Finite Group Databasebeta

← #27539262

Subgroups — #27539262

Order 13568.

556 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285367C211yesmaximal, characteristic, complemented
2#20283567C211yesmaximal, characteristic
3#20276924C211yesmaximal, characteristic, complemented
4#20276920C211yesmaximal, characteristic, complemented
5#20276874C211yesmaximal, characteristic, complemented
6#20276870C211yesmaximal, characteristic, complemented
7#20276758C211yesmaximal, characteristic, complemented
8#20276758C211yesmaximal, characteristic, complemented
9#20276644C211yesmaximal, characteristic
10#20276580C211yesmaximal, characteristic, complemented
11#20276576C211yesmaximal, characteristic, complemented
12#20276570C211yesmaximal, characteristic, complemented
13SD32×Dic53C211yesmaximal, characteristic, complemented
14#20276519C211yesmaximal, characteristic, complemented
15C53×#2187C211yesmaximal, characteristic
16#16811233—21no—
17#16811098—21no—
18C2×#13133161C2211yescharacteristic
19C2×#13133161C2211yescharacteristic
20#16811024C2211yescharacteristic, complemented
21#16811001C2211yescharacteristic, complemented
22#16810990—21no—
23#16810879—21no—
24#16810864C2211yescharacteristic
25#16810735—21no—
26#16810735—21no—
27#16810727—21no—
28Q16×Dic53C2211yescharacteristic, complemented
29#16810717C2211yescharacteristic, complemented
30D8×Dic53C2211yescharacteristic
31D8×Dic53C2211yescharacteristic
32D8×Dic53C2211yescharacteristic
33D8×Dic53C2211yescharacteristic
34#16810661C2211yescharacteristic
35#16810628—21no—
36#16810578C2211yescharacteristic
37#16810451C2211yescharacteristic, complemented
38#16810450C2211yescharacteristic, complemented
39#16810449C2211yescharacteristic, complemented
40#16810448C2211yescharacteristic, complemented
41#16810441C2211yescharacteristic, complemented
42#16810441C2211yescharacteristic, complemented
43#16810428C2211yescharacteristic, complemented
44#16810428C2211yescharacteristic, complemented
45#16810427C2211yescharacteristic, complemented
46#16810427C2211yescharacteristic, complemented
47#16810424C2211yescharacteristic
48#16810424C2211yescharacteristic
49C22⋊#11719908C2211yescharacteristic
50#16810398C2211yescharacteristic, complemented
51C16×Dic53C2211yescharacteristic, complemented
52C22×D8×C53C2211yescharacteristic
53C2×SD32×C53C2211yescharacteristic
54C2×SD32×C53C2211yescharacteristic
55C53×#467C2211yescharacteristic
56C53×#357C2211yescharacteristic
57C53×#357C2211yescharacteristic
58C53×C22⋊C16C2211yescharacteristic
59C2×Dic212C2311yescharacteristic
60C2×#11719925—21no—
61C2×#11719917D411yescharacteristic
62#13133215—21no—
63#13133210—41no—
64#13133210—41no—
65#13133208—21no—
66#13133208—41no—
67#13133207—41no—
68#13133196—21no—
69Q8×Dic53—21no—
70#13133182—21no—
71D4×Dic53—21no—
72D4×Dic53—21no—
73D4×Dic53—21no—
74D4×Dic53—21no—
75D4×Dic53—41no—
76D4×Dic53—41no—
77#13133176—21no—
78#13133176D411yescharacteristic
79C2×C22⋊Dic53—41no—
80C2×C22⋊Dic53—41no—
81#13133161—22no—
82#13133161—22no—
83C2×#11719908C2311yescharacteristic
84C2×#11719908C2311yescharacteristic
85#13133151C2311yescharacteristic
