Finite Group Databasebeta

← #17046360

Subgroups — #17046360

Order 4352.

848 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763087C211yesmaximal, characteristic
2C4×#10587574C211yesmaximal, characteristic, complemented
3#14531246C211yesmaximal, characteristic
4#14529734C211yesmaximal, characteristic, complemented
5#14529539C211yesmaximal, characteristic, complemented
6#14529516C211yesmaximal, characteristic, complemented
7#14529515C211yesmaximal, characteristic, complemented
8#14529469C211yesmaximal, characteristic
9#14529464C211yesmaximal, characteristic
10#14529401C211yesmaximal, characteristic
11#14529356C211yesmaximal, characteristic, complemented
12#14529355C211yesmaximal, characteristic, complemented
13#14528782C211yesmaximal, characteristic
14#14528632C211yesmaximal, characteristic, complemented
15C17×#1925C211yesmaximal, characteristic
16C2×#10587574C2211yescharacteristic
17C2×C4×Dic34—21no—
18C2×D4×Dic17C2211yescharacteristic
19C2×#10587563—21no—
20#11763087C418yescomplemented
21C2×#10587489C2211yescharacteristic
22C2×#10587478C2211yescharacteristic
23C2×#10587478C2211yescharacteristic
24C2×#10587485C2211yescharacteristic
25C2×#10587485C2211yescharacteristic
26C2×#10587503C2211yescharacteristic
27C2×#10587503C2211yescharacteristic
28C2×C8×Dic17C2211yescharacteristic
29C2×C8×Dic17C2211yescharacteristic
30#11762713—21no—
31#11762712C2211yescharacteristic
32#11762708C2211yescharacteristic
33#11762707C2211yescharacteristic
34#11762706—21no—
35#11762705C2211yescharacteristic
36#11762705C2211yescharacteristic
37#11762703—21no—
38#11762697—21no—
39#11762678C2211yescharacteristic
40C4⋊C4×Dic17C2211yescharacteristic
41C4×#63225—21no—
42C4×C4⋊Dic17C2211yescharacteristic
43C22⋊C4×Dic17—21no—
44C42×Dic17C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762564C2211yescharacteristic
53C2×C17×#107C2211yescharacteristic
54C2×C17×#102C2211yescharacteristic
55C2×C17×#102C2211yescharacteristic
56C17×#390C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#336C2211yescharacteristic
60C22×D4×C17C2311yescharacteristic
61C23×Dic17—21no—
62C22×Dic34—21no—
63C22×C8×C17C2311yescharacteristic
64C22×C8×C17C2311yescharacteristic
65C2×C4⋊C4×C17C2311yescharacteristic
66C2×C4⋊C4×C17C2311yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C22⋊C4×C17—21no—
69C2×C22⋊C4×C17—21no—
70C2×C42×C17—21no—
71C2×C42×C17C2311yescharacteristic
72C2×C22⋊Dic17—21no—
73C2×C22⋊Dic17—21no—
74C2×C22⋊Dic17—21no—
75D4×Dic17—22no—
76D4×Dic17C2×C414yes—
77C2×C4⋊Dic17—21no—
78C2×C4⋊Dic17C2311yescharacteristic
79C2×C4⋊Dic17C2311yescharacteristic
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×#63225—21no—
83C2×#63225—21no—
84C2×#63225—21no—
85C2×#63225—21no—
86C2×#63225—21no—
87C2×#63225—21no—
88C2×C4×Dic17—21no—
89C2×C4×Dic17—21no—
90C2×C4×Dic17D411yescharacteristic
91C2×C4×Dic17D411yescharacteristic
92C2×C4×Dic17D411yescharacteristic
93C2×C4×Dic17D411yescharacteristic
94C2×C4×Dic17C2311yescharacteristic
95C2×C4×Dic17C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97C22×#3622C2311yescharacteristic
98#10587574—22no—
99#10587574C2×C414yes—
100#10587563—44no—
101C4×Dic34—44no—
102C17×#107—41no—
103C17×#102—41no—
104C17×#102C2×C414yes—
105C17×#95—21no—
106#10587503—41no—
107#10587503C2×C414yes—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587489C2×C414yes—
