Finite Group Databasebeta

← #17073141

Subgroups — #17073141

Order 4352.

896 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762964C211yesmaximal, characteristic
2C2×#11762870C211yesmaximal, characteristic
3#14531368C211yesmaximal, characteristic
4#14530176C211yesmaximal, characteristic, complemented
5#14530162C211yesmaximal, characteristic, complemented
6#14530106C211yesmaximal, characteristic, complemented
7#14530106C211yesmaximal, characteristic, complemented
8#14529654C211yesmaximal, characteristic, complemented
9#14529644C211yesmaximal, characteristic, complemented
10#14529473C211yesmaximal, characteristic, complemented
11#14529459C211yesmaximal, characteristic, complemented
12#14529097C211yesmaximal, characteristic
13#14528981C211yesmaximal, characteristic
14#14528718C211yesmaximal, characteristic
15C17×#2625C211yesmaximal, characteristic
16C23×Dic34C2211yescharacteristic
17C22×#63231C2211yescharacteristic
18C22×#63231C2211yescharacteristic
19C2×#10587612C2211yescharacteristic
20#11762964—24no—
21#11762870—24no—
22C2×#10587478C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×C4⋊#3622C2211yescharacteristic
25C2×C4⋊#3622C2211yescharacteristic
26C2×#10587506C2211yescharacteristic
27C2×#10587506C2211yescharacteristic
28C2×#10587481C2211yescharacteristic
29C2×#10587481C2211yescharacteristic
30C2×#10587481C2211yescharacteristic
31C2×#10587481C2211yescharacteristic
32C2×#10587481C2211yescharacteristic
33C2×#10587481C2211yescharacteristic
34C2×C22⋊#3622C2211yescharacteristic
35C2×C22⋊#3622C2211yescharacteristic
36C2×#10587474C2211yescharacteristic
37C2×C4×#3622C2211yescharacteristic
38#11762715C2211yescharacteristic
39#11762715C2211yescharacteristic
40#11762715C2211yescharacteristic
41#11762715C2211yescharacteristic
42#11762714—21no—
43#11762714—21no—
44#11762670—21no—
45#11762670—21no—
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50C2×C17×C22⋊Q8C2211yescharacteristic
51C2×C17×#117C2211yescharacteristic
52C17×#400—21no—
53C17×#398—21no—
54C17×#397—21no—
55C17×#397—21no—
56C17×#391C2211yescharacteristic
57C17×#391C2211yescharacteristic
58C17×#389C2211yescharacteristic
59C17×#389C2211yescharacteristic
60C17×#380C2211yescharacteristic
61C22×Q8×C17C2311yescharacteristic
62C23×C4×C17C2311yescharacteristic
63C23×Dic17—21no—
64C22×Dic34—21no—
65C22×Dic34—22no—
66C22×Dic34—22no—
67C22×Dic34—22no—
68C22×Dic34—22no—
69C22×Dic34—41no—
70C22×Dic34—41no—
71C22×Dic34C2311yescharacteristic
72C22×Dic34C2311yescharacteristic
73C17×C22⋊Q8—22no—
74C17×C22⋊Q8—22no—
75C17×#117—22no—
76C17×#117—22no—
77C2×C4⋊C4×C17—21no—
78C2×C4⋊C4×C17—21no—
79C2×C4⋊C4×C17—21no—
80C2×C4⋊C4×C17D411yescharacteristic
81C2×C4⋊C4×C17D411yescharacteristic
82C2×C4⋊C4×C17C2311yescharacteristic
83C2×C4⋊C4×C17C2311yescharacteristic
84C2×C4⋊C4×C17C2311yescharacteristic
85C2×C22⋊C4×C17—21no—
