Finite Group Databasebeta

← #17097704

Subgroups — #17097704

Order 4352.

858 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763102C211yesmaximal, characteristic, complemented
2#14532628C211yesmaximal, characteristic
3Q8×#63225C211yesmaximal, characteristic, complemented
4#14530310C211yesmaximal, characteristic
5#14530258C211yesmaximal, characteristic, complemented
6#14530156C211yesmaximal, characteristic
7#14530110C211yesmaximal, characteristic, complemented
8#14529527C211yesmaximal, characteristic, complemented
9#14529348C211yesmaximal, characteristic, complemented
10#14529141C211yesmaximal, characteristic
11#14529141C211yesmaximal, characteristic
12#14529054C211yesmaximal, characteristic, complemented
13#14529038C211yesmaximal, characteristic, complemented
14#14529031C211yesmaximal, characteristic
15C2×C17×#440C211yesmaximal, characteristic
16C2×#10587574C2211yescharacteristic
17C2×#10587610—21no—
18C2×Q8×Dic17C2211yescharacteristic
19C2×Q8×Dic17C2211yescharacteristic
20#11763102—24no—
21C2×#10587489C2211yescharacteristic
22C2×#10587478C2211yescharacteristic
23C2×#10587478C2211yescharacteristic
24C2×#10587488C2211yescharacteristic
25C2×#10587485C2211yescharacteristic
26C2×#10587506C2211yescharacteristic
27C2×#10587506C2211yescharacteristic
28C2×#10587503C2211yescharacteristic
29C2×#10587503C2211yescharacteristic
30#11762732—21no—
31#11762721—21no—
32#11762712C2211yescharacteristic
33#11762712C2211yescharacteristic
34#11762708C2211yescharacteristic
35#11762707—21no—
36#11762707C2211yescharacteristic
37#11762707C2211yescharacteristic
38#11762705—21no—
39#11762705C2211yescharacteristic
40#11762682—21no—
41#11762678—21no—
42#11762678C2211yescharacteristic
43C4×#63225—21no—
44C4×#63225C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762564C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52#11762563C2211yescharacteristic
53C22×SD16×C17C2211yescharacteristic
54C2×C4×Q8×C17C2211yescharacteristic
55C2×C4×D4×C17C2211yescharacteristic
56C17×#440—24no—
57C2×C17×#107C2211yescharacteristic
58C2×C17×#103C2211yescharacteristic
59C2×C17×#102C2211yescharacteristic
60C2×C17×#97C2211yescharacteristic
61C22×Q8×C17C2311yescharacteristic
62C22×D4×C17C2311yescharacteristic
63C23×C4×C17—21no—
64C22×Dic34—21no—
65C2×SD16×C17—22no—
66C2×SD16×C17—22no—
67C2×SD16×C17Q814yescomplemented
68C22×C8×C17C2311yescharacteristic
69C22×C8×C17C2311yescharacteristic
70C4×Q8×C17—22no—
71C4×Q8×C17—41no—
72C4×D4×C17—22no—
73C4×D4×C17—41no—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C22⋊C4×C17—21no—
78C2×C42×C17—21no—
79C2×C42×C17C2311yescharacteristic
80C2×C22⋊Dic17—21no—
81C2×C22⋊Dic17—21no—
82C2×C22⋊Dic17—21no—
83C2×C22⋊Dic17—21no—
84Q8×Dic17—22no—
85Q8×Dic17—22no—
86Q8×Dic17—22no—
87Q8×Dic17—41no—
88C2×C4⋊Dic17—21no—
89C2×C4⋊Dic17—21no—
