Finite Group Databasebeta

← #17060864

Subgroups — #17060864

Order 4352.

702 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14532071C211yesmaximal, characteristic
2#14529723C211yesmaximal, characteristic, complemented
3#14529722C211yesmaximal, characteristic, complemented
4#14529400C211yesmaximal, characteristic, complemented
5#14529399C211yesmaximal, characteristic, complemented
6#14529398C211yesmaximal, characteristic, complemented
7#14529395C211yesmaximal, characteristic, complemented
8#14529081C211yesmaximal, characteristic
9#14529073C211yesmaximal, characteristic
10#14528967C211yesmaximal, characteristic, complemented
11#14528960C211yesmaximal, characteristic, complemented
12#14528960C211yesmaximal, characteristic, complemented
13#14528903C211yesmaximal, characteristic, complemented
14#14528902C211yesmaximal, characteristic, complemented
15C17×#1851C211yesmaximal, characteristic, complemented
16C22×C8×D17C2211yescharacteristic
17C2×C4×Dic34C2211yescharacteristic
18C2×C4×C22⋊D17C2211yescharacteristic
19C2×C4⋊#3622C2211yescharacteristic
20C2×#10587485C2211yescharacteristic
21C2×#10587485C2211yescharacteristic
22C2×#10587485C2211yescharacteristic
23C2×C22⋊#3622C2211yescharacteristic
24C2×C22⋊#3622C2211yescharacteristic
25C2×#10587491C2211yescharacteristic
26C2×#10587491C2211yescharacteristic
27C2×#10587491C2211yescharacteristic
28C2×#10587474C2211yescharacteristic
29C2×#10587486C2211yescharacteristic
30#11762751C411yescharacteristic
31#11762750C411yescharacteristic
32#11762738C411yescharacteristic
33#11762737C411yescharacteristic
34#11762725C411yescharacteristic
35#11762724C411yescharacteristic
36#11762722C411yescharacteristic
37#11762721C411yescharacteristic
38C4×#63227C2211yescharacteristic
39C4×#63227C2211yescharacteristic
40C4×C22⋊Dic17C2211yescharacteristic
41#11762655C2211yescharacteristic
42#11762564C2211yescharacteristic
43#11762564C2211yescharacteristic
44#11762563C2211yescharacteristic
45#11762563C2211yescharacteristic
46#11762563C2211yescharacteristic
47#11762563C2211yescharacteristic
48#11762563C2211yescharacteristic
49#11762563C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C22×M16×C17C2211yescharacteristic
53C2×C17×C4⋊C8C2211yescharacteristic
54C2×C17×C4⋊C8C2211yescharacteristic
55C2×C17×C22⋊C8C2211yescharacteristic
56C4×C22⋊C4×C17C2211yescharacteristic
57C17×#336C2211yescharacteristic
58C17×#336C2211yescharacteristic
59C23×C4×C17C2311yescharacteristic
60C22×C22⋊D17C2×C411yescharacteristic
61C22×C4×D17C2311yescharacteristic
62C22×Dic34C2×C411yescharacteristic
63C2×M16×C17—41no—
64C2×M16×C17—41no—
65C2×M16×C17—41no—
66C22×C8×C17C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C22×C8×C17C2311yescharacteristic
70C2×C22⋊C4×C17C2×C411yescharacteristic
71C2×C22⋊C4×C17C2×C411yescharacteristic
72C2×C42×C17C2311yescharacteristic
73C2×C42×C17C2311yescharacteristic
74C2×C22⋊Dic17C2×C411yescharacteristic
