Finite Group Databasebeta

← #17091843

Subgroups — #17091843

Order 4352.

920 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763113C211yesmaximal, characteristic, complemented
2C2×#11762817C211yesmaximal, characteristic
3C22⋊C4×Dic34C211yesmaximal, characteristic
4#14530323C211yesmaximal, characteristic, complemented
5#14530162C211yesmaximal, characteristic, complemented
6#14530162C211yesmaximal, characteristic, complemented
7#14529932C211yesmaximal, characteristic, complemented
8#14529051C211yesmaximal, characteristic, complemented
9#14529037C211yesmaximal, characteristic, complemented
10#14528983C211yesmaximal, characteristic
11#14528969C211yesmaximal, characteristic
12#14528815C211yesmaximal, characteristic
13#14528786C211yesmaximal, characteristic, complemented
14#14528718C211yesmaximal, characteristic
15C17×#2441C211yesmaximal, characteristic
16C23×Dic34C2211yescharacteristic
17C22×#63223C2211yescharacteristic
18C2×C4×Dic34C2211yescharacteristic
19C2×C4×Dic34C2211yescharacteristic
20#11763113—24no—
21C2×#10587478C2211yescharacteristic
22C2×#10587478C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×#10587479C2211yescharacteristic
25C2×C4⋊#3622C2211yescharacteristic
26C2×C4⋊#3622C2211yescharacteristic
27#11762817—24no—
28C2×#10587481C2211yescharacteristic
29C2×#10587481C2211yescharacteristic
30C2×#10587481C2211yescharacteristic
31C2×#10587481C2211yescharacteristic
32C2×#10587481C2211yescharacteristic
33C2×#10587481C2211yescharacteristic
34C2×#10587481C2211yescharacteristic
35C2×#10587481C2211yescharacteristic
36C2×C22⋊#3622C2211yescharacteristic
37#11762715C2211yescharacteristic
38#11762715C2211yescharacteristic
39#11762713—21no—
40#11762713—21no—
41#11762669C2211yescharacteristic
42#11762667—21no—
43C22⋊C4×Dic17—21no—
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762564C2211yescharacteristic
49#11762564C2211yescharacteristic
50C2×C17×#125C2211yescharacteristic
51C2×C17×#117C2211yescharacteristic
52C17×#399—21no—
53C17×#396—21no—
54C17×#389—21no—
55C17×#389C2211yescharacteristic
56C17×#389C2211yescharacteristic
57C17×#387—21no—
58C17×#384C2211yescharacteristic
59C17×#380C2211yescharacteristic
60C4×C22⋊C4×C17C2211yescharacteristic
61C23×C4×C17C2311yescharacteristic
62C23×Dic17—21no—
63C22×Dic34—22no—
64C22×Dic34—22no—
65C22×Dic34—21no—
66C22×Dic34—22no—
67C22×Dic34—41no—
68C22×Dic34—41no—
69C22×Dic34Q814yes—
70C22×Dic34C2311yescharacteristic
71C22×Dic34C2311yescharacteristic
72C17×#125—22no—
73C17×#125—41no—
74C17×#117—22no—
75C17×#117—41no—
76C2×C4⋊C4×C17—21no—
77C2×C4⋊C4×C17—21no—
78C2×C4⋊C4×C17—21no—
79C2×C4⋊C4×C17—21no—
80C2×C4⋊C4×C17—21no—
81C2×C4⋊C4×C17—21no—
82C2×C4⋊C4×C17C2311yescharacteristic
83C2×C4⋊C4×C17C2311yescharacteristic
84C2×C4⋊C4×C17C2311yescharacteristic
85C2×C4⋊C4×C17C2311yescharacteristic
