Finite Group Databasebeta

← #27301453

Subgroups — #27301453

Order 13568.

578 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C16×D53C211yesmaximal, characteristic, complemented
2C2×#16810860C211yesmaximal, characteristic
3C2×#16810425C211yesmaximal, characteristic, complemented
4C2×#16810423C211yesmaximal, characteristic, complemented
5#20274741C211yesmaximal, characteristic, complemented
6#20272806C211yesmaximal, characteristic, complemented
7C53×#1380C211yesmaximal, characteristic, complemented
8C22×D424C2211yescharacteristic
9C22×C8×D53C2211yescharacteristic
10C2×#13133192—21no—
11C22×#11719908C2211yescharacteristic
12C2×C16×D53—22no—
13C2×C16×D53—22no—
14C2×C16×D53—22no—
15#16810860—22no—
16#16810860C414yescomplemented
17C2×#13133151C2211yescharacteristic
18C2×#13133137—21no—
19#16810480—21no—
20#16810425—22no—
21#16810423—22no—
22#16810385—21no—
23C22×C16×C53C2211yescharacteristic
24C2×#107×C53C2211yescharacteristic
25C2×#107×C53C2211yescharacteristic
26C22×D212—21no—
27C22×D212—41no—
28C22×C4×D53D411yescharacteristic
29C2×D424—21no—
30C2×D424—21no—
31C2×D424—21no—
32C2×D424—21no—
33C2×D424—21no—
34C2×D424C2×C412yes—
35C22×#88322D411yescharacteristic
36C2×C8×D53—21no—
37C2×C8×D53—21no—
38C2×C8×D53—21no—
39C2×C8×D53—21no—
40C2×C8×D53C2×C412yes—
41C2×C8×D53D412yes—
42#13133192—24no—
43#13133192—42no—
44C2×C4⋊Dic53—21no—
45C2×#11719921—41no—
46C2×#11719921—41no—
47C2×#11719908—21no—
48C2×#11719908—21no—
49C2×#11719908D412yescomplemented
50C16×D53—28no—
51#13133151—21no—
52#13133151—21no—
53#13133137—24no—
54#13133124—41no—
55C22×C8×C53C2311yescharacteristic
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53—21no—
58C2×C16×C53—21no—
59C2×C16×C53—21no—
60C2×C16×C53D412yescomplemented
61#107×C53—21no—
62#107×C53—21no—
63#107×C53—21no—
64#107×C53C2×C412yes—
65C23×D53—41no—
66C23×D53D811yescharacteristic
67C22×C4×C53—41no—
68C22×C4×C53—41no—
69C22×C4×C53C2×D411yescharacteristic
70C22×Dic53—41no—
71C22×Dic53D811yescharacteristic
72C2×D212—22no—
73C2×D212—41no—
74C2×D212—41no—
75C2×D212—41no—
76C2×D212—41no—
77C2×D212—44no—
78C2×D212—41no—
79C2×D212—82no—
80C2×D212—82no—
81C2×C4×D53—21no—
82C2×C4×D53—21no—
83C2×C4×D53—21no—
84C2×C4×D53—21no—
85C2×C4×D53C22⋊C412yes—
86C2×C4×D53SD1612yes—
87C2×C8×C53C22×C411yescharacteristic
88C2×C8×C53C2×D411yescharacteristic
89C2×C8×C53C2×D411yescharacteristic
90C2×C8×C53C2×D411yescharacteristic
91C2×C8×C53#4111yescharacteristic
92C2×C8×C53#4111yescharacteristic
93C4⋊C4×C53—22no—
94C4⋊C4×C53—41no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53—41no—
97#11719921—44no—
98#11719921—82no—
99C4⋊Dic53—41no—
100C4⋊Dic53—41no—
101C2×#88322—21no—
102C2×#88322—21no—
103C2×#88322C22⋊C412yes—
104D424—22no—
105D424—22no—
106D424—41no—
107D424—41no—
108C8×D53—22no—
109C8×D53—22no—
110C8×D53—22no—
111C8×D53—22no—
112C16×C53—22no—
113#11719908—22no—
114C23×C53C2×D811yescharacteristic
115C22×D53—21no—
116C22×D53—21no—
117C22×D53—21no—
118C22×D53—21no—
119C22×D53—42no—
120C22×D53—81no—
121C22×D53—81no—
122C22×D53—81no—
123C22×D53—81no—
124C22×D53—81no—
125C22×D53—81no—
126C22×D53#10212yes—
127C22×D53D1612yes—
128C22×D53SD3212yes—
129C2×C4×C53—42no—
130C2×C4×C53—81no—
131C2×C4×C53—81no—
132C2×C4×C53—81no—
133C2×C4×C53—81no—
