Finite Group Databasebeta

← #17194476

Subgroups — #17194476

Order 4352.

2342 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763277C211yesmaximal, characteristic
2#14531754C214yesmaximal, complemented
3#14531746C214yesmaximal, complemented
4#14531723C214yesmaximal, complemented
5#14531719C214yesmaximal, complemented
6#14531394C211yesmaximal, characteristic, complemented
7#14531393C211yesmaximal, characteristic, complemented
8#14531392C211yesmaximal, characteristic, complemented
9#14531373C212yesmaximal
10#14531189C211yesmaximal, characteristic, complemented
11#14531180C212yesmaximal, complemented
12#14531179C212yesmaximal, complemented
13D17×#382C211yesmaximal, characteristic, complemented
14#14531138C212yesmaximal, complemented
15C17×#2314C211yesmaximal, characteristic, complemented
16C22×#63252C2211yescharacteristic
17#11763277C4116yescomplemented
18C2×#10587573C2212yes—
19C2×#10587579C2212yes—
20C2×#10587567C2212yes—
21C2×#10587567C2212yes—
22C2×#10587578C2212yes—
23C2×#10587558C2211yescharacteristic
24C2×#10587558C2211yescharacteristic
25C2×C4⋊C4×D17C2212yes—
26C22×C4×Dic17C2211yescharacteristic
27C2×C42×D17C2211yescharacteristic
28C2×C42×D17C2211yescharacteristic
29#11762713C2214yes—
30#11762709C2214yes—
31#11762702C2214yes—
32#11762701C2214yes—
33#11762699C2214yes—
34#11762696C2214yes—
35#11762695C2214yes—
36#11762690C2214yes—
37#11762687C2214yes—
38#11762685C2214yes—
39#11762679C2211yescharacteristic
40#11762676C2214yes—
41#11762676C2214yes—
42#11762675C2211yescharacteristic
43#11762674C2212yes—
44#11762674C2212yes—
45#11762673C2214yes—
46#11762671C2214yes—
47#11762670C2214yes—
48#11762669C2212yes—
49#11762666C2212yes—
50#11762666C2212yes—
51#11762666C2214yes—
52C4⋊C4×Dic17C2212yes—
53C4⋊C4×Dic17C2212yes—
54C4⋊C4×Dic17C2214yes—
55C4×#63225C2214yes—
56C4×C4⋊Dic17C2212yes—
57#11762660C2214yes—
58C22⋊C4×Dic17C2214yes—
59C4×#63227C2214yes—
60C4×#63227C2214yes—
61#11762656C2214yes—
62#11762655C2214yes—
63#11762654C2214yes—
64#95×D17C2214yes—
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×D17C2311yescharacteristic
81C22×C4×D17C2314yes—
82C22×C4×D17C2312yes—
83C22×C4×D17C2312yes—
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×#63225C2312yes—
107C2×#63225C2312yes—
108C2×C4×Dic17C2×C418yescomplemented
109C2×C4×Dic17C2314yes—
110C2×C4×Dic17C2314yes—
111C2×C4×Dic17C2314yes—
112C2×C4×Dic17C2314yes—
113C2×C4×Dic17C2311yescharacteristic
114C2×C4×Dic17C2312yes—
115C2×C4×Dic17C2312yes—
116C2×C4×Dic17C2311yescharacteristic
117C2×C4×Dic17C2312yes—
118#10587579C2×C4116yes—
119#10587578—28no—
120C4⋊C4×D17—28no—
121#10587573C2×C4116yescomplemented
122#10587567C2×C4116yes—
123#10587567C2×C4116yescomplemented
124#10587558C2×C418yes—
125#10587558C2×C418yescomplemented
126C42×D17C2×C418yes—
