Finite Group Databasebeta

← C2×#20287521 #28263806

Subgroups — C2×#20287521 #28263806

Order 13568.

1778 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811351C211yesmaximal, characteristic
2C2×#16811350C211yesmaximal, characteristic
3#20287521C2116yesmaximal, direct product factor
4C2×#16810993C211yesmaximal, characteristic
5C2×#16810990C211yesmaximal, characteristic
6C2×#16810972C211yesmaximal, characteristic
7C2×#16810929C211yesmaximal, characteristic
8C2×#16810907C211yesmaximal, characteristic
9C2×#16810834C211yesmaximal, characteristic
10C2×#16810832C211yesmaximal, characteristic
11C2×#16810832C211yesmaximal, characteristic
12C2×#16810752C211yesmaximal, characteristic
13C2×#16810729C211yesmaximal, characteristic
14C2×#16810728C211yesmaximal, characteristic
15C2×#16810717C211yesmaximal, characteristic
16C2×C53×#479C211yesmaximal, characteristic
17C22×Dic212C2211yescharacteristic
18C22×Dic212C2211yescharacteristic
19C22×#11719925C2211yescharacteristic
20#16811351—24no—
21#16811351C2218yes—
22#16811350—24no—
23#16811350C2218yes—
24C2×#13133217C2211yescharacteristic
25C2×#13133216—21no—
26C2×#13133216C2211yescharacteristic
27C2×#13133215C2211yescharacteristic
28C2×#13133215C2211yescharacteristic
29C2×#13133209—21no—
30C2×#13133208C2211yescharacteristic
31C2×#13133207—21no—
32C2×#13133207—21no—
33C2×#13133207—21no—
34C2×#13133207—21no—
35C2×C4×Dic106—21no—
36C2×C4×Dic106C2211yescharacteristic
37C2×C4×Dic106C2211yescharacteristic
38C2×#13133182—21no—
39C2×#13133182C2211yescharacteristic
40C2×#13133182C2211yescharacteristic
41C2×#13133182C2211yescharacteristic
42C2×#13133182C2211yescharacteristic
43#16810993C2218yes—
44#16810990C2218yes—
45#16810972C2218yes—
46#16810929C2218yes—
47#16810907C2218yes—
48#16810834C2218yes—
49#16810832C2218yes—
50#16810832C2218yes—
51#16810752C2218yes—
52#16810729C2218yes—
53#16810728C2218yes—
54#16810717C2218yes—
55C2×#13133148C2211yescharacteristic
56C2×#13133146C2211yescharacteristic
57C2×#13133146C2211yescharacteristic
58C2×#13133142C2211yescharacteristic
59C2×#13133142C2211yescharacteristic
60C2×#13133141C2211yescharacteristic
61C2×#13133140C2211yescharacteristic
62C2×#13133140C2211yescharacteristic
63C2×#13133140C2211yescharacteristic
64C2×#13133140C2211yescharacteristic
65C2×#13133140C2211yescharacteristic
66C2×#13133128C2211yescharacteristic
67C2×#13133127C2211yescharacteristic
68C2×C8×Dic53C2211yescharacteristic
69C2×#125×C53C2211yescharacteristic
70C2×C4×Q8×C53C2211yescharacteristic
71C53×#479C2218yes—
72C2×#107×C53C2211yescharacteristic
73C2×#106×C53C2211yescharacteristic
