Finite Group Databasebeta

← #27539224

Subgroups — #27539224

Order 13568.

534 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283592C211yesmaximal, characteristic
2#20276915C211yesmaximal, characteristic, complemented
3#20276910C211yesmaximal, characteristic, complemented
4#20276876C211yesmaximal, characteristic, complemented
5#20276869C211yesmaximal, characteristic, complemented
6#20276762C211yesmaximal, characteristic, complemented
7#20276757C211yesmaximal, characteristic, complemented
8#20276653C211yesmaximal, characteristic
9#20276653C211yesmaximal, characteristic
10#20276584C211yesmaximal, characteristic, complemented
11#20276577C211yesmaximal, characteristic, complemented
12C4×#13133163C211yesmaximal, characteristic, complemented
13C4×#13133160C211yesmaximal, characteristic, complemented
14#20276522C211yesmaximal, characteristic, complemented
15C53×#3177C211yesmaximal, characteristic, complemented
16C2×#13133239C2211yescharacteristic
17#16811103—21no—
18C2×#13133163C2211yescharacteristic
19C2×#13133163C2211yescharacteristic
20C2×#13133160C2211yescharacteristic
21C2×#13133160C2211yescharacteristic
22#16810725—21no—
23#16810725—21no—
24C4×Dic212C2211yescharacteristic
25C4×Dic212C2211yescharacteristic
26C4×D424C2211yescharacteristic
27C4×D424C2211yescharacteristic
28C2×#13133151C2211yescharacteristic
29#16810625—21no—
30#16810450C2211yescharacteristic
31#16810450C2211yescharacteristic
32#16810448C2211yescharacteristic
33#16810448C2211yescharacteristic
34C4⋊#11719908C2211yescharacteristic
35C4⋊#11719908C2211yescharacteristic
36#16810427C2211yescharacteristic
37#16810427C2211yescharacteristic
38#16810427C2211yescharacteristic
39#16810427C2211yescharacteristic
40#16810426C2211yescharacteristic
41#16810426C2211yescharacteristic
42#16810423C2211yescharacteristic
43#16810423C2211yescharacteristic
44C22⋊#11719908C2211yescharacteristic
45C22⋊#11719908C2211yescharacteristic
46#16810399C2211yescharacteristic
47C4×#11719908C2211yescharacteristic
48C53×#541—21no—
49C53×#499C2211yescharacteristic
50C53×#494C2211yescharacteristic
51C53×#491—21no—
52C53×#470—21no—
53C53×#467C2211yescharacteristic
54C53×#467C2211yescharacteristic
55C53×#441—21no—
56C4×Q16×C53C2211yescharacteristic
57C4×Q16×C53C2211yescharacteristic
58C53×#405C2211yescharacteristic
59C2×#11719945—21no—
60#13133239—22no—
61#13133239—22no—
62C2×Dic212C2311yescharacteristic
63C2×#11719913—21no—
64C2×D424C2311yescharacteristic
65C4×Dic106—21no—
66C4×Dic106—21no—
67C4×D212—21no—
68C4×D212—21no—
69#13133177—41no—
70#13133177—41no—
71#13133175—41no—
72C2×C4⋊Dic53—21no—
73C4⋊C4×D53—41no—
74#13133163—22no—
75#13133163—22no—
76#13133160—22no—
77#13133160—22no—
78C2×#11719908C2311yescharacteristic
79C2×#11719908C2311yescharacteristic
80C2×#11719908C2311yescharacteristic
81C2×#11719908C2311yescharacteristic
82#13133151—21no—
83#13133151C2311yescharacteristic
84#13133151C2311yescharacteristic
85#13133142—21no—
