Finite Group Databasebeta

← #15880718

Subgroups — #15880718

Order 3328.

744 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13117239C211yesmaximal, characteristic
2#13115553C211yesmaximal, characteristic, complemented
3#13115553C211yesmaximal, characteristic, complemented
4#13115403C211yesmaximal, characteristic, complemented
5#13115400C211yesmaximal, characteristic, complemented
6#13115396C211yesmaximal, characteristic, complemented
7#13115393C211yesmaximal, characteristic, complemented
8#13114820C211yesmaximal, characteristic
9#13114818C211yesmaximal, characteristic
10#13114814C211yesmaximal, characteristic
11#13114704C211yesmaximal, characteristic, complemented
12C8×#6101C211yesmaximal, characteristic, complemented
13#13114604C211yesmaximal, characteristic
14C22⋊C8×Dic13C211yesmaximal, characteristic
15C13×#1750C211yesmaximal, characteristic
16C2×C13×#117C2211yescharacteristic
17C2×#88180C2211yescharacteristic
18C22×C4×Dic13C2211yescharacteristic
19C2×C13×C22⋊C8C2211yescharacteristic
20C2×C13×C22⋊C8C2211yescharacteristic
21C2×C13×#97C2211yescharacteristic
22C2×C4×C8×C13C2211yescharacteristic
23C2×C22⋊#1055C2211yescharacteristic
24C2×C22⋊#1055C2211yescharacteristic
25C2×#88055C2211yescharacteristic
26C2×#88054C2211yescharacteristic
27C2×#88054C2211yescharacteristic
28C2×#88054C2211yescharacteristic
29C2×#88054C2211yescharacteristic
30C2×C8×Dic13C2211yescharacteristic
31C2×C8×Dic13C2211yescharacteristic
32C2×C8×Dic13C2211yescharacteristic
33#11718638C411yescharacteristic
34#11718638C411yescharacteristic
35#11718637C411yescharacteristic
36#11718637C411yescharacteristic
37#11718603C411yescharacteristic
38#11718603C411yescharacteristic
39#11718603C411yescharacteristic
40#11718603C411yescharacteristic
41C4×#6101C2211yescharacteristic
42C4×#6101C2211yescharacteristic
43C4×#6101C2211yescharacteristic
44C4×#6101C2211yescharacteristic
45C2×C4⋊#1055C2211yescharacteristic
46C2×C4⋊#1055C2211yescharacteristic
47C13×#336C2211yescharacteristic
48C13×#336C2211yescharacteristic
49#11718142C2211yescharacteristic
50#11718142C2211yescharacteristic
51#11718142C2211yescharacteristic
52#11718142C2211yescharacteristic
53#11718142C2211yescharacteristic
54#11718142C2211yescharacteristic
55#11718142C2211yescharacteristic
56#11718142C2211yescharacteristic
57#11718116C2211yescharacteristic
58#11718116C2211yescharacteristic
59C23×C4×C13C2311yescharacteristic
60C23×Dic13C2×C411yescharacteristic
61C23×Dic13C2×C411yescharacteristic
62C22×C8×C13C2311yescharacteristic
63C22×C8×C13C2311yescharacteristic
64C22×C8×C13C2311yescharacteristic
65C22×C8×C13C2311yescharacteristic
66C13×#117—41no—
67C13×#117—41no—
68C2×C13×C4⋊C4C2×C411yescharacteristic
69C2×C13×C4⋊C4C2×C411yescharacteristic
70C2×C13×C22⋊C4C2×C411yescharacteristic
71C2×C13×C22⋊C4C2×C411yescharacteristic
72C2×C42×C13C2311yescharacteristic
73C2×C42×C13C2311yescharacteristic
