Finite Group Databasebeta

← #27287473

Subgroups — #27287473

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#20279717C211yesmaximal, characteristic
2#20275520C211yesmaximal, characteristic
3#20274717C211yesmaximal, characteristic, complemented
4C2×#16810366C211yesmaximal, characteristic
5C2×#16810353C211yesmaximal, characteristic, complemented
6#20272556C211yesmaximal, characteristic, complemented
7C53×#1947C211yesmaximal, characteristic, complemented
8C2×#13133214C2211yescharacteristic
9C2×#13133205—21no—
10C2×C4×Dic106C2211yescharacteristic
11C2×C4⋊#88322C2211yescharacteristic
12C2×#13133140—21no—
13C2×#13133137—22no—
14C2×C4×#88322C2211yescharacteristic
15#16810541—22no—
16#16810525—21no—
17#16810524—21no—
18#16810524—21no—
19#16810385—21no—
20#16810366—22no—
21#16810353—22no—
22C2×#107×C53—21no—
23C2×C4⋊C8×C53C2211yescharacteristic
24C53×#384C2211yescharacteristic
25C4×C4⋊C4×C53C2211yescharacteristic
26C53×#340—21no—
27C22×Dic106D411yescharacteristic
28C22×D212—21no—
29C22×D212—21no—
30C22×D212D412yes—
31C22×#88322—21no—
32C22×#88322—21no—
33#13133214—21no—
34#13133214—21no—
35#13133214—21no—
36#13133214—21no—
37#13133205—42no—
38#13133205—42no—
39C4×Dic106—21no—
40C4×Dic106—21no—
41C4×Dic106—21no—
42C4×Dic106—21no—
43C2×C4⋊Dic53D411yescharacteristic
44C2×#11719919—21no—
45C2×#11719921—22no—
46C2×#11719921—22no—
47C2×#11719921—22no—
48C2×#11719921—22no—
49C2×C4×Dic53—21no—
50C4⋊#88322—21no—
51C4⋊#88322—21no—
52#13133140—42no—
53#13133137—44no—
54C4×#88322—21no—
55C4×#88322—21no—
56C22×C8×C53—21no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53D411yescharacteristic
61C2×C4⋊C4×C53D411yescharacteristic
62C2×C42×C53—21no—
63C2×C42×C53C2311yescharacteristic
64#107×C53—42no—
65C4⋊C8×C53—21no—
66C4⋊C8×C53D412yes—
67#95×C53—21no—
68C23×D53—22no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—41no—
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×Dic53—21no—
76C22×Dic53—21no—
77C2×D212—22no—
78C2×D212—22no—
79C2×D212—42no—
80C2×D212—42no—
81C2×D212—41no—
82C2×D212—41no—
83C2×D212—41no—
84C2×D212—41no—
85C2×D212—41no—
86C2×D212—41no—
87C2×D212—41no—
88C2×D212—41no—
89C2×D212—41no—
90C2×D212—41no—
91C2×D212—41no—
92C2×D212—41no—
93C2×Dic106—21no—
94C2×Dic106—21no—
95C2×Dic106—21no—
96C2×Dic106—21no—
97C2×Dic106—21no—
98C2×Dic106D812yes—
99C2×C8×C53—22no—
100C2×C8×C53—41no—
101C2×C8×C53—41no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—42no—
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—
123#11719919—41no—
124#11719919—41no—
125C4×Dic53—41no—
126C4×Dic53—41no—
127C2×#88322—22no—
128C2×#88322—41no—
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—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—81no—
150C2×C4×C53—81no—
151C2×C4×C53—81no—
152C2×C4×C53#12111yescharacteristic
153C2×C4×C53#12111yescharacteristic
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53#12311yescharacteristic
156C2×C4×C53#12311yescharacteristic
157C2×C4×C53#12411yescharacteristic
158C2×C4×C53C2×D811yescharacteristic
159C2×C4×C53#13511yescharacteristic
160C2×C4×C53#13511yescharacteristic
161C2×C4×C53#13511yescharacteristic
162C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C53#13711yescharacteristic
165C2×C4×C53#13711yescharacteristic
166C2×Dic53—22no—
167C2×Dic53—41no—
168C2×Dic53—41no—
169C2×Dic53—41no—
170C2×Dic53—41no—
171C2×Dic53—41no—
172D212—44no—
173D212—82no—
174D212—81no—
175D212—81no—
176D212—81no—
177D212—81no—
178D212—81no—
179D212—81no—
180D212—81no—
181D212—81no—
182Dic106—22no—
183Dic106—22no—
184Dic106—41no—
185Dic106—41no—
186C8×C53—42no—
187#88322—42no—
188#88322—81no—
189#10411—531nomaximal, Sylow
190C22×C53#39411yescharacteristic
191C22×C53#44711yescharacteristic
192C22×C53#44911yescharacteristic
193C22×C53#45211yescharacteristic
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—81no—
207C4×C53—81no—
208C4×C53—161no—
209C4×C53#46311yescharacteristic
210C4×C53#46411yescharacteristic
211C4×C53#46611yescharacteristic
212C4×C53#46811yescharacteristic
213C4×C53#48011yescharacteristic
214C4×C53#48211yescharacteristic
215C4×C53#48811yescharacteristic
216C4×C53#48911yescharacteristic
217Dic53—42no—
218Dic53—81no—
219#2548—531no—
220#1957—531no—
221#1947D5311yescharacteristic, complemented
222#1803—531no—
223C2×#331—531no—
224C2×Q8⋊C8—531no—
225#1279—531no—
226C2×C53#162111yescharacteristic
