Finite Group Databasebeta

← #16929693

Subgroups — #16929693

Order 4352.

1136 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊#3622C211yesmaximal, characteristic, complemented
2C2×#11762688C211yesmaximal, characteristic
3C2×#11762532C212yesmaximal, complemented
4#14526626C212yesmaximal, complemented
5C17×#1779C211yesmaximal, characteristic, complemented
6C23×C22⋊D17C411yescharacteristic
7C23×#3622—21no—
8C22×C22⋊Dic17C411yescharacteristic
9C22×#63227C411yescharacteristic
10C22×#63227C2211yescharacteristic
11C2×C22⋊#3622—24no—
12C2×C22⋊#3622—21no—
13C2×C22⋊#3622C2212yes—
14#11762688—21no—
15#11762688—21no—
16#11762688—21no—
17#11762688—21no—
18C2×#10587502C411yescharacteristic
19C2×#10587502C2211yescharacteristic
20#11762532—24no—
21C24×C4×C17C2211yescharacteristic
22C2×C17×C22⋊C8—21no—
23C2×C17×C22⋊C8—21no—
24C2×C17×C22⋊C8C2212yes—
25C25×C17C2×C411yescharacteristic
26C24×D17C2×C411yescharacteristic
27C23×C4×C17—22no—
28C23×C4×C17—22no—
29C23×C4×C17—42no—
30C23×C4×C17—42no—
31C23×C4×C17C2×C411yescharacteristic
32C23×C4×C17D411yescharacteristic
33C23×C4×C17D411yescharacteristic
34C23×C4×C17D411yescharacteristic
35C23×C4×C17D411yescharacteristic
36C23×C4×C17C2311yescharacteristic
37C22×C22⋊D17—21no—
38C22×C22⋊D17—21no—
39C22×C22⋊D17—22no—
40C22×C22⋊D17—42no—
41C22×C22⋊D17—42no—
42C22×C22⋊D17C814yescomplemented
43C23×Dic17C2×C411yescharacteristic
44C23×Dic17C2×C411yescharacteristic
45C23×Dic17C2×C411yescharacteristic
46C22×C8×C17—22no—
47C2×C22⋊Dic17—21no—
48C2×C22⋊Dic17—21no—
49C2×C22⋊Dic17—42no—
50C2×#63227—21no—
51C2×#63227—21no—
52C2×#63227—21no—
53C2×#63227—21no—
54C2×#63227—22no—
55C2×#63227—22no—
56C2×#63227—42no—
57C22×#3622—24no—
58C22×#3622—22no—
59C22×#3622—42no—
60C22×#3622—42no—
61C17×C22⋊C8—22no—
62C17×C22⋊C8—22no—
63C17×C22⋊C8—42no—
64C17×C22⋊C8—42no—
65#10587502—21no—
66#10587502—21no—
67#10587502—21no—
68#10587502—21no—
69C22⋊#3622—24no—
70C22⋊#3622—22no—
71C22⋊#3622D414yescomplemented
72C24×C17—22no—
73C24×C17—22no—
74C24×C17—22no—
75C24×C17—22no—
76C24×C17—41no—
77C24×C17—41no—
78C24×C17C22⋊C411yescharacteristic
79C24×C17C22⋊C411yescharacteristic
80C24×C17C2×C812yes—
81C24×C17M1612yes—
82C24×C17C22×C411yescharacteristic
83C23×D17—21no—
84C23×D17—21no—
85C23×D17—22no—
86C23×D17—22no—
87C23×D17—41no—
88C23×D17—41no—
89C23×D17—41no—
90C23×D17—41no—
91C23×D17C2×C812yescomplemented
92C22×C4×C17—22no—
93C22×C4×C17—22no—
94C22×C4×C17—21no—
95C22×C4×C17—21no—
96C22×C4×C17—22no—
97C22×C4×C17—22no—
98C22×C4×C17—22no—
99C22×C4×C17—22no—
100C22×C4×C17—21no—
101C22×C4×C17—22no—
102C22×C4×C17—22no—
103C22×C4×C17—21no—
104C22×C4×C17—41no—
105C22×C4×C17—41no—
106C22×C4×C17—41no—
107C22×C4×C17—41no—
108C22×C4×C17—41no—
109C22×C4×C17—41no—
110C22×C4×C17—41no—
111C22×C4×C17—48no—
112C22×C4×C17—48no—
113C22×C4×C17C22⋊C411yescharacteristic
114C22×C4×C17C22⋊C411yescharacteristic
115C22×C4×C17C2×D411yescharacteristic
116C22×C4×C17C2×D412yes—
117C22×C4×C17C2×D411yescharacteristic
118C22×C4×C17#4112yes—
119C2×C22⋊D17—24no—
120C2×C22⋊D17—24no—
