Finite Group Databasebeta

← #16965458

Subgroups — #16965458

Order 4352.

1486 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762670C211yesmaximal, characteristic
2C2×#11762667C212yesmaximal
3#14527048C212yesmaximal, complemented
4#14527047C212yesmaximal, complemented
5#14527042C212yesmaximal, complemented
6#14527039C212yesmaximal, complemented
7#14527029C211yesmaximal, characteristic
8#14527027C212yesmaximal
9C17×#1439C211yesmaximal, characteristic
10C24×Dic17C2211yescharacteristic
11C22×C4⋊Dic17C2211yescharacteristic
12C22×#63225C2212yes—
13C22×#63225C2212yes—
14C22×#63225C2212yes—
15C22×C22⋊Dic17C2212yes—
16C22×C22⋊Dic17C2211yescharacteristic
17#11762670C418yescomplemented
18#11762667—28no—
19C2×#10587502C2212yes—
20C2×#10587502C2212yes—
21C2×#10587502C2212yes—
22C2×#10587502C2212yes—
23C2×#10587502C2212yes—
24C2×#10587502C2211yescharacteristic
25C2×#10587502C2212yes—
26C2×#10587502C2212yes—
27C2×#10587502C2212yes—
28C22×C4⋊C4×C17C2211yescharacteristic
29C22×C22⋊C4×C17C2212yes—
30C22×C22⋊C4×C17C2211yescharacteristic
31C2×C17×#95C2212yes—
32C2×C17×#95C2211yescharacteristic
33C25×C17C2311yescharacteristic
34C23×C4×C17C2312yes—
35C23×C4×C17C2312yes—
36C23×C4×C17C2311yescharacteristic
37C23×C4×C17C2311yescharacteristic
38C23×Dic17—41no—
39C23×Dic17—41no—
40C23×Dic17—41no—
41C23×Dic17—41no—
42C23×Dic17C2×C414yes—
43C23×Dic17D414yes—
44C23×Dic17Q814yes—
45C23×Dic17C2312yes—
46C23×Dic17C2312yes—
47C23×Dic17C2312yes—
48C23×Dic17C2312yes—
49C2×C4⋊C4×C17—42no—
50C2×C4⋊C4×C17—41no—
51C2×C22⋊C4×C17—24no—
52C2×C22⋊C4×C17—41no—
53C2×C22⋊C4×C17—41no—
54C2×C22⋊C4×C17—42no—
55C2×C22⋊C4×C17—42no—
56C2×C22⋊C4×C17C2×C414yes—
57C2×C22⋊Dic17—24no—
58C2×C22⋊Dic17—41no—
59C2×C22⋊Dic17—41no—
60C2×C22⋊Dic17—42no—
61C2×C22⋊Dic17—42no—
62C2×C22⋊Dic17C2×C414yes—
63C2×C4⋊Dic17—41no—
64C2×C4⋊Dic17—42no—
65C2×#63225—24no—
66C2×#63225—24no—
67C2×#63225—42no—
68C2×#63225—42no—
69C2×#63225—42no—
70C2×#63225—42no—
71C2×#63225—42no—
72C2×#63225—42no—
73C2×#63225—42no—
74C17×#95—42no—
75C17×#95—41no—
76#10587502—24no—
77#10587502—24no—
78#10587502—42no—
79#10587502—42no—
80#10587502—42no—
81#10587502—42no—
82#10587502—41no—
83#10587502C2×C418yes—
84#10587502C2×C418yes—
85C24×C17—21no—
86C24×C17—21no—
87C24×C17—21no—
88C24×C17—21no—
89C24×C17—22no—
90C24×C17—22no—
91C24×C17—22no—
92C24×C17—22no—
93C24×C17C22×C412yes—
94C24×C17C2×D412yes—
95C24×C17C2×Q812yes—
96C24×C17C2411yescharacteristic
97C22×C4×C17—21no—
98C22×C4×C17—22no—
99C22×C4×C17—22no—
100C22×C4×C17—21no—
101C22×C4×C17—21no—
102C22×C4×C17—21no—
103C22×C4×C17—22no—
104C22×C4×C17—22no—
105C22×C4×C17—22no—
106C22×C4×C17—22no—
107C22×C4×C17—22no—
108C22×C4×C17—22no—
109C22×C4×C17—22no—
110C22×C4×C17—22no—
111C22×C4×C17—22no—
112C22×C4×C17—22no—
113C22×C4×C17—22no—
114C22×C4×C17—22no—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—22no—
118C22×C4×C17—22no—
119C22×C4×C17C22×C412yes—
120C22×C4×C17C22×C412yes—
121C22×Dic17—22no—
122C22×Dic17—22no—
123C22×Dic17—22no—
124C22×Dic17—22no—
125C22×Dic17—22no—
126C22×Dic17—22no—
127C22×Dic17—22no—
128C22×Dic17—22no—
129C22×Dic17—22no—
130C22×Dic17—22no—
131C22×Dic17—22no—
