Finite Group Databasebeta

← #27541147

Subgroups — #27541147

Order 13568.

536 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×Dic212C211yesmaximal, characteristic
2#20276962C211yesmaximal, characteristic, complemented
3#20276923C211yesmaximal, characteristic, complemented
4#20276912C211yesmaximal, characteristic, complemented
5#20276876C211yesmaximal, characteristic, complemented
6#20276821C211yesmaximal, characteristic, complemented
7#20276804C211yesmaximal, characteristic, complemented
8#20276765C211yesmaximal, characteristic, complemented
9#20276648C211yesmaximal, characteristic, complemented
10#20276648C211yesmaximal, characteristic, complemented
11#20276584C211yesmaximal, characteristic, complemented
12#20276541C211yesmaximal, characteristic
13#20276434C211yesmaximal, characteristic, complemented
14C2×#16810426C211yesmaximal, characteristic, complemented
15C53×#3336C211yesmaximal, characteristic
16C22×Dic212C2211yescharacteristic
17C22×Dic212C2211yescharacteristic
18D4×Dic106—21no—
19C2×#13133163C2211yescharacteristic
20C2×#13133163C2211yescharacteristic
21C2×#13133163C2211yescharacteristic
22C2×#13133157C2211yescharacteristic
23C22×#11719908C2211yescharacteristic
24#16811003C2211yescharacteristic
25#16810864C2211yescharacteristic
26#16810864C2211yescharacteristic
27#16810834—21no—
28#16810781—21no—
29#16810781—21no—
30C4×Dic212C2211yescharacteristic
31#16810450C2211yescharacteristic
32#16810450C2211yescharacteristic
33#16810448C2211yescharacteristic
34C4⋊#11719908C2211yescharacteristic
35#16810427C2211yescharacteristic
36#16810426C2211yescharacteristic
37#16810426C2211yescharacteristic
38#16810426C2211yescharacteristic
39#16810426C2211yescharacteristic
40#16810426C2211yescharacteristic
41#16810426C2211yescharacteristic
42#16810426C2211yescharacteristic
43#16810426C2211yescharacteristic
44#16810426C2211yescharacteristic
45C22⋊#11719908C2211yescharacteristic
46C22⋊#11719908C2211yescharacteristic
47#16810399C2211yescharacteristic
48C53×#552—21no—
49C53×#499C2211yescharacteristic
50C53×#485—21no—
51C53×#483—21no—
52C53×#479—21no—
53C53×#467C2211yescharacteristic
54C53×#467C2211yescharacteristic
55C4×Q16×C53C2211yescharacteristic
56C8×D4×C53C2211yescharacteristic
57C53×#427C2211yescharacteristic
58C2×#107×C53C2211yescharacteristic
59C22×Dic106—21no—
60C22×Dic106—21no—
61C2×Dic212—21no—
62C2×Dic212—21no—
63C2×Dic212—21no—
64C2×Dic212—22no—
65C2×Dic212—22no—
66C2×Dic212—41no—
67C2×Dic212—41no—
68C2×Dic212C2311yescharacteristic
69C2×Dic212C2311yescharacteristic
70C2×Dic212C2311yescharacteristic
71#13133217—21no—
72#13133216—41no—
73#13133196—21no—
74#13133196—21no—
75#13133194—41no—
76#13133194—41no—
77C4×Dic106—21no—
78D4×Dic53—41no—
79#13133163—22no—
80#13133163—41no—
81#13133163—41no—
82#13133157—41no—
83C2×#11719908—41no—
84C2×#11719908C2311yescharacteristic
85C2×#11719908C2311yescharacteristic
86C2×#11719908C2311yescharacteristic
87C2×#11719908C2311yescharacteristic
88#13133151C2311yescharacteristic
