Finite Group Databasebeta

← #27278859

Subgroups — #27278859

Order 13568.

616 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810824C211yesmaximal, characteristic
2#20277670C211yesmaximal, characteristic, complemented
3#20275642C211yesmaximal, characteristic
4#20275521C211yesmaximal, characteristic
5C2×#16810355C211yesmaximal, characteristic, complemented
6#20272599C211yesmaximal, characteristic, complemented
7C53×#1279C211yesmaximal, characteristic, complemented
8C22×#11719913—21no—
9C2×#13133214C2211yescharacteristic
10C2×C4×Dic106C2211yescharacteristic
11#16810824—22no—
12#16810824C414yescomplemented
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133139—21no—
15C2×#13133139—22no—
16C2×#13133137—21no—
17C2×#13133137—21no—
18#16810529—21no—
19#16810510—22no—
20#16810493C2211yescharacteristic
21#16810488—21no—
22C4⋊C4×Dic53—21no—
23C4×C4⋊Dic53—21no—
24#16810355—22no—
25C2×C4⋊C8×C53C2211yescharacteristic
26C2×C4×C8×C53C2211yescharacteristic
27C4×C4⋊C4×C53C2211yescharacteristic
28C22×Dic106D411yescharacteristic
29C22×D212—21no—
30C22×D212—21no—
31C22×D212—21no—
32C2×#11719913—24no—
33C2×#11719913—42no—
34C2×#11719913—42no—
35C22×#88322—21no—
36#13133214—21no—
37#13133214—21no—
38#13133214—21no—
39#13133214C2×C412yes—
40C4×Dic106—21no—
41C4×Dic106—21no—
42C4×Dic106C2×C412yes—
43C4×Dic106D412yes—
44C2×C4⋊Dic53—21no—
45C2×C4⋊Dic53—21no—
46C2×C4⋊Dic53D411yescharacteristic
47C2×C4⋊Dic53D412yes—
48C2×#11719919—21no—
49C2×C4×Dic53—22no—
50C2×C4×Dic53—21no—
51C4⋊#88322—21no—
52C4⋊#88322—21no—
53#13133139—24no—
54#13133139—44no—
55#13133137—42no—
56#13133137—42no—
57#13133124—22no—
58#13133124—22no—
59#13133124—22no—
60C22×C8×C53—21no—
61C22×C8×C53—21no—
62C2×C4⋊C4×C53D411yescharacteristic
63C2×C4⋊C4×C53D411yescharacteristic
64C2×C42×C53—21no—
65C2×C42×C53C2311yescharacteristic
66C4⋊C8×C53—21no—
67C4⋊C8×C53C2×C412yes—
68C4×C8×C53—21no—
69C4×C8×C53—21no—
70#95×C53—21no—
71C23×D53—41no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—22no—
80C2×D212—22no—
81C2×D212—22no—
82C2×D212—22no—
83C2×D212—42no—
84C2×D212—42no—
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×D212—41no—
94C2×D212—41no—
95C2×D212—41no—
96C2×Dic106—21no—
97C2×Dic106—21no—
98C2×Dic106—21no—
99C2×Dic106—21no—
100C2×Dic106—21no—
101C2×Dic106C22⋊C412yes—
102C2×C8×C53—22no—
103C2×C8×C53—22no—
104C2×C8×C53—41no—
105C2×C8×C53—41no—
106C2×C8×C53—41no—
107C2×C8×C53—41no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53SD1612yescomplemented
112C42×C53—41no—
113C42×C53—41no—
114C42×C53C22×C411yescharacteristic
115C42×C53C2×D411yescharacteristic
116C42×C53#4111yescharacteristic
117C42×C53#4111yescharacteristic
118C4⋊Dic53—21no—
119C4⋊Dic53—41no—
120C4⋊Dic53—41no—
121C4⋊Dic53—41no—
122C4⋊Dic53—41no—
123C4⋊Dic53—42no—
124C4⋊Dic53C22⋊C412yes—
125#11719919—22no—
126#11719919—22no—
127C4×Dic53—22no—
128C4×Dic53—22no—
129C4×Dic53—82no—
130C2×#88322—41no—
131C2×#88322—41no—
132C2×#88322—41no—
133#11719913—48no—
134C23×C53#12011yescharacteristic
135C22×D53—42no—
136C22×D53—81no—
137C22×D53—81no—
138C22×D53—81no—
139C22×D53—81no—
140C22×D53—81no—
141C22×D53—81no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—22no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53C2×C22⋊C411yescharacteristic
151C2×C4×C53C4×D411yescharacteristic
152C2×C4×C53C4×D411yescharacteristic
153C2×C4×C53#12111yescharacteristic
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53#12311yescharacteristic
156C2×C4×C53C2×SD1611yescharacteristic
