Finite Group Databasebeta

← #15766731

Subgroups — #15766731

Order 3328.

522 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718688C211yesmaximal, characteristic
2#13115642C211yesmaximal, characteristic
3#13114841C211yesmaximal, characteristic, complemented
4C2×#11718049C211yesmaximal, characteristic, complemented
5C2×#11718077C211yesmaximal, characteristic
6#13112720C211yesmaximal, characteristic, complemented
7C13×#1984C211yesmaximal, characteristic, complemented
8C2×C4×Q8×C13C2211yescharacteristic
9C22×#6106—21no—
10C2×#88128C2211yescharacteristic
11C2×C13×C4⋊C8C2211yescharacteristic
12C2×C4×C8×C13C2211yescharacteristic
13C2×#88075—21no—
14C2×#88061—21no—
15C2×#88061—21no—
16#11718688—22no—
17#11718688—22no—
18C2×#88049—21no—
19#11718596C2211yescharacteristic
20C2×C4⋊#1055C2211yescharacteristic
21C2×C4⋊#1055C2211yescharacteristic
22C13×#336—21no—
23C13×#336—21no—
24#11718121—21no—
25#11718077—22no—
26#11718049—22no—
27C22×Q8×C13D411yescharacteristic
28C22×D52—21no—
29C22×D52—21no—
30C22×D52—21no—
31C22×C8×C13—21no—
32C22×C8×C13D411yescharacteristic
33C22×C8×C13D411yescharacteristic
34C4×Q8×C13—21no—
35C4×Q8×C13—21no—
36C4×Q8×C13—21no—
37C4×Q8×C13—21no—
38C2×C13×C4⋊C4—21no—
39C2×C13×C4⋊C4D411yescharacteristic
40C2×C42×C13—21no—
41C2×C42×C13C2311yescharacteristic
42C2×#6106—24no—
43C2×#6106—42no—
44C2×#6106—42no—
45C2×C4⋊Dic13—21no—
46C2×C4⋊Dic13—21no—
47C2×C4⋊Dic13—21no—
48C22×#1055—21no—
49C22×#1055—21no—
50#88128—21no—
51#88128—21no—
52#88128—21no—
53#88128D412yes—
54C13×C4⋊C8—21no—
55C13×C4⋊C8—21no—
56C4×C8×C13—21no—
57C4×C8×C13—21no—
58#88075—42no—
59#88061—42no—
60#88061—42no—
61#88049—24no—
62C4⋊#1055—21no—
63C4⋊#1055—21no—
64C4⋊#1055—21no—
65C4⋊#1055D412yes—
66#10475—131nomaximal, Sylow
67C23×D13—41no—
68C2×Q8×C13—21no—
69C2×Q8×C13—21no—
70C2×Q8×C13—21no—
71C2×Q8×C13—21no—
72C2×Q8×C13—21no—
73C2×Q8×C13SD1612yes—
74C22×C4×C13—21no—
75C22×C4×C13—21no—
76C22×C4×C13C2×D411yescharacteristic
77C22×C4×C13C2×D411yescharacteristic
78C22×C4×C13#4111yescharacteristic
79C22×Dic13—41no—
80C2×D52—22no—
81C2×D52—22no—
82C2×D52—22no—
83C2×D52—41no—
84C2×D52—41no—
85C2×D52—41no—
86C2×D52—41no—
87C2×D52—41no—
88C2×D52—41no—
89C2×D52—41no—
90C2×D52—41no—
91C2×D52—41no—
92C2×D52—41no—
93C2×D52—41no—
94C2×D52—81no—
95C2×D52—81no—
96C2×C8×C13—21no—
97C2×C8×C13—21no—
98C2×C8×C13—21no—
99C2×C8×C13—21no—
100C2×C8×C13—21no—
101C2×C8×C13—41no—
102C2×C8×C13—41no—
103C2×C8×C13—41no—
104C2×C8×C13SD1612yescomplemented
105C13×C4⋊C4—21no—
106C13×C4⋊C4—21no—
107C13×C4⋊C4—41no—
108C13×C4⋊C4—41no—
109C42×C13—41no—
110C42×C13—41no—
111C42×C13C2×D411yescharacteristic
112C42×C13C2×D411yescharacteristic
113C42×C13#4111yescharacteristic
114C42×C13#4111yescharacteristic
115#6106—48no—
116C4⋊Dic13—41no—
117C4⋊Dic13—41no—
118C4⋊Dic13—41no—
119C4⋊Dic13—41no—
120C4⋊Dic13—81no—
121C2×#1055—22no—
122C2×#1055—41no—
123C2×#1055—41no—
124C2×#1055—41no—
125C2×#1055—41no—
126C2×#1055—41no—
127C2×#460—131no—
128#1984D1311yescharacteristic, complemented
129#1957—131no—
130#1803—131no—
131C2×#334—131no—
132C2×#331—131no—
133#1296—131no—
134C23×C13#12111yescharacteristic
135C22×D13—42no—
136C22×D13—81no—
137C22×D13—81no—
138C22×D13—81no—
139C22×D13—81no—
140C22×D13—81no—
141C22×D13—81no—
142Q8×C13—22no—
143Q8×C13—22no—
144Q8×C13—41no—
145Q8×C13—41no—
146C2×C4×C13—21no—
147C2×C4×C13—21no—
