Finite Group Databasebeta

← #17097707

Subgroups — #17097707

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×#11763101C211yesmaximal, characteristic, complemented
2C2×#11762909C211yesmaximal, characteristic
3C4⋊C4×Dic34C211yesmaximal, characteristic, complemented
4#14530312C211yesmaximal, characteristic
5#14530255C211yesmaximal, characteristic, complemented
6#14530158C211yesmaximal, characteristic
7#14530109C211yesmaximal, characteristic, complemented
8#14529524C211yesmaximal, characteristic, complemented
9#14529350C211yesmaximal, characteristic, complemented
10#14529142C211yesmaximal, characteristic
11#14529142C211yesmaximal, characteristic
12#14529051C211yesmaximal, characteristic, complemented
13#14529037C211yesmaximal, characteristic, complemented
14#14529031C211yesmaximal, characteristic
15C17×#2452C211yesmaximal, characteristic
16C22×#63229C2211yescharacteristic
17C2×C4×Dic34C2211yescharacteristic
18C2×C4×Dic34C2211yescharacteristic
19C2×C4×Dic34C2211yescharacteristic
20#11763101—24no—
21#11762909—24no—
22C2×#10587478C2211yescharacteristic
23C2×#10587478C2211yescharacteristic
24C2×#10587478C2211yescharacteristic
25C2×#10587478C2211yescharacteristic
26C2×#10587479C2211yescharacteristic
27C2×C4⋊#3622C2211yescharacteristic
28C2×#10587481C2211yescharacteristic
29C2×#10587481C2211yescharacteristic
30C2×#10587481C2211yescharacteristic
31C2×#10587503C2211yescharacteristic
32C2×#10587503C2211yescharacteristic
33C2×#10587503C2211yescharacteristic
34C2×#10587474C2211yescharacteristic
35#11762707C2211yescharacteristic
36#11762707C2211yescharacteristic
37#11762706—21no—
38#11762706—21no—
39#11762679C2211yescharacteristic
40#11762678—21no—
41C4⋊C4×Dic17—21no—
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48#11762564C2211yescharacteristic
49#11762564C2211yescharacteristic
50C2×C17×C4⋊Q8C2211yescharacteristic
51C2×C17×C22⋊Q8—21no—
52C2×C4×D4×C17C2211yescharacteristic
53C17×#396—21no—
54C17×#393—21no—
55C17×#390C2211yescharacteristic
56C17×#390C2211yescharacteristic
57C17×#389C2211yescharacteristic
58C17×#389C2211yescharacteristic
59C17×#385—21no—
60C4×C4⋊C4×C17C2211yescharacteristic
61C22×Q8×C17—21no—
62C22×D4×C17C2311yescharacteristic
63C23×C4×C17—21no—
64C22×Dic34C2311yescharacteristic
65C17×C4⋊Q8—22no—
66C17×C4⋊Q8—41no—
67C17×C22⋊Q8—44no—
68C4×D4×C17—22no—
69C4×D4×C17—41no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17D411yescharacteristic
74C2×C4⋊C4×C17D411yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C22⋊C4×C17—21no—
79C2×C22⋊C4×C17—21no—
80C2×C22⋊C4×C17—21no—
81C2×C22⋊C4×C17—21no—
82C2×C22⋊C4×C17—21no—
83C2×C42×C17C2311yescharacteristic
84C2×C42×C17C2311yescharacteristic
85C2×C42×C17C2311yescharacteristic
86C2×#63229—22no—
87C2×#63229—22no—
88C2×#63229Q814yescomplemented
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×C4×Dic17—21no—
100C2×C4×Dic17—21no—
101C2×C4×Dic17—21no—
102C22×#3622C2311yescharacteristic
103C22×#3622C2311yescharacteristic
104C22×#3622C2311yescharacteristic
105C22×#3622C2311yescharacteristic
106C4×Dic34—22no—
107C4×Dic34—22no—
