Finite Group Databasebeta

← #17075967

Subgroups — #17075967

Order 4352.

924 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763031C211yesmaximal, characteristic
2C2×#11763031C211yesmaximal, characteristic
3C2×#11763019C211yesmaximal, characteristic, complemented
4C2×C4×#63231C211yesmaximal, characteristic, complemented
5#14533764C211yesmaximal, characteristic
6#14530176C211yesmaximal, characteristic, complemented
7#14530169C211yesmaximal, characteristic
8#14530030C211yesmaximal, characteristic, complemented
9#14530025C211yesmaximal, characteristic
10#14529593C211yesmaximal, characteristic
11#14529503C211yesmaximal, characteristic, complemented
12#14529139C211yesmaximal, characteristic
13#14529125C211yesmaximal, characteristic, complemented
14#14528796C211yesmaximal, characteristic, complemented
15C17×#2351C211yesmaximal, characteristic, complemented
16C22×#63231C2211yescharacteristic
17C22×#63231C2211yescharacteristic
18C22×#63230C2211yescharacteristic
19C2×#10587555C2211yescharacteristic
20C2×#10587561C2211yescharacteristic
21C2×#10587561C2211yescharacteristic
22C2×C4×Dic34C2211yescharacteristic
23#11763031—24no—
24#11763031—24no—
25#11763019—24no—
26C4×#63231—24no—
27C2×#10587478C2211yescharacteristic
28C2×C4⋊#3622C2211yescharacteristic
29C2×C4⋊#3622C2211yescharacteristic
30C2×C4⋊#3622C2211yescharacteristic
31C2×#10587506C2211yescharacteristic
32C2×#10587481C2211yescharacteristic
33C2×#10587481C2211yescharacteristic
34C2×#10587481C2211yescharacteristic
35C2×#10587481C2211yescharacteristic
36C2×#10587481C2211yescharacteristic
37C2×#10587481C2211yescharacteristic
38C2×#10587480C2211yescharacteristic
39C2×#10587480C2211yescharacteristic
40C2×#10587480C2211yescharacteristic
41C2×C4×#3622C2211yescharacteristic
42#11762731—21no—
43#11762731—21no—
44#11762715C2211yescharacteristic
45#11762711C2211yescharacteristic
46#11762701—21no—
47#11762701—21no—
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762564C2211yescharacteristic
51#11762564C2211yescharacteristic
52C2×C4×Q8×C17C2211yescharacteristic
53C2×C4×Q8×C17C2211yescharacteristic
54C2×C4×Q8×C17C2211yescharacteristic
55C17×#389—21no—
56C17×#389—21no—
57C17×#389C2211yescharacteristic
58C17×#389C2211yescharacteristic
59C17×#382—21no—
60C17×#382C2211yescharacteristic
61C4×C4⋊C4×C17—21no—
62C4×C4⋊C4×C17C2211yescharacteristic
63C22×Q8×C17C2311yescharacteristic
64C22×D68C2311yescharacteristic
65C22×Dic34C2311yescharacteristic
66C22×Dic34C2311yescharacteristic
67C4×Q8×C17—22no—
68C4×Q8×C17—22no—
69C4×Q8×C17—22no—
70C4×Q8×C17—41no—
71C4×Q8×C17—41no—
72C4×Q8×C17D414yes—
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×#63231—22no—
93C2×#63231—22no—
94C2×#63231D414yes—
95C2×#63230—22no—
96C2×#63230—22no—
97C2×#63230—22no—
98C2×#63227—21no—
99C2×#63227—21no—
100C2×#63227—21no—
101C2×#63227—21no—
102C2×#63227—21no—
