Finite Group Databasebeta

← C22×#11718125 #15860371

Subgroups — C22×#11718125 #15860371

Order 3328.

2190 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#6099C211yesmaximal, characteristic
2C23×#6099C211yesmaximal, characteristic
3C2×#11718125C2112yesmaximal, direct product factor
4C22×C13×#117C211yesmaximal, characteristic
5C24×C4×C13C2211yescharacteristic
6C2×C13×#117—23no—
7C2×C13×#117—23no—
8C2×C13×#117C416yes—
9C2×C13×#117C2216yes—
10C22×C13×C4⋊C4—21no—
11C22×C13×C22⋊C4C412yes—
12C22×C42×C13—21no—
13C22×#6099—21no—
14C22×#6099—23no—
15C22×#6099—23no—
16C22×#6099—21no—
17C22×#6099—23no—
18C22×#6099—21no—
19C22×#6099—23no—
20C22×#6099C412yescomplemented
21C22×#6099C416yescomplemented
22C22×#6099C416yescomplemented
23C22×#6099C2216yes—
24C22×#6099C2216yes—
25C23×#1055—21no—
26C23×#1055C412yescomplemented
27#11718125C22116yesdirect product factor
28C25×C13C2×C411yescharacteristic
29C23×C4×C13—22no—
30C23×C4×C13C2×C411yescharacteristic
31C23×C4×C13C2×C411yescharacteristic
32C23×C4×C13C2×C413yes—
33C23×C4×C13C2×C413yes—
34C23×C4×C13C2×C413yes—
35C23×C4×C13D411yescharacteristic
36C23×C4×C13D413yes—
37C23×C4×C13D411yescharacteristic
38C23×C4×C13D411yescharacteristic
39C23×C4×C13D413yes—
40C23×C4×C13D413yes—
41C23×C4×C13Q811yescharacteristic
42C23×C4×C13Q813yes—
43C23×C4×C13C2313yes—
44C13×#117—212no—
45C13×#117—26no—
46C13×#117—26no—
47C13×#117C2×C4112yes—
48C13×#117C2314yes—
49C2×C13×C4⋊C4—26no—
50C2×C13×C4⋊C4—26no—
51C2×C13×C22⋊C4—26no—
52C2×C13×C22⋊C4C2×C4112yes—
53C2×C42×C13—26no—
54C2×C42×C13—26no—
55C2×#6099—26no—
56C2×#6099—26no—
57C2×#6099—212no—
58C2×#6099—26no—
59C2×#6099—26no—
60C2×#6099—212no—
61C2×#6099—26no—
62C2×#6099—26no—
63C2×#6099—26no—
64C2×#6099—26no—
65C2×#6099—26no—
66C2×#6099—26no—
67C2×#6099—26no—
68C2×#6099C2×C4112yescomplemented
69C2×#6099C2×C4112yescomplemented
70C2×#6099C2×C4112yescomplemented
71C2×#6099C2314yes—
72C2×#6099C2314yes—
73C22×#1055—26no—
74C22×#1055—22no—
75C22×#1055—26no—
76C22×#1055—22no—
77C22×#1055—26no—
78C22×#1055C2×C4112yescomplemented
79C22×#344—131nomaximal, Sylow
80C24×C13—44no—
81C24×C13C4211yescharacteristic
82C24×C13C4213yes—
83C24×C13C22⋊C411yescharacteristic
84C24×C13C22⋊C413yes—
85C24×C13C4⋊C411yescharacteristic
86C24×C13C4⋊C413yes—
87C24×C13C22×C413yes—
88C22×C4×C13—212no—
89C22×C4×C13—48no—
90C22×C4×C13C4213yes—
91C22×C4×C13C4213yes—
92C22×C4×C13C4216yes—
93C22×C4×C13C22⋊C413yes—
94C22×C4×C13C22⋊C413yes—
95C22×C4×C13C22⋊C411yescharacteristic
96C22×C4×C13C22⋊C413yes—
97C22×C4×C13C22⋊C413yes—
