Finite Group Databasebeta

← C2×#13131525 #16808192

Subgroups — C2×#13131525 #16808192

Order 3328.

2254 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13131525C2116yesmaximal, direct product factor
2C2×#11719068C212yesmaximal, complemented
3C2×#11719068C212yesmaximal, complemented
4C2×#11719130C211yesmaximal, characteristic, complemented
5C2×#11719128C211yesmaximal, characteristic, complemented
6C2×#11719125C211yesmaximal, characteristic, complemented
7C2×#11719119C211yesmaximal, characteristic, complemented
8C2×#11719067C212yesmaximal, complemented
9C2×#11719169C211yesmaximal, characteristic
10C22×C22⋊#1057C211yesmaximal, characteristic
11C2×#11719062C212yesmaximal, complemented
12C22×#88131C211yesmaximal, characteristic
13C24×Dic13C2211yescharacteristic
14C22×#6122C2212yes—
15C23×#1057C2212yes—
16C22×C13×C22⋊C4C411yescharacteristic
17C22×C22⋊Dic13C411yescharacteristic
18C22×#6101C411yescharacteristic
19C22×#6101C411yescharacteristic
20C22×C4×Dic13C2211yescharacteristic
21C22×C4×Dic13C2211yescharacteristic
22C2×#88131C411yescharacteristic
23C2×#88131C411yescharacteristic
24C2×#88131C411yescharacteristic
25C2×#88131C411yescharacteristic
26C2×#88131C418yes—
27C2×#88131C2218yes—
28C2×#88131C2211yescharacteristic
29C2×#88131C2211yescharacteristic
30C2×#88131C2211yescharacteristic
31C2×#88131C2211yescharacteristic
32#11719169C2218yes—
33C2×C22⋊#1057C2218yes—
34C2×C22⋊#1057C2211yescharacteristic
35C2×C22⋊#1057C2211yescharacteristic
36C2×C22⋊#1057C2212yes—
37C2×C22⋊#1057C2211yescharacteristic
38C2×C22⋊#1057C2211yescharacteristic
39C2×C22⋊#1057C2212yes—
40#11719130C2218yescomplemented
41#11719128C2218yescomplemented
42#11719125C2218yescomplemented
43C2×#88112C2212yescomplemented
44C2×#88112C2212yescomplemented
45C2×#88110C2212yescomplemented
46C2×#88110C2212yescomplemented
47#11719119C2218yescomplemented
48C2×C4⋊#1057C2212yescomplemented
49C2×C4⋊#1057C2212yescomplemented
50C2×C4×#1057C2212yescomplemented
51C2×C4×#1057C2212yescomplemented
52#11719068C22116yescomplemented
53#11719068C22116yescomplemented
54#11719067C22116yescomplemented
55#11719062C22116yescomplemented
56C25×C13C2×C411yescharacteristic
57C23×C4×C13C2×C411yescharacteristic
58C23×C4×C13C2×C411yescharacteristic
59C23×Dic13—28no—
60C23×Dic13C2×C411yescharacteristic
61C23×Dic13C2×C411yescharacteristic
62C23×Dic13C2×C414yes—
63C23×Dic13C2×C411yescharacteristic
64C23×Dic13C2×C411yescharacteristic
65C23×Dic13C2×C411yescharacteristic
66C23×Dic13C2311yescharacteristic
67C23×Dic13C2311yescharacteristic
68C23×Dic13C2314yes—
69C23×Dic13C2311yescharacteristic
70C2×#6122—28no—
71C2×#6122C2318yes—
