Finite Group Databasebeta

← #17097239

Subgroups — #17097239

Order 4352.

896 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763109C211yesmaximal, characteristic, complemented
2C2×#11762849C211yesmaximal, characteristic, complemented
3#14532497C211yesmaximal, characteristic, complemented
4#14530317C211yesmaximal, characteristic, complemented
5#14530286C211yesmaximal, characteristic, complemented
6#14530172C211yesmaximal, characteristic, complemented
7#14530122C211yesmaximal, characteristic, complemented
8#14529547C211yesmaximal, characteristic, complemented
9#14529355C211yesmaximal, characteristic, complemented
10#14529223C211yesmaximal, characteristic
11#14529107C211yesmaximal, characteristic
12#14529045C211yesmaximal, characteristic, complemented
13#14528977C211yesmaximal, characteristic, complemented
14#14528969C211yesmaximal, characteristic, complemented
15C17×#2361C211yesmaximal, characteristic
16C22×#63223C2211yescharacteristic
17C2×#10587556C2211yescharacteristic
18C2×#10587612C2211yescharacteristic
19#11763109—24no—
20#11762849—24no—
21C2×#10587478C2211yescharacteristic
22C2×#10587478C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×#10587479C2211yescharacteristic
25C2×#10587479C2211yescharacteristic
26C2×#10587479C2211yescharacteristic
27C2×C4⋊#3622C2211yescharacteristic
28C2×C4⋊#3622C2211yescharacteristic
29C2×#10587503C2211yescharacteristic
30C2×#10587503C2211yescharacteristic
31C2×#10587503C2211yescharacteristic
32C2×#10587503C2211yescharacteristic
33C2×C22⋊#3622C2211yescharacteristic
34#11762745—21no—
35#11762732—21no—
36#11762707C2211yescharacteristic
37#11762707C2211yescharacteristic
38#11762706—21no—
39#11762692—21no—
40#11762679C2211yescharacteristic
41#11762669C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762564C2211yescharacteristic
49#11762564C2211yescharacteristic
50C2×C17×#121C2211yescharacteristic
51C2×C4×D4×C17—21no—
52C2×C4×D4×C17C2211yescharacteristic
53C22×C4⋊C4×C17C2211yescharacteristic
54C17×#390C2211yescharacteristic
55C17×#389C2211yescharacteristic
56C17×#388—21no—
57C17×#386—21no—
58C17×#385—21no—
59C4×C4⋊C4×C17C2211yescharacteristic
60C4×C22⋊C4×C17C2211yescharacteristic
61C22×D4×C17—21no—
62C22×D4×C17C2311yescharacteristic
63C23×C4×C17—21no—
64C23×C4×C17—21no—
65C23×C4×C17C2311yescharacteristic
66C17×#121—22no—
67C17×#121Q814yes—
68C4×D4×C17—22no—
69C4×D4×C17—41no—
70C4×D4×C17—44no—
71C2×C4⋊C4×C17—22no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17—41no—
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×C17D411yescharacteristic
83C2×C22⋊C4×C17D411yescharacteristic
84C2×C42×C17—21no—
85C2×C42×C17C2311yescharacteristic
86C2×C42×C17C2311yescharacteristic
87C2×C22⋊Dic17—21no—
88C2×C22⋊Dic17—21no—
89#10587612—22no—
90#10587612—41no—
91C2×C4⋊Dic17—21no—
92C2×C4⋊Dic17—21no—
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96C2×C4⋊Dic17C2311yescharacteristic
97C2×#63225—21no—
98C2×#63225—21no—
99C2×#63225—21no—
100C2×#63225—21no—
101C2×C4×Dic17—21no—
102C2×#63223—22no—
103C2×#63223—22no—
104C2×#63223—22no—
105C22×#3622C2311yescharacteristic
106C22×#3622C2311yescharacteristic
107C22×#3622C2311yescharacteristic
108C22×#3622C2311yescharacteristic
109#10587556—22no—
110#10587556—41no—
111C17×#95—21no—
112C17×#95—21no—
113C17×#95D411yescharacteristic
114C17×#95D411yescharacteristic
115#10587503—22no—
116#10587503—22no—
117#10587503—22no—
118#10587503—41no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122C22⋊#3622—22no—