74C2×C4⋊C8×C53C2211yescharacteristic
75C2×#103×C53C2211yescharacteristic
76C2×#103×C53C2211yescharacteristic
77C22×Dic106—21no—
78C22×Dic106D411yescharacteristic
79C22×Dic106C2311yescharacteristic
80C22×Dic106C2311yescharacteristic
81C22×Dic106C2311yescharacteristic
82C2×Dic212—24no—
83C2×Dic212—24no—
84C2×Dic212C2314yes—
85C2×Dic212C2314yes—
86C2×#11719925—24no—
87C2×#11719925C2314yes—
88C22×#88322C2311yescharacteristic
89C22×#88322C2311yescharacteristic
90#13133217—22no—
91#13133217C2314yes—
92#13133216—22no—
93#13133216—28no—
94#13133216C2314yes—
95#13133215—22no—
96#13133215—22no—
97#13133215C2314yes—
98#13133215C2314yes—
99#13133209—28no—
100#13133208—22no—
101#13133208C2314yes—
102#13133207—28no—
103#13133207—28no—
104#13133207—28no—
105#13133207—28no—
106C4×Dic106—22no—
107C4×Dic106—28no—
108C4×Dic106D414yes—
109C4×Dic106C2314yes—
110C4×Dic106C2314yes—
111#13133182—28no—
112#13133182—22no—
113#13133182—22no—
114#13133182—22no—
115#13133182—22no—
116#13133182C2314yes—
117#13133182C2314yes—
118#13133182C2314yes—
119#13133182C2314yes—
120C2×C4⋊Dic53—21no—
121C2×C4⋊Dic53—21no—
122C2×C4⋊Dic53—21no—
123C2×C4⋊Dic53D411yescharacteristic
124C2×C4⋊Dic53C2311yescharacteristic
125C2×#11719919—21no—
126C2×#11719919—21no—
127C2×#11719919—21no—
128C2×#11719919—21no—
129C2×#11719919—21no—
130C2×#11719919—21no—
131C2×#11719919—21no—
132C2×#11719919—21no—
133C2×#11719919—21no—
134C2×#11719919—21no—
135C2×#11719919—21no—
136C2×#11719919D411yescharacteristic
137C2×#11719919D411yescharacteristic
138C2×C4×Dic53—21no—
139C2×C4×Dic53—21no—
140C2×C4×Dic53—21no—
141C2×C4×Dic53—21no—
142C2×C4×Dic53—21no—
143C2×C4×Dic53C2311yescharacteristic
144C2×C4×Dic53C2311yescharacteristic
145#13133148C2314yes—
146#13133146C2314yes—
147#13133146C2314yes—
148#13133142C2314yes—
149#13133142C2314yes—
150#13133141C2314yes—
151#13133140C2314yes—
152#13133140C2314yes—
153#13133140C2314yes—
154#13133140C2314yes—
155#13133140C2314yes—
156#13133128C2314yes—
157#13133127C2314yes—
158C8×Dic53C2314yes—
159C22×Q8×C53C2311yescharacteristic
160C22×C8×C53C2311yescharacteristic
161C22×C8×C53C2311yescharacteristic
162#125×C53—22no—
163#125×C53C2314yes—
164C4×Q8×C53—22no—
165C4×Q8×C53C2314yes—
166C2×C4⋊C4×C53—21no—
167C2×C4⋊C4×C53—21no—
168C2×C4⋊C4×C53—21no—
169C2×C4⋊C4×C53C2311yescharacteristic
170C2×C4⋊C4×C53C2311yescharacteristic
171C2×C4⋊C4×C53C2311yescharacteristic
172C2×C42×C53—21no—
173C2×C42×C53C2311yescharacteristic
174#107×C53C2314yes—
175#106×C53C2314yes—
176C4⋊C8×C53C2314yes—
177#103×C53C2314yes—
178#103×C53C2314yes—
179C2×Q8×C53—21no—
180C2×Q8×C53—22no—
181C2×Q8×C53—22no—
