Finite Group Databasebeta

← #17052675

Subgroups — #17052675

Order 4352.

1010 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762907C211yesmaximal, characteristic
2C2×#11762907C211yesmaximal, characteristic
3C2×C4×#63230C211yesmaximal, characteristic
4C4⋊C4×D68C211yesmaximal, characteristic
5#14529519C211yesmaximal, characteristic, complemented
6#14529514C211yesmaximal, characteristic, complemented
7#14529052C211yesmaximal, characteristic, complemented
8#14529044C211yesmaximal, characteristic
9#14529038C211yesmaximal, characteristic, complemented
10#14529030C211yesmaximal, characteristic
11#14528953C211yesmaximal, characteristic, complemented
12#14528791C211yesmaximal, characteristic
13#14528789C211yesmaximal, characteristic, complemented
14#14528659C211yesmaximal, characteristic, complemented
15C17×#2351C211yesmaximal, characteristic, complemented
16C22×#63230C2211yescharacteristic
17C2×C4×D68C2211yescharacteristic
18C2×C4×D68C2211yescharacteristic
19C2×C4×D68C2211yescharacteristic
20C2×C4⋊C4×D17—21no—
21#11762907—24no—
22#11762907C418yescomplemented
23C4×#63230—24no—
24C2×#10587478C2211yescharacteristic
25C2×#10587478C2211yescharacteristic
26C2×#10587478C2211yescharacteristic
27C2×#10587479C2211yescharacteristic
28C2×#10587479C2211yescharacteristic
29C2×C4⋊#3622C2211yescharacteristic
30C2×#10587506C2211yescharacteristic
31C2×#10587506C2211yescharacteristic
32C2×#10587506C2211yescharacteristic
33C2×#10587480C2211yescharacteristic
34C2×#10587480C2211yescharacteristic
35C2×#10587480C2211yescharacteristic
36C2×#10587474C2211yescharacteristic
37C2×#10587474C2211yescharacteristic
38C2×C4×#3622C2211yescharacteristic
39#11762707C2211yescharacteristic
40#11762707C2211yescharacteristic
41#11762683—21no—
42#11762683—21no—
43#11762679C2211yescharacteristic
44#11762666—21no—
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762564C2211yescharacteristic
50#11762564C2211yescharacteristic
51C2×C4×Q8×C17C2211yescharacteristic
52C2×C4×Q8×C17C2211yescharacteristic
53C2×C4×Q8×C17C2211yescharacteristic
54C17×#389—21no—
55C17×#389—21no—
56C17×#389C2211yescharacteristic
57C17×#389C2211yescharacteristic
58C17×#382—21no—
59C17×#382C2211yescharacteristic
60C4×C4⋊C4×C17—21no—
61C4×C4⋊C4×C17C2211yescharacteristic
62C22×Q8×C17C2311yescharacteristic
63C22×D68C2311yescharacteristic
64C22×C4×D17—21no—
65C22×C4×D17—21no—
66C22×C4×D17—21no—
67C4×Q8×C17—22no—
68C4×Q8×C17—22no—
69C4×Q8×C17—22no—
70C4×Q8×C17—41no—
71C4×Q8×C17—41no—
72C4×Q8×C17C2×C414yescomplemented
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17—21no—
76C2×C4⋊C4×C17—21no—
77C2×C4⋊C4×C17—21no—
78C2×C4⋊C4×C17D411yescharacteristic
79C2×C4⋊C4×C17D411yescharacteristic
80C2×C4⋊C4×C17C2311yescharacteristic
81C2×C4⋊C4×C17C2311yescharacteristic
82C2×C4⋊C4×C17C2311yescharacteristic
83C2×C42×C17—21no—
84C2×C42×C17—21no—
85C2×C42×C17—21no—
86C2×C42×C17C2311yescharacteristic
87C2×C42×C17C2311yescharacteristic
