Finite Group Databasebeta

← #17194466

Subgroups — #17194466

Order 4352.

2542 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763277C211yesmaximal, characteristic
2#14531738C214yesmaximal, complemented
3#14531730C214yesmaximal, complemented
4#14531712C214yesmaximal, complemented
5#14531708C214yesmaximal, complemented
6#14531394C211yesmaximal, characteristic, complemented
7#14531393C211yesmaximal, characteristic, complemented
8#14531392C211yesmaximal, characteristic, complemented
9#14531373C212yesmaximal
10#14531190C211yesmaximal, characteristic, complemented
11#14531188C212yesmaximal, complemented
12#14531186C212yesmaximal, complemented
13D17×#384C211yesmaximal, characteristic, complemented
14#14531138C212yesmaximal, complemented
15C17×#2314C211yesmaximal, characteristic, complemented
16C22×#63252C2211yescharacteristic
17#11763277C4116yescomplemented
18C2×#10587573C2212yes—
19C2×#10587579C2212yes—
20C2×#10587567C2212yes—
21C2×#10587567C2212yes—
22C2×#10587578C2212yes—
23C2×#10587558C2214yes—
24C2×#10587558C2211yescharacteristic
25C2×#10587558C2211yescharacteristic
26C2×#10587563C2214yes—
27C2×C4⋊C4×D17C2212yes—
28C2×C4⋊C4×D17C2214yes—
29C22×C4×Dic17C2211yescharacteristic
30C2×C42×D17C2211yescharacteristic
31C2×C42×D17C2211yescharacteristic
32#11762713C2214yes—
33#11762709C2214yes—
34#11762702C2214yes—
35#11762701C2214yes—
36#11762699C2214yes—
37#11762696C2214yes—
38#11762695C2214yes—
39#11762690C2214yes—
40#11762687C2214yes—
41#11762685C2214yes—
42#11762679C2211yescharacteristic
43#11762678C2212yes—
44#11762678C2214yes—
45#11762678C2212yes—
46#11762675C2211yescharacteristic
47#11762673C2214yes—
48#11762673C2214yes—
49#11762669C2212yes—
50#11762668C2214yes—
51#11762668C2214yes—
52#11762667C2214yes—
53#11762666C2212yes—
54#11762666C2212yes—
55C4⋊C4×Dic17C2212yes—
56C4⋊C4×Dic17C2212yes—
57C4×#63225C2214yes—
58C4×#63225C2214yes—
59C4×C4⋊Dic17C2212yes—
60#11762660C2214yes—
61#11762660C2214yes—
62#11762660C2214yes—
63C4×#63227C2214yes—
64#11762656C2214yes—
65C2×C17×#117C2211yescharacteristic
66C17×#388C2214yes—
67C17×#387C2214yes—
68C17×#384C2211yescharacteristic
69C17×#382C2211yescharacteristic
70C17×#380C2212yes—
71C4×C4⋊C4×C17C2212yes—
72C23×C4×C17C2311yescharacteristic
73C22×C22⋊D17C2311yescharacteristic
74C22×C22⋊D17C2311yescharacteristic
75C23×Dic17C2312yes—
76C2×#63252—24no—
77C2×#63252—24no—
78C2×#63252C2×C418yes—
79C22×D68C2311yescharacteristic
80C22×C4×D17C2312yes—
81C22×C4×D17C2312yes—
82C22×C4×D17C2311yescharacteristic
83C22×C4×D17C2314yes—
