Finite Group Databasebeta

← #27541938

Subgroups — #27541938

Order 13568.

468 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287607C211yesmaximal, characteristic, complemented
2#20276960C211yesmaximal, characteristic, complemented
3#20276951C211yesmaximal, characteristic
4#20276946C211yesmaximal, characteristic
5#20276879C211yesmaximal, characteristic, complemented
6#20276872C211yesmaximal, characteristic
7#20276867C211yesmaximal, characteristic, complemented
8#20276774C211yesmaximal, characteristic
9#20276774C211yesmaximal, characteristic
10#20276762C211yesmaximal, characteristic, complemented
11#20276624C211yesmaximal, characteristic, complemented
12#20276584C211yesmaximal, characteristic, complemented
13#20276579C211yesmaximal, characteristic
14C4×#13133163C211yesmaximal, characteristic, complemented
15C53×#3393C211yesmaximal, characteristic
16#16811351—21no—
17C2×#13133163C2211yescharacteristic
18C2×#13133163C2211yescharacteristic
19C2×#13133161C2211yescharacteristic
20#16811028C2211yescharacteristic
21#16811025C2211yescharacteristic
22#16811025C2211yescharacteristic
23#16810931—21no—
24#16810931—21no—
25#16810909—21no—
26C4×Dic212C2211yescharacteristic
27C4×Dic212C2211yescharacteristic
28C4×Dic212C2211yescharacteristic
29#16810450C2211yescharacteristic
30#16810448C2211yescharacteristic
31#16810448C2211yescharacteristic
32C4⋊#11719908C2211yescharacteristic
33C4⋊#11719908C2211yescharacteristic
34C4⋊#11719908C2211yescharacteristic
35#16810427C2211yescharacteristic
36#16810427C2211yescharacteristic
37#16810427C2211yescharacteristic
38#16810427C2211yescharacteristic
39#16810427C2211yescharacteristic
40#16810427C2211yescharacteristic
41#16810426C2211yescharacteristic
42#16810424C2211yescharacteristic
43#16810424C2211yescharacteristic
44#16810424C2211yescharacteristic
45#16810399C2211yescharacteristic
46#16810399C2211yescharacteristic
47C4×#11719908C2211yescharacteristic
48C53×#556—21no—
49C53×#495C2211yescharacteristic
50C53×#495C2211yescharacteristic
51C53×#494C2211yescharacteristic
52C53×#464—21no—
53C53×#464—21no—
54C53×#461—21no—
55C8×Q8×C53C2211yescharacteristic
56C4×Q16×C53C2211yescharacteristic
57C4×Q16×C53C2211yescharacteristic
58C4×D8×C53C2211yescharacteristic
59C2×Dic212C2311yescharacteristic
60#13133217—21no—
61#13133215—41no—
62#13133209—21no—
63#13133209—21no—
64#13133207—41no—
65#13133207—41no—
66C4×Dic106—21no—
67C4×Dic106—21no—
68C4×Dic106—21no—
69#13133182—41no—
70#13133163—22no—
71#13133163—22no—
72#13133161—22no—
73C2×#11719908C2311yescharacteristic
74C2×#11719908C2311yescharacteristic
75C2×#11719908C2311yescharacteristic
76C2×#11719908C2311yescharacteristic
77#13133151—21no—
78#13133151C2311yescharacteristic
79#13133151C2311yescharacteristic
80#13133151C2311yescharacteristic
