Finite Group Databasebeta

← #15799689

Subgroups — #15799689

Order 3328.

1710 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×#11718335C214yesmaximal
2C2×#11718596C211yesmaximal, characteristic, complemented
3C2×#11718648C212yesmaximal
4C2×C4×#6103C211yesmaximal, characteristic
5#13113105C214yesmaximal, complemented
6#13113065C212yesmaximal, complemented
7C13×C4⋊#95C211yesmaximal, characteristic, complemented
8C23×C4×D13C2211yescharacteristic
9C22×C13×C4⋊C4C2212yes—
10C22×C42×C13C2211yescharacteristic
11C22×#6103C2214yes—
12C22×#6103C2212yes—
13C22×C4⋊Dic13C2214yes—
14C22×C4⋊Dic13C2212yes—
15C22×C4×Dic13C2211yescharacteristic
16C2×C13×#95C2214yes—
17C2×#88071C2214yes—
18C2×#88071C2212yes—
19C2×#88071C2218yes—
20#11718648—28no—
21C4×#6103—24no—
22#11718596—24no—
23#11718335—216no—
24C24×D13C2311yescharacteristic
25C23×C4×C13C2314yes—
26C23×C4×C13C2312yes—
27C23×C4×C13C2311yescharacteristic
28C23×Dic13C2314yes—
29C23×Dic13C2312yes—
30C23×Dic13C2311yescharacteristic
31C22×C4×D13—22no—
32C22×C4×D13—24no—
33C22×C4×D13—24no—
34C22×C4×D13—22no—
35C22×C4×D13D414yes—
36C2×C13×C4⋊C4—24no—
37C2×C13×C4⋊C4—24no—
38C2×C13×C4⋊C4—24no—
39C2×C42×C13—24no—
40C2×C42×C13D414yes—
41C2×#6103—24no—
42C2×#6103—24no—
43C2×#6103—216no—
44C2×#6103—28no—
45C2×#6103D418yes—
46C2×C4⋊Dic13—28no—
47C2×C4⋊Dic13—28no—
48C2×C4⋊Dic13—24no—
49C2×C4⋊Dic13—24no—
50C2×C4⋊Dic13—28no—
51C2×C4⋊Dic13—24no—
52C2×C4×Dic13—22no—
53C2×C4×Dic13—24no—
54C13×#95—28no—
55#88071—24no—
56#88071—216no—
57#88071—28no—
58#15150—131nomaximal, Sylow
59C24×C13C2411yescharacteristic
60C23×D13—21no—
61C23×D13—22no—
62C23×D13—22no—
63C23×D13—22no—
64C23×D13—21no—
65C23×D13—24no—
66C23×D13C2×D414yes—
67C23×D13C2×D412yes—
68C22×C4×C13—24no—
69C22×C4×C13—24no—
70C22×C4×C13—28no—
71C22×C4×C13—24no—
72C22×C4×C13—24no—
73C22×C4×C13—24no—
74C22×C4×C13—28no—
75C22×C4×C13C2×D414yes—
76C22×C4×C13C2×D412yes—
77C22×C4×C13C2×D414yes—
78C22×C4×C13C2×D412yes—
79C22×C4×C13#4114yes—
80C22×C4×C13#4114yes—
81C22×C4×C13#4114yes—
82C22×C4×C13#4112yes—
83C22×Dic13—21no—
84C22×Dic13—22no—
85C22×Dic13—24no—
86C22×Dic13—22no—
87C22×Dic13—21no—
88C22×Dic13—22no—
89C22×Dic13—22no—
90C22×Dic13—24no—
91C22×Dic13—24no—
92C22×Dic13—28no—
93C22×Dic13—24no—
94C22×Dic13—24no—
95C22×Dic13—28no—
96C22×Dic13C2×D414yes—
97C22×Dic13C2×D412yes—
98C2×C4×D13—28no—
99C2×C4×D13—216no—
100C2×C4×D13—44no—
101C2×C4×D13—44no—
102C2×C4×D13—48no—
103C13×C4⋊C4—48no—
104C42×C13—28no—
105#6103—216no—
106#6103—416no—
107C4⋊Dic13—416no—
108C4⋊Dic13—48no—
109C4×Dic13—44no—
110C2×#397—134no—
111C2×#384—131no—
112C2×#380—132no—
113C2×C4×C22⋊C4—131no—
114#1445—134no—
115C4⋊#95—132no—
116C4⋊#95D1311yescharacteristic, complemented
117C23×C13C22×D411yescharacteristic
118C23×C13C22×D411yescharacteristic
119C23×C13C22×D412yes—
120C23×C13C2×#4111yescharacteristic
121C23×C13C2×#4112yes—
122C23×C13C2×#4112yes—
123C23×C13C2×#4112yes—
124C23×C132−1+414yes—
125C22×D13—24no—
126C22×D13—22no—
127C22×D13—24no—
128C22×D13—24no—
129C22×D13—22no—
130C22×D13—24no—
