Finite Group Databasebeta

← #16019323

Subgroups — #16019323

Order 3328.

476 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13125495C211yesmaximal, characteristic, complemented
2#13117000C211yesmaximal, characteristic, complemented
3#13116993C211yesmaximal, characteristic, complemented
4#13116985C211yesmaximal, characteristic, complemented
5#13116975C211yesmaximal, characteristic, complemented
6#13116947C211yesmaximal, characteristic
7#13116942C211yesmaximal, characteristic, complemented
8#13116930C211yesmaximal, characteristic
9#13116925C211yesmaximal, characteristic, complemented
10#13116895C211yesmaximal, characteristic, complemented
11#13116895C211yesmaximal, characteristic, complemented
12#13116886C211yesmaximal, characteristic, complemented
13#13116881C211yesmaximal, characteristic, complemented
14#13116595C211yesmaximal, characteristic, complemented
15C13×#3175C211yesmaximal, characteristic
16C13×#542—21no—
17#11719223—21no—
18C13×#500C2211yescharacteristic
19C13×#495C2211yescharacteristic
20C13×#489—21no—
21C13×#468—21no—
22C13×#467C2211yescharacteristic
23C13×#466C2211yescharacteristic
24C13×#440—21no—
25C4×C13×Q16C2211yescharacteristic
26C4×C13×D8C2211yescharacteristic
27C13×#405C2211yescharacteristic
28C2×#88069C2211yescharacteristic
29C2×#88069C2211yescharacteristic
30C2×#88067C2211yescharacteristic
31C2×#88067C2211yescharacteristic
32#11718798—21no—
33#11718777C2211yescharacteristic
34#11718777C2211yescharacteristic
35C2×#88052C2211yescharacteristic
36C22×#6090C2211yescharacteristic
37#11718377—21no—
38#11718364—21no—
39#11718360C2211yescharacteristic
40C4×Dic52C2211yescharacteristic
41C4×Dic52C2211yescharacteristic
42#11718344C2211yescharacteristic
43#11718156C2211yescharacteristic
44#11718156C2211yescharacteristic
45#11718154C2211yescharacteristic
46#11718154C2211yescharacteristic
47#11718084C2211yescharacteristic
48#11718084C2211yescharacteristic
49#11718084C2211yescharacteristic
50#11718084C2211yescharacteristic
51#11718082C2211yescharacteristic
52#11718082C2211yescharacteristic
53#11718081C2211yescharacteristic
54#11718081C2211yescharacteristic
55C4⋊#6090C2211yescharacteristic
56C4⋊#6090C2211yescharacteristic
57C4⋊#6090C2211yescharacteristic
58C4⋊#6090C2211yescharacteristic
59C2×C13×Q16C2311yescharacteristic
60C2×C13×SD16—21no—
61C2×C13×D8C2311yescharacteristic
62C2×C13×M16D411yescharacteristic
63C22×C8×C13C2311yescharacteristic
64C13×C4⋊Q8—21no—
65C13×#126—41no—
66C13×#124—21no—
67C13×#124—41no—
68C13×#123—41no—
69C13×C22⋊Q8—21no—
70C13×C22⋊Q8—41no—
71C13×#121—21no—
72C4×Q8×C13—21no—
73C4×D4×C13—21no—
74C13×#117D411yescharacteristic
75#88178—21no—
76#88178—21no—
77C2×Dic52—21no—
78C2×Dic52C2311yescharacteristic
79C2×Dic52C2311yescharacteristic
80#88143—41no—
81#88143—41no—
82#88133—41no—
83#88133—41no—
84#88124—21no—
85#88123—21no—
86C4×Dic26—21no—
