Finite Group Databasebeta

← #17094346

Subgroups — #17094346

Order 4352.

840 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763031C211yesmaximal, characteristic
2C2×C4×#63230C211yesmaximal, characteristic
3#14532571C211yesmaximal, characteristic
4#14530328C211yesmaximal, characteristic, complemented
5#14530285C211yesmaximal, characteristic, complemented
6#14530188C211yesmaximal, characteristic
7#14530025C211yesmaximal, characteristic
8#14529459C211yesmaximal, characteristic, complemented
9#14529343C211yesmaximal, characteristic, complemented
10#14529139C211yesmaximal, characteristic
11#14529125C211yesmaximal, characteristic, complemented
12#14529125C211yesmaximal, characteristic, complemented
13#14529052C211yesmaximal, characteristic, complemented
14#14529044C211yesmaximal, characteristic
15Q8×C4⋊C4×C17C211yesmaximal, characteristic, complemented
16C22×#63231C2211yescharacteristic
17C22×#63230C2211yescharacteristic
18C2×#10587561C2211yescharacteristic
19C2×C4×D68C2211yescharacteristic
20#11763031—24no—
21C4×#63230—24no—
22C2×#10587479C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×C4⋊#3622C2211yescharacteristic
25C2×#10587506C2211yescharacteristic
26C2×#10587506C2211yescharacteristic
27C2×#10587481C2211yescharacteristic
28C2×#10587481C2211yescharacteristic
29C2×#10587481C2211yescharacteristic
30C2×#10587480C2211yescharacteristic
31C2×#10587480C2211yescharacteristic
32C2×#10587480C2211yescharacteristic
33C2×C4×#3622C2211yescharacteristic
34#11762738—21no—
35#11762735—21no—
36#11762715C2211yescharacteristic
37#11762715C2211yescharacteristic
38#11762711C2211yescharacteristic
39#11762707C2211yescharacteristic
40#11762680—21no—
41#11762671—21no—
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48#11762564C2211yescharacteristic
49#11762564C2211yescharacteristic
50C2×C4×Q8×C17C2211yescharacteristic
51C2×C4×Q8×C17C2211yescharacteristic
52C2×C4×Q8×C17C2211yescharacteristic
53C17×#389—21no—
54C17×#389—21no—
55C17×#389C2211yescharacteristic
56C17×#389C2211yescharacteristic
57C17×#384—21no—
58C17×#384C2211yescharacteristic
59C4×C4⋊C4×C17—21no—
60C4×C4⋊C4×C17C2211yescharacteristic
61C22×Q8×C17C2311yescharacteristic
62C22×D68C2311yescharacteristic
63C22×C4×D17—21no—
64C22×Dic34C2311yescharacteristic
65C4×Q8×C17—22no—
66C4×Q8×C17—22no—
67C4×Q8×C17—22no—
68C4×Q8×C17—22no—
69C4×Q8×C17—41no—
70C4×Q8×C17—41no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17—21no—
76C2×C4⋊C4×C17—21no—
77C2×C4⋊C4×C17—21no—
78C2×C4⋊C4×C17D411yescharacteristic
79C2×C4⋊C4×C17D411yescharacteristic
80C2×C4⋊C4×C17C2311yescharacteristic
81C2×C4⋊C4×C17C2311yescharacteristic
82C2×C4⋊C4×C17C2311yescharacteristic
83C2×C42×C17—21no—
84C2×C42×C17—21no—
85C2×C42×C17—21no—
86C2×C42×C17C2311yescharacteristic
87C2×C42×C17C2311yescharacteristic
88C2×C42×C17C2311yescharacteristic
89C2×#63231—22no—
90C2×#63231—22no—
91C2×#63231—22no—
92C2×#63230—22no—
93C2×#63230—22no—
94C2×#63230D414yes—
95C2×#63227—21no—
96C2×#63227—21no—
97C2×#63227—21no—
98C2×#63227—21no—
99C2×#63227—21no—
100C2×C4⋊Dic17C2311yescharacteristic
101C2×C4⋊Dic17C2311yescharacteristic
102C2×#63225—21no—
103C22×#3622C2311yescharacteristic
104C22×#3622C2311yescharacteristic
105C22×#3622C2311yescharacteristic
106C22×#3622C2311yescharacteristic
107#10587561—22no—
108#10587561—41no—
109C4×D68—22no—
110C4×D68—41no—
111C17×#95—21no—
112C17×#95—21no—
113C17×#95D411yescharacteristic
