Finite Group Databasebeta

← #15841526

Subgroups — #15841526

Order 3328.

932 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13125957C211yesmaximal, characteristic
2#13124867C211yesmaximal, characteristic
3#13124308C211yesmaximal, characteristic
4#13114078C214yesmaximal, complemented
5#13113932C214yesmaximal, complemented
6#13113508C211yesmaximal, characteristic, complemented
7#13113265C211yesmaximal, characteristic, complemented
8#13113231C211yesmaximal, characteristic, complemented
9C2×C13×#557C211yesmaximal, characteristic, complemented
10C13×#557—22no—
11C13×#557—22no—
12C13×#557—22no—
13C13×#557C2214yes—
14C2×C13×C4⋊Q8—21no—
15C2×C13×C4⋊Q8C2211yescharacteristic
16C2×C13×#125—22no—
17C2×C13×#125—21no—
18C2×C4×Q8×C13—21no—
19C2×C4×Q8×C13—21no—
20C2×C4×Q8×C13C2211yescharacteristic
21C2×C4×Q8×C13C2211yescharacteristic
22C2×#88196—21no—
23#11719182C2211yescharacteristic
24#11718888—21no—
25#11718749C2212yes—
26#11718748—22no—
27#11718739C2212yes—
28#11718692C2212yes—
29#11718689—22no—
30#11718688C2212yes—
31#11718688C2212yes—
32C4×#6106—22no—
33#11718681C2212yes—
34#11718629—21no—
35#11718628—22no—
36#11718627—21no—
37#11718586C2211yescharacteristic
38#11718586C2211yescharacteristic
39#11718585C2211yescharacteristic
40#11718139C2214yes—
41Q8⋊#1055C2214yes—
42Q8⋊#1055C2214yes—
43C22×Q8×C13—21no—
44C22×Q8×C13D411yescharacteristic
45C22×Q8×C13D411yescharacteristic
46C2×#6127—21no—
47C13×C4⋊Q8—24no—
48C13×C4⋊Q8—24no—
49C13×C4⋊Q8—21no—
50C13×C4⋊Q8—21no—
51C13×C4⋊Q8D412yes—
52C13×C4⋊Q8C2312yes—
53C13×#125—28no—
54C13×#125—24no—
55C13×#125—28no—
56C13×#125—24no—
57C4×Q8×C13—24no—
58C4×Q8×C13—24no—
59C4×Q8×C13—24no—
60C4×Q8×C13—24no—
61C4×Q8×C13—21no—
62C4×Q8×C13—21no—
63C4×Q8×C13—21no—
64C4×Q8×C13—21no—
65C4×Q8×C13D412yes—
66C4×Q8×C13D412yes—
67C4×Q8×C13C2312yes—
68C4×Q8×C13C2312yes—
69C2×C13×C4⋊C4—22no—
70C2×C13×C4⋊C4—22no—
71C2×C13×C4⋊C4—22no—
72C2×C13×C4⋊C4—21no—
73C2×C13×C4⋊C4—21no—
74C2×C13×C4⋊C4—21no—
75C2×C13×C4⋊C4D411yescharacteristic
76C2×C13×C4⋊C4D411yescharacteristic
77C2×C13×C4⋊C4D412yes—
78C2×C42×C13—22no—
79C2×C42×C13—21no—
80C2×C42×C13—21no—
81C2×C42×C13C2311yescharacteristic
82C2×#6107—22no—
83#88196—44no—
84C2×#6106—22no—
85C2×#6106—22no—
86#88184—21no—
87#88184—21no—
88C2×#6099—21no—
89#88129—21no—
90#88128C2311yescharacteristic
91C4×D52—21no—
92C4×D52—21no—
93#88123C2311yescharacteristic
94#88075—22no—
95C22⋊#1055—21no—
96C22⋊#1055—21no—
97#88050—22no—
98#88049—22no—
99#88049—22no—
100#88049—24no—
101#88048—22no—
102#88047—24no—
103C4⋊#1055—21no—