87C4×Dic26—21no—
88C13×#107—21no—
89C13×#107C2311yescharacteristic
90C13×#107C2311yescharacteristic
91C13×#103—21no—
92C13×#103—21no—
93C13×#102—21no—
94C13×#102—21no—
95C13×#97D411yescharacteristic
96C13×#97D411yescharacteristic
97C4×C8×C13C2311yescharacteristic
98C4×C8×C13C2311yescharacteristic
99#88069—41no—
100#88069—41no—
101#88067—41no—
102#88067—41no—
103#88058C2311yescharacteristic
104#88058C2311yescharacteristic
105#88057—21no—
106#88056—21no—
107#88056—21no—
108#88056—21no—
109#88056—21no—
110#88052—41no—
111C2×#6090—41no—
112C2×#6090C2311yescharacteristic
113C2×#6090C2311yescharacteristic
114C2×#6090C2311yescharacteristic
115C2×#6090C2311yescharacteristic
116#33824—131nomaximal, Sylow
117C2×Q8×C13—21no—
118C2×Q8×C13—41no—
119C2×D4×C13—21no—
120C2×D4×C13—41no—
121C22×C4×C13—41no—
122C22×C4×C13C2×D411yescharacteristic
123C2×Dic26—21no—
124C2×Dic26—21no—
125C13×Q16—21no—
126C13×Q16—21no—
127C13×SD16—42no—
128C13×D8—21no—
129C13×D8—21no—
130C13×M16—41no—
131C2×C8×C13—21no—
132C2×C8×C13—21no—
133C2×C8×C13C2×D411yescharacteristic
134C2×C8×C13C2×D411yescharacteristic
135C2×C8×C13C2×D411yescharacteristic
136C2×C8×C13C2411yescharacteristic
137C13×C4⋊C4—21no—
138C13×C4⋊C4—21no—
139C13×C4⋊C4—41no—
140C13×C4⋊C4—41no—
141C13×C4⋊C4—41no—
142C13×C4⋊C4D811yescharacteristic
143C13×C4⋊C4D811yescharacteristic
144C13×C22⋊C4—41no—
145C13×C22⋊C4—41no—
146C13×C22⋊C4—41no—
147C13×C22⋊C4—41no—
148C13×C22⋊C4—41no—
149C13×C22⋊C4D811yescharacteristic
150C13×C22⋊C4D811yescharacteristic
151C42×C13—41no—
152C42×C13C2×D411yescharacteristic
153C42×C13C2×D411yescharacteristic
154C22⋊Dic13—41no—
155C22⋊Dic13—41no—
156C4⋊Dic13—21no—
157C4⋊Dic13—21no—
158C4⋊Dic13—41no—
159C4⋊Dic13—41no—
160#6101—41no—
161#6101—41no—
162#6101—41no—
163#6101—41no—
164#6101—41no—
165#6101—41no—
166C4×Dic13—41no—
167C4×Dic13—41no—
168Dic52—21no—
169Dic52—21no—
170Dic52—21no—
171Dic52—21no—
172Dic52—42no—
173#6090—21no—
174#6090—21no—
175#6090—21no—
176#6090—21no—
177#3177—131no—
178#3175D1311yescharacteristic
179#2229—131no—
180#2228—131no—
181#2227—131no—
182#2225—131no—
183#2220—131no—
184#2219—131no—
185#2217—131no—
186#2216—131no—
187#2211—131no—
188#2211—131no—
189#2209—131no—
190#2208—131no—
191#2151—131no—
192C23×C13—41no—
193C23×C13C2×D811yescharacteristic
194Q8×C13—41no—
195Q8×C13—41no—
196Q8×C13—161no—
197D4×C13—41no—
198D4×C13—41no—
199D4×C13—161no—
200C2×C4×C13—21no—
201C2×C4×C13—21no—
202C2×C4×C13—41no—
203C2×C4×C13—41no—
204C2×C4×C13—41no—
205C2×C4×C13—161no—
206C2×C4×C13C2×D811yescharacteristic
207C2×C4×C13C2×D811yescharacteristic
208C2×C4×C13C2×D811yescharacteristic
209C2×C4×C13C2×D811yescharacteristic
210C2×C4×C13C2×D811yescharacteristic
211C2×C4×C13C22×D411yescharacteristic
212C2×Dic13—41no—
213C2×Dic13—41no—
214C2×Dic13—41no—
215C2×Dic13—41no—
216Dic26—41no—
217Dic26—41no—
218Dic26—41no—
219Dic26—41no—
