Finite Group Databasebeta

← #17045113

Subgroups — #17045113

Order 4352.

752 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531245C211yesmaximal, characteristic
2#14529397C211yesmaximal, characteristic, complemented
3#14529388C211yesmaximal, characteristic, complemented
4#14529343C211yesmaximal, characteristic, complemented
5#14529080C211yesmaximal, characteristic
6#14529075C211yesmaximal, characteristic
7#14528968C211yesmaximal, characteristic, complemented
8#14528960C211yesmaximal, characteristic, complemented
9#14528926C211yesmaximal, characteristic, complemented
10#14528899C211yesmaximal, characteristic, complemented
11#14528854C211yesmaximal, characteristic, complemented
12#14528765C211yesmaximal, characteristic, complemented
13#14528553C211yesmaximal, characteristic, complemented
14#14528479C211yesmaximal, characteristic, complemented
15C17×#1917C211yesmaximal, characteristic, complemented
16C22×#63218C2211yescharacteristic
17C22×C8×D17C2211yescharacteristic
18C2×C4×D68C2211yescharacteristic
19C2×#10587558C2211yescharacteristic
20C2×#10587558C2211yescharacteristic
21C2×C4⋊#3622C2211yescharacteristic
22C2×#10587485C2211yescharacteristic
23C2×#10587485C2211yescharacteristic
24C2×#10587491C2211yescharacteristic
25C2×#10587491C2211yescharacteristic
26C2×#10587491C2211yescharacteristic
27C2×#10587491C2211yescharacteristic
28C2×#10587491C2211yescharacteristic
29C2×#10587491C2211yescharacteristic
30C2×C4×#3622C2211yescharacteristic
31#11762710C411yescharacteristic
32#11762710C411yescharacteristic
33#11762705C411yescharacteristic
34#11762705C411yescharacteristic
35#11762698C411yescharacteristic
36#11762698C411yescharacteristic
37#11762698C411yescharacteristic
38#11762698C411yescharacteristic
39C4×#63227C2211yescharacteristic
40C4×#63227C2211yescharacteristic
41C42×Dic17C2211yescharacteristic
42#11762564C2211yescharacteristic
43#11762564C2211yescharacteristic
44#11762564C2211yescharacteristic
45#11762564C2211yescharacteristic
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×D68C2×C411yescharacteristic
60C22×C4×D17C2311yescharacteristic
61C22×C4×D17C2311yescharacteristic
62C22×C8×C17C2311yescharacteristic
63C22×C8×C17C2311yescharacteristic
64C22×C8×C17C2311yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C2×C4⋊C4×C17C2×C411yescharacteristic
67C2×C4⋊C4×C17C2×C411yescharacteristic
68C2×C42×C17C2311yescharacteristic
69C2×C42×C17C2311yescharacteristic
70C2×C42×C17C2311yescharacteristic
71C2×#63227C2×C411yescharacteristic
72C2×#63227C2×C411yescharacteristic
73C2×#63227C2×C411yescharacteristic
74C2×#63227C2×C411yescharacteristic
75C2×#63227C2×C411yescharacteristic
76C2×#63227C2×C411yescharacteristic
77C2×#63227C2×C411yescharacteristic
78C2×#63227C2×C411yescharacteristic
79C2×#63227C2×C411yescharacteristic
80C2×#63227C2×C411yescharacteristic
81C2×C4⋊Dic17C2×C411yescharacteristic
82C2×#63225C2×C411yescharacteristic
83C2×#63225C2×C411yescharacteristic
84C2×#63225C2×C411yescharacteristic
85C2×#63225C2×C411yescharacteristic
