Finite Group Databasebeta

← #15880163

Subgroups — #15880163

Order 3328.

982 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13117713C211yesmaximal, characteristic, complemented
2#13117677C211yesmaximal, characteristic
3#13115535C211yesmaximal, characteristic, complemented
4#13115521C211yesmaximal, characteristic, complemented
5#13115465C211yesmaximal, characteristic, complemented
6#13115458C211yesmaximal, characteristic, complemented
7#13115106C211yesmaximal, characteristic, complemented
8#13115092C211yesmaximal, characteristic, complemented
9#13114922C211yesmaximal, characteristic, complemented
10#13114869C211yesmaximal, characteristic, complemented
11#13114776C211yesmaximal, characteristic
12#13114668C211yesmaximal, characteristic, complemented
13#103×Dic13C211yesmaximal, characteristic, complemented
14#13114617C211yesmaximal, characteristic, complemented
15C2×C13×#417C211yesmaximal, characteristic
16C22×C13×#41C2211yescharacteristic
17C22×C13×M16C2211yescharacteristic
18C22×C13×C4⋊C4C2211yescharacteristic
19C2×Q8×Dic13C2211yescharacteristic
20C2×Q8×Dic13C2211yescharacteristic
21C13×#417C418yescomplemented
22C2×C13×#103C2211yescharacteristic
23C2×C13×#103C2211yescharacteristic
24C2×C13×#102C2211yescharacteristic
25C2×C13×#102C2211yescharacteristic
26C2×#88075C2211yescharacteristic
27C2×#88075C2211yescharacteristic
28#11718882—21no—
29#11718882—21no—
30#11718882C2211yescharacteristic
31#11718882C2211yescharacteristic
32C2×#88072C2211yescharacteristic
33C2×#88072C2211yescharacteristic
34#11718873—21no—
35#11718873—21no—
36#11718873C2211yescharacteristic
37#11718873C2211yescharacteristic
38C4×C22⋊Dic13—21no—
39C4×C22⋊Dic13—21no—
40C4×C22⋊Dic13C2211yescharacteristic
41C2×C22⋊#1055C2211yescharacteristic
42C2×#88058C2211yescharacteristic
43C2×#88057C2211yescharacteristic
44C2×#88055C2211yescharacteristic
45C2×C8×Dic13C2211yescharacteristic
46#11718646C2211yescharacteristic
47#11718646C2211yescharacteristic
48#11718642—21no—
49#11718642—21no—
50C4⋊C4×Dic13C2211yescharacteristic
51C4⋊C4×Dic13C2211yescharacteristic
52#11718143C2211yescharacteristic
53#11718143C2211yescharacteristic
54#11718142C2211yescharacteristic
55#11718142C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60C2×C13×#41—22no—
61C2×C13×#41—22no—
62C2×C13×#41—22no—
63C2×C13×#41—41no—
64C2×C13×#41—41no—
65C2×C13×#41C2×C414yes—
66C22×Q8×C13C2311yescharacteristic
67C22×D4×C13—21no—
68C22×D4×C13C2311yescharacteristic
69C23×C4×C13—21no—
70C23×C4×C13—21no—
71C23×C4×C13C2311yescharacteristic
72C2×C13×M16—41no—
73C2×C13×M16—41no—
74C2×C13×M16C2×C414yes—
75C22×C8×C13C2311yescharacteristic
76C22×C8×C13C2311yescharacteristic
77C2×C13×C4⋊C4—21no—
78C2×C13×C4⋊C4—22no—
79C2×C13×C4⋊C4C2×C414yes—
80C2×C13×C4⋊C4C2311yescharacteristic
81C2×C13×C4⋊C4C2311yescharacteristic
82C2×C22⋊Dic13—21no—
83C2×C22⋊Dic13—21no—
84C2×C22⋊Dic13—21no—
85C2×C22⋊Dic13—21no—
86C2×C22⋊Dic13—21no—
87C2×C22⋊Dic13—21no—
88C2×C22⋊Dic13D411yescharacteristic
89C2×C22⋊Dic13D411yescharacteristic
90Q8×Dic13—22no—
91Q8×Dic13—22no—
92Q8×Dic13—41no—
93Q8×Dic13—41no—