84C22×C4×D17C2311yescharacteristic
85C22×Dic34C2311yescharacteristic
86C17×#117C2×C418yescomplemented
87C2×C4⋊C4×C17C2312yes—
88C2×C4⋊C4×C17C2312yes—
89C2×C4⋊C4×C17C2314yes—
90C2×C4⋊C4×C17C2312yes—
91C2×C22⋊C4×C17C2314yes—
92C2×C22⋊C4×C17C2312yes—
93C2×C42×C17C2314yes—
94C2×C42×C17C2311yescharacteristic
95C2×C42×C17C2311yescharacteristic
96C2×C22⋊Dic17C2314yes—
97C2×#63227C2314yes—
98C2×#63227C2314yes—
99C2×#63227C2314yes—
100C2×#63227C2312yes—
101C2×#63227C2312yes—
102C2×C4⋊Dic17C2312yes—
103C2×C4⋊Dic17C2312yes—
104C2×C4⋊Dic17C2314yes—
105C2×#63225C2314yes—
106C2×#63225C2314yes—
107C2×#63225C2314yes—
108C2×#63225C2314yes—
109C2×#63225C2314yes—
110C2×#63225C2312yes—
111C2×#63225C2312yes—
112C2×C4×Dic17C2×C418yescomplemented
113C2×C4×Dic17C2314yes—
114C2×C4×Dic17C2314yes—
115C2×C4×Dic17C2314yes—
116C2×C4×Dic17C2314yes—
117C2×C4×Dic17C2311yescharacteristic
118C2×C4×Dic17C2312yes—
119C2×C4×Dic17C2312yes—
120C2×C4×Dic17C2311yescharacteristic
121C2×C4×Dic17C2312yes—
122#10587579C2×C4116yes—
123#10587578—28no—
124C4⋊C4×D17—28no—
125C4⋊C4×D17—48no—
126#10587573C2×C4116yescomplemented
127#10587567C2×C4116yes—
128#10587567C2×C4116yescomplemented
129#10587563—48no—
130#10587558—48no—
131#10587558C2×C418yes—
132#10587558C2×C418yescomplemented
133C42×D17C2×C418yes—
134C42×D17C2×C418yescomplemented
135C17×#95C2314yes—
136C17×#95C2314yes—
137C17×#95C2314yes—
138#10587502C2314yes—
139#10587502C2314yes—
140#10587502C2314yes—
141#10587502C2314yes—
142#10587502C2314yes—
143#10587502C2314yes—
144#10587502C2314yes—
145C24×C17C2411yescharacteristic
146C23×D17C2411yescharacteristic
147C23×D17C2411yescharacteristic
148C22×C4×C17—22no—
149C22×C4×C17—22no—
150C22×C4×C17C22×C414yes—
151C22×C4×C17C2414yes—
152C22×C4×C17C2414yes—
153C22×C4×C17C2411yescharacteristic
154C22×C4×C17C2412yes—
155C22×C4×C17C2412yes—
156C22×C4×C17C2411yescharacteristic
157C2×C22⋊D17—22no—
158C2×C22⋊D17—22no—
159C2×C22⋊D17—22no—
160C2×C22⋊D17—22no—
161C2×C22⋊D17C22×C414yes—
162C2×C22⋊D17C22×C414yes—
163C22×Dic17—24no—
164C22×Dic17—24no—
165C22×Dic17C22×C418yes—
166C22×Dic17C2414yes—
167C22×Dic17C2414yes—
168C22×Dic17C2411yescharacteristic
169C22×Dic17C2412yes—
170C22×Dic17C2411yescharacteristic
171C22×Dic17C2412yes—
172#63252—232no—
173C2×D68—22no—
174C2×D68—22no—
175C2×D68C22×C414yes—
176C2×C4×D17—22no—
177C2×C4×D17—22no—
178C2×C4×D17—24no—
179C2×C4×D17—24no—
180C2×C4×D17—24no—
181C2×C4×D17—24no—
182C2×C4×D17—28no—
183C2×C4×D17—44no—
184C2×C4×D17—44no—
185C2×C4×D17C22×C414yes—
186C2×C4×D17C22×C418yes—
