Finite Group Databasebeta

← #17052671

Subgroups — #17052671

Order 4352.

1010 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762911C211yesmaximal, characteristic
2C2×#11762908C211yesmaximal, characteristic
3C2×#11762896C211yesmaximal, characteristic
4#14531701C211yesmaximal, characteristic, complemented
5#14531684C211yesmaximal, characteristic
6#14529520C211yesmaximal, characteristic, complemented
7#14529515C211yesmaximal, characteristic, complemented
8#14529049C211yesmaximal, characteristic, complemented
9#14529042C211yesmaximal, characteristic
10#14529035C211yesmaximal, characteristic, complemented
11#14529028C211yesmaximal, characteristic
12#14528791C211yesmaximal, characteristic
13#14528786C211yesmaximal, characteristic, complemented
14#14528656C211yesmaximal, characteristic, complemented
15C17×#1847C211yesmaximal, characteristic, complemented
16C22×#63219C2211yescharacteristic
17C2×C4×Dic34C2211yescharacteristic
18C2×#10587573C2211yescharacteristic
19C2×#10587573C2211yescharacteristic
20C2×C4×D68C2211yescharacteristic
21C2×#10587579C2211yescharacteristic
22C2×#10587579C2211yescharacteristic
23C2×C4⋊C4×D17—21no—
24#11762911C418yescomplemented
25#11762908—24no—
26#11762896—24no—
27C2×#10587489C2211yescharacteristic
28C2×C4⋊#3622C2211yescharacteristic
29C2×#10587487C2211yescharacteristic
30C2×#10587481C2211yescharacteristic
31C2×#10587481C2211yescharacteristic
32C2×#10587492C2211yescharacteristic
33C2×#10587480C2211yescharacteristic
34C2×#10587480C2211yescharacteristic
35C2×#10587486C2211yescharacteristic
36C2×C8×Dic17C2211yescharacteristic
37#11762706—21no—
38#11762706—21no—
39#11762705C2211yescharacteristic
40#11762705C2211yescharacteristic
41#11762705C2211yescharacteristic
42#11762705C2211yescharacteristic
43#11762680—21no—
44#11762680—21no—
45#11762673—21no—
46#11762666—21no—
47C4×C4⋊Dic17C2211yescharacteristic
48#11762660—21no—
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762571C2211yescharacteristic
53#11762563C2211yescharacteristic
54#11762563C2211yescharacteristic
55C2×C17×#107C2211yescharacteristic
56C2×C17×#107C2211yescharacteristic
57C2×C17×#106C2211yescharacteristic
58C2×C17×#106C2211yescharacteristic
59C2×C17×#97C2211yescharacteristic
60C17×#384C2211yescharacteristic
61C17×#382C2211yescharacteristic
62C22×D68C2311yescharacteristic
63C22×C4×D17—21no—
64C22×C4×D17—21no—
65C22×C4×D17—21no—
66C22×Dic34C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C42×C17C2311yescharacteristic
76C2×#63227—21no—
77C2×#63227—21no—
78C2×#63227—21no—
79C2×C4⋊Dic17D411yescharacteristic
80C2×C4⋊Dic17D411yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×#63225—21no—
84C2×#63225—21no—
85C2×#63225—21no—
86C2×#63225—21no—
87C2×#63225—21no—
88C2×#63225—21no—
89C2×#63225—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C2×C4×Dic17—21no—
93C2×C4×Dic17C2311yescharacteristic
94C2×C4×Dic17C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97C2×#63219C2×C414yes—