89#13133142—21no—
90#13133142—21no—
91#13133142—21no—
92C2×Q16×C53C2311yescharacteristic
93C22×C8×C53C2311yescharacteristic
94C22×C8×C53C2311yescharacteristic
95#126×C53—41no—
96#125×C53—21no—
97#125×C53—41no—
98#123×C53—41no—
99C22⋊Q8×C53—21no—
100C22⋊Q8×C53—21no—
101C4×Q8×C53—21no—
102C4×D4×C53D411yescharacteristic
103#117×C53—21no—
104#117×C53—41no—
105C2×C4⋊C4×C53—21no—
106#107×C53—21no—
107#107×C53C2311yescharacteristic
108#107×C53C2311yescharacteristic
109#107×C53C2311yescharacteristic
110#106×C53—21no—
111#106×C53—21no—
112C4⋊C8×C53D411yescharacteristic
113#103×C53—21no—
114#103×C53—21no—
115#103×C53—21no—
116C22⋊C8×C53D411yescharacteristic
117C22⋊C8×C53D411yescharacteristic
118C4×C8×C53C2311yescharacteristic
119C2×Q8×C53—21no—
120C2×D4×C53D811yescharacteristic
121C22×C4×C53—41no—
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×Dic53—41no—
125C22×Dic53—41no—
126C2×Dic106—21no—
127C2×Dic106—21no—
128C2×Dic106—21no—
129C2×Dic106—42no—
130C2×Dic106—42no—
131C2×Dic106—41no—
132C2×Dic106—41no—
133C2×Dic106—81no—
134C2×Dic106—81no—
135Q16×C53—21no—
136Q16×C53—21no—
137C2×C8×C53—21no—
138C2×C8×C53—21no—
139C2×C8×C53—21no—
140C2×C8×C53—21no—
141C2×C8×C53C2×D411yescharacteristic
142C2×C8×C53C2×D411yescharacteristic
143C2×C8×C53C2×D411yescharacteristic
144C2×C8×C53C2411yescharacteristic
145C4⋊C4×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—41no—
149C4⋊C4×C53—41no—
150C4⋊C4×C53—41no—
151C4⋊C4×C53—41no—
152C4⋊C4×C53—41no—
153C4⋊C4×C53—41no—
154C4⋊C4×C53D811yescharacteristic
155C22⋊C4×C53—41no—
156C22⋊C4×C53—41no—
157C22⋊C4×C53—41no—
158C22⋊C4×C53D811yescharacteristic
159C22⋊C4×C53D811yescharacteristic
160C42×C53—41no—
161C42×C53—41no—
162C42×C53C2×D411yescharacteristic
163C22⋊Dic53—41no—
164C22⋊Dic53—41no—
165C4⋊Dic53—21no—
166C4⋊Dic53—41no—
167C4⋊Dic53—41no—
168#11719919—41no—
169#11719919—41no—
170#11719919—41no—
171C4×Dic53—41no—
172Dic212—21no—
173Dic212—21no—
174Dic212—21no—
175Dic212—21no—
176Dic212—21no—
177Dic212—42no—
178Dic212—42no—
179Dic212—81no—
180Dic212#4112yes—
181#11719908—21no—
182#11719908—21no—
183#11719908—21no—
184#11719908—21no—
185C23×C53C2×D811yescharacteristic
186C23×C53C2×D811yescharacteristic
187Q8×C53—41no—
188Q8×C53—41no—
189D4×C53—22no—
190C2×C4×C53—21no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—41no—
195C2×C4×C53—41no—
196C2×C4×C53—41no—
197C2×C4×C53—41no—
198C2×C4×C53—82no—
199C2×C4×C53C2×D811yescharacteristic
200C2×C4×C53C2×D811yescharacteristic
201C2×C4×C53C2×D811yescharacteristic
202C2×C4×C53C2×D811yescharacteristic
203C2×C4×C53C22×D411yescharacteristic
204C2×Dic53—41no—
205C2×Dic53—41no—
206C2×Dic53—41no—
207C2×Dic53—41no—
208C2×Dic53—82no—
209C2×Dic53—82no—
210Dic106—22no—
211Dic106—41no—
212Dic106—41no—
213Dic106—41no—
214Dic106—41no—
215Dic106—41no—
216Dic106—82no—
217Dic106—82no—
218C8×C53—21no—
219C8×C53—21no—
220C8×C53—21no—
221C8×C53C2×#4111yescharacteristic
222C8×C53H(5,𝔽2)11yescharacteristic
223#33879—531nomaximal, Sylow
224C22×C53—21no—
225C22×C53—21no—
226C22×C53—21no—
