Finite Group Databasebeta

← #17004431

Subgroups — #17004431

Order 4352.

598 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14537895C211yesmaximal, characteristic
2#14537879C211yesmaximal, characteristic, complemented
3#14527656C211yesmaximal, characteristic, complemented
4#14527642C211yesmaximal, characteristic, complemented
5#14527635C211yesmaximal, characteristic, complemented
6#14527621C211yesmaximal, characteristic, complemented
7#14527603C211yesmaximal, characteristic, complemented
8SD16⋊#3622C211yesmaximal, characteristic, complemented
9Q16×#3622C211yesmaximal, characteristic, complemented
10SD16×#3622C211yesmaximal, characteristic, complemented
11#14527569C211yesmaximal, characteristic, complemented
12#14527218C211yesmaximal, characteristic
13#14527187C211yesmaximal, characteristic, complemented
14#14527084C211yesmaximal, characteristic, complemented
15C17×#3107C211yesmaximal, characteristic
16#11762940—21no—
17#11762940—21no—
18#11762940C2211yescharacteristic
19#11762940C2211yescharacteristic
20Q8×#3622C2211yescharacteristic
21Q8×#3622C2211yescharacteristic
22Q8×#3622C2211yescharacteristic
23Q8×#3622C2211yescharacteristic
24D4×#3622—21no—
25D4×#3622—21no—
26D4×#3622C2211yescharacteristic
27D4×#3622C2211yescharacteristic
28#11762870—21no—
29#11762870—21no—
30#11762866C2211yescharacteristic
31#11762862—21no—
32#11762862C2211yescharacteristic
33#11762862C2211yescharacteristic
34C2×C4⋊#3622—21no—
35C2×C4⋊#3622C2211yescharacteristic
36#11762562C2211yescharacteristic
37#11762562C2211yescharacteristic
38#11762560C2211yescharacteristic
39#11762560C2211yescharacteristic
40Q8⋊#3622C2211yescharacteristic
41Q8⋊#3622C2211yescharacteristic
42Q8⋊#3622C2211yescharacteristic
43Q8⋊#3622C2211yescharacteristic
44Q8⋊#3622C2211yescharacteristic
45Q8⋊#3622C2211yescharacteristic
46D4⋊#3622C2211yescharacteristic
47D4⋊#3622C2211yescharacteristic
48#11762531C2211yescharacteristic
49#11762531C2211yescharacteristic
50#11762530C2211yescharacteristic
51C8×#3622C2211yescharacteristic
52C17×#533C2211yescharacteristic
53C4×#41×C17C2211yescharacteristic
54C17×#485C411yescharacteristic
55C17×#485C411yescharacteristic
56C17×#475C411yescharacteristic
57C17×#474C411yescharacteristic
58C17×#462C411yescharacteristic
59C17×#461C411yescharacteristic
60C17×#452C411yescharacteristic
61C17×#452C411yescharacteristic
62C4×Q16×C17C2211yescharacteristic
63C4×Q16×C17C2211yescharacteristic
64C4×SD16×C17C2211yescharacteristic
65C4×SD16×C17C2211yescharacteristic
66C17×#431C2211yescharacteristic
67C2×#41×C17C2×C411yescharacteristic
68C2×Q16×C17C2×C411yescharacteristic
69C2×Q16×C17C2×C411yescharacteristic
70C2×SD16×C17C2×C411yescharacteristic
71C2×SD16×C17C2×C411yescharacteristic
72C17×C4⋊Q8C2×C411yescharacteristic
73C17×C4⋊Q8C2×C411yescharacteristic
74C17×#125—21no—
75C17×C22⋊Q8—21no—
