Finite Group Databasebeta

← #17052465

Subgroups — #17052465

Order 4352.

982 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×#11762872C211yesmaximal, characteristic
2C2×#11762845C211yesmaximal, characteristic, complemented
3C2×#11762844C211yesmaximal, characteristic, complemented
4#14531734C211yesmaximal, characteristic
5#14531728C211yesmaximal, characteristic
6#14529546C211yesmaximal, characteristic, complemented
7#14529544C211yesmaximal, characteristic, complemented
8#14528986C211yesmaximal, characteristic, complemented
9#14528985C211yesmaximal, characteristic, complemented
10#14528980C211yesmaximal, characteristic, complemented
11#14528942C211yesmaximal, characteristic, complemented
12#14528812C211yesmaximal, characteristic, complemented
13#14528811C211yesmaximal, characteristic, complemented
14#14528676C211yesmaximal, characteristic, complemented
15C17×#1854C211yesmaximal, characteristic, complemented
16C23×#3622C2211yescharacteristic
17C22×C8×D17C2211yescharacteristic
18C2×C4×C22⋊D17C2211yescharacteristic
19C2×#10587612C2211yescharacteristic
20C2×#10587578C2211yescharacteristic
21C2×#10587578C2211yescharacteristic
22#11762872C418yescomplemented
23#11762845—24no—
24#11762844—24no—
25C2×#10587489C2211yescharacteristic
26C2×#10587478C2211yescharacteristic
27C2×#10587478C2211yescharacteristic
28C2×#10587478C2211yescharacteristic
29C2×#10587478C2211yescharacteristic
30C2×#10587488C2211yescharacteristic
31C2×#10587479C2211yescharacteristic
32C2×#10587479C2211yescharacteristic
33C2×#10587485C2211yescharacteristic
34C2×#10587491C2211yescharacteristic
35C2×C8×Dic17C2211yescharacteristic
36#11762709—21no—
37#11762709—21no—
38#11762703—21no—
39#11762700—21no—
40#11762698—21no—
41#11762697—21no—
42#11762678C2211yescharacteristic
43#11762678C2211yescharacteristic
44#11762669C2211yescharacteristic
45#11762666C2211yescharacteristic
46#11762666C2211yescharacteristic
47C4⋊C4×Dic17C2211yescharacteristic
48C4⋊C4×Dic17C2211yescharacteristic
49C4×#63225—21no—
50#11762660—21no—
51#11762572C2211yescharacteristic
52#11762572C2211yescharacteristic
53#11762571C2211yescharacteristic
54#11762571C2211yescharacteristic
55C2×C17×#117C2211yescharacteristic
56C2×C17×#107C2211yescharacteristic
57C2×C17×#106C2211yescharacteristic
58C2×C17×C22⋊C8C2211yescharacteristic
59C17×#380C2211yescharacteristic
60C17×#340C2211yescharacteristic
61C17×#340C2211yescharacteristic
62C23×C4×C17C2311yescharacteristic
63C22×C22⋊D17D411yescharacteristic
64C22×C4×D17C2311yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C17×#117—22no—
68C17×#117—41no—
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C22⋊C4×C17—21no—
74C2×C22⋊C4×C17—21no—
75C2×C42×C17—21no—
76C2×C22⋊Dic17—21no—
77C2×C22⋊Dic17—21no—
78C2×C22⋊Dic17D411yescharacteristic
79C4×C22⋊D17C2×C414yes—
80C4×C22⋊D17Q814yes—
81C2×#63227—21no—
82C2×#63227—21no—
83C2×#63227—21no—
84C2×#63227—21no—
85C2×#63227D411yescharacteristic
