Finite Group Databasebeta

← #17899014

Subgroups — #17899014

Order 4352.

1950 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14542382C211yesmaximal, characteristic
2#14537804C211yesmaximal, characteristic, complemented
3#14537803C211yesmaximal, characteristic, complemented
4#14537785C213yesmaximal, complemented
5#14537782C213yesmaximal, complemented
6#14537759C213yesmaximal, complemented
7#14537756C213yesmaximal, complemented
8C8×#63256C211yesmaximal, characteristic, complemented
9Q8×#63218C211yesmaximal, characteristic, complemented
10C8×Q8×D17C211yesmaximal, characteristic, complemented
11#14537523C213yesmaximal, complemented
12#14537491C213yesmaximal, complemented
13#14537445C213yesmaximal, complemented
14C4×#10587587C213yesmaximal, complemented
15C8×#41×C17C211yesmaximal, characteristic, complemented
16C2×#10587587C2213yes—
17C2×#10587587C2213yes—
18#11763542C413yes—
19#11763535C411yescharacteristic
20#11763527C413yes—
21#11763509C413yes—
22#11763503C413yes—
23#11763478C411yescharacteristic
24#11763471C411yescharacteristic
25#11763466C411yescharacteristic
26#11763356C2213yes—
27#11763343C2213yes—
28#11763341C2213yes—
29C4×#63256C2211yescharacteristic
30C4×#63252C2213yes—
31C4×Q8×D17C2211yescharacteristic
32#11762939C2213yes—
33#11762936C2213yes—
34Q8×#3622C2211yescharacteristic
35Q8×#3622C2211yescharacteristic
36C8×Dic34C2213yes—
37#11762933C2213yes—
38#11762888C2213yes—
39#11762887C2213yes—
40#11762887C2213yes—
41#11762886C2213yes—
42#11762884C2216yes—
43#11762883C2216yes—
44#11762880C2216yes—
45C8×D68C2213yes—
46#11762874C2213yes—
47C8×C22⋊D17C2216yes—
48#11762868C2216yes—
49#11762867C2216yes—
50#11762866C2213yes—
51#11762865C2213yes—
52#11762861C2213yes—
53#11762831C2213yes—
54C4⋊C8×D17C2213yes—
55#11762774C2216yes—
56#11762764C2213yes—
57#11762764C2213yes—
58#11762763C2213yes—
59#11762763C2213yes—
60#11762763C2213yes—
61#11762763C2211yescharacteristic
62C4×#63223C2213yes—
63C4×#63218C2213yes—
64C4×#63218C2213yes—
65C4×C8×D17C2213yes—
66C4×C8×D17C2213yes—
67C4×#41×C17C2211yescharacteristic
68C8×Q8×C17C2211yescharacteristic
69C8×Q8×C17C2211yescharacteristic
70C8×D4×C17C2213yes—
71C8×D4×C17C2213yes—
72C17×#431C2213yes—
73C2×C4×C8×C17C2213yes—
74C2×#41×C17C2×C411yescharacteristic
75C2×#63256C2×C411yescharacteristic
76C2×Q8×D17C2×C411yescharacteristic
77C2×#63252C2313yes—
78C22×C8×C17C2313yes—
79C22×C8×C17C2313yes—
80C4×Q8×C17C2×C411yescharacteristic
81C4×Q8×C17C2311yescharacteristic
82C4×D4×C17C2×C413yes—
83C4×D4×C17C2313yes—
84C17×#117C2×C413yes—
85C2×C42×C17C2313yes—
86Q8×Dic17C2×C411yescharacteristic
87Q8×Dic17C2×C411yescharacteristic
88#10587626C2×C413yes—
89#10587624C2×C413yes—
90#10587624C2×C413yes—
91#10587622C2×C413yes—
92#10587616C2×C413yes—
93#10587615C2×C416yes—
94C4×C22⋊D17C2×C413yes—
95C4×C22⋊D17C2×C413yes—
96C4×C22⋊D17C2×C413yes—
97C4×C22⋊D17C2×C413yes—
98C4×C22⋊D17C2316yes—
99#10587612C2313yes—
100#10587610C2×C413yes—
