Finite Group Databasebeta

← #17056419

Subgroups — #17056419

Order 4352.

692 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14537733C211yesmaximal, characteristic, complemented
2#14537352C211yesmaximal, characteristic
3#14529766C211yesmaximal, characteristic, complemented
4#14529560C211yesmaximal, characteristic, complemented
5#14529538C211yesmaximal, characteristic, complemented
6#14529533C211yesmaximal, characteristic, complemented
7#14529385C211yesmaximal, characteristic, complemented
8#14529385C211yesmaximal, characteristic, complemented
9#14529061C211yesmaximal, characteristic
10#14529061C211yesmaximal, characteristic
11#14528643C211yesmaximal, characteristic, complemented
12#14528631C211yesmaximal, characteristic
13#14528624C211yesmaximal, characteristic
14#14528594C211yesmaximal, characteristic
15C17×#3076C211yesmaximal, characteristic
16C2×#10587634C2211yescharacteristic
17C2×#10587634C2211yescharacteristic
18#11762940—21no—
19#11762940—21no—
20#11762940—21no—
21Q8×#3622—21no—
22C2×#10587490C2211yescharacteristic
23C2×#10587490C2211yescharacteristic
24#11762836—21no—
25#11762836—21no—
26#11762836C2211yescharacteristic
27#11762836C2211yescharacteristic
28#11762836C2211yescharacteristic
29#11762836C2211yescharacteristic
30#11762833—21no—
31#11762833C2211yescharacteristic
32C2×C4⋊#3622—21no—
33C2×C4⋊#3622C2211yescharacteristic
34C2×#10587508C2211yescharacteristic
35C2×#10587508C2211yescharacteristic
36#11762765C2211yescharacteristic
37#11762765C2211yescharacteristic
38C4×#63223C2211yescharacteristic
39C2×#10587486C2211yescharacteristic
40C2×C8×Dic17C2211yescharacteristic
41#11762655C2211yescharacteristic
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—41no—
93#10587634—41no—
94C2×C4×Dic17C2311yescharacteristic
95C2×C4×Dic17C2311yescharacteristic
96C2×#63223—21no—
97C2×#63223—21no—
98C2×#63223C2311yescharacteristic
99C2×#63223C2311yescharacteristic
100C2×#63223C2311yescharacteristic
101C2×#63223C2311yescharacteristic
102C22×#3622—21no—
103C22×#3622—21no—
104C22×#3622C2311yescharacteristic
105C22×#3622C2311yescharacteristic
106C17×#107C2×C411yescharacteristic
107C17×#107C2×C411yescharacteristic
108C17×#107C2×C411yescharacteristic
109C17×#107C2×C411yescharacteristic
110C17×#106C2×C411yescharacteristic
111C17×#106C2×C411yescharacteristic
112C17×#106C2×C411yescharacteristic
113C17×#106C2×C411yescharacteristic
114C17×C4⋊C8C2×C411yescharacteristic
115C17×C4⋊C8C2×C411yescharacteristic
116C17×C4⋊C8C2×C411yescharacteristic
117C17×C4⋊C8C2×C411yescharacteristic
118C17×#103C2×C411yescharacteristic
119C17×#103C2×C411yescharacteristic
120C17×#103C2×C411yescharacteristic
121C17×#103C2×C411yescharacteristic
122C17×#102C2×C411yescharacteristic
123C17×#102C2×C411yescharacteristic
124C17×#102C2×C411yescharacteristic
125C17×#102C2×C411yescharacteristic
126#10587508—22no—
127#10587508Q814yescomplemented
128#10587502—21no—
129#10587502—21no—
130#10587490—41no—
131#10587490—41no—
132#10587486—41no—
133C8×Dic17—41no—
134C4⋊#3622—21no—
135C4⋊#3622—21no—
136C4⋊#3622—21no—
137C4⋊#3622—21no—
138C4⋊#3622—21no—
139C4⋊#3622—21no—
140C4⋊#3622—21no—
141C4⋊#3622—21no—
142C4⋊#3622—21no—
143C4⋊#3622—21no—
144C4⋊#3622D411yescharacteristic
145C4⋊#3622D411yescharacteristic
146C4⋊#3622D411yescharacteristic
