Finite Group Databasebeta

← #17042177

Subgroups — #17042177

Order 4352.

730 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14537725C211yesmaximal, characteristic, complemented
2#14537212C211yesmaximal, characteristic
3#14529729C211yesmaximal, characteristic, complemented
4#14529560C211yesmaximal, characteristic, complemented
5#14529385C211yesmaximal, characteristic, complemented
6#14529385C211yesmaximal, characteristic, complemented
7#14529061C211yesmaximal, characteristic
8#14529061C211yesmaximal, characteristic
9#14528804C211yesmaximal, characteristic
10#14528670C211yesmaximal, characteristic, complemented
11#14528665C211yesmaximal, characteristic, complemented
12#14528661C211yesmaximal, characteristic, complemented
13#14528631C211yesmaximal, characteristic
14#14528624C211yesmaximal, characteristic
15C17×#3076C211yesmaximal, characteristic
16C2×#10587634C2211yescharacteristic
17C2×#10587634C2211yescharacteristic
18#11762940—21no—
19#11762940—21no—
20#11762940—21no—
21#11762940—21no—
22C2×#10587490C2211yescharacteristic
23C2×#10587490C2211yescharacteristic
24#11762836C2211yescharacteristic
25#11762836C2211yescharacteristic
26#11762833C2211yescharacteristic
27C2×#10587508C2211yescharacteristic
28C2×#10587508C2211yescharacteristic
29#11762765—21no—
30#11762765—21no—
31#11762765C2211yescharacteristic
32#11762765C2211yescharacteristic
33#11762765C2211yescharacteristic
34#11762765C2211yescharacteristic
35C4×#63223—21no—
36C4×#63223C2211yescharacteristic
37C2×#10587474—21no—
38C2×#10587474C2211yescharacteristic
39C2×#10587486C2211yescharacteristic
40C2×C8×Dic17C2211yescharacteristic
41C4×C4⋊Dic17C2211yescharacteristic
42#11762574C2211yescharacteristic
43#11762574C2211yescharacteristic
44#11762568C2211yescharacteristic
45#11762568C2211yescharacteristic
46#11762568C2211yescharacteristic
47#11762568C2211yescharacteristic
48#11762568C2211yescharacteristic
49#11762568C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C17×#533C2211yescharacteristic
53C4×#41×C17C2211yescharacteristic
54C17×#479C411yescharacteristic, complemented
55C17×#479C411yescharacteristic, complemented
56C17×#478C411yescharacteristic, complemented
57C17×#478C411yescharacteristic, complemented
58C17×#477C411yescharacteristic, complemented
59C17×#476C411yescharacteristic, complemented
60C17×#475C411yescharacteristic, complemented
61C17×#474C411yescharacteristic, complemented
62C17×#427C2211yescharacteristic
63C17×#427C2211yescharacteristic
64C2×C17×C4⋊C8C2211yescharacteristic
65C17×#416C2211yescharacteristic
66C17×#416C2211yescharacteristic
67C2×#41×C17C2311yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C22×C8×C17C2311yescharacteristic
70C17×C4⋊Q8C2×C411yescharacteristic
71C17×C4⋊Q8C2×C411yescharacteristic
72C17×#125C2×C411yescharacteristic
73C17×#125C2×C411yescharacteristic
74C17×C22⋊Q8—21no—
75C17×C22⋊Q8—21no—
76C4×Q8×C17—21no—
77C4×Q8×C17—21no—
78C4×Q8×C17C2×C411yescharacteristic
79C4×Q8×C17C2×C411yescharacteristic
80C4×D4×C17—21no—
81C4×D4×C17—21no—
82C4×D4×C17C2×C411yescharacteristic
83C4×D4×C17C2×C411yescharacteristic
84C17×#117—21no—
85C17×#117C2311yescharacteristic
86C17×#117C2311yescharacteristic
87C17×#117C2311yescharacteristic
88C2×C42×C17—21no—
89C2×C42×C17C2311yescharacteristic