98C2×#63219D414yes—
99C2×#63219Q814yes—
100#10587579—22no—
101#10587579—22no—
102#10587579—41no—
103#10587579—41no—
104C4⋊C4×D17—44no—
105#10587573—22no—
106#10587573—22no—
107#10587573—41no—
108#10587573—41no—
109C4×D68—22no—
110C4×D68C2×C414yes—
111C4×Dic34—22no—
112C4×Dic34C2×C414yescomplemented
113C17×#107—22no—
114C17×#107—41no—
115C17×#106—22no—
116C17×#106—41no—
117C17×#97C2×C414yescomplemented
118C17×#95—21no—
119C17×#95—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587502D411yescharacteristic
125#10587502D411yescharacteristic
126#10587492C2×C414yes—
127#10587489C2×C414yescomplemented
128#10587487C2×C414yescomplemented
129#10587486—22no—
130C8×Dic17—22no—
131#10587481—22no—
132#10587481—22no—
133#10587480—22no—
134#10587480—22no—
135C4⋊#3622—22no—
136C23×D17—21no—
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2411yescharacteristic
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17C2×D411yescharacteristic
148C22×Dic17C2×D411yescharacteristic
149C22×Dic17C2×D411yescharacteristic
150C22×Dic17C2×D411yescharacteristic
151C2×D68—21no—
152C2×D68—21no—
153C2×D68—21no—
154C2×D68C22×C412yes—
155C2×D68C2×D412yes—
156C2×D68C2×Q812yes—
157C2×C4×D17—24no—
158C2×C4×D17—42no—
159C2×C4×D17—42no—
160C2×C4×D17—42no—
161C2×C4×D17—42no—
162C2×C4×D17—42no—
163C2×C4×D17—42no—
164C2×C4×D17—81no—
165C2×C4×D17—81no—
166C2×Dic34—21no—
167C2×Dic34—21no—
168C2×Dic34—21no—
169C2×Dic34C22×C412yes—
170C2×Dic34C2×D412yes—
171C2×Dic34C2×Q812yes—
172C2×C8×C17—21no—
173C2×C8×C17—21no—
174C2×C8×C17C22×C412yes—
175C2×C8×C17C22×C412yes—
176C2×C8×C17C2×D412yes—
177C2×C8×C17C2×Q812yes—
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—21no—
185C4⋊C4×C17—21no—
186C4⋊C4×C17—42no—
187C4⋊C4×C17—81no—
188C42×C17C22×C412yes—
189C42×C17#4112yes—
190#63227—24no—
191#63227—42no—
192#63227—42no—
193C4⋊Dic17—21no—
194C4⋊Dic17—21no—
195C4⋊Dic17—22no—
196C4⋊Dic17—41no—
197C4⋊Dic17C22×C412yes—
198C4⋊Dic17C22×C412yes—
199#63225—24no—
200#63225—42no—
201#63225—42no—
202#63225—42no—
203#63225—42no—
204#63225—42no—
205#63225—42no—
206C4×Dic17—24no—
207C4×Dic17—21no—
208C4×Dic17—21no—
209C4×Dic17—21no—
210C4×Dic17—21no—
211C4×Dic17—42no—
212C4×Dic17—42no—
213C2×#3622—21no—
214C2×#3622—21no—
215C2×#3622—21no—
216C2×#3622—21no—
217C2×#3622—21no—
218C2×#3622—21no—
219#63219C4⋊C4116yescomplemented
220#22677—171nomaximal, Sylow
221C23×C17C22×D411yescharacteristic
222C22×D17—22no—
223C22×D17—22no—
224C22×D17—22no—
225C22×D17—41no—
226C22×D17—41no—
227C22×D17—41no—
228C22×D17—41no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—41no—
234C2×C4×C17—41no—
235C2×C4×C17—41no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—41no—
239C2×C4×C17—41no—
240C2×C4×C17—41no—
241C2×C4×C17—41no—
242C2×C4×C17—41no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
245C2×C4×C17C4×D412yes—
246C2×C4×C17C4×D412yes—
247C2×C4×C17C23×C411yescharacteristic
248C2×C4×C17C22×D411yescharacteristic
