Finite Group Databasebeta

← #27537397

Subgroups — #27537397

Order 13568.

682 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811054C211yesmaximal, characteristic
2C4×#13133240C211yesmaximal, characteristic, complemented
3C4×#13133236C211yesmaximal, characteristic, complemented
4#20276589C211yesmaximal, characteristic, complemented
5#20276588C211yesmaximal, characteristic, complemented
6#20276584C211yesmaximal, characteristic, complemented
7#20276583C211yesmaximal, characteristic, complemented
8#20276581C211yesmaximal, characteristic, complemented
9#20276580C211yesmaximal, characteristic, complemented
10#20276578C211yesmaximal, characteristic, complemented
11#20276496C211yesmaximal, characteristic, complemented
12#20276421C211yesmaximal, characteristic, complemented
13#20276419C211yesmaximal, characteristic, complemented
14#346×D53C211yesmaximal, characteristic, complemented
15C53×#2175C211yesmaximal, characteristic, complemented
16C2×#13133240C2211yescharacteristic
17C2×#13133236C2211yescharacteristic
18C4×#11719949—21no—
19C4×#11719947—21no—
20#16811054C418yescomplemented
21C2×#13133163C2211yescharacteristic
22C2×#13133162C2211yescharacteristic
23C2×#13133158C2211yescharacteristic
24C2×C16×D53C2211yescharacteristic
25C2×C16×D53C2211yescharacteristic
26C4×Dic212C2211yescharacteristic
27Q16×Dic53C2211yescharacteristic
28#16810717C2211yescharacteristic
29#16810714—21no—
30C4×#11719924—21no—
31C4×#11719923—21no—
32#16810708—21no—
33C4×D424C2211yescharacteristic
34D8×Dic53C2211yescharacteristic
35#16810705C2211yescharacteristic
36#16810660C2211yescharacteristic
37#16810660C2211yescharacteristic
38#16810618—21no—
39#16810616—21no—
40C4×C8×D53C2211yescharacteristic
41#16810450C2211yescharacteristic
42#16810450C2211yescharacteristic
43#16810449C2211yescharacteristic
44#16810428C2211yescharacteristic
45#16810427C2211yescharacteristic
46#16810426C2211yescharacteristic
47#16810425C2211yescharacteristic
48#16810424C2211yescharacteristic
49#16810423C2211yescharacteristic
50#16810399C2211yescharacteristic
51#16810398C2211yescharacteristic
52#16810398C2211yescharacteristic
53C2×SD32×C53C2211yescharacteristic
54C4×Q16×C53C2211yescharacteristic
55C4×D8×C53C2211yescharacteristic
56C53×#366C2211yescharacteristic
57C53×#358C2211yescharacteristic
58C53×#357C2211yescharacteristic
59C53×#346C2211yescharacteristic
60C2×#11719949—21no—
61C2×#11719947—21no—
62#13133240—22no—
63#13133240C2×C414yescomplemented
64#13133236—22no—
65#13133236C2×C414yescomplemented
66C2×Dic212C2311yescharacteristic
67C2×#11719923—21no—
68C2×#11719924—21no—
69C2×D424C2311yescharacteristic
70C2×C8×D53D411yescharacteristic
71C2×C8×D53C2311yescharacteristic
72Q8×Dic53—21no—
73C4×Dic106—21no—
74#13133182—21no—
75C4×D212—21no—
76C4×D212—41no—
77D4×Dic53—21no—
78#13133179—21no—
79#13133179—41no—
80C4×C22⋊D53—41no—
81#13133177—41no—
82#13133175—21no—
83#13133175—21no—
84#13133175—41no—
85#13133173—41no—
86C2×C4×Dic53—41no—
87C42×D53—41no—
88C42×D53D411yescharacteristic
89#13133163C2×C414yescomplemented
90#13133162C2×C414yescomplemented
91#13133158C2×C414yescomplemented
92C2×#11719908C2311yescharacteristic
93C2×#11719908C2311yescharacteristic