74C2×C22⋊Dic13C2×C411yescharacteristic
75C2×C22⋊Dic13C2×C411yescharacteristic
76#88180—41no—
77#88180—41no—
78C2×C4⋊Dic13C2×C411yescharacteristic
79C2×C4⋊Dic13C2×C411yescharacteristic
80C2×#6101C2×C411yescharacteristic
81C2×#6101C2×C411yescharacteristic
82C2×#6101C2×C411yescharacteristic
83C2×#6101C2×C411yescharacteristic
84C2×#6101C2×C411yescharacteristic
85C2×#6101C2×C411yescharacteristic
86C2×#6101C2×C411yescharacteristic
87C2×#6101C2×C411yescharacteristic
88C2×C4×Dic13—22no—
89C2×C4×Dic13—22no—
90C2×C4×Dic13C2×C411yescharacteristic
91C2×C4×Dic13C2×C411yescharacteristic
92C2×C4×Dic13C2311yescharacteristic
93C2×C4×Dic13C2311yescharacteristic
94C2×C4×Dic13C2311yescharacteristic
95C2×C4×Dic13C2311yescharacteristic
96C22×#1055C2311yescharacteristic
97C22×#1055C2311yescharacteristic
98C22×#1055C2311yescharacteristic
99C22×#1055C2311yescharacteristic
100C13×C22⋊C8—22no—
101C13×C22⋊C8Q814yes—
102C13×#97—41no—
103C4×C8×C13—41no—
104#88071C2×C411yescharacteristic
105#88071C2×C411yescharacteristic
106#88071C2×C411yescharacteristic
107#88071C2×C411yescharacteristic
108#88071C2×C411yescharacteristic
109#88071C2×C411yescharacteristic
110#88071C2×C411yescharacteristic
111#88071C2×C411yescharacteristic
112C22⋊#1055—22no—
113C22⋊#1055—22no—
114#88055—41no—
115#88054—22no—
116#88054—22no—
117#88054—41no—
118#88054—41no—
119C8×Dic13—22no—
120C8×Dic13—22no—
121C8×Dic13—41no—
122C4⋊#1055—41no—
123C4⋊#1055—41no—
124#22626—131nomaximal, Sylow
125C24×C13C22×C411yescharacteristic
126C22×C4×C13—21no—
127C22×C4×C13—21no—
128C22×C4×C13—21no—
129C22×C4×C13—21no—
130C22×C4×C13C22×C411yescharacteristic
131C22×C4×C13C22×C411yescharacteristic
132C22×C4×C13C22×C411yescharacteristic
133C22×C4×C13C22×C411yescharacteristic
134C22×C4×C13C22×C411yescharacteristic
135C22×C4×C13C2×Q812yes—
136C22×C4×C13#4112yes—
137C22×C4×C13C2411yescharacteristic
138C22×Dic13—22no—
139C22×Dic13—22no—
140C22×Dic13—22no—
141C22×Dic13—22no—
142C22×Dic13—22no—
143C22×Dic13—41no—
144C22×Dic13C22×C411yescharacteristic
145C22×Dic13C22×C411yescharacteristic
146C22×Dic13C22×C411yescharacteristic
147C22×Dic13C22×C411yescharacteristic
148C22×Dic13C22×C411yescharacteristic
149C22×Dic13C22×C411yescharacteristic
150C22×Dic13C22×C411yescharacteristic
151C22×Dic13C22×C411yescharacteristic
152C2×C8×C13—21no—
153C2×C8×C13—21no—
154C2×C8×C13—21no—
155C2×C8×C13—21no—
156C2×C8×C13—21no—
157C2×C8×C13—21no—
158C2×C8×C13—21no—
159C2×C8×C13—21no—
160C2×C8×C13—21no—
161C2×C8×C13—21no—
162C2×C8×C13C2×Q812yes—
163C2×C8×C13C2×Q812yes—
164C13×C4⋊C4—41no—
165C13×C4⋊C4—41no—
166C13×C22⋊C4—41no—
167C13×C22⋊C4—41no—
168C42×C13—21no—
169C42×C13—21no—
170C42×C13—21no—
171C42×C13—21no—
172C22⋊Dic13—41no—
173C22⋊Dic13—41no—
174C4⋊Dic13—41no—
175C4⋊Dic13—41no—
176#6101—22no—
177#6101—22no—
178#6101—22no—
179#6101—22no—
180#6101—22no—
181#6101—22no—
182#6101—41no—