76C17×C22⋊Q8C2×C411yescharacteristic
77C17×C22⋊Q8C2×C411yescharacteristic
78C4×Q8×C17—21no—
79C4×Q8×C17C2×C411yescharacteristic
80C4×Q8×C17C2311yescharacteristic
81C4×Q8×C17C2311yescharacteristic
82C4×Q8×C17C2311yescharacteristic
83C4×D4×C17—21no—
84C4×D4×C17—21no—
85C4×D4×C17C2×C411yescharacteristic
86C4×D4×C17C2311yescharacteristic
87C17×#117—21no—
88C17×#117—21no—
89C17×#117C2×C411yescharacteristic
90C2×C42×C17—21no—
91C2×C42×C17C2311yescharacteristic
92C22×#3622—21no—
93C22×#3622—21no—
94C22×#3622—21no—
95C17×#107C2×C411yescharacteristic
96C17×#107C2×C411yescharacteristic
97C17×#106C2×C411yescharacteristic
98C17×#106C2×C411yescharacteristic
99C17×C4⋊C8C2×C411yescharacteristic
100C17×C4⋊C8C2×C411yescharacteristic
101C17×#103C2×C411yescharacteristic
102C17×#103C2×C411yescharacteristic
103C17×#103C2×C411yescharacteristic
104C17×#103C2×C411yescharacteristic
105C17×#103C2×C411yescharacteristic
106C17×#103C2×C411yescharacteristic
107C17×#102C2×C411yescharacteristic
108C17×#102C2×C411yescharacteristic
109C17×C22⋊C8C2×C411yescharacteristic
110C17×C22⋊C8C2×C411yescharacteristic
111C4×C8×C17C2311yescharacteristic
112C4×C8×C17C2311yescharacteristic
113C22⋊#3622—21no—
114C22⋊#3622—21no—
115C22⋊#3622—21no—
116C22⋊#3622—21no—
117C22⋊#3622D411yescharacteristic
118C22⋊#3622D411yescharacteristic
119C4⋊#3622—21no—
120C4⋊#3622—21no—
121C4⋊#3622—21no—
122C4⋊#3622—21no—
123C4⋊#3622—21no—
124C4⋊#3622—21no—
125C4⋊#3622—21no—
126C4⋊#3622—21no—
127C4⋊#3622D411yescharacteristic
128C4⋊#3622D411yescharacteristic
129C4⋊#3622C2311yescharacteristic
130C4⋊#3622C2311yescharacteristic
131C4⋊#3622C2311yescharacteristic
132C4⋊#3622C2311yescharacteristic
133#10587474—21no—
134#10587474—21no—
135#10587474—21no—
136C4×#3622—21no—
137C4×#3622—21no—
138C4×#3622—21no—
139C4×#3622—21no—
140C4×#3622C2311yescharacteristic
141C4×#3622C2311yescharacteristic
142C4×#3622C2311yescharacteristic
143C4×#3622C2311yescharacteristic
144#41×C17—22no—
145#41×C17—22no—
146C2×Q8×C17—21no—
147C2×Q8×C17C22×C411yescharacteristic
148C2×Q8×C17C22×C411yescharacteristic
149C2×Q8×C17C22×C411yescharacteristic
150C2×D4×C17—21no—
151C2×D4×C17C22×C411yescharacteristic
152C22×C4×C17—21no—
153C22×C4×C17—21no—
154C22×C4×C17—21no—
155C22×C4×C17C22×C411yescharacteristic
156C22×C4×C17C2×D411yescharacteristic
157C22×C4×C17#4111yescharacteristic
158Q16×C17—22no—
159Q16×C17—22no—
160SD16×C17—22no—
161SD16×C17—22no—
162C2×C8×C17C22×C411yescharacteristic
163C2×C8×C17C22×C411yescharacteristic
164C2×C8×C17C22×C411yescharacteristic
165C2×C8×C17C22×C411yescharacteristic
166C4⋊C4×C17—21no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17—21no—
171C4⋊C4×C17—21no—
172C4⋊C4×C17—21no—
173C4⋊C4×C17—21no—
174C4⋊C4×C17—21no—
175C4⋊C4×C17C22×C411yescharacteristic
176C4⋊C4×C17C22×C411yescharacteristic
177C4⋊C4×C17C22×C411yescharacteristic
