Finite Group Databasebeta

← #17097198

Subgroups — #17097198

Order 4352.

860 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763110C211yesmaximal, characteristic, complemented
2C2×#11762851C211yesmaximal, characteristic, complemented
3#14532508C211yesmaximal, characteristic, complemented
4#14531820C211yesmaximal, characteristic
5#14530304C211yesmaximal, characteristic, complemented
6#14530282C211yesmaximal, characteristic, complemented
7#14530157C211yesmaximal, characteristic, complemented
8#14530123C211yesmaximal, characteristic, complemented
9#14529554C211yesmaximal, characteristic, complemented
10#14529356C211yesmaximal, characteristic, complemented
11#14529222C211yesmaximal, characteristic
12#14529099C211yesmaximal, characteristic
13#14529034C211yesmaximal, characteristic, complemented
14#14528985C211yesmaximal, characteristic, complemented
15C17×#1851C211yesmaximal, characteristic, complemented
16C2×#10587556C2211yescharacteristic
17C2×#10587616C2211yescharacteristic
18C2×C4×D68C2211yescharacteristic
19C2×#10587612C2211yescharacteristic
20C22×C4⋊Dic17C2211yescharacteristic
21#11763110—24no—
22#11762851—24no—
23C2×#10587489C2211yescharacteristic
24C2×#10587489C2211yescharacteristic
25C2×#10587489C2211yescharacteristic
26C2×#10587488C2211yescharacteristic
27C2×#10587488C2211yescharacteristic
28C2×#10587488C2211yescharacteristic
29C2×#10587492C2211yescharacteristic
30C2×#10587492C2211yescharacteristic
31C2×#10587492C2211yescharacteristic
32C2×#10587492C2211yescharacteristic
33#11762751—21no—
34#11762743—21no—
35#11762711C2211yescharacteristic
36#11762709—21no—
37#11762707C2211yescharacteristic
38#11762707C2211yescharacteristic
39#11762707C2211yescharacteristic
40#11762698—21no—
41#11762685—21no—
42#11762681—21no—
43#11762675C2211yescharacteristic
44#11762669C2211yescharacteristic
45#11762667—21no—
46C4×C4⋊Dic17C2211yescharacteristic
47#11762656—21no—
48#11762572C2211yescharacteristic
49#11762572C2211yescharacteristic
50#11762572C2211yescharacteristic
51#11762572C2211yescharacteristic
52#11762572C2211yescharacteristic
53#11762572C2211yescharacteristic
54C22×M16×C17C2211yescharacteristic
55C2×C17×C4⋊C8C2211yescharacteristic
56C2×C17×C4⋊C8C2211yescharacteristic
57C2×C17×C22⋊C8C2211yescharacteristic
58C4×C22⋊C4×C17C2211yescharacteristic
59C17×#336C2211yescharacteristic
60C17×#336C2211yescharacteristic
61C23×C4×C17C2311yescharacteristic
62C22×C22⋊D17—21no—
63C23×Dic17—21no—
64C22×D68C2311yescharacteristic
65C22×C4×D17—21no—
66C2×M16×C17—22no—
67C2×M16×C17—22no—
68C2×M16×C17—22no—
69C22×C8×C17C2311yescharacteristic
70C22×C8×C17C2311yescharacteristic
71C22×C8×C17C2311yescharacteristic
72C22×C8×C17C2311yescharacteristic
73C2×C22⋊C4×C17D411yescharacteristic
74C2×C22⋊C4×C17D411yescharacteristic
75C2×C42×C17C2311yescharacteristic
76C2×C42×C17C2311yescharacteristic
77C2×C22⋊Dic17—21no—
78C2×C22⋊Dic17—21no—