72C22×#1057—28no—
73C22×#1057C2318yes—
74C22×#1057C2312yes—
75C22×#1057C2312yes—
76C22×#1057C2312yes—
77C22×#1057C2312yes—
78C2×C13×C22⋊C4C2×C411yescharacteristic
79C2×C13×C22⋊C4C2×C411yescharacteristic
80C2×C13×C22⋊C4C2×C411yescharacteristic
81C2×C13×C22⋊C4C2×C411yescharacteristic
82C2×C13×C22⋊C4C2×C418yes—
83C2×C22⋊Dic13C2×C411yescharacteristic
84C2×C22⋊Dic13C2×C411yescharacteristic
85C2×C22⋊Dic13C2×C411yescharacteristic
86C2×C22⋊Dic13C2×C411yescharacteristic
87C2×C22⋊Dic13C2×C418yes—
88C2×#6101C2×C418yes—
89C2×#6101C2×C411yescharacteristic
90C2×#6101C2×C411yescharacteristic
91C2×#6101C2×C411yescharacteristic
92C2×#6101C2×C411yescharacteristic
93C2×#6101C2×C411yescharacteristic
94C2×#6101C2×C411yescharacteristic
95C2×#6101C2×C411yescharacteristic
96C2×#6101C2×C411yescharacteristic
97C2×#6101C2×C418yes—
98C2×C4×Dic13C2×C411yescharacteristic
99C2×C4×Dic13C2×C411yescharacteristic
100C2×C4×Dic13C2×C411yescharacteristic
101C2×C4×Dic13C2×C411yescharacteristic
102C2×C4×Dic13C2×C414yes—
103C2×C4×Dic13C2×C414yes—
104C2×C4×Dic13C2311yescharacteristic
105C2×C4×Dic13C2311yescharacteristic
106C2×C4×Dic13C2311yescharacteristic
107C2×C4×Dic13C2311yescharacteristic
108C2×C4×Dic13C2314yes—
109C2×C4×Dic13C2314yes—
110#88131C2×C418yes—
111#88131C2×C418yes—
112#88131C2×C418yes—
113#88131C2×C418yes—
114#88131C2×C414yes—
115#88131C2×C414yes—
116#88131C2×C414yes—
117#88131C2×C414yes—
118#88131C2314yes—
119#88131C2314yes—
120#88131C2314yes—
121#88131C2314yes—
122C22⋊#1057C2318yes—
123C22⋊#1057C2314yes—
124C22⋊#1057C2314yes—
125C22⋊#1057C2318yes—
126C22⋊#1057C2314yes—
127C22⋊#1057C2314yes—
128#88112C2318yescomplemented
129#88112C2318yescomplemented
130#88110C2318yescomplemented
131#88110C2318yescomplemented
132C4⋊#1057C2318yescomplemented
133C4⋊#1057C2318yescomplemented
134C4×#1057C2318yescomplemented
135C4×#1057C2318yescomplemented
136C2×#2924—131nomaximal, Sylow
137C24×C13—24no—
138C24×C13—24no—
139C24×C13—42no—
140C24×C13C22×C411yescharacteristic
141C24×C13C22×C414yes—
142C24×C13C22×C411yescharacteristic
143C24×C13C22×C411yescharacteristic
144C22×C4×C13—24no—
145C22×C4×C13—42no—
146C22×C4×C13C22×C414yes—
147C22×C4×C13C22×C411yescharacteristic
148C22×C4×C13C22×C411yescharacteristic
149C22×C4×C13C22×C411yescharacteristic
150C22×C4×C13C22×C411yescharacteristic
151C22×C4×C13C22×C414yes—
152C22×Dic13—24no—
153C22×Dic13—24no—
154C22×Dic13—24no—
155C22×Dic13—216no—
156C22×Dic13—24no—
157C22×Dic13—24no—
158C22×Dic13—24no—
159C22×Dic13—216no—
160C22×Dic13—42no—
161C22×Dic13C22×C414yes—
162C22×Dic13C22×C414yes—
163C22×Dic13C22×C412yes—
164C22×Dic13C22×C411yescharacteristic
