Finite Group Databasebeta

← C2×#13126106 #16735126

Subgroups — C2×#13126106 #16735126

Order 3328.

1678 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719224C211yesmaximal, characteristic
2C2×#11719223C211yesmaximal, characteristic
3#13126106C2116yesmaximal, direct product factor
4C2×#11718363C211yesmaximal, characteristic, complemented
5C2×#11718344C211yesmaximal, characteristic, complemented
6C2×#11718364C211yesmaximal, characteristic, complemented
7C2×#11718342C211yesmaximal, characteristic, complemented
8C2×#11718383C211yesmaximal, characteristic
9C2×#11718384C211yesmaximal, characteristic
10C2×#11718384C211yesmaximal, characteristic
11C2×#11718382C211yesmaximal, characteristic
12C2×#11718490C211yesmaximal, characteristic, complemented
13C2×#11718490C211yesmaximal, characteristic, complemented
14C2×#11718511C211yesmaximal, characteristic, complemented
15C2×#11718512C211yesmaximal, characteristic, complemented
16C2×C13×#433C211yesmaximal, characteristic
17C2×C13×#117C2211yescharacteristic
18C2×#88180C2211yescharacteristic
19C22×C4⋊Dic13C2211yescharacteristic
20C2×#88178C2211yescharacteristic
21C2×#88178C2211yescharacteristic
22#11719224—24no—
23#11719224C2218yes—
24#11719223—24no—
25#11719223C2218yes—
26C2×#88144—21no—
27C2×#88143—21no—
28C2×#88143—21no—
29C2×#88142—21no—
30C2×#88133—21no—
31C2×#88133—21no—
32C2×#88132—21no—
33C2×#88132—21no—
34C2×#88124C2211yescharacteristic
35C2×#88124C2211yescharacteristic
36C2×#88124C2211yescharacteristic
37C2×#88123C2211yescharacteristic
38C2×#88123C2211yescharacteristic
39C2×#88123C2211yescharacteristic
40C2×C4×Dic26C2211yescharacteristic
41C2×C4×Dic26C2211yescharacteristic
42C13×#433C2218yes—
43C2×C13×C4⋊C8C2211yescharacteristic
44C2×C13×C4⋊C8C2211yescharacteristic
45C2×C13×C22⋊C8C2211yescharacteristic
46C2×C13×C22⋊C8C2211yescharacteristic
47C2×C13×#97C2211yescharacteristic
48C2×C4×C8×C13C2211yescharacteristic
49C2×#88058C2211yescharacteristic
50C2×#88058C2211yescharacteristic
51C2×#88058C2211yescharacteristic
52C2×#88058C2211yescharacteristic
53C2×#88057C2211yescharacteristic
54C2×#88057C2211yescharacteristic
55C2×#88057C2211yescharacteristic
56C2×#88057C2211yescharacteristic
57C2×#88056C2211yescharacteristic
58C2×#88056C2211yescharacteristic
59C2×#88056C2211yescharacteristic
60C2×#88056C2211yescharacteristic
61C2×#88056C2211yescharacteristic
62C2×#88056C2211yescharacteristic
63C2×#88056C2211yescharacteristic
64C2×#88056C2211yescharacteristic
65#11718512C2218yescomplemented
66#11718511C2218yescomplemented
67#11718490C2218yescomplemented
68#11718490C2218yescomplemented
69#11718384C2218yes—
70#11718384C2218yes—