103C2×#63227—21no—
104C2×C4⋊Dic17—21no—
105C2×C4⋊Dic17—21no—
106C2×C4⋊Dic17C2311yescharacteristic
107C2×#63225—21no—
108C2×C4×Dic17—21no—
109C22×#3622C2311yescharacteristic
110C22×#3622C2311yescharacteristic
111C22×#3622C2311yescharacteristic
112C22×#3622C2311yescharacteristic
113#10587561—22no—
114#10587561—22no—
115#10587561—22no—
116#10587561—41no—
117#10587555—22no—
118#10587555—41no—
119C4×Dic34—22no—
120C4×Dic34—22no—
121C17×#95—21no—
122C17×#95—21no—
123C17×#95—21no—
124C17×#95—21no—
125C17×#95D411yescharacteristic
126C17×#95D411yescharacteristic
127#10587506—22no—
128#10587502—21no—
129#10587502—21no—
130#10587481—22no—
131#10587481—22no—
132#10587481—22no—
133#10587481—22no—
134#10587481—22no—
135#10587481—41no—
136#10587480—22no—
137#10587480—22no—
138#10587480—41no—
139#10587478—22no—
140C4⋊#3622—22no—
141C4⋊#3622—22no—
142C4⋊#3622—22no—
143C4×#3622—22no—
144C23×D17—21no—
145C2×Q8×C17—21no—
146C2×Q8×C17—21no—
147C2×Q8×C17—21no—
148C2×Q8×C17C2×D412yes—
149C2×Q8×C17C2×D412yes—
150C2×Q8×C17#4112yes—
151C22×C4×C17—21no—
152C22×C4×C17—21no—
153C22×C4×C17—21no—
154C22×C4×C17—21no—
155C22×C4×C17C2×D411yescharacteristic
156C22×C4×C17C2×D411yescharacteristic
157C22×C4×C17C2×D411yescharacteristic
158C22×C4×C17C2×D411yescharacteristic
159C22×C4×C17C2×D411yescharacteristic
160C22×C4×C17C2×D411yescharacteristic
161C22×C4×C17C2411yescharacteristic
162C22×Dic17—21no—
163C22×Dic17—21no—
164C22×Dic17—21no—
165C2×D68—21no—
166C2×D68—21no—
167C2×D68—21no—
168C2×D68—21no—
169C2×D68—21no—
170C2×D68—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×Dic34C2×D412yes—
183C4⋊C4×C17—24no—
184C4⋊C4×C17—22no—
185C4⋊C4×C17—21no—
186C4⋊C4×C17—21no—
187C4⋊C4×C17—21no—
188C4⋊C4×C17—21no—
189C4⋊C4×C17—21no—
190C4⋊C4×C17—41no—
191C4⋊C4×C17—42no—
192C4⋊C4×C17—42no—
193C4⋊C4×C17—42no—
194C4⋊C4×C17—42no—
195C4⋊C4×C17C2×D412yes—
196C42×C17—24no—
197C42×C17—21no—
198C42×C17—21no—
199C42×C17—21no—
200C42×C17—21no—
201C42×C17—42no—
202C42×C17—42no—
203C42×C17C2×D412yes—
204C42×C17#4112yes—
205#63231—28no—
206#63231—44no—
207#63230—44no—
208#63227—42no—
209#63227—42no—
210#63227—42no—
211#63227—42no—
212#63227—42no—
213#63227—42no—
214C4⋊Dic17—21no—
215C4⋊Dic17—21no—
216C4⋊Dic17—42no—
217C4⋊Dic17—42no—
218#63225—42no—
219C4×Dic17—42no—
220C2×#3622—21no—
221C2×#3622—21no—
222C2×#3622—21no—
223C2×#3622—21no—
224C2×#3622—21no—
225C2×#3622—21no—
226C2×#3622—21no—
227C2×#3622—21no—
228C2×#3622—21no—
229C2×#3622—21no—
230C2×#3622—21no—
231C2×#3622C2×D412yes—
232#24736—171nomaximal, Sylow
233C23×C17C22×D411yescharacteristic
234C22×D17—41no—
235C22×D17—41no—
236C22×D17—41no—
237C22×D17—41no—
238C22×D17—41no—
239C22×D17—41no—
240C22×D17—41no—
241Q8×C17—22no—
242Q8×C17—22no—
243Q8×C17—22no—
244Q8×C17#12114yes—
245C2×C4×C17—22no—
246C2×C4×C17—21no—
247C2×C4×C17—22no—
248C2×C4×C17—22no—
249C2×C4×C17—21no—
250C2×C4×C17—21no—
