Finite Group Databasebeta

← #27272303

Subgroups — #27272303

Order 13568.

506 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274783C211yesmaximal, characteristic
2#20274477C211yesmaximal, characteristic
3#20272541C212yesmaximal, complemented
4#20272538C212yesmaximal, complemented
5C4×C22⋊Q8×C53C211yesmaximal, characteristic
6C2×C22⋊#88322—21no—
7C2×C22⋊#88322—21no—
8C2×#13133128—21no—
9#16810378—21no—
10#16810378C2212yes—
11#16810378C2212yes—
12C2×C22⋊Q8×C53C411yescharacteristic
13C2×C4×Q8×C53C411yescharacteristic
14C22×C42×C53C2211yescharacteristic
15C53×#391C2211yescharacteristic
16C53×#389C2211yescharacteristic
17C53×#387—21no—
18C53×#385—21no—
19C53×#380C411yescharacteristic
20C4×C4⋊C4×C53—21no—
21C4×C4⋊C4×C53C411yescharacteristic
22C4×C22⋊C4×C53—21no—
23C22×#88322—22no—
24C22×#88322—22no—
25C22⋊#88322—24no—
26C22⋊#88322—42no—
27#13133128—42no—
28C22×Q8×C53C2×C411yescharacteristic
29C23×C4×C53C2×C411yescharacteristic
30C23×C4×C53D411yescharacteristic
31C23×C4×C53D411yescharacteristic
32C22⋊Q8×C53—22no—
33C22⋊Q8×C53—22no—
34C4×Q8×C53—42no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C22⋊C4×C53—21no—
42C2×C22⋊C4×C53—21no—
43C2×C42×C53—21no—
44C2×C42×C53—21no—
45C2×C42×C53—42no—
46C2×C42×C53C2×C411yescharacteristic
47C2×C42×C53D411yescharacteristic
48C2×C42×C53D411yescharacteristic
49C2×C42×C53C2311yescharacteristic
50#95×C53—21no—
51#95×C53—21no—
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55C24×C53C22⋊C411yescharacteristic
56C2×Q8×C53—21no—
57C2×Q8×C53—41no—
58C2×Q8×C53—41no—
59C2×Q8×C53M1612yes—
60C22×C4×C53—22no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—22no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—22no—
67C22×C4×C53—22no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—41no—
71C22×C4×C53—41no—
72C22×C4×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22×C411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C4⋊C4×C53—24no—
81C4⋊C4×C53—21no—
82C4⋊C4×C53—42no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—42no—
86C4⋊C4×C53M1612yes—
87C22⋊C4×C53—24no—
88C22⋊C4×C53—42no—
89C42×C53—22no—
90C42×C53—22no—
91C42×C53—41no—
92C42×C53—41no—
93C42×C53—42no—
94C42×C53—42no—
95C2×#88322—24no—
96C2×#88322—42no—
97C2×#88322—42no—
98C2×#88322—42no—
99C2×#88322—42no—
100C2×#88322—42no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53#10412yes—
108C23×C53C2×C22⋊C411yescharacteristic
109Q8×C53—42no—
110Q8×C53—42no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—22no—
121C2×C4×C53—22no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—44no—
128C2×C4×C53—44no—
129C2×C4×C53—42no—
130C2×C4×C53—42no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53#9912yes—
142C2×C4×C53C2×M1611yescharacteristic
143C2×C4×C53C2×M1611yescharacteristic
144C2×C4×C53#13512yes—
145C2×C4×C53#13712yes—
146#88322—44no—
147#88322—82no—
148#7623—531nomaximal, Sylow
149C22×C53—22no—
150C22×C53—22no—
151C22×C53—22no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53#40711yescharacteristic
157C22×C53C2×#9911yescharacteristic
158C22×C53#41311yescharacteristic
159C22×C53#41611yescharacteristic
160C22×C53#41911yescharacteristic
161C22×C53C2×#10411yescharacteristic
162C22×C53#42111yescharacteristic
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—22no—
166C4×C53—22no—
167C4×C53—22no—
168C4×C53—41no—
169C4×C53—42no—
170C4×C53—42no—
171C4×C53—41no—
172C4×C53—81no—
173C4×C53—81no—
174C4×C22⋊Q8D5311yescharacteristic
175#1817—531no—
176#1749—531no—
177#1272—532no—
178#1271—532no—
179C2×C53—42no—
180C2×C53#146511yescharacteristic