249C2×C4×C17C22×Q811yescharacteristic
250C2×C4×C17C2×#4111yescharacteristic
251C2×C4×C17H(5,𝔽2)11yescharacteristic
252C2×C4×C172−1+411yescharacteristic
253C2×Dic17—22no—
254C2×Dic17—22no—
255C2×Dic17—21no—
256C2×Dic17—22no—
257C2×Dic17—21no—
258C2×Dic17—22no—
259C2×Dic17—22no—
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—21no—
263C2×Dic17—21no—
264C2×Dic17—21no—
265C2×Dic17—21no—
266C2×Dic17—21no—
267C2×Dic17—21no—
268C2×Dic17—21no—
269C2×Dic17—41no—
270C2×Dic17—41no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17AGL(1,ℤ/8)12yes—
274D68—22no—
275D68—22no—
276D68—22no—
277D68C2×C4⋊C414yes—
278C4×D17—48no—
279C4×D17—84no—
280C4×D17—84no—
281Dic34—22no—
282Dic34—22no—
283Dic34—22no—
284Dic34C2×C4⋊C414yescomplemented
285C8×C17—22no—
286C8×C17C2×C4⋊C414yescomplemented
287#3622—41no—
288#3622—41no—
289C2×#440—171no—
290C2×#440—171no—
291C2×C4×SD16—171no—
292#2351—171no—
293D4×C4⋊C4—171no—
294#1943—171no—
295#1942—171no—
296#1865—171no—
297#1864—171no—
298#1863—171no—
299#1862—171no—
300#1847D1711yescharacteristic, complemented
301#1804—171no—
302#1803—171no—
303#1771—171no—
304C22×C17C2×C4×D411yescharacteristic
305C22×C17D4211yescharacteristic
306C22×C17D4×Q811yescharacteristic
307C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
308C22×C17#57511yescharacteristic
309C22×C17#57711yescharacteristic
310C22×C17#57811yescharacteristic
311C2×D17—24no—
312C2×D17—42no—
313C2×D17—42no—
314C2×D17—42no—
315C2×D17—42no—
316C2×D17—42no—
317C2×D17—42no—
318C4×C17—22no—
319C4×C17—22no—
320C4×C17—81no—
321C4×C17—81no—
322C4×C17—81no—
323C4×C17—81no—
324C4×C17C22×C4⋊C411yescharacteristic
325C4×C17#52011yescharacteristic
326C4×C17#54711yescharacteristic
327C4×C17#55411yescharacteristic
328Dic17—24no—
329Dic17—22no—
330Dic17—42no—
331Dic17—42no—
332Dic17—41no—
333Dic17—41no—
334Dic17—41no—
335C22×SD16—171no—
336C2×C4×Q8—171no—
337C2×C4×Q8—171no—
338C2×C4×Q8—171no—
339C2×C4×D4—171no—
340C2×C4×D4—171no—
341C2×C4×D4—171no—
342C22×C4⋊C4—341no—
343#440—178no—
344#440—344no—
345C4×SD16—344no—
346C2×#107—171no—
347C2×#107C2×D1711yescharacteristic
348C2×#107C2×D1711yescharacteristic
349C2×#106C2×D1711yescharacteristic
350C2×#106C2×D1711yescharacteristic
351C2×C4⋊C8—171no—
352C2×#103—171no—
353C2×#103—171no—
354C2×#103—171no—
355C2×#102—171no—
356C2×#102—171no—
357C2×#102—171no—
358C2×#97—171no—
359C2×#97C2×D1711yescharacteristic
360C2×C4×C8—171no—
361#389—171no—
362#389—171no—
363#389—171no—
364#389—171no—
365#389—341no—
366#389—341no—
367#385—341no—
368#385—341no—
369#384C2×D1711yescharacteristic
370#382—341no—
371#382C2×D1711yescharacteristic
372#380—341no—
373C4×C4⋊C4—171no—
374C4×C4⋊C4—341no—
375#340—171no—
376#340—171no—
377#340—171no—
378#340—171no—
379#336—171no—
380#336—171no—
381C2×C17D4×C4⋊C411yescharacteristic
382C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
383C2×C17#294711yescharacteristic
384C2×C17#326511yescharacteristic
