Finite Group Databasebeta

← #16727260

Subgroups — #16727260

Order 3328.

2604 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13129607C211yesmaximal, characteristic
2#13129549C211yesmaximal, characteristic, complemented
3#13129548C211yesmaximal, characteristic, complemented
4#13123749C216yesmaximal, complemented
5#13123742C216yesmaximal, complemented
6#13123735C216yesmaximal, complemented
7C2×#11718484C216yesmaximal, complemented
8#13123445C211yesmaximal, characteristic, complemented
9#13123384C211yesmaximal, characteristic, complemented
10#13123264C211yesmaximal, characteristic, complemented
11C13×#2901C211yesmaximal, characteristic, complemented
12C13×#585C2211yescharacteristic
13#11719648C2211yescharacteristic
14C13×#568C2211yescharacteristic
15C4×C13×#41C2211yescharacteristic
16#11719543C414yescomplemented
17#11719542C414yescomplemented
18#11719541C414yescomplemented
19#11719540C414yescomplemented
20#11719527C2216yes—
21C2×#88204C2216yes—
22#11719524—24no—
23#11719524C2211yescharacteristic
24#11719524C2216yescomplemented
25C2×#88202—24no—
26C2×#88202C2211yescharacteristic
27C2×#88202C2211yescharacteristic
28C2×#88202C2216yescomplemented
29#11719455—24no—
30D13×#131—24no—
31#11719450C2216yescomplemented
32C2×#88161C2216yescomplemented
33C13×#421C2216yescomplemented
34C2×C13×#104C2216yescomplemented
35#11718785C2216yescomplemented
36C2×#88062C2216yescomplemented
37#11718666C2216yescomplemented
38#11718658C2213yescomplemented
39#11718658C2213yescomplemented
40C2×#88045C2216yescomplemented
41C4×#6099C2213yescomplemented
42C2×#88043C2213yescomplemented
43#11718485C22124yescomplemented
44#11718484C2214yescomplemented
45#11718484C22112yescomplemented
46#11718484C22112yescomplemented
47#11718484C22112yescomplemented
48C13×2−1+4—21no—
49C13×2−1+4C2×C414yes—
50C13×H(5,𝔽2)—23no—
51C13×H(5,𝔽2)C2×C414yes—
52C2×C13×#41—24no—
53C2×C13×#41C2316yes—
54C2×C13×#41C2311yescharacteristic
55#88257C2×C414yes—
56#88256C2×C414yes—
57#41×D13—24no—
58#88254—21no—
59#88251—23no—
60C2×#6127C2316yes—
61C13×#131—24no—
62C2×C13×M16C2316yes—
63C4×Q8×C13—21no—
64C4×D4×C13—23no—
65C13×#117C2313yes—
66C2×C42×C13C2313yes—
67#88205C2×C4112yescomplemented
68#88204C2×C4112yescomplemented
69#88204C2×C4112yescomplemented
70#88204C2314yes—
71#88204C23112yes—
72#88203C2×C4112yescomplemented
73#88202—24no—
74#88202—212no—
75#88202—26no—
76#88202—26no—
77#88202C2×C414yes—
78#88202C2×C414yes—
79#88202D414yes—
80#88202D414yes—
81#88202C2314yes—
82#88202C23112yes—
83C2×#6107C2×C416yescomplemented
84#88196C2×C4112yescomplemented
85C2×#6106C2×C416yescomplemented
86C2×#6105C2×C416yescomplemented
87#88186C2×C4112yescomplemented
88C2×#6104C2×C416yescomplemented
89C2×#6099—212no—
90C2×#6099—26no—
91C2×#6099C2316yes—
92C2×#6099C2313yes—
93C2×#6099C2313yes—
94C22×#1055—26no—
95C22×#1055C2313yes—
96C22×#1055C2313yes—
97#88161—26no—
98#88161—26no—
99#88161C2314yes—
100#88161C23112yes—
101M16×D13—212no—
102#88155—212no—
103C2×#6094—26no—
104C2×C8×D13—26no—
105C13×#104C2314yes—
106C13×#104C23112yes—
107#88062C2314yes—
108#88062C23112yes—
109#88045C23112yes—
110#88045C2314yes—
111#88043C2313yes—
112#88043C2313yes—
113#88043C2311yescharacteristic
114#88043C2313yes—
115C4×#1055C2316yes—
116#60184—131nomaximal, Sylow
117C13×#41—24no—
118C13×#41—212no—
119C13×#41—26no—
120C13×#41—26no—
121C13×#41C22×C4112yes—
122C13×#41C22×C414yes—
123C13×#41C2×D414yes—
124C13×#41C2414yes—
125C2×Q8×C13—24no—