121C2×C22⋊D17—48no—
122C2×C22⋊D17—42no—
123C2×C22⋊D17—42no—
124C2×C22⋊D17—42no—
125C2×C22⋊D17—42no—
126C2×C22⋊D17—48no—
127C22×Dic17—21no—
128C22×Dic17—21no—
129C22×Dic17—21no—
130C22×Dic17—21no—
131C22×Dic17—22no—
132C22×Dic17—22no—
133C22×Dic17—41no—
134C22×Dic17—41no—
135C22×Dic17—41no—
136C22×Dic17—41no—
137C22×Dic17—41no—
138C22×Dic17—41no—
139C22×Dic17C2×C812yescomplemented
140C2×C8×C17—42no—
141C2×C8×C17—42no—
142C2×C8×C17—42no—
143C22⋊Dic17—42no—
144C22⋊Dic17—42no—
145#63227—42no—
146#63227—42no—
147#63227—41no—
148#63227—42no—
149#63227—41no—
150C2×#3622—24no—
151C2×#3622—48no—
152C2×#3622—42no—
153C2×#3622—42no—
154#9079—171nomaximal, Sylow
155C23×C17—21no—
156C23×C17—21no—
157C23×C17—21no—
158C23×C17—21no—
159C23×C17—21no—
160C23×C17—21no—
161C23×C17—21no—
162C23×C17—21no—
163C23×C17—24no—
164C23×C17—24no—
165C23×C17—21no—
166C23×C17—21no—
167C23×C17—21no—
168C23×C17—21no—
169C23×C17—21no—
170C23×C17—21no—
171C23×C17—24no—
172C23×C17—24no—
173C23×C17—42no—
174C23×C17—42no—
175C23×C17—44no—
176C23×C17—44no—
177C23×C17—44no—
178C23×C17—44no—
179C23×C17C22⋊C812yes—
180C23×C17C22⋊C812yes—
181C23×C17#9912yes—
182C23×C17#10012yes—
183C23×C17C2×C22⋊C411yescharacteristic
184C23×C17C4×D412yes—
185C23×C17C22×C811yescharacteristic
186C23×C17C2×M1611yescharacteristic
187C23×C17#13112yes—
188C22×D17—22no—
189C22×D17—24no—
190C22×D17—42no—
191C22×D17—42no—
192C22×D17—41no—
193C22×D17—41no—
194C22×D17—41no—
195C22×D17—41no—
196C22×D17—41no—
197C22×D17—41no—
198C22×D17—42no—
199C22×D17—42no—
200C22×D17—42no—
201C22×D17—82no—
202C22×D17—82no—
203C22×D17—82no—
204C22×D17—82no—
205C2×C4×C17—24no—
206C2×C4×C17—24no—
207C2×C4×C17—21no—
208C2×C4×C17—21no—
209C2×C4×C17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17—42no—
217C2×C4×C17—42no—
218C2×C4×C17—44no—
219C2×C4×C17—44no—
220C2×C4×C17—44no—
221C2×C4×C17—42no—
222C2×C4×C17—42no—
223C2×C4×C17#9912yes—
224C2×C4×C17#12011yescharacteristic
225C2×C4×C17#12112yes—
226C2×C4×C17#12311yescharacteristic
227C22⋊D17—48no—
228C22⋊D17—48no—
229C2×Dic17—22no—
230C2×Dic17—42no—
231C2×Dic17—41no—
232C2×Dic17—41no—
233C2×Dic17—42no—
234C2×Dic17—41no—
235C2×Dic17—41no—
236C2×Dic17—41no—
237C2×Dic17—41no—
238C2×Dic17—41no—
239C2×Dic17—41no—
240C2×Dic17—82no—
241C2×Dic17—82no—
242C8×C17—82no—
243#3622—44no—
244C22×C22⋊C8—171no—
245C2×#386—171no—
246#1779D1711yescharacteristic, complemented
247C2×#323—172no—
248#1280—172no—
249C22×C17—22no—
250C22×C17—22no—
251C22×C17—22no—
252C22×C17—22no—
253C22×C17—22no—
254C22×C17—21no—
255C22×C17—21no—
256C22×C17—21no—
257C22×C17—21no—
258C22×C17—22no—
259C22×C17—22no—
260C22×C17—22no—
261C22×C17—22no—
262C22×C17—22no—
263C22×C17—22no—
264C22×C17—22no—
265C22×C17—22no—
266C22×C17—48no—
267C22×C17—48no—
268C22×C17—41no—
269C22×C17—41no—
270C22×C17—41no—
271C22×C17—41no—
272C22×C17#32314yescomplemented
273C22×C17#38611yescharacteristic
274C22×C17C2×C22⋊C811yescharacteristic
275C22×C17#40811yescharacteristic
276C22×C17C2×#9911yescharacteristic
277C22×C17C2×#9911yescharacteristic