86C2×C22⋊C4×C17—21no—
87C2×C22⋊C4×C17—21no—
88C2×C22⋊C4×C17D411yescharacteristic
89C2×C22⋊C4×C17D411yescharacteristic
90C2×C42×C17—21no—
91C2×C42×C17C2311yescharacteristic
92C2×C42×C17C2311yescharacteristic
93C2×C22⋊Dic17—21no—
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96C2×#63225—21no—
97C2×#63225—21no—
98C2×C4×Dic17—21no—
99C2×C4×Dic17—21no—
100C2×#63223—22no—
101C2×#63223—22no—
102C2×#63223—41no—
103C22×#3622C2311yescharacteristic
104C22×#3622C2311yescharacteristic
105C22×#3622C2311yescharacteristic
106C22×#3622C2311yescharacteristic
107C4×Dic34—22no—
108C4×Dic34—22no—
109C4×Dic34—41no—
110C4×Dic34—41no—
111C17×#95—21no—
112C17×#95—21no—
113C17×#95—21no—
114C17×#95D411yescharacteristic
115C17×#95D411yescharacteristic
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121C22⋊#3622—22no—
122#10587481—22no—
123#10587481—22no—
124#10587481—22no—
125#10587481—22no—
126#10587481—22no—
127#10587481—41no—
128#10587481—41no—
129#10587481D414yes—
130#10587479—22no—
131#10587479—41no—
132#10587478—22no—
133#10587478—41no—
134C4⋊#3622—22no—
135C4⋊#3622—41no—
136C24×C17C2×D411yescharacteristic
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17—21no—
143C22×C4×C17—21no—
144C22×C4×C17—21no—
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2×Q812yes—
151C22×C4×C17#4112yes—
152C22×C4×C17C2411yescharacteristic
153C22×Dic17—21no—
154C22×Dic17—24no—
155C22×Dic17—21no—
156C22×Dic17—21no—
157C22×Dic17—21no—
158C22×Dic17—42no—
159C22×Dic17—42no—
160C2×Dic34—21no—
161C2×Dic34—21no—
162C2×Dic34—21no—
163C2×Dic34—21no—
164C2×Dic34—21no—
165C2×Dic34—21no—
166C2×Dic34—21no—
167C2×Dic34—28no—
168C2×Dic34—24no—
169C2×Dic34—21no—
170C2×Dic34—44no—
171C2×Dic34—44no—
172C2×Dic34—44no—
173C2×Dic34—42no—
174C2×Dic34—42no—
175C2×Dic34C2×D412yes—
176C2×Dic34C2×Q812yes—
177C2×Dic34C2×Q812yes—
178C2×Dic34#4112yes—
179C4⋊C4×C17—21no—
180C4⋊C4×C17—21no—
181C4⋊C4×C17—21no—
182C4⋊C4×C17—21no—
183C4⋊C4×C17—21no—
184C4⋊C4×C17—21no—
185C4⋊C4×C17—21no—
186C4⋊C4×C17—42no—
187C4⋊C4×C17—42no—
188C4⋊C4×C17—42no—
189C4⋊C4×C17—42no—
190C4⋊C4×C17—81no—
191C4⋊C4×C17—81no—
192C4⋊C4×C17C2×D412yes—
193C22⋊C4×C17—22no—
194C22⋊C4×C17—41no—
195C22⋊C4×C17—42no—
196C22⋊C4×C17—81no—
197C42×C17—21no—
198C42×C17—21no—
199C42×C17—21no—
200C42×C17—21no—
201C42×C17—42no—
202C22⋊Dic17—42no—
203C4⋊Dic17—21no—
204C4⋊Dic17—21no—
205C4⋊Dic17—21no—
206C4⋊Dic17—21no—
207#63225—42no—
208#63225—42no—
209C4×Dic17—42no—
210C4×Dic17—42no—
211#63223—44no—
212C2×#3622—21no—
213C2×#3622—21no—
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×#3622C2×D412yes—
224#25981—171nomaximal, Sylow
225C23×C17—21no—
226C23×C17—21no—
227C23×C17—21no—
228C23×C17—21no—
229C23×C17—21no—
230C23×C17—21no—
231C23×C17C22⋊Q812yes—
232C23×C17C22×D411yescharacteristic
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—22no—
