Finite Group Databasebeta

← C2×#20276378 #27536870

Subgroups — C2×#20276378 #27536870

Order 13568.

1338 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811169C211yesmaximal, characteristic
2C2×#16810597C211yesmaximal, characteristic, complemented
3#20276378C218yesmaximal, direct product factor
4C2×#16810417C212yesmaximal, complemented
5C2×#16810415C212yesmaximal, complemented
6C2×C53×#410C211yesmaximal, characteristic, complemented
7C22×#11719945C411yescharacteristic, complemented
8#16811169—22no—
9#16811169—22no—
10#16811169C414yescomplemented
11#16811169C2214yes—
12C2×#13133215C411yescharacteristic, complemented
13C2×#13133213C2211yescharacteristic
14C2×#13133206C411yescharacteristic, complemented
15C2×#13133204C2211yescharacteristic
16C2×#13133189—21no—
17C2×#13133189—21no—
18C2×C4×C22⋊D53—21no—
19C2×C4×C22⋊D53—21no—
20C2×#13133176—21no—
21C22×C4×Dic53C411yescharacteristic, complemented
22#16810597—22no—
23#16810597C2214yescomplemented
24C2×#13133132—21no—
25C2×#13133132C2212yes—
26#16810417C2218yescomplemented
27#16810415C2218yescomplemented
28C22×#41×C53C2211yescharacteristic
29C2×#117×C53—21no—
30C53×#410—22no—
31C53×#410C2214yescomplemented
32C2×#99×C53—21no—
33C2×#99×C53C2212yes—
34C2×#11719945—22no—
35C2×#11719945—42no—
36C2×#11719945—41no—
37C2×#11719945—41no—
38C2×#11719945C2×C414yescomplemented
39C22×Dic106C2×C411yescharacteristic
40C22×D212C2×C411yescharacteristic
41C22×C22⋊D53—21no—
42C22×C22⋊D53—21no—
43C22×C22⋊D53—21no—
44C23×Dic53—21no—
45C22×C4×D53—21no—
46#13133215—22no—
47#13133215C2×C414yescomplemented
48#13133213—21no—
49#13133213—21no—
50#13133213C2×C412yes—
51#13133213C2312yes—
52#13133206—22no—
53#13133206C2×C414yescomplemented
54#13133204—21no—
55#13133204—21no—
56#13133204C2×C412yes—
57#13133204C2312yes—
58#13133189—24no—
59#13133189—24no—
60#13133189—24no—
61#13133189—24no—
62C4×C22⋊D53—24no—
63C4×C22⋊D53—24no—
64C4×C22⋊D53—24no—
65C4×C22⋊D53—24no—
66#13133176—24no—
67#13133176—42no—
68C2×C4⋊Dic53—41no—
69C2×#11719919—21no—
70C2×#11719919—21no—
71C2×C22⋊Dic53—21no—
72C2×C22⋊Dic53—21no—
73C2×C22⋊Dic53—22no—
74C2×#11719921—21no—
75C2×#11719921—21no—
76C2×C4×Dic53—21no—
77C2×C4×Dic53—22no—
78C2×C4×Dic53—41no—
79C2×C4×Dic53C2×C411yescharacteristic
80C2×C4×Dic53C2×C411yescharacteristic
81C2×C4×Dic53C2×C414yescomplemented
82#13133132—24no—
83#13133132—22no—
84#13133132C2314yes—
85C2×#41×C53—41no—
86C2×#41×C53—41no—
87C2×#41×C53—42no—
88C2×#41×C53C2×C412yes—
89C2×#41×C53D412yes—
90C2×#41×C53D412yes—
91C2×#41×C53C2312yes—
92C22×Q8×C53C2×C411yescharacteristic
93C22×D4×C53D411yescharacteristic
94C22×D4×C53D411yescharacteristic
95C22×D4×C53C2311yescharacteristic
96C23×C4×C53C2×C411yescharacteristic
97C23×C4×C53D411yescharacteristic
98C23×C4×C53D411yescharacteristic
99#117×C53—24no—
100#117×C53—42no—
101C2×C4⋊C4×C53—41no—
102C2×C22⋊C4×C53—22no—
103C2×C42×C53—41no—