86C2×C4×Dic17C2×C411yescharacteristic
87C2×C4×Dic17C2×C411yescharacteristic
88C2×C4×Dic17C2×C411yescharacteristic
89C2×C4×Dic17C2×C411yescharacteristic
90C2×C4×Dic17C2311yescharacteristic
91C2×C4×Dic17C2311yescharacteristic
92C22×#3622C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96C2×#63218—22no—
97C2×#63218—41no—
98C2×#63218—41no—
99C2×C8×D17—22no—
100C2×C8×D17—41no—
101C2×C8×D17—41no—
102C4×D68—22no—
103C4×D68—22no—
104#10587558—41no—
105#10587558—41no—
106#10587558—41no—
107#10587558—41no—
108C17×C4⋊C8—41no—
109C17×#97—41no—
110C17×#95C2×C411yescharacteristic
111C17×#95C2×C411yescharacteristic
112#10587502C2×C411yescharacteristic
113#10587502C2×C411yescharacteristic
114#10587502C2×C411yescharacteristic
115#10587502C2×C411yescharacteristic
116#10587491—22no—
117#10587491—22no—
118#10587491—41no—
119#10587491—41no—
120#10587491—41no—
121#10587491—41no—
122#10587485—41no—
123#10587485—41no—
124C4⋊#3622—41no—
125C4×#3622—41no—
126C23×D17C22×C411yescharacteristic
127C23×D17C22×C411yescharacteristic
128C22×C4×C17C22×C411yescharacteristic
129C22×C4×C17C22×C411yescharacteristic
130C22×C4×C17C22×C411yescharacteristic
131C22×C4×C17C22×C411yescharacteristic
132C22×C4×C17C22×C411yescharacteristic
133C22×C4×C17C22×C411yescharacteristic
134C22×C4×C17C2411yescharacteristic
135C22×Dic17C22×C411yescharacteristic
136C22×Dic17C22×C411yescharacteristic
137C22×Dic17C22×C411yescharacteristic
138C22×Dic17C22×C411yescharacteristic
139C22×Dic17C22×C411yescharacteristic
140C22×Dic17C22×C411yescharacteristic
141C2×D68—22no—
142C2×D68—22no—
143C2×D68—41no—
144C2×C4×D17—21no—
145C2×C4×D17—21no—
146C2×C4×D17—21no—
147C2×C4×D17—21no—
148C2×C4×D17—21no—
149C2×C4×D17—21no—
150C2×C4×D17—21no—
151C2×C4×D17—21no—
152C2×C4×D17—21no—
153C2×C4×D17—21no—
154C2×C4×D17#4112yes—
155C2×C4×D17#4112yes—
156C2×C8×C17—21no—
157C2×C8×C17—21no—
158C2×C8×C17—21no—
159C2×C8×C17—21no—
160C2×C8×C17—21no—
161C2×C8×C17—21no—
162C2×C8×C17—21no—
163C2×C8×C17—21no—
164C2×C8×C17—21no—
165C2×C8×C17—21no—
166C2×C8×C17—21no—
167C2×C8×C17—21no—
168C4⋊C4×C17—22no—
169C4⋊C4×C17—41no—
170C42×C17—21no—
171C42×C17—21no—
172C42×C17—21no—
173C42×C17—21no—
174C42×C17—21no—
175C42×C17—21no—
176#63227—22no—
177#63227—22no—
178#63227—22no—
179#63227—22no—
180#63227—41no—
181#63227—41no—
182#63227—41no—
183#63227—41no—
184#63227—41no—
185#63227—41no—
186C4⋊Dic17—22no—
187#63225—41no—
188#63225—41no—
189#63225—41no—
190#63225—41no—
191C4×Dic17—22no—
192C4×Dic17—21no—
193C4×Dic17—21no—
194C4×Dic17—21no—
195C4×Dic17—21no—
196C4×Dic17—22no—
197C4×Dic17—41no—
198C4×Dic17—41no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622—21no—
208C2×#3622—21no—
209C2×#3622—21no—
210C2×#3622—21no—
211#63218—44no—
212C8×D17—44no—
213#22007—171nomaximal, Sylow
214C23×C17C23×C411yescharacteristic
215C22×D17—21no—
216C22×D17—21no—
217C22×D17—21no—
218C22×D17—21no—
219C22×D17—21no—
220C22×D17—21no—
221C22×D17—21no—
222C22×D17—21no—
223C22×D17—21no—
224C22×D17—21no—
225C22×D17—21no—