114C17×#95D411yescharacteristic
115#10587506—22no—
116#10587506—22no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587481—22no—
123#10587481—22no—
124#10587481—41no—
125#10587480—22no—
126#10587480—22no—
127#10587480—22no—
128#10587479—22no—
129#10587479—41no—
130C4⋊#3622—22no—
131C4×#3622—22no—
132C23×D17—21no—
133C2×Q8×C17—21no—
134C2×Q8×C17—21no—
135C2×Q8×C17—21no—
136C2×Q8×C17C2×D412yes—
137C2×Q8×C17#4112yes—
138C2×Q8×C17#4112yes—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17—21no—
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2411yescharacteristic
150C22×Dic17—21no—
151C22×Dic17—21no—
152C22×Dic17—21no—
153C2×D68—21no—
154C2×D68—21no—
155C2×D68—21no—
156C2×D68—21no—
157C2×D68—21no—
158C2×D68C2×D412yes—
159C2×C4×D17—42no—
160C2×C4×D17—42no—
161C2×C4×D17—81no—
162C2×Dic34—21no—
163C2×Dic34—21no—
164C2×Dic34—21no—
165C2×Dic34—21no—
166C2×Dic34—21no—
167C2×Dic34—21no—
168C4⋊C4×C17—24no—
169C4⋊C4×C17—22no—
170C4⋊C4×C17—22no—
171C4⋊C4×C17—21no—
172C4⋊C4×C17—21no—
173C4⋊C4×C17—21no—
174C4⋊C4×C17—21no—
175C4⋊C4×C17—21no—
176C4⋊C4×C17—42no—
177C4⋊C4×C17—42no—
178C4⋊C4×C17—42no—
179C4⋊C4×C17—81no—
180C4⋊C4×C17—81no—
181C4⋊C4×C17—81no—
182C4⋊C4×C17#4112yes—
183C42×C17—24no—
184C42×C17—21no—
185C42×C17—21no—
186C42×C17—21no—
187C42×C17—21no—
188C42×C17—21no—
189C42×C17—21no—
190C42×C17—42no—
191C42×C17—42no—
192#63231—44no—
193#63230—28no—
194#63227—42no—
195#63227—42no—
196#63227—42no—
197#63227—42no—
198#63227—81no—
199C4⋊Dic17—21no—
200C4⋊Dic17—21no—
201C4⋊Dic17—21no—
202C4⋊Dic17—21no—
203#63225—81no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622—21no—
208C2×#3622—21no—
209C2×#3622—21no—
210C2×#3622—21no—
211C2×#3622—21no—
212C2×#3622—21no—
213C2×#3622—21no—
214C2×#3622—21no—
215C2×#3622C2×D412yes—
216#26161—171nomaximal, Sylow
217C23×C17C22×D411yescharacteristic
218C22×D17—22no—
219C22×D17—41no—
220C22×D17—41no—
221C22×D17—41no—
222C22×D17—41no—
223C22×D17—41no—
224C22×D17—41no—
225Q8×C17—22no—
226Q8×C17—22no—
227Q8×C17—22no—
228Q8×C17#12314yes—
229C2×C4×C17—21no—
230C2×C4×C17—22no—
231C2×C4×C17—21no—
232C2×C4×C17—22no—
233C2×C4×C17—22no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—22no—
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×C17—21no—
248C2×C4×C17—21no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17—41no—
253C2×C4×C17—41no—
254C2×C4×C17—41no—
255C2×C4×C17—41no—
256C2×C4×C17—41no—
257C2×C4×C17#13512yes—
258C2×C4×C17#13712yes—
259C2×C4×C17C22×D411yescharacteristic
260C2×C4×C17C2×#4111yescharacteristic
261C2×C4×C17C2×#4111yescharacteristic
262C2×C4×C17C2×#4111yescharacteristic
263C2×C4×C17H(5,𝔽2)11yescharacteristic
264C2×C4×C172−1+411yescharacteristic
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—
274D68—22no—
275D68—22no—
276D68—41no—
277D68—41no—
278C4×D17—84no—
279Dic34—41no—
280Dic34—41no—
281Dic34—41no—
282Dic34—41no—
283#3622—22no—
284#3622—41no—
285#3622—41no—
286#3622—41no—
287C2×#464—171no—
288C2×C4×SD16—171no—
289#2447—171no—
290Q8×C4⋊C4D1711yescharacteristic, complemented
291#2096—171no—
292#2088—171no—
293#2071—171no—
294#2042—171no—
295#1933—171no—
296#1916—171no—
297#1880—171no—
298#1878—171no—
299#1878—171no—
300#1865—171no—
301#1864—171no—
302C22×C17#52511yescharacteristic
303C22×C17D4211yescharacteristic
304C22×C17#54811yescharacteristic
305C22×C17C2×#13511yescharacteristic
306C22×C17C2×#13711yescharacteristic
307C22×C17#57711yescharacteristic
