Finite Group Databasebeta

← #17060784

Subgroups — #17060784

Order 4352.

622 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14532060C211yesmaximal, characteristic
2#14529724C211yesmaximal, characteristic, complemented
3#14529718C211yesmaximal, characteristic, complemented
4#14529508C211yesmaximal, characteristic, complemented
5#14529504C211yesmaximal, characteristic, complemented
6#14529398C211yesmaximal, characteristic
7#14529397C211yesmaximal, characteristic
8#14529395C211yesmaximal, characteristic
9#14529391C211yesmaximal, characteristic
10#14529351C211yesmaximal, characteristic, complemented
11#14529346C211yesmaximal, characteristic, complemented
12#14529344C211yesmaximal, characteristic, complemented
13#14528854C211yesmaximal, characteristic
14#14528765C211yesmaximal, characteristic
15C17×#1917C211yesmaximal, characteristic, complemented
16C2×C4×Dic34C2211yescharacteristic
17C2×#10587558C2211yescharacteristic
18C2×C4⋊#3622C2211yescharacteristic
19C2×C4⋊#3622C2211yescharacteristic
20C2×C4⋊#3622C2211yescharacteristic
21C2×#10587485C2211yescharacteristic
22C2×#10587485C2211yescharacteristic
23C2×#10587491C2211yescharacteristic
24C2×#10587491C2211yescharacteristic
25C2×#10587491C2211yescharacteristic
26C2×#10587491C2211yescharacteristic
27C2×#10587474C2211yescharacteristic
28C2×#10587486C2211yescharacteristic
29C2×C8×Dic17C2211yescharacteristic
30#11762750C411yescharacteristic
31#11762744C411yescharacteristic
32#11762743C411yescharacteristic
33#11762740C411yescharacteristic
34#11762733C411yescharacteristic
35#11762731C411yescharacteristic
36#11762729C411yescharacteristic
37#11762722C411yescharacteristic
38C4×#63225C2211yescharacteristic
39C4×#63225C2211yescharacteristic
40C4×#63227C2211yescharacteristic
41C4×#63227C2211yescharacteristic
42#11762564C2211yescharacteristic
43#11762564C2211yescharacteristic
44#11762563C2211yescharacteristic
45#11762563C2211yescharacteristic
46#11762563C2211yescharacteristic
47#11762563C2211yescharacteristic
48#11762563C2211yescharacteristic
49#11762563C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C2×C17×C4⋊C8C2211yescharacteristic
53C2×C17×#97C2211yescharacteristic
54C4×C4⋊C4×C17C2211yescharacteristic
55C17×#336C2211yescharacteristic
56C17×#336C2211yescharacteristic
57C17×#336C2211yescharacteristic
58C17×#336C2211yescharacteristic
59C22×C4×D17C2311yescharacteristic
60C22×Dic34C2×C411yescharacteristic
61C22×C8×C17C2311yescharacteristic
62C22×C8×C17C2311yescharacteristic
63C22×C8×C17C2311yescharacteristic
64C22×C8×C17C2311yescharacteristic
65C2×C4⋊C4×C17C2×C411yescharacteristic
66C2×C4⋊C4×C17C2×C411yescharacteristic
67C2×C42×C17C2311yescharacteristic
68C2×C42×C17C2311yescharacteristic
69C2×C42×C17C2311yescharacteristic
70C2×#63227C2×C411yescharacteristic
71C2×#63227C2×C411yescharacteristic
72C2×#63227C2×C411yescharacteristic
