Finite Group Databasebeta

← #16806682

Subgroups — #16806682

Order 3328.

642 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13130938C211yesmaximal, characteristic, complemented
2#13130938C211yesmaximal, characteristic, complemented
3#13130937C211yesmaximal, characteristic, complemented
4#13130912C211yesmaximal, characteristic
5#13130912C211yesmaximal, characteristic
6#13130911C211yesmaximal, characteristic
7#13130896C211yesmaximal, characteristic, complemented
8#13130895C211yesmaximal, characteristic, complemented
9#13130893C211yesmaximal, characteristic, complemented
10#13130784C211yesmaximal, characteristic
11#13130784C211yesmaximal, characteristic
12#13130782C211yesmaximal, characteristic
13#13130769C211yesmaximal, characteristic, complemented
14#13130764C211yesmaximal, characteristic, complemented
15#13117386C211yesmaximal, characteristic
16C22×#6118C2211yescharacteristic
17C2×#88152C411yescharacteristic
18C2×#88112C2211yescharacteristic
19C2×#88112C2211yescharacteristic
20C2×#88112C2211yescharacteristic
21C2×#88112C2211yescharacteristic
22C2×#88112C2211yescharacteristic
23C2×#88112C2211yescharacteristic
24C2×#88111C2211yescharacteristic
25C2×#88111C2211yescharacteristic
26C2×#88111C2211yescharacteristic
27C2×#88110C2211yescharacteristic
28C2×#88110C2211yescharacteristic
29C2×#88110C2211yescharacteristic
30C2×C4⋊#1057C2211yescharacteristic
31C2×C4⋊#1057C2211yescharacteristic
32C2×C4⋊#1057C2211yescharacteristic
33C13×#387C411yescharacteristic
34#11718651C411yescharacteristic
35C4⋊C4×Dic13C2211yescharacteristic
36C4⋊C4×Dic13C2211yescharacteristic
37#11718601C411yescharacteristic
38C42×Dic13C2211yescharacteristic
39#11718331C411yescharacteristic
40#11718327C2211yescharacteristic
41#11718327C2211yescharacteristic
42#11718327C2211yescharacteristic
43#11718316C411yescharacteristic
44#11718314C411yescharacteristic
45#11718312C411yescharacteristic
46#11718310C2211yescharacteristic
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×C4×D13C2311yescharacteristic
60C22×#1057C2311yescharacteristic
61C22×#1057C2311yescharacteristic
62C22×#1057C2311yescharacteristic
63C22×#1057C2311yescharacteristic
64C22×#1057C2311yescharacteristic
65C22×#1057C2311yescharacteristic
66C22×#1057C2311yescharacteristic
67C22×#1057C2311yescharacteristic
68C2×#6118—41no—
69C2×#6118—41no—
70C2×#6118—41no—
71C2×C13×C4⋊C4C2×C411yescharacteristic
72C2×C13×C4⋊C4C2×C411yescharacteristic
73C2×C42×C13C2×C411yescharacteristic
74C2×#6103C2×C411yescharacteristic
75C2×#6103C2×C411yescharacteristic
76C2×#6103C2×C411yescharacteristic
77C2×#6103C2×C411yescharacteristic
78C2×#6103C2×C411yescharacteristic
79C2×#6103C2×C411yescharacteristic
80C2×C4⋊Dic13C2×C411yescharacteristic
