Finite Group Databasebeta

← #27301027

Subgroups — #27301027

Order 13568.

626 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810856C211yesmaximal, characteristic
2C2×#107×D53C211yesmaximal, characteristic, complemented
3C2×#16810424C211yesmaximal, characteristic, complemented
4C2×#16810402C211yesmaximal, characteristic, complemented
5#20272807C211yesmaximal, characteristic, complemented
6#20272806C211yesmaximal, characteristic, complemented
7C2×C53×#357C211yesmaximal, characteristic, complemented
8C22×C8×D53C2211yescharacteristic
9C2×#13133188—21no—
10C2×C4⋊C4×D53—21no—
11C22×#11719908C2211yescharacteristic
12#16810856—22no—
13#16810856C414yescomplemented
14C2×#13133151C2211yescharacteristic
15C2×#13133151C2211yescharacteristic
16C2×#13133150—21no—
17#107×D53—22no—
18#107×D53—22no—
19C2×#13133138—21no—
20#16810424—22no—
21#16810402—22no—
22C22×D8×C53C2211yescharacteristic
23C22×C16×C53C2211yescharacteristic
24C2×#107×C53C2211yescharacteristic
25C53×#357—22no—
26C22×C22⋊D53—41no—
27C22×C4×D53—41no—
28C22×C4×D53D411yescharacteristic
29C22×#88322D411yescharacteristic
30C2×C8×D53—21no—
31C2×C8×D53—21no—
32C2×C8×D53—21no—
33C2×C8×D53—21no—
34C2×C8×D53C2×C412yes—
35C2×C8×D53D412yes—
36#13133188—24no—
37#13133188—42no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53—21no—
40C2×#11719919—41no—
41C4⋊C4×D53—24no—
42C4⋊C4×D53—42no—
43C2×C22⋊Dic53—41no—
44C2×#11719908—21no—
45C2×#11719908—21no—
46C2×#11719908—21no—
47#13133151—21no—
48#13133151—21no—
49#13133151C2×C412yescomplemented
50#13133151D412yescomplemented
51#13133150—24no—
52#13133138—24no—
53C22×D4×C53—21no—
54C2×D8×C53—21no—
55C2×D8×C53—21no—
56C2×D8×C53—21no—
57C2×D8×C53—21no—
58C2×D8×C53—21no—
59C2×D8×C53C2×C412yescomplemented
60C22×C8×C53C2311yescharacteristic
61C2×C4⋊C4×C53—21no—
62C2×C16×C53—21no—
63C2×C16×C53—21no—
64C2×C16×C53—21no—
65#107×C53—21no—
66#107×C53D412yescomplemented
67C24×C53—41no—
68C23×D53D811yescharacteristic
69C2×D4×C53—22no—
70C2×D4×C53—41no—
71C2×D4×C53—41no—
72C2×D4×C53—41no—
73C2×D4×C53—41no—
74C2×D4×C53—41no—
75C22×C4×C53—41no—
76C22×C4×C53C2×D411yescharacteristic
77C2×C22⋊D53—44no—
78C2×C22⋊D53—82no—
79C2×C22⋊D53—82no—
80C22×Dic53—41no—
81C22×Dic53—41no—
82C22×Dic53D811yescharacteristic
83C2×C4×D53—21no—
84C2×C4×D53—21no—
85C2×C4×D53—21no—
86C2×C4×D53—21no—
87C2×C4×D53—44no—
88C2×C4×D53—82no—
89C2×C4×D53—82no—
90C2×C4×D53C22⋊C412yes—
91C2×C4×D53SD1612yes—
92D8×C53—22no—
93D8×C53—22no—
94D8×C53—41no—
95D8×C53—41no—
96C2×C8×C53C22×C411yescharacteristic
97C2×C8×C53C2×D411yescharacteristic
98C2×C8×C53C2×D411yescharacteristic
99C2×C8×C53C2×D411yescharacteristic
100C2×C8×C53#4111yescharacteristic
101C2×C8×C53#4111yescharacteristic
102C4⋊C4×C53—22no—
103C4⋊C4×C53—41no—
104C22⋊Dic53—44no—
105C4⋊Dic53—22no—
106C4⋊Dic53—22no—
107C4⋊Dic53—41no—
108C4⋊Dic53—41no—
109#11719919—44no—
110C2×#88322—21no—
111C2×#88322—21no—
112C2×#88322C22⋊C412yes—
113C8×D53—22no—
114C8×D53—22no—
115C8×D53—22no—
116C8×D53—22no—
117C16×C53—41no—
118#11719908—41no—
119C23×C53—42no—
120C23×C53—81no—
121C23×C53—81no—
122C23×C53—81no—
123C23×C53—81no—
124C23×C53—81no—
125C23×C53—81no—
126C23×C53C2×D811yescharacteristic
127C22×D53—21no—
128C22×D53—21no—
129C22×D53—21no—
130C22×D53—21no—
131C22×D53#10212yes—
132C22×D53D1612yes—
133C22×D53SD3212yes—
134D4×C53—42no—
135D4×C53—42no—
136D4×C53—81no—
137D4×C53—81no—
