Finite Group Databasebeta

← #17052722

Subgroups — #17052722

Order 4352.

1030 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#63230C211yesmaximal, characteristic
2C2×#11762894C211yesmaximal, characteristic
3C2×#11762894C211yesmaximal, characteristic
4Q8×#63225C211yesmaximal, characteristic, complemented
5#14531685C211yesmaximal, characteristic
6#14529511C211yesmaximal, characteristic, complemented
7#14529509C211yesmaximal, characteristic, complemented
8#14529054C211yesmaximal, characteristic, complemented
9#14529048C211yesmaximal, characteristic
10#14529042C211yesmaximal, characteristic
11#14528939C211yesmaximal, characteristic, complemented
12#14528787C211yesmaximal, characteristic, complemented
13#14528783C211yesmaximal, characteristic
14#14528645C211yesmaximal, characteristic, complemented
15C17×#1865C211yesmaximal, characteristic, complemented
16C22×#63230C2211yescharacteristic
17C2×Q8×Dic17C2211yescharacteristic
18C2×Q8×Dic17C2211yescharacteristic
19C2×C4×D68C2211yescharacteristic
20C2×#10587579C2211yescharacteristic
21C2×#10587579C2211yescharacteristic
22C2×C4⋊C4×D17—21no—
23C4×#63230—24no—
24#11762894—24no—
25#11762894C418yescomplemented
26C2×#10587488C2211yescharacteristic
27C2×#10587479C2211yescharacteristic
28C2×#10587479C2211yescharacteristic
29C2×#10587479C2211yescharacteristic
30C2×#10587485C2211yescharacteristic
31C2×#10587506C2211yescharacteristic
32C2×#10587492C2211yescharacteristic
33C2×#10587492C2211yescharacteristic
34C2×#10587480C2211yescharacteristic
35C2×C4×#3622C2211yescharacteristic
36C2×C8×Dic17C2211yescharacteristic
37C2×C8×Dic17C2211yescharacteristic
38#11762708C2211yescharacteristic
39#11762707—21no—
40#11762707C2211yescharacteristic
41#11762705—21no—
42#11762705C2211yescharacteristic
43#11762683—21no—
44#11762680—21no—
45#11762678—21no—
46#11762678C2211yescharacteristic
47#11762678C2211yescharacteristic
48#11762668—21no—
49C4×#63225—21no—
50C4×#63225C2211yescharacteristic
51#11762572C2211yescharacteristic
52#11762571C2211yescharacteristic
53#11762571C2211yescharacteristic
54#11762563C2211yescharacteristic
55C2×C4×Q8×C17C2211yescharacteristic
56C2×C17×#106C2211yescharacteristic
57C2×C17×#103C2211yescharacteristic
58C2×C17×#103C2211yescharacteristic
59C17×#389C2211yescharacteristic
60C17×#340C2211yescharacteristic
61C17×#336C2211yescharacteristic
62C22×Q8×C17C2311yescharacteristic
63C22×D68C2311yescharacteristic
64C22×C4×D17—21no—
65C22×C4×D17—21no—
66C22×C4×D17—21no—
67C22×C8×C17C2311yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C4×Q8×C17—22no—
70C4×Q8×C17—41no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C42×C17—21no—
77C2×C42×C17C2311yescharacteristic
78Q8×Dic17—22no—
79Q8×Dic17—22no—
80Q8×Dic17—41no—
81Q8×Dic17C2×C414yescomplemented
82C2×#63230C2×C414yes—
83C2×#63230D414yes—
84C2×#63230Q814yes—
85C2×#63227—21no—
86C2×#63227—21no—
87C2×#63227—21no—
88C2×C4⋊Dic17—21no—
89C2×C4⋊Dic17—21no—
90C2×C4⋊Dic17—21no—