147C4⋊#3622D411yescharacteristic
148#10587474—21no—
149#10587474—21no—
150#10587474—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—21no—
208C4×Dic17—21no—
209C4×Dic17—21no—
210C4×Dic17C2×Q812yes—
211#63223—21no—
212#63223—21no—
213#63223—21no—
214#63223—21no—
215#63223—21no—
216#63223—21no—
217#63223—21no—
218#63223—42no—
219#63223—42no—
220#63223C2×Q812yes—
221C2×#3622—21no—
222C2×#3622—21no—
223C2×#3622—21no—
224C2×#3622—21no—
225C2×#3622—21no—
226C2×#3622—21no—
227C2×#3622—21no—
228C2×#3622—21no—
229C2×#3622—42no—
230C2×#3622—42no—
231C2×#3622C2×D411yescharacteristic
232C2×#3622C2×D411yescharacteristic
233C2×#3622C2×D411yescharacteristic
234C2×#3622C2×D411yescharacteristic
235#22984—171nomaximal, Sylow
236C23×C17—21no—
237C23×C17C2×#4111yescharacteristic
238Q8×C17—21no—
239Q8×C17—42no—
240Q8×C17—81no—
241Q8×C17C4×Q812yes—
242D4×C17—22no—
243D4×C17—22no—
244D4×C17—21no—
245D4×C17C4×Q812yes—
246C2×C4×C17—21no—
247C2×C4×C17—22no—
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—41no—
256C2×C4×C17—41no—
257C2×C4×C17—42no—
258C2×C4×C17—81no—
259C2×C4×C17C4×D411yescharacteristic
260C2×C4×C17C4×D411yescharacteristic
261C2×C4×C17C4×D411yescharacteristic
262C2×C4×C17C4×D411yescharacteristic
263C2×C4×C17C23×C411yescharacteristic
264C2×C4×C17C22×D411yescharacteristic
265C2×C4×C17C22×Q811yescharacteristic
266C2×C4×C17C2×#4111yescharacteristic
267C2×C4×C17H(5,𝔽2)11yescharacteristic
268C2×C4×C172−1+411yescharacteristic
269C2×Dic17—22no—
270C2×Dic17—41no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275C8×C17—21no—
276C8×C17—21no—
277C8×C17—21no—
278C8×C17—21no—
279#3622—21no—
280#3622—21no—
281#3622—21no—
282#3622—21no—
283#3622—22no—
284#3622—41no—
285#3622—41no—
286#3622—41no—
287#3076D1711yescharacteristic
288#2962—171no—
289#2906—171no—
290#1995—171no—
291#1950—171no—
292#1946—171no—
293#1945—171no—
294#1922—171no—
295#1922—171no—
296#1866—171no—
297#1866—171no—
298#1768—171no—
299#1765—171no—
300#1764—171no—
301#1759—171no—
302C22×C17—21no—
303C22×C17—21no—
304C22×C17—22no—
305C22×C17—41no—
306C22×C17—41no—
307C22×C17C2×C4×D411yescharacteristic
308C22×C17#55211yescharacteristic
309C22×C17#55411yescharacteristic
310C4×C17—22no—
311C4×C17—21no—
312C4×C17—21no—
313C4×C17—21no—
314C4×C17—21no—
315C4×C17—41no—
316C4×C17—41no—
317C4×C17—41no—
318C4×C17—41no—
319C4×C17—41no—
320C4×C17—41no—
321C4×C17C2×C4×Q811yescharacteristic
322C4×C17#52011yescharacteristic
323C4×C17#54611yescharacteristic
324C4×C17D4×Q811yescharacteristic
325Dic17—42no—
326Dic17—81no—
327C2×#131—171no—
328C2×#131—171no—
329#533C2×D1711yescharacteristic
330C4×#41C2×D1711yescharacteristic
331#479Dic1711yescharacteristic, complemented
332#479Dic1711yescharacteristic, complemented
333#478Dic1711yescharacteristic, complemented
334#478Dic1711yescharacteristic, complemented
335#477Dic1711yescharacteristic, complemented
336#476Dic1711yescharacteristic, complemented
337#475Dic1711yescharacteristic, complemented
338#474Dic1711yescharacteristic, complemented
339#446—341no—
340#446—341no—
341#446—341no—
342C8×Q8—341no—
343C2×#108—171no—
344C2×#108—171no—
345#427C2×D1711yescharacteristic
346#427C2×D1711yescharacteristic
347#424—171no—
348#424—171no—
349#424—171no—
350#424—171no—
351#424—341no—
352#424—341no—
353#423—171no—
354#423—341no—