165C22×Dic13C22×C412yes—
166C22×Dic13C22×C414yes—
167C22×Dic13C22×C411yescharacteristic
168C22×Dic13C22×C411yescharacteristic
169C22×Dic13C22×C411yescharacteristic
170C22×Dic13C22×C411yescharacteristic
171C22×Dic13C22×C411yescharacteristic
172C22×Dic13C22×C414yes—
173C22×Dic13C22×C414yes—
174C22×Dic13C22×C412yes—
175C22×Dic13C22×C411yescharacteristic
176C22×Dic13#4118yes—
177C22×Dic13C2412yes—
178C22×Dic13C2412yes—
179C22×Dic13C2412yes—
180C22×Dic13C2411yescharacteristic
181#6122—216no—
182C2×#1057—24no—
183C2×#1057—24no—
184C2×#1057—216no—
185C2×#1057—24no—
186C2×#1057—24no—
187C2×#1057C2414yes—
188C2×#1057C2414yes—
189C2×#1057C2414yes—
190C2×#1057C2414yes—
191C13×C22⋊C4C22×C414yes—
192C13×C22⋊C4C22×C414yes—
193C13×C22⋊C4C22×C414yes—
194C13×C22⋊C4C22×C414yes—
195C22⋊Dic13C22×C414yes—
196C22⋊Dic13C22×C414yes—
197C22⋊Dic13C22×C414yes—
198C22⋊Dic13C22×C414yes—
199#6101C22×C414yes—
200#6101C22×C414yes—
201#6101C22×C414yes—
202#6101C22×C414yes—
203#6101C22×C414yes—
204#6101C22×C414yes—
205#6101C22×C414yes—
206#6101C22×C414yes—
207C4×Dic13C22×C414yes—
208C4×Dic13C22×C414yes—
209C4×Dic13C22×C414yes—
210C4×Dic13C22×C414yes—
211C4×Dic13C22×C412yes—
212C4×Dic13C22×C412yes—
213C4×Dic13C22×C412yes—
214C4×Dic13C22×C412yes—
215C4×Dic13C2412yes—
216C4×Dic13C2412yes—
217C4×Dic13C2412yes—
218C4×Dic13C2412yes—
219C22×#117—131no—
220#2924—1316no—
221C2×#433—132no—
222C2×#433—132no—
223C2×#432—131no—
224C2×#432—131no—
225C2×#431—131no—
226C2×#431—131no—
227C2×#424—132no—
228C2×#407—131no—
229C22×C22⋊C8—131no—
230C2×#405—132no—
231C23×C13—22no—
232C23×C13—22no—
233C23×C13—22no—
234C23×C13—22no—
235C23×C13—28no—
236C23×C13—22no—
237C23×C13—28no—
238C23×C13—22no—
239C23×C13—22no—
240C23×C13—44no—
241C23×C13—416no—
242C23×C13#11714yes—
243C23×C13#13114yes—
244C23×C13C23×C412yes—
245C23×C13C23×C412yes—
246C23×C13C23×C412yes—
247C23×C13C23×C411yescharacteristic
248C2×C4×C13—28no—
249C2×C4×C13—22no—
250C2×C4×C13—22no—
251C2×C4×C13—22no—
252C2×C4×C13—22no—
253C2×C4×C13—44no—
254C2×C4×C13C23×C412yes—
255C2×C4×C13C23×C412yes—
256C2×C4×C13C23×C412yes—
257C2×C4×C13C23×C412yes—
258C2×Dic13—28no—
259C2×Dic13—22no—
260C2×Dic13—22no—
261C2×Dic13—28no—
262C2×Dic13—22no—
263C2×Dic13—22no—
264C2×Dic13—22no—
265C2×Dic13—22no—
266C2×Dic13—24no—
267C2×Dic13—28no—
268C2×Dic13—28no—
269C2×Dic13—24no—
270C2×Dic13—24no—
271C2×Dic13—24no—
272C2×Dic13—44no—
273C2×Dic13#11718yes—
274C2×Dic13#11714yes—
275C2×Dic13C23×C412yes—
276C2×Dic13C23×C412yes—
277C2×Dic13C23×C412yes—
278C2×Dic13C23×C412yes—
279C2×Dic13C23×C412yes—
280C2×Dic13C23×C412yes—
281C2×Dic13C23×C411yescharacteristic
282C2×Dic13C23×C412yes—