187C2×C4×D17C22×C418yes—
188C2×C4×D17C22×C414yes—
189C2×C4×D17#4114yes—
190C2×C4×D17#4114yes—
191C2×Dic34—22no—
192C2×Dic34—22no—
193C2×Dic34C22×C414yes—
194C4⋊C4×C17—24no—
195C4⋊C4×C17—24no—
196C4⋊C4×C17—44no—
197C4⋊C4×C17C22×C418yes—
198C22⋊C4×C17—44no—
199C22⋊C4×C17C22×C418yes—
200C42×C17—44no—
201C42×C17C22×C414yes—
202C42×C17C22×C414yes—
203C22⋊Dic17—44no—
204#63227—28no—
205#63227—44no—
206#63227—44no—
207#63227C22×C418yes—
208#63227C22×C418yes—
209C4⋊Dic17—24no—
210C4⋊Dic17—24no—
211C4⋊Dic17—44no—
212#63225—28no—
213#63225—28no—
214#63225—28no—
215#63225—28no—
216#63225—28no—
217#63225C22×C418yes—
218#63225C22×C418yes—
219C4×Dic17—28no—
220C4×Dic17—44no—
221C4×Dic17—44no—
222C4×Dic17—44no—
223C4×Dic17C22×C414yes—
224C4×Dic17C22×C418yes—
225C4×Dic17C22×C418yes—
226C4×Dic17C22×C414yes—
227C4×Dic17C22×C418yes—
228#34188—171nomaximal, Sylow
229C23×C17—21no—
230C23×C17—21no—
231C23×C17—22no—
232C23×C17—22no—
233C23×C17C23×C412yes—
234C23×C17C2511yescharacteristic
235C22×D17—22no—
236C22×D17—22no—
237C22×D17—21no—
238C22×D17—21no—
239C22×D17—22no—
240C22×D17—22no—
241C22×D17C23×C412yes—
242C22×D17C23×C412yes—
243C22×D17C2×#4112yes—
244C22×D17C2×#4112yes—
245C2×C4×C17—21no—
246C2×C4×C17—21no—
247C2×C4×C17—22no—
248C2×C4×C17—22no—
249C2×C4×C17—22no—
250C2×C4×C17—22no—
251C2×C4×C17—28no—
252C2×C4×C17—24no—
253C2×C4×C17—24no—
254C2×C4×C17—24no—
255C2×C4×C17—24no—
256C2×C4×C17—24no—
257C2×C4×C17—24no—
258C2×C4×C17C23×C414yes—
259C2×C4×C17C23×C414yes—
260C2×C4×C17C23×C412yes—
261C2×C4×C17C23×C412yes—
262C2×C4×C17C2×#4112yes—
263C2×C4×C17C2×#4112yes—
264C22⋊D17—28no—
265C22⋊D17—28no—
266C2×Dic17—21no—
267C2×Dic17—21no—
268C2×Dic17—22no—
269C2×Dic17—22no—
270C2×Dic17—216no—
271C2×Dic17—24no—
272C2×Dic17—24no—
273C2×Dic17—24no—
274C2×Dic17—24no—
275C2×Dic17—24no—
276C2×Dic17—24no—
277C2×Dic17C23×C412yes—
278C2×Dic17C23×C414yes—
279C2×Dic17C23×C412yes—
280C2×Dic17C23×C414yes—
281C2×Dic17C2×#4112yes—
282C2×Dic17C2×#4112yes—
283C2×Dic17H(5,𝔽2)14yes—
284C2×Dic172−1+414yes—
285D68—28no—
286C4×D17—28no—
287C4×D17—216no—
288C4×D17—216no—
289C4×D17—88no—
290C4×D17#117116yescomplemented
291Dic34—28no—
292C2×C4×#41—171no—
293#2355—174no—
294#2354—174no—
295#2352—174no—
296#2352—174no—
297#2314—171no—
298#2314—171no—
299#2314—171no—
300#2314D1711yescharacteristic, complemented
301#2311—172no—
302#2285—171no—
303#2285—172no—
304#2285—172no—
305C2×#384—171no—
306#2274—172no—
307C22×C17—22no—
308C22×C17—24no—
309C22×C17—41no—
310C22×C17—41no—
311C22×C17—42no—
