Finite Group Databasebeta

← C2×#20291223 #28328740

Subgroups — C2×#20291223 #28328740

Order 13568.

3526 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133296C212yesmaximal
2#20291223C2116yesmaximal, direct product factor
3C2×#16811574C212yesmaximal, complemented
4C2×#16811572C214yesmaximal, complemented
5C2×#16811566C212yesmaximal
6C2×#16811565C211yesmaximal, characteristic
7C2×C4⋊C4×#6157C212yesmaximal, complemented
8C2×C4×C22⋊#6157C211yesmaximal, characteristic
9C22×C4⋊C4×D53C211yesmaximal, characteristic
10C2×#13133296—216no—
11C2×#13133296C22116yes—
12C22×C4⋊#6157C2212yes—
13C22×C4⋊#6157C2214yes—
14C22×C22⋊#6157C2211yescharacteristic
15C22×C22⋊#6157C2212yes—
16C22×C22⋊#6157C2211yescharacteristic
17C22×C4×#6157C2212yes—
18C22×C4×#6157C2214yes—
19#16811574C22116yescomplemented
20#16811572C22132yescomplemented
21#16811566C22116yes—
22#16811565C2218yes—
23C4⋊C4×#6157C22116yescomplemented
24C4×C22⋊#6157C2218yes—
25C2×(C2×#6157)⋊C4C2212yes—
26C2×(C2×#6157)⋊C4C2214yes—
27C2×(C2×#6157)⋊C4C2212yes—
28C2×(C2×#6157)⋊C4C2214yes—
29C23×C4×D53C2211yescharacteristic
30C23×C4×D53C2212yes—
31C22×C4⋊Dic53C411yescharacteristic, complemented
32C22×#11719919C412yescomplemented
33C2×C4⋊C4×D53C412yescomplemented
34C2×C4⋊C4×D53C411yescharacteristic, complemented
35C2×C4⋊C4×D53C411yescharacteristic, complemented
36C2×C4⋊C4×D53C418yescomplemented
37C2×C4⋊C4×D53C2211yescharacteristic
38C2×C4⋊C4×D53C2212yes—
39C2×C4⋊C4×D53C2211yescharacteristic
40C2×C4⋊C4×D53C2218yes—
41C22×C4⋊C4×C53C411yescharacteristic, complemented
42C23×#6157C2312yes—
43C23×#6157C2314yes—
44C23×#6157C2312yes—
45#13133296—264no—
46C2×C4⋊#6157—216no—
47C2×C4⋊#6157—28no—
48C2×C4⋊#6157C2318yes—
49C2×C4⋊#6157C23116yes—
50C2×C22⋊#6157—28no—
51C2×C22⋊#6157—24no—
52C2×C22⋊#6157D418yes—
53C2×C22⋊#6157C2314yes—
54C2×C22⋊#6157C2318yes—
55C2×C22⋊#6157C2314yes—
56C2×C4×#6157—216no—
57C2×C4×#6157—28no—
58C2×C4×#6157C2318yes—
59C2×C4×#6157C23116yes—
60(C2×#6157)⋊C4C2318yes—
61(C2×#6157)⋊C4C23116yes—
62(C2×#6157)⋊C4C2318yes—
63(C2×#6157)⋊C4C23116yes—
64C24×D53C2311yescharacteristic
65C23×Dic53C2×C411yescharacteristic
66C23×Dic53C2×C412yes—
67C22×C4×D53—28no—
68C22×C4×D53—28no—
69C22×C4×D53—28no—
70C22×C4×D53C2×C418yes—
71C22×C4×D53C2×C412yes—
72C22×C4×D53C2×C412yes—
73C22×C4×D53C2×C411yescharacteristic
74C22×C4×D53C2×C411yescharacteristic
75C22×C4×D53C2×C414yes—
76C22×C4×D53C2311yescharacteristic
77C22×C4×D53C2312yes—
78C22×C4×D53C2312yes—