90#10587634—22no—
91#10587634—22no—
92#10587634—22no—
93#10587634Q814yescomplemented
94C2×C4⋊Dic17—21no—
95C2×C4×Dic17C2311yescharacteristic
96C2×C4×Dic17C2311yescharacteristic
97C2×#63223—21no—
98C2×#63223—21no—
99C2×#63223C2311yescharacteristic
100C2×#63223C2311yescharacteristic
101C2×#63223C2311yescharacteristic
102C2×#63223C2311yescharacteristic
103C22×#3622—21no—
104C22×#3622—21no—
105C22×#3622C2311yescharacteristic
106C22×#3622C2311yescharacteristic
107C17×#107C2×C411yescharacteristic
108C17×#107C2×C411yescharacteristic
109C17×#107C2×C411yescharacteristic
110C17×#107C2×C411yescharacteristic
111C17×#106C2×C411yescharacteristic
112C17×#106C2×C411yescharacteristic
113C17×#106C2×C411yescharacteristic
114C17×#106C2×C411yescharacteristic
115C17×C4⋊C8C2×C411yescharacteristic
116C17×C4⋊C8C2×C411yescharacteristic
117C17×C4⋊C8C2×C411yescharacteristic
118C17×C4⋊C8C2×C411yescharacteristic
119C17×#103C2×C411yescharacteristic
120C17×#103C2×C411yescharacteristic
121C17×#103C2×C411yescharacteristic
122C17×#103C2×C411yescharacteristic
123C17×#102C2×C411yescharacteristic
124C17×#102C2×C411yescharacteristic
125C17×#102C2×C411yescharacteristic
126C17×#102C2×C411yescharacteristic
127#10587508—22no—
128#10587508—22no—
129#10587502—21no—
130#10587490—22no—
131#10587490—22no—
132#10587486—22no—
133C8×Dic17—22no—
134C4⋊#3622—21no—
135C4⋊#3622—21no—
136C4⋊#3622—21no—
137C4⋊#3622—21no—
138C4⋊#3622—21no—
139C4⋊#3622—21no—
140#10587474—21no—
141#10587474—21no—
142#10587474—21no—
143#10587474—21no—
144#10587474—21no—
145#10587474D411yescharacteristic
146#10587474D411yescharacteristic
147#10587474D411yescharacteristic
148#10587474D411yescharacteristic
149C4×#3622—21no—
150C4×#3622—21no—
151C4×#3622—21no—
152C4×#3622—21no—
153C4×#3622—21no—
154#41×C17—21no—
155#41×C17—21no—
156#41×C17C2×Q812yes—
157#41×C17#4112yes—
158C2×Q8×C17—21no—
159C2×Q8×C17—21no—
160C2×Q8×C17C22×C411yescharacteristic
161C2×D4×C17—21no—
162C2×D4×C17C22×C411yescharacteristic
163C22×C4×C17—21no—
164C22×C4×C17—21no—
165C22×C4×C17—21no—
166C22×C4×C17#4111yescharacteristic
167C22×C4×C17#4111yescharacteristic
168C22×C4×C17C2411yescharacteristic
169C22×Dic17—21no—
170C22×Dic17—21no—
171C2×C8×C17—21no—
172C2×C8×C17—21no—
173C2×C8×C17—21no—
174C2×C8×C17—21no—
175C2×C8×C17C22×C411yescharacteristic
176C2×C8×C17C22×C411yescharacteristic
177C2×C8×C17C22×C411yescharacteristic
178C2×C8×C17C22×C411yescharacteristic
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—21no—
187C4⋊C4×C17C22×C411yescharacteristic
188C4⋊C4×C17C22×C411yescharacteristic
189C4⋊C4×C17C22×C411yescharacteristic
190C4⋊C4×C17C22×C411yescharacteristic
191C4⋊C4×C17C22×C411yescharacteristic
192C4⋊C4×C17C22×C411yescharacteristic
193C22⋊C4×C17—21no—
194C22⋊C4×C17—21no—
195C22⋊C4×C17—21no—
196C22⋊C4×C17—21no—
197C22⋊C4×C17—21no—
198C42×C17—21no—
199C42×C17—21no—
200C42×C17—21no—
201C42×C17—21no—
202C42×C17—21no—
203C42×C17C22×C411yescharacteristic
204C42×C17C22×C411yescharacteristic
205C42×C17C2×D411yescharacteristic
206C42×C17C2×D411yescharacteristic
207C4⋊Dic17—42no—
208C4×Dic17—21no—
209C4×Dic17—21no—
210C4×Dic17—21no—
211C4×Dic17—21no—
212#63223—24no—
213#63223—21no—
214#63223—21no—
215#63223—24no—
216#63223—21no—
217#63223—21no—
218#63223—21no—
