Finite Group Databasebeta

← #27272703

Subgroups — #27272703

Order 13568.

464 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283054C211yesmaximal, characteristic
2#20274695C211yesmaximal, characteristic
3#20272540C212yesmaximal, complemented
4#20272537C212yesmaximal, complemented
5C53×#1797C211yesmaximal, characteristic
6C2×#13133196C411yescharacteristic
7C2×C4⋊#88322—21no—
8C2×C22⋊#88322—21no—
9#16810529C411yescharacteristic
10#16810529C2211yescharacteristic
11#16810521C411yescharacteristic
12#16810521C2211yescharacteristic
13#16810507—21no—
14#16810507—21no—
15#16810496—21no—
16#16810486—21no—
17#16810483C411yescharacteristic
18#16810378C2212yes—
19C22×C42×C53C2211yescharacteristic
20C2×C4⋊C8×C53—21no—
21C2×C22⋊C8×C53—21no—
22C53×#336C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C22×#88322—22no—
25#13133196—42no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34C4⋊#88322—42no—
35C22⋊#88322—42no—
36#13133124—21no—
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43C23×C4×C53C2×C411yescharacteristic
44C23×C4×C53D411yescharacteristic
45C23×C4×C53D411yescharacteristic
46C22×C8×C53—22no—
47C2×C42×C53—42no—
48C2×C42×C53C2×C411yescharacteristic
49C2×C42×C53D411yescharacteristic
50C2×C42×C53D411yescharacteristic
51C2×C42×C53C2311yescharacteristic
52C4⋊C8×C53—42no—
53C22⋊C8×C53—42no—
54C24×C53C22⋊C411yescharacteristic
55C22×C4×C53—22no—
56C22×C4×C53—22no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—41no—
60C22×C4×C53—41no—
61C22×C4×C53—41no—
62C22×C4×C53—41no—
63C22×C4×C53—41no—
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53SD1612yes—
68C22×C4×C53Q1612yes—
69C22×C4×C53C22×C411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76C2×Dic106—22no—
77C2×Dic106—41no—
78C2×Dic106—41no—
79C2×C8×C53—42no—
80C2×C8×C53—42no—
81C2×C8×C53—42no—
82C42×C53—22no—
83C42×C53—22no—
84C42×C53—41no—
85C42×C53—41no—
86C22⋊Dic53—42no—
87C22⋊Dic53—42no—
88C4⋊Dic53—22no—
89C4⋊Dic53—41no—
90C4⋊Dic53—41no—
91#11719919—42no—
92#11719919—42no—
93#11719919—41no—
94#11719919—41no—
95C2×#88322—42no—
96C2×#88322—42no—
97C2×#88322—42no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53#10312yes—
105C23×C53C2×C22⋊C411yescharacteristic
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
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—42no—
120C2×C4×C53—42no—
121C2×C4×C53—44no—
122C2×C4×C53—42no—
123C2×C4×C53—42no—
124C2×C4×C53#9912yes—
125C2×C4×C53#10312yes—
126C2×C4×C53#13112yes—
127C2×C4×C53C2×SD1611yescharacteristic
128C2×C4×C53C2×Q1611yescharacteristic
129C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
130C2×C4×C53#13711yescharacteristic
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143Dic106—82no—
144C8×C53—82no—
145#88322—82no—
146#7715—531nomaximal, Sylow
147C22×C53—22no—
148C22×C53—22no—
149C22×C53—22no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53#40811yescharacteristic
155C22×C53C2×#9911yescharacteristic
156C22×C53#41311yescharacteristic
157C22×C53C2×#10311yescharacteristic
158C22×C53#41711yescharacteristic
159C22×C53#42111yescharacteristic
160C22×C53#42111yescharacteristic
161C4×C53—22no—
162C4×C53—22no—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—21no—
166C4×C53—22no—
167C4×C53—21no—
168C4×C53—41no—
169Dic53—81no—
170Dic53—81no—
171Dic53—81no—
172Dic53—81no—
173#2855—531no—
174#1797—531no—
175#1797D5311yescharacteristic
176#1272—532no—
177#1271—532no—
178C2×C53—42no—
179C2×C53#146811yescharacteristic
180C2×C53#148111yescharacteristic
181C2×C53#149511yescharacteristic
182C2×C53#151311yescharacteristic
183C2×C53#152011yescharacteristic
184C2×C53#153511yescharacteristic
185C2×C53#155011yescharacteristic