75C2×C22⋊Dic17C2×C411yescharacteristic
76C2×C22⋊Dic17C2×C411yescharacteristic
77C4×C22⋊D17—41no—
78C4×C22⋊D17—41no—
79C2×#63227C2×C411yescharacteristic
80C2×#63227C2×C411yescharacteristic
81C2×#63227C2×C411yescharacteristic
82C2×#63227C2×C411yescharacteristic
83C2×#63227C2×C411yescharacteristic
84C2×C4⋊Dic17C2×C411yescharacteristic
85C2×#63225C2×C411yescharacteristic
86C2×#63225C2×C411yescharacteristic
87C2×#63225C2×C411yescharacteristic
88C2×C4×Dic17C2311yescharacteristic
89C2×C4×Dic17C2311yescharacteristic
90C2×C4×Dic17C2311yescharacteristic
91C22×#3622C2311yescharacteristic
92C22×#3622C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95C2×C8×D17—41no—
96C2×C8×D17—41no—
97C2×C8×D17—41no—
98C4×Dic34—41no—
99C4×Dic34—41no—
100C17×C4⋊C8—41no—
101C17×C4⋊C8—41no—
102C17×C22⋊C8—41no—
103C17×#95C2×C411yescharacteristic
104C17×#95C2×C411yescharacteristic
105#10587502C2×C411yescharacteristic
106#10587502C2×C411yescharacteristic
107#10587502C2×C411yescharacteristic
108#10587502C2×C411yescharacteristic
109#10587502C2×C411yescharacteristic
110#10587502C2×C411yescharacteristic
111#10587502C2×C411yescharacteristic
112#10587502C2×C411yescharacteristic
113#10587502C2×C411yescharacteristic
114#10587502C2×C411yescharacteristic
115C22⋊#3622—41no—
116C22⋊#3622—41no—
117#10587491—41no—
118#10587491—41no—
119#10587491—41no—
120#10587486—41no—
121#10587485—41no—
122#10587485—41no—
123#10587485—41no—
124C4⋊#3622—41no—
125#10587474—41no—
126C24×C17C22×C411yescharacteristic
127C23×D17C22×C411yescharacteristic
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17C22×C411yescharacteristic
135C22×C4×C17C22×C411yescharacteristic
136C22×C4×C17C22×C411yescharacteristic
137C22×C4×C17C22×C411yescharacteristic
138C22×C4×C17C22×C411yescharacteristic
139C22×C4×C17C2411yescharacteristic
140C2×C22⋊D17—22no—
141C2×C22⋊D17—41no—
142C2×C22⋊D17—41no—
143C22×Dic17C22×C411yescharacteristic
144C22×Dic17C22×C411yescharacteristic
145C22×Dic17C22×C411yescharacteristic
146C22×Dic17C22×C411yescharacteristic
147C22×Dic17C22×C411yescharacteristic
148C22×Dic17C22×C411yescharacteristic
149C22×Dic17C22×C411yescharacteristic
150C2×C4×D17—21no—
151C2×C4×D17—21no—
152C2×C4×D17—21no—
153C2×C4×D17—21no—
154C2×C4×D17—21no—
155C2×C4×D17—21no—
156C2×Dic34—41no—
157C2×Dic34—41no—
158C2×Dic34—41no—
159M16×C17—44no—
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—
168C2×C8×C17—21no—
169C2×C8×C17—21no—
170C2×C8×C17—21no—
171C2×C8×C17—21no—
172C22⋊C4×C17—41no—
173C22⋊C4×C17—41no—
174C42×C17—21no—
175C42×C17—21no—
176C42×C17—21no—
177C42×C17—21no—
178C22⋊Dic17—41no—
179C22⋊Dic17—41no—
180C22⋊Dic17—41no—
181#63227—22no—
182#63227—22no—
183#63227—41no—
184#63227—41no—
185#63227—41no—
186C4⋊Dic17—41no—
187#63225—41no—
188#63225—41no—
189#63225—41no—
190C4×Dic17—21no—
191C4×Dic17—21no—
192C4×Dic17—21no—
193C4×Dic17—21no—
194C4×Dic17—21no—
195C4×Dic17—21no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622—21no—
