Finite Group Databasebeta

← #16802973

Subgroups — #16802973

Order 3328.

738 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13131009C211yesmaximal, characteristic, complemented
2#13131008C211yesmaximal, characteristic, complemented
3#13130910C211yesmaximal, characteristic, complemented
4#13130909C211yesmaximal, characteristic, complemented
5#13130906C211yesmaximal, characteristic, complemented
6#13130905C211yesmaximal, characteristic, complemented
7#13130809C211yesmaximal, characteristic
8#13130809C211yesmaximal, characteristic
9#13130780C211yesmaximal, characteristic, complemented
10#13130779C211yesmaximal, characteristic, complemented
11#13130765C211yesmaximal, characteristic, complemented
12#13130624C211yesmaximal, characteristic, complemented
13#13130573C211yesmaximal, characteristic, complemented
14#13130573C211yesmaximal, characteristic, complemented
15#13118124C211yesmaximal, characteristic
16C23×#1057C2211yescharacteristic
17C23×#1057C2211yescharacteristic
18C2×D4×Dic13C2211yescharacteristic
19C2×#88141C2211yescharacteristic
20C2×C22⋊#1057C2211yescharacteristic
21C2×C22⋊#1057C2211yescharacteristic
22C2×C22⋊#1057C2211yescharacteristic
23C2×C22⋊#1057C2211yescharacteristic
24C2×C22⋊#1057C2211yescharacteristic
25C2×C22⋊#1057C2211yescharacteristic
26C2×#88112C2211yescharacteristic
27C2×#88112C2211yescharacteristic
28C2×#88110C2211yescharacteristic
29C2×#88110C2211yescharacteristic
30C2×#88110C2211yescharacteristic
31C2×C4⋊#1057C2211yescharacteristic
32C2×C4⋊#1057C2211yescharacteristic
33C2×C4×#1057C2211yescharacteristic
34C13×#394C411yescharacteristic
35#11718883C411yescharacteristic
36#11718647C411yescharacteristic
37#11718613C411yescharacteristic
38#11718613C411yescharacteristic
39#11718608C411yescharacteristic
40#11718608C411yescharacteristic
41C22⋊C4×Dic13C2211yescharacteristic
42C22⋊C4×Dic13C2211yescharacteristic
43C22⋊C4×Dic13C2211yescharacteristic
44C22⋊C4×Dic13C2211yescharacteristic
45#11718320C411yescharacteristic
46#11718309C2211yescharacteristic
47#11718290C2211yescharacteristic
48#11718290C2211yescharacteristic
49#11718290C2211yescharacteristic
50#11718290C2211yescharacteristic
51#11718290C2211yescharacteristic
52#11718290C2211yescharacteristic
53#11718290C2211yescharacteristic
54#11718290C2211yescharacteristic
55#11718290C2211yescharacteristic
56#11718290C2211yescharacteristic
57#11718290C2211yescharacteristic
58#11718290C2211yescharacteristic
59C22×D4×C13C2×C411yescharacteristic
60C23×Dic13C2311yescharacteristic
61C23×Dic13C2311yescharacteristic
62C22×Dic26C2×C411yescharacteristic
63C22×#1057—22no—
64C22×#1057—22no—
65C22×#1057—41no—
66C22×#1057—41no—
67C22×#1057—41no—
68C22×#1057—41no—
69C22×#1057C2311yescharacteristic
70C22×#1057C2311yescharacteristic
71C22×#1057C2311yescharacteristic
72C22×#1057C2311yescharacteristic
73C22×#1057C2311yescharacteristic
74C22×#1057C2311yescharacteristic
75C22×#1057C2311yescharacteristic
76C22×#1057C2311yescharacteristic
