Finite Group Databasebeta

← #17097183

Subgroups — #17097183

Order 4352.

812 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763117C211yesmaximal, characteristic, complemented
2C2×#11762851C211yesmaximal, characteristic
3#14532497C211yesmaximal, characteristic
4#14531787C211yesmaximal, characteristic
5#14530321C211yesmaximal, characteristic, complemented
6#14530272C211yesmaximal, characteristic, complemented
7#14530164C211yesmaximal, characteristic, complemented
8#14530123C211yesmaximal, characteristic, complemented
9#14529543C211yesmaximal, characteristic, complemented
10#14529363C211yesmaximal, characteristic, complemented
11#14529217C211yesmaximal, characteristic
12#14529094C211yesmaximal, characteristic
13#14529041C211yesmaximal, characteristic, complemented
14#14528974C211yesmaximal, characteristic
15C17×#1851C211yesmaximal, characteristic
16C2×#10587556C2211yescharacteristic
17C2×#10587610C2211yescharacteristic
18C2×C4×Dic34C2211yescharacteristic
19C2×#10587612C2211yescharacteristic
20C22×C4⋊Dic17C2211yescharacteristic
21#11763117—24no—
22#11762851—24no—
23C2×#10587489C2211yescharacteristic
24C2×#10587489C2211yescharacteristic
25C2×#10587489C2211yescharacteristic
26C2×#10587489C2211yescharacteristic
27C2×#10587488C2211yescharacteristic
28C2×#10587488C2211yescharacteristic
29C2×#10587487C2211yescharacteristic
30C2×#10587487C2211yescharacteristic
31C2×#10587487C2211yescharacteristic
32C2×#10587487C2211yescharacteristic
33#11762745—21no—
34#11762732—21no—
35#11762715C2211yescharacteristic
36#11762707C2211yescharacteristic
37#11762707C2211yescharacteristic
38#11762707C2211yescharacteristic
39#11762706—21no—
40#11762706—21no—
41#11762696—21no—
42#11762692—21no—
43#11762681—21no—
44#11762679C2211yescharacteristic
45#11762669C2211yescharacteristic
46C4×C4⋊Dic17C2211yescharacteristic
47C22⋊C4×Dic17—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
62C23×Dic17—21no—
63C22×Dic34C2311yescharacteristic
64C2×M16×C17—22no—
65C2×M16×C17—22no—
66C2×M16×C17—22no—
67C22×C8×C17C2311yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C22×C8×C17C2311yescharacteristic
70C22×C8×C17C2311yescharacteristic
71C2×C22⋊C4×C17D411yescharacteristic
72C2×C22⋊C4×C17D411yescharacteristic
73C2×C42×C17C2311yescharacteristic
74C2×C42×C17C2311yescharacteristic
75C2×C22⋊Dic17—21no—
76C2×C22⋊Dic17—21no—
77C2×C22⋊Dic17—21no—
78#10587612—22no—
79#10587612—41no—
80C2×C4⋊Dic17—21no—
81C2×C4⋊Dic17—21no—
82C2×C4⋊Dic17—21no—
83C2×C4⋊Dic17—22no—
84C2×C4⋊Dic17—41no—
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92#10587610—22no—
93#10587610Q814yes—
94C2×#63225—21no—
95C2×#63225—21no—
96C2×#63225—21no—
97C2×#63225—21no—
98C2×#63225—21no—
99C2×#63225—21no—
100C2×#63225—21no—
101C2×C4×Dic17—21no—
102C2×C4×Dic17—21no—
103C2×C4×Dic17—21no—
104#10587556—22no—
105#10587556—41no—
106C4×Dic34—22no—
107C4×Dic34—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#10587489—22no—
121#10587489—22no—
122#10587489—22no—
123#10587489—41no—
124#10587488—22no—
125#10587488D414yes—
126#10587487—22no—
127#10587487—22no—
128#10587487—22no—
129#10587487—41no—
130C24×C17C2×D411yescharacteristic
