Finite Group Databasebeta

← C2×#20290904 #28320566

Subgroups — C2×#20290904 #28320566

Order 13568.

1698 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811565C211yesmaximal, characteristic
2C2×C4×C22⋊#6157C211yesmaximal, characteristic
3#20290904C218yesmaximal, direct product factor
4C2×#16811501C214yesmaximal
5C22×#13133193C211yesmaximal, characteristic
6C22×C4⋊#6157—21no—
7C22×C22⋊#6157C2212yes—
8C22×C22⋊#6157C2212yes—
9C22×C4×#6157—21no—
10#16811565—22no—
11#16811565C2214yes—
12C4×C22⋊#6157—22no—
13C4×C22⋊#6157C2214yes—
14C2×(C2×#6157)⋊C4—21no—
15C2×(C2×#6157)⋊C4—21no—
16#16811501C22116yes—
17C23×C4×D53C2211yescharacteristic
18C2×#13133193—21no—
19C2×#13133193—21no—
20C2×#13133193—21no—
21C2×#13133193—21no—
22C2×#13133193—22no—
23C2×#13133193—22no—
24C2×#13133193C414yescomplemented
25C2×#13133193C2214yes—
26C22×C4⋊Dic53C411yescharacteristic, complemented
27C22×#11719919C412yescomplemented
28C22×#11719921C2212yes—
29C23×Q8×C53C411yescharacteristic, complemented
30C23×#6157—22no—
31C23×#6157—22no—
32C2×C4⋊#6157—24no—
33C2×C4⋊#6157—44no—
34C2×C22⋊#6157—22no—
35C2×C22⋊#6157—24no—
36C2×C22⋊#6157—28no—
37C2×C22⋊#6157—22no—
38C2×C22⋊#6157Q818yescomplemented
39C2×C22⋊#6157C2314yes—
40C2×C22⋊#6157C2314yes—
41C2×C4×#6157—24no—
42C2×C4×#6157—44no—
43(C2×#6157)⋊C4—24no—
44(C2×#6157)⋊C4—24no—
45C24×D53C2311yescharacteristic
46C23×Dic53C2×C412yes—
47C23×Dic53C2×C411yescharacteristic
48C22×C4×D53—44no—
49C22×C4×D53C2×C412yes—
50C22×C4×D53D411yescharacteristic
51C22×C4×D53D411yescharacteristic
52C22×C4×D53D411yescharacteristic
53C22×C4×D53D411yescharacteristic
54C22×C4×D53D412yes—
55C22×C4×D53D412yes—
56C22×C4×D53C2312yes—
57#13133193—24no—
58#13133193—24no—
59#13133193—24no—
60#13133193—24no—
61#13133193—24no—
62#13133193—24no—
63#13133193—24no—
64#13133193—24no—
65C2×C4⋊Dic53—22no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53C2×C414yescomplemented
69C2×#11719919—22no—
70C2×#11719919—22no—
71C2×#11719919—24no—
72C2×#11719919C2×C418yescomplemented
73C2×#11719921—22no—
74C2×#11719921—22no—
75C2×#11719921—22no—
76C2×#11719921—22no—
77C2×#11719921C2×C414yes—
78C2×#11719921C2314yes—
79C22×Q8×C53—22no—
80C22×Q8×C53—21no—
81C22×Q8×C53—21no—
82C22×Q8×C53—44no—
83C22×Q8×C53C2×C414yescomplemented
84C23×C4×C53C2×C411yescharacteristic
85C23×C4×C53C2×C412yes—
86C24×C53C22×C411yescharacteristic
87C23×D53—42no—
88C23×D53—42no—
89C23×D53C22×C411yescharacteristic
90C23×D53C2×D411yescharacteristic
91C23×D53C2×D411yescharacteristic