73C2×#63227C2×C411yescharacteristic
74C2×#63227C2×C411yescharacteristic
75C2×#63227C2×C411yescharacteristic
76C2×C4⋊Dic17C2×C411yescharacteristic
77C2×#63225C2×C411yescharacteristic
78C2×#63225C2×C411yescharacteristic
79C2×#63225C2×C411yescharacteristic
80C2×#63225C2×C411yescharacteristic
81C2×#63225C2×C411yescharacteristic
82C2×#63225C2×C411yescharacteristic
83C2×#63225C2×C411yescharacteristic
84C2×#63225C2×C411yescharacteristic
85C2×C4×Dic17C2311yescharacteristic
86C2×C4×Dic17C2311yescharacteristic
87C2×C4×Dic17C2311yescharacteristic
88C22×#3622C2311yescharacteristic
89C22×#3622C2311yescharacteristic
90C22×#3622C2311yescharacteristic
91C22×#3622C2311yescharacteristic
92#10587558—41no—
93#10587558—41no—
94C4×Dic34—41no—
95C4×Dic34—41no—
96C17×C4⋊C8—41no—
97C17×#97—41no—
98C17×#95C2×C411yescharacteristic
99C17×#95C2×C411yescharacteristic
100#10587502C2×C411yescharacteristic
101#10587502C2×C411yescharacteristic
102#10587502C2×C411yescharacteristic
103#10587502C2×C411yescharacteristic
104#10587502C2×C411yescharacteristic
105#10587502C2×C411yescharacteristic
106#10587502C2×C411yescharacteristic
107#10587502C2×C411yescharacteristic
108#10587491—41no—
109#10587491—41no—
110#10587491—41no—
111#10587491—41no—
112#10587486—41no—
113#10587485—41no—
114#10587485—41no—
115C8×Dic17—41no—
116C4⋊#3622—41no—
117C4⋊#3622—41no—
118C4⋊#3622—41no—
119#10587474—41no—
120C23×D17C22×C411yescharacteristic
121C22×C4×C17C22×C411yescharacteristic
122C22×C4×C17C22×C411yescharacteristic
123C22×C4×C17C22×C411yescharacteristic
124C22×C4×C17C22×C411yescharacteristic
125C22×C4×C17C22×C411yescharacteristic
126C22×C4×C17C22×C411yescharacteristic
127C22×C4×C17C2411yescharacteristic
128C22×Dic17C22×C411yescharacteristic
129C22×Dic17C22×C411yescharacteristic
130C22×Dic17C22×C411yescharacteristic
131C22×Dic17C22×C411yescharacteristic
132C22×Dic17C22×C411yescharacteristic
133C22×Dic17C22×C411yescharacteristic
134C22×Dic17C22×C411yescharacteristic
135C2×C4×D17—21no—
136C2×C4×D17—21no—
137C2×C4×D17—21no—
138C2×C4×D17—21no—
139C2×C4×D17—21no—
140C2×C4×D17—21no—
141C2×Dic34—22no—
142C2×Dic34—41no—
143C2×Dic34—41no—
144C2×C8×C17—21no—
145C2×C8×C17—21no—
146C2×C8×C17—21no—
147C2×C8×C17—21no—
148C2×C8×C17—21no—
149C2×C8×C17—21no—
150C2×C8×C17—21no—
151C2×C8×C17—21no—
152C2×C8×C17—21no—
153C2×C8×C17—21no—
154C2×C8×C17—21no—
155C2×C8×C17—21no—
156C4⋊C4×C17—41no—
157C4⋊C4×C17—41no—
158C42×C17—21no—
159C42×C17—21no—
160C42×C17—21no—
161C42×C17—21no—
162C42×C17—21no—
163C42×C17—21no—
164#63227—22no—
165#63227—41no—
166#63227—41no—
167#63227—41no—
168#63227—41no—
169#63227—41no—
170C4⋊Dic17—41no—
171#63225—22no—
172#63225—41no—
173#63225—41no—
174#63225—41no—
175#63225—41no—
176#63225—41no—
177#63225—41no—
178#63225—41no—
179C4×Dic17—21no—
180C4×Dic17—21no—
181C4×Dic17—21no—
182C4×Dic17—21no—
183C4×Dic17—21no—
184C4×Dic17—21no—
185C2×#3622—21no—
