Finite Group Databasebeta

← #15784306

Subgroups — #15784306

Order 3328.

2782 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#88146C211yesmaximal, characteristic
2C22×#88126C211yesmaximal, characteristic
3C22×C4⋊C4×D13C211yesmaximal, characteristic
4#13113066C211yesmaximal, characteristic, complemented
5#13113065C211yesmaximal, characteristic, complemented
6#13113065C211yesmaximal, characteristic, complemented
7#13113064C214yesmaximal, complemented
8#13113057C214yesmaximal
9C13×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×C4×D13C2212yes—
11C23×C4×D13C2211yescharacteristic
12C22×C13×C4⋊C4C2211yescharacteristic
13C22×C13×C4⋊C4C2211yescharacteristic
14C22×C42×C13C2211yescharacteristic
15C22×#6103C2212yes—
16C22×#6103C2212yes—
17C22×C4⋊Dic13C2211yescharacteristic
18C22×C4⋊Dic13C2211yescharacteristic
19C22×#6101C2212yes—
20C22×#6101C2212yes—
21C22×C4×Dic13C2211yescharacteristic
22C22×C4×Dic13C2212yes—
23C2×#88146—24no—
24C2×#88146—24no—
25C2×#88146C418yescomplemented
26C2×C4⋊C4×D13—24no—
27C2×C4⋊C4×D13—24no—
28C2×C4⋊C4×D13C418yescomplemented
29C2×#88126—24no—
30C2×#88126—24no—
31C2×#88126C418yescomplemented
32C2×C13×#95C2214yes—
33C2×#88071C2214yes—
34C2×#88071C2214yes—
35C2×#88071C2214yes—
36C24×D13C2311yescharacteristic
37C23×C4×C13C2312yes—
38C23×C4×C13C2312yes—
39C23×C4×C13C2312yes—
40C23×C4×C13C2311yescharacteristic
41C23×Dic13C2312yes—
42C23×Dic13C2312yes—
43C23×Dic13C2312yes—
44C23×Dic13C2311yescharacteristic
45C22×C4×D13—24no—
46C22×C4×D13—24no—
47C22×C4×D13—44no—
48C22×C4×D13—44no—
49C22×C4×D13C2×C418yes—
50C22×C4×D13C2×C414yes—
51C22×C4×D13C2×C414yes—
52C22×C4×D13C2×C414yes—
53C22×C4×D13D414yes—
54C22×C4×D13D414yes—
55C22×C4×D13D414yes—
56C22×C4×D13Q814yes—
57C2×C13×C4⋊C4—22no—
58C2×C13×C4⋊C4—22no—
59C2×C13×C4⋊C4—22no—
60C2×C13×C4⋊C4—22no—
61C2×C13×C4⋊C4C2×C414yes—
62C2×C13×C4⋊C4C2×C414yescomplemented
63C2×C42×C13—22no—
64C2×C42×C13—22no—
65C2×C42×C13C2×C414yescomplemented
66C2×#6103—24no—
67C2×#6103—24no—
68C2×#6103—24no—
69C2×#6103—24no—
70C2×#6103C2×C418yes—
71C2×#6103C2×C418yes—
72C2×C4⋊Dic13—22no—
73C2×C4⋊Dic13—22no—
74C2×C4⋊Dic13—22no—
75C2×C4⋊Dic13—22no—
76C2×C4⋊Dic13C2×C414yes—
77C2×C4⋊Dic13C2×C414yescomplemented
78C2×#6101—24no—
79C2×#6101—24no—
80C2×#6101—24no—
81C2×#6101—24no—
82C2×#6101C2×C418yes—
83C2×#6101C2×C418yescomplemented
84C2×C4×Dic13—22no—
85C2×C4×Dic13—22no—
86C2×C4×Dic13—24no—
87C2×C4×Dic13—24no—
88C2×C4×Dic13C2×C414yescomplemented
89C2×C4×Dic13C2×C418yescomplemented
90#88146—232no—
91C4⋊C4×D13—232no—
92#88126—232no—
93C13×#95—44no—
94#88071—44no—
95#88071—44no—
96#88071—44no—
97#13795—131nomaximal, Sylow
98C24×C13C2411yescharacteristic
99C23×D13—22no—
100C23×D13—22no—
101C23×D13—22no—
102C23×D13—22no—
103C23×D13C22×C412yes—
104C23×D13C22×C412yes—
105C23×D13C22×C412yes—
106C23×D13C2×D412yes—
107C23×D13C2×D412yes—
108C23×D13C2×D412yes—
