Finite Group Databasebeta

← #27278906

Subgroups — #27278906

Order 13568.

680 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810825C211yesmaximal, characteristic
2#20277615C211yesmaximal, 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×#13133179—21no—
11C2×C4⋊C4×D53—21no—
12#16810825—22no—
13#16810825C414yescomplemented
14C2×C4⋊#88322C2211yescharacteristic
15C2×C4⋊#88322C2211yescharacteristic
16C2×#13133141—21no—
17#16810519—21no—
18#16810494—22no—
19#16810480—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—
30C22×C4×D53—22no—
31C2×#11719924—24no—
32C2×#11719924—42no—
33C2×#11719924—42no—
34C22×#88322—21no—
35C22×#88322—21no—
36C4×D212—21no—
37C4×D212—21no—
38C4×D212C2×C412yes—
39C4×D212D412yes—
40#13133179—42no—
41#13133179—42no—
42C2×C4⋊Dic53D411yescharacteristic
43C2×C4⋊Dic53D411yescharacteristic
44C2×C4⋊Dic53D411yescharacteristic
45C2×#11719919—22no—
46C4⋊C4×D53—42no—
47C4⋊C4×D53—42no—
48C2×#11719921—22no—
49C2×#11719921—21no—
50C2×C4×Dic53—21no—
51C4⋊#88322—21no—
52C4⋊#88322—21no—
53C4⋊#88322—21no—
54C4⋊#88322C2×C412yescomplemented
55#13133141—24no—
56#13133124—22no—
57#13133124—21no—
58C22×Q8×C53—21no—
59C22×C8×C53—21no—
60C4⋊Q8×C53—21no—
61C4⋊Q8×C53—21no—
62C4⋊Q8×C53—21no—
63C4⋊Q8×C53C2×C412yescomplemented
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53D412yes—
69C2×C42×C53C2311yescharacteristic
70C4×C8×C53—21no—
71C4×C8×C53—21no—
72C23×D53—21no—
73C2×Q8×C53—22no—
74C2×Q8×C53—41no—
75C2×Q8×C53—41no—
76C2×Q8×C53—41no—
77C2×Q8×C53—41no—
78C2×Q8×C53—41no—
79C22×C4×C53—22no—
80C22×C4×C53—41no—
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×Dic53—21no—
85C22×Dic53—21no—
86C22×Dic53—21no—
87C2×D212—21no—
88C2×D212—21no—
89C2×D212—21no—
90C2×D212C22⋊C412yes—
91C2×D212D812yes—
92C2×D212SD1612yes—
93C2×C4×D53—22no—
94C2×C4×D53—22no—
95C2×C4×D53—41no—
96C2×C4×D53—41no—
97C2×C4×D53—41no—
98C2×C4×D53—41no—
99C2×C4×D53—44no—
100C2×C4×D53—44no—
101C2×C4×D53—82no—
102C2×C8×C53—22no—
103C2×C8×C53—41no—
104C2×C8×C53—41no—
105C4⋊C4×C53—22no—
106C4⋊C4×C53—22no—
107C4⋊C4×C53—42no—
108C4⋊C4×C53—41no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53—41no—
111C4⋊C4×C53—41no—
112C4⋊C4×C53—41no—
113C4⋊C4×C53—41no—
114C42×C53C22×C411yescharacteristic
115C42×C53C2×D411yescharacteristic
116C42×C53#4111yescharacteristic
117C42×C53#4111yescharacteristic
118#11719924—48no—
119#11719921—22no—
120#11719921—22no—
121#11719921—44no—
122C4⋊Dic53—21no—
123C4⋊Dic53—21no—
124C4⋊Dic53—21no—
125C4⋊Dic53—21no—
126C4⋊Dic53C22⋊C412yes—
127C4⋊Dic53SD1612yescomplemented
128#11719919—44no—
129C4×Dic53—41no—
130C4×Dic53—41no—
131C2×#88322—22no—
132C2×#88322—41no—
133C2×#88322—41no—
134C2×#88322—41no—
135C2×#88322—41no—
136C2×#88322—41no—
137C23×C53#12011yescharacteristic
138C22×D53—22no—
139C22×D53—22no—
140C22×D53—22no—
141C22×D53—41no—
142C22×D53—41no—
143C22×D53—41no—
144C22×D53—41no—
145Q8×C53—42no—
146Q8×C53—42no—
147Q8×C53—81no—
148Q8×C53—81no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—42no—
152C2×C4×C53—42no—
153C2×C4×C53—42no—
154C2×C4×C53—42no—
155C2×C4×C53—81no—
156C2×C4×C53—81no—
157C2×C4×C53C2×C22⋊C411yescharacteristic
158C2×C4×C53C4×D411yescharacteristic
159C2×C4×C53C4×D411yescharacteristic
160C2×C4×C53#12111yescharacteristic
161C2×C4×C53#12111yescharacteristic
162C2×C4×C53#12311yescharacteristic
