Finite Group Databasebeta

← #27279106

Subgroups — #27279106

Order 13568.

928 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810540C211yesmaximal, characteristic
2#20274645C212yesmaximal
3C2×#16810349C211yesmaximal, characteristic, complemented
4#20272593C212yesmaximal, complemented
5C53×#2313C211yesmaximal, characteristic, complemented
6C23×D212C2211yescharacteristic
7C22×#11719921—21no—
8C22×#11719921—21no—
9C2×#13133137—24no—
10C2×C22⋊#88322C2212yes—
11#16810540—22no—
12#16810540C414yescomplemented
13C2×#13133124C2211yescharacteristic
14#16810349—22no—
15C22×C4⋊C4×C53C2211yescharacteristic
16C53×#387—24no—
17C53×#381—21no—
18C53×#380C2212yes—
19C4×C22⋊C4×C53—21no—
20C24×D53—21no—
21C22×D212—22no—
22C22×D212—22no—
23C22×D212—21no—
24C22×D212—44no—
25C22×D212C2×C412yes—
26C23×Dic53—21no—
27C22×#88322—22no—
28C2×#11719921—24no—
29C2×#11719921—24no—
30C2×#11719921—44no—
31C2×#11719921—44no—
32#13133137—48no—
33C22⋊#88322—22no—
34C22⋊#88322—22no—
35#13133124—21no—
36#13133124C2×C412yes—
37C23×C4×C53D411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C23×C4×C53C2311yescharacteristic
40C2×C4⋊C4×C53—22no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53C2×C412yes—
43C2×C4⋊C4×C53D414yes—
44C2×C4⋊C4×C53D412yes—
45C2×C22⋊C4×C53—22no—
46C2×C42×C53—22no—
47#95×C53—22no—
48#95×C53—24no—
49#95×C53—22no—
50C24×C53C2×D411yescharacteristic
51C23×D53—24no—
52C23×D53—22no—
53C23×D53—42no—
54C23×D53—42no—
55C23×D53—42no—
56C23×D53—41no—
57C23×D53—41no—
58C23×D53—42no—
59C23×D53—42no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—22no—
64C22×C4×C53—22no—
65C22×C4×C53—21no—
66C22×C4×C53—24no—
67C22×C4×C53—42no—
68C22×C4×C53—42no—
69C22×C4×C53C22⋊C412yes—
70C22×C4×C53C22⋊C412yes—
71C22×C4×C53C22×C411yescharacteristic
72C22×C4×C53C2×D412yes—
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53#4112yes—
75C22×Dic53—22no—
76C22×Dic53—42no—
77C22×Dic53—42no—
78C22×Dic53—42no—
79C2×D212—28no—
80C2×D212—24no—
81C2×D212—44no—
82C2×D212—48no—
83C2×D212—44no—
84C2×D212—44no—
85C2×D212—42no—
86C4⋊C4×C53—24no—
87C4⋊C4×C53—24no—
88C4⋊C4×C53—44no—
89C22⋊C4×C53—42no—
90C22⋊C4×C53—42no—
91C42×C53—82no—
92#11719921—48no—
93#11719921—48no—
94C2×#88322—44no—
95C2×#88322—42no—
96C23×C53—22no—
97C23×C53—22no—
98C23×C53C2×C22⋊C411yescharacteristic
99C23×C53#12011yescharacteristic
100C23×C53#12311yescharacteristic
101C23×C53#13512yes—
102C23×C53AGL(1,ℤ/8)12yes—
103C22×D53—24no—
104C22×D53—42no—
105C22×D53—44no—
106C22×D53—48no—
107C22×D53—48no—
108C22×D53—42no—
109C22×D53—42no—
110C22×D53—42no—
111C22×D53—42no—
112C22×D53—42no—
113C22×D53—44no—
114C22×D53—84no—
115C22×D53—84no—
116C22×D53—82no—
117C22×D53—82no—
118C22×D53—82no—
119C22×D53—82no—
120C2×C4×C53—24no—
121C2×C4×C53—22no—
122C2×C4×C53—22no—
123C2×C4×C53—22no—
124C2×C4×C53—22no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—42no—
128C2×C4×C53—42no—
129C2×C4×C53—42no—
130C2×C4×C53—42no—
131C2×C4×C53—42no—
132C2×C4×C53—44no—
133C2×C4×C53—44no—
134C2×C4×C53—44no—
135C2×C4×C53—81no—
136C2×C4×C53C4×D412yes—
137C2×C4×C53#12112yes—
138C2×Dic53—42no—
139C2×Dic53—42no—
140C2×Dic53—42no—
141C2×Dic53—84no—
142D212—28no—
143D212—48no—
144D212—44no—
145#88322—82no—
146#9018—531nomaximal, Sylow
147C22×C53—24no—
148C22×C53—24no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53#38611yescharacteristic
154C22×C53#41611yescharacteristic
155C22×C53#41811yescharacteristic
156C22×C53#44912yes—
157C22×C53#45412yes—
158C22×C53B(4,𝔽2)12yes—
159C22×C53#48212yes—
160C2×D53—44no—
161C2×D53—44no—
162C2×D53—44no—
163C2×D53—82no—
164C2×D53—82no—
165C2×D53—88no—
166C2×D53—88no—
167C2×D53—82no—
168C2×D53—82no—
169C4×C53—22no—
170C4×C53—22no—
171C4×C53—42no—
172C4×C53—42no—
173C4×C53—84no—
174Dic53—82no—
175C2×#394—531no—
176#2313D5311yescharacteristic, complemented
177#1786—532no—
178C2×#324—531no—
179#1294—532no—
180C2×C53—21no—
181C2×C53—21no—
182C2×C53—22no—