109C23×D13C2×Q812yes—
110C22×C4×C13—22no—
111C22×C4×C13—22no—
112C22×C4×C13—22no—
113C22×C4×C13—22no—
114C22×C4×C13—22no—
115C22×C4×C13—22no—
116C22×C4×C13—22no—
117C22×C4×C13—24no—
118C22×C4×C13—24no—
119C22×C4×C13—22no—
120C22×C4×C13—24no—
121C22×C4×C13—22no—
122C22×C4×C13—24no—
123C22×C4×C13—22no—
124C22×C4×C13C22×C414yes—
125C22×C4×C13C22×C414yes—
126C22×C4×C13C22×C414yes—
127C22×C4×C13C22×C412yes—
128C22×C4×C13C22×C412yes—
129C22×C4×C13C22×C412yes—
130C22×C4×C13C2×D412yes—
131C22×C4×C13C2×D412yes—
132C22×C4×C13C2×D412yes—
133C22×C4×C13C2×Q812yes—
134C22×Dic13—22no—
135C22×Dic13—22no—
136C22×Dic13—22no—
137C22×Dic13—22no—
138C22×Dic13—22no—
139C22×Dic13—22no—
140C22×Dic13—22no—
141C22×Dic13—22no—
142C22×Dic13—24no—
143C22×Dic13—24no—
144C22×Dic13—24no—
145C22×Dic13—22no—
146C22×Dic13—24no—
147C22×Dic13—22no—
148C22×Dic13C22×C414yes—
149C22×Dic13C22×C414yes—
150C22×Dic13C22×C414yes—
151C22×Dic13C22×C412yes—
152C22×Dic13C22×C412yes—
153C22×Dic13C22×C412yes—
154C22×Dic13C2×D412yes—
155C22×Dic13C2×D412yes—
156C22×Dic13C2×D412yes—
157C22×Dic13C2×Q812yes—
158C2×C4×D13—216no—
159C2×C4×D13—416no—
160C2×C4×D13—416no—
161C2×C4×D13—48no—
162C2×C4×D13—48no—
163C2×C4×D13C42116yes—
164C2×C4×D13C22⋊C4116yes—
165C2×C4×D13C22⋊C4116yes—
166C2×C4×D13C22⋊C4116yes—
167C2×C4×D13C4⋊C4116yes—
168C2×C4×D13C4⋊C4116yes—
169C2×C4×D13C4⋊C4116yes—
170C13×C4⋊C4—28no—
171C13×C4⋊C4—28no—
172C42×C13—28no—
173#6103—216no—
174#6103—216no—
175C4⋊Dic13—28no—
176C4⋊Dic13—28no—
177#6101—216no—
178#6101—216no—
179C4×Dic13—28no—
180C4×Dic13—216no—
181C22×#117—131no—
182C22×#117—131no—
183C23×C4⋊C4—131no—
184C4⋊#95—131no—
185C4⋊#95—134no—
186C4⋊#95—131no—
187C4⋊#95—131no—
188C4⋊#95D1311yescharacteristic, complemented
189C22⋊#95—134no—
190C23×C13C23×C411yescharacteristic
191C23×C13C23×C411yescharacteristic
192C23×C13C23×C411yescharacteristic
193C23×C13C22×D411yescharacteristic
194C23×C13C22×D411yescharacteristic
195C23×C13C22×D411yescharacteristic
196C23×C13C22×Q811yescharacteristic
197C23×C13H(5,𝔽2)12yes—
198C23×C13H(5,𝔽2)12yes—
199C23×C132−1+412yes—
200C23×C132−1+412yes—
201C22×D13—24no—
202C22×D13—24no—
203C22×D13—24no—
204C22×D13—24no—
205C22×D13—24no—
206C22×D13—24no—
207C22×D13—24no—
208C22×D13—22no—
209C22×D13—22no—
210C22×D13—22no—
211C22×D13—24no—
212C22×D13—22no—
213C22×D13—24no—
214C22×D13—24no—
215C22×D13—24no—
216C22×D13—24no—
217C22×D13C2×C4214yes—
218C22×D13C2×C22⋊C414yes—
219C22×D13C2×C22⋊C414yes—
220C22×D13C2×C22⋊C414yes—
221C22×D13C2×C4⋊C414yes—
222C22×D13C2×C4⋊C414yes—
223C22×D13C2×C4⋊C414yes—
224C2×C4×C13—24no—
225C2×C4×C13—24no—
226C2×C4×C13—24no—
227C2×C4×C13—28no—
228C2×C4×C13—24no—
229C2×C4×C13—24no—
230C2×C4×C13—28no—
231C2×C4×C13—44no—
232C2×C4×C13—42no—
233C2×C4×C13—42no—
234C2×C4×C13—42no—
235C2×C4×C13—42no—
236C2×C4×C13—42no—
237C2×C4×C13—42no—
238C2×C4×C13—44no—
239C2×C4×C13—42no—
240C2×C4×C13—42no—