101C2×#63223C2313yes—
102C2×#63223C2313yes—
103#10587587—212no—
104#10587587—212no—
105C2×#63218C2313yes—
106C2×#63218C2313yes—
107C2×#63218C2313yes—
108C2×#63218C2311yescharacteristic
109C2×C8×D17C2313yes—
110C2×C8×D17C2313yes—
111C2×C8×D17C2313yes—
112C2×C8×D17C2311yescharacteristic
113#10587583C2×C413yes—
114#10587581C2×C413yes—
115#10587579C2×C413yes—
116#10587578C2×C413yes—
117C4⋊C4×D17C2×C413yes—
118#10587576C2×C413yes—
119#10587574C2×C413yes—
120#10587573C2×C413yes—
121#10587571C2×C416yes—
122#10587570C2×C416yes—
123#10587569C2×C416yes—
124#10587565C2×C416yes—
125#10587562C2×C416yes—
126#10587561C2×C413yes—
127C4×D68C2×C413yes—
128C4×D68C2×C413yes—
129C4×D68C2313yes—
130#10587558C2311yescharacteristic
131#10587558C2313yes—
132C42×D17C2313yes—
133C42×D17C2313yes—
134#10587556C2×C413yes—
135C4×Dic34C2×C413yes—
136C4×Dic34C2×C413yes—
137C4×Dic34C2313yes—
138C17×C4⋊C8C2313yes—
139C17×C4⋊C8C2313yes—
140C17×C22⋊C8C2316yes—
141C4×C8×C17C2313yes—
142C4×C8×C17C2313yes—
143C4×C8×C17C2311yescharacteristic
144C4×C8×C17C2313yes—
145C22⋊#3622C2316yes—
146#10587491C2316yes—
147#10587491C2316yes—
148#10587486C2311yescharacteristic
149#10587486C2313yes—
150#10587486C2313yes—
151#10587486C2313yes—
152#10587485C2316yes—
153#10587485C2316yes—
154C8×Dic17C2311yescharacteristic
155C8×Dic17C2313yes—
156C8×Dic17C2313yes—
157C8×Dic17C2313yes—
158C4⋊#3622C2313yes—
159C4⋊#3622C2313yes—
160#10587474C2311yescharacteristic
161#10587474C2313yes—
162C4×#3622C2313yes—
163C4×#3622C2313yes—
164#41×C17—24no—
165C2×Q8×C17C22×C411yescharacteristic
166C2×D4×C17C22×C413yes—
167C22×C4×C17C22×C413yes—
168C22×C4×C17C22×C413yes—
169C22×C4×C17C2413yes—
170C2×C22⋊D17C22×C413yes—
171C2×C22⋊D17C22×C413yes—
172C2×C22⋊D17C22×C416yes—
173#63256—24no—
174Q8×D17—24no—
175#63252—26no—
176#63252—26no—
177C2×D68C22×C413yes—
178C2×D68C22×C413yes—
179C2×C4×D17C22×C413yes—
180C2×C4×D17C22×C413yes—
181C2×C4×D17C22×C413yes—
182C2×C4×D17C22×C413yes—
183C2×C4×D17C2413yes—
184C2×C4×D17C2411yescharacteristic
185C2×Dic34C22×C413yes—
186C2×Dic34C22×C413yes—
187C2×C8×C17—26no—
188C2×C8×C17—26no—
189C2×C8×C17C2413yes—
190C2×C8×C17C2413yes—
191C2×C8×C17C2411yescharacteristic
192C2×C8×C17C2411yescharacteristic
193C4⋊C4×C17C22×C413yes—
194C4⋊C4×C17C22×C413yes—
195C22⋊C4×C17C22×C416yes—
196C42×C17C22×C413yes—
197C42×C17C22×C413yes—
198C42×C17C2413yes—
199C42×C17C2411yescharacteristic
200C22⋊Dic17C22×C413yes—
201C22⋊Dic17C22×C416yes—
202C22⋊Dic17C22×C413yes—
203#63227C22×C411yescharacteristic
204#63227C22×C411yescharacteristic
205#63227C22×C413yes—
206#63227C22×C413yes—
207#63227C22×C416yes—
208#63227C22×C416yes—
209C4⋊Dic17C22×C413yes—
210C4⋊Dic17C22×C413yes—
211C4⋊Dic17C22×C413yes—
212C4⋊Dic17C22×C413yes—
213#63225C22×C411yescharacteristic
214#63225C22×C411yescharacteristic
215#63225C22×C413yes—
216#63225C22×C413yes—
217#63225C22×C416yes—
218#63225C22×C416yes—
219C4×Dic17C22×C413yes—
