Finite Group Databasebeta

← #27475360

Subgroups — #27475360

Order 13568.

1402 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810996C211yesmaximal, characteristic, complemented
2C2×#16810995C211yesmaximal, characteristic, complemented
3C2×#16810822C214yesmaximal, complemented
4#20283109C211yesmaximal, characteristic
5#20277282C211yesmaximal, characteristic, complemented
6#20275804C214yesmaximal, complemented
7#20275520C211yesmaximal, characteristic, complemented
8C4×#13133146C211yesmaximal, characteristic, complemented
9C53×#1957C211yesmaximal, characteristic, complemented
10C22×#11719913C2214yes—
11C22×#11719923C2214yes—
12C2×#13133217C2211yescharacteristic
13C2×#13133214C2211yescharacteristic
14C2×#13133213—22no—
15C2×#13133213—22no—
16C2×#13133213C2212yes—
17C2×#13133213C2212yes—
18C2×#13133182C2212yes—
19#16810996—24no—
20#16810995—24no—
21#16810822—216no—
22C2×#13133146C2212yes—
23C2×#13133137C2214yes—
24C2×C4×#88322C2211yescharacteristic
25#16810528C2212yes—
26#16810519—22no—
27C4⋊C4×Dic53—22no—
28C42×Dic53C2211yescharacteristic
29#16810377C2212yes—
30C2×#127×C53C2211yescharacteristic
31C2×#102×C53C2214yes—
32C2×C4×C8×C53C2211yescharacteristic
33C53×#384C2211yescharacteristic
34C22×Dic106C2312yes—
35C22×D212—22no—
36C22×D212C2312yes—
37C22×C22⋊D53—22no—
38C22×C22⋊D53—24no—
39C22×C22⋊D53—22no—
40C2×#11719913—28no—
41C2×#11719913—28no—
42C2×#11719913—44no—
43C2×#11719923—28no—
44C2×#11719923—28no—
45C2×#11719923—44no—
46C22×#88322C2312yes—
47#13133217—22no—
48#13133217D414yescomplemented
49#13133214—22no—
50#13133214—22no—
51#13133213—28no—
52#13133213—44no—
53#13133213—48no—
54#13133213—48no—
55#13133182—28no—
56C2×C4⋊Dic53—22no—
57C2×#11719919—22no—
58C2×C4×Dic53—22no—
59C2×C4×Dic53D412yes—
60C2×C4×Dic53D412yes—
61C2×C4×Dic53C2311yescharacteristic
62C2×C4×Dic53C2311yescharacteristic
63#13133146D418yescomplemented
64#13133137—28no—
65C4×#88322—22no—
66#13133124—24no—
67C22×D4×C53—22no—
68C22×D4×C53C2312yes—
69C22×C8×C53C2312yes—
70#127×C53—22no—
71#127×C53—22no—
72C2×C4⋊C4×C53—22no—
73C2×C4⋊C4×C53C2312yes—
74C2×C42×C53C2311yescharacteristic
75#102×C53—28no—
76C4×C8×C53—22no—
77C24×C53—22no—
78C23×D53—22no—
79C2×D4×C53—24no—
80C2×D4×C53—22no—
81C2×D4×C53—22no—
82C2×D4×C53—24no—
83C2×D4×C53—44no—
84C2×D4×C53—44no—
85C2×D4×C53—44no—
86C22×C4×C53—22no—
87C22×C4×C53C2×D412yes—
88C22×C4×C53C2411yescharacteristic
89C2×C22⋊D53—48no—
90C2×C22⋊D53—48no—
91C2×C22⋊D53—48no—
92C2×C22⋊D53—44no—
93C2×C22⋊D53—44no—
94C2×C22⋊D53—48no—
95C2×C22⋊D53—48no—
96C22×Dic53—22no—
97C22×Dic53C2×D411yescharacteristic
98C22×Dic53C2×D412yes—
99C22×Dic53C2×D411yescharacteristic
100C2×D212—24no—
101C2×D212—22no—
102C2×D212—22no—
103C2×D212—24no—
104C2×D212—44no—
105C2×D212—44no—
106C2×D212—44no—
107C2×Dic106—24no—
108C2×Dic106—24no—
109C2×Dic106—22no—
110C2×Dic106C2×D414yes—
111C2×C8×C53—22no—
112C2×C8×C53—22no—
113C2×C8×C53C2×D414yes—
114C4⋊C4×C53—24no—
115C4⋊C4×C53—44no—
116C42×C53C2×D412yes—
117C42×C53#4112yes—
118#11719923—416no—
119C4⋊Dic53—28no—
120#11719919—44no—
121C4×Dic53—24no—
122C4×Dic53—22no—
123C4×Dic53—44no—
124C4×Dic53SD1618yescomplemented
125C4×Dic53C2×D414yes—
126C2×#88322—22no—
127C2×#88322—22no—
