Finite Group Databasebeta

← #17052545

Subgroups — #17052545

Order 4352.

978 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762885C211yesmaximal, characteristic
2C2×#11762850C211yesmaximal, characteristic, complemented
3#14531714C211yesmaximal, characteristic
4#14531709C211yesmaximal, characteristic
5#14529553C211yesmaximal, characteristic, complemented
6#14529549C211yesmaximal, characteristic, complemented
7#14528983C211yesmaximal, characteristic, complemented
8#14528982C211yesmaximal, characteristic, complemented
9#14528972C211yesmaximal, characteristic, complemented
10#14528970C211yesmaximal, characteristic, complemented
11#14528945C211yesmaximal, characteristic, complemented
12#14528817C211yesmaximal, characteristic, complemented
13#14528814C211yesmaximal, characteristic, complemented
14#107×Dic17C211yesmaximal, characteristic, complemented
15C2×C17×#428C211yesmaximal, characteristic, complemented
16C22×#63218C2211yescharacteristic
17C2×C4×C22⋊D17C2211yescharacteristic
18C2×#10587578C2211yescharacteristic
19C2×C4⋊C4×D17C2211yescharacteristic
20#11762885C418yescomplemented
21#11762850—24no—
22C2×#10587489C2211yescharacteristic
23C2×#10587478C2211yescharacteristic
24C2×#10587478C2211yescharacteristic
25C2×#10587478C2211yescharacteristic
26C2×#10587488C2211yescharacteristic
27C2×#10587479C2211yescharacteristic
28C2×#10587485C2211yescharacteristic
29C2×C22⋊#3622C2211yescharacteristic
30C2×#10587491C2211yescharacteristic
31C2×C8×Dic17C2211yescharacteristic
32#11762703—21no—
33#11762699—21no—
34#11762693—21no—
35#11762690—21no—
36#11762688—21no—
37#11762685—21no—
38#11762674C2211yescharacteristic
39#11762674C2211yescharacteristic
40#11762669C2211yescharacteristic
41#11762669C2211yescharacteristic
42#11762666C2211yescharacteristic
43#11762666C2211yescharacteristic
44C4⋊C4×Dic17C2211yescharacteristic
45C4⋊C4×Dic17C2211yescharacteristic
46C4×#63227—21no—
47C4×C22⋊Dic17—21no—
48#11762572C2211yescharacteristic
49#11762572C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762571C2211yescharacteristic
53#11762571C2211yescharacteristic
54C22×M16×C17C2211yescharacteristic
55C2×C17×#117C2211yescharacteristic
56C22×C4⋊C4×C17C2211yescharacteristic
57C17×#428—24no—
58C2×C17×#107C2211yescharacteristic
59C2×C17×#107C2211yescharacteristic
60C2×C17×#106C2211yescharacteristic
61C2×C17×#106C2211yescharacteristic
62C23×C4×C17—21no—
63C23×C4×C17C2311yescharacteristic
64C22×C22⋊D17D411yescharacteristic
65C22×C4×D17—21no—
66C22×C4×D17C2311yescharacteristic
67C2×M16×C17—22no—
68C2×M16×C17—22no—
69C2×M16×C17C2×C414yescomplemented
70C22×C8×C17C2311yescharacteristic
71C22×C8×C17C2311yescharacteristic
72C17×#117—22no—
73C17×#117—41no—
74C2×C4⋊C4×C17—22no—
75C2×C4⋊C4×C17—21no—
76C2×C4⋊C4×C17—41no—
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C4⋊C4×C17C2311yescharacteristic
79C2×C4⋊C4×C17C2311yescharacteristic