220C4×Dic17C22×C413yes—
221C4×Dic17C22×C413yes—
222C4×Dic17C22×C413yes—
223C4×Dic17C2411yescharacteristic
224C4×Dic17C2413yes—
225#63223—26no—
226#63223—26no—
227C2×#3622C2413yes—
228C2×#3622C2413yes—
229C2×#3622C2411yescharacteristic
230C2×#3622C2411yescharacteristic
231#63218—26no—
232#63218—26no—
233#63218—26no—
234#63218#4114yescomplemented
235C8×D17—26no—
236C8×D17—26no—
237C8×D17—26no—
238C8×D17#4114yescomplemented
239#60218—171nomaximal, Sylow
240C23×C17C23×C413yes—
241C22×D17C23×C413yes—
242C22×D17C23×C411yescharacteristic
243Q8×C17#13114yescomplemented
244D4×C17—26no—
245C2×C4×C17—26no—
246C2×C4×C17—23no—
247C2×C4×C17—23no—
248C2×C4×C17—43no—
249C2×C4×C17C23×C411yescharacteristic
250C2×C4×C17C23×C411yescharacteristic
251C2×C4×C17C23×C413yes—
252C2×C4×C17C23×C413yes—
253C2×C4×C17C23×C413yes—
254C2×C4×C17C2511yescharacteristic
255C22⋊D17—26no—
256C22⋊D17—26no—
257C22⋊D17—46no—
258C2×Dic17C23×C413yes—
259C2×Dic17C23×C411yescharacteristic
260C2×Dic17C23×C411yescharacteristic
261C2×Dic17C23×C413yes—
262C2×Dic17C23×C413yes—
263C2×Dic17C23×C411yescharacteristic
264D68—26no—
265D68—26no—
266C4×D17—26no—
267C4×D17—26no—
268C4×D17—23no—
269C4×D17—23no—
270C4×D17—43no—
271C4×D17—43no—
272C4×D17C2×#4112yes—
273C4×D17C2×#4112yes—
274Dic34—26no—
275Dic34—26no—
276C8×C17—23no—
277C8×C17—23no—
278C8×C17C2×#4112yes—
279C8×C17C2×#4112yes—
280#3622—23no—
281#3622—23no—
282#3622C2×#4112yes—
283#3622C2×#4112yes—
284#3480—171no—
285#2971—171no—
286#2971—171no—
287#2969—173no—
288#2969—173no—
289#2966—173no—
290#2966—173no—
291C8×#41—171no—
292C8×#41D1711yescharacteristic, complemented
293Q8×M16—171no—
294C2×C8×Q8—171no—
295#2935—173no—
296#2931—173no—
297#2925—173no—
298C4×#131—173no—
299C22×C17—23no—
300C22×C17—23no—
301C22×C17—23no—
302C22×C17C24×C411yescharacteristic
303C2×D17—21no—
304C2×D17—21no—
305C2×D17—23no—
306C2×D17—23no—
307C2×D17—23no—
308C2×D17C4×#4112yes—
309C4×C17—23no—
310C4×C17—21no—
311C4×C17—23no—
312C4×C17—21no—
313C4×C17C2×#13116yes—
314C4×C17C22×#4111yescharacteristic
315C4×C17C22×#4111yescharacteristic
316Dic17—23no—
317Dic17—23no—
318Dic17—21no—
319Dic17—21no—
320Dic17—23no—
321Dic17C4×#4112yes—
322C2×#131—173no—
323C2×#131—173no—
324#559—173no—
325#557—171no—
326#556—173no—
327#553—173no—
328#553—173no—
329#550—171no—
330#548—171no—
331#548—171no—
332#532—173no—
333#529—173no—
334#529—173no—
335C4×#41—171no—
336C4×#41—173no—
337C4×#41C2×D1711yescharacteristic
338C2×C4×Q8—171no—
339#446—173no—
340#446—173no—
341C8×Q8—171no—
342C8×Q8—171no—
343C8×Q8—173no—
344C8×Q8—173no—
345C8×Q8C2×D1711yescharacteristic
346C8×Q8C2×D1711yescharacteristic
347#436—173no—
348#436—173no—
349#436—173no—
350#436—176no—
351#436—173no—
352#435—176no—
353#435—176no—
354C8×D4—176no—
355C8×D4—173no—
356C8×D4—173no—
357C8×D4C2×D1713yes—
358C8×D4C2×D1713yes—
359#433—176no—
360#433—176no—
361#432—173no—
362#432—173no—
363#431—173no—
364#431C2×D1713yes—
365#423—173no—
