Finite Group Databasebeta

← #15966180

Subgroups — #15966180

Order 3328.

788 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13117300C211yesmaximal, characteristic, complemented
2#13116338C211yesmaximal, characteristic, complemented
3#13116224C211yesmaximal, characteristic, complemented
4#13116207C211yesmaximal, characteristic, complemented
5#13116146C211yesmaximal, characteristic
6#13115711C211yesmaximal, characteristic
7#13115571C211yesmaximal, characteristic, complemented
8#13115536C211yesmaximal, characteristic, complemented
9#13115532C211yesmaximal, characteristic, complemented
10#13115460C211yesmaximal, characteristic, complemented
11#13115411C211yesmaximal, characteristic, complemented
12#13115187C211yesmaximal, characteristic
13#13115082C211yesmaximal, characteristic, complemented
14#13114689C211yesmaximal, characteristic, complemented
15C13×#2860C211yesmaximal, characteristic, complemented
16C2×C13×#126—21no—
17C22×#6104C2211yescharacteristic
18C2×C4×D52C2211yescharacteristic
19C2×#88126—21no—
20C13×#390C2211yescharacteristic
21C13×#390C2211yescharacteristic
22C13×#390C2211yescharacteristic
23C13×#389C2211yescharacteristic
24C13×#389C2211yescharacteristic
25C13×#389C2211yescharacteristic
26C13×#388—21no—
27C13×#388—21no—
28C13×#388—21no—
29C43×C13C2211yescharacteristic
30C2×#88072C2211yescharacteristic
31C2×#88072C2211yescharacteristic
32C2×#88072C2211yescharacteristic
33C2×#88049C2211yescharacteristic
34C2×#88049C2211yescharacteristic
35C2×#88049C2211yescharacteristic
36C2×#88048C2211yescharacteristic
37C2×#88048C2211yescharacteristic
38C2×#88047C2211yescharacteristic
39#11718601—21no—
40#11718601—21no—
41#11718600C2211yescharacteristic
42#11718600C2211yescharacteristic
43C4×#6103—21no—
44C4×C4⋊Dic13C2211yescharacteristic
45#11718589C2211yescharacteristic
46#11718589C2211yescharacteristic
47C2×C4⋊#1055C2211yescharacteristic
48C2×C4⋊#1055C2211yescharacteristic
49C2×#88043C2211yescharacteristic
50C2×#88043C2211yescharacteristic
51#11718121C2211yescharacteristic
52#11718121C2211yescharacteristic
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718116C2211yescharacteristic
58#11718116C2211yescharacteristic
59C22×D4×C13C2311yescharacteristic
60C22×D52C2311yescharacteristic
61C22×C4×D13—21no—
62C13×#126—44no—
63C2×C13×C4⋊C4—21no—
64C2×C13×C4⋊C4—21no—
65C2×C13×C4⋊C4—21no—
66C2×C13×C4⋊C4C2311yescharacteristic
67C2×C13×C4⋊C4C2311yescharacteristic
68C2×C13×C4⋊C4C2311yescharacteristic
69C2×C13×C22⋊C4—21no—
70C2×C13×C22⋊C4—21no—
71C2×C13×C22⋊C4—21no—
72C2×C13×C22⋊C4—21no—
73C2×C13×C22⋊C4—21no—
74C2×C13×C22⋊C4—21no—
75C2×C42×C13D411yescharacteristic
76C2×C42×C13D411yescharacteristic
77C2×C42×C13D411yescharacteristic
78C2×C42×C13D411yescharacteristic
79C2×C42×C13C2311yescharacteristic
80C2×C42×C13C2311yescharacteristic
