Finite Group Databasebeta

← #27301150

Subgroups — #27301150

Order 13568.

590 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D848C211yesmaximal, characteristic
2C2×#16810705C211yesmaximal, characteristic, complemented
3C2×#16810423C211yesmaximal, characteristic, complemented
4C2×#16810423C211yesmaximal, characteristic, complemented
5#20275579C211yesmaximal, characteristic, complemented
6#20272792C211yesmaximal, characteristic, complemented
7C53×#1380C211yesmaximal, characteristic, complemented
8C22×D424C2211yescharacteristic
9C22×D424C2211yescharacteristic
10C2×#13133179—21no—
11C2×D848—22no—
12C2×D848—22no—
13C2×D848C414yescomplemented
14C22×#11719908C2211yescharacteristic
15#16810705—22no—
16#16810705—22no—
17C2×#13133137—21no—
18C2×#13133137—21no—
19C2×C8×Dic53C2211yescharacteristic
20#16810524—21no—
21#16810423—22no—
22#16810423—22no—
23C22×C16×C53C2211yescharacteristic
24C2×#107×C53C2211yescharacteristic
25C2×#107×C53C2211yescharacteristic
26C22×D212—21no—
27C22×D212—21no—
28C22×C4×D53—41no—
29C2×D424—21no—
30C2×D424—21no—
31C2×D424—21no—
32C2×D424—21no—
33C2×D424—21no—
34C2×D424—21no—
35C2×D424—21no—
36C2×D424—21no—
37C2×D424C2×C412yes—
38C2×D424C2×C412yescomplemented
39C2×D424D412yes—
40C2×D424D412yes—
41C22×#88322D411yescharacteristic
42#13133179—24no—
43#13133179—42no—
44C2×#11719921—41no—
45C2×#11719921—41no—
46C2×#11719921—41no—
47C2×C4×Dic53D411yescharacteristic
48D848—28no—
49C2×#11719908—21no—
50C2×#11719908—21no—
51C2×#11719908—21no—
52#13133137—24no—
53#13133137—42no—
54C8×Dic53—21no—
55C8×Dic53—21no—
56C22×C8×C53C2311yescharacteristic
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C16×C53—21no—
60C2×C16×C53—21no—
61C2×C16×C53C2×C412yescomplemented
62#107×C53—21no—
63#107×C53—21no—
64#107×C53—21no—
65#107×C53D412yes—
66C23×D53—41no—
67C23×D53—41no—
68C22×C4×C53—41no—
69C22×C4×C53—41no—
70C22×C4×C53C2×D411yescharacteristic
71C22×Dic53D811yescharacteristic
72C22×Dic53D811yescharacteristic
73C2×D212—22no—
74C2×D212—22no—
75C2×D212—22no—
76C2×D212—22no—
77C2×D212—41no—
78C2×D212—41no—
79C2×D212—41no—
80C2×D212—41no—
81C2×D212—41no—
82C2×D212—41no—
83C2×D212—41no—
84C2×D212—41no—
85C2×C4×D53—44no—
86C2×C4×D53—82no—
87C2×C4×D53—82no—
88C2×C8×C53C22×C411yescharacteristic
89C2×C8×C53C2×D411yescharacteristic
90C2×C8×C53C2×D411yescharacteristic
91C2×C8×C53C2×D411yescharacteristic
92C2×C8×C53#4111yescharacteristic
93C2×C8×C53#4111yescharacteristic
94C4⋊C4×C53—22no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53—41no—
98#11719921—44no—
99#11719921—82no—
100#11719921—82no—
101C4×Dic53—21no—
102C4×Dic53—21no—
103C2×#88322—21no—
104C2×#88322—21no—
105C2×#88322—21no—
106D424—22no—
107D424—22no—
108D424—22no—
109D424—22no—
110D424—22no—
111D424—41no—
112D424—41no—
113D424C22⋊C414yescomplemented
114C16×C53—22no—
115#11719908—41no—
116C23×C53C2×D811yescharacteristic
117C22×D53—42no—
118C22×D53—42no—
119C22×D53—42no—
120C22×D53—42no—
121C22×D53—81no—
122C22×D53—81no—
123C22×D53—81no—
124C22×D53—81no—
125C22×D53—81no—
126C22×D53—81no—
127C22×D53—81no—
128C22×D53—81no—
129C22×D53—81no—
130C22×D53—81no—
131C2×C4×C53—42no—
132C2×C4×C53—81no—
133C2×C4×C53—81no—
134C2×C4×C53—81no—
135C2×C4×C53—81no—
