Finite Group Databasebeta

← #27310860

Subgroups — #27310860

Order 13568.

1830 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810534C211yesmaximal, characteristic
2C2×#16810514C213yesmaximal
3#20272986C211yesmaximal, characteristic, complemented
4#20272978C213yesmaximal, complemented
5#20272957C213yesmaximal
6#20272942C213yesmaximal, complemented
7C22×C4×Q8×C53C211yesmaximal, characteristic, complemented
8C22×#11719919C2213yes—
9C22×#11719921C2211yescharacteristic
10C22×#11719921C2213yes—
11C22×#11719921C2213yes—
12#16810534C418yescomplemented
13#16810514—212no—
14C2×#13133124C2213yes—
15C2×#13133124C2213yes—
16C2×#13133124C2213yes—
17C2×#13133124C2216yes—
18C2×#13133124C2213yes—
19C23×Q8×C53C2211yescharacteristic
20C2×C4×Q8×C53—212no—
21C22×C4⋊C4×C53C2213yes—
22C22×C42×C53C2213yes—
23C24×D53C2311yescharacteristic
24C23×Dic53C2311yescharacteristic
25C23×Dic53C2313yes—
26C23×Dic53C2313yes—
27C2×#11719919—26no—
28C2×#11719919—46no—
29C2×#11719921—26no—
30C2×#11719921—26no—
31C2×#11719921—26no—
32C2×#11719921—26no—
33C2×#11719921—46no—
34C2×#11719921C2×C4112yes—
35#13133124—26no—
36#13133124—26no—
37#13133124—26no—
38#13133124—46no—
39#13133124C2×C4112yes—
40C22×Q8×C53—26no—
41C22×Q8×C53—26no—
42C22×Q8×C53C2×C414yes—
43C23×C4×C53C2313yes—
44C23×C4×C53C2313yes—
45C23×C4×C53C2311yescharacteristic
46C4×Q8×C53—232no—
47C2×C4⋊C4×C53—212no—
48C2×C4⋊C4×C53—26no—
49C2×C42×C53—212no—
50C2×C42×C53D4112yes—
51C24×C53C2411yescharacteristic
52C23×D53—23no—
53C23×D53—23no—
54C23×D53—23no—
55C23×D53—21no—
56C23×D53—21no—
57C23×D53—23no—
58C23×D53C22×C412yes—
59C2×Q8×C53—28no—
60C2×Q8×C53—224no—
61C2×Q8×C53—224no—
62C22×C4×C53—26no—
63C22×C4×C53—26no—
64C22×C4×C53—21no—
65C22×C4×C53—23no—
66C22×C4×C53—26no—
67C22×C4×C53—23no—
68C22×C4×C53—26no—
69C22×C4×C53—23no—
70C22×C4×C53C22×C416yes—
71C22×C4×C53C2×D416yes—
72C22×C4×C53C2×D416yes—
73C22×C4×C53C2×D416yes—
74C22×C4×C53#4116yes—
75C22×Dic53—23no—
76C22×Dic53—23no—
77C22×Dic53—26no—
78C22×Dic53—26no—
79C22×Dic53—23no—
80C22×Dic53—23no—
81C22×Dic53—26no—
82C22×Dic53—26no—
83C22×Dic53—23no—
84C22×Dic53—23no—
85C22×Dic53—23no—
86C22×Dic53—21no—
87C22×Dic53C22×C416yes—
88C4⋊C4×C53—224no—
89C42×C53—224no—
90#11719921—224no—
91#11719921—412no—
92#11719921—44no—
93#11719919—412no—
94C23×C53C23×C411yescharacteristic
95C23×C53C22×D413yes—
96C23×C53C2×#4113yes—
97C23×C53H(5,𝔽2)13yes—
98C23×C532−1+411yescharacteristic
99C23×C532−1+411yescharacteristic
100C23×C532−1+413yes—
101C22×D53—26no—
102C22×D53—22no—
103C22×D53—26no—
104C22×D53—41no—
105C22×D53—43no—
106C22×D53—43no—
107C22×D53—43no—
108C22×D53—43no—
109C22×D53—43no—