104#99×C53—24no—
105#99×C53—22no—
106#99×C53C2314yes—
107C24×C53C22⋊C411yescharacteristic
108C24×C53C2×D411yescharacteristic
109C24×C53C2×D411yescharacteristic
110C23×D53—21no—
111#41×C53—42no—
112#41×C53—42no—
113#41×C53—42no—
114#41×C53—42no—
115#41×C53—42no—
116#41×C53—42no—
117#41×C53—42no—
118#41×C53—42no—
119C2×Q8×C53—41no—
120C2×Q8×C53—41no—
121C2×Q8×C53C22⋊C412yes—
122C2×Q8×C53C22×C412yes—
123C2×D4×C53—41no—
124C2×D4×C53—41no—
125C2×D4×C53—41no—
126C2×D4×C53—41no—
127C2×D4×C53—41no—
128C2×D4×C53—41no—
129C2×D4×C53C22⋊C412yes—
130C2×D4×C53C22⋊C412yes—
131C2×D4×C53C22×C411yescharacteristic
132C2×D4×C53C2×D412yes—
133C2×D4×C53C2×D411yescharacteristic
134C2×D4×C53C2×D412yes—
135C2×D4×C53C2×D411yescharacteristic
136C2×D4×C53C2411yescharacteristic
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—22no—
140C22×C4×C53—22no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—22no—
146C22×C4×C53—42no—
147C22×C4×C53C22⋊C412yes—
148C22×C4×C53C22⋊C411yescharacteristic
149C22×C4×C53C22⋊C411yescharacteristic
150C22×C4×C53C22⋊C412yes—
151C22×C4×C53C22⋊C412yes—
152C22×C4×C53C22⋊C411yescharacteristic
153C22×C4×C53C22×C411yescharacteristic
154C22×C4×C53C22×C412yes—
155C22×C4×C53C2×D412yes—
156C22×C4×C53C2×D412yes—
157C2×C22⋊D53—22no—
158C2×C22⋊D53—22no—
159C2×C22⋊D53—24no—
160C2×C22⋊D53—22no—
161C2×C22⋊D53—22no—
162C2×C22⋊D53—44no—
163C2×C22⋊D53—44no—
164C2×C22⋊D53—42no—
165C2×C22⋊D53—42no—
166C22×Dic53—21no—
167C22×Dic53—21no—
168C22×Dic53—24no—
169C22×Dic53—21no—
170C22×Dic53—44no—
171C22×Dic53—42no—
172#11719945—44no—
173#11719945—44no—
174#11719945—44no—
175#11719945—44no—
176C2×D212—21no—
177C2×D212—41no—
178C2×D212—41no—
179C2×D212C22×C412yes—
180C2×C4×D53—24no—
181C2×C4×D53—42no—
182C2×C4×D53—41no—
183C2×C4×D53—41no—
184C2×Dic106—21no—
185C2×Dic106—41no—
186C2×Dic106—41no—
187C2×Dic106C22×C412yes—
188C4⋊C4×C53—44no—
189C22⋊C4×C53—24no—
190C22⋊C4×C53—42no—
191C42×C53—44no—
192C22⋊Dic53—22no—
193C22⋊Dic53—22no—
194C22⋊Dic53—22no—
195C22⋊Dic53—22no—
196C22⋊Dic53—24no—
197C22⋊Dic53—42no—
198#11719921—24no—
199#11719921—24no—
200C4⋊Dic53—44no—
201#11719919—24no—
202#11719919—24no—
203C4×Dic53—21no—
204C4×Dic53—21no—
205C4×Dic53—24no—
206C4×Dic53—44no—
207C4×Dic53C22×C412yes—
208C4×Dic53C22×C412yes—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—42no—
216C23×C53—42no—
217C23×C53—42no—
218C23×C53C2×C22⋊C411yescharacteristic
219C23×C53C2×C22⋊C412yes—
220C23×C53C2×C22⋊C411yescharacteristic
221C23×C53C2×C22⋊C411yescharacteristic
222C23×C53C22×D411yescharacteristic
223C23×C53C22×D411yescharacteristic
224C22×D53—22no—
225C22×D53—42no—
226C22×D53—42no—
227C22×D53—41no—
228C22×D53—41no—
229Q8×C53—42no—
230Q8×C53—42no—
231D4×C53—42no—
232D4×C53—42no—
233D4×C53—41no—
234D4×C53—41no—
235D4×C53—42no—
236D4×C53—41no—
237D4×C53—42no—
238D4×C53—41no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—22no—