385C2×C17#326911yescharacteristic
386C2×C17#338611yescharacteristic
387C2×C17#338811yescharacteristic
388D17—44no—
389C22×Q8—171no—
390C22×D4—171no—
391C23×C4—341no—
392C23×C4—341no—
393C23×C4—341no—
394C2×SD16—174no—
395C2×SD16—174no—
396C2×SD16—174no—
397C22×C8—171no—
398C22×C8—171no—
399C22×C8C22×D1711yescharacteristic
400C22×C8C22×D1711yescharacteristic
401C4×Q8—174no—
402C4×Q8—342no—
403C4×Q8—342no—
404C4×Q8—342no—
405C4×Q8—681no—
406C4×Q8—681no—
407C4×D4—174no—
408C4×D4—342no—
409C4×D4—342no—
410C4×D4—342no—
411C4×D4—681no—
412C4×D4—681no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—171no—
416C2×C4⋊C4—171no—
417C2×C4⋊C4—171no—
418C2×C4⋊C4—171no—
419C2×C4⋊C4—341no—
420C2×C4⋊C4—341no—
421C2×C4⋊C4—341no—
422C2×C4⋊C4—341no—
423C2×C4⋊C4—341no—
424C2×C4⋊C4—341no—
425C2×C4⋊C4—341no—
426C2×C4⋊C4—684no—
427C2×C4⋊C4C22×D1711yescharacteristic
428C2×C4⋊C4C22×D1711yescharacteristic
429C2×C4⋊C4C22×D1711yescharacteristic
430C2×C4⋊C4C22×D1711yescharacteristic
431C2×C22⋊C4—341no—
432C2×C22⋊C4—341no—
433C2×C22⋊C4—341no—
434C2×C42—171no—
435C2×C42—171no—
436C2×C42—341no—
437C2×C42—341no—
438C2×C42—341no—
439C2×C42C22×D1711yescharacteristic
440#107—22no—
441#107—41no—
442#107—174no—
443#106—22no—
444#106—41no—
445C4⋊C8—342no—
446#103—174no—
447#103—342no—
448#103—342no—
449#102—174no—
450#102—342no—
451#102—342no—
452#97—342no—
453#97C4×D1714yescomplemented
454C4×C8—342no—
455#95—21no—
456#95—21no—
457#95—171no—
458#95—171no—
459#95—341no—
460#95—341no—
461#95—341no—
462#95—341no—
463C17#2267711yescharacteristic, complemented, Sylow
464C24—341no—
465C2×Q8—172no—
466C2×Q8—172no—
467C2×Q8—172no—
468C2×Q8—341no—
469C2×Q8—341no—
470C2×Q8—341no—
471C2×D4—172no—
472C2×D4—172no—
473C2×D4—172no—
474C2×D4—341no—
475C2×D4—341no—
476C2×D4—341no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—171no—
482C22×C4—171no—
483C22×C4—171no—
484C22×C4—171no—
485C22×C4—341no—
486C22×C4—341no—
487C22×C4—341no—
488C22×C4—344no—
489C22×C4—682no—
490C22×C4—682no—
491C22×C4—682no—
492C22×C4—682no—
493C22×C4—682no—
494C22×C4—682no—
495C22×C4—1361no—
496C22×C4—1361no—
497C22×C4D4×D1711yescharacteristic
498C22×C4D4×D1711yescharacteristic
499C22×C4C23×D1711yescharacteristic
500SD16—1716no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—341no—
504C2×C8—341no—
505C2×C8—341no—
506C2×C8—341no—
507C2×C8—341no—
508C2×C8—341no—
509C2×C8C2×C4×D1712yes—
510C2×C8C2×C4×D1712yes—
511C2×C8D4×D1712yes—
512C2×C8Q8×D1712yes—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—42no—
522C4⋊C4—81no—
523C4⋊C4—172no—
524C4⋊C4—172no—
525C4⋊C4—342no—
526C4⋊C4—341no—
527C4⋊C4—341no—
528C4⋊C4—344no—
529C4⋊C4—681no—
530C4⋊C4—682no—
531C4⋊C4—682no—
532C4⋊C4—682no—
533C4⋊C4—682no—
534C4⋊C4—682no—
535C4⋊C4—682no—
536C22⋊C4—344no—
537C22⋊C4—682no—
538C22⋊C4—682no—
539C42—341no—
540C42—341no—
541C42—341no—
542C42—341no—
543C42—344no—
544C42—682no—
545C42—682no—
546C42C2×C4×D1712yes—
547C42#6325612yes—
548C23—342no—
