Finite Group Databasebeta

← C2×#20289616 #28301323

Subgroups — C2×#20289616 #28301323

Order 13568.

986 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811593C211yesmaximal, characteristic
2C2×#16811593C211yesmaximal, characteristic
3#20289616C214yesmaximal, direct product factor
4C2×#16810481C211yesmaximal, characteristic
5C22×#11719940—21no—
6C22×#11719940C412yescomplemented
7#16811593—21no—
8#16811593—21no—
9#16811593—21no—
10#16811593—21no—
11#16811593C412yescomplemented
12#16811593C412yescomplemented
13#16811593C2212yes—
14#16811593C2212yes—
15C2×C22⋊#88324—21no—
16C2×C22⋊#88324—21no—
17C2×C22⋊#88324—21no—
18C2×C22⋊#88324C412yescomplemented
19C24×Dic53C2211yescharacteristic
20C22×#11719919—21no—
21C22×C22⋊Dic53C411yescharacteristic
22#16810481—21no—
23#16810481—21no—
24#16810481C412yes—
25#16810481C2212yes—
26C2×#13133124—21no—
27C22×C22⋊C4×C53C411yescharacteristic
28C2×#11719940—22no—
29C2×#11719940—22no—
30C2×#11719940—22no—
31C2×#11719940—44no—
32C2×#11719940—44no—
33C2×#11719940C2×C414yescomplemented
34C22×#88324—22no—
35C22×#88324—22no—
36C22⋊#88324—22no—
37C22⋊#88324—22no—
38C22⋊#88324—22no—
39C22⋊#88324—22no—
40C22⋊#88324—22no—
41C22⋊#88324—22no—
42C22⋊#88324—22no—
43C22⋊#88324C2×C414yescomplemented
44C23×Dic53—21no—
45C23×Dic53—44no—
46C23×Dic53C2×C411yescharacteristic
47C23×Dic53C2×C411yescharacteristic
48C23×Dic53C2×C411yescharacteristic
49C23×Dic53C2×C411yescharacteristic
50C23×Dic53C2×C411yescharacteristic
51C23×Dic53D411yescharacteristic
52C23×Dic53D411yescharacteristic
53C23×Dic53D411yescharacteristic
54C23×Dic53D411yescharacteristic
55C23×Dic53D411yescharacteristic
56C23×Dic53D411yescharacteristic
57C23×Dic53Q811yescharacteristic
58C23×Dic53Q811yescharacteristic
59C23×Dic53C2311yescharacteristic
60C2×#11719919—22no—
61C2×#11719919—22no—
62C2×#11719919—44no—
63C2×C22⋊Dic53—21no—
64C2×C22⋊Dic53—42no—
65C2×C22⋊Dic53C2×C412yes—
66#13133124—22no—
67#13133124—22no—
68C25×C53C2×C411yescharacteristic
69C23×C4×C53—21no—
70C2×C22⋊C4×C53—21no—
71C2×C22⋊C4×C53—42no—
72C2×C22⋊C4×C53C2×C412yes—
73C24×C53—22no—
74C24×C53—22no—
75C24×C53—42no—
76C24×C53—42no—
77C24×C53C4211yescharacteristic
78C24×C53C4211yescharacteristic
79C24×C53C22⋊C411yescharacteristic
80C24×C53C22⋊C411yescharacteristic
81C24×C53C4⋊C411yescharacteristic
82C24×C53C4⋊C411yescharacteristic
83C24×C53C22×C411yescharacteristic
84C22×C4×C53—22no—
85C22×C4×C53—42no—
86C22×C4×C53—42no—
87C22×C4×C53—42no—
88C22×Dic53—24no—
89C22×Dic53—24no—
90C22×Dic53—24no—
91C22×Dic53—22no—
92C22×Dic53—42no—
93C22×Dic53—42no—
94C22×Dic53—42no—
95C22×Dic53—42no—
96C22×Dic53—42no—
97C22×Dic53—42no—
98C22×Dic53—42no—
99C22×Dic53—42no—
100C22×Dic53—42no—
101C22×Dic53—42no—
102C22×Dic53—42no—
103C22×Dic53—42no—
104C22×Dic53—42no—
105C22×Dic53—42no—
106C22×Dic53C4211yescharacteristic
107C22×Dic53C4211yescharacteristic
108C22×Dic53C22⋊C411yescharacteristic
109C22×Dic53C22⋊C411yescharacteristic
110C22×Dic53C22⋊C411yescharacteristic
111C22×Dic53C22⋊C411yescharacteristic
112C22×Dic53C22⋊C411yescharacteristic
113C22×Dic53C22⋊C411yescharacteristic