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4⋊Dic17C2311yescharacteristic
94C2×#63225—21no—
95C2×#63225—21no—
96C2×#63225—21no—
97C2×#63225—21no—
98C2×#63225—21no—
99C2×#63225—21no—
100C2×#63225D411yescharacteristic
101C2×#63225D411yescharacteristic
102C2×C4×Dic17—21no—
103C2×C4×Dic17—21no—
104C2×C4×Dic17C2311yescharacteristic
105C2×C4×Dic17C2311yescharacteristic
106C22×#3622C2311yescharacteristic
107C22×#3622C2311yescharacteristic
108#10587579—22no—
109#10587579—22no—
110#10587579—41no—
111#10587579C2×C414yes—
112C4⋊C4×D17—44no—
113C4×D68—22no—
114C4×D68—41no—
115C17×#106—41no—
116C17×#103—22no—
117C17×#103C2×C414yescomplemented
118C17×#95—21no—
119#10587506—22no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502D411yescharacteristic
124#10587502D411yescharacteristic
125#10587492—22no—
126#10587492C2×C414yes—
127#10587488—41no—
128#10587485—22no—
129C8×Dic17—22no—
130C8×Dic17C2×C414yescomplemented
131#10587480—22no—
132#10587479—22no—
133#10587479—22no—
134#10587479C2×C414yescomplemented
135C4×#3622—22no—
136C23×D17—21no—
137C2×Q8×C17—21no—
138C2×Q8×C17—21no—
139C2×Q8×C17—21no—
140C2×Q8×C17C22×C412yes—
141C2×Q8×C17C2×D412yes—
142C2×Q8×C17C2×Q812yes—
143C22×C4×C17—21no—
144C22×C4×C17—21no—
145C22×C4×C17—21no—
146C22×C4×C17—21no—
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2411yescharacteristic
150C22×Dic17—21no—
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17C2×D411yescharacteristic
154C22×Dic17C2×D411yescharacteristic
155C22×Dic17C2×D411yescharacteristic
156C22×Dic17C2×D411yescharacteristic
157C2×D68—21no—
158C2×D68—21no—
159C2×D68—21no—
160C2×D68C22×C412yes—
161C2×D68C2×D412yes—
162C2×D68C2×Q812yes—
163C2×C4×D17—24no—
164C2×C4×D17—42no—
165C2×C4×D17—42no—
166C2×C4×D17—42no—
167C2×C4×D17—42no—
168C2×C4×D17—42no—
169C2×C4×D17—42no—
170C2×C4×D17—81no—
171C2×C4×D17—81no—
172C2×C8×C17—21no—
173C2×C8×C17—21no—
174C2×C8×C17—21no—
175C2×C8×C17—21no—
176C2×C8×C17—21no—
177C2×C8×C17C22×C412yes—
178C4⋊C4×C17—21no—
179C4⋊C4×C17—21no—
180C4⋊C4×C17—21no—
181C4⋊C4×C17—21no—
182C4⋊C4×C17—21no—
183C4⋊C4×C17—42no—
184C4⋊C4×C17—42no—
185C4⋊C4×C17C22×C412yes—
186C42×C17—21no—
187C42×C17—21no—
188C42×C17—42no—
189#63230C4⋊C4116yescomplemented
190#63227—24no—
191#63227—42no—
192#63227—42no—
193C4⋊Dic17—21no—
194C4⋊Dic17—24no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199C4⋊Dic17—42no—
200C4⋊Dic17—42no—
201C4⋊Dic17C22×C412yes—
202#63225—22no—
203#63225—42no—
204#63225—42no—
205#63225—81no—
206#63225—81no—
207#63225—81no—
208#63225—81no—
209#63225SD1614yescomplemented
210C4×Dic17—21no—
211C4×Dic17—21no—
212C4×Dic17—24no—
213C4×Dic17—42no—
214C4×Dic17C22×C412yes—
215C4×Dic17#4112yes—
216C2×#3622—21no—
217C2×#3622—21no—
218C2×#3622—21no—
219C2×#3622C22×C412yes—
220C2×#3622C2×D412yes—
221C2×#3622C2×Q812yes—
222C4⋊C4×SD16—171nomaximal, Sylow
223C23×C17C22×D411yescharacteristic
224C22×D17—22no—
225C22×D17—22no—
226C22×D17—22no—
227C22×D17—41no—
228C22×D17—41no—
229C22×D17—41no—
230C22×D17—41no—
231Q8×C17—22no—