182C2×Q8×C53C2412yes—
183C22×C4×C53—21no—
184C22×C4×C53—21no—
185C22×C4×C53—21no—
186C22×C4×C53—21no—
187C22×C4×C53C2×D411yescharacteristic
188C22×C4×C53C2×D411yescharacteristic
189C22×C4×C53C2411yescharacteristic
190C22×Dic53—21no—
191C22×Dic53—21no—
192C22×Dic53—21no—
193C22×Dic53—21no—
194C22×Dic53C2×D411yescharacteristic
195C22×Dic53C2×D411yescharacteristic
196C22×Dic53C2×D411yescharacteristic
197C22×Dic53C2×D411yescharacteristic
198C2×Dic106—21no—
199C2×Dic106—22no—
200C2×Dic106—24no—
201C2×Dic106—22no—
202C2×Dic106—22no—
203C2×Dic106—22no—
204C2×Dic106—22no—
205C2×Dic106—21no—
206C2×Dic106—21no—
207C2×Dic106—24no—
208C2×Dic106—44no—
209C2×Dic106C2×D414yes—
210C2×Dic106#4114yes—
211C2×Dic106C2412yes—
212C2×Dic106C2412yes—
213C2×Dic106C2412yes—
214C2×C8×C53—22no—
215C2×C8×C53—22no—
216C2×C8×C53C2412yes—
217C2×C8×C53C2412yes—
218C4⋊C4×C53—24no—
219C4⋊C4×C53—24no—
220C4⋊C4×C53—24no—
221C4⋊C4×C53—21no—
222C4⋊C4×C53—21no—
223C4⋊C4×C53—21no—
224C4⋊C4×C53C2412yes—
225C4⋊C4×C53C2412yes—
226C4⋊C4×C53C2412yes—
227C42×C53—24no—
228C42×C53C2×D412yes—
229C42×C53C2412yes—
230#11719925—28no—
231C4⋊Dic53—24no—
232C4⋊Dic53—24no—
233C4⋊Dic53—21no—
234C4⋊Dic53—24no—
235C4⋊Dic53C2×D414yes—
236C4⋊Dic53C2412yes—
237#11719919—24no—
238#11719919—24no—
239#11719919—24no—
240#11719919—24no—
241#11719919—24no—
242#11719919—24no—
243#11719919—24no—
244#11719919—24no—
245#11719919—24no—
246#11719919—24no—
247#11719919—24no—
248#11719919C2×D414yes—
249#11719919C2×D414yes—
250C4×Dic53—24no—
251C4×Dic53—24no—
252C4×Dic53—24no—
253C4×Dic53—24no—
254C4×Dic53—24no—
255C4×Dic53C2×D412yes—
256C4×Dic53C2×D412yes—
257C4×Dic53C2412yes—
258C4×Dic53C2412yes—
259C2×#88322—22no—
260C2×#88322—22no—
261C2×#88322C2412yes—
262C2×#88322C2412yes—
263Dic212—28no—
264Dic212—28no—
265C23×C53C22×D411yescharacteristic
266Q8×C53—22no—
267Q8×C53—22no—
268Q8×C53—42no—
269C2×C4×C53—22no—
270C2×C4×C53—22no—
271C2×C4×C53—22no—
272C2×C4×C53—22no—
273C2×C4×C53—22no—
274C2×C4×C53—22no—
275C2×C4×C53—42no—
276C2×C4×C53—42no—
277C2×C4×C53—42no—
278C2×C4×C53—42no—
279C2×C4×C53C22×D412yes—
280C2×C4×C53C22×D412yes—
281C2×C4×C53C22×D411yescharacteristic
282C2×C4×C53C2×#4112yes—
283C2×C4×C532−1+412yes—
284C2×C4×C53C2511yescharacteristic
285C2×Dic53—24no—
286C2×Dic53—22no—
287C2×Dic53—22no—
288C2×Dic53—22no—
289C2×Dic53—22no—
290C2×Dic53—22no—
291C2×Dic53—22no—
292C2×Dic53—22no—
293C2×Dic53—42no—
294C2×Dic53—42no—
295C2×Dic53—42no—
296C2×Dic53#13714yes—
297C2×Dic53C22×D412yes—
298C2×Dic53C22×D412yes—
299C2×Dic53C22×D412yes—
300C2×Dic53C22×D412yes—
301Dic106—24no—
302Dic106—22no—
