Finite Group Databasebeta

← #27541273

Subgroups — #27541273

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
1#20287118C211yesmaximal, characteristic
2#20276960C211yesmaximal, characteristic, complemented
3#20276920C211yesmaximal, characteristic, complemented
4#20276915C211yesmaximal, characteristic, complemented
5#20276879C211yesmaximal, characteristic, complemented
6#20276830C211yesmaximal, characteristic, complemented
7#20276813C211yesmaximal, characteristic, complemented
8#20276757C211yesmaximal, characteristic, complemented
9#20276632C211yesmaximal, characteristic, complemented
10#20276632C211yesmaximal, characteristic, complemented
11#20276577C211yesmaximal, characteristic, complemented
12#20276541C211yesmaximal, characteristic, complemented
13#20276435C211yesmaximal, characteristic, complemented
14C2×#16810427C211yesmaximal, characteristic, complemented
15D4×Q16×C53C211yesmaximal, characteristic, complemented
16#16811319—21no—
17C2×#13133160C2211yescharacteristic
18C2×#13133160C2211yescharacteristic
19C2×#13133160C2211yescharacteristic
20C2×#13133157C2211yescharacteristic
21C22×#11719908C2211yescharacteristic
22#16811025C2211yescharacteristic
23#16810948—21no—
24#16810942—21no—
25#16810924—21no—
26#16810859C2211yescharacteristic
27#16810859C2211yescharacteristic
28C4×D424C2211yescharacteristic
29#16810661C2211yescharacteristic
30C2×#13133151C2211yescharacteristic
31#16810450C2211yescharacteristic
32#16810448C2211yescharacteristic
33#16810448C2211yescharacteristic
34C4⋊#11719908C2211yescharacteristic
35#16810427C2211yescharacteristic
36#16810427C2211yescharacteristic
37#16810427C2211yescharacteristic
38#16810427C2211yescharacteristic
39#16810427C2211yescharacteristic
40#16810427C2211yescharacteristic
41#16810427C2211yescharacteristic
42#16810427C2211yescharacteristic
43#16810427C2211yescharacteristic
44#16810423C2211yescharacteristic
45C22⋊#11719908C2211yescharacteristic
46C22⋊#11719908C2211yescharacteristic
47#16810399C2211yescharacteristic
48C22×Q16×C53C2211yescharacteristic
49C22×Q16×C53C2211yescharacteristic
50D4×Q8×C53—21no—
51C53×#494C2211yescharacteristic
52C53×#467C2211yescharacteristic
53C53×#467C2211yescharacteristic
54C53×#462—21no—
55C53×#451—21no—
56C53×#451—21no—
57C4×Q16×C53C2211yescharacteristic
58C8×D4×C53C2211yescharacteristic
59C2×D424C2311yescharacteristic
60#13133212—41no—
61#13133209—21no—
62#13133199—41no—
63#13133191—21no—
64#13133191—21no—
65#13133189—41no—
66C4×D212—21no—
67#13133177—41no—
68#13133176—21no—
69C2×C4⋊Dic53—21no—
70#13133160—22no—
71#13133160—41no—
72#13133160—41no—
73#13133157—41no—
74C2×#11719908—41no—
75C2×#11719908C2311yescharacteristic
76C2×#11719908C2311yescharacteristic
77C2×#11719908C2311yescharacteristic
78C2×#11719908C2311yescharacteristic
79#13133151—21no—
80#13133151C2311yescharacteristic
81#13133151C2311yescharacteristic
82#13133151C2311yescharacteristic
83#13133149—21no—
84#13133149—21no—
85#13133139—21no—
86#13133139—21no—
87#13133139—21no—
88C22×Q8×C53—21no—
89C22×Q8×C53—21no—
90C2×Q16×C53—21no—
91C2×Q16×C53—21no—
92C2×Q16×C53—21no—
93C2×Q16×C53—22no—
94C2×Q16×C53—22no—
95C2×Q16×C53—41no—
96C2×Q16×C53—41no—
97C2×Q16×C53C2311yescharacteristic
98C2×Q16×C53C2311yescharacteristic
99C2×Q16×C53C2311yescharacteristic
100C22×C8×C53C2311yescharacteristic
101C22×C8×C53C2311yescharacteristic
102C4⋊Q8×C53—21no—
103C4⋊Q8×C53—41no—
104C22⋊Q8×C53—21no—
105C22⋊Q8×C53—21no—
106C22⋊Q8×C53—41no—
107C22⋊Q8×C53—41no—
108C4×Q8×C53—21no—
109C4×D4×C53—41no—
110C4×D4×C53D411yescharacteristic
111#107×C53C2311yescharacteristic
