Finite Group Databasebeta

← #17899338

Subgroups — #17899338

Order 4352.

1702 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#63252C211yesmaximal, characteristic
2#14537885C213yesmaximal, complemented
3#14537884C213yesmaximal, complemented
4#14537878C213yesmaximal, complemented
5#14537875C213yesmaximal, complemented
6#14537873C211yesmaximal, characteristic, complemented
7#14537872C211yesmaximal, characteristic, complemented
8#14537863C211yesmaximal, characteristic, complemented
9Q8×#63218C211yesmaximal, characteristic, complemented
10#14537726C213yesmaximal, complemented
11#14537713C213yesmaximal, complemented
12C8×Q8×D17C211yesmaximal, characteristic, complemented
13#14537445C213yesmaximal
14#14537423C213yesmaximal
15C17×#2982C211yesmaximal, characteristic, complemented
16Q8×Dic34C411yescharacteristic
17#11763537C413yes—
18#11763519C413yes—
19#11763514C413yes—
20#11763514C413yes—
21Q8×D68C411yescharacteristic
22Q8×C22⋊D17C411yescharacteristic
23Q8×C22⋊D17C411yescharacteristic
24#11763366C2213yes—
25#11763363C2213yes—
26#11763363C2213yes—
27C4×#63252C2213yes—
28C4×Q8×D17C2211yescharacteristic
29C4×Q8×D17C2211yescharacteristic
30#11762940C2213yes—
31#11762940C2213yes—
32#11762939C2213yes—
33#11762939C2213yes—
34#11762939C2216yes—
35#11762938C2216yes—
36#11762938C2216yes—
37#11762936C2213yes—
38#11762936C2213yes—
39Q8×#3622C2211yescharacteristic
40Q8×#3622C2211yescharacteristic
41C8×Dic34C2213yes—
42C8×Dic34C2213yes—
43C8×Dic34C2216yes—
44#11762933C2213yes—
45#11762933C2213yes—
46#11762869C2213yes—
47#11762869C2213yes—
48#11762868C2216yes—
49#11762867C2213yes—
50#11762867C2216yes—
51#11762867C2213yes—
52#11762866C2213yes—
53#11762865C2213yes—
54#11762862C2213yes—
55#11762861C2213yes—
56#11762860C2213yes—
57#11762860C2213yes—
58#11762834C2213yes—
59#11762834C2213yes—
60#11762831C2213yes—
61C4⋊C8×D17C2213yes—
62#11762774C2211yescharacteristic
63#11762774C2213yes—
64#11762763C2216yes—
65C4×#63218C2213yes—
66C4×C8×D17C2213yes—
67C2×C4×Q8×C17C2211yescharacteristic
68C8×Q8×C17C2211yescharacteristic
69C8×Q8×C17C2211yescharacteristic
70C8×Q8×C17C2213yes—
71C8×Q8×C17C2213yes—
72C17×#431C2213yes—
73C17×#431C2213yes—
74C22×Q8×C17C2×C411yescharacteristic
75C2×Q8×D17C2×C411yescharacteristic
76C2×Q8×D17C2×C411yescharacteristic
77C2×#63252C2311yescharacteristic
78C4×Q8×C17C2×C411yescharacteristic
79C4×Q8×C17C2×C413yes—
80C4×Q8×C17C2311yescharacteristic
81C4×Q8×C17C2313yes—
82C2×C4⋊C4×C17C2×C413yes—
83C2×C42×C17C2313yes—
84#10587632C2×C413yes—
85#10587632C2×C413yes—
86Q8×Dic17C2×C411yescharacteristic
87Q8×Dic17C2×C411yescharacteristic
88#10587630C2×C413yes—
89#10587630C2×C413yes—
90C4×C22⋊D17C2×C413yes—
91C4×C22⋊D17C2×C413yes—
92C4×C22⋊D17C2×C413yes—
93C4×C22⋊D17C2×C413yes—
94#10587612C2313yes—
95#10587610C2×C413yes—