79C22×C4×D53C2314yes—
80C22×C4×D53C2318yes—
81C22×C4×D53C2311yescharacteristic
82C2×C4⋊Dic53C2×C412yes—
83C2×C4⋊Dic53C2×C411yescharacteristic
84C2×C4⋊Dic53C2×C411yescharacteristic
85C2×C4⋊Dic53C2×C418yescomplemented
86C2×#11719919C2×C412yes—
87C2×#11719919C2×C412yes—
88C2×#11719919C2×C412yes—
89C2×#11719919C2×C412yes—
90C2×#11719919C2×C4116yescomplemented
91C4⋊C4×D53C2×C418yes—
92C4⋊C4×D53C2×C414yes—
93C4⋊C4×D53C2×C414yes—
94C4⋊C4×D53C2×C418yescomplemented
95C4⋊C4×D53C2×C418yescomplemented
96C4⋊C4×D53C2×C4116yescomplemented
97C4⋊C4×D53C2314yes—
98C4⋊C4×D53C2318yes—
99C4⋊C4×D53C2314yes—
100C23×C4×C53C2×C412yes—
101C23×C4×C53C2×C411yescharacteristic
102C2×C4⋊C4×C53C2×C412yes—
103C2×C4⋊C4×C53C2×C411yescharacteristic
104C2×C4⋊C4×C53C2×C411yescharacteristic
105C2×C4⋊C4×C53C2×C418yescomplemented
106C24×C53C22×C411yescharacteristic
107C23×D53—24no—
108C23×D53—24no—
109C23×D53C22×C411yescharacteristic
110C23×D53C22×C412yes—
111C23×D53C2×D414yes—
112C23×D53#4114yes—
113C23×D53C2412yes—
114C23×D53C2411yescharacteristic
115C22×#6157—24no—
116C22×#6157—24no—
117C22×#6157—28no—
118C22×#6157—24no—
119C22×#6157—24no—
120C22×#6157—28no—
121C22×#6157—28no—
122C22×#6157—24no—
123C22×#6157C2×D418yes—
124C22×#6157C2414yes—
125C22×#6157C2418yes—
126C22×#6157C2414yes—
127C22×C4×C53—28no—
128C22×C4×C53—24no—
129C22×C4×C53C22×C412yes—
130C22×C4×C53C22×C412yes—
131C22×C4×C53C22×C411yescharacteristic
132C22×C4×C53C22×C418yes—
133C22×C4×C53C22×C414yes—
134C22×C4×C53C22×C411yescharacteristic
135C22×Dic53—28no—
136C22×Dic53—24no—
137C22×Dic53C22×C411yescharacteristic
138C22×Dic53C22×C412yes—
139C22×Dic53C22×C412yes—
140C22×Dic53C22×C414yes—
141C22×Dic53C22×C418yes—
142C22×Dic53C22×C411yescharacteristic
143C2×C4×D53—28no—
144C2×C4×D53—28no—
145C2×C4×D53—232no—
146C2×C4×D53—24no—
147C2×C4×D53—24no—
148C2×C4×D53—216no—
149C2×C4×D53—24no—
150C2×C4×D53—216no—
151C2×C4×D53—24no—
152C2×C4×D53—24no—
153C2×C4×D53—24no—
154C2×C4×D53—24no—
155C2×C4×D53—216no—
156C2×C4×D53—216no—
157C2×C4×D53C22×C412yes—
158C2×C4×D53C22×C414yes—
159C2×C4×D53C22×C414yes—
160C2×C4×D53C22×C418yes—
161C2×C4×D53C22×C418yes—
162C2×C4×D53C22×C414yes—
163C2×C4×D53C22×C414yes—
164C2×C4×D53C22×C412yes—
165C2×C4×D53#4118yes—
166C2×C4×D53C2412yes—
167C2×C4×D53C2414yes—
168C2×C4×D53C2414yes—
169C2×C4×D53C2412yes—
170C22⋊#6157—216no—
171C22⋊#6157—28no—
172C22⋊#6157C2×D4116yes—
173C4⋊#6157—232no—
174C4⋊#6157—216no—
175C4×#6157—232no—
176C4×#6157—216no—