86#13133151C2311yescharacteristic
87#13133151C2311yescharacteristic
88#13133149—21no—
89#13133148—21no—
90#13133142—21no—
91#13133141—21no—
92#13133140—21no—
93#13133138—21no—
94#13133138—21no—
95#13133138—21no—
96#13133138—21no—
97#13133127D411yescharacteristic
98#13133127D411yescharacteristic
99C8×Dic53C2311yescharacteristic
100C8×Dic53C2311yescharacteristic
101C22×D4×C53—21no—
102C2×Q16×C53C2311yescharacteristic
103C2×D8×C53—22no—
104C2×D8×C53—22no—
105C2×D8×C53—21no—
106C2×D8×C53C2311yescharacteristic
107C2×D8×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C22⋊Q8×C53—21no—
110SD32×C53—22no—
111SD32×C53—22no—
112C2×C16×C53C2311yescharacteristic
113C2×C16×C53C2311yescharacteristic
114#107×C53C2311yescharacteristic
115#103×C53—21no—
116C24×C53—41no—
117C2×Q8×C53—21no—
118C2×D4×C53—24no—
119C2×D4×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—42no—
122C2×D4×C53—41no—
123C22×C4×C53C2×D411yescharacteristic
124C22×Dic53—41no—
125C22×Dic53—41no—
126C22×Dic53—41no—
127C22×Dic53—41no—
128C2×Dic106—21no—
129Q16×C53—21no—
130Q16×C53—21no—
131D8×C53—24no—
132D8×C53—21no—
133D8×C53—21no—
134D8×C53#4112yes—
135D8×C53#4112yes—
136C2×C8×C53—21no—
137C2×C8×C53—21no—
138C2×C8×C53C2×D411yescharacteristic
139C2×C8×C53C2411yescharacteristic
140C4⋊C4×C53—21no—
141C4⋊C4×C53—41no—
142C22⋊C4×C53—41no—
143C22⋊Dic53—41no—
144C22⋊Dic53—41no—
145C22⋊Dic53—41no—
146C22⋊Dic53—41no—
147C22⋊Dic53—41no—
148C22⋊Dic53—41no—
149C22⋊Dic53D811yescharacteristic
150C22⋊Dic53D811yescharacteristic
151#11719925—42no—
152C4⋊Dic53—21no—
153C4⋊Dic53—21no—
154C4⋊Dic53—21no—
155C4⋊Dic53—41no—
156C4⋊Dic53—41no—
157C4⋊Dic53D811yescharacteristic
158C4⋊Dic53D811yescharacteristic
159#11719919—41no—
160#11719919—41no—
161#11719919—41no—
162#11719919—41no—
163#11719919—41no—
164C4×Dic53—41no—
165C4×Dic53—41no—
166C4×Dic53—41no—
167C4×Dic53C2×D411yescharacteristic
168C4×Dic53C2×D411yescharacteristic
169#11719917—41no—
170C2×#88322C2×D411yescharacteristic
171C2×#88322C2×D411yescharacteristic
172Dic212—21no—
173Dic212—21no—
174C16×C53—21no—
175C16×C53—21no—
176#11719908—21no—
177#11719908—21no—
178C23×C53—44no—
179C23×C53—41no—
180C23×C53—41no—
181C23×C53—161no—
182C23×C53—161no—
183C23×C53C2×D811yescharacteristic
184Q8×C53—41no—
185Q8×C53—41no—
186D4×C53—22no—
187D4×C53—22no—
188D4×C53—44no—
189D4×C53—41no—
190D4×C53—41no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—41no—
194C2×C4×C53—41no—
195C2×C4×C53C2×D811yescharacteristic
196C2×C4×C53C22×D411yescharacteristic
197C2×Dic53—41no—
198C2×Dic53—41no—
199C2×Dic53—41no—
200C2×Dic53—41no—
201C2×Dic53—161no—
202C2×Dic53—161no—
203C2×Dic53—161no—
204C2×Dic53—161no—
205C2×Dic53C2×D811yescharacteristic
206C2×Dic53C2×D811yescharacteristic
207C2×Dic53C2×D811yescharacteristic
208C2×Dic53C2×D811yescharacteristic
209Dic106—41no—
210Dic106—41no—
211C8×C53—21no—
212C8×C53C2×#4111yescharacteristic
213C8×C53C2×#4111yescharacteristic
214#88322—21no—
215#88322—21no—
216#33683—531nomaximal, Sylow
217C22×C53—21no—
218C22×C53—21no—
219C22×C53—21no—
220C22×C53—42no—
221C22×C53—42no—
222C22×C53—81no—
223C22×C53—81no—