126C2×Q8×C13C22×C416yes—
127C2×Q8×C13#4111yescharacteristic
128C2×D4×C13—26no—
129C2×D4×C13—212no—
130C2×D4×C13C22×C416yes—
131C2×D4×C13#4113yes—
132C22×C4×C13—26no—
133C22×C4×C13—23no—
134C22×C4×C13C2413yes—
135C2×C22⋊D13—26no—
136#6131—24no—
137Q8×D13—24no—
138#6129—212no—
139D4×D13—212no—
140#6127—26no—
141#6127—26no—
142#6127C22×C4112yes—
143#6127C2414yes—
144C2×D52C22×C416yes—
145C2×C4×D13—26no—
146C2×Dic26C22×C416yes—
147C13×M16—26no—
148C13×M16C2414yes—
149C2×C8×C13—26no—
150C13×C4⋊C4—23no—
151C13×C22⋊C4—23no—
152C42×C13—23no—
153C42×C13C2411yescharacteristic
154C42×C13C2413yes—
155#6107C22×C4112yes—
156#6107C22×C414yes—
157#6106C22×C414yes—
158#6106C22×C4112yes—
159#6105C22×C4112yes—
160#6105C22×C414yes—
161#6104C22×C414yes—
162#6104C22×C4112yes—
163#6099—26no—
164#6099—24no—
165#6099—212no—
166#6099C22×C4112yes—
167#6099C2×D4112yes—
168#6099C2414yes—
169C2×#1055—26no—
170C2×#1055—24no—
171C2×#1055—212no—
172C2×#1055C22×C416yes—
173C2×#1055C22×C416yes—
174C2×#1055C2×D416yes—
175C2×#1055C2×D416yes—
176C2×#1055C2411yescharacteristic
177C2×#1055C2413yes—
178C2×#1055C2411yescharacteristic
179C2×#1055C2413yes—
180#6094—24no—
181#6094—212no—
182C8×D13—24no—
183C8×D13—212no—
184#3584—131no—
185#3573—131no—
186#3573—131no—
187#2957—136no—
188#2956—136no—
189#2955—136no—
190C2×#444—136no—
191#2916—131no—
192#2901—131no—
193#2901D1311yescharacteristic, complemented
194#2875—131no—
195C23×C13—26no—
196C23×C13C2×#4113yes—
197C22×D13—26no—
198Q8×C13—26no—
199Q8×C13C4×D414yes—
200Q8×C13C23×C414yes—
201D4×C13—212no—
202D4×C13—24no—
203D4×C13—26no—
204D4×C13C4×D4112yes—
205D4×C13C23×C414yes—
206C2×C4×C13—26no—
207C2×C4×C13—212no—
208C2×C4×C13—24no—
209C2×C4×C13—21no—
210C2×C4×C13—23no—
211C2×C4×C13—83no—
212C2×C4×C13C23×C416yes—
213C2×C4×C13C22×D416yes—
214C2×C4×C13C2×#4113yes—
215C2×C4×C13C2511yescharacteristic
216C22⋊D13—212no—
217C22⋊D13—24no—
218C2×Dic13—26no—
219D52—26no—
220D52C23×C414yes—
221C4×D13—212no—
222C4×D13—24no—
223Dic26—26no—
224Dic26C23×C414yes—
225C8×C13—24no—
226#1055—24no—
227#1055C23×C411yescharacteristic
228#1055C23×C411yescharacteristic
229#1055C23×C416yes—
230#1055C22×D411yescharacteristic
231#1055C22×D416yes—
232#1055C22×D411yescharacteristic
233#585—131no—
234#585C2×D1311yescharacteristic
235#578—134no—
236#577—134no—
237#577—134no—
238#576—134no—
239#575—136no—
240C2×#135—136no—
241#568—131no—
242#568—136no—
243#568—136no—
244#568—264no—
245#568—264no—
246#568C2×D1311yescharacteristic
247C2×#131—131no—
248C2×#131—131no—
249C2×#131—136no—
250C2×#131—136no—
251C2×#131—264no—
252C2×#131—264no—
253C4×#41C2×D1311yescharacteristic
254#444—134no—
255#444—1312no—
256#444—1312no—
257#444—1312no—
258#443—1324no—
259#430—136no—
260C2×#108—136no—
261#421—136no—
262#421C2×D1316yescomplemented
263C2×#104—136no—
264C2×#104C2×D1316yescomplemented
265#405—133no—
266#405—133no—
267C4×M16—133no—
268C2×#97—133no—
269C22×C13—24no—
270C22×C13—43no—
271C22×C13—86no—
272C22×C13C2×C4×D416yes—
273C22×C13C22×#4111yescharacteristic
274C2×D13—24no—
275C2×D13—86no—
276C4×C13—24no—
277C4×C13—43no—
278C4×C13—41no—
279C4×C13C2×C4×D416yes—
280C4×C13C24×C411yescharacteristic
281C4×C13C23×D411yescharacteristic
282Dic13—24no—
2832−1+4—21no—
2842−1+4—134no—
2852−1+4—261no—
2862−1+4C4×D1314yes—
287H(5,𝔽2)—23no—
288H(5,𝔽2)—134no—
289H(5,𝔽2)—263no—
290H(5,𝔽2)C4×D1314yes—
291C2×#41—24no—
292C2×#41—136no—
293C2×#41—264no—
294C2×#41C22×D1316yes—
295C2×#41C22×D1311yescharacteristic
