Finite Group Databasebeta

← #27306804

Subgroups — #27306804

Order 13568.

2230 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810491C211yesmaximal, characteristic, complemented
2C2×#16810469C211yesmaximal, characteristic, complemented
3#20272983C212yesmaximal, complemented
4#20272956C212yesmaximal
5#20272952C214yesmaximal, complemented
6#20272938C214yesmaximal, complemented
7C22×C4×D4×C53C211yesmaximal, characteristic
8C22×C22⋊Dic53C2218yes—
9C22×C4×Dic53C2212yes—
10#16810491—24no—
11#16810469—24no—
12C2×#13133124C2212yes—
13C2×#13133124C2212yes—
14C2×#13133124C2212yes—
15C2×#13133124C2212yes—
16C2×#13133124C2212yes—
17C2×#13133124C2218yes—
18C23×D4×C53C2211yescharacteristic
19C24×C4×C53C2212yes—
20C2×C4×D4×C53—24no—
21C2×C4×D4×C53—24no—
22C2×C4×D4×C53C418yescomplemented
23C22×C4⋊C4×C53C2211yescharacteristic
24C22×C22⋊C4×C53C2212yes—
25C22×C42×C53C2211yescharacteristic
26C23×Dic53C2314yes—
27C23×Dic53C2314yes—
28C2×C22⋊Dic53—216no—
29C2×C22⋊Dic53—216no—
30C2×C22⋊Dic53—48no—
31C2×C4×Dic53—24no—
32C2×C4×Dic53—28no—
33#13133124—24no—
34#13133124—24no—
35#13133124—24no—
36#13133124—24no—
37#13133124—24no—
38#13133124—48no—
39C25×C53C2312yes—
40C22×D4×C53—28no—
41C22×D4×C53—22no—
42C22×D4×C53—22no—
43C22×D4×C53C2×C414yes—
44C23×C4×C53—24no—
45C23×C4×C53—24no—
46C23×C4×C53—48no—
47C23×C4×C53C2×C418yes—
48C23×C4×C53C2312yes—
49C23×C4×C53C2311yescharacteristic
50C23×C4×C53C2311yescharacteristic
51C23×C4×C53C2311yescharacteristic
52C4×D4×C53—232no—
53C2×C4⋊C4×C53—22no—
54C2×C4⋊C4×C53—22no—
55C2×C4⋊C4×C53C2×C414yes—
56C2×C22⋊C4×C53—24no—
57C2×C22⋊C4×C53—24no—
58C2×C22⋊C4×C53C2×C418yes—
59C2×C42×C53C2×C414yes—
60C2×C42×C53D414yes—
61C2×C42×C53D414yes—
62C24×C53—22no—
63C24×C53—22no—
64C24×C53—24no—
65C24×C53—28no—
66C24×C53—28no—
67C24×C53C22×C414yes—
68C24×C53#4118yes—
69C24×C53C2411yescharacteristic
70C2×D4×C53—216no—
71C2×D4×C53—216no—
72C2×D4×C53—216no—
73C2×D4×C53—28no—
74C22×C4×C53—24no—
75C22×C4×C53—24no—
76C22×C4×C53—24no—
77C22×C4×C53—22no—
78C22×C4×C53—22no—
79C22×C4×C53—28no—
80C22×C4×C53—216no—
81C22×C4×C53—416no—
82C22×C4×C53—416no—
83C22×C4×C53—416no—
84C22×C4×C53C22×C414yes—
85C22×C4×C53C22×C412yes—
86C22×C4×C53C22×C412yes—
87C22×C4×C53C22×C412yes—
88C22×C4×C53C2×D412yes—
89C22×C4×C53C2×D412yes—
90C22×C4×C53C2×D412yes—
91C22×C4×C53C2×D412yes—
92C22×C4×C53C2×D412yes—
93C22×C4×C53C2×D412yes—
94C22×Dic53—24no—
95C22×Dic53—24no—
96C22×Dic53—28no—
97C22×Dic53—28no—
98C22×Dic53—24no—
99C22×Dic53—28no—
100C22×Dic53—28no—
101C22×Dic53—24no—
102C22×Dic53—24no—
103C22×Dic53#4118yes—
104C4⋊C4×C53—28no—
105C22⋊C4×C53—216no—
106C42×C53C22⋊C4116yescomplemented
107C22⋊Dic53—432no—
108C4×Dic53—216no—
109C23×C53—24no—
110C23×C53—24no—
111C23×C53—28no—
112C23×C53—28no—
113C23×C53—28no—
114C23×C53—28no—
115C23×C53—216no—
116C23×C53—216no—
117C23×C53—44no—
118C23×C53—44no—
119C23×C53—44no—
120C23×C53—44no—