96#10587610C2×C413yes—
97#10587610C2×C416yes—
98C2×#63218C2311yescharacteristic
99C2×#63218C2313yes—
100C2×C8×D17C2311yescharacteristic
101C2×C8×D17C2313yes—
102#10587583C2×C413yes—
103#10587583C2×C413yes—
104#10587582C2×C416yes—
105#10587582C2×C416yes—
106C4⋊C4×D17C2×C413yes—
107C4⋊C4×D17C2×C413yes—
108#10587575C2×C416yes—
109#10587575C2×C416yes—
110#10587574C2×C413yes—
111#10587574C2×C413yes—
112#10587573C2×C413yes—
113#10587573C2×C413yes—
114C4×D68C2×C413yes—
115C4×D68C2×C413yes—
116#10587558C2313yes—
117#10587558C2313yes—
118C42×D17C2313yes—
119C42×D17C2313yes—
120#10587556C2×C416yes—
121#10587555C2×C413yes—
122#10587555C2×C413yes—
123C4×Dic34C2×C413yes—
124C4×Dic34C2×C413yes—
125C4×Dic34C2313yes—
126C4×Dic34C2313yes—
127C4×Dic34C2316yes—
128C17×C4⋊C8C2313yes—
129C17×C4⋊C8C2313yes—
130C17×C4⋊C8C2313yes—
131C17×C4⋊C8C2313yes—
132C17×C22⋊C8C2311yescharacteristic
133C17×C22⋊C8C2313yes—
134C4×C8×C17C2313yes—
135C4×C8×C17C2313yes—
136C4×C8×C17C2316yes—
137C22⋊#3622C2311yescharacteristic
138C22⋊#3622C2313yes—
139#10587491C2311yescharacteristic
140#10587491C2311yescharacteristic
141#10587491C2313yes—
142#10587491C2313yes—
143#10587486C2313yes—
144#10587486C2313yes—
145#10587486C2316yes—
146#10587485C2316yes—
147#10587485C2316yes—
148#10587485C2313yes—
149#10587485C2313yes—
150#10587485C2313yes—
151#10587485C2313yes—
152C8×Dic17C2313yes—
153C8×Dic17C2313yes—
154C8×Dic17C2316yes—
155C4⋊#3622C2313yes—
156C4⋊#3622C2313yes—
157C4⋊#3622C2313yes—
158C4⋊#3622C2313yes—
159#10587474C2316yes—
160C4×#3622C2313yes—
161C4×#3622C2313yes—
162C2×Q8×C17—24no—
163C2×Q8×C17C22×C413yes—
164C2×Q8×C17C22×C411yescharacteristic
165C22×C4×C17C22×C413yes—
166C22×C4×C17C22×C413yes—
167C22×C4×C17C2411yescharacteristic
168C2×C22⋊D17C22×C411yescharacteristic
169C2×C22⋊D17C22×C411yescharacteristic
170Q8×D17—24no—
171Q8×D17—24no—
172#63252—22no—
173#63252—22no—
174C2×D68C22×C411yescharacteristic
175C2×C4×D17C22×C413yes—
176C2×C4×D17C22×C413yes—
177C2×C4×D17C22×C413yes—
178C2×C4×D17C22×C413yes—
179C2×C4×D17C2411yescharacteristic
180C2×C4×D17C2411yescharacteristic
181C2×Dic34C22×C411yescharacteristic
182C2×Dic34C22×C413yes—
183C2×Dic34C22×C413yes—
184C2×Dic34C22×C416yes—
185C2×C8×C17C2413yes—
186C2×C8×C17C2411yescharacteristic
187C2×C8×C17C2413yes—
188C2×C8×C17C2411yescharacteristic
189C4⋊C4×C17C22×C413yes—
190C4⋊C4×C17C22×C413yes—
191C4⋊C4×C17C22×C416yes—
192C42×C17C22×C413yes—
193C42×C17C22×C413yes—
194C42×C17C2413yes—
195C42×C17C2413yes—
196C22⋊Dic17C22×C413yes—
197C22⋊Dic17C22×C413yes—
198#63227C22×C413yes—
199#63227C22×C413yes—
200#63227C22×C413yes—
201#63227C22×C413yes—
202C4⋊Dic17C22×C413yes—
203C4⋊Dic17C22×C413yes—
204C4⋊Dic17C22×C413yes—
205C4⋊Dic17C22×C413yes—
206C4⋊Dic17C22×C416yes—
207#63225C22×C413yes—