86#10587612—22no—
87#10587612—41no—
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×#63225—21no—
93C2×#63225—21no—
94C2×#63225—21no—
95C2×#63225—21no—
96C2×#63225D411yescharacteristic
97C2×C4×Dic17—21no—
98C2×C4×Dic17—21no—
99C2×C4×Dic17—21no—
100C2×C4×Dic17—21no—
101C2×C4×Dic17—21no—
102C2×C4×Dic17C2311yescharacteristic
103C22×#3622—22no—
104C22×#3622—22no—
105C22×#3622C2×C414yes—
106C22×#3622C2311yescharacteristic
107C22×#3622C2311yescharacteristic
108C2×C8×D17—22no—
109C2×C8×D17—22no—
110C2×C8×D17C2×C414yes—
111#10587578—22no—
112#10587578—22no—
113#10587578—41no—
114#10587578—41no—
115C17×#107—41no—
116C17×#106—22no—
117C17×C22⋊C8C2×C414yes—
118C17×#95—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587491C2×C414yes—
125#10587489—41no—
126#10587488—22no—
127#10587485C2×C414yes—
128C8×Dic17C2×C414yes—
129#10587479—22no—
130#10587479—22no—
131#10587478—22no—
132#10587478—22no—
133#10587478—22no—
134#10587478D414yescomplemented
135C24×C17C2×D411yescharacteristic
136C23×D17C2×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×C17C22×C412yes—
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×Q812yes—
148C22×C4×C17C2411yescharacteristic
149C2×C22⋊D17—22no—
150C2×C22⋊D17—22no—
151C2×C22⋊D17C4⋊C414yes—
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—21no—
155C22×Dic17—21no—
156C22×Dic17C2×D411yescharacteristic
157C22×Dic17C2×D411yescharacteristic
158C22×Dic17C2×D411yescharacteristic
159C2×C4×D17—21no—
160C2×C4×D17—21no—
161C2×C4×D17—21no—
162C2×C4×D17—21no—
163C2×C4×D17C22×C412yes—
164C2×C4×D17C2×Q812yes—
165C2×C8×C17—21no—
166C2×C8×C17—21no—
167C2×C8×C17—21no—
168C2×C8×C17—21no—
169C2×C8×C17C22×C412yes—
170C2×C8×C17C22×C412yes—
171C4⋊C4×C17—21no—
172C4⋊C4×C17—21no—
173C4⋊C4×C17—21no—
174C4⋊C4×C17—21no—
175C4⋊C4×C17—21no—
176C4⋊C4×C17—21no—
177C4⋊C4×C17—21no—
178C4⋊C4×C17C2×D412yes—
179C22⋊C4×C17—42no—
180C22⋊C4×C17—81no—
181C42×C17—42no—
182C22⋊Dic17—42no—
183C22⋊Dic17—81no—
184C22⋊Dic17C4⋊C414yes—
185#63227—42no—
186#63227—42no—
187#63227—81no—
188#63227—81no—
189#63227C4⋊C414yes—
190C4⋊Dic17—21no—
191C4⋊Dic17—21no—
192C4⋊Dic17—21no—
193C4⋊Dic17—21no—
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17C2×D412yes—
198#63225—42no—
199#63225—42no—
200#63225—81no—
201#63225—81no—
202#63225C4⋊C414yes—
203C4×Dic17—42no—
204C4×Dic17—42no—
205C4×Dic17—42no—
206C4×Dic17—42no—
207C4×Dic17—42no—
208C4×Dic17C22×C412yes—
209C4×Dic17C2×Q812yes—
210C2×#3622—21no—
211C2×#3622—21no—
212C2×#3622—28no—
213C2×#3622C22×C412yes—
214C2×#3622C22×C412yes—
215C2×#3622C2×D412yes—
216C2×#3622#4112yes—
217C8×D17—28no—
218#22658—171nomaximal, Sylow
219C23×C17—21no—
220C23×C17—21no—
221C23×C17—21no—
222C23×C17—21no—
223C23×C17—21no—
224C23×C17—21no—
225C23×C17C2×C4⋊C412yes—
226C23×C17C22×D411yescharacteristic
227C22×D17—21no—
228C22×D17—21no—
229C22×D17—21no—
230C22×D17—21no—
231C22×D17—21no—
232C22×D17—21no—