81#13133149—21no—
82#13133149—21no—
83#13133142—21no—
84#13133142—21no—
85#13133142—21no—
86C2×Q16×C53—21no—
87C2×Q16×C53C2311yescharacteristic
88C2×Q16×C53C2311yescharacteristic
89C2×SD16×C53—21no—
90C2×SD16×C53—21no—
91C2×D8×C53C2311yescharacteristic
92C4⋊Q8×C53—21no—
93#126×C53—41no—
94#126×C53—41no—
95#124×C53—21no—
96#124×C53—21no—
97C22⋊Q8×C53—41no—
98C4×Q8×C53—21no—
99C4×Q8×C53—21no—
100C4×Q8×C53D411yescharacteristic
101C4×D4×C53—21no—
102#117×C53—41no—
103#107×C53C2311yescharacteristic
104C4⋊C8×C53D411yescharacteristic
105C4⋊C8×C53D411yescharacteristic
106C4⋊C8×C53D411yescharacteristic
107#103×C53—21no—
108#103×C53—21no—
109#102×C53—21no—
110C4×C8×C53C2311yescharacteristic
111C4×C8×C53C2311yescharacteristic
112C4×C8×C53C2311yescharacteristic
113C2×Q8×C53—21no—
114C2×Q8×C53—21no—
115C2×Q8×C53D811yescharacteristic
116C2×D4×C53—21no—
117C22×C4×C53—41no—
118C2×Dic106—21no—
119C2×Dic106—41no—
120Q16×C53—21no—
121Q16×C53—21no—
122Q16×C53—21no—
123Q16×C53—21no—
124Q16×C53—81no—
125SD16×C53—81no—
126SD16×C53—81no—
127D8×C53—21no—
128D8×C53—21no—
129C2×C8×C53C2×D411yescharacteristic
130C2×C8×C53C2×D411yescharacteristic
131C2×C8×C53C2×D411yescharacteristic
132C2×C8×C53C2411yescharacteristic
133C4⋊C4×C53—21no—
134C4⋊C4×C53—41no—
135C4⋊C4×C53—41no—
136C4⋊C4×C53—41no—
137C4⋊C4×C53—41no—
138C4⋊C4×C53D811yescharacteristic
139C4⋊C4×C53D811yescharacteristic
140C4⋊C4×C53D811yescharacteristic
141C22⋊C4×C53—41no—
142C22⋊C4×C53—41no—
143C22⋊C4×C53—41no—
144C22⋊C4×C53—41no—
145C22⋊C4×C53—41no—
146C42×C53—41no—
147C42×C53—41no—
148C42×C53C2×D411yescharacteristic
149C42×C53C2×D411yescharacteristic
150C42×C53C2×D411yescharacteristic
151C4⋊Dic53—21no—
152C4⋊Dic53—21no—
153C4⋊Dic53—21no—
154C4⋊Dic53—41no—
155#11719919—41no—
156#11719919—41no—
157#11719919—41no—
158#11719919—41no—
159#11719919—41no—
160#11719919—41no—
161#11719919—41no—
162C4×Dic53—41no—
163C4×Dic53—41no—
164C4×Dic53—41no—
165Dic212—21no—
166Dic212#4112yes—
167#11719908—21no—
168#11719908—21no—
169#11719908—21no—
170#11719908—21no—
171C23×C53—41no—
172Q8×C53—41no—
173Q8×C53—41no—
174Q8×C53—41no—
175Q8×C53—41no—
176Q8×C53—41no—
177D4×C53—41no—
178D4×C53—41no—
179C2×C4×C53—41no—
180C2×C4×C53—41no—
181C2×C4×C53—41no—
182C2×C4×C53—82no—
183C2×C4×C53C2×D811yescharacteristic
184C2×C4×C53C2×D811yescharacteristic
185C2×C4×C53C2×D811yescharacteristic
186C2×C4×C53C2×D811yescharacteristic
187C2×C4×C53C2×D811yescharacteristic
188C2×C4×C53C2×D811yescharacteristic
189C2×C4×C53C22×D411yescharacteristic
190C2×Dic53—41no—
191C2×Dic53—41no—
192C2×Dic53—41no—
193C2×Dic53—41no—
194Dic106—22no—
195Dic106—41no—
196Dic106—82no—
197C8×C53—21no—
198C8×C53—21no—
199C8×C53—21no—
200C8×C53C2×#4111yescharacteristic
201C8×C532−1+411yescharacteristic
202#33944—531nomaximal, Sylow
203C22×C53—81no—