157C2×C4×C53#13511yescharacteristic
158C2×C4×C53#13511yescharacteristic
159C2×C4×C53#13511yescharacteristic
160C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
162C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C53#13711yescharacteristic
164C2×Dic53—22no—
165C2×Dic53—22no—
166C2×Dic53—41no—
167C2×Dic53—41no—
168C2×Dic53—41no—
169C2×Dic53—41no—
170C2×Dic53—42no—
171C2×Dic53—42no—
172C2×Dic53—42no—
173D212—44no—
174D212—42no—
175D212—42no—
176D212—42no—
177D212—42no—
178D212—82no—
179D212—81no—
180D212—81no—
181D212—81no—
182D212—81no—
183Dic106—22no—
184Dic106—22no—
185Dic106—41no—
186Dic106—41no—
187C8×C53—42no—
188C8×C53—42no—
189#88322—81no—
190#8978—531nomaximal, Sylow
191C22×C53#38611yescharacteristic
192C22×C53#44811yescharacteristic
193C22×C53#44911yescharacteristic
194C22×C53#44911yescharacteristic
195C22×C53#44911yescharacteristic
196C22×C53#45311yescharacteristic
197C22×C53#45411yescharacteristic
198C2×D53—82no—
199C2×D53—82no—
200C2×D53—82no—
201C2×D53—161no—
202C2×D53—161no—
203C2×D53—161no—
204C2×D53—161no—
205C4×C53—21no—
206C4×C53—21no—
207C4×C53—42no—
208C4×C53—81no—
209C4×C53#41611yescharacteristic
210C4×C53#41811yescharacteristic
211C4×C53C4×SD1611yescharacteristic
212C4×C53#44011yescharacteristic
213C4×C53#46011yescharacteristic
214C4×C53#46411yescharacteristic
215C4×C53#48211yescharacteristic
216C4×C53#48211yescharacteristic
217Dic53—42no—
218Dic53—42no—
219Dic53—82no—
220C2×#460—531no—
221#2355—531no—
222#1982—531no—
223#1957—531no—
224C2×Q8⋊C8—531no—
225#1296—531no—
226#1279D5311yescharacteristic, complemented
227C2×C53#162011yescharacteristic
228C2×C53#162211yescharacteristic
229C2×C53#162611yescharacteristic
230C2×C53#162811yescharacteristic
231C2×C53#187511yescharacteristic
232C2×C53#188011yescharacteristic
233C2×C53#188311yescharacteristic
234D53—162no—
235C22×SD16—1061no—
236C2×#127—531no—
237C2×C4×Q8—531no—
238#460—534no—
239#460—1062no—
240C2×C4⋊C8—531no—
241C2×C4⋊C8C2×D5311yescharacteristic
242C2×#102—1061no—
243C2×#102—1062no—
244C2×#102—1061no—
245C2×#102—1061no—
246C2×C4×C8C2×D5311yescharacteristic
247#391—1061no—
248#387—1062no—
249#384—531no—
250#382—1061no—
251C4×C4⋊C4—1061no—
252C4×C4⋊C4—1061no—
253C4×C4⋊C4C2×D5311yescharacteristic
254Q8⋊C8—1062no—
255C53#897811yescharacteristic, complemented, Sylow
256C22×Q8—531no—
257C22×D4—1061no—
258C22×D4—1061no—
259C22×D4—1061no—
260C2×SD16—1064no—
261C2×SD16—2122no—
262C2×SD16—2122no—
263C22×C8—21no—
264C22×C8—21no—
265C22×C8—1061no—
266#127—532no—
267#127—1061no—
268#127—1061no—
269#127—1061no—
270C4×Q8—532no—
271C4×Q8—532no—
272C4×Q8—1061no—
273C4×Q8—1061no—
274C2×C4⋊C4—531no—
275C2×C4⋊C4—532no—
276C2×C4⋊C4—1061no—
277C2×C4⋊C4—1061no—
278C2×C4⋊C4—1061no—
279C2×C4⋊C4D21211yescharacteristic
280C2×C4⋊C4D21211yescharacteristic
281C2×C42—21no—
282C2×C42—1061no—
283C2×C42—1062no—
284C2×C42C22×D5311yescharacteristic
285C4⋊C8—21no—
286C4⋊C8—1061no—
287C4⋊C8—1061no—
288C4⋊C8C4×D5312yes—
289#102—1064no—
290#102—2122no—
291#102—2124no—
292#102—2122no—
293C4×C8—21no—
294C4×C8—21no—
295#95—21no—
296#95—1062no—
297#95—1062no—
298#95—1062no—
299C24—2121no—
300C2×Q8—532no—
301C2×Q8—1061no—
302C2×Q8—1061no—
303C2×Q8—1061no—
304C2×Q8—1061no—
305C2×Q8—1061no—
306C2×D4—1062no—
307C2×D4—1062no—
308C2×D4—1062no—
309C2×D4—2122no—
310C2×D4—2122no—
311C2×D4—2121no—
312C2×D4—2121no—
313C2×D4—2121no—
314C2×D4—2121no—
315C2×D4—2121no—
316C2×D4—2121no—
317C2×D4—2121no—
318C2×D4—2121no—
319C2×D4—2121no—
320C2×D4—2121no—