71#11718383C2218yes—
72#11718382C2218yes—
73#11718364C2218yescomplemented
74#11718363C2218yescomplemented
75#11718344C2218yescomplemented
76#11718342C2218yescomplemented
77C23×C4×C13C2311yescharacteristic
78C23×Dic13—21no—
79C22×Dic26—21no—
80C22×Dic26—21no—
81C22×Dic26C2311yescharacteristic
82C22×Dic26C2311yescharacteristic
83C22×C8×C13C2311yescharacteristic
84C22×C8×C13C2311yescharacteristic
85C22×C8×C13C2311yescharacteristic
86C22×C8×C13C2311yescharacteristic
87C13×#117D414yes—
88C13×#117C2314yes—
89C2×C13×C4⋊C4D411yescharacteristic
90C2×C13×C4⋊C4D411yescharacteristic
91C2×C13×C22⋊C4D411yescharacteristic
92C2×C13×C22⋊C4D411yescharacteristic
93C2×C42×C13C2311yescharacteristic
94C2×C42×C13C2311yescharacteristic
95C2×C22⋊Dic13—21no—
96C2×C22⋊Dic13—21no—
97C2×C22⋊Dic13—21no—
98#88180—22no—
99#88180C2314yes—
100C2×C4⋊Dic13—21no—
101C2×C4⋊Dic13—21no—
102C2×C4⋊Dic13—21no—
103C2×C4⋊Dic13—21no—
104C2×C4⋊Dic13—21no—
105C2×C4⋊Dic13—22no—
106C2×C4⋊Dic13C2311yescharacteristic
107C2×C4⋊Dic13C2311yescharacteristic
108C2×C4⋊Dic13C2311yescharacteristic
109C2×C4⋊Dic13C2311yescharacteristic
110C2×C4⋊Dic13C2311yescharacteristic
111C2×C4⋊Dic13C2311yescharacteristic
112C2×C4⋊Dic13C2314yes—
113#88178—22no—
114#88178—22no—
115#88178C2314yes—
116#88178C2314yes—
117C2×#6101—21no—
118C2×#6101—21no—
119C2×#6101—21no—
120C2×#6101—21no—
121C2×#6101—21no—
122C2×#6101—21no—
123C2×#6101—21no—
124C2×#6101—21no—
125C2×#6101—21no—
126C2×#6101—21no—
127C2×C4×Dic13—21no—
128C2×C4×Dic13—21no—
129C2×C4×Dic13—21no—
130#88144—28no—
131#88143—28no—
132#88143—28no—
133#88142—28no—
134#88133—28no—
135#88133—28no—
136#88132—28no—
137#88132—28no—
138#88124—22no—
139#88124—22no—
140#88124—22no—
141#88124C2314yes—
142#88124C2314yes—
143#88124C2314yes—
144#88123—22no—
145#88123—22no—
146#88123—22no—
147#88123C2314yes—
148#88123C2314yes—
149#88123C2314yes—
150C4×Dic26—22no—
151C4×Dic26—22no—
152C4×Dic26C2314yes—
153C4×Dic26C2314yes—
154C13×C4⋊C8C2314yes—
155C13×C4⋊C8C2314yes—
156C13×C22⋊C8C2314yes—
157C13×C22⋊C8C2314yes—
158C13×#97C2314yes—
159C4×C8×C13C2314yes—
160#88058C2314yes—
161#88058C2314yes—
162#88058C2314yes—
163#88058C2314yes—
164#88057C2314yes—
165#88057C2314yes—
166#88057C2314yes—
167#88057C2314yes—
168#88056C2314yes—
169#88056C2314yes—
170#88056C2314yes—
171#88056C2314yes—
172#88056C2314yes—
173#88056C2314yes—
174#88056C2314yes—
175#88056C2314yes—
176C2×#3242—131nomaximal, Sylow
177C24×C13C2×D411yescharacteristic
178C22×C4×C13—22no—
179C22×C4×C13—22no—
180C22×C4×C13C2×D411yescharacteristic
181C22×C4×C13C2×D411yescharacteristic
182C22×C4×C13C2×D411yescharacteristic
183C22×C4×C13C2×D411yescharacteristic
