Finite Group Databasebeta

← #27301032

Subgroups — #27301032

Order 13568.

582 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810856C211yesmaximal, characteristic
2C2×#16810660C211yesmaximal, 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×#13133175—21no—
11C22×#11719908C2211yescharacteristic
12#16810856—22no—
13#16810856C414yescomplemented
14#16810660—22no—
15#16810660—22no—
16C2×#13133151C2211yescharacteristic
17C2×#13133151C2211yescharacteristic
18C2×#13133148—21no—
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×D53D411yescharacteristic
28C22×#88322D411yescharacteristic
29C2×C8×D53—21no—
30C2×C8×D53—21no—
31C2×C8×D53—21no—
32C2×C8×D53—21no—
33C2×C8×D53C2×C412yes—
34C2×C8×D53D412yes—
35#13133188—24no—
36#13133188—42no—
37#13133175—24no—
38#13133175—42no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53—21no—
41C2×C22⋊Dic53—41no—
42C2×#11719921—41no—
43C2×C4×Dic53—41no—
44C2×#11719908—21no—
45C2×#11719908—21no—
46C2×#11719908—21no—
47#13133151—21no—
48#13133151—21no—
49#13133151C2×C412yes—
50#13133151D412yescomplemented
51#13133148—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×C412yes—
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×D53C22⋊C412yes—
88C2×C4×D53SD1612yes—
89D8×C53—22no—
90D8×C53—22no—
91D8×C53—41no—
92D8×C53—41no—
93C2×C8×C53C22×C411yescharacteristic
94C2×C8×C53C2×D411yescharacteristic
95C2×C8×C53C2×D411yescharacteristic
96C2×C8×C53C2×D411yescharacteristic
97C2×C8×C53#4111yescharacteristic
98C2×C8×C53#4111yescharacteristic
99C4⋊C4×C53—22no—
100C4⋊C4×C53—41no—
101C22⋊Dic53—44no—
102#11719921—44no—
103C4⋊Dic53—22no—
104C4⋊Dic53—22no—
105C4⋊Dic53—41no—
106C4⋊Dic53—41no—
107C4×Dic53—44no—
108C2×#88322—21no—
109C2×#88322—21no—
110C2×#88322C22⋊C412yes—
111C8×D53—22no—
112C8×D53—22no—
113C8×D53—22no—
114C8×D53—22no—
115C16×C53—41no—
116#11719908—41no—
117C23×C53—42no—
118C23×C53—81no—
119C23×C53—81no—
120C23×C53—81no—
121C23×C53—81no—
122C23×C53—81no—
123C23×C53—81no—
124C23×C53C2×D811yescharacteristic
125C22×D53—21no—
126C22×D53—21no—
127C22×D53—21no—
128C22×D53—21no—
129C22×D53—21no—
130C22×D53—21no—
131C22×D53#10212yes—
132D4×C53—42no—
133D4×C53—42no—
134D4×C53—81no—
135D4×C53—81no—
136C2×C4×C53—42no—
137C2×C4×C53—81no—
138C2×C4×C53—81no—
139C2×C4×C53C2×C22⋊C411yescharacteristic
140C2×C4×C53#12011yescharacteristic
141C2×C4×C53#12311yescharacteristic
142C2×C4×C53C2×SD1611yescharacteristic
143C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C53#13711yescharacteristic
145C22⋊D53—88no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—42no—
149C2×Dic53—42no—
150C2×Dic53—81no—
151C2×Dic53—81no—
152C2×Dic53—81no—
153C2×Dic53—81no—
154C2×Dic53#10212yes—
155C4×D53—22no—
156C4×D53—22no—
157C4×D53—22no—
158C4×D53—22no—
159C8×C53C4×D411yescharacteristic
160C8×C53C4×D411yescharacteristic
161C8×C53#12111yescharacteristic
162C8×C53#12111yescharacteristic
163#88322—22no—
164#12992—531nomaximal, Sylow
165C22×C53—82no—
166C22×C53—82no—
167C22×C53—82no—
168C22×C53—161no—
169C22×C53—161no—
170C22×C53—161no—
171C22×C53—161no—
172C22×C53C2×#10211yescharacteristic
173C22×C53#44711yescharacteristic
174C22×C53#48011yescharacteristic
175C22×C53#50811yescharacteristic
176C22×C53#50811yescharacteristic
177C22×C53#50911yescharacteristic
178C22×C53#50911yescharacteristic
179C2×D53—22no—
180C2×D53—22no—
181C2×D53—22no—
182C2×D53—41no—
183C2×D53—41no—
184C2×D53—41no—
185C2×D53—41no—
186C4×C53—82no—
187C4×C53#38611yescharacteristic
188C4×C53#41711yescharacteristic
189C4×C53#44811yescharacteristic
190C4×C53#48111yescharacteristic
191Dic53—22no—
192Dic53—82no—
193Dic53—82no—
194C2×#466—531no—
195C2×#427—531no—
196C2×#357—531no—
197C2×#357D5311yescharacteristic, complemented
198C2×C22⋊C16—531no—
199#1380—531no—
200#1380—531no—
201C2×C53—162no—
202C2×C53#187411yescharacteristic
203C2×C53#213811yescharacteristic
204C2×C53#214011yescharacteristic
205C2×C53#218611yescharacteristic
206C2×C53#218611yescharacteristic
207C2×C53#223411yescharacteristic
208C2×C53#223411yescharacteristic
209D53—42no—
210C22×D8C2×D5311yescharacteristic
211C23×C8—531no—
212C2×#121—1061no—
213C2×#117—1061no—
214C22×C16—531no—
215C22×C16C2×D5311yescharacteristic
216#466—534no—
217#466—1062no—
218#427—1062no—
219#427—1062no—
220C2×#107—531no—
221C2×#107—531no—