79#10587616—22no—
80#10587616Q814yes—
81C2×#63227—21no—
82C2×#63227—21no—
83C2×#63227—21no—
84C2×#63227—21no—
85#10587612—22no—
86#10587612—41no—
87C2×C4⋊Dic17—21no—
88C2×C4⋊Dic17—22no—
89C2×C4⋊Dic17—41no—
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96C2×C4⋊Dic17C2311yescharacteristic
97C2×#63225—21no—
98C2×#63225—21no—
99C2×#63225—21no—
100C2×#63225—21no—
101C2×#63225—21no—
102C2×C4×Dic17—21no—
103C2×C4×Dic17—21no—
104C4×D68—22no—
105C4×D68—41no—
106#10587556—22no—
107#10587556—41no—
108C17×C4⋊C8—22no—
109C17×C4⋊C8—22no—
110C17×C22⋊C8—22no—
111C17×#95D411yescharacteristic
112C17×#95D411yescharacteristic
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587492—22no—
121#10587492—22no—
122#10587492—22no—
123#10587492—41no—
124#10587489—22no—
125#10587489—41no—
126#10587489D414yescomplemented
127#10587488—22no—
128#10587488—22no—
129#10587488—22no—
130C24×C17C2×D411yescharacteristic
131C23×D17—21no—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2×Q812yes—
142C22×C4×C17#4112yes—
143C22×C4×C17C2411yescharacteristic
144C2×C22⋊D17—24no—
145C2×C22⋊D17—81no—
146C2×C22⋊D17—81no—
147C22×Dic17—21no—
148C22×Dic17—21no—
149C22×Dic17—21no—
150C22×Dic17—21no—
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—42no—
155C22×Dic17—42no—
156C22×Dic17—81no—
157C2×D68—21no—
158C2×D68—21no—
159C2×D68—21no—
160C2×D68—21no—
161C2×D68—21no—
162C2×D68C2×Q812yes—
163C2×C4×D17—42no—
164C2×C4×D17—42no—
165C2×C4×D17—81no—
166M16×C17—28no—
167C2×C8×C17—21no—
168C2×C8×C17—21no—
169C2×C8×C17—21no—
170C2×C8×C17—21no—
171C2×C8×C17—21no—
172C2×C8×C17—21no—
173C2×C8×C17—21no—
174C2×C8×C17—21no—
175C2×C8×C17—21no—
176C2×C8×C17—21no—
177C2×C8×C17C2×D412yes—
178C2×C8×C17#4112yes—
179C22⋊C4×C17—41no—
180C22⋊C4×C17—41no—
181C42×C17—21no—
182C42×C17—21no—
183C42×C17—21no—
184C42×C17—21no—
185C22⋊Dic17—42no—
186C22⋊Dic17—42no—
187#63227—24no—
188#63227—42no—
189#63227—42no—
190#63227—81no—
191C4⋊Dic17—21no—
192C4⋊Dic17—21no—
193C4⋊Dic17—21no—
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199C4⋊Dic17—21no—
200C4⋊Dic17—21no—
201C4⋊Dic17—21no—
202C4⋊Dic17—42no—
203C4⋊Dic17C2×D412yes—
204C4⋊Dic17C2×D412yes—
205C4⋊Dic17C2×Q812yes—
206#63225—42no—
207#63225—42no—
208#63225—81no—
209#63225—81no—
210#63225—81no—
211C4×Dic17—42no—
212C4×Dic17—81no—
213#26363—171nomaximal, Sylow
214C23×C17—21no—
215C23×C17—21no—
216C23×C17—21no—
217C23×C17—21no—
218C23×C17—21no—
219C23×C17—21no—
220C23×C17C22⋊Q812yes—
221C23×C17C22×D411yescharacteristic
222C22×D17—22no—
223C22×D17—41no—
224C22×D17—41no—
225C22×D17—41no—
226C22×D17—41no—
227C22×D17—41no—
228C22×D17—41no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—22no—
232C2×C4×C17—22no—
233C2×C4×C17—22no—
234C2×C4×C17—22no—
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—21no—
247C2×C4×C17C22⋊Q812yes—
248C2×C4×C17C22×D411yescharacteristic
249C2×C4×C17C22×Q811yescharacteristic
