Finite Group Databasebeta

← #17037713

Subgroups — #17037713

Order 4352.

1058 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762850C211yesmaximal, characteristic
2C2×#11762848C211yesmaximal, characteristic
3C2×#11762761C211yesmaximal, characteristic
4#14531265C211yesmaximal, characteristic
5D17×#389C211yesmaximal, characteristic
6#14529524C211yesmaximal, characteristic, complemented
7#14529522C211yesmaximal, characteristic, complemented
8#14529519C211yesmaximal, characteristic, complemented
9#14529516C211yesmaximal, characteristic, complemented
10#14529515C211yesmaximal, characteristic, complemented
11#14529513C211yesmaximal, characteristic, complemented
12#14529511C211yesmaximal, characteristic, complemented
13#14528502C211yesmaximal, characteristic
14#14528502C211yesmaximal, characteristic
15C17×#1944C211yesmaximal, characteristic, complemented
16C22×#63218C2211yescharacteristic
17C22×#63218C2211yescharacteristic
18C2×#10587578C2211yescharacteristic
19C2×#10587578C2211yescharacteristic
20C2×C4⋊C4×D17—21no—
21C2×C4⋊C4×D17C2211yescharacteristic
22C2×C4⋊C4×D17C2211yescharacteristic
23C2×C42×D17C2211yescharacteristic
24#11762850—24no—
25#11762848C418yescomplemented
26C2×#10587489C2211yescharacteristic
27C2×#10587478C2211yescharacteristic
28C2×#10587478C2211yescharacteristic
29C2×#10587488C2211yescharacteristic
30C2×#10587479C2211yescharacteristic
31C2×#10587479C2211yescharacteristic
32#11762761—24no—
33C2×#10587474C2211yescharacteristic
34C2×C8×Dic17C2211yescharacteristic
35C2×C8×Dic17C2211yescharacteristic
36#11762708C2211yescharacteristic
37#11762708C2211yescharacteristic
38#11762707C2211yescharacteristic
39#11762707C2211yescharacteristic
40#11762706—21no—
41#11762705C2211yescharacteristic
42#11762705C2211yescharacteristic
43#11762704—21no—
44#11762683—21no—
45#11762680—21no—
46C4⋊C4×Dic17—21no—
47#11762660—21no—
48#95×D17—21no—
49#11762572C2211yescharacteristic
50#11762572C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762571C2211yescharacteristic
53#11762571C2211yescharacteristic
54#11762571C2211yescharacteristic
55C2×C17×#107C2211yescharacteristic
56C2×C17×#106C2211yescharacteristic
57C2×C17×#97C2211yescharacteristic
58C17×#389C2211yescharacteristic
59C17×#389C2211yescharacteristic
60C17×#340C2211yescharacteristic
61C17×#340C2211yescharacteristic
62C22×C4×D17—21no—
63C22×C4×D17—21no—
64C22×C4×D17D411yescharacteristic
65C22×C4×D17D411yescharacteristic
66C22×C4×D17C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C42×C17C2311yescharacteristic
76C2×#63227—21no—
77C2×#63227—21no—
78C2×C4⋊Dic17—21no—
79C2×C4⋊Dic17—21no—
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×#63225—21no—
85C2×#63225—21no—
86C2×#63225—21no—
87C2×#63225—21no—
88C2×#63225—21no—
89C2×#63225—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C2×C4×Dic17D411yescharacteristic
93C2×C4×Dic17D411yescharacteristic
94C2×C4×Dic17C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97C2×#63218—22no—