132C22×Dic17—22no—
133C22×Dic17—22no—
134C22×Dic17—22no—
135C22×Dic17—22no—
136C22×Dic17—22no—
137C22×Dic17—22no—
138C22×Dic17—22no—
139C22×Dic17—22no—
140C22×Dic17—22no—
141C22×Dic17—22no—
142C22×Dic17—22no—
143C22×Dic17—22no—
144C22×Dic17—22no—
145C22×Dic17—44no—
146C22×Dic17—44no—
147C22×Dic17—44no—
148C22×Dic17—44no—
149C22×Dic17—44no—
150C22×Dic17—44no—
151C22×Dic17C4⋊C4116yescomplemented
152C22×Dic17C22×C414yes—
153C22×Dic17C22×C414yes—
154C22×Dic17C2×D414yes—
155C22×Dic17C2×Q814yes—
156C4⋊C4×C17—44no—
157C22⋊C4×C17—44no—
158C22⋊C4×C17—48no—
159C22⋊Dic17—44no—
160C22⋊Dic17—48no—
161C4⋊Dic17—44no—
162#63225—48no—
163#63225—48no—
164#63225—48no—
165#14674—171nomaximal, Sylow
166C23×C17—22no—
167C23×C17—22no—
168C23×C17—24no—
169C23×C17—24no—
170C23×C17—22no—
171C23×C17—22no—
172C23×C17—24no—
173C23×C17—24no—
174C23×C17—22no—
175C23×C17—22no—
176C23×C17—24no—
177C23×C17—24no—
178C23×C17—41no—
179C23×C17—42no—
180C23×C17—41no—
181C23×C17—42no—
182C23×C17—42no—
183C23×C17—42no—
184C23×C17—42no—
185C23×C17—41no—
186C23×C17—42no—
187C23×C17—41no—
188C23×C17C2×C4⋊C414yes—
189C23×C17C23×C411yescharacteristic
190C23×C17C22×D411yescharacteristic
191C23×C17C22×Q811yescharacteristic
192C23×C17H(5,𝔽2)11yescharacteristic
193C23×C17H(5,𝔽2)11yescharacteristic
194C23×C17H(5,𝔽2)12yes—
195C23×C17H(5,𝔽2)12yes—
196C23×C17H(5,𝔽2)11yescharacteristic
197C23×C17H(5,𝔽2)11yescharacteristic
198C23×C172−1+412yes—
199C23×C172−1+412yes—
200C2×C4×C17—22no—
201C2×C4×C17—22no—
202C2×C4×C17—24no—
203C2×C4×C17—24no—
204C2×C4×C17—41no—
205C2×C4×C17—41no—
206C2×C4×C17—42no—
207C2×C4×C17—42no—
208C2×C4×C17—41no—
209C2×C4×C17—41no—
210C2×C4×C17—42no—
211C2×C4×C17—42no—
212C2×C4×C17—42no—
213C2×C4×C17—42no—
214C2×C4×C17—42no—
215C2×C4×C17—42no—
216C2×C4×C17—42no—
217C2×C4×C17—42no—
218C2×C4×C17—42no—
219C2×C4×C17—42no—
220C2×C4×C17—42no—
221C2×C4×C17—42no—
222C2×C4×C17—42no—
223C2×C4×C17—42no—
224C2×Dic17—24no—
225C2×Dic17—24no—
226C2×Dic17—24no—
227C2×Dic17—24no—
228C2×Dic17—24no—
229C2×Dic17—24no—
230C2×Dic17—24no—
231C2×Dic17—24no—
232C2×Dic17—24no—
233C2×Dic17—416no—
234C2×Dic17—42no—
235C2×Dic17—42no—
236C2×Dic17—42no—
237C2×Dic17—42no—
238C2×Dic17—42no—
239C2×Dic17—42no—
240C2×Dic17—42no—
241C2×Dic17—42no—
242C2×Dic17—42no—
243C2×Dic17—42no—
244C2×Dic17—42no—
245C2×Dic17—42no—
246C2×Dic17—42no—
247C2×Dic17—42no—
248C2×Dic17—42no—
249C2×Dic17—42no—
250C2×Dic17—42no—
251C2×Dic17—42no—
252C2×Dic17C2×C4⋊C418yes—
253C2×#381—171no—
254C2×#380—172no—
255#1444—172no—
256#1444—172no—
257#1442—172no—
258#1442—172no—
259#1439—172no—
260#1439—171no—
261#1439D1711yescharacteristic
262C22×C17—24no—
263C22×C17—28no—
264C22×C17—44no—
265C22×C17—42no—
266C22×C17—42no—
267C22×C17—44no—
268C22×C17—42no—
269C22×C17—42no—
270C22×C17—44no—
271C22×C17—42no—
272C22×C17—42no—
273C22×C17C22×C4⋊C411yescharacteristic
274C22×C17#51811yescharacteristic
275C22×C17#51811yescharacteristic
276C22×C17#52012yes—
277C22×C17#52012yes—
278C22×C17#53411yescharacteristic
279C22×C17#53411yescharacteristic
280C22×C17#53512yes—
281C22×C17#53612yes—
282C22×C17#54311yescharacteristic
283C22×C17#54311yescharacteristic
284C22×C17#54412yes—