232Q8×C17—22no—
233Q8×C17—22no—
234Q8×C17C2×C4⋊C414yescomplemented
235C2×C4×C17—22no—
236C2×C4×C17—21no—
237C2×C4×C17—22no—
238C2×C4×C17—22no—
239C2×C4×C17—21no—
240C2×C4×C17—22no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—21no—
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×SD1612yes—
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
259C2×Dic17—21no—
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—21no—
263C2×Dic17—22no—
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×Dic17C4×D412yes—
276C2×Dic17C4×D412yes—
277C2×Dic17C2×SD1612yes—
278C2×Dic17C2×SD1612yes—
279C2×Dic17#13512yes—
280D68—22no—
281D68—22no—
282D68—22no—
283D68C2×C4⋊C414yes—
284C4×D17—48no—
285C4×D17—84no—
286C4×D17—84no—
287C8×C17—22no—
288C8×C17—41no—
289#3622—41no—
290#3622C2×C4⋊C414yescomplemented
291C2×C4×SD16—171no—
292C2×C4×SD16—171no—
293C2×C4×SD16—171no—
294Q8×C4⋊C4—171no—
295D4×C4⋊C4—171no—
296#1941—171no—
297#1941—171no—
298#1865—171no—
299#1865D1711yescharacteristic, complemented
300#1864—171no—
301#1864—171no—
302#1844—171no—
303#1803—171no—
304#1802—171no—
305C8×C4⋊C4—171no—
306C22×C17C2×C4×D411yescharacteristic
307C22×C17D4211yescharacteristic
308C22×C17D4×Q811yescharacteristic
309C22×C17C22×SD1611yescharacteristic
310C22×C17C2×#13511yescharacteristic
311C22×C17#57711yescharacteristic
312C22×C17#57711yescharacteristic
313C2×D17—24no—
314C2×D17—42no—
315C2×D17—42no—
316C2×D17—42no—
317C2×D17—42no—
318C2×D17—42no—
319C2×D17—42no—
320C4×C17—24no—
321C4×C17—22no—
322C4×C17—42no—
323C4×C17—41no—
324C4×C17—81no—
325C4×C17—81no—
326C4×C17C22×C4⋊C411yescharacteristic
327C4×C17#52011yescharacteristic
328C4×C17#54711yescharacteristic
329C4×C17#55411yescharacteristic
330Dic17—22no—
331Dic17—22no—
332Dic17—42no—
333Dic17—41no—
334Dic17—81no—
335Dic17—81no—
336Dic17C4×SD1614yescomplemented
337C22×SD16—171no—
338C2×C4×Q8—171no—
339C2×C4×Q8—171no—
340C2×C4×Q8C2×D1711yescharacteristic
341C2×C4×D4—171no—
342C2×C4×D4—171no—
343C2×C4×D4—171no—
344C22×C4⋊C4—341no—
345C4×SD16—178no—
346C4×SD16—344no—
347C4×SD16—344no—
348C2×#106—171no—
349C2×#106—171no—
350C2×#106—171no—
351C2×#106—171no—
352C2×#106C2×D1711yescharacteristic
353C2×C4⋊C8—171no—
354C2×#103—171no—
355C2×#103C2×D1711yescharacteristic
356C2×#103C2×D1711yescharacteristic
357C2×#102—171no—
358C2×#102—171no—
359C2×#102—171no—
360C2×C4×C8—171no—
361C2×C4×C8—171no—
362C2×C4×C8—171no—
363#389—171no—
364#389—171no—
365#389—171no—
366#389—341no—
367#389—341no—
368#389C2×D1711yescharacteristic
369#385—341no—
370#385—341no—
371#384—171no—
372#384—171no—
373#384—341no—
374#380—341no—
375C4×C4⋊C4—171no—
376C4×C4⋊C4—341no—
377#340—171no—
378#340—171no—
379#340—171no—
380#340C2×D1711yescharacteristic
381#336—171no—
382#336C2×D1711yescharacteristic
383C2×C17D4×C4⋊C411yescharacteristic
384C2×C17C2×C4×SD1611yescharacteristic
385C2×C17#295211yescharacteristic
386C2×C17D4×SD1611yescharacteristic
387C2×C17#328411yescharacteristic
388C2×C17Q8×SD1611yescharacteristic