131C22×D13—28no—
132C22×D13—28no—
133C22×D13—41no—
134C22×D13—41no—
135C22×D13—42no—
136C22×D13—42no—
137C22×D13—42no—
138C22×D13—44no—
139C22×D13—44no—
140C22×D13#12714yes—
141C2×C4×C13—24no—
142C2×C4×C13—24no—
143C2×C4×C13—28no—
144C2×C4×C13—28no—
145C2×C4×C13—44no—
146C2×C4×C13—44no—
147C2×C4×C13—44no—
148C2×C4×C13—48no—
149C2×C4×C13—48no—
150C2×C4×C13#12118yes—
151C2×C4×C13#12114yes—
152C2×C4×C13#12318yes—
153C2×C4×C13#12314yes—
154C2×C4×C13#12414yes—
155C2×C4×C13#12618yes—
156C2×Dic13—24no—
157C2×Dic13—22no—
158C2×Dic13—24no—
159C2×Dic13—28no—
160C2×Dic13—41no—
161C2×Dic13—42no—
162C2×Dic13—41no—
163C2×Dic13—42no—
164C2×Dic13—42no—
165C2×Dic13—44no—
166C2×Dic13—44no—
167C2×Dic13—48no—
168C2×Dic13—44no—
169C2×Dic13—48no—
170C2×Dic13—44no—
171C4×D13—416no—
172C24×C4—131no—
173C22×C4⋊C4—134no—
174C22×C4⋊C4—132no—
175C22×C4⋊C4C2×D1312yes—
176C22×C22⋊C4—132no—
177C22×C22⋊C4—134no—
178C22×C42—131no—
179C22×C42C2×D1311yescharacteristic
180#397—2616no—
181#384—264no—
182#380—268no—
183C4×C22⋊C4—264no—
184C2×#95—138no—
185C2×#95—132no—
186C2×#95—134no—
187C2×#95C2×D1314yes—
188C22×C13C2×#12111yescharacteristic
189C22×C13C2×#12112yes—
190C22×C13C2×#12312yes—
191C22×C13C2×#12311yescharacteristic
192C22×C13C2×#12411yescharacteristic
193C22×C13C2×#12612yes—
194C22×C13C2×#12711yescharacteristic
195C22×C13#53212yes—
196C22×C13#53211yescharacteristic
197C22×C13#53612yes—
198C22×C13#53612yes—
199C22×C13#54112yes—
200C22×C13#54714yes—
201C22×C13#54714yes—
202C22×C13#55214yes—
203C22×C13#55614yes—
204C2×D13—24no—
205C2×D13—28no—
206C2×D13—42no—
207C2×D13—42no—
208C2×D13—44no—
209C2×D13—48no—
210C2×D13—44no—
211C2×D13—44no—
212C4×C13—28no—
213C4×C13—84no—
214C4×C13#39718yescomplemented
215Dic13—42no—
216Dic13—44no—
217Dic13—84no—
218C25—131no—
219C23×C4—132no—
220C23×C4—131no—
221C23×C4—134no—
222C23×C4—134no—
223C23×C4—262no—
224C23×C4—264no—
225C23×C4—264no—
226C23×C4—262no—
227C23×C4C22×D1314yes—
228C23×C4C22×D1312yes—
229C23×C4C22×D1311yescharacteristic
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—268no—
234C2×C4⋊C4—264no—
235C2×C4⋊C4—264no—
236C2×C4⋊C4—268no—
237C2×C4⋊C4—268no—
238C2×C4⋊C4—264no—
239C2×C22⋊C4—138no—
240C2×C22⋊C4—264no—
241C2×C22⋊C4—264no—
242C2×C22⋊C4—268no—
243C2×C22⋊C4—2616no—
244C2×C42—24no—
245C2×C42—262no—
246C2×C42—264no—
247C2×C42D5214yes—
248#95—28no—
249#95—268no—
250#95—2616no—
251#95—264no—
252C2×C13C2×#39711yescharacteristic
253C2×C13#248612yes—
254C2×C13#249614yes—
255C2×C13#249912yes—
256C2×C13#250112yes—
257C2×C13#254712yes—
258C2×C13#262111yescharacteristic
259C2×C13#285611yescharacteristic
260D13—44no—
261C24—134no—
262C24—132no—
263C24—261no—
264C24—262no—
265C24—262no—
266C24—262no—
267C24—261no—
268C24—264no—
269C24C23×D1311yescharacteristic
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—28no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—28no—
277C22×C4—134no—
278C22×C4—132no—
279C22×C4—261no—
280C22×C4—262no—
281C22×C4—264no—
282C22×C4—262no—
283C22×C4—262no—
284C22×C4—262no—