81C2×C4⋊Dic13C2×C411yescharacteristic
82C2×#6101C2×C411yescharacteristic
83C2×#6101C2×C411yescharacteristic
84C2×C4×Dic13C2×C411yescharacteristic
85C2×C4×Dic13C2×C411yescharacteristic
86C2×C4×Dic13C2×C411yescharacteristic
87C2×C4×Dic13C2311yescharacteristic
88C2×C4×Dic13C2311yescharacteristic
89C2×C4×Dic13C2311yescharacteristic
90C2×C4×Dic13C2311yescharacteristic
91C2×C4×Dic13C2311yescharacteristic
92C2×C4×Dic13C2311yescharacteristic
93#88152—24no—
94#88112—41no—
95#88112—41no—
96#88112—41no—
97#88112—41no—
98#88112—41no—
99#88112—41no—
100#88111—41no—
101#88111—41no—
102#88111—41no—
103#88110—41no—
104#88110—41no—
105#88110—41no—
106C4⋊#1057—41no—
107C4⋊#1057—41no—
108C4⋊#1057—41no—
109C13×#95C2×C411yescharacteristic
110C13×#95C2×C411yescharacteristic
111C13×#95C2×C411yescharacteristic
112C13×#95C2×C411yescharacteristic
113#88071C2×C411yescharacteristic
114#88071C2×C411yescharacteristic
115#88071C2×C411yescharacteristic
116#88071C2×C411yescharacteristic
117#88071C2×C411yescharacteristic
118#88071C2×C411yescharacteristic
119#88071C2×C411yescharacteristic
120#88071C2×C411yescharacteristic
121#23378—131nomaximal, Sylow
122C23×D13C22×C411yescharacteristic
123C22×C4×C13C22×C411yescharacteristic
124C22×C4×C13C22×C411yescharacteristic
125C22×C4×C13C22×C411yescharacteristic
126C22×C4×C13C22×C411yescharacteristic
127C22×C4×C13C22×C411yescharacteristic
128C22×C4×C13C22×C411yescharacteristic
129C22×C4×C13C22×C411yescharacteristic
130C22×Dic13C22×C411yescharacteristic
131C22×Dic13C22×C411yescharacteristic
132C22×Dic13C22×C411yescharacteristic
133C22×Dic13C22×C411yescharacteristic
134C22×Dic13C22×C411yescharacteristic
135C22×Dic13C22×C411yescharacteristic
136C22×Dic13C2411yescharacteristic
137C2×C4×D13—21no—
138C2×C4×D13—21no—
139C2×C4×D13—21no—
140C2×C4×D13—21no—
141C2×C4×D13—21no—
142C2×C4×D13—21no—
143C2×#1057—21no—
144C2×#1057—21no—
145C2×#1057—21no—
146C2×#1057—21no—
147C2×#1057—21no—
148C2×#1057—21no—
149C2×#1057—21no—
150C2×#1057—21no—
151C2×#1057—21no—
152C2×#1057—21no—
153C2×#1057—21no—
154C2×#1057—21no—
155C2×#1057—21no—
156C2×#1057—21no—
157C2×#1057—21no—
158C2×#1057—21no—
159C2×#1057—21no—
160C2×#1057—21no—
161C2×#1057—21no—
162C2×#1057—21no—
163C2×#1057—21no—
164C2×#1057—21no—
165C2×#1057—21no—
166C2×#1057—21no—
167#6118—44no—
168C13×C4⋊C4—22no—
169C13×C4⋊C4—41no—
170C42×C13—41no—
171#6103—22no—
172#6103—22no—
173#6103—22no—
174#6103—41no—
175#6103—41no—
176#6103—41no—
177C4⋊Dic13—22no—
178C4⋊Dic13—41no—
179#6101—22no—
180#6101—41no—
181C4×Dic13—21no—
182C4×Dic13—21no—
183C4×Dic13—21no—
184C4×Dic13—21no—
185C4×Dic13—21no—
186C4×Dic13—21no—
187C4×Dic13—21no—
188C4×Dic13—21no—
189C4×Dic13—21no—
190C4×Dic13—21no—
191C4×Dic13—21no—
