C8×C53C2×D411yescharacteristic
139C2×C8×C53C2×D412yes—
140C2×C8×C53#4112yes—
141C2×C8×C53C2411yescharacteristic
142C4⋊C4×C53—22no—
143C4⋊C4×C53—41no—
144C22⋊C4×C53—42no—
145C22⋊C4×C53—42no—
146C22⋊Dic53—41no—
147C22⋊Dic53—41no—
148C22⋊Dic53—41no—
149#11719921—41no—
150#11719921—41no—
151C4⋊Dic53—21no—
152C4⋊Dic53—21no—
153#11719919—41no—
154#11719919—41no—
155C4×Dic53—41no—
156Dic212—21no—
157Dic212—21no—
158D424—21no—
159D424—21no—
160#11719913—42no—
161#11719908—22no—
162#11719908C2×D414yescomplemented
163C23×C53—22no—
164C23×C53—22no—
165C23×C53—41no—
166C23×C53—42no—
167C23×C53—82no—
168C23×C53—82no—
169C23×C53—82no—
170C23×C53C2×D811yescharacteristic
171C23×C53C2×D811yescharacteristic
172C23×C53C2×D811yescharacteristic
173C22×D53—41no—
174D4×C53—42no—
175D4×C53—42no—
176D4×C53—84no—
177D4×C53—84no—
178D4×C53—82no—
179C2×C4×C53—22no—
180C2×C4×C53—21no—
181C2×C4×C53—21no—
182C2×C4×C53—21no—
183C2×C4×C53—21no—
184C2×C4×C53—22no—
185C2×C4×C53—22no—
186C2×C4×C53—42no—
187C2×C4×C53—82no—
188C2×C4×C53#12112yes—
189C2×C4×C53C2×D811yescharacteristic
190C2×C4×C53C2×D811yescharacteristic
191C2×C4×C53C2×D811yescharacteristic
192C2×C4×C53#13512yes—
193C2×C4×C53C22×D411yescharacteristic
194C22⋊D53—82no—
195C22⋊D53—82no—
196C22⋊D53—82no—
197C2×Dic53—41no—
198C2×Dic53—41no—
199C2×Dic53—41no—
200D212—41no—
201D212—41no—
202C4×D53—82no—
203Dic106—41no—
204Dic106—41no—
205C8×C53—21no—
206C8×C53—21no—
207C8×C53#12112yes—
208C8×C53C22×D411yescharacteristic
209C8×C53C2×#4111yescharacteristic
210#33605—531nomaximal, Sylow
211C22×C53—21no—
212C22×C53—21no—
213C22×C53—21no—
214C22×C53—21no—
215C22×C53—21no—
216C22×C53—21no—
217C22×C53—21no—
218C22×C53—44no—
219C22×C53—82no—
220C22×C53—82no—
221C22×C53—82no—
222C22×C53—164no—
223C22×C53#46612yes—
224C22×C53#50912yescomplemented
225C22×C53C22×D811yescharacteristic
226C2×D53—81no—
227C2×D53—81no—
228C2×D53—81no—
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—82no—
232C4×C53#46612yes—
233C4×C53C2×#12111yescharacteristic
234C4×C53C2×#13511yescharacteristic
235Dic53—81no—
236Dic53—81no—
237Dic53—81no—
238C2×#509—531no—
239C22×M32—531no—
240#3050—531no—
241C2×#466D5311yescharacteristic, complemented
242#2220—532no—
243#2218—532no—
244#2217—532no—
245#2215—532no—
246#2159—531no—
247#2140—531no—
248#2139—531no—
249C2×C53—22no—
250C2×C53—22no—
251C2×C53—41no—
252C2×C53—162no—
253C2×C53C2×#46611yescharacteristic
254C2×C53C2×#50911yescharacteristic
255C2×C53#341411yescharacteristic
256D53—161no—
257C2×#135—531no—
258C22×D8C2×D5311yescharacteristic
259C23×C8C2×D5311yescharacteristic
260#525—1061no—
261C2×#121—21no—
262#509—1064no—
263C2×SD32—532no—
264C2×D16—532no—
265C2×M32—531no—
266C2×M32—531no—
267C2×M32—532no—
268C2×M32—1064no—
269C22×C16—532no—
270#467—531no—
271#467—531no—
272#466—21no—
273#466—21no—
274#466—531no—
275#466—531no—
276#466C2×D5312yes—
277#466C2×D5312yes—
278#465—1061no—
279#465—1061no—
280#427—531no—
281C2×#107C2×D5311yescharacteristic
282#416—1061no—
283C2×#102—21no—
284#367—532no—
285#366—532no—
286#358—532no—
287#357—532no—
288#357—532no—
289#357—532no—
290C53#3360511yescharacteristic, complemented, Sylow
291C2×#41—1061no—
292C22×D4—21no—
293C22×D4—41no—
294C23×C4C22⋊D5311yescharacteristic
295#135—1062no—
296#135—1062no—
297C2×Q16—531no—
298C2×SD16—1061no—
299C2×D8—22no—
300C2×D8—22no—
301C2×D8—21no—
302C2×D8—531no—
303C2×D8C22×D5312yes—
304C22×C8—22no—
305C22×C8—22no—
306C22×C8C22⋊D5311yescharacteristic
307C22×C8C22⋊D5311yescharacteristic
308C22×C8C22⋊D5311yescharacteristic
309C22×C8C22×D5311yescharacteristic
310C22×C8C22×D5311yescharacteristic
311C22×C8C22×D5311yescharacteristic
312#123—2121no—
313C22⋊Q8—1061no—
314C22⋊Q8—1061no—
315#121—22no—
316#121—22no—
317#121—41no—
318#121—41no—
319#121—1061no—
320#121—1061no—
321#120—2121no—
322C4×D4—2121no—
323C4×D4—2121no—
324#117—1061no—
325C2×C4⋊C4—21no—
326C2×C22⋊C4—41no—
327SD32—1064no—
328D16—1064no—
329M32—534no—
330M32—1064no—
331M32—1062no—
332C2×C16—532no—
333C2×C16—532no—
334C2×C16—1064no—
335#107—21no—
336#107—531no—
337#107—531no—
338#107C22×D5312yes—
339#106—1061no—
340#103—1061no—
341#103—1061no—
342#102—22no—
343#102—22no—
344#102—1061no—
345#102—1061no—
346C24—41no—
347C24#1171992211yescharacteristic
348#41—2122no—
349#41—2122no—
350C2×Q8—1061no—
351C2×D4—22no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—41no—
356C2×D4—42no—
357C2×D4—84no—
358C2×D4—1061no—
359C2×D4—2121no—
360C2×D4—2121no—
361C2×D4—2121no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—41no—
365C22×C4—2121no—
366C22×C4#1171992211yescharacteristic
367C22×C4#1171992211yescharacteristic
368C22×C4#1171992211yescharacteristic
369C22×C4C2×C22⋊D5311yescharacteristic
370C22×C4C2×C22⋊D5311yescharacteristic
371C22×C4C2×C22⋊D5311yescharacteristic
372Q16—1061no—
373Q16—1061no—
374SD16—2122no—
375D8—22no—
376D8—22no—
377D8—42no—
378D8—1061no—
379D8—1061no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8D4×D5312yes—
388C2×C8#1171994712yes—
389C2×C8C2×C22⋊D5311yescharacteristic
390C2×C8C2×C22⋊D5311yescharacteristic
391C2×C8C2×C22⋊D5311yescharacteristic
392C2×C8C23×D5311yescharacteristic
393C4⋊C4—22no—
394C4⋊C4—41no—
395C4⋊C4—1061no—
396C4⋊C4—1061no—
397C4⋊C4—2121no—
398C4⋊C4—2121no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—2121no—
402C22⋊C4—2121no—
403C22⋊C4—2121no—
404C22⋊C4—2121no—
405C22⋊C4—2121no—
406C42—2121no—
407C16—534no—
408C16—1062no—
409C23—22no—
410C23—22no—
411C23—41no—
412C23—42no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23—2121no—
417C23C2×#1171992211yescharacteristic
418C23C2×#1171992211yescharacteristic
419C23C2×#1171992211yescharacteristic
420Q8—2121no—
421Q8—2121no—
422D4—42no—
423D4—42no—
424D4—84no—
425D4—84no—
426D4—82no—
427D4—4242no—
428D4—4242no—
429D4—4242no—
430D4—2121no—
431D4—2121no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—42no—
440C2×C4—82no—
441C2×C4—4242no—
442C2×C4—2121no—
443C2×C4—2121no—
444C2×C4—2121no—
445C2×C4#1313318912yes—
446C2×C4C2×#1171992211yescharacteristic
447C2×C4C2×#1171992211yescharacteristic
448C2×C4C2×#1171992211yescharacteristic
449C2×C4#1313323712yes—
450C2×C4C22×C22⋊D5311yescharacteristic
451C8—21no—
452C8—21no—
453C8#1313318912yes—
454C8C2×D4×D5311yescharacteristic
455C8C2×#1171994711yescharacteristic
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—44no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—164no—
468C22—4241no—
469C22—4241no—
470C22—4241no—
471C22#1681085712yes—
472C22#1681105912yescomplemented
473C22C22×#1171992211yescharacteristic
474C4—21no—
475C4—21no—
476C4—82no—
477C4—4241no—
478C4—4241no—
479C4—4241no—
480C4#1681085712yes—
481C4C2×#1313318911yescharacteristic
482C4C2×#1313323711yescharacteristic
483C2—22no—
484C2—22no—
485C2—41no—
486C2—162no—
487C2—8481no—
488C2C2×#1681085711yescharacteristic
489C2C2×#1681105911yescharacteristic
490C2#2028784911yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC2×C53×#466C2
socleC22×C53C22×D8
hypercenterC2×#466D53
hypocenterC53#33605
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C2×#466