98C22×C4×C13C22⋊C413yes—
99C22×C4×C13C22⋊C413yes—
100C22×C4×C13C22⋊C416yes—
101C22×C4×C13C22⋊C416yes—
102C22×C4×C13C22⋊C411yescharacteristic
103C22×C4×C13C22⋊C413yes—
104C22×C4×C13C22⋊C413yes—
105C22×C4×C13C22⋊C416yes—
106C22×C4×C13C4⋊C413yes—
107C22×C4×C13C4⋊C413yes—
108C22×C4×C13C4⋊C413yes—
109C22×C4×C13C4⋊C413yes—
110C22×C4×C13C4⋊C413yes—
111C22×C4×C13C4⋊C413yes—
112C22×C4×C13C4⋊C416yes—
113C22×C4×C13C4⋊C416yes—
114C22×C4×C13C4⋊C411yescharacteristic
115C22×C4×C13C4⋊C411yescharacteristic
116C22×C4×C13C4⋊C413yes—
117C22×C4×C13C4⋊C413yes—
118C22×C4×C13C4⋊C416yes—
119C22×C4×C13C22×C413yes—
120C22×C4×C13C22×C413yes—
121C22×C4×C13C22×C413yes—
122C22×C4×C13C22×C413yes—
123C22×C4×C13C22×C413yes—
124C22×C4×C13C2×D413yes—
125C22×C4×C13C2×D413yes—
126C22×C4×C13C2×D413yes—
127C22×C4×C13C2×D413yes—
128C22×C4×C13C2×D413yes—
129C22×C4×C13C2×D413yes—
130C22×C4×C13C2×Q813yes—
131C22×C4×C13C2×Q813yes—
132C22×C4×C13C2411yescharacteristic
133C13×C4⋊C4—212no—
134C13×C4⋊C4—24no—
135C13×C22⋊C4—212no—
136C13×C22⋊C4C22×C418yes—
137C42×C13—212no—
138C42×C13—24no—
139#6099—212no—
140#6099—212no—
141#6099—212no—
142#6099—212no—
143#6099—24no—
144#6099—24no—
145#6099—24no—
146#6099C22×C418yescomplemented
147C2×#1055—212no—
148C2×#1055—212no—
149C2×#1055—212no—
150C2×#1055—212no—
151C2×#1055—24no—
152C2×#1055C22×C418yescomplemented
153C23×M16—131no—
154C23×M16—131no—
155C22×#117D1311yescharacteristic
156C2×#344—1312no—
157C23×C13—24no—
158C23×C13—24no—
159C23×C13—24no—
160C23×C13—46no—
161C23×C13—46no—
162C23×C13—46no—
163C23×C13—46no—
164C23×C13#9513yes—
165C23×C13#9513yes—
166C23×C13#9513yes—
167C23×C13#9516yes—
168C23×C13#9511yescharacteristic
169C23×C13C2×C4213yes—
170C23×C13C2×C4213yes—
171C23×C13C2×C22⋊C413yes—
172C23×C13C2×C22⋊C413yes—
173C23×C13C2×C4⋊C413yes—
174C23×C13C2×C4⋊C413yes—
175C23×C13C23×C411yescharacteristic
176C2×C4×C13—28no—
177C2×C4×C13—424no—
178C2×C4×C13#9516yes—
179C2×C4×C13#9516yes—
180C2×C4×C13#9516yes—
181C2×C4×C13#9516yes—
182C2×C4×C13#9516yes—
183C2×C4×C13#9516yes—
184C2×C4×C13#9513yes—
185C2×C4×C13#9513yes—
186C2×C4×C13#9513yes—
187C2×C4×C13#9513yes—
188C2×C4×C13C2×C4213yes—
189C2×C4×C13C2×C4213yes—
190C2×C4×C13C2×C22⋊C413yes—
191C2×C4×C13C2×C22⋊C413yes—
192C2×C4×C13C2×C22⋊C413yes—
193C2×C4×C13C2×C22⋊C413yes—
194C2×C4×C13C2×C22⋊C413yes—
195C2×C4×C13C2×C22⋊C413yes—
196C2×C4×C13C2×C22⋊C413yes—
197C2×C4×C13C2×C22⋊C413yes—
198C2×C4×C13C2×C22⋊C413yes—
199C2×C4×C13C2×C22⋊C413yes—
200C2×C4×C13C2×C4⋊C413yes—
201C2×C4×C13C2×C4⋊C413yes—