86#13133142—21no—
87#13133139—21no—
88#13133139—21no—
89C2×Q16×C53—21no—
90C2×Q16×C53C2311yescharacteristic
91C2×Q16×C53C2311yescharacteristic
92C2×M16×C53D411yescharacteristic
93C22×C8×C53C2311yescharacteristic
94C4⋊Q8×C53—21no—
95#126×C53—41no—
96#126×C53—41no—
97#125×C53—21no—
98#125×C53—41no—
99#125×C53—41no—
100C22⋊Q8×C53—21no—
101C22⋊Q8×C53—21no—
102C4×Q8×C53—21no—
103C4×Q8×C53—21no—
104#117×C53D411yescharacteristic
105#107×C53C2311yescharacteristic
106#107×C53C2311yescharacteristic
107#106×C53—21no—
108#103×C53—21no—
109#103×C53—21no—
110#103×C53—21no—
111#103×C53—21no—
112#97×C53D411yescharacteristic
113#97×C53D411yescharacteristic
114C4×C8×C53C2311yescharacteristic
115C4×C8×C53C2311yescharacteristic
116C2×Q8×C53—21no—
117C2×Q8×C53—21no—
118C22×C4×C53C2×D411yescharacteristic
119C2×C22⋊D53—41no—
120C22×Dic53—41no—
121#11719945—24no—
122#11719945—42no—
123C2×D212—21no—
124C2×C4×D53—41no—
125C2×C4×D53—41no—
126C2×C4×D53—41no—
127C2×Dic106—21no—
128Q16×C53—21no—
129Q16×C53—21no—
130Q16×C53—21no—
131Q16×C53—21no—
132Q16×C53—42no—
133M16×C53—41no—
134C2×C8×C53—21no—
135C2×C8×C53—21no—
136C2×C8×C53C2×D411yescharacteristic
137C2×C8×C53C2×D411yescharacteristic
138C2×C8×C53C2×D411yescharacteristic
139C2×C8×C53C2411yescharacteristic
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—41no—
143C4⋊C4×C53—41no—
144C4⋊C4×C53—41no—
145C4⋊C4×C53—41no—
146C4⋊C4×C53—41no—
147C4⋊C4×C53—41no—
148C4⋊C4×C53—41no—
149C4⋊C4×C53—41no—
150C4⋊C4×C53D811yescharacteristic
151C4⋊C4×C53D811yescharacteristic
152C22⋊C4×C53—41no—
153C22⋊C4×C53—41no—
154C22⋊C4×C53D811yescharacteristic
155C22⋊C4×C53D811yescharacteristic
156C42×C53—41no—
157C42×C53—41no—
158C42×C53C2×D411yescharacteristic
159C42×C53C2×D411yescharacteristic
160#11719921—41no—
161#11719921—41no—
162C4⋊Dic53—21no—
163C4⋊Dic53—21no—
164C4⋊Dic53—41no—
165#11719919—41no—
166#11719919—41no—
167C4×Dic53—41no—
168C4×Dic53—41no—
169Dic212—21no—
170Dic212#4112yes—
171D424—21no—
172D424#4112yes—
173#11719913—24no—
174#11719908—21no—
175#11719908—21no—
176#11719908—21no—
177#11719908—21no—
178C23×C53C2×D811yescharacteristic
179C22×D53—41no—
180Q8×C53—41no—
181Q8×C53—41no—
182Q8×C53—41no—
183Q8×C53—41no—
184C2×C4×C53—21no—
185C2×C4×C53—21no—
186C2×C4×C53—41no—
187C2×C4×C53—41no—
188C2×C4×C53—41no—
189C2×C4×C53—41no—
190C2×C4×C53C2×D811yescharacteristic
191C2×C4×C53C2×D811yescharacteristic
192C2×C4×C53C2×D811yescharacteristic
193C2×C4×C53C2×D811yescharacteristic
194C2×C4×C53C2×D811yescharacteristic
195C2×C4×C53C22×D411yescharacteristic
196C22⋊D53—44no—
197C2×Dic53—41no—
198C2×Dic53—41no—
199C2×Dic53—41no—
200C2×Dic53—161no—
201D212—22no—
202D212—41no—
203C4×D53—44no—
204C4×D53—161no—
205C4×D53—161no—
206Dic106—22no—
207Dic106—41no—
208C8×C53—21no—
209C8×C53—21no—
210C8×C53—21no—
211C8×C53C2×#4111yescharacteristic
212C8×C53C2×#4111yescharacteristic
213#33680—531nomaximal, Sylow
214C22×C53—21no—
215C22×C53—21no—
216C22×C53—21no—
