Finite Group Databasebeta

← #27541267

Subgroups — #27541267

Order 13568.

548 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×Dic212C211yesmaximal, characteristic, complemented
2#20276962C211yesmaximal, characteristic, complemented
3#20276923C211yesmaximal, characteristic, complemented
4#20276917C211yesmaximal, characteristic, complemented
5#20276876C211yesmaximal, characteristic, complemented
6#20276826C211yesmaximal, characteristic, complemented
7#20276809C211yesmaximal, characteristic, complemented
8#20276760C211yesmaximal, characteristic, complemented
9#20276631C211yesmaximal, characteristic, complemented
10#20276631C211yesmaximal, characteristic, complemented
11#20276579C211yesmaximal, characteristic, complemented
12#20276541C211yesmaximal, characteristic, complemented
13#20276434C211yesmaximal, characteristic, complemented
14C2×#16810426C211yesmaximal, characteristic, complemented
15C53×#3339C211yesmaximal, characteristic
16C22×Dic212C2211yescharacteristic
17C22×Dic212C2211yescharacteristic
18D4×Dic106—21no—
19C2×#13133161C2211yescharacteristic
20C2×#13133161C2211yescharacteristic
21C2×#13133161C2211yescharacteristic
22C2×#13133157C2211yescharacteristic
23C22×#11719908C2211yescharacteristic
24#16811003C2211yescharacteristic
25#16810864C2211yescharacteristic
26#16810864C2211yescharacteristic
27#16810834—21no—
28#16810781—21no—
29#16810781—21no—
30C4×Dic212C2211yescharacteristic
31#16810450C2211yescharacteristic
32#16810448C2211yescharacteristic
33#16810448C2211yescharacteristic
34C4⋊#11719908C2211yescharacteristic
35#16810426C2211yescharacteristic
36#16810426C2211yescharacteristic
37#16810426C2211yescharacteristic
38#16810426C2211yescharacteristic
39#16810426C2211yescharacteristic
40#16810426C2211yescharacteristic
41#16810426C2211yescharacteristic
42#16810426C2211yescharacteristic
43#16810426C2211yescharacteristic
44#16810424C2211yescharacteristic
45C22⋊#11719908C2211yescharacteristic
46C22⋊#11719908C2211yescharacteristic
47#16810399C2211yescharacteristic
48C53×#553—21no—
49C53×#499C2211yescharacteristic
50C53×#482—21no—
51C53×#481—21no—
52C53×#478—21no—
53C53×#466C2211yescharacteristic
54C53×#466C2211yescharacteristic
55C4×D8×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#13133161—22no—
80#13133161—41no—
81#13133161—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×D8×C53C2311yescharacteristic
93C22×C8×C53C2311yescharacteristic
94C22×C8×C53C2311yescharacteristic
95#126×C53—41no—
96#125×C53—21no—
97#123×C53—41no—
98#121×C53—21no—
99#121×C53—21no—
100#121×C53—41no—
101C4×D4×C53—21no—
102C4×D4×C53—41no—
103C4×D4×C53D411yescharacteristic
104#117×C53—21no—
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#102×C53—21no—
114#102×C53—21no—
115#102×C53—21no—
116C22⋊C8×C53D411yescharacteristic
117C22⋊C8×C53D411yescharacteristic
118C4×C8×C53C2311yescharacteristic
119C2×D4×C53—21no—
120C2×D4×C53—41no—
121C2×D4×C53—41no—
122C2×D4×C53D811yescharacteristic
123C22×C4×C53—41no—
124C22×C4×C53—41no—
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×Dic53—41no—
128C22×Dic53—41no—
129C2×Dic106—21no—
130C2×Dic106—21no—
131C2×Dic106—21no—
132C2×Dic106—42no—
133C2×Dic106—42no—
134C2×Dic106—41no—
135C2×Dic106—41no—
136C2×Dic106—81no—
137C2×Dic106—81no—
138D8×C53—21no—
139D8×C53—21no—
140C2×C8×C53—21no—
141C2×C8×C53—21no—
142C2×C8×C53—21no—
143C2×C8×C53—21no—
144C2×C8×C53C2×D411yescharacteristic
145C2×C8×C53C2×D411yescharacteristic
146C2×C8×C53C2×D411yescharacteristic
147C2×C8×C53C2411yescharacteristic
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
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×C53—41no—