220C8×C13—21no—
221C8×C13—21no—
222C8×C13—21no—
223C8×C13H(5,𝔽2)11yescharacteristic
224C8×C132−1+411yescharacteristic
225#542—21no—
226#541—261no—
227C2×Q32—131no—
228C2×Q32—131no—
229C2×SD32—131no—
230C2×SD32—131no—
231C2×M32—131no—
232C22×C16—131no—
233#500C2×D1311yescharacteristic
234#499—131no—
235#495C2×D1311yescharacteristic
236#494—131no—
237#491—261no—
238#489—21no—
239#470—261no—
240#468—21no—
241#467—131no—
242#467—131no—
243#467C2×D1311yescharacteristic
244#466C2×D1311yescharacteristic
245#441—261no—
246#440—21no—
247C4×Q16—131no—
248C4×Q16—131no—
249C4×Q16C2×D1311yescharacteristic
250C4×D8C2×D1311yescharacteristic
251#405C2×D1311yescharacteristic
252#367—131no—
253#367—131no—
254#366—131no—
255#366—131no—
256C4⋊C16—131no—
257C4⋊C16—131no—
258C4⋊C16—131no—
259C4⋊C16—131no—
260#358—131no—
261#358—131no—
262#358—131no—
263#358—131no—
264#358—131no—
265#358—131no—
266#357—131no—
267#357—131no—
268C22×C13—21no—
269C22×C13—21no—
270C22×C13—21no—
271C22×C13—81no—
272C22×C13—81no—
273C22×C13—81no—
274C22×C13C22×D811yescharacteristic
275C4×C13—21no—
276C4×C13—21no—
277C4×C13—21no—
278C4×C13—21no—
279C4×C13—21no—
280C4×C13—81no—
281C4×C13—81no—
282C4×C13—81no—
283C4×C13#53711yescharacteristic
284C4×C13#57511yescharacteristic
285Dic13—81no—
286Dic13—81no—
287Dic13—81no—
288Dic13—81no—
289C2×Q16—131no—
290C2×Q16—131no—
291C2×Q16—261no—
292C2×Q16C22×D1311yescharacteristic
293C2×SD16—21no—
294C2×D8C22×D1311yescharacteristic
295C2×M16C22⋊D1311yescharacteristic
296C22×C8C22×D1311yescharacteristic
297C4⋊Q8—21no—
298C4⋊Q8—261no—
299#126—41no—
300#126—521no—
301#126—521no—
302#125—261no—
303#125—521no—
304#125—521no—
305#124—21no—
306#124—41no—
307#123—41no—
308C22⋊Q8—21no—
309C22⋊Q8—41no—
310C22⋊Q8—261no—
311C22⋊Q8—261no—
312#121—21no—
313C4×Q8—21no—
314C4×Q8—261no—
315C4×Q8—261no—
316C4×D4—21no—
317#117C22⋊D1311yescharacteristic
318Q32—521no—
319Q32—521no—
320SD32—521no—
321SD32—521no—
322M32—521no—
323C2×C16—131no—
324C2×C16—131no—
325C2×C16—131no—
326C2×C16—131no—
327C2×C16—521no—
328#107—21no—
329#107—131no—
330#107—131no—
331#107C22×D1311yescharacteristic
332#107C22×D1311yescharacteristic
333#106—261no—
334#103—21no—
335#103—21no—
336#103—261no—
337#103—261no—
338#103—261no—
339#103—261no—
340#102—21no—
341#102—21no—
342#97C22⋊D1311yescharacteristic
343#97C22⋊D1311yescharacteristic
344C4×C8C22×D1311yescharacteristic
345C4×C8C22×D1311yescharacteristic
346C2×C13—41no—
347C2×C13—161no—
348C2×C13#324511yescharacteristic
349C2×C13#341611yescharacteristic
350C2×C13#341711yescharacteristic
351C2×Q8—21no—
352C2×Q8—41no—
353C2×Q8—261no—
354C2×Q8—261no—
355C2×D4—21no—
356C2×D4—41no—
357C22×C4—41no—
358C22×C4C2×C22⋊D1311yescharacteristic
359Q16—21no—
360Q16—21no—
361Q16—261no—
362Q16—261no—
363Q16—261no—
364Q16—261no—
365Q16—522no—
366SD16—42no—
367D8—21no—
368D8—21no—
