Finite Group Databasebeta

← #27278249

Subgroups — #27278249

Order 13568.

618 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277615C211yesmaximal, characteristic
2#20275471C211yesmaximal, characteristic, complemented
3#20274712C211yesmaximal, characteristic
4C2×#16810352C211yesmaximal, characteristic
5#20272598C211yesmaximal, characteristic, complemented
6#20272555C211yesmaximal, characteristic, complemented
7C53×#1845C211yesmaximal, characteristic, complemented
8C2×C4×D212C2211yescharacteristic
9C2×#13133179—21no—
10C2×C4⋊C4×D53—21no—
11C2×#13133150—21no—
12C2×C4⋊#88322C2211yescharacteristic
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133137—21no—
15#16810519—21no—
16#16810494—22no—
17#16810480—21no—
18C4×C4⋊Dic53C2211yescharacteristic
19#16810385—21no—
20#16810385—21no—
21#16810352C414yescomplemented
22C2×#107×C53—22no—
23C2×C4×C8×C53C2211yescharacteristic
24C53×#384C2211yescharacteristic
25C53×#384C2211yescharacteristic
26C22×D212D411yescharacteristic
27C22×C4×D53—21no—
28C22×C4×D53—22no—
29C22×#88322—21no—
30C22×#88322—21no—
31C4×D212—21no—
32C4×D212—21no—
33C4×D212C2×C412yes—
34C4×D212Q812yes—
35#13133179—42no—
36#13133179—42no—
37C2×C4⋊Dic53D411yescharacteristic
38C2×C4⋊Dic53D411yescharacteristic
39C2×C4⋊Dic53D411yescharacteristic
40C2×#11719919—22no—
41C4⋊C4×D53—42no—
42C4⋊C4×D53—42no—
43C2×#11719921—22no—
44C2×#11719921—21no—
45C2×C4×Dic53—21no—
46#13133150—42no—
47C4⋊#88322—21no—
48C4⋊#88322—21no—
49C4⋊#88322—21no—
50C4⋊#88322C2×C412yescomplemented
51#13133137—42no—
52#13133124—22no—
53#13133124—21no—
54C22×C8×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53D412yes—
61C2×C42×C53C2311yescharacteristic
62#107×C53—44no—
63C4×C8×C53—21no—
64C4×C8×C53C2×C412yescomplemented
65C23×D53—21no—
66C22×C4×C53—22no—
67C22×C4×C53—41no—
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C2×D212—21no—
75C2×D212—21no—
76C2×D212—21no—
77C2×D212C4⋊C412yes—
78C2×D212D812yes—
79C2×D212Q1612yes—
80C2×C4×D53—22no—
81C2×C4×D53—22no—
82C2×C4×D53—41no—
83C2×C4×D53—41no—
84C2×C4×D53—41no—
85C2×C4×D53—41no—
86C2×C4×D53—44no—
87C2×C4×D53—44no—
88C2×C4×D53—82no—
89C2×C8×C53—22no—
90C2×C8×C53—22no—
91C2×C8×C53—22no—
92C4⋊C4×C53—42no—
93C4⋊C4×C53—42no—
94C4⋊C4×C53—41no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53—41no—
98C4⋊C4×C53—41no—
99C4⋊C4×C53—41no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—41no—
102C42×C53C22×C411yescharacteristic
103C42×C53C2×Q811yescharacteristic
104C42×C53#4111yescharacteristic
105C42×C53#4111yescharacteristic
106#11719921—22no—
107#11719921—22no—
108#11719921—44no—
109C4⋊Dic53—21no—
110C4⋊Dic53—21no—
111C4⋊Dic53—21no—
112C4⋊Dic53—21no—
113C4⋊Dic53C4⋊C412yes—
114C4⋊Dic53D812yes—
115#11719919—44no—
116C4×Dic53—41no—
117C4×Dic53—41no—
118C2×#88322—22no—
119C2×#88322—41no—
120C2×#88322—41no—
121C2×#88322—41no—
122C2×#88322—41no—
123C2×#88322—41no—
124C23×C53#12011yescharacteristic
125C22×D53—22no—
126C22×D53—22no—
127C22×D53—22no—
128C22×D53—41no—
129C22×D53—41no—
130C22×D53—41no—
131C22×D53—41no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—42no—
135C2×C4×C53—42no—
136C2×C4×C53—42no—
137C2×C4×C53—81no—
138C2×C4×C53—81no—
139C2×C4×C53—81no—
140C2×C4×C53C2×C4⋊C411yescharacteristic
141C2×C4×C53C4×D411yescharacteristic
142C2×C4×C53C4×D411yescharacteristic
143C2×C4×C53C22⋊Q811yescharacteristic
144C2×C4×C53C22⋊Q811yescharacteristic