138C2×C4×C53—42no—
139C2×C4×C53—81no—
140C2×C4×C53—81no—
141C2×C4×C53C2×C22⋊C411yescharacteristic
142C2×C4×C53#12011yescharacteristic
143C2×C4×C53#12311yescharacteristic
144C2×C4×C53C2×SD1611yescharacteristic
145C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C53#13711yescharacteristic
147C22⋊D53—88no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—42no—
151C2×Dic53—42no—
152C2×Dic53—81no—
153C2×Dic53—81no—
154C2×Dic53—81no—
155C2×Dic53—81no—
156C2×Dic53#10212yes—
157C4×D53—22no—
158C4×D53—22no—
159C4×D53—22no—
160C4×D53—22no—
161C4×D53—88no—
162C8×C53C4×D411yescharacteristic
163C8×C53C4×D411yescharacteristic
164C8×C53#12111yescharacteristic
165C8×C53#12111yescharacteristic
166#88322—22no—
167#12991—531nomaximal, Sylow
168C22×C53—82no—
169C22×C53—82no—
170C22×C53—82no—
171C22×C53—161no—
172C22×C53—161no—
173C22×C53—161no—
174C22×C53—161no—
175C22×C53C2×#10211yescharacteristic
176C22×C53#44711yescharacteristic
177C22×C53#48011yescharacteristic
178C22×C53C2×D1611yescharacteristic
179C22×C53C2×SD3211yescharacteristic
180C22×C53#50911yescharacteristic
181C22×C53#51011yescharacteristic
182C2×D53—22no—
183C2×D53—22no—
184C2×D53—22no—
185C2×D53—22no—
186C2×D53—22no—
187C2×D53—22no—
188C2×D53#35714yescomplemented
189C4×C53—82no—
190C4×C53#38611yescharacteristic
191C4×C53#41711yescharacteristic
192C4×C53#44811yescharacteristic
193C4×C53#48111yescharacteristic
194Dic53—22no—
195Dic53—82no—
196Dic53—82no—
197C2×#466—531no—
198C22×#107—531no—
199C2×#357—531no—
200C2×#357D5311yescharacteristic, complemented
201C2×C22⋊C16—531no—
202#1380—531no—
203#1380—531no—
204C2×C53—162no—
205C2×C53#187411yescharacteristic
206C2×C53C2×#35711yescharacteristic
207C2×C53#213911yescharacteristic
208C2×C53#218411yescharacteristic
209C2×C53#218511yescharacteristic
210C2×C53#223211yescharacteristic
211C2×C53#223311yescharacteristic
212D53—24no—
213C22×D8C2×D5311yescharacteristic
214C23×C8—531no—
215C2×#121—1061no—
216C22×C4⋊C4—1061no—
217C22×C16—531no—
218C22×C16C2×D5311yescharacteristic
219#466—534no—
220#466—1062no—
221C2×#107—531no—
222C2×#107—531no—
223C2×#107—1062no—
224C2×#107—1061no—
225C2×#107—1062no—
226C2×#107C2×D5311yescharacteristic
227C2×#102—1061no—
228#357—22no—
229#357—1062no—
230C22⋊C16—1062no—
231C53#1299111yescharacteristic, complemented, Sylow
232C22×D4—21no—
233C22×D4—2121no—
234C23×C4—531no—
235C23×C4—2121no—
236C2×D8—21no—
237C2×D8—21no—
238C2×D8—21no—
239C2×D8—21no—
240C2×D8—21no—
241C2×D8C4×D5312yescomplemented
242C22×C8—531no—
243C22×C8—532no—
244C22×C8—532no—
245C22×C8—1061no—
246C22×C8—1061no—
247C22×C8—1061no—
248C22×C8—1061no—
249C22×C8C22×D5311yescharacteristic
250#121—1064no—
251#121—2122no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—1064no—
254C2×C4⋊C4—1061no—
255C2×C4⋊C4—1061no—
256C2×C4⋊C4—2122no—
257C2×C4⋊C4—2121no—
258C2×C22⋊C4—2121no—
259C2×C16—21no—
260C2×C16—21no—
261C2×C16—21no—
262C2×C16—1061no—
263C2×C16—1061no—
264C2×C16—1061no—
265#107—21no—
266#107—532no—
267#107—532no—
268#107—1064no—
269#107—1061no—
270#107—1061no—
271#107C22⋊D5312yescomplemented
272#102—1064no—
273C24—41no—
274C24—531no—
275C2×D4—22no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—4242no—
282C2×D4—4242no—
283C2×D4—2124no—
284C22×C4—41no—
285C22×C4—531no—
286C22×C4—532no—
287C22×C4—532no—
288C22×C4—1061no—
289C22×C4—1061no—
290C22×C4—1061no—