251C2×C4×C17—21no—
252C2×C4×C17—21no—
253C2×C4×C17—21no—
254C2×C4×C17—22no—
255C2×C4×C17—21no—
256C2×C4×C17—21no—
257C2×C4×C17—21no—
258C2×C4×C17—21no—
259C2×C4×C17—21no—
260C2×C4×C17—21no—
261C2×C4×C17—21no—
262C2×C4×C17—21no—
263C2×C4×C17—21no—
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—41no—
272C2×C4×C17#12112yes—
273C2×C4×C17#12112yes—
274C2×C4×C17#13512yes—
275C2×C4×C17C22×D411yescharacteristic
276C2×C4×C17C22×D411yescharacteristic
277C2×C4×C17C2×#4111yescharacteristic
278C2×C4×C17C2×#4111yescharacteristic
279C2×C4×C17H(5,𝔽2)11yescharacteristic
280C2×C4×C172−1+411yescharacteristic
281C2×Dic17—22no—
282C2×Dic17—41no—
283C2×Dic17—41no—
284C2×Dic17—41no—
285C2×Dic17—41no—
286C2×Dic17—41no—
287C2×Dic17—41no—
288C2×Dic17—41no—
289C2×Dic17—41no—
290D68—41no—
291D68—41no—
292D68—41no—
293D68—41no—
294Dic34—22no—
295Dic34—22no—
296Dic34—41no—
297Dic34—41no—
298Dic34—41no—
299Dic34—41no—
300Dic34—41no—
301Dic34—41no—
302#3622—22no—
303#3622—41no—
304#3622—41no—
305#3622—41no—
306C2×#464—171no—
307C2×#464—171no—
308C2×#462—171no—
309C2×C4×Q16—171no—
310#2545—171no—
311#2351D1711yescharacteristic, complemented
312#2068—171no—
313#2067—171no—
314#2043—171no—
315#2042—171no—
316#1958—171no—
317#1939—171no—
318#1880—171no—
319#1878—171no—
320#1805—171no—
321C22×C17C2×#12111yescharacteristic
322C22×C17D4211yescharacteristic
323C22×C17#54811yescharacteristic
324C22×C17C2×#13511yescharacteristic
325C22×C17#57511yescharacteristic
326C22×C17#57811yescharacteristic
327C22×C17#57811yescharacteristic
328C2×D17—81no—
329C2×D17—81no—
330C2×D17—81no—
331C2×D17—81no—
332C2×D17—81no—
333C2×D17—81no—
334C2×D17—81no—
335C4×C17—24no—
336C4×C17—22no—
337C4×C17—22no—
338C4×C17—22no—
339C4×C17—41no—
340C4×C17—41no—
341C4×C17—41no—
342C4×C17—42no—
343C4×C17—81no—
344C4×C17—81no—
345C4×C17C2×#12111yescharacteristic
346C4×C17#53711yescharacteristic
347C4×C17#54711yescharacteristic
348C4×C17#55311yescharacteristic
349Dic17—42no—
350Dic17—81no—
351Dic17—81no—
352C22×Q16—171no—
353C22×Q16—171no—
354C22×SD16—171no—
355C2×C4⋊Q8—171no—
356C2×#124—171no—
357C2×#124—171no—
358C2×C4×Q8—171no—
359C2×C4×Q8C2×D1711yescharacteristic
360C2×C4×Q8C2×D1711yescharacteristic
361C2×C4×Q8C2×D1711yescharacteristic
362#464—344no—
363#464—344no—
364#462—344no—
365C4×Q16—344no—
366C2×#107—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×#102—171no—
378C2×#102—171no—
379C2×#102—171no—
380C2×C4×C8—171no—
381#396—341no—
382#396—341no—
383#391—171no—
384#390—171no—
385#389—21no—
386#389—21no—
387#389C2×D1711yescharacteristic
388#389C2×D1711yescharacteristic
389#388—341no—
390#388—341no—
391#382—21no—
392#382C2×D1711yescharacteristic
393C4×C4⋊C4—21no—
394C4×C4⋊C4C2×D1711yescharacteristic
395#340—171no—
396#340—171no—
397#336—171no—
398#336—171no—
399C2×C17#241611yescharacteristic
400C2×C17#303511yescharacteristic