108C4×Dic34—22no—
109C4×Dic34—22no—
110C4×Dic34—41no—
111C4×Dic34—41no—
112C17×#95—21no—
113C17×#95—21no—
114C17×#95D411yescharacteristic
115C17×#95D411yescharacteristic
116#10587503—22no—
117#10587503—22no—
118#10587503—41no—
119#10587502—21no—
120#10587502—21no—
121#10587481—22no—
122#10587481—22no—
123#10587481D414yes—
124#10587479—22no—
125#10587478—22no—
126#10587478—22no—
127#10587478—22no—
128#10587478—41no—
129C4⋊#3622—22no—
130#10587474—22no—
131C24×C17—21no—
132C2×Q8×C17—42no—
133C2×Q8×C17—42no—
134C2×Q8×C17—81no—
135C2×D4×C17—21no—
136C2×D4×C17—21no—
137C2×D4×C17—21no—
138C2×D4×C17—21no—
139C2×D4×C17—21no—
140C2×D4×C17C2×Q812yes—
141C22×C4×C17—21no—
142C22×C4×C17—21no—
143C22×C4×C17—21no—
144C22×C4×C17—42no—
145C22×C4×C17—42no—
146C22×C4×C17—81no—
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2×D411yescharacteristic
151C22×C4×C17C2×D411yescharacteristic
152C22×C4×C17C2×D411yescharacteristic
153C22×C4×C17C2411yescharacteristic
154C22×Dic17—21no—
155C22×Dic17—21no—
156C22×Dic17—21no—
157C22×Dic17—21no—
158C2×Dic34—21no—
159C2×Dic34—21no—
160C2×Dic34—21no—
161C2×Dic34C2×D412yes—
162C2×Dic34C2×Q812yes—
163C2×Dic34#4112yes—
164C4⋊C4×C17—22no—
165C4⋊C4×C17—22no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17—21no—
171C4⋊C4×C17—42no—
172C4⋊C4×C17—42no—
173C4⋊C4×C17—42no—
174C4⋊C4×C17C2×D412yes—
175C22⋊C4×C17—42no—
176C22⋊C4×C17—42no—
177C22⋊C4×C17—42no—
178C22⋊C4×C17—42no—
179C22⋊C4×C17—81no—
180C42×C17—21no—
181C42×C17—21no—
182C42×C17—21no—
183C42×C17—21no—
184C42×C17—21no—
185C42×C17—21no—
186#63229—28no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—21no—
192C4⋊Dic17—42no—
193C4⋊Dic17—81no—
194C4⋊Dic17#4112yes—
195#63225—24no—
196#63225—42no—
197#63225—42no—
198#63225—81no—
199#63225—81no—
200C4×Dic17—24no—
201C4×Dic17—42no—
202C4×Dic17—42no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622—21no—
208C2×#3622—21no—
209C2×#3622—21no—
210C2×#3622—21no—
211C2×#3622—21no—
212C2×#3622—21no—
213C2×#3622C2×D412yes—
214C2×#3622C2×Q812yes—
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—84no—
225D4×C17—22no—
226D4×C17—22no—
227D4×C17—41no—
228D4×C17—41no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
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—21no—
243C2×C4×C17—21no—
244C2×C4×C17—21no—
245C2×C4×C17—21no—
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—41no—
253C2×C4×C17—41no—
254C2×C4×C17—84no—
255C2×C4×C17AGL(1,ℤ/8)12yes—
256C2×C4×C17#13712yes—
257C2×C4×C17C22×D411yescharacteristic
258C2×C4×C17C22×Q811yescharacteristic
259C2×C4×C17C2×#4111yescharacteristic
260C2×C4×C17C2×#4111yescharacteristic
261C2×C4×C17H(5,𝔽2)11yescharacteristic
262C2×C4×C172−1+411yescharacteristic
263C2×Dic17—22no—
264C2×Dic17—22no—
265C2×Dic17—22no—
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—
275Dic34—22no—
276Dic34—22no—
277Dic34—22no—
278Dic34C22⋊Q814yescomplemented
279#3622—22no—
280#3622—22no—
281#3622—41no—
282#3622—41no—
283C2×#477—171no—
284C2×#440—171no—
285#2452D1711yescharacteristic
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—22no—