80C2×C4⋊C4×C17C2311yescharacteristic
81C2×C22⋊C4×C17—21no—
82C2×C42×C17—21no—
83C2×C22⋊Dic17—21no—
84C2×C22⋊Dic17—21no—
85C2×C22⋊Dic17D411yescharacteristic
86C4×C22⋊D17C2×C414yes—
87C4×C22⋊D17Q814yes—
88C2×#63227—21no—
89C2×#63227—21no—
90C2×#63227—21no—
91C2×#63227D411yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96C2×#63225—21no—
97C2×#63225D411yescharacteristic
98C2×C4×Dic17—21no—
99C2×C4×Dic17—21no—
100C2×C4×Dic17—21no—
101C2×C4×Dic17C2311yescharacteristic
102C22×#3622C2311yescharacteristic
103C22×#3622C2311yescharacteristic
104C2×#63218—22no—
105C2×#63218—22no—
106C2×#63218C2×C414yescomplemented
107#10587578—22no—
108#10587578—41no—
109C4⋊C4×D17—22no—
110C4⋊C4×D17—41no—
111C17×#107—22no—
112C17×#107D414yescomplemented
113C17×#106—22no—
114C17×#106—22no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587502—21no—
125C22⋊#3622C2×C414yescomplemented
126#10587491C2×C414yescomplemented
127#10587489—22no—
128#10587488—22no—
129#10587485C2×C414yescomplemented
130C8×Dic17C2×C414yescomplemented
131#10587479—22no—
132#10587478—22no—
133#10587478—41no—
134#10587478—41no—
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×C17—42no—
146C22×C4×C17—42no—
147C22×C4×C17—81no—
148C22×C4×C17C22×C412yes—
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2×Q812yes—
151C22×C4×C17C2411yescharacteristic
152C2×C22⋊D17—41no—
153C2×C22⋊D17—41no—
154C2×C22⋊D17C4⋊C414yes—
155C22×Dic17—21no—
156C22×Dic17—21no—
157C22×Dic17—21no—
158C22×Dic17—21no—
159C22×Dic17C2×D411yescharacteristic
160C22×Dic17C2×D411yescharacteristic
161C22×Dic17C2×D411yescharacteristic
162C2×C4×D17—21no—
163C2×C4×D17—21no—
164C2×C4×D17—21no—
165C2×C4×D17—21no—
166C2×C4×D17—42no—
167C2×C4×D17—42no—
168C2×C4×D17—81no—
169C2×C4×D17C22×C412yes—
170C2×C4×D17C2×Q812yes—
171M16×C17—28no—
172C2×C8×C17—21no—
173C2×C8×C17—21no—
174C2×C8×C17C22×C412yes—
175C2×C8×C17C22×C412yes—
176C2×C8×C17C2×D412yes—
177C2×C8×C17#4112yes—
178C4⋊C4×C17—21no—
179C4⋊C4×C17—21no—
180C4⋊C4×C17—21no—
181C4⋊C4×C17—21no—
182C4⋊C4×C17—21no—
183C4⋊C4×C17—21no—
184C4⋊C4×C17—42no—
185C4⋊C4×C17C2×D412yes—
186C4⋊C4×C17C2×D412yes—
187C22⋊C4×C17—42no—
188C42×C17—42no—
189C22⋊Dic17—42no—
190C22⋊Dic17—81no—
191C22⋊Dic17C4⋊C414yes—
192#63227—42no—
193#63227—42no—
194#63227—81no—
195#63227C4⋊C414yes—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199C4⋊Dic17—21no—
200C4⋊Dic17—21no—
201C4⋊Dic17—21no—
202C4⋊Dic17—21no—
203C4⋊Dic17—21no—
204#63225—42no—
205#63225C4⋊C414yes—
206C4×Dic17—42no—
207C4×Dic17—81no—
208C4×Dic17—81no—
209C4×Dic17C22×C412yes—
210C4×Dic17C2×Q812yes—
211C2×#3622—21no—
212C2×#3622—21no—
213C2×#3622—21no—
214C2×#3622—21no—
215C2×#3622C22×C412yes—
216C2×#3622C22×C412yes—
217#63218—28no—
218#22664—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—22no—
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×C17—84no—
253C2×C4×C17C2×C4⋊C412yes—
254C2×C4×C17C23×C411yescharacteristic