178C4⋊C4×C17C22×C411yescharacteristic
179C22⋊C4×C17—21no—
180C22⋊C4×C17—21no—
181C22⋊C4×C17—21no—
182C22⋊C4×C17—21no—
183C22⋊C4×C17—21no—
184C42×C17—21no—
185C42×C17—21no—
186C42×C17—21no—
187C42×C17—21no—
188C42×C17—21no—
189C42×C17C22×C411yescharacteristic
190C42×C17C2×D411yescharacteristic
191C42×C17#4111yescharacteristic
192C42×C17C2411yescharacteristic
193C2×#3622—21no—
194C2×#3622—21no—
195C2×#3622—21no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—81no—
200C2×#3622—81no—
201C2×#3622—81no—
202C2×#3622C2×D411yescharacteristic
203C2×#3622C2×D411yescharacteristic
204C2×#3622C2×D411yescharacteristic
205C2×#3622C2×D411yescharacteristic
206#17921—171nomaximal, Sylow
207C23×C17—21no—
208C23×C17C4×D411yescharacteristic
209Q8×C17—21no—
210Q8×C17—21no—
211Q8×C17—21no—
212Q8×C17—21no—
213Q8×C17—21no—
214Q8×C17—81no—
215Q8×C17#13112yes—
216D4×C17—24no—
217D4×C17—21no—
218D4×C17#13112yes—
219C2×C4×C17—24no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—41no—
232C2×C4×C17—41no—
233C2×C4×C17—41no—
234C2×C4×C17—81no—
235C2×C4×C17C4×D411yescharacteristic
236C2×C4×C17C4×D411yescharacteristic
237C2×C4×C17C4×D411yescharacteristic
238C2×C4×C17C23×C411yescharacteristic
239C2×C4×C17C22×D411yescharacteristic
240C2×C4×C17C2×#4111yescharacteristic
241C8×C17—21no—
242C8×C17—21no—
243C8×C17—21no—
244C8×C17—21no—
245#3622—21no—
246#3622—21no—
247#3622—21no—
248#3622—21no—
249#3622—41no—
250#3622—41no—
251#3622—41no—
252#3622—41no—
253#3622—41no—
254#3622—41no—
255#3107D1711yescharacteristic
256#2988—171no—
257#2985—171no—
258#1587—171no—
259#1585—171no—
260#1584—171no—
261#1582—171no—
262#1579—171no—
263SD16⋊C8—171no—
264C8×Q16—171no—
265C8×SD16—171no—
266#1573—171no—
267#1488—171no—
268#1481—171no—
269#1454—171no—
270C22×C17—21no—
271C22×C17—22no—
272C22×C17—21no—
273C22×C17—21no—
274C22×C17—41no—
275C22×C17—41no—
276C22×C17C2×C4×D411yescharacteristic
277C4×C17—21no—
278C4×C17—22no—
279C4×C17—21no—
280C4×C17—21no—
281C4×C17—21no—
282C4×C17—41no—
283C4×C17—41no—
284C4×C17—41no—
285C4×C17—41no—
286C4×C17—41no—
287C4×C17—41no—
288C4×C17C2×#13111yescharacteristic
289C4×C17C2×#13111yescharacteristic
290C4×C17#57711yescharacteristic
291C4×C17#57811yescharacteristic
292#533C2×D1711yescharacteristic
293C4×#41C2×D1711yescharacteristic
294#485Dic1711yescharacteristic
295#485Dic1711yescharacteristic
296#475Dic1711yescharacteristic
297#474Dic1711yescharacteristic
298#462Dic1711yescharacteristic
299#461Dic1711yescharacteristic
300#452Dic1711yescharacteristic
301#452Dic1711yescharacteristic
302#446—171no—
303#446—171no—
304#446—341no—
305#446—341no—
306C8×Q8—171no—
307C8×Q8—171no—
308C8×Q8—171no—
309C8×Q8—171no—
310C4×Q16C2×D1711yescharacteristic
311C4×Q16C2×D1711yescharacteristic
312C4×SD16C2×D1711yescharacteristic
313C4×SD16C2×D1711yescharacteristic
314C8×D4—171no—
315C8×D4—171no—