86C2×C22⋊C4×C17—21no—
87C2×C22⋊C4×C17D411yescharacteristic
88C2×C22⋊C4×C17D411yescharacteristic
89C2×C42×C17C2311yescharacteristic
90C2×C42×C17C2311yescharacteristic
91C2×C22⋊Dic17—21no—
92C2×#63231—22no—
93C2×#63231—22no—
94C2×#63231—22no—
95C2×#63231—22no—
96C2×#63231—41no—
97C2×#63231—41no—
98#10587612—41no—
99#10587612—41no—
100C2×C4⋊Dic17C2311yescharacteristic
101C2×C4⋊Dic17C2311yescharacteristic
102C2×C4×Dic17—21no—
103C22×#3622C2311yescharacteristic
104C22×#3622C2311yescharacteristic
105C22×#3622C2311yescharacteristic
106C22×#3622C2311yescharacteristic
107C17×#95—21no—
108C17×#95—21no—
109C17×#95—21no—
110C17×#95—21no—
111C17×#95—21no—
112C17×#95—21no—
113C17×#95—21no—
114#10587506—41no—
115#10587506—41no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124C22⋊#3622—22no—
125C22⋊#3622D414yes—
126#10587481—22no—
127#10587481—22no—
128#10587481—22no—
129#10587481—22no—
130#10587481—41no—
131#10587481—41no—
132#10587479—41no—
133#10587478—41no—
134C4⋊#3622—22no—
135C4⋊#3622—22no—
136#10587474—22no—
137C4×#3622—22no—
138C24×C17C2×D411yescharacteristic
139C2×Q8×C17—21no—
140C2×Q8×C17—21no—
141C2×Q8×C17—21no—
142C2×Q8×C17—21no—
143C2×Q8×C17—21no—
144C2×Q8×C17—21no—
145C22×C4×C17—21no—
146C22×C4×C17—21no—
147C22×C4×C17—21no—
148C22×C4×C17—21no—
149C22×C4×C17—21no—
150C22×C4×C17—21no—
151C22×C4×C17—21no—
152C22×C4×C17—21no—
153C22×C4×C17C2×D412yes—
154C22×C4×C17C2×D411yescharacteristic
155C22×C4×C17C2×D411yescharacteristic
156C22×C4×C17C2×D411yescharacteristic
157C22×C4×C17C2×D411yescharacteristic
158C22×C4×C17C2×D411yescharacteristic
159C22×C4×C17#4112yes—
160C22×C4×C17C2411yescharacteristic
161C22×Dic17—24no—
162C22×Dic17—21no—
163C22×Dic17—21no—
164C22×Dic17—21no—
165C22×Dic17—21no—
166C22×Dic17—42no—
167C22×Dic17—42no—
168C2×Dic34—21no—
169C2×Dic34—21no—
170C2×Dic34—21no—
171C2×Dic34—21no—
172C2×Dic34—21no—
173C2×Dic34—21no—
174C2×Dic34—21no—
175C2×Dic34—21no—
176C2×Dic34—24no—
177C2×Dic34—21no—
178C2×Dic34—21no—
179C2×Dic34—44no—
180C2×Dic34—44no—
181C2×Dic34—44no—
182C2×Dic34—44no—
183C2×Dic34—42no—
184C2×Dic34—42no—
185C2×Dic34#4112yes—
186C2×Dic34#4112yes—
187C4⋊C4×C17—22no—
188C4⋊C4×C17—22no—
189C4⋊C4×C17—21no—
190C4⋊C4×C17—21no—
191C4⋊C4×C17—21no—
192C4⋊C4×C17—21no—
193C4⋊C4×C17—21no—
194C4⋊C4×C17—21no—
195C4⋊C4×C17—42no—
196C4⋊C4×C17—81no—
197C4⋊C4×C17—81no—
198C22⋊C4×C17—22no—
199C22⋊C4×C17—22no—
200C22⋊C4×C17—42no—
201C22⋊C4×C17—81no—
202C42×C17—21no—
203C42×C17—21no—
204C42×C17—21no—
205C42×C17—21no—
206C22⋊Dic17—42no—
207#63231—44no—
208#63231—44no—
209C4⋊Dic17—21no—
210C4⋊Dic17—21no—
211C4⋊Dic17—21no—
212C4⋊Dic17—21no—
213C4×Dic17—42no—
214C2×#3622—21no—
215C2×#3622—21no—
216C2×#3622—21no—
217C2×#3622—21no—
218C2×#3622—21no—
219C2×#3622—21no—
220C2×#3622—21no—
221C2×#3622—21no—
222C2×#3622—21no—
223C2×#3622—21no—
224C2×#3622C2×D412yes—