389C2×C17#338111yescharacteristic
390D17—44no—
391C22×Q8C22×D1711yescharacteristic
392C22×D4—171no—
393C23×C4—341no—
394C23×C4—341no—
395C23×C4—341no—
396C2×SD16—174no—
397C2×SD16—174no—
398C2×SD16—174no—
399C22×C8—171no—
400C22×C8—171no—
401C22×C8C22×D1711yescharacteristic
402C22×C8C22×D1711yescharacteristic
403C4×Q8—22no—
404C4×Q8—41no—
405C4×Q8—174no—
406C4×Q8—342no—
407C4×Q8—342no—
408C4×Q8—681no—
409C4×D4—174no—
410C4×D4—342no—
411C4×D4—342no—
412C4×D4—342no—
413C4×D4—681no—
414C4×D4—681no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—171no—
418C2×C4⋊C4—171no—
419C2×C4⋊C4—171no—
420C2×C4⋊C4—171no—
421C2×C4⋊C4—171no—
422C2×C4⋊C4—341no—
423C2×C4⋊C4—341no—
424C2×C4⋊C4—341no—
425C2×C4⋊C4—341no—
426C2×C4⋊C4—341no—
427C2×C4⋊C4—341no—
428C2×C4⋊C4—341no—
429C2×C4⋊C4—341no—
430C2×C4⋊C4—341no—
431C2×C4⋊C4—684no—
432C2×C4⋊C4C22×D1711yescharacteristic
433C2×C4⋊C4C22×D1711yescharacteristic
434C2×C4⋊C4C22×D1711yescharacteristic
435C2×C22⋊C4—341no—
436C2×C22⋊C4—341no—
437C2×C22⋊C4—341no—
438C2×C42—21no—
439C2×C42—171no—
440C2×C42—171no—
441C2×C42—341no—
442C2×C42—341no—
443C2×C42C22×D1711yescharacteristic
444#106—41no—
445#106—174no—
446#106—342no—
447#106—342no—
448#106—681no—
449C4⋊C8—342no—
450#103—22no—
451#103—342no—
452#103C4×D1714yescomplemented
453#102—174no—
454#102—342no—
455#102—342no—
456C4×C8—174no—
457C4×C8—342no—
458C4×C8—342no—
459#95—21no—
460#95—171no—
461#95—171no—
462#95—341no—
463#95—341no—
464#95—341no—
465C17C4⋊C4×SD1611yescharacteristic, complemented, Sylow
466C24—341no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8C2×C4×D1712yes—
471C2×Q8D4×D1712yes—
472C2×Q8Q8×D1712yes—
473C2×D4—172no—
474C2×D4—172no—
475C2×D4—172no—
476C2×D4—341no—
477C2×D4—341no—
478C2×D4—341no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—171no—
484C22×C4—171no—
485C22×C4—171no—
486C22×C4—171no—
487C22×C4—344no—
488C22×C4—341no—
489C22×C4—341no—
490C22×C4—341no—
491C22×C4—682no—
492C22×C4—682no—
493C22×C4—682no—
494C22×C4—682no—
495C22×C4—682no—
496C22×C4—682no—
497C22×C4—1361no—
498C22×C4—1361no—
499C22×C4D4×D1711yescharacteristic
500C22×C4D4×D1711yescharacteristic
501C22×C4C23×D1711yescharacteristic
502SD16—1716no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—172no—
509C2×C8—172no—
510C2×C8—172no—
511C2×C8—341no—
512C2×C8—341no—
513C2×C8—341no—
514C2×C8C2×C4×D1712yes—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—174no—
523C4⋊C4—172no—
524C4⋊C4—342no—
525C4⋊C4—344no—
526C4⋊C4—341no—
527C4⋊C4—341no—
528C4⋊C4—341no—
529C4⋊C4—341no—
530C4⋊C4—341no—
531C4⋊C4—682no—
532C4⋊C4—682no—
533C4⋊C4—682no—
534C4⋊C4—682no—
535C4⋊C4—1361no—
536C4⋊C4—1361no—
537C4⋊C4—1361no—
538C4⋊C4—1361no—
539C4⋊C4C2×C4×D1712yes—
540C22⋊C4—344no—
541C22⋊C4—682no—
542C22⋊C4—682no—
543C42—21no—
544C42—21no—
545C42—42no—
546C42—172no—
547C42—172no—
548C42—344no—
549C42—341no—
550C42—341no—
551C42—682no—
552C23—342no—
553C23—342no—
554C23—342no—
555C23—681no—
556C23—681no—
557C23—681no—