222C2×#107C2×D5311yescharacteristic
223C2×#106—1061no—
224C2×#102—1061no—
225#357—22no—
226#357—1062no—
227C22⋊C16—1062no—
228C53#1299211yescharacteristic, complemented, Sylow
229C22×D4—21no—
230C22×D4—2121no—
231C23×C4—531no—
232C2×D8—21no—
233C2×D8—21no—
234C2×D8—21no—
235C2×D8—21no—
236C2×D8—21no—
237C2×D8C4×D5312yes—
238C22×C8—531no—
239C22×C8—532no—
240C22×C8—532no—
241C22×C8—1061no—
242C22×C8—1061no—
243C22×C8—1061no—
244C22×C8—1061no—
245C22×C8C22×D5311yescharacteristic
246#121—1064no—
247#121—2122no—
248#117—1064no—
249#117—2122no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—1061no—
252C2×C4⋊C4—1061no—
253C2×C22⋊C4—2121no—
254C2×C22⋊C4—2121no—
255C2×C42—2121no—
256C2×C16—21no—
257C2×C16—21no—
258C2×C16—21no—
259C2×C16—1061no—
260C2×C16—1061no—
261C2×C16—1061no—
262#107—21no—
263#107—532no—
264#107—532no—
265#107—1061no—
266#107—1061no—
267#107C22⋊D5312yescomplemented
268#106—1064no—
269#102—1064no—
270C24—41no—
271C24—531no—
272C2×D4—22no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—4242no—
279C2×D4—4242no—
280C2×D4—2124no—
281C22×C4—41no—
282C22×C4—531no—
283C22×C4—532no—
284C22×C4—532no—
285C22×C4—1061no—
286C22×C4—1061no—
287C22×C4—1061no—
288C22×C4—1061no—
289C22×C4—2121no—
290C22×C4—2121no—
291C22×C4D4×D5311yescharacteristic
292D8—22no—
293D8—22no—
294D8—41no—
295D8—41no—
296C2×C8—532no—
297C2×C8—1061no—
298C2×C8—1061no—
299C2×C8—1062no—
300C2×C8—1062no—
301C2×C8—1062no—
302C2×C8—1062no—
303C2×C8C2×C4×D5311yescharacteristic
304C2×C8#1171994511yescharacteristic
305C2×C8D4×D5311yescharacteristic
306C2×C8D4×D5311yescharacteristic
307C2×C8#1171994711yescharacteristic
308C2×C8C2×C22⋊D5311yescharacteristic
309C4⋊C4—22no—
310C4⋊C4—41no—
311C4⋊C4—1062no—
312C4⋊C4—1062no—
313C4⋊C4—2121no—
314C4⋊C4—2121no—
315C22⋊C4—2124no—
316C22⋊C4—2124no—
317C42—2124no—
318C16—41no—
319C16—2121no—
320C23—42no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—81no—
326C23—81no—
327C23—532no—
328C23—1061no—
329C23—1061no—
330C23—1061no—
331C23—1061no—
332C23—1061no—
333C23—1061no—
334C23D8×D5311yescharacteristic
335D4—42no—
336D4—42no—
337D4—81no—
338D4—81no—
339D4—4248no—
340C2×C4—42no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—532no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—1062no—
347C2×C4—1062no—
348C2×C4—1062no—
349C2×C4—1062no—
350C2×C4—4241no—
351C2×C4—4241no—
352C2×C4—4241no—
353C2×C4—4241no—
354C2×C4—2122no—
355C2×C4—2122no—
356C2×C4C22⋊C4×D5311yescharacteristic
357C2×C4#1313318511yescharacteristic
358C2×C4#1313319911yescharacteristic
359C2×C4SD16×D5311yescharacteristic
360C2×C4#1313324111yescharacteristic
361C2×C4#1313324611yescharacteristic
362C8—1062no—
363C8#1313317711yescharacteristic
364C8C4×C22⋊D5311yescharacteristic
365C8#1313318811yescharacteristic
366C8#1313319011yescharacteristic
367C22—82no—
368C22—82no—
369C22—82no—
370C22—161no—
371C22—161no—
372C22—161no—
373C22—161no—
374C22—1062no—
375C22—1062no—
376C22—1062no—
377C22—2121no—
378C22—2121no—
379C22—2121no—
380C22—2121no—
381C22#102×D5311yescharacteristic
382C22#1681075511yescharacteristic
383C22#1681093511yescharacteristic
384C22#1681105211yescharacteristic
385C22#1681105411yescharacteristic
386C22#1681105711yescharacteristic
387C22#1681106011yescharacteristic
388C4—82no—
389C4—1062no—
390C4—4242no—
391C4—4242no—
392C4#1681049911yescharacteristic
393C4#1681062111yescharacteristic
394C4#1681076011yescharacteristic
395C4#1681094011yescharacteristic
396C2—162no—
397C2—2122no—
398C2#2027502911yescharacteristic
399C2#2027641911yescharacteristic
400C2#2027642911yescharacteristic
401C2#2027663511yescharacteristic
402C2#2027663711yescharacteristic
403C2#2027690411yescharacteristic
404C2#2027690511yescharacteristic
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#12992
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×#16810660C2semidirect
3. C2×#16810424C2semidirect
4. C2×#16810402C2semidirect
5. #20272807C2semidirect
6. #20272806C2semidirect
7. C2×C53×#357C2semidirect
13. #16810856C4semidirect
50. #13133151D4semidirect
66. #107×C53D4semidirect
197. C2×#357D53semidirect
228. C53#12992semidirect
267. #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×#168106602
C2×#168104242
C2×#168104022
#202728072
#202728062
C2×C53×#3572
#1299253