186C2×#3622—21no—
187C2×#3622—21no—
188C2×#3622—21no—
189C2×#3622—21no—
190C2×#3622—21no—
191C2×#3622—21no—
192C2×#3622—21no—
193C2×#3622—21no—
194C2×#3622—21no—
195C2×#3622—21no—
196C2×#3622—21no—
197#23345—171nomaximal, Sylow
198C23×C17C23×C411yescharacteristic
199C22×D17—21no—
200C22×D17—21no—
201C22×D17—21no—
202C22×D17—21no—
203C22×D17—21no—
204C22×D17—21no—
205C22×D17—21no—
206C2×C4×C17—21no—
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—21no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17H(5,𝔽2)11yescharacteristic
225C2×C4×C17H(5,𝔽2)11yescharacteristic
226C2×C4×C17H(5,𝔽2)11yescharacteristic
227C2×C4×C172−1+411yescharacteristic
228C2×C4×C172−1+411yescharacteristic
229C2×C4×C172−1+411yescharacteristic
230C2×Dic17—21no—
231C2×Dic17—21no—
232C2×Dic17—21no—
233C2×Dic17—21no—
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—21no—
244C2×Dic17—21no—
245C2×Dic17—21no—
246C2×Dic17—21no—
247C2×Dic17—21no—
248C2×Dic17—21no—
249C2×Dic17—21no—
250C2×Dic17#13112yes—
251C4×D17—41no—
252C4×D17—41no—
253C4×D17—41no—
254C4×D17—41no—
255Dic34—44no—
256C8×C17—41no—
257C8×C17—41no—
258C8×C17—41no—
259C8×C17—41no—
260#3622—41no—
261#3622—41no—
262#3622—41no—
263#3622—41no—
264#2396—171no—
265#1985—171no—
266#1984—171no—
267#1940—171no—
268#1940—171no—
269#1924—171no—
270#1924—171no—
271#1924—171no—
272#1923—171no—
273#1917—171no—
274#1917—171no—
275#1917D1711yescharacteristic, complemented
276#1916—171no—
277#1818—171no—
278#1798—171no—
279C22×C17#51811yescharacteristic
280C22×C17#51911yescharacteristic
281C22×C17#52011yescharacteristic
282C22×C17C2×#13111yescharacteristic
283C22×C17#56811yescharacteristic
284C22×C17#56811yescharacteristic
285C22×C17#56811yescharacteristic
286C2×D17—41no—
287C2×D17—41no—
288C2×D17—41no—
289C2×D17—41no—
290C2×D17—41no—
291C2×D17—41no—
292C2×D17—41no—
293C4×C17—41no—
294C4×C17—41no—
295C4×C17—41no—
296C4×C17—41no—
297C4×C17—41no—
298C4×C17—41no—
299C4×C17#56211yescharacteristic
300C4×C17#56211yescharacteristic
301C4×C17#56311yescharacteristic
302C4×C17#56311yescharacteristic
303Dic17—22no—
304Dic17—41no—
305Dic17—41no—
306Dic17—41no—
307Dic17—41no—
308Dic17—41no—
309Dic17—41no—
310C2×C4×Q8—171no—
311C2×#117—171no—
312C2×C4⋊C8—171no—
313C2×C4⋊C8—171no—
314C2×C4⋊C8—171no—
315C2×C4⋊C8—171no—
316C2×C4⋊C8—171no—
317C2×C4⋊C8C2×D1711yescharacteristic
318C2×C22⋊C8—171no—
319C2×C22⋊C8—171no—
320C2×C22⋊C8—171no—
321C2×C22⋊C8—171no—
322C2×#97—171no—
323C2×#97—171no—
324C2×#97C2×D1711yescharacteristic
325C2×C4×C8—171no—
326#400—171no—
327#399—171no—
328#398—171no—
329#398—171no—
330#396—171no—
331#396—171no—
332#395—171no—
333#393—171no—
334C4×C4⋊C4—171no—
335C4×C4⋊C4—171no—
336C4×C4⋊C4C2×D1711yescharacteristic
337C4×C22⋊C4—171no—
338C4×C22⋊C4—171no—
339#336—171no—
340#336—171no—
341#336—171no—
342#336—171no—
343#336—171no—
344#336—171no—
345#336—171no—