366C2×C4⋊C8—173no—
367#408—176no—
368#405—173no—
369#405—173no—
370#405—171no—
371#405—173no—
372#405—173no—
373#405—173no—
374C4×M16—173no—
375C4×M16—173no—
376C4×M16—173no—
377C2×C4×C8—173no—
378C2×C4×C8—173no—
379C2×C4×C8C2×D1713yes—
380C2×C17—43no—
381C2×C17C2×C4×#4111yescharacteristic
382C2×C17C22×#13111yescharacteristic
383C2×C17#357311yescharacteristic
384D17—22no—
385D17—43no—
386C2×#41—171no—
387C2×#41—173no—
388C2×#41C4×D1711yescharacteristic
389C22×Q8—171no—
390#131—3412no—
391#131—3412no—
392C2×M16—173no—
393C2×M16—173no—
394C2×M16—171no—
395C2×M16—173no—
396C2×M16—173no—
397C2×M16—173no—
398C22×C8—171no—
399C22×C8—173no—
400C22×C8—173no—
401C22×C8—173no—
402C22×C8C22×D1713yes—
403C22×C8C22×D1713yes—
404C4⋊Q8—173no—
405#127—173no—
406#126—176no—
407#126—176no—
408#125—173no—
409#125—173no—
410#124—173no—
411#124—176no—
412#124—173no—
413#123—176no—
414#123—176no—
415C22⋊Q8—173no—
416C22⋊Q8—173no—
417#121—173no—
418#121—173no—
419#121—173no—
420#121—173no—
421#121—176no—
422C4×Q8—171no—
423C4×Q8—171no—
424C4×Q8—173no—
425C4×Q8—173no—
426C4×Q8—173no—
427C4×Q8—173no—
428C4×Q8C4×D1711yescharacteristic
429C4×Q8C22×D1711yescharacteristic
430C4×D4—173no—
431C4×D4—173no—
432C4×D4—173no—
433C4×D4—173no—
434C4×D4—176no—
435C4×D4—173no—
436C4×D4—173no—
437C4×D4—173no—
438C4×D4—173no—
439C4×D4C4×D1713yes—
440C4×D4C22×D1713yes—
441#117—173no—
442#117—173no—
443#117—171no—
444#117—173no—
445#117C4×D1713yes—
446C2×C4⋊C4—173no—
447C2×C42—173no—
448C2×C42—173no—
449C2×C42C22×D1713yes—
450C4⋊C8—176no—
451C4⋊C8—176no—
452C4⋊C8—173no—
453C4⋊C8—173no—
454C4⋊C8C22×D1713yes—
455C4⋊C8C22×D1713yes—
456C22⋊C8—176no—
457C22⋊C8—176no—
458C22⋊C8—176no—
459C22⋊C8C22×D1716yes—
460#97—171no—
461#97—173no—
462#97—173no—
463#97—173no—
464#97—171no—
465#97—173no—
466C4×C8—173no—
467C4×C8—173no—
468C4×C8—171no—
469C4×C8—173no—
470C4×C8—173no—
471C4×C8—173no—
472C4×C8C22×D1713yes—
473C4×C8C22×D1713yes—
474C4×C8C22×D1713yes—
475C4×C8C22×D1711yescharacteristic
476C17#6021811yescharacteristic, complemented, Sylow
477#41—24no—
478#41—346no—
479#41—346no—
480#41—344no—
481C2×Q8—173no—
482C2×Q8—173no—
483C2×Q8—344no—
484C2×Q8C2×C4×D1711yescharacteristic
485C2×D4—173no—
486C2×D4—173no—
487C2×D4—176no—
488C2×D4—173no—
489C2×D4—173no—
490C2×D4C2×C4×D1713yes—
491C22×C4—171no—
492C22×C4—173no—
493C22×C4—173no—
494C22×C4—173no—
495C22×C4—173no—
496C22×C4—173no—
497C22×C4C2×C4×D1713yes—
498C22×C4C2×C4×D1713yes—
499C22×C4C23×D1713yes—
500M16—174no—
501M16—346no—
502M16—346no—
503M16—346no—
504M16—346no—
505M16—346no—
506C2×C8—26no—
507C2×C8—26no—
508C2×C8—171no—
509C2×C8—171no—
510C2×C8—174no—
511C2×C8—173no—
512C2×C8—173no—
513C2×C8—346no—
514C2×C8—346no—
515C2×C8—346no—
516C2×C8C23×D1713yes—
517C2×C8C23×D1713yes—
518C2×C8C23×D1711yescharacteristic
519C2×C8C23×D1711yescharacteristic
520C4⋊C4—173no—
521C4⋊C4—176no—
522C4⋊C4—171no—
523C4⋊C4—171no—
524C4⋊C4—173no—