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—
208C8×D17—44no—
209#23354—171nomaximal, Sylow
210C23×C17—21no—
211C23×C17—21no—
212C23×C17—21no—
213C23×C17—21no—
214C23×C17—21no—
215C23×C17—21no—
216C23×C17—21no—
217C23×C17C23×C411yescharacteristic
218C22×D17—21no—
219C22×D17—21no—
220C22×D17—21no—
221C22×D17—21no—
222C22×D17—21no—
223C22×D17—21no—
224C22×D17#13112yes—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
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—41no—
241C2×C4×C17—41no—
242C2×C4×C17—41no—
243C2×C4×C17—41no—
244C2×C4×C17H(5,𝔽2)11yescharacteristic
245C2×C4×C17H(5,𝔽2)11yescharacteristic
246C2×C4×C17H(5,𝔽2)11yescharacteristic
247C2×C4×C17H(5,𝔽2)11yescharacteristic
248C2×C4×C172−1+411yescharacteristic
249C2×C4×C172−1+411yescharacteristic
250C22⋊D17—44no—
251C2×Dic17—21no—
252C2×Dic17—21no—
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—21no—
270C2×Dic17—21no—
271C2×Dic17—21no—
272C4×D17—41no—
273C4×D17—41no—
274C4×D17—41no—
275C4×D17—41no—
276Dic34—44no—
277C8×C17—41no—
278C8×C17—41no—
279C8×C17—41no—
280C8×C17—41no—
281#3622—41no—
282#3622—41no—
283#3622—41no—
284#3622—41no—
285#2398—171no—
286#1985—171no—
287#1985—171no—
288#1924—171no—
289#1924—171no—
290#1924—171no—
291#1924—171no—
292#1870—171no—
293#1869—171no—
294#1851—171no—
295#1851D1711yescharacteristic, complemented
296#1850—171no—
297#1850—171no—
298#1832—171no—
299#1832—171no—
300C22×C17—41no—
301C22×C17—41no—
302C22×C17—41no—
303C22×C17—41no—
304C22×C17—41no—
305C22×C17—41no—
306C22×C17—41no—
307C22×C17#51811yescharacteristic
308C22×C17#52011yescharacteristic
309C22×C17#52011yescharacteristic
310C22×C17C2×#13111yescharacteristic
311C22×C17#56811yescharacteristic
312C22×C17#56811yescharacteristic
313C22×C17#56811yescharacteristic
314C2×D17—22no—
315C2×D17—22no—
316C2×D17—22no—
317C2×D17—41no—
318C2×D17—41no—
319C2×D17—41no—
320C2×D17—41no—
321C4×C17—41no—
322C4×C17—41no—
323C4×C17—41no—
324C4×C17—41no—
325C4×C17—41no—
326C4×C17H(3,𝔽4)11yescharacteristic
327C4×C17#56211yescharacteristic
328C4×C17#56211yescharacteristic
329C4×C17#56311yescharacteristic
330Dic17—41no—
331Dic17—41no—
332Dic17—41no—
333Dic17—41no—
334Dic17—41no—
335Dic17—41no—
336Dic17—41no—
337C22×M16C2×D1711yescharacteristic
338C23×C8—171no—
339C2×C4×Q8—171no—
340C2×C4×D4—171no—
341C2×C4⋊C8—171no—
342C2×C4⋊C8—171no—
343C2×C4⋊C8—171no—
344C2×C4⋊C8—171no—
345C2×C4⋊C8C2×D1711yescharacteristic
346C2×C4⋊C8C2×D1711yescharacteristic
347C2×C22⋊C8—171no—
348C2×C22⋊C8—171no—
349C2×C22⋊C8—171no—
350C2×C22⋊C8—171no—
351C2×C22⋊C8—171no—
352C2×C22⋊C8C2×D1711yescharacteristic
353C2×#97—171no—
354C2×#97—171no—
355#400—171no—
356#400—171no—
357#397—171no—
358#397—171no—
359#394—171no—
360#394—171no—
361#393—171no—
362#393—171no—
363C4×C22⋊C4—171no—
364C4×C22⋊C4—171no—
365C4×C22⋊C4—171no—
366C4×C22⋊C4C2×D1711yescharacteristic
367#376—171no—