159C22⋊C4×C53D811yescharacteristic
160C22⋊C4×C53D811yescharacteristic
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×C53—41no—
186C23×C53C2×D811yescharacteristic
187C23×C53C2×D811yescharacteristic
188D4×C53—22no—
189D4×C53—41no—
190D4×C53—41no—
191D4×C53—82no—
192D4×C53—161no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—41no—
198C2×C4×C53—41no—
199C2×C4×C53—41no—
200C2×C4×C53—161no—
201C2×C4×C53—161no—
202C2×C4×C53C2×D811yescharacteristic
203C2×C4×C53C2×D811yescharacteristic
204C2×C4×C53C2×D811yescharacteristic
205C2×C4×C53C2×D811yescharacteristic
206C2×C4×C53C22×D411yescharacteristic
207C2×Dic53—41no—
208C2×Dic53—41no—
209C2×Dic53—41no—
210C2×Dic53—41no—
211C2×Dic53—82no—
212C2×Dic53—82no—
213Dic106—22no—
214Dic106—41no—
215Dic106—41no—
216Dic106—41no—
217Dic106—41no—
218Dic106—41no—
219Dic106—82no—
220Dic106—82no—
221C8×C53—21no—
222C8×C53—21no—
223C8×C53—21no—
224C8×C53C2×#4111yescharacteristic
225C8×C53H(5,𝔽2)11yescharacteristic
226#33887—531nomaximal, Sylow
227C22×C53—21no—
228C22×C53—21no—
229C22×C53—21no—
230C22×C53—21no—
231C22×C53—21no—
232C22×C53—81no—
233C22×C53—81no—
234C22×C53—81no—
235C22×C53#51012yes—
236C22×C53C22×D811yescharacteristic
237C4×C53—21no—
238C4×C53—21no—
239C4×C53—21no—
240C4×C53—21no—
241C4×C53—81no—
242C4×C53—81no—
243C4×C53—81no—
244C4×C53#54611yescharacteristic
245C4×C53#57711yescharacteristic
246Dic53—42no—
247Dic53—81no—
248Dic53—81no—
249Dic53—81no—
250#3339D5311yescharacteristic
251D4×Q16—531no—
252#2245—531no—
253#2237—531no—
254#2236—531no—
255#2229—531no—
256#2220—531no—
257#2217—531no—
258#2208—531no—
259#2185—531no—
260#2185—531no—
261#2175—531no—
262#2168—531no—
263#2141—531no—
264C2×#358—531no—
265C2×C53—22no—
266C2×C53—41no—
267C2×C53—161no—
268C2×C53#329411yescharacteristic
269C2×C53C2×#51011yescharacteristic
270C2×C53#341611yescharacteristic
271C22×Q16—531no—
272C22×Q16—531no—
273#553—21no—
274D4×Q8—1061no—
275C2×SD32—531no—
276C2×SD32—531no—
277C2×SD32—531no—
278C2×M32—531no—
279C22×C16—531no—
280#499C2×D5311yescharacteristic
281#494—531no—
282#482—21no—
283#481—21no—
284#478—21no—
285#467—531no—
286#467—531no—
287#466C2×D5311yescharacteristic
288#466C2×D5311yescharacteristic
289#462—1061no—
290#451—1061no—
291#451—1061no—
292C4×Q16—531no—
293C4×D8C2×D5311yescharacteristic
294C8×D4C2×D5311yescharacteristic
295#427C2×D5311yescharacteristic
296C2×#107C2×D5311yescharacteristic
297#367—531no—
298#366—531no—
299#366—531no—
300C4⋊C16—531no—
301#358—531no—
302#358—531no—
303#358—531no—
304#358—531no—
305#358—531no—
306#358—531no—
307#358—531no—
308#358—531no—
309#358—531no—
310#357—531no—
311C22⋊C16—531no—
312C22⋊C16—531no—
313#346—531no—
314C53#3388711yescharacteristic, complemented, Sylow
315C22×Q8—1061no—
316C22×Q8—1061no—
317C2×Q16—531no—
318C2×Q16—531no—
319C2×Q16—531no—
320C2×Q16—1062no—
321C2×Q16—1062no—
322C2×Q16—1061no—
323C2×Q16—1061no—
324C2×Q16—1061no—
325C2×Q16—2121no—
326C2×Q16—2121no—
327C2×D8C22×D5311yescharacteristic
328C22×C8C22×D5311yescharacteristic
329C22×C8C22×D5311yescharacteristic
330C4⋊Q8—1061no—
331C4⋊Q8—2121no—
332#126—41no—
333#125—21no—
334#123—41no—
335C22⋊Q8—1061no—
336C22⋊Q8—1061no—
337C22⋊Q8—2121no—
338C22⋊Q8—2121no—
339#121—21no—
340#121—21no—
341#121—41no—
342C4×Q8—1061no—
343C4×D4—21no—
344C4×D4—41no—
345C4×D4—2121no—
346C4×D4C22⋊D5311yescharacteristic
347#117—21no—
348C2×C4⋊C4—21no—
349SD32—1062no—
350SD32—2121no—
351SD32—2121no—
352M32—2121no—
353C2×C16—531no—
354C2×C16—531no—
355C2×C16—531no—
356C2×C16—531no—
357C2×C16—2121no—
358#107—21no—
359#107—531no—
360#107C22×D5311yescharacteristic
361#107C22×D5311yescharacteristic