184C22×C4×C13C2×D411yescharacteristic
185C22×C4×C13C2×D412yes—
186C22×C4×C13C2412yes—
187C22×C4×C13C2411yescharacteristic
188C22×Dic13—21no—
189C22×Dic13—21no—
190C22×Dic13—21no—
191C22×Dic13—21no—
192C22×Dic13—21no—
193C22×Dic13—21no—
194C22×Dic13—21no—
195C22×Dic13—21no—
196C22×Dic13—24no—
197C22×Dic13—82no—
198C2×Dic26—22no—
199C2×Dic26—22no—
200C2×Dic26—22no—
201C2×Dic26—22no—
202C2×Dic26—21no—
203C2×Dic26—21no—
204C2×Dic26—24no—
205C2×Dic26—24no—
206C2×Dic26—82no—
207C2×Dic26—82no—
208C2×Dic26C2412yes—
209C2×Dic26C2412yes—
210C2×C8×C13—22no—
211C2×C8×C13—22no—
212C2×C8×C13—22no—
213C2×C8×C13—22no—
214C2×C8×C13C2412yes—
215C2×C8×C13C2412yes—
216C2×C8×C13C2412yes—
217C2×C8×C13C2412yes—
218C13×C4⋊C4C2×D414yes—
219C13×C4⋊C4C2×D414yes—
220C13×C22⋊C4C2×D414yes—
221C13×C22⋊C4C2×D414yes—
222C42×C13C2×D412yes—
223C42×C13C2×D412yes—
224C42×C13C2412yes—
225C42×C13C2412yes—
226C22⋊Dic13—24no—
227C22⋊Dic13—24no—
228C22⋊Dic13—24no—
229C4⋊Dic13—21no—
230C4⋊Dic13—21no—
231C4⋊Dic13—21no—
232C4⋊Dic13—21no—
233C4⋊Dic13—21no—
234C4⋊Dic13—24no—
235C4⋊Dic13—24no—
236C4⋊Dic13—24no—
237C4⋊Dic13—24no—
238C4⋊Dic13—21no—
239C4⋊Dic13—24no—
240C4⋊Dic13C2412yes—
241C4⋊Dic13C2412yes—
242C4⋊Dic13C2412yes—
243C4⋊Dic13C2412yes—
244C4⋊Dic13C2412yes—
245C4⋊Dic13C2412yes—
246#6101—24no—
247#6101—24no—
248#6101—24no—
249#6101—24no—
250#6101—24no—
251#6101—24no—
252#6101—24no—
253#6101—24no—
254#6101—24no—
255#6101—24no—
256C4×Dic13—24no—
257C4×Dic13—24no—
258C4×Dic13—24no—
259C2×#544—131no—
260C2×#541—131no—
261#3242—1316no—
262C2×#501—131no—
263C2×#499—131no—
264C2×#491—131no—
265C2×#487—131no—
266C2×#484—131no—
267C2×#485—131no—
268C2×#485—131no—
269C2×#483—131no—
270C2×#479—131no—
271C2×#479—131no—
272C2×#477—131no—
273C2×#475—131no—
274C2×#433D1311yescharacteristic
275C23×C13—22no—
276C23×C13—22no—
277C23×C13—22no—
278C23×C13C22×D412yes—
279C23×C13C22×D411yescharacteristic
280C2×C4×C13—22no—
281C2×C4×C13—22no—
282C2×C4×C13—22no—
283C2×C4×C13—22no—
284C2×C4×C13—22no—
285C2×C4×C13—24no—
286C2×C4×C13—22no—
287C2×C4×C13—22no—
288C2×C4×C13C22×D412yes—
289C2×C4×C13C22×D412yes—
290C2×C4×C13C22×D412yes—
291C2×C4×C13C22×D412yes—
292C2×C4×C13C22×D412yes—
293C2×C4×C13C22×D411yescharacteristic
294C2×C4×C132−1+412yes—
295C2×C4×C132−1+412yes—
296C2×C4×C13C2511yescharacteristic
297C2×Dic13—22no—
298C2×Dic13—22no—
299C2×Dic13—22no—
300C2×Dic13—22no—
301C2×Dic13—22no—
302C2×Dic13—22no—
303C2×Dic13—22no—
304C2×Dic13—22no—
305C2×Dic13—42no—
306C2×Dic13—42no—
307C2×Dic13—42no—
308C2×Dic13—42no—
309C2×Dic13—42no—
310C2×Dic13—42no—
311C2×Dic13—42no—
312C2×Dic13—42no—
313C2×Dic13—84no—
314Dic26—22no—
315Dic26—22no—
