Finite Group Databasebeta

← #27279643

Subgroups — #27279643

Order 13568.

560 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283744C211yesmaximal, characteristic
2#20273033C211yesmaximal, characteristic
3C2×#16810349C212yesmaximal, complemented
4#20272563C212yesmaximal, complemented
5C53×#2322C211yesmaximal, characteristic
6#16810695—24no—
7#16810680—21no—
8#16810587—21no—
9C2×C22⋊#88322C2212yes—
10C2×C22⋊#88322C2212yes—
11#16810349—24no—
12#16810349—21no—
13#16810347—21no—
14C2×#123×C53C411yescharacteristic
15C2×C4×D4×C53C2211yescharacteristic
16C22×C4⋊C4×C53C2211yescharacteristic
17C53×#389—21no—
18C53×#388—21no—
19C53×#387—21no—
20C53×#386C411yescharacteristic
21C53×#385—21no—
22C53×#385C411yescharacteristic
23C53×#381C411yescharacteristic
24C4×C22⋊C4×C53C2211yescharacteristic
25C2×#11719917—22no—
26C22×#88322—22no—
27C22×#88322—22no—
28C4⋊#88322—22no—
29C22⋊#88322—22no—
30C22⋊#88322—22no—
31C22⋊#88322—22no—
32C22⋊#88322—22no—
33C22⋊#88322D414yes—
34#13133128—22no—
35C22×D4×C53C2×C411yescharacteristic
36C23×C4×C53C2×C411yescharacteristic
37C23×C4×C53C2×C411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C23×C4×C53C2311yescharacteristic
40#123×C53—22no—
41#123×C53—22no—
42C4×D4×C53—22no—
43C4×D4×C53—22no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—22no—
47C2×C4⋊C4×C53—21no—
48C2×C4⋊C4×C53—21no—
49C2×C4⋊C4×C53—22no—
50C2×C4⋊C4×C53D411yescharacteristic
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53—21no—
53C2×C22⋊C4×C53C2×C411yescharacteristic
54C2×C22⋊C4×C53C2×C411yescharacteristic
55C2×C22⋊C4×C53C2×C411yescharacteristic
56C2×C22⋊C4×C53D411yescharacteristic
57C2×C42×C53—21no—
58C2×C42×C53D411yescharacteristic
59#95×C53—21no—
60#95×C53—21no—
61#95×C53—21no—
62#95×C53—21no—
63#95×C53—21no—
64C24×C53C22⋊C411yescharacteristic
65C24×C53C22×C411yescharacteristic
66C2×D4×C53—22no—
67C2×D4×C53—21no—
68C2×D4×C53—21no—
69C2×D4×C53—22no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—22no—
73C22×C4×C53—21no—
74C22×C4×C53—22no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53—21no—
79C22×C4×C53—22no—
80C22×C4×C53—81no—
81C22×C4×C53—81no—
82C22×C4×C53—81no—
83C22×C4×C53C22⋊C411yescharacteristic
84C22×C4×C53C22⋊C411yescharacteristic
85C22×C4×C53C22⋊C411yescharacteristic
86C22×C4×C53M1612yescomplemented
87C22×C4×C53C2×D412yes—
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53#4112yes—
91C4⋊C4×C53—24no—
92C4⋊C4×C53—22no—
93C4⋊C4×C53—41no—
94C4⋊C4×C53—42no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53—81no—
97C22⋊C4×C53—24no—
98C22⋊C4×C53—22no—
99C22⋊C4×C53—21no—
100C22⋊C4×C53—22no—
101C22⋊C4×C53—22no—
102C22⋊C4×C53—81no—
103C22⋊C4×C53M1612yescomplemented
104C42×C53—21no—
105C42×C53—21no—
106C42×C53—81no—
107#11719917—82no—
108C2×#88322—24no—
109C2×#88322—42no—
110C2×#88322—82no—
111C2×#88322—82no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53C2×C22⋊C411yescharacteristic
120C23×C53C4×D412yes—
121C23×C53C2×M1611yescharacteristic
122C23×C53C2×M1611yescharacteristic
123C23×C53#13112yes—
124D4×C53—42no—
125D4×C53—42no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—22no—
130C2×C4×C53—22no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—24no—
134C2×C4×C53—41no—
135C2×C4×C53—42no—
136C2×C4×C53—41no—
137C2×C4×C53—42no—
138C2×C4×C53—42no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—42no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—81no—
153C2×C4×C53—82no—
154C2×C4×C53#12011yescharacteristic
155C2×C4×C53#12112yes—
156C2×C4×C53#12311yescharacteristic
157#88322—44no—
158#88322—82no—
159#9110—531nomaximal, Sylow
160C22×C53—21no—
161C22×C53—22no—
162C22×C53—21no—
163C22×C53—22no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—42no—
167C22×C53—42no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53—81no—
172C22×C53#38611yescharacteristic
173C22×C53#40711yescharacteristic
174C22×C53#40811yescharacteristic
175C22×C53#43512yes—
176C22×C53#43612yes—
177C4×C53—21no—
178C4×C53—22no—
179C4×C53—21no—
180C4×C53—42no—
181C4×C53—42no—
182C4×C53—41no—
183C4×C53—41no—
184C4×C53—41no—
185C4×C53—81no—
186C4×C53—81no—