219#63223—21no—
220#63223C2×Q812yes—
221#63223#4112yes—
222C2×#3622—24no—
223C2×#3622—21no—
224C2×#3622—21no—
225C2×#3622—21no—
226C2×#3622—21no—
227C2×#3622—24no—
228C2×#3622—21no—
229C2×#3622—21no—
230C2×#3622C2×D411yescharacteristic
231C2×#3622C2×D411yescharacteristic
232C2×#3622C2×D411yescharacteristic
233C2×#3622C2×D411yescharacteristic
234C2×#3622C2×Q812yes—
235C2×#3622#4112yes—
236#21629—171nomaximal, Sylow
237C23×C17—21no—
238C23×C17C2×#4111yescharacteristic
239Q8×C17—21no—
240Q8×C17—81no—
241Q8×C17—81no—
242Q8×C17C4×Q812yes—
243D4×C17—22no—
244D4×C17—22no—
245D4×C17—21no—
246D4×C17C4×Q812yes—
247C2×C4×C17—21no—
248C2×C4×C17—22no—
249C2×C4×C17—22no—
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—81no—
259C2×C4×C17—81no—
260C2×C4×C17C4×D411yescharacteristic
261C2×C4×C17C4×D411yescharacteristic
262C2×C4×C17C4×D411yescharacteristic
263C2×C4×C17C4×D411yescharacteristic
264C2×C4×C17C23×C411yescharacteristic
265C2×C4×C17C22×D411yescharacteristic
266C2×C4×C17C22×Q811yescharacteristic
267C2×C4×C17C2×#4111yescharacteristic
268C2×C4×C17C2×#4111yescharacteristic
269C2×C4×C17C2×#4111yescharacteristic
270C2×Dic17—22no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C8×C17—21no—
277C8×C17—21no—
278C8×C17—21no—
279C8×C17—21no—
280#3622—22no—
281#3622—21no—
282#3622—22no—
283#3622—21no—
284#3622—41no—
285#3622—41no—
286#3622C22⋊Q812yes—
287#3622C22⋊Q812yes—
288#3076D1711yescharacteristic
289#2961—171no—
290#2875—171no—
291#1986—171no—
292#1950—171no—
293#1922—171no—
294#1922—171no—
295#1866—171no—
296#1866—171no—
297#1806—171no—
298#1773—171no—
299#1772—171no—
300#1771—171no—
301#1765—171no—
302#1764—171no—
303C22×C17—21no—
304C22×C17—21no—
305C22×C17—22no—
306C22×C17—41no—
307C22×C17—41no—
308C22×C17C2×C4×D411yescharacteristic
309C22×C17#52911yescharacteristic
310C22×C17#53311yescharacteristic
311C4×C17—21no—
312C4×C17—21no—
313C4×C17—21no—
314C4×C17—21no—
315C4×C17—22no—
316C4×C17—41no—
317C4×C17—41no—
318C4×C17—41no—
319C4×C17—41no—
320C4×C17—41no—
321C4×C17—41no—
322C4×C17C2×C4×Q811yescharacteristic
323C4×C17C4×#4111yescharacteristic
324C4×C17C2×C22⋊Q811yescharacteristic
325C4×C17#52511yescharacteristic
326Dic17—42no—
327Dic17—81no—
328C2×#131—171no—
329C2×#131—171no—
330#533C2×D1711yescharacteristic
331C4×#41C2×D1711yescharacteristic
332#479Dic1711yescharacteristic, complemented
333#479Dic1711yescharacteristic, complemented
334#478Dic1711yescharacteristic, complemented
335#478Dic1711yescharacteristic, complemented
336#477Dic1711yescharacteristic, complemented
337#476Dic1711yescharacteristic, complemented
338#475Dic1711yescharacteristic, complemented
339#474Dic1711yescharacteristic, complemented
340#446—341no—
341#446—341no—
342#446—341no—
343#446—341no—
344C2×#108—171no—
345C2×#108—171no—
346#427C2×D1711yescharacteristic
347#427C2×D1711yescharacteristic
348#424—171no—
349#424—171no—
350#423—171no—
351C2×C4⋊C8C2×D1711yescharacteristic
352C2×#104—171no—
353C2×#104—171no—
354#416C2×D1711yescharacteristic
355#416C2×D1711yescharacteristic
356#405—171no—
357#405—171no—
358#405—171no—
359#405—171no—
360#405—341no—
361#405—341no—
362C4×M16—171no—
363C4×M16—341no—
364C2×#97—171no—
365C2×#97—171no—
366C2×#97—341no—
367C2×C4×C8—171no—
368C4×C4⋊C4—171no—
369#341—171no—