525C4⋊C4—173no—
526C4⋊C4—173no—
527C4⋊C4—173no—
528C4⋊C4—173no—
529C4⋊C4—176no—
530C4⋊C4C2×C4×D1713yes—
531C4⋊C4C2×C4×D1713yes—
532C22⋊C4—173no—
533C22⋊C4—173no—
534C22⋊C4—176no—
535C22⋊C4—171no—
536C22⋊C4—171no—
537C22⋊C4—173no—
538C22⋊C4—176no—
539C22⋊C4—173no—
540C22⋊C4—176no—
541C22⋊C4C2×C4×D1716yes—
542C42—173no—
543C42—173no—
544C42—173no—
545C42—173no—
546C42—171no—
547C42—173no—
548C42C2×C4×D1713yes—
549C42C2×C4×D1713yes—
550C42C23×D1713yes—
551C42C23×D1711yescharacteristic
552C23—171no—
553C23—173no—
554C23C22×C4×D1713yes—
555Q8—346no—
556Q8—346no—
557Q8#1058758714yescomplemented
558D4—26no—
559D4—346no—
560D4—346no—
561D4—346no—
562D4—346no—
563D4—686no—
564C2×C4—26no—
565C2×C4—23no—
566C2×C4—23no—
567C2×C4—43no—
568C2×C4—171no—
569C2×C4—172no—
570C2×C4—172no—
571C2×C4—171no—
572C2×C4—171no—
573C2×C4—173no—
574C2×C4—173no—
575C2×C4—173no—
576C2×C4—346no—
577C2×C4—346no—
578C2×C4—343no—
579C2×C4—343no—
580C2×C4—683no—
581C2×C4—683no—
582C2×C4C22×C4×D1713yes—
583C2×C4C22×C4×D1711yescharacteristic
584C2×C4C22×C4×D1711yescharacteristic
585C2×C4C22×C4×D1713yes—
586C2×C4C22×C4×D1713yes—
587C2×C4C24×D1711yescharacteristic
588C8—23no—
589C8—23no—
590C8—172no—
591C8—172no—
592C8—343no—
593C8—343no—
594C8#41×D1712yes—
595C8#41×D1712yes—
596C22—23no—
597C22—23no—
598C22—23no—
599C22—172no—
600C22—341no—
601C22—341no—
602C22—343no—
603C22—343no—
604C22—343no—
605C22C23×C4×D1711yescharacteristic
606C4—21no—
607C4—21no—
608C4—23no—
609C4—23no—
610C4—172no—
611C4—341no—
612C4—343no—
613C4—341no—
614C4—343no—
615C4—343no—
616C4C2×#1058758716yes—
617C4C2×#41×D1711yescharacteristic
618C4C2×#41×D1711yescharacteristic
619C2—43no—
620C2—342no—
621C2—683no—
622C2C4×#41×D1711yescharacteristic
623C2C22×#1058758711yescharacteristic
624C2#1454349511yescharacteristic
subgroupG/H
centerC2×C8C23×D17
commutator, socleC22×C17C24×C4
FrattiniC2×C4C24×D17
FittingC8×#41×C17C2
prime radical#14542382C2
cosocleC2×C4×C17C25
hypercenterC8×#41D17
hypocenterC17#60218
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C8 ◁ C8×#41

Complements

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

NH ≅ G/Ntype
2. #14537804C2semidirect
3. #14537803C2semidirect
4. #14537785C2semidirect
5. #14537782C2semidirect
6. #14537759C2semidirect
7. #14537756C2semidirect
8. C8×#63256C2semidirect
9. Q8×#63218C2semidirect
10. C8×Q8×D17C2semidirect
11. #14537523C2semidirect
12. #14537491C2semidirect
13. #14537445C2semidirect
14. C4×#10587587C2semidirect
15. C8×#41×C17C2semidirect
234. #63218#41semidirect
238. C8×D17#41semidirect
243. Q8×C17#131semidirect
292. C8×#41D17semidirect
476. C17#60218semidirect
557. Q8#10587587semidirect

Maximal subgroups

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

subgroupindex
#145423822
#145378042
#145378032
#145377852
#145377822
#145377592
#145377562
C8×#632562
Q8×#632182
C8×Q8×D172
#145375232
#145374912
#145374452
C4×#105875872
C8×#41×C172
#6021817