88C2×C42×C17C2311yescharacteristic
89C2×#63230C2×C414yes—
90C2×#63230D414yes—
91C2×#63230Q814yes—
92C2×#63227—21no—
93C2×#63227—21no—
94C2×#63227—21no—
95C2×C4⋊Dic17—21no—
96C2×C4⋊Dic17—21no—
97C2×C4⋊Dic17C2311yescharacteristic
98C2×C4⋊Dic17C2311yescharacteristic
99C2×C4⋊Dic17C2311yescharacteristic
100C2×#63225—21no—
101C2×#63225—21no—
102C22×#3622C2311yescharacteristic
103C22×#3622C2311yescharacteristic
104C22×#3622C2311yescharacteristic
105C22×#3622C2311yescharacteristic
106C4⋊C4×D17—44no—
107C4×D68—22no—
108C4×D68—22no—
109C4×D68—22no—
110C4×D68—41no—
111C4×D68—41no—
112C4×D68C2×C414yes—
113C17×#95—21no—
114C17×#95—21no—
115C17×#95—21no—
116C17×#95—21no—
117C17×#95D411yescharacteristic
118C17×#95D411yescharacteristic
119#10587506—22no—
120#10587506—22no—
121#10587506C2×C414yescomplemented
122#10587502—21no—
123#10587502—21no—
124#10587480—22no—
125#10587480—22no—
126#10587480C2×C414yes—
127#10587479—22no—
128#10587479—41no—
129#10587478—22no—
130#10587478—41no—
131#10587478C2×C414yescomplemented
132C4⋊#3622—22no—
133#10587474—22no—
134#10587474C2×C414yescomplemented
135C4×#3622—22no—
136C23×D17—21no—
137C2×Q8×C17—21no—
138C2×Q8×C17—21no—
139C2×Q8×C17—21no—
140C2×Q8×C17C22×C412yes—
141C2×Q8×C17C2×D412yes—
142C2×Q8×C17C2×Q812yes—
143C22×C4×C17—21no—
144C22×C4×C17—21no—
145C22×C4×C17—21no—
146C22×C4×C17—21no—
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—
157C2×D68—21no—
158C2×D68—21no—
159C2×D68—21no—
160C2×D68C22×C412yes—
161C2×D68C2×D412yes—
162C2×D68C2×Q812yes—
163C2×C4×D17—24no—
164C2×C4×D17—42no—
165C2×C4×D17—42no—
166C2×C4×D17—42no—
167C2×C4×D17—42no—
168C2×C4×D17—42no—
169C2×C4×D17—42no—
170C2×C4×D17—81no—
171C2×C4×D17—81no—
172C4⋊C4×C17—24no—
173C4⋊C4×C17—22no—
174C4⋊C4×C17—21no—
175C4⋊C4×C17—21no—
176C4⋊C4×C17—21no—
177C4⋊C4×C17—21no—
178C4⋊C4×C17—21no—
179C4⋊C4×C17—41no—
180C4⋊C4×C17—42no—
181C4⋊C4×C17—42no—
182C4⋊C4×C17—42no—
183C4⋊C4×C17—42no—
184C4⋊C4×C17C22×C412yes—
185C42×C17—24no—
186C42×C17—21no—
187C42×C17—21no—
188C42×C17—21no—
189C42×C17—21no—
190C42×C17—42no—
191C42×C17—42no—
192C42×C17C22×C412yes—
193C42×C17#4112yes—
194#63230C4⋊C4116yescomplemented
195#63227—24no—
196#63227—42no—
197#63227—42no—
198C4⋊Dic17—21no—
199C4⋊Dic17—21no—
200C4⋊Dic17—21no—
201C4⋊Dic17—21no—
202C4⋊Dic17—21no—
203C4⋊Dic17—42no—
204C4⋊Dic17—81no—
205C4⋊Dic17C22×C412yes—
206#63225—42no—
207#63225—42no—
208C2×#3622—21no—
209C2×#3622—21no—
210C2×#3622—21no—
211C2×#3622—21no—
212C2×#3622—21no—
213C2×#3622—21no—
214C2×#3622—21no—
215C2×#3622—21no—
216C2×#3622C22×C412yes—
217C2×#3622C22×C412yes—
218C2×#3622C2×D412yes—
219C2×#3622C2×Q812yes—
220#22677—171nomaximal, Sylow
221C23×C17C22×D411yescharacteristic
222C22×D17—22no—
223C22×D17—22no—
224C22×D17—22no—
225C22×D17—41no—
226C22×D17—41no—
227C22×D17—41no—
228C22×D17—41no—
229Q8×C17—22no—
230Q8×C17—22no—
231Q8×C17—22no—
232Q8×C17C2×C4⋊C414yescomplemented
233C2×C4×C17—22no—
