Finite Group Databasebeta

← #27278927

Subgroups — #27278927

Order 13568.

608 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810825C211yesmaximal, characteristic
2#20277666C211yesmaximal, characteristic
3#20275668C211yesmaximal, characteristic, complemented
4C2×#16810352C211yesmaximal, characteristic
5#20272598C211yesmaximal, characteristic, complemented
6#20272555C211yesmaximal, characteristic, complemented
7C53×#1990C211yesmaximal, characteristic, complemented
8C22×#11719924—21no—
9C2×C4×D212C2211yescharacteristic
10C2×#13133175—21no—
11#16810825—22no—
12#16810825C414yescomplemented
13C2×C4⋊#88322C2211yescharacteristic
14C2×C4⋊#88322C2211yescharacteristic
15C2×#13133141—21no—
16#16810524—21no—
17#16810513—22no—
18#16810480—21no—
19C4⋊C4×Dic53—21no—
20C4×C4⋊Dic53C2211yescharacteristic
21#16810385—21no—
22#16810385—21no—
23#16810352—22no—
24C2×C4⋊Q8×C53C2211yescharacteristic
25C2×C4×C8×C53C2211yescharacteristic
26C53×#384C2211yescharacteristic
27C53×#340—22no—
28C22×D212D411yescharacteristic
29C22×C4×D53—21no—
30C2×#11719924—24no—
31C2×#11719924—42no—
32C2×#11719924—42no—
33C22×#88322—21no—
34C22×#88322—21no—
35C4×D212—21no—
36C4×D212—21no—
37C4×D212C2×C412yes—
38C4×D212D412yes—
39#13133175—42no—
40#13133175—42no—
41C2×C4⋊Dic53D411yescharacteristic
42C2×C4⋊Dic53D411yescharacteristic
43C2×C4⋊Dic53D411yescharacteristic
44C2×#11719921—22no—
45C2×#11719921—22no—
46C2×#11719921—21no—
47C2×C4×Dic53—22no—
48C2×C4×Dic53—21no—
49C4⋊#88322—21no—
50C4⋊#88322—21no—
51C4⋊#88322—21no—
52C4⋊#88322C2×C412yes—
53#13133141—24no—
54#13133124—22no—
55#13133124—21no—
56C22×Q8×C53—21no—
57C22×C8×C53—21no—
58C4⋊Q8×C53—21no—
59C4⋊Q8×C53—21no—
60C4⋊Q8×C53—21no—
61C4⋊Q8×C53C2×C412yes—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53D412yes—
67C2×C42×C53C2311yescharacteristic
68C4×C8×C53—21no—
69C4×C8×C53—21no—
70C23×D53—21no—
71C2×Q8×C53—22no—
72C2×Q8×C53—41no—
73C2×Q8×C53—41no—
74C2×Q8×C53—41no—
75C2×Q8×C53—41no—
76C2×Q8×C53—41no—
77C22×C4×C53—22no—
78C22×C4×C53—41no—
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C2×D212—21no—
86C2×D212—21no—
87C2×D212—21no—
88C2×D212—21no—
89C2×D212—21no—
90C2×D212C22⋊C412yes—
91C2×C4×D53—22no—
92C2×C4×D53—22no—
93C2×C4×D53—41no—
94C2×C4×D53—41no—
95C2×C4×D53—41no—
96C2×C4×D53—41no—
97C2×C8×C53—22no—
98C2×C8×C53—41no—
99C2×C8×C53—41no—
100C4⋊C4×C53—22no—
101C4⋊C4×C53—22no—
102C4⋊C4×C53—42no—
103C4⋊C4×C53—41no—
104C4⋊C4×C53—41no—
105C4⋊C4×C53—41no—
106C4⋊C4×C53—41no—
107C4⋊C4×C53—41no—
108C4⋊C4×C53—41no—
109C42×C53C22×C411yescharacteristic
110C42×C53C2×D411yescharacteristic
111C42×C53#4111yescharacteristic
112C42×C53#4111yescharacteristic
113#11719924—48no—
114#11719921—22no—
115#11719921—22no—
116#11719921—44no—
117#11719921—82no—
118C4⋊Dic53—21no—
119C4⋊Dic53—21no—
120C4⋊Dic53—21no—
121C4⋊Dic53—21no—
122C4⋊Dic53C22⋊C412yes—
123C4⋊Dic53SD1612yescomplemented
124C4×Dic53—44no—
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—22no—
135C22×D53—41no—
136C22×D53—41no—
137C22×D53—41no—
138C22×D53—41no—
139C22×D53—41no—
140C22×D53—41no—
141Q8×C53—42no—
142Q8×C53—42no—
143Q8×C53—81no—
144Q8×C53—81no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—42no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53—81no—
152C2×C4×C53—81no—
153C2×C4×C53C2×C22⋊C411yescharacteristic
154C2×C4×C53C4×D411yescharacteristic
155C2×C4×C53C4×D411yescharacteristic
156C2×C4×C53#12111yescharacteristic