308C22×C17#57711yescharacteristic
309C2×D17—42no—
310C2×D17—42no—
311C2×D17—42no—
312C2×D17—81no—
313C2×D17—81no—
314C2×D17—81no—
315C2×D17—81no—
316C4×C17—22no—
317C4×C17—24no—
318C4×C17—22no—
319C4×C17—41no—
320C4×C17—42no—
321C4×C17—41no—
322C4×C17—41no—
323C4×C17—41no—
324C4×C17—81no—
325C4×C17—81no—
326C4×C17C2×#12311yescharacteristic
327C4×C17#53911yescharacteristic
328C4×C17#54711yescharacteristic
329C4×C17#55311yescharacteristic
330Dic17—81no—
331Dic17—81no—
332Dic17—81no—
333C22×Q16—171no—
334C22×SD16—171no—
335C2×#124—171no—
336C2×C4×Q8C2×D1711yescharacteristic
337C2×C4×Q8C2×D1711yescharacteristic
338C2×C4×Q8C2×D1711yescharacteristic
339C2×C4×D4—171no—
340#464—344no—
341C4×SD16—344no—
342C2×#106—171no—
343C2×#106—171no—
344C2×C4⋊C8—171no—
345C2×#103—171no—
346C2×#103—171no—
347C2×#103—171no—
348C2×#103—171no—
349C2×#103—171no—
350C2×#102—171no—
351C2×#102—171no—
352C2×#102—171no—
353C2×C4×C8—171no—
354#397—341no—
355#397—341no—
356#391—171no—
357#391—171no—
358#390—171no—
359#389—21no—
360#389—21no—
361#389—171no—
362#389C2×D1711yescharacteristic
363#389C2×D1711yescharacteristic
364#385—341no—
365#384—21no—
366#384C2×D1711yescharacteristic
367#381—341no—
368C4×C4⋊C4—21no—
369C4×C4⋊C4C2×D1711yescharacteristic
370#340—171no—
371#340—171no—
372#340—171no—
373#340—171no—
374#336—171no—
375#336—171no—
376#336—171no—
377#336—171no—
378C2×C17#244311yescharacteristic
379C2×C17#310811yescharacteristic
380C2×C17#312511yescharacteristic
381C2×C17#326611yescharacteristic
382C2×C17#328411yescharacteristic
383C2×C17#336511yescharacteristic
384C2×C17#336911yescharacteristic
385D17—82no—
386C22×Q8—171no—
387C22×Q8C22×D1711yescharacteristic
388C22×D4—171no—
389C23×C4—341no—
390C2×Q16—342no—
391C2×Q16—342no—
392C2×Q16—342no—
393C2×SD16—174no—
394C2×SD16—342no—
395C2×SD16—342no—
396C22×C8—171no—
397C22×C8—171no—
398C22×C8—171no—
399C22×C8—171no—
400#124—342no—
401#124—681no—
402C4×Q8—22no—
403C4×Q8—22no—
404C4×Q8—22no—
405C4×Q8—22no—
406C4×Q8—41no—
407C4×Q8—41no—
408C4×D4—342no—
409C4×D4—681no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—171no—
418C2×C4⋊C4—171no—
419C2×C4⋊C4—341no—
420C2×C4⋊C4C22⋊D1711yescharacteristic
421C2×C4⋊C4C22⋊D1711yescharacteristic
422C2×C4⋊C4C22×D1711yescharacteristic
423C2×C4⋊C4C22×D1711yescharacteristic
424C2×C4⋊C4C22×D1711yescharacteristic
425C2×C22⋊C4—341no—
426C2×C22⋊C4—341no—
427C2×C22⋊C4—341no—
428C2×C22⋊C4—341no—
429C2×C22⋊C4—341no—
430C2×C42—21no—
431C2×C42—21no—
432C2×C42—21no—
433C2×C42C22×D1711yescharacteristic
434C2×C42C22×D1711yescharacteristic
435C2×C42C22×D1711yescharacteristic
436#106—342no—
437#106—681no—
438C4⋊C8—342no—
439#103—342no—
440#103—342no—
441#103—342no—
442#103—342no—
443#103—681no—
444#102—342no—
445#102—342no—
446#102—342no—
447C4×C8—342no—
448#95—21no—
449#95—21no—
450#95—341no—
451#95—341no—
452#95—341no—
453#95—341no—
454#95—341no—
455#95C22⋊D1711yescharacteristic
456#95C22⋊D1711yescharacteristic
457C17#2616111yescharacteristic, complemented, Sylow
458C24—341no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—341no—
463C2×Q8—341no—
464C2×Q8—341no—
465C2×Q8—341no—
466C2×Q8—341no—
467C2×Q8—341no—
468C2×Q8#6325212yes—
469C2×Q8D4×D1712yes—
470C2×Q8#6325612yes—
471C2×D4—172no—
472C2×D4—341no—
473C2×D4—341no—
474C2×D4—341no—
475C2×D4—341no—
476C2×D4—341no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—341no—
482C22×C4—341no—
483C22×C4—341no—