217C22×C53C22×D811yescharacteristic
218C2×D53—42no—
219C2×D53—81no—
220C2×D53—81no—
221C4×C53—21no—
222C4×C53—21no—
223C4×C53—21no—
224C4×C53—21no—
225C4×C53—21no—
226C4×C53—81no—
227C4×C53—81no—
228C4×C53—81no—
229C4×C53—81no—
230C4×C53#52511yescharacteristic
231C4×C53#57511yescharacteristic
232Dic53—42no—
233Dic53—81no—
234Dic53—81no—
235#3177D5311yescharacteristic, complemented
236#2951—531no—
237#2236—531no—
238#2235—531no—
239#2229—531no—
240#2228—531no—
241#2209—531no—
242#2208—531no—
243#2189—531no—
244#2189—531no—
245#2176—531no—
246#2175—531no—
247C4×Q32—531no—
248C4×SD32—531no—
249#2164—531no—
250C2×C53—41no—
251C2×C53#310011yescharacteristic
252C2×C53#341611yescharacteristic
253C2×C53#341711yescharacteristic
254D53—82no—
255C2×#135—531no—
256#541—21no—
257#520—1061no—
258C2×Q32—531no—
259C2×Q32—531no—
260C2×SD32—531no—
261C2×SD32—531no—
262#499C2×D5311yescharacteristic
263#494C2×D5311yescharacteristic
264#491—21no—
265#470—21no—
266#467C2×D5311yescharacteristic
267#467C2×D5311yescharacteristic
268#441—21no—
269#440—1061no—
270#440—1061no—
271C4×Q16—531no—
272C4×Q16—531no—
273C4×Q16C2×D5311yescharacteristic
274C4×Q16C2×D5311yescharacteristic
275C4×D8—531no—
276C4×D8—531no—
277C2×#107—531no—
278#417—1061no—
279#405C2×D5311yescharacteristic
280#367—531no—
281#367—531no—
282#366—531no—
283#366—531no—
284C4⋊C16—531no—
285C4⋊C16—531no—
286#358—531no—
287#358—531no—
288#358—531no—
289#358—531no—
290#358—531no—
291#358—531no—
292#357—531no—
293#357—531no—
294C22⋊C16—531no—
295C22⋊C16—531no—
296#346—531no—
297C4×C16—531no—
298C53#3368011yescharacteristic, complemented, Sylow
299C2×#41—1061no—
300#135—1062no—
301#135—1062no—
302C2×Q16—21no—
303C2×Q16—531no—
304C2×Q16C22×D5311yescharacteristic
305C2×Q16C22×D5311yescharacteristic
306C2×SD16—1061no—
307C2×D8—531no—
308C2×M16C22⋊D5311yescharacteristic
309C22×C8C22×D5311yescharacteristic
310C4⋊Q8—21no—
311#126—41no—
312#126—41no—
313#125—21no—
314#125—41no—
315#125—41no—
316C22⋊Q8—21no—
317C22⋊Q8—21no—
318C4×Q8—21no—
319C4×Q8—21no—
320C4×Q8—1061no—
321C4×Q8—1061no—
322C4×D4—1061no—
323C4×D4—1061no—
324C4×D4—2121no—
325C4×D4—2121no—
326#117—2121no—
327#117C22⋊D5311yescharacteristic
328C2×C4⋊C4—1061no—
329C2×C4⋊C4—2121no—
330Q32—1062no—
331Q32—1062no—
332SD32—1062no—
333SD32—1062no—
334C2×C16—531no—
335C2×C16—531no—
336C2×C16—531no—
337C2×C16—531no—
338#107—531no—
339#107—531no—
340#107—1061no—
341#107C22×D5311yescharacteristic
342#107C22×D5311yescharacteristic
343#106—21no—
344#103—21no—
345#103—21no—
346#103—21no—
347#103—21no—
348#103—1061no—
349#103—1061no—
350#102—1061no—
351#102—1061no—
352#97C22⋊D5311yescharacteristic
353#97C22⋊D5311yescharacteristic
354C4×C8C22×D5311yescharacteristic
355C4×C8C22×D5311yescharacteristic
356#41—1064no—
357#41—2122no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—1061no—
361C2×D4—1061no—
362C2×D4—2121no—
363C22×C4—2121no—
364C22×C4—2121no—
365C22×C4—2121no—
366C22×C4—2121no—