192C4×Dic13—21no—
193C4×Dic13—41no—
194C4×Dic13—41no—
195C4×Dic13M1614yescomplemented
196C4×#126—131no—
197#1940—131no—
198#1940—131no—
199#1940—131no—
200#1924—131no—
201#1924—131no—
202#1924—131no—
203#1917—131no—
204#1917—131no—
205#1917—131no—
206#1851—131no—
207#1851—131no—
208#1851—131no—
209#1842—131no—
210#1839—131no—
211C23×C13C23×C411yescharacteristic
212C22×D13—21no—
213C22×D13—21no—
214C22×D13—21no—
215C22×D13—21no—
216C22×D13—21no—
217C22×D13—21no—
218C22×D13—21no—
219C2×C4×C13—21no—
220C2×C4×C13—21no—
221C2×C4×C13—21no—
222C2×C4×C13—21no—
223C2×C4×C13—21no—
224C2×C4×C13—21no—
225C2×C4×C13—21no—
226C2×C4×C13—21no—
227C2×C4×C13—21no—
228C2×C4×C13—21no—
229C2×C4×C13—21no—
230C2×C4×C13—21no—
231C2×C4×C13—21no—
232C2×C4×C13—21no—
233C2×C4×C13—21no—
234C2×C4×C13—21no—
235C2×C4×C13—21no—
236C2×C4×C13—21no—
237C2×C4×C13—21no—
238C2×C4×C13—21no—
239C2×C4×C13C2×M1612yes—
240C2×Dic13—21no—
241C2×Dic13—21no—
242C2×Dic13—21no—
243C2×Dic13—21no—
244C2×Dic13—21no—
245C2×Dic13—21no—
246C2×Dic13—21no—
247C2×Dic13—21no—
248C2×Dic13—21no—
249C2×Dic13—21no—
250C2×Dic13—21no—
251C2×Dic13—21no—
252C2×Dic13—21no—
253C2×Dic13—21no—
254C2×Dic13—21no—
255C2×Dic13—21no—
256C2×Dic13C2×M1612yes—
257C2×Dic13C2×M1612yes—
258C2×Dic13H(5,𝔽2)11yescharacteristic
259C2×Dic13H(5,𝔽2)11yescharacteristic
260C2×Dic13H(5,𝔽2)11yescharacteristic
261C2×Dic132−1+411yescharacteristic
262C2×Dic132−1+411yescharacteristic
263C2×Dic132−1+411yescharacteristic
264C4×D13—41no—
265C4×D13—41no—
266C4×D13—41no—
267C4×D13—41no—
268#1057—41no—
269#1057—41no—
270#1057—41no—
271#1057—41no—
272#1057—41no—
273#1057—41no—
274#1057—41no—
275#1057—41no—
276C22×M16—131no—
277C2×#126—131no—
278C2×C4⋊C8—131no—
279C2×C4⋊C8—131no—
280C2×C4⋊C8—131no—
281C2×C4⋊C8—131no—
282C2×C4⋊C8—131no—
283C2×C4⋊C8—131no—
284C2×C4⋊C8—131no—
285C2×C4⋊C8—131no—
286C2×C4⋊C8—131no—
287C2×C22⋊C8—131no—
288C2×C22⋊C8—131no—
289C2×C22⋊C8—131no—
290C2×#97—131no—
291C2×#97—131no—
292C2×#97—131no—
293#388—131no—
294#388—131no—
295#388—131no—
296#387—131no—
297#387—131no—
298#387#26111yescharacteristic
299#383—131no—
300C4×C4⋊C4—131no—
301C4×C4⋊C4—131no—
302C4×C4⋊C4—131no—
303C4×C22⋊C4—131no—
304C4×C22⋊C4—131no—
305C4×C22⋊C4—131no—
306C43—131no—
307#336—131no—
308#336—131no—
309#336—131no—
310#336—131no—
311#336—131no—
312#336—131no—
313#336—131no—
314#336—131no—
315#336—131no—
316#336—131no—
317#336—131no—
318#336—131no—
319C22×C13#52011yescharacteristic
320C22×C13#52011yescharacteristic
321C22×C13#52011yescharacteristic
322C22×C13C22×M1611yescharacteristic
323C22×C13#56811yescharacteristic
324C22×C13#56811yescharacteristic
325C22×C13#56811yescharacteristic
326C2×D13—41no—
327C2×D13—41no—
328C2×D13—41no—