401C2×C17#304411yescharacteristic
402C2×C17#327711yescharacteristic
403C2×C17#328511yescharacteristic
404C2×C17#336411yescharacteristic
405C2×C17#337711yescharacteristic
406D17—161no—
407C22×Q8—171no—
408C22×Q8—171no—
409C22×Q8C22×D1711yescharacteristic
410C22×D4—171no—
411C2×Q16—174no—
412C2×Q16—342no—
413C2×Q16—342no—
414C2×Q16—342no—
415C2×Q16—342no—
416C2×Q16—342no—
417C2×SD16—342no—
418C2×SD16—342no—
419C2×SD16—342no—
420C22×C8—171no—
421C22×C8—171no—
422C22×C8—171no—
423C22×C8—171no—
424C4⋊Q8—342no—
425C4⋊Q8—681no—
426#124—342no—
427#124—342no—
428#124—342no—
429#124—681no—
430C4×Q8—22no—
431C4×Q8—22no—
432C4×Q8—22no—
433C4×Q8—41no—
434C4×Q8—41no—
435C4×Q8—342no—
436C4×Q8—342no—
437C4×Q8D6814yes—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—21no—
441C2×C4⋊C4—21no—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—171no—
444C2×C4⋊C4—341no—
445C2×C4⋊C4—341no—
446C2×C4⋊C4—341no—
447C2×C4⋊C4C22⋊D1711yescharacteristic
448C2×C4⋊C4C22⋊D1711yescharacteristic
449C2×C4⋊C4C22×D1711yescharacteristic
450C2×C4⋊C4C22×D1711yescharacteristic
451C2×C4⋊C4C22×D1711yescharacteristic
452C2×C22⋊C4—341no—
453C2×C22⋊C4—341no—
454C2×C22⋊C4—341no—
455C2×C22⋊C4—341no—
456C2×C22⋊C4—341no—
457C2×C22⋊C4—341no—
458C2×C42—21no—
459C2×C42—21no—
460C2×C42—21no—
461C2×C42—341no—
462C2×C42C22×D1711yescharacteristic
463C2×C42C22×D1711yescharacteristic
464C2×C42C22×D1711yescharacteristic
465#107—342no—
466C4⋊C8—342no—
467C4⋊C8—342no—
468C4⋊C8—342no—
469#103—342no—
470#103—342no—
471#103—342no—
472#103—342no—
473#103—342no—
474#103—342no—
475#103—681no—
476#102—342no—
477#102—342no—
478#102—681no—
479C4×C8—342no—
480#95—21no—
481#95—21no—
482#95—21no—
483#95—21no—
484#95—341no—
485#95—341no—
486#95C22⋊D1711yescharacteristic
487#95C22⋊D1711yescharacteristic
488C17#2473611yescharacteristic, complemented, Sylow
489C24—341no—
490C2×Q8—21no—
491C2×Q8—21no—
492C2×Q8—21no—
493C2×Q8—172no—
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—
506C2×Q8D4×D1712yes—
507C2×Q8#6325612yes—
508C2×D4—341no—
509C2×D4—341no—
510C2×D4—341no—
511C2×D4—341no—
512C2×D4—341no—
513C2×D4—341no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—341no—
519C22×C4—341no—
520C22×C4—341no—
521C22×C4C2×C22⋊D1711yescharacteristic
522C22×C4C2×C22⋊D1711yescharacteristic
523C22×C4C2×C22⋊D1711yescharacteristic
524C22×C4C2×C22⋊D1711yescharacteristic
525C22×C4C2×C22⋊D1711yescharacteristic
526C22×C4C2×C22⋊D1711yescharacteristic
527C22×C4C23×D1711yescharacteristic
528Q16—348no—
529Q16—684no—
530SD16—684no—
531C2×C8—172no—
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—341no—
542C2×C8—341no—
543C4⋊C4—21no—
544C4⋊C4—24no—
545C4⋊C4—22no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—41no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—341no—
556C4⋊C4—341no—
557C4⋊C4—682no—
558C4⋊C4—682no—
559C4⋊C4—682no—
560C4⋊C4C2×D6812yes—