234C2×C4×C17—21no—
235C2×C4×C17—22no—
236C2×C4×C17—22no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—22no—
243C2×C4×C17—21no—
244C2×C4×C17—21no—
245C2×C4×C17—21no—
246C2×C4×C17—21no—
247C2×C4×C17—21no—
248C2×C4×C17—21no—
249C2×C4×C17—21no—
250C2×C4×C17—21no—
251C2×C4×C17—21no—
252C2×C4×C17—41no—
253C2×C4×C17—41no—
254C2×C4×C17—41no—
255C2×C4×C17—41no—
256C2×C4×C17—41no—
257C2×C4×C17—41no—
258C2×C4×C17—41no—
259C2×C4×C17—41no—
260C2×C4×C17C4×D412yes—
261C2×C4×C17C4×D412yes—
262C2×C4×C17AGL(1,ℤ/8)12yes—
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—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278D68—22no—
279D68—22no—
280D68—22no—
281D68C2×C4⋊C414yes—
282C4×D17—48no—
283C4×D17—84no—
284C4×D17—84no—
285#3622—22no—
286#3622—41no—
287#3622—41no—
288#3622C2×C4⋊C414yescomplemented
289C2×#440—171no—
290C2×#440—171no—
291C2×C4×SD16—171no—
292#2351D1711yescharacteristic, complemented
293D4×C4⋊C4—171no—
294#1943—171no—
295#1942—171no—
296#1865—171no—
297#1864—171no—
298#1863—171no—
299#1862—171no—
300#1847—171no—
301#1804—171no—
302#1803—171no—
303#1771—171no—
304C22×C17C2×C4×D411yescharacteristic
305C22×C17D4211yescharacteristic
306C22×C17D4×Q811yescharacteristic
307C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
308C22×C17#57511yescharacteristic
309C22×C17#57711yescharacteristic
310C22×C17#57811yescharacteristic
311C2×D17—24no—
312C2×D17—42no—
313C2×D17—42no—
314C2×D17—42no—
315C2×D17—42no—
316C2×D17—42no—
317C2×D17—42no—
318C4×C17—24no—
319C4×C17—22no—
320C4×C17—22no—
321C4×C17—22no—
322C4×C17—42no—
323C4×C17—41no—
324C4×C17—41no—
325C4×C17—41no—
326C4×C17—81no—
327C4×C17—81no—
328C4×C17C22×C4⋊C411yescharacteristic
329C4×C17#52011yescharacteristic
330C4×C17#54711yescharacteristic
331C4×C17#55411yescharacteristic
332Dic17—42no—
333Dic17—81no—
334Dic17—81no—
335C22×SD16—171no—
336C2×C4×Q8C2×D1711yescharacteristic
337C2×C4×Q8C2×D1711yescharacteristic
338C2×C4×Q8C2×D1711yescharacteristic
339C2×C4×D4—171no—
340C2×C4×D4—171no—
341C2×C4×D4—171no—
342C22×C4⋊C4—341no—
343#440—178no—
344#440—344no—
345C4×SD16—344no—
346C2×#107—171no—
347C2×#107—171no—
348C2×#107—171no—
349C2×#106—171no—
350C2×#106—171no—
351C2×C4⋊C8—171no—
352C2×#103—171no—
353C2×#103—171no—
354C2×#103—171no—
355C2×#102—171no—
356C2×#102—171no—
357C2×#102—171no—
358C2×#97—171no—
359C2×#97—171no—
360C2×C4×C8—171no—
361#389—21no—
362#389—21no—
363#389—171no—
364#389—171no—
365#389C2×D1711yescharacteristic
366#389C2×D1711yescharacteristic
367#385—341no—
368#385—341no—
369#384—171no—
370#382—21no—
371#382C2×D1711yescharacteristic
372#380—341no—
373C4×C4⋊C4—21no—
374C4×C4⋊C4C2×D1711yescharacteristic
375#340—171no—
376#340—171no—
377#340—171no—
378#340—171no—
379#336—171no—
380#336—171no—
381C2×C17D4×C4⋊C411yescharacteristic
382C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
383C2×C17#294711yescharacteristic
384C2×C17#326511yescharacteristic