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×Q8×C53—21no—
121C2×Q8×C53—21no—
122C2×Q8×C53—42no—
123C2×Q8×C53—42no—
124C2×Q8×C53—41no—
125C2×Q8×C53—41no—
126C2×Q8×C53—81no—
127C2×Q8×C53—81no—
128C2×D4×C53D811yescharacteristic
129C22×C4×C53—41no—
130C22×C4×C53—41no—
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C2×C22⋊D53—41no—
134C2×C22⋊D53—41no—
135C22×Dic53—41no—
136C2×D212—21no—
137C2×C4×D53—41no—
138Q16×C53—21no—
139Q16×C53—21no—
140Q16×C53—21no—
141Q16×C53—21no—
142Q16×C53—21no—
143Q16×C53—42no—
144Q16×C53—42no—
145Q16×C53—81no—
146Q16×C53#4112yes—
147C2×C8×C53—21no—
148C2×C8×C53—21no—
149C2×C8×C53—21no—
150C2×C8×C53—21no—
151C2×C8×C53C2×D411yescharacteristic
152C2×C8×C53C2×D411yescharacteristic
153C2×C8×C53C2×D411yescharacteristic
154C2×C8×C53C2411yescharacteristic
155C4⋊C4×C53—21no—
156C4⋊C4×C53—41no—
157C4⋊C4×C53—41no—
158C4⋊C4×C53—41no—
159C4⋊C4×C53—41no—
160C4⋊C4×C53—41no—
161C4⋊C4×C53D811yescharacteristic
162C22⋊C4×C53—41no—
163C22⋊C4×C53—41no—
164C22⋊C4×C53D811yescharacteristic
165C22⋊C4×C53D811yescharacteristic
166C42×C53—41no—
167C42×C53C2×D411yescharacteristic
168C22⋊Dic53—41no—
169#11719921—41no—
170#11719921—41no—
171#11719921—41no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—41no—
176#11719919—41no—
177#11719919—41no—
178C4×Dic53—41no—
179D424—21no—
180D424—21no—
181#11719908—21no—
182#11719908—21no—
183#11719908—21no—
184#11719908—21no—
185C23×C53C2×D811yescharacteristic
186C23×C53C2×D811yescharacteristic
187C22×D53—41no—
188Q8×C53—22no—
189Q8×C53—41no—
190Q8×C53—41no—
191Q8×C53—41no—
192Q8×C53—41no—
193Q8×C53—41no—
194Q8×C53—82no—
195Q8×C53—82no—
196D4×C53—22no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—41no—
202C2×C4×C53—41no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—82no—
206C2×C4×C53—82no—
207C2×C4×C53C2×D811yescharacteristic
208C2×C4×C53C2×D811yescharacteristic
209C2×C4×C53C2×D811yescharacteristic
210C2×C4×C53C2×D811yescharacteristic
211C2×C4×C53C22×D411yescharacteristic
212C22⋊D53—82no—
213C22⋊D53—161no—
214C2×Dic53—41no—
215C2×Dic53—41no—
216C2×Dic53—41no—
217C2×Dic53—161no—
218D212—41no—
219D212—41no—
220C4×D53—161no—
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#51012yes—
233C22×C53C22×D811yescharacteristic
234C2×D53—81no—
235C2×D53—81no—
236C2×D53—81no—
237C4×C53—21no—
238C4×C53—21no—
239C4×C53—21no—
240C4×C53—21no—
241C4×C53—42no—
242C4×C53—81no—
243C4×C53—81no—
244C4×C53—81no—
245C4×C53#54611yescharacteristic
246C4×C53#57711yescharacteristic
247Dic53—81no—
248Dic53—81no—
249Dic53—81no—
250#3339—531no—
251D4×Q16D5311yescharacteristic, complemented
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#329411yescharacteristic
268C2×C53C2×#51011yescharacteristic
269C2×C53#341611yescharacteristic
270D53—161no—
271C22×Q16C2×D5311yescharacteristic
272C22×Q16C2×D5311yescharacteristic
273#553—1061no—
274D4×Q8—21no—
275C2×SD32—531no—
276C2×SD32—531no—
277C2×SD32—531no—
278C2×M32—531no—
279C22×C16—531no—
280#499—531no—
281#494C2×D5311yescharacteristic
282#482—1061no—
283#481—1061no—
284#478—1061no—
285#467C2×D5311yescharacteristic
286#467C2×D5311yescharacteristic
287#466—531no—
288#466—531no—
289#462—21no—
290#451—21no—
291#451—21no—
292C4×Q16C2×D5311yescharacteristic
293C4×D8—531no—
294C8×D4C2×D5311yescharacteristic
295#427—531no—
296C2×#107—531no—
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—21no—
316C22×Q8—21no—
317C2×Q16—22no—
318C2×Q16—22no—
319C2×Q16—21no—
320C2×Q16—21no—
321C2×Q16—21no—
322C2×Q16—41no—
323C2×Q16—41no—
324C2×Q16C22×D5311yescharacteristic
325C2×Q16C22×D5311yescharacteristic
326C2×Q16C22×D5311yescharacteristic
327C2×D8—531no—
