Finite Group Databasebeta

← #27287464

Subgroups — #27287464

Order 13568.

572 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279714C211yesmaximal, characteristic
2#20275520C211yesmaximal, characteristic
3#20275476C211yesmaximal, characteristic, complemented
4C2×#16810366C211yesmaximal, characteristic
5C2×Q8⋊#88322C211yesmaximal, characteristic, complemented
6#20272554C211yesmaximal, characteristic, complemented
7C53×#1803C211yesmaximal, characteristic, complemented
8C2×#13133214C2211yescharacteristic
9C2×#13133206—21no—
10C2×#13133148—21no—
11C2×C4⋊#88322C2211yescharacteristic
12C2×#13133137—22no—
13C2×C4×#88322C2211yescharacteristic
14#16810540—22no—
15#16810525—21no—
16#16810525—21no—
17#16810524—21no—
18C4×C4⋊Dic53C2211yescharacteristic
19#16810385—21no—
20#16810366—22no—
21Q8⋊#88322—22no—
22C2×C4×Q8×C53C2211yescharacteristic
23C2×C4⋊C8×C53C2211yescharacteristic
24C2×#103×C53—21no—
25C53×#384C2211yescharacteristic
26C53×#340—21no—
27C22×D212—21no—
28C22×D212—21no—
29C22×D212D412yes—
30C22×#88322—21no—
31C22×#88322—21no—
32#13133214—21no—
33#13133214—21no—
34#13133214—21no—
35#13133214—21no—
36#13133206—42no—
37#13133206—42no—
38C2×C4⋊Dic53D411yescharacteristic
39C2×C4⋊Dic53D411yescharacteristic
40C2×#11719921—22no—
41C2×#11719921—22no—
42C2×#11719921—22no—
43C2×#11719921—22no—
44C2×C4×Dic53—21no—
45#13133148—42no—
46C4⋊#88322—21no—
47C4⋊#88322D412yes—
48#13133137—44no—
49C4×#88322—21no—
50C4×#88322—21no—
51#13133124—21no—
52C22×Q8×C53D411yescharacteristic
53C22×C8×C53—21no—
54C4×Q8×C53—21no—
55C4×Q8×C53—21no—
56C4×Q8×C53—21no—
57C4×Q8×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53D411yescharacteristic
63C2×C42×C53—21no—
64C2×C42×C53C2311yescharacteristic
65C4⋊C8×C53—21no—
66C4⋊C8×C53—21no—
67#103×C53—42no—
68C23×D53—22no—
69C2×Q8×C53—21no—
70C2×Q8×C53—21no—
71C2×Q8×C53—21no—
72C2×Q8×C53—21no—
73C2×Q8×C53—21no—
74C2×Q8×C53SD1612yes—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—41no—
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×Dic53—21no—
82C22×Dic53—21no—
83C2×D212—22no—
84C2×D212—22no—
85C2×D212—42no—
86C2×D212—42no—
87C2×D212—41no—
88C2×D212—41no—
89C2×D212—41no—
90C2×D212—41no—
91C2×D212—41no—
92C2×D212—41no—
93C2×D212—41no—
94C2×D212—41no—
95C2×D212—41no—
96C2×D212—41no—
97C2×D212—41no—
98C2×D212—41no—
99C2×C8×C53—41no—
100C2×C8×C53—41no—
101C2×C8×C53—41no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—42no—
105C4⋊C4×C53—41no—
106C4⋊C4×C53—41no—
107C4⋊C4×C53—41no—
108C4⋊C4×C53—41no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53—41no—
111C42×C53—41no—
112C42×C53—41no—
113C42×C53C2×D411yescharacteristic
114C42×C53#4111yescharacteristic
115C42×C53#4111yescharacteristic
116C42×C53#4111yescharacteristic
117#11719921—44no—
118#11719921—44no—
119#11719921—82no—
120#11719921—82no—
121C4⋊Dic53—21no—
122C4⋊Dic53—21no—
123C4⋊Dic53—21no—
124C4⋊Dic53—21no—
125C4×Dic53—41no—
126C4×Dic53—41no—
127C2×#88322—22no—
128C2×#88322—22no—
129C2×#88322—41no—
130C2×#88322—41no—
131C2×#88322—41no—
132C2×#88322—41no—
133C23×C53#12011yescharacteristic
134C22×D53—42no—
135C22×D53—42no—
136C22×D53—42no—
137C22×D53—42no—
138C22×D53—42no—
139C22×D53—42no—
140C22×D53—42no—
141Q8×C53—22no—
142Q8×C53—22no—
143Q8×C53—41no—
144Q8×C53—41no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—22no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—81no—
154C2×C4×C53—81no—