558C23—681no—
559C23C2×D4×D1711yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—22no—
563Q8C4⋊C4×D1714yescomplemented
564D4—174no—
565D4—342no—
566D4—342no—
567D4—342no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
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—172no—
586C2×C4—172no—
587C2×C4—172no—
588C2×C4—172no—
589C2×C4—172no—
590C2×C4—341no—
591C2×C4—341no—
592C2×C4—341no—
593C2×C4—341no—
594C2×C4—342no—
595C2×C4—341no—
596C2×C4—341no—
597C2×C4—341no—
598C2×C4—688no—
599C2×C4—681no—
600C2×C4—681no—
601C2×C4—681no—
602C2×C4—681no—
603C2×C4—681no—
604C2×C4—681no—
605C2×C4—681no—
606C2×C4—681no—
607C2×C4—1364no—
608C2×C4—1364no—
609C2×C4SD16×D1712yes—
610C2×C4C22×C4×D1711yescharacteristic
611C2×C4C2×D4×D1711yescharacteristic
612C2×C4#1058769111yescharacteristic
613C2×C4C2×Q8×D1711yescharacteristic
614C2×C4#41×D1711yescharacteristic
615C2×C4#1058769711yescharacteristic
616C8—22no—
617C8—41no—
618C8—174no—
619C8—681no—
620C22—344no—
621C22—682no—
622C22—682no—
623C22—682no—
624C22—682no—
625C22—682no—
626C22—682no—
627C22C4×D4×D1711yescharacteristic
628C22#1176344111yescharacteristic
629C22#1176347211yescharacteristic
630C22C2×SD16×D1711yescharacteristic
631C22#135×D1711yescharacteristic
632C22#1176361811yescharacteristic
633C22#1176361811yescharacteristic
634C4—24no—
635C4—22no—
636C4—42no—
637C4—41no—
638C4—81no—
639C4—81no—
640C4—174no—
641C4—342no—
642C4—342no—
643C4—682no—
644C4—681no—
645C4—1361no—
646C4—1361no—
647C4C2×C4⋊C4×D1711yescharacteristic
648C4#1176328811yescharacteristic
649C4#1176345711yescharacteristic
650C4#1176351311yescharacteristic
651C2—684no—
652C2#1453168211yescharacteristic
653C2C4×SD16×D1711yescharacteristic
654C2#1453766411yescharacteristic
655C2#1454047311yescharacteristic
656C2#1454050711yescharacteristic
657C2#1454164311yescharacteristic
658C2#1454166211yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1865C2
socleC23×C17C22×D4
prime radicalC22×D68C23
cosocleC22×C4×C17C24
hypercenter#1865D17
hypocenterC17C4⋊C4×SD16
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1865

Complements

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

NH ≅ G/Ntype
4. Q8×#63225C2semidirect
6. #14529511C2semidirect
7. #14529509C2semidirect
8. #14529054C2semidirect
11. #14528939C2semidirect
12. #14528787C2semidirect
14. #14528645C2semidirect
15. C17×#1865C2semidirect
25. #11762894C4semidirect
81. Q8×Dic17C2×C4semidirect
117. C17×#103C2×C4semidirect
130. C8×Dic17C2×C4semidirect
134. #10587479C2×C4semidirect
189. #63230C4⋊C4semidirect
209. #63225SD16semidirect
234. Q8×C17C2×C4⋊C4semidirect
290. #3622C2×C4⋊C4semidirect
299. #1865D17semidirect
336. Dic17C4×SD16semidirect
452. #103C4×D17semidirect
465. C17C4⋊C4×SD16semidirect
563. Q8C4⋊C4×D17semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#632302
C2×#117628942
Q8×#632252
#145316852
#145295112
#145295092
#145290542
#145290482
#145290422
#145289392
#145287872
#145287832
#145286452
C17×#18652
C4⋊C4×SD1617