98C2×#63218—22no—
99C2×#63218C2×C414yes—
100C2×#63218D414yes—
101C2×#63218Q814yes—
102C2×#63218Q814yes—
103#10587578—22no—
104#10587578—22no—
105#10587578—22no—
106#10587578C2×C414yes—
107C4⋊C4×D17—22no—
108C4⋊C4×D17—22no—
109C4⋊C4×D17—22no—
110C4⋊C4×D17—44no—
111C4⋊C4×D17C2×C414yes—
112C42×D17—22no—
113C42×D17—22no—
114C17×#107—22no—
115C17×#106C2×C414yescomplemented
116C17×#97—22no—
117C17×#95—21no—
118C17×#95—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587502—21no—
125#10587489C2×C414yescomplemented
126#10587488—22no—
127C8×Dic17—22no—
128C8×Dic17—22no—
129#10587479—22no—
130#10587479C2×C414yescomplemented
131#10587478—22no—
132#10587478C2×C414yescomplemented
133#10587474—22no—
134C23×D17C2×D411yescharacteristic
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2411yescharacteristic
142C22×Dic17—21no—
143C22×Dic17—21no—
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17C2×D411yescharacteristic
147C22×Dic17C2×D411yescharacteristic
148C22×Dic17C2×D411yescharacteristic
149C2×C4×D17—24no—
150C2×C4×D17—22no—
151C2×C4×D17—22no—
152C2×C4×D17—22no—
153C2×C4×D17—22no—
154C2×C4×D17—41no—
155C2×C4×D17—41no—
156C2×C4×D17—42no—
157C2×C4×D17—42no—
158C2×C4×D17—42no—
159C2×C4×D17—42no—
160C2×C4×D17—42no—
161C2×C4×D17C22×C412yes—
162C2×C4×D17C2×D412yes—
163C2×C4×D17C2×Q812yes—
164C2×C4×D17C2×Q812yes—
165C2×C4×D17#4112yes—
166C2×C4×D17#4112yes—
167C2×C8×C17—21no—
168C2×C8×C17—21no—
169C2×C8×C17C22×C412yes—
170C2×C8×C17C2×D412yes—
171C2×C8×C17C2×Q812yes—
172C2×C8×C17C2×Q812yes—
173C4⋊C4×C17—21no—
174C4⋊C4×C17—21no—
175C4⋊C4×C17—21no—
176C4⋊C4×C17—21no—
177C4⋊C4×C17—21no—
178C4⋊C4×C17—21no—
179C4⋊C4×C17—42no—
180C4⋊C4×C17—42no—
181C4⋊C4×C17C22×C412yes—
182C4⋊C4×C17C22×C412yes—
183C42×C17—21no—
184C42×C17—21no—
185#63227—24no—
186#63227—42no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—21no—
192C4⋊Dic17—21no—
193C4⋊Dic17—42no—
194C4⋊Dic17—42no—
195C4⋊Dic17C22×C412yes—
196C4⋊Dic17C22×C412yes—
197#63225—24no—
198#63225—42no—
199#63225—42no—
200#63225—42no—
201#63225—42no—
202#63225—42no—
203C4×Dic17—21no—
204C4×Dic17—21no—
205C4×Dic17—24no—
206C4×Dic17—22no—
207C4×Dic17—22no—
208C4×Dic17—42no—
209C2×#3622—21no—
210C2×#3622—21no—
211C2×#3622C22×C412yes—
212C2×#3622C2×D412yes—
213C2×#3622C2×Q812yes—
214C2×#3622C2×Q812yes—
215#63218—28no—
216#63218C4⋊C4116yescomplemented
217#21159—171nomaximal, Sylow
218C23×C17C22×D411yescharacteristic
219C22×D17—21no—
220C22×D17—21no—
221C22×D17—21no—
222C22×D17—21no—
223C22×D17C4×D412yes—
224C22×D17#12112yes—
225C22×D17C4⋊Q812yes—
226C2×C4×C17—22no—
227C2×C4×C17—22no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—41no—
235C2×C4×C17—41no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—41no—
239C2×C4×C17—41no—
240C2×C4×C17—41no—
241C2×C4×C17—41no—
242C2×C4×C17—41no—
243C2×C4×C17—41no—
244C2×C4×C17C23×C411yescharacteristic
245C2×C4×C17C22×D411yescharacteristic
246C2×C4×C17C22×Q811yescharacteristic
247C2×C4×C17C22×Q811yescharacteristic