127C42×D17C2×C418yescomplemented
128C17×#95C2314yes—
129C17×#95C2314yes—
130C17×#95C2314yes—
131#10587502C2314yes—
132#10587502C2314yes—
133#10587502C2314yes—
134#10587502C2314yes—
135#10587502C2314yes—
136#10587502C2314yes—
137#10587502C2314yes—
138#10587502C2314yes—
139#10587502C2314yes—
140#10587502C2314yes—
141#10587502C2314yes—
142C24×C17C2411yescharacteristic
143C23×D17C2411yescharacteristic
144C23×D17C2411yescharacteristic
145C22×C4×C17—22no—
146C22×C4×C17—22no—
147C22×C4×C17C22×C414yes—
148C22×C4×C17C2414yes—
149C22×C4×C17C2414yes—
150C22×C4×C17C2411yescharacteristic
151C22×C4×C17C2412yes—
152C22×C4×C17C2412yes—
153C22×C4×C17C2411yescharacteristic
154C2×C22⋊D17—22no—
155C2×C22⋊D17—22no—
156C2×C22⋊D17—22no—
157C2×C22⋊D17—22no—
158C2×C22⋊D17C22×C414yes—
159C2×C22⋊D17C22×C414yes—
160C22×Dic17—42no—
161C22×Dic17—42no—
162C22×Dic17C22×C418yes—
163C22×Dic17C2414yes—
164C22×Dic17C2414yes—
165C22×Dic17C2411yescharacteristic
166C22×Dic17C2412yes—
167C22×Dic17C2412yes—
168C22×Dic17C2411yescharacteristic
169#63252—232no—
170C2×D68—22no—
171C2×D68—22no—
172C2×D68C22×C414yes—
173C2×C4×D17—22no—
174C2×C4×D17—22no—
175C2×C4×D17—28no—
176C2×C4×D17—42no—
177C2×C4×D17—42no—
178C2×C4×D17—42no—
179C2×C4×D17—42no—
180C2×C4×D17—44no—
181C2×C4×D17—44no—
182C2×C4×D17C22×C414yes—
183C2×C4×D17C22×C418yes—
184C2×C4×D17C22×C418yes—
185C2×C4×D17C22×C414yes—
186C2×C4×D17#4114yes—
187C2×C4×D17#4114yes—
188C2×Dic34—22no—
189C2×Dic34—22no—
190C2×Dic34C22×C414yes—
191C4⋊C4×C17—24no—
192C4⋊C4×C17—24no—
193C4⋊C4×C17—44no—
194C4⋊C4×C17C22×C418yes—
195C22⋊C4×C17—44no—
196C22⋊C4×C17C22×C418yes—
197C42×C17—44no—
198C42×C17C22×C414yes—
199C42×C17C22×C414yes—
200C22⋊Dic17—44no—
201#63227—28no—
202#63227—44no—
203#63227—44no—
204#63227C22×C418yes—
205#63227C22×C418yes—
206C4⋊Dic17—24no—
207C4⋊Dic17—24no—
208C4⋊Dic17—44no—
209#63225—28no—
210#63225C22×C418yes—
211#63225C22×C418yes—
212C4×Dic17—28no—
213C4×Dic17—44no—
214C4×Dic17—44no—
215C4×Dic17—44no—
216C4×Dic17C22×C414yes—
217C4×Dic17C22×C418yes—
218C4×Dic17C22×C418yes—
219C4×Dic17C22×C414yes—
220C4×Dic17C22×C418yes—
221#34189—171nomaximal, Sylow
222C23×C17—21no—
223C23×C17—21no—
224C23×C17—22no—
225C23×C17—22no—
226C23×C17C23×C412yes—
227C23×C17C2511yescharacteristic
228C22×D17—22no—
229C22×D17—22no—
230C22×D17—21no—
231C22×D17—21no—
232C22×D17—22no—
233C22×D17—22no—
234C22×D17C23×C412yes—
235C22×D17C23×C412yes—
236C22×D17C2×#4112yes—
237C22×D17C2×#4112yes—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—22no—
241C2×C4×C17—22no—
242C2×C4×C17—22no—
243C2×C4×C17—22no—
244C2×C4×C17—28no—
245C2×C4×C17—24no—
246C2×C4×C17—24no—
247C2×C4×C17—24no—
248C2×C4×C17—24no—
249C2×C4×C17—24no—
250C2×C4×C17—24no—
251C2×C4×C17C23×C414yes—
252C2×C4×C17C23×C414yes—
253C2×C4×C17C23×C412yes—