250C2×C4×C17C2×#4111yescharacteristic
251C2×C4×C17C2×#4111yescharacteristic
252C2×C4×C17H(5,𝔽2)11yescharacteristic
253C2×C4×C172−1+411yescharacteristic
254C22⋊D17—84no—
255C2×Dic17—22no—
256C2×Dic17—22no—
257C2×Dic17—22no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17—41no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268C2×Dic17—41no—
269C2×Dic17—41no—
270C2×Dic17—41no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C2×Dic17—84no—
277D68—22no—
278D68—22no—
279D68—41no—
280D68—41no—
281C4×D17—84no—
282C8×C17—22no—
283C8×C17—22no—
284C8×C17—41no—
285C8×C17—41no—
286C2×#478—171no—
287C2×#428—171no—
288#2442—171no—
289#2363—171no—
290#2092—171no—
291#2088—171no—
292#2065—171no—
293#2059—171no—
294#1949—171no—
295#1918—171no—
296#1895—171no—
297#1874—171no—
298#1862—171no—
299#1854—171no—
300#1851D1711yescharacteristic, complemented
301C22×C17—22no—
302C22×C17—22no—
303C22×C17—22no—
304C22×C17—41no—
305C22×C17—41no—
306C22×C17—41no—
307C22×C17—41no—
308C22×C17C2×C22⋊Q811yescharacteristic
309C22×C17D4211yescharacteristic
310C22×C17#54711yescharacteristic
311C22×C17#57511yescharacteristic
312C22×C17#57511yescharacteristic
313C22×C17#57611yescharacteristic
314C22×C17#57711yescharacteristic
315C2×D17—42no—
316C2×D17—42no—
317C2×D17—42no—
318C2×D17—81no—
319C2×D17—81no—
320C2×D17—81no—
321C2×D17—81no—
322C4×C17—22no—
323C4×C17—41no—
324C4×C17—41no—
325C4×C17—41no—
326C4×C17—41no—
327C4×C17#52511yescharacteristic
328C4×C17#53911yescharacteristic
329C4×C17D4×Q811yescharacteristic
330C4×C17#55411yescharacteristic
331Dic17—42no—
332Dic17—42no—
333Dic17—42no—
334Dic17—81no—
335Dic17—81no—
336Dic17—81no—
337Dic17—81no—
338C22×M16C2×D1711yescharacteristic
339C2×#125—171no—
340C2×#121—171no—
341C2×C4×D4—171no—
342C2×#117—171no—
343C22×C4⋊C4—171no—
344#478—344no—
345#428—344no—
346C2×#107—171no—
347C2×#107—171no—
348C2×#107—171no—
349C2×#106—171no—
350C2×#106—171no—
351C2×#106—171no—
352C2×C4⋊C8C2×D1711yescharacteristic
353C2×C4⋊C8C2×D1711yescharacteristic
354C2×#102—171no—
355C2×#102—171no—
356C2×#102—171no—
357C2×#102—171no—
358C2×C22⋊C8C2×D1711yescharacteristic
359#400—341no—
360#398—341no—
361#390—171no—
362#390—341no—
363#389—171no—
364#389—171no—
365#389—171no—
366#388—341no—
367#386—341no—
368#385—341no—
369#382—171no—
370#380—171no—
371#380—341no—
372C4×C4⋊C4—171no—
373C4×C22⋊C4—341no—
374C4×C22⋊C4C2×D1711yescharacteristic
375#340—171no—
376#340—171no—
377#340—171no—
378#340—171no—
379#340—171no—
380#340—171no—
381#336C2×D1711yescharacteristic
382#336C2×D1711yescharacteristic
383C2×C17—42no—
384C2×C17#244911yescharacteristic
385C2×C17#308011yescharacteristic
386C2×C17#308111yescharacteristic
387C2×C17#327111yescharacteristic
388C2×C17#327311yescharacteristic
389C2×C17#334811yescharacteristic
390C2×C17#335511yescharacteristic
391D17—82no—
392C22×D4—171no—
393C22×D4—341no—
394C23×C4—341no—