303Dic106—28no—
304Dic106—22no—
305Dic106—22no—
306Dic106—22no—
307Dic106—22no—
308Dic106—42no—
309Dic106—42no—
310Dic106—48no—
311Dic106C2×#4114yes—
312C8×C53—22no—
313C8×C53—22no—
314#88322—22no—
315#88322—22no—
316C2×#3374—531nomaximal, Sylow
317C22×C53#54812yes—
318C22×C53C2×#13712yes—
319C22×C53#57812yes—
320C22×C53C23×D411yescharacteristic
321C4×C53—22no—
322C4×C53—22no—
323C4×C53—42no—
324C4×C53—42no—
325C4×C53—42no—
326C4×C53—42no—
327C4×C53#54812yes—
328C4×C53C22×#4111yescharacteristic
329C4×C53C2×2−1+411yescharacteristic
330Dic53—24no—
331Dic53—22no—
332Dic53—22no—
333Dic53—22no—
334Dic53—42no—
335Dic53—42no—
336Dic53—42no—
337Dic53C2×#13714yes—
338C2×#557—531no—
339C2×#557—531no—
340#3374—5316no—
341C2×#491—531no—
342C2×#491—531no—
343C2×#487—531no—
344C2×#479—531no—
345C2×#479D5311yescharacteristic
346C2×#475—531no—
347C2×#462—531no—
348C2×#462—531no—
349C2×#462—531no—
350C2×#446—531no—
351C2×#441—531no—
352C2×#441—531no—
353C2×C4×Q16—531no—
354C2×C53#337414yescomplemented
355C2×C53C2×#54811yescharacteristic
356C2×C53C22×#13711yescharacteristic
357C2×C53C2×#57811yescharacteristic
358C22×Q16—531no—
359C22×Q16—531no—
360C22×Q16—531no—
361#557—538no—
362#557—538no—
363#557—1064no—
364#557—1064no—
365C2×C4⋊Q8—531no—
366C2×C4⋊Q8—531no—
367C2×C4⋊Q8—531no—
368C2×C4⋊Q8—531no—
369C2×C4⋊Q8—1061no—
370C2×#125—531no—
371C2×#125—1061no—
372C2×#125—1061no—
373C2×#125—1061no—
374C2×#125—1061no—
375C2×#125—1061no—
376C2×#125C2×D5311yescharacteristic
377C2×C4×Q8—531no—
378C2×C4×Q8—531no—
379C2×C4×Q8—531no—
380C2×C4×Q8—531no—
381C2×C4×Q8—531no—
382C2×C4×Q8—531no—
383C2×C4×Q8—1061no—
384C2×C4×Q8—1061no—
385C2×C4×Q8C2×D5311yescharacteristic
386#491—538no—
387#491—538no—
388#487—538no—
389#479—538no—
390#479C2×D5318yes—
391#475—538no—
392#462—538no—
393#462—538no—
394#462—538no—
395#446—538no—
396#441—538no—
397#441—538no—
398C4×Q16—538no—
399C2×#107C2×D5311yescharacteristic
400C2×#106—531no—
401C2×#106C2×D5311yescharacteristic
402C2×C4⋊C8—531no—
403C2×C4⋊C8—531no—
404C2×C4⋊C8C2×D5311yescharacteristic
405C2×#103—531no—
406C2×#103—531no—
407C2×#103—531no—
408C2×#103—531no—
409C2×#103—531no—
410C2×#103—531no—
411C2×#103—531no—
412C2×#103—531no—
413C2×#103C2×D5311yescharacteristic
414C2×#103C2×D5311yescharacteristic
415C2×#97—531no—
416C2×#97—531no—
417C2×C4×C8—531no—
418C53C2×#337411yescharacteristic, complemented, Sylow
419C22×Q8—531no—
420C22×Q8—531no—
421C22×Q8—531no—
422C22×Q8—531no—
423C22×Q8—1061no—
424C22×Q8C22×D5311yescharacteristic
425C2×Q16—534no—
426C2×Q16—534no—
427C2×Q16—534no—
428C2×Q16—1064no—
429C2×Q16—1064no—