204C22×C53—81no—
205C22×C53—81no—
206C22×C53C22×D811yescharacteristic
207C4×C53—21no—
208C4×C53—21no—
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—21no—
213C4×C53—81no—
214C4×C53—81no—
215C4×C53—81no—
216C4×C53#54811yescharacteristic
217C4×C53#57711yescharacteristic
218Dic53—42no—
219Dic53—81no—
220Dic53—81no—
221Dic53—81no—
222#3393D5311yescharacteristic
223#3383—531no—
224#2245—531no—
225#2243—531no—
226#2242—531no—
227#2229—531no—
228#2228—531no—
229#2227—531no—
230#2211—531no—
231#2211—531no—
232#2209—531no—
233#2183—531no—
234#2176—531no—
235#2175—531no—
236C4×Q32—531no—
237C2×C53—161no—
238C2×C53#335811yescharacteristic
239C2×C53#341411yescharacteristic
240C2×C53#341711yescharacteristic
241#557—1061no—
242#556—21no—
243C2×Q32—531no—
244C2×Q32—531no—
245C2×SD32—531no—
246#500—531no—
247#499—531no—
248#499—531no—
249#495C2×D5311yescharacteristic
250#495C2×D5311yescharacteristic
251#494C2×D5311yescharacteristic
252#479—1061no—
253#479—1061no—
254#475—1061no—
255#464—21no—
256#464—21no—
257#461—21no—
258C8×Q8C2×D5311yescharacteristic
259C4×Q16—531no—
260C4×Q16—531no—
261C4×Q16—531no—
262C4×Q16C2×D5311yescharacteristic
263C4×Q16C2×D5311yescharacteristic
264C4×D8C2×D5311yescharacteristic
265#367—531no—
266#366—531no—
267#366—531no—
268C4⋊C16—531no—
269C4⋊C16—531no—
270C4⋊C16—531no—
271#358—531no—
272#358—531no—
273#358—531no—
274#358—531no—
275#358—531no—
276#358—531no—
277#358—531no—
278#357—531no—
279#357—531no—
280#357—531no—
281#346—531no—
282#346—531no—
283C4×C16—531no—
284C53#3394411yescharacteristic, complemented, Sylow
285C2×Q16—21no—
286C2×Q16—531no—
287C2×Q16C22×D5311yescharacteristic
288C2×Q16C22×D5311yescharacteristic
289C2×SD16—21no—
290C2×SD16—21no—
291C2×D8C22×D5311yescharacteristic
292C4⋊Q8—21no—
293C4⋊Q8—1061no—
294C4⋊Q8—2121no—
295#126—41no—
296#126—41no—
297#125—1061no—
298#125—1061no—
299#125—2121no—
300#125—2121no—
301#124—21no—
302#124—21no—
303C22⋊Q8—41no—
304C4×Q8—21no—
305C4×Q8—21no—
306C4×Q8—1061no—
307C4×Q8—1061no—
308C4×Q8—1061no—
309C4×Q8—2121no—
310C4×Q8C22⋊D5311yescharacteristic
311C4×D4—21no—
312#117—41no—
313Q32—1062no—
314Q32—1062no—
315SD32—1062no—
316C2×C16—531no—
317C2×C16—531no—
318C2×C16—531no—
319C2×C16—531no—
320#107—531no—
321#107—531no—
322#107—531no—
323#107—1061no—
324#107C22×D5311yescharacteristic
325#106—1061no—
326#106—1061no—
327C4⋊C8C22⋊D5311yescharacteristic
328C4⋊C8C22⋊D5311yescharacteristic
329C4⋊C8C22⋊D5311yescharacteristic
330#103—21no—
331#103—21no—
332#103—1061no—
333#103—1061no—
334#103—1061no—
335#102—21no—
336C4×C8C22×D5311yescharacteristic
337C4×C8C22×D5311yescharacteristic
338C4×C8C22×D5311yescharacteristic
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—1061no—
342C2×Q8—2121no—
343C2×Q8#1171992211yescharacteristic
344C2×D4—21no—