163C2×C4×C53C2×D811yescharacteristic
164C2×C4×C53C2×SD1611yescharacteristic
165C2×C4×C53C2×SD1611yescharacteristic
166C2×C4×C53#13511yescharacteristic
167C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C53#13711yescharacteristic
170C2×C4×C53#13711yescharacteristic
171C2×Dic53—22no—
172C2×Dic53—22no—
173C2×Dic53—41no—
174C2×Dic53—41no—
175C2×Dic53—41no—
176C2×Dic53—41no—
177C2×Dic53—41no—
178C2×Dic53—41no—
179C2×Dic53—41no—
180D212—22no—
181D212—22no—
182D212—22no—
183D212#10214yes—
184C4×D53—42no—
185C4×D53—42no—
186C4×D53—42no—
187C4×D53—42no—
188C4×D53—88no—
189C8×C53—42no—
190#88322—42no—
191#88322—81no—
192#8985—531nomaximal, Sylow
193C22×C53#38611yescharacteristic
194C22×C53#44711yescharacteristic
195C22×C53#44811yescharacteristic
196C22×C53#45011yescharacteristic
197C22×C53#45211yescharacteristic
198C22×C53#45511yescharacteristic
199C22×C53#45611yescharacteristic
200C2×D53—24no—
201C2×D53—42no—
202C2×D53—42no—
203C2×D53—42no—
204C2×D53—42no—
205C2×D53—42no—
206C2×D53—42no—
207C4×C53—21no—
208C4×C53—21no—
209C4×C53—82no—
210C4×C53—82no—
211C4×C53C2×#10211yescharacteristic
212C4×C53#41711yescharacteristic
213C4×C53C4×SD1611yescharacteristic
214C4×C53#44011yescharacteristic
215C4×C53#46111yescharacteristic
216C4×C53#46311yescharacteristic
217C4×C53#48011yescharacteristic
218C4×C53#48111yescharacteristic
219Dic53—42no—
220Dic53—42no—
221Dic53—81no—
222C2×#461—531no—
223D4×C4⋊C4—531no—
224#1990D5311yescharacteristic, complemented
225#1988—531no—
226C2×D4⋊C8—531no—
227#1296—531no—
228#1279—531no—
229C2×C53#163911yescharacteristic
230C2×C53#164311yescharacteristic
231C2×C53#164711yescharacteristic
232C2×C53#165111yescharacteristic
233C2×C53#187411yescharacteristic
234C2×C53#187711yescharacteristic
235C2×C53#188211yescharacteristic
236D53—44no—
237C22×SD16—1061no—
238C2×C4⋊Q8C2×D5311yescharacteristic
239C2×C4×D4—531no—
240C2×C4×D4—1061no—
241C22×C4⋊C4—1061no—
242#461—534no—
243#461—1062no—
244C2×C4⋊C8—531no—
245C2×C4⋊C8—531no—
246C2×#103—1061no—
247C2×C4×C8C2×D5311yescharacteristic
248#389—1061no—
249#385—1062no—
250#384C2×D5311yescharacteristic
251#380—1061no—
252C4×C4⋊C4—531no—
253#340—22no—
254#340—1061no—
255#340—1061no—
256D4⋊C8—1062no—
257C53#898511yescharacteristic, complemented, Sylow
258C22×Q8—21no—
259C22×D4—531no—
260C23×C4—1062no—
261C23×C4—1061no—
262C2×SD16—1064no—
263C2×SD16—2122no—
264C2×SD16—2122no—
265C22×C8—21no—
266C22×C8—1061no—
267C22×C8—1061no—
268C4⋊Q8—21no—
269C4⋊Q8—21no—
270C4⋊Q8—21no—
271C4⋊Q8C4×D5312yescomplemented
272C4×D4—532no—
273C4×D4—532no—
274C4×D4—1061no—
275C4×D4—1061no—
276C4×D4—2122no—
277C4×D4—2122no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—531no—
283C2×C4⋊C4—531no—
284C2×C4⋊C4—531no—
285C2×C4⋊C4—1062no—
286C2×C4⋊C4—2122no—
287C2×C4⋊C4—2122no—
288C2×C4⋊C4C22⋊D5312yes—
289C2×C22⋊C4—1061no—
290C2×C22⋊C4—1062no—
291C2×C42—1061no—
292C2×C42C22×D5311yescharacteristic
293C4⋊C8—532no—
294C4⋊C8—1061no—
295C4⋊C8—1061no—
296C4⋊C8—1061no—
297#103—1064no—
298C4×C8—21no—
299C4×C8—21no—
300#95—1062no—
301#95—1061no—
302C24—1061no—
303C2×Q8—22no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×D4—532no—
310C2×D4—532no—
311C2×D4—532no—
312C2×D4—1061no—
313C2×D4—1061no—
314C2×D4—1061no—
315C22×C4—22no—
316C22×C4—41no—
317C22×C4—1061no—
318C22×C4—1061no—
319C22×C4—1062no—