115#10587485—22no—
116#10587485C2×C414yes—
117C8×Dic17—41no—
118C8×Dic17—41no—
119#10587478—41no—
120#10587478C2×C414yes—
121C24×C17—21no—
122C2×D4×C17—21no—
123C2×D4×C17—21no—
124C2×D4×C17—21no—
125C2×D4×C17—21no—
126C2×D4×C17—21no—
127C2×D4×C17C22×C412yes—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17C2×D411yescharacteristic
132C22×C4×C17C2×D411yescharacteristic
133C22×C4×C17C2411yescharacteristic
134C22×Dic17—21no—
135C22×Dic17—21no—
136C22×Dic17—24no—
137C22×Dic17—21no—
138C22×Dic17—21no—
139C22×Dic17—42no—
140C22×Dic17—42no—
141C22×Dic17C2×D411yescharacteristic
142C22×Dic17C2×D411yescharacteristic
143C22×Dic17C2×D411yescharacteristic
144C22×Dic17C2×D411yescharacteristic
145C2×Dic34—24no—
146C2×Dic34—42no—
147C2×Dic34—42no—
148C2×C8×C17—21no—
149C2×C8×C17—21no—
150C2×C8×C17—21no—
151C2×C8×C17—21no—
152C2×C8×C17—21no—
153C2×C8×C17C22×C412yes—
154C4⋊C4×C17—21no—
155C4⋊C4×C17—21no—
156C4⋊C4×C17—21no—
157C4⋊C4×C17—21no—
158C4⋊C4×C17—21no—
159C4⋊C4×C17C22×C412yes—
160C22⋊C4×C17—42no—
161C22⋊C4×C17—42no—
162C42×C17—21no—
163C42×C17—21no—
164C42×C17—42no—
165C22⋊Dic17—24no—
166C22⋊Dic17—42no—
167C22⋊Dic17—42no—
168C4⋊Dic17—21no—
169C4⋊Dic17—21no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—42no—
175C4⋊Dic17C22×C412yes—
176C4⋊Dic17C22×C412yes—
177#63225—24no—
178#63225—42no—
179#63225—42no—
180#63225—42no—
181#63225—42no—
182#63225—42no—
183C4×Dic17—21no—
184C4×Dic17—21no—
185C4×Dic17—22no—
186C4×Dic17—22no—
187C4×Dic17—41no—
188C4×Dic17—42no—
189C4×Dic17—42no—
190C4×Dic17D814yes—
191C4×Dic17C22×C412yes—
192C4×Dic17#4112yes—
193C2×#3622—21no—
194C2×#3622—21no—
195C2×#3622—21no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622C22×C412yes—
199#22110—171nomaximal, Sylow
200C23×C17—22no—
201C23×C17—41no—
202C23×C17—41no—
203C23×C17—41no—
204C23×C17—41no—
205C23×C17—41no—
206C23×C17—41no—
207C23×C17C22×D411yescharacteristic
208D4×C17—22no—
209D4×C17—22no—
210D4×C17—41no—
211D4×C17—41no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—22no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—41no—
219C2×C4×C17—41no—
220C2×C4×C17—41no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17C2×D812yes—
227C2×C4×C17C23×C411yescharacteristic
228C2×C4×C17C2×#4111yescharacteristic
229C2×C4×C17H(5,𝔽2)11yescharacteristic
230C2×C4×C172−1+411yescharacteristic
231C2×C4×C172−1+411yescharacteristic
232C2×C4×C172−1+411yescharacteristic
233C2×Dic17—21no—
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—22no—
240C2×Dic17—22no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—48no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17C4×D412yes—
253C2×Dic17C4×D412yes—
254C2×Dic17C2×D812yes—
255C2×Dic17C2×D812yes—
256C2×Dic17#13512yes—
257C2×Dic17#13712yes—
258Dic34—48no—
259C8×C17—22no—
260C8×C17—41no—
261#3622—22no—
262#3622—41no—
263C2×#474—171no—
264C4×C4⋊Q8—171no—
265#2293—171no—
266#1988—171no—
267#1947—171no—
268#1942—171no—
269#1942—171no—
270#1934—171no—
271#1934—171no—
272#1925—171no—
273#1925D1711yescharacteristic
274#1918—171no—
275#1918—171no—
276#1802—171no—
277C4×C4⋊C8—171no—
278C22×C17—42no—
279C22×C17—42no—
280C22×C17—42no—
281C22×C17—81no—
282C22×C17—81no—
283C22×C17—81no—
284C22×C17—81no—
285C22×C17C2×C4×D411yescharacteristic
286C22×C17#53611yescharacteristic