227C22×C53—21no—
228C22×C53—21no—
229C22×C53#51012yes—
230C22×C53C22×D811yescharacteristic
231C4×C53—21no—
232C4×C53—21no—
233C4×C53—21no—
234C4×C53—21no—
235C4×C53—81no—
236C4×C53—81no—
237C4×C53—81no—
238C4×C53—81no—
239C4×C53#54611yescharacteristic
240C4×C53#57711yescharacteristic
241Dic53—42no—
242Dic53—81no—
243Dic53—81no—
244Dic53—81no—
245#3336D5311yescharacteristic
246D4×Q16—531no—
247#2245—531no—
248#2237—531no—
249#2235—531no—
250#2229—531no—
251#2219—531no—
252#2216—531no—
253#2209—531no—
254#2188—531no—
255#2188—531no—
256#2176—531no—
257#2168—531no—
258#2141—531no—
259C2×#358—531no—
260C2×C53—22no—
261C2×C53—41no—
262C2×C53#329411yescharacteristic
263C2×C53C2×#51011yescharacteristic
264C2×C53#341711yescharacteristic
265C22×Q16—531no—
266C22×Q16—531no—
267#552—21no—
268D4×Q8—1061no—
269C2×Q32—531no—
270C2×Q32—531no—
271C2×Q32—531no—
272C2×M32—531no—
273C22×C16—531no—
274#499C2×D5311yescharacteristic
275#494—531no—
276#485—21no—
277#483—21no—
278#479—21no—
279#467—531no—
280#467—531no—
281#467C2×D5311yescharacteristic
282#467C2×D5311yescharacteristic
283#462—1061no—
284#451—1061no—
285#451—1061no—
286C4×Q16—531no—
287C4×Q16C2×D5311yescharacteristic
288C8×D4C2×D5311yescharacteristic
289#427C2×D5311yescharacteristic
290C2×#107C2×D5311yescharacteristic
291#367—531no—
292#367—531no—
293#366—531no—
294C4⋊C16—531no—
295#358—531no—
296#358—531no—
297#358—531no—
298#358—531no—
299#358—531no—
300#358—531no—
301#358—531no—
302#358—531no—
303#358—531no—
304#358—531no—
305C22⋊C16—531no—
306C22⋊C16—531no—
307#346—531no—
308C53#3387911yescharacteristic, complemented, Sylow
309C22×Q8—1061no—
310C22×Q8—1061no—
311C2×Q16—531no—
312C2×Q16—531no—
313C2×Q16—531no—
314C2×Q16—1062no—
315C2×Q16—1062no—
316C2×Q16—1061no—
317C2×Q16—1061no—
318C2×Q16—1061no—
319C2×Q16—2121no—
320C2×Q16—2121no—
321C2×Q16C22×D5311yescharacteristic
322C22×C8C22×D5311yescharacteristic
323C22×C8C22×D5311yescharacteristic
324C4⋊Q8—1061no—
325C4⋊Q8—2121no—
326#126—41no—
327#125—21no—
328#125—41no—
329#123—41no—
330C22⋊Q8—21no—
331C22⋊Q8—21no—
332C22⋊Q8—1061no—
333C22⋊Q8—1061no—
334C22⋊Q8—2121no—
335C22⋊Q8—2121no—
336C4×Q8—21no—
337C4×Q8—1061no—
338C4×D4—2121no—
339C4×D4C22⋊D5311yescharacteristic
340#117—21no—
341#117—41no—
342C2×C4⋊C4—21no—
343Q32—1062no—
344Q32—2121no—
345Q32—2121no—
346M32—2121no—
347C2×C16—531no—
348C2×C16—531no—
349C2×C16—531no—
350C2×C16—531no—
351C2×C16—2121no—
352#107—21no—
353#107—531no—
354#107C22×D5311yescharacteristic
355#107C22×D5311yescharacteristic
356#107C22×D5311yescharacteristic
357#106—21no—
358#106—21no—
359C4⋊C8C22⋊D5311yescharacteristic
360#103—21no—
361#103—21no—
362#103—21no—
363#103—1061no—
364#103—1061no—
365#103—1061no—
366C22⋊C8C22⋊D5311yescharacteristic
367C22⋊C8C22⋊D5311yescharacteristic
368C4×C8C22×D5311yescharacteristic
369C2×Q8—21no—
370C2×Q8—1061no—
371C2×Q8—1061no—
372C2×Q8—1061no—
373C2×Q8—4241no—
374C2×Q8—4241no—
375C2×Q8—2122no—
376C2×Q8—2122no—
377C2×Q8—2121no—
378C2×Q8—2121no—
379C2×D4#1171992211yescharacteristic