313C4×C17—22no—
314C4×C17—41no—
315C4×C17—41no—
316C4×C17—41no—
317C4×C17—41no—
318C4×C17—42no—
319C4×C17—81no—
320C4×C17—81no—
321C4×C17C2×C22⋊Q811yescharacteristic
322C4×C17#53911yescharacteristic
323C4×C17#54711yescharacteristic
324C4×C17#55411yescharacteristic
325Dic17—24no—
326Dic17—42no—
327Dic17—81no—
328Dic17—81no—
329C22×SD16—171no—
330C2×C4⋊Q8C2×D1711yescharacteristic
331C2×C22⋊Q8—21no—
332C2×C4×Q8—171no—
333C2×C4×Q8—171no—
334C2×C4×Q8—171no—
335C2×C4×D4C2×D1711yescharacteristic
336#477—344no—
337#440—344no—
338C2×#107—171no—
339C2×#107—171no—
340C2×#107—171no—
341C2×#107—171no—
342C2×#106—171no—
343C2×C4⋊C8—171no—
344C2×#103—171no—
345C2×#103—171no—
346C2×#103—171no—
347C2×#102—171no—
348C2×#102—171no—
349C2×#102—171no—
350C2×#97—171no—
351#396—21no—
352#393—21no—
353#390C2×D1711yescharacteristic
354#390C2×D1711yescharacteristic
355#389—171no—
356#389—171no—
357#389—341no—
358#389—341no—
359#389C2×D1711yescharacteristic
360#389C2×D1711yescharacteristic
361#385—21no—
362#384—171no—
363#384—341no—
364C4×C4⋊C4—341no—
365C4×C4⋊C4C2×D1711yescharacteristic
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—21no—
383C22×Q8—171no—
384C22×D4C22×D1711yescharacteristic
385C23×C4—21no—
386C2×SD16—174no—
387C2×SD16—342no—
388C2×SD16—342no—
389C22×C8—171no—
390C22×C8—171no—
391C22×C8—171no—
392C22×C8—171no—
393C4⋊Q8—22no—
394C4⋊Q8—41no—
395C22⋊Q8—44no—
396C4×Q8—342no—
397C4×Q8—342no—
398C4×Q8—342no—
399C4×Q8—342no—
400C4×Q8—681no—
401C4×Q8—681no—
402C4×D4—22no—
403C4×D4—41no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—171no—
408C2×C4⋊C4—171no—
409C2×C4⋊C4—171no—
410C2×C4⋊C4—341no—
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⋊C4C22⋊D1711yescharacteristic
418C2×C4⋊C4C22⋊D1711yescharacteristic
419C2×C4⋊C4C22×D1711yescharacteristic
420C2×C4⋊C4C22×D1711yescharacteristic
421C2×C4⋊C4C22×D1711yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C42—341no—
428C2×C42—341no—
429C2×C42—341no—
430C2×C42C22×D1711yescharacteristic
431C2×C42C22×D1711yescharacteristic
432C2×C42C22×D1711yescharacteristic
433#107—342no—
434#107—342no—
435#107—342no—
436#107—681no—
437#106—342no—
438C4⋊C8—342no—
439#103—174no—
440#103—342no—
441#103—342no—
442#102—342no—
443#102—342no—
444#102—681no—
445#97—342no—
446#95—21no—
447#95—21no—
448#95—341no—
449#95—341no—
450#95C22⋊D1711yescharacteristic
451#95C22⋊D1711yescharacteristic
452C17#2639711yescharacteristic, complemented, Sylow
453C24—21no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—81no—
457C2×Q8—172no—
458C2×Q8—172no—
459C2×Q8—172no—
460C2×Q8—341no—
461C2×Q8—341no—
462C2×Q8—341no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4Q8×D1712yes—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—81no—
475C22×C4—341no—
476C22×C4—341no—
477C22×C4—341no—
478C22×C4—341no—
479C22×C4C2×C22⋊D1711yescharacteristic
480C22×C4C2×C22⋊D1711yescharacteristic
481C22×C4C2×C22⋊D1711yescharacteristic
482C22×C4C2×C22⋊D1711yescharacteristic
483C22×C4C2×C22⋊D1711yescharacteristic
484C22×C4C2×C22⋊D1711yescharacteristic
485C22×C4C23×D1711yescharacteristic