241C2×C4×C13C2×C4214yes—
242C2×C4×C13C2×C22⋊C414yes—
243C2×C4×C13C2×C22⋊C414yes—
244C2×C4×C13C2×C22⋊C414yes—
245C2×C4×C13C2×C4⋊C414yes—
246C2×C4×C13C2×C4⋊C414yes—
247C2×C4×C13C2×C4⋊C414yes—
248C2×Dic13—24no—
249C2×Dic13—24no—
250C2×Dic13—24no—
251C2×Dic13—28no—
252C2×Dic13—24no—
253C2×Dic13—24no—
254C2×Dic13—28no—
255C2×Dic13—44no—
256C2×Dic13—42no—
257C2×Dic13—42no—
258C2×Dic13—42no—
259C2×Dic13—42no—
260C2×Dic13—42no—
261C2×Dic13—42no—
262C2×Dic13—42no—
263C2×Dic13—42no—
264C2×Dic13—44no—
265C2×Dic13C2×C4214yes—
266C2×Dic13C2×C22⋊C414yes—
267C2×Dic13C2×C22⋊C414yes—
268C2×Dic13C2×C22⋊C414yes—
269C2×Dic13C2×C4⋊C414yes—
270C2×Dic13C2×C4⋊C414yes—
271C2×Dic13C2×C4⋊C414yes—
272C4×D13—432no—
273C4×D13#95164yescomplemented
274C24×C4—131no—
275C24×C4—132no—
276C2×#117—138no—
277C2×#117—138no—
278C2×#117—264no—
279C2×#117—264no—
280C2×#117—264no—
281C2×#117—264no—
282C22×C4⋊C4—132no—
283C22×C4⋊C4—132no—
284C22×C4⋊C4—131no—
285C22×C4⋊C4—131no—
286C22×C4⋊C4—138no—
287C22×C4⋊C4—264no—
288C22×C4⋊C4—264no—
289C22×C4⋊C4C2×D1311yescharacteristic
290C22×C4⋊C4C2×D1311yescharacteristic
291C22×C22⋊C4—132no—
292C22×C22⋊C4—132no—
293C22×C42—132no—
294C22×C42—131no—
295C22×C42C2×D1311yescharacteristic
296C2×#95—134no—
297C2×#95—134no—
298C2×#95—134no—
299C2×#95C2×D1314yes—
300C22×C13C22×C4211yescharacteristic
301C22×C13C22×C22⋊C411yescharacteristic
302C22×C13C22×C22⋊C411yescharacteristic
303C22×C13C22×C22⋊C411yescharacteristic
304C22×C13C22×C4⋊C411yescharacteristic
305C22×C13C22×C4⋊C411yescharacteristic
306C22×C13C22×C4⋊C411yescharacteristic
307C22×C13#51812yes—
308C22×C13#51912yes—
309C22×C13#52012yes—
310C22×C13#52012yes—
311C22×C13#52012yes—
312C22×C13#52012yes—
313C22×C13#53512yes—
314C22×C13#53511yescharacteristic
315C22×C13#53511yescharacteristic
316C22×C13#53611yescharacteristic
317C22×C13#53612yes—
318C22×C13#53611yescharacteristic
319C22×C13#53712yes—
320C22×C13#53712yes—
321C22×C13#54412yes—
322C22×C13#54412yes—
323C2×D13—24no—
324C2×D13—24no—
325C2×D13—28no—
326C2×D13—24no—
327C2×D13—24no—
328C2×D13—28no—
329C2×D13—24no—
330C2×D13—28no—
331C2×D13—24no—
332C2×D13—28no—
333C2×D13C2×#9518yes—
334C4×C13—44no—
335C4×C13—44no—
336C4×C13—44no—
337C4×C13C2×#9518yes—
338Dic13—44no—
339Dic13—44no—
340Dic13—44no—
341Dic13C2×#9518yes—
342C25—131no—
343C23×C4—131no—
344C23×C4—134no—
345C23×C4—134no—
346C23×C4—134no—
347C23×C4—134no—
348C23×C4—134no—
349C23×C4—134no—
350C23×C4—134no—
351C23×C4—132no—
352C23×C4—132no—
353C23×C4—132no—
354C23×C4—138no—
355C23×C4—264no—
356C23×C4—264no—
357C23×C4—524no—
358C23×C4—524no—
359C23×C4C22×D1312yes—
360C23×C4C22×D1312yes—
361C23×C4C22×D1312yes—
362C23×C4C22×D1311yescharacteristic
363#117—2632no—
364#117—2632no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—138no—
370C2×C4⋊C4—134no—
371C2×C4⋊C4—138no—
372C2×C4⋊C4—134no—
373C2×C4⋊C4—2632no—
374C2×C4⋊C4—264no—