233C22×D17C2×C4⋊C412yes—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—22no—
237C2×C4×C17—22no—
238C2×C4×C17—22no—
239C2×C4×C17—22no—
240C2×C4×C17—22no—
241C2×C4×C17—41no—
242C2×C4×C17—41no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17C2×C4⋊C412yes—
253C2×C4×C17C23×C411yescharacteristic
254C2×C4×C17C22×D411yescharacteristic
255C2×C4×C17C22×Q811yescharacteristic
256C2×C4×C17C2×#4111yescharacteristic
257C2×C4×C17H(5,𝔽2)11yescharacteristic
258C2×C4×C172−1+411yescharacteristic
259C22⋊D17—28no—
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—21no—
263C2×Dic17—21no—
264C2×Dic17—22no—
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—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C2×Dic17C2×C4⋊C412yes—
277C2×Dic17C2×C4⋊C412yes—
278C2×Dic17C2×C4⋊C412yes—
279C2×Dic17#13512yes—
280C2×Dic17#13512yes—
281C4×D17—22no—
282C4×D17—22no—
283C4×D17—22no—
284C4×D17—22no—
285C8×C17—22no—
286C8×C17—22no—
287#3622—22no—
288#3622C4×D414yes—
289C2×C8×D4—171no—
290C2×#427—171no—
291C2×#427—171no—
292#2354—171no—
293#2354—171no—
294#1948—171no—
295#1948—171no—
296#1854—171no—
297#1854—171no—
298#1854—171no—
299#1854D1711yescharacteristic, complemented
300#1845—171no—
301#1808—171no—
302#1808—171no—
303C4×#107—171no—
304C22×C17—22no—
305C22×C17—22no—
306C22×C17—22no—
307C22×C17—41no—
308C22×C17—41no—
309C22×C17—41no—
310C22×C17—41no—
311C22×C17C22×C4⋊C411yescharacteristic
312C22×C17D4211yescharacteristic
313C22×C17#54711yescharacteristic
314C22×C17C2×#13511yescharacteristic
315C22×C17C2×#13511yescharacteristic
316C22×C17#57611yescharacteristic
317C22×C17#57811yescharacteristic
318C2×D17—22no—
319C2×D17—22no—
320C2×D17—22no—
321C2×D17—41no—
322C2×D17—41no—
323C2×D17—41no—
324C2×D17—41no—
325C4×C17—22no—
326C4×C17—42no—
327C4×C17—81no—
328C4×C17—81no—
329C4×C17—81no—
330C4×C17C2×C4×D411yescharacteristic
331C4×C17#52011yescharacteristic
332C4×C17D4×Q811yescharacteristic
333C4×C17#55411yescharacteristic
334Dic17—22no—
335Dic17—22no—
336Dic17—42no—
337Dic17—81no—
338Dic17—81no—
339Dic17—81no—
340Dic17#42714yescomplemented
341C23×C8—171no—
342C23×C8—171no—
343C2×C4×D4—171no—
344C2×#117—171no—
345C2×#117—171no—
346C2×#117—171no—
347C2×#117C2×D1711yescharacteristic
348C8×D4—178no—
349#427—344no—
350#427—344no—
351C2×#107—171no—
352C2×#107—171no—
353C2×#107—171no—
354C2×#107—171no—
355C2×#107—171no—
356C2×#107C2×D1711yescharacteristic
357C2×#106—171no—
358C2×#106—171no—
359C2×#106—171no—
360C2×#106C2×D1711yescharacteristic
361C2×C4⋊C8—171no—
362C2×C22⋊C8—171no—
363C2×C22⋊C8C2×D1711yescharacteristic
364C2×C4×C8—171no—
365#390—341no—
366#390—341no—
367#388—341no—
368#388—341no—
369#388—341no—
370#388—341no—
371#384—171no—
372#384—171no—
373#380—171no—
374#380—171no—
375#380—171no—
376#380C2×D1711yescharacteristic
377C4×C4⋊C4—171no—
378C4×C4⋊C4—171no—
379C4×C4⋊C4—341no—
380C4×C4⋊C4—341no—
381#340—171no—
382#340—171no—
383#340—171no—
384#340—171no—
385#340C2×D1711yescharacteristic