278C22×C17C2×#10011yescharacteristic
279C22×C17#41311yescharacteristic
280C22×C17C8×D412yes—
281C22×C17#43512yes—
282C2×D17—42no—
283C2×D17—42no—
284C2×D17—42no—
285C2×D17—84no—
286C2×D17—81no—
287C2×D17—81no—
288C2×D17—81no—
289C2×D17—81no—
290C2×D17—84no—
291C4×C17—21no—
292C4×C17—22no—
293C4×C17—22no—
294C4×C17—21no—
295C4×C17—41no—
296Dic17—42no—
297Dic17—81no—
298Dic17—81no—
299C23×D4—171no—
300C24×C4C2×D1711yescharacteristic
301C23×C8—341no—
302C22×C22⋊C4—171no—
303C22×C22⋊C4—171no—
304C22×C22⋊C4—171no—
305C2×C22⋊C8—21no—
306C2×C22⋊C8—21no—
307C2×C22⋊C8—172no—
308C2×C22⋊C8—344no—
309C2×C22⋊C8—341no—
310C2×C22⋊C8C2×D1712yes—
311#386—341no—
312#386—341no—
313#386—341no—
314#386—341no—
315C2×#95—171no—
316C2×#95—171no—
317#323—344no—
318C2×C17—22no—
319C2×C17—22no—
320C2×C17—24no—
321C2×C17—42no—
322C2×C17C2×#32311yescharacteristic
323C2×C17#145911yescharacteristic
324C2×C17#146811yescharacteristic
325C2×C17#147111yescharacteristic
326C2×C17#186911yescharacteristic
327C2×C17#189811yescharacteristic
328C2×C17#190011yescharacteristic
329D17—82no—
330C25—171no—
331C25C4×D1711yescharacteristic
332C22×D4—174no—
333C22×D4—342no—
334C22×D4—341no—
335C22×D4—341no—
336C22×D4—682no—
337C22×D4—682no—
338C23×C4—22no—
339C23×C4—22no—
340C23×C4—42no—
341C23×C4—42no—
342C23×C4—171no—
343C23×C4—171no—
344C23×C4—171no—
345C23×C4C4×D1711yescharacteristic
346C23×C4D6811yescharacteristic
347C23×C4D6811yescharacteristic
348C23×C4C22⋊D1711yescharacteristic
349C23×C4C22⋊D1711yescharacteristic
350C23×C4C22×D1711yescharacteristic
351C22×C8—22no—
352C22×C8—344no—
353C22×C8—342no—
354C22×C8—682no—
355C22×C8—682no—
356C2×C22⋊C4—341no—
357C2×C22⋊C4—341no—
358C2×C22⋊C4—341no—
359C2×C22⋊C4—342no—
360C2×C22⋊C4—342no—
361C2×C22⋊C4—341no—
362C2×C22⋊C4—341no—
363C2×C22⋊C4—341no—
364C2×C22⋊C4—682no—
365C2×C22⋊C4—682no—
366C22⋊C8—22no—
367C22⋊C8—22no—
368C22⋊C8—42no—
369C22⋊C8—42no—
370C22⋊C8—174no—
371C22⋊C8—344no—
372C22⋊C8—342no—
373#95—341no—
374#95—341no—
375#95—341no—
376#95—341no—
377C17#907911yescharacteristic, complemented, Sylow
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—41no—
383C24—41no—
384C24—172no—
385C24—341no—
386C24—341no—
387C24—342no—
388C24—342no—
389C24—681no—
390C24—681no—
391C24—681no—
392C24—681no—
393C24C8×D1712yes—
394C24#6321812yes—
395C24#6322711yescharacteristic
396C24#6322711yescharacteristic
397C24C2×C4×D1711yescharacteristic
398C2×D4—344no—
399C2×D4—344no—
400C2×D4—688no—
401C2×D4—682no—
402C2×D4—682no—
403C2×D4—682no—
404C2×D4—682no—
405C2×D4—688no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—172no—
428C22×C4—341no—
429C22×C4—341no—
430C22×C4—341no—
431C22×C4—341no—
432C22×C4—342no—
433C22×C4—342no—
434C22×C4—681no—
435C22×C4—681no—
436C22×C4—681no—
437C22×C4—681no—
438C22×C4—681no—
439C22×C4—681no—
440C22×C4#6322711yescharacteristic
441C22×C4#6322711yescharacteristic
442C22×C4C2×D6811yescharacteristic
443C22×C4D4×D1712yes—
444C22×C4#6325412yes—
445C22×C4C2×C22⋊D1711yescharacteristic
