Finite Group Databasebeta

← #27284274

Subgroups — #27284274

Order 13568.

592 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283067C211yesmaximal, characteristic
2#20274695C211yesmaximal, characteristic
3#20272594C214yesmaximal
4C53×#1434C211yesmaximal, characteristic
5C2×#13133217C411yescharacteristic, complemented
6C2×#13133196—21no—
7C2×C4⋊#88322—21no—
8C2×C22⋊#88322C2212yes—
9#16810529—21no—
10#16810521—21no—
11#16810521C412yescomplemented
12#16810493C411yescharacteristic, complemented
13#16810483—21no—
14#16810483C2212yes—
15#16810378—21no—
16C22×C42×C53C2211yescharacteristic
17C2×#95×C53—21no—
18C2×#95×C53—21no—
19C2×#95×C53C2212yes—
20C22×Dic106—21no—
21C22×#88322—22no—
22#13133217—42no—
23#13133196—44no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53C2×C412yes—
28C2×C4⋊Dic53C2×C412yes—
29C2×#11719919—22no—
30C2×C22⋊Dic53—22no—
31C4⋊#88322—42no—
32C22⋊#88322—24no—
33#13133124—22no—
34#13133124—22no—
35C23×C4×C53—22no—
36C23×C4×C53D411yescharacteristic
37C23×C4×C53D411yescharacteristic
38C23×C4×C53C2311yescharacteristic
39C2×C42×C53—42no—
40C2×C42×C53C2×C411yescharacteristic
41C2×C42×C53C2×C411yescharacteristic
42C2×C42×C53D411yescharacteristic
43C2×C42×C53D411yescharacteristic
44#95×C53—22no—
45#95×C53—42no—
46#95×C53—42no—
47#95×C53Q814yescomplemented
48C24×C53C2×D411yescharacteristic
49C22×C4×C53—22no—
50C22×C4×C53—22no—
51C22×C4×C53—24no—
52C22×C4×C53—41no—
53C22×C4×C53—41no—
54C22×C4×C53—41no—
55C22×C4×C53—41no—
56C22×C4×C53—41no—
57C22×C4×C53—42no—
58C22×C4×C53—42no—
59C22×C4×C53—42no—
60C22×C4×C53—42no—
61C22×C4×C53—42no—
62C22×C4×C53—42no—
63C22×C4×C53C22⋊C411yescharacteristic
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×Q812yes—
72C22×C4×C53#4112yes—
73C22×Dic53—22no—
74C22×Dic53—22no—
75C2×Dic106—42no—
76C2×Dic106—42no—
77C2×Dic106—42no—
78C42×C53—22no—
79C42×C53—22no—
80C42×C53—41no—
81C42×C53—41no—
82C22⋊Dic53—44no—
83C4⋊Dic53—42no—
84C4⋊Dic53—42no—
85C4⋊Dic53—42no—
86C4⋊Dic53—42no—
87C4⋊Dic53—42no—
88#11719919—44no—
89C2×#88322—42no—
90C2×#88322—42no—
91C2×#88322—42no—
92C23×C53—21no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53C2×C22⋊C411yescharacteristic
97C23×C53C22⋊Q812yes—
98C23×C53C2×SD1611yescharacteristic
99C23×C53C2×Q1611yescharacteristic
100C23×C53AGL(1,ℤ/8)12yes—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
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—42no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—44no—
119C2×C4×C53—44no—
120C2×C4×C53—44no—
121C2×C4×C53—44no—
122C2×C4×C53—82no—
123C2×C4×C53—82no—
124C2×C4×C53—82no—
125C2×C4×C53—82no—
126C2×C4×C53#9912yes—
127C2×C4×C53#10312yes—
128C2×C4×C53#10312yes—
129C2×C4×C53C4×Q812yes—
130C2×C4×C53#12411yescharacteristic
131C2×C4×C53C4⋊Q811yescharacteristic
132C2×Dic53—42no—
133C2×Dic53—42no—
134C2×Dic53—42no—
135C2×Dic53—42no—
136C2×Dic53—42no—
137C2×Dic53—42no—
138Dic106—84no—
139#88322—82no—
140#9911—531nomaximal, Sylow
141C22×C53—22no—
142C22×C53—22no—
143C22×C53—22no—
144C22×C53—22no—
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—21no—
149C22×C53#39111yescharacteristic
150C22×C53C2×#9911yescharacteristic
151C22×C53#41011yescharacteristic
152C22×C53C2×#10311yescharacteristic
153C22×C53#41811yescharacteristic
154C22×C53#42111yescharacteristic
155C22×C53#42111yescharacteristic
156C22×C53#47612yes—
157C22×C53#47712yes—
158C4×C53—21no—
159C4×C53—22no—
160C4×C53—21no—
161C4×C53—22no—
162C4×C53—22no—
163C4×C53—22no—
164C4×C53—41no—
165C4×C53—84no—
166Dic53—82no—
167Dic53—82no—
168#2857—531no—
169#1797—531no—
170#1434D5311yescharacteristic
171#1294—534no—
172C2×C53—22no—
173C2×C53—22no—
174C2×C53#151011yescharacteristic
175C2×C53#151411yescharacteristic
176C2×C53#151811yescharacteristic