380C22×C4—41no—
381C22×C4—2121no—
382C22×C4—2121no—
383C22×C4C2×C22⋊D5311yescharacteristic
384C22×C4C2×C22⋊D5311yescharacteristic
385Q16—21no—
386Q16—21no—
387Q16—532no—
388Q16—1061no—
389Q16—1061no—
390Q16—1061no—
391Q16—1061no—
392Q16—1061no—
393Q16—4241no—
394Q16—2122no—
395Q16—2122no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8C2×C22⋊D5311yescharacteristic
401C2×C8C2×C22⋊D5311yescharacteristic
402C2×C8C2×C22⋊D5311yescharacteristic
403C2×C8C23×D5311yescharacteristic
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—1061no—
414C4⋊C4—2121no—
415C4⋊C4—2121no—
416C4⋊C4—2121no—
417C4⋊C4—2121no—
418C4⋊C4—2121no—
419C4⋊C4#1171992211yescharacteristic
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—41no—
423C22⋊C4—2121no—
424C22⋊C4—2121no—
425C22⋊C4#1171992211yescharacteristic
426C22⋊C4#1171992211yescharacteristic
427C42—41no—
428C42—41no—
429C42—2121no—
430C42C2×C22⋊D5311yescharacteristic
431C16—1061no—
432C16—1061no—
433C16—1061no—
434C16—1061no—
435C23C2×#1171992211yescharacteristic
436C23C2×#1171992211yescharacteristic
437Q8—41no—
438Q8—41no—
439Q8—1062no—
440Q8—4242no—
441Q8—4242no—
442Q8—2121no—
443Q8—2121no—
444Q8—2121no—
445Q8—2121no—
446Q8—2121no—
447D4—22no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—82no—
457C2×C4—4242no—
458C2×C4—4242no—
459C2×C4—2121no—
460C2×C4—2121no—
461C2×C4—2121no—
462C2×C4—2121no—
463C2×C4C2×#1171992211yescharacteristic
464C2×C4C2×#1171992211yescharacteristic
465C2×C4C2×#1171992211yescharacteristic
466C2×C4C2×#1171992211yescharacteristic
467C2×C4C22×C22⋊D5311yescharacteristic
468C8—21no—
469C8—21no—
470C8—21no—
471C8#41×D5311yescharacteristic
472C8#1313326311yescharacteristic
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22#1681106312yes—
479C22C22×#1171992211yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—81no—
485C4—81no—
486C4—81no—
487C4—81no—
488C4—4241no—
489C4—4241no—
490C4—4241no—
491C4—2122no—
492C4#1681126111yescharacteristic
493C4#1681143111yescharacteristic
494C2—22no—
495C2—41no—
496C2#2028652211yescharacteristic
497C2C2×#1681106311yescharacteristic
498C2#2028786611yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3336C2
socleC22×C53C22×D8
prime radicalC2×D4×C53D8
hypercenter#3336D53
hypocenterC53#33879
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3336

Complements

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

NH ≅ G/Ntype
2. #20276962C2semidirect
3. #20276923C2semidirect
4. #20276912C2semidirect
5. #20276876C2semidirect
6. #20276821C2semidirect
7. #20276804C2semidirect
8. #20276765C2semidirect
9. #20276648C2semidirect
10. #20276648C2semidirect
11. #20276584C2semidirect
13. #20276434C2semidirect
14. C2×#16810426C2semidirect
308. C53#33879semidirect

Maximal subgroups

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

subgroupindex
D4×Dic2122
#202769622
#202769232
#202769122
#202768762
#202768212
#202768042
#202767652
#202766482
#202765842
#202765412
#202764342
C2×#168104262
C53×#33362
#3387953