Finite Group Databasebeta

← #17076329

Subgroups — #17076329

Order 4352.

904 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763065C211yesmaximal, characteristic
2C2×#11763048C211yesmaximal, characteristic
3C2×#11763019C211yesmaximal, characteristic, complemented
4C2×#11762836C211yesmaximal, characteristic
5#14534550C211yesmaximal, characteristic
6#14530176C211yesmaximal, characteristic, complemented
7#14530162C211yesmaximal, characteristic, complemented
8#14530106C211yesmaximal, characteristic, complemented
9#14530106C211yesmaximal, characteristic, complemented
10#14529707C211yesmaximal, characteristic, complemented
11#14529552C211yesmaximal, characteristic, complemented
12#14529215C211yesmaximal, characteristic
13#14529205C211yesmaximal, characteristic
14#14528799C211yesmaximal, characteristic, complemented
15C17×#2323C211yesmaximal, characteristic
16C22×#63231C2211yescharacteristic
17C22×#63231C2211yescharacteristic
18C22×#63223C2211yescharacteristic
19C23×#3622C2211yescharacteristic
20C2×#10587555C2211yescharacteristic
21C2×#10587610C2211yescharacteristic
22C2×#10587610C2211yescharacteristic
23#11763065—24no—
24#11763048—24no—
25#11763019—24no—
26C2×#10587478C2211yescharacteristic
27C2×#10587479C2211yescharacteristic
28#11762836—24no—
29C2×C4⋊#3622C2211yescharacteristic
30C2×C4⋊#3622C2211yescharacteristic
31C2×C4⋊#3622C2211yescharacteristic
32C2×C4⋊#3622C2211yescharacteristic
33C2×#10587506C2211yescharacteristic
34C2×#10587506C2211yescharacteristic
35C2×#10587481C2211yescharacteristic
36C2×#10587481C2211yescharacteristic
37C2×#10587481C2211yescharacteristic
38C2×#10587481C2211yescharacteristic
39C2×#10587481C2211yescharacteristic
40C2×#10587481C2211yescharacteristic
41#11762739—21no—
42#11762739—21no—
43#11762734—21no—
44#11762734—21no—
45#11762715C2211yescharacteristic
46#11762715C2211yescharacteristic
47#11762675C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52C2×C17×C22⋊Q8C2211yescharacteristic
53C2×C4×Q8×C17C2211yescharacteristic
54C2×C17×#117C2211yescharacteristic
55C17×#391C2211yescharacteristic
56C17×#389—21no—
57C17×#389C2211yescharacteristic
58C17×#388—21no—
59C17×#388—21no—
60C17×#387—21no—
61C17×#380C2211yescharacteristic
62C4×C4⋊C4×C17C2211yescharacteristic
63C22×Q8×C17C2311yescharacteristic
64C23×C4×C17C2311yescharacteristic
65C22×Dic34C2311yescharacteristic
66C22×Dic34C2311yescharacteristic
67C17×C22⋊Q8—22no—
68C17×C22⋊Q8D414yes—
69C4×Q8×C17—22no—
70C4×Q8×C17—22no—
71C17×#117—22no—
72C17×#117—22no—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17—21no—
76C2×C4⋊C4×C17D411yescharacteristic
77C2×C4⋊C4×C17D411yescharacteristic
78C2×C4⋊C4×C17C2311yescharacteristic
79C2×C4⋊C4×C17C2311yescharacteristic
80C2×C4⋊C4×C17C2311yescharacteristic
81C2×C22⋊C4×C17—21no—
82C2×C22⋊C4×C17—21no—
83C2×C22⋊C4×C17D411yescharacteristic
84C2×C22⋊C4×C17D411yescharacteristic
85C2×C42×C17—21no—
86C2×C42×C17—21no—
87C2×C42×C17C2311yescharacteristic
88C2×C42×C17C2311yescharacteristic