131C22×C4×C17—21no—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17C2×D411yescharacteristic
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2×Q812yes—
141C22×C4×C17#4112yes—
142C22×C4×C17C2411yescharacteristic
143C22×Dic17—21no—
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17—21no—
148C22×Dic17—21no—
149C22×Dic17—21no—
150C22×Dic17—21no—
151C22×Dic17—42no—
152C22×Dic17—42no—
153C22×Dic17—81no—
154C2×Dic34—21no—
155C2×Dic34—21no—
156C2×Dic34—21no—
157C2×Dic34—21no—
158C2×Dic34—21no—
159C2×Dic34C2×Q812yes—
160M16×C17—28no—
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×C17C2×D412yes—
172C2×C8×C17#4112yes—
173C22⋊C4×C17—22no—
174C22⋊C4×C17—41no—
175C42×C17—21no—
176C42×C17—21no—
177C42×C17—21no—
178C42×C17—21no—
179C22⋊Dic17—24no—
180C22⋊Dic17—42no—
181C22⋊Dic17—81no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—21no—
185C4⋊Dic17—21no—
186C4⋊Dic17—21no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—21no—
192C4⋊Dic17—21no—
193C4⋊Dic17—42no—
194C4⋊Dic17—42no—
195C4⋊Dic17—81no—
196C4⋊Dic17C2×D412yes—
197C4⋊Dic17C2×D412yes—
198C4⋊Dic17C2×Q812yes—
199#63225—24no—
200#63225—42no—
201#63225—42no—
202#63225—42no—
203#63225—42no—
204#63225—81no—
205#63225—81no—
206C4×Dic17—42no—
207C4×Dic17—42no—
208C4×Dic17—81no—
209#26362—171nomaximal, Sylow
210C23×C17—21no—
211C23×C17—21no—
212C23×C17—21no—
213C23×C17—21no—
214C23×C17—21no—
215C23×C17—21no—
216C23×C17C22⋊Q812yes—
217C23×C17C22×D411yescharacteristic
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—22no—
223C2×C4×C17—22no—
224C2×C4×C17—22no—
225C2×C4×C17—22no—
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×C17C22⋊Q812yes—
236C2×C4×C17#13712yes—
237C2×C4×C17C22×D411yescharacteristic
238C2×C4×C17C22×Q811yescharacteristic
239C2×C4×C17C2×#4111yescharacteristic
240C2×C4×C17C2×#4111yescharacteristic
241C2×C4×C17H(5,𝔽2)11yescharacteristic
242C2×C4×C172−1+411yescharacteristic
243C2×Dic17—22no—
244C2×Dic17—22no—
245C2×Dic17—22no—
246C2×Dic17—22no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256C2×Dic17—41no—
257C2×Dic17—41no—
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—84no—
268Dic34—22no—
269Dic34—22no—
270Dic34—41no—
271Dic34—41no—
272C8×C17—22no—
273C8×C17—22no—
274C8×C17—41no—
275C8×C17—41no—
276C2×#479—171no—
277C2×#428—171no—
278#2441—171no—
279#2360—171no—
280#2095—171no—
281#2086—171no—
282#2066—171no—
283#2059—171no—
284#1947—171no—
285#1919—171no—
286#1894—171no—
287#1873—171no—
288#1863—171no—
289#1852—171no—
290#1851D1711yescharacteristic
291C22×C17—22no—
292C22×C17—22no—
293C22×C17—22no—
294C22×C17—41no—
295C22×C17—41no—
296C22×C17—41no—
297C22×C17—41no—
298C22×C17C2×C22⋊Q811yescharacteristic
299C22×C17D4211yescharacteristic
300C22×C17#54711yescharacteristic
301C22×C17C2×#13711yescharacteristic
302C22×C17#57511yescharacteristic
303C22×C17#57611yescharacteristic
304C22×C17#57811yescharacteristic
305C4×C17—22no—
306C4×C17—22no—
307C4×C17—41no—
308C4×C17—41no—
309C4×C17—41no—
310C4×C17#52511yescharacteristic
311C4×C17#53911yescharacteristic
312C4×C17D4×Q811yescharacteristic
313C4×C17#55411yescharacteristic
314Dic17—42no—
315Dic17—42no—
316Dic17—42no—
317Dic17—42no—
318Dic17—81no—
319Dic17—81no—
320Dic17—81no—
321Dic17—81no—