128C2×#88322C2×D414yes—
129#11719913—416no—
130C23×C53—42no—
131C23×C53—42no—
132C23×C53—42no—
133C23×C53—44no—
134C23×C53—44no—
135C23×C53C22×D411yescharacteristic
136C22×D53—42no—
137C22×D53—42no—
138C22×D53—42no—
139C22×D53—44no—
140C22×D53—44no—
141D4×C53—44no—
142D4×C53—44no—
143D4×C53—88no—
144C2×C4×C53—42no—
145C2×C4×C53—44no—
146C2×C4×C53#12114yes—
147C2×C4×C53C2×SD1614yes—
148C2×C4×C53AGL(1,ℤ/8)14yes—
149C2×C4×C53C22×D411yescharacteristic
150C2×C4×C53C22×D412yes—
151C2×C4×C53C2×#4111yescharacteristic
152C2×C4×C53H(5,𝔽2)12yes—
153C22⋊D53—88no—
154C22⋊D53—816no—
155C22⋊D53—88no—
156C2×Dic53—22no—
157C2×Dic53—22no—
158C2×Dic53—22no—
159C2×Dic53—24no—
160C2×Dic53—42no—
161C2×Dic53—42no—
162C2×Dic53#12712yes—
163C2×Dic53#12712yes—
164C2×Dic53C2×SD1614yes—
165C2×Dic53C2×SD1614yes—
166D212—44no—
167D212—44no—
168D212—88no—
169Dic106—28no—
170Dic106—44no—
171C8×C53—24no—
172#88322—24no—
173#28441—531nomaximal, Sylow
174C22×C53—82no—
175C22×C53—82no—
176C22×C53—84no—
177C22×C53—84no—
178C22×C53—82no—
179C22×C53C2×#12111yescharacteristic
180C22×C53C2×#12711yescharacteristic
181C22×C53#53511yescharacteristic
182C22×C53C22×SD1612yes—
183C22×C53C2×AGL(1,ℤ/8)12yes—
184C2×D53—82no—
185C2×D53—82no—
186C2×D53—84no—
187C2×D53—84no—
188C2×D53—82no—
189C4×C53—82no—
190C4×C53#46018yescomplemented
191C4×C53D4212yes—
192C4×C53#55012yes—
193Dic53—22no—
194Dic53—24no—
195Dic53—44no—
196Dic53#49214yescomplemented
197C2×#492—531no—
198C2×#492—531no—
199C2×#460—534no—
200#2867—531no—
201C4×C4⋊Q8—531no—
202#2014—534no—
203#1957—531no—
204#1957D5311yescharacteristic, complemented
205C4×C4⋊C8—531no—
206C2×C53—162no—
207C2×C53#254111yescharacteristic
208C2×C53C2×#46012yes—
209C2×C53C2×#49211yescharacteristic
210C2×C53#315111yescharacteristic
211C2×C53#320812yes—
212D53—162no—
213C22×SD16—534no—
214C22×SD16—534no—
215C2×C4⋊Q8—531no—
216C2×#127—532no—
217C2×#127—532no—
218C2×#127—531no—
219C2×#127—1062no—
220C2×#127—1062no—
221C2×#127C2×D5311yescharacteristic
222C2×C4×Q8—532no—
223#492—1064no—
224#492—1064no—
225#460—10616no—
226C2×C4⋊C8—532no—
227C2×#102—534no—
228C2×#102C2×D5314yes—
229C2×C4×C8—531no—
230C2×C4×C8C2×D5311yescharacteristic
231#391—532no—
232#389—1062no—
233#384C2×D5311yescharacteristic
234C4×C4⋊C4—1062no—
235C43—531no—
236#336—532no—
237C53#2844111yescharacteristic, complemented, Sylow
238C22×Q8—532no—
239C22×D4—22no—
240C22×D4—532no—
241C22×D4—1062no—
242C22×D4—1064no—
243C22×D4—1062no—
244C22×D4—1062no—
245C22×D4C22×D5312yes—
246C2×SD16—1068no—
247C2×SD16—1068no—
248C2×SD16—1068no—
249C2×SD16—1068no—
250C2×SD16—2124no—
251C2×SD16—2124no—
252C22×C8—532no—
253C22×C8C22×D5312yes—
254C4⋊Q8—534no—
255C4⋊Q8—1062no—
256#127—22no—
257#127—22no—
258#127—1068no—
259#127—1062no—
260#127—1062no—
261#127—2128no—
262#127—2124no—
263#127—2128no—
264C4×Q8—1068no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—1062no—
267C2×C4⋊C4—1062no—
268C2×C4⋊C4C22×D5312yes—
269C2×C42—532no—
270C2×C42—531no—
271C2×C42—532no—
272C2×C42—531no—
273C2×C42—1062no—
274C2×C42C22×D5311yescharacteristic
275C4⋊C8—538no—
276#102—28no—
277#102—1068no—
278C4×C8—22no—
279C4×C8—1062no—
280#95—1064no—
281C24—22no—
282C24—1062no—
283C2×Q8—534no—