375C2×C4⋊C4—264no—
376C2×C4⋊C4—264no—
377C2×C4⋊C4—264no—
378C2×C4⋊C4—262no—
379C2×C4⋊C4—262no—
380C2×C4⋊C4—262no—
381C2×C4⋊C4—262no—
382C2×C4⋊C4C4×D1314yes—
383C2×C4⋊C4C4×D1314yescomplemented
384C2×C22⋊C4—138no—
385C2×C22⋊C4—138no—
386C2×C22⋊C4—264no—
387C2×C22⋊C4—264no—
388C2×C22⋊C4—264no—
389C2×C22⋊C4—264no—
390C2×C42—22no—
391C2×C42—22no—
392C2×C42—138no—
393C2×C42—134no—
394C2×C42—264no—
395C2×C42—264no—
396C2×C42—262no—
397C2×C42—262no—
398C2×C42C4×D1314yescomplemented
399#95—44no—
400#95—524no—
401#95—524no—
402#95—524no—
403C2×C13C22×#9511yescharacteristic
404C2×C13#227412yes—
405C2×C13#230911yescharacteristic
406C2×C13#231011yescharacteristic
407C2×C13#231112yes—
408C2×C13#231111yescharacteristic
409C2×C13#231111yescharacteristic
410C2×C13#231612yes—
411C2×C13#231711yescharacteristic
412C2×C13#231712yes—
413C2×C13#231711yescharacteristic
414D13—28no—
415C24—132no—
416C24—132no—
417C24—132no—
418C24—132no—
419C24—132no—
420C24—132no—
421C24—132no—
422C24—262no—
423C24—262no—
424C24—262no—
425C24—262no—
426C24C23×D1311yescharacteristic
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—132no—
442C22×C4—132no—
443C22×C4—132no—
444C22×C4—132no—
445C22×C4—132no—
446C22×C4—132no—
447C22×C4—132no—
448C22×C4—1316no—
449C22×C4—1316no—
450C22×C4—1316no—
451C22×C4—1316no—
452C22×C4—1316no—
453C22×C4—1316no—
454C22×C4—1316no—
455C22×C4—134no—
456C22×C4—134no—
457C22×C4—134no—
458C22×C4—2616no—
459C22×C4—262no—
460C22×C4—262no—
461C22×C4—262no—
462C22×C4—262no—
463C22×C4—262no—
464C22×C4—262no—
465C22×C4—262no—
466C22×C4—264no—
467C22×C4—262no—
468C22×C4—264no—
469C22×C4—262no—
470C22×C4—264no—
471C22×C4—264no—
472C22×C4—262no—
473C22×C4—5216no—
474C22×C4—5216no—
475C22×C4—528no—
476C22×C4—528no—
477C22×C4C2×C4×D1314yes—
478C22×C4C2×C4×D1314yes—
479C22×C4C2×C4×D1314yes—
480C22×C4C2×C4×D1312yes—
481C22×C4C2×C4×D1312yes—
482C22×C4C2×C4×D1312yes—
483C22×C4D4×D1312yes—
484C22×C4D4×D1312yes—
485C22×C4D4×D1312yes—
486C22×C4Q8×D1312yes—
487C4⋊C4—28no—
488C4⋊C4—28no—
489C4⋊C4—2616no—
490C4⋊C4—2616no—
491C4⋊C4—268no—
492C4⋊C4—268no—
493C22⋊C4—2616no—
494C22⋊C4—2616no—
495C42—28no—
496C42—2616no—
497C42—268no—
498C13#1379511yescharacteristic, complemented, Sylow
499C23—134no—
500C23—134no—
501C23—134no—
502C23—134no—
503C23—134no—
504C23—134no—
505C23—134no—
506C23—262no—
507C23—264no—
508C23—264no—
509C23—264no—
510C23—264no—
511C23—264no—
512C23—264no—
513C23—264no—
514C23—264no—
515C23—264no—
516C23—264no—
517C23—264no—
518C23—262no—
519C23—262no—
520C23—264no—
521C23—262no—
522C23C22×C4×D1311yescharacteristic
523C23C22×C4×D1311yescharacteristic
524C23C22×C4×D1311yescharacteristic
525C23C2×D4×D1311yescharacteristic
526C23C2×D4×D1311yescharacteristic
527C23C2×D4×D1311yescharacteristic
528C23#8825112yes—
529C23C2×Q8×D1311yescharacteristic
530C23#8825412yes—
531C23#8825612yes—