92C23×D53C2×D411yescharacteristic
93C23×D53C2×D411yescharacteristic
94C23×D53C2×Q814yes—
95C23×D53#4114yes—
96C23×D53C2411yescharacteristic
97C22×#6157—28no—
98C22×#6157—24no—
99C22×#6157—24no—
100C22×#6157—44no—
101C22×#6157—44no—
102C22×#6157—44no—
103C22×#6157—44no—
104C22×#6157—44no—
105C2×Q8×C53—24no—
106C2×Q8×C53—24no—
107C2×Q8×C53—44no—
108C2×Q8×C53—48no—
109C2×Q8×C53—48no—
110C2×Q8×C53—44no—
111C22×C4×C53—24no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—44no—
115C22×C4×C53C22⋊C411yescharacteristic
116C22×C4×C53C22⋊C412yes—
117C22×C4×C53C22⋊C411yescharacteristic
118C22×C4×C53C22×C412yes—
119C22×C4×C53C22×C414yes—
120C22×Dic53—22no—
121C22×Dic53—22no—
122C22×Dic53—42no—
123C22×Dic53—44no—
124C22×Dic53C22⋊C411yescharacteristic
125C22×Dic53C22⋊C411yescharacteristic
126C22×Dic53C22⋊C412yes—
127C22×Dic53C22×C414yes—
128C22×Dic53C22×C412yes—
129C2×C4×D53—24no—
130C2×C4×D53—24no—
131C2×C4×D53—44no—
132C2×C4×D53—44no—
133C2×C4×D53—44no—
134C2×C4×D53—44no—
135C2×C4×D53—42no—
136C2×C4×D53—42no—
137C2×C4×D53C22⋊C412yes—
138C2×C4×D53C22⋊C412yes—
139C2×C4×D53C22⋊C412yes—
140C2×C4×D53C22⋊C412yes—
141C2×C4×D53C2×D412yes—
142C2×C4×D53C2×D412yes—
143C2×C4×D53C2×D412yes—
144C2×C4×D53C2×D412yes—
145C22⋊#6157—28no—
146C22⋊#6157—24no—
147C22⋊#6157—28no—
148C22⋊#6157—44no—
149C22⋊#6157C2×Q818yescomplemented
150C4⋊#6157—48no—
151C4×#6157—48no—
152#11719921—24no—
153#11719921—24no—
154#11719921—24no—
155#11719921—24no—
156C4⋊Dic53—24no—
157C4⋊Dic53—24no—
158#11719919—28no—
159#11719919—28no—
160C23×C53—22no—
161C23×C53—22no—
162C23×C53C2×C22⋊C411yescharacteristic
163C23×C53C2×C22⋊C411yescharacteristic
164C23×C53C4×Q814yes—
165C23×C53C23×C411yescharacteristic
166C22×D53—22no—
167C22×D53—22no—
168C22×D53—22no—
169C22×D53—22no—
170C22×D53—41no—
171C22×D53—41no—
172C22×D53—41no—
173C22×D53—41no—
174C22×D53—41no—
175C22×D53—41no—
176C22×D53—41no—
177C22×D53—48no—
178C22×D53—48no—
179C22×D53—41no—
180C22×D53C2×C22⋊C411yescharacteristic
181C22×D53C2×C22⋊C411yescharacteristic
182C22×D53C4×Q814yes—
183C22×D53C22⋊Q814yes—
184C22×D53C22⋊Q814yes—
185C22×D53#12412yes—
186C22×D53#12412yes—
187C22×D53C4⋊Q812yes—
188C22×D53C4⋊Q812yes—
189C22×D53C22×D411yescharacteristic
190C22×D53C22×D411yescharacteristic
191C22×D53C22×Q812yes—
192C22×D53C2×#4112yes—
193C2×#6157—28no—
194C2×#6157—44no—
195C2×#6157—44no—
196C2×#6157—416no—
197C2×#6157—44no—
198C2×#6157—44no—
199C2×#6157—44no—
200C2×#6157—88no—
201Q8×C53—48no—
202Q8×C53—48no—
203C2×C4×C53—22no—
204C2×C4×C53—24no—