355C2×C4⋊C8—171no—
356C2×C4⋊C8—341no—
357C2×C4⋊C8C2×D1711yescharacteristic
358C2×#104—171no—
359C2×#104—171no—
360#416C2×D1711yescharacteristic
361#416C2×D1711yescharacteristic
362#405—171no—
363#405—171no—
364C4×M16—171no—
365C2×#97—171no—
366C2×C4×C8—171no—
367#376—171no—
368#341—171no—
369#341—171no—
370#339—171no—
371#339—171no—
372#339—171no—
373#339—171no—
374#339—171no—
375#339—171no—
376#336—171no—
377#336—171no—
378C2×C17—21no—
379C2×C17—21no—
380C2×C17—42no—
381C2×C17#236011yescharacteristic
382C2×C17#295511yescharacteristic
383C2×C17#295711yescharacteristic
384C2×#41C22×D1711yescharacteristic
385#131—342no—
386#131—342no—
387#131—681no—
388#131—681no—
389C2×M16—171no—
390C2×M16—171no—
391C2×M16—171no—
392C2×M16—171no—
393C2×M16—341no—
394C2×M16—341no—
395C22×C8—171no—
396C22×C8—171no—
397C22×C8—341no—
398C22×C8—341no—
399C22×C8C22×D1711yescharacteristic
400C22×C8C22×D1711yescharacteristic
401C4⋊Q8C2×Dic1711yescharacteristic
402C4⋊Q8C2×Dic1711yescharacteristic
403#125C2×Dic1711yescharacteristic
404#125C2×Dic1711yescharacteristic
405C22⋊Q8—21no—
406C22⋊Q8—21no—
407C4×Q8—21no—
408C4×Q8—21no—
409C4×Q8C2×Dic1711yescharacteristic
410C4×Q8C2×Dic1711yescharacteristic
411C4×D4—21no—
412C4×D4—21no—
413C4×D4C2×Dic1711yescharacteristic
414C4×D4C2×Dic1711yescharacteristic
415#117—21no—
416#117C22×D1711yescharacteristic
417#117C22×D1711yescharacteristic
418#117C22×D1711yescharacteristic
419C2×C42—21no—
420C2×C42—171no—
421C2×C42—171no—
422C2×C42C22×D1711yescharacteristic
423#108—681no—
424#108—681no—
425#107C2×Dic1711yescharacteristic
426#107C2×Dic1711yescharacteristic
427#107C2×Dic1711yescharacteristic
428#107C2×Dic1711yescharacteristic
429#106C2×Dic1711yescharacteristic
430#106C2×Dic1711yescharacteristic
431#106C2×Dic1711yescharacteristic
432#106C2×Dic1711yescharacteristic
433C4⋊C8—171no—
434C4⋊C8—171no—
435C4⋊C8—171no—
436C4⋊C8—171no—
437C4⋊C8—341no—
438C4⋊C8—341no—
439C4⋊C8—341no—
440C4⋊C8—341no—
441C4⋊C8—341no—
442C4⋊C8—341no—
443C4⋊C8—341no—
444C4⋊C8—341no—
445C4⋊C8—341no—
446C4⋊C8—341no—
447C4⋊C8C2×Dic1711yescharacteristic
448C4⋊C8C2×Dic1711yescharacteristic
449C4⋊C8C2×Dic1711yescharacteristic
450C4⋊C8C2×Dic1711yescharacteristic
451#104—174no—
452#104—342no—
453#103C2×Dic1711yescharacteristic
454#103C2×Dic1711yescharacteristic
455#103C2×Dic1711yescharacteristic
456#103C2×Dic1711yescharacteristic
457#102C2×Dic1711yescharacteristic
458#102C2×Dic1711yescharacteristic
459#102C2×Dic1711yescharacteristic
460#102C2×Dic1711yescharacteristic
461#97—341no—
462#97—341no—
463#97—341no—
464#97—681no—
465C4×C8—341no—
466C4×C8—341no—
467C4×C8—341no—
468C4×C8—681no—
469#95—341no—
470#95—341no—
471C17#2298411yescharacteristic, complemented, Sylow
472#41—21no—
473#41—21no—
474#41Q8×D1712yes—
475#41#6325612yes—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8C22×Dic1711yescharacteristic
479C2×D4—21no—
480C2×D4C22×Dic1711yescharacteristic
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—341no—
485C22×C4—341no—
486C22×C4#6325411yescharacteristic
487C22×C4#6325411yescharacteristic
488C22×C4C23×D1711yescharacteristic
489M16—172no—
490M16—341no—
491M16—341no—
492M16—341no—
493M16—341no—
494M16—341no—
495M16—341no—
496M16—341no—
497M16—682no—
498M16—682no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—171no—