225C2×#3622C2×D412yes—
226#24514—171nomaximal, Sylow
227C23×C17—21no—
228C23×C17—21no—
229C23×C17—21no—
230C23×C17—21no—
231C23×C17—21no—
232C23×C17—21no—
233C23×C17#12112yes—
234C23×C17C22×D411yescharacteristic
235Q8×C17—41no—
236Q8×C17—41no—
237Q8×C17—41no—
238Q8×C17—41no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—22no—
242C2×C4×C17—22no—
243C2×C4×C17—22no—
244C2×C4×C17—22no—
245C2×C4×C17—21no—
246C2×C4×C17—21no—
247C2×C4×C17—21no—
248C2×C4×C17—21no—
249C2×C4×C17—21no—
250C2×C4×C17—21no—
251C2×C4×C17—21no—
252C2×C4×C17—21no—
253C2×C4×C17—21no—
254C2×C4×C17—21no—
255C2×C4×C17—21no—
256C2×C4×C17—21no—
257C2×C4×C17—41no—
258C2×C4×C17—41no—
259C2×C4×C17—41no—
260C2×C4×C17—41no—
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#12112yes—
270C2×C4×C17C22×D411yescharacteristic
271C2×C4×C17C2×#4111yescharacteristic
272C2×C4×C17C2×#4111yescharacteristic
273C2×C4×C17C2×#4111yescharacteristic
274C2×C4×C17H(5,𝔽2)11yescharacteristic
275C2×C4×C17H(5,𝔽2)11yescharacteristic
276C2×Dic17—22no—
277C2×Dic17—22no—
278C2×Dic17—48no—
279C2×Dic17—41no—
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—
289Dic34—22no—
290Dic34—22no—
291Dic34—22no—
292Dic34—22no—
293Dic34—48no—
294Dic34—41no—
295Dic34—41no—
296Dic34—41no—
297Dic34—41no—
298#3622—22no—
299#3622—22no—
300#3622—41no—
301#3622—41no—
302C2×#451—171no—
303C2×#433—171no—
304#2625D1711yescharacteristic
305#2310—171no—
306#2068—171no—
307#2066—171no—
308#2056—171no—
309#2056—171no—
310#1971—171no—
311#1969—171no—
312#1935—171no—
313#1933—171no—
314#1873—171no—
315#1854—171no—
316#1787—171no—
317C22×C17—22no—
318C22×C17—22no—
319C22×C17—22no—
320C22×C17—41no—
321C22×C17—41no—
322C22×C17—41no—
323C22×C17—41no—
324C22×C17C2×#12111yescharacteristic
325C22×C17#52511yescharacteristic
326C22×C17#53511yescharacteristic
327C22×C17#57711yescharacteristic
328C22×C17#57811yescharacteristic
329C22×C17#57811yescharacteristic
330C22×C17#57811yescharacteristic
331C4×C17—22no—
332C4×C17—41no—
333C4×C17—41no—
334C4×C17—41no—
335C4×C17—41no—
336C4×C17—81no—
337C4×C17—81no—
338C4×C17—81no—
339C4×C17—81no—
340C4×C17#54611yescharacteristic
341C4×C17#54611yescharacteristic
342C4×C17#55111yescharacteristic
343C4×C17#55111yescharacteristic
344Dic17—42no—
345Dic17—42no—
346Dic17—81no—
347Dic17—81no—
348C23×Q8—171no—
349C22×Q16—171no—
350C22×Q16—171no—
351C2×C22⋊Q8C2×D1711yescharacteristic
352C2×#117—171no—
353C2×#117C2×D1711yescharacteristic
354#451—344no—
355#433—344no—
356C2×#107—171no—
357C2×#106—171no—
358C2×C4⋊C8—171no—
359C2×C4⋊C8—171no—
360C2×#103—171no—
361C2×#103—171no—
362C2×#103—171no—
363C2×#103—171no—
364C2×#103—171no—
365C2×#103—171no—
366C2×#103—171no—
367C2×#103—171no—
368C2×C22⋊C8—171no—
369C2×C22⋊C8—171no—
370C2×#97—171no—
371C2×C4×C8—171no—
372#400—21no—
373#398—21no—
374#397—21no—
375#397—21no—