246C2×C4×C53—22no—
247C2×C4×C53—22no—
248C2×C4×C53—22no—
249C2×C4×C53—22no—
250C2×C4×C53—22no—
251C2×C4×C53—22no—
252C2×C4×C53—22no—
253C2×C4×C53—22no—
254C2×C4×C53—22no—
255C2×C4×C53—22no—
256C2×C4×C53—22no—
257C2×C4×C53—44no—
258C2×C4×C53—84no—
259C2×C4×C53#9912yes—
260C2×C4×C53#9911yescharacteristic
261C2×C4×C53#9911yescharacteristic
262C2×C4×C53C2×C22⋊C411yescharacteristic
263C2×C4×C53C2×C22⋊C412yes—
264C2×C4×C53C2×C22⋊C412yes—
265C2×C4×C53C2×C22⋊C412yes—
266C2×C4×C53C23×C411yescharacteristic
267C22⋊D53—44no—
268C22⋊D53—44no—
269C22⋊D53—44no—
270C22⋊D53—44no—
271C22⋊D53—48no—
272C2×Dic53—24no—
273C2×Dic53—22no—
274C2×Dic53—22no—
275C2×Dic53—22no—
276C2×Dic53—48no—
277C2×Dic53—42no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—44no—
281C2×Dic53—84no—
282D212—42no—
283D212—42no—
284C4×D53—44no—
285C4×D53—44no—
286Dic106—42no—
287Dic106—42no—
288C2×#2126—531nomaximal, Sylow
289C22×C53—22no—
290C22×C53—21no—
291C22×C53—22no—
292C22×C53—21no—
293C22×C53—21no—
294C22×C53—21no—
295C22×C53—42no—
296C22×C53—41no—
297C22×C53—41no—
298C22×C53—41no—
299C22×C53—41no—
300C22×C53—42no—
301C22×C53—42no—
302C22×C53—42no—
303C22×C53—42no—
304C22×C53—81no—
305C22×C53—81no—
306C22×C53C2×#9911yescharacteristic
307C22×C53C2×#9911yescharacteristic
308C22×C53C22×C22⋊C411yescharacteristic
309C2×D53—42no—
310C2×D53—42no—
311C2×D53—42no—
312C2×D53—82no—
313C2×D53—82no—
314C4×C53—21no—
315C4×C53—21no—
316C4×C53—22no—
317C4×C53—22no—
318C4×C53—84no—
319C4×C53C2×#9912yes—
320C4×C53C2×#9912yes—
321Dic53—24no—
322Dic53—42no—
323Dic53—42no—
324Dic53—84no—
325C2×#532—531no—
326C2×#410—531no—
327C2×#410D5311yescharacteristic, complemented
328#2126—538no—
329C2×#354—532no—
330C2×#353—532no—
331C2×C53—21no—
332C2×C53—21no—
333C2×C53—41no—
334C2×C53—41no—
335C2×C53—42no—
336C2×C53—41no—
337C2×C53—41no—
338C2×C53#212612yescomplemented
339C2×C53C22×#9911yescharacteristic
340D53—82no—
341C22×#41—531no—
342C22×#41C2×D5311yescharacteristic
343#532—534no—
344#532—534no—
345#532—1062no—
346#532—1062no—
347C2×C4⋊Q8—531no—
348C2×#127—531no—
349C2×#124—531no—
350C2×#124—531no—
351C2×#121—1061no—
352C2×#121—1061no—
353C2×C4×D4—1061no—
354C2×C4×D4—1061no—
355C2×#117—21no—
356C2×#117—1061no—
357C22×C42—531no—
358#410—22no—
359#410—534no—
360#410—1062no—
361#410C2×D5314yescomplemented
362C2×#99—21no—
363C2×#99—532no—
364C2×#99—1061no—
365C2×#99C2×D5312yes—
366#354—538no—
367#353—538no—
368C53C2×#212611yescharacteristic, complemented, Sylow
369C2×#41—41no—
370C2×#41—42no—
371C2×#41—41no—
372C2×#41—534no—
373C2×#41—1062no—
374C2×#41—2122no—
375C2×#41—2121no—
376C2×#41—2121no—
377C2×#41C4×D5312yes—
378C2×#41D21212yes—
379C2×#41C22⋊D5312yes—
380C2×#41C22×D5312yes—
381C22×Q8—531no—
382C22×Q8C4×D5311yescharacteristic
383C22×D4—531no—
384C22×D4—1061no—
385C22×D4—1061no—
386C22×D4—1061no—
387C22×D4D21211yescharacteristic
388C22×D4C22⋊D5311yescharacteristic