114C22×Dic53C22⋊C411yescharacteristic
115C22×Dic53C22⋊C411yescharacteristic
116C22×Dic53C22⋊C411yescharacteristic
117C22×Dic53C22⋊C411yescharacteristic
118C22×Dic53C4⋊C411yescharacteristic
119C22×Dic53C4⋊C411yescharacteristic
120C22×Dic53C4⋊C411yescharacteristic
121C22×Dic53C4⋊C411yescharacteristic
122C22×Dic53C4⋊C411yescharacteristic
123C22×Dic53C4⋊C411yescharacteristic
124C22×Dic53C4⋊C411yescharacteristic
125C22×Dic53C4⋊C411yescharacteristic
126C22×Dic53C4⋊C411yescharacteristic
127C22×Dic53C4⋊C411yescharacteristic
128C22×Dic53C22×C411yescharacteristic
129C22×Dic53C22×C411yescharacteristic
130C22×Dic53C2×D411yescharacteristic
131C22×Dic53C2×D411yescharacteristic
132C22×Dic53C2×D411yescharacteristic
133C22×Dic53C2×Q811yescharacteristic
134#11719940—44no—
135#11719940—44no—
136#11719940—44no—
137#11719940—44no—
138C2×#88324—24no—
139C2×#88324—24no—
140C2×#88324—44no—
141C2×#88324—44no—
142C22⋊C4×C53—42no—
143C22⋊C4×C53—42no—
144C22⋊Dic53—42no—
145C22⋊Dic53—42no—
146#11719919—44no—
147#11719919—44no—
148C23×C53—22no—
149C23×C53—22no—
150C23×C53—22no—
151C23×C53—22no—
152C23×C53—22no—
153C23×C53—22no—
154C23×C53—22no—
155C23×C53—22no—
156C23×C53—22no—
157C23×C53—22no—
158C23×C53—21no—
159C23×C53—21no—
160C23×C53—21no—
161C23×C53—21no—
162C23×C53—41no—
163C23×C53—41no—
164C23×C53—41no—
165C23×C53—41no—
166C23×C53—41no—
167C23×C53—41no—
168C23×C53—41no—
169C23×C53—41no—
170C23×C53—48no—
171C23×C53—48no—
172C23×C53#9511yescharacteristic
173C23×C53#9511yescharacteristic
174C23×C53#9511yescharacteristic
175C23×C53#9511yescharacteristic
176C23×C53#10012yes—
177C23×C53#10112yes—
178C23×C53C2×C4211yescharacteristic
179C23×C53C2×C22⋊C411yescharacteristic
180C23×C53C2×C4⋊C411yescharacteristic
181C2×C4×C53—42no—
182C2×C4×C53—42no—
183C2×C4×C53—42no—
184C2×C4×C53—84no—
185C2×Dic53—22no—
186C2×Dic53—22no—
187C2×Dic53—22no—
188C2×Dic53—22no—
189C2×Dic53—22no—
190C2×Dic53—22no—
191C2×Dic53—22no—
192C2×Dic53—22no—
193C2×Dic53—22no—
194C2×Dic53—22no—
195C2×Dic53—22no—
196C2×Dic53—22no—
197C2×Dic53—22no—
198C2×Dic53—22no—
199C2×Dic53—22no—
200C2×Dic53—41no—
201C2×Dic53—41no—
202C2×Dic53—41no—
203C2×Dic53—41no—
204C2×Dic53—41no—
205C2×Dic53—41no—
206C2×Dic53—41no—
207C2×Dic53—41no—
208C2×Dic53—41no—
209C2×Dic53—41no—
210C2×Dic53—41no—
211C2×Dic53—41no—
212C2×Dic53—42no—
213C2×Dic53—42no—
214C2×Dic53—42no—
215C2×Dic53—84no—
216C2×Dic53#9511yescharacteristic
217C2×Dic53#9511yescharacteristic
218C2×Dic53#9511yescharacteristic
219C2×Dic53#9511yescharacteristic
220C2×Dic53#9914yes—
221C2×Dic53C2×C22⋊C411yescharacteristic
222C2×Dic53C2×C22⋊C411yescharacteristic
223C2×Dic53C2×C4⋊C411yescharacteristic
224C2×Dic53C2×C4⋊C411yescharacteristic
225#88324—44no—
226#88324—44no—
227C2×#1305—531nomaximal, Sylow
228C22×C53—24no—
229C22×C53—24no—
230C22×C53—21no—
231C22×C53—21no—
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—21no—
235C22×C53—21no—
236C22×C53—21no—
237C22×C53—21no—
238C22×C53—21no—
239C22×C53—21no—
240C22×C53—21no—
241C22×C53—21no—
242C22×C53—21no—
243C22×C53—21no—
244C22×C53—21no—
245C22×C53—21no—