370#341—171no—
371#339—171no—
372#339—171no—
373#339—171no—
374#339—171no—
375#339—171no—
376#339—171no—
377#336—171no—
378#336—171no—
379C2×C17—21no—
380C2×C17—21no—
381C2×C17—42no—
382C2×C17C4×C22⋊Q811yescharacteristic
383C2×C17#295611yescharacteristic
384C2×C17#295711yescharacteristic
385C2×#41C22×D1711yescharacteristic
386#131—174no—
387#131—342no—
388#131—342no—
389#131—342no—
390C2×M16—171no—
391C2×M16—171no—
392C2×M16—171no—
393C2×M16—171no—
394C2×M16—341no—
395C2×M16—341no—
396C22×C8—171no—
397C22×C8—171no—
398C22×C8—341no—
399C22×C8—341no—
400C22×C8C22×D1711yescharacteristic
401C22×C8C22×D1711yescharacteristic
402C4⋊Q8C2×Dic1711yescharacteristic
403C4⋊Q8C2×Dic1711yescharacteristic
404#125C2×Dic1711yescharacteristic
405#125C2×Dic1711yescharacteristic
406C22⋊Q8—21no—
407C22⋊Q8—21no—
408C4×Q8—21no—
409C4×Q8—21no—
410C4×Q8C2×Dic1711yescharacteristic
411C4×Q8C2×Dic1711yescharacteristic
412C4×D4—21no—
413C4×D4—21no—
414C4×D4C2×Dic1711yescharacteristic
415C4×D4C2×Dic1711yescharacteristic
416#117—21no—
417#117C22×D1711yescharacteristic
418#117C22×D1711yescharacteristic
419#117C22×D1711yescharacteristic
420C2×C4⋊C4—341no—
421C2×C42—21no—
422C2×C42—171no—
423C2×C42—171no—
424C2×C42C22×D1711yescharacteristic
425#108—342no—
426#108—342no—
427#107C2×Dic1711yescharacteristic
428#107C2×Dic1711yescharacteristic
429#107C2×Dic1711yescharacteristic
430#107C2×Dic1711yescharacteristic
431#106C2×Dic1711yescharacteristic
432#106C2×Dic1711yescharacteristic
433#106C2×Dic1711yescharacteristic
434#106C2×Dic1711yescharacteristic
435C4⋊C8—341no—
436C4⋊C8—341no—
437C4⋊C8—341no—
438C4⋊C8—341no—
439C4⋊C8—341no—
440C4⋊C8—341no—
441C4⋊C8C2×Dic1711yescharacteristic
442C4⋊C8C2×Dic1711yescharacteristic
443C4⋊C8C2×Dic1711yescharacteristic
444C4⋊C8C2×Dic1711yescharacteristic
445#104—342no—
446#104—342no—
447#103C2×Dic1711yescharacteristic
448#103C2×Dic1711yescharacteristic
449#103C2×Dic1711yescharacteristic
450#103C2×Dic1711yescharacteristic
451#102C2×Dic1711yescharacteristic
452#102C2×Dic1711yescharacteristic
453#102C2×Dic1711yescharacteristic
454#102C2×Dic1711yescharacteristic
455#97—171no—
456#97—171no—
457#97—171no—
458#97—171no—
459#97—341no—
460#97—341no—
461#97—341no—
462#97—341no—
463#97—342no—
464#97—341no—
465C4×C8—341no—
466C4×C8—341no—
467C4×C8—341no—
468C4×C8—341no—
469C4×C8—342no—
470C4×C8—341no—
471#95—341no—
472C17#2162911yescharacteristic, complemented, Sylow
473#41—21no—
474#41—21no—
475#41Q8×D1712yes—
476#41#6325612yes—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8C22×Dic1711yescharacteristic
480C2×D4—21no—
481C2×D4C22×Dic1711yescharacteristic
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—341no—
486C22×C4—341no—
487C22×C4#6325411yescharacteristic
488C22×C4#6325411yescharacteristic
489C22×C4C23×D1711yescharacteristic
490M16—172no—
491M16—172no—
492M16—344no—
493M16—344no—
494M16—341no—
495M16—341no—
496M16—341no—
497M16—341no—
498M16—341no—
499M16—341no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—171no—
505C2×C8—171no—
506C2×C8—172no—
507C2×C8—172no—
508C2×C8—171no—
509C2×C8—171no—
510C2×C8—344no—
511C2×C8—344no—
512C2×C8—341no—
513C2×C8—341no—
514C2×C8—341no—
515C2×C8—341no—
516C2×C8—341no—
517C2×C8—341no—
518C2×C8C22×Dic1711yescharacteristic