430C2×Q16—1064no—
431C22×C8—531no—
432C22×C8—531no—
433C22×C8C22×D5311yescharacteristic
434C22×C8C22×D5311yescharacteristic
435C4⋊Q8—534no—
436C4⋊Q8—534no—
437C4⋊Q8—534no—
438C4⋊Q8—534no—
439C4⋊Q8—1062no—
440C4⋊Q8—1062no—
441C4⋊Q8—1062no—
442C4⋊Q8—1068no—
443C4⋊Q8—1062no—
444#125—22no—
445#125—534no—
446#125—1068no—
447#125—1062no—
448#125—1068no—
449#125—1068no—
450#125—1068no—
451#125—1068no—
452#125C22×D5314yes—
453C4×Q8—22no—
454C4×Q8—534no—
455C4×Q8—534no—
456C4×Q8—534no—
457C4×Q8—534no—
458C4×Q8—534no—
459C4×Q8—534no—
460C4×Q8—534no—
461C4×Q8—1068no—
462C4×Q8—1062no—
463C4×Q8—1062no—
464C4×Q8—1062no—
465C4×Q8—1062no—
466C4×Q8—1068no—
467C4×Q8—1062no—
468C4×Q8C22×D5314yes—
469C2×C4⋊C4—21no—
470C2×C4⋊C4—21no—
471C2×C4⋊C4—21no—
472C2×C4⋊C4—531no—
473C2×C4⋊C4—531no—
474C2×C4⋊C4—531no—
475C2×C4⋊C4—531no—
476C2×C4⋊C4—1061no—
477C2×C4⋊C4—1061no—
478C2×C4⋊C4—1061no—
479C2×C4⋊C4—1061no—
480C2×C4⋊C4—1061no—
481C2×C4⋊C4—1061no—
482C2×C4⋊C4—1061no—
483C2×C4⋊C4—1061no—
484C2×C4⋊C4—1061no—
485C2×C4⋊C4—1061no—
486C2×C4⋊C4—1061no—
487C2×C4⋊C4—1061no—
488C2×C4⋊C4—1061no—
489C2×C4⋊C4—1061no—
490C2×C4⋊C4C22×D5311yescharacteristic
491C2×C4⋊C4C22×D5311yescharacteristic
492C2×C4⋊C4C22×D5311yescharacteristic
493C2×C42—21no—
494C2×C42—531no—
495C2×C42—531no—
496C2×C42—1061no—
497C2×C42—1061no—
498C2×C42—1061no—
499C2×C42—1061no—
500C2×C42—1061no—
501C2×C42C22×D5311yescharacteristic
502#107C22×D5314yes—
503#106—534no—
504#106C22×D5314yes—
505C4⋊C8—534no—
506C4⋊C8—534no—
507C4⋊C8C22×D5314yes—
508#103—534no—
509#103—534no—
510#103—534no—
511#103—534no—
512#103—534no—
513#103—534no—
514#103—534no—
515#103—534no—
516#103C22×D5314yes—
517#103C22×D5314yes—
518#97—534no—
519#97—534no—
520C4×C8—534no—
521C2×Q8—21no—
522C2×Q8—22no—
523C2×Q8—22no—
524C2×Q8—534no—
525C2×Q8—534no—
526C2×Q8—532no—
527C2×Q8—532no—
528C2×Q8—532no—
529C2×Q8—1064no—
530C2×Q8—1064no—
531C2×Q8—1061no—
532C2×Q8—1061no—
533C2×Q8—1061no—
534C2×Q8—1062no—
535C2×Q8—1062no—
536C2×Q8—1062no—
537C2×Q8—1062no—
538C2×Q8—1062no—
539C2×Q8—2124no—
540C2×Q8C23×D5312yes—
541C22×C4—21no—
542C22×C4—21no—
543C22×C4—21no—
544C22×C4—21no—
545C22×C4—531no—
546C22×C4—531no—
547C22×C4—531no—
548C22×C4—531no—
549C22×C4—1061no—
550C22×C4—1061no—
551C22×C4—1061no—
552C22×C4—1061no—
553C22×C4D4×D5311yescharacteristic
554C22×C4D4×D5311yescharacteristic
555C22×C4C23×D5311yescharacteristic
556Q16—1068no—
557Q16—1068no—
558Q16—1068no—
559C2×C8—22no—
560C2×C8—22no—
561C2×C8—532no—
562C2×C8—532no—
563C2×C8—1062no—
564C2×C8—1062no—
565C2×C8C23×D5312yes—
566C2×C8C23×D5312yes—
567C4⋊C4—24no—
568C4⋊C4—24no—
569C4⋊C4—24no—
