Finite Group Databasebeta

← #17052549

Subgroups — #17052549

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×#11762884C211yesmaximal, characteristic
2C2×#11762852C211yesmaximal, characteristic, complemented
3C2×#11762848C211yesmaximal, characteristic, complemented
4#14531715C211yesmaximal, characteristic
5#14531714C211yesmaximal, characteristic
6#14529555C211yesmaximal, characteristic, complemented
7#14529553C211yesmaximal, characteristic, complemented
8#14528980C211yesmaximal, characteristic, complemented
9#14528974C211yesmaximal, characteristic, complemented
10#14528973C211yesmaximal, characteristic, complemented
11#14528947C211yesmaximal, characteristic, complemented
12#14528816C211yesmaximal, characteristic, complemented
13#14528815C211yesmaximal, characteristic, complemented
14#14528676C211yesmaximal, characteristic, complemented
15C17×#1854C211yesmaximal, characteristic, complemented
16C22×#63223C2211yescharacteristic
17C22×#63218C2211yescharacteristic
18C2×C4×C22⋊D17C2211yescharacteristic
19C2×#10587578C2211yescharacteristic
20C22×C4⋊Dic17C2211yescharacteristic
21C2×C4⋊C4×D17C2211yescharacteristic
22#11762884C418yescomplemented
23#11762852—24no—
24#11762848—24no—
25C2×#10587489C2211yescharacteristic
26C2×#10587489C2211yescharacteristic
27C2×#10587478C2211yescharacteristic
28C2×#10587478C2211yescharacteristic
29C2×#10587478C2211yescharacteristic
30C2×#10587479C2211yescharacteristic
31C2×#10587479C2211yescharacteristic
32C2×#10587479C2211yescharacteristic
33C2×#10587485C2211yescharacteristic
34C2×#10587491C2211yescharacteristic
35C2×C8×Dic17C2211yescharacteristic
36#11762703—21no—
37#11762701—21no—
38#11762696—21no—
39#11762693—21no—
40#11762686—21no—
41#11762685—21no—
42#11762674C2211yescharacteristic
43#11762674C2211yescharacteristic
44#11762669C2211yescharacteristic
45#11762666C2211yescharacteristic
46#11762666C2211yescharacteristic
47C4⋊C4×Dic17C2211yescharacteristic
48C4⋊C4×Dic17C2211yescharacteristic
49C22⋊C4×Dic17—21no—
50C4×#63227—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
64C23×Dic17—21no—
65C22×C4×D17—21no—
66C22×C4×D17C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C17×#117—22no—
70C17×#117—41no—
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C22⋊C4×C17—21no—
76C2×C22⋊C4×C17—21no—
77C2×C42×C17—21no—
78C2×C22⋊Dic17—21no—
79C2×C22⋊Dic17D411yescharacteristic
80C4×C22⋊D17C2×C414yes—
81C4×C22⋊D17Q814yes—
82C2×#63227—21no—
83C2×#63227—21no—
84C2×#63227—21no—
85C2×#63227D411yescharacteristic
86C2×C4⋊Dic17—21no—
87C2×C4⋊Dic17—22no—
88C2×C4⋊Dic17—41no—
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×#63225—21no—
94C2×#63225D411yescharacteristic
95C2×C4×Dic17—21no—
96C2×C4×Dic17—21no—
97C2×C4×Dic17—21no—
98C2×C4×Dic17C2311yescharacteristic
99C2×#63223—22no—
100C2×#63223—22no—
101C2×#63223C2×C414yescomplemented
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—41no—
112C17×#106—22no—
113C17×C22⋊C8C2×C414yescomplemented
114C17×#95—21no—
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#10587491C2×C414yescomplemented
125#10587489—22no—
126#10587489—41no—
127#10587485C2×C414yescomplemented
128C8×Dic17C2×C414yescomplemented
129#10587479—22no—
130#10587479—22no—
131#10587479—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—41no—
150C2×C22⋊D17—41no—
151C2×C22⋊D17C4⋊C414yes—