77C2×C13×C4⋊C4C2×C411yescharacteristic
78C2×C13×C22⋊C4C2×C411yescharacteristic
79C2×C13×C22⋊C4C2×C411yescharacteristic
80C2×C13×C22⋊C4C2×C411yescharacteristic
81C2×C13×C22⋊C4C2×C411yescharacteristic
82C2×C22⋊Dic13C2×C411yescharacteristic
83C2×C22⋊Dic13C2×C411yescharacteristic
84C2×C22⋊Dic13C2×C411yescharacteristic
85C2×C22⋊Dic13C2×C411yescharacteristic
86C2×C22⋊Dic13C2×C411yescharacteristic
87C2×C22⋊Dic13C2×C411yescharacteristic
88D4×Dic13—41no—
89D4×Dic13—41no—
90C2×C4⋊Dic13C2×C411yescharacteristic
91C2×#6101C2×C411yescharacteristic
92C2×#6101C2×C411yescharacteristic
93C2×C4×Dic13C2311yescharacteristic
94C2×C4×Dic13C2311yescharacteristic
95C2×C4×Dic13C2311yescharacteristic
96C2×C4×Dic13C2311yescharacteristic
97C2×C4×Dic13C2311yescharacteristic
98#88141—41no—
99#88141—41no—
100C22⋊#1057—22no—
101C22⋊#1057—22no—
102C22⋊#1057—41no—
103C22⋊#1057—41no—
104C22⋊#1057—41no—
105C22⋊#1057—41no—
106#88112—22no—
107#88112—41no—
108#88110—41no—
109#88110—41no—
110#88110—41no—
111C4⋊#1057—22no—
112C4⋊#1057—41no—
113C4×#1057—41no—
114C13×#95C2×C411yescharacteristic
115#88071C2×C411yescharacteristic
116#88071C2×C411yescharacteristic
117#88071C2×C411yescharacteristic
118#88071C2×C411yescharacteristic
119#88071C2×C411yescharacteristic
120#88071C2×C411yescharacteristic
121#88071C2×C411yescharacteristic
122#88071C2×C411yescharacteristic
123#88071C2×C411yescharacteristic
124#88071C2×C411yescharacteristic
125#88071C2×C411yescharacteristic
126#22396—131nomaximal, Sylow
127C24×C13C22×C411yescharacteristic
128C24×C13C22×C411yescharacteristic
129C2×D4×C13—22no—
130C2×D4×C13—22no—
131C2×D4×C13—41no—
132C22×C4×C13C22×C411yescharacteristic
133C22×C4×C13C22×C411yescharacteristic
134C22×C4×C13C22×C411yescharacteristic
135C22×C4×C13C22×C411yescharacteristic
136C22×C4×C13C22×C411yescharacteristic
137C22×Dic13—21no—
138C22×Dic13—21no—
139C22×Dic13—21no—
140C22×Dic13—21no—
141C22×Dic13—21no—
142C22×Dic13—21no—
143C22×Dic13—21no—
144C22×Dic13—21no—
145C22×Dic13—21no—
146C22×Dic13—21no—
147C22×Dic13—21no—
148C22×Dic13—21no—
149C22×Dic13C22×C411yescharacteristic
150C22×Dic13C22×C411yescharacteristic
151C22×Dic13C22×C411yescharacteristic
152C22×Dic13C22×C411yescharacteristic
153C22×Dic13C22×C411yescharacteristic
154C22×Dic13C22×C411yescharacteristic
155C22×Dic13C22×C411yescharacteristic
156C22×Dic13C2411yescharacteristic
157C2×Dic26—41no—
158C2×Dic26—41no—
159C2×Dic26—41no—
160C2×#1057—21no—
161C2×#1057—21no—
162C2×#1057—21no—
163C2×#1057—21no—
164C2×#1057—21no—
165C2×#1057—21no—
166C2×#1057—21no—
167C2×#1057—21no—
168C2×#1057—21no—
169C2×#1057—21no—
170C2×#1057—21no—
171C2×#1057—21no—
172C2×#1057—21no—
173C2×#1057—21no—
174C2×#1057—21no—
175C2×#1057—21no—
176C2×#1057—21no—
177C2×#1057—21no—
178C2×#1057—21no—
179C2×#1057—21no—
180C2×#1057—21no—
181C2×#1057—21no—
182C2×#1057—44no—
183C2×#1057—44no—
184C2×#1057#4112yes—
185C2×#1057#4112yes—
186C13×C4⋊C4—41no—
187C13×C22⋊C4—22no—
188C13×C22⋊C4—22no—
189C13×C22⋊C4—41no—
190C13×C22⋊C4—41no—