94C16×D53—22no—
95C16×D53C2×C414yescomplemented
96#13133151C2311yescharacteristic
97#13133151C2311yescharacteristic
98#13133148—21no—
99#13133148—21no—
100#13133142—21no—
101#13133141—21no—
102#13133140—21no—
103#13133139—21no—
104#13133138—21no—
105#13133137—21no—
106C4×#88322D411yescharacteristic
107C8×Dic53D411yescharacteristic
108C8×Dic53C2311yescharacteristic
109C2×Q16×C53C2311yescharacteristic
110C2×D8×C53C2311yescharacteristic
111C4×Q8×C53—21no—
112C4×D4×C53—21no—
113SD32×C53C2×C414yescomplemented
114C2×C16×C53C2311yescharacteristic
115C2×C16×C53C2311yescharacteristic
116#107×C53C2311yescharacteristic
117#107×C53C2311yescharacteristic
118#103×C53—21no—
119#102×C53—21no—
120C4×C8×C53C2311yescharacteristic
121C2×Q8×C53—21no—
122C2×D4×C53—21no—
123C22×C4×C53—41no—
124C2×C22⋊D53—41no—
125C22×Dic53—41no—
126C22×Dic53—41no—
127#11719949—24no—
128#11719949—42no—
129#11719947—24no—
130#11719947—42no—
131C2×D212—21no—
132C2×D212—41no—
133C2×C4×D53—41no—
134C2×C4×D53—41no—
135C2×C4×D53—41no—
136C2×C4×D53D811yescharacteristic
137C2×C4×D53D811yescharacteristic
138C2×C4×D53C2×D411yescharacteristic
139C2×Dic106—21no—
140Q16×C53—21no—
141Q16×C53C22×C412yes—
142D8×C53—21no—
143D8×C53C22×C412yes—
144C2×C8×C53C2×D411yescharacteristic
145C2×C8×C53C2411yescharacteristic
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—41no—
149C22⋊C4×C53—41no—
150C42×C53—41no—
151C42×C53C2×D411yescharacteristic
152C22⋊Dic53—41no—
153#11719924—24no—
154#11719923—24no—
155#11719921—41no—
156#11719921—41no—
157#11719921—41no—
158#11719921—41no—
159C4⋊Dic53—21no—
160C4⋊Dic53—21no—
161C4⋊Dic53—41no—
162#11719919—41no—
163#11719919—41no—
164C4×Dic53—41no—
165C4×Dic53—41no—
166C4×Dic53—41no—
167C4×Dic53—41no—
168C4×Dic53—41no—
169C4×Dic53D811yescharacteristic
170C4×Dic53D811yescharacteristic
171C4×Dic53C2×D411yescharacteristic
172C2×#88322C2×D411yescharacteristic
173C2×#88322C2×D411yescharacteristic
174Dic212—21no—
175Dic212C22×C412yes—
176D424—21no—
177D424C22×C412yes—
178C8×D53—22no—
179C8×D53C22×C412yes—
180C8×D53#4112yes—
181C16×C53—21no—
182C16×C53C22×C412yes—
183#11719908—21no—
184#11719908C22×C412yes—
185C23×C53—41no—
186C22×D53—41no—
187C22×D53C2×D811yescharacteristic
188Q8×C53—22no—
189Q8×C53—41no—
190D4×C53—22no—
191D4×C53—41no—
192C2×C4×C53—41no—
193C2×C4×C53—41no—
194C2×C4×C53—41no—
195C2×C4×C53—82no—
196C2×C4×C53C2×D811yescharacteristic
197C2×C4×C53C2×D811yescharacteristic
198C2×C4×C53C22×D411yescharacteristic
199C22⋊D53—44no—
200C2×Dic53—41no—
201C2×Dic53—44no—
202C2×Dic53—41no—
203C2×Dic53—41no—
204C2×Dic53—82no—
205C2×Dic53C2×D811yescharacteristic
206C2×Dic53C2×D811yescharacteristic
207C2×Dic53C2×D811yescharacteristic
208D212—22no—
209D212—44no—
210D212—41no—
211C4×D53—41no—
212C4×D53—41no—
213C4×D53—44no—
214C4×D53—82no—
215C4×D53—82no—
216C4×D53C4×D412yes—
217C4×D53#13512yes—
218Dic106—22no—
219Dic106—41no—
220C8×C53—21no—
221C8×C53C23×C411yescharacteristic
222C8×C53C2×#4111yescharacteristic
223#88322—21no—
224#88322C4×D412yes—
225#33451—531nomaximal, Sylow
226C22×C53—42no—
227C22×C53—81no—
228C22×C53—81no—
229C22×C53C22×D811yescharacteristic
230C2×D53—21no—
231C2×D53—21no—
232C2×D53—42no—