148C2×C4×C13—22no—
149C2×C4×C13—41no—
150C2×C4×C13—41no—
151C2×C4×C13—41no—
152C2×C4×C13—41no—
153C2×C4×C13—41no—
154C2×C4×C13#12011yescharacteristic
155C2×C4×C13#12111yescharacteristic
156C2×C4×C13#12111yescharacteristic
157C2×C4×C13#12311yescharacteristic
158C2×C4×C13#12311yescharacteristic
159C2×C4×C13#12411yescharacteristic
160C2×C4×C13C2×SD1611yescharacteristic
161C2×C4×C13C2×SD1611yescharacteristic
162C2×C4×C13#13511yescharacteristic
163C2×C4×C13#13511yescharacteristic
164C2×C4×C13AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C13AGL(1,ℤ/8)11yescharacteristic
166C2×C4×C13AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C13#13711yescharacteristic
168C2×Dic13—81no—
169C2×Dic13—81no—
170C2×Dic13—81no—
171D52—42no—
172D52—42no—
173D52—42no—
174D52—42no—
175D52—82no—
176D52—82no—
177D52—81no—
178D52—81no—
179D52—81no—
180D52—81no—
181C8×C13—22no—
182C8×C13—41no—
183C8×C13—81no—
184#1055—42no—
185#1055—81no—
186C22×SD16—261no—
187C2×#127—131no—
188C2×C4×Q8C2×D1311yescharacteristic
189#460—262no—
190#460—262no—
191C2×C4⋊C8—131no—
192C2×C4⋊C8—131no—
193C2×C4⋊C8C2×D1311yescharacteristic
194C2×#103—261no—
195C2×#102—261no—
196C2×#102—261no—
197C2×#102—261no—
198C2×C4×C8C2×D1311yescharacteristic
199#384—131no—
200#340—261no—
201#336—21no—
202#336—21no—
203#334—262no—
204#331—262no—
205C22×C13#39411yescharacteristic
206C22×C13#46011yescharacteristic
207C22×C13#46411yescharacteristic
208C22×C13#46511yescharacteristic
209C22×C13#46911yescharacteristic
210C22×C13#47111yescharacteristic
211C22×C13#47211yescharacteristic
212C2×D13—82no—
213C2×D13—82no—
214C2×D13—82no—
215C2×D13—161no—
216C2×D13—161no—
217C2×D13—161no—
218C2×D13—161no—
219C4×C13—21no—
220C4×C13—21no—
221C4×C13—42no—
222C4×C13—81no—
223C4×C13#44911yescharacteristic
224C4×C13#45011yescharacteristic
225C4×C13#46111yescharacteristic
226C4×C13#46311yescharacteristic
227C4×C13#46511yescharacteristic
228C4×C13#46811yescharacteristic
229C4×C13#48111yescharacteristic
230C4×C13#48211yescharacteristic
231Dic13—161no—
232C22×Q8D5211yescharacteristic
233C22×D4—261no—
234C22×D4—261no—
235C22×D4—261no—
236C2×SD16—264no—
237C2×SD16—522no—
238C2×SD16—522no—
239C22×C8—21no—
240C22×C8—261no—
241C22×C8—261no—
242C22×C8C22⋊D1311yescharacteristic
243C22×C8C22⋊D1311yescharacteristic
244#127—132no—
245#127—261no—
246#127—261no—
247#127—261no—
248C4×Q8—21no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—261no—
254C2×C4⋊C4—261no—
255C2×C4⋊C4—261no—
256C2×C4⋊C4D5211yescharacteristic
257C2×C42—21no—
258C2×C42C22×D1311yescharacteristic
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—132no—
262C4⋊C8—261no—
263C4⋊C8—261no—
264C4⋊C8—261no—
265#103—522no—
266#102—264no—
267#102—522no—
268#102—522no—
269C4×C8—21no—
270C4×C8—21no—
271C2×C13#166611yescharacteristic
272C2×C13#167811yescharacteristic
273C2×C13#168811yescharacteristic
274C2×C13#170211yescharacteristic
275C2×C13#203411yescharacteristic
276C2×C13#205111yescharacteristic
277C2×C13#205511yescharacteristic
278D13—162no—
279C24—521no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8#609512yes—
286C2×D4—262no—
287C2×D4—262no—
288C2×D4—262no—
289C2×D4—521no—
290C2×D4—521no—
291C2×D4—521no—
292C2×D4—521no—
293C2×D4—521no—
294C2×D4—521no—
295C2×D4—521no—
296C2×D4—521no—
297C2×D4—521no—
298C2×D4—521no—
299C2×D4—521no—
300C2×D4—1041no—