386#340C2×D1711yescharacteristic
387C2×C17—42no—
388C2×C17D4×C4⋊C411yescharacteristic
389C2×C17C2×#42711yescharacteristic
390C2×C17#291311yescharacteristic
391C2×C17#328311yescharacteristic
392C2×C17#328711yescharacteristic
393C2×C17#334111yescharacteristic
394C2×C17#334211yescharacteristic
395D17—42no—
396C22×D4—171no—
397C23×C4—171no—
398C23×C4C22×D1711yescharacteristic
399C22×C8—171no—
400C22×C8—174no—
401C22×C8—171no—
402C22×C8—174no—
403C22×C8—342no—
404C22×C8—342no—
405C22×C8—342no—
406C22×C8—342no—
407C22×C8C22×D1711yescharacteristic
408C22×C8C22×D1711yescharacteristic
409C4×D4—174no—
410C4×D4—174no—
411#117—22no—
412#117—41no—
413#117—342no—
414#117—342no—
415#117—342no—
416#117—681no—
417#117—681no—
418#117—681no—
419C2×C4⋊C4—171no—
420C2×C4⋊C4—171no—
421C2×C4⋊C4—171no—
422C2×C4⋊C4—171no—
423C2×C4⋊C4—171no—
424C2×C4⋊C4—341no—
425C2×C4⋊C4—341no—
426C2×C4⋊C4—341no—
427C2×C4⋊C4—341no—
428C2×C4⋊C4C22×D1711yescharacteristic
429C2×C4⋊C4C22×D1711yescharacteristic
430C2×C4⋊C4C22×D1711yescharacteristic
431C2×C4⋊C4C22×D1711yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—171no—
435C2×C22⋊C4—171no—
436C2×C22⋊C4—341no—
437C2×C22⋊C4—341no—
438C2×C22⋊C4—341no—
439C2×C22⋊C4—341no—
440C2×C22⋊C4—341no—
441C2×C22⋊C4—341no—
442C2×C42—21no—
443C2×C42—171no—
444C2×C42—341no—
445C2×C42—341no—
446C2×C42—341no—
447C2×C42—341no—
448C2×C42—341no—
449#107—41no—
450#107—174no—
451#107—342no—
452#107—342no—
453#107—342no—
454#107—681no—
455#106—22no—
456#106—342no—
457#106—342no—
458#106—342no—
459C4⋊C8—174no—
460C22⋊C8—174no—
461C22⋊C8C4×D1714yes—
462C4×C8—174no—
463#95—21no—
464#95—341no—
465#95—341no—
466#95—341no—
467#95—341no—
468#95—341no—
469C17#2265811yescharacteristic, complemented, Sylow
470C24—171no—
471C24D4×D1711yescharacteristic
472C2×D4—174no—
473C2×D4—342no—
474C2×D4—342no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—171no—
484C22×C4—171no—
485C22×C4—171no—
486C22×C4—172no—
487C22×C4—172no—
488C22×C4—341no—
489C22×C4—341no—
490C22×C4—341no—
491C22×C4—341no—
492C22×C4—341no—
493C22×C4—341no—
494C22×C4—341no—
495C22×C4—341no—
496C22×C4C2×C4×D1712yes—
497C22×C4D4×D1711yescharacteristic
498C22×C4Q8×D1712yes—
499C22×C4C23×D1711yescharacteristic
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—172no—
505C2×C8—172no—
506C2×C8—172no—
507C2×C8—172no—
508C2×C8—348no—
509C2×C8—348no—
510C2×C8—341no—
511C2×C8—341no—
512C2×C8C2×C4×D1712yes—
513C2×C8C2×C4×D1712yes—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—174no—
522C4⋊C4—172no—
523C4⋊C4—341no—
524C4⋊C4—341no—
525C4⋊C4—341no—
526C4⋊C4—341no—
527C4⋊C4—341no—
528C4⋊C4—341no—
529C4⋊C4—341no—
530C4⋊C4—682no—
531C4⋊C4—682no—
532C4⋊C4—1361no—
533C4⋊C4—1361no—
534C4⋊C4D4×D1712yes—
535C22⋊C4—42no—
536C22⋊C4—81no—
537C22⋊C4—174no—
538C22⋊C4—174no—
539C22⋊C4—682no—
540C22⋊C4—682no—
541C22⋊C4—682no—
542C22⋊C4—1361no—
543C22⋊C4—1361no—
544C22⋊C4—1361no—
545C42—42no—
546C42—172no—