285C22×C4—261no—
286C22×C4—264no—
287C22×C4—2616no—
288C22×C4—268no—
289C22×C4—264no—
290C22×C4—264no—
291C22×C4—268no—
292C22×C4—264no—
293C22×C4—268no—
294C22×C4—524no—
295C22×C4—524no—
296C22×C4—528no—
297C22×C4C2×D5214yes—
298C22×C4C2×D5212yes—
299C22×C4#612714yes—
300C22×C4D4×D1314yes—
301C22×C4D4×D1312yes—
302C22×C4#612914yes—
303C22×C4#613114yes—
304C22×C4#613112yes—
305C4⋊C4—48no—
306C4⋊C4—5216no—
307C4⋊C4—528no—
308C22⋊C4—2616no—
309C22⋊C4—5216no—
310C42—28no—
311C42—524no—
312C13#1515011yescharacteristic, complemented, Sylow
313C23—134no—
314C23—264no—
315C23—262no—
316C23—264no—
317C23—264no—
318C23—262no—
319C23—268no—
320C23—268no—
321C23—264no—
322C23—521no—
323C23—521no—
324C23—522no—
325C23—522no—
326C23—524no—
327C23—524no—
328C23—522no—
329C23C22×D5211yescharacteristic
330C23C2×#612712yes—
331C23C2×D4×D1311yescharacteristic
332C23C2×D4×D1312yes—
333C23C2×#612912yes—
334C23C2×#613111yescharacteristic
335C23C2×#613112yes—
336C23#8825714yes—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—28no—
340C2×C4—28no—
341C2×C4—44no—
342C2×C4—48no—
343C2×C4—48no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—264no—
347C2×C4—264no—
348C2×C4—262no—
349C2×C4—268no—
350C2×C4—522no—
351C2×C4—521no—
352C2×C4—521no—
353C2×C4—522no—
354C2×C4—522no—
355C2×C4—524no—
356C2×C4—5216no—
357C2×C4—528no—
358C2×C4—524no—
359C2×C4—528no—
360C2×C4—524no—
361C2×C4—524no—
362C2×C4#8812914yes—
363C2×C4#8813718yes—
364C2×C4#8814014yes—
365C2×C4#8814918yes—
366C2×C4#8814914yes—
367C2×C4#8815218yes—
368C22—268no—
369C22—264no—
370C22—528no—
371C22—524no—
372C22—524no—
373C22—522no—
374C22—522no—
375C22—524no—
376C22C2×#8812911yescharacteristic
377C22C2×#8813712yes—
378C22C2×#8814011yescharacteristic
379C22C2×#8814912yes—
380C22C2×#8814911yescharacteristic
381C22C2×#8815212yes—
382C22#1171923112yes—
383C22#1171925614yes—
384C22#1171928914yes—
385C22#1171931614yes—
386C22#1171934714yes—
387C22#1171938012yes—
388C22#1171940212yes—
389C22D13×#12711yescharacteristic
390C22#1171943012yes—
391C22#1171943011yescharacteristic
392C4—28no—
393C4—84no—
394C4—524no—
395C4—522no—
396C4—1044no—
397C4#1171833518yescomplemented
398C2—524no—
399C2C2×#1171833511yescharacteristic
400C2#1311914512yes—
401C2#1311925614yes—
402C2#1311929112yes—
403C2#1311930612yes—
404C2#1311983112yes—
405C2#1312060511yescharacteristic
406C2#1312318211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D13
commutator, socle, cosocleC24×C13C24
FittingC13×C4⋊#95C2
prime radicalC24×D13C23
hypercenterC4⋊#95D13
hypocenterC13#15150
LCSG ▷ C24×C13 ▷ C13
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
2. C2×#11718596C2semidirect
5. #13113105C2semidirect
6. #13113065C2semidirect
7. C13×C4⋊#95C2semidirect
116. C4⋊#95D13semidirect
214. C4×C13#397semidirect
312. C13#15150semidirect
397. C4#11718335semidirect

Maximal subgroups

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

subgroupindex
C2×#117183352
C2×#117185962
C2×#117186482
C2×C4×#61032
#131131052
#131130652
C13×C4⋊#952
#1515013