395C23×C4—341no—
396C23×C4C22×D1711yescharacteristic
397C2×M16—22no—
398C2×M16—22no—
399C2×M16—22no—
400C22×C8C22×D1711yescharacteristic
401C22×C8C22×D1711yescharacteristic
402C22×C8C22×D1711yescharacteristic
403C22×C8C22×D1711yescharacteristic
404#125—342no—
405#125—681no—
406#121—174no—
407#121—342no—
408C4×D4—342no—
409C4×D4—681no—
410#117—342no—
411#117—681no—
412C2×C4⋊C4—171no—
413C2×C4⋊C4—171no—
414C2×C4⋊C4—171no—
415C2×C4⋊C4—171no—
416C2×C4⋊C4—171no—
417C2×C4⋊C4—171no—
418C2×C4⋊C4—171no—
419C2×C4⋊C4—342no—
420C2×C4⋊C4—341no—
421C2×C4⋊C4—341no—
422C2×C4⋊C4—341no—
423C2×C4⋊C4—341no—
424C2×C4⋊C4—341no—
425C2×C4⋊C4—341no—
426C2×C4⋊C4—681no—
427C2×C22⋊C4—341no—
428C2×C22⋊C4—341no—
429C2×C22⋊C4—341no—
430C2×C22⋊C4—341no—
431C2×C22⋊C4—341no—
432C2×C22⋊C4—341no—
433C2×C22⋊C4D6811yescharacteristic
434C2×C22⋊C4D6811yescharacteristic
435C2×C42—341no—
436C2×C42—341no—
437C2×C42C22×D1711yescharacteristic
438C2×C42C22×D1711yescharacteristic
439#107—174no—
440#107—342no—
441#107—681no—
442#106—342no—
443#106—342no—
444#106—342no—
445C4⋊C8—22no—
446C4⋊C8—22no—
447#102—342no—
448#102—342no—
449#102—342no—
450#102—681no—
451C22⋊C8—22no—
452#95—341no—
453#95—341no—
454#95—341no—
455#95—341no—
456#95—341no—
457#95—341no—
458#95—341no—
459#95D6811yescharacteristic
460#95D6811yescharacteristic
461C17#2636311yescharacteristic, complemented, Sylow
462C24—341no—
463C24C2×D6811yescharacteristic
464C2×D4—172no—
465C2×D4—344no—
466C2×D4—341no—
467C2×D4—341no—
468C2×D4—341no—
469C2×D4—341no—
470C2×D4—341no—
471C2×D4—1361no—
472C2×D4—1361no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—341no—
478C22×C4—341no—
479C22×C4—341no—
480C22×C4—341no—
481C22×C4—341no—
482C22×C4—341no—
483C22×C4—341no—
484C22×C4—682no—
485C22×C4—682no—
486C22×C4—682no—
487C22×C4—682no—
488C22×C4—1361no—
489C22×C4—1361no—
490C22×C4C2×D6811yescharacteristic
491C22×C4C2×D6811yescharacteristic
492C22×C4C2×D6811yescharacteristic
493C22×C4C2×D6811yescharacteristic
494C22×C4C2×D6811yescharacteristic
495C22×C4#6325412yes—
496C22×C4Q8×D1712yes—
497C22×C4C23×D1711yescharacteristic
498M16—28no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8D4×D1712yes—
510C2×C8#6325412yes—
511C4⋊C4—172no—
512C4⋊C4—172no—
513C4⋊C4—172no—
514C4⋊C4—341no—
515C4⋊C4—341no—
516C4⋊C4—341no—
517C4⋊C4—341no—
518C4⋊C4—341no—
519C4⋊C4—341no—
520C4⋊C4—341no—
521C4⋊C4—341no—
522C4⋊C4—341no—
523C4⋊C4—341no—
524C4⋊C4—341no—
525C4⋊C4—682no—
526C4⋊C4—682no—
527C4⋊C4—682no—
528C4⋊C4—1361no—
529C4⋊C4—1361no—
530C4⋊C4—1361no—
531C22⋊C4—41no—
532C22⋊C4—41no—
533C22⋊C4—344no—
534C22⋊C4—682no—
535C22⋊C4—682no—
536C22⋊C4—682no—
537C22⋊C4—682no—
538C22⋊C4—1361no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—682no—
544C42—1361no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—342no—
552C23—681no—
553C23—681no—
554C23—681no—
555C23—681no—
556C23—681no—
557C23—681no—