94C2×C4⋊Dic13—21no—
95C2×C4⋊Dic13—21no—
96C2×C4⋊Dic13—21no—
97C2×C4⋊Dic13—21no—
98C2×C4⋊Dic13C2311yescharacteristic
99C2×C4⋊Dic13C2311yescharacteristic
100C2×C4⋊Dic13C2311yescharacteristic
101C2×C4⋊Dic13C2311yescharacteristic
102C2×C4×Dic13—21no—
103C2×C4×Dic13—21no—
104C2×C4×Dic13—21no—
105C2×C4×Dic13—21no—
106C2×C4×Dic13C2311yescharacteristic
107C2×C4×Dic13C2311yescharacteristic
108C22×#1055C2311yescharacteristic
109C22×#1055C2311yescharacteristic
110C13×#103C2×C414yes—
111C13×#103C2×C414yes—
112C13×#102C2×C414yes—
113C13×#102C2×C414yes—
114#88075—22no—
115#88075—22no—
116#88072—41no—
117#88072—41no—
118#88071—21no—
119#88071—21no—
120#88071—21no—
121#88071—21no—
122#88071—21no—
123#88071—21no—
124#88071D411yescharacteristic
125#88071D411yescharacteristic
126C22⋊#1055—22no—
127#88058—41no—
128#88057—41no—
129#88055—41no—
130C8×Dic13—41no—
131#22579—131nomaximal, Sylow
132C24×C13—21no—
133C24×C13C2×D411yescharacteristic
134C13×#41—28no—
135C13×#41—44no—
136C13×#41—44no—
137C13×#41—44no—
138C2×Q8×C13—21no—
139C2×Q8×C13—21no—
140C2×Q8×C13—21no—
141C2×Q8×C13C22×C412yes—
142C2×Q8×C13#4112yes—
143C2×Q8×C13#4112yes—
144C2×D4×C13—24no—
145C2×D4×C13—21no—
146C2×D4×C13—21no—
147C2×D4×C13—21no—
148C2×D4×C13—21no—
149C2×D4×C13—21no—
150C2×D4×C13—42no—
151C2×D4×C13—42no—
152C2×D4×C13C22×C412yes—
153C22×C4×C13—21no—
154C22×C4×C13—21no—
155C22×C4×C13—21no—
156C22×C4×C13—21no—
157C22×C4×C13—21no—
158C22×C4×C13—21no—
159C22×C4×C13—21no—
160C22×C4×C13—24no—
161C22×C4×C13—24no—
162C22×C4×C13—42no—
163C22×C4×C13—42no—
164C22×C4×C13—81no—
165C22×C4×C13—81no—
166C22×C4×C13C22×C412yes—
167C22×C4×C13C2×D411yescharacteristic
168C22×C4×C13#4112yes—
169C22×C4×C13C2411yescharacteristic
170C22×Dic13—21no—
171C22×Dic13—21no—
172C22×Dic13—21no—
173C22×Dic13—21no—
174C22×Dic13C2×D411yescharacteristic
175C22×Dic13C2×D411yescharacteristic
176C22×Dic13C2×D411yescharacteristic
177C22×Dic13C2×D411yescharacteristic
178C13×M16—44no—
179C2×C8×C13—21no—
180C2×C8×C13—21no—
181C2×C8×C13—21no—
182C2×C8×C13—21no—
183C2×C8×C13C22×C412yes—
184C2×C8×C13C22×C412yes—
185C13×C4⋊C4—24no—
186C13×C4⋊C4—21no—
187C13×C4⋊C4—21no—
188C13×C4⋊C4C22×C412yes—
189C13×C4⋊C4C22×C412yes—
190C22⋊Dic13—22no—
191C22⋊Dic13—41no—
192C22⋊Dic13—42no—
193C22⋊Dic13—42no—
194C22⋊Dic13—42no—
195C22⋊Dic13—81no—
196C22⋊Dic13—81no—
197C22⋊Dic13—81no—
198C4⋊Dic13—21no—
199C4⋊Dic13—21no—
200C4⋊Dic13—21no—
201C4⋊Dic13—21no—
202C4⋊Dic13—21no—
203C4⋊Dic13—21no—
204C4⋊Dic13—21no—
205C4⋊Dic13—21no—
206C4⋊Dic13—42no—
207C4⋊Dic13—42no—
208C4⋊Dic13—81no—
209C4⋊Dic13—81no—
210C4×Dic13—21no—
211C4×Dic13—21no—
212C4×Dic13—21no—
213C4×Dic13—21no—
214C4×Dic13—42no—
215C4×Dic13—42no—
216C4×Dic13—81no—
217C4×Dic13—81no—
218C2×#1055—21no—
219C2×#1055—21no—
220C2×#1055—21no—
221C2×#1055—21no—
222C2×#1055—21no—
223C2×#1055—21no—
224C2×#417D1311yescharacteristic
225#2345—131no—
226#2341—131no—
227#1936—131no—
228#1934—131no—
229#1926—131no—
230#1925—131no—
231#1865—131no—
232#1863—131no—
233#1823—131no—
234#1809—131no—
235#1788—131no—
236#1763—131no—
237C4×#103—131no—
238#1753—131no—