89C2×C22⋊Dic17—21no—
90C2×C22⋊Dic17—21no—
91C2×#63231—22no—
92C2×#63231—22no—
93C2×#63231—22no—
94C2×#63231—22no—
95C2×#63231—41no—
96C2×#63231—41no—
97C2×C4⋊Dic17—21no—
98C2×C4⋊Dic17—21no—
99C2×C4⋊Dic17C2311yescharacteristic
100C2×C4⋊Dic17C2311yescharacteristic
101#10587610—22no—
102#10587610—22no—
103#10587610—22no—
104#10587610—22no—
105C2×#63225—21no—
106C2×#63225—21no—
107C2×#63223—22no—
108C2×#63223—22no—
109C2×#63223—41no—
110C22×#3622—22no—
111C22×#3622—22no—
112C22×#3622—41no—
113C22×#3622C2311yescharacteristic
114C22×#3622C2311yescharacteristic
115C22×#3622C2311yescharacteristic
116C22×#3622C2311yescharacteristic
117#10587555—22no—
118#10587555—22no—
119C17×#95—21no—
120C17×#95—21no—
121C17×#95—21no—
122C17×#95—21no—
123C17×#95—21no—
124#10587506—22no—
125#10587506—22no—
126#10587502—21no—
127#10587502—21no—
128#10587502—21no—
129#10587502—21no—
130#10587502—21no—
131#10587502—21no—
132#10587481—22no—
133#10587481—22no—
134#10587481—22no—
135#10587481—22no—
136#10587481—41no—
137#10587481—41no—
138#10587479—22no—
139#10587478—22no—
140C4⋊#3622—22no—
141C4⋊#3622—22no—
142C4⋊#3622—22no—
143C4⋊#3622D414yes—
144C24×C17C2×D411yescharacteristic
145C2×Q8×C17—21no—
146C2×Q8×C17—21no—
147C2×Q8×C17—21no—
148C2×Q8×C17—21no—
149C2×Q8×C17—21no—
150C2×Q8×C17C2×D412yes—
151C22×C4×C17—21no—
152C22×C4×C17—21no—
153C22×C4×C17—21no—
154C22×C4×C17—21no—
155C22×C4×C17—21no—
156C22×C4×C17—21no—
157C22×C4×C17—21no—
158C22×C4×C17—21no—
159C22×C4×C17C2×D412yes—
160C22×C4×C17C2×D411yescharacteristic
161C22×C4×C17C2×D411yescharacteristic
162C22×C4×C17C2×D411yescharacteristic
163C22×C4×C17C2×D411yescharacteristic
164C22×C4×C17C2×D411yescharacteristic
165C22×C4×C17#4112yes—
166C22×C4×C17C2411yescharacteristic
167C22×Dic17—21no—
168C22×Dic17—21no—
169C22×Dic17—21no—
170C22×Dic17—21no—
171C2×Dic34—21no—
172C2×Dic34—21no—
173C2×Dic34—21no—
174C2×Dic34—21no—
175C2×Dic34—21no—
176C2×Dic34—21no—
177C2×Dic34—21no—
178C2×Dic34—21no—
179C2×Dic34—21no—
180C2×Dic34—21no—
181C2×Dic34—21no—
182C2×Dic34—21no—
183C4⋊C4×C17—24no—
184C4⋊C4×C17—22no—
185C4⋊C4×C17—22no—
186C4⋊C4×C17—21no—
187C4⋊C4×C17—21no—
188C4⋊C4×C17—21no—
189C4⋊C4×C17—21no—
190C4⋊C4×C17—21no—
191C4⋊C4×C17—42no—
192C4⋊C4×C17—81no—
193C4⋊C4×C17C2×D412yes—
194C22⋊C4×C17—24no—
195C22⋊C4×C17—22no—
196C22⋊C4×C17—22no—
197C22⋊C4×C17—81no—
198C42×C17—21no—
199C42×C17—21no—
200C42×C17—42no—
201C42×C17—81no—
202C42×C17C2×D412yes—
203C42×C17#4112yes—
204C22⋊Dic17—42no—
205C22⋊Dic17—42no—
206#63231—44no—
207#63231—44no—
208C4⋊Dic17—21no—
209C4⋊Dic17—21no—
210C4⋊Dic17—21no—
211C4⋊Dic17—21no—
212C4⋊Dic17—42no—
213C4⋊Dic17—42no—
214#63225—42no—
215#63225—42no—
216#63223—44no—
217C2×#3622—21no—
218C2×#3622—21no—
219C2×#3622—21no—
220C2×#3622—21no—
221C2×#3622—21no—
222C2×#3622—21no—
223C2×#3622—21no—
224C2×#3622—21no—
225C2×#3622—21no—
226C2×#3622—21no—
227C2×#3622—44no—
228C2×#3622C2×D412yes—
229C2×#3622C2×D412yes—
230#24764—171nomaximal, Sylow
231C23×C17—21no—