296#137—1312no—
297#137—1312no—
298AGL(1,ℤ/8)—1312no—
299AGL(1,ℤ/8)—1312no—
300#135—1312no—
301#135—1312no—
302#135—134no—
303#135—1312no—
304C2×Q16—136no—
305C2×SD16—136no—
306C2×SD16—136no—
307C2×D8—136no—
308#131—24no—
309#131—134no—
310#131—134no—
311#131—134no—
312#131—134no—
313#131—134no—
314#131—1312no—
315#131—134no—
316#131—1312no—
317#131—2612no—
318#131—266no—
319#131—266no—
320#131—266no—
321#131—266no—
322#131—2612no—
323#131—264no—
324C2×M16—133no—
325C2×M16—133no—
326C2×M16—136no—
327C2×M16—266no—
328C2×M16—266no—
329C2×M16—2612no—
330C2×M16—2612no—
331C2×M16C22×D1316yes—
332C22×C8—133no—
333C22×C8—133no—
334C22×C8—266no—
335C22×C8—266no—
336C4×Q8—21no—
337C4×D4—23no—
338#117C22×D1313yes—
339C2×C42C22×D1313yes—
340#108—134no—
341#108—1312no—
342#104—134no—
343#104—1312no—
344#104C22×D1314yes—
345#104C22×D13112yes—
346#97—131no—
347#97—133no—
348#97—133no—
349#97—133no—
350C4×C8—136no—
351C2×C13—21no—
352C2×C13—26no—
353C2×C13—44no—
354C2×C13C22×C4×D411yescharacteristic
355D13—44no—
356#41—24no—
357#41—212no—
358#41—26no—
359#41—26no—
360#41—1312no—
361#41—134no—
362#41—264no—
363#41—2612no—
364#41—266no—
365#41—266no—
366#41C2×C4×D13112yes—
367#41C2×C4×D1314yes—
368#41D4×D1314yes—
369#41C23×D1314yes—
370C2×Q8—24no—
371C2×Q8—136no—
372C2×Q8—264no—
373C2×Q8C2×C4×D1316yes—
374C2×Q8#612911yescharacteristic
375C2×D4—26no—
376C2×D4—212no—
377C2×D4—136no—
378C2×D4—266no—
379C2×D4—2612no—
380C2×D4C2×C4×D1316yes—
381C2×D4#612913yes—
382C22×C4—26no—
383C22×C4—23no—
384C22×C4—266no—
385C22×C4C23×D1313yes—
386Q16—134no—
387Q16—1312no—
388SD16—134no—
389SD16—1312no—
390SD16—134no—
391SD16—1312no—
392D8—134no—
393D8—1312no—
394M16—26no—
395M16—1312no—
396M16—1312no—
397M16—134no—
398M16—264no—
399M16—2612no—
400M16—264no—
401M16—2612no—
402M16—266no—
403M16C23×D1314yes—
404C2×C8—26no—
405C2×C8—136no—
406C2×C8—136no—
407C2×C8—136no—
408C2×C8—136no—
409C2×C8—133no—
410C2×C8—133no—
411C2×C8—131no—
412C2×C8—131no—
413C2×C8—264no—
414C2×C8—2612no—
415C2×C8—264no—
416C2×C8—2612no—
417C2×C8—266no—
418C4⋊C4—23no—
419C22⋊C4—23no—
420C42—23no—
421C42C23×D1313yes—
422C42C23×D1311yescharacteristic
423C13#6018411yescharacteristic, complemented, Sylow
424C23—26no—
425C23—266no—
426C23C2×#612913yes—
427Q8—26no—
428Q8—134no—
429Q8—266no—
430Q8#8813514yes—
431Q8C22×C4×D1314yes—
432D4—212no—
433D4—24no—
434D4—26no—
435D4—134no—
436D4—2612no—
437D4—264no—
438D4—266no—
439D4#88135112yes—
440D4C22×C4×D1314yes—
441C2×C4—26no—
442C2×C4—212no—
443C2×C4—24no—
444C2×C4—21no—
445C2×C4—23no—
446C2×C4—83no—
447C2×C4—266no—
448C2×C4—2612no—
449C2×C4—264no—
450C2×C4C22×C4×D1316yes—
451C2×C4C2×D4×D1316yes—
452C2×C4C2×#612913yes—
453C2×C4C24×D1311yescharacteristic
454C8—24no—
455C8—131no—
456C8—131no—
457C8—136no—
458C8—131no—
459C8—131no—
460C8—136no—
461C8—264no—
462C22—24no—
463C22—43no—
464C22—86no—
465C22—264no—
466C22—1046no—
467C22C2×#8813516yes—
468C22C22×#612911yescharacteristic
469C4—24no—
470C4—43no—
471C4—41no—
472C4—264no—
473C4C2×#8813516yes—
474C4C23×C4×D1311yescharacteristic
475C4C22×D4×D1311yescharacteristic
476C2—21no—
477C2—26no—
478C2—44no—
479C2—524no—
480C2C22×#8813511yescharacteristic
subgroupG/H
centerC4C22×D4×D13
commutatorC4×C13C24×C4
FrattiniC2×C4C24×D13
FittingC13×#2901C2
socleC2×C13C22×C4×D4
prime radical#13129607C2
cosocleC2×C4×C13C25
hypercenter#2901D13
hypocenterC13#60184
LCSG ▷ C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C4 ◁ C2×#41 ◁ #2901