328C22×C8C22×D5311yescharacteristic
329C22×C8C22×D5311yescharacteristic
330C4⋊Q8—21no—
331C4⋊Q8—41no—
332#126—2121no—
333#125—1061no—
334#123—2121no—
335C22⋊Q8—21no—
336C22⋊Q8—21no—
337C22⋊Q8—41no—
338C22⋊Q8—41no—
339#121—1061no—
340#121—1061no—
341#121—2121no—
342C4×Q8—21no—
343C4×D4—41no—
344C4×D4—1061no—
345C4×D4—2121no—
346C4×D4C22⋊D5311yescharacteristic
347#117—1061no—
348C2×C4⋊C4—1061no—
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—531no—
359#107—531no—
360#107—531no—
361#107—1061no—
362#107C22×D5311yescharacteristic
363#106—1061no—
364#106—1061no—
365C4⋊C8C22⋊D5311yescharacteristic
366#103—21no—
367#103—21no—
368#103—21no—
369#102—1061no—
370#102—1061no—
371#102—1061no—
372C22⋊C8C22⋊D5311yescharacteristic
373C22⋊C8C22⋊D5311yescharacteristic
374C4×C8C22×D5311yescharacteristic
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×Q8—41no—
381C2×Q8—41no—
382C2×Q8—81no—
383C2×Q8—81no—
384C2×D4—1061no—
385C2×D4—2121no—
386C2×D4—2121no—
387C2×D4#1171992211yescharacteristic
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—2121no—
391C22×C4—2121no—
392C22×C4C2×C22⋊D5311yescharacteristic
393C22×C4C2×C22⋊D5311yescharacteristic
394Q16—21no—
395Q16—21no—
396Q16—21no—
397Q16—21no—
398Q16—21no—
399Q16—42no—
400Q16—42no—
401Q16—81no—
402Q16#1171994712yes—
403D8—1061no—
404D8—1061no—
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—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—1061no—
420C4⋊C4—1061no—
421C4⋊C4—1061no—
422C4⋊C4—2121no—
423C4⋊C4—2121no—
424C4⋊C4—2121no—
425C4⋊C4#1171992211yescharacteristic
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—2121no—
429C22⋊C4—2121no—
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—2121no—
442C23C2×#1171992211yescharacteristic
443C23C2×#1171992211yescharacteristic
444Q8—22no—
445Q8—41no—
446Q8—41no—
447Q8—41no—
448Q8—41no—
449Q8—41no—
450Q8—82no—
451Q8—82no—
452D4—22no—
453D4—8481no—
454D4—4242no—
455D4—2121no—
456D4—2121no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—8481no—
468C2×C4—8481no—
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—
480C8C2×#1171994711yescharacteristic
481C8#1313326211yescharacteristic
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—4241no—
488C22—4241no—
489C22—4241no—
490C22#1681106412yes—
491C22C22×#1171992211yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—42no—
497C4—81no—
498C4—81no—
499C4—81no—
500C4—4241no—
501C4—4241no—
502C4—4241no—
503C4#1681126711yescharacteristic
504C4#1681143411yescharacteristic
505C2—22no—
506C2—41no—
507C2—8481no—
508C2#2028652811yescharacteristic
509C2C2×#1681106411yescharacteristic
510C2#2028785911yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingD4×Q16×C53C2
socleC22×C53C22×D8
prime radical#20287118C2
hypercenterD4×Q16D53
hypocenterC53#33887
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×Q16

Complements

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

NH ≅ G/Ntype
2. #20276960C2semidirect
3. #20276920C2semidirect
4. #20276915C2semidirect
5. #20276879C2semidirect
6. #20276830C2semidirect
7. #20276813C2semidirect
8. #20276757C2semidirect
9. #20276632C2semidirect
10. #20276632C2semidirect
11. #20276577C2semidirect
12. #20276541C2semidirect
13. #20276435C2semidirect
14. C2×#16810427C2semidirect
15. D4×Q16×C53C2semidirect
251. D4×Q16D53semidirect
314. C53#33887semidirect

Maximal subgroups

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

subgroupindex
#202871182
#202769602
#202769202
#202769152
#202768792
#202768302
#202768132
#202767572
#202766322
#202765772
#202765412
#202764352
C2×#168104272
D4×Q16×C532
#3388753