504C2×C8—171no—
505C2×C8—171no—
506C2×C8—171no—
507C2×C8—341no—
508C2×C8—341no—
509C2×C8—341no—
510C2×C8—341no—
511C2×C8—341no—
512C2×C8—341no—
513C2×C8—341no—
514C2×C8—341no—
515C2×C8—682no—
516C2×C8—682no—
517C2×C8C22×Dic1711yescharacteristic
518C2×C8C22×Dic1711yescharacteristic
519C2×C8C22×Dic1711yescharacteristic
520C2×C8C22×Dic1711yescharacteristic
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4C22×Dic1711yescharacteristic
530C4⋊C4C22×Dic1711yescharacteristic
531C4⋊C4C22×Dic1711yescharacteristic
532C4⋊C4C22×Dic1711yescharacteristic
533C4⋊C4C22×Dic1711yescharacteristic
534C4⋊C4C22×Dic1711yescharacteristic
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—172no—
546C42—341no—
547C42—341no—
548C42—341no—
549C42D4×D1711yescharacteristic
550C42D4×D1711yescharacteristic
551C42C22×Dic1711yescharacteristic
552C42C22×Dic1711yescharacteristic
553C23—21no—
554C23C2×#6325411yescharacteristic
555Q8—21no—
556Q8—42no—
557Q8—81no—
558Q8Q8×Dic1712yes—
559D4—22no—
560D4—22no—
561D4—21no—
562D4Q8×Dic1712yes—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—81no—
576C2×C4—342no—
577C2×C4—681no—
578C2×C4—681no—
579C2×C4—681no—
580C2×C4—681no—
581C2×C4—681no—
582C2×C4D4×Dic1711yescharacteristic
583C2×C4D4×Dic1711yescharacteristic
584C2×C4D4×Dic1711yescharacteristic
585C2×C4D4×Dic1711yescharacteristic
586C2×C4C2×D4×D1711yescharacteristic
587C2×C4C2×Q8×D1711yescharacteristic
588C2×C4C2×#6325611yescharacteristic
589C2×C4#1058769611yescharacteristic
590C2×C4#1058769711yescharacteristic
591C2×C4C23×Dic1711yescharacteristic
592C8—21no—
593C8—21no—
594C8—21no—
595C8—21no—
596C8—342no—
597C8—341no—
598C8—341no—
599C8—341no—
600C8—341no—
601C8—681no—
602C8—681no—
603C8—681no—
604C22—21no—
605C22—21no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22C2×D4×Dic1711yescharacteristic
610C22#1176350111yescharacteristic
611C22#1176351711yescharacteristic
612C4—22no—
613C4—21no—
614C4—21no—
615C4—21no—
616C4—21no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—682no—
624C4—1361no—
625C4C2×Q8×Dic1711yescharacteristic
626C4#1176329211yescharacteristic
627C4#1176345011yescharacteristic
628C4#1176347211yescharacteristic
629C2—21no—
630C2—21no—
631C2—42no—
632C2#1453179311yescharacteristic
633C2#1453768711yescharacteristic
634C2#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#22984
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. #14537733C2semidirect
3. #14529766C2semidirect
4. #14529560C2semidirect
5. #14529538C2semidirect
6. #14529533C2semidirect
7. #14529385C2semidirect
8. #14529385C2semidirect
11. #14528643C2semidirect
54. C17×#479C4semidirect
55. C17×#479C4semidirect
56. C17×#478C4semidirect
57. C17×#478C4semidirect
58. C17×#477C4semidirect
59. C17×#476C4semidirect
60. C17×#475C4semidirect
61. C17×#474C4semidirect
127. #10587508Q8semidirect
331. #479Dic17semidirect
332. #479Dic17semidirect
333. #478Dic17semidirect
334. #478Dic17semidirect
335. #477Dic17semidirect
336. #476Dic17semidirect
337. #475Dic17semidirect
338. #474Dic17semidirect
471. C17#22984semidirect

Maximal subgroups

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

subgroupindex
#145377332
#145373522
#145297662
#145295602
#145295382
#145295332
#145293852
#145290612
#145286432
#145286312
#145286242
#145285942
C17×#30762
#2298417