177C4⋊C4×C53C22×C418yes—
178C4⋊C4×C53C22×C414yes—
179C4⋊C4×C53C22×C414yes—
180C4⋊Dic53C22×C418yes—
181C4⋊Dic53C22×C414yes—
182C4⋊Dic53C22×C414yes—
183#11719919C22×C418yes—
184#11719919C22×C418yes—
185#11719919C22×C418yes—
186#11719919C22×C418yes—
187C23×C53—22no—
188C23×C53—22no—
189C23×C53C4×D414yes—
190C23×C53C23×C412yes—
191C23×C53C23×C411yescharacteristic
192C22×D53—22no—
193C22×D53—22no—
194C22×D53—24no—
195C22×D53—24no—
196C22×D53—216no—
197C22×D53—216no—
198C22×D53—24no—
199C22×D53—24no—
200C22×D53C4×D414yes—
201C22×D53C4×D418yes—
202C22×D53C23×C412yes—
203C22×D53C23×C411yescharacteristic
204C22×D53C22×D414yes—
205C22×D53C22×D412yes—
206C22×D53C2×#4114yes—
207C22×D53C2×#4114yes—
208C22×D53C2×#4112yes—
209C22×D532−1+414yes—
210C22×D53C2511yescharacteristic
211C2×#6157—216no—
212C2×#6157—24no—
213C2×#6157—24no—
214C2×#6157—28no—
215C2×#6157—24no—
216C2×#6157—28no—
217C2×#6157—24no—
218C2×#6157—28no—
219C2×#6157—24no—
220C2×#6157—416no—
221C2×#6157—48no—
222C2×#6157C22×D418yes—
223C2×C4×C53—22no—
224C2×C4×C53—216no—
225C2×C4×C53—24no—
226C2×C4×C53—24no—
227C2×C4×C53—28no—
228C2×C4×C53#11714yescomplemented
229C2×C4×C53C23×C414yes—
230C2×C4×C53C23×C414yes—
231C2×C4×C53C23×C412yes—
232C2×C4×C53C23×C412yes—
233C2×Dic53—22no—
234C2×Dic53—216no—
235C2×Dic53—24no—
236C2×Dic53—24no—
237C2×Dic53—28no—
238C2×Dic53#11714yescomplemented
239C2×Dic53C23×C412yes—
240C2×Dic53C23×C414yes—
241C2×Dic53C23×C414yes—
242C2×Dic53C23×C412yes—
243C4×D53—24no—
244C4×D53—216no—
245C4×D53—216no—
246C4×D53—28no—
247C4×D53—28no—
248C4×D53—24no—
249C4×D53—24no—
250C4×D53—24no—
251C4×D53—24no—
252C4×D53—24no—
253C4×D53—28no—
254C4×D53—28no—
255C4×D53#11718yescomplemented
256C4×D53C2×#4118yes—
257C2×#2344—531nomaximal, Sylow
258C22×C53—28no—
259C22×C53—22no—
260C22×C53—22no—
261C22×C53C2×#11712yes—
262C22×C53C2×C4×D414yes—
263C22×C53C2×C4×D412yes—
264C22×C53#51912yes—
265C22×C53C24×C411yescharacteristic
266C2×D53—28no—
267C2×D53—22no—
268C2×D53—22no—
269C2×D53—216no—
270C2×D53—28no—
271C2×D53—28no—
272C2×D53C2×#11712yes—
273C2×D53C2×C4×D414yes—
274C2×D53C2×C4×D412yes—
275C2×D53#51912yes—
276C2×D53#54718yes—
277C2×D53#55218yes—
278C2×D53C23×D411yescharacteristic
279C2×D53C22×#4112yes—
280C2×D53C22×#4111yescharacteristic
281C2×D53C2×2−1+412yes—
282#6157—28no—
283#6157—48no—
284#6157—44no—
285C4×C53—22no—
286C4×C53—24no—
287C4×C53—24no—
288C4×C53C2×#11714yescomplemented
289Dic53—22no—
290Dic53—24no—
291Dic53—24no—