254C2×C4×C17C23×C412yes—
255C2×C4×C17C2×#4112yes—
256C2×C4×C17C2×#4112yes—
257C22⋊D17—28no—
258C22⋊D17—28no—
259C2×Dic17—21no—
260C2×Dic17—21no—
261C2×Dic17—22no—
262C2×Dic17—22no—
263C2×Dic17—22no—
264C2×Dic17—22no—
265C2×Dic17—24no—
266C2×Dic17—24no—
267C2×Dic17—24no—
268C2×Dic17—24no—
269C2×Dic17—24no—
270C2×Dic17—24no—
271C2×Dic17—48no—
272C2×Dic17C23×C412yes—
273C2×Dic17C23×C414yes—
274C2×Dic17C23×C414yes—
275C2×Dic17C23×C412yes—
276C2×Dic17C2×#4112yes—
277C2×Dic17C2×#4112yes—
278D68—28no—
279C4×D17—28no—
280C4×D17—48no—
281C4×D17—48no—
282C4×D17—88no—
283C4×D17#117116yes—
284Dic34—28no—
285C2×C4×#41—171no—
286#2357—174no—
287#2356—174no—
288#2353—174no—
289#2353—174no—
290#2314—171no—
291#2314—171no—
292#2314—171no—
293#2314D1711yescharacteristic, complemented
294#2311—172no—
295#2285—171no—
296#2283—172no—
297#2283—172no—
298C2×#382—171no—
299#2274—172no—
300C22×C17—22no—
301C22×C17—24no—
302C22×C17—42no—
303C22×C17—41no—
304C22×C17—41no—
305C22×C17C24×C411yescharacteristic
306C22×C17C22×#4111yescharacteristic
307C22×C17C22×#4111yescharacteristic
308C22×C17#58512yes—
309C22×C17#58512yes—
310C2×D17—24no—
311C2×D17—24no—
312C2×D17—22no—
313C2×D17—22no—
314C2×D17—22no—
315C2×D17—24no—
316C2×D17—42no—
317C2×D17—41no—
318C2×D17—41no—
319C2×D17C2×#11714yes—
320C4×C17—22no—
321C4×C17—24no—
322C4×C17—24no—
323C4×C17—44no—
324C4×C17—44no—
325C4×C17C2×#11714yes—
326Dic17—22no—
327Dic17—24no—
328Dic17—24no—
329Dic17—44no—
330Dic17—44no—
331Dic17C2×#11714yes—
332C22×#41—171no—
333C4×#41—1716no—
334C2×C4×Q8—172no—
335C2×C4×D4—172no—
336C2×C4×D4—172no—
337C2×C4×D4—172no—
338C2×#117—172no—
339C2×#117—171no—
340C2×#117—171no—
341C2×#117C2×D1711yescharacteristic
342C22×C4⋊C4—172no—
343C22×C42—171no—
344C22×C42—171no—
345C22×C42—171no—
346#391—174no—
347#390—174no—
348#388—174no—
349#388—174no—
350#388—174no—
351#388C2×D1714yes—
352#387—174no—
353#387—174no—
354#387—174no—
355#387C2×D1714yes—
356#386—174no—
357#386—174no—
358#384—171no—
359#384C2×D1711yescharacteristic
360#383—174no—
361#383—174no—
362#382—172no—
363#382—172no—
364#382—171no—
365#382—174no—
366#382C2×D1711yescharacteristic
367#381—174no—
368#381—174no—
369#380—172no—
370#380—172no—
371#380—172no—
372#380—174no—
373#380C2×D1712yes—
374C4×C4⋊C4—172no—
375C4×C4⋊C4—172no—
376C4×C4⋊C4—174no—
377C4×C4⋊C4—172no—
378C4×C4⋊C4—174no—
379C4×C4⋊C4—174no—
380C4×C4⋊C4C2×D1712yes—
381C4×C22⋊C4—174no—
382C4×C22⋊C4—174no—
383C4×C22⋊C4—174no—
384C4×C22⋊C4—174no—
385#376—174no—
386#376—174no—
387C2×#95—174no—
388C2×C17—42no—
389C2×C17C22×#11711yescharacteristic
390C2×C17#342812yes—
391C2×C17#351012yes—
392C2×C17#351211yescharacteristic
393C2×C17#351211yescharacteristic
394D17—24no—
395D17—42no—
396C2×#41—178no—
397C2×#41—344no—
398C2×#41—344no—
399C22×Q8—171no—