547C42—172no—
548C42—682no—
549C42—682no—
550C42—682no—
551C42—682no—
552C42—682no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—172no—
560C23—341no—
561C23—341no—
562C23—341no—
563C23—341no—
564C23—341no—
565C23—341no—
566C23C4⋊C4×D1712yes—
567C23C2×D4×D1711yescharacteristic
568D4—348no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—172no—
588C2×C4—172no—
589C2×C4—172no—
590C2×C4—172no—
591C2×C4—172no—
592C2×C4—341no—
593C2×C4—341no—
594C2×C4—341no—
595C2×C4—341no—
596C2×C4—342no—
597C2×C4—342no—
598C2×C4—342no—
599C2×C4—342no—
600C2×C4—342no—
601C2×C4—681no—
602C2×C4—681no—
603C2×C4—681no—
604C2×C4—681no—
605C2×C4—681no—
606C2×C4—681no—
607C2×C4—681no—
608C2×C4—681no—
609C2×C4—681no—
610C2×C4—681no—
611C2×C4—681no—
612C2×C4C4⋊C4×D1712yes—
613C2×C4C22×C4×D1711yescharacteristic
614C2×C4C2×D4×D1711yescharacteristic
615C2×C4#1058769111yescharacteristic
616C2×C4C2×Q8×D1711yescharacteristic
617C2×C4#41×D1711yescharacteristic
618C2×C4#1058769711yescharacteristic
619C8—22no—
620C8—22no—
621C8—174no—
622C8—342no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—342no—
631C22—342no—
632C22—342no—
633C22—681no—
634C22—681no—
635C22—681no—
636C22—681no—
637C22C2×C4⋊C4×D1711yescharacteristic
638C22#1176344111yescharacteristic
639C22#1176345711yescharacteristic
640C22#135×D1711yescharacteristic
641C22#135×D1711yescharacteristic
642C22#1176361311yescharacteristic
643C22#1176362511yescharacteristic
644C4—22no—
645C4—42no—
646C4—81no—
647C4—81no—
648C4—81no—
649C4—174no—
650C4—342no—
651C4—342no—
652C4—682no—
653C4—1361no—
654C4—1361no—
655C4—1361no—
656C4C4×D4×D1711yescharacteristic
657C4#1176328811yescharacteristic
658C4#1176347211yescharacteristic
659C4#1176351311yescharacteristic
660C2—42no—
661C2—682no—
662C2#1453168211yescharacteristic
663C2D17×#42711yescharacteristic
664C2#1453738311yescharacteristic
665C2#1454049911yescharacteristic
666C2#1454053711yescharacteristic
667C2#1454120111yescharacteristic
668C2#1454120911yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1854C2
socleC23×C17C22×D4
prime radicalC22×C22⋊D17D4
cosocleC22×C4×C17C24
hypercenter#1854D17
hypocenterC17#22658
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
2. C2×#11762845C2semidirect
3. C2×#11762844C2semidirect
6. #14529546C2semidirect
7. #14529544C2semidirect
8. #14528986C2semidirect
9. #14528985C2semidirect
10. #14528980C2semidirect
11. #14528942C2semidirect
12. #14528812C2semidirect
13. #14528811C2semidirect
14. #14528676C2semidirect
15. C17×#1854C2semidirect
22. #11762872C4semidirect
134. #10587478D4semidirect
299. #1854D17semidirect
340. Dic17#427semidirect
469. C17#22658semidirect

Maximal subgroups

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

subgroupindex
C2×#117628722
C2×#117628452
C2×#117628442
#145317342
#145317282
#145295462
#145295442
#145289862
#145289852
#145289802
#145289422
#145288122
#145288112
#145286762
C17×#18542
#2265817