233C2×D53—81no—
234C2×D53—81no—
235C2×D53C4×D812yes—
236C4×C53—21no—
237C4×C53—21no—
238C4×C53—42no—
239C4×C53—81no—
240C4×C53—81no—
241C4×C53C2×C4×D411yescharacteristic
242C4×C53C2×#13511yescharacteristic
243Dic53—21no—
244Dic53—21no—
245Dic53—42no—
246Dic53—81no—
247Dic53—81no—
248Dic53C4×D812yes—
249C2×#508—531no—
250C4×#135—531no—
251C4×#135—531no—
252#2177—531no—
253#2177—531no—
254#2176—531no—
255#2176—531no—
256#2175—531no—
257#2175—531no—
258#2175—531no—
259#2175D5311yescharacteristic, complemented
260#2158—531no—
261#2138—531no—
262#2138—531no—
263C2×#346—531no—
264C2×C53—82no—
265C2×C53C2×C4×D811yescharacteristic
266C2×C53#341411yescharacteristic
267C2×C53#341411yescharacteristic
268D53—22no—
269D53—82no—
270C2×#135—531no—
271C2×#135—531no—
272C4×#41—1061no—
273C4×#41—1061no—
274#508—538no—
275C2×Q32—531no—
276C2×SD32—531no—
277C2×SD32C2×D5311yescharacteristic
278C2×D16—531no—
279C22×C16—531no—
280C22×C16—531no—
281C4×Q16—531no—
282C4×Q16—531no—
283C4×Q16—531no—
284C4×Q16C2×D5311yescharacteristic
285C4×SD16—1061no—
286C4×SD16—1061no—
287C4×SD16—1061no—
288C4×SD16—1061no—
289C4×D8—531no—
290C4×D8—531no—
291C4×D8—531no—
292C4×D8C2×D5311yescharacteristic
293#427—531no—
294#427—531no—
295#416—1061no—
296#416—1061no—
297C2×C4×C8—531no—
298#367—531no—
299#367—531no—
300#366—531no—
301#366C2×D5311yescharacteristic
302#358—531no—
303#358—531no—
304#358—531no—
305#358C2×D5311yescharacteristic
306#357—531no—
307#357—531no—
308#357—531no—
309#357C2×D5311yescharacteristic
310#346—531no—
311#346—531no—
312#346—531no—
313#346C2×D5311yescharacteristic
314C53#3345111yescharacteristic, complemented, Sylow
315C2×#41—1061no—
316C2×#41—1061no—
317#135—534no—
318#135—534no—
319#135—1062no—
320#135—1062no—
321C2×Q16—531no—
322C2×Q16C22×D5311yescharacteristic
323C2×SD16—1061no—
324C2×SD16—1061no—
325C2×D8—531no—
326C2×D8C22×D5311yescharacteristic
327C22×C8—531no—
328C22×C8—531no—
329C4×Q8—21no—
330C4×Q8—1061no—
331C4×Q8—1061no—
332C4×Q8—1061no—
333C4×D4—21no—
334C4×D4—1061no—
335C4×D4—1061no—
336C4×D4—1061no—
337C4×D4—2121no—
338C4×D4—2121no—
339C4×D4—2121no—
340C4×D4—2121no—
341#117—1061no—
342#117—1061no—
343#117—2121no—
344#117—2121no—
345C2×C42—531no—
346C2×C42—2121no—
347C2×C42—2121no—
348Q32—534no—
349SD32—534no—
350SD32C4×D5314yescomplemented
351D16—534no—
352C2×C16—531no—
353C2×C16—534no—
354C2×C16—531no—
355C2×C16—1062no—
356C2×C16C22×D5311yescharacteristic
357C2×C16C22×D5311yescharacteristic
358#107—531no—
359#107—531no—
360#107C22×D5311yescharacteristic
361#107C22×D5311yescharacteristic
362#106—1061no—
363#106—1061no—
364#103—21no—
365#103—1061no—
366#103—1061no—
367#103—1061no—
368#102—21no—
369#102—1061no—
370#102—1061no—
371#102—1061no—
372C4×C8—531no—
373C4×C8—531no—
374C4×C8—531no—
375C4×C8C22×D5311yescharacteristic
376#41—1064no—
377#41—1064no—
378#41—2122no—
379#41—2122no—
380C2×Q8—21no—
381C2×Q8—1061no—
382C2×D4—21no—
383C2×D4—1061no—
384C2×D4—2121no—
385C2×D4—2121no—
386C22×C4—41no—
387C22×C4—531no—
388C22×C4—531no—
389C22×C4—531no—
390C22×C4—2121no—
391C22×C4—2121no—
392C22×C4—2121no—
393C22×C4—2121no—
394C22×C4—2121no—
395Q16—21no—
396Q16—532no—
397Q16—1061no—
398Q16C2×C4×D5312yes—