369M16—41no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8C2×C22⋊D1311yescharacteristic
373C2×C8C2×C22⋊D1311yescharacteristic
374C2×C8C2×C22⋊D1311yescharacteristic
375C2×C8C23×D1311yescharacteristic
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—261no—
382C4⋊C4—261no—
383C4⋊C4—521no—
384C4⋊C4—521no—
385C4⋊C4—521no—
386C4⋊C4—521no—
387C4⋊C4—521no—
388C4⋊C4—521no—
389C4⋊C4—521no—
390C4⋊C4—521no—
391C4⋊C4#610411yescharacteristic
392C4⋊C4#610411yescharacteristic
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—521no—
399C22⋊C4—521no—
400C22⋊C4#610411yescharacteristic
401C22⋊C4#610411yescharacteristic
402C42—41no—
403C42—521no—
404C42—521no—
405C42C2×C22⋊D1311yescharacteristic
406C42C2×C22⋊D1311yescharacteristic
407C16—261no—
408C16—261no—
409C16—261no—
410C16—261no—
411C13#3382411yescharacteristic, complemented, Sylow
412C23—41no—
413C23C2×#610411yescharacteristic
414Q8—41no—
415Q8—41no—
416Q8—161no—
417Q8—521no—
418Q8—521no—
419Q8—521no—
420Q8—521no—
421D4—41no—
422D4—41no—
423D4—161no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—161no—
430C2×C4—521no—
431C2×C4—521no—
432C2×C4—521no—
433C2×C4—521no—
434C2×C4C2×#610411yescharacteristic
435C2×C4C2×#610411yescharacteristic
436C2×C4C2×#610411yescharacteristic
437C2×C4C2×#610411yescharacteristic
438C2×C4C2×#610411yescharacteristic
439C2×C4C22×C22⋊D1311yescharacteristic
440C8—21no—
441C8—21no—
442C8—21no—
443C8#8825611yescharacteristic
444C8#8825711yescharacteristic
445C22—21no—
446C22—21no—
447C22—21no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22C22×#610411yescharacteristic
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—81no—
458C4—81no—
459C4—81no—
460C4—1041no—
461C4—1041no—
462C4—1041no—
463C4—1041no—
464C4#1171952711yescharacteristic
465C4#1171953811yescharacteristic
466C2—41no—
467C2—161no—
468C2#1312614311yescharacteristic
469C2#1312798311yescharacteristic
470C2#1312798711yescharacteristic
subgroupG/H
centerC22C22×#6104
commutator, cosocleC2×C8×C13C24
FrattiniC2×C8C23×D13
FittingC13×#3175C2
socleC22×C13C22×D8
prime radicalC13×#466C22
hypercenter#3175D13
hypocenterC13#33824
LCSG ▷ C2×C8×C13 ▷ C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3175

Complements

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

NH ≅ G/Ntype
1. #13125495C2semidirect
2. #13117000C2semidirect
3. #13116993C2semidirect
4. #13116985C2semidirect
5. #13116975C2semidirect
7. #13116942C2semidirect
9. #13116925C2semidirect
10. #13116895C2semidirect
11. #13116895C2semidirect
12. #13116886C2semidirect
13. #13116881C2semidirect
14. #13116595C2semidirect
411. C13#33824semidirect

Maximal subgroups

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

subgroupindex
#131254952
#131170002
#131169932
#131169852
#131169752
#131169472
#131169422
#131169302
#131169252
#131168952
#131168862
#131168812
#131165952
C13×#31752
#3382413