238C2×C4×C17—22no—
239C2×C4×C17—22no—
240C2×C4×C17—22no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—22no—
244C2×C4×C17—21no—
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—41no—
252C2×C4×C17—41no—
253C2×C4×C17—41no—
254C2×C4×C17—41no—
255C2×C4×C17—41no—
256C2×C4×C17—41no—
257C2×C4×C17—41no—
258C2×C4×C17—41no—
259C2×C4×C17—41no—
260C2×C4×C17—41no—
261C2×C4×C17—41no—
262C2×C4×C17C22⋊Q812yes—
263C2×C4×C17#13712yes—
264C2×C4×C17C22×D411yescharacteristic
265C2×C4×C17C22×Q811yescharacteristic
266C2×C4×C17C2×#4111yescharacteristic
267C2×C4×C17C2×#4111yescharacteristic
268C2×C4×C17H(5,𝔽2)11yescharacteristic
269C2×C4×C17H(5,𝔽2)11yescharacteristic
270C2×Dic17—22no—
271C2×Dic17—22no—
272C2×Dic17—22no—
273C2×Dic17—22no—
274C2×Dic17—48no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281C2×Dic17—41no—
282C2×Dic17—41no—
283Dic34—22no—
284Dic34—22no—
285Dic34—22no—
286Dic34—22no—
287Dic34—22no—
288Dic34—48no—
289Dic34—41no—
290Dic34—41no—
291Dic34C22⋊Q814yes—
292#3622—22no—
293#3622—41no—
294#3622—41no—
295#3622—41no—
296C2×#479—171no—
297C2×#419—171no—
298#2441D1711yescharacteristic
299Q8×C22⋊C4—171no—
300#2095—171no—
301#2066—171no—
302#2066—171no—
303#2026—171no—
304#1865—171no—
305#1863—171no—
306#1854—171no—
307#1851—171no—
308#1809—171no—
309#1803—171no—
310#1787—171no—
311C22×C17—22no—
312C22×C17—22no—
313C22×C17—22no—
314C22×C17—41no—
315C22×C17—41no—
316C22×C17—41no—
317C22×C17—41no—
318C22×C17C2×C22⋊Q811yescharacteristic
319C22×C17D4211yescharacteristic
320C22×C17#54611yescharacteristic
321C22×C17C2×#13711yescharacteristic
322C22×C17#57511yescharacteristic
323C22×C17#57711yescharacteristic
324C22×C17#57811yescharacteristic
325C4×C17—22no—
326C4×C17—22no—
327C4×C17—41no—
328C4×C17—41no—
329C4×C17—42no—
330C4×C17—41no—
331C4×C17—81no—
332C4×C17—81no—
333C4×C17—81no—
334C4×C17C2×C22⋊Q811yescharacteristic
335C4×C17#54311yescharacteristic
336C4×C17#54611yescharacteristic
337C4×C17#55111yescharacteristic
338Dic17—24no—
339Dic17—42no—
340Dic17—81no—
341Dic17—81no—
342C23×Q8—171no—
343C22×M16—171no—
344C2×#125C2×D1711yescharacteristic
345C2×C4×Q8—171no—
346C2×C4×Q8—171no—
347C2×#117C2×D1711yescharacteristic
348#479—344no—
349C2×#107—171no—
350C2×#107—171no—
351C2×#106—171no—
352C2×#106—171no—
353C2×C4⋊C8—171no—
354C2×C4⋊C8—171no—
355#419—344no—
356C2×#103—171no—
357C2×#103—171no—
358C2×#103—171no—
359C2×#103—171no—
360C2×#103—171no—
361C2×#103—171no—
362C2×#103—171no—
363C2×#103—171no—
364C2×C22⋊C8—171no—
365#399—21no—
366#396—21no—
367#391—171no—
368#391—171no—
369#391—341no—
370#391—341no—
371#389—21no—
372#389C2×D1711yescharacteristic
373#389C2×D1711yescharacteristic
374#387—21no—
375#384C2×D1711yescharacteristic
376#380—171no—
377#380—341no—
378#380C2×D1711yescharacteristic
379C4×C22⋊C4—341no—