283C2×Dic13C2×#4114yes—
284C2×Dic13C2×#4118yes—
285C2×Dic13C2511yescharacteristic
286#1057—24no—
287#1057—24no—
288#1057—24no—
289#1057—24no—
290C24×C4—131no—
291C22×M16—132no—
292C23×C8—132no—
293C2×#117—138no—
294C2×#117—131no—
295C2×#117—131no—
296C2×#117—131no—
297C2×#117—131no—
298C2×#117—131no—
299C2×#117—131no—
300C2×#117—131no—
301C2×#117—131no—
302C2×#117—138no—
303C22×C4⋊C4—131no—
304C22×C4⋊C4—131no—
305C22×C22⋊C4—131no—
306C22×C22⋊C4#26111yescharacteristic
307C22×C42—131no—
308C22×C42—131no—
309#433—1316no—
310#433—1316no—
311#432—138no—
312#432—138no—
313#431—138no—
314#431—138no—
315#424—1316no—
316C2×C4⋊C8—132no—
317C2×C4⋊C8—132no—
318C2×C4⋊C8—132no—
319C2×C4⋊C8—132no—
320#407—138no—
321C2×C22⋊C8—131no—
322C2×C22⋊C8—131no—
323C2×C22⋊C8—132no—
324C2×C22⋊C8—138no—
325C2×C22⋊C8—131no—
326C2×C22⋊C8—131no—
327C2×C22⋊C8—132no—
328#405—1316no—
329C2×#97—132no—
330C2×#97—132no—
331C2×C4×C8—132no—
332C2×C4×C8—132no—
333C22×C13—22no—
334C22×C13—28no—
335C22×C13—28no—
336C22×C13—22no—
337C22×C13—22no—
338C22×C13—22no—
339C22×C13—22no—
340C22×C13—22no—
341C22×C13—22no—
342C22×C13—416no—
343C22×C13—44no—
344C22×C13C2×#11714yes—
345C22×C13C2×#11712yes—
346C22×C13C2×#13114yes—
347C22×C13C2×#13112yes—
348C22×C13#56812yes—
349C22×C13C24×C411yescharacteristic
350C4×C13—22no—
351C4×C13—22no—
352C4×C13—22no—
353C4×C13—22no—
354Dic13—22no—
355Dic13—22no—
356Dic13—22no—
357Dic13—22no—
358Dic13—22no—
359Dic13—22no—
360Dic13C2×#11712yes—
361Dic13C2×#11714yes—
362Dic13C22×#4112yes—
363C25C2×#26111yescharacteristic
364C23×C4—131no—
365C23×C4—131no—
366C23×C4—131no—
367C23×C4—131no—
368C23×C4—134no—
369C23×C4—134no—
370C23×C4—131no—
371C23×C4—131no—
372C23×C4—131no—
373C23×C4—131no—
374C23×C4—268no—
375C23×C4C2×#26111yescharacteristic
376C23×C4C2×#26111yescharacteristic
377C2×M16—138no—
378C2×M16—268no—
379C22×C8—132no—
380C22×C8—132no—
381C22×C8—138no—
382C22×C8—132no—
383C22×C8—132no—
384C22×C8—268no—
385#117—134no—
386#117—134no—
387#117—134no—
388#117—134no—
389#117—138no—
390#117—138no—
391#117—138no—
392#117—138no—
393#117—134no—
394#117—134no—
395#117—134no—
396#117—134no—
397C2×C4⋊C4—131no—
398C2×C4⋊C4—131no—
399C2×C4⋊C4—131no—
400C2×C4⋊C4—131no—
401C2×C4⋊C4—138no—
402C2×C4⋊C4—138no—
403C2×C4⋊C4—131no—
404C2×C4⋊C4—131no—
405C2×C4⋊C4—131no—
406C2×C4⋊C4—131no—
407C2×C22⋊C4—138no—
408C2×C22⋊C4—131no—
409C2×C22⋊C4—131no—
410C2×C22⋊C4—131no—
411C2×C22⋊C4—131no—
412C2×C22⋊C4C2×#26118yes—
413C2×C22⋊C4C2×#26111yescharacteristic
414C2×C22⋊C4C2×#26111yescharacteristic
415C2×C22⋊C4C2×#26111yescharacteristic
416C2×C22⋊C4C2×#26111yescharacteristic
417C2×C42—134no—