376#391—171no—
377#391—171no—
378#391—171no—
379#391—171no—
380#391—341no—
381#391—341no—
382#391C2×D1711yescharacteristic
383#391C2×D1711yescharacteristic
384#389C2×D1711yescharacteristic
385#389C2×D1711yescharacteristic
386#381—341no—
387#381—341no—
388#380C2×D1711yescharacteristic
389#340—171no—
390#340—171no—
391#340—171no—
392#340—171no—
393C2×C17—42no—
394C2×C17#241111yescharacteristic
395C2×C17#305911yescharacteristic
396C2×C17#306311yescharacteristic
397C2×C17#312111yescharacteristic
398C2×C17#312711yescharacteristic
399C2×C17#322311yescharacteristic
400C2×C17#322511yescharacteristic
401C22×Q8—171no—
402C22×Q8—171no—
403C22×Q8—342no—
404C22×Q8—342no—
405C22×Q8—341no—
406C22×Q8—342no—
407C22×Q8—342no—
408C22×Q8—681no—
409C22×Q8—681no—
410C22×Q8C22×D1711yescharacteristic
411C23×C4—341no—
412C23×C4C22×D1711yescharacteristic
413C2×Q16—342no—
414C2×Q16—342no—
415C2×Q16—342no—
416C2×Q16—342no—
417C2×Q16—681no—
418C2×Q16—681no—
419C22×C8—171no—
420C22×C8—171no—
421C22×C8—171no—
422C22×C8—171no—
423C22⋊Q8—22no—
424C22⋊Q8—22no—
425#117—22no—
426#117—22no—
427#117—681no—
428#117—681no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—171no—
433C2×C4⋊C4—171no—
434C2×C4⋊C4C22⋊D1711yescharacteristic
435C2×C4⋊C4C22⋊D1711yescharacteristic
436C2×C4⋊C4C22×D1711yescharacteristic
437C2×C4⋊C4C22×D1711yescharacteristic
438C2×C4⋊C4C22×D1711yescharacteristic
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4—341no—
442C2×C22⋊C4C22⋊D1711yescharacteristic
443C2×C22⋊C4C22⋊D1711yescharacteristic
444C2×C42—341no—
445C2×C42C22×D1711yescharacteristic
446C2×C42C22×D1711yescharacteristic
447#107—681no—
448#106—681no—
449C4⋊C8—342no—
450C4⋊C8—342no—
451#103—342no—
452#103—342no—
453#103—342no—
454#103—342no—
455#103—681no—
456#103—681no—
457#103—681no—
458#103—681no—
459C22⋊C8—174no—
460C22⋊C8—342no—
461#97—342no—
462C4×C8—342no—
463#95—21no—
464#95—21no—
465#95—21no—
466#95—21no—
467#95—21no—
468#95—21no—
469#95—21no—
470#95—341no—
471#95—341no—
472#95—341no—
473#95—341no—
474#95—341no—
475#95—341no—
476#95—341no—
477#95—341no—
478C17#2451411yescharacteristic, complemented, Sylow
479C24C2×C22⋊D1711yescharacteristic
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—172no—
487C2×Q8—172no—
488C2×Q8—344no—
489C2×Q8—341no—
490C2×Q8—341no—
491C2×Q8—341no—
492C2×Q8—341no—
493C2×Q8—341no—
494C2×Q8—341no—
495C2×Q8—341no—
496C2×Q8—341no—
497C2×Q8—341no—
498C2×Q8—341no—
499C2×Q8—684no—
500C2×Q8—684no—
501C2×Q8—684no—
502C2×Q8—684no—
503C2×Q8—682no—
504C2×Q8—682no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—344no—
514C22×C4—341no—
515C22×C4—341no—
516C22×C4—341no—
517C22×C4—341no—
518C22×C4—682no—
519C22×C4—682no—
520C22×C4D4×D1712yes—
521C22×C4#6325412yes—
522C22×C4C2×C22⋊D1711yescharacteristic
523C22×C4C2×C22⋊D1711yescharacteristic
524C22×C4C2×C22⋊D1711yescharacteristic