90C2×C4⋊Dic17—21no—
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95#10587610—44no—
96C2×#63225—21no—
97C2×#63225—21no—
98C2×#63225—21no—
99C2×#63225—21no—
100C2×#63225—21no—
101C2×#63225—21no—
102C2×#63225D411yescharacteristic
103C2×#63225D411yescharacteristic
104C2×C4×Dic17—21no—
105C2×C4×Dic17—21no—
106C2×C4×Dic17C2311yescharacteristic
107C2×C4×Dic17C2311yescharacteristic
108C22×#3622C2311yescharacteristic
109C22×#3622C2311yescharacteristic
110#10587574—22no—
111#10587574—41no—
112C17×#107—22no—
113C17×#103D414yes—
114C17×#102—22no—
115C17×#97—22no—
116#10587506—22no—
117#10587506—22no—
118#10587503—22no—
119#10587503—41no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587502D411yescharacteristic
125#10587502D411yescharacteristic
126#10587489—22no—
127#10587488—22no—
128#10587485—22no—
129#10587478—22no—
130#10587478—41no—
131C24×C17—21no—
132C2×Q8×C17—21no—
133C2×Q8×C17—21no—
134C2×Q8×C17—21no—
135C2×Q8×C17C2×D412yes—
136C2×Q8×C17C2×Q812yes—
137C2×Q8×C17#4112yes—
138C2×D4×C17—21no—
139C2×D4×C17—21no—
140C2×D4×C17—21no—
141C2×D4×C17—21no—
142C2×D4×C17—21no—
143C2×D4×C17C2×Q812yes—
144C22×C4×C17—21no—
145C22×C4×C17—21no—
146C22×C4×C17—21no—
147C22×C4×C17—42no—
148C22×C4×C17—42no—
149C22×C4×C17—81no—
150C22×C4×C17C2×D411yescharacteristic
151C22×C4×C17C2×D411yescharacteristic
152C22×C4×C17C2411yescharacteristic
153C22×Dic17—21no—
154C22×Dic17—21no—
155C22×Dic17—21no—
156C22×Dic17—21no—
157C22×Dic17C2×D411yescharacteristic
158C22×Dic17C2×D411yescharacteristic
159C22×Dic17C2×D411yescharacteristic
160C22×Dic17C2×D411yescharacteristic
161C2×Dic34—42no—
162C2×Dic34—42no—
163C2×Dic34—81no—
164SD16×C17—28no—
165C2×C8×C17—21no—
166C2×C8×C17—21no—
167C2×C8×C17—21no—
168C2×C8×C17—21no—
169C2×C8×C17C2×D412yes—
170C2×C8×C17C2×Q812yes—
171C4⋊C4×C17—21no—
172C4⋊C4×C17—21no—
173C4⋊C4×C17—21no—
174C4⋊C4×C17—42no—
175C4⋊C4×C17C2×D412yes—
176C22⋊C4×C17—42no—
177C42×C17—21no—
178C42×C17—21no—
179C42×C17—42no—
180C22⋊Dic17—42no—
181C22⋊Dic17—42no—
182C22⋊Dic17—42no—
183C22⋊Dic17—81no—
184C4⋊Dic17—24no—
185C4⋊Dic17—21no—
186C4⋊Dic17—21no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—21no—
192C4⋊Dic17—42no—
193C4⋊Dic17—42no—
194C4⋊Dic17#4112yes—
195#63225—22no—
196#63225—22no—
197#63225—42no—
198#63225—42no—
199#63225—42no—
200#63225—81no—
201#63225—81no—
202#63225—81no—
203C4×Dic17—24no—
204C4×Dic17—21no—
205C4×Dic17—21no—
206C4×Dic17—21no—
207C4×Dic17—21no—
208C4×Dic17—42no—
209C2×#3622—21no—
210C2×#3622—21no—
211C2×#3622—21no—
212C2×#3622—21no—
213C2×#3622—21no—
214C2×#3622—21no—
215#26397—171nomaximal, Sylow
216C23×C17—22no—
217C23×C17—41no—
218C23×C17—41no—
219C23×C17—41no—
220C23×C17—41no—
221C23×C17—41no—
222C23×C17—41no—
223C23×C17C22×D411yescharacteristic
224Q8×C17—22no—
225Q8×C17—22no—
226Q8×C17—22no—
227Q8×C17C22⋊Q814yescomplemented