239C23×C13—21no—
240C23×C13—21no—
241C23×C13—21no—
242C23×C13—21no—
243C23×C13—21no—
244C23×C13—21no—
245C23×C13—22no—
246C23×C13—41no—
247C23×C13—41no—
248C23×C13—41no—
249C23×C13—41no—
250C23×C13—41no—
251C23×C13—41no—
252C23×C13C4×D412yes—
253C23×C13C22×D411yescharacteristic
254Q8×C13—22no—
255Q8×C13—22no—
256Q8×C13—22no—
257Q8×C13#11714yes—
258D4×C13—22no—
259D4×C13—22no—
260D4×C13—48no—
261D4×C13—41no—
262D4×C13—41no—
263C2×C4×C13—22no—
264C2×C4×C13—22no—
265C2×C4×C13—22no—
266C2×C4×C13—21no—
267C2×C4×C13—21no—
268C2×C4×C13—22no—
269C2×C4×C13—22no—
270C2×C4×C13—22no—
271C2×C4×C13—22no—
272C2×C4×C13—22no—
273C2×C4×C13—22no—
274C2×C4×C13—48no—
275C2×C4×C13—41no—
276C2×C4×C13—41no—
277C2×C4×C13—41no—
278C2×C4×C13—41no—
279C2×C4×C13—84no—
280C2×C4×C13C4×D412yes—
281C2×C4×C13C23×C411yescharacteristic
282C2×C4×C13C2×#4111yescharacteristic
283C2×C4×C13C2×#4111yescharacteristic
284C2×C4×C13C2×#4111yescharacteristic
285C2×C4×C13H(5,𝔽2)11yescharacteristic
286C2×C4×C13H(5,𝔽2)11yescharacteristic
287C2×Dic13—21no—
288C2×Dic13—21no—
289C2×Dic13—21no—
290C2×Dic13—21no—
291C2×Dic13—21no—
292C2×Dic13—21no—
293C2×Dic13—21no—
294C2×Dic13—21no—
295C2×Dic13—21no—
296C2×Dic13—21no—
297C2×Dic13—21no—
298C2×Dic13—41no—
299C2×Dic13—41no—
300C2×Dic13—41no—
301C2×Dic13—41no—
302C2×Dic13—41no—
303C2×Dic13—41no—
304C2×Dic13—41no—
305C2×Dic13—41no—
306C2×Dic13—41no—
307C2×Dic13—41no—
308C2×Dic13—41no—
309C2×Dic13—41no—
310C2×Dic13#13712yes—
311C8×C13—22no—
312C8×C13—22no—
313#1055—41no—
314#1055—41no—
315C22×#41C2×D1311yescharacteristic
316C22×M16C2×D1311yescharacteristic
317C2×C4×Q8—131no—
318C2×C4×Q8—131no—
319C22×C4⋊C4C2×D1311yescharacteristic
320C2×#107—131no—
321C2×#106—131no—
322#417Dic1318yescomplemented
323C2×#103—131no—
324C2×#103—131no—
325C2×#103C2×D1311yescharacteristic
326C2×#103C2×D1311yescharacteristic
327C2×#102—131no—
328C2×#102—131no—
329C2×#102C2×D1311yescharacteristic
330C2×#102C2×D1311yescharacteristic
331C2×C22⋊C8—131no—
332C2×#97—131no—
333C2×C4×C8—131no—
334#390—131no—
335#390—131no—
336#390—261no—
337#390—261no—
338#389—131no—
339#389—131no—
340#387—261no—
341#387—261no—
342#380—131no—
343#380—131no—
344#380—261no—
345#380—261no—
346C4×C4⋊C4—131no—
347C4×C4⋊C4—131no—
348C4×C22⋊C4—131no—
349C4×C22⋊C4—261no—
350C4×C22⋊C4—261no—
351#340—131no—
352#340—131no—
353#340—131no—
354#340—131no—
355#340—131no—
356#340—131no—
357#336—131no—
358#336—131no—
359C22×C13—22no—
360C22×C13—22no—
361C22×C13—22no—
362C22×C13—41no—
363C22×C13—41no—
364C22×C13—41no—
365C22×C13—41no—
366C22×C13—42no—
367C22×C13—42no—
368C22×C13—42no—
369C22×C13—81no—
370C22×C13—81no—
371C22×C13—81no—
372C22×C13—81no—
373C22×C13C2×C4×D411yescharacteristic
374C22×C13#54611yescharacteristic
375C22×C13#54611yescharacteristic
376C22×C13C2×#13711yescharacteristic
377C22×C13#57511yescharacteristic
378C22×C13#57611yescharacteristic
379C22×C13#57711yescharacteristic
380C4×C13—22no—
381C4×C13—24no—
382C4×C13—42no—
383C4×C13—42no—
384C4×C13C2×#11711yescharacteristic
385C4×C13#51811yescharacteristic
386C4×C13#55111yescharacteristic