110C22×D53—43no—
111C22×D53—43no—
112C22×D53—46no—
113Q8×C53—232no—
114C2×C4×C53—212no—
115C2×C4×C53—26no—
116C2×C4×C53—26no—
117C2×C4×C53—212no—
118C2×C4×C53—26no—
119C2×C4×C53—26no—
120C2×C4×C53—26no—
121C2×C4×C53—26no—
122C2×C4×C53—26no—
123C2×C4×C53—46no—
124C2×C4×C53—46no—
125C2×C4×C53C4×D4112yes—
126C2×C4×C53#12014yes—
127C2×Dic53—26no—
128C2×Dic53—212no—
129C2×Dic53—43no—
130C2×Dic53—43no—
131C2×Dic53—46no—
132C2×Dic53—41no—
133C2×Dic53—43no—
134C2×Dic53—43no—
135C2×Dic53—43no—
136C2×Dic53—43no—
137C2×Dic53—43no—
138C2×Dic53—46no—
139C2×Dic53—46no—
140#14301—531nomaximal, Sylow
141C22×C53C2×C4×D413yes—
142C22×C53#51911yescharacteristic
143C22×C53#52013yes—
144C22×C53C2×#12011yescharacteristic
145C22×C53#52513yes—
146C22×C53#53513yes—
147C22×C53#53613yes—
148C22×C53#53613yes—
149C22×C53#53713yes—
150C22×C53#54113yes—
151C22×C53#54213yes—
152C22×C53#54216yes—
153C2×D53—46no—
154C2×D53—46no—
155C2×D53—42no—
156C2×D53—81no—
157C2×D53—81no—
158C2×D53—83no—
159C2×D53—83no—
160C4×C53—24no—
161C4×C53—212no—
162C4×C53—46no—
163Dic53—46no—
164Dic53—81no—
165Dic53—83no—
166C22×C4×Q8D5311yescharacteristic, complemented
167C2×#393—531no—
168C2×#388—533no—
169#1446—531no—
170#1444—533no—
171#1439—533no—
172#1434—533no—
173C2×C53C4×#12011yescharacteristic
174C2×C53#232313yes—
175C2×C53#232613yes—
176C2×C53#261411yescharacteristic
177C2×C53#261511yescharacteristic
178C2×C53#263213yes—
179C2×C53#263713yes—
180D53—82no—
181C23×Q8C2×D5311yescharacteristic
182C2×C4×Q8—212no—
183C22×C4⋊C4—533no—
184C22×C4⋊C4C2×D5313yes—
185C22×C22⋊C4—533no—
186C22×C22⋊C4—531no—
187C22×C22⋊C4—533no—
188C22×C42C2×D5313yes—
189#393—538no—
190#388—10612no—
191C2×#95—533no—
192C2×#95—533no—
193C2×#95—533no—
194C2×#95—533no—
195C2×#95—536no—
196C53#1430111yescharacteristic, complemented, Sylow
197C25—531no—
198C22×Q8—26no—
199C22×Q8—26no—
200C22×Q8C4×D5314yes—
201C23×C4—533no—
202C23×C4—531no—
203C23×C4—533no—
204C23×C4C22×D5313yes—
205C23×C4C22×D5313yes—
206C23×C4C22×D5311yescharacteristic
207C4×Q8—232no—
208C2×C4⋊C4—212no—
209C2×C4⋊C4—26no—
210C2×C4⋊C4—1066no—
211C2×C4⋊C4—2126no—
212C2×C22⋊C4—5312no—
213C2×C22⋊C4—1066no—
214C2×C22⋊C4—1066no—
215C2×C22⋊C4—1066no—
216C2×C22⋊C4—1066no—
217C2×C22⋊C4—2126no—
218C2×C42—212no—
219C2×C42C22⋊D53112yes—
220#95—5312no—
221#95—1066no—
222#95—1066no—
223#95—1066no—
224#95—2126no—
225C24—532no—
226C24—1063no—
227C24—1063no—
228C24—1063no—
229C24—1061no—
230C24—1061no—
231C24—1063no—
232C24C23×D5311yescharacteristic
233C2×Q8—28no—
234C2×Q8—224no—
235C2×Q8—224no—
236C22×C4—26no—
237C22×C4—26no—
238C22×C4—21no—
239C22×C4—23no—
240C22×C4—26no—