549C23—342no—
550C23—342no—
551C23—681no—
552C23—681no—
553C23—681no—
554C23—681no—
555C23C2×D4×D1711yescharacteristic
556Q8—174no—
557Q8—342no—
558Q8—342no—
559Q8—342no—
560D4—174no—
561D4—342no—
562D4—342no—
563D4—342no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—172no—
581C2×C4—342no—
582C2×C4—342no—
583C2×C4—341no—
584C2×C4—341no—
585C2×C4—342no—
586C2×C4—342no—
587C2×C4—342no—
588C2×C4—341no—
589C2×C4—341no—
590C2×C4—341no—
591C2×C4—341no—
592C2×C4—341no—
593C2×C4—341no—
594C2×C4—341no—
595C2×C4—341no—
596C2×C4—341no—
597C2×C4—688no—
598C2×C4—681no—
599C2×C4—681no—
600C2×C4—681no—
601C2×C4—681no—
602C2×C4—1364no—
603C2×C4—1364no—
604C2×C4#1058757912yes—
605C2×C4#1058757912yes—
606C2×C4C22×C4×D1711yescharacteristic
607C2×C4C2×D4×D1711yescharacteristic
608C2×C4#1058769111yescharacteristic
609C2×C4C2×Q8×D1711yescharacteristic
610C2×C4C2×#6325611yescharacteristic
611C2×C4#1058769411yescharacteristic
612C8—22no—
613C8—681no—
614C8—681no—
615C8C4⋊C4×D1714yescomplemented
616C22—344no—
617C22—682no—
618C22—682no—
619C22—682no—
620C22—682no—
621C22—682no—
622C22—682no—
623C22C2×#1058757911yescharacteristic
624C22#1176344411yescharacteristic
625C22#1176347411yescharacteristic
626C22AGL(1,ℤ/8)×D1711yescharacteristic
627C22#1176360811yescharacteristic
628C22#1176361911yescharacteristic
629C22#1176362511yescharacteristic
630C4—22no—
631C4—22no—
632C4—81no—
633C4—81no—
634C4—81no—
635C4—81no—
636C4—344no—
637C4—342no—
638C4—681no—
639C4—681no—
640C4—682no—
641C4—682no—
642C4—681no—
643C4C2×C4⋊C4×D1711yescharacteristic
644C4#1176329011yescharacteristic
645C4#1176346211yescharacteristic
646C4#1176352011yescharacteristic
647C2—684no—
648C2#1453168411yescharacteristic
649C2#1453759111yescharacteristic
650C2#1453762511yescharacteristic
651C2#1454027811yescharacteristic
652C2#1454033811yescharacteristic
653C2#1454170211yescharacteristic
654C2#1454172811yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1847C2
socleC23×C17C22×D4
prime radicalC22×D68C23
cosocleC22×C4×C17C24
hypercenter#1847D17
hypocenterC17#22677
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
4. #14531701C2semidirect
6. #14529520C2semidirect
7. #14529515C2semidirect
8. #14529049C2semidirect
10. #14529035C2semidirect
13. #14528786C2semidirect
14. #14528656C2semidirect
15. C17×#1847C2semidirect
24. #11762911C4semidirect
112. C4×Dic34C2×C4semidirect
117. C17×#97C2×C4semidirect
127. #10587489C2×C4semidirect
128. #10587487C2×C4semidirect
219. #63219C4⋊C4semidirect
284. Dic34C2×C4⋊C4semidirect
286. C8×C17C2×C4⋊C4semidirect
300. #1847D17semidirect
453. #97C4×D17semidirect
463. C17#22677semidirect
615. C8C4⋊C4×D17semidirect

Maximal subgroups

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

subgroupindex
C2×#117629112
C2×#117629082
C2×#117628962
#145317012
#145316842
#145295202
#145295152
#145290492
#145290422
#145290352
#145290282
#145287912
#145287862
#145286562
C17×#18472
#2267717