389C22×D4C22×D5311yescharacteristic
390C23×C4—1061no—
391C23×C4—1061no—
392C23×C4C4×D5311yescharacteristic
393C23×C4D21211yescharacteristic
394C23×C4C22⋊D5311yescharacteristic
395C4⋊Q8—534no—
396C4⋊Q8—1062no—
397#127—532no—
398#127—532no—
399#127—1061no—
400#127—1061no—
401#124—534no—
402#124—532no—
403#124—532no—
404#124—1062no—
405#124—1061no—
406#124—1061no—
407#121—1064no—
408#121—1064no—
409#121—1064no—
410#121—1064no—
411C4×D4—1064no—
412C4×D4—1064no—
413C4×D4—1064no—
414C4×D4—1064no—
415#117—24no—
416#117—42no—
417#117—1064no—
418#117—2122no—
419C2×C4⋊C4—41no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4—2121no—
423C2×C22⋊C4—22no—
424C2×C22⋊C4—1061no—
425C2×C22⋊C4—1061no—
426C2×C22⋊C4—1061no—
427C2×C22⋊C4—1061no—
428C2×C22⋊C4—1062no—
429C2×C42—41no—
430C2×C42—531no—
431C2×C42—531no—
432C2×C42—534no—
433C2×C42—1062no—
434C2×C42—1061no—
435C2×C42—2121no—
436#99—24no—
437#99—22no—
438#99—534no—
439#99—1062no—
440#99—1064no—
441#99C22×D5314yes—
442C24—1061no—
443C24#1171992111yescharacteristic
444C24C2×D21211yescharacteristic
445C24C2×C22⋊D5311yescharacteristic
446#41—42no—
447#41—42no—
448#41—42no—
449#41—42no—
450#41—42no—
451#41—42no—
452#41—42no—
453#41—42no—
454#41—2124no—
455#41—2124no—
456#41—2124no—
457#41—2124no—
458C2×Q8—41no—
459C2×Q8—41no—
460C2×Q8—532no—
461C2×Q8—1061no—
462C2×Q8—2121no—
463C2×Q8—2121no—
464C2×Q8#1171992112yes—
465C2×Q8C2×C4×D5312yes—
466C2×D4—41no—
467C2×D4—41no—
468C2×D4—41no—
469C2×D4—41no—
470C2×D4—41no—
471C2×D4—41no—
472C2×D4—532no—
473C2×D4—1062no—
474C2×D4—1062no—
475C2×D4—1064no—
476C2×D4—1062no—
477C2×D4—1062no—
478C2×D4—1061no—
479C2×D4—2121no—
480C2×D4—2121no—
481C2×D4—2124no—
482C2×D4—2124no—
483C2×D4—2122no—
484C2×D4—2122no—
485C2×D4#1171992112yes—
486C2×D4#1171992112yes—
487C2×D4C2×C4×D5311yescharacteristic
488C2×D4C2×D21212yes—
489C2×D4C2×D21211yescharacteristic
490C2×D4C2×C22⋊D5312yes—
491C2×D4C2×C22⋊D5311yescharacteristic
492C2×D4C23×D5311yescharacteristic
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—42no—
503C22×C4—1061no—
504C22×C4—1061no—
505C22×C4—1064no—
506C22×C4—1061no—
507C22×C4—1064no—
508C22×C4—2124no—
509C22×C4—2122no—
510C22×C4—2122no—
511C22×C4—2121no—
512C22×C4—2121no—
513C22×C4#1171992112yes—
514C22×C4#1171992111yescharacteristic
515C22×C4#1171992111yescharacteristic
516C22×C4#1171992112yes—
517C22×C4#1171992112yes—
518C22×C4#1171992111yescharacteristic
519C22×C4C2×C4×D5311yescharacteristic
520C22×C4C2×C4×D5312yes—
521C22×C4C2×D21212yes—
522C22×C4C2×C22⋊D5312yes—
523C4⋊C4—44no—
524C4⋊C4—1064no—
525C4⋊C4—1064no—
526C4⋊C4—2124no—
527C22⋊C4—24no—
528C22⋊C4—42no—
529C22⋊C4—1062no—
530C22⋊C4—1062no—
531C22⋊C4—1062no—
532C22⋊C4—1062no—
533C22⋊C4—1064no—
534C22⋊C4—1064no—
535C22⋊C4—1064no—
536C22⋊C4—2122no—
537C42—44no—
538C42—532no—
539C42—532no—
540C42—1064no—
541C42—1061no—
542C42—1061no—