385C2×C17#326911yescharacteristic
386C2×C17#338611yescharacteristic
387C2×C17#338811yescharacteristic
388D17—44no—
389C22×Q8C22×D1711yescharacteristic
390C22×D4—171no—
391C23×C4—341no—
392C23×C4—341no—
393C23×C4—341no—
394C2×SD16—174no—
395C2×SD16—174no—
396C2×SD16—174no—
397C22×C8—171no—
398C22×C8—171no—
399C22×C8—171no—
400C22×C8—171no—
401C4×Q8—22no—
402C4×Q8—22no—
403C4×Q8—22no—
404C4×Q8—41no—
405C4×Q8—41no—
406C4×Q8C4×D1714yescomplemented
407C4×D4—174no—
408C4×D4—342no—
409C4×D4—342no—
410C4×D4—342no—
411C4×D4—681no—
412C4×D4—681no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—171no—
419C2×C4⋊C4—171no—
420C2×C4⋊C4—171no—
421C2×C4⋊C4—341no—
422C2×C4⋊C4—341no—
423C2×C4⋊C4—341no—
424C2×C4⋊C4—341no—
425C2×C4⋊C4—684no—
426C2×C4⋊C4C22⋊D1711yescharacteristic
427C2×C4⋊C4C22⋊D1711yescharacteristic
428C2×C4⋊C4C22×D1711yescharacteristic
429C2×C4⋊C4C22×D1711yescharacteristic
430C2×C4⋊C4C22×D1711yescharacteristic
431C2×C22⋊C4—341no—
432C2×C22⋊C4—341no—
433C2×C22⋊C4—341no—
434C2×C42—21no—
435C2×C42—21no—
436C2×C42—21no—
437C2×C42C22×D1711yescharacteristic
438C2×C42C22×D1711yescharacteristic
439C2×C42C22×D1711yescharacteristic
440#107—174no—
441#107—342no—
442#107—681no—
443#106—342no—
444#106—681no—
445C4⋊C8—342no—
446#103—174no—
447#103—342no—
448#103—342no—
449#102—174no—
450#102—342no—
451#102—342no—
452#97—174no—
453#97—342no—
454C4×C8—342no—
455#95—21no—
456#95—21no—
457#95—21no—
458#95—21no—
459#95—341no—
460#95—341no—
461#95C22⋊D1711yescharacteristic
462#95C22⋊D1711yescharacteristic
463C17#2267711yescharacteristic, complemented, Sylow
464C24—341no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8C2×C4×D1712yes—
469C2×Q8D4×D1712yes—
470C2×Q8Q8×D1712yes—
471C2×D4—172no—
472C2×D4—172no—
473C2×D4—172no—
474C2×D4—341no—
475C2×D4—341no—
476C2×D4—341no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—341no—
482C22×C4—344no—
483C22×C4—341no—
484C22×C4—341no—
485C22×C4—682no—
486C22×C4—682no—
487C22×C4—682no—
488C22×C4—682no—
489C22×C4—682no—
490C22×C4—682no—
491C22×C4—1361no—
492C22×C4—1361no—
493C22×C4C2×C22⋊D1711yescharacteristic
494C22×C4C2×C22⋊D1711yescharacteristic
495C22×C4C2×C22⋊D1711yescharacteristic
496C22×C4C2×C22⋊D1711yescharacteristic
497C22×C4C2×C22⋊D1711yescharacteristic
498C22×C4C2×C22⋊D1711yescharacteristic
499C22×C4C23×D1711yescharacteristic
500SD16—1716no—
501C2×C8—172no—
502C2×C8—172no—
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—
513C4⋊C4—21no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—41no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—172no—
526C4⋊C4—341no—
527C4⋊C4—341no—
528C4⋊C4—341no—
529C4⋊C4—341no—
530C4⋊C4—341no—
531C4⋊C4—682no—
532C4⋊C4—682no—
533C4⋊C4—682no—
534C4⋊C4—1361no—
535C4⋊C4C2×C4×D1712yes—
536C22⋊C4—344no—
537C22⋊C4—682no—
538C22⋊C4—682no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—24no—
544C42—42no—
545C42—42no—
546C42C2×C4×D1712yes—
547C42#6325212yes—