183#6101—41no—
184C4×Dic13—21no—
185C4×Dic13—21no—
186C4×Dic13—21no—
187C4×Dic13—21no—
188C4×Dic13—21no—
189C4×Dic13—21no—
190C4×Dic13—22no—
191C4×Dic13—22no—
192C4×Dic13#4112yes—
193C4×Dic13#4112yes—
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—21no—
204C2×#1055—21no—
205C2×#1055—21no—
206#2285—131no—
207#1939—131no—
208#1939—131no—
209#1917—131no—
210#1917—131no—
211#1916—131no—
212#1916—131no—
213#1798—131no—
214#1798—131no—
215#1797—131no—
216#1770—131no—
217C8×C4⋊C4—131no—
218#1750—131no—
219#1750D1311yescharacteristic
220C4×C22⋊C8—131no—
221C23×C13—21no—
222C23×C13—21no—
223C23×C13—21no—
224C23×C13—21no—
225C23×C13—21no—
226C23×C13—21no—
227C23×C13C4×Q812yes—
228C23×C13C23×C411yescharacteristic
229C2×C4×C13—21no—
230C2×C4×C13—21no—
231C2×C4×C13—22no—
232C2×C4×C13—22no—
233C2×C4×C13—22no—
234C2×C4×C13—22no—
235C2×C4×C13—21no—
236C2×C4×C13—21no—
237C2×C4×C13—21no—
238C2×C4×C13—21no—
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×C13C4×Q812yes—
248C2×C4×C13C22×Q811yescharacteristic
249C2×C4×C13C2×#4111yescharacteristic
250C2×C4×C13C2×#4111yescharacteristic
251C2×C4×C13C2×#4111yescharacteristic
252C2×C4×C13H(5,𝔽2)11yescharacteristic
253C2×C4×C13H(5,𝔽2)11yescharacteristic
254C2×Dic13—21no—
255C2×Dic13—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—44no—
274C2×Dic13—44no—
275C2×Dic13#11712yes—
276C2×Dic13#11712yes—
277C2×Dic13#13112yes—
278C2×Dic13#13112yes—
279C2×Dic13#13112yes—
280C8×C13—22no—
281C8×C13—22no—
282C8×C13—41no—
283C8×C13—41no—
284#1055—41no—
285#1055—41no—
286#1055—41no—
287#1055—41no—
288C2×#117—131no—
289C2×#117C2×D1311yescharacteristic
290C22×C42—131no—
291C2×C4⋊C8—131no—
292C2×C4⋊C8—131no—
293C2×C4⋊C8—131no—
294C2×C4⋊C8—131no—
295C2×C4⋊C8—131no—
296C2×C4⋊C8—131no—
297C2×C22⋊C8—131no—
298C2×C22⋊C8—131no—
299C2×C22⋊C8C2×D1311yescharacteristic
300C2×C22⋊C8C2×D1311yescharacteristic
301C2×#97—131no—
302C2×#97C2×D1311yescharacteristic
303C2×C4×C8—131no—
304C2×C4×C8—131no—
305C2×C4×C8—131no—
306C2×C4×C8C2×D1311yescharacteristic
307#384—131no—
308#384—131no—
309#382—131no—
310#382—131no—
311#380—131no—
312#380—131no—
313#380—131no—
314#380—131no—
315C4×C4⋊C4—131no—
316C4×C4⋊C4—131no—
317C4×C4⋊C4—131no—
318C4×C4⋊C4—131no—
319#336—131no—
320#336—131no—
321#336—131no—
322#336—131no—
323#336—131no—
324#336—131no—
325#336—131no—
326#336—131no—
327#336—131no—
328#336—131no—
329#336C2×D1311yescharacteristic
330#336C2×D1311yescharacteristic
331C22×C13—22no—
332C22×C13—22no—
333C22×C13—22no—
334C22×C13—41no—
335C22×C13—41no—
336C22×C13—41no—
337C22×C13—41no—
338C22×C13C2×#11711yescharacteristic
339C22×C13C2×C4×Q811yescharacteristic
340C22×C13#51811yescharacteristic
341C22×C13C2×#13111yescharacteristic
342C22×C13C2×#13111yescharacteristic
343C22×C13C2×#13111yescharacteristic
344C22×C13#56811yescharacteristic
345C4×C13—22no—
346C4×C13—41no—
347C4×C13—41no—