291C22×C4—1061no—
292C22×C4—4242no—
293C22×C4—4242no—
294C22×C4—2124no—
295C22×C4—2121no—
296C22×C4—2121no—
297C22×C4D4×D5311yescharacteristic
298D8—22no—
299D8—22no—
300D8—41no—
301D8—41no—
302C2×C8—532no—
303C2×C8—1061no—
304C2×C8—1061no—
305C2×C8—1062no—
306C2×C8—1062no—
307C2×C8—1062no—
308C2×C8—1062no—
309C2×C8C2×C4×D5311yescharacteristic
310C2×C8#1171994511yescharacteristic
311C2×C8D4×D5311yescharacteristic
312C2×C8D4×D5311yescharacteristic
313C2×C8#1171994711yescharacteristic
314C2×C8C2×C22⋊D5311yescharacteristic
315C4⋊C4—22no—
316C4⋊C4—41no—
317C4⋊C4—1062no—
318C4⋊C4—1062no—
319C4⋊C4—2124no—
320C4⋊C4—2121no—
321C4⋊C4—2121no—
322C22⋊C4—2124no—
323C16—41no—
324C16—2121no—
325C23—42no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23—532no—
333C23—532no—
334C23—532no—
335C23—1061no—
336C23—1061no—
337C23—1061no—
338C23—1061no—
339C23D8×D5311yescharacteristic
340D4—42no—
341D4—42no—
342D4—81no—
343D4—81no—
344D4—4248no—
345C2×C4—42no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—532no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4—1062no—
352C2×C4—1062no—
353C2×C4—1062no—
354C2×C4—1062no—
355C2×C4—4248no—
356C2×C4—4241no—
357C2×C4—4241no—
358C2×C4—4241no—
359C2×C4—4241no—
360C2×C4—2122no—
361C2×C4—2122no—
362C2×C4C22⋊C4×D5311yescharacteristic
363C2×C4#1313318511yescharacteristic
364C2×C4#1313319911yescharacteristic
365C2×C4SD16×D5311yescharacteristic
366C2×C4#1313324111yescharacteristic
367C2×C4#1313324611yescharacteristic
368C8—1062no—
369C8#1313317711yescharacteristic
370C8C4×C22⋊D5311yescharacteristic
371C8#1313318811yescharacteristic
372C8#1313319011yescharacteristic
373C22—82no—
374C22—82no—
375C22—82no—
376C22—161no—
377C22—161no—
378C22—161no—
379C22—161no—
380C22—534no—
381C22—1062no—
382C22—1062no—
383C22—1062no—
384C22—1062no—
385C22—1062no—
386C22—1062no—
387C22#102×D5311yescharacteristic
388C22#1681075511yescharacteristic
389C22#1681093511yescharacteristic
390C22D16×D5311yescharacteristic
391C22SD32×D5311yescharacteristic
392C22#1681105711yescharacteristic
393C22#1681106211yescharacteristic
394C4—82no—
395C4—1062no—
396C4—4242no—
397C4—4242no—
398C4#1681049911yescharacteristic
399C4#1681062111yescharacteristic
400C4#1681076011yescharacteristic
401C4#1681094011yescharacteristic
402C2—162no—
403C2—1064no—
404C2#2027502911yescharacteristic
405C2#357×D5311yescharacteristic
406C2#2027642411yescharacteristic
407C2#2027662511yescharacteristic
408C2#2027663011yescharacteristic
409C2#2027689411yescharacteristic
410C2#2027689911yescharacteristic
subgroupG/H
centerC23D8×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC2×C53×#357C2
socleC23×C53C2×D8
prime radicalC2×#16810856C2
cosocleC22×C8×C53C23
hypercenterC2×#357D53
hypocenterC53#12991
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
2. C2×#107×D53C2semidirect
3. C2×#16810424C2semidirect
4. C2×#16810402C2semidirect
5. #20272807C2semidirect
6. #20272806C2semidirect
7. C2×C53×#357C2semidirect
13. #16810856C4semidirect
49. #13133151C2×C4semidirect
50. #13133151D4semidirect
59. C2×D8×C53C2×C4semidirect
66. #107×C53D4semidirect
188. C2×D53#357semidirect
200. C2×#357D53semidirect
231. C53#12991semidirect
241. C2×D8C4×D53semidirect
271. #107C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168108562
C2×#107×D532
C2×#168104242
C2×#168104022
#202728072
#202728062
C2×C53×#3572
#1299153