155C2×C4×C53—81no—
156C2×C4×C53#12111yescharacteristic
157C2×C4×C53#12111yescharacteristic
158C2×C4×C53#12111yescharacteristic
159C2×C4×C53#12311yescharacteristic
160C2×C4×C53#12311yescharacteristic
161C2×C4×C53#12411yescharacteristic
162C2×C4×C53C2×SD1611yescharacteristic
163C2×C4×C53#13511yescharacteristic
164C2×C4×C53#13511yescharacteristic
165C2×C4×C53#13511yescharacteristic
166C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C53#13711yescharacteristic
170C2×Dic53—41no—
171C2×Dic53—41no—
172C2×Dic53—41no—
173C2×Dic53—41no—
174C2×Dic53—41no—
175C2×Dic53—41no—
176D212—44no—
177D212—82no—
178D212—81no—
179D212—81no—
180D212—81no—
181D212—81no—
182D212—81no—
183D212—81no—
184D212—81no—
185D212—81no—
186C8×C53—81no—
187#88322—42no—
188#88322—42no—
189#10410—531nomaximal, Sylow
190C22×C53#39411yescharacteristic
191C22×C53#44911yescharacteristic
192C22×C53#44911yescharacteristic
193C22×C53#45011yescharacteristic
194C22×C53#45211yescharacteristic
195C22×C53#45311yescharacteristic
196C22×C53#45511yescharacteristic
197C2×D53—82no—
198C2×D53—82no—
199C2×D53—82no—
200C2×D53—82no—
201C2×D53—82no—
202C2×D53—82no—
203C2×D53—82no—
204C4×C53—21no—
205C4×C53—21no—
206C4×C53—42no—
207C4×C53—81no—
208C4×C53—161no—
209C4×C53#46311yescharacteristic
210C4×C53#46411yescharacteristic
211C4×C53#46511yescharacteristic
212C4×C53#46911yescharacteristic
213C4×C53#48111yescharacteristic
214C4×C53#48211yescharacteristic
215C4×C53#48811yescharacteristic
216C4×C53#48911yescharacteristic
217Dic53—81no—
218Dic53—81no—
219#2548—531no—
220#1957—531no—
221#1948—531no—
222#1803D5311yescharacteristic, complemented
223C2×#331—531no—
224C2×Q8⋊C8—531no—
225#1279—531no—
226C2×C53#162011yescharacteristic
227C2×C53#163111yescharacteristic
228C2×C53#164011yescharacteristic
229C2×C53#164211yescharacteristic
230C2×C53#203411yescharacteristic
231C2×C53#203711yescharacteristic
232C2×C53#204211yescharacteristic
233D53—162no—
234C2×#127—531no—
235C2×#124—1061no—
236C2×C4×Q8C2×D5311yescharacteristic
237C2×#106—1061no—
238C2×C4⋊C8—531no—
239C2×C4⋊C8C2×D5311yescharacteristic
240C2×#103—21no—
241C2×#102—1062no—
242C2×C4×C8—531no—
243#394—1062no—
244#390—1061no—
245#390—1061no—
246#390—1061no—
247#384C2×D5311yescharacteristic
248C4×C4⋊C4—531no—
249#340—21no—
250#340—1061no—
251#331—1062no—
252Q8⋊C8—1062no—
253C53#1041011yescharacteristic, complemented, Sylow
254C22×Q8C22⋊D5311yescharacteristic
255C22×D4—532no—
256C22×D4—1061no—
257C22×D4—1061no—
258C22×C8—21no—
259C22×C8—1061no—
260C22×C8—1061no—
261#127—1061no—
262#127—1061no—
263#127—1061no—
264#127—1061no—
265#124—2122no—
266#124—2122no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—531no—
276C2×C4⋊C4—531no—
277C2×C4⋊C4C22⋊D5311yescharacteristic
278C2×C22⋊C4—1062no—
279C2×C22⋊C4—1062no—
280C2×C22⋊C4—1062no—
281C2×C22⋊C4—1062no—
282C2×C42—21no—
283C2×C42—1061no—
284C2×C42C22×D5311yescharacteristic
285#106—2122no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—532no—
289C4⋊C8—1061no—
290#103—42no—
291#102—2124no—
292C4×C8—1061no—
293C4×C8—1061no—
294#95—1061no—
295C24—1062no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8#1171992412yes—
302C2×D4—1062no—
303C2×D4—1062no—
304C2×D4—2122no—
305C2×D4—2122no—
306C2×D4—2121no—
307C2×D4—2121no—
308C2×D4—2121no—
309C2×D4—2121no—
310C2×D4—2121no—
311C2×D4—2121no—
312C2×D4—2121no—
313C2×D4—2121no—
314C2×D4—2121no—