316C8×D4—341no—
317C8×D4—341no—
318#433—341no—
319#433—341no—
320#432—171no—
321#431—171no—
322#431—171no—
323#431—341no—
324#431C2×D1711yescharacteristic
325C2×C4⋊C8—171no—
326C2×C4⋊C8—341no—
327#335—171no—
328#335—171no—
329#334—171no—
330#334—171no—
331Q8⋊C8—171no—
332Q8⋊C8—171no—
333Q8⋊C8—171no—
334Q8⋊C8—171no—
335Q8⋊C8—171no—
336Q8⋊C8—171no—
337D4⋊C8—171no—
338D4⋊C8—171no—
339#322—171no—
340#322—171no—
341#322—171no—
342C82—171no—
343C2×C17—41no—
344C2×C17—42no—
345C2×C17#292911yescharacteristic
346C2×C17#294711yescharacteristic
347C2×C17#295711yescharacteristic
348C2×#41C2×Dic1711yescharacteristic
349C2×Q16C2×Dic1711yescharacteristic
350C2×Q16C2×Dic1711yescharacteristic
351C2×SD16C2×Dic1711yescharacteristic
352C2×SD16C2×Dic1711yescharacteristic
353C22×C8—341no—
354C22×C8—341no—
355C22×C8—341no—
356C4⋊Q8C2×Dic1711yescharacteristic
357C4⋊Q8C2×Dic1711yescharacteristic
358#125—21no—
359C22⋊Q8—21no—
360C22⋊Q8C2×Dic1711yescharacteristic
361C22⋊Q8C2×Dic1711yescharacteristic
362C4×Q8—21no—
363C4×Q8C2×Dic1711yescharacteristic
364C4×Q8C22×D1711yescharacteristic
365C4×Q8C22×D1711yescharacteristic
366C4×Q8C22×D1711yescharacteristic
367C4×D4—21no—
368C4×D4—21no—
369C4×D4C2×Dic1711yescharacteristic
370C4×D4C22×D1711yescharacteristic
371#117—21no—
372#117—21no—
373#117C2×Dic1711yescharacteristic
374C2×C42—21no—
375C2×C42C22×D1711yescharacteristic
376#107C2×Dic1711yescharacteristic
377#107C2×Dic1711yescharacteristic
378#106C2×Dic1711yescharacteristic
379#106C2×Dic1711yescharacteristic
380C4⋊C8—171no—
381C4⋊C8—171no—
382C4⋊C8—171no—
383C4⋊C8—171no—
384C4⋊C8—171no—
385C4⋊C8—171no—
386C4⋊C8—341no—
387C4⋊C8—341no—
388C4⋊C8—341no—
389C4⋊C8—341no—
390C4⋊C8—341no—
391C4⋊C8—341no—
392C4⋊C8—341no—
393C4⋊C8—341no—
394C4⋊C8C2×Dic1711yescharacteristic
395C4⋊C8C2×Dic1711yescharacteristic
396#103C2×Dic1711yescharacteristic
397#103C2×Dic1711yescharacteristic
398#103C2×Dic1711yescharacteristic
399#103C2×Dic1711yescharacteristic
400#103C2×Dic1711yescharacteristic
401#103C2×Dic1711yescharacteristic
402#102C2×Dic1711yescharacteristic
403#102C2×Dic1711yescharacteristic
404C22⋊C8—171no—
405C22⋊C8—171no—
406C22⋊C8—341no—
407C22⋊C8—341no—
408C22⋊C8—341no—
409C22⋊C8—341no—
410C22⋊C8C2×Dic1711yescharacteristic
411C22⋊C8C2×Dic1711yescharacteristic
412#97—341no—
413#97—341no—
414#97—341no—
415C4×C8—171no—
416C4×C8—171no—
417C4×C8—171no—
418C4×C8—171no—
419C4×C8—341no—
420C4×C8—341no—
421C4×C8—341no—
422C4×C8—341no—
423C4×C8C22×D1711yescharacteristic
424C4×C8C22×D1711yescharacteristic
425C17#1792111yescharacteristic, complemented, Sylow
426#41—22no—
427#41—22no—
428C2×Q8—21no—
429C2×Q8C22×Dic1711yescharacteristic
430C2×Q8C22×Dic1711yescharacteristic
431C2×Q8C22×Dic1711yescharacteristic
432C2×D4—21no—
433C2×D4C22×Dic1711yescharacteristic
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4D4×D1711yescharacteristic