191C22⋊Dic13—22no—
192C22⋊Dic13—22no—
193C22⋊Dic13—41no—
194C22⋊Dic13—41no—
195C22⋊Dic13—41no—
196C22⋊Dic13—41no—
197C4⋊Dic13—41no—
198#6101—41no—
199#6101—41no—
200C4×Dic13—21no—
201C4×Dic13—21no—
202C4×Dic13—21no—
203C4×Dic13—21no—
204C4×Dic13—21no—
205C4×Dic13—21no—
206C4×Dic13—21no—
207C4×Dic13—21no—
208C4×Dic13—21no—
209C4×Dic13—21no—
210#2398—131no—
211#1985—131no—
212#1985—131no—
213#1924—131no—
214#1924—131no—
215#1923—131no—
216#1923—131no—
217#1869—131no—
218#1869—131no—
219#1850—131no—
220#1850—131no—
221#1840—131no—
222#1768—131no—
223#1752—131no—
224#1752—131no—
225C23×C13—21no—
226C23×C13—21no—
227C23×C13—21no—
228C23×C13—21no—
229C23×C13—21no—
230C23×C13—21no—
231C23×C13—21no—
232C23×C13—21no—
233C23×C13—21no—
234C23×C13—21no—
235C23×C13—21no—
236C23×C13—21no—
237C23×C13#13112yes—
238C23×C13#13112yes—
239C23×C13C23×C411yescharacteristic
240D4×C13—44no—
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—21no—
256C2×Dic13—21no—
257C2×Dic13—21no—
258C2×Dic13—21no—
259C2×Dic13—21no—
260C2×Dic13—21no—
261C2×Dic13—21no—
262C2×Dic13—21no—
263C2×Dic13—21no—
264C2×Dic13—21no—
265C2×Dic13—21no—
266C2×Dic13—21no—
267C2×Dic13—21no—
268C2×Dic13—21no—
269C2×Dic13—21no—
270C2×Dic13—21no—
271C2×Dic13—21no—
272C2×Dic13—21no—
273C2×Dic13—21no—
274C2×Dic13—21no—
275C2×Dic13—21no—
276C2×Dic13—21no—
277C2×Dic13—41no—
278C2×Dic13—41no—
279C2×Dic13—41no—
280C2×Dic13—41no—
281C2×Dic13—41no—
282C2×Dic13—41no—
283C2×Dic13—41no—
284C2×Dic13—41no—
285C2×Dic13C2×#4111yescharacteristic
286C2×Dic13C2×#4111yescharacteristic
287C2×Dic13H(5,𝔽2)11yescharacteristic
288C2×Dic13H(5,𝔽2)11yescharacteristic
289C2×Dic13H(5,𝔽2)11yescharacteristic
290C2×Dic132−1+411yescharacteristic
291Dic26—44no—
292#1057—22no—
293#1057—22no—
294#1057—41no—
295#1057—41no—
296#1057—41no—
297#1057—41no—
298#1057—41no—
299#1057—41no—
300C23×C8—131no—
301C23×C8—131no—
302C2×C4×Q8—131no—
303C2×C4×D4—131no—
304C2×C4⋊C8—131no—
305C2×C4⋊C8—131no—
306C2×C4⋊C8—131no—
307C2×C4⋊C8—131no—
308C2×C22⋊C8—131no—
309C2×C22⋊C8—131no—
310C2×C22⋊C8—131no—
311C2×C22⋊C8—131no—
312C2×C22⋊C8—131no—
313C2×C22⋊C8—131no—
314C2×#97—131no—
315C2×#97—131no—
316C2×#97—131no—
317C2×C4×C8—131no—
318#400—131no—
319#400—131no—
320#397—131no—
321#397—131no—
322#394—131no—
323#394#26111yescharacteristic
324#393—131no—
325#393—131no—
326C4×C22⋊C4—131no—
327C4×C22⋊C4—131no—
328C4×C22⋊C4—131no—
329C4×C22⋊C4—131no—
330#376—131no—
331#336—131no—
332#336—131no—
333#336—131no—
334#336—131no—
335#336—131no—
336#336—131no—
337#336—131no—
338#336—131no—
339#336—131no—
340#336—131no—
341#336—131no—
342#336—131no—
343C22×C13—22no—
344C22×C13—22no—
345C22×C13—22no—
346C22×C13—22no—
347C22×C13—22no—
348C22×C13—22no—
349C22×C13—41no—
350C22×C13—41no—
351C22×C13—41no—
352C22×C13—41no—
353C22×C13—41no—
354C22×C13—41no—
355C22×C13—41no—
356C22×C13—41no—
357C22×C13C4×#4111yescharacteristic
358C22×C13#51811yescharacteristic
359C22×C13#52011yescharacteristic
360C22×C13C2×#13111yescharacteristic