228D4×C17—22no—
229D4×C17—22no—
230D4×C17—41no—
231D4×C17—41no—
232C2×C4×C17—22no—
233C2×C4×C17—22no—
234C2×C4×C17—22no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—22no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—41no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—84no—
248C2×C4×C17C22×D411yescharacteristic
249C2×C4×C17C22×Q811yescharacteristic
250C2×C4×C17C2×#4111yescharacteristic
251C2×C4×C17C2×#4111yescharacteristic
252C2×C4×C17H(5,𝔽2)11yescharacteristic
253C2×C4×C172−1+411yescharacteristic
254C2×Dic17—21no—
255C2×Dic17—21no—
256C2×Dic17—21no—
257C2×Dic17—21no—
258C2×Dic17—22no—
259C2×Dic17—21no—
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—21no—
263C2×Dic17—21no—
264C2×Dic17—21no—
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×Dic17AGL(1,ℤ/8)12yes—
277C2×Dic17#13712yes—
278Dic34—84no—
279C8×C17—22no—
280C8×C17—22no—
281#3622—41no—
282#3622—41no—
283C2×#477—171no—
284C2×#440D1711yescharacteristic
285#2452—171no—
286Q8×C4⋊C4—171no—
287#2093—171no—
288#2083—171no—
289#2065—171no—
290#2057—171no—
291#1944—171no—
292#1917—171no—
293#1880—171no—
294#1880—171no—
295#1865—171no—
296#1863—171no—
297#1862—171no—
298C22×C17—42no—
299C22×C17—42no—
300C22×C17—42no—
301C22×C17—81no—
302C22×C17—81no—
303C22×C17—81no—
304C22×C17—81no—
305C22×C17#52511yescharacteristic
306C22×C17D4211yescharacteristic
307C22×C17D4×Q811yescharacteristic
308C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
309C22×C17C2×#13711yescharacteristic
310C22×C17#57611yescharacteristic
311C22×C17#57811yescharacteristic
312C4×C17—24no—
313C4×C17—41no—
314C4×C17—42no—
315C4×C17—41no—
316C4×C17—81no—
317C4×C17C2×C22⋊Q811yescharacteristic
318C4×C17#53911yescharacteristic
319C4×C17#54711yescharacteristic
320C4×C17#55411yescharacteristic
321Dic17—22no—
322Dic17—22no—
323Dic17—41no—
324Dic17—42no—
325Dic17—41no—
326Dic17—81no—
327Dic17—81no—
328Dic17—81no—
329C22×SD16C2×D1711yescharacteristic
330C2×C4⋊Q8—171no—
331C2×C22⋊Q8—341no—
332C2×C4×Q8—171no—
333C2×C4×Q8—171no—
334C2×C4×Q8C2×D1711yescharacteristic
335C2×C4×D4C2×D1711yescharacteristic
336#477—344no—
337#440—24no—
338C2×#107—171no—
339C2×#107—171no—
340C2×#107—171no—
341C2×#107C2×D1711yescharacteristic
342C2×#106—171no—
343C2×C4⋊C8—171no—
344C2×#103—171no—
345C2×#103—171no—
346C2×#103C2×D1711yescharacteristic
347C2×#102—171no—
348C2×#102—171no—
349C2×#102C2×D1711yescharacteristic
350C2×#97C2×D1711yescharacteristic
351#396—341no—
352#393—341no—
353#390—171no—
354#390—171no—
355#389—171no—
356#389—171no—
357#389—171no—
358#389—171no—
359#389—341no—
360#389—341no—
361#385—341no—
362#384—171no—
363#384—341no—
364C4×C4⋊C4—171no—
365C4×C4⋊C4—341no—
366#340—171no—
367#340—171no—
368#340—171no—
369#340—171no—
370#336—171no—
371#336—171no—
372#336—171no—
373#336—171no—
374C2×C17—82no—