Complements

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

NH ≅ G/Ntype
2. #13129549C2semidirect
3. #13129548C2semidirect
4. #13123749C2semidirect
5. #13123742C2semidirect
6. #13123735C2semidirect
7. C2×#11718484C2semidirect
8. #13123445C2semidirect
9. #13123384C2semidirect
10. #13123264C2semidirect
11. C13×#2901C2semidirect
16. #11719543C4semidirect
17. #11719542C4semidirect
18. #11719541C4semidirect
19. #11719540C4semidirect
24. #11719524C22semidirect
28. C2×#88202C22semidirect
31. #11719450C22semidirect
32. C2×#88161C22semidirect
33. C13×#421C22semidirect
34. C2×C13×#104C22semidirect
35. #11718785C22semidirect
36. C2×#88062C22semidirect
37. #11718666C22semidirect
38. #11718658C22semidirect
39. #11718658C22semidirect
40. C2×#88045C22semidirect
41. C4×#6099C22semidirect
42. C2×#88043C22semidirect
43. #11718485C22semidirect
44. #11718484C22semidirect
45. #11718484C22semidirect
46. #11718484C22semidirect
47. #11718484C22semidirect
67. #88205C2×C4semidirect
68. #88204C2×C4semidirect
69. #88204C2×C4semidirect
72. #88203C2×C4semidirect
83. C2×#6107C2×C4semidirect
84. #88196C2×C4semidirect
85. C2×#6106C2×C4semidirect
86. C2×#6105C2×C4semidirect
87. #88186C2×C4semidirect
88. C2×#6104C2×C4semidirect
193. #2901D13semidirect
262. #421C2×D13semidirect
264. C2×#104C2×D13semidirect
423. C13#60184semidirect

Maximal subgroups

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

subgroupindex
#131296072
#131295492
#131295482
#131237492
#131237422
#131237352
C2×#117184842
#131234452
#131233842
#131232642
C13×#29012
#6018413