361C22×C13C2×#13111yescharacteristic
362C22×C13#56811yescharacteristic
363C22×C13#56811yescharacteristic
364C4×C13—41no—
365C4×C13—41no—
366C4×C13—41no—
367C4×C13—41no—
368C4×C13—41no—
369Dic13—41no—
370Dic13—41no—
371Dic13—41no—
372Dic13—41no—
373Dic13—41no—
374Dic13—41no—
375Dic13—41no—
376Dic13#53811yescharacteristic
377Dic13#53811yescharacteristic
378Dic13#54211yescharacteristic
379Dic13#54211yescharacteristic
380C22×Q8—131no—
381C22×D4C2×#26111yescharacteristic
382C23×C4—131no—
383C23×C4—131no—
384C22×C8—131no—
385C22×C8—131no—
386C22×C8—131no—
387C22×C8—131no—
388C22×C8—131no—
389C22×C8—131no—
390C22×C8—131no—
391C22×C8—131no—
392C22×C8—262no—
393C22×C8—262no—
394C22×C8—521no—
395C22×C8—521no—
396C22×C8—521no—
397C22×C8—521no—
398C4×Q8—521no—
399C4×Q8—521no—
400C4×D4—521no—
401C4×D4—521no—
402C2×C4⋊C4—131no—
403C2×C4⋊C4—131no—
404C2×C4⋊C4—131no—
405C2×C4⋊C4C2×#26111yescharacteristic
406C2×C22⋊C4—131no—
407C2×C22⋊C4—131no—
408C2×C22⋊C4—131no—
409C2×C22⋊C4—131no—
410C2×C22⋊C4—131no—
411C2×C22⋊C4—131no—
412C2×C22⋊C4C2×#26111yescharacteristic
413C2×C22⋊C4C2×#26111yescharacteristic
414C2×C22⋊C4C2×#26111yescharacteristic
415C2×C22⋊C4C2×#26111yescharacteristic
416C2×C42—131no—
417C2×C42—131no—
418C2×C42—131no—
419C2×C42—131no—
420C2×C42—131no—
421C4⋊C8—262no—
422C4⋊C8—262no—
423C4⋊C8—521no—
424C4⋊C8—521no—
425C22⋊C8—262no—
426C22⋊C8—262no—
427C22⋊C8—521no—
428C22⋊C8—521no—
429C22⋊C8—521no—
430C22⋊C8—521no—
431#97—521no—
432#97—521no—
433#97—521no—
434C4×C8—521no—
435#95—131no—
436#95—131no—
437#95—131no—
438#95—131no—
439#95—131no—
440#95—131no—
441#95—131no—
442#95—131no—
443#95—131no—
444#95—131no—
445#95—131no—
446#95C2×#26111yescharacteristic
447C2×C13—42no—
448C2×C13—42no—
449C2×C13#233711yescharacteristic
450C2×C13#296811yescharacteristic
451C2×C13#296811yescharacteristic
452C2×C13#297711yescharacteristic
453C2×C13#297811yescharacteristic
454C2×C13#299211yescharacteristic
455C2×C13#299211yescharacteristic
456C24C22×#26111yescharacteristic
457C24C22×#26111yescharacteristic
458C2×Q8—521no—
459C2×Q8—521no—
460C2×Q8—521no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—41no—
464C22×C4—131no—
465C22×C4—131no—
466C22×C4—131no—
467C22×C4—131no—
468C22×C4—131no—
469C22×C4—131no—
470C22×C4—131no—
471C22×C4—131no—
472C22×C4—261no—
473C22×C4—261no—
474C22×C4—261no—
475C22×C4—261no—
476C22×C4—261no—
477C22×C4—261no—
478C22×C4—261no—
479C22×C4—261no—
480C22×C4—261no—
481C22×C4—261no—
482C22×C4—261no—
483C22×C4—261no—
484C22×C4C22×#26111yescharacteristic
485C22×C4C22×#26111yescharacteristic
486C22×C4C22×#26111yescharacteristic
487C22×C4C22×#26111yescharacteristic
488C22×C4C22×#26111yescharacteristic
489C2×C8—132no—
490C2×C8—132no—
491C2×C8—261no—
492C2×C8—261no—
493C2×C8—261no—
494C2×C8—261no—
495C2×C8—261no—
496C2×C8—261no—
497C2×C8—261no—
498C2×C8—261no—
499C2×C8—261no—
500C2×C8—261no—
501C2×C8—261no—
502C2×C8—261no—
503C2×C8—261no—
504C2×C8—261no—
505C2×C8—261no—
506C2×C8—261no—
507C2×C8—261no—