316Dic26—22no—
317Dic26—22no—
318Dic26—42no—
319Dic26—42no—
320Dic26—84no—
321Dic26—84no—
322C8×C13—22no—
323C8×C13—22no—
324C8×C13—22no—
325C8×C13—22no—
326#544—138no—
327#544—264no—
328#541—138no—
329#541—264no—
330C2×C4⋊Q8—131no—
331C2×C4⋊Q8—131no—
332C2×C4⋊Q8—131no—
333C2×C4⋊Q8—261no—
334C2×#126—261no—
335C2×#126—261no—
336C2×#125—131no—
337C2×#125—131no—
338C2×#125—131no—
339C2×#125—261no—
340C2×#125—261no—
341C2×#125—261no—
342C2×C22⋊Q8—131no—
343C2×C22⋊Q8—131no—
344C2×C22⋊Q8—261no—
345C2×C22⋊Q8—261no—
346C2×C4×Q8—131no—
347C2×C4×Q8—131no—
348C2×#117—131no—
349C2×#117C2×D1311yescharacteristic
350C22×C4⋊C4—131no—
351#501—138no—
352#499—138no—
353#491—138no—
354#487—138no—
355#485—138no—
356#485—138no—
357#484—138no—
358#483—138no—
359#479—138no—
360#479—138no—
361#477—138no—
362#475—138no—
363#433C2×D1318yes—
364C2×#107—131no—
365C2×#107—131no—
366C2×#107—131no—
367C2×#107—131no—
368C2×#106—131no—
369C2×#106—131no—
370C2×#106—131no—
371C2×#106—131no—
372C2×C4⋊C8C2×D1311yescharacteristic
373C2×C4⋊C8C2×D1311yescharacteristic
374C2×#103—131no—
375C2×#103—131no—
376C2×#103—131no—
377C2×#103—131no—
378C2×#103—131no—
379C2×#103—131no—
380C2×#103—131no—
381C2×#103—131no—
382C2×C22⋊C8C2×D1311yescharacteristic
383C2×C22⋊C8C2×D1311yescharacteristic
384C2×#97C2×D1311yescharacteristic
385C2×C4×C8C2×D1311yescharacteristic
386C22×C13—22no—
387C22×C13—22no—
388C22×C13—22no—
389C22×C13—44no—
390C22×C13#53612yes—
391C22×C13#57712yes—
392C22×C13#57812yes—
393C22×C13C23×D411yescharacteristic
394C4×C13—22no—
395C4×C13—22no—
396C4×C13—22no—
397C4×C13—22no—
398C4×C13—22no—
399C4×C13#53612yes—
400C4×C13C2×2−1+411yescharacteristic
401C4×C13C2×2−1+411yescharacteristic
402Dic13—42no—
403Dic13—42no—
404Dic13—42no—
405Dic13—42no—
406Dic13—42no—
407Dic13—42no—
408Dic13—42no—
409Dic13—42no—
410C22×Q8—131no—
411C22×Q8—131no—
412C22×Q8—261no—
413C22×Q8—261no—
414C23×C4—261no—
415C23×C4C22×D1311yescharacteristic
416C22×C8C22×D1311yescharacteristic
417C22×C8C22×D1311yescharacteristic
418C22×C8C22×D1311yescharacteristic
419C22×C8C22×D1311yescharacteristic
420C4⋊Q8—134no—
421C4⋊Q8—134no—
422C4⋊Q8—134no—
423C4⋊Q8—268no—
424C4⋊Q8—262no—
425C4⋊Q8—262no—
426C4⋊Q8—262no—
427#126—268no—
428#126—268no—
429#125—134no—
430#125—134no—
431#125—134no—
432#125—268no—
433#125—268no—
434#125—268no—
435#125—262no—
436#125—262no—
437#125—262no—
438C22⋊Q8—134no—
439C22⋊Q8—134no—
440C22⋊Q8—262no—
441C22⋊Q8—262no—
442C22⋊Q8—268no—
443C22⋊Q8—268no—
444C4×Q8—134no—
445C4×Q8—134no—
446C4×Q8—262no—
447C4×Q8—262no—
448#117—134no—
449#117—262no—
450#117D5214yes—
451#117C22×D1314yes—
452C2×C4⋊C4—131no—
453C2×C4⋊C4—131no—