292Dic53C2×#11714yescomplemented
293C22×#117—532no—
294C23×C4⋊C4—531no—
295#2344—5316no—
296C2×#389—532no—
297C2×#387—534no—
298C2×#381—532no—
299C2×#380—531no—
300C2×C4×C4⋊C4—532no—
301C2×C4×C22⋊C4—531no—
302C2×C53—24no—
303C2×C53#234414yescomplemented
304C2×C53C22×#11711yescharacteristic
305C2×C53C22×C4×D411yescharacteristic
306C2×C53C2×#51911yescharacteristic
307D53—24no—
308D53#234414yescomplemented
309D53C2×#54712yes—
310D53C2×#55212yes—
311C24×C4—532no—
312C24×C4—531no—
313C2×#117—5316no—
314C2×#117—10616no—
315C22×C4⋊C4—532no—
316C22×C4⋊C4—531no—
317C22×C4⋊C4—532no—
318C22×C4⋊C4—531no—
319C22×C4⋊C4—531no—
320C22×C4⋊C4—531no—
321C22×C4⋊C4—532no—
322C22×C4⋊C4—531no—
323C22×C4⋊C4—538no—
324C22×C4⋊C4—538no—
325C22×C4⋊C4—534no—
326C22×C4⋊C4—532no—
327C22×C4⋊C4#615711yescharacteristic, complemented
328C22×C22⋊C4—531no—
329C22×C22⋊C4—532no—
330C22×C22⋊C4—531no—
331C22×C42—534no—
332C22×C42—532no—
333#389—5316no—
334#387—5332no—
335#381—5316no—
336#380—538no—
337C4×C4⋊C4—5316no—
338C4×C22⋊C4—538no—
339C2×#95—532no—
340C2×#95—534no—
341C2×#95—532no—
342C2×#95—534no—
343C53C2×#234411yescharacteristic, complemented, Sylow
344C25—531no—
345C23×C4—531no—
346C23×C4—532no—
347C23×C4—531no—
348C23×C4—532no—
349C23×C4—532no—
350C23×C4—538no—
351C23×C4—531no—
352C23×C4—532no—
353C23×C4—534no—
354C23×C4—531no—
355C23×C4—534no—
356C23×C4—538no—
357C23×C4—532no—
358C23×C4—532no—
359C23×C4—531no—
360C23×C4—534no—
361C23×C4—532no—
362C23×C4—1068no—
363C23×C4—1068no—
364C23×C4—1068no—
365C23×C4C2×#615712yes—
366C23×C4C2×#615711yescharacteristic
367#117—10664no—
368C2×C4⋊C4—532no—
369C2×C4⋊C4—532no—
370C2×C4⋊C4—532no—
371C2×C4⋊C4—532no—
372C2×C4⋊C4—5316no—
373C2×C4⋊C4—532no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—538no—
377C2×C4⋊C4—534no—
378C2×C4⋊C4—534no—
379C2×C4⋊C4—538no—
380C2×C4⋊C4—5316no—
381C2×C4⋊C4—538no—
382C2×C4⋊C4—538no—
383C2×C4⋊C4—534no—
384C2×C4⋊C4—538no—
385C2×C4⋊C4—534no—
386C2×C4⋊C4—5316no—
387C2×C4⋊C4—538no—
388C2×C4⋊C4—10616no—
389C2×C4⋊C4—1068no—
390C2×C4⋊C4C2×#615712yes—
391C2×C4⋊C4C2×#615711yescharacteristic
392C2×C4⋊C4C2×#615711yescharacteristic
393C2×C4⋊C4C2×#615718yescomplemented
394C2×C22⋊C4—534no—
395C2×C22⋊C4—538no—
396C2×C22⋊C4—538no—
397C2×C22⋊C4—534no—
398C2×C22⋊C4—1068no—
399C2×C22⋊C4—1064no—
400C2×C42—5316no—
401C2×C42—538no—
402C2×C42—10616no—
403C2×C42—1068no—
404#95—538no—
405#95—5316no—
406#95—538no—
407#95—5316no—
408C24—531no—
409C24—534no—