205C2×C4×C53—24no—
206C2×C4×C53—24no—
207C2×C4×C53—44no—
208C2×C4×C53—44no—
209C2×C4×C53—44no—
210C2×C4×C53#9914yes—
211C2×C4×C53C2×C22⋊C412yes—
212C2×C4×C53C2×C22⋊C412yes—
213C2×Dic53—22no—
214C2×Dic53—22no—
215C2×Dic53—24no—
216C2×Dic53—42no—
217C2×Dic53—42no—
218C2×Dic53—44no—
219C2×Dic53—44no—
220C2×Dic53—44no—
221C2×Dic53C2×C22⋊C412yes—
222C2×Dic53C2×C22⋊C412yes—
223C4×D53—24no—
224C4×D53—24no—
225C4×D53—24no—
226C4×D53—24no—
227C4×D53—42no—
228C4×D53—42no—
229C4×D53—42no—
230C4×D53—42no—
231C2×#1996—531nomaximal, Sylow
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—21no—
235C22×C53—21no—
236C22×C53—28no—
237C22×C53#39112yes—
238C22×C53#39112yes—
239C22×C53C2×#9912yes—
240C22×C53#41012yes—
241C22×C53C22×C22⋊C411yescharacteristic
242C22×C53C2×C4×Q812yes—
243C2×D53—21no—
244C2×D53—21no—
245C2×D53—21no—
246C2×D53—21no—
247C2×D53—21no—
248C2×D53—21no—
249C2×D53—21no—
250C2×D53—21no—
251C2×D53—28no—
252C2×D53—24no—
253C2×D53—24no—
254C2×D53—44no—
255C2×D53—44no—
256C2×D53—42no—
257C2×D53—42no—
258C2×D53—42no—
259C2×D53—42no—
260C2×D53#39112yes—
261C2×D53#39112yes—
262C2×D53C2×C22⋊Q812yes—
263C2×D53C2×#12411yescharacteristic
264C2×D53C2×C4⋊Q811yescharacteristic
265#6157—48no—
266#6157—84no—
267C4×C53—22no—
268C4×C53—44no—
269C4×C53C2×#9914yes—
270Dic53—22no—
271Dic53—22no—
272Dic53—44no—
273C22×C22⋊Q8—531no—
274C2×#380—531no—
275C2×C4×C22⋊C4—531no—
276#1996—538no—
277C2×#342—534no—
278C2×C53—22no—
279C2×C53—22no—
280C2×C53#199614yescomplemented
281C2×C53C2×#39111yescharacteristic
282C2×C53C22×#9911yescharacteristic
283C2×C53C2×#41011yescharacteristic
284D53—22no—
285D53—21no—
286D53—21no—
287D53—22no—
288D53—22no—
289C23×Q8#615711yescharacteristic, complemented
290C24×C4—531no—
291C2×C22⋊Q8—534no—
292C2×C22⋊Q8—534no—
293C2×C22⋊Q8—1061no—
294C2×C22⋊Q8—1061no—
295C2×C22⋊Q8—1061no—
296C2×C22⋊Q8—1061no—
297C2×C22⋊Q8—1062no—
298C2×C22⋊Q8—1062no—
299C22×C4⋊C4—532no—
300C22×C4⋊C4—531no—
301C22×C4⋊C4—1061no—
302C22×C22⋊C4—532no—
303C22×C22⋊C4—532no—
304C22×C22⋊C4—532no—
305C22×C42—1061no—
306#380—534no—
307#380—1062no—
308C4×C22⋊C4—534no—
309C4×C22⋊C4—1062no—
310C2×#95—1061no—
311C2×#95—1061no—
312#342—5316no—
313C53C2×#199611yescharacteristic, complemented, Sylow
314C25—531no—
315C22×Q8—22no—
316C22×Q8—21no—
317C22×Q8—21no—
318C22×Q8—44no—
319C22×Q8C2×#615714yescomplemented
320C23×C4—531no—
321C23×C4—531no—
322C23×C4—531no—
323C23×C4—531no—
324C23×C4—531no—
325C23×C4—532no—
326C23×C4—532no—
327C23×C4—532no—
328C23×C4—532no—