81C2×C42×C13C2311yescharacteristic
82C2×#6104—22no—
83C2×#6104—22no—
84C2×#6104—41no—
85C2×#6103—21no—
86C2×#6103—21no—
87C2×#6103—21no—
88C2×#6103—21no—
89C2×#6103—21no—
90C2×C4⋊Dic13C2311yescharacteristic
91C2×C4⋊Dic13C2311yescharacteristic
92C2×C4⋊Dic13C2311yescharacteristic
93C2×#6101—21no—
94C2×#6101—21no—
95C2×C4×Dic13—21no—
96C22×#1055C2311yescharacteristic
97C22×#1055C2311yescharacteristic
98C22×#1055C2311yescharacteristic
99C22×#1055C2311yescharacteristic
100C4×D52—22no—
101C4×D52—22no—
102#88126—44no—
103C13×#95—21no—
104C13×#95—21no—
105C13×#95—21no—
106#88072—22no—
107#88072—22no—
108#88072—41no—
109#88071—21no—
110#88071—21no—
111#88071—21no—
112#88049—22no—
113#88049—22no—
114#88049—41no—
115#88048—22no—
116#88048—41no—
117#88047—22no—
118C4⋊#1055—22no—
119C4⋊#1055—22no—
120#88043—41no—
121#88043—41no—
122#29444—131nomaximal, Sylow
123C24×C13—21no—
124C23×D13—21no—
125C2×D4×C13—21no—
126C2×D4×C13—21no—
127C2×D4×C13—21no—
128C2×D4×C13—21no—
129C2×D4×C13—21no—
130C2×D4×C13—21no—
131C22×C4×C13—21no—
132C22×C4×C13—21no—
133C22×C4×C13—21no—
134C22×C4×C13C2×D411yescharacteristic
135C22×C4×C13C2×D411yescharacteristic
136C22×C4×C13C2×D411yescharacteristic
137C22×C4×C13C2×D411yescharacteristic
138C22×C4×C13C2×D411yescharacteristic
139C22×C4×C13C2×D411yescharacteristic
140C22×C4×C13C2411yescharacteristic
141C22×Dic13—21no—
142C22×Dic13—21no—
143C22×Dic13—21no—
144C2×D52—21no—
145C2×D52—21no—
146C2×D52—21no—
147C2×D52—21no—
148C2×D52—21no—
149C2×D52#4112yes—
150C2×C4×D13—24no—
151C2×C4×D13—42no—
152C2×C4×D13—42no—
153C13×C4⋊C4—21no—
154C13×C4⋊C4—21no—
155C13×C4⋊C4—21no—
156C13×C4⋊C4—21no—
157C13×C4⋊C4—21no—
158C13×C4⋊C4—21no—
159C13×C4⋊C4—42no—
160C13×C4⋊C4—42no—
161C13×C4⋊C4—42no—
162C13×C22⋊C4—42no—
163C13×C22⋊C4—42no—
164C13×C22⋊C4—42no—
165C13×C22⋊C4—42no—
166C13×C22⋊C4—42no—
167C13×C22⋊C4—42no—
168C42×C13—22no—
169C42×C13—22no—
170C42×C13—21no—
171C42×C13—21no—
172C42×C13—21no—
173C42×C13—21no—
174C42×C13—21no—
175C42×C13—21no—
176C42×C13—41no—
177C42×C13—41no—
178#6104—44no—
179#6103—24no—
180#6103—42no—
181#6103—42no—
182#6103—42no—
183#6103—42no—
184C4⋊Dic13—21no—
185C4⋊Dic13—21no—
186C4⋊Dic13—21no—
187C4⋊Dic13—21no—
188C4⋊Dic13—21no—
189C4⋊Dic13#4112yes—
190#6101—42no—
191#6101—42no—
192C4×Dic13—42no—
193C2×#1055—21no—
194C2×#1055—21no—
195C2×#1055—21no—
196C2×#1055—21no—
197C2×#1055—21no—
198C2×#1055—21no—
199C2×#1055—21no—
200C2×#1055—21no—
201C2×#1055—21no—
202C2×#1055—21no—
203C2×#1055#4112yes—
204C2×#1055#4112yes—
205#2860D1311yescharacteristic, complemented
206#2293—131no—
207#2088—131no—
208#2067—131no—
209#2065—131no—
210#2054—131no—
211#1972—131no—
212#1943—131no—
213#1936—131no—
214#1936—131no—
215#1926—131no—
216#1919—131no—
217#1879—131no—
218#1862—131no—
219#1767—131no—
220C23×C13—41no—
221C23×C13—41no—