321C2×D4—2121no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—1061no—
325C22×C4—1061no—
326C22×C4—1062no—
327C22×C4C2×D21211yescharacteristic
328C22×C4D4×D5311yescharacteristic
329C22×C4D4×D5311yescharacteristic
330SD16—2128no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—2121no—
338C2×C8—2121no—
339C2×C8—2121no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—532no—
344C4⋊C4—1062no—
345C4⋊C4—1062no—
346C4⋊C4—1061no—
347C4⋊C4—2122no—
348C4⋊C4—2121no—
349C4⋊C4—2121no—
350C4⋊C4—2121no—
351C4⋊C4—2121no—
352C4⋊C4#1171991312yescomplemented
353C42—41no—
354C42—41no—
355C42—1062no—
356C42—1062no—
357C42—4242no—
358C42C2×C4×D5311yescharacteristic
359C42#1171994511yescharacteristic
360C42D4×D5311yescharacteristic
361C42#1171994911yescharacteristic
362C23—4241no—
363C23—4241no—
364C23—4241no—
365C23—4241no—
366C23—4241no—
367C23—4241no—
368C23—2122no—
369C23#1313318711yescharacteristic
370Q8—1062no—
371Q8—1062no—
372Q8—2121no—
373Q8—2121no—
374D4—4242no—
375D4—4241no—
376D4—4241no—
377D4—4241no—
378D4—4241no—
379D4—2124no—
380D4—2122no—
381D4—2122no—
382D4—2122no—
383D4—2122no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—1062no—
393C2×C4—1062no—
394C2×C4—2122no—
395C2×C4—2122no—
396C2×C4—2122no—
397C2×C4—2121no—
398C2×C4—2121no—
399C2×C4—2121no—
400C2×C4—2121no—
401C2×C4C22⋊C4×D5311yescharacteristic
402C2×C4#1313317911yescharacteristic
403C2×C4C4×D21211yescharacteristic
404C2×C4#1313319011yescharacteristic
405C2×C4#1313319211yescharacteristic
406C2×C4#1313320211yescharacteristic
407C2×C4C2×#1171991311yescharacteristic
408C2×C4#1313323811yescharacteristic
409C2×C4#1313323911yescharacteristic
410C2×C4#1313324011yescharacteristic
411C2×C4#1313324111yescharacteristic
412C2×C4#1313324411yescharacteristic
413C2×C4#1313324511yescharacteristic
414C2×C4#1313325011yescharacteristic
415C8—42no—
416C8—42no—
417C8—4241no—
418C22—8481no—
419C22—8481no—
420C22—8481no—
421C22—8481no—
422C22—4242no—
423C22—4242no—
424C22—4242no—
425C22#1681050111yescharacteristic
426C22#1681076411yescharacteristic
427C22#1681076911yescharacteristic
428C22#1681077011yescharacteristic
429C22#1681077111yescharacteristic
430C22#1681079211yescharacteristic
431C22#1681079711yescharacteristic
432C4—21no—
433C4—21no—
434C4—42no—
435C4—81no—
436C4—4242no—
437C4—2122no—
438C4—2122no—
439C4#1681061811yescharacteristic
440C4#1681062611yescharacteristic
441C4C4×#1171991311yescharacteristic
442C4#1681072511yescharacteristic
443C4#1681082411yescharacteristic
444C4#1681084611yescharacteristic
445C4#1681094911yescharacteristic
446C4#1681095011yescharacteristic
447C2—8482no—
448C2#2027377611yescharacteristic
449C2#2027378911yescharacteristic
450C2#2027381411yescharacteristic
451C2#2027382711yescharacteristic
452C2#2027503611yescharacteristic
453C2#2027506911yescharacteristic
454C2#2027509211yescharacteristic
subgroupG/H
centerC23#13133187
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1279C2
socleC23×C53#120
prime radicalC2×#13133214C22
cosocleC2×C42×C53C23
hypercenter#1279D53
hypocenterC53#8978
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
2. #20277670C2semidirect
5. C2×#16810355C2semidirect
6. #20272599C2semidirect
7. C53×#1279C2semidirect
12. #16810824C4semidirect
111. C4⋊C4×C53SD16semidirect
226. #1279D53semidirect
255. C53#8978semidirect
352. C4⋊C4#11719913semidirect

Maximal subgroups

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

subgroupindex
C2×#168108242
#202776702
#202756422
#202755212
C2×#168103552
#202725992
C53×#12792
#897853