315C2×D4—2121no—
316C2×D4—2121no—
317C2×D4—2121no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—41no—
321C22×C4—1061no—
322C22×C4—1061no—
323C22×C4D4×D5311yescharacteristic
324C22×C4D4×D5311yescharacteristic
325C22×C4C2×C22⋊D5311yescharacteristic
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—1062no—
330C2×C8—1062no—
331C2×C8—2121no—
332C2×C8—2121no—
333C2×C8—2121no—
334C2×C8—2121no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—42no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—1061no—
345C4⋊C4—1061no—
346C4⋊C4—1061no—
347C4⋊C4—1061no—
348C22⋊C4—4242no—
349C22⋊C4—4242no—
350C22⋊C4—2124no—
351C22⋊C4—2124no—
352C42—41no—
353C42—41no—
354C42—2121no—
355C42—2121no—
356C42#1171994511yescharacteristic
357C42#1171994511yescharacteristic
358C42D4×D5311yescharacteristic
359C42#1171994711yescharacteristic
360C23—2122no—
361C23—2122no—
362C23—2122no—
363C23—2122no—
364C23—2122no—
365C23—2122no—
366C23—2122no—
367C23#1313318511yescharacteristic
368Q8—22no—
369Q8—22no—
370Q8—41no—
371Q8—41no—
372D4—4242no—
373D4—4241no—
374D4—4241no—
375D4—4241no—
376D4—4241no—
377D4—4241no—
378D4—4241no—
379D4—4241no—
380D4—4241no—
381D4—2124no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—2121no—
394C2×C4—2121no—
395C2×C4—2121no—
396C2×C4—2121no—
397C2×C4—2121no—
398C2×C4—2121no—
399C2×C4#1313318911yescharacteristic
400C2×C4#1313319011yescharacteristic
401C2×C4#1313319011yescharacteristic
402C2×C4#1313319911yescharacteristic
403C2×C4#1313320011yescharacteristic
404C2×C4#1313320311yescharacteristic
405C2×C4C2×#1171992411yescharacteristic
406C2×C4#1313323711yescharacteristic
407C2×C4#1313323811yescharacteristic
408C2×C4#1313324011yescharacteristic
409C2×C4#1313324211yescharacteristic
410C2×C4#1313324311yescharacteristic
411C2×C4#1313324411yescharacteristic
412C2×C4#1313324911yescharacteristic
413C8—81no—
414C8—2122no—
415C8—2122no—
416C22—4242no—
417C22—4242no—
418C22—4242no—
419C22—4242no—
420C22—4242no—
421C22—4242no—
422C22—4242no—
423C22#1681053811yescharacteristic
424C22#1681076611yescharacteristic
425C22#1681076711yescharacteristic
426C22#1681077311yescharacteristic
427C22#1681078411yescharacteristic
428C22#1681078911yescharacteristic
429C22#1681079911yescharacteristic
430C4—21no—
431C4—21no—
432C4—42no—
433C4—81no—
434C4—161no—
435C4—4241no—
436C4—4241no—
437C4#1681084111yescharacteristic
438C4#1681084811yescharacteristic
439C4#1681085411yescharacteristic
440C4#1681087411yescharacteristic
441C4#1681094111yescharacteristic
442C4#1681094711yescharacteristic
443C4#1681097911yescharacteristic
444C4#1681098411yescharacteristic
445C2—8482no—
446C2#2027377311yescharacteristic
447C2#2027384011yescharacteristic
448C2#2027389911yescharacteristic
449C2#2027391411yescharacteristic
450C2#2027590411yescharacteristic
451C2#2027592311yescharacteristic
452C2#2027595211yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1803C2
socleC23×C53#120
prime radicalC2×#13133214C22
hypercenter#1803D53
hypocenterC53#10410
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
3. #20275476C2semidirect
5. C2×Q8⋊#88322C2semidirect
6. #20272554C2semidirect
7. C53×#1803C2semidirect
222. #1803D53semidirect
253. C53#10410semidirect

Maximal subgroups

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

subgroupindex
#202797142
#202755202
#202754762
C2×#168103662
C2×Q8⋊#883222
#202725542
C53×#18032
#1041053