222C23×C13—41no—
223C23×C13—41no—
224C23×C13—41no—
225C23×C13—41no—
226C23×C13—41no—
227C23×C13C22×D411yescharacteristic
228C22×D13—22no—
229C22×D13—41no—
230C22×D13—41no—
231C22×D13—41no—
232C22×D13—41no—
233C22×D13—41no—
234C22×D13—41no—
235D4×C13—41no—
236D4×C13—41no—
237D4×C13—41no—
238D4×C13—41no—
239C2×C4×C13—21no—
240C2×C4×C13—21no—
241C2×C4×C13—21no—
242C2×C4×C13—21no—
243C2×C4×C13—21no—
244C2×C4×C13—21no—
245C2×C4×C13—21no—
246C2×C4×C13—21no—
247C2×C4×C13—21no—
248C2×C4×C13—21no—
249C2×C4×C13—21no—
250C2×C4×C13—21no—
251C2×C4×C13—21no—
252C2×C4×C13—21no—
253C2×C4×C13—21no—
254C2×C4×C13—21no—
255C2×C4×C13—41no—
256C2×C4×C13—41no—
257C2×C4×C13—41no—
258C2×C4×C13—41no—
259C2×C4×C13—41no—
260C2×C4×C13—41no—
261C2×C4×C13—41no—
262C2×C4×C13—41no—
263C2×C4×C13—41no—
264C2×C4×C13AGL(1,ℤ/8)12yes—
265C2×C4×C13AGL(1,ℤ/8)12yes—
266C2×C4×C13C2×#4111yescharacteristic
267C2×C4×C13C2×#4111yescharacteristic
268C2×C4×C13C2×#4111yescharacteristic
269C2×C4×C13C2×#4111yescharacteristic
270C2×C4×C13H(5,𝔽2)11yescharacteristic
271C2×C4×C132−1+411yescharacteristic
272C2×Dic13—22no—
273C2×Dic13—41no—
274C2×Dic13—41no—
275C2×Dic13—41no—
276C2×Dic13—41no—
277C2×Dic13—41no—
278C2×Dic13—41no—
279C2×Dic13—41no—
280C2×Dic13—41no—
281D52—22no—
282D52—22no—
283D52—41no—
284D52—41no—
285C4×D13—48no—
286#1055—22no—
287#1055—22no—
288#1055—41no—
289#1055—41no—
290C22×D8—131no—
291C2×#126—21no—
292C2×C4×D4—131no—
293C2×#117—261no—
294C2×#107—131no—
295C2×#106—131no—
296C2×#106—131no—
297C2×C4⋊C8—131no—
298C2×C4⋊C8—131no—
299C2×#102—131no—
300C2×#102—131no—
301C2×#102—131no—
302C2×#102—131no—
303C2×#102—131no—
304C2×#102—131no—
305C2×#97—131no—
306C2×#97—131no—
307#390—131no—
308#390—131no—
309#390C2×D1311yescharacteristic
310#390C2×D1311yescharacteristic
311#390C2×D1311yescharacteristic
312#389—131no—
313#389—131no—
314#389C2×D1311yescharacteristic
315#389C2×D1311yescharacteristic
316#389C2×D1311yescharacteristic
317#388—21no—
318#388—21no—
319#388—21no—
320#388—261no—
321#388—261no—
322C4×C4⋊C4—131no—
323C4×C22⋊C4—261no—
324C43C2×D1311yescharacteristic
325#340—131no—
326#340—131no—
327#340—131no—
328#340—131no—
329#340—131no—
330#340—131no—
331#336—131no—
332#336—131no—
333C22×C13—81no—
334C22×C13—81no—
335C22×C13—81no—
336C22×C13—81no—
337C22×C13—81no—
338C22×C13—81no—
339C22×C13—81no—
340C22×C13#52511yescharacteristic
341C22×C13#53211yescharacteristic
342C22×C13#53711yescharacteristic
343C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
344C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
345C22×C13#57511yescharacteristic
346C22×C13#57511yescharacteristic
347C2×D13—42no—
348C2×D13—42no—
349C2×D13—42no—
350C2×D13—81no—
351C2×D13—81no—
352C2×D13—81no—
353C2×D13—81no—
354C4×C13—22no—
355C4×C13—22no—
356C4×C13—41no—
357C4×C13—41no—
358C4×C13—41no—