227C2×C53#163011yescharacteristic
228C2×C53#164011yescharacteristic
229C2×C53#164111yescharacteristic
230C2×C53#203311yescharacteristic
231C2×C53#203711yescharacteristic
232C2×C53#204211yescharacteristic
233D53—162no—
234C2×#127—531no—
235C2×#124—1061no—
236C2×C4×Q8—531no—
237C2×#107—21no—
238C2×C4⋊C8—531no—
239C2×C4⋊C8C2×D5311yescharacteristic
240C2×#103—1061no—
241C2×#102—1062no—
242C2×C4×C8—531no—
243#394—1062no—
244#390—1061no—
245#390—1061no—
246#390—1061no—
247#384C2×D5311yescharacteristic
248C4×C4⋊C4C2×D5311yescharacteristic
249#340—21no—
250#340—1061no—
251#331—1062no—
252Q8⋊C8—1062no—
253C53#1041111yescharacteristic, complemented, Sylow
254C22×Q8—531no—
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—1061no—
268C4×Q8—1061no—
269C4×Q8—1061no—
270C4×Q8—1061no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—531no—
275C2×C4⋊C4—1061no—
276C2×C4⋊C4C22⋊D5311yescharacteristic
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#107—42no—
286C4⋊C8—21no—
287C4⋊C8—1061no—
288C4⋊C8—1061no—
289C4⋊C8D21212yes—
290#103—2122no—
291#102—2124no—
292C4×C8—1061no—
293C4×C8—1061no—
294#95—21no—
295C24—1062no—
296C2×Q8—532no—
297C2×Q8—1061no—
298C2×Q8—1061no—
299C2×Q8—1061no—
300C2×Q8—1061no—
301C2×Q8—1061no—
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—22no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—1062no—
330C2×C8—2121no—
331C2×C8—2121no—
332C2×C8—2121no—
333C2×C8—2121no—
334C2×C8—2121no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—42no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—1061no—
345C4⋊C4—1061no—
346C4⋊C4—2121no—
347C4⋊C4—2121no—
348C22⋊C4—4242no—
349C22⋊C4—4242no—
350C22⋊C4—2124no—
351C22⋊C4—2124no—
352C42—41no—
353C42—41no—
354C42—2121no—
355C42—2121no—
356C42C2×D21211yescharacteristic
357C42#1171994511yescharacteristic
358C42#1171994911yescharacteristic
359C42#1171994911yescharacteristic
360C23—2122no—
361C23—2122no—
362C23—2122no—
363C23—2122no—
364C23—2122no—
365C23—2122no—
366C23—2122no—
367C23#1313318511yescharacteristic
368Q8—1062no—
369Q8—1062no—
370Q8—2121no—
371Q8—2121no—
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—41no—
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—1062no—
394C2×C4—2121no—
395C2×C4—2121no—
396C2×C4—2121no—
397C2×C4—2121no—
398C2×C4—2121no—
399C2×C4#1313319111yescharacteristic
400C2×C4#1313319211yescharacteristic
401C2×C4#1313319211yescharacteristic
402C2×C4#1313320211yescharacteristic
403C2×C4#1313320211yescharacteristic
404C2×C4#1313320611yescharacteristic
405C2×C4D8×D5311yescharacteristic
406C2×C4#1313323711yescharacteristic
407C2×C4#1313323711yescharacteristic
408C2×C4#1313324011yescharacteristic
409C2×C4#1313324111yescharacteristic
410C2×C4#1313324311yescharacteristic
411C2×C4#1313324811yescharacteristic
412C2×C4#1313324911yescharacteristic
413C8—42no—
414C8—4241no—
415C8—2122no—
416C22—4242no—
417C22—4242no—
418C22—4242no—
419C22—4242no—
420C22—4242no—
421C22—4242no—
422C22—4242no—
423C22#1681054011yescharacteristic
424C22#1681075511yescharacteristic
425C22#1681076611yescharacteristic
426C22#1681078311yescharacteristic
427C22#1681078411yescharacteristic
428C22#1681078911yescharacteristic
429C22#1681079911yescharacteristic
430C4—21no—
431C4—21no—
432C4—81no—
433C4—81no—
434C4—161no—
435C4—4241no—
436C4—2122no—
437C4#1681083811yescharacteristic
438C4#1681084511yescharacteristic
439C4#1681086011yescharacteristic
440C4#1681087011yescharacteristic
441C4#1681093811yescharacteristic
442C4#1681095011yescharacteristic
443C4#1681098011yescharacteristic
444C4#1681098511yescharacteristic
445C2—8482no—
446C2#2027377911yescharacteristic
447C2#2027383511yescharacteristic
448C2#2027389911yescharacteristic
449C2#2027390711yescharacteristic
450C2#2027589911yescharacteristic
451C2#2027592711yescharacteristic
452C2#2027595611yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1947C2
socleC23×C53#120
prime radicalC2×#13133214C22
hypercenter#1947D53
hypocenterC53#10411
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
3. #20274717C2semidirect
5. C2×#16810353C2semidirect
6. #20272556C2semidirect
7. C53×#1947C2semidirect
221. #1947D53semidirect
253. C53#10411semidirect

Maximal subgroups

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

subgroupindex
#202797172
#202755202
#202747172
C2×#168103662
C2×#168103532
#202725562
C53×#19472
#1041153