387C4×C13#55111yescharacteristic
388Dic13—22no—
389Dic13—41no—
390Dic13—41no—
391Dic13—41no—
392Dic13—81no—
393Dic13—81no—
394Dic13—81no—
395Dic13—81no—
396C2×#41—22no—
397C2×#41—22no—
398C2×#41—22no—
399C2×#41—41no—
400C2×#41—41no—
401C2×#41C2×Dic1314yes—
402C22×Q8C22×D1311yescharacteristic
403C22×D4—21no—
404C22×D4C22×D1311yescharacteristic
405C23×C4—21no—
406C23×C4—21no—
407C23×C4C22×D1311yescharacteristic
408C2×M16—41no—
409C2×M16—41no—
410C2×M16C2×Dic1314yes—
411C22×C8—131no—
412C22×C8—131no—
413C22×C8C22×D1311yescharacteristic
414C22×C8C22×D1311yescharacteristic
415C4×Q8—262no—
416C4×Q8—262no—
417C4×Q8—521no—
418C4×Q8—521no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—22no—
421C2×C4⋊C4—131no—
422C2×C4⋊C4—131no—
423C2×C4⋊C4—131no—
424C2×C4⋊C4—131no—
425C2×C4⋊C4—261no—
426C2×C4⋊C4—261no—
427C2×C4⋊C4—261no—
428C2×C4⋊C4—261no—
429C2×C4⋊C4C2×Dic1314yes—
430C2×C4⋊C4C22×D1311yescharacteristic
431C2×C4⋊C4C22×D1311yescharacteristic
432C2×C22⋊C4—131no—
433C2×C22⋊C4—131no—
434C2×C22⋊C4—261no—
435C2×C22⋊C4—261no—
436C2×C22⋊C4—261no—
437C2×C22⋊C4—261no—
438C2×C22⋊C4—261no—
439C2×C22⋊C4—261no—
440C2×C42—131no—
441C2×C42—131no—
442C2×C42—261no—
443C2×C42—261no—
444C2×C42—261no—
445C2×C42—261no—
446#107—521no—
447#106—521no—
448#103—262no—
449#103—262no—
450#103C2×Dic1314yes—
451#103C2×Dic1314yes—
452#102—521no—
453#102—521no—
454#102C2×Dic1314yes—
455#102C2×Dic1314yes—
456C22⋊C8—262no—
457#97—521no—
458C4×C8—521no—
459#95—131no—
460#95—131no—
461#95—261no—
462#95—261no—
463#95—261no—
464#95—261no—
465#95—261no—
466#95—261no—
467C2×C13—42no—
468C2×C13—82no—
469C2×C13#234211yescharacteristic
470C2×C13C4×#13711yescharacteristic
471C2×C13#294611yescharacteristic
472C2×C13#330911yescharacteristic
473C2×C13#331611yescharacteristic
474C2×C13#332611yescharacteristic
475C2×C13#333311yescharacteristic
476C24—21no—
477C24D4×D1311yescharacteristic
478#41—28no—
479#41—44no—
480#41—44no—
481#41—44no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8#612712yes—
486C2×Q8#612712yes—
487C2×Q8C22×Dic1312yes—
488C2×D4—24no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—42no—
495C2×D4—42no—
496C2×D4C22×Dic1312yes—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—24no—
505C22×C4—24no—
506C22×C4—42no—
507C22×C4—42no—
508C22×C4—81no—
509C22×C4—81no—
510C22×C4—131no—
511C22×C4—131no—
512C22×C4—131no—
513C22×C4—131no—
514C22×C4—261no—
515C22×C4—261no—
516C22×C4—261no—
517C22×C4—261no—
518C22×C4D4×D1311yescharacteristic
519C22×C4#612912yes—
520C22×C4C22×Dic1312yes—
521C22×C4C23×D1311yescharacteristic
522M16—44no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—261no—
528C2×C8—261no—
529C2×C8—261no—
530C2×C8—261no—
531C2×C8—261no—
532C2×C8—261no—
533C2×C8C22×Dic1312yes—
534C2×C8C22×Dic1312yes—
535C4⋊C4—24no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—261no—
539C4⋊C4—261no—
540C4⋊C4—261no—
541C4⋊C4—261no—
542C4⋊C4—261no—
543C4⋊C4—261no—
544C4⋊C4—261no—
545C4⋊C4—261no—
546C4⋊C4—522no—
547C4⋊C4—522no—
548C4⋊C4—1041no—
549C4⋊C4—1041no—
550C4⋊C4C22×Dic1312yes—
551C4⋊C4C22×Dic1312yes—
552C22⋊C4—262no—
553C22⋊C4—522no—