255C2×C4×C17C22×D411yescharacteristic
256C2×C4×C17C22×Q811yescharacteristic
257C2×C4×C17C2×#4111yescharacteristic
258C2×C4×C17H(5,𝔽2)11yescharacteristic
259C2×C4×C172−1+411yescharacteristic
260C22⋊D17—44no—
261C2×Dic17—21no—
262C2×Dic17—21no—
263C2×Dic17—21no—
264C2×Dic17—21no—
265C2×Dic17—21no—
266C2×Dic17—21no—
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×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17C2×C4⋊C412yes—
280C2×Dic17C2×C4⋊C412yes—
281C2×Dic17C2×C4⋊C412yes—
282C4×D17—22no—
283C4×D17—22no—
284C4×D17—22no—
285C4×D17—22no—
286C4×D17—84no—
287C8×C17—22no—
288C8×C17C4×D414yes—
289#3622—22no—
290#3622—22no—
291C2×#436—171no—
292C2×#428—171no—
293C2×#428D1711yescharacteristic, complemented
294#2352—171no—
295#2352—171no—
296#1949—171no—
297#1949—171no—
298#1854—171no—
299#1854—171no—
300#1852—171no—
301#1852—171no—
302#1846—171no—
303#1809—171no—
304#1809—171no—
305C4×#107—171no—
306C22×C17—22no—
307C22×C17—22no—
308C22×C17—22no—
309C22×C17—41no—
310C22×C17—41no—
311C22×C17—41no—
312C22×C17—41no—
313C22×C17C22×C4⋊C411yescharacteristic
314C22×C17D4211yescharacteristic
315C22×C17#54711yescharacteristic
316C22×C17#57511yescharacteristic
317C22×C17#57511yescharacteristic
318C22×C17#57611yescharacteristic
319C22×C17#57811yescharacteristic
320C2×D17—22no—
321C2×D17—22no—
322C2×D17—22no—
323C2×D17—41no—
324C2×D17—41no—
325C2×D17—41no—
326C2×D17—41no—
327C4×C17—22no—
328C4×C17—42no—
329C4×C17—42no—
330C4×C17—81no—
331C4×C17—81no—
332C4×C17C2×C4×D411yescharacteristic
333C4×C17#52011yescharacteristic
334C4×C17D4×Q811yescharacteristic
335C4×C17#55411yescharacteristic
336Dic17—22no—
337Dic17—22no—
338Dic17—22no—
339Dic17—81no—
340Dic17—81no—
341Dic17—81no—
342Dic17—81no—
343C22×M16—171no—
344C22×M16C2×D1711yescharacteristic
345C2×C4×D4—171no—
346C2×#117—171no—
347C2×#117C2×D1711yescharacteristic
348C22×C4⋊C4—171no—
349C22×C4⋊C4C2×D1711yescharacteristic
350#436—178no—
351#428—24no—
352#428—344no—
353C2×#107—171no—
354C2×#107—171no—
355C2×#107—171no—
356C2×#107—171no—
357C2×#107C2×D1711yescharacteristic
358C2×#107C2×D1711yescharacteristic
359C2×#106—171no—
360C2×#106—171no—
361C2×#106C2×D1711yescharacteristic
362C2×#106C2×D1711yescharacteristic
363C2×C4⋊C8—171no—
364C2×C22⋊C8—171no—
365C2×C22⋊C8—171no—
366C2×C4×C8—171no—
367#388—341no—
368#388—341no—
369#387—341no—
370#387—341no—
371#386—341no—
372#386—341no—
373#382—171no—
374#382—171no—
375#380—171no—
376#380—171no—
377#380—171no—
378#380—171no—
379C4×C4⋊C4—171no—
380C4×C4⋊C4—171no—
381C4×C22⋊C4—341no—
382C4×C22⋊C4—341no—
383#340—171no—
384#340—171no—
385#340—171no—
386#340—171no—
387#340—171no—
388#340—171no—
389C2×C17—42no—
390C2×C17D4×C4⋊C411yescharacteristic
391C2×C17#291111yescharacteristic
392C2×C17#291311yescharacteristic
393C2×C17#327311yescharacteristic
394C2×C17#327711yescharacteristic
395C2×C17#335211yescharacteristic
396C2×C17#335511yescharacteristic