181C2×C53#148811yescharacteristic
182C2×C53#149111yescharacteristic
183C2×C53#150111yescharacteristic
184C2×C53#151111yescharacteristic
185C2×C53#153211yescharacteristic
186C2×C53#155411yescharacteristic
187C2×C22⋊Q8Dic5311yescharacteristic
188C2×C4×Q8Dic5311yescharacteristic
189C22×C42C2×D5311yescharacteristic
190C2×C22⋊C8—1061no—
191C2×C22⋊C8—1061no—
192C2×#97—1061no—
193#391C2×D5311yescharacteristic
194#389C2×D5311yescharacteristic
195#387—21no—
196#385—21no—
197#380Dic5311yescharacteristic
198C4×C4⋊C4—21no—
199C4×C4⋊C4Dic5311yescharacteristic
200C4×C22⋊C4—21no—
201#336—532no—
202#336—532no—
203#336—1061no—
204C53#762311yescharacteristic, complemented, Sylow
205C22×Q8C2×Dic5311yescharacteristic
206C23×C4C2×Dic5311yescharacteristic
207C23×C4C22⋊D5311yescharacteristic
208C23×C4C22⋊D5311yescharacteristic
209C22×C8—1062no—
210C22×C8—1062no—
211C22⋊Q8—22no—
212C22⋊Q8—22no—
213C4×Q8—42no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4C2×Dic5311yescharacteristic
218C2×C4⋊C4C2×Dic5311yescharacteristic
219C2×C4⋊C4C2×Dic5311yescharacteristic
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—42no—
225C2×C42C2×Dic5311yescharacteristic
226C2×C42C22⋊D5311yescharacteristic
227C2×C42C22⋊D5311yescharacteristic
228C2×C42C22×D5311yescharacteristic
229C22⋊C8—1064no—
230C22⋊C8—2122no—
231#97—2122no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237C24C22⋊Dic5311yescharacteristic
238C2×Q8—21no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8#1171991712yes—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4C22⋊Dic5311yescharacteristic
257C22×C4C22⋊Dic5311yescharacteristic
258C22×C4C22⋊Dic5311yescharacteristic
259C22×C4C22×Dic5311yescharacteristic
260C22×C4C2×C22⋊D5311yescharacteristic
261C22×C4C2×C22⋊D5311yescharacteristic
262C2×C8—1064no—
263C2×C8—2122no—
264C2×C8—2122no—
265C2×C8—2122no—
266C2×C8—2122no—
267C2×C8—2122no—
268C4⋊C4—24no—
269C4⋊C4—21no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4#1171991712yes—
275C22⋊C4—24no—
276C22⋊C4—42no—
277C42—22no—
278C42—22no—
279C42—41no—
280C42—41no—
281C42—42no—
282C42—42no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23#1313314412yes—
290C23C2×C22⋊Dic5311yescharacteristic
291Q8—42no—
292Q8—42no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4#1313313212yes—
324C2×C4C2×#1171991711yescharacteristic
325C2×C4C2×#1171991711yescharacteristic
326C2×C4#1313323712yes—
327C2×C4#1313324812yes—
328C8—4242no—
329C8—2124no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22#1681058711yescharacteristic
338C22C2×#1313313211yescharacteristic
339C22#1681060711yescharacteristic
340C22#1681061911yescharacteristic
341C22#1681063211yescharacteristic
342C22C2×#1313314411yescharacteristic
343C22#1681064011yescharacteristic
344C4—22no—
345C4—22no—
346C4—22no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—42no—
352C4—42no—
353C4—81no—
354C4—81no—
355C2—42no—
356C2#2027305311yescharacteristic
357C2#2027314911yescharacteristic
358C2#2027316211yescharacteristic
359C2#2027320611yescharacteristic
360C2#2027324811yescharacteristic
361C2#2027333511yescharacteristic
362C2#2027342511yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC4×C22⋊Q8×C53C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenterC4×C22⋊Q8D53
hypocenterC53#7623
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20272541C2semidirect
4. #20272538C2semidirect
204. C53#7623semidirect

Maximal subgroups

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

subgroupindex
#202747832
#202744772
#202725412
#202725382
C4×C22⋊Q8×C532
#762353