157C2×C4×C53#12111yescharacteristic
158C2×C4×C53#12311yescharacteristic
159C2×C4×C53C2×SD1611yescharacteristic
160C2×C4×C53#13511yescharacteristic
161C2×C4×C53#13511yescharacteristic
162C2×C4×C53#13511yescharacteristic
163C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
166C2×C4×C53#13711yescharacteristic
167C2×Dic53—22no—
168C2×Dic53—22no—
169C2×Dic53—41no—
170C2×Dic53—41no—
171C2×Dic53—41no—
172C2×Dic53—41no—
173C2×Dic53—41no—
174C2×Dic53—41no—
175C2×Dic53—41no—
176D212—22no—
177D212—22no—
178D212—41no—
179D212—41no—
180C4×D53—42no—
181C4×D53—42no—
182C4×D53—42no—
183C4×D53—42no—
184C8×C53—42no—
185#88322—42no—
186#88322—81no—
187#8988—531nomaximal, Sylow
188C22×C53#38611yescharacteristic
189C22×C53#44911yescharacteristic
190C22×C53#44911yescharacteristic
191C22×C53#45011yescharacteristic
192C22×C53#45211yescharacteristic
193C22×C53#45511yescharacteristic
194C22×C53#45611yescharacteristic
195C2×D53—42no—
196C2×D53—42no—
197C2×D53—42no—
198C2×D53—81no—
199C2×D53—81no—
200C2×D53—81no—
201C2×D53—81no—
202C4×C53—21no—
203C4×C53—21no—
204C4×C53—82no—
205C4×C53—82no—
206C4×C53#41611yescharacteristic
207C4×C53#41811yescharacteristic
208C4×C53C4×SD1611yescharacteristic
209C4×C53#44011yescharacteristic
210C4×C53#46111yescharacteristic
211C4×C53#46311yescharacteristic
212C4×C53#48211yescharacteristic
213C4×C53#48211yescharacteristic
214Dic53—42no—
215Dic53—42no—
216Dic53—81no—
217C2×#461—531no—
218#2354—531no—
219#1990D5311yescharacteristic, complemented
220#1988—531no—
221C2×D4⋊C8—531no—
222#1296—531no—
223#1279—531no—
224C2×C53#164211yescharacteristic
225C2×C53#164611yescharacteristic
226C2×C53#165011yescharacteristic
227C2×C53#165411yescharacteristic
228C2×C53#187511yescharacteristic
229C2×C53#187711yescharacteristic
230C2×C53#188211yescharacteristic
231D53—82no—
232C22×SD16—1061no—
233C2×C4⋊Q8C2×D5311yescharacteristic
234C2×C4×D4—531no—
235C2×#117—1061no—
236#461—534no—
237#461—1062no—
238C2×C4⋊C8—531no—
239C2×C4⋊C8—531no—
240C2×#103—1061no—
241C2×C4×C8C2×D5311yescharacteristic
242#390—1061no—
243#388—1062no—
244#384C2×D5311yescharacteristic
245#380—1061no—
246C4×C4⋊C4—531no—
247C4×C4⋊C4—1061no—
248#340—22no—
249#340—1061no—
250#340—1061no—
251D4⋊C8—1062no—
252C53#898811yescharacteristic, complemented, Sylow
253C22×Q8—21no—
254C22×D4—531no—
255C23×C4—1061no—
256C2×SD16—1064no—
257C2×SD16—2122no—
258C2×SD16—2122no—
259C22×C8—21no—
260C22×C8—1061no—
261C22×C8—1061no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4⋊Q8—21no—
265C4⋊Q8C4×D5312yes—
266C4×D4—532no—
267C4×D4—532no—
268C4×D4—1061no—
269C4×D4—1061no—
270#117—2122no—
271#117—2122no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—531no—
277C2×C4⋊C4—531no—
278C2×C4⋊C4—531no—
279C2×C4⋊C4C22⋊D5312yes—
280C2×C22⋊C4—1061no—
281C2×C22⋊C4—1062no—
282C2×C22⋊C4—1062no—
283C2×C42—1062no—
284C2×C42—1061no—
285C2×C42C22×D5311yescharacteristic
286C4⋊C8—532no—
287C4⋊C8—1061no—
288C4⋊C8—1061no—
289C4⋊C8—1061no—
290#103—1064no—
291C4×C8—21no—
292C4×C8—21no—
293#95—1062no—
294#95—1061no—
295C24—1061no—
296C2×Q8—22no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×D4—532no—
303C2×D4—1061no—
304C2×D4—1061no—
305C2×D4—1061no—
306C2×D4—1061no—
307C2×D4—1061no—
308C22×C4—22no—
309C22×C4—41no—
310C22×C4—1061no—
311C22×C4—1061no—
312C22×C4—1062no—
313C22×C4—1062no—
314C22×C4—1061no—
315C22×C4—2121no—
316C22×C4—2121no—
317C22×C4—2121no—
318C22×C4—2121no—