400C22×D4—171no—
401C22×D4—171no—
402C22×D4—171no—
403C23×C4—171no—
404C23×C4—172no—
405C23×C4—171no—
406C23×C4—172no—
407C23×C4—172no—
408C23×C4—174no—
409C23×C4C22×D1711yescharacteristic
410C4×Q8—1716no—
411C4×D4—1716no—
412C4×D4—1716no—
413C4×D4—1716no—
414#117—178no—
415#117—178no—
416#117—348no—
417#117C4×D1718yescomplemented
418C2×C4⋊C4—172no—
419C2×C4⋊C4—172no—
420C2×C4⋊C4—172no—
421C2×C4⋊C4—174no—
422C2×C4⋊C4—172no—
423C2×C4⋊C4—174no—
424C2×C4⋊C4—348no—
425C2×C4⋊C4C22×D1712yes—
426C2×C4⋊C4C22×D1712yes—
427C2×C4⋊C4C22×D1714yes—
428C2×C4⋊C4C22×D1712yes—
429C2×C22⋊C4—172no—
430C2×C22⋊C4—172no—
431C2×C22⋊C4—174no—
432C2×C22⋊C4—174no—
433C2×C22⋊C4—174no—
434C2×C22⋊C4—174no—
435C2×C22⋊C4C22×D1714yes—
436C2×C22⋊C4C22×D1712yes—
437C2×C42—178no—
438C2×C42—171no—
439C2×C42—172no—
440C2×C42—172no—
441C2×C42—172no—
442C2×C42—171no—
443C2×C42—178no—
444C2×C42—178no—
445C2×C42—174no—
446C2×C42—174no—
447C2×C42—174no—
448C2×C42—174no—
449C2×C42C22×D1714yes—
450C2×C42C22×D1711yescharacteristic
451C2×C42C22×D1711yescharacteristic
452#95—174no—
453#95—174no—
454#95—174no—
455#95—174no—
456#95—174no—
457#95—174no—
458#95—174no—
459#95—174no—
460#95—174no—
461#95—174no—
462#95—174no—
463#95C22×D1714yes—
464#95C22×D1714yes—
465#95C22×D1714yes—
466C17#3418911yescharacteristic, complemented, Sylow
467C24—171no—
468C24—171no—
469C24C23×D1711yescharacteristic
470#41—3432no—
471C2×Q8—174no—
472C2×Q8—342no—
473C2×Q8—342no—
474C2×D4—174no—
475C2×D4—174no—
476C2×D4—174no—
477C2×D4—342no—
478C2×D4—342no—
479C2×D4—342no—
480C2×D4—342no—
481C2×D4—342no—
482C2×D4—342no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—171no—
486C22×C4—174no—
487C22×C4—174no—
488C22×C4—174no—
489C22×C4—172no—
490C22×C4—171no—
491C22×C4—172no—
492C22×C4—178no—
493C22×C4—174no—
494C22×C4—178no—
495C22×C4—178no—
496C22×C4—174no—
497C22×C4—174no—
498C22×C4—348no—
499C22×C4—342no—
500C22×C4—342no—
501C22×C4—682no—
502C22×C4—682no—
503C22×C4—682no—
504C22×C4—682no—
505C22×C4—684no—
506C22×C4—684no—
507C22×C4—682no—
508C22×C4—682no—
509C22×C4C2×C4×D1714yes—
510C22×C4C23×D1714yes—
511C22×C4C23×D1714yes—
512C22×C4C23×D1711yescharacteristic
513C22×C4C23×D1712yes—
514C22×C4C23×D1712yes—
515C22×C4C23×D1711yescharacteristic
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—44no—
519C4⋊C4—178no—
520C4⋊C4—178no—
521C4⋊C4—348no—
522C4⋊C4—344no—
523C4⋊C4—344no—
524C4⋊C4—684no—
525C4⋊C4C2×C4×D1718yes—
526C22⋊C4—44no—
527C22⋊C4—178no—
528C22⋊C4—178no—
529C22⋊C4—348no—
530C22⋊C4—684no—
531C22⋊C4—684no—
532C22⋊C4—684no—
533C22⋊C4C2×C4×D1718yes—
534C42—44no—
535C42—174no—
536C42—174no—
537C42—178no—
538C42—178no—
539C42—178no—
540C42—348no—
541C42—684no—
542C42—684no—
543C42—684no—
544C42C2×C4×D1714yes—
545C42C2×C4×D1714yes—
546C23—21no—