241C22×C4—23no—
242C22×C4—26no—
243C22×C4—23no—
244C22×C4—536no—
245C22×C4—1063no—
246C22×C4—1063no—
247C22×C4—1066no—
248C22×C4—1066no—
249C22×C4—1063no—
250C22×C4—1063no—
251C22×C4—1066no—
252C22×C4—1066no—
253C22×C4—1063no—
254C22×C4—1061no—
255C22×C4—1063no—
256C22×C4—1063no—
257C22×C4C2×C4×D5316yes—
258C22×C4#1171994516yes—
259C22×C4C2×C22⋊D5316yes—
260C22×C4C2×C22⋊D5316yes—
261C22×C4C2×C22⋊D5316yes—
262C4⋊C4—224no—
263C4⋊C4—21212no—
264C22⋊C4—10624no—
265C22⋊C4—21212no—
266C22⋊C4—2124no—
267C42—224no—
268C23—1066no—
269C23—1062no—
270C23—1066no—
271C23—2121no—
272C23—2123no—
273C23—2123no—
274C23—2123no—
275C23—2123no—
276C23—2123no—
277C23—2123no—
278C23—2126no—
279C23—2123no—
280C23C22×C4×D5311yescharacteristic
281C23C22×C22⋊D5313yes—
282C23C2×#1171994513yes—
283C23#1313326313yes—
284C23#1313326411yescharacteristic
285C23#1313326511yescharacteristic
286C23#1313326513yes—
287Q8—232no—
288C2×C4—212no—
289C2×C4—26no—
290C2×C4—212no—
291C2×C4—26no—
292C2×C4—26no—
293C2×C4—26no—
294C2×C4—26no—
295C2×C4—26no—
296C2×C4—26no—
297C2×C4—46no—
298C2×C4—46no—
299C2×C4—1066no—
300C2×C4—10612no—
301C2×C4—2121no—
302C2×C4—2123no—
303C2×C4—2123no—
304C2×C4—2123no—
305C2×C4—2123no—
306C2×C4—2123no—
307C2×C4—2126no—
308C2×C4—2126no—
309C2×C4—2123no—
310C2×C4—2123no—
311C2×C4—2126no—
312C2×C4C4×C22⋊D53112yes—
313C2×C4#1313318614yes—
314C22—4243no—
315C22—4241no—
316C22—4241no—
317C22—4243no—
318C22—2126no—
319C22—2126no—
320C22—2122no—
321C22C2×C4×C22⋊D5313yes—
322C22#1681109911yescharacteristic
323C22#1681110513yes—
324C22C2×#1313318611yescharacteristic
325C22#1681113413yes—
326C22#1681118713yes—
327C22#1681119713yes—
328C22#1681119713yes—
329C22#1681120113yes—
330C22#1681123113yes—
331C22#1681124516yes—
332C22#1681124513yes—
333C4—24no—
334C4—212no—
335C4—46no—
336C4—4241no—
337C4—4243no—
338C4—2126no—
339C2—4242no—
340C2C4×#1313318611yescharacteristic
341C2#2027739113yes—
342C2#2027742213yes—
343C2#2028043411yescharacteristic
344C2#2028043911yescharacteristic
345C2#2028057513yes—
346C2#2028062113yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC22×C4×Q8×C53C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenterC22×C4×Q8D53
hypocenterC53#14301
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
3. #20272986C2semidirect
4. #20272978C2semidirect
6. #20272942C2semidirect
7. C22×C4×Q8×C53C2semidirect
12. #16810534C4semidirect
166. C22×C4×Q8D53semidirect
196. C53#14301semidirect

Maximal subgroups

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

subgroupindex
C2×#168105342
C2×#168105142
#202729862
#202729782
#202729572
#202729422
C22×C4×Q8×C532
#1430153