285C22×C17#54412yes—
286C22×C17#56011yescharacteristic
287C22×C17#56012yes—
288C22×C17#56011yescharacteristic
289C22×C17#56212yes—
290C22×C17#56212yes—
291C22×C17#56212yes—
292C22×C17#56212yes—
293C22×C17#56311yescharacteristic
294C22×C17#56312yes—
295C22×C17#56311yescharacteristic
296C4×C17—42no—
297C4×C17—42no—
298C4×C17—82no—
299C4×C17—82no—
300Dic17—28no—
301Dic17—44no—
302Dic17—82no—
303Dic17—82no—
304C24×C4—171no—
305C22×C4⋊C4—172no—
306C22×C4⋊C4—172no—
307C22×C4⋊C4—172no—
308C22×C4⋊C4—171no—
309C22×C4⋊C4C2×D1711yescharacteristic
310C22×C22⋊C4—171no—
311C22×C22⋊C4—172no—
312C22×C22⋊C4C2×D1712yes—
313C22×C22⋊C4C2×D1711yescharacteristic
314#381—178no—
315#380—348no—
316C2×#95—172no—
317C2×#95—172no—
318C2×#95—172no—
319C2×#95—172no—
320C2×#95—172no—
321C2×#95—171no—
322C2×#95—172no—
323C2×#95—172no—
324C2×#95—172no—
325C2×#95C2×D1711yescharacteristic
326C2×#95C2×D1712yes—
327C2×C17—44no—
328C2×C17#231511yescharacteristic
329C2×C17#231712yes—
330C2×C17#237611yescharacteristic
331C2×C17#237612yes—
332C2×C17#237611yescharacteristic
333C2×C17#281511yescharacteristic
334C2×C17#281612yes—
335C2×C17#281711yescharacteristic
336C2×C17#284111yescharacteristic
337C2×C17#284112yes—
338C2×C17#284111yescharacteristic
339C25C22×D1711yescharacteristic
340C23×C4—172no—
341C23×C4—174no—
342C23×C4—174no—
343C23×C4—174no—
344C23×C4—172no—
345C23×C4—172no—
346C23×C4—172no—
347C23×C4—681no—
348C23×C4—681no—
349C23×C4—681no—
350C23×C4—681no—
351C23×C4C22×D1712yes—
352C23×C4C22×D1712yes—
353C23×C4C22×D1711yescharacteristic
354C23×C4C22×D1711yescharacteristic
355C2×C4⋊C4—42no—
356C2×C4⋊C4—41no—
357C2×C4⋊C4—344no—
358C2×C4⋊C4—344no—
359C2×C4⋊C4—682no—
360C2×C4⋊C4—682no—
361C2×C4⋊C4—682no—
362C2×C4⋊C4—682no—
363C2×C4⋊C4—682no—
364C2×C4⋊C4—682no—
365C2×C4⋊C4—682no—
366C2×C4⋊C4—682no—
367C2×C4⋊C4—681no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—41no—
370C2×C22⋊C4—41no—
371C2×C22⋊C4—42no—
372C2×C22⋊C4—42no—
373C2×C22⋊C4—174no—
374C2×C22⋊C4—344no—
375C2×C22⋊C4—681no—
376C2×C22⋊C4—681no—
377C2×C22⋊C4—682no—
378C2×C22⋊C4—682no—
379C2×C22⋊C4C4×D1714yes—
380#95—41no—
381#95—42no—
382#95—178no—
383#95—178no—
384#95—344no—
385#95—344no—
386#95—681no—
387#95—682no—
388#95—682no—
389#95—682no—
390#95—682no—
391C17#1467411yescharacteristic, complemented, Sylow
392C24—22no—
393C24—22no—
394C24—21no—
395C24—21no—
396C24—21no—
397C24—21no—
398C24—22no—
399C24—22no—
400C24C2×C4×D1712yes—
401C24D4×D1712yes—
402C24Q8×D1712yes—
403C24C23×D1711yescharacteristic
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—174no—
427C22×C4—174no—
428C22×C4—174no—
429C22×C4—1716no—
430C22×C4—174no—
431C22×C4—342no—
432C22×C4—342no—
433C22×C4—342no—
434C22×C4—342no—
435C22×C4—342no—
436C22×C4—342no—
437C22×C4—342no—
438C22×C4—342no—
439C22×C4—342no—
440C22×C4—342no—
441C22×C4—342no—
442C22×C4—342no—
443C22×C4—342no—
444C22×C4—342no—
445C22×C4—342no—
446C22×C4—342no—
447C22×C4—342no—
448C22×C4—342no—
449C22×C4—342no—
450C22×C4—342no—
451C22×C4—342no—
452C22×C4—342no—
453C22×C4—342no—
454C22×C4—342no—
455C22×C4—684no—
456C22×C4—684no—
457C22×C4—684no—
458C22×C4—684no—
459C22×C4—684no—
460C22×C4—684no—