319C22×C4D4×D5311yescharacteristic
320C22×C4D4×D5311yescharacteristic
321C22×C4C2×C22⋊D5311yescharacteristic
322SD16—2128no—
323C2×C8—22no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—1062no—
327C2×C8—2121no—
328C2×C8—2121no—
329C2×C8—2121no—
330C2×C8—2121no—
331C2×C8—2121no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—42no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—532no—
342C4⋊C4—532no—
343C4⋊C4—1061no—
344C4⋊C4—1061no—
345C4⋊C4—1061no—
346C4⋊C4—1061no—
347C22⋊C4—1062no—
348C22⋊C4—1062no—
349C22⋊C4—4242no—
350C22⋊C4—2124no—
351C42—2124no—
352C42—2121no—
353C42—2121no—
354C42C2×C4×D5311yescharacteristic
355C42#1171994511yescharacteristic
356C42D4×D5311yescharacteristic
357C42#1171994711yescharacteristic
358C23—1062no—
359C23—2121no—
360C23—2121no—
361C23—2121no—
362C23—2121no—
363C23—2121no—
364C23—2121no—
365C23#1313318511yescharacteristic
366Q8—42no—
367Q8—42no—
368Q8—81no—
369Q8—81no—
370D4—1062no—
371D4—1062no—
372D4—2121no—
373D4—2121no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—1062no—
383C2×C4—1062no—
384C2×C4—2121no—
385C2×C4—2121no—
386C2×C4—2121no—
387C2×C4—2121no—
388C2×C4—2121no—
389C2×C4—2121no—
390C2×C4—2121no—
391C2×C4—2122no—
392C2×C4—2122no—
393C2×C4—2122no—
394C2×C4—2122no—
395C2×C4C22⋊C4×D5311yescharacteristic
396C2×C4#1313317711yescharacteristic
397C2×C4C4×C22⋊D5311yescharacteristic
398C2×C4#1313318811yescharacteristic
399C2×C4#1313319011yescharacteristic
400C2×C4#1313319911yescharacteristic
401C2×C4SD16×D5311yescharacteristic
402C2×C4#1313323611yescharacteristic
403C2×C4#1313323811yescharacteristic
404C2×C4#1313324011yescharacteristic
405C2×C4#1313324111yescharacteristic
406C2×C4#1313324411yescharacteristic
407C2×C4#1313324411yescharacteristic
408C2×C4#1313324911yescharacteristic
409C8—42no—
410C8—4241no—
411C8—2122no—
412C22—4241no—
413C22—4241no—
414C22—4241no—
415C22—4241no—
416C22—2122no—
417C22—2122no—
418C22—2122no—
419C22#1681049911yescharacteristic
420C22#1681076511yescharacteristic
421C22#1681076711yescharacteristic
422C22#1681077211yescharacteristic
423C22#1681078411yescharacteristic
424C22#1681080011yescharacteristic
425C22#1681080511yescharacteristic
426C4—21no—
427C4—21no—
428C4—82no—
429C4—82no—
430C4—4241no—
431C4—2122no—
432C4—2122no—
433C4#1681061611yescharacteristic
434C4#1681062611yescharacteristic
435C4#1681070811yescharacteristic
436C4#1681072611yescharacteristic
437C4#1681082711yescharacteristic
438C4#1681084111yescharacteristic
439C4#1681094511yescharacteristic
440C4#1681094611yescharacteristic
441C2—4242no—
442C2#2027391211yescharacteristic
443C2#2027394011yescharacteristic
444C2#2027396811yescharacteristic
445C2#2027399611yescharacteristic
446C2#2027503411yescharacteristic
447C2#2027504611yescharacteristic
448C2#2027508711yescharacteristic
subgroupG/H
centerC23#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1990C2
socleC23×C53#120
prime radicalC22×D212D4
cosocleC2×C42×C53C23
hypercenter#1990D53
hypocenterC53#8988
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
3. #20275668C2semidirect
5. #20272598C2semidirect
6. #20272555C2semidirect
7. C53×#1990C2semidirect
12. #16810825C4semidirect
123. C4⋊Dic53SD16semidirect
219. #1990D53semidirect
252. C53#8988semidirect

Maximal subgroups

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

subgroupindex
C2×#168108252
#202776662
#202756682
C2×#168103522
#202725982
#202725552
C53×#19902
#898853