543C42—2124no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—1062no—
554C23—2122no—
555C23—2122no—
556C23—2121no—
557C23—2121no—
558C23C2×#1171992111yescharacteristic
559C23C2×#1171992111yescharacteristic
560C23C2×#1171992111yescharacteristic
561C23C2×#1171992112yes—
562C23C22×C22⋊D5311yescharacteristic
563C23C22×D21211yescharacteristic
564Q8—42no—
565Q8—42no—
566Q8—2122no—
567Q8—2122no—
568D4—41no—
569D4—41no—
570D4—42no—
571D4—41no—
572D4—42no—
573D4—41no—
574D4—42no—
575D4—42no—
576D4—2122no—
577D4—2122no—
578D4—2124no—
579D4—2124no—
580D4—2124no—
581D4—2124no—
582D4—2128no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—44no—
602C2×C4—84no—
603C2×C4—1064no—
604C2×C4—1062no—
605C2×C4—1062no—
606C2×C4—1062no—
607C2×C4—4244no—
608C2×C4—2128no—
609C2×C4—2124no—
610C2×C4—2122no—
611C2×C4—2121no—
612C2×C4—2121no—
613C2×C4—2124no—
614C2×C4—2124no—
615C2×C4#1313313111yescharacteristic
616C2×C4#1313313112yes—
617C2×C4#1313313111yescharacteristic
618C2×C4C2×#1171992111yescharacteristic
619C2×C4C2×#1171992112yes—
620C2×C4C2×#1171992112yes—
621C2×C4C2×#1171992112yes—
622C2×C4C22×C4×D5311yescharacteristic
623C22—22no—
624C22—21no—
625C22—21no—
626C22—22no—
627C22—21no—
628C22—21no—
629C22—42no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—81no—
639C22—81no—
640C22—4242no—
641C22—4242no—
642C22—2122no—
643C22—2122no—
644C22—2122no—
645C22C2×#1313313111yescharacteristic
646C22C2×#1313313111yescharacteristic
647C22C22×#1171992111yescharacteristic
648C4—21no—
649C4—22no—
650C4—22no—
651C4—21no—
652C4—84no—
653C4—1064no—
654C4—4244no—
655C4—2122no—
656C4—2122no—
657C4C2×#1313313112yes—
658C4C2×#1313313112yes—
659C2—21no—
660C2—21no—
661C2—41no—
662C2—41no—
663C2—42no—
664C2—41no—
665C2—41no—
666C2—4242no—
667C2#2027637812yesdirect product factor
668C2C22×#1313313111yescharacteristic
subgroupG/H
centerC2×C4#13133131
commutatorC2×C4×C53C23×C4
FrattiniC2×D4C23×D53
FittingC2×C53×#410C2
socleC22×C53C2×#99
prime radicalC2×#16811169C2
cosocleC2×D4×C53C24
hypercenterC2×#410D53
hypocenterC53C2×#2126
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
2. C2×#16810597C2semidirect
3. #20276378C2direct (N × H)
4. C2×#16810417C2semidirect
5. C2×#16810415C2semidirect
6. C2×C53×#410C2semidirect
7. C22×#11719945C4semidirect
10. #16811169C4semidirect
12. C2×#13133215C4semidirect
14. C2×#13133206C4semidirect
21. C22×C4×Dic53C4semidirect
23. #16810597C22semidirect
26. #16810417C22semidirect
27. #16810415C22semidirect
31. C53×#410C22semidirect
38. C2×#11719945C2×C4semidirect
47. #13133215C2×C4semidirect
53. #13133206C2×C4semidirect
81. C2×C4×Dic53C2×C4semidirect
327. C2×#410D53semidirect
338. C2×C53#2126semidirect
361. #410C2×D53semidirect
368. C53C2×#2126semidirect
667. C2#20276378direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111692
C2×#168105972
#202763782
C2×#168104172
C2×#168104152
C2×C53×#4102
C2×#212653