486SD16—348no—
487C2×C8—172no—
488C2×C8—172no—
489C2×C8—341no—
490C2×C8—341no—
491C2×C8—341no—
492C2×C8—341no—
493C2×C8—341no—
494C2×C8—341no—
495C2×C8—341no—
496C2×C8—341no—
497C2×C8—341no—
498C2×C8—341no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—172no—
510C4⋊C4—344no—
511C4⋊C4—341no—
512C4⋊C4—341no—
513C4⋊C4—341no—
514C4⋊C4—341no—
515C4⋊C4—341no—
516C4⋊C4—682no—
517C4⋊C4—682no—
518C4⋊C4—682no—
519C4⋊C4—1361no—
520C4⋊C4—1361no—
521C4⋊C4—1361no—
522C4⋊C4D4×D1712yes—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—81no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—344no—
535C42—682no—
536C42—682no—
537C23—22no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23C22×C22⋊D1711yescharacteristic
545Q8—84no—
546Q8—174no—
547Q8—342no—
548Q8—342no—
549Q8—342no—
550D4—22no—
551D4—22no—
552D4—41no—
553D4—41no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
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—84no—
580C2×C4—342no—
581C2×C4—342no—
582C2×C4—342no—
583C2×C4—342no—
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#1058761812yes—
593C2×C4#1058763712yes—
594C2×C4C2×D4×D1711yescharacteristic
595C2×C4#1058769111yescharacteristic
596C2×C4C2×Q8×D1711yescharacteristic
597C2×C4#1058769411yescharacteristic
598C2×C4#41×D1711yescharacteristic
599C2×C4#41×D1711yescharacteristic
600C8—342no—
601C8—342no—
602C8—681no—
603C8—681no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22#1176331611yescharacteristic
612C22D4×C22⋊D1711yescharacteristic
613C22Q8×C22⋊D1711yescharacteristic
614C22C2×#1058761811yescharacteristic
615C22C2×#1058763711yescharacteristic
616C22#1176361211yescharacteristic
617C22#1176362411yescharacteristic
618C4—22no—
619C4—22no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—42no—
625C4—81no—
626C4—81no—
627C4—344no—
628C4—682no—
629C4—1361no—
630C4—1361no—
631C4C22⋊Q8×D1711yescharacteristic
632C4#1176340511yescharacteristic
633C4#1176346111yescharacteristic
634C4#1176351311yescharacteristic
635C2—82no—
636C2#1453259011yescharacteristic
637C2#1453890211yescharacteristic
638C2#1453904011yescharacteristic
639C2#1454035011yescharacteristic
640C2#1454040011yescharacteristic
641C2#1454169411yescharacteristic
642C2#1454170511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2452C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
hypercenter#2452D17
hypocenterC17#26397
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2452

Complements

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

NH ≅ G/Ntype
1. C2×#11763101C2semidirect
3. C4⋊C4×Dic34C2semidirect
5. #14530255C2semidirect
7. #14530109C2semidirect
8. #14529524C2semidirect
9. #14529350C2semidirect
12. #14529051C2semidirect
13. #14529037C2semidirect
88. C2×#63229Q8semidirect
278. Dic34C22⋊Q8semidirect
452. C17#26397semidirect

Maximal subgroups

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

subgroupindex
C2×#117631012
C2×#117629092
C4⋊C4×Dic342
#145303122
#145302552
#145301582
#145301092
#145295242
#145293502
#145291422
#145290512
#145290372
#145290312
C17×#24522
#2639717