202C2×C4×C13C2×C4⋊C413yes—
203C2×C4×C13C2×C4⋊C413yes—
204C2×C4×C13C2×C4⋊C413yes—
205C2×C4×C13C2×C4⋊C413yes—
206C2×C4×C13C2×C4⋊C413yes—
207C2×C4×C13C2×C4⋊C413yes—
208C2×C4×C13C2×C4⋊C413yes—
209C2×C4×C13C2×C4⋊C413yes—
210C2×C4×C13C23×C411yescharacteristic
211C2×C4×C13C23×C411yescharacteristic
212C2×C4×C13C22×D411yescharacteristic
213C2×C4×C13C22×D411yescharacteristic
214C2×C4×C13C22×D411yescharacteristic
215C2×C4×C13C22×Q811yescharacteristic
216#1055—28no—
217#1055—28no—
218C24×C4C2×D1311yescharacteristic
219C22×M16—136no—
220C22×M16—136no—
221C22×M16—132no—
222C22×M16—136no—
223C22×M16—136no—
224C22×M16—263no—
225C22×M16—261no—
226C22×M16—263no—
227C22×M16—263no—
228C22×M16—261no—
229C22×M16—263no—
230C22×M16—261no—
231C23×C8—132no—
232C23×C8—261no—
233C2×#117—23no—
234C2×#117—23no—
235C2×#117Dic1316yes—
236C2×#117C2×D1316yes—
237C22×C4⋊C4—21no—
238C22×C22⋊C4Dic1312yes—
239C22×C42—21no—
240#344—1316no—
241C22×C13—26no—
242C22×C13—26no—
243C22×C13—26no—
244C22×C13—26no—
245C22×C13—26no—
246C22×C13—26no—
247C22×C13—46no—
248C22×C13—43no—
249C22×C13—43no—
250C22×C13—43no—
251C22×C13—41no—
252C22×C13#34414yescomplemented
253C22×C13C2×#9513yes—
254C22×C13C2×#9513yes—
255C22×C13C2×#9513yes—
256C22×C13C2×#9513yes—
257C22×C13C22×C4211yescharacteristic
258C22×C13C22×C22⋊C411yescharacteristic
259C22×C13C22×C4⋊C411yescharacteristic
260C4×C13—48no—
261C4×C13C2×#9513yes—
262C4×C13C2×#9513yes—
263C4×C13C2×#9513yes—
264C4×C13C2×#9513yes—
265C4×C13C22×C22⋊C411yescharacteristic
266C4×C13C22×C22⋊C411yescharacteristic
267C4×C13C22×C4⋊C411yescharacteristic
268C4×C13C22×C4⋊C411yescharacteristic
269C25C2×Dic1311yescharacteristic
270C23×C4—22no—
271C23×C4Dic2611yescharacteristic
272C23×C4Dic2613yes—
273C23×C4C4×D1311yescharacteristic
274C23×C4C4×D1313yes—
275C23×C4C4×D1313yes—
276C23×C4C4×D1311yescharacteristic
277C23×C4D5213yes—
278C23×C4D5211yescharacteristic
279C23×C4C2×Dic1313yes—
280C23×C4C22⋊D1311yescharacteristic
281C23×C4C22⋊D1313yes—
282C23×C4C22⋊D1313yes—
283C23×C4C22⋊D1311yescharacteristic
284C23×C4C22×D1313yes—
285C2×M16—1312no—
286C2×M16—134no—
287C2×M16—1312no—
288C2×M16—1312no—
289C2×M16—134no—
290C2×M16—2612no—
291C2×M16—266no—
292C2×M16—266no—
293C2×M16—266no—
294C2×M16—266no—
295C2×M16—2612no—
296C2×M16—266no—
297C2×M16—266no—
298C2×M16—266no—
299C2×M16—266no—
300C2×M16—266no—
301C2×M16—266no—
302C2×M16—266no—
303C22×C8—1312no—
304C22×C8—266no—
305C22×C8—262no—
306C22×C8—266no—
307C22×C8—262no—
308C22×C8—266no—
309#117—212no—
310#117—26no—
311#117—26no—