121C23×C53—48no—
122C23×C53—48no—
123C23×C53#11718yes—
124C23×C53C23×C411yescharacteristic
125C23×C53C22×D411yescharacteristic
126C23×C53C22×D411yescharacteristic
127C23×C53C2×#4114yes—
128C23×C53C2×#4114yes—
129C23×C53H(5,𝔽2)14yes—
130D4×C53—232no—
131C2×C4×C53—24no—
132C2×C4×C53—24no—
133C2×C4×C53—24no—
134C2×C4×C53—24no—
135C2×C4×C53—24no—
136C2×C4×C53—24no—
137C2×C4×C53—24no—
138C2×C4×C53—24no—
139C2×C4×C53—24no—
140C2×C4×C53—24no—
141C2×C4×C53—28no—
142C2×C4×C53—432no—
143C2×C4×C53—44no—
144C2×C4×C53—44no—
145C2×C4×C53C2×C22⋊C414yes—
146C2×C4×C53C2×C22⋊C414yes—
147C2×C4×C53C2×C22⋊C414yes—
148C2×Dic53—28no—
149C2×Dic53—216no—
150C2×Dic53—44no—
151C2×Dic53—44no—
152C2×Dic53—48no—
153C2×Dic53—44no—
154C2×Dic53—44no—
155C2×Dic53—44no—
156C2×Dic53—48no—
157C2×Dic53—48no—
158#13959—531nomaximal, Sylow
159C22×C53—216no—
160C22×C53—28no—
161C22×C53—44no—
162C22×C53—416no—
163C22×C53—416no—
164C22×C53—48no—
165C22×C53—44no—
166C22×C53C22×C22⋊C411yescharacteristic
167C22×C53C2×#11712yes—
168C22×C53C4×#4112yes—
169C22×C53#51812yes—
170C22×C53#52512yes—
171C22×C53#52512yes—
172C22×C53#52512yes—
173C22×C53#53212yes—
174C22×C53#53512yes—
175C22×C53#53512yes—
176C22×C53#55118yes—
177C22×C53#55318yes—
178C4×C53—24no—
179C4×C53—24no—
180C4×C53—24no—
181C4×C53—44no—
182Dic53—48no—
183Dic53—84no—
184C22×C4×D4D5311yescharacteristic
185C2×#383—531no—
186C2×#376—531no—
187#1445—532no—
188#1438—532no—
189#1437—534no—
190C22⋊#95—534no—
191C2×C53—48no—
192C2×C53#228011yescharacteristic
193C2×C53#229711yescharacteristic
194C2×C53#230911yescharacteristic
195C2×C53#235314yes—
196C2×C53#259814yes—
197C2×C53#260314yes—
198C23×D4C2×D5311yescharacteristic
199C24×C4C2×D5312yes—
200C2×C4×D4—24no—
201C2×C4×D4—24no—
202C2×C4×D4Dic5318yescomplemented
203C22×C4⋊C4C2×D5311yescharacteristic
204C22×C22⋊C4—538no—
205C22×C22⋊C4C2×D5312yes—
206C22×C42—532no—
207C22×C42C2×D5311yescharacteristic
208#383—1064no—
209#376—1064no—
210C2×#95—532no—
211C2×#95—532no—
212C2×#95—532no—
213C2×#95—532no—
214C2×#95—532no—
215C2×#95—538no—
216C53#1395911yescharacteristic, complemented, Sylow
217C25C22×D5312yes—
218C22×D4—28no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4C2×Dic5314yes—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—48no—
225C23×C4—534no—
226C23×C4—534no—
227C23×C4C2×Dic5318yes—
228C23×C4C22×D5312yes—
229C23×C4C22×D5311yescharacteristic
230C23×C4C22×D5311yescharacteristic
231C23×C4C22×D5311yescharacteristic
232C4×D4—232no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4C2×Dic5314yes—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—10616no—
239C2×C22⋊C4—10616no—
240C2×C22⋊C4—2128no—
241C2×C22⋊C4C2×Dic5318yes—
242C2×C42—1068no—
243C2×C42—1064no—
244C2×C42C2×Dic5314yes—
245C2×C42C22⋊D5314yes—
246C2×C42C22⋊D5314yes—
247#95—1064no—
248#95—1064no—
249#95—1064no—
250#95—1064no—
251#95—1064no—
252#95—2128no—
253C24—22no—