322C22×M16C2×D1711yescharacteristic
323C2×#125—171no—
324C2×C22⋊Q8—171no—
325C2×C4×Q8—171no—
326C2×#117—171no—
327C22×C4⋊C4—171no—
328#479—344no—
329#428—344no—
330C2×#107—171no—
331C2×#107—171no—
332C2×#107—171no—
333C2×#107—171no—
334C2×#106—171no—
335C2×#106—171no—
336C2×C4⋊C8C2×D1711yescharacteristic
337C2×C4⋊C8C2×D1711yescharacteristic
338C2×#103—171no—
339C2×#103—171no—
340C2×#103—171no—
341C2×#103—171no—
342C2×C22⋊C8C2×D1711yescharacteristic
343#399—341no—
344#396—341no—
345#391—171no—
346#389—171no—
347#389—171no—
348#389—171no—
349#389—341no—
350#389—341no—
351#387—341no—
352#387—341no—
353#385—341no—
354#384—171no—
355#380—171no—
356C4×C4⋊C4—171no—
357C4×C22⋊C4—341no—
358C4×C22⋊C4C2×D1711yescharacteristic
359#340—171no—
360#340—171no—
361#340—171no—
362#340—171no—
363#340—171no—
364#340—171no—
365#336C2×D1711yescharacteristic
366#336C2×D1711yescharacteristic
367C2×C17—42no—
368C2×C17#244911yescharacteristic
369C2×C17#307811yescharacteristic
370C2×C17#308311yescharacteristic
371C2×C17#326811yescharacteristic
372C2×C17#327611yescharacteristic
373C2×C17#334611yescharacteristic
374C2×C17#335211yescharacteristic
375C22×Q8—171no—
376C23×C4—341no—
377C23×C4C22×D1711yescharacteristic
378C2×M16—22no—
379C2×M16—22no—
380C2×M16—22no—
381C22×C8C22×D1711yescharacteristic
382C22×C8C22×D1711yescharacteristic
383C22×C8C22×D1711yescharacteristic
384C22×C8C22×D1711yescharacteristic
385#125—342no—
386#125—681no—
387C22⋊Q8—174no—
388C22⋊Q8—342no—
389C4×Q8—342no—
390C4×Q8—681no—
391#117—342no—
392#117—681no—
393C2×C4⋊C4—171no—
394C2×C4⋊C4—171no—
395C2×C4⋊C4—171no—
396C2×C4⋊C4—171no—
397C2×C4⋊C4—171no—
398C2×C4⋊C4—171no—
399C2×C4⋊C4—171no—
400C2×C4⋊C4—342no—
401C2×C4⋊C4—341no—
402C2×C4⋊C4—341no—
403C2×C4⋊C4—341no—
404C2×C4⋊C4—341no—
405C2×C4⋊C4—341no—
406C2×C4⋊C4—341no—
407C2×C4⋊C4—341no—
408C2×C4⋊C4—341no—
409C2×C4⋊C4—341no—
410C2×C4⋊C4—341no—
411C2×C4⋊C4—681no—
412C2×C22⋊C4—341no—
413C2×C22⋊C4—341no—
414C2×C22⋊C4—341no—
415C2×C22⋊C4D6811yescharacteristic
416C2×C22⋊C4D6811yescharacteristic
417C2×C42—341no—
418C2×C42—341no—
419C2×C42—341no—
420C2×C42C22×D1711yescharacteristic
421C2×C42C22×D1711yescharacteristic
422#107—342no—
423#107—342no—
424#107—342no—
425#107—681no—
426#106—174no—
427#106—342no—
428C4⋊C8—22no—
429C4⋊C8—22no—
430#103—342no—
431#103—342no—
432#103—342no—
433#103—681no—
434C22⋊C8—22no—
435#95—341no—
436#95—341no—
437#95—341no—
438#95—341no—
439#95—341no—
440#95—341no—
441#95—341no—
442#95D6811yescharacteristic
443#95D6811yescharacteristic
444C17#2636211yescharacteristic, complemented, Sylow
445C24C2×D6811yescharacteristic
446C2×Q8—172no—
447C2×Q8—341no—
448C2×Q8—341no—
449C2×Q8—341no—
450C2×Q8—341no—
451C2×Q8—341no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—341no—
457C22×C4—341no—
458C22×C4—341no—
459C22×C4—341no—
460C22×C4—341no—
461C22×C4—341no—
462C22×C4—341no—
463C22×C4—341no—
464C22×C4—682no—
465C22×C4—682no—
466C22×C4—1361no—
467C22×C4C2×D6811yescharacteristic
468C22×C4C2×D6811yescharacteristic
469C22×C4C2×D6811yescharacteristic
470C22×C4C2×D6811yescharacteristic
471C22×C4C2×D6811yescharacteristic
472C22×C4#6325412yes—
473C22×C4Q8×D1712yes—
474C22×C4C23×D1711yescharacteristic