104C4⋊#1055—21no—
105C4⋊#1055C2312yes—
106C4⋊#1055C2312yes—
107C4×#1055—21no—
108C4×#1055C2312yes—
109#18760—131nomaximal, Sylow
110C2×Q8×C13—24no—
111C2×Q8×C13—21no—
112C2×Q8×C13—21no—
113C2×Q8×C13—22no—
114C2×Q8×C13—22no—
115C2×Q8×C13—22no—
116C2×Q8×C13—22no—
117C2×Q8×C13—42no—
118C2×Q8×C13—42no—
119C2×Q8×C13C2×D412yes—
120C2×Q8×C13C2×D412yes—
121C22×C4×C13—21no—
122C22×C4×C13—21no—
123C22×C4×C13—21no—
124C22×C4×C13—21no—
125C22×C4×C13—22no—
126C22×C4×C13C2×D411yescharacteristic
127C22×C4×C13C2×D411yescharacteristic
128C22×C4×C13C2×D411yescharacteristic
129C2×C22⋊D13—41no—
130#6127—42no—
131#6127—42no—
132C2×D52—21no—
133C2×D52—21no—
134C2×D52—21no—
135C2×C4×D13—41no—
136C2×Dic26—21no—
137C13×C4⋊C4—28no—
138C13×C4⋊C4—28no—
139C13×C4⋊C4—28no—
140C13×C4⋊C4—22no—
141C13×C4⋊C4—22no—
142C13×C4⋊C4—22no—
143C13×C4⋊C4—21no—
144C13×C4⋊C4—21no—
145C13×C4⋊C4—22no—
146C13×C4⋊C4—22no—
147C13×C4⋊C4—22no—
148C13×C4⋊C4—22no—
149C13×C4⋊C4C2×D412yes—
150C13×C4⋊C4C2×D412yes—
151C13×C4⋊C4C2×D414yes—
152C42×C13—28no—
153C42×C13—22no—
154C42×C13—22no—
155C42×C13—22no—
156C42×C13—22no—
157C42×C13C2×D411yescharacteristic
158C42×C13C2×D411yescharacteristic
159C42×C13C2×D411yescharacteristic
160C42×C13C2411yescharacteristic
161#6107—44no—
162#6106—44no—
163#6106—44no—
164#6103—41no—
165#6103—41no—
166C4⋊Dic13—21no—
167C4⋊Dic13—21no—
168#6099—42no—
169C2×#1055—22no—
170C2×#1055—22no—
171C2×#1055—22no—
172C2×#557D1311yescharacteristic, complemented
173#3226—131no—
174#3103—131no—
175#3037—131no—
176#1648—134no—
177#1626—134no—
178#1546—131no—
179#1487—131no—
180#1479—131no—
181C23×C13#12011yescharacteristic
182C22×D13—41no—
183Q8×C13—24no—
184Q8×C13—22no—
185Q8×C13—22no—
186Q8×C13—22no—
187Q8×C13—42no—
188Q8×C13—42no—
189Q8×C13—42no—
190Q8×C13—44no—
191Q8×C13#13714yescomplemented
192C2×C4×C13—24no—
193C2×C4×C13—22no—
194C2×C4×C13—22no—
195C2×C4×C13—22no—
196C2×C4×C13—22no—
197C2×C4×C13—22no—
198C2×C4×C13—24no—
199C2×C4×C13—41no—
200C2×C4×C13—42no—
201C2×C4×C13—42no—
202C2×C4×C13—42no—
203C2×C4×C13—44no—
204C2×C4×C13#12011yescharacteristic
205C2×C4×C13#12011yescharacteristic
206C2×C4×C13#12011yescharacteristic
207C2×C4×C13#13512yes—
208C2×C4×C13AGL(1,ℤ/8)12yes—
209C2×C4×C13C22×D411yescharacteristic
210C2×C4×C13C22×D411yescharacteristic
211C2×C4×C13C22×D411yescharacteristic
212C22⋊D13—82no—
213C2×Dic13—41no—
214D52—42no—
215D52—41no—
216D52—41no—
217D52—42no—
218C4×D13—82no—
219Dic26—41no—
220Dic26—41no—
221#1055—42no—
222#1055—42no—
223#1055—42no—
224C2×#137—261no—
225#557—22no—
226#557—22no—