438C22×C4#6325411yescharacteristic
439C22×C4C22×Dic1711yescharacteristic
440Q16—22no—
441Q16—22no—
442SD16—22no—
443SD16—22no—
444C2×C8—171no—
445C2×C8—171no—
446C2×C8—171no—
447C2×C8—171no—
448C2×C8—341no—
449C2×C8—341no—
450C2×C8—341no—
451C2×C8—341no—
452C2×C8—341no—
453C2×C8—341no—
454C2×C8—1361no—
455C2×C8—1361no—
456C2×C8—1361no—
457C2×C8C22×Dic1711yescharacteristic
458C2×C8C22×Dic1711yescharacteristic
459C2×C8C22×Dic1711yescharacteristic
460C2×C8C22×Dic1711yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4C22×Dic1711yescharacteristic
471C4⋊C4C22×Dic1711yescharacteristic
472C4⋊C4C22×Dic1711yescharacteristic
473C4⋊C4C22×Dic1711yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42D4×D1711yescharacteristic
485C42#6325411yescharacteristic
486C42C22×Dic1711yescharacteristic
487C42C23×D1711yescharacteristic
488C23—21no—
489C23D4×Dic1711yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8—81no—
496Q8#1058763412yes—
497D4—24no—
498D4—21no—
499D4#1058763412yes—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—81no—
516C2×C4D4×Dic1711yescharacteristic
517C2×C4D4×Dic1711yescharacteristic
518C2×C4D4×Dic1711yescharacteristic
519C2×C4C2×D4×D1711yescharacteristic
520C2×C4C2×#6325411yescharacteristic
521C2×C4C23×Dic1711yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—341no—
527C8—341no—
528C8—341no—
529C8—341no—
530C8—681no—
531C8—681no—
532C8—681no—
533C8—681no—
534C8—681no—
535C8—681no—
536C22—21no—
537C22—22no—
538C22—21no—
539C22—21no—
540C22—41no—
541C22—41no—
542C22C2×D4×Dic1711yescharacteristic
543C4—21no—
544C4—22no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4C2×#1058763411yescharacteristic
555C4C2×#1058763411yescharacteristic
556C4#1176361811yescharacteristic
557C4#1176362511yescharacteristic
558C2—41no—
559C2—42no—
560C2#1453747911yescharacteristic
561C2#1453762811yescharacteristic
562C2#1453770111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
FittingC17×#3107C2
socleC22×C17C2×C4×D4
prime radicalC2×#41×C17C2×C4
cosocleC42×C17C24
hypercenter#3107D17
hypocenterC17#17921
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3107

Complements

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

NH ≅ G/Ntype
2. #14537879C2semidirect
3. #14527656C2semidirect
4. #14527642C2semidirect
5. #14527635C2semidirect
6. #14527621C2semidirect
7. #14527603C2semidirect
8. SD16⋊#3622C2semidirect
9. Q16×#3622C2semidirect
10. SD16×#3622C2semidirect
11. #14527569C2semidirect
13. #14527187C2semidirect
14. #14527084C2semidirect
425. C17#17921semidirect

Maximal subgroups

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

subgroupindex
#145378952
#145378792
#145276562
#145276422
#145276352
#145276212
#145276032
SD16⋊#36222
Q16×#36222
SD16×#36222
#145275692
#145272182
#145271872
#145270842
C17×#31072
#1792117