570C4⋊C4—21no—
571C4⋊C4—21no—
572C4⋊C4—21no—
573C4⋊C4—534no—
574C4⋊C4—534no—
575C4⋊C4—534no—
576C4⋊C4—532no—
577C4⋊C4—1064no—
578C4⋊C4—1064no—
579C4⋊C4—1064no—
580C4⋊C4—1064no—
581C4⋊C4—1064no—
582C4⋊C4—1064no—
583C4⋊C4—1064no—
584C4⋊C4—1064no—
585C4⋊C4—1064no—
586C4⋊C4—1064no—
587C4⋊C4—1064no—
588C4⋊C4—1061no—
589C4⋊C4—1064no—
590C4⋊C4—1064no—
591C4⋊C4—1064no—
592C4⋊C4C23×D5312yes—
593C4⋊C4C23×D5312yes—
594C4⋊C4C23×D5312yes—
595C42—24no—
596C42—532no—
597C42—532no—
598C42—532no—
599C42—532no—
600C42—1064no—
601C42—1064no—
602C42—1064no—
603C42—1064no—
604C42—1064no—
605C42D4×D5312yes—
606C42C23×D5312yes—
607C23C2×D4×D5311yescharacteristic
608Q8—22no—
609Q8—22no—
610Q8—42no—
611Q8—534no—
612Q8—1068no—
613Q8—1064no—
614Q8—1062no—
615Q8—1062no—
616Q8—1062no—
617Q8—1062no—
618Q8—1062no—
619Q8—2128no—
620Q8—2122no—
621Q8—2122no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—534no—
633C2×C4—532no—
634C2×C4—532no—
635C2×C4—532no—
636C2×C4—532no—
637C2×C4—1064no—
638C2×C4—1062no—
639C2×C4—1062no—
640C2×C4—1062no—
641C2×C4—1062no—
642C2×C4—1062no—
643C2×C4—1062no—
644C2×C4—1062no—
645C2×C4—2122no—
646C2×C4—2122no—
647C2×C4—2122no—
648C2×C4C2×D4×D5312yes—
649C2×C4C2×D4×D5312yes—
650C2×C4C2×D4×D5311yescharacteristic
651C2×C4#41×D5312yes—
652C2×C4#1313326512yes—
653C2×C4C24×D5311yescharacteristic
654C8—22no—
655C8—22no—
656C8—1062no—
657C8—1062no—
658C22#1681128312yes—
659C22#137×D5312yes—
660C22#1681143712yes—
661C22C22×D4×D5311yescharacteristic
662C4—22no—
663C4—22no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—534no—
669C4—1062no—
670C4—1064no—
671C4—1062no—
672C4—1062no—
673C4—2122no—
674C4—2122no—
675C4—2122no—
676C4#1681128312yes—
677C4C2×#41×D5311yescharacteristic
678C4C2×#1313326511yescharacteristic
679C2#2028752114yesdirect product factor
680C2C2×#1681128311yescharacteristic
681C2C2×#137×D5311yescharacteristic
682C2C2×#1681143711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC2×C4×C53C25
FrattiniC2×C4C24×D53
FittingC2×C53×#479C2
socle, prime radicalC23×C53C22×D4
hypercenterC2×#479D53
hypocenterC53C2×#3374
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#479

Complements

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

NH ≅ G/Ntype
3. #20287521C2direct (N × H)
354. C2×C53#3374semidirect
418. C53C2×#3374semidirect
679. C2#20287521direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168113512
C2×#168113502
#202875212
C2×#168109932
C2×#168109902
C2×#168109722
C2×#168109292
C2×#168109072
C2×#168108342
C2×#168108322
C2×#168107522
C2×#168107292
C2×#168107282
C2×#168107172
C2×C53×#4792
C2×#337453