547C23—21no—
548C23—22no—
549C23—22no—
550C23—172no—
551C23—172no—
552C23—172no—
553C23—172no—
554C23—342no—
555C23—342no—
556C23—341no—
557C23—341no—
558C23—342no—
559C23—342no—
560C23C22×C4×D1712yes—
561C23C24×D1711yescharacteristic
562Q8—348no—
563D4—348no—
564D4—348no—
565D4—348no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—172no—
580C2×C4—172no—
581C2×C4—172no—
582C2×C4—1716no—
583C2×C4—174no—
584C2×C4—174no—
585C2×C4—172no—
586C2×C4—341no—
587C2×C4—341no—
588C2×C4—342no—
589C2×C4—342no—
590C2×C4—342no—
591C2×C4—342no—
592C2×C4—348no—
593C2×C4—344no—
594C2×C4—344no—
595C2×C4—344no—
596C2×C4—344no—
597C2×C4—344no—
598C2×C4—344no—
599C2×C4—688no—
600C2×C4—688no—
601C2×C4—688no—
602C2×C4—1368no—
603C2×C4C22×C4×D1712yes—
604C2×C4C22×C4×D1714yes—
605C2×C4C22×C4×D1714yes—
606C2×C4C22×C4×D1712yes—
607C2×C4#41×D1712yes—
608C2×C4#41×D1712yes—
609C22—22no—
610C22—24no—
611C22—42no—
612C22—41no—
613C22—41no—
614C22—174no—
615C22—344no—
616C22—342no—
617C22—342no—
618C22—344no—
619C22—342no—
620C22—344no—
621C22—682no—
622C22—681no—
623C22—681no—
624C22C23×C4×D1711yescharacteristic
625C22C2×#41×D1711yescharacteristic
626C22C2×#41×D1711yescharacteristic
627C22#1176364412yes—
628C22#1176364612yes—
629C4—22no—
630C4—24no—
631C4—24no—
632C4—44no—
633C4—44no—
634C4—174no—
635C4—342no—
636C4—344no—
637C4—344no—
638C4—684no—
639C4—684no—
640C4#117×D1714yes—
641C2—42no—
642C2—344no—
643C2—682no—
644C2C2×#117×D1711yescharacteristic
645C2#1454198412yes—
646C2#1454267612yes—
647C2#1454270311yescharacteristic
648C2#1454270711yescharacteristic
subgroupG/H
center, FrattiniC23C24×D17
commutatorC22×C17C24×C4
FittingC17×#2314C2
socle, cosocleC23×C17C25
prime radicalC22×#63252C22
hypercenter#2314D17
hypocenterC17#34189
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ #2314

Complements

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

NH ≅ G/Ntype
2. #14531754C2semidirect
3. #14531746C2semidirect
4. #14531723C2semidirect
5. #14531719C2semidirect
6. #14531394C2semidirect
7. #14531393C2semidirect
8. #14531392C2semidirect
10. #14531189C2semidirect
11. #14531180C2semidirect
12. #14531179C2semidirect
13. D17×#382C2semidirect
14. #14531138C2semidirect
15. C17×#2314C2semidirect
17. #11763277C4semidirect
86. C17×#117C2×C4semidirect
108. C2×C4×Dic17C2×C4semidirect
121. #10587573C2×C4semidirect
123. #10587567C2×C4semidirect
125. #10587558C2×C4semidirect
127. C42×D17C2×C4semidirect
293. #2314D17semidirect
417. #117C4×D17semidirect
466. C17#34189semidirect

Maximal subgroups

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

subgroupindex
C2×#117632772
#145317542
#145317462
#145317232
#145317192
#145313942
#145313932
#145313922
#145313732
#145311892
#145311802
#145311792
D17×#3822
#145311382
C17×#23142
#3418917