375C2×C17#244911yescharacteristic
376C2×C17#311211yescharacteristic
377C2×C17#312411yescharacteristic
378C2×C17#327011yescharacteristic
379C2×C17#327411yescharacteristic
380C2×C17#338511yescharacteristic
381C2×C17#338611yescharacteristic
382C22×Q8—341no—
383C22×Q8C22×D1711yescharacteristic
384C22×D4C22×D1711yescharacteristic
385C23×C4—21no—
386C2×SD16—22no—
387C2×SD16—22no—
388C2×SD16Dic3414yescomplemented
389C22×C8—171no—
390C22×C8—171no—
391C22×C8C22×D1711yescharacteristic
392C22×C8C22×D1711yescharacteristic
393C4⋊Q8—342no—
394C4⋊Q8—681no—
395C22⋊Q8—684no—
396C4×Q8—22no—
397C4×Q8—41no—
398C4×Q8—342no—
399C4×Q8—342no—
400C4×Q8—342no—
401C4×Q8—681no—
402C4×D4—22no—
403C4×D4—41no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—171no—
406C2×C4⋊C4—171no—
407C2×C4⋊C4—171no—
408C2×C4⋊C4—171no—
409C2×C4⋊C4—171no—
410C2×C4⋊C4—171no—
411C2×C4⋊C4—341no—
412C2×C4⋊C4—341no—
413C2×C4⋊C4—341no—
414C2×C4⋊C4—341no—
415C2×C4⋊C4—341no—
416C2×C4⋊C4—341no—
417C2×C4⋊C4—341no—
418C2×C4⋊C4—341no—
419C2×C4⋊C4—341no—
420C2×C4⋊C4C22×D1711yescharacteristic
421C2×C4⋊C4C22×D1711yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—341no—
424C2×C22⋊C4—341no—
425C2×C22⋊C4—341no—
426C2×C22⋊C4—341no—
427C2×C42—21no—
428C2×C42—171no—
429C2×C42—171no—
430C2×C42—341no—
431C2×C42—341no—
432C2×C42C22×D1711yescharacteristic
433#107—22no—
434#107—342no—
435#107—342no—
436#107—681no—
437#106—342no—
438C4⋊C8—342no—
439#103—342no—
440#103—342no—
441#103C22⋊D1714yes—
442#102—22no—
443#102—342no—
444#102—681no—
445#97—22no—
446#95—171no—
447#95—171no—
448#95—341no—
449#95—341no—
450#95—341no—
451#95—341no—
452C17#2639711yescharacteristic, complemented, Sylow
453C24—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—682no—
458C2×Q8—682no—
459C2×Q8—1361no—
460C2×Q8C2×Dic3412yes—
461C2×Q8#6325212yes—
462C2×Q8C2×C22⋊D1712yes—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4C2×Dic3412yes—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—81no—
475C22×C4—171no—
476C22×C4—171no—
477C22×C4—171no—
478C22×C4—171no—
479C22×C4—341no—
480C22×C4—341no—
481C22×C4—341no—
482C22×C4—341no—
483C22×C4D4×D1711yescharacteristic
484C22×C4D4×D1711yescharacteristic
485C22×C4C23×D1711yescharacteristic
486SD16—28no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—341no—
492C2×C8—341no—
493C2×C8—341no—
494C2×C8—341no—
495C2×C8—341no—
496C2×C8—341no—
497C2×C8C2×Dic3412yes—
498C2×C8C2×C22⋊D1712yes—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—42no—
503C4⋊C4—172no—
504C4⋊C4—342no—
505C4⋊C4—342no—
506C4⋊C4—344no—
507C4⋊C4—341no—
508C4⋊C4—341no—
509C4⋊C4—341no—
510C4⋊C4—341no—
511C4⋊C4—341no—
512C4⋊C4—341no—
513C4⋊C4—341no—
514C4⋊C4—682no—
515C4⋊C4—682no—
516C4⋊C4—682no—
517C4⋊C4—682no—
518C4⋊C4—682no—
519C4⋊C4—1361no—
520C4⋊C4—1361no—
521C4⋊C4—1361no—
522C4⋊C4C2×C22⋊D1712yes—