461C22×C4C2×C4×D1712yes—
462C22×C4C2×C4×D1712yes—
463C4⋊C4—44no—
464C4⋊C4—688no—
465C4⋊C4—688no—
466C4⋊C4—688no—
467C4⋊C4—684no—
468C22⋊C4—44no—
469C22⋊C4—48no—
470C22⋊C4—684no—
471C22⋊C4—688no—
472C23—22no—
473C23—22no—
474C23—24no—
475C23—24no—
476C23—22no—
477C23—22no—
478C23—24no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—41no—
485C23—42no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—41no—
492C23—42no—
493C23—41no—
494C23C4⋊C4×D1714yes—
495C23C22×C4×D1711yescharacteristic
496C23C2×D4×D1711yescharacteristic
497C23#1058769111yescharacteristic
498C23#1058769111yescharacteristic
499C23#1058769112yes—
500C23#1058769112yes—
501C23C2×Q8×D1711yescharacteristic
502C23#1058769611yescharacteristic
503C23#1058769611yescharacteristic
504C23#1058769712yes—
505C23#1058769712yes—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—178no—
531C2×C4—344no—
532C2×C4—344no—
533C2×C4—344no—
534C2×C4—344no—
535C2×C4—344no—
536C2×C4—344no—
537C2×C4—344no—
538C2×C4—344no—
539C2×C4—344no—
540C2×C4—6816no—
541C2×C4—682no—
542C2×C4—682no—
543C2×C4—682no—
544C2×C4—682no—
545C2×C4—682no—
546C2×C4—682no—
547C2×C4—682no—
548C2×C4—682no—
549C2×C4—682no—
550C2×C4—682no—
551C2×C4—682no—
552C2×C4—682no—
553C2×C4—682no—
554C2×C4—682no—
555C2×C4—682no—
556C2×C4—682no—
557C2×C4—682no—
558C2×C4—682no—
559C22—24no—
560C22—28no—
561C22—42no—
562C22—44no—
563C22—42no—
564C22—42no—
565C22—44no—
566C22—42no—
567C22—42no—
568C22—44no—
569C22—42no—
570C22C2×C4⋊C4×D1711yescharacteristic
571C22#1176328211yescharacteristic
572C22#1176328311yescharacteristic
573C22#1176328812yes—
574C22#1176328812yes—
575C22#1176336711yescharacteristic
576C22#1176336711yescharacteristic
577C22#1176337412yes—
578C22#1176338112yes—
579C22#1176343411yescharacteristic
580C22#1176343411yescharacteristic
581C22#1176343812yes—
582C22#1176343812yes—
583C22#1176354611yescharacteristic
584C22#1176354712yes—
585C22#1176354811yescharacteristic
586C22#1176355312yes—
587C22#1176355412yes—
588C22#1176355412yes—
589C22#1176355612yes—
590C22#1176355911yescharacteristic
591C22#1176356112yes—
592C22#1176356211yescharacteristic
593C4—42no—
594C4—42no—
595C4—82no—
596C4—82no—
597C4—348no—
598C4—684no—
599C4—1362no—
600C4—1362no—
601C2—44no—
602C2#1453139711yescharacteristic
603C2#1453140312yes—
604C2#1453192711yescharacteristic
605C2#1453193012yes—
606C2#1453193411yescharacteristic
607C2#1453680911yescharacteristic
608C2#1453681612yes—
609C2#1453682211yescharacteristic
610C2#1453704411yescharacteristic
611C2#1453704512yes—
612C2#1453704611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC17×#1439C2
socle, cosocleC24×C17C24
prime radicalC25×C17C23
hypercenter#1439D17
hypocenterC17#14674
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
3. #14527048C2semidirect
4. #14527047C2semidirect
5. #14527042C2semidirect
6. #14527039C2semidirect
17. #11762670C4semidirect
151. C22×Dic17C4⋊C4semidirect
391. C17#14674semidirect

Maximal subgroups

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

subgroupindex
C2×#117626702
C2×#117626672
#145270482
#145270472
#145270422
#145270392
#145270292
#145270272
C17×#14392
#1467417