348C4×C13—41no—
349C4×C13—41no—
350C4×C13#55311yescharacteristic
351C4×C13#55311yescharacteristic
352C4×C13#55411yescharacteristic
353C4×C13#55411yescharacteristic
354Dic13—22no—
355Dic13—22no—
356Dic13—41no—
357Dic13—41no—
358Dic13—41no—
359Dic13—41no—
360Dic13—41no—
361Dic13#43314yes—
362C23×C4—131no—
363C23×C4—131no—
364C23×C4C22×D1311yescharacteristic
365C22×C8—131no—
366C22×C8—131no—
367C22×C8—131no—
368C22×C8—131no—
369C22×C8C22×D1311yescharacteristic
370C22×C8C22×D1311yescharacteristic
371C22×C8C22×D1311yescharacteristic
372C22×C8C22×D1311yescharacteristic
373#117—41no—
374#117—41no—
375#117—521no—
376#117—521no—
377C2×C4⋊C4—131no—
378C2×C4⋊C4—131no—
379C2×C4⋊C4—131no—
380C2×C4⋊C4—131no—
381C2×C4⋊C4—131no—
382C2×C4⋊C4—131no—
383C2×C4⋊C4—131no—
384C2×C4⋊C4—131no—
385C2×C4⋊C4—131no—
386C2×C4⋊C4—131no—
387C2×C4⋊C4C4×D1311yescharacteristic
388C2×C4⋊C4C4×D1311yescharacteristic
389C2×C22⋊C4—131no—
390C2×C22⋊C4—131no—
391C2×C22⋊C4C4×D1311yescharacteristic
392C2×C22⋊C4C4×D1311yescharacteristic
393C2×C42—131no—
394C2×C42—131no—
395C2×C42—131no—
396C2×C42—131no—
397C2×C42—131no—
398C2×C42—131no—
399C2×C42—262no—
400C2×C42—262no—
401C2×C42C22×D1311yescharacteristic
402C2×C42C22×D1311yescharacteristic
403C4⋊C8—262no—
404C4⋊C8—262no—
405C4⋊C8—521no—
406C4⋊C8—521no—
407C4⋊C8—521no—
408C4⋊C8—521no—
409C22⋊C8—22no—
410C22⋊C8—262no—
411C22⋊C8—262no—
412C22⋊C8Dic2614yes—
413#97—41no—
414#97—521no—
415C4×C8—41no—
416C4×C8—262no—
417C4×C8—262no—
418C4×C8—521no—
419#95—131no—
420#95—131no—
421#95—131no—
422#95—131no—
423#95—131no—
424#95—131no—
425#95—131no—
426#95—131no—
427C2×C13—42no—
428C2×C13#234511yescharacteristic
429C2×C13C2×#43311yescharacteristic
430C2×C13#292411yescharacteristic
431C2×C13#296011yescharacteristic
432C2×C13#296211yescharacteristic
433C2×C13#297911yescharacteristic
434C2×C13#298111yescharacteristic
435C24C2×C4×D1311yescharacteristic
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—131no—
441C22×C4—131no—
442C22×C4—131no—
443C22×C4—131no—
444C22×C4—131no—
445C22×C4—131no—
446C22×C4—131no—
447C22×C4—131no—
448C22×C4—262no—
449C22×C4—262no—
450C22×C4—262no—
451C22×C4—262no—
452C22×C4—262no—
453C22×C4—521no—
454C22×C4C2×Dic2612yes—
455C22×C4C2×C4×D1311yescharacteristic
456C22×C4C2×C4×D1311yescharacteristic
457C22×C4C2×C4×D1311yescharacteristic
458C22×C4C2×C4×D1311yescharacteristic
459C22×C4C2×C4×D1311yescharacteristic
460C22×C4#612712yes—
461C22×C4C23×D1311yescharacteristic
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—261no—
473C2×C8—261no—
474C2×C8—261no—
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×C8C2×Dic2612yes—
485C2×C8C2×Dic2612yes—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—262no—
489C4⋊C4—262no—
490C4⋊C4—262no—
491C4⋊C4—262no—
492C4⋊C4—262no—
493C4⋊C4—262no—
494C4⋊C4—521no—
495C4⋊C4—521no—
496C4⋊C4—521no—
497C4⋊C4—521no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C22⋊C4—521no—