558C23#1058758312yes—
559C23C22×D6811yescharacteristic
560D4—342no—
561D4—342no—
562D4—681no—
563D4—681no—
564D4—1364no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
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—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—342no—
584C2×C4—342no—
585C2×C4—342no—
586C2×C4—681no—
587C2×C4—681no—
588C2×C4—681no—
589C2×C4—681no—
590C2×C4—681no—
591C2×C4—681no—
592C2×C4—681no—
593C2×C4—681no—
594C2×C4—681no—
595C2×C4—681no—
596C2×C4—681no—
597C2×C4—681no—
598C2×C4—681no—
599C2×C4—681no—
600C2×C4—681no—
601C2×C4—681no—
602C2×C4—681no—
603C2×C4—681no—
604C2×C4—1364no—
605C2×C4—1364no—
606C2×C4#1058758312yes—
607C2×C4C2×D4×D1711yescharacteristic
608C2×C4C2×#6325411yescharacteristic
609C2×C4C2×#6325411yescharacteristic
610C2×C4C2×Q8×D1711yescharacteristic
611C2×C4#1058769611yescharacteristic
612C2×C4#1058769711yescharacteristic
613C8—22no—
614C8—22no—
615C8—41no—
616C8—41no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—682no—
625C22—682no—
626C22—682no—
627C22—1361no—
628C22—1361no—
629C22—1361no—
630C22—1361no—
631C22C2×#1058758311yescharacteristic
632C22D4×D6811yescharacteristic
633C22#1176346311yescharacteristic
634C22#1176360911yescharacteristic
635C22#1176360911yescharacteristic
636C22#1176361411yescharacteristic
637C22#1176362111yescharacteristic
638C4—22no—
639C4—41no—
640C4—41no—
641C4—41no—
642C4—41no—
643C4—682no—
644C4—682no—
645C4—682no—
646C4—1361no—
647C4—1361no—
648C4—1361no—
649C4—1361no—
650C4#1176331511yescharacteristic
651C4#1176340311yescharacteristic
652C4#1176347211yescharacteristic
653C4#1176351711yescharacteristic
654C2—42no—
655C2—1362no—
656C2#1453259111yescharacteristic
657C2#1453860711yescharacteristic
658C2#1453861611yescharacteristic
659C2#1454037411yescharacteristic
660C2#1454039611yescharacteristic
661C2#1454130111yescharacteristic
662C2#1454138911yescharacteristic
subgroupG/H
centerC23C22×D68
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1851C2
socleC23×C17C22×D4
prime radicalC2×#10587616C22
hypercenter#1851D17
hypocenterC17#26363
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1851

Complements

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

NH ≅ G/Ntype
1. C2×#11763110C2semidirect
2. C2×#11762851C2semidirect
3. #14532508C2semidirect
5. #14530304C2semidirect
6. #14530282C2semidirect
7. #14530157C2semidirect
8. #14530123C2semidirect
9. #14529554C2semidirect
10. #14529356C2semidirect
13. #14529034C2semidirect
14. #14528985C2semidirect
15. C17×#1851C2semidirect
126. #10587489D4semidirect
300. #1851D17semidirect
461. C17#26363semidirect

Maximal subgroups

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

subgroupindex
C2×#117631102
C2×#117628512
#145325082
#145318202
#145303042
#145302822
#145301572
#145301232
#145295542
#145293562
#145292222
#145290992
#145290342
#145289852
C17×#18512
#2636317