224C22×C53—81no—
225C22×C53—81no—
226C22×C53—164no—
227C22×C53C22×D811yescharacteristic
228C4×C53—21no—
229C4×C53—81no—
230C4×C53—81no—
231C4×C53#52511yescharacteristic
232C4×C53#57511yescharacteristic
233Dic53—21no—
234Dic53—21no—
235Dic53—21no—
236Dic53—21no—
237Dic53—81no—
238Dic53—81no—
239Dic53—81no—
240Dic53—81no—
241#3177—531no—
242#2948—531no—
243#2237—531no—
244#2237—531no—
245#2228—531no—
246#2228—531no—
247#2208—531no—
248#2208—531no—
249#2187—531no—
250#2187D5311yescharacteristic
251#2175—531no—
252#2175—531no—
253C4×SD32—531no—
254C4×SD32—531no—
255#2164—531no—
256C2×C53—41no—
257C2×C53—82no—
258C2×C53—82no—
259C2×C53#310011yescharacteristic
260C2×C53#341611yescharacteristic
261C2×C53#341611yescharacteristic
262C22×D8C2×D5311yescharacteristic
263#541—1061no—
264#518—1061no—
265C2×SD32—531no—
266C2×SD32—531no—
267C2×SD32C2×D5311yescharacteristic
268C2×SD32C2×D5311yescharacteristic
269#499—531no—
270#494—531no—
271#491—1061no—
272#470—1061no—
273#467—531no—
274#467C2×D5311yescharacteristic
275#442—1061no—
276#442—1061no—
277#441—1061no—
278C4×Q16—531no—
279C4×Q16—531no—
280C4×D8—531no—
281C4×D8—531no—
282C4×D8—531no—
283C4×D8—531no—
284#427—531no—
285#418—1061no—
286#405—531no—
287#367—531no—
288#367—531no—
289#366—531no—
290#366—531no—
291C4⋊C16—531no—
292C4⋊C16—531no—
293#358—531no—
294#358—531no—
295#358—531no—
296#358—531no—
297#357—531no—
298#357—531no—
299#357C2×D5311yescharacteristic
300#357C2×D5311yescharacteristic
301C22⋊C16—531no—
302C22⋊C16C2×D5311yescharacteristic
303#346—531no—
304C4×C16—531no—
305C53#3368311yescharacteristic, complemented, Sylow
306C22×D4—21no—
307C2×Q16—531no—
308C2×Q16—1061no—
309C2×Q16C22×D5311yescharacteristic
310C2×D8—22no—
311C2×D8—22no—
312C2×D8—21no—
313C2×D8C22×D5311yescharacteristic
314C2×D8C22×D5311yescharacteristic
315C2×M16—531no—
316C22×C8C22×D5311yescharacteristic
317C4⋊Q8—1061no—
318#126—2121no—
319#126—2121no—
320#125—1061no—
321#125—2121no—
322#125—2121no—
323C22⋊Q8—21no—
324C22⋊Q8—1061no—
325C4×Q8—1061no—
326C4×Q8—1061no—
327C4×D4—1061no—
328C4×D4—1061no—
329C4×D4—1061no—
330C4×D4—1061no—
331C4×D4—2121no—
332C4×D4—2121no—
333#117—531no—
334#117—1061no—
335C2×C22⋊C4—2121no—
336C2×C22⋊C4—2121no—
337SD32—22no—
338SD32—22no—
339SD32—1062no—
340SD32—1062no—
341C2×C16—531no—
342C2×C16—531no—
343C2×C16C22×D5311yescharacteristic
344C2×C16C22×D5311yescharacteristic
345#107—531no—
346#107—531no—
347#107—531no—
348#107C22×D5311yescharacteristic
349#106—1061no—
350#106—1061no—
351#103—21no—
352#103—1061no—
353#103—1061no—
354#103—1061no—
355#102—1061no—
356#102—1061no—
357#102—1061no—
358#102—1061no—
359#97—531no—
360#97—531no—
361C4×C8—531no—
362C4×C8—531no—
363C24—41no—
364C2×Q8—21no—
365C2×Q8—1061no—
366C2×D4—24no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—42no—
370C2×D4—41no—
371C22×C4—2121no—
372C22×C4—2121no—
373C22×C4—2121no—
374C22×C4—2121no—
375C22×C4D4×D5311yescharacteristic
376Q16—21no—
377Q16—21no—
378Q16—1061no—
379Q16—1061no—
380Q16—2122no—
381D8—24no—
382D8—21no—
383D8—21no—
384D8#1171994712yes—
385D8#1171994712yes—
386M16—2121no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—531no—