519C2×C8C22×Dic1711yescharacteristic
520C2×C8C22×Dic1711yescharacteristic
521C2×C8C22×Dic1711yescharacteristic
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—682no—
531C4⋊C4C22×Dic1711yescharacteristic
532C4⋊C4C22×Dic1711yescharacteristic
533C4⋊C4C22×Dic1711yescharacteristic
534C4⋊C4C22×Dic1711yescharacteristic
535C4⋊C4C22×Dic1711yescharacteristic
536C4⋊C4C22×Dic1711yescharacteristic
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—341no—
548C42—341no—
549C42—341no—
550C42—341no—
551C42D4×D1711yescharacteristic
552C42D4×D1711yescharacteristic
553C42C22×Dic1711yescharacteristic
554C42C22×Dic1711yescharacteristic
555C23—21no—
556C23C2×#6325411yescharacteristic
557Q8—21no—
558Q8—81no—
559Q8—81no—
560Q8Q8×Dic1712yes—
561D4—22no—
562D4—22no—
563D4—21no—
564D4Q8×Dic1712yes—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—342no—
579C2×C4—681no—
580C2×C4—681no—
581C2×C4—681no—
582C2×C4—681no—
583C2×C4—681no—
584C2×C4D4×Dic1711yescharacteristic
585C2×C4D4×Dic1711yescharacteristic
586C2×C4D4×Dic1711yescharacteristic
587C2×C4D4×Dic1711yescharacteristic
588C2×C4C2×D4×D1711yescharacteristic
589C2×C4C2×Q8×D1711yescharacteristic
590C2×C4C2×#6325611yescharacteristic
591C2×C4#41×D1711yescharacteristic
592C2×C4#41×D1711yescharacteristic
593C2×C4C23×Dic1711yescharacteristic
594C8—21no—
595C8—21no—
596C8—21no—
597C8—21no—
598C8—172no—
599C8—172no—
600C8—342no—
601C8—341no—
602C8—341no—
603C8—342no—
604C8—681no—
605C8—681no—
606C22—21no—
607C22—21no—
608C22—22no—
609C22—41no—
610C22—41no—
611C22C2×D4×Dic1711yescharacteristic
612C22#1176334511yescharacteristic
613C22#1176336111yescharacteristic
614C4—21no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—22no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—682no—
626C4—1361no—
627C4C2×Q8×Dic1711yescharacteristic
628C4#41×Dic1711yescharacteristic
629C4C22⋊Q8×D1711yescharacteristic
630C4#1176332211yescharacteristic
631C2—21no—
632C2—21no—
633C2—42no—
634C2C22⋊Q8×Dic1711yescharacteristic
635C2#1453769411yescharacteristic
636C2#1453770111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#3076C2
socleC22×C17C2×C4×D4
prime radicalC2×#41×C17C23
cosocleC22×C4×C17C24
hypercenter#3076D17
hypocenterC17#21629
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3076

Complements

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

NH ≅ G/Ntype
1. #14537725C2semidirect
3. #14529729C2semidirect
4. #14529560C2semidirect
5. #14529385C2semidirect
6. #14529385C2semidirect
10. #14528670C2semidirect
11. #14528665C2semidirect
12. #14528661C2semidirect
54. C17×#479C4semidirect
55. C17×#479C4semidirect
56. C17×#478C4semidirect
57. C17×#478C4semidirect
58. C17×#477C4semidirect
59. C17×#476C4semidirect
60. C17×#475C4semidirect
61. C17×#474C4semidirect
93. #10587634Q8semidirect
332. #479Dic17semidirect
333. #479Dic17semidirect
334. #478Dic17semidirect
335. #478Dic17semidirect
336. #477Dic17semidirect
337. #476Dic17semidirect
338. #475Dic17semidirect
339. #474Dic17semidirect
472. C17#21629semidirect

Maximal subgroups

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

subgroupindex
#145377252
#145372122
#145297292
#145295602
#145293852
#145290612
#145288042
#145286702
#145286652
#145286612
#145286312
#145286242
C17×#30762
#2162917