454C2×C4⋊C4—134no—
455C2×C4⋊C4—131no—
456C2×C4⋊C4—131no—
457C2×C4⋊C4—131no—
458C2×C4⋊C4—131no—
459C2×C4⋊C4—261no—
460C2×C4⋊C4—262no—
461C2×C4⋊C4—261no—
462C2×C4⋊C4—261no—
463C2×C4⋊C4—261no—
464C2×C4⋊C4—261no—
465C2×C4⋊C4—261no—
466C2×C4⋊C4—261no—
467C2×C4⋊C4—261no—
468C2×C4⋊C4—261no—
469C2×C4⋊C4—261no—
470C2×C4⋊C4—261no—
471C2×C4⋊C4—261no—
472C2×C4⋊C4—261no—
473C2×C4⋊C4—261no—
474C2×C4⋊C4—261no—
475C2×C4⋊C4D5211yescharacteristic
476C2×C4⋊C4D5211yescharacteristic
477C2×C22⋊C4—261no—
478C2×C22⋊C4—261no—
479C2×C22⋊C4—261no—
480C2×C22⋊C4D5211yescharacteristic
481C2×C22⋊C4D5211yescharacteristic
482C2×C42—261no—
483C2×C42—261no—
484C2×C42—261no—
485C2×C42C22×D1311yescharacteristic
486C2×C42C22×D1311yescharacteristic
487#107—134no—
488#107—134no—
489#107—134no—
490#107—134no—
491#106—134no—
492#106—134no—
493#106—134no—
494#106—134no—
495C4⋊C8C22×D1314yes—
496C4⋊C8C22×D1314yes—
497#103—134no—
498#103—134no—
499#103—134no—
500#103—134no—
501#103—134no—
502#103—134no—
503#103—134no—
504#103—134no—
505C22⋊C8C22×D1314yes—
506C22⋊C8C22×D1314yes—
507#97C22×D1314yes—
508C4×C8C22×D1314yes—
509C2×C13—42no—
510C2×C13#324214yescomplemented
511C2×C13C2×#53611yescharacteristic
512C2×C13C2×#57711yescharacteristic
513C2×C13C2×#57811yescharacteristic
514C24C2×D5211yescharacteristic
515C2×Q8—132no—
516C2×Q8—132no—
517C2×Q8—261no—
518C2×Q8—261no—
519C2×Q8—264no—
520C2×Q8—264no—
521C2×Q8—262no—
522C2×Q8—262no—
523C2×Q8—262no—
524C2×Q8—262no—
525C2×Q8—1042no—
526C2×Q8—1042no—
527C22×C4—22no—
528C22×C4—22no—
529C22×C4—261no—
530C22×C4—261no—
531C22×C4—261no—
532C22×C4—261no—
533C22×C4—261no—
534C22×C4—261no—
535C22×C4—261no—
536C22×C4—261no—
537C22×C4—264no—
538C22×C4—1042no—
539C22×C4C2×D5212yes—
540C22×C4C2×D5211yescharacteristic
541C22×C4C2×D5211yescharacteristic
542C22×C4C2×D5211yescharacteristic
543C22×C4C2×D5211yescharacteristic
544C22×C4C2×D5211yescharacteristic
545C22×C4C23×D1312yes—
546C22×C4C23×D1311yescharacteristic
547C2×C8—22no—
548C2×C8—22no—
549C2×C8—22no—
550C2×C8—22no—
551C2×C8C23×D1312yes—
552C2×C8C23×D1312yes—
553C2×C8C23×D1312yes—
554C2×C8C23×D1312yes—
555C4⋊C4—132no—
556C4⋊C4—132no—
557C4⋊C4—132no—
558C4⋊C4—132no—
559C4⋊C4—132no—
560C4⋊C4—132no—
561C4⋊C4—264no—
562C4⋊C4—261no—
563C4⋊C4—261no—
564C4⋊C4—261no—
565C4⋊C4—261no—
566C4⋊C4—261no—
567C4⋊C4—261no—
568C4⋊C4—264no—
569C4⋊C4—264no—
570C4⋊C4—264no—
571C4⋊C4—264no—
572C4⋊C4—264no—
573C4⋊C4—264no—
574C4⋊C4—264no—
575C4⋊C4—264no—
576C4⋊C4—264no—
577C4⋊C4—264no—
578C4⋊C4—264no—
579C4⋊C4—264no—
580C4⋊C4—264no—
581C4⋊C4—264no—
582C4⋊C4C2×D5214yes—
583C4⋊C4C2×D5214yes—
584C22⋊C4—264no—
585C22⋊C4—264no—
586C22⋊C4—264no—
587C22⋊C4C2×D5214yes—