232C23×C17—21no—
233C23×C17—21no—
234C23×C17—21no—
235C23×C17—21no—
236C23×C17—21no—
237C23×C17#12112yes—
238C23×C17C22×D411yescharacteristic
239Q8×C17—22no—
240Q8×C17—22no—
241Q8×C17—41no—
242Q8×C17—41no—
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—22no—
250C2×C4×C17—22no—
251C2×C4×C17—22no—
252C2×C4×C17—22no—
253C2×C4×C17—21no—
254C2×C4×C17—21no—
255C2×C4×C17—22no—
256C2×C4×C17—22no—
257C2×C4×C17—21no—
258C2×C4×C17—21no—
259C2×C4×C17—21no—
260C2×C4×C17—21no—
261C2×C4×C17—41no—
262C2×C4×C17—41no—
263C2×C4×C17—41no—
264C2×C4×C17—41no—
265C2×C4×C17—41no—
266C2×C4×C17—41no—
267C2×C4×C17—41no—
268C2×C4×C17—41no—
269C2×C4×C17—41no—
270C2×C4×C17—41no—
271C2×C4×C17#12112yes—
272C2×C4×C17#12112yes—
273C2×C4×C17#12112yes—
274C2×C4×C17C22×D411yescharacteristic
275C2×C4×C17C22×D411yescharacteristic
276C2×C4×C17C2×#4111yescharacteristic
277C2×C4×C17C2×#4111yescharacteristic
278C2×C4×C17H(5,𝔽2)11yescharacteristic
279C2×C4×C172−1+411yescharacteristic
280C2×Dic17—41no—
281C2×Dic17—41no—
282C2×Dic17—41no—
283C2×Dic17—41no—
284C2×Dic17—41no—
285C2×Dic17—41no—
286C2×Dic17—41no—
287C2×Dic17—41no—
288C2×Dic17—41no—
289C2×Dic17—41no—
290C2×Dic17—41no—
291C2×Dic17—41no—
292Dic34—41no—
293Dic34—41no—
294Dic34—41no—
295Dic34—41no—
296Dic34—41no—
297Dic34—41no—
298Dic34—41no—
299Dic34—41no—
300#3622—22no—
301#3622—22no—
302#3622—41no—
303#3622—41no—
304C2×#470—171no—
305C2×#467—171no—
306C2×#462—171no—
307C2×#424—171no—
308#2621—171no—
309#2323D1711yescharacteristic
310#2068—171no—
311#2066—171no—
312#2056—171no—
313#2056—171no—
314#1981—171no—
315#1949—171no—
316#1894—171no—
317#1892—171no—
318#1805—171no—
319C22×C17—22no—
320C22×C17—22no—
321C22×C17—22no—
322C22×C17—41no—
323C22×C17—41no—
324C22×C17—41no—
325C22×C17—41no—
326C22×C17C2×#12111yescharacteristic
327C22×C17C2×#12111yescharacteristic
328C22×C17#53711yescharacteristic
329C22×C17#57711yescharacteristic
330C22×C17#57811yescharacteristic
331C22×C17#57811yescharacteristic
332C22×C17#57811yescharacteristic
333C4×C17—22no—
334C4×C17—22no—
335C4×C17—22no—
336C4×C17—42no—
337C4×C17—42no—
338C4×C17—41no—
339C4×C17—41no—
340C4×C17—81no—
341C4×C17—81no—
342C4×C17D4211yescharacteristic
343C4×C17#54711yescharacteristic
344C4×C17#54811yescharacteristic
345C4×C17#55311yescharacteristic
346Dic17—81no—
347Dic17—81no—
348Dic17—81no—
349Dic17—81no—
350C22×Q16—171no—
351C22×Q16—171no—
352C22×M16—171no—
353C23×C8—171no—
354C2×C4⋊Q8—171no—
355C2×C22⋊Q8—171no—
356C2×C22⋊Q8—171no—
357C2×C22⋊Q8C2×D1711yescharacteristic
358C2×C4×Q8C2×D1711yescharacteristic
359C2×#117C2×D1711yescharacteristic
360#470—344no—
361#467—344no—
362#462—344no—
363C2×#107—171no—
364C2×#106—171no—
365#424—344no—
366C2×C4⋊C8—171no—
367C2×C4⋊C8—171no—
368C2×C4⋊C8—171no—
369C2×C4⋊C8—171no—
370C2×#103—171no—
371C2×#103—171no—
372C2×#103—171no—
373C2×#103—171no—
374C2×#103—171no—
375C2×#103—171no—
376C2×#103—171no—
377C2×#103—171no—
378#398—341no—
379#398—341no—
380#397—341no—
381#397—341no—
382#391—171no—