123#10587479—22no—
124#10587479—22no—
125#10587479—22no—
126#10587479—41no—
127#10587478—22no—
128#10587478D414yescomplemented
129C4⋊#3622—22no—
130C4⋊#3622—22no—
131C24×C17—21no—
132C24×C17C2×D411yescharacteristic
133C2×D4×C17—24no—
134C2×D4×C17—21no—
135C2×D4×C17—21no—
136C2×D4×C17—21no—
137C2×D4×C17—21no—
138C2×D4×C17—21no—
139C2×D4×C17—42no—
140C2×D4×C17—42no—
141C2×D4×C17C2×Q812yes—
142C22×C4×C17—21no—
143C22×C4×C17—21no—
144C22×C4×C17—21no—
145C22×C4×C17—21no—
146C22×C4×C17—21no—
147C22×C4×C17—21no—
148C22×C4×C17—21no—
149C22×C4×C17—42no—
150C22×C4×C17—42no—
151C22×C4×C17—42no—
152C22×C4×C17—42no—
153C22×C4×C17—81no—
154C22×C4×C17—81no—
155C22×C4×C17C2×D411yescharacteristic
156C22×C4×C17C2×D411yescharacteristic
157C22×C4×C17C2×D411yescharacteristic
158C22×C4×C17C2×D411yescharacteristic
159C22×C4×C17C2×D411yescharacteristic
160C22×C4×C17C2×Q812yes—
161C22×C4×C17#4112yes—
162C22×C4×C17C2411yescharacteristic
163C22×Dic17—21no—
164C22×Dic17—21no—
165C22×Dic17—21no—
166C22×Dic17—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×C17C2×D412yes—
174C4⋊C4×C17C2×Q812yes—
175C22⋊C4×C17—24no—
176C22⋊C4×C17—22no—
177C22⋊C4×C17—41no—
178C22⋊C4×C17—42no—
179C22⋊C4×C17—42no—
180C22⋊C4×C17—81no—
181C42×C17—21no—
182C42×C17—21no—
183C42×C17—21no—
184C42×C17—21no—
185C42×C17—42no—
186C22⋊Dic17—42no—
187C22⋊Dic17—81no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—21no—
192C4⋊Dic17—21no—
193C4⋊Dic17—21no—
194C4⋊Dic17—21no—
195C4⋊Dic17—42no—
196C4⋊Dic17—81no—
197C4⋊Dic17C2×D412yes—
198#63225—42no—
199#63225—42no—
200#63225—42no—
201#63225—81no—
202C4×Dic17—42no—
203#63223—28no—
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×#3622—21no—
214C2×#3622C2×D412yes—
215C2×#3622#4112yes—
216#26366—171nomaximal, Sylow
217C23×C17—21no—
218C23×C17—21no—
219C23×C17—21no—
220C23×C17—21no—
221C23×C17—21no—
222C23×C17—21no—
223C23×C17—22no—
224C23×C17—41no—
225C23×C17—41no—
226C23×C17—41no—
227C23×C17—41no—
228C23×C17—41no—
229C23×C17—41no—
230C23×C17C22⋊Q812yes—
231C23×C17C22×D411yescharacteristic
232D4×C17—22no—
233D4×C17—22no—
234D4×C17—48no—
235D4×C17—41no—
236D4×C17—41no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—22no—
242C2×C4×C17—22no—
243C2×C4×C17—22no—
244C2×C4×C17—22no—
245C2×C4×C17—22no—
246C2×C4×C17—21no—
247C2×C4×C17—21no—
248C2×C4×C17—22no—
249C2×C4×C17—21no—
250C2×C4×C17—21no—
251C2×C4×C17—21no—
252C2×C4×C17—21no—
253C2×C4×C17—21no—
254C2×C4×C17—21no—
255C2×C4×C17—21no—
256C2×C4×C17—41no—
257C2×C4×C17—41no—
258C2×C4×C17—41no—
259C2×C4×C17—41no—
260C2×C4×C17—41no—
261C2×C4×C17—41no—
262C2×C4×C17—41no—
263C2×C4×C17—84no—
264C2×C4×C17—84no—
265C2×C4×C17C22⋊Q812yes—
266C2×C4×C17AGL(1,ℤ/8)12yes—
267C2×C4×C17C22×D411yescharacteristic
268C2×C4×C17C22×Q811yescharacteristic
269C2×C4×C17C2×#4111yescharacteristic
270C2×C4×C17C2×#4111yescharacteristic
271C2×C4×C17H(5,𝔽2)11yescharacteristic
272C2×C4×C172−1+411yescharacteristic
273C2×Dic17—22no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281C2×Dic17—41no—
282C2×Dic17—41no—
283C2×Dic17—41no—
284C2×Dic17—41no—
285#3622—22no—
286#3622—22no—
287#3622—41no—
288#3622—41no—
289C2×#478—171no—
290C2×#428—171no—
291#2441—171no—
292#2361D1711yescharacteristic
293#2094—171no—
294#2088—171no—
295#2067—171no—
296#2059—171no—
297#1948—171no—
298#1918—171no—
299#1895—171no—
300#1875—171no—
301#1864—171no—
302#1853—171no—
303#1851—171no—
304C22×C17—22no—
305C22×C17—22no—
306C22×C17—22no—
307C22×C17—41no—
308C22×C17—41no—
309C22×C17—41no—
310C22×C17—41no—