312#117C2×Dic13112yes—
313#117C22×D1314yes—
314C2×C4⋊C4—26no—
315C2×C4⋊C4—26no—
316C2×C22⋊C4—26no—
317C2×C22⋊C4C2×Dic13112yes—
318C2×C42—26no—
319C2×C42—26no—
320C2×C13—21no—
321C2×C13—21no—
322C2×C13—23no—
323C2×C13—23no—
324C2×C13—23no—
325C2×C13—21no—
326C2×C13C2×#34416yescomplemented
327C2×C13C22×#9511yescharacteristic
328C24—44no—
329C24C4×Dic1311yescharacteristic
330C24C4×Dic1313yes—
331C24C4⋊Dic1311yescharacteristic
332C24C4⋊Dic1313yes—
333C24C22⋊Dic1311yescharacteristic
334C24C22⋊Dic1313yes—
335C24C22×Dic1313yes—
336C22×C4—212no—
337C22×C4—48no—
338C22×C4C4×Dic1313yes—
339C22×C4C4×Dic1316yes—
340C22×C4C4×Dic1313yes—
341C22×C4#610113yes—
342C22×C4#610113yes—
343C22×C4#610113yes—
344C22×C4#610113yes—
345C22×C4#610116yes—
346C22×C4#610116yes—
347C22×C4#610111yescharacteristic
348C22×C4#610111yescharacteristic
349C22×C4#610113yes—
350C22×C4#610113yes—
351C22×C4C4⋊Dic1313yes—
352C22×C4C4⋊Dic1316yes—
353C22×C4C4⋊Dic1313yes—
354C22×C4#610311yescharacteristic
355C22×C4#610313yes—
356C22×C4#610313yes—
357C22×C4#610313yes—
358C22×C4#610313yes—
359C22×C4#610316yes—
360C22×C4#610316yes—
361C22×C4#610313yes—
362C22×C4#610313yes—
363C22×C4#610311yescharacteristic
364C22×C4C22⋊Dic1313yes—
365C22×C4C22⋊Dic1313yes—
366C22×C4C22⋊Dic1316yes—
367C22×C4C2×Dic2613yes—
368C22×C4C2×Dic2613yes—
369C22×C4C2×C4×D1313yes—
370C22×C4C2×C4×D1313yes—
371C22×C4C2×C4×D1313yes—
372C22×C4C2×C4×D1313yes—
373C22×C4C2×D5213yes—
374C22×C4C2×D5213yes—
375C22×C4C22×Dic1313yes—
376C22×C4C2×C22⋊D1313yes—
377C22×C4C2×C22⋊D1313yes—
378C22×C4C2×C22⋊D1313yes—
379C22×C4C2×C22⋊D1313yes—
380C22×C4C23×D1311yescharacteristic
381M16—138no—
382M16—2612no—
383M16—2612no—
384M16—2612no—
385M16—2612no—
386M16—264no—
387M16—264no—
388M16—264no—
389C2×C8—138no—
390C2×C8—2612no—
391C2×C8—2612no—
392C2×C8—2612no—
393C2×C8—2612no—
394C2×C8—264no—
395C4⋊C4—212no—
396C4⋊C4—24no—
397C22⋊C4—212no—
398C22⋊C4C22×Dic1318yes—
399C42—212no—
400C42—24no—
401C13C22×#34411yescharacteristic, complemented, Sylow
402C23—24no—
403C23—24no—
404C23—24no—
405C23—46no—
406C23—46no—
407C23—46no—
408C23—46no—
409C23#8807113yes—
410C23#8807113yes—
411C23#8807113yes—
412C23#8807116yes—
413C23#8807111yescharacteristic
414C23C2×C4×Dic1313yes—
415C23C2×C4×Dic1313yes—
416C23C2×C4⋊Dic1313yes—
417C23C2×C4⋊Dic1313yes—
418C23C2×C22⋊Dic1313yes—
419C23C2×C22⋊Dic1313yes—
420C23C23×Dic1311yescharacteristic
421C2×C4—28no—
422C2×C4—424no—
423C2×C4#8807116yes—
424C2×C4#8807116yes—
425C2×C4#8807116yes—
426C2×C4#8807116yes—
427C2×C4#8807116yes—