329C2×D13—41no—
330C2×D13—41no—
331C2×D13—41no—
332C2×D13—41no—
333C4×C13—22no—
334C4×C13—41no—
335C4×C13—41no—
336C4×C13—41no—
337C4×C13—41no—
338C4×C13—41no—
339C4×C13—41no—
340Dic13—22no—
341Dic13—22no—
342Dic13—41no—
343Dic13—41no—
344Dic13—41no—
345Dic13—41no—
346Dic13#56011yescharacteristic
347Dic13#56311yescharacteristic
348Dic13#56311yescharacteristic
349Dic13#56311yescharacteristic
350C23×C4—131no—
351C2×M16—521no—
352C2×M16—521no—
353C2×M16—521no—
354C22×C8—131no—
355C22×C8—131no—
356C22×C8—131no—
357C22×C8—131no—
358C22×C8—131no—
359C22×C8—131no—
360C22×C8—131no—
361C22×C8—131no—
362#126—264no—
363C2×C4⋊C4—131no—
364C2×C4⋊C4—131no—
365C2×C4⋊C4—131no—
366C2×C4⋊C4—131no—
367C2×C4⋊C4C2×#26111yescharacteristic
368C2×C4⋊C4C2×#26111yescharacteristic
369C2×C22⋊C4—131no—
370C2×C22⋊C4—131no—
371C2×C22⋊C4—131no—
372C2×C22⋊C4—131no—
373C2×C22⋊C4—131no—
374C2×C22⋊C4—131no—
375C2×C42—131no—
376C2×C42—131no—
377C2×C42—131no—
378C2×C42—131no—
379C2×C42—131no—
380C2×C42—131no—
381C2×C42—131no—
382C2×C42—131no—
383C2×C42—131no—
384C2×C42C2×#26111yescharacteristic
385C4⋊C8—521no—
386C4⋊C8—521no—
387C4⋊C8—521no—
388C4⋊C8—521no—
389C4⋊C8—521no—
390C4⋊C8—521no—
391C4⋊C8—521no—
392C4⋊C8—521no—
393C4⋊C8—521no—
394C22⋊C8—521no—
395C22⋊C8—521no—
396C22⋊C8—521no—
397#97—521no—
398#97—521no—
399#97—521no—
400#95—131no—
401#95—131no—
402#95—131no—
403#95—131no—
404#95—131no—
405#95—131no—
406#95—131no—
407#95—131no—
408#95C2×#26111yescharacteristic
409#95C2×#26111yescharacteristic
410#95C2×#26111yescharacteristic
411#95C2×#26111yescharacteristic
412C2×C13#238311yescharacteristic
413C2×C13#298911yescharacteristic
414C2×C13#298911yescharacteristic
415C2×C13#298911yescharacteristic
416C2×C13#299511yescharacteristic
417C2×C13#299511yescharacteristic
418C2×C13#299511yescharacteristic
419D13—81no—
420C24—131no—
421C22×C4—131no—
422C22×C4—131no—
423C22×C4—131no—
424C22×C4—131no—
425C22×C4—131no—
426C22×C4—131no—
427C22×C4—131no—
428C22×C4—261no—
429C22×C4—261no—
430C22×C4—261no—
431C22×C4—261no—
432C22×C4—261no—
433C22×C4—261no—
434C22×C4C22×#26111yescharacteristic
435C22×C4C22×#26111yescharacteristic
436C22×C4C22×#26111yescharacteristic
437C22×C4C22×#26111yescharacteristic
438C22×C4C22×#26111yescharacteristic
439C22×C4C22×#26111yescharacteristic
440C22×C4C22×#26111yescharacteristic
441M16—524no—
442C2×C8—261no—
443C2×C8—261no—
444C2×C8—261no—
445C2×C8—261no—
446C2×C8—261no—
447C2×C8—261no—
448C2×C8—261no—
449C2×C8—261no—
450C2×C8—261no—
451C2×C8—261no—
452C2×C8—261no—
453C2×C8—261no—
454C2×C8—261no—
455C2×C8—261no—
456C2×C8—261no—
457C2×C8—261no—
458C2×C8—261no—
459C2×C8—261no—
460C2×C8—261no—
461C2×C8—261no—
462C2×C8—261no—
463C2×C8—261no—
464C2×C8—261no—