301C2×D4—1041no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—521no—
305C22×C4C2×D5211yescharacteristic
306C22×C4#612711yescharacteristic
307C22×C4C2×C22⋊D1311yescharacteristic
308SD16—528no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—262no—
318C2×C8—521no—
319C2×C8—521no—
320C2×C8—521no—
321C2×C8—521no—
322C2×C8—521no—
323C2×C8#610612yescomplemented
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—521no—
329C4⋊C4—521no—
330C4⋊C4—521no—
331C4⋊C4—521no—
332C4⋊C4—1041no—
333C42—41no—
334C42—41no—
335C42D4×D1311yescharacteristic
336C42D4×D1311yescharacteristic
337C42#612911yescharacteristic
338C42#612911yescharacteristic
339C13#1047511yescharacteristic, complemented, Sylow
340C23—522no—
341C23—1041no—
342C23—1041no—
343C23—1041no—
344C23—1041no—
345C23—1041no—
346C23—1041no—
347C23#8818411yescharacteristic
348Q8—22no—
349Q8—22no—
350Q8—41no—
351Q8—41no—
352D4—522no—
353D4—522no—
354D4—522no—
355D4—522no—
356D4—1042no—
357D4—1042no—
358D4—1041no—
359D4—1041no—
360D4—1041no—
361D4—1041no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—1041no—
371C2×C4—1041no—
372C2×C4—1041no—
373C2×C4#8813611yescharacteristic
374C2×C4#8813711yescharacteristic
375C2×C4#8813911yescharacteristic
376C2×C4#8814011yescharacteristic
377C2×C4C2×#609511yescharacteristic
378C2×C4#8815811yescharacteristic
379C2×C4#8816211yescharacteristic
380C2×C4#8816211yescharacteristic
381C2×C4#8819211yescharacteristic
382C2×C4#8819311yescharacteristic
383C2×C4C2×#610611yescharacteristic
384C2×C4#8819611yescharacteristic
385C2×C4#8820311yescharacteristic
386C2×C4#8820411yescharacteristic
387C8—22no—
388C8—41no—
389C8—81no—
390C8—522no—
391C8—1041no—
392C22—1042no—
393C22—1042no—
394C22—1042no—
395C22—2081no—
396C22—2081no—
397C22—2081no—
398C22—2081no—
399C22#1171862211yescharacteristic
400C22#1171868811yescharacteristic
401C22#1171868911yescharacteristic
402C22#1171877511yescharacteristic
403C22#1171879611yescharacteristic
404C22#1171880211yescharacteristic
405C22#1171880311yescharacteristic
406C4—21no—
407C4—21no—
408C4—42no—
409C4—81no—
410C4—2081no—
411C4#1171839211yescharacteristic
412C4#1171839511yescharacteristic
413C4#1171839611yescharacteristic
414C4#1171839811yescharacteristic
415C4#1171870911yescharacteristic
416C4#1171871011yescharacteristic
417C4#1171871811yescharacteristic
418C4#1171874411yescharacteristic
419C2—2082no—
420C2#1311418411yescharacteristic
421C2#1311425211yescharacteristic
422C2#1311432011yescharacteristic
423C2#1311440411yescharacteristic
424C2#1311602611yescharacteristic
425C2#1311612711yescharacteristic
426C2#1311615111yescharacteristic
subgroupG/H
centerC23#88184
commutator, cosocleC2×C42×C13C23
FrattiniC2×C42C22×D13
FittingC13×#1984C2
socleC23×C13#121
prime radicalC2×#88128C22
hypercenter#1984D13
hypocenterC13#10475
LCSG ▷ C2×C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1984

Complements

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

NH ≅ G/Ntype
3. #13114841C2semidirect
4. C2×#11718049C2semidirect
6. #13112720C2semidirect
7. C13×#1984C2semidirect
104. C2×C8×C13SD16semidirect
128. #1984D13semidirect
323. C2×C8#6106semidirect
339. C13#10475semidirect

Maximal subgroups

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

subgroupindex
C2×#117186882
#131156422
#131148412
C2×#117180492
C2×#117180772
#131127202
C13×#19842
#1047513