311C22×C17—42no—
312C22×C17—42no—
313C22×C17—42no—
314C22×C17—81no—
315C22×C17—81no—
316C22×C17—81no—
317C22×C17—81no—
318C22×C17C2×C22⋊Q811yescharacteristic
319C22×C17D4211yescharacteristic
320C22×C17#54711yescharacteristic
321C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
322C22×C17#57511yescharacteristic
323C22×C17#57711yescharacteristic
324C22×C17#57711yescharacteristic
325C4×C17—22no—
326C4×C17—22no—
327C4×C17—42no—
328C4×C17—41no—
329C4×C17—41no—
330C4×C17—41no—
331C4×C17—42no—
332C4×C17—81no—
333C4×C17#52511yescharacteristic
334C4×C17#53911yescharacteristic
335C4×C17D4×Q811yescharacteristic
336C4×C17#55411yescharacteristic
337Dic17—42no—
338Dic17—81no—
339Dic17—81no—
340Dic17—81no—
341C22×M16—171no—
342C2×#125—171no—
343C2×#121C2×D1711yescharacteristic
344C2×C4×D4—21no—
345C2×C4×D4C2×D1711yescharacteristic
346C2×#117—171no—
347C22×C4⋊C4C2×D1711yescharacteristic
348#478—344no—
349#428—344no—
350C2×#107—171no—
351C2×#107—171no—
352C2×#106—171no—
353C2×#106—171no—
354C2×#106—171no—
355C2×#106—171no—
356C2×C4⋊C8—171no—
357C2×C4⋊C8—171no—
358C2×#102—171no—
359C2×#102—171no—
360C2×#102—171no—
361C2×#102—171no—
362C2×C22⋊C8—171no—
363#399—341no—
364#396—341no—
365#390C2×D1711yescharacteristic
366#389—171no—
367#389—171no—
368#389—341no—
369#389C2×D1711yescharacteristic
370#388—21no—
371#387—341no—
372#386—21no—
373#385—21no—
374#384—171no—
375#380—171no—
376C4×C4⋊C4C2×D1711yescharacteristic
377C4×C22⋊C4C2×D1711yescharacteristic
378#340—171no—
379#340—171no—
380#340—171no—
381#340—171no—
382#340—171no—
383#340—171no—
384#336—171no—
385#336—171no—
386C2×C17—42no—
387C2×C17—82no—
388C2×C17#244911yescharacteristic
389C2×C17#307711yescharacteristic
390C2×C17#308411yescharacteristic
391C2×C17#327111yescharacteristic
392C2×C17#327211yescharacteristic
393C2×C17#334511yescharacteristic
394C2×C17#335111yescharacteristic
395C22×D4—21no—
396C22×D4C22×D1711yescharacteristic
397C23×C4—21no—
398C23×C4—21no—
399C23×C4C22×D1711yescharacteristic
400C2×M16—342no—
401C2×M16—342no—
402C2×M16—342no—
403C22×C8—171no—
404C22×C8—171no—
405C22×C8—171no—
406C22×C8—171no—
407#125—342no—
408#125—681no—
409#121—22no—
410#121Dic3414yes—
411C4×D4—22no—
412C4×D4—41no—
413C4×D4—44no—
414#117—342no—
415#117—681no—
416C2×C4⋊C4—22no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—41no—
420C2×C4⋊C4—171no—
421C2×C4⋊C4—171no—
422C2×C4⋊C4—171no—
423C2×C4⋊C4—171no—
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⋊C4C22×D1711yescharacteristic
431C2×C4⋊C4C22×D1711yescharacteristic
432C2×C4⋊C4C22×D1711yescharacteristic
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—341no—
438C2×C22⋊C4—341no—
439C2×C22⋊C4C22⋊D1711yescharacteristic
440C2×C22⋊C4C22⋊D1711yescharacteristic
441C2×C42—21no—
442C2×C42—341no—
443C2×C42C22×D1711yescharacteristic
444C2×C42C22×D1711yescharacteristic
445#107—174no—
446#107—342no—
447#106—342no—
448#106—342no—
449#106—342no—
450#106—681no—
451C4⋊C8—342no—
452C4⋊C8—342no—
453#102—342no—
454#102—342no—
455#102—342no—
456#102—681no—
457C22⋊C8—342no—
458#95—21no—
459#95—21no—
460#95—341no—
461#95—341no—
462#95—341no—
463#95C22⋊D1711yescharacteristic
464#95C22⋊D1711yescharacteristic
465C17#2636611yescharacteristic, complemented, Sylow
466C24—21no—
467C24C2×C22⋊D1711yescharacteristic
468C2×D4—24no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4C2×Dic3412yes—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—81no—
489C22×C4—81no—
490C22×C4—341no—
491C22×C4—341no—
492C22×C4—341no—
493C22×C4—341no—