134C2×C4×C53—81no—
135C2×C4×C53C2×C22⋊C411yescharacteristic
136C2×C4×C53#12111yescharacteristic
137C2×C4×C53#12111yescharacteristic
138C2×C4×C53C2×SD1611yescharacteristic
139C2×C4×C53#13511yescharacteristic
140C2×C4×C53#13511yescharacteristic
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—81no—
144C2×Dic53—81no—
145C2×Dic53—81no—
146C2×Dic53#10212yes—
147D212—42no—
148D212—42no—
149D212—88no—
150D212—81no—
151D212—81no—
152C4×D53—22no—
153C4×D53—22no—
154C4×D53—22no—
155C4×D53—22no—
156C8×C53C4×D411yescharacteristic
157C8×C53C4×D411yescharacteristic
158C8×C53#12011yescharacteristic
159C8×C53#12311yescharacteristic
160#88322—22no—
161#13059—531nomaximal, Sylow
162C22×C53C2×#10211yescharacteristic
163C22×C53#46611yescharacteristic
164C22×C53#46611yescharacteristic
165C22×C53C2×D1611yescharacteristic
166C22×C53C2×SD3211yescharacteristic
167C22×C53#50811yescharacteristic
168C22×C53#50811yescharacteristic
169C2×D53—22no—
170C2×D53—22no—
171C2×D53—22no—
172C2×D53—22no—
173C2×D53—22no—
174C2×D53—22no—
175C2×D53—82no—
176C2×D53—82no—
177C2×D53—82no—
178C2×D53—161no—
179C2×D53—161no—
180C2×D53—161no—
181C2×D53—161no—
182C2×D53#35714yescomplemented
183C4×C53—82no—
184C4×C53—161no—
185C4×C53#38611yescharacteristic
186C4×C53#41611yescharacteristic
187C4×C53#46511yescharacteristic
188C4×C53#46511yescharacteristic
189Dic53—22no—
190Dic53—161no—
191C23×C16—531no—
192C2×#466—531no—
193C2×#357—531no—
194C2×#357—531no—
195#1808—531no—
196#1380—531no—
197#1380D5311yescharacteristic, complemented
198C2×C53#189311yescharacteristic
199C2×C53C2×#35711yescharacteristic
200C2×C53#213811yescharacteristic
201C2×C53#221511yescharacteristic
202C2×C53#221511yescharacteristic
203C2×C53#221711yescharacteristic
204C2×C53#221711yescharacteristic
205D53—24no—
206D53—162no—
207C22×D8—531no—
208C23×C8—531no—
209C2×#121—1061no—
210C22×C16—531no—
211C22×C16—1062no—
212C22×C16—1062no—
213C22×C16—1062no—
214C22×C16C2×D5311yescharacteristic
215#466—534no—
216#466—1062no—
217C2×#107—531no—
218C2×#107C2×D5311yescharacteristic
219C2×#107C2×D5311yescharacteristic
220C2×#102—1061no—
221#380—1061no—
222#357—1062no—
223#357—1062no—
224#340—1061no—
225C53#1305911yescharacteristic, complemented, Sylow
226C22×D4—1061no—
227C22×D4—2121no—
228C23×C4—531no—
229C2×D8—532no—
230C2×D8—1061no—
231C2×D8—1061no—
232C2×D8—1061no—
233C2×D8—1061no—
234C2×D8—1061no—
235C22×C8—531no—
236C22×C8—532no—
237C22×C8—532no—
238C22×C8—1061no—
239C22×C8—1061no—
240C22×C8—1061no—
241C22×C8—1061no—
242C22×C8C22×D5311yescharacteristic
243#121—1064no—
244#121—2122no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—1061no—
248C2×C22⋊C4—2121no—
249C2×C22⋊C4—2121no—
250C2×C16—21no—
251C2×C16—21no—
252C2×C16—532no—
253C2×C16—1068no—
254C2×C16—1061no—
255C2×C16—1061no—
256C2×C16D21212yescomplemented
257#107—21no—
258#107—21no—
259#107—21no—
260#107—1061no—
261#107—1061no—
262#107C4×D5312yes—
263#102—1064no—
264#95—2121no—
265C24—531no—
266C24—2121no—
267C2×D4—1062no—
268C2×D4—4242no—
269C2×D4—4242no—
270C2×D4—2124no—
271C2×D4—2121no—
272C2×D4—2121no—
273C2×D4—2121no—
274C2×D4—2121no—
275C2×D4—2121no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—531no—
279C22×C4—532no—
280C22×C4—532no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4—1061no—