548C23—342no—
549C23—342no—
550C23—342no—
551C23—681no—
552C23—681no—
553C23—681no—
554C23—681no—
555C23C22×C22⋊D1711yescharacteristic
556Q8—22no—
557Q8—22no—
558Q8—22no—
559Q8C4⋊C4×D1714yescomplemented
560D4—174no—
561D4—342no—
562D4—342no—
563D4—342no—
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—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—342no—
592C2×C4—688no—
593C2×C4—681no—
594C2×C4—681no—
595C2×C4—681no—
596C2×C4—681no—
597C2×C4—681no—
598C2×C4—681no—
599C2×C4—681no—
600C2×C4—681no—
601C2×C4—1364no—
602C2×C4—1364no—
603C2×C4C4×C22⋊D1712yes—
604C2×C4C4×C22⋊D1712yes—
605C2×C4#1058763512yes—
606C2×C4C22×C4×D1711yescharacteristic
607C2×C4C2×#6325211yescharacteristic
608C2×C4C2×D4×D1711yescharacteristic
609C2×C4#1058769111yescharacteristic
610C2×C4C2×Q8×D1711yescharacteristic
611C2×C4#1058769411yescharacteristic
612C8—174no—
613C8—342no—
614C8—681no—
615C8—681no—
616C22—344no—
617C22—682no—
618C22—682no—
619C22—682no—
620C22—682no—
621C22—682no—
622C22—682no—
623C22C2×C4×C22⋊D1711yescharacteristic
624C22D4×C22⋊D1711yescharacteristic
625C22Q8×C22⋊D1711yescharacteristic
626C22C2×#1058763511yescharacteristic
627C22#1176360711yescharacteristic
628C22#1176361711yescharacteristic
629C22#1176362411yescharacteristic
630C4—24no—
631C4—22no—
632C4—22no—
633C4—22no—
634C4—41no—
635C4—41no—
636C4—42no—
637C4—41no—
638C4—81no—
639C4—81no—
640C4—682no—
641C4—1361no—
642C4—1361no—
643C4C2×C4⋊C4×D1711yescharacteristic
644C4#1176329011yescharacteristic
645C4#1176346011yescharacteristic
646C4#1176351411yescharacteristic
647C2—684no—
648C2C4⋊C4×C22⋊D1711yescharacteristic
649C2C4×#1058763511yescharacteristic
650C2#1453762711yescharacteristic
651C2#1454028011yescharacteristic
652C2#1454034011yescharacteristic
653C2#1454170111yescharacteristic
654C2#1454172311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2351C2
socleC23×C17C22×D4
prime radicalC22×D68C23
cosocleC22×C4×C17C24
hypercenter#2351D17
hypocenterC17#22677
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2351

Complements

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

NH ≅ G/Ntype
5. #14529519C2semidirect
6. #14529514C2semidirect
7. #14529052C2semidirect
9. #14529038C2semidirect
11. #14528953C2semidirect
13. #14528789C2semidirect
14. #14528659C2semidirect
15. C17×#2351C2semidirect
22. #11762907C4semidirect
72. C4×Q8×C17C2×C4semidirect
121. #10587506C2×C4semidirect
131. #10587478C2×C4semidirect
134. #10587474C2×C4semidirect
194. #63230C4⋊C4semidirect
232. Q8×C17C2×C4⋊C4semidirect
288. #3622C2×C4⋊C4semidirect
292. #2351D17semidirect
406. C4×Q8C4×D17semidirect
463. C17#22677semidirect
559. Q8C4⋊C4×D17semidirect

Maximal subgroups

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

subgroupindex
C2×#117629072
C2×C4×#632302
C4⋊C4×D682
#145295192
#145295142
#145290522
#145290442
#145290382
#145290302
#145289532
#145287912
#145287892
#145286592
C17×#23512
#2267717