484C22×C4—682no—
485C22×C4—682no—
486C22×C4—1361no—
487C22×C4C2×C22⋊D1711yescharacteristic
488C22×C4C2×C22⋊D1711yescharacteristic
489C22×C4C2×C22⋊D1711yescharacteristic
490C22×C4C2×C22⋊D1711yescharacteristic
491C22×C4C2×C22⋊D1711yescharacteristic
492C22×C4C2×C22⋊D1711yescharacteristic
493C22×C4C23×D1711yescharacteristic
494Q16—684no—
495SD16—348no—
496C2×C8—172no—
497C2×C8—341no—
498C2×C8—341no—
499C2×C8—341no—
500C2×C8—341no—
501C2×C8—341no—
502C2×C8—341no—
503C2×C8—341no—
504C2×C8—341no—
505C2×C8—341no—
506C2×C8—341no—
507C2×C8—341no—
508C4⋊C4—24no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—81no—
520C4⋊C4—81no—
521C4⋊C4—81no—
522C4⋊C4—341no—
523C4⋊C4—341no—
524C4⋊C4—341no—
525C4⋊C4—341no—
526C4⋊C4—1361no—
527C4⋊C4#6325212yes—
528C22⋊C4—682no—
529C22⋊C4—682no—
530C22⋊C4—682no—
531C22⋊C4—682no—
532C22⋊C4—1361no—
533C42—24no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—42no—
541C42—42no—
542C23—342no—
543C23—681no—
544C23—681no—
545C23—681no—
546C23—681no—
547C23—681no—
548C23—681no—
549C23C22×C22⋊D1711yescharacteristic
550Q8—22no—
551Q8—22no—
552Q8—22no—
553Q8—681no—
554Q8—681no—
555Q8—681no—
556Q8—681no—
557Q8#1058758014yes—
558D4—342no—
559D4—342no—
560D4—681no—
561D4—681no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
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—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
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—1364no—
600C2×C4#1058762812yes—
601C2×C4#1058763612yes—
602C2×C4C2×#6325211yescharacteristic
603C2×C4C2×#6325211yescharacteristic
604C2×C4C2×D4×D1711yescharacteristic
605C2×C4#1058769111yescharacteristic
606C2×C4C2×#6325611yescharacteristic
607C2×C4#1058769411yescharacteristic
608C8—342no—
609C8—681no—
610C8—681no—
611C8—681no—
612C22—682no—
613C22—682no—
614C22—682no—
615C22—1361no—
616C22—1361no—
617C22—1361no—
618C22—1361no—
619C22#1176332011yescharacteristic
620C22D4×C22⋊D1711yescharacteristic
621C22#1176347011yescharacteristic
622C22C2×#1058763611yescharacteristic
623C22C2×#1058762811yescharacteristic
624C22#1176361711yescharacteristic
625C22#1176362011yescharacteristic
626C4—22no—
627C4—22no—
628C4—24no—
629C4—41no—
630C4—41no—
631C4—42no—
632C4—41no—
633C4—41no—
634C4—81no—
635C4—81no—
636C4—1361no—
637C4—1361no—
638C4—1361no—
639C4C2×#1058758011yescharacteristic
640C4#1176340611yescharacteristic
641C4#1176346011yescharacteristic
642C4#1176350711yescharacteristic
643C2—1362no—
644C2#1453251911yescharacteristic
645C2#1453886411yescharacteristic
646C2#1453906111yescharacteristic
647C2#1454029511yescharacteristic
648C2#1454051411yescharacteristic
649C2#1454148011yescharacteristic
650C2#1454152411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingQ8×C4⋊C4×C17C2
socleC23×C17C22×D4
prime radicalC22×D68C23
hypercenterQ8×C4⋊C4D17
hypocenterC17#26161
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #14530328C2semidirect
5. #14530285C2semidirect
8. #14529459C2semidirect
9. #14529343C2semidirect
11. #14529125C2semidirect
12. #14529125C2semidirect
13. #14529052C2semidirect
15. Q8×C4⋊C4×C17C2semidirect
290. Q8×C4⋊C4D17semidirect
457. C17#26161semidirect

Maximal subgroups

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

subgroupindex
C2×#117630312
C2×C4×#632302
#145325712
#145303282
#145302852
#145301882
#145300252
#145294592
#145293432
#145291392
#145291252
#145290522
#145290442
Q8×C4⋊C4×C172
#2616117