410C24—534no—
411C24—532no—
412C24—532no—
413C24—531no—
414C24—1064no—
415C24—1064no—
416C24C22×#615711yescharacteristic
417C22×C4—28no—
418C22×C4—24no—
419C22×C4—531no—
420C22×C4—538no—
421C22×C4—532no—
422C22×C4—532no—
423C22×C4—538no—
424C22×C4—534no—
425C22×C4—534no—
426C22×C4—538no—
427C22×C4—531no—
428C22×C4—532no—
429C22×C4—532no—
430C22×C4—534no—
431C22×C4—534no—
432C22×C4—532no—
433C22×C4—534no—
434C22×C4—538no—
435C22×C4—538no—
436C22×C4—534no—
437C22×C4—532no—
438C22×C4—534no—
439C22×C4—534no—
440C22×C4—534no—
441C22×C4—538no—
442C22×C4—1064no—
443C22×C4—10616no—
444C22×C4—10616no—
445C22×C4—1064no—
446C22×C4—1064no—
447C22×C4—10616no—
448C22×C4—1064no—
449C22×C4—1064no—
450C22×C4—1064no—
451C22×C4—10632no—
452C22×C4—1068no—
453C22×C4—1068no—
454C22×C4—10616no—
455C22×C4—1064no—
456C22×C4—1064no—
457C22×C4—1064no—
458C22×C4—1068no—
459C22×C4—1064no—
460C22×C4—1068no—
461C22×C4—1064no—
462C22×C4—1064no—
463C22×C4—1068no—
464C22×C4—1068no—
465C22×C4—1064no—
466C22×C4C22×#615718yes—
467C22×C4C22×#615714yes—
468C22×C4C22×#615711yescharacteristic
469C22×C4C22×#615712yes—
470C22×C4C22×#615712yes—
471C22×C4C22×#615711yescharacteristic
472C4⋊C4—538no—
473C4⋊C4—538no—
474C4⋊C4—538no—
475C4⋊C4—538no—
476C4⋊C4—538no—
477C4⋊C4—534no—
478C4⋊C4—534no—
479C4⋊C4—10616no—
480C4⋊C4—10632no—
481C4⋊C4C22×#615718yes—
482C4⋊C4C22×#615714yes—
483C4⋊C4C22×#615714yes—
484C22⋊C4—5316no—
485C22⋊C4—1068no—
486C22⋊C4—10616no—
487C42—10616no—
488C42—10632no—
489C23—22no—
490C23—22no—
491C23—534no—
492C23—532no—
493C23—532no—
494C23—534no—
495C23—531no—
496C23—531no—
497C23—538no—
498C23—534no—
499C23—534no—
500C23—534no—
501C23—532no—
502C23—1064no—
503C23—1064no—
504C23—1064no—
505C23—1064no—
506C23—10616no—
507C23—10616no—
508C23—1062no—
509C23—1062no—
510C23D4×#615714yes—
511C23C23×#615712yes—
512C23C23×#615711yescharacteristic
513C2×C4—22no—
514C2×C4—216no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—534no—
519C2×C4—532no—
520C2×C4—538no—
521C2×C4—538no—
522C2×C4—538no—
523C2×C4—532no—
524C2×C4—534no—
525C2×C4—534no—
526C2×C4—1068no—
527C2×C4—1064no—
528C2×C4—1064no—
529C2×C4—1064no—
530C2×C4—1064no—
531C2×C4—1064no—
532C2×C4—1064no—
533C2×C4—1068no—
534C2×C4—1068no—
535C2×C4—10616no—
536C2×C4—10616no—
537C2×C4—1068no—
538C2×C4—1064no—
539C2×C4—1064no—
540C2×C4—10616no—
541C2×C4—1062no—
542C2×C4—10616no—
543C2×C4—1064no—
544C2×C4—1064no—
545C2×C4—1068no—
546C2×C4—1068no—
547C2×C4—1064no—