312C22×C17C24×C411yescharacteristic
313C22×C17C22×#4111yescharacteristic
314C22×C17C22×#4111yescharacteristic
315C22×C17C2×H(5,𝔽2)12yes—
316C22×C17C2×2−1+412yes—
317C2×D17—24no—
318C2×D17—22no—
319C2×D17—22no—
320C2×D17—24no—
321C2×D17—22no—
322C2×D17—24no—
323C2×D17—41no—
324C2×D17—41no—
325C2×D17—42no—
326C2×D17C2×#11714yes—
327C4×C17—22no—
328C4×C17—24no—
329C4×C17—24no—
330C4×C17—44no—
331C4×C17—44no—
332C4×C17C2×#11714yes—
333Dic17—22no—
334Dic17—24no—
335Dic17—44no—
336Dic17—44no—
337Dic17C2×#11714yes—
338Dic17#52018yes—
339C22×#41—171no—
340C4×#41—1716no—
341C2×C4×Q8—172no—
342C2×C4×D4—172no—
343C2×C4×D4—172no—
344C2×C4×D4—172no—
345C2×#117—172no—
346C2×#117—174no—
347C2×#117—171no—
348C2×#117—171no—
349C2×#117—174no—
350C2×#117C2×D1711yescharacteristic
351C22×C4⋊C4—172no—
352C22×C4⋊C4—174no—
353C22×C42—171no—
354C22×C42—171no—
355C22×C42—171no—
356#391—174no—
357#390—174no—
358#388—174no—
359#388—174no—
360#388—174no—
361#388C2×D1714yes—
362#387—174no—
363#387—174no—
364#387—174no—
365#387C2×D1714yes—
366#386—174no—
367#386—174no—
368#384—174no—
369#384—172no—
370#384—172no—
371#384—171no—
372#384C2×D1711yescharacteristic
373#382—171no—
374#382—174no—
375#382—174no—
376#382C2×D1711yescharacteristic
377#380—172no—
378#380—172no—
379#380—172no—
380#380—174no—
381#380—174no—
382#380—174no—
383#380C2×D1712yes—
384C4×C4⋊C4—172no—
385C4×C4⋊C4—172no—
386C4×C4⋊C4—172no—
387C4×C4⋊C4—174no—
388C4×C4⋊C4—174no—
389C4×C4⋊C4—174no—
390C4×C4⋊C4—174no—
391C4×C4⋊C4—174no—
392C4×C4⋊C4C2×D1712yes—
393C4×C22⋊C4—174no—
394C4×C22⋊C4—174no—
395C2×C17—42no—
396C2×C17C22×#11711yescharacteristic
397C2×C17C2×#52012yes—
398C2×C17#350811yescharacteristic
399C2×C17#350911yescharacteristic
400C2×C17#351112yes—
401D17—24no—
402D17—42no—
403C2×#41—178no—
404C2×#41—344no—
405C2×#41—344no—
406C22×Q8—171no—
407C22×D4—171no—
408C22×D4—171no—
409C22×D4—171no—
410C23×C4—171no—
411C23×C4—172no—
412C23×C4—172no—
413C23×C4—172no—
414C23×C4—171no—
415C23×C4—174no—
416C23×C4C22×D1711yescharacteristic
417C4×Q8—1716no—
418C4×D4—1716no—
419C4×D4—1716no—
420C4×D4—1716no—
421#117—178no—
422#117—178no—
423#117—348no—
424#117—688no—
425#117—688no—
426#117C4×D1718yescomplemented
427C2×C4⋊C4—172no—
428C2×C4⋊C4—172no—
429C2×C4⋊C4—174no—
430C2×C4⋊C4—174no—
431C2×C4⋊C4—172no—
432C2×C4⋊C4—174no—
433C2×C4⋊C4—174no—
434C2×C4⋊C4—172no—
435C2×C4⋊C4—174no—
436C2×C4⋊C4—174no—
437C2×C4⋊C4—348no—
438C2×C4⋊C4—688no—
439C2×C4⋊C4C22×D1712yes—
440C2×C4⋊C4C22×D1712yes—
441C2×C4⋊C4C22×D1714yes—
442C2×C4⋊C4C22×D1712yes—
443C2×C22⋊C4—172no—
