Finite Group Databasebeta

← #27469836

Subgroups — #27469836

Order 13568.

796 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#20283016C211yesmaximal, characteristic
2#20277122C211yesmaximal, characteristic, complemented
3#20276135C212yesmaximal, complemented
4#20276130C212yesmaximal, complemented
5#20275531C212yesmaximal, complemented
6#20275035C212yesmaximal
7#20275025C212yesmaximal, complemented
8#20274849C212yesmaximal, complemented
9C53×#1749C211yesmaximal, characteristic, complemented
10C2×#13133205C2211yescharacteristic
11C2×#13133196C2211yescharacteristic
12C2×#13133191C2211yescharacteristic
13C2×#13133176C2212yes—
14C2×#13133142C2214yes—
15C2×#13133139C2214yes—
16#16810529C2211yescharacteristic
17#16810525C2211yescharacteristic
18#16810514—22no—
19#16810502—22no—
20#16810493C2212yes—
21#16810489C2212yes—
22#16810489C2211yescharacteristic
23#16810483—22no—
24C4×C4⋊Dic53—22no—
25#16810386C2214yes—
26#16810386C2214yes—
27C22×C42×C53C2211yescharacteristic
28C2×C22⋊C8×C53C2212yes—
29C2×#97×C53C2212yes—
30C53×#336C2212yes—
31C22×Dic106C2311yescharacteristic
32C22×D212C2311yescharacteristic
33C22×C22⋊D53—21no—
34#13133205—24no—
35#13133196—24no—
36#13133191—24no—
37#13133176—44no—
38C2×C4⋊Dic53—22no—
39C2×C4⋊Dic53—22no—
40C2×C4⋊Dic53C2311yescharacteristic
41C2×C4⋊Dic53C2311yescharacteristic
42C2×C4⋊Dic53C2314yes—
43C2×#11719919—21no—
44C2×C22⋊Dic53—21no—
45C2×C22⋊Dic53—22no—
46C2×#11719921—22no—
47C2×#11719921—22no—
48C2×#11719921—21no—
49C2×C4×Dic53—22no—
50#13133142—44no—
51#13133139—44no—
52#13133124—22no—
53#13133124—22no—
54#13133124—24no—
55C23×C4×C53D412yes—
56C23×C4×C53C2311yescharacteristic
57C22×C8×C53C2314yes—
58C2×C42×C53—24no—
59C2×C42×C53D412yes—
60C2×C42×C53C2311yescharacteristic
61C2×C42×C53C2311yescharacteristic
62C22⋊C8×C53—24no—
63#97×C53—24no—
64C24×C53C2×D411yescharacteristic
65C23×D53—21no—
66C22×C4×C53—22no—
67C22×C4×C53—22no—
68C22×C4×C53—24no—
69C22×C4×C53—42no—
70C22×C4×C53—42no—
71C22×C4×C53C2×D412yes—
72C22×C4×C53C2×D412yes—
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53#4114yes—
75C22×C4×C53C2411yescharacteristic
76C2×C22⋊D53—44no—
77C2×C22⋊D53—42no—
78C22×Dic53—21no—
79C22×Dic53—24no—
80C22×Dic53—21no—
81C22×Dic53—21no—
82C2×D212—22no—
83C2×D212—22no—
84C2×D212—22no—
85C2×Dic106—22no—
86C2×Dic106—22no—
87C2×Dic106—22no—
88C2×C8×C53—24no—
89C2×C8×C53—24no—
90C2×C8×C53—24no—
91C42×C53—24no—
92C42×C53—22no—
93C42×C53#4114yes—
94C22⋊Dic53—44no—
95C22⋊Dic53—42no—
96#11719921—44no—
97#11719921—44no—
98#11719921—42no—
99C4⋊Dic53—22no—
100C4⋊Dic53—22no—
101C4⋊Dic53—24no—
102C4⋊Dic53—24no—
103C4⋊Dic53—44no—
104C4⋊Dic53—44no—
105#11719919—42no—
106C4×Dic53—44no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—22no—
110C23×C53—22no—
111C23×C53#12412yes—
112C23×C53C22×D411yescharacteristic
113C22×D53—41no—
114C22×D53—41no—
115C22×D53—41no—
116C22×D53—42no—
117C22×D53—42no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—22no—
121C2×C4×C53—22no—
122C2×C4×C53—28no—
123C2×C4×C53—22no—
124C2×C4×C53—22no—
125C2×C4×C53—22no—
126C2×C4×C53—48no—
127C2×C4×C53#12314yes—
128C2×C4×C53#12412yes—
129C2×C4×C53C2×#4112yes—
130C2×C4×C53C2×#4112yes—
131C2×C4×C53H(5,𝔽2)12yes—
132C22⋊D53—84no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—44no—
139C2×Dic53—42no—
140C2×Dic53—44no—
141C2×Dic53—44no—
142C2×Dic53—42no—
143D212—44no—
144Dic106—44no—
145C8×C53—44no—
146#27979—531nomaximal, Sylow
147C22×C53—22no—
148C22×C53—22no—
149C22×C53—22no—
150C22×C53—42no—
151C22×C53—42no—
152C22×C53C2×#12311yescharacteristic
153C22×C53C2×#12411yescharacteristic
154C22×C53#53511yescharacteristic
155C22×C53#57512yes—
156C22×C53#57512yes—
157C2×D53—82no—
158C2×D53—81no—
159C2×D53—81no—
160C2×D53—82no—
161C2×D53—81no—
162C4×C53—24no—
163C4×C53—22no—
164C4×C53—42no—
165C4×C53#55314yes—
166Dic53—81no—
167Dic53—81no—
168Dic53—84no—
169Dic53—81no—
170#2850—531no—
171#2285—531no—
172#2074—532no—
173#2073—532no—
174#1959—532no—
175#1875—532no—
176#1873—532no—
177#1836—532no—