588C22⋊C4C2×D5214yes—
589C42—264no—
590C42—264no—
591C42—264no—
592C42C2×D5212yes—
593C42C2×D5212yes—
594C42C23×D1312yes—
595C42C23×D1312yes—
596C13C2×#324211yescharacteristic, complemented, Sylow
597C23—22no—
598C23—22no—
599C23—22no—
600C23C22×D5212yes—
601C23C22×D5211yescharacteristic
602Q8—262no—
603Q8—262no—
604Q8—262no—
605Q8—262no—
606Q8—522no—
607Q8—522no—
608Q8—1044no—
609Q8—1044no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—262no—
619C2×C4—262no—
620C2×C4—262no—
621C2×C4—262no—
622C2×C4—262no—
623C2×C4—262no—
624C2×C4—262no—
625C2×C4—262no—
626C2×C4—522no—
627C2×C4—522no—
628C2×C4—522no—
629C2×C4—522no—
630C2×C4—522no—
631C2×C4—522no—
632C2×C4—522no—
633C2×C4—522no—
634C2×C4—1044no—
635C2×C4C22×D5212yes—
636C2×C4C22×D5212yes—
637C2×C4C22×D5212yes—
638C2×C4C22×D5212yes—
639C2×C4C22×D5212yes—
640C2×C4C22×D5211yescharacteristic
641C2×C4#8825712yes—
642C2×C4#8825712yes—
643C2×C4C24×D1311yescharacteristic
644C8—22no—
645C8—22no—
646C8—22no—
647C8—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—44no—
652C22#1171923112yes—
653C22#1171945612yes—
654C22#1171945812yes—
655C22C23×D5211yescharacteristic
656C4—22no—
657C4—22no—
658C4—22no—
659C4—22no—
660C4—22no—
661C4—522no—
662C4—522no—
663C4—522no—
664C4—522no—
665C4—522no—
666C4—522no—
667C4—522no—
668C4—522no—
669C4#1171923112yes—
670C4C2×#8825711yescharacteristic
671C4C2×#8825711yescharacteristic
672C2—42no—
673C2#1312610614yesdirect product factor
674C2C2×#1171923111yescharacteristic
675C2C2×#1171945611yescharacteristic
676C2C2×#1171945811yescharacteristic
subgroupG/H
centerC23C22×D52
commutator, cosocleC2×C4×C13C25
FrattiniC2×C4C24×D13
FittingC2×C13×#433C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
hypercenterC2×#433D13
hypocenterC13C2×#3242
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
3. #13126106C2direct (N × H)
4. C2×#11718363C2semidirect
5. C2×#11718344C2semidirect
6. C2×#11718364C2semidirect
7. C2×#11718342C2semidirect
12. C2×#11718490C2semidirect
13. C2×#11718490C2semidirect
14. C2×#11718511C2semidirect
15. C2×#11718512C2semidirect
65. #11718512C22semidirect
66. #11718511C22semidirect
67. #11718490C22semidirect
68. #11718490C22semidirect
73. #11718364C22semidirect
74. #11718363C22semidirect
75. #11718344C22semidirect
76. #11718342C22semidirect
510. C2×C13#3242semidirect
596. C13C2×#3242semidirect
673. C2#13126106direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117192242
C2×#117192232
#131261062
C2×#117183632
C2×#117183442
C2×#117183642
C2×#117183422
C2×#117183832
C2×#117183842
C2×#117183822
C2×#117184902
C2×#117185112
C2×#117185122
C2×C13×#4332
C2×#324213