359C4×C13—41no—
360C4×C13—81no—
361C4×C13—81no—
362C4×C13—81no—
363C4×C13#55111yescharacteristic
364C4×C13#55211yescharacteristic
365C4×C13#55311yescharacteristic
366C4×C13#55611yescharacteristic
367Dic13—42no—
368Dic13—81no—
369Dic13—81no—
370C22×D4—131no—
371C22×D4C22×D1311yescharacteristic
372C23×C4—261no—
373C2×D8—262no—
374C2×D8—262no—
375C2×D8—521no—
376C22×C8—131no—
377C22×C8—131no—
378C22×C8—131no—
379C22×C8—131no—
380#126—44no—
381C4×D4—262no—
382C4×D4—262no—
383#117—524no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—131no—
388C2×C4⋊C4—131no—
389C2×C4⋊C4—131no—
390C2×C4⋊C4—261no—
391C2×C4⋊C4—261no—
392C2×C4⋊C4C22×D1311yescharacteristic
393C2×C4⋊C4C22×D1311yescharacteristic
394C2×C4⋊C4C22×D1311yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—261no—
402C2×C22⋊C4—261no—
403C2×C22⋊C4—261no—
404C2×C22⋊C4—261no—
405C2×C22⋊C4—261no—
406C2×C42—261no—
407C2×C42C22⋊D1311yescharacteristic
408C2×C42C22⋊D1311yescharacteristic
409C2×C42C22⋊D1311yescharacteristic
410C2×C42C22⋊D1311yescharacteristic
411C2×C42C22×D1311yescharacteristic
412C2×C42C22×D1311yescharacteristic
413C2×C42C22×D1311yescharacteristic
414#107—262no—
415#106—262no—
416#106—521no—
417C4⋊C8—262no—
418C4⋊C8—262no—
419#102—262no—
420#102—262no—
421#102—262no—
422#102—262no—
423#102—521no—
424#102—521no—
425#97—521no—
426#97—521no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—261no—
431#95—261no—
432#95—261no—
433C2×C13—161no—
434C2×C13#260111yescharacteristic
435C2×C13#310911yescharacteristic
436C2×C13#311111yescharacteristic
437C2×C13#317111yescharacteristic
438C2×C13#318011yescharacteristic
439C2×C13#325111yescharacteristic
440C2×C13#325111yescharacteristic
441D13—82no—
442C24—21no—
443C24—261no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—132no—
451C2×D4—261no—
452C2×D4—261no—
453C2×D4—261no—
454C2×D4—261no—
455C2×D4—261no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—264no—
460C22×C4—261no—
461C22×C4—261no—
462C22×C4—261no—
463C22×C4—522no—
464C22×C4—522no—
465C22×C4C2×C22⋊D1311yescharacteristic
466C22×C4C2×C22⋊D1311yescharacteristic
467C22×C4C2×C22⋊D1311yescharacteristic
468C22×C4C2×C22⋊D1311yescharacteristic
469C22×C4C2×C22⋊D1311yescharacteristic
470C22×C4C2×C22⋊D1311yescharacteristic
471C22×C4C23×D1311yescharacteristic
472D8—524no—
473C2×C8—132no—
474C2×C8—132no—
475C2×C8—261no—
476C2×C8—261no—
477C2×C8—261no—
478C2×C8—261no—
479C2×C8—261no—
480C2×C8—261no—
481C2×C8—261no—
482C2×C8—261no—
483C2×C8—261no—
484C2×C8—261no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—132no—
495C4⋊C4—261no—
496C4⋊C4—261no—
497C4⋊C4—261no—
498C4⋊C4—261no—
499C4⋊C4—261no—
500C4⋊C4—522no—
501C4⋊C4—522no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—264no—
509C22⋊C4—522no—
510C22⋊C4—522no—
511C22⋊C4—522no—
512C22⋊C4—522no—