177C2×C53#152011yescharacteristic
178C2×C53#198711yescharacteristic
179C2×C53#199511yescharacteristic
180C2×C53#199611yescharacteristic
181C2×C4⋊Q8—531no—
182C2×C22⋊Q8—1061no—
183C22×C42C2×D5311yescharacteristic
184C2×C4⋊C8—1061no—
185C2×C22⋊C8—532no—
186#391—1061no—
187#389—532no—
188#389—1061no—
189#384—531no—
190#380—532no—
191#380—1061no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95C2×D5312yes—
195#336—1061no—
196C53#991111yescharacteristic, complemented, Sylow
197C22×Q8—1061no—
198C23×C4—22no—
199C23×C4C22⋊D5311yescharacteristic
200C23×C4C22⋊D5311yescharacteristic
201C23×C4C22×D5311yescharacteristic
202C22×C8—1062no—
203C4⋊Q8—2122no—
204C22⋊Q8—2124no—
205C2×C4⋊C4—532no—
206C2×C4⋊C4—532no—
207C2×C4⋊C4—1062no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4—1061no—
210C2×C4⋊C4—1061no—
211C2×C22⋊C4—1062no—
212C2×C42—42no—
213C2×C42C4×D5311yescharacteristic
214C2×C42C4×D5311yescharacteristic
215C2×C42D21211yescharacteristic
216C2×C42D21211yescharacteristic
217C4⋊C8—2122no—
218C22⋊C8—1064no—
219#95—22no—
220#95—42no—
221#95—42no—
222#95—1062no—
223#95—1062no—
224#95Dic10614yescomplemented
225C24C2×C22⋊D5311yescharacteristic
226C2×Q8—2122no—
227C2×Q8—2122no—
228C2×Q8—2122no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—1062no—
244C22×C4—1062no—
245C22×C4#1171992111yescharacteristic
246C22×C4#1171992111yescharacteristic
247C22×C4#1171992111yescharacteristic
248C22×C4#1171992111yescharacteristic
249C22×C4#1171992312yes—
250C22×C4#1171992512yes—
251C22×C4C2×Dic10612yes—
252C22×C4C2×C4×D5311yescharacteristic
253C22×C4C2×D21211yescharacteristic
254C22×C4#1171994512yes—
255C2×C8—2122no—
256C2×C8—2122no—
257C2×C8—2122no—
258C4⋊C4—2122no—
259C4⋊C4—2122no—
260C4⋊C4—2124no—
261C4⋊C4—2122no—
262C4⋊C4—2122no—
263C4⋊C4—2122no—
264C22⋊C4—2124no—
265C42—22no—
266C42—22no—
267C42—41no—
268C42—41no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23C2×#1171992111yescharacteristic
274C23#1313319612yes—
275C23C2×#1171992311yescharacteristic
276C23C2×#1171992511yescharacteristic
277C23#1313324212yes—
278Q8—4244no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—2122no—
305C2×C4—2122no—
306C2×C4—2122no—
307C2×C4—2122no—
308C2×C4—2122no—
309C2×C4—2122no—
310C2×C4#1313313112yes—
311C2×C4#1313314012yes—
312C2×C4#1313314012yes—
313C2×C4C4×Dic10612yes—
314C2×C4#1313320511yescharacteristic
315C2×C4#1313321711yescharacteristic
316C8—4242no—
317C22—22no—
318C22—22no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—22no—
324C22—22no—
325C22#1681052911yescharacteristic
326C22C2×#1313313111yescharacteristic
327C22#1681059611yescharacteristic
328C22C2×#1313314011yescharacteristic
329C22#1681062711yescharacteristic
330C22#1681064111yescharacteristic
331C22#1681064111yescharacteristic
332C22#1681091312yes—
333C22#1681091812yes—
334C4—22no—
335C4—22no—
336C4—22no—
337C4—21no—
338C4—22no—
339C4—21no—
340C4—41no—
341C4—84no—
342C4—4242no—
343C4—4242no—
344C2—22no—
345C2—22no—
346C2#2027324311yescharacteristic
347C2#2027326111yescharacteristic
348C2#2027327811yescharacteristic
349C2#2027328611yescharacteristic
350C2#2027566311yescharacteristic
351C2#2027569611yescharacteristic
352C2#2027570011yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1434C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#1434D53
hypocenterC53#9911
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1434

Complements

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

NH ≅ G/Ntype
5. C2×#13133217C4semidirect
11. #16810521C4semidirect
12. #16810493C4semidirect
47. #95×C53Q8semidirect
196. C53#9911semidirect
224. #95Dic106semidirect

Maximal subgroups

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

subgroupindex
#202830672
#202746952
#202725942
C53×#14342
#991153