397D17—42no—
398C22×D4—171no—
399C23×C4—21no—
400C23×C4—171no—
401C23×C4—341no—
402C23×C4C22×D1711yescharacteristic
403C2×M16—22no—
404C2×M16—22no—
405C2×M16—174no—
406C2×M16—342no—
407C2×M16—342no—
408C2×M16C4×D1714yescomplemented
409C22×C8—171no—
410C22×C8—171no—
411C22×C8C22×D1711yescharacteristic
412C22×C8C22×D1711yescharacteristic
413C4×D4—174no—
414C4×D4—174no—
415#117—22no—
416#117—41no—
417#117—342no—
418#117—681no—
419C2×C4⋊C4—22no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—41no—
422C2×C4⋊C4—171no—
423C2×C4⋊C4—171no—
424C2×C4⋊C4—171no—
425C2×C4⋊C4—171no—
426C2×C4⋊C4—171no—
427C2×C4⋊C4—342no—
428C2×C4⋊C4—341no—
429C2×C4⋊C4—681no—
430C2×C4⋊C4C22×D1711yescharacteristic
431C2×C4⋊C4C22×D1711yescharacteristic
432C2×C4⋊C4C22×D1711yescharacteristic
433C2×C4⋊C4C22×D1711yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—171no—
436C2×C22⋊C4—171no—
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—
447#107—22no—
448#107—342no—
449#107—342no—
450#107—681no—
451#107—681no—
452#107C22⋊D1714yescomplemented
453#106—22no—
454#106—22no—
455#106—342no—
456#106—342no—
457C4⋊C8—174no—
458C22⋊C8—174no—
459C22⋊C8—174no—
460C4×C8—174no—
461#95—341no—
462#95—341no—
463#95—341no—
464#95—341no—
465#95—341no—
466#95—341no—
467#95—341no—
468#95—341no—
469#95—341no—
470#95—341no—
471C17#2266411yescharacteristic, complemented, Sylow
472C24—171no—
473C24D4×D1711yescharacteristic
474C2×D4—174no—
475C2×D4—681no—
476C2×D4—681no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—81no—
488C22×C4—171no—
489C22×C4—171no—
490C22×C4—171no—
491C22×C4—172no—
492C22×C4—172no—
493C22×C4—341no—
494C22×C4—341no—
495C22×C4—341no—
496C22×C4—341no—
497C22×C4—341no—
498C22×C4—341no—
499C22×C4—341no—
500C22×C4—341no—
501C22×C4—682no—
502C22×C4—682no—
503C22×C4—1361no—
504C22×C4C2×C4×D1712yes—
505C22×C4D4×D1711yescharacteristic
506C22×C4Q8×D1712yes—
507C22×C4C23×D1711yescharacteristic
508M16—28no—
509M16—348no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—172no—
513C2×C8—172no—
514C2×C8—341no—
515C2×C8—341no—
516C2×C8—341no—
517C2×C8—341no—
518C2×C8C2×C4×D1712yes—
519C2×C8C2×C4×D1712yes—
520C2×C8#6325212yes—
521C2×C8C2×C22⋊D1712yes—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—42no—
529C4⋊C4—174no—
530C4⋊C4—341no—
531C4⋊C4—341no—
532C4⋊C4—341no—
533C4⋊C4—341no—
534C4⋊C4—341no—
535C4⋊C4—341no—
536C4⋊C4—341no—
537C4⋊C4—341no—
538C4⋊C4—682no—
539C4⋊C4C2×C22⋊D1712yes—
540C4⋊C4C2×C22⋊D1712yes—
541C22⋊C4—42no—
542C22⋊C4—174no—
543C22⋊C4—174no—
544C22⋊C4—682no—
545C22⋊C4—682no—
546C22⋊C4—682no—
547C22⋊C4—1361no—
548C22⋊C4—1361no—
549C42—42no—
550C42—172no—
551C42—172no—
552C42—682no—
553C42—1361no—
554C42—1361no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—172no—
562C23—341no—
563C23—341no—
564C23—341no—
565C23—341no—
566C23—341no—
567C23—341no—
568C23C4⋊C4×D1712yes—
569C23C2×D4×D1711yescharacteristic