428C2×C4#8807116yes—
429C2×C4#8807113yes—
430C2×C4#8807113yes—
431C2×C4#8807113yes—
432C2×C4#8807113yes—
433C2×C4C2×C4×Dic1313yes—
434C2×C4C2×C4×Dic1313yes—
435C2×C4C2×#610113yes—
436C2×C4C2×#610113yes—
437C2×C4C2×#610113yes—
438C2×C4C2×#610113yes—
439C2×C4C2×#610113yes—
440C2×C4C2×#610113yes—
441C2×C4C2×#610113yes—
442C2×C4C2×#610113yes—
443C2×C4C2×C4⋊Dic1313yes—
444C2×C4C2×C4⋊Dic1313yes—
445C2×C4C2×#610313yes—
446C2×C4C2×#610313yes—
447C2×C4C2×#610313yes—
448C2×C4C2×#610313yes—
449C2×C4C2×#610313yes—
450C2×C4C2×#610313yes—
451C2×C4C2×#610313yes—
452C2×C4C2×#610313yes—
453C2×C4C2×C22⋊Dic1313yes—
454C2×C4C2×C22⋊Dic1313yes—
455C2×C4C22×Dic2611yescharacteristic
456C2×C4C22×C4×D1311yescharacteristic
457C2×C4C22×C4×D1311yescharacteristic
458C2×C4C22×D5211yescharacteristic
459C2×C4C22×C22⋊D1311yescharacteristic
460C2×C4C22×C22⋊D1311yescharacteristic
461C8—268no—
462C8—268no—
463C22—26no—
464C22—26no—
465C22—26no—
466C22—26no—
467C22—26no—
468C22—26no—
469C22—46no—
470C22—43no—
471C22—43no—
472C22—43no—
473C22—41no—
474C22#1171812514yesdirect product factor
475C22C2×#8807113yes—
476C22C2×#8807113yes—
477C22C2×#8807113yes—
478C22C2×#8807113yes—
479C22C22×C4×Dic1311yescharacteristic
480C22C22×C4⋊Dic1311yescharacteristic
481C22C22×C22⋊Dic1311yescharacteristic
482C4—48no—
483C4C2×#8807113yes—
484C4C2×#8807113yes—
485C4C2×#8807113yes—
486C4C2×#8807113yes—
487C4C22×#610111yescharacteristic
488C4C22×#610111yescharacteristic
489C4C22×#610311yescharacteristic
490C4C22×#610311yescharacteristic
491C2—21no—
492C2—21no—
493C2—21no—
494C2—23no—
495C2—23no—
496C2—23no—
497C2C2×#1171812516yesdirect product factor
498C2C22×#8807111yescharacteristic
subgroupG/H
centerC22×C4#6103
commutatorC22×C13C22×C42
FrattiniC22×C4C23×D13
FittingC22×C13×#117C2
socleC23×C13#95
prime radicalC25×C13C2×C4
cosocleC22×C4×C13C24
hypercenterC22×#117D13
hypocenterC13C22×#344
LCSG ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
3. C2×#11718125C2direct (N × H)
20. C22×#6099C4semidirect
21. C22×#6099C4semidirect
22. C22×#6099C4semidirect
26. C23×#1055C4semidirect
27. #11718125C22direct (N × H)
68. C2×#6099C2×C4semidirect
69. C2×#6099C2×C4semidirect
70. C2×#6099C2×C4semidirect
78. C22×#1055C2×C4semidirect
146. #6099C22×C4semidirect
152. C2×#1055C22×C4semidirect
252. C22×C13#344semidirect
326. C2×C13C2×#344semidirect
401. C13C22×#344semidirect
474. C22#11718125direct (N × H)
497. C2C2×#11718125direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#60992
C2×#117181252
C22×C13×#1172
C22×#34413