362#107C22×D5311yescharacteristic
363#106—21no—
364#106—21no—
365C4⋊C8C22⋊D5311yescharacteristic
366#103—1061no—
367#103—1061no—
368#103—1061no—
369#102—21no—
370#102—21no—
371#102—21no—
372C22⋊C8C22⋊D5311yescharacteristic
373C22⋊C8C22⋊D5311yescharacteristic
374C4×C8C22×D5311yescharacteristic
375C2×Q8—1061no—
376C2×Q8—1061no—
377C2×Q8—1061no—
378C2×Q8—4241no—
379C2×Q8—4241no—
380C2×Q8—2122no—
381C2×Q8—2122no—
382C2×Q8—2121no—
383C2×Q8—2121no—
384C2×D4—21no—
385C2×D4—41no—
386C2×D4—41no—
387C2×D4#1171992211yescharacteristic
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—2121no—
391C22×C4—2121no—
392C22×C4C2×C22⋊D5311yescharacteristic
393C22×C4C2×C22⋊D5311yescharacteristic
394Q16—532no—
395Q16—1061no—
396Q16—1061no—
397Q16—1061no—
398Q16—1061no—
399Q16—1061no—
400Q16—4241no—
401Q16—2122no—
402Q16—2122no—
403D8—21no—
404D8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8C2×C22⋊D5311yescharacteristic
410C2×C8C2×C22⋊D5311yescharacteristic
411C2×C8C2×C22⋊D5311yescharacteristic
412C2×C8C23×D5311yescharacteristic
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—1061no—
420C4⋊C4—2121no—
421C4⋊C4—2121no—
422C4⋊C4—2121no—
423C4⋊C4—2121no—
424C4⋊C4—2121no—
425C4⋊C4#1171992211yescharacteristic
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—2121no—
431C22⋊C4—2121no—
432C22⋊C4#1171992211yescharacteristic
433C22⋊C4#1171992211yescharacteristic
434C42—41no—
435C42—2121no—
436C42C2×C22⋊D5311yescharacteristic
437C16—1061no—
438C16—1061no—
439C16—1061no—
440C16—1061no—
441C23—41no—
442C23C2×#1171992211yescharacteristic
443C23C2×#1171992211yescharacteristic
444Q8—1062no—
445Q8—4242no—
446Q8—4242no—
447Q8—2121no—
448Q8—2121no—
449Q8—2121no—
450Q8—2121no—
451Q8—2121no—
452D4—22no—
453D4—41no—
454D4—41no—
455D4—82no—
456D4—161no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—161no—
465C2×C4—161no—
466C2×C4—4242no—
467C2×C4—4242no—
468C2×C4—2121no—
469C2×C4—2121no—
470C2×C4—2121no—
471C2×C4—2121no—
472C2×C4C2×#1171992211yescharacteristic
473C2×C4C2×#1171992211yescharacteristic
474C2×C4C2×#1171992211yescharacteristic
475C2×C4C2×#1171992211yescharacteristic
476C2×C4C22×C22⋊D5311yescharacteristic
477C8—21no—
478C8—21no—
479C8—21no—
480C8#41×D5311yescharacteristic
481C8#1313326311yescharacteristic
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22#1681106312yes—
491C22C22×#1171992211yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—81no—
497C4—81no—
498C4—81no—
499C4—4241no—
500C4—4241no—
501C4—4241no—
502C4—2122no—
503C4#1681126111yescharacteristic
504C4#1681143111yescharacteristic
505C2—22no—
506C2—41no—
507C2—161no—
508C2#2028652211yescharacteristic
509C2C2×#1681106311yescharacteristic
510C2#2028786211yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
Fitting, prime radicalC53×#3339C2
socleC22×C53C22×D8
hypercenter#3339D53
hypocenterC53#33887
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3339

Complements

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

NH ≅ G/Ntype
1. D4×Dic212C2semidirect
2. #20276962C2semidirect
3. #20276923C2semidirect
4. #20276917C2semidirect
5. #20276876C2semidirect
6. #20276826C2semidirect
7. #20276809C2semidirect
8. #20276760C2semidirect
9. #20276631C2semidirect
10. #20276631C2semidirect
11. #20276579C2semidirect
12. #20276541C2semidirect
13. #20276434C2semidirect
14. C2×#16810426C2semidirect
314. C53#33887semidirect

Maximal subgroups

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

subgroupindex
D4×Dic2122
#202769622
#202769232
#202769172
#202768762
#202768262
#202768092
#202767602
#202766312
#202765792
#202765412
#202764342
C2×#168104262
C53×#33392
#3388753