368#336—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#336C2×D1711yescharacteristic
379#336C2×D1711yescharacteristic
380C2×C17—81no—
381C2×C17#237911yescharacteristic
382C2×C17#297611yescharacteristic
383C2×C17#297811yescharacteristic
384C2×C17#299211yescharacteristic
385C2×C17#299211yescharacteristic
386C2×C17#299511yescharacteristic
387C2×C17#299511yescharacteristic
388D17—42no—
389C22×Q8—171no—
390C22×D4—171no—
391C23×C4—171no—
392C23×C4C22×D1711yescharacteristic
393C2×M16—41no—
394C2×M16—41no—
395C2×M16—41no—
396C22×C8—171no—
397C22×C8—171no—
398C22×C8—171no—
399C22×C8—171no—
400C22×C8—681no—
401C22×C8—681no—
402C22×C8—681no—
403C22×C8C22×D1711yescharacteristic
404C22×C8C22×D1711yescharacteristic
405C22×C8C22×D1711yescharacteristic
406C22×C8C22×D1711yescharacteristic
407C4×Q8—681no—
408C4×Q8—681no—
409C4×D4—681no—
410C4×D4—681no—
411C2×C4⋊C4—171no—
412C2×C4⋊C4—171no—
413C2×C4⋊C4—171no—
414C2×C4⋊C4—171no—
415C2×C22⋊C4—171no—
416C2×C22⋊C4—171no—
417C2×C22⋊C4—171no—
418C2×C22⋊C4—171no—
419C2×C22⋊C4—171no—
420C2×C22⋊C4—171no—
421C2×C22⋊C4—171no—
422C2×C22⋊C4—171no—
423C2×C22⋊C4C4×D1711yescharacteristic
424C2×C22⋊C4C4×D1711yescharacteristic
425C2×C42—171no—
426C2×C42—171no—
427C2×C42—171no—
428C2×C42C22×D1711yescharacteristic
429C2×C42C22×D1711yescharacteristic
430C4⋊C8—41no—
431C4⋊C8—41no—
432C4⋊C8—681no—
433C4⋊C8—681no—
434C4⋊C8—681no—
435C4⋊C8—681no—
436C22⋊C8—41no—
437C22⋊C8—681no—
438C22⋊C8—681no—
439C22⋊C8—681no—
440C22⋊C8—681no—
441C22⋊C8—681no—
442#97—681no—
443#97—681no—
444#95—171no—
445#95—171no—
446#95—171no—
447#95—171no—
448#95—171no—
449#95—171no—
450#95—171no—
451#95—171no—
452#95—171no—
453#95—171no—
454#95C4×D1711yescharacteristic
455#95C4×D1711yescharacteristic
456C17#2335411yescharacteristic, complemented, Sylow
457C24—171no—
458C24C2×C4×D1711yescharacteristic
459C2×Q8—681no—
460C2×Q8—681no—
461C2×Q8—681no—
462C2×D4—342no—
463C2×D4—681no—
464C2×D4—681no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—171no—
472C22×C4—171no—
473C22×C4—171no—
474C22×C4—171no—
475C22×C4—171no—
476C22×C4—171no—
477C22×C4—171no—
478C22×C4—341no—
479C22×C4—341no—
480C22×C4—341no—
481C22×C4—341no—
482C22×C4—341no—
483C22×C4—341no—
484C22×C4C2×C4×D1711yescharacteristic
485C22×C4C2×C4×D1711yescharacteristic
486C22×C4C2×C4×D1711yescharacteristic
487C22×C4C2×C4×D1711yescharacteristic
488C22×C4C2×C4×D1711yescharacteristic
489C22×C4C23×D1711yescharacteristic
490M16—44no—
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—681no—
517C4⋊C4—681no—
518C4⋊C4—681no—
519C4⋊C4—681no—
520C22⋊C4—41no—
521C22⋊C4—41no—
522C22⋊C4—342no—
523C22⋊C4—342no—
524C22⋊C4—681no—
525C22⋊C4—681no—
526C22⋊C4—681no—
527C22⋊C4—681no—
528C22⋊C4—681no—
529C22⋊C4—681no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—341no—
535C42—341no—
536C42—341no—
537C42—341no—
538C42—341no—
539C42—341no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—172no—