383#391—171no—
384#391C2×D1711yescharacteristic
385#389—21no—
386#389C2×D1711yescharacteristic
387#388—21no—
388#388—21no—
389#387—21no—
390#382—171no—
391#380C2×D1711yescharacteristic
392C4×C4⋊C4C2×D1711yescharacteristic
393#340—171no—
394#340—171no—
395#340—171no—
396#340—171no—
397C2×C17—42no—
398C2×C17#241611yescharacteristic
399C2×C17#304411yescharacteristic
400C2×C17#304611yescharacteristic
401C2×C17#305911yescharacteristic
402C2×C17#306311yescharacteristic
403C2×C17#325811yescharacteristic
404C2×C17#326111yescharacteristic
405C22×Q8—171no—
406C22×Q8—171no—
407C22×Q8C22×D1711yescharacteristic
408C23×C4C22×D1711yescharacteristic
409C2×Q16—342no—
410C2×Q16—342no—
411C2×Q16—342no—
412C2×Q16—342no—
413C2×Q16—681no—
414C2×Q16—681no—
415C2×M16—342no—
416C2×M16—342no—
417C2×M16—681no—
418C22×C8—171no—
419C22×C8—171no—
420C22×C8—171no—
421C22×C8—171no—
422C22×C8—342no—
423C22×C8—342no—
424C22×C8—681no—
425C4⋊Q8—342no—
426C4⋊Q8—342no—
427C22⋊Q8—22no—
428C22⋊Q8—342no—
429C22⋊Q8—342no—
430C22⋊Q8—342no—
431C22⋊Q8—342no—
432C22⋊Q8D6814yes—
433C4×Q8—22no—
434C4×Q8—22no—
435#117—22no—
436#117—22no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—171no—
441C2×C4⋊C4—171no—
442C2×C4⋊C4—341no—
443C2×C4⋊C4—341no—
444C2×C4⋊C4—341no—
445C2×C4⋊C4—341no—
446C2×C4⋊C4C22⋊D1711yescharacteristic
447C2×C4⋊C4C22⋊D1711yescharacteristic
448C2×C4⋊C4C22×D1711yescharacteristic
449C2×C4⋊C4C22×D1711yescharacteristic
450C2×C4⋊C4C22×D1711yescharacteristic
451C2×C22⋊C4—21no—
452C2×C22⋊C4—21no—
453C2×C22⋊C4—341no—
454C2×C22⋊C4—341no—
455C2×C22⋊C4C22⋊D1711yescharacteristic
456C2×C22⋊C4C22⋊D1711yescharacteristic
457C2×C42—21no—
458C2×C42—21no—
459C2×C42C22×D1711yescharacteristic
460C2×C42C22×D1711yescharacteristic
461#107—342no—
462#106—342no—
463C4⋊C8—174no—
464C4⋊C8—342no—
465C4⋊C8—342no—
466C4⋊C8—342no—
467#103—342no—
468#103—342no—
469#103—342no—
470#103—342no—
471#103—342no—
472#103—342no—
473#103—681no—
474#103—681no—
475#95—21no—
476#95—21no—
477#95—21no—
478#95—21no—
479#95—21no—
480#95—341no—
481#95—341no—
482#95—341no—
483#95—341no—
484#95—341no—
485#95—341no—
486C17#2476411yescharacteristic, complemented, Sylow
487C24C2×C22⋊D1711yescharacteristic
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—21no—
492C2×Q8—21no—
493C2×Q8—341no—
494C2×Q8—341no—
495C2×Q8—341no—
496C2×Q8—341no—
497C2×Q8—341no—
498C2×Q8—341no—
499C2×Q8—341no—
500C2×Q8—341no—
501C2×Q8—341no—
502C2×Q8—341no—
503C2×Q8—341no—
504C2×Q8—341no—
505C2×Q8C2×D6812yes—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—341no—
515C22×C4—341no—
516C22×C4—341no—
517C22×C4—341no—
518C22×C4C2×D6812yes—
519C22×C4#6325212yes—
520C22×C4C2×C22⋊D1711yescharacteristic
521C22×C4C2×C22⋊D1711yescharacteristic
522C22×C4C2×C22⋊D1711yescharacteristic
523C22×C4C2×C22⋊D1711yescharacteristic
524C22×C4C2×C22⋊D1711yescharacteristic
525C22×C4C23×D1711yescharacteristic
526Q16—684no—
527Q16—684no—
528M16—684no—
529C2×C8—172no—
530C2×C8—172no—
531C2×C8—341no—
532C2×C8—341no—