246C22×C53—42no—
247C22×C53—42no—
248C22×C53—44no—
249C22×C53—44no—
250C22×C53—42no—
251C22×C53—42no—
252C22×C53—44no—
253C22×C53—44no—
254C22×C53#34212yes—
255C22×C53#34312yes—
256C22×C53#34312yes—
257C22×C53#34412yes—
258C22×C53C2×#9511yescharacteristic
259C22×C53C2×#10011yescharacteristic
260C22×C53C2×#10111yescharacteristic
261C4×C53—82no—
262Dic53—21no—
263Dic53—21no—
264Dic53—21no—
265Dic53—21no—
266Dic53—21no—
267Dic53—21no—
268Dic53—82no—
269Dic53#34212yes—
270Dic53C2×#9912yes—
271C2×#407—531no—
272C2×#407—531no—
273C2×#380—531no—
274#1305—534no—
275C2×C53—21no—
276C2×C53—22no—
277C2×C53—21no—
278C2×C53—41no—
279C2×C53—41no—
280C2×C53—41no—
281C2×C53—41no—
282C2×C53#130514yescomplemented
283C2×C53C2×#34211yescharacteristic
284C2×C53C2×#34311yescharacteristic
285C2×C53C2×#34411yescharacteristic
286C24×C4—531no—
287C22×M16—532no—
288C22×M16—1061no—
289C22×C4⋊C4—1061no—
290C22×C22⋊C4—531no—
291C22×C22⋊C4#615711yescharacteristic
292#407—532no—
293#407—532no—
294#407—532no—
295#407—532no—
296#407—1061no—
297#407—1061no—
298#407—1061no—
299#407—1061no—
300C2×C22⋊C8—532no—
301C2×C22⋊C8—1061no—
302C2×C22⋊C8—1061no—
303C2×C22⋊C8—1061no—
304#380—532no—
305#380—532no—
306#380—1061no—
307#380—1061no—
308C2×#95—1061no—
309C53C2×#130511yescharacteristic, complemented, Sylow
310C25C2×#615711yescharacteristic
311C23×C4—21no—
312C23×C4—531no—
313C23×C4—531no—
314C23×C4—531no—
315C23×C4—531no—
316C23×C4—531no—
317C23×C4—531no—
318C23×C4—531no—
319C23×C4—531no—
320C23×C4—531no—
321C23×C4—531no—
322C23×C4—531no—
323C23×C4—531no—
324C23×C4—531no—
325C23×C4—531no—
326C23×C4—1061no—
327C23×C4—2124no—
328C2×M16—534no—
329C2×M16—1062no—
330C2×M16—1062no—
331C2×M16—1062no—
332C2×M16—2124no—
333C2×M16—2124no—
334C22×C8—1062no—
335C22×C8—1062no—
336C2×C4⋊C4—1062no—
337C2×C4⋊C4—1062no—
338C2×C4⋊C4—2124no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—42no—
341C2×C22⋊C4—532no—
342C2×C22⋊C4—1061no—
343C2×C22⋊C4—2122no—
344C2×C22⋊C4C2×#615712yes—
345C22⋊C8—534no—
346C22⋊C8—1062no—
347C22⋊C8—1062no—
348C22⋊C8—1062no—
349C22⋊C8—1062no—
350C22⋊C8—1062no—
351C22⋊C8—1062no—
352C22⋊C8—1062no—
353#95—1062no—
354#95—1062no—
355C24—22no—
356C24—22no—
357C24—42no—
358C24—42no—
359C24C4×#615711yescharacteristic
360C24C4×#615711yescharacteristic
361C24C4⋊#615711yescharacteristic
362C24C4⋊#615711yescharacteristic
363C24C22⋊#615711yescharacteristic
364C24C22⋊#615711yescharacteristic
365C24C22×#615711yescharacteristic
366C22×C4—22no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—531no—
371C22×C4—531no—
372C22×C4—531no—
373C22×C4—531no—
374C22×C4—531no—
375C22×C4—531no—
376C22×C4—531no—
377C22×C4—531no—
378C22×C4—531no—
379C22×C4—531no—
380C22×C4—531no—
381C22×C4—531no—
382C22×C4—531no—
383C22×C4—531no—
384C22×C4—531no—
385C22×C4—531no—
386C22×C4—531no—
387C22×C4—531no—
388C22×C4—531no—
389C22×C4—531no—
390C22×C4—531no—
391C22×C4—531no—
392C22×C4—531no—
393C22×C4—531no—
394C22×C4—531no—
395C22×C4—531no—
396C22×C4—531no—
397C22×C4—531no—
398C22×C4—1064no—
399C22×C4—1064no—
400C22×C4—1064no—
401C22×C4—1062no—
402C22×C4—2122no—