548C23—341no—
549C23—341no—
550C23—341no—
551C23—341no—
552C23—341no—
553C23—341no—
554C23C22×C4×D1711yescharacteristic
555Q8—684no—
556D4—684no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
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—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—341no—
577C2×C4—341no—
578C2×C4—341no—
579C2×C4—341no—
580C2×C4—341no—
581C2×C4—341no—
582C2×C4—341no—
583C2×C4—341no—
584C2×C4—341no—
585C2×C4—341no—
586C2×C4—341no—
587C2×C4—341no—
588C2×C4—341no—
589C2×C4—341no—
590C2×C4—341no—
591C2×C4—341no—
592C2×C4—341no—
593C2×C4—341no—
594C2×C4—341no—
595C2×C4—341no—
596C2×C4—341no—
597C2×C4—681no—
598C2×C4—681no—
599C2×C4—681no—
600C2×C4—681no—
601C2×C4#1058769111yescharacteristic
602C2×C4#1058769111yescharacteristic
603C2×C4#1058769111yescharacteristic
604C2×C4#1058769411yescharacteristic
605C2×C4#1058769611yescharacteristic
606C2×C4#1058769711yescharacteristic
607C8—41no—
608C8—41no—
609C8—41no—
610C8—41no—
611C8—681no—
612C8—681no—
613C8—681no—
614C8—681no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—342no—
623C22—342no—
624C22—342no—
625C22—681no—
626C22—681no—
627C22—681no—
628C22—681no—
629C22#1176328111yescharacteristic
630C22#1176328811yescharacteristic
631C22#1176329011yescharacteristic
632C22#131×D1711yescharacteristic
633C22#1176357411yescharacteristic
634C22#1176357511yescharacteristic
635C22#1176357511yescharacteristic
636C4—41no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—681no—
642C4—681no—
643C4—681no—
644C4—681no—
645C4—681no—
646C4—681no—
647C4—681no—
648C4#1176355211yescharacteristic
649C4#1176355511yescharacteristic
650C4#1176355611yescharacteristic
651C4#1176356011yescharacteristic
652C2—81no—
653C2—682no—
654C2#1453195711yescharacteristic
655C2#1453783111yescharacteristic
656C2#1453784111yescharacteristic
657C2#1453791911yescharacteristic
658C2#1453792111yescharacteristic
659C2#1453794811yescharacteristic
660C2#1453795111yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D17
commutator, socleC23×C17C23×C4
FittingC17×#1851C2
prime radicalC22×C22⋊D17C2×C4
cosocleC22×C4×C17C24
hypercenter#1851D17
hypocenterC17#23354
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ #1851

Complements

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

NH ≅ G/Ntype
2. #14529723C2semidirect
3. #14529722C2semidirect
4. #14529400C2semidirect
5. #14529399C2semidirect
6. #14529398C2semidirect
7. #14529395C2semidirect
10. #14528967C2semidirect
11. #14528960C2semidirect
12. #14528960C2semidirect
13. #14528903C2semidirect
14. #14528902C2semidirect
15. C17×#1851C2semidirect
295. #1851D17semidirect
456. C17#23354semidirect

Maximal subgroups

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

subgroupindex
#145320712
#145297232
#145297222
#145294002
#145293992
#145293982
#145293952
#145290812
#145290732
#145289672
#145289602
#145289032
#145289022
C17×#18512
#2335417