183C2×C53#159714yes—
184C2×C53#187511yescharacteristic
185C2×C53#188811yescharacteristic
186C2×C53#188911yescharacteristic
187D53—84no—
188C23×D4—531no—
189C22×C4⋊C4C2×D5311yescharacteristic
190C22×C22⋊C4—1061no—
191C22×C22⋊C4—1061no—
192C2×#102—1064no—
193C2×C22⋊C8—532no—
194#394—534no—
195#394—1062no—
196#387—24no—
197#381—21no—
198#380C2×D5312yes—
199C4×C22⋊C4—21no—
200C2×#95—531no—
201#324—1062no—
202C53#901811yescharacteristic, complemented, Sylow
203C25—1061no—
204C22×D4—532no—
205C22×D4—1061no—
206C22×D4—1062no—
207C22×D4—1062no—
208C22×D4—2124no—
209C23×C4—1061no—
210C23×C4C22⋊D5311yescharacteristic
211C23×C4C22⋊D5311yescharacteristic
212C23×C4C22×D5311yescharacteristic
213C22×C8—1062no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4C4×D5312yes—
217C2×C4⋊C4D21212yes—
218C2×C4⋊C4C22⋊D5314yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—1064no—
221C2×C22⋊C4—1064no—
222C2×C22⋊C4—2124no—
223C2×C22⋊C4—2124no—
224C2×C42—22no—
225#102—2128no—
226C22⋊C8—1062no—
227C22⋊C8—1062no—
228#95—22no—
229#95—24no—
230#95—22no—
231#95—532no—
232#95—1061no—
233C24—1064no—
234C24—1062no—
235C24—2122no—
236C24—2122no—
237C24—2122no—
238C24—2121no—
239C24—2121no—
240C24—2122no—
241C24—2122no—
242C24C2×C22⋊D5311yescharacteristic
243C2×D4—1068no—
244C2×D4—1064no—
245C2×D4—2124no—
246C2×D4—2124no—
247C2×D4—2128no—
248C2×D4—2124no—
249C2×D4—2122no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—24no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—1062no—
260C22×C4—2122no—
261C22×C4—2122no—
262C22×C4—2122no—
263C22×C4#1171992112yes—
264C22×C4#1171992112yes—
265C22×C4C2×C4×D5311yescharacteristic
266C22×C4C2×D21211yescharacteristic
267C22×C4#1171994512yes—
268C22×C4C2×C22⋊D5312yes—
269C2×C8—2124no—
270C2×C8—2122no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—44no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—2128no—
277C22⋊C4—2128no—
278C42—82no—
279C23—22no—
280C23—22no—
281C23—1064no—
282C23—4242no—
283C23—4244no—
284C23—4244no—
285C23—4242no—
286C23—4242no—
287C23—4242no—
288C23—2122no—
289C23—2124no—
290C23—2122no—
291C23—2122no—
292C23—2128no—
293C23—2128no—
294C23—2122no—
295C23—2124no—
296C23—2122no—
297C23—2122no—
298C23C2×#1171992111yescharacteristic
299C23#1313318611yescharacteristic
300C23#1313320011yescharacteristic
301C23#1313323712yes—
302C23#1313324212yes—
303D4—1068no—
304D4—2128no—
305D4—2124no—
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—81no—
322C2×C4—4244no—
323C2×C4—2122no—
324C2×C4—2122no—
325C2×C4—2122no—
326C2×C4C4×C22⋊D5312yes—
327C2×C4#1313319112yes—
328C8—4242no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—24no—
334C22—24no—
335C22—4242no—
336C22—4248no—
337C22—4242no—
338C22—4242no—
339C22—4242no—
340C22—4248no—
341C22—2124no—
342C22—2124no—
343C22—2124no—
344C22#1681050211yescharacteristic
345C22#1681061711yescharacteristic
346C22#1681062711yescharacteristic
347C22#1681076812yes—
348C22#1681079612yes—
349C22#1681081012yes—
350C22#1681094712yes—
351C4—22no—
352C4—22no—
353C4—42no—
354C4—42no—
355C4—84no—
356C4—4242no—
357C2—21no—
358C2—22no—
359C2—21no—
360C2—4244no—
361C2#2027364414yes—
362C2#2027503711yescharacteristic
363C2#2027511811yescharacteristic
364C2#2027512311yescharacteristic
subgroupG/H
centerC23#13133186
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2313C2
socleC23×C53#120
prime radicalC23×D212C22
cosocleC23×C4×C53C23
hypercenter#2313D53
hypocenterC53#9018
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2313

Complements

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

NH ≅ G/Ntype
3. C2×#16810349C2semidirect
4. #20272593C2semidirect
5. C53×#2313C2semidirect
12. #16810540C4semidirect
176. #2313D53semidirect
202. C53#9018semidirect

Maximal subgroups

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

subgroupindex
C2×#168105402
#202746452
C2×#168103492
#202725932
C53×#23132
#901853