145C2×C4×C53#12311yescharacteristic
146C2×C4×C53C2×D811yescharacteristic
147C2×C4×C53C2×D811yescharacteristic
148C2×C4×C53C2×Q1611yescharacteristic
149C2×C4×C53#13511yescharacteristic
150C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C53#13711yescharacteristic
153C2×C4×C53#13711yescharacteristic
154C2×Dic53—22no—
155C2×Dic53—22no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53—41no—
162C2×Dic53—41no—
163D212—22no—
164D212—22no—
165D212—22no—
166D212#10714yescomplemented
167C4×D53—42no—
168C4×D53—42no—
169C4×D53—42no—
170C4×D53—42no—
171C4×D53—88no—
172C8×C53—24no—
173#88322—42no—
174#88322—81no—
175#8882—531nomaximal, Sylow
176C22×C53#38511yescharacteristic
177C22×C53#44711yescharacteristic
178C22×C53#44711yescharacteristic
179C22×C53#45111yescharacteristic
180C22×C53#45211yescharacteristic
181C22×C53#45311yescharacteristic
182C22×C53#45611yescharacteristic
183C2×D53—24no—
184C2×D53—42no—
185C2×D53—42no—
186C2×D53—42no—
187C2×D53—42no—
188C2×D53—42no—
189C2×D53—42no—
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—82no—
193C4×C53—161no—
194C4×C53C2×#10711yescharacteristic
195C4×C53#42811yescharacteristic
196C4×C53C4×D811yescharacteristic
197C4×C53#44011yescharacteristic
198C4×C53#47411yescharacteristic
199C4×C53#47811yescharacteristic
200C4×C53#48011yescharacteristic
201C4×C53#48411yescharacteristic
202Dic53—42no—
203Dic53—42no—
204Dic53—81no—
205D4×C4⋊C4—531no—
206#1947—531no—
207#1845D5311yescharacteristic, complemented
208#1802—531no—
209C2×D4⋊C8—531no—
210#1296—531no—
211#1279—531no—
212C2×C53#161911yescharacteristic
213C2×C53#163211yescharacteristic
214C2×C53#164711yescharacteristic
215C2×C53#164911yescharacteristic
216C2×C53#185211yescharacteristic
217C2×C53#185911yescharacteristic
218C2×C53#186211yescharacteristic
219D53—44no—
220C2×C4×D4—531no—
221C2×C4×D4—1061no—
222C22×C4⋊C4—1061no—
223C2×#107—22no—
224C2×#107—1061no—
225C2×C4⋊C8—531no—
226C2×C4⋊C8—531no—
227C2×#102—1061no—
228C2×C4×C8C2×D5311yescharacteristic
229#389—1061no—
230#385—1062no—
231#384C2×D5311yescharacteristic
232#384C2×D5311yescharacteristic
233#380—1061no—
234C4×C4⋊C4—531no—
235#340—1061no—
236#340—1061no—
237D4⋊C8—534no—
238C53#888211yescharacteristic, complemented, Sylow
239C22×D4—531no—
240C23×C4—1062no—
241C23×C4—1061no—
242C22×C8—21no—
243C22×C8—1061no—
244C22×C8—1061no—
245C4×D4—532no—
246C4×D4—532no—
247C4×D4—1061no—
248C4×D4—1061no—
249C4×D4—2122no—
250C4×D4—2122no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—531no—
257C2×C4⋊C4—531no—
258C2×C4⋊C4—531no—
259C2×C4⋊C4—1062no—
260C2×C4⋊C4—2122no—
261C2×C4⋊C4—2122no—
262C2×C4⋊C4C22⋊D5312yes—
263C2×C22⋊C4—1061no—
264C2×C22⋊C4—1062no—
265C2×C42—1061no—
266C2×C42C22×D5311yescharacteristic
267#107—44no—
268#107—2122no—
269C4⋊C8—532no—
270C4⋊C8—1061no—
271C4⋊C8—1061no—
272C4⋊C8—1061no—
273#102—2122no—
274C4×C8—21no—
275C4×C8C4×D5312yescomplemented
276#95—1062no—
277#95—1061no—
278C24—1061no—
279C2×D4—532no—
280C2×D4—532no—
281C2×D4—532no—
282C2×D4—1061no—
283C2×D4—1061no—
284C2×D4—1061no—
285C22×C4—22no—
286C22×C4—41no—
287C22×C4—1061no—
288C22×C4—1061no—
289C22×C4—1062no—
290C22×C4—1062no—
291C22×C4—1061no—
292C22×C4—4242no—
293C22×C4—2124no—
294C22×C4—2124no—
295C22×C4—2121no—
296C22×C4—2121no—
297C22×C4—2121no—
298C22×C4—2121no—
299C22×C4D4×D5311yescharacteristic
300C22×C4D4×D5311yescharacteristic