525C22×C4C2×C22⋊D1711yescharacteristic
526C22×C4C2×C22⋊D1711yescharacteristic
527C22×C4C23×D1711yescharacteristic
528Q16—684no—
529Q16—684no—
530C2×C8—172no—
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—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—42no—
551C4⋊C4—81no—
552C4⋊C4—81no—
553C4⋊C4—341no—
554C4⋊C4—341no—
555C4⋊C4—341no—
556C4⋊C4—341no—
557C22⋊C4—22no—
558C22⋊C4—22no—
559C22⋊C4—42no—
560C22⋊C4—81no—
561C22⋊C4—682no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—682no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23#1058762512yes—
574C23C22×C22⋊D1711yescharacteristic
575Q8—41no—
576Q8—41no—
577Q8—41no—
578Q8—41no—
579Q8—342no—
580Q8—342no—
581Q8—342no—
582Q8—342no—
583Q8—688no—
584Q8—681no—
585Q8—681no—
586Q8—681no—
587Q8—681no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—342no—
619C2×C4—342no—
620C2×C4—688no—
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#1058762512yes—
632C2×C4C2×D4×D1711yescharacteristic
633C2×C4C2×#6325411yescharacteristic
634C2×C4#41×D1711yescharacteristic
635C2×C4#41×D1711yescharacteristic
636C2×C4#1058769611yescharacteristic
637C2×C4#1058769611yescharacteristic
638C8—342no—
639C8—342no—
640C8—681no—
641C8—681no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22C2×#1058762511yescharacteristic
650C22#1176331611yescharacteristic
651C22#1176337311yescharacteristic
652C22#1176362011yescharacteristic
653C22#1176362411yescharacteristic
654C22#1176362411yescharacteristic
655C22#1176362411yescharacteristic
656C4—22no—
657C4—41no—
658C4—41no—
659C4—41no—
660C4—41no—
661C4—81no—
662C4—81no—
663C4—81no—
664C4—81no—
665C4—682no—
666C4—682no—
667C4—1361no—
668C4—1361no—
669C4#1176345111yescharacteristic
670C4#1176345111yescharacteristic
671C4#1176348211yescharacteristic
672C4#1176348211yescharacteristic
673C2—42no—
674C2#1453217011yescharacteristic
675C2#1453843611yescharacteristic
676C2#1453847711yescharacteristic
677C2#1453900711yescharacteristic
678C2#1453907911yescharacteristic
679C2#1453986111yescharacteristic
680C2#1453988711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2625C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#2625D17
hypocenterC17#24514
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2625

Complements

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

NH ≅ G/Ntype
4. #14530176C2semidirect
5. #14530162C2semidirect
6. #14530106C2semidirect
7. #14530106C2semidirect
8. #14529654C2semidirect
9. #14529644C2semidirect
10. #14529473C2semidirect
11. #14529459C2semidirect
478. C17#24514semidirect

Maximal subgroups

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

subgroupindex
C2×#117629642
C2×#117628702
#145313682
#145301762
#145301622
#145301062
#145296542
#145296442
#145294732
#145294592
#145290972
#145289812
#145287182
C17×#26252
#2451417