390C2×C8—531no—
391C2×C8D4×D5311yescharacteristic
392C2×C8C23×D5311yescharacteristic
393C4⋊C4—21no—
394C4⋊C4—41no—
395C4⋊C4—531no—
396C4⋊C4—531no—
397C4⋊C4—1061no—
398C4⋊C4—1061no—
399C4⋊C4—1061no—
400C4⋊C4—2121no—
401C4⋊C4—2121no—
402C4⋊C4—2121no—
403C4⋊C4—2121no—
404C4⋊C4—2121no—
405C4⋊C4—2121no—
406C4⋊C4—2121no—
407C22⋊C4—41no—
408C22⋊C4—531no—
409C22⋊C4—531no—
410C22⋊C4—2121no—
411C22⋊C4—2121no—
412C22⋊C4—2121no—
413C22⋊C4—2121no—
414C22⋊C4—2121no—
415C22⋊C4—2121no—
416C42—531no—
417C42—531no—
418C42—2121no—
419C42—2121no—
420C42—2121no—
421C16—21no—
422C16—21no—
423C16—1061no—
424C16—1061no—
425C23—44no—
426C23—41no—
427C23—41no—
428C23—161no—
429C23—161no—
430C23D8×D5311yescharacteristic
431Q8—41no—
432Q8—41no—
433Q8—2121no—
434Q8—2121no—
435D4—22no—
436D4—22no—
437D4—44no—
438D4—41no—
439D4—41no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—531no—
445C2×C4—531no—
446C2×C4—531no—
447C2×C4—531no—
448C2×C4—8481no—
449C2×C4—8481no—
450C2×C4—8481no—
451C2×C4—8481no—
452C2×C4—2121no—
453C2×C4—2121no—
454C2×C4—2121no—
455C2×C4—2121no—
456C2×C4D8×D5311yescharacteristic
457C2×C4C2×D4×D5311yescharacteristic
458C8—21no—
459C8—1061no—
460C8—1061no—
461C8C2×#1171994711yescharacteristic
462C8C2×#1171994711yescharacteristic
463C22—21no—
464C22—21no—
465C22—21no—
466C22—42no—
467C22—42no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22—164no—
473C22C2×D8×D5311yescharacteristic
474C4—21no—
475C4—81no—
476C4—81no—
477C4—1061no—
478C4—1061no—
479C4—1061no—
480C4—1061no—
481C4—4241no—
482C4—4241no—
483C4—4241no—
484C4—4241no—
485C4#1681112911yescharacteristic
486C4#1681141911yescharacteristic
487C2—41no—
488C2—82no—
489C2—82no—
490C2#2028471011yescharacteristic
491C2#2028785711yescharacteristic
492C2#2028786111yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2187C2
socleC22×C53C22×D8
prime radicalC22×D8×C53C22
hypercenter#2187D53
hypocenterC53#33683
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2187

Complements

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

NH ≅ G/Ntype
1. #20285367C2semidirect
3. #20276924C2semidirect
4. #20276920C2semidirect
5. #20276874C2semidirect
6. #20276870C2semidirect
7. #20276758C2semidirect
8. #20276758C2semidirect
10. #20276580C2semidirect
11. #20276576C2semidirect
12. #20276570C2semidirect
13. SD32×Dic53C2semidirect
14. #20276519C2semidirect
20. #16811024C22semidirect
21. #16811001C22semidirect
28. Q16×Dic53C22semidirect
29. #16810717C22semidirect
37. #16810451C22semidirect
38. #16810450C22semidirect
39. #16810449C22semidirect
40. #16810448C22semidirect
41. #16810441C22semidirect
42. #16810441C22semidirect
43. #16810428C22semidirect
44. #16810428C22semidirect
45. #16810427C22semidirect
46. #16810427C22semidirect
50. #16810398C22semidirect
51. C16×Dic53C22semidirect
305. C53#33683semidirect

Maximal subgroups

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

subgroupindex
#202853672
#202835672
#202769242
#202769202
#202768742
#202768702
#202767582
#202766442
#202765802
#202765762
#202765702
SD32×Dic532
#202765192
C53×#21872
#3368353