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—21no—
155C22×Dic17—21no—
156C22×Dic17—42no—
157C22×Dic17—42no—
158C22×Dic17—81no—
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—
171C2×C8×C17—21no—
172C2×C8×C17—21no—
173C2×C8×C17—21no—
174C2×C8×C17—21no—
175C2×C8×C17C22×C412yes—
176C2×C8×C17C22×C412yes—
177C4⋊C4×C17—21no—
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×C17C2×D412yes—
185C22⋊C4×C17—42no—
186C22⋊C4×C17—42no—
187C42×C17—42no—
188C22⋊Dic17—81no—
189C22⋊Dic17C4⋊C414yes—
190#63227—42no—
191#63227—42no—
192#63227—81no—
193#63227C4⋊C414yes—
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199C4⋊Dic17—21no—
200C4⋊Dic17—21no—
201C4⋊Dic17—42no—
202C4⋊Dic17C2×D412yes—
203#63225—42no—
204#63225C4⋊C414yes—
205C4×Dic17—42no—
206C4×Dic17—81no—
207C4×Dic17—81no—
208C4×Dic17C22×C412yes—
209C4×Dic17C2×Q812yes—
210#63223—28no—
211C2×#3622—21no—
212C2×#3622—21no—
213C2×#3622C22×C412yes—
214C2×#3622C22×C412yes—
215C2×#3622C2×D412yes—
216C2×#3622#4112yes—
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—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—44no—
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—21no—
263C2×Dic17—21no—
264C2×Dic17—21no—
265C2×Dic17—21no—
266C2×Dic17—22no—
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—84no—
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×C17—22no—
289#3622—22no—
290#3622C4×D414yes—
291C2×#436—171no—
292C2×#428—171no—
293C2×#428—171no—
294#2352—171no—
295#2352—171no—
296#1949—171no—
297#1949—171no—
298#1854—171no—
299#1854D1711yescharacteristic, complemented
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—81no—
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—42no—
340Dic17—81no—
341Dic17—81no—
342Dic17—81no—
343C22×M16—171no—
344C22×M16—171no—
345C2×C4×D4—171no—
346C2×#117—171no—
347C2×#117C2×D1711yescharacteristic
348C22×C4⋊C4—171no—
349C22×C4⋊C4—171no—
350#436—178no—
351#428—344no—
352#428—344no—
353C2×#107—171no—
354C2×#107—171no—
355C2×#107—171no—
356C2×#107—171no—
357C2×#107—171no—
358C2×#107C2×D1711yescharacteristic
359C2×#106—171no—
360C2×#106—171no—
361C2×#106—171no—
362C2×#106C2×D1711yescharacteristic
363C2×C4⋊C8—171no—
364C2×C22⋊C8—171no—
365C2×C22⋊C8C2×D1711yescharacteristic
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#380C2×D1711yescharacteristic
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#340C2×D1711yescharacteristic
388#340C2×D1711yescharacteristic
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—171no—
400C23×C4—341no—
401C23×C4—341no—
402C23×C4C22×D1711yescharacteristic
403C2×M16—174no—
404C2×M16—174no—
405C2×M16—342no—
406C2×M16—342no—
407C2×M16—342no—
408C2×M16—342no—
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—171no—
420C2×C4⋊C4—171no—
421C2×C4⋊C4—171no—
422C2×C4⋊C4—171no—
423C2×C4⋊C4—171no—
424C2×C4⋊C4—342no—
425C2×C4⋊C4—342no—
426C2×C4⋊C4—341no—
427C2×C4⋊C4—341no—
428C2×C4⋊C4—681no—
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—21no—
436C2×C22⋊C4—171no—
437C2×C22⋊C4—171no—
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—41no—
448#107—174no—
449#107—342no—
450#107—342no—
451#107—342no—
452#107—681no—
453#106—22no—
454#106—342no—
455#106—342no—
456#106—342no—
457C4⋊C8—174no—
458C22⋊C8—174no—
459C22⋊C8C4×D1714yescomplemented
460C4×C8—174no—
461#95—21no—
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—171no—