418C2×C42—134no—
419C2×C42—131no—
420C2×C42—131no—
421C2×C42—131no—
422C2×C42—131no—
423C2×C42—131no—
424C2×C42—131no—
425C2×C42—131no—
426C2×C42—131no—
427C2×C42—134no—
428C2×C42—134no—
429C4⋊C8—138no—
430C4⋊C8—138no—
431C4⋊C8—138no—
432C4⋊C8—138no—
433C22⋊C8—134no—
434C22⋊C8—134no—
435C22⋊C8—138no—
436C22⋊C8—134no—
437C22⋊C8—134no—
438C22⋊C8—138no—
439#97—138no—
440#97—138no—
441C4×C8—138no—
442C4×C8—138no—
443C2×C13—24no—
444C2×C13—24no—
445C2×C13—42no—
446C2×C13#292414yescomplemented
447C2×C13C22×#11711yescharacteristic
448C2×C13C22×#13111yescharacteristic
449C2×C13C2×#56811yescharacteristic
450C24—24no—
451C24—24no—
452C24—42no—
453C24C22×#26111yescharacteristic
454C24C22×#26114yes—
455C24C22×#26111yescharacteristic
456C24C22×#26111yescharacteristic
457C22×C4—24no—
458C22×C4—42no—
459C22×C4—138no—
460C22×C4—132no—
461C22×C4—132no—
462C22×C4—131no—
463C22×C4—131no—
464C22×C4—132no—
465C22×C4—132no—
466C22×C4—132no—
467C22×C4—134no—
468C22×C4—134no—
469C22×C4—132no—
470C22×C4—134no—
471C22×C4—134no—
472C22×C4—131no—
473C22×C4—131no—
474C22×C4—131no—
475C22×C4—131no—
476C22×C4—131no—
477C22×C4—131no—
478C22×C4—134no—
479C22×C4—264no—
480C22×C4—264no—
481C22×C4—2616no—
482C22×C4—264no—
483C22×C4—2616no—
484C22×C4—264no—
485C22×C4—264no—
486C22×C4—264no—
487C22×C4—522no—
488C22×C4C22×#26114yes—
489C22×C4C22×#26111yescharacteristic
490C22×C4C22×#26111yescharacteristic
491C22×C4C22×#26111yescharacteristic
492C22×C4C22×#26111yescharacteristic
493C22×C4C22×#26114yes—
494M16—2616no—
495C2×C8—134no—
496C2×C8—134no—
497C2×C8—134no—
498C2×C8—134no—
499C2×C8—264no—
500C2×C8—264no—
501C2×C8—2616no—
502C2×C8—264no—
503C2×C8—264no—
504C4⋊C4—134no—
505C4⋊C4—134no—
506C4⋊C4—134no—
507C4⋊C4—134no—
508C4⋊C4—134no—
509C4⋊C4—134no—
510C4⋊C4—134no—
511C4⋊C4—134no—
512C22⋊C4—134no—
513C22⋊C4—134no—
514C22⋊C4—134no—
515C22⋊C4—134no—
516C22⋊C4C22×#26114yes—
517C22⋊C4C22×#26114yes—
518C22⋊C4C22×#26114yes—
519C22⋊C4C22×#26114yes—
520C42—132no—
521C42—132no—
522C42—132no—
523C42—132no—
524C42—132no—
525C42—132no—
526C42—132no—
527C42—132no—
528C42—134no—
529C42—134no—
530C42—134no—
531C42—134no—
532C13C2×#292411yescharacteristic, complemented, Sylow
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—28no—
538C23—22no—
539C23—28no—
540C23—22no—
541C23—22no—
542C23—44no—
543C23—416no—
544C23#8823714yes—
545C23#8823814yes—
546C23C23×#26112yes—
547C23C23×#26112yes—
548C23C23×#26112yes—
549C23C23×#26111yescharacteristic
550C2×C4—28no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—44no—
556C2×C4—134no—
557C2×C4—138no—
558C2×C4—138no—
559C2×C4—134no—
560C2×C4—131no—