287C22×C17#53711yescharacteristic
288C22×C17C22×D811yescharacteristic
289C22×C17C2×#13511yescharacteristic
290C22×C17C2×#13711yescharacteristic
291C22×C17#57511yescharacteristic
292C4×C17—22no—
293C4×C17—42no—
294C4×C17—41no—
295C4×C17—81no—
296C4×C17—81no—
297C4×C17#51911yescharacteristic
298C4×C17#52011yescharacteristic
299C4×C17#54111yescharacteristic
300C4×C17#54211yescharacteristic
301Dic17—22no—
302Dic17—22no—
303Dic17—22no—
304Dic17—42no—
305Dic17—42no—
306Dic17—81no—
307Dic17—81no—
308Dic17C4×D814yes—
309C2×C4⋊Q8—171no—
310C2×C4×Q8—341no—
311C2×C4×D4—171no—
312C2×#117—341no—
313#474—178no—
314C2×#107—171no—
315C2×#107—171no—
316C2×#107—171no—
317C2×#107C2×D1711yescharacteristic
318C2×C4⋊C8—171no—
319C2×C4⋊C8—171no—
320C2×#102—171no—
321C2×#102—171no—
322C2×#102C2×D1711yescharacteristic
323C2×#102C2×D1711yescharacteristic
324C2×C4×C8—171no—
325C2×C4×C8—171no—
326#391—341no—
327#390—171no—
328#390C2×D1711yescharacteristic
329#389—171no—
330#389—171no—
331#389—171no—
332#389—171no—
333#389—341no—
334#388—341no—
335#388—341no—
336#384—171no—
337C4×C4⋊C4—171no—
338C4×C4⋊C4—171no—
339C4×C4⋊C4—341no—
340C4×C4⋊C4C2×D1711yescharacteristic
341C4×C22⋊C4—341no—
342C43—171no—
343#340—171no—
344#340—171no—
345#340—171no—
346#340—171no—
347#340—171no—
348#340—171no—
349#340—171no—
350#340C2×D1711yescharacteristic
351#336—171no—
352#336C2×D1711yescharacteristic
353C2×C17—82no—
354C2×C17#232311yescharacteristic
355C2×C17C2×C4×D811yescharacteristic
356C2×C17C4×#13711yescharacteristic
357C2×C17#322611yescharacteristic
358C2×C17#323311yescharacteristic
359C2×C17#324511yescharacteristic
360C2×C17#324811yescharacteristic
361C22×Q8—341no—
362C22×D4C22×D1711yescharacteristic
363C23×C4—341no—
364C22×C8—171no—
365C22×C8—171no—
366C22×C8C22×D1711yescharacteristic
367C22×C8C22×D1711yescharacteristic
368C4⋊Q8—174no—
369C4⋊Q8—342no—
370C4×Q8—684no—
371C4×D4—174no—
372C4×D4—342no—
373#117—684no—
374C2×C4⋊C4—171no—
375C2×C4⋊C4—171no—
376C2×C4⋊C4—171no—
377C2×C4⋊C4—171no—
378C2×C4⋊C4—341no—
379C2×C4⋊C4—341no—
380C2×C4⋊C4—341no—
381C2×C4⋊C4—341no—
382C2×C4⋊C4—341no—
383C2×C4⋊C4—341no—
384C2×C4⋊C4—341no—
385C2×C4⋊C4C22×D1711yescharacteristic
386C2×C4⋊C4C22×D1711yescharacteristic
387C2×C4⋊C4C22×D1711yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—341no—
391C2×C22⋊C4—341no—
392C2×C22⋊C4—341no—
393C2×C42—21no—
394C2×C42—171no—
395C2×C42—171no—
396C2×C42—171no—
397C2×C42—171no—
398C2×C42—171no—
399C2×C42—171no—
400C2×C42—341no—
401C2×C42—341no—
402C2×C42C22×D1711yescharacteristic
403#107—41no—
404#107—174no—
405#107—174no—
406#107—681no—
407C4⋊C8—174no—
408C4⋊C8—342no—
409#102—41no—
410#102—174no—
411#102—681no—
412#102C4×D1714yes—
413C4×C8—681no—
414C4×C8—681no—
415#95—21no—
416#95—341no—
417#95—341no—
418#95—341no—
419#95—341no—
420#95—341no—
421#95—341no—
422C17#2211011yescharacteristic, complemented, Sylow
423C24—21no—
424C2×Q8—344no—
425C2×Q8—682no—
426C2×Q8—682no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4C2×C4×D1712yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—171no—
437C22×C4—171no—
438C22×C4—171no—
439C22×C4—171no—
440C22×C4—344no—
441C22×C4—341no—
442C22×C4—341no—
443C22×C4—341no—
444C22×C4—341no—
445C22×C4—682no—
446C22×C4—682no—
447C22×C4D4×D1711yescharacteristic
448C22×C4D4×D1711yescharacteristic
449C22×C4C23×D1711yescharacteristic