329C23×C4—532no—
330C23×C4—1062no—
331C23×C4—1062no—
332C23×C4—2124no—
333C23×C4C2×#615712yes—
334C23×C4C2×#615711yescharacteristic
335C22⋊Q8—1064no—
336C22⋊Q8—1064no—
337C22⋊Q8—1064no—
338C22⋊Q8—1064no—
339C22⋊Q8—1064no—
340C22⋊Q8—1064no—
341C22⋊Q8—1064no—
342C22⋊Q8—1064no—
343C2×C4⋊C4—538no—
344C2×C4⋊C4—534no—
345C2×C4⋊C4—1064no—
346C2×C4⋊C4—1062no—
347C2×C4⋊C4—1062no—
348C2×C4⋊C4—1062no—
349C2×C4⋊C4—1061no—
350C2×C4⋊C4—1061no—
351C2×C4⋊C4—1064no—
352C2×C4⋊C4—2124no—
353C2×C22⋊C4—534no—
354C2×C22⋊C4—538no—
355C2×C22⋊C4—534no—
356C2×C22⋊C4—534no—
357C2×C22⋊C4—534no—
358C2×C22⋊C4—1062no—
359C2×C22⋊C4—1062no—
360C2×C22⋊C4—1062no—
361C2×C22⋊C4—1062no—
362C2×C22⋊C4—1062no—
363C2×C22⋊C4—1064no—
364C2×C22⋊C4—1068no—
365C2×C22⋊C4—1062no—
366C2×C42—1064no—
367C2×C42—2124no—
368#95—1064no—
369#95—1064no—
370C24—531no—
371C24—531no—
372C24—531no—
373C24—531no—
374C24—534no—
375C24—534no—
376C24—531no—
377C24—531no—
378C24—2122no—
379C24—2122no—
380C24C22×#615711yescharacteristic
381C2×Q8—24no—
382C2×Q8—24no—
383C2×Q8—48no—
384C2×Q8—44no—
385C2×Q8—48no—
386C2×Q8—44no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—44no—
391C22×C4—531no—
392C22×C4—531no—
393C22×C4—532no—
394C22×C4—532no—
395C22×C4—532no—
396C22×C4—532no—
397C22×C4—532no—
398C22×C4—532no—
399C22×C4—532no—
400C22×C4—532no—
401C22×C4—532no—
402C22×C4—532no—
403C22×C4—534no—
404C22×C4—1064no—
405C22×C4—1064no—
406C22×C4—1064no—
407C22×C4—1068no—
408C22×C4—1062no—
409C22×C4—1062no—
410C22×C4—1064no—
411C22×C4—2124no—
412C22×C4—2124no—
413C22×C4—2124no—
414C22×C4—2124no—
415C22×C4—2122no—
416C22×C4—2122no—
417C22×C4—2122no—
418C22×C4—2124no—
419C22×C4—2124no—
420C22×C4—2124no—
421C22×C4—2124no—
422C22×C4—2124no—
423C22×C4—2124no—
424C22×C4C22⋊#615712yes—
425C22×C4C22⋊#615711yescharacteristic
426C22×C4C22⋊#615711yescharacteristic
427C22×C4C22×#615712yes—
428C22×C4C22×#615714yes—
429C4⋊C4—1068no—
430C4⋊C4—1068no—
431C4⋊C4—1064no—
432C4⋊C4—1064no—
433C4⋊C4—2128no—
434C22⋊C4—538no—
435C22⋊C4—1064no—
436C22⋊C4—1064no—
437C22⋊C4—1064no—
438C22⋊C4—1064no—
439C22⋊C4—1068no—
440C22⋊C4—1064no—
441C22⋊C4—1068no—
442C22⋊C4—2124no—
443C42—2128no—
444C23—22no—
445C23—22no—
446C23—534no—
447C23—532no—
448C23—532no—
449C23—531no—
450C23—531no—
451C23—531no—
452C23—531no—
453C23—534no—
454C23—532no—
455C23—532no—
456C23—534no—
457C23—532no—
458C23—532no—
459C23—1062no—
460C23—1062no—
461C23—1062no—
462C23—1062no—
463C23—2121no—
464C23—2121no—
465C23—2121no—
466C23—2121no—
467C23—2128no—
468C23—2128no—