136C2×C4×C53—81no—
137C2×C4×C53C4×D411yescharacteristic
138C2×C4×C53#12011yescharacteristic
139C2×C4×C53#12111yescharacteristic
140C2×C4×C53#13511yescharacteristic
141C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53D1612yescomplemented
149D212—24no—
150D212—42no—
151D212—42no—
152D212—42no—
153D212—42no—
154D212—42no—
155D212—81no—
156D212—81no—
157C4×D53—88no—
158C8×C53C2×C22⋊C411yescharacteristic
159C8×C53C4×D411yescharacteristic
160C8×C53#12111yescharacteristic
161C8×C53#12311yescharacteristic
162#88322—41no—
163#13012—531nomaximal, Sylow
164C22×C53C4×D811yescharacteristic
165C22×C53#44711yescharacteristic
166C22×C53#45911yescharacteristic
167C22×C53C2×D1611yescharacteristic
168C22×C53#50811yescharacteristic
169C22×C53#50911yescharacteristic
170C22×C53#50911yescharacteristic
171C2×D53—44no—
172C2×D53—82no—
173C2×D53—82no—
174C2×D53—82no—
175C2×D53—82no—
176C2×D53—82no—
177C2×D53—82no—
178C2×D53—82no—
179C2×D53—82no—
180C2×D53—82no—
181C2×D53—161no—
182C2×D53—161no—
183C2×D53—161no—
184C2×D53—161no—
185C4×C53—82no—
186C4×C53—161no—
187C4×C53#38611yescharacteristic
188C4×C53#44211yescharacteristic
189C4×C53#44911yescharacteristic
190C4×C53#46311yescharacteristic
191Dic53—22no—
192Dic53—41no—
193C22×D16—531no—
194C2×C4×D8—531no—
195C2×#357—531no—
196C2×#357—531no—
197#1970—531no—
198#1380D5311yescharacteristic, complemented
199#1374—531no—
200C2×C53#187911yescharacteristic
201C2×C53C4×D1611yescharacteristic
202C2×C53#217711yescharacteristic
203C2×C53#218411yescharacteristic
204C2×C53#218611yescharacteristic
205C2×C53#220611yescharacteristic
206C2×C53#221011yescharacteristic
207D53—84no—
208D53—162no—
209C22×D8—531no—
210C22×D8—531no—
211C2×C4×D4—1061no—
212C2×D16—534no—
213C2×D16—1062no—
214C2×D16—1062no—
215C22×C16—531no—
216C22×C16C2×D5311yescharacteristic
217C4×D8—1062no—
218C4×D8—1062no—
219C2×#107C2×D5311yescharacteristic
220C2×#107C2×D5311yescharacteristic
221C2×#102—1061no—
222C2×#102—1061no—
223C2×C4×C8—531no—
224#390—1061no—
225#357—1062no—
226#357—1062no—
227C53#1301211yescharacteristic, complemented, Sylow
228C22×D4—1061no—
229C22×D4—1061no—
230C23×C4—2121no—
231C2×D8—532no—
232C2×D8—532no—
233C2×D8—532no—
234C2×D8—532no—
235C2×D8—1061no—
236C2×D8—1061no—
237C2×D8—1061no—
238C2×D8—1061no—
239C2×D8—1061no—
240C2×D8—1061no—
241C2×D8—1061no—
242C2×D8—1061no—
243C22×C8—531no—
244C22×C8C22×D5311yescharacteristic
245C4×D4—1064no—
246C4×D4—2122no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C22⋊C4—2121no—
250C2×C22⋊C4—2121no—
251C2×C22⋊C4—2121no—
252C2×C42—531no—
253D16—1068no—
254C2×C16—21no—
255C2×C16—21no—
256C2×C16—1061no—
257C2×C16—1061no—
258C2×C16—1061no—
259C2×C16C4×D5312yescomplemented
260#107—21no—
261#107—21no—
262#107—21no—
263#107D21212yes—
264#102—1064no—
265#102—2122no—
266C4×C8—1061no—
267C4×C8—1061no—
268C24—2121no—
269C24—2121no—
270C2×D4—1062no—
271C2×D4—1062no—
272C2×D4—1062no—
273C2×D4—1062no—
274C2×D4—2121no—
275C2×D4—2121no—
276C2×D4—2121no—
277C2×D4—2121no—
278C2×D4—2121no—
279C2×D4—2121no—
280C2×D4—2121no—
281C2×D4—2121no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—531no—
285C22×C4—531no—
286C22×C4—4242no—
287C22×C4—4242no—
288C22×C4—2124no—