475M16—28no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8D4×D1712yes—
487C2×C8#6325412yes—
488C4⋊C4—172no—
489C4⋊C4—172no—
490C4⋊C4—172no—
491C4⋊C4—341no—
492C4⋊C4—341no—
493C4⋊C4—341no—
494C4⋊C4—344no—
495C4⋊C4—341no—
496C4⋊C4—341no—
497C4⋊C4—341no—
498C4⋊C4—341no—
499C4⋊C4—341no—
500C4⋊C4—341no—
501C4⋊C4—341no—
502C4⋊C4—341no—
503C4⋊C4—682no—
504C4⋊C4—682no—
505C4⋊C4—682no—
506C4⋊C4—682no—
507C4⋊C4—682no—
508C4⋊C4—682no—
509C4⋊C4—1361no—
510C4⋊C4—1361no—
511C4⋊C4—1361no—
512C22⋊C4—22no—
513C22⋊C4—41no—
514C22⋊C4—344no—
515C22⋊C4—682no—
516C22⋊C4—1361no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—682no—
522C42—682no—
523C42—1361no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23#1058758312yes—
531C23C22×D6811yescharacteristic
532Q8—342no—
533Q8—342no—
534Q8—681no—
535Q8—681no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—342no—
554C2×C4—342no—
555C2×C4—342no—
556C2×C4—342no—
557C2×C4—681no—
558C2×C4—681no—
559C2×C4—681no—
560C2×C4—681no—
561C2×C4—681no—
562C2×C4—681no—
563C2×C4—681no—
564C2×C4—681no—
565C2×C4—681no—
566C2×C4—681no—
567C2×C4—681no—
568C2×C4—681no—
569C2×C4—681no—
570C2×C4—681no—
571C2×C4—681no—
572C2×C4—681no—
573C2×C4—681no—
574C2×C4—681no—
575C2×C4—681no—
576C2×C4—681no—
577C2×C4—1364no—
578C2×C4#1058758312yes—
579C2×C4#1058759512yes—
580C2×C4C2×D4×D1711yescharacteristic
581C2×C4C2×#6325411yescharacteristic
582C2×C4C2×#6325411yescharacteristic
583C2×C4C2×Q8×D1711yescharacteristic
584C2×C4#1058769611yescharacteristic
585C2×C4#1058769711yescharacteristic
586C8—22no—
587C8—22no—
588C8—41no—
589C8—41no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22C2×#1058758311yescharacteristic
598C22D4×D6811yescharacteristic
599C22#1176346311yescharacteristic
600C22C2×#1058759511yescharacteristic
601C22#1176360911yescharacteristic
602C22#1176361411yescharacteristic
603C22#1176362611yescharacteristic
604C4—22no—
605C4—22no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—682no—
610C4—682no—
611C4—682no—
612C4—682no—
613C4—1361no—
614C4—1361no—
615C4—1361no—
616C4—1361no—
617C4#1176331511yescharacteristic
618C4#1176340311yescharacteristic
619C4#1176347211yescharacteristic
620C4#1176351711yescharacteristic
621C2—42no—
622C2#1453259111yescharacteristic
623C2#1453859111yescharacteristic
624C2#1453863711yescharacteristic
625C2#1454032911yescharacteristic
626C2#1454043211yescharacteristic
627C2#1454127111yescharacteristic
628C2#1454135811yescharacteristic
subgroupG/H
centerC23C22×D68
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1851C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#1851D17
hypocenterC17#26362
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×#11763117C2semidirect
5. #14530321C2semidirect
6. #14530272C2semidirect
7. #14530164C2semidirect
8. #14530123C2semidirect
9. #14529543C2semidirect
10. #14529363C2semidirect
13. #14529041C2semidirect
444. C17#26362semidirect

Maximal subgroups

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

subgroupindex
C2×#117631172
C2×#117628512
#145324972
#145317872
#145303212
#145302722
#145301642
#145301232
#145295432
#145293632
#145292172
#145290942
#145290412
#145289742
C17×#18512
#2636217