532C23#8825712yes—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—134no—
551C2×C4—134no—
552C2×C4—134no—
553C2×C4—134no—
554C2×C4—134no—
555C2×C4—134no—
556C2×C4—134no—
557C2×C4—1364no—
558C2×C4—264no—
559C2×C4—264no—
560C2×C4—264no—
561C2×C4—264no—
562C2×C4—264no—
563C2×C4—268no—
564C2×C4—268no—
565C2×C4—5232no—
566C2×C4—522no—
567C2×C4—522no—
568C2×C4—524no—
569C2×C4—522no—
570C2×C4—522no—
571C2×C4—522no—
572C2×C4—524no—
573C2×C4—522no—
574C2×C4—522no—
575C2×C4—522no—
576C2×C4C42×D1314yes—
577C2×C4C22⋊C4×D1314yes—
578C2×C4C22⋊C4×D1314yes—
579C2×C4C22⋊C4×D1314yes—
580C2×C4C4⋊C4×D1314yes—
581C2×C4C4⋊C4×D1314yes—
582C2×C4C4⋊C4×D1314yes—
583C22—138no—
584C22—268no—
585C22—268no—
586C22—264no—
587C22—268no—
588C22—264no—
589C22—264no—
590C22—264no—
591C22—268no—
592C22—264no—
593C22—264no—
594C22C2×C42×D1311yescharacteristic
595C22C2×C22⋊C4×D1311yescharacteristic
596C22C2×C22⋊C4×D1311yescharacteristic
597C22C2×C22⋊C4×D1311yescharacteristic
598C22C2×C4⋊C4×D1311yescharacteristic
599C22C2×C4⋊C4×D1311yescharacteristic
600C22C2×C4⋊C4×D1311yescharacteristic
601C22#1171920912yes—
602C22#1171922812yes—
603C22#1171925012yes—
604C22#1171925112yes—
605C22#1171927712yes—
606C22#1171927912yes—
607C22#1171931212yes—
608C22#1171931312yes—
609C22#1171933312yes—
610C22#1171933412yes—
611C22#1171937911yescharacteristic
612C22#1171938011yescharacteristic
613C22#1171939412yes—
614C22#1171942011yescharacteristic
615C22#1171942911yescharacteristic
616C22#1171943412yes—
617C4—44no—
618C4—44no—
619C4—44no—
620C4—138no—
621C4—524no—
622C4—524no—
623C4—524no—
624C4D13×#9518yes—
625C2—268no—
626C2C2×D13×#9511yescharacteristic
627C2#1311719012yes—
628C2#1311741511yescharacteristic
629C2#1311741911yescharacteristic
630C2#1311742512yes—
631C2#1311742611yescharacteristic
632C2#1311742711yescharacteristic
633C2#1311745112yes—
634C2#1311745711yescharacteristic
635C2#1311745812yes—
636C2#1311746111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D13
commutatorC22×C13C22×C42
FittingC13×C4⋊#95C2
socle, cosocleC24×C13C24
prime radicalC24×D13C23
hypercenterC4⋊#95D13
hypocenterC13#13795
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
4. #13113066C2semidirect
5. #13113065C2semidirect
6. #13113065C2semidirect
7. #13113064C2semidirect
9. C13×C4⋊#95C2semidirect
25. C2×#88146C4semidirect
28. C2×C4⋊C4×D13C4semidirect
31. C2×#88126C4semidirect
62. C2×C13×C4⋊C4C2×C4semidirect
65. C2×C42×C13C2×C4semidirect
77. C2×C4⋊Dic13C2×C4semidirect
83. C2×#6101C2×C4semidirect
88. C2×C4×Dic13C2×C4semidirect
89. C2×C4×Dic13C2×C4semidirect
188. C4⋊#95D13semidirect
273. C4×D13#95semidirect
383. C2×C4⋊C4C4×D13semidirect
398. C2×C42C4×D13semidirect
498. C13#13795semidirect

Maximal subgroups

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

subgroupindex
C22×#881462
C22×#881262
C22×C4⋊C4×D132
#131130662
#131130652
#131130642
#131130572
C13×C4⋊#952
#1379513