289C22×C4D4×D5311yescharacteristic
290D8—534no—
291D8—1062no—
292D8—1062no—
293D8—1062no—
294D8—1062no—
295D8—1062no—
296D8—2121no—
297D8—2121no—
298C2×C8—1061no—
299C2×C8—1061no—
300C2×C8—1061no—
301C2×C8C2×C4×D5311yescharacteristic
302C2×C8C2×D21211yescharacteristic
303C2×C8#1171994511yescharacteristic
304C2×C8D4×D5311yescharacteristic
305C2×C8D4×D5311yescharacteristic
306C2×C8#1171994911yescharacteristic
307C4⋊C4—22no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C22⋊C4—4242no—
312C22⋊C4—4242no—
313C22⋊C4—2124no—
314C42—1061no—
315C42—1061no—
316C16—22no—
317C16—2121no—
318C23—4241no—
319C23—4241no—
320C23—4241no—
321C23—4241no—
322C23—4241no—
323C23—4241no—
324C23—4241no—
325C23—4241no—
326C23—4241no—
327C23—4241no—
328C23—2122no—
329C23—2122no—
330C23—2122no—
331C23—2122no—
332C23D8×D5311yescharacteristic
333D4—1064no—
334D4—4241no—
335D4—4241no—
336D4—2122no—
337D4—2122no—
338D4—2122no—
339D4—2122no—
340D4—2122no—
341C2×C4—42no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—532no—
348C2×C4—1061no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4—1061no—
353C2×C4—4248no—
354C2×C4#1313317911yescharacteristic
355C2×C4#1313318711yescharacteristic
356C2×C4#1313319011yescharacteristic
357C2×C4#1313324011yescharacteristic
358C2×C4#1313324411yescharacteristic
359C2×C4#1313324411yescharacteristic
360C8—2121no—
361C8C22⋊C4×D5311yescharacteristic
362C8C4×D21211yescharacteristic
363C8#1313319211yescharacteristic
364C8#1313320211yescharacteristic
365C22—8481no—
366C22—8481no—
367C22—8481no—
368C22—8481no—
369C22—4242no—
370C22—4242no—
371C22—4242no—
372C22—4242no—
373C22—4242no—
374C22—4242no—
375C22—4242no—
376C22—4242no—
377C22—4242no—
378C22—2124no—
379C22#1681070511yescharacteristic
380C22#1681075811yescharacteristic
381C22#1681081711yescharacteristic
382C22D16×D5311yescharacteristic
383C22#1681105611yescharacteristic
384C22#1681106011yescharacteristic
385C22#1681106011yescharacteristic
386C4—82no—
387C4—161no—
388C4—1062no—
389C4—2121no—
390C4#1681050111yescharacteristic
391C4#1681073411yescharacteristic
392C4#1681077111yescharacteristic
393C4#1681084111yescharacteristic
394C2—8482no—
395C2—4244no—
396C2#2027506411yescharacteristic
397C2#2027656111yescharacteristic
398C2#2027659011yescharacteristic
399C2#2027662811yescharacteristic
400C2#2027664111yescharacteristic
401C2#2027674911yescharacteristic
402C2#2027677311yescharacteristic
subgroupG/H
centerC23D8×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#1380C2
socleC23×C53C2×D8
prime radicalC22×D848C2
cosocleC22×C8×C53C23
hypercenter#1380D53
hypocenterC53#13012
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
2. C2×#16810705C2semidirect
3. C2×#16810423C2semidirect
4. C2×#16810423C2semidirect
5. #20275579C2semidirect
6. #20272792C2semidirect
7. C53×#1380C2semidirect
13. C2×D848C4semidirect
38. C2×D424C2×C4semidirect
61. C2×C16×C53C2×C4semidirect
113. D424C22⋊C4semidirect
148. C2×Dic53D16semidirect
198. #1380D53semidirect
227. C53#13012semidirect
259. C2×C16C4×D53semidirect

Maximal subgroups

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

subgroupindex
C22×D8482
C2×#168107052
C2×#168104232
#202755792
#202727922
C53×#13802
#1301253