346#336—171no—
347#336—171no—
348#336—171no—
349#336C2×D1711yescharacteristic
350#336C2×D1711yescharacteristic
351#336C2×D1711yescharacteristic
352#336C2×D1711yescharacteristic
353C2×C17#237811yescharacteristic
354C2×C17#297611yescharacteristic
355C2×C17#298011yescharacteristic
356C2×C17#298511yescharacteristic
357C2×C17#298711yescharacteristic
358C2×C17#299311yescharacteristic
359C2×C17#299511yescharacteristic
360D17—81no—
361C22×Q8—171no—
362C23×C4—171no—
363C22×C8—171no—
364C22×C8—171no—
365C22×C8—171no—
366C22×C8—171no—
367C22×C8C22×D1711yescharacteristic
368C22×C8C22×D1711yescharacteristic
369C22×C8C22×D1711yescharacteristic
370C22×C8C22×D1711yescharacteristic
371C4×Q8—681no—
372C4×Q8—681no—
373#117—681no—
374#117—681no—
375C2×C4⋊C4—171no—
376C2×C4⋊C4—171no—
377C2×C4⋊C4—171no—
378C2×C4⋊C4—171no—
379C2×C4⋊C4—171no—
380C2×C4⋊C4—171no—
381C2×C4⋊C4—171no—
382C2×C4⋊C4—171no—
383C2×C4⋊C4—171no—
384C2×C4⋊C4C4×D1711yescharacteristic
385C2×C4⋊C4C4×D1711yescharacteristic
386C2×C22⋊C4—171no—
387C2×C22⋊C4—171no—
388C2×C22⋊C4—171no—
389C2×C22⋊C4—171no—
390C2×C22⋊C4—171no—
391C2×C22⋊C4—171no—
392C2×C42—171no—
393C2×C42—171no—
394C2×C42—171no—
395C2×C42C22×D1711yescharacteristic
396C2×C42C22×D1711yescharacteristic
397C2×C42C22×D1711yescharacteristic
398C4⋊C8—41no—
399C4⋊C8—681no—
400C4⋊C8—681no—
401C4⋊C8—681no—
402C4⋊C8—681no—
403C4⋊C8—681no—
404C22⋊C8—681no—
405C22⋊C8—681no—
406C22⋊C8—681no—
407C22⋊C8—681no—
408#97—41no—
409#97—681no—
410#97—681no—
411C4×C8—681no—
412#95—171no—
413#95—171no—
414#95—171no—
415#95—171no—
416#95—171no—
417#95—171no—
418#95—171no—
419#95—171no—
420#95C4×D1711yescharacteristic
421#95C4×D1711yescharacteristic
422C17#2334511yescharacteristic, complemented, Sylow
423C24—171no—
424C2×Q8—342no—
425C2×Q8—681no—
426C2×Q8—681no—
427C22×C4—171no—
428C22×C4—171no—
429C22×C4—171no—
430C22×C4—171no—
431C22×C4—171no—
432C22×C4—171no—
433C22×C4—171no—
434C22×C4—341no—
435C22×C4—341no—
436C22×C4—341no—
437C22×C4—341no—
438C22×C4—341no—
439C22×C4—341no—
440C22×C4C2×C4×D1711yescharacteristic
441C22×C4C2×C4×D1711yescharacteristic
442C22×C4C2×C4×D1711yescharacteristic
443C22×C4C2×C4×D1711yescharacteristic
444C22×C4C2×C4×D1711yescharacteristic
445C22×C4C2×C4×D1711yescharacteristic
446C22×C4C23×D1711yescharacteristic
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—341no—
460C2×C8—341no—
461C2×C8—341no—
462C2×C8—341no—
463C2×C8—341no—
464C2×C8—341no—
465C2×C8—341no—
466C2×C8—341no—
467C2×C8—341no—
468C2×C8—341no—
469C2×C8—341no—
470C2×C8—341no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—342no—
474C4⋊C4—681no—
475C4⋊C4—681no—
476C4⋊C4—681no—
477C4⋊C4—681no—
478C4⋊C4—681no—
479C4⋊C4—681no—
480C4⋊C4—681no—
481C4⋊C4—681no—
482C22⋊C4—342no—
483C22⋊C4—681no—
484C22⋊C4—681no—
485C22⋊C4—681no—
486C22⋊C4—681no—
487C22⋊C4—681no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—341no—