494C22×C4C2×Dic3412yes—
495C22×C4#6325212yes—
496C22×C4C2×C22⋊D1711yescharacteristic
497C22×C4C2×C22⋊D1711yescharacteristic
498C22×C4C2×C22⋊D1711yescharacteristic
499C22×C4C2×C22⋊D1711yescharacteristic
500C22×C4C2×C22⋊D1711yescharacteristic
501C22×C4C23×D1711yescharacteristic
502M16—348no—
503C2×C8—172no—
504C2×C8—172no—
505C2×C8—341no—
506C2×C8—341no—
507C2×C8—341no—
508C2×C8—341no—
509C2×C8—341no—
510C2×C8—341no—
511C2×C8—341no—
512C2×C8—341no—
513C2×C8—341no—
514C2×C8—341no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—172no—
522C4⋊C4—341no—
523C4⋊C4—341no—
524C4⋊C4—341no—
525C4⋊C4—341no—
526C4⋊C4—341no—
527C4⋊C4—341no—
528C4⋊C4—341no—
529C4⋊C4—682no—
530C4⋊C4—682no—
531C4⋊C4—682no—
532C4⋊C4—682no—
533C4⋊C4—1361no—
534C4⋊C4—1361no—
535C4⋊C4C2×Dic3412yes—
536C4⋊C4D4×D1712yes—
537C22⋊C4—24no—
538C22⋊C4—22no—
539C22⋊C4—41no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—81no—
543C22⋊C4—682no—
544C22⋊C4—1361no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—42no—
550C42—682no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—22no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23#1058761012yes—
565C23C22×C22⋊D1711yescharacteristic
566D4—22no—
567D4—22no—
568D4—48no—
569D4—41no—
570D4—41no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—342no—
600C2×C4—681no—
601C2×C4—681no—
602C2×C4—681no—
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#1058761012yes—
612C2×C4#1058761812yes—
613C2×C4C22×Dic3411yescharacteristic
614C2×C4C2×#6325211yescharacteristic
615C2×C4C2×D4×D1711yescharacteristic
616C2×C4#1058769111yescharacteristic
617C2×C4#41×D1711yescharacteristic
618C2×C4#1058769711yescharacteristic
619C8—342no—
620C8—342no—
621C8—681no—
622C8—681no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22C2×#1058761011yescharacteristic
638C22D4×C22⋊D1711yescharacteristic
639C22#1176345811yescharacteristic
640C22C2×#1058761811yescharacteristic
641C22#1176360711yescharacteristic
642C22#1176361711yescharacteristic
643C22#1176362011yescharacteristic
644C4—22no—
645C4—22no—
646C4—41no—
647C4—41no—
648C4—42no—
649C4—41no—
650C4—42no—
651C4—81no—
652C4—682no—
653C4—1361no—
654C4—1361no—
655C4—1361no—
656C4#1176331811yescharacteristic
657C4#1176340111yescharacteristic
658C4D4×Dic3411yescharacteristic
659C4#1176351811yescharacteristic
660C2—42no—
661C2—82no—
662C2#1453258711yescharacteristic
663C2#1453858111yescharacteristic
664C2#1453864611yescharacteristic
665C2#1454037011yescharacteristic
666C2#1454038011yescharacteristic
667C2#1454125411yescharacteristic
668C2#1454133811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2361C2
socleC23×C17C22×D4
prime radicalC2×C17×#121C22
hypercenter#2361D17
hypocenterC17#26366
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2361

Complements

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

NH ≅ G/Ntype
1. C2×#11763109C2semidirect
2. C2×#11762849C2semidirect
3. #14532497C2semidirect
4. #14530317C2semidirect
5. #14530286C2semidirect
6. #14530172C2semidirect
7. #14530122C2semidirect
8. #14529547C2semidirect
9. #14529355C2semidirect
12. #14529045C2semidirect
13. #14528977C2semidirect
14. #14528969C2semidirect
128. #10587478D4semidirect
465. C17#26366semidirect

Maximal subgroups

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

subgroupindex
C2×#117631092
C2×#117628492
#145324972
#145303172
#145302862
#145301722
#145301222
#145295472
#145293552
#145292232
#145291072
#145290452
#145289772
#145289692
C17×#23612
#2636617