226C22×D17—21no—
227C22×D17—21no—
228C22×D17#11712yes—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—21no—
244C2×C4×C17—21no—
245C2×C4×C17—21no—
246C2×C4×C17#13112yes—
247C2×C4×C17C2×#4111yescharacteristic
248C2×C4×C17C2×#4111yescharacteristic
249C2×C4×C17H(5,𝔽2)11yescharacteristic
250C2×C4×C17H(5,𝔽2)11yescharacteristic
251C2×C4×C17H(5,𝔽2)11yescharacteristic
252C2×C4×C172−1+411yescharacteristic
253C2×Dic17—21no—
254C2×Dic17—21no—
255C2×Dic17—21no—
256C2×Dic17—21no—
257C2×Dic17—21no—
258C2×Dic17—21no—
259C2×Dic17—21no—
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—21no—
263C2×Dic17—21no—
264C2×Dic17—21no—
265C2×Dic17—21no—
266C2×Dic17—21no—
267C2×Dic17—21no—
268C2×Dic17—21no—
269C2×Dic17#11712yes—
270C2×Dic17#13112yes—
271D68—44no—
272C4×D17—22no—
273C4×D17—22no—
274C4×D17—22no—
275C4×D17—22no—
276C4×D17—41no—
277C4×D17—41no—
278C4×D17—41no—
279C4×D17—41no—
280C8×C17—41no—
281C8×C17—41no—
282C8×C17—41no—
283C8×C17—41no—
284#3622—41no—
285#3622—41no—
286#3622—41no—
287#3622—41no—
288#2293—171no—
289#1924—171no—
290#1923—171no—
291#1917D1711yescharacteristic, complemented
292#1916—171no—
293#1870—171no—
294#1869—171no—
295#1851—171no—
296#1850—171no—
297#1839—171no—
298#1831—171no—
299#1818—171no—
300#1798—171no—
301#1750—171no—
302#1729—171no—
303C22×C17C2×#11711yescharacteristic
304C22×C17#51811yescharacteristic
305C22×C17#52011yescharacteristic
306C22×C17C2×#13111yescharacteristic
307C22×C17C2×#13111yescharacteristic
308C22×C17#56811yescharacteristic
309C22×C17#56811yescharacteristic
310C2×D17—22no—
311C2×D17—22no—
312C2×D17—22no—
313C2×D17—41no—
314C2×D17—41no—
315C2×D17—41no—
316C2×D17—41no—
317C2×D17—41no—
318C2×D17—41no—
319C2×D17—41no—
320C2×D17—41no—
321C2×D17—41no—
322C2×D17—41no—
323C2×D17—41no—
324C4×C17—22no—
325C4×C17—41no—
326C4×C17—41no—
327C4×C17—41no—
328C4×C17—41no—
329C4×C17—41no—
330C4×C17#53811yescharacteristic
331C4×C17#53811yescharacteristic
332C4×C17#53911yescharacteristic
333C4×C17#53911yescharacteristic
334Dic17—22no—
335Dic17—22no—
336Dic17—41no—
337Dic17—41no—
338Dic17—41no—
339Dic17—41no—
340C22×M16—171no—
341C23×C8—171no—
342C2×C4×D4—171no—
343C2×#117—171no—
344C2×#117—171no—
345C2×C4⋊C8—171no—
346C2×C4⋊C8—171no—
347C2×C4⋊C8—171no—
348C2×C4⋊C8C2×D1711yescharacteristic
349C2×C22⋊C8—171no—
350C2×C22⋊C8—171no—
351C2×C22⋊C8—171no—
352C2×C22⋊C8—171no—
353C2×C22⋊C8—171no—
354C2×C22⋊C8—171no—
355C2×#97C2×D1711yescharacteristic
356C2×C4×C8—171no—
357#390—171no—
358#390—171no—
359#389—171no—
360#389—171no—
361#388—171no—
362#388—171no—
363#388—171no—
364#388—171no—
365C4×C4⋊C4C2×D1711yescharacteristic
366C4×C22⋊C4—171no—
367C4×C22⋊C4—171no—
368C43—171no—
369#336—171no—
370#336—171no—
371#336—171no—
372#336—171no—
373#336—171no—
374#336—171no—
375#336—171no—
376#336—171no—
377#336—171no—
378#336—171no—
379#336C2×D1711yescharacteristic
380#336C2×D1711yescharacteristic
381#336C2×D1711yescharacteristic
382#336C2×D1711yescharacteristic
383C2×C17#231811yescharacteristic
384C2×C17#292411yescharacteristic