227#557—22no—
228#557C2×D1314yes—
229#532—131no—
230C2×C4⋊Q8—21no—
231C2×C4⋊Q8C2×D1311yescharacteristic
232C2×#125—22no—
233C2×#125—21no—
234C2×C4×Q8—21no—
235C2×C4×Q8—21no—
236C2×C4×Q8C2×D1311yescharacteristic
237C2×C4×Q8C2×D1311yescharacteristic
238#500—132no—
239#486—132no—
240#481—261no—
241#480—262no—
242#475—132no—
243#474—262no—
244#464—262no—
245#462—132no—
246#460—132no—
247#460—132no—
248#448—261no—
249C4×SD16—262no—
250#431—131no—
251#431—131no—
252#423—131no—
253#417—261no—
254#332—134no—
255Q8⋊C8—134no—
256Q8⋊C8—134no—
257C22×C13—21no—
258C22×C13—21no—
259C22×C13#44912yes—
260C22×C13C2×#12011yescharacteristic
261C2×D13—81no—
262C2×D13—81no—
263C2×D13—81no—
264C4×C13—24no—
265C4×C13—21no—
266C4×C13—21no—
267C4×C13—21no—
268C4×C13—42no—
269C4×C13—44no—
270C4×C13—42no—
271C4×C13—42no—
272C4×C13#44912yes—
273C4×C13C2×#13511yescharacteristic
274C4×C13C2×AGL(1,ℤ/8)11yescharacteristic
275C4×C13C2×#13712yes—
276Dic13—81no—
277C2×#41—261no—
278C22×Q8—21no—
279C22×Q8C22⋊D1311yescharacteristic
280C22×Q8C22⋊D1311yescharacteristic
281#137—524no—
282C2×Q16—262no—
283C2×SD16—262no—
284C2×SD16—262no—
285C2×M16—261no—
286C4⋊Q8—24no—
287C4⋊Q8—24no—
288C4⋊Q8—21no—
289C4⋊Q8—21no—
290C4⋊Q8—131no—
291C4⋊Q8C22⋊D1312yes—
292C4⋊Q8C22×D1312yes—
293#127—131no—
294#125—28no—
295#125—24no—
296#125—28no—
297#125—24no—
298#124—261no—
299#121—261no—
300#121—261no—
301C4×Q8—24no—
302C4×Q8—24no—
303C4×Q8—24no—
304C4×Q8—24no—
305C4×Q8—21no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×Q8—21no—
309C4×Q8C22⋊D1312yes—
310C4×Q8C22⋊D1312yes—
311C4×Q8C22×D1312yes—
312C4×Q8C22×D1312yes—
313C4×D4—261no—
314C4×D4—261no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4C22⋊D1311yescharacteristic
322C2×C4⋊C4C22⋊D1311yescharacteristic
323C2×C4⋊C4C22⋊D1312yes—
324C2×C42—22no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42C22×D1311yescharacteristic
328#107—264no—
329#106—262no—
330C4⋊C8—132no—
331C4⋊C8—132no—
332C4⋊C8—261no—
333C4⋊C8—261no—
334#103—262no—
335#103—262no—
336#102—262no—
337#102—262no—
338#102—264no—
339C22⋊C8—261no—
340C22⋊C8—261no—
341C4×C8—132no—
342C4×C8—261no—
343C2×C13—22no—
344C2×C13C2×#44911yescharacteristic
345C2×C13#300411yescharacteristic
346C2×C13#302011yescharacteristic
347D13—161no—
348#41—522no—
349#41—522no—
350C2×Q8—24no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—261no—
360C2×Q8C2×C22⋊D1312yes—
361C2×Q8C2×C22⋊D1312yes—
362C2×D4—261no—
363C2×D4—261no—
364C2×D4—261no—
365C2×D4—521no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—521no—