533C2×C8—341no—
534C2×C8—341no—
535C2×C8—341no—
536C2×C8—341no—
537C2×C8—341no—
538C2×C8—341no—
539C2×C8—341no—
540C2×C8—341no—
541C2×C8—684no—
542C4⋊C4—21no—
543C4⋊C4—24no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—42no—
551C4⋊C4—81no—
552C4⋊C4—341no—
553C4⋊C4—341no—
554C4⋊C4—341no—
555C4⋊C4—341no—
556C4⋊C4—682no—
557C4⋊C4—682no—
558C4⋊C4—682no—
559C4⋊C4—682no—
560C4⋊C4C2×D6812yes—
561C22⋊C4—24no—
562C22⋊C4—22no—
563C22⋊C4—22no—
564C22⋊C4—81no—
565C22⋊C4—682no—
566C22⋊C4—682no—
567C42—21no—
568C42—21no—
569C42—42no—
570C42—81no—
571C42D4×D1712yes—
572C42#6325412yes—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23#1058761612yes—
580C23C22×C22⋊D1711yescharacteristic
581Q8—22no—
582Q8—22no—
583Q8—41no—
584Q8—41no—
585Q8—681no—
586Q8—681no—
587Q8—681no—
588Q8—681no—
589Q8—681no—
590Q8—681no—
591Q8—681no—
592Q8—681no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—681no—
622C2×C4—681no—
623C2×C4—681no—
624C2×C4—681no—
625C2×C4—681no—
626C2×C4—681no—
627C2×C4—681no—
628C2×C4—681no—
629C2×C4—681no—
630C2×C4—681no—
631C2×C4—681no—
632C2×C4—681no—
633C2×C4#1058761612yes—
634C2×C4#1058762512yes—
635C2×C4#1058762512yes—
636C2×C4C22×D6811yescharacteristic
637C2×C4C2×#6325211yescharacteristic
638C2×C4C2×D4×D1711yescharacteristic
639C2×C4C2×#6325411yescharacteristic
640C2×C4#1058769611yescharacteristic
641C2×C4#1058769711yescharacteristic
642C8—342no—
643C8—342no—
644C8—681no—
645C8—681no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22C2×#1058762511yescharacteristic
654C22C2×#1058761611yescharacteristic
655C22#1176338711yescharacteristic
656C22#1176362011yescharacteristic
657C22#1176362411yescharacteristic
658C22#1176362411yescharacteristic
659C22#1176362411yescharacteristic
660C4—22no—
661C4—22no—
662C4—22no—
663C4—42no—
664C4—42no—
665C4—41no—
666C4—41no—
667C4—81no—
668C4—81no—
669C4—1361no—
670C4—1361no—
671C4—1361no—
672C4—1361no—
673C4D4×D6811yescharacteristic
674C4#1176346311yescharacteristic
675C4#1176346811yescharacteristic
676C4#1176350511yescharacteristic
677C2—42no—
678C2#1453222411yescharacteristic
679C2#1453831411yescharacteristic
680C2#1453833411yescharacteristic
681C2#1453843611yescharacteristic
682C2#1453847711yescharacteristic
683C2#1454021011yescharacteristic
684C2#1454023411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2323C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#2323D17
hypocenterC17#24764
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
3. C2×#11763019C2semidirect
6. #14530176C2semidirect
7. #14530162C2semidirect
8. #14530106C2semidirect
9. #14530106C2semidirect
10. #14529707C2semidirect
11. #14529552C2semidirect
14. #14528799C2semidirect
486. C17#24764semidirect

Maximal subgroups

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

subgroupindex
C2×#117630652
C2×#117630482
C2×#117630192
C2×#117628362
#145345502
#145301762
#145301622
#145301062
#145297072
#145295522
#145292152
#145292052
#145287992
C17×#23232
#2476417