367C22×C4C2×C22⋊D5311yescharacteristic
368Q16—21no—
369Q16—21no—
370Q16—21no—
371Q16—21no—
372Q16—42no—
373Q16—532no—
374Q16—1061no—
375SD16—1064no—
376D8—532no—
377D8—1061no—
378M16—41no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8C2×C22⋊D5311yescharacteristic
382C2×C8C2×C22⋊D5311yescharacteristic
383C2×C8C2×C22⋊D5311yescharacteristic
384C2×C8C23×D5311yescharacteristic
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—1061no—
396C4⋊C4—1061no—
397C4⋊C4—2121no—
398C4⋊C4—2121no—
399C4⋊C4—2121no—
400C4⋊C4#1171992211yescharacteristic
401C4⋊C4#1171992211yescharacteristic
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—2121no—
405C22⋊C4—2121no—
406C22⋊C4#1171992211yescharacteristic
407C22⋊C4#1171992211yescharacteristic
408C42—41no—
409C42—41no—
410C42—2121no—
411C42—2121no—
412C42C2×C22⋊D5311yescharacteristic
413C42C2×C22⋊D5311yescharacteristic
414C16—1061no—
415C16—1061no—
416C16—1061no—
417C16—1061no—
418C23—2121no—
419C23C2×#1171992211yescharacteristic
420Q8—41no—
421Q8—41no—
422Q8—41no—
423Q8—41no—
424Q8—1062no—
425Q8—2121no—
426D4—1062no—
427D4—2124no—
428D4—2121no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—8481no—
436C2×C4—8481no—
437C2×C4—8481no—
438C2×C4—2124no—
439C2×C4—2121no—
440C2×C4—2121no—
441C2×C4—2121no—
442C2×C4C2×#1171992211yescharacteristic
443C2×C4C2×#1171992211yescharacteristic
444C2×C4C2×#1171992211yescharacteristic
445C2×C4C2×#1171992211yescharacteristic
446C2×C4C2×#1171992211yescharacteristic
447C2×C4C22×C22⋊D5311yescharacteristic
448C8—21no—
449C8—21no—
450C8—21no—
451C8#41×D5311yescharacteristic
452C8#41×D5311yescharacteristic
453C22—21no—
454C22—21no—
455C22—21no—
456C22—4241no—
457C22—4241no—
458C22—2122no—
459C22C22×#1171992211yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—81no—
466C4—81no—
467C4—81no—
468C4—81no—
469C4—4241no—
470C4—4241no—
471C4—2122no—
472C4#1681113011yescharacteristic
473C4#1681142111yescharacteristic
474C2—41no—
475C2—4242no—
476C2#2028471111yescharacteristic
477C2#2028785911yescharacteristic
478C2#2028786611yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3177C2
socleC22×C53C22×D8
prime radicalC2×#13133239C22
hypercenter#3177D53
hypocenterC53#33680
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3177

Complements

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

NH ≅ G/Ntype
2. #20276915C2semidirect
3. #20276910C2semidirect
4. #20276876C2semidirect
5. #20276869C2semidirect
6. #20276762C2semidirect
7. #20276757C2semidirect
10. #20276584C2semidirect
11. #20276577C2semidirect
12. C4×#13133163C2semidirect
13. C4×#13133160C2semidirect
14. #20276522C2semidirect
15. C53×#3177C2semidirect
235. #3177D53semidirect
298. C53#33680semidirect

Maximal subgroups

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

subgroupindex
#202835922
#202769152
#202769102
#202768762
#202768692
#202767622
#202767572
#202766532
#202765842
#202765772
C4×#131331632
C4×#131331602
#202765222
C53×#31772
#3368053