284C22×C4—1061no—
285C22×C4—2121no—
286C22×C4D4×D5311yescharacteristic
287D8—1062no—
288D8—1062no—
289D8—2121no—
290D8—2121no—
291C2×C8—532no—
292C2×C8—1061no—
293C2×C8—1061no—
294C2×C8—1062no—
295C2×C8—1062no—
296C2×C8—1062no—
297C2×C8—1062no—
298C2×C8C2×C4×D5311yescharacteristic
299C2×C8C2×D21211yescharacteristic
300C2×C8#1171994511yescharacteristic
301C2×C8D4×D5311yescharacteristic
302C2×C8D4×D5311yescharacteristic
303C2×C8#1171994911yescharacteristic
304C4⋊C4—22no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—2121no—
309C4⋊C4—2121no—
310C22⋊C4—4242no—
311C22⋊C4—2124no—
312C16—22no—
313C16—1062no—
314C23—532no—
315C23—532no—
316C23—532no—
317C23—1061no—
318C23—1061no—
319C23—1061no—
320C23—1061no—
321C23—4241no—
322C23—4241no—
323C23—4241no—
324C23—4241no—
325C23—4241no—
326C23—4241no—
327C23—2122no—
328C23D8×D5311yescharacteristic
329D4—4248no—
330D4—4241no—
331D4—4241no—
332D4—2122no—
333D4—2122no—
334C2×C4—42no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—532no—
341C2×C4—1061no—
342C2×C4—1061no—
343C2×C4—1062no—
344C2×C4—1062no—
345C2×C4—1062no—
346C2×C4—1062no—
347C2×C4—4241no—
348C2×C4—4241no—
349C2×C4—4241no—
350C2×C4C22⋊C4×D5311yescharacteristic
351C2×C4#1313319011yescharacteristic
352C2×C4#1313319211yescharacteristic
353C2×C4SD16×D5311yescharacteristic
354C2×C4#1313323811yescharacteristic
355C2×C4#1313324011yescharacteristic
356C8—1062no—
357C8#1313317911yescharacteristic
358C8C4×D21211yescharacteristic
359C8#1313318711yescharacteristic
360C8#1313320211yescharacteristic
361C22—534no—
362C22—8481no—
363C22—8481no—
364C22—8481no—
365C22—8481no—
366C22—1062no—
367C22—1062no—
368C22—1062no—
369C22—1062no—
370C22—1062no—
371C22—1062no—
372C22—4242no—
373C22—4242no—
374C22—4242no—
375C22#102×D5311yescharacteristic
376C22#1681085811yescharacteristic
377C22#1681086011yescharacteristic
378C22D16×D5311yescharacteristic
379C22SD32×D5311yescharacteristic
380C22#1681105411yescharacteristic
381C22#1681105611yescharacteristic
382C4—82no—
383C4—161no—
384C4—8481no—
385C4—1062no—
386C4#1681050111yescharacteristic
387C4#1681061811yescharacteristic
388C4#1681085311yescharacteristic
389C4#1681085511yescharacteristic
390C2—8482no—
391C2—1064no—
392C2#2027514211yescharacteristic
393C2#357×D5311yescharacteristic
394C2#2027642111yescharacteristic
395C2#2027680011yescharacteristic
396C2#2027680211yescharacteristic
397C2#2027681211yescharacteristic
398C2#2027681411yescharacteristic
subgroupG/H
centerC23D8×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#1380C2
socleC23×C53C2×D8
prime radicalC2×#16810860C2
cosocleC22×C8×C53C23
hypercenter#1380D53
hypocenterC53#13059
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
1. C22×C16×D53C2semidirect
3. C2×#16810425C2semidirect
4. C2×#16810423C2semidirect
5. #20274741C2semidirect
6. #20272806C2semidirect
7. C53×#1380C2semidirect
16. #16810860C4semidirect
49. C2×#11719908D4semidirect
60. C2×C16×C53D4semidirect
182. C2×D53#357semidirect
197. #1380D53semidirect
225. C53#13059semidirect
256. C2×C16D212semidirect

Maximal subgroups

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

subgroupindex
C22×C16×D532
C2×#168108602
C2×#168104252
C2×#168104232
#202747412
#202728062
C53×#13802
#1305953