372C22×C4C2×C22⋊D1311yescharacteristic
373C22×C4C2×C22⋊D1311yescharacteristic
374C22×C4C2×C22⋊D1311yescharacteristic
375Q16—524no—
376SD16—524no—
377SD16—524no—
378M16—522no—
379C2×C8—262no—
380C2×C8—262no—
381C2×C8—262no—
382C4⋊C4—28no—
383C4⋊C4—28no—
384C4⋊C4—28no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—261no—
395C4⋊C4—261no—
396C4⋊C4C2×C22⋊D1312yes—
397C4⋊C4C2×C22⋊D1312yes—
398C4⋊C4C2×C22⋊D1314yes—
399C22⋊C4—521no—
400C22⋊C4—521no—
401C42—28no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42C2×C22⋊D1311yescharacteristic
407C42C2×C22⋊D1311yescharacteristic
408C42C2×C22⋊D1311yescharacteristic
409C42C23×D1311yescharacteristic
410C13#1876011yescharacteristic, complemented, Sylow
411C23—521no—
412C23#8820711yescharacteristic
413Q8—24no—
414Q8—22no—
415Q8—22no—
416Q8—22no—
417Q8—42no—
418Q8—42no—
419Q8—42no—
420Q8—44no—
421Q8—521no—
422Q8—521no—
423Q8#8819614yescomplemented
424D4—522no—
425D4—522no—
426D4—521no—
427D4—521no—
428D4—1042no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—521no—
442C2×C4—1042no—
443C2×C4#8820312yes—
444C2×C4#8820412yes—
445C2×C4#8820711yescharacteristic
446C2×C4#8820711yescharacteristic
447C2×C4#8820711yescharacteristic
448C2×C4C22×C22⋊D1311yescharacteristic
449C2×C4C22×C22⋊D1311yescharacteristic
450C2×C4C22×C22⋊D1311yescharacteristic
451C8—522no—
452C8—522no—
453C8—522no—
454C22—21no—
455C22—21no—
456C22—1041no—
457C22—1041no—
458C22—1041no—
459C22#1171889912yes—
460C22C2×#8820711yescharacteristic
461C4—21no—
462C4—21no—
463C4—24no—
464C4—21no—
465C4—42no—
466C4—42no—
467C4—42no—
468C4—44no—
469C4—1041no—
470C4#1171889912yes—
471C4C2×#8819612yes—
472C4C2×#8820311yescharacteristic
473C4C2×#8820411yescharacteristic
474C2—22no—
475C2—2081no—
476C2C2×#1171889911yescharacteristic
477C2#1312407011yescharacteristic
478C2#1312420011yescharacteristic
subgroupG/H
centerC22C2×#88207
commutator, cosocleC42×C13C24
FrattiniC42C23×D13
FittingC2×C13×#557C2
socleC22×C13C2×#120
prime radical#11719182C22
hypercenterC2×#557D13
hypocenterC13#18760
LCSG ▷ C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
4. #13114078C2semidirect
5. #13113932C2semidirect
6. #13113508C2semidirect
7. #13113265C2semidirect
8. #13113231C2semidirect
9. C2×C13×#557C2semidirect
172. C2×#557D13semidirect
191. Q8×C13#137semidirect
410. C13#18760semidirect
423. Q8#88196semidirect

Maximal subgroups

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

subgroupindex
#131259572
#131248672
#131243082
#131140782
#131139322
#131135082
#131132652
#131132312
C2×C13×#5572
#1876013