399SD16—1064no—
400SD16—1064no—
401D8—21no—
402D8—532no—
403D8—1061no—
404D8C2×C4×D5312yes—
405C2×C8—531no—
406C2×C8—532no—
407C2×C8—532no—
408C2×C8—531no—
409C2×C8—1062no—
410C2×C8D4×D5311yescharacteristic
411C2×C8C23×D5311yescharacteristic
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—41no—
415C4⋊C4—1061no—
416C4⋊C4—1061no—
417C4⋊C4—2121no—
418C4⋊C4—2121no—
419C4⋊C4—2121no—
420C22⋊C4—41no—
421C22⋊C4—2121no—
422C22⋊C4—2121no—
423C22⋊C4—2121no—
424C22⋊C4—2121no—
425C22⋊C4—2121no—
426C42—41no—
427C42—531no—
428C42—531no—
429C42—531no—
430C42—2121no—
431C42—2121no—
432C42—2121no—
433C42—2121no—
434C42—2121no—
435C42D4×D5311yescharacteristic
436C16—21no—
437C16—532no—
438C16—1061no—
439C16C2×C4×D5312yes—
440C23—41no—
441C23—531no—
442C23—2121no—
443Q8—22no—
444Q8—41no—
445Q8—1062no—
446Q8—2121no—
447D4—22no—
448D4—41no—
449D4—1062no—
450D4—2124no—
451D4—2124no—
452D4—2121no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—82no—
457C2×C4—531no—
458C2×C4—532no—
459C2×C4—532no—
460C2×C4—531no—
461C2×C4—531no—
462C2×C4—4242no—
463C2×C4—4242no—
464C2×C4—4242no—
465C2×C4—2121no—
466C2×C4—2121no—
467C2×C4—2124no—
468C2×C4—2124no—
469C2×C4—2121no—
470C2×C4—2121no—
471C2×C4—2121no—
472C2×C4D8×D5311yescharacteristic
473C2×C4D8×D5311yescharacteristic
474C2×C4C2×D4×D5311yescharacteristic
475C8—21no—
476C8—532no—
477C8—1061no—
478C8C22×C4×D5311yescharacteristic
479C8#41×D5311yescharacteristic
480C22—42no—
481C22—81no—
482C22—81no—
483C22—532no—
484C22—1061no—
485C22—1061no—
486C22—4241no—
487C22—4241no—
488C22—2122no—
489C22C2×D8×D5311yescharacteristic
490C4—21no—
491C4—21no—
492C4—42no—
493C4—81no—
494C4—81no—
495C4—532no—
496C4—1061no—
497C4—1061no—
498C4—4241no—
499C4—4241no—
500C4—2122no—
501C4C4×D4×D5311yescharacteristic
502C4#135×D5311yescharacteristic
503C2—82no—
504C2—1062no—
505C2—4242no—
506C2C4×D8×D5311yescharacteristic
507C2#2028784711yescharacteristic
508C2#2028785011yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutatorC8×C53C23×C4
FrattiniC2×C8C23×D53
FittingC53×#2175C2
socleC22×C53C22×D8
prime radicalC2×#16811054C2
cosocleC2×C8×C53C24
hypercenter#2175D53
hypocenterC53#33451
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2175

Complements

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

NH ≅ G/Ntype
2. C4×#13133240C2semidirect
3. C4×#13133236C2semidirect
4. #20276589C2semidirect
5. #20276588C2semidirect
6. #20276584C2semidirect
7. #20276583C2semidirect
8. #20276581C2semidirect
9. #20276580C2semidirect
10. #20276578C2semidirect
11. #20276496C2semidirect
12. #20276421C2semidirect
13. #20276419C2semidirect
14. #346×D53C2semidirect
15. C53×#2175C2semidirect
20. #16811054C4semidirect
63. #13133240C2×C4semidirect
65. #13133236C2×C4semidirect
89. #13133163C2×C4semidirect
90. #13133162C2×C4semidirect
91. #13133158C2×C4semidirect
95. C16×D53C2×C4semidirect
113. SD32×C53C2×C4semidirect
259. #2175D53semidirect
314. C53#33451semidirect
350. SD32C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168110542
C4×#131332402
C4×#131332362
#202765892
#202765882
#202765842
#202765832
#202765812
#202765802
#202765782
#202764962
#202764212
#202764192
#346×D532
C53×#21752
#3345153