508C2×C8—261no—
509C2×C8—261no—
510C2×C8—261no—
511C2×C8—261no—
512C2×C8—261no—
513C2×C8—524no—
514C2×C8—524no—
515C4⋊C4—41no—
516C4⋊C4—521no—
517C4⋊C4—521no—
518C4⋊C4—521no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—41no—
522C22⋊C4—41no—
523C22⋊C4—262no—
524C22⋊C4—262no—
525C22⋊C4—521no—
526C22⋊C4—521no—
527C22⋊C4—521no—
528C22⋊C4—521no—
529C42—261no—
530C42—261no—
531C42—261no—
532C42—261no—
533C42—261no—
534C42—261no—
535C42—261no—
536C42—261no—
537C42—261no—
538C42—261no—
539C13#2239611yescharacteristic, complemented, Sylow
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23#8823812yes—
553C23#8823812yes—
554C23C23×#26111yescharacteristic
555Q8—524no—
556D4—44no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—131no—
573C2×C4—131no—
574C2×C4—131no—
575C2×C4—131no—
576C2×C4—131no—
577C2×C4—131no—
578C2×C4—261no—
579C2×C4—261no—
580C2×C4—261no—
581C2×C4—261no—
582C2×C4—261no—
583C2×C4—261no—
584C2×C4—261no—
585C2×C4—261no—
586C2×C4—261no—
587C2×C4—261no—
588C2×C4—261no—
589C2×C4—261no—
590C2×C4—261no—
591C2×C4—261no—
592C2×C4—261no—
593C2×C4—261no—
594C2×C4—261no—
595C2×C4—261no—
596C2×C4—261no—
597C2×C4—261no—
598C2×C4—261no—
599C2×C4—521no—
600C2×C4—521no—
601C2×C4—521no—
602C2×C4—521no—
603C2×C4—521no—
604C2×C4—521no—
605C2×C4—521no—
606C2×C4—521no—
607C8—262no—
608C8—262no—
609C8—521no—
610C8—521no—
611C8—521no—
612C8—521no—
613C8—521no—
614C8—521no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22C2×#8823811yescharacteristic
630C22C2×#8823811yescharacteristic
631C22#1171962111yescharacteristic
632C22#1171962311yescharacteristic
633C22#1171962911yescharacteristic
634C22#41×#26111yescharacteristic
635C22#1171963111yescharacteristic
636C4—41no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—131no—
642C4—131no—
643C4—131no—
644C4—131no—
645C4—521no—
646C4—521no—
647C4—521no—
648C4—521no—
649C4—521no—
650C4—521no—
651C4—521no—
652C2—42no—
653C2—42no—
654C2#1313133711yescharacteristic
655C2#1313166111yescharacteristic
656C2#1313166311yescharacteristic
657C2#1313170811yescharacteristic
658C2#1313171311yescharacteristic
659C2#1313177111yescharacteristic
660C2#1313177211yescharacteristic
subgroupG/H
center, FrattiniC23C23×#261
commutator, socleC23×C13C23×C4
FittingC13×#394C4
prime radicalC22×D4×C13C2×C4
cosocleC22×Dic13C24
hypercenter#394#261
hypocenterC13#22396
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #13131009C2semidirect
2. #13131008C2semidirect
3. #13130910C2semidirect
4. #13130909C2semidirect
5. #13130906C2semidirect
6. #13130905C2semidirect
9. #13130780C2semidirect
10. #13130779C2semidirect
11. #13130765C2semidirect
12. #13130624C2semidirect
13. #13130573C2semidirect
14. #13130573C2semidirect
539. C13#22396semidirect

Maximal subgroups

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

subgroupindex
#131310092
#131310082
#131309102
#131309092
#131309062
#131309052
#131308092
#131307802
#131307792
#131307652
#131306242
#131305732
#131181242
#2239613