501C22⋊C4—521no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—132no—
507C42—132no—
508C42—262no—
509C42—262no—
510C42—261no—
511C42—261no—
512C42—261no—
513C42—261no—
514C42—261no—
515C42—261no—
516C13#2262611yescharacteristic, complemented, Sylow
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23C4×Dic2612yes—
524C23C22×C4×D1311yescharacteristic
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—132no—
544C2×C4—132no—
545C2×C4—132no—
546C2×C4—132no—
547C2×C4—132no—
548C2×C4—261no—
549C2×C4—261no—
550C2×C4—261no—
551C2×C4—261no—
552C2×C4—261no—
553C2×C4—261no—
554C2×C4—261no—
555C2×C4—261no—
556C2×C4—261no—
557C2×C4—261no—
558C2×C4—261no—
559C2×C4—261no—
560C2×C4—261no—
561C2×C4—261no—
562C2×C4—261no—
563C2×C4—261no—
564C2×C4—261no—
565C2×C4—261no—
566C2×C4—261no—
567C2×C4—524no—
568C2×C4—524no—
569C2×C4C4×Dic2612yes—
570C2×C4C22×Dic2611yescharacteristic
571C2×C4C2×#612711yescharacteristic
572C2×C4#8825111yescharacteristic
573C2×C4#8825111yescharacteristic
574C2×C4#41×D1311yescharacteristic
575C2×C4#41×D1311yescharacteristic
576C8—22no—
577C8—22no—
578C8—41no—
579C8—41no—
580C8—521no—
581C8—521no—
582C8—521no—
583C8—521no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22C2×C4×Dic2611yescharacteristic
592C22#1171919111yescharacteristic
593C22D13×#11711yescharacteristic
594C22D13×#13111yescharacteristic
595C22D13×#13111yescharacteristic
596C22D13×#13111yescharacteristic
597C22#1171945511yescharacteristic
598C4—22no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—134no—
604C4—262no—
605C4—262no—
606C4—521no—
607C4—521no—
608C4—521no—
609C4—521no—
610C4—521no—
611C4#1171924411yescharacteristic
612C4#1171924411yescharacteristic
613C4#1171924511yescharacteristic
614C4#1171924511yescharacteristic
615C2—42no—
616C2#1311771411yescharacteristic
617C2D13×#43311yescharacteristic
618C2#1312349211yescharacteristic
619C2#1312376811yescharacteristic
620C2#1312378011yescharacteristic
621C2#1312390011yescharacteristic
622C2#1312390911yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D13
commutator, socleC23×C13C23×C4
FittingC13×#1750C2
prime radicalC24×C13C22×C4
cosocleC22×C4×C13C24
hypercenter#1750D13
hypocenterC13#22626
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C22×C4 ◁ #1750

Complements

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

NH ≅ G/Ntype
2. #13115553C2semidirect
3. #13115553C2semidirect
4. #13115403C2semidirect
5. #13115400C2semidirect
6. #13115396C2semidirect
7. #13115393C2semidirect
11. #13114704C2semidirect
12. C8×#6101C2semidirect
516. C13#22626semidirect

Maximal subgroups

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

subgroupindex
#131172392
#131155532
#131154032
#131154002
#131153962
#131153932
#131148202
#131148182
#131148142
#131147042
C8×#61012
#131146042
C22⋊C8×Dic132
C13×#17502
#2262613