186C2×C22⋊Q8—531no—
187C22×C42C2×D5311yescharacteristic
188C2×C4⋊C8—21no—
189C2×C4⋊C8—1061no—
190C2×C22⋊C8—21no—
191C2×C22⋊C8—1061no—
192#391—531no—
193#391—531no—
194#389—531no—
195#389—531no—
196#387—1061no—
197#387—1061no—
198#385—1061no—
199#381—1061no—
200#380—531no—
201#336—532no—
202#336C2×D5312yes—
203C53#771511yescharacteristic, complemented, Sylow
204C22×Q8—531no—
205C23×C4C4×D5311yescharacteristic
206C23×C4D21211yescharacteristic
207C23×C4C22⋊D5311yescharacteristic
208C22×C8—22no—
209C22×C8—1062no—
210C22⋊Q8—2122no—
211C2×C4⋊C4—531no—
212C2×C4⋊C4—531no—
213C2×C4⋊C4—531no—
214C2×C4⋊C4—1061no—
215C2×C4⋊C4—1061no—
216C2×C4⋊C4—1061no—
217C2×C22⋊C4—1061no—
218C2×C22⋊C4—1061no—
219C2×C42—42no—
220C2×C42C4×D5311yescharacteristic
221C2×C42D21211yescharacteristic
222C2×C42C22⋊D5311yescharacteristic
223C2×C42C22×D5311yescharacteristic
224C4⋊C8—42no—
225C4⋊C8—2122no—
226C22⋊C8—42no—
227C22⋊C8—2122no—
228#95—1061no—
229#95—1061no—
230#95—1061no—
231#95—1061no—
232#95—1061no—
233#95—1061no—
234#95—1061no—
235C24#1171992111yescharacteristic
236C2×Q8—1062no—
237C2×Q8—2121no—
238C2×Q8—2121no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—1061no—
249C22×C4—1061no—
250C22×C4—1061no—
251C22×C4—1061no—
252C22×C4#1171992111yescharacteristic
253C22×C4#1171992111yescharacteristic
254C22×C4#1171992111yescharacteristic
255C22×C4#1171992312yes—
256C22×C4#1171992512yes—
257C22×C4C2×C4×D5311yescharacteristic
258C22×C4C2×D21211yescharacteristic
259C22×C4C2×C22⋊D5311yescharacteristic
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—2122no—
264C2×C8—2122no—
265C2×C8—2122no—
266C4⋊C4—1062no—
267C4⋊C4—2121no—
268C4⋊C4—2121no—
269C4⋊C4—2122no—
270C4⋊C4—2122no—
271C4⋊C4—2121no—
272C4⋊C4—2121no—
273C22⋊C4—2122no—
274C22⋊C4—2122no—
275C42—22no—
276C42—22no—
277C42—41no—
278C42—41no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23#1313314012yes—
286C23C2×#1171992111yescharacteristic
287Q8—4242no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
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—42no—
302C2×C4—42no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4—2121no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4—2121no—
313C2×C4—2121no—
314C2×C4—2121no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4—2121no—
318C2×C4#1313313112yes—
319C2×C4#1313314012yes—
320C2×C4#1313322312yes—
321C2×C4C2×#1171992311yescharacteristic
322C2×C4C2×#1171992511yescharacteristic
323C2×C4#1313324511yescharacteristic
324C2×C4#1313325011yescharacteristic
325C8—82no—
326C8—4242no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22#1681058911yescharacteristic
335C22C2×#1313313111yescharacteristic
336C22#1681060511yescharacteristic
337C22C2×#1313314011yescharacteristic
338C22#1681062511yescharacteristic
339C22#1681063911yescharacteristic
340C22#1681064111yescharacteristic
341C4—21no—
342C4—21no—
343C4—21no—
344C4—22no—
345C4—21no—
346C4—22no—
347C4—22no—
348C4—41no—
349C4—4241no—
350C4—4241no—
351C4—4241no—
352C4—4241no—
353C2—42no—
354C2#2027306111yescharacteristic
355C2#2027311711yescharacteristic
356C2#2027318111yescharacteristic
357C2#2027325911yescharacteristic
358C2#2027328611yescharacteristic
359C2#2027334711yescharacteristic
360C2#2027340611yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1797C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1797D53
hypocenterC53#7715
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
3. #20272540C2semidirect
4. #20272537C2semidirect
203. C53#7715semidirect

Maximal subgroups

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

subgroupindex
#202830542
#202746952
#202725402
#202725372
C53×#17972
#771553