554C22⋊C4—521no—
555C22⋊C4—522no—
556C22⋊C4—522no—
557C22⋊C4—1041no—
558C22⋊C4—1041no—
559C22⋊C4—1041no—
560C42—261no—
561C42—261no—
562C42—261no—
563C42—261no—
564C42—522no—
565C42—522no—
566C42—1041no—
567C42—1041no—
568C13#2257911yescharacteristic, complemented, Sylow
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—22no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23D4×Dic1312yes—
583C23C2×D4×D1311yescharacteristic
584Q8—22no—
585Q8—22no—
586Q8—22no—
587Q8#8818014yes—
588D4—22no—
589D4—22no—
590D4—48no—
591D4—41no—
592D4—41no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—48no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—84no—
610C2×C4—132no—
611C2×C4—261no—
612C2×C4—261no—
613C2×C4—261no—
614C2×C4—261no—
615C2×C4—261no—
616C2×C4—261no—
617C2×C4—261no—
618C2×C4—261no—
619C2×C4—261no—
620C2×C4—261no—
621C2×C4—261no—
622C2×C4—521no—
623C2×C4—521no—
624C2×C4—521no—
625C2×C4—521no—
626C2×C4—521no—
627C2×C4—521no—
628C2×C4—521no—
629C2×C4—521no—
630C2×C4—521no—
631C2×C4—521no—
632C2×C4—521no—
633C2×C4—521no—
634C2×C4D4×Dic1312yes—
635C2×C4C2×#612711yescharacteristic
636C2×C4C2×#612711yescharacteristic
637C2×C4C2×#612911yescharacteristic
638C2×C4#8825111yescharacteristic
639C2×C4#8825111yescharacteristic
640C2×C4C23×Dic1311yescharacteristic
641C8—22no—
642C8—22no—
643C8—521no—
644C8—521no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22#1171919511yescharacteristic
660C22#1171919511yescharacteristic
661C22#1171947911yescharacteristic
662C22#1171948011yescharacteristic
663C22D13×#13711yescharacteristic
664C22#1171948211yescharacteristic
665C22C2×D4×Dic1311yescharacteristic
666C4—22no—
667C4—24no—
668C4—42no—
669C4—42no—
670C4—262no—
671C4—521no—
672C4—521no—
673C4—521no—
674C4—1041no—
675C4—1041no—
676C4—1041no—
677C4—1041no—
678C4#1171920011yescharacteristic
679C4#1171920011yescharacteristic
680C4C2×#8818011yescharacteristic
681C4#1171950311yescharacteristic
682C2—42no—
683C2—82no—
684C2#1311768511yescharacteristic
685C2#137×Dic1311yescharacteristic
686C2#1312366911yescharacteristic
687C2#1312684311yescharacteristic
688C2#1312694811yescharacteristic
689C2#1312709011yescharacteristic
690C2#1312717011yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×C13×#417C2
socleC23×C13C22×D4
prime radicalC22×C13×#41C22
cosocleC22×C4×C13C24
hypercenterC2×#417D13
hypocenterC13#22579
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
1. #13117713C2semidirect
3. #13115535C2semidirect
4. #13115521C2semidirect
5. #13115465C2semidirect
6. #13115458C2semidirect
7. #13115106C2semidirect
8. #13115092C2semidirect
9. #13114922C2semidirect
10. #13114869C2semidirect
12. #13114668C2semidirect
13. #103×Dic13C2semidirect
14. #13114617C2semidirect
21. C13×#417C4semidirect
322. #417Dic13semidirect
568. C13#22579semidirect

Maximal subgroups

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

subgroupindex
#131177132
#131176772
#131155352
#131155212
#131154652
#131154582
#131151062
#131150922
#131149222
#131148692
#131147762
#131146682
#103×Dic132
#131146172
C2×C13×#4172
#2257913