465C2×C8—261no—
466C4⋊C4—22no—
467C4⋊C4—41no—
468C4⋊C4—262no—
469C4⋊C4—262no—
470C4⋊C4—521no—
471C4⋊C4—521no—
472C22⋊C4—262no—
473C22⋊C4—262no—
474C22⋊C4—262no—
475C22⋊C4—521no—
476C22⋊C4—521no—
477C22⋊C4—521no—
478C42—41no—
479C42—134no—
480C42—261no—
481C42—261no—
482C42—261no—
483C42—261no—
484C42—261no—
485C42—261no—
486C42—261no—
487C42—261no—
488C42—261no—
489C42—261no—
490C42—261no—
491C42—261no—
492C42—521no—
493C42—521no—
494C13#2337811yescharacteristic, complemented, Sylow
495C23—261no—
496C23—261no—
497C23—261no—
498C23—261no—
499C23—261no—
500C23—261no—
501C23—261no—
502C23C23×#26111yescharacteristic
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—131no—
524C2×C4—131no—
525C2×C4—131no—
526C2×C4—131no—
527C2×C4—131no—
528C2×C4—131no—
529C2×C4—132no—
530C2×C4—132no—
531C2×C4—261no—
532C2×C4—261no—
533C2×C4—261no—
534C2×C4—261no—
535C2×C4—261no—
536C2×C4—261no—
537C2×C4—261no—
538C2×C4—261no—
539C2×C4—261no—
540C2×C4—261no—
541C2×C4—261no—
542C2×C4—261no—
543C2×C4—261no—
544C2×C4—261no—
545C2×C4—261no—
546C2×C4—261no—
547C2×C4—521no—
548C2×C4—521no—
549C2×C4—521no—
550C2×C4—521no—
551C2×C4D13⋊M1612yes—
552C8—521no—
553C8—521no—
554C8—521no—
555C8—521no—
556C8—521no—
557C8—521no—
558C8—521no—
559C8—521no—
560C22—521no—
561C22—521no—
562C22—521no—
563C22—521no—
564C22—521no—
565C22—521no—
566C22—521no—
567C22C2×D13⋊M1611yescharacteristic
568C22#1171962111yescharacteristic
569C22#1171962511yescharacteristic
570C22#1171962911yescharacteristic
571C22#1171963111yescharacteristic
572C22#1171963111yescharacteristic
573C22#1171963111yescharacteristic
574C4—22no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—131no—
582C4—131no—
583C4—131no—
584C4—131no—
585C4—262no—
586C4—262no—
587C4—521no—
588C4—521no—
589C4—521no—
590C4—521no—
591C2—1041no—
592C2#1313138311yescharacteristic
593C2#1313175611yescharacteristic
594C2#1313175811yescharacteristic
595C2#1313176011yescharacteristic
596C2#1313179011yescharacteristic
597C2#1313179311yescharacteristic
598C2#1313179411yescharacteristic
subgroupG/H
center, FrattiniC23C23×#261
commutator, socleC23×C13C23×C4
FittingC13×#387C4
prime radicalC23×D13C22×C4
cosocleC22×Dic13C24
hypercenter#387#261
hypocenterC13#23378
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
1. #13130938C2semidirect
2. #13130938C2semidirect
3. #13130937C2semidirect
7. #13130896C2semidirect
8. #13130895C2semidirect
9. #13130893C2semidirect
13. #13130769C2semidirect
14. #13130764C2semidirect
195. C4×Dic13M16semidirect
494. C13#23378semidirect

Maximal subgroups

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

subgroupindex
#131309382
#131309372
#131309122
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