513C42—22no—
514C42—22no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—41no—
522C42—41no—
523C42—522no—
524C13#2944411yescharacteristic, complemented, Sylow
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—262no—
533C23—521no—
534C23—521no—
535C23—521no—
536C23—521no—
537C23—521no—
538C23—521no—
539C23C22×C22⋊D1311yescharacteristic
540D4—41no—
541D4—41no—
542D4—41no—
543D4—41no—
544D4—262no—
545D4—262no—
546D4—521no—
547D4—521no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—262no—
574C2×C4—528no—
575C2×C4—521no—
576C2×C4—521no—
577C2×C4—521no—
578C2×C4—521no—
579C2×C4—521no—
580C2×C4—521no—
581C2×C4—521no—
582C2×C4—521no—
583C2×C4#8818612yes—
584C2×C4#8820312yes—
585C2×C4#41×D1311yescharacteristic
586C2×C4#41×D1311yescharacteristic
587C2×C4#41×D1311yescharacteristic
588C2×C4#41×D1311yescharacteristic
589C2×C4#8825611yescharacteristic
590C2×C4#8825711yescharacteristic
591C8—262no—
592C8—262no—
593C8—521no—
594C8—521no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—522no—
603C22—522no—
604C22—522no—
605C22—1041no—
606C22—1041no—
607C22—1041no—
608C22—1041no—
609C22C2×#8818611yescharacteristic
610C22C2×#8820311yescharacteristic
611C22#1171952711yescharacteristic
612C22#1171952711yescharacteristic
613C22#1171953311yescharacteristic
614C22#1171953711yescharacteristic
615C22#1171953811yescharacteristic
616C4—22no—
617C4—22no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—81no—
623C4—81no—
624C4—81no—
625C4—522no—
626C4—1041no—
627C4—1041no—
628C4#1171940711yescharacteristic
629C4#1171941011yescharacteristic
630C4#1171941311yescharacteristic
631C4#1171941411yescharacteristic
632C2—161no—
633C2—1042no—
634C2#1312043511yescharacteristic
635C2#1312492311yescharacteristic
636C2#1312493911yescharacteristic
637C2#1312542511yescharacteristic
638C2#1312553511yescharacteristic
639C2#1312619211yescharacteristic
640C2#1312619811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#2860C2
socleC23×C13C22×D4
prime radicalC22×#6104C22
hypercenter#2860D13
hypocenterC13#29444
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C43 ◁ #2860

Complements

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

NH ≅ G/Ntype
1. #13117300C2semidirect
2. #13116338C2semidirect
3. #13116224C2semidirect
4. #13116207C2semidirect
7. #13115571C2semidirect
8. #13115536C2semidirect
9. #13115532C2semidirect
10. #13115460C2semidirect
11. #13115411C2semidirect
13. #13115082C2semidirect
14. #13114689C2semidirect
15. C13×#2860C2semidirect
205. #2860D13semidirect
524. C13#29444semidirect

Maximal subgroups

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

subgroupindex
#131173002
#131163382
#131162242
#131162072
#131161462
#131157112
#131155712
#131155362
#131155322
#131154602
#131154112
#131151872
#131150822
#131146892
C13×#28602
#2944413