523C22⋊C4—42no—
524C22⋊C4—682no—
525C22⋊C4—682no—
526C22⋊C4—682no—
527C22⋊C4—1361no—
528C42—21no—
529C42—21no—
530C42—42no—
531C42—344no—
532C42—341no—
533C42—341no—
534C42—341no—
535C42—341no—
536C42—682no—
537C23—22no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23C2×D4×D1711yescharacteristic
545Q8—22no—
546Q8—22no—
547Q8—22no—
548Q8—1364no—
549Q8#1058761014yescomplemented
550D4—22no—
551D4—22no—
552D4—41no—
553D4—41no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—84no—
570C2×C4—172no—
571C2×C4—172no—
572C2×C4—341no—
573C2×C4—341no—
574C2×C4—341no—
575C2×C4—341no—
576C2×C4—342no—
577C2×C4—341no—
578C2×C4—341no—
579C2×C4—341no—
580C2×C4—341no—
581C2×C4—341no—
582C2×C4—341no—
583C2×C4—681no—
584C2×C4—681no—
585C2×C4—681no—
586C2×C4—681no—
587C2×C4—681no—
588C2×C4—681no—
589C2×C4—681no—
590C2×C4—681no—
591C2×C4—681no—
592C2×C4—681no—
593C2×C4—681no—
594C2×C4C22×Dic3411yescharacteristic
595C2×C4C2×#6325211yescharacteristic
596C2×C4#1058769111yescharacteristic
597C2×C4#41×D1711yescharacteristic
598C2×C4#1058769711yescharacteristic
599C2×C4C22×C22⋊D1711yescharacteristic
600C8—22no—
601C8—22no—
602C8—681no—
603C8—681no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22#1176331811yescharacteristic
612C22D4×C22⋊D1711yescharacteristic
613C22D4×Dic3411yescharacteristic
614C22AGL(1,ℤ/8)×D1711yescharacteristic
615C22#137×D1711yescharacteristic
616C22#1176361311yescharacteristic
617C22#1176362511yescharacteristic
618C4—24no—
619C4—41no—
620C4—41no—
621C4—42no—
622C4—81no—
623C4—342no—
624C4—342no—
625C4—681no—
626C4—682no—
627C4—681no—
628C4—1361no—
629C4—1361no—
630C4—1361no—
631C4C2×#1058761011yescharacteristic
632C4#1176340111yescharacteristic
633C4#1176345811yescharacteristic
634C4#1176351811yescharacteristic
635C2—82no—
636C2#1453258711yescharacteristic
637C2#1453890411yescharacteristic
638C2#1453904411yescharacteristic
639C2#1454034811yescharacteristic
640C2#1454039811yescharacteristic
641C2#1454169511yescharacteristic
642C2#1454170611yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#440C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
hypercenterC2×#440D17
hypocenterC17#26397
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
1. C2×#11763102C2semidirect
3. Q8×#63225C2semidirect
5. #14530258C2semidirect
7. #14530110C2semidirect
8. #14529527C2semidirect
9. #14529348C2semidirect
12. #14529054C2semidirect
13. #14529038C2semidirect
67. C2×SD16×C17Q8semidirect
227. Q8×C17C22⋊Q8semidirect
388. C2×SD16Dic34semidirect
452. C17#26397semidirect
549. Q8#10587610semidirect

Maximal subgroups

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

subgroupindex
C2×#117631022
#145326282
Q8×#632252
#145303102
#145302582
#145301562
#145301102
#145295272
#145293482
#145291412
#145290542
#145290382
#145290312
C2×C17×#4402
#2639717