380C4×C22⋊C4C2×D1711yescharacteristic
381#340—171no—
382#340—171no—
383#340—171no—
384#340—171no—
385#336—171no—
386#336—171no—
387C2×C17—42no—
388C2×C17#243711yescharacteristic
389C2×C17#307811yescharacteristic
390C2×C17#308211yescharacteristic
391C2×C17#327111yescharacteristic
392C2×C17#327611yescharacteristic
393C2×C17#332011yescharacteristic
394C2×C17#332711yescharacteristic
395C22×Q8—174no—
396C22×Q8—171no—
397C22×Q8—171no—
398C22×Q8—342no—
399C22×Q8—342no—
400C22×Q8—342no—
401C22×Q8—341no—
402C22×Q8—681no—
403C22×Q8—681no—
404C23×C4—341no—
405C23×C4C22×D1711yescharacteristic
406C2×M16—342no—
407C2×M16—342no—
408C2×M16—681no—
409C22×C8—171no—
410C22×C8—171no—
411C22×C8—171no—
412C22×C8—171no—
413#125—22no—
414#125—41no—
415C4×Q8—342no—
416C4×Q8—342no—
417C4×Q8—681no—
418C4×Q8—681no—
419#117—22no—
420#117—41no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—171no—
428C2×C4⋊C4—171no—
429C2×C4⋊C4—341no—
430C2×C4⋊C4—341no—
431C2×C4⋊C4C22×D1711yescharacteristic
432C2×C4⋊C4C22×D1711yescharacteristic
433C2×C4⋊C4C22×D1711yescharacteristic
434C2×C4⋊C4C22×D1711yescharacteristic
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—341no—
438C2×C22⋊C4C22⋊D1711yescharacteristic
439C2×C22⋊C4C22⋊D1711yescharacteristic
440C2×C42—21no—
441C2×C42—341no—
442C2×C42—341no—
443C2×C42C22×D1711yescharacteristic
444C2×C42C22×D1711yescharacteristic
445#107—342no—
446#107—681no—
447#106—342no—
448#106—681no—
449C4⋊C8—342no—
450C4⋊C8—681no—
451#103—174no—
452#103—342no—
453#103—342no—
454#103—342no—
455#103—342no—
456#103—342no—
457#103—681no—
458#103—681no—
459C22⋊C8—342no—
460#95—21no—
461#95—21no—
462#95—21no—
463#95—341no—
464#95—341no—
465#95—341no—
466#95—341no—
467#95—341no—
468#95C22⋊D1711yescharacteristic
469#95C22⋊D1711yescharacteristic
470C17#2598111yescharacteristic, complemented, Sylow
471C24C2×C22⋊D1711yescharacteristic
472C2×Q8—172no—
473C2×Q8—172no—
474C2×Q8—172no—
475C2×Q8—172no—
476C2×Q8—348no—
477C2×Q8—341no—
478C2×Q8—341no—
479C2×Q8—341no—
480C2×Q8—341no—
481C2×Q8—344no—
482C2×Q8—341no—
483C2×Q8—341no—
484C2×Q8—341no—
485C2×Q8—341no—
486C2×Q8—682no—
487C2×Q8—682no—
488C2×Q8—684no—
489C2×Q8—684no—
490C2×Q8—684no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—341no—
500C22×C4—341no—
501C22×C4—344no—
502C22×C4—341no—
503C22×C4—341no—
504C22×C4—682no—
505C22×C4—682no—
506C22×C4Q8×D1712yes—
507C22×C4#6325612yes—
508C22×C4C2×C22⋊D1711yescharacteristic
509C22×C4C2×C22⋊D1711yescharacteristic
510C22×C4C2×C22⋊D1711yescharacteristic
511C22×C4C2×C22⋊D1711yescharacteristic
512C22×C4C2×C22⋊D1711yescharacteristic
513C22×C4C23×D1711yescharacteristic
514M16—684no—
515C2×C8—172no—
516C2×C8—341no—
517C2×C8—341no—
518C2×C8—341no—
519C2×C8—341no—
520C2×C8—341no—
521C2×C8—341no—
522C2×C8—341no—
523C2×C8—341no—
524C2×C8—341no—