301C22×C4C2×C22⋊D5311yescharacteristic
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—1062no—
306C2×C8—2121no—
307C2×C8—2121no—
308C2×C8—2121no—
309C2×C8—2121no—
310C2×C8—2121no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—532no—
322C4⋊C4—532no—
323C4⋊C4—1061no—
324C4⋊C4—1061no—
325C4⋊C4—1061no—
326C4⋊C4—1061no—
327C4⋊C4—2124no—
328C22⋊C4—1062no—
329C22⋊C4—1062no—
330C22⋊C4—2124no—
331C42—2121no—
332C42—2121no—
333C42C2×C4×D5311yescharacteristic
334C42#1171994511yescharacteristic
335C42Q8×D5311yescharacteristic
336C42#1171994911yescharacteristic
337C23—1062no—
338C23—1062no—
339C23—1062no—
340C23—2121no—
341C23—2121no—
342C23—2121no—
343C23—2121no—
344C23#1313318511yescharacteristic
345D4—534no—
346D4—1062no—
347D4—1062no—
348D4—1062no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—1062no—
358C2×C4—1062no—
359C2×C4—4248no—
360C2×C4—2121no—
361C2×C4—2121no—
362C2×C4—2121no—
363C2×C4—2121no—
364C2×C4—2121no—
365C2×C4—2121no—
366C2×C4—2121no—
367C2×C4—2122no—
368C2×C4—2122no—
369C2×C4—2122no—
370C2×C4—2122no—
371C2×C4C4⋊C4×D5311yescharacteristic
372C2×C4C4×C22⋊D5311yescharacteristic
373C2×C4#1313317911yescharacteristic
374C2×C4#1313319311yescharacteristic
375C2×C4#1313319511yescharacteristic
376C2×C4#1313320211yescharacteristic
377C2×C4D8×D5311yescharacteristic
378C2×C4D8×D5311yescharacteristic
379C2×C4Q16×D5311yescharacteristic
380C2×C4#1313324011yescharacteristic
381C2×C4#1313324111yescharacteristic
382C2×C4#1313324111yescharacteristic
383C2×C4#1313324911yescharacteristic
384C2×C4#1313324911yescharacteristic
385C8—24no—
386C8—4241no—
387C8—2122no—
388C22—1064no—
389C22—2122no—
390C22—2122no—
391C22—2122no—
392C22—2122no—
393C22—2122no—
394C22—2122no—
395C22#1681049411yescharacteristic
396C22#1681075511yescharacteristic
397C22#1681075511yescharacteristic
398C22#1681077711yescharacteristic
399C22#1681078411yescharacteristic
400C22#1681079011yescharacteristic
401C22#1681080511yescharacteristic
402C4—21no—
403C4—21no—
404C4—82no—
405C4—161no—
406C4—4241no—
407C4—2122no—
408C4—2122no—
409C4#107×D5311yescharacteristic
410C4#1681066411yescharacteristic
411C4#1681070511yescharacteristic
412C4#1681072211yescharacteristic
413C4#1681090211yescharacteristic
414C4#1681092611yescharacteristic
415C4#1681093811yescharacteristic
416C4#1681096011yescharacteristic
417C2—2124no—
418C2#2027376711yescharacteristic
419C2#2027384611yescharacteristic
420C2#2027394711yescharacteristic
421C2#2027396111yescharacteristic
422C2#2027490111yescharacteristic
423C2#2027494011yescharacteristic
424C2#2027495911yescharacteristic
subgroupG/H
centerC23#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1845C2
socleC23×C53#120
prime radicalC22×D212D4
cosocleC2×C42×C53C23
hypercenter#1845D53
hypocenterC53#8882
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1845

Complements

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

NH ≅ G/Ntype
2. #20275471C2semidirect
5. #20272598C2semidirect
6. #20272555C2semidirect
7. C53×#1845C2semidirect
21. #16810352C4semidirect
50. C4⋊#88322C2×C4semidirect
64. C4×C8×C53C2×C4semidirect
166. D212#107semidirect
207. #1845D53semidirect
238. C53#8882semidirect
275. C4×C8C4×D53semidirect

Maximal subgroups

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

subgroupindex
#202776152
#202754712
#202747122
C2×#168103522
#202725982
#202725552
C53×#18452
#888253