403C22×C4—2122no—
404C22×C4—2122no—
405C22×C4—2122no—
406C22×C4—2122no—
407C22×C4—2122no—
408C22×C4—2122no—
409C22×C4—2122no—
410C22×C4—2122no—
411C22×C4—2122no—
412C22×C4—2122no—
413C22×C4—2122no—
414C22×C4—2122no—
415C22×C4—2122no—
416M16—2124no—
417M16—2124no—
418M16—2124no—
419M16—2124no—
420C2×C8—1064no—
421C2×C8—1064no—
422C2×C8—2124no—
423C2×C8—2124no—
424C4⋊C4—2124no—
425C4⋊C4—2124no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—2122no—
429C22⋊C4—2122no—
430C23—22no—
431C23—22no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—48no—
453C23—48no—
454C23(C2×#6157)⋊C411yescharacteristic
455C23(C2×#6157)⋊C411yescharacteristic
456C23(C2×#6157)⋊C411yescharacteristic
457C23(C2×#6157)⋊C411yescharacteristic
458C23#1313327712yes—
459C23#1313327812yes—
460C23C2×C4×#615711yescharacteristic
461C23C2×C22⋊#615711yescharacteristic
462C23C2×C4⋊#615711yescharacteristic
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—84no—
467C2×C4—531no—
468C2×C4—531no—
469C2×C4—534no—
470C2×C4—531no—
471C2×C4—531no—
472C2×C4—531no—
473C2×C4—531no—
474C2×C4—531no—
475C2×C4—531no—
476C2×C4—1062no—
477C2×C4—1062no—
478C2×C4—1062no—
479C2×C4—1062no—
480C2×C4—1062no—
481C2×C4—1062no—
482C2×C4—1062no—
483C2×C4—1062no—
484C2×C4—1062no—
485C2×C4—1062no—
486C2×C4—1062no—
487C2×C4—1062no—
488C2×C4—1062no—
489C2×C4—1062no—
490C2×C4—1062no—
491C2×C4—4244no—
492C2×C4—2121no—
493C2×C4—2121no—
494C2×C4—2121no—
495C2×C4—2121no—
496C2×C4—2121no—
497C2×C4—2121no—
498C2×C4—2121no—
499C2×C4—2121no—
500C2×C4—2121no—
501C2×C4—2121no—
502C2×C4—2121no—
503C2×C4—2121no—
504C2×C4—2122no—
505C2×C4—2122no—
506C2×C4—2122no—
507C8—2124no—
508C8—2124no—
509C22—24no—
510C22—24no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—44no—
532C22—44no—
533C22—44no—
534C22—44no—
535C22#1681150012yes—
536C22#1681150312yes—
537C22#1681150312yes—
538C22#1681150412yes—
539C22C2×(C2×#6157)⋊C411yescharacteristic
540C22C2×#1313327711yescharacteristic
541C22C2×#1313327811yescharacteristic
542C4—82no—
543C4—532no—
544C4—532no—
545C4—1061no—
546C4—1061no—
547C4—1061no—
548C4—1061no—
549C4—1061no—
550C4—1061no—
551C4—4242no—
552C2—21no—
553C2—21no—
554C2—22no—
555C2—41no—
556C2—41no—
557C2—41no—
558C2—41no—
559C2#2028961614yesdirect product factor
560C2C2×#1681150011yescharacteristic
561C2C2×#1681150311yescharacteristic
562C2C2×#1681150411yescharacteristic
subgroupG/H
centerC23(C2×#6157)⋊C4
commutatorC23×C53C2×C42
FrattiniC24C22×#6157
FittingC22×C22⋊C4×C53C4
socleC23×C53#95
prime radicalC25×C53C2×C4
cosocleC23×Dic53C23
hypercenterC22×C22⋊C4#6157
hypocenterC53C2×#1305
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C25 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #20289616C2direct (N × H)
6. C22×#11719940C4semidirect
11. #16811593C4semidirect
12. #16811593C4semidirect
18. C2×C22⋊#88324C4semidirect
33. C2×#11719940C2×C4semidirect
43. C22⋊#88324C2×C4semidirect
282. C2×C53#1305semidirect
309. C53C2×#1305semidirect
559. C2#20289616direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168115932
#202896162
C2×#168104812
C2×#130553