345C22×C4—41no—
346Q16—21no—
347Q16—21no—
348Q16—21no—
349Q16—21no—
350Q16—81no—
351Q16—532no—
352Q16—1061no—
353SD16—81no—
354SD16—81no—
355D8—21no—
356D8—21no—
357C2×C8C2×C22⋊D5311yescharacteristic
358C2×C8C2×C22⋊D5311yescharacteristic
359C2×C8C2×C22⋊D5311yescharacteristic
360C2×C8C23×D5311yescharacteristic
361C4⋊C4—21no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—1061no—
367C4⋊C4—1061no—
368C4⋊C4—1061no—
369C4⋊C4—2121no—
370C4⋊C4—2121no—
371C4⋊C4—2121no—
372C4⋊C4—2121no—
373C4⋊C4—2121no—
374C4⋊C4—2121no—
375C4⋊C4—2121no—
376C4⋊C4—2121no—
377C4⋊C4#1171992211yescharacteristic
378C4⋊C4#1171992211yescharacteristic
379C4⋊C4#1171992211yescharacteristic
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—41no—
384C22⋊C4—41no—
385C42—41no—
386C42—41no—
387C42—2121no—
388C42—2121no—
389C42—2121no—
390C42C2×C22⋊D5311yescharacteristic
391C42C2×C22⋊D5311yescharacteristic
392C42C2×C22⋊D5311yescharacteristic
393C16—1061no—
394C16—1061no—
395C16—1061no—
396C16—1061no—
397C23—41no—
398Q8—41no—
399Q8—41no—
400Q8—41no—
401Q8—41no—
402Q8—41no—
403Q8—1062no—
404Q8—4242no—
405Q8—2121no—
406D4—41no—
407D4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—82no—
412C2×C4—2121no—
413C2×C4—2121no—
414C2×C4—2121no—
415C2×C4—2121no—
416C2×C4C2×#1171992211yescharacteristic
417C2×C4C2×#1171992211yescharacteristic
418C2×C4C2×#1171992211yescharacteristic
419C2×C4C2×#1171992211yescharacteristic
420C2×C4C2×#1171992211yescharacteristic
421C2×C4C2×#1171992211yescharacteristic
422C2×C4C22×C22⋊D5311yescharacteristic
423C8—21no—
424C8—21no—
425C8—21no—
426C8#41×D5311yescharacteristic
427C8#1313326411yescharacteristic
428C22—81no—
429C22—81no—
430C22—81no—
431C22C22×#1171992211yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—81no—
439C4—81no—
440C4—81no—
441C4—4241no—
442C4—4241no—
443C4—4241no—
444C4—2122no—
445C4#1681128011yescharacteristic
446C4#1681143411yescharacteristic
447C2—161no—
448C2#2028734611yescharacteristic
449C2#2028784911yescharacteristic
450C2#2028786611yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3393C2
socleC22×C53C22×D8
prime radicalC2×D8×C53C23
hypercenter#3393D53
hypocenterC53#33944
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ #3393

Complements

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

NH ≅ G/Ntype
1. #20287607C2semidirect
2. #20276960C2semidirect
5. #20276879C2semidirect
7. #20276867C2semidirect
10. #20276762C2semidirect
11. #20276624C2semidirect
12. #20276584C2semidirect
14. C4×#13133163C2semidirect
284. C53#33944semidirect

Maximal subgroups

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

subgroupindex
#202876072
#202769602
#202769512
#202769462
#202768792
#202768722
#202768672
#202767742
#202767622
#202766242
#202765842
#202765792
C4×#131331632
C53×#33932
#3394453