208#63225C22×C413yes—
209#63225C22×C413yes—
210#63225C22×C413yes—
211#63225C22×C416yes—
212#63225C22×C416yes—
213#63225C22×C416yes—
214#63225C22×C416yes—
215C4×Dic17C22×C413yes—
216C4×Dic17C22×C413yes—
217C4×Dic17C22×C413yes—
218C4×Dic17C22×C413yes—
219C4×Dic17C2413yes—
220C4×Dic17C2413yes—
221C2×#3622C2411yescharacteristic
222C2×#3622C2411yescharacteristic
223C2×#3622C2413yes—
224C2×#3622C2413yes—
225#63218—22no—
226#63218—26no—
227C8×D17—26no—
228C8×D17—22no—
229#60253—171nomaximal, Sylow
230C23×C17C23×C411yescharacteristic
231C22×D17C23×C411yescharacteristic
232C22×D17C23×C411yescharacteristic
233Q8×C17—26no—
234Q8×C17#13114yes—
235C2×C4×C17—26no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—43no—
239C2×C4×C17C23×C411yescharacteristic
240C2×C4×C17C23×C413yes—
241C2×C4×C17C23×C413yes—
242C2×C4×C17C23×C413yes—
243C2×C4×C17C23×C413yes—
244C2×C4×C17C2511yescharacteristic
245C22⋊D17—22no—
246C22⋊D17—22no—
247C2×Dic17C23×C413yes—
248C2×Dic17C23×C413yes—
249C2×Dic17C23×C413yes—
250C2×Dic17C23×C413yes—
251C2×Dic17C23×C411yescharacteristic
252C2×Dic17C23×C411yescharacteristic
253D68—22no—
254C4×D17—26no—
255C4×D17—26no—
256C4×D17—21no—
257C4×D17—21no—
258C4×D17—43no—
259C4×D17—43no—
260C4×D172−1+412yes—
261C4×D172−1+412yes—
262Dic34—26no—
263Dic34—26no—
264Dic34—22no—
265Dic34—46no—
266C8×C17—23no—
267C8×C17—21no—
268C8×C17—21no—
269C8×C17—23no—
270#3622—21no—
271#3622—23no—
272#3622—21no—
273#3622—23no—
274Q8×#41—171no—
275#2986—173no—
276#2986—173no—
277#2985—173no—
278#2985—173no—
279#2984—171no—
280#2984—171no—
281#2982—171no—
282#2982D1711yescharacteristic, complemented
283Q8×M16—171no—
284#2962—173no—
285#2960—173no—
286C2×C8×Q8—171no—
287#2925—173no—
288#2922—173no—
289C22×C17—21no—
290C22×C17—21no—
291C22×C17—21no—
292C22×C17C24×C411yescharacteristic
293C2×D17—21no—
294C2×D17—21no—
295C2×D17—21no—
296C2×D17—21no—
297C2×D17—21no—
298C2×D17#51912yes—
299C4×C17—21no—
300C4×C17—23no—
301C4×C17—23no—
302C4×C17—23no—
303C4×C17C2×#13116yes—
304C4×C17C2×2−1+411yescharacteristic
305C4×C17C2×2−1+411yescharacteristic
306Dic17—21no—
307Dic17—23no—
308Dic17—23no—
309Dic17—23no—
310Dic17—23no—
311Dic17#51912yes—
312Q82—171no—
313#557—173no—
314#554—173no—
315#554—173no—
316#554—173no—
317D4×Q8—171no—
318D4×Q8—171no—
319D4×Q8—171no—
320#533—173no—
321#533—173no—
322#533—173no—
323C4×#41—173no—
324C2×C4×Q8—171no—
325C2×C4×Q8—171no—
326C2×C4×Q8C2×D1711yescharacteristic
327#446—173no—
328#446—173no—
329#446—173no—
330#446—176no—
331#446—176no—
332#446—173no—
333#446—173no—
334#446—176no—
335#446—173no—
336C8×Q8—171no—
337C8×Q8—171no—
338C8×Q8—173no—
339C8×Q8—173no—
340C8×Q8—173no—
341C8×Q8—173no—
342C8×Q8—176no—
343C8×Q8C2×D1711yescharacteristic
344C8×Q8C2×D1713yes—
345C8×Q8C2×D1713yes—
346C8×Q8C2×D1711yescharacteristic