444C2×C22⋊C4—172no—
445C2×C22⋊C4—174no—
446C2×C22⋊C4—174no—
447C2×C22⋊C4—174no—
448C2×C22⋊C4—174no—
449C2×C22⋊C4C22×D1714yes—
450C2×C22⋊C4C22×D1712yes—
451C2×C42—178no—
452C2×C42—171no—
453C2×C42—172no—
454C2×C42—172no—
455C2×C42—172no—
456C2×C42—171no—
457C2×C42—178no—
458C2×C42—178no—
459C2×C42—174no—
460C2×C42—174no—
461C2×C42—174no—
462C2×C42—174no—
463C2×C42C22×D1714yes—
464C2×C42C22×D1711yescharacteristic
465C2×C42C22×D1711yescharacteristic
466#95—174no—
467#95—174no—
468#95—174no—
469#95—174no—
470#95—174no—
471#95—174no—
472#95—174no—
473#95C22×D1714yes—
474#95C22×D1714yes—
475#95C22×D1714yes—
476C17#3418811yescharacteristic, complemented, Sylow
477C24—171no—
478C24—171no—
479C24C23×D1711yescharacteristic
480#41—3432no—
481C2×Q8—174no—
482C2×Q8—342no—
483C2×Q8—342no—
484C2×D4—174no—
485C2×D4—174no—
486C2×D4—174no—
487C2×D4—342no—
488C2×D4—342no—
489C2×D4—342no—
490C2×D4—342no—
491C2×D4—342no—
492C2×D4—342no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—171no—
496C22×C4—172no—
497C22×C4—174no—
498C22×C4—174no—
499C22×C4—174no—
500C22×C4—178no—
501C22×C4—171no—
502C22×C4—172no—
503C22×C4—174no—
504C22×C4—178no—
505C22×C4—178no—
506C22×C4—174no—
507C22×C4—174no—
508C22×C4—344no—
509C22×C4—344no—
510C22×C4—344no—
511C22×C4—344no—
512C22×C4—348no—
513C22×C4—344no—
514C22×C4—344no—
515C22×C4—342no—
516C22×C4—342no—
517C22×C4—684no—
518C22×C4—684no—
519C22×C4C2×C4×D1714yes—
520C22×C4C23×D1714yes—
521C22×C4C23×D1714yes—
522C22×C4C23×D1711yescharacteristic
523C22×C4C23×D1712yes—
524C22×C4C23×D1712yes—
525C22×C4C23×D1711yescharacteristic
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—44no—
529C4⋊C4—178no—
530C4⋊C4—178no—
531C4⋊C4—348no—
532C4⋊C4—348no—
533C4⋊C4—348no—
534C4⋊C4—348no—
535C4⋊C4—348no—
536C4⋊C4—344no—
537C4⋊C4—344no—
538C4⋊C4—684no—
539C4⋊C4C2×C4×D1718yes—
540C22⋊C4—44no—
541C22⋊C4—178no—
542C22⋊C4—178no—
543C22⋊C4—348no—
544C22⋊C4—684no—
545C22⋊C4—684no—
546C22⋊C4—684no—
547C22⋊C4C2×C4×D1718yes—
548C42—44no—
549C42—174no—
550C42—174no—
551C42—178no—
552C42—178no—
553C42—178no—
554C42—348no—
555C42—684no—
556C42—684no—
557C42—684no—
558C42C2×C4×D1714yes—
559C42C2×C4×D1714yes—
560C23—21no—
561C23—21no—
562C23—22no—
563C23—22no—
564C23—172no—
565C23—172no—
566C23—172no—
567C23—172no—
568C23—342no—
569C23—342no—
570C23—341no—
571C23—341no—
572C23—342no—
573C23—342no—
574C23C22×C4×D1712yes—
575C23C24×D1711yescharacteristic
576Q8—348no—
577D4—348no—
578D4—348no—