570D4—684no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
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—41no—
588C2×C4—41no—
589C2×C4—84no—
590C2×C4—172no—
591C2×C4—172no—
592C2×C4—172no—
593C2×C4—341no—
594C2×C4—341no—
595C2×C4—341no—
596C2×C4—341no—
597C2×C4—341no—
598C2×C4—341no—
599C2×C4—342no—
600C2×C4—342no—
601C2×C4—342no—
602C2×C4—342no—
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×C4—681no—
613C2×C4—681no—
614C2×C4—681no—
615C2×C4—1364no—
616C2×C4C4⋊C4×D1712yes—
617C2×C4C22×C4×D1711yescharacteristic
618C2×C4C2×#6325211yescharacteristic
619C2×C4#1058769111yescharacteristic
620C2×C4C2×Q8×D1711yescharacteristic
621C2×C4#1058769411yescharacteristic
622C2×C4C22×C22⋊D1711yescharacteristic
623C8—22no—
624C8—342no—
625C8—342no—
626C8C4×C22⋊D1714yes—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—342no—
635C22—342no—
636C22—342no—
637C22—681no—
638C22—681no—
639C22—681no—
640C22—681no—
641C22C2×C4⋊C4×D1711yescharacteristic
642C22D4×C22⋊D1711yescharacteristic
643C22#1176346011yescharacteristic
644C22#1176360511yescharacteristic
645C22#1176360811yescharacteristic
646C22#1176361311yescharacteristic
647C22#1176362511yescharacteristic
648C4—22no—
649C4—42no—
650C4—42no—
651C4—81no—
652C4—81no—
653C4—342no—
654C4—342no—
655C4—342no—
656C4—1361no—
657C4—1361no—
658C4—1361no—
659C4—1361no—
660C4C2×C4×C22⋊D1711yescharacteristic
661C4#1176329011yescharacteristic
662C4Q8×C22⋊D1711yescharacteristic
663C4#1176351411yescharacteristic
664C2—42no—
665C2—682no—
666C2C4⋊C4×C22⋊D1711yescharacteristic
667C2#1453737111yescharacteristic
668C2#1453738311yescharacteristic
669C2#1454038911yescharacteristic
670C2#1454043511yescharacteristic
671C2#1454135511yescharacteristic
672C2#1454138611yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C17×#428C2
socleC23×C17C22×D4
prime radicalC22×C22⋊D17D4
cosocleC22×C4×C17C24
hypercenterC2×#428D17
hypocenterC17#22664
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C2×#11762850C2semidirect
5. #14529553C2semidirect
6. #14529549C2semidirect
7. #14528983C2semidirect
8. #14528982C2semidirect
9. #14528972C2semidirect
10. #14528970C2semidirect
11. #14528945C2semidirect
12. #14528817C2semidirect
13. #14528814C2semidirect
14. #107×Dic17C2semidirect
15. C2×C17×#428C2semidirect
20. #11762885C4semidirect
69. C2×M16×C17C2×C4semidirect
106. C2×#63218C2×C4semidirect
112. C17×#107D4semidirect
125. C22⋊#3622C2×C4semidirect
126. #10587491C2×C4semidirect
129. #10587485C2×C4semidirect
130. C8×Dic17C2×C4semidirect
293. C2×#428D17semidirect
408. C2×M16C4×D17semidirect
452. #107C22⋊D17semidirect
471. C17#22664semidirect

Maximal subgroups

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

subgroupindex
C2×#117628852
C2×#117628502
#145317142
#145317092
#145295532
#145295492
#145289832
#145289822
#145289722
#145289702
#145289452
#145288172
#145288142
#107×Dic172
C2×C17×#4282
#2266417