320C22×C4—1062no—
321C22×C4—1061no—
322C22×C4—4242no—
323C22×C4—2124no—
324C22×C4—2124no—
325C22×C4—2121no—
326C22×C4—2121no—
327C22×C4—2121no—
328C22×C4—2121no—
329C22×C4D4×D5311yescharacteristic
330C22×C4D4×D5311yescharacteristic
331C22×C4C2×C22⋊D5311yescharacteristic
332SD16—2128no—
333C2×C8—22no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—1062no—
337C2×C8—2121no—
338C2×C8—2121no—
339C2×C8—2121no—
340C2×C8—2121no—
341C2×C8—2121no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—532no—
352C4⋊C4—532no—
353C4⋊C4—1061no—
354C4⋊C4—1061no—
355C4⋊C4—1061no—
356C4⋊C4—1061no—
357C4⋊C4—2124no—
358C22⋊C4—1062no—
359C22⋊C4—1062no—
360C22⋊C4—2124no—
361C42—2121no—
362C42—2121no—
363C42C2×C4×D5311yescharacteristic
364C42#1171994511yescharacteristic
365C42D4×D5311yescharacteristic
366C42#1171994711yescharacteristic
367C23—1062no—
368C23—1062no—
369C23—1062no—
370C23—2121no—
371C23—2121no—
372C23—2121no—
373C23—2121no—
374C23#1313318511yescharacteristic
375Q8—42no—
376Q8—42no—
377Q8—81no—
378Q8—81no—
379D4—534no—
380D4—1062no—
381D4—1062no—
382D4—1062no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—1062no—
392C2×C4—1062no—
393C2×C4—4248no—
394C2×C4—2121no—
395C2×C4—2121no—
396C2×C4—2121no—
397C2×C4—2121no—
398C2×C4—2121no—
399C2×C4—2121no—
400C2×C4—2121no—
401C2×C4—2122no—
402C2×C4—2122no—
403C2×C4—2122no—
404C2×C4—2122no—
405C2×C4C22⋊C4×D5311yescharacteristic
406C2×C4#1313317711yescharacteristic
407C2×C4C4×C22⋊D5311yescharacteristic
408C2×C4#1313318811yescharacteristic
409C2×C4#1313319011yescharacteristic
410C2×C4#1313319911yescharacteristic
411C2×C4D8×D5311yescharacteristic
412C2×C4SD16×D5311yescharacteristic
413C2×C4SD16×D5311yescharacteristic
414C2×C4#1313324011yescharacteristic
415C2×C4#1313324111yescharacteristic
416C2×C4#1313324411yescharacteristic
417C2×C4#1313324611yescharacteristic
418C2×C4#1313324911yescharacteristic
419C8—42no—
420C8—4241no—
421C8—2122no—
422C22—1064no—
423C22—2122no—
424C22—2122no—
425C22—2122no—
426C22—2122no—
427C22—2122no—
428C22—2122no—
429C22#1681049911yescharacteristic
430C22#1681075511yescharacteristic
431C22#1681076011yescharacteristic
432C22#1681077211yescharacteristic
433C22#1681078411yescharacteristic
434C22#1681080011yescharacteristic
435C22#1681080511yescharacteristic
436C4—21no—
437C4—21no—
438C4—82no—
439C4—82no—
440C4—4241no—
441C4—2122no—
442C4—2122no—
443C4#102×D5311yescharacteristic
444C4#1681062111yescharacteristic
445C4#1681070811yescharacteristic
446C4#1681072611yescharacteristic
447C4#1681082711yescharacteristic
448C4#1681084111yescharacteristic
449C4#1681093511yescharacteristic
450C4#1681094011yescharacteristic
451C2—2124no—
452C2#2027389111yescharacteristic
453C2#2027391911yescharacteristic
454C2#2027394711yescharacteristic
455C2#2027397511yescharacteristic
456C2#2027502911yescharacteristic
457C2#2027504611yescharacteristic
458C2#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#8985
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
13. #16810825C4semidirect
54. C4⋊#88322C2×C4semidirect
63. C4⋊Q8×C53C2×C4semidirect
127. C4⋊Dic53SD16semidirect
224. #1990D53semidirect
257. C53#8985semidirect
271. C4⋊Q8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168108252
#202776152
#202756682
C2×#168103522
#202725982
#202725552
C53×#19902
#898553