178#1749D5311yescharacteristic, complemented
179C2×C53—42no—
180C2×C53#250411yescharacteristic
181C2×C53#309312yes—
182C2×C53#313511yescharacteristic
183C2×C53#314011yescharacteristic
184C2×C53#321912yes—
185D53—161no—
186C2×#124—531no—
187C2×C22⋊Q8—531no—
188C2×#121—531no—
189C2×#117—532no—
190C22×C42C2×D5311yescharacteristic
191C2×#103—534no—
192C2×#102—534no—
193C2×C22⋊C8C2×D5312yes—
194C2×#97C2×D5312yes—
195#391—531no—
196#390—531no—
197#388—1062no—
198#386—1062no—
199#384—532no—
200#382—532no—
201#382—531no—
202#380—1062no—
203C4×C4⋊C4—1062no—
204#340—534no—
205#340—534no—
206#336C2×D5312yes—
207C53#2797911yescharacteristic, complemented, Sylow
208C22×Q8—531no—
209C22×D4—531no—
210C22×D4—1061no—
211C23×C4D21212yes—
212C23×C4C22×D5311yescharacteristic
213C22×C8C22×D5314yes—
214#124—1064no—
215C22⋊Q8—1064no—
216#121—1064no—
217#117—2124no—
218C2×C4⋊C4—534no—
219C2×C4⋊C4—531no—
220C2×C4⋊C4—531no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4—1062no—
223C2×C4⋊C4—1062no—
224C2×C22⋊C4—1062no—
225C2×C22⋊C4—1061no—
226C2×C22⋊C4—1061no—
227C2×C22⋊C4—1062no—
228C2×C22⋊C4—1062no—
229C2×C42—24no—
230C2×C42—1062no—
231C2×C42D21212yes—
232C2×C42C22×D5311yescharacteristic
233C2×C42C22×D5311yescharacteristic
234#103—2124no—
235#102—2124no—
236C22⋊C8—24no—
237#97—24no—
238#95—1064no—
239#95—1062no—
240#95—1062no—
241C24—1061no—
242C24C2×D21211yescharacteristic
243C2×Q8—1062no—
244C2×Q8—1062no—
245C2×Q8—1062no—
246C2×D4—1062no—
247C2×D4—1062no—
248C2×D4—1062no—
249C2×D4—2124no—
250C2×D4—2122no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—1061no—
257C22×C4—1064no—
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4C2×D21212yes—
261C22×C4C2×D21212yes—
262C22×C4C2×D21211yescharacteristic
263C22×C4#1171994514yes—
264C22×C4C23×D5311yescharacteristic
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—24no—
268C4⋊C4—1062no—
269C4⋊C4—1062no—
270C4⋊C4—1064no—
271C4⋊C4—1064no—
272C4⋊C4—2124no—
273C4⋊C4—2124no—
274C4⋊C4—2122no—
275C22⋊C4—2124no—
276C22⋊C4—2124no—
277C22⋊C4—2122no—
278C22⋊C4—2124no—
279C22⋊C4—2122no—
280C42—24no—
281C42—22no—
282C42—2124no—
283C42#1171994714yes—
284C23—21no—
285C23—22no—
286C23—22no—
287C23—21no—
288C23—2121no—
289C23—2121no—
290C23—2122no—
291C23—2122no—
292C23—2121no—
293C23#1313320512yes—
294C23C22×D21211yescharacteristic
295Q8—2124no—
296D4—4244no—
297D4—2124no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—28no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—48no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4—2121no—
310C2×C4—2121no—
311C2×C4—2122no—
312C2×C4—2124no—
313C2×C4—2124no—
314C2×C4—2122no—
315C2×C4—2124no—
316C2×C4—2121no—
317C2×C4#1313320114yes—
318C2×C4#1313320512yes—
319C2×C4C2×#1171994712yes—
320C2×C4C2×#1171994512yes—
321C2×C4#1313326312yes—
322C8—44no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—42no—
327C22—42no—
328C22—4242no—
329C22—4241no—
330C22—4241no—
331C22—4241no—
332C22—4242no—
333C22C2×#1313320111yescharacteristic
334C22C2×#1313320511yescharacteristic
335C22#1681119211yescharacteristic
336C22#1681142312yes—
337C22#1681142312yes—
338C4—22no—
339C4—24no—
340C4—42no—
341C4—4244no—
342C4—4241no—
343C4—4241no—
344C4—4241no—
345C4#1681131914yes—
346C2—42no—
347C2—8481no—
348C2#2027920711yescharacteristic
349C2#2028464812yes—
350C2#2028506111yescharacteristic
351C2#2028510211yescharacteristic
352C2#2028575412yes—
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1749C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#1749D53
hypocenterC53#27979
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
2. #20277122C2semidirect
3. #20276135C2semidirect
4. #20276130C2semidirect
5. #20275531C2semidirect
7. #20275025C2semidirect
8. #20274849C2semidirect
9. C53×#1749C2semidirect
178. #1749D53semidirect
207. C53#27979semidirect

Maximal subgroups

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

subgroupindex
#202830162
#202771222
#202761352
#202761302
#202755312
#202750352
#202750252
#202748492
C53×#17492
#2797953