284C2×Q8—1062no—
285C2×Q8—1064no—
286C2×Q8—1064no—
287C2×D4—24no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—24no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—1064no—
295C2×D4—1062no—
296C2×D4—1062no—
297C2×D4—1064no—
298C2×D4—2124no—
299C2×D4—2124no—
300C2×D4—2128no—
301C2×D4—2128no—
302C2×D4—2128no—
303C2×D4—2128no—
304C2×D4—2128no—
305C2×D4—2124no—
306C2×D4—2124no—
307C2×D4—2124no—
308C22×C4—22no—
309C22×C4—531no—
310C22×C4—532no—
311C22×C4—531no—
312C22×C4—1062no—
313C22×C4D4×D5312yes—
314C22×C4C23×D5311yescharacteristic
315SD16—21216no—
316SD16—21216no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—534no—
320C2×C8—1062no—
321C2×C8—1062no—
322C2×C8D4×D5314yes—
323C4⋊C4—24no—
324C4⋊C4—44no—
325C4⋊C4—1068no—
326C4⋊C4—2124no—
327C42—538no—
328C42—534no—
329C42—1064no—
330C42—1062no—
331C42—2124no—
332C42#1171994512yes—
333C42D4×D5312yes—
334C23—42no—
335C23—44no—
336C23—44no—
337C23—42no—
338C23—42no—
339C23—2122no—
340C23—2124no—
341C23—2124no—
342C23—2122no—
343C23—2122no—
344C23C2×D4×D5311yescharacteristic
345Q8—1068no—
346Q8—2124no—
347D4—44no—
348D4—44no—
349D4—88no—
350D4—4248no—
351D4—42416no—
352D4—4248no—
353D4—4248no—
354D4—2124no—
355D4—2124no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—534no—
359C2×C4—532no—
360C2×C4—534no—
361C2×C4—532no—
362C2×C4—1062no—
363C2×C4—1062no—
364C2×C4—1062no—
365C2×C4—1064no—
366C2×C4—2122no—
367C2×C4—2122no—
368C2×C4#1313319014yes—
369C2×C4SD16×D5314yes—
370C2×C4#1313324114yes—
371C2×C4C2×D4×D5312yes—
372C2×C4C2×D4×D5311yescharacteristic
373C2×C4C2×#1171994511yescharacteristic
374C2×C4#1313326312yes—
375C8—24no—
376C8—1064no—
377C22—82no—
378C22—82no—
379C22—84no—
380C22—82no—
381C22—84no—
382C22—4242no—
383C22—4242no—
384C22—4244no—
385C22—4242no—
386C22—4244no—
387C22C2×#1313319011yescharacteristic
388C22#127×D5311yescharacteristic
389C22#1681119111yescharacteristic
390C22C2×SD16×D5312yes—
391C22C2×#1313324112yes—
392C4—82no—
393C4—534no—
394C4—1064no—
395C4—1062no—
396C4—2124no—
397C4#1681082218yescomplemented
398C4#1681125712yes—
399C4#1681129412yes—
400C2—162no—
401C2—8482no—
402C2#2027966011yescharacteristic
403C2C2×#1681082212yes—
404C2#492×D5311yescharacteristic
405C2#2028516611yescharacteristic
406C2#2028565712yes—
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1957C2
socleC23×C53C22×D4
prime radical#20283109C2
hypercenter#1957D53
hypocenterC53#28441
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1957

Complements

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

NH ≅ G/Ntype
1. C2×#16810996C2semidirect
2. C2×#16810995C2semidirect
3. C2×#16810822C2semidirect
5. #20277282C2semidirect
6. #20275804C2semidirect
7. #20275520C2semidirect
8. C4×#13133146C2semidirect
9. C53×#1957C2semidirect
48. #13133217D4semidirect
63. #13133146D4semidirect
124. C4×Dic53SD16semidirect
190. C4×C53#460semidirect
196. Dic53#492semidirect
204. #1957D53semidirect
237. C53#28441semidirect
397. C4#16810822semidirect

Maximal subgroups

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

subgroupindex
C2×#168109962
C2×#168109952
C2×#168108222
#202831092
#202772822
#202758042
#202755202
C4×#131331462
C53×#19572
#2844153