347#433—173no—
348#433—173no—
349#433—173no—
350#433—173no—
351#433—176no—
352#433—176no—
353#432—173no—
354#432—173no—
355#431—173no—
356#431—173no—
357#431—173no—
358#431—173no—
359#431C2×D1713yes—
360#431C2×D1713yes—
361#424—173no—
362#424—173no—
363#423—173no—
364C2×C4⋊C8—173no—
365#408—171no—
366#408—173no—
367#405—176no—
368C4×M16—173no—
369C2×C4×C8—173no—
370C2×C17—41no—
371C2×C17C2×#51911yescharacteristic
372C2×C17C22×#13111yescharacteristic
373C2×C17#357311yescharacteristic
374D17—22no—
375D17—41no—
376C2×#41—171no—
377C22×Q8—171no—
378C22×Q8—171no—
379C22×Q8C4×D1711yescharacteristic
380C2×M16—171no—
381C2×M16—173no—
382C22×C8—171no—
383C22×C8—173no—
384C4⋊Q8—173no—
385C4⋊Q8—173no—
386C4⋊Q8—173no—
387C4⋊Q8—173no—
388C4⋊Q8—173no—
389C4⋊Q8—173no—
390#125—176no—
391#125—176no—
392#125—176no—
393C22⋊Q8—173no—
394C22⋊Q8—173no—
395C22⋊Q8—173no—
396C22⋊Q8—173no—
397C22⋊Q8—176no—
398C22⋊Q8—173no—
399C22⋊Q8—173no—
400C22⋊Q8—176no—
401C22⋊Q8—176no—
402C4×Q8—171no—
403C4×Q8—171no—
404C4×Q8—173no—
405C4×Q8—173no—
406C4×Q8—173no—
407C4×Q8—173no—
408C4×Q8—173no—
409C4×Q8—173no—
410C4×Q8—176no—
411C4×Q8C4×D1711yescharacteristic
412C4×Q8C4×D1713yes—
413C4×Q8C22×D1711yescharacteristic
414C4×Q8C22×D1713yes—
415C4×D4—173no—
416C4×D4—173no—
417C4×D4—173no—
418C4×D4—173no—
419C4×D4—173no—
420C4×D4—173no—
421#117—173no—
422#117—173no—
423#117—173no—
424C2×C4⋊C4—173no—
425C2×C4⋊C4—173no—
426C2×C4⋊C4C4×D1713yes—
427C2×C42—173no—
428C2×C42—173no—
429C2×C42C22×D1713yes—
430C4⋊C8—176no—
431C4⋊C8—176no—
432C4⋊C8—173no—
433C4⋊C8—173no—
434C4⋊C8—173no—
435C4⋊C8—173no—
436C4⋊C8—173no—
437C4⋊C8—173no—
438C4⋊C8—173no—
439C4⋊C8—173no—
440C4⋊C8C22×D1713yes—
441C4⋊C8C22×D1713yes—
442C4⋊C8C22×D1713yes—
443C4⋊C8C22×D1713yes—
444C22⋊C8—171no—
445C22⋊C8—173no—
446C22⋊C8—171no—
447C22⋊C8—171no—
448C22⋊C8—173no—
449C22⋊C8—173no—
450C22⋊C8C22×D1711yescharacteristic
451C22⋊C8C22×D1713yes—
452#97—173no—
453#97—173no—
454#97—176no—
455#97—176no—
456C4×C8—173no—
457C4×C8—173no—
458C4×C8—176no—
459C4×C8—173no—
460C4×C8—173no—
461C4×C8C22×D1713yes—
462C4×C8C22×D1713yes—
463C4×C8C22×D1716yes—
464C17#6025311yescharacteristic, complemented, Sylow
465#41—342no—
466#41—342no—
467C2×Q8—24no—
468C2×Q8—171no—
469C2×Q8—173no—
470C2×Q8—173no—
471C2×Q8—176no—
472C2×Q8—344no—
473C2×Q8—344no—
474C2×Q8C2×C4×D1713yes—
475C2×Q8C2×C4×D1711yescharacteristic
476C2×D4—171no—
477C2×D4—171no—
478C2×D4—171no—
479C22×C4—171no—
480C22×C4—173no—
481C22×C4—173no—
482C22×C4—171no—
483C22×C4—173no—
484C22×C4—173no—
485C22×C4C2×C4×D1713yes—
486C22×C4C2×C4×D1713yes—
487C22×C4C23×D1711yescharacteristic
488M16—342no—
489M16—346no—
490C2×C8—171no—
491C2×C8—173no—
492C2×C8—171no—
493C2×C8—173no—
494C2×C8—346no—
495C2×C8—342no—
496C2×C8C23×D1713yes—
497C2×C8C23×D1711yescharacteristic