548C2×C4—1064no—
549C2×C4—1068no—
550C2×C4—1064no—
551C2×C4—1068no—
552C2×C4—21216no—
553C2×C4—2128no—
554C2×C4#1313329614yescomplemented
555C2×C4C23×#615712yes—
556C2×C4C23×#615714yes—
557C2×C4C23×#615714yes—
558C2×C4C23×#615712yes—
559C22—22no—
560C22—22no—
561C22—28no—
562C22—532no—
563C22—532no—
564C22—531no—
565C22—531no—
566C22—532no—
567C22—532no—
568C22—532no—
569C22—538no—
570C22—538no—
571C22—534no—
572C22—1068no—
573C22—1068no—
574C22—10616no—
575C22—1062no—
576C22—1062no—
577C22—1068no—
578C22C2×#1313329612yes—
579C22C2×D4×#615714yes—
580C22C2×D4×#615712yes—
581C22#1681169112yes—
582C22C24×#615711yescharacteristic
583C4—22no—
584C4—24no—
585C4—24no—
586C4—534no—
587C4—1068no—
588C4—1062no—
589C4—1064no—
590C4—1064no—
591C4—2128no—
592C4—2124no—
593C4C2×#1313329614yescomplemented
594C2—24no—
595C2—532no—
596C2—532no—
597C2—534no—
598C2—1064no—
599C2#2029122314yesdirect product factor
600C2C22×#1313329611yescharacteristic
601C2C22×D4×#615711yescharacteristic
602C2C2×#1681169111yescharacteristic
subgroupG/H
centerC23C23×#6157
commutatorC22×C53C24×C4
FrattiniC22C24×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C23×C4
prime radicalC24×D53C23
cosocleC22×D53C25
hypercenterC22×C4⋊C4#6157
hypocenterC53C2×#2344
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #20291223C2direct (N × H)
3. C2×#16811574C2semidirect
4. C2×#16811572C2semidirect
7. C2×C4⋊C4×#6157C2semidirect
19. #16811574C22semidirect
20. #16811572C22semidirect
23. C4⋊C4×#6157C22semidirect
31. C22×C4⋊Dic53C4semidirect
32. C22×#11719919C4semidirect
33. C2×C4⋊C4×D53C4semidirect
34. C2×C4⋊C4×D53C4semidirect
35. C2×C4⋊C4×D53C4semidirect
36. C2×C4⋊C4×D53C4semidirect
41. C22×C4⋊C4×C53C4semidirect
85. C2×C4⋊Dic53C2×C4semidirect
90. C2×#11719919C2×C4semidirect
94. C4⋊C4×D53C2×C4semidirect
95. C4⋊C4×D53C2×C4semidirect
96. C4⋊C4×D53C2×C4semidirect
105. C2×C4⋊C4×C53C2×C4semidirect
228. C2×C4×C53#117semidirect
238. C2×Dic53#117semidirect
255. C4×D53#117semidirect
288. C4×C53C2×#117semidirect
292. Dic53C2×#117semidirect
303. C2×C53#2344semidirect
308. D53#2344semidirect
327. C22×C4⋊C4#6157semidirect
343. C53C2×#2344semidirect
393. C2×C4⋊C4C2×#6157semidirect
554. C2×C4#13133296semidirect
593. C4C2×#13133296semidirect
599. C2#20291223direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131332962
#202912232
C2×#168115742
C2×#168115722
C2×#168115662
C2×#168115652
C2×C4⋊C4×#61572
C2×C4×C22⋊#61572
C22×C4⋊C4×D532
C2×#234453