561C2×C4—131no—
562C2×C4—132no—
563C2×C4—132no—
564C2×C4—132no—
565C2×C4—132no—
566C2×C4—132no—
567C2×C4—132no—
568C2×C4—132no—
569C2×C4—264no—
570C2×C4—268no—
571C2×C4—264no—
572C2×C4—268no—
573C2×C4—264no—
574C2×C4—264no—
575C2×C4—268no—
576C2×C4—262no—
577C2×C4—262no—
578C2×C4—268no—
579C2×C4—262no—
580C2×C4—262no—
581C2×C4—262no—
582C2×C4—262no—
583C2×C4—524no—
584C2×C4C23×#26112yes—
585C2×C4C23×#26112yes—
586C2×C4C23×#26112yes—
587C2×C4C23×#26112yes—
588C8—264no—
589C8—264no—
590C8—264no—
591C8—264no—
592C22—28no—
593C22—28no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—416no—
602C22—44no—
603C22C2×#8823714yes—
604C22C2×#8823712yes—
605C22C2×#8823814yes—
606C22C2×#8823812yes—
607C22#1171962112yes—
608C22C24×#26111yescharacteristic
609C4—22no—
610C4—22no—
611C4—22no—
612C4—22no—
613C4—132no—
614C4—134no—
615C4—132no—
616C4—262no—
617C4—262no—
618C4—262no—
619C4—262no—
620C4—262no—
621C4—262no—
622C2—24no—
623C2—24no—
624C2—42no—
625C2#1313152514yesdirect product factor
626C2C22×#8823711yescharacteristic
627C2C22×#8823811yescharacteristic
628C2C2×#1171962111yescharacteristic
subgroupG/H
centerC23C23×#261
commutatorC22×C13C24×C4
FrattiniC22C24×#261
FittingC22×C13×C22⋊C4C4
socleC23×C13C23×C4
prime radicalC25×C13C2×C4
cosocleC2×Dic13C25
hypercenterC22×C22⋊C4#261
hypocenterC13C2×#2924
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #13131525C2direct (N × H)
2. C2×#11719068C2semidirect
3. C2×#11719068C2semidirect
4. C2×#11719130C2semidirect
5. C2×#11719128C2semidirect
6. C2×#11719125C2semidirect
7. C2×#11719119C2semidirect
8. C2×#11719067C2semidirect
11. C2×#11719062C2semidirect
40. #11719130C22semidirect
41. #11719128C22semidirect
42. #11719125C22semidirect
43. C2×#88112C22semidirect
44. C2×#88112C22semidirect
45. C2×#88110C22semidirect
46. C2×#88110C22semidirect
47. #11719119C22semidirect
48. C2×C4⋊#1057C22semidirect
49. C2×C4⋊#1057C22semidirect
50. C2×C4×#1057C22semidirect
51. C2×C4×#1057C22semidirect
52. #11719068C22semidirect
53. #11719068C22semidirect
54. #11719067C22semidirect
55. #11719062C22semidirect
128. #88112C23semidirect
129. #88112C23semidirect
130. #88110C23semidirect
131. #88110C23semidirect
132. C4⋊#1057C23semidirect
133. C4⋊#1057C23semidirect
134. C4×#1057C23semidirect
135. C4×#1057C23semidirect
446. C2×C13#2924semidirect
532. C13C2×#2924semidirect
625. C2#13131525direct (N × H)

Maximal subgroups

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

subgroupindex
#131315252
C2×#117190682
C2×#117191302
C2×#117191282
C2×#117191252
C2×#117191192
C2×#117190672
C2×#117191692
C22×C22⋊#10572
C2×#117190622
C22×#881312
C2×#292413