498C2×C8C23×D1711yescharacteristic
499C2×C8C23×D1713yes—
500C4⋊C4—176no—
501C4⋊C4—176no—
502C4⋊C4—173no—
503C4⋊C4—173no—
504C4⋊C4—173no—
505C4⋊C4—173no—
506C4⋊C4—173no—
507C4⋊C4—173no—
508C4⋊C4—173no—
509C4⋊C4—173no—
510C4⋊C4—176no—
511C4⋊C4—176no—
512C4⋊C4—176no—
513C4⋊C4C2×C4×D1713yes—
514C4⋊C4C2×C4×D1716yes—
515C4⋊C4C2×C4×D1713yes—
516C22⋊C4—173no—
517C22⋊C4—173no—
518C22⋊C4—173no—
519C22⋊C4—173no—
520C22⋊C4—173no—
521C22⋊C4—173no—
522C42—173no—
523C42—173no—
524C42—173no—
525C42—173no—
526C42—173no—
527C42—173no—
528C42C2×C4×D1713yes—
529C42C2×C4×D1713yes—
530C42C23×D1713yes—
531C42C23×D1713yes—
532C23—171no—
533C23—171no—
534C23C22×C4×D1711yescharacteristic
535Q8—26no—
536Q8—342no—
537Q8—346no—
538Q8—346no—
539Q8—686no—
540Q8#1058758714yes—
541D4—342no—
542D4—342no—
543D4—342no—
544C2×C4—26no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—43no—
548C2×C4—171no—
549C2×C4—172no—
550C2×C4—172no—
551C2×C4—171no—
552C2×C4—173no—
553C2×C4—173no—
554C2×C4—173no—
555C2×C4—173no—
556C2×C4—346no—
557C2×C4—346no—
558C2×C4—341no—
559C2×C4—341no—
560C2×C4—683no—
561C2×C4—683no—
562C2×C4C22×C4×D1713yes—
563C2×C4C22×C4×D1711yescharacteristic
564C2×C4C22×C4×D1713yes—
565C2×C4C22×C4×D1713yes—
566C2×C4C22×C4×D1713yes—
567C2×C4C24×D1711yescharacteristic
568C8—23no—
569C8—21no—
570C8—21no—
571C8—23no—
572C8—341no—
573C8—343no—
574C8—341no—
575C8—343no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—172no—
580C22—341no—
581C22—341no—
582C22—341no—
583C22—341no—
584C22—341no—
585C22C23×C4×D1711yescharacteristic
586C4—23no—
587C4—23no—
588C4—23no—
589C4—21no—
590C4—172no—
591C4—343no—
592C4—343no—
593C4—343no—
594C4—343no—
595C4—341no—
596C4C2×#1058758716yes—
597C42−1+4×D1711yescharacteristic
598C42−1+4×D1711yescharacteristic
599C2—41no—
600C2—342no—
601C2—681no—
602C2D17×#51911yescharacteristic
603C2C22×#1058758711yescharacteristic
604C2#1454349711yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D17
commutator, socleC22×C17C24×C4
FittingC17×#2982C2
prime radicalC2×#63252C23
cosocleC2×C4×C17C25
hypercenter#2982D17
hypocenterC17#60253
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. #14537885C2semidirect
3. #14537884C2semidirect
4. #14537878C2semidirect
5. #14537875C2semidirect
6. #14537873C2semidirect
7. #14537872C2semidirect
8. #14537863C2semidirect
9. Q8×#63218C2semidirect
10. #14537726C2semidirect
11. #14537713C2semidirect
12. C8×Q8×D17C2semidirect
15. C17×#2982C2semidirect
282. #2982D17semidirect
464. C17#60253semidirect

Maximal subgroups

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

subgroupindex
Q8×#632522
#145378852
#145378842
#145378782
#145378752
#145378732
#145378722
#145378632
Q8×#632182
#145377262
#145377132
C8×Q8×D172
#145374452
#145374232
C17×#29822
#6025317