561C22⋊C4—682no—
562C22⋊C4—682no—
563C22⋊C4—682no—
564C22⋊C4—682no—
565C22⋊C4—682no—
566C22⋊C4—682no—
567C42—21no—
568C42—21no—
569C42—21no—
570C42—21no—
571C42—24no—
572C42—42no—
573C42—42no—
574C42—682no—
575C42C2×D6812yes—
576C42#6325212yes—
577C23—681no—
578C23—681no—
579C23—681no—
580C23—681no—
581C23—681no—
582C23—681no—
583C23—681no—
584C23C22×C22⋊D1711yescharacteristic
585Q8—22no—
586Q8—22no—
587Q8—22no—
588Q8—342no—
589Q8—342no—
590Q8—681no—
591Q8—681no—
592Q8—681no—
593Q8—681no—
594Q8—681no—
595Q8—681no—
596Q8#1058758114yes—
597D4—681no—
598D4—681no—
599D4—681no—
600D4—681no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—342no—
629C2×C4—681no—
630C2×C4—681no—
631C2×C4—681no—
632C2×C4—681no—
633C2×C4—681no—
634C2×C4—681no—
635C2×C4—681no—
636C2×C4—681no—
637C2×C4#1058761612yes—
638C2×C4#1058761612yes—
639C2×C4#1058763612yes—
640C2×C4C22×D6811yescharacteristic
641C2×C4C2×#6325211yescharacteristic
642C2×C4C2×D4×D1711yescharacteristic
643C2×C4#1058769111yescharacteristic
644C2×C4C2×#6325611yescharacteristic
645C2×C4#1058769411yescharacteristic
646C8—342no—
647C8—681no—
648C8—681no—
649C8—681no—
650C22—1361no—
651C22—1361no—
652C22—1361no—
653C22—1361no—
654C22—1361no—
655C22—1361no—
656C22—1361no—
657C22C2×#1058761611yescharacteristic
658C22D4×C22⋊D1711yescharacteristic
659C22#1176347011yescharacteristic
660C22C2×#1058763611yescharacteristic
661C22#1176360711yescharacteristic
662C22#1176362411yescharacteristic
663C22#1176362411yescharacteristic
664C4—22no—
665C4—24no—
666C4—22no—
667C4—22no—
668C4—41no—
669C4—41no—
670C4—41no—
671C4—42no—
672C4—81no—
673C4—81no—
674C4—682no—
675C4—1361no—
676C4—1361no—
677C4C2×#1058758111yescharacteristic
678C4#1176339011yescharacteristic
679C4#1176346011yescharacteristic
680C4#1176350711yescharacteristic
681C2—2721no—
682C2#1453221811yescharacteristic
683C2#1453823911yescharacteristic
684C2#1453831411yescharacteristic
685C2#1454043711yescharacteristic
686C2#1454052111yescharacteristic
687C2#1454146911yescharacteristic
688C2#1454162611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2351C2
socleC23×C17C22×D4
prime radicalC22×D68C23
hypercenter#2351D17
hypocenterC17#24736
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2351

Complements

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

NH ≅ G/Ntype
3. C2×#11763019C2semidirect
4. C2×C4×#63231C2semidirect
6. #14530176C2semidirect
8. #14530030C2semidirect
11. #14529503C2semidirect
13. #14529125C2semidirect
14. #14528796C2semidirect
15. C17×#2351C2semidirect
311. #2351D17semidirect
488. C17#24736semidirect

Maximal subgroups

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

subgroupindex
C2×#117630312
C2×#117630192
C2×C4×#632312
#145337642
#145301762
#145301692
#145300302
#145300252
#145295932
#145295032
#145291392
#145291252
#145287962
C17×#23512
#2473617