525C2×C8—341no—
526C2×C8—341no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—81no—
539C4⋊C4—81no—
540C4⋊C4—341no—
541C4⋊C4—341no—
542C4⋊C4—341no—
543C4⋊C4—341no—
544C4⋊C4—682no—
545C4⋊C4—682no—
546C4⋊C4D4×D1712yes—
547C22⋊C4—22no—
548C22⋊C4—41no—
549C22⋊C4—42no—
550C22⋊C4—81no—
551C22⋊C4—682no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—42no—
557C42—682no—
558C42—682no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23#1058763212yes—
566C23C22×C22⋊D1711yescharacteristic
567Q8—174no—
568Q8—342no—
569Q8—342no—
570Q8—342no—
571Q8—342no—
572Q8—342no—
573Q8—688no—
574Q8—681no—
575Q8—681no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—342no—
606C2×C4—342no—
607C2×C4—342no—
608C2×C4—342no—
609C2×C4—688no—
610C2×C4—681no—
611C2×C4—681no—
612C2×C4—681no—
613C2×C4—681no—
614C2×C4—681no—
615C2×C4—681no—
616C2×C4—681no—
617C2×C4—681no—
618C2×C4#1058763212yes—
619C2×C4#1058763712yes—
620C2×C4C2×D4×D1711yescharacteristic
621C2×C4#1058769111yescharacteristic
622C2×C4C2×Q8×D1711yescharacteristic
623C2×C4C2×#6325611yescharacteristic
624C2×C4#41×D1711yescharacteristic
625C2×C4#1058769611yescharacteristic
626C8—342no—
627C8—681no—
628C8—681no—
629C8—681no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22C2×#1058763211yescharacteristic
638C22D4×C22⋊D1711yescharacteristic
639C22#1176344711yescharacteristic
640C22C2×#1058763711yescharacteristic
641C22#1176360711yescharacteristic
642C22#1176362011yescharacteristic
643C22#1176362411yescharacteristic
644C4—22no—
645C4—22no—
646C4—41no—
647C4—41no—
648C4—42no—
649C4—41no—
650C4—81no—
651C4—81no—
652C4—81no—
653C4—344no—
654C4—682no—
655C4—1361no—
656C4—1361no—
657C4C22⋊Q8×D1711yescharacteristic
658C4#1176343411yescharacteristic
659C4#1176345011yescharacteristic
660C4#1176348611yescharacteristic
661C2—42no—
662C2#1453244711yescharacteristic
663C2#1453858511yescharacteristic
664C2#1453862011yescharacteristic
665C2#1454036511yescharacteristic
666C2#1454042611yescharacteristic
667C2#1454094911yescharacteristic
668C2#1454104611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2441C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#2441D17
hypocenterC17#25981
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2441

Complements

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

NH ≅ G/Ntype
1. C2×#11763113C2semidirect
4. #14530323C2semidirect
5. #14530162C2semidirect
6. #14530162C2semidirect
7. #14529932C2semidirect
8. #14529051C2semidirect
9. #14529037C2semidirect
13. #14528786C2semidirect
470. C17#25981semidirect

Maximal subgroups

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

subgroupindex
C2×#117631132
C2×#117628172
C22⋊C4×Dic342
#145303232
#145301622
#145299322
#145290512
#145290372
#145289832
#145289692
#145288152
#145287862
#145287182
C17×#24412
#2598117