385C2×C17#292511yescharacteristic
386C2×C17#298111yescharacteristic
387C2×C17#298111yescharacteristic
388C2×C17#299411yescharacteristic
389C2×C17#299511yescharacteristic
390D17—42no—
391D17—81no—
392C22×D4—171no—
393C23×C4—171no—
394C23×C4—171no—
395C2×M16—342no—
396C2×M16—681no—
397C2×M16—681no—
398C22×C8—171no—
399C22×C8—171no—
400C22×C8—171no—
401C22×C8—171no—
402C22×C8—342no—
403C22×C8—681no—
404C22×C8—681no—
405C22×C8C22×D1711yescharacteristic
406C22×C8C22×D1711yescharacteristic
407C22×C8C22×D1711yescharacteristic
408C22×C8C22×D1711yescharacteristic
409C4×D4—342no—
410C4×D4—342no—
411#117—681no—
412#117—681no—
413#117—681no—
414#117—681no—
415C2×C4⋊C4—171no—
416C2×C4⋊C4—171no—
417C2×C4⋊C4—171no—
418C2×C4⋊C4—171no—
419C2×C4⋊C4—171no—
420C2×C4⋊C4C4×D1711yescharacteristic
421C2×C4⋊C4C4×D1711yescharacteristic
422C2×C22⋊C4—171no—
423C2×C22⋊C4—171no—
424C2×C22⋊C4—171no—
425C2×C22⋊C4—171no—
426C2×C22⋊C4—171no—
427C2×C22⋊C4—171no—
428C2×C22⋊C4—171no—
429C2×C22⋊C4—171no—
430C2×C22⋊C4—171no—
431C2×C22⋊C4—171no—
432C2×C42—171no—
433C2×C42—171no—
434C2×C42—171no—
435C2×C42—171no—
436C2×C42—171no—
437C2×C42—171no—
438C2×C42C22×D1711yescharacteristic
439C2×C42C22×D1711yescharacteristic
440C2×C42C22×D1711yescharacteristic
441C4⋊C8—41no—
442C4⋊C8—681no—
443C4⋊C8—681no—
444C4⋊C8—681no—
445C22⋊C8—342no—
446C22⋊C8—342no—
447C22⋊C8—681no—
448C22⋊C8—681no—
449C22⋊C8—681no—
450C22⋊C8—681no—
451#97—41no—
452C4×C8—681no—
453#95—171no—
454#95—171no—
455#95—171no—
456#95—171no—
457#95C4×D1711yescharacteristic
458#95C4×D1711yescharacteristic
459C17#2200711yescharacteristic, complemented, Sylow
460C24—171no—
461C24—171no—
462C2×D4—342no—
463C2×D4—342no—
464C2×D4—681no—
465C22×C4—171no—
466C22×C4—171no—
467C22×C4—172no—
468C22×C4—172no—
469C22×C4—171no—
470C22×C4—171no—
471C22×C4—171no—
472C22×C4—171no—
473C22×C4—341no—
474C22×C4—341no—
475C22×C4—341no—
476C22×C4—341no—
477C22×C4—341no—
478C22×C4—341no—
479C22×C4—341no—
480C22×C4—341no—
481C22×C4—341no—
482C22×C4—341no—
483C22×C4C2×C4×D1711yescharacteristic
484C22×C4C2×C4×D1711yescharacteristic
485C22×C4C2×C4×D1711yescharacteristic
486C22×C4C2×C4×D1711yescharacteristic
487C22×C4C2×C4×D1711yescharacteristic
488C22×C4C2×C4×D1711yescharacteristic
489C22×C4C23×D1711yescharacteristic
490M16—684no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—341no—
504C2×C8—341no—
505C2×C8—341no—
506C2×C8—341no—
507C2×C8—341no—
508C2×C8—341no—
509C2×C8—341no—
510C2×C8—341no—
511C2×C8—341no—
512C2×C8—341no—
513C2×C8—341no—
514C2×C8—341no—
515C2×C8—684no—
516C4⋊C4—22no—
517C4⋊C4—41no—
518C4⋊C4—342no—
519C4⋊C4—681no—
520C4⋊C4—681no—
521C4⋊C4—681no—
522C4⋊C4—681no—
523C22⋊C4—342no—
524C22⋊C4—342no—
525C22⋊C4—342no—
526C22⋊C4—342no—
527C22⋊C4—681no—
528C22⋊C4—681no—
529C22⋊C4—681no—
530C22⋊C4—681no—
531C22⋊C4—681no—
532C22⋊C4—681no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—342no—
540C42—341no—
541C42—341no—
542C42—341no—
543C42—341no—
544C42—342no—
545C42—681no—
546C42—681no—