495C42—341no—
496C42—341no—
497C42—341no—
498C42—341no—
499C42—341no—
500C23—341no—
501C23—341no—
502C23—341no—
503C23—341no—
504C23—341no—
505C23—341no—
506C23—341no—
507C23C22×C4×D1711yescharacteristic
508Q8—684no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
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—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—172no—
528C2×C4—341no—
529C2×C4—341no—
530C2×C4—341no—
531C2×C4—341no—
532C2×C4—341no—
533C2×C4—341no—
534C2×C4—341no—
535C2×C4—341no—
536C2×C4—341no—
537C2×C4—341no—
538C2×C4—341no—
539C2×C4—341no—
540C2×C4—341no—
541C2×C4—341no—
542C2×C4—341no—
543C2×C4—341no—
544C2×C4—341no—
545C2×C4—341no—
546C2×C4—341no—
547C2×C4—341no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4—681no—
551C2×C4—681no—
552C2×C4#1058769111yescharacteristic
553C2×C4#1058769411yescharacteristic
554C2×C4#1058769611yescharacteristic
555C2×C4#1058769611yescharacteristic
556C2×C4#1058769711yescharacteristic
557C2×C4#1058769711yescharacteristic
558C8—41no—
559C8—41no—
560C8—41no—
561C8—41no—
562C8—681no—
563C8—681no—
564C8—681no—
565C8—681no—
566C22—681no—
567C22—681no—
568C22—681no—
569C22—681no—
570C22—681no—
571C22—681no—
572C22—681no—
573C22#1176328111yescharacteristic
574C22#1176328511yescharacteristic
575C22#1176328911yescharacteristic
576C22#131×D1711yescharacteristic
577C22#1176357411yescharacteristic
578C22#1176357511yescharacteristic
579C22#1176357511yescharacteristic
580C4—41no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—342no—
587C4—681no—
588C4—681no—
589C4—681no—
590C4—681no—
591C4—681no—
592C4—681no—
593C4#1176355611yescharacteristic
594C4#1176355711yescharacteristic
595C4#1176356011yescharacteristic
596C4#1176356111yescharacteristic
597C2—1361no—
598C2#1453194911yescharacteristic
599C2#1453782811yescharacteristic
600C2#1453785211yescharacteristic
601C2#1453787711yescharacteristic
602C2#1453788711yescharacteristic
603C2#1453792711yescharacteristic
604C2#1453795611yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D17
commutator, socleC23×C17C23×C4
FittingC17×#1917C2
prime radicalC23×D17C22×C4
cosocleC22×C4×C17C24
hypercenter#1917D17
hypocenterC17#23345
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ #1917

Complements

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

NH ≅ G/Ntype
2. #14529724C2semidirect
3. #14529718C2semidirect
4. #14529508C2semidirect
5. #14529504C2semidirect
10. #14529351C2semidirect
11. #14529346C2semidirect
12. #14529344C2semidirect
15. C17×#1917C2semidirect
275. #1917D17semidirect
422. C17#23345semidirect

Maximal subgroups

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

subgroupindex
#145320602
#145297242
#145297182
#145295082
#145295042
#145293982
#145293972
#145293952
#145293912
#145293512
#145293462
#145293442
#145288542
#145287652
C17×#19172
#2334517