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—344no—
450C2×C8—688no—
451C2×C8—682no—
452C2×C8—682no—
453C22⋊C4—682no—
454C22⋊C4—682no—
455C22⋊C4—682no—
456C22⋊C4—682no—
457C22⋊C4—681no—
458C22⋊C4—681no—
459C22⋊C4—682no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—24no—
473C23—24no—
474C23—21no—
475C23—24no—
476C23—24no—
477C23—21no—
478C23—42no—
479C23—42no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—342no—
485C23—344no—
486C23—682no—
487C23—682no—
488C23—682no—
489C23—682no—
490C23—681no—
491C23—681no—
492C23—681no—
493C23—681no—
494C23—681no—
495C23—682no—
496C23—681no—
497C23—1362no—
498C23—1362no—
499C23—1362no—
500C23—1362no—
501C23#1058747712yes—
502C23#1058749112yes—
503C23#1058749112yes—
504C23#1058749412yes—
505C23#1058756712yes—
506C23C2×C8×D1711yescharacteristic
507C23C2×#6321811yescharacteristic
508C23#1058759312yes—
509C23C2×#6322711yescharacteristic
510D4—688no—
511D4—688no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—342no—
531C2×C4—682no—
532C2×C4—681no—
533C2×C4—681no—
534C2×C4—682no—
535C2×C4—681no—
536C2×C4—681no—
537C2×C4—681no—
538C2×C4—681no—
539C2×C4—681no—
540C2×C4—681no—
541C2×C4—1362no—
542C2×C4—1362no—
543C2×C4#1058747712yes—
544C2×C4#1058756811yescharacteristic
545C2×C4#1058757211yescharacteristic
546C2×C4#1058762512yes—
547C8—82no—
548C8—684no—
549C22—22no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—48no—
567C22—48no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—682no—
573C22—682no—
574C22—682no—
575C22—1364no—
576C22—1364no—
577C22—1361no—
578C22—1361no—
579C22—1361no—
580C22—1361no—
581C22#1176253214yescomplemented
582C22#1176268611yescharacteristic
583C22C2×#1058749111yescharacteristic
584C22#1176277711yescharacteristic
585C22C2×#1058747711yescharacteristic
586C22C2×#1058747711yescharacteristic
587C22C2×#1058749411yescharacteristic
588C22#1176279111yescharacteristic
589C22#1176287212yes—
590C22#1176287712yes—
591C4—22no—
592C4—21no—
593C4—22no—
594C4—21no—
595C4—41no—
596C4—682no—
597C4—1361no—
598C4—1361no—
599C2—22no—
600C2—22no—
601C2—24no—
602C2—42no—
603C2—1362no—
604C2C2×#1176253211yescharacteristic
605C2#1452710011yescharacteristic
606C2#1452713311yescharacteristic
607C2#1452714511yescharacteristic
608C2#1452907411yescharacteristic
609C2#1452923511yescharacteristic
610C2#1452924511yescharacteristic
subgroupG/H
centerC23C2×#63227
commutatorC23×C17C22×C8
FrattiniC23×C4C22×D17
FittingC17×#1779C2
socleC23×C17C2×C22⋊C4
prime radicalC23×C22⋊D17C4
cosocleC23×C4×C17C23
hypercenter#1779D17
hypocenterC17#9079
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊#3622C2semidirect
3. C2×#11762532C2semidirect
4. #14526626C2semidirect
5. C17×#1779C2semidirect
42. C22×C22⋊D17C8semidirect
71. C22⋊#3622D4semidirect
91. C23×D17C2×C8semidirect
139. C22×Dic17C2×C8semidirect
246. #1779D17semidirect
272. C22×C17#323semidirect
377. C17#9079semidirect
581. C22#11762532semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊#36222
C2×#117626882
C2×#117625322
#145266262
C17×#17792
#907917