547C23—172no—
548C23—341no—
549C23—341no—
550C23—341no—
551C23—341no—
552C23—341no—
553C23—341no—
554C23—341no—
555C23—341no—
556C23—341no—
557C23—341no—
558C23—341no—
559C23—341no—
560C23—341no—
561C23C22×C4×D1711yescharacteristic
562D4—684no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—172no—
581C2×C4—172no—
582C2×C4—341no—
583C2×C4—341no—
584C2×C4—341no—
585C2×C4—341no—
586C2×C4—342no—
587C2×C4—342no—
588C2×C4—342no—
589C2×C4—342no—
590C2×C4—341no—
591C2×C4—341no—
592C2×C4—341no—
593C2×C4—341no—
594C2×C4—341no—
595C2×C4—341no—
596C2×C4—341no—
597C2×C4—341no—
598C2×C4—341no—
599C2×C4—341no—
600C2×C4—341no—
601C2×C4—341no—
602C2×C4—681no—
603C2×C4—681no—
604C2×C4—681no—
605C2×C4—681no—
606C2×C4#1058759312yes—
607C2×C4#1058769111yescharacteristic
608C2×C4#1058769411yescharacteristic
609C2×C4#41×D1711yescharacteristic
610C2×C4#41×D1711yescharacteristic
611C2×C4#1058769611yescharacteristic
612C2×C4#1058769611yescharacteristic
613C8—41no—
614C8—41no—
615C8—41no—
616C8—41no—
617C8—681no—
618C8—681no—
619C8—681no—
620C8—681no—
621C22—342no—
622C22—342no—
623C22—342no—
624C22—681no—
625C22—681no—
626C22—681no—
627C22—681no—
628C22—681no—
629C22—681no—
630C22—681no—
631C22—681no—
632C22—681no—
633C22—681no—
634C22—681no—
635C22#117×D1711yescharacteristic
636C22#1176328111yescharacteristic
637C22#1176329111yescharacteristic
638C22#131×D1711yescharacteristic
639C22C2×#1058759311yescharacteristic
640C22#1176357411yescharacteristic
641C22#1176357511yescharacteristic
642C4—22no—
643C4—41no—
644C4—41no—
645C4—41no—
646C4—41no—
647C4—41no—
648C4—342no—
649C4—342no—
650C4—681no—
651C4—681no—
652C4—681no—
653C4—681no—
654C4#1176339511yescharacteristic
655C4#1176339811yescharacteristic
656C4#1176340511yescharacteristic
657C4#1176340711yescharacteristic
658C2—682no—
659C2—1361no—
660C2#1453141611yescharacteristic
661C2#1453744311yescharacteristic
662C2#1453744711yescharacteristic
663C2#1453785611yescharacteristic
664C2#1453786111yescharacteristic
665C2#1453794711yescharacteristic
666C2#1453795311yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D17
commutator, socleC23×C17C23×C4
FittingC17×#1917C2
prime radicalC22×D68C2×C4
cosocleC22×C4×C17C24
hypercenter#1917D17
hypocenterC17#22007
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ #1917

Complements

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

NH ≅ G/Ntype
2. #14529397C2semidirect
3. #14529388C2semidirect
4. #14529343C2semidirect
7. #14528968C2semidirect
8. #14528960C2semidirect
9. #14528926C2semidirect
10. #14528899C2semidirect
11. #14528854C2semidirect
12. #14528765C2semidirect
13. #14528553C2semidirect
14. #14528479C2semidirect
15. C17×#1917C2semidirect
291. #1917D17semidirect
459. C17#22007semidirect

Maximal subgroups

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

subgroupindex
#145312452
#145293972
#145293882
#145293432
#145290802
#145290752
#145289682
#145289602
#145289262
#145288992
#145288542
#145287652
#145285532
#145284792
C17×#19172
#2200717