254C24—22no—
255C24—28no—
256C24—24no—
257C24—28no—
258C24#1171994518yes—
259C24C22×Dic5314yes—
260C24C23×D5311yescharacteristic
261C2×D4—216no—
262C2×D4—216no—
263C2×D4—216no—
264C2×D4—28no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—28no—
271C22×C4—216no—
272C22×C4—416no—
273C22×C4—416no—
274C22×C4—416no—
275C22×C4—538no—
276C22×C4—1064no—
277C22×C4—1064no—
278C22×C4—1068no—
279C22×C4—1068no—
280C22×C4—1064no—
281C22×C4—1068no—
282C22×C4—1068no—
283C22×C4—1064no—
284C22×C4—1064no—
285C22×C4C22×Dic5314yes—
286C22×C4C22×Dic5312yes—
287C22×C4C22×Dic5312yes—
288C22×C4C22×Dic5312yes—
289C22×C4C2×C22⋊D5312yes—
290C22×C4C2×C22⋊D5312yes—
291C22×C4C2×C22⋊D5312yes—
292C22×C4C2×C22⋊D5312yes—
293C22×C4C2×C22⋊D5312yes—
294C22×C4C2×C22⋊D5312yes—
295C4⋊C4—28no—
296C22⋊C4—216no—
297C22⋊C4—21232no—
298C42—10616no—
299C42C22⋊Dic53116yescomplemented
300C23—216no—
301C23—216no—
302C23—24no—
303C23—24no—
304C23—28no—
305C23—28no—
306C23—28no—
307C23—28no—
308C23—48no—
309C23—48no—
310C23—44no—
311C23—44no—
312C23—44no—
313C23—44no—
314C23#1313317618yes—
315C23C23×Dic5311yescharacteristic
316C23C22×C22⋊D5311yescharacteristic
317C23C22×C22⋊D5311yescharacteristic
318C23#41×D5314yes—
319C23C2×#1171994514yes—
320C23#1313326214yes—
321D4—232no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—28no—
333C2×C4—432no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—10616no—
337C2×C4—1068no—
338C2×C4—2128no—
339C2×C4—2128no—
340C2×C4—2124no—
341C2×C4—2124no—
342C2×C4—2124no—
343C2×C4—2128no—
344C2×C4—2124no—
345C2×C4—2124no—
346C2×C4C2×C22⋊Dic5314yes—
347C2×C4C2×C22⋊Dic5314yes—
348C2×C4C2×C22⋊Dic5314yes—
349C22—28no—
350C22—216no—
351C22—416no—
352C22—44no—
353C22—48no—
354C22—416no—
355C22—44no—
356C22C22×C22⋊Dic5311yescharacteristic
357C22C2×#1313317612yes—
358C22#41×Dic5312yes—
359C22#1681109812yes—
360C22#1681113012yes—
361C22#1681113412yes—
362C22#1681113412yes—
363C22#1681116912yes—
364C22#1681119012yes—
365C22#1681119012yes—
366C22#1681130218yes—
367C22#1681132218yes—
368C4—24no—
369C4—24no—
370C4—24no—
371C4—44no—
372C4—4244no—
373C4—2128no—
374C2—48no—
375C2#2027709811yescharacteristic
376C2#2027720611yescharacteristic
377C2#2027729711yescharacteristic
378C2#2027765514yes—
379C2#2028027514yes—
380C2#2028033614yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC22×C4×D4×C53C2
socle, cosocleC24×C53C24
prime radicalC23×D4×C53C22
hypercenterC22×C4×D4D53
hypocenterC53#13959
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16810491C2semidirect
2. C2×#16810469C2semidirect
3. #20272983C2semidirect
5. #20272952C2semidirect
6. #20272938C2semidirect
22. C2×C4×D4×C53C4semidirect
106. C42×C53C22⋊C4semidirect
202. C2×C4×D4Dic53semidirect
216. C53#13959semidirect
299. C42C22⋊Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168104912
C2×#168104692
#202729832
#202729562
#202729522
#202729382
C22×C4×D4×C532
#1395953