Complements

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

NH ≅ G/Ntype
1. C2×#16811059C2semidirect
2. C22×#13133157C2semidirect
3. #20284296C2semidirect
4. #20276826C2semidirect
5. #20276816C2semidirect
6. #20276809C2semidirect
7. #20276799C2semidirect
8. #20276501C2semidirect
9. #20276431C2semidirect
10. #20276425C2semidirect
11. C2×C53×#466C2semidirect
15. C2×#13133161C22semidirect
16. C2×#13133158C22semidirect
20. C2×#13133157C22semidirect
21. C22×#11719908C22semidirect
22. #16810864C22semidirect
23. #16810864C22semidirect
24. #16810859C22semidirect
25. #16810859C22semidirect
30. #16810450C22semidirect
31. #16810448C22semidirect
32. #16810426C22semidirect
33. #16810424C22semidirect
34. #16810424C22semidirect
35. #16810423C22semidirect
64. #13133157D4semidirect
162. #11719908C2×D4semidirect
224. C22×C53#509semidirect
241. C2×#466D53semidirect
290. C53#33605semidirect
472. C22#16811059semidirect

Maximal subgroups

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

subgroupindex
C2×#168110592
C22×#131331572
#202842962
#202768262
#202768162
#202768092
#202767992
#202765012
#202764312
#202764252
C2×C53×#4662
#3360553