469C23—2121no—
470C23—2121no—
471C23—2121no—
472C23—2121no—
473C23C2×C22⋊#615711yescharacteristic
474C23C2×C22⋊#615711yescharacteristic
475C23Q8×#615714yes—
476C23C23×#615711yescharacteristic
477Q8—48no—
478Q8—48no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—532no—
487C2×C4—532no—
488C2×C4—1062no—
489C2×C4—1062no—
490C2×C4—1064no—
491C2×C4—1064no—
492C2×C4—1064no—
493C2×C4—1064no—
494C2×C4—1068no—
495C2×C4—1064no—
496C2×C4—4248no—
497C2×C4—2122no—
498C2×C4—2122no—
499C2×C4—2122no—
500C2×C4—2122no—
501C2×C4—2122no—
502C2×C4—2122no—
503C2×C4—2124no—
504C2×C4—2124no—
505C2×C4—2124no—
506C2×C4—2124no—
507C2×C4—2124no—
508C2×C4—21216no—
509C2×C4—2124no—
510C2×C4—2124no—
511C2×C4—2124no—
512C2×C4#1313327514yes—
513C2×C4C2×C22⋊#615712yes—
514C2×C4C2×C22⋊#615712yes—
515C22—28no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—532no—
521C22—531no—
522C22—531no—
523C22—532no—
524C22—532no—
525C22—1064no—
526C22—1068no—
527C22—1064no—
528C22—1061no—
529C22—1061no—
530C22—1061no—
531C22—1061no—
532C22—1061no—
533C22—1061no—
534C22—1061no—
535C22—1061no—
536C22—2122no—
537C22—2124no—
538C22—2122no—
539C22—2122no—
540C22—2122no—
541C22—2124no—
542C22(C2×#6157)⋊Q812yes—
543C22(C2×#6157)⋊Q812yes—
544C22C2×#1313327512yes—
545C22#1681160412yes—
546C22C22×C22⋊#615711yescharacteristic
547C22C2×Q8×#615712yes—
548C4—22no—
549C4—44no—
550C4—1062no—
551C4—1062no—
552C4—4244no—
553C4—2128no—
554C4—2124no—
555C4C2×#1313327514yes—
556C2—22no—
557C2—22no—
558C2—1061no—
559C2—1062no—
560C2—1061no—
561C2—1062no—
562C2—1062no—
563C2#2029090414yesdirect product factor
564C2C2×(C2×#6157)⋊Q811yescharacteristic
565C2C22×#1313327511yescharacteristic
566C2C2×#1681160411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC23×Q8×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×D53C23
cosocleC23×D53C24
hypercenterC23×Q8#6157
hypocenterC53C2×#1996
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
3. #20290904C2direct (N × H)
24. C2×#13133193C4semidirect
26. C22×C4⋊Dic53C4semidirect
27. C22×#11719919C4semidirect
29. C23×Q8×C53C4semidirect
38. C2×C22⋊#6157Q8semidirect
68. C2×C4⋊Dic53C2×C4semidirect
72. C2×#11719919C2×C4semidirect
83. C22×Q8×C53C2×C4semidirect
149. C22⋊#6157C2×Q8semidirect
280. C2×C53#1996semidirect
289. C23×Q8#6157semidirect
313. C53C2×#1996semidirect
319. C22×Q8C2×#6157semidirect
563. C2#20290904direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168115652
C2×C4×C22⋊#61572
#202909042
C2×#168115012
C22×#131331932
C2×#199653