486C22×C4—171no—
487C22×C4—171no—
488C22×C4—172no—
489C22×C4—172no—
490C22×C4—341no—
491C22×C4—341no—
492C22×C4—341no—
493C22×C4—341no—
494C22×C4—341no—
495C22×C4—341no—
496C22×C4—341no—
497C22×C4—341no—
498C22×C4—682no—
499C22×C4—682no—
500C22×C4—682no—
501C22×C4—682no—
502C22×C4—1361no—
503C22×C4—1361no—
504C22×C4C2×C4×D1712yes—
505C22×C4D4×D1711yescharacteristic
506C22×C4Q8×D1712yes—
507C22×C4C23×D1711yescharacteristic
508M16—348no—
509M16—348no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—172no—
515C2×C8—172no—
516C2×C8—172no—
517C2×C8—172no—
518C2×C8—341no—
519C2×C8—341no—
520C2×C8C2×C4×D1712yes—
521C2×C8C2×C4×D1712yes—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—174no—
530C4⋊C4—172no—
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⋊C4—682no—
540C4⋊C4D4×D1712yes—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—174no—
544C22⋊C4—174no—
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—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—41no—
588C2×C4—41no—
589C2×C4—172no—
590C2×C4—172no—
591C2×C4—172no—
592C2×C4—341no—
593C2×C4—341no—
594C2×C4—341no—
595C2×C4—341no—
596C2×C4—341no—
597C2×C4—341no—
598C2×C4—342no—
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—1364no—
615C2×C4—1364no—
616C2×C4C4⋊C4×D1712yes—
617C2×C4C22×C4×D1711yescharacteristic
618C2×C4C2×D4×D1711yescharacteristic
619C2×C4#1058769111yescharacteristic
620C2×C4C2×Q8×D1711yescharacteristic
621C2×C4#41×D1711yescharacteristic
622C2×C4#1058769711yescharacteristic
623C8—22no—
624C8—22no—
625C8—174no—
626C8—342no—
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
642C22#1176344111yescharacteristic
643C22#1176345711yescharacteristic
644C22#1176360511yescharacteristic
645C22#1176360811yescharacteristic
646C22#1176361111yescharacteristic
647C22#1176362311yescharacteristic
648C4—22no—
649C4—42no—
650C4—81no—
651C4—81no—
652C4—81no—
653C4—342no—
654C4—342no—
655C4—342no—
656C4—682no—
657C4—1361no—
658C4—1361no—
659C4—1361no—
660C4C4×D4×D1711yescharacteristic
661C4#1176328811yescharacteristic
662C4#1176347211yescharacteristic
663C4#1176351311yescharacteristic
664C2—42no—
665C2—682no—
666C2#1453168211yescharacteristic
667C2#1453737111yescharacteristic
668C2#1453738111yescharacteristic
669C2#1454038811yescharacteristic
670C2#1454043411yescharacteristic
671C2#1454135211yescharacteristic
672C2#1454138311yescharacteristic
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#22664
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×#11762852C2semidirect
3. C2×#11762848C2semidirect
6. #14529555C2semidirect
7. #14529553C2semidirect
8. #14528980C2semidirect
9. #14528974C2semidirect
10. #14528973C2semidirect
11. #14528947C2semidirect
12. #14528816C2semidirect
13. #14528815C2semidirect
14. #14528676C2semidirect
15. C17×#1854C2semidirect
22. #11762884C4semidirect
101. C2×#63223C2×C4semidirect
106. C2×#63218C2×C4semidirect
113. C17×C22⋊C8C2×C4semidirect
124. #10587491C2×C4semidirect
127. #10587485C2×C4semidirect
128. C8×Dic17C2×C4semidirect
134. #10587478D4semidirect
299. #1854D17semidirect
459. C22⋊C8C4×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×#117628842
C2×#117628522
C2×#117628482
#145317152
#145317142
#145295552
#145295532
#145289802
#145289742
#145289732
#145289472
#145288162
#145288152
#145286762
C17×#18542
#2266417