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—172no—
456C2×C8—341no—
457C2×C8—341no—
458C2×C8—341no—
459C2×C8—341no—
460C2×C8—341no—
461C2×C8C2×C4×D1712yes—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—172no—
468C4⋊C4—172no—
469C4⋊C4—344no—
470C4⋊C4—341no—
471C4⋊C4—341no—
472C4⋊C4—341no—
473C4⋊C4—341no—
474C4⋊C4—341no—
475C4⋊C4—341no—
476C4⋊C4—682no—
477C4⋊C4—682no—
478C4⋊C4—682no—
479C4⋊C4—682no—
480C4⋊C4—682no—
481C4⋊C4—682no—
482C4⋊C4C2×C4×D1712yes—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—344no—
486C22⋊C4—682no—
487C22⋊C4—682no—
488C42—21no—
489C42—21no—
490C42—42no—
491C42—172no—
492C42—172no—
493C42—174no—
494C42—342no—
495C42—342no—
496C42—341no—
497C42—341no—
498C42—681no—
499C42—682no—
500C42—682no—
501C23—22no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23C2×D4×D1711yescharacteristic
509Q8—688no—
510D4—22no—
511D4—22no—
512D4—41no—
513D4—41no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—172no—
529C2×C4—172no—
530C2×C4—172no—
531C2×C4—172no—
532C2×C4—172no—
533C2×C4—172no—
534C2×C4—341no—
535C2×C4—341no—
536C2×C4—341no—
537C2×C4—341no—
538C2×C4—341no—
539C2×C4—341no—
540C2×C4—342no—
541C2×C4—342no—
542C2×C4—688no—
543C2×C4—681no—
544C2×C4—681no—
545C2×C4—681no—
546C2×C4—681no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4—681no—
551C2×C4—681no—
552C2×C4—681no—
553C2×C4D8×D1712yes—
554C2×C4C22×C4×D1711yescharacteristic
555C2×C4#1058769111yescharacteristic
556C2×C4#41×D1711yescharacteristic
557C2×C4#1058769711yescharacteristic
558C2×C4#1058769711yescharacteristic
559C2×C4#1058769711yescharacteristic
560C8—22no—
561C8—41no—
562C8—342no—
563C8—681no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22C4×D4×D1711yescharacteristic
572C22#1176338111yescharacteristic
573C22#1176338811yescharacteristic
574C22C2×D8×D1711yescharacteristic
575C22#135×D1711yescharacteristic
576C22#137×D1711yescharacteristic
577C22#1176360511yescharacteristic
578C4—22no—
579C4—42no—
580C4—41no—
581C4—81no—
582C4—81no—
583C4—174no—
584C4—342no—
585C4—342no—
586C4—342no—
587C4—682no—
588C4—682no—
589C4—1361no—
590C4—1361no—
591C4#1176328611yescharacteristic
592C4#1176328811yescharacteristic
593C4#1176341611yescharacteristic
594C4#1176342311yescharacteristic
595C2—82no—
596C2#1453146311yescharacteristic
597C2C4×D8×D1711yescharacteristic
598C2#1453760611yescharacteristic
599C2#1453990311yescharacteristic
600C2#1453996011yescharacteristic
601C2#1454009211yescharacteristic
602C2#1454011811yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1925C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenter#1925D17
hypocenterC17#22110
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1925

Complements

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

NH ≅ G/Ntype
2. C4×#10587574C2semidirect
4. #14529734C2semidirect
5. #14529539C2semidirect
6. #14529516C2semidirect
7. #14529515C2semidirect
11. #14529356C2semidirect
12. #14529355C2semidirect
14. #14528632C2semidirect
20. #11763087C4semidirect
422. C17#22110semidirect

Maximal subgroups

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

subgroupindex
C2×#117630872
C4×#105875742
#145312462
#145297342
#145295392
#145295162
#145295152
#145294692
#145294642
#145294012
#145293562
#145293552
#145287822
#145286322
C17×#19252
#2211017