579D4—348no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—28no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—174no—
594C2×C4—172no—
595C2×C4—172no—
596C2×C4—172no—
597C2×C4—174no—
598C2×C4—174no—
599C2×C4—1716no—
600C2×C4—174no—
601C2×C4—172no—
602C2×C4—3416no—
603C2×C4—3416no—
604C2×C4—3416no—
605C2×C4—341no—
606C2×C4—341no—
607C2×C4—342no—
608C2×C4—342no—
609C2×C4—348no—
610C2×C4—344no—
611C2×C4—344no—
612C2×C4—344no—
613C2×C4—344no—
614C2×C4—344no—
615C2×C4—344no—
616C2×C4—1368no—
617C2×C4C22×C4×D1712yes—
618C2×C4C22×C4×D1714yes—
619C2×C4C22×C4×D1714yes—
620C2×C4C22×C4×D1712yes—
621C2×C4#41×D1712yes—
622C2×C4#41×D1712yes—
623C22—22no—
624C22—24no—
625C22—41no—
626C22—41no—
627C22—42no—
628C22—174no—
629C22—342no—
630C22—342no—
631C22—344no—
632C22—344no—
633C22—342no—
634C22—344no—
635C22—681no—
636C22—681no—
637C22—682no—
638C22C23×C4×D1711yescharacteristic
639C22C2×#41×D1711yescharacteristic
640C22C2×#41×D1711yescharacteristic
641C22H(5,𝔽2)×D1712yes—
642C222−1+4×D1712yes—
643C4—22no—
644C4—24no—
645C4—24no—
646C4—44no—
647C4—44no—
648C4—174no—
649C4—178no—
650C4—342no—
651C4—344no—
652C4—684no—
653C4—684no—
654C4#117×D1714yes—
655C2—42no—
656C2—344no—
657C2—682no—
658C2C2×#117×D1711yescharacteristic
659C2D17×#52012yes—
660C2#1454265611yescharacteristic
661C2#1454266611yescharacteristic
662C2#1454268912yes—
subgroupG/H
center, FrattiniC23C24×D17
commutatorC22×C17C24×C4
FittingC17×#2314C2
socle, cosocleC23×C17C25
prime radicalC22×#63252C22
hypercenter#2314D17
hypocenterC17#34188
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ #2314

Complements

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

NH ≅ G/Ntype
2. #14531738C2semidirect
3. #14531730C2semidirect
4. #14531712C2semidirect
5. #14531708C2semidirect
6. #14531394C2semidirect
7. #14531393C2semidirect
8. #14531392C2semidirect
10. #14531190C2semidirect
11. #14531188C2semidirect
12. #14531186C2semidirect
13. D17×#384C2semidirect
14. #14531138C2semidirect
15. C17×#2314C2semidirect
17. #11763277C4semidirect
86. C17×#117C2×C4semidirect
112. C2×C4×Dic17C2×C4semidirect
126. #10587573C2×C4semidirect
128. #10587567C2×C4semidirect
132. #10587558C2×C4semidirect
134. C42×D17C2×C4semidirect
290. C4×D17#117semidirect
300. #2314D17semidirect
426. #117C4×D17semidirect
476. C17#34188semidirect

Maximal subgroups

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

subgroupindex
C2×#117632772
#145317382
#145317302
#145317122
#145317082
#145313942
#145313932
#145313922
#145313732
#145311902
#145311882
#145311862
D17×#3842
#145311382
C17×#23142
#3418817