187#2973—531no—
188#2322D5311yescharacteristic
189#1459—531no—
190C2×#324—532no—
191#1282—532no—
192C2×C53—21no—
193C2×C53—21no—
194C2×C53—22no—
195C2×C53—22no—
196C2×C53—81no—
197C2×C53#187011yescharacteristic
198C2×C53#191311yescharacteristic
199C2×C53#191511yescharacteristic
200C2×#123Dic5311yescharacteristic
201C2×C4×D4C2×D5311yescharacteristic
202C22×C4⋊C4C2×D5311yescharacteristic
203#435—1064no—
204#432—1061no—
205#407—1061no—
206C2×C22⋊C8—532no—
207C2×C22⋊C8—532no—
208#389—21no—
209#388—21no—
210#387—21no—
211#386Dic5311yescharacteristic
212#385—21no—
213#385Dic5311yescharacteristic
214#381Dic5311yescharacteristic
215C4×C22⋊C4C2×D5311yescharacteristic
216#324—1064no—
217#324—1061no—
218#323—1061no—
219C53#911011yescharacteristic, complemented, Sylow
220C22×D4C2×Dic5311yescharacteristic
221C23×C4C2×Dic5311yescharacteristic
222C23×C4C2×Dic5311yescharacteristic
223C23×C4C22⋊D5311yescharacteristic
224C23×C4C22×D5311yescharacteristic
225C2×M16—1062no—
226C22×C8—1062no—
227C22×C8—1062no—
228#123—22no—
229#123—22no—
230C4×D4—22no—
231C4×D4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4C22⋊D5311yescharacteristic
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4C2×Dic5311yescharacteristic
242C2×C22⋊C4C2×Dic5311yescharacteristic
243C2×C22⋊C4C2×Dic5311yescharacteristic
244C2×C22⋊C4C22⋊D5311yescharacteristic
245C2×C42—21no—
246C2×C42C22⋊D5311yescharacteristic
247C4⋊C8—1062no—
248C22⋊C8—534no—
249C22⋊C8—1062no—
250C22⋊C8—1062no—
251C22⋊C8—1062no—
252C22⋊C8—1062no—
253#97—1062no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259C24C22⋊Dic5311yescharacteristic
260C24C22×Dic5311yescharacteristic
261C2×D4—22no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—81no—
276C22×C4—81no—
277C22×C4—81no—
278C22×C4#1171991712yescomplemented
279C22×C4C22⋊Dic5311yescharacteristic
280C22×C4C22⋊Dic5311yescharacteristic
281C22×C4C22⋊Dic5311yescharacteristic
282C22×C4D4×D5312yes—
283C22×C4#1171994712yes—
284C22×C4C2×C22⋊D5311yescharacteristic
285C22×C4C2×C22⋊D5311yescharacteristic
286M16—4242no—
287C2×C8—1064no—
288C2×C8—4242no—
289C2×C8—4242no—
290C2×C8—2122no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—81no—
297C22⋊C4—24no—
298C22⋊C4—22no—
299C22⋊C4—21no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—81no—
303C22⋊C4#1171991712yescomplemented
304C42—21no—
305C42—21no—
306C42—81no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23C2×C22⋊Dic5311yescharacteristic
315C23D4×Dic5312yes—
316C23C2×#1171991711yescharacteristic
317C23C2×#1171991711yescharacteristic
318C23#1313322412yes—
319D4—42no—
320D4—42no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—24no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—81no—
348C2×C4—82no—
349C2×C4#1313318511yescharacteristic
350C2×C4#1313318912yes—
351C2×C4#1313319811yescharacteristic
352C8—4242no—
353C8—2124no—
354C22—21no—
355C22—22no—
356C22—21no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—42no—
361C22—42no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—81no—
366C22#1681050311yescharacteristic
367C22#1681058711yescharacteristic
368C22#1681059011yescharacteristic
369C22#1681069512yes—
370C22#1681070112yes—
371C4—22no—
372C4—21no—
373C4—21no—
374C4—42no—
375C4—41no—
376C4—41no—
377C4—42no—
378C4—41no—
379C4—81no—
380C4—81no—
381C2—21no—
382C2—21no—
383C2—22no—
384C2—22no—
385C2—81no—
386C2#2027501311yescharacteristic
387C2#2027525811yescharacteristic
388C2#2027526811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2322C2
socleC22×C53#386
prime radicalC22×D4×C53C2×C4
cosocleC23×C4×C53C23
hypercenter#2322D53
hypocenterC53#9110
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2322

Complements

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

NH ≅ G/Ntype
3. C2×#16810349C2semidirect
4. #20272563C2semidirect
86. C22×C4×C53M16semidirect
103. C22⋊C4×C53M16semidirect
219. C53#9110semidirect
278. C22×C4#11719917semidirect
303. C22⋊C4#11719917semidirect

Maximal subgroups

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

subgroupindex
#202837442
#202730332
C2×#168103492
#202725632
C53×#23222
#911053