248C2×C4×C17C2×#4111yescharacteristic
249C2×C4×C17C2×#4111yescharacteristic
250C2×Dic17—21no—
251C2×Dic17—21no—
252C2×Dic17—21no—
253C2×Dic17—21no—
254C2×Dic17—22no—
255C2×Dic17—22no—
256C2×Dic17—41no—
257C2×Dic17—41no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17—41no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17C4×D412yes—
267C2×Dic17#12112yes—
268C2×Dic17C4⋊Q812yes—
269C2×Dic17AGL(1,ℤ/8)12yes—
270C2×Dic17#13712yes—
271C4×D17—44no—
272C4×D17—44no—
273C4×D17—48no—
274C4×D17—84no—
275C4×D17C2×C4⋊C414yes—
276C4×D17C4×Q814yes—
277C4×D17C22⋊Q814yes—
278C4×D17#12514yes—
279C8×C17—22no—
280C8×C17C2×C4⋊C414yescomplemented
281#3622—22no—
282#3622C2×C4⋊C414yescomplemented
283C2×#428—171no—
284C2×#428—171no—
285C2×C4×M16—171no—
286#2296—171no—
287C2×#389—171no—
288#1944—171no—
289#1944D1711yescharacteristic, complemented
290#1943—171no—
291#1943—171no—
292#1942—171no—
293#1942—171no—
294#1941—171no—
295#1941—171no—
296#1736—171no—
297#1736—171no—
298C22×C17C2×C4×D411yescharacteristic
299C22×C17C2×#12111yescharacteristic
300C22×C17C2×C4⋊Q811yescharacteristic
301C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
302C22×C17C2×#13711yescharacteristic
303C22×C17#57511yescharacteristic
304C22×C17#57511yescharacteristic
305C2×D17—22no—
306C2×D17—22no—
307C2×D17—22no—
308C2×D17—22no—
309C2×D17—22no—
310C2×D17—22no—
311C2×D17#38914yes—
312C4×C17—42no—
313C4×C17—42no—
314C4×C17—41no—
315C4×C17—41no—
316C4×C17—81no—
317C4×C17—81no—
318C4×C17C22×C4⋊C411yescharacteristic
319C4×C17C2×C4×Q811yescharacteristic
320C4×C17C2×C22⋊Q811yescharacteristic
321C4×C17C2×#12511yescharacteristic
322Dic17—22no—
323Dic17—22no—
324Dic17—42no—
325Dic17—42no—
326Dic17—81no—
327Dic17—81no—
328Dic17#38914yes—
329C22×M16—171no—
330C22×M16—171no—
331C2×#117—171no—
332C2×#117—171no—
333C22×C4⋊C4—171no—
334C22×C4⋊C4—171no—
335C22×C4⋊C4—341no—
336C22×C42—171no—
337#428—178no—
338#428—344no—
339C2×#107—171no—
340C2×#107—171no—
341C2×#107—171no—
342C2×#107C2×D1711yescharacteristic
343C2×#106—171no—
344C2×#106—171no—
345C2×#106—171no—
346C2×#106C2×D1711yescharacteristic
347C4×M16—344no—
348C2×#97—171no—
349C2×#97C2×D1711yescharacteristic
350C2×C4×C8—171no—
351C2×C4×C8—171no—
352#389—171no—
353#389—171no—
354#389—171no—
355#389—171no—
356#389—171no—
357#389—171no—
358#389—341no—
359#389—341no—
360#389C2×D1711yescharacteristic
361#389C2×D1711yescharacteristic
362#385—341no—
363#385—341no—
364C4×C4⋊C4—341no—
365C4×C4⋊C4—341no—
366C2×#95—341no—
367#340—171no—
368#340—171no—
369#340—171no—
370#340—171no—
371#340—171no—
372#340—171no—
373#340C2×D1711yescharacteristic
374#340C2×D1711yescharacteristic
375C2×C17C2×#38911yescharacteristic
376C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
377C2×C17C4×#13711yescharacteristic
378C2×C17#305511yescharacteristic
379C2×C17#305611yescharacteristic
380C2×C17#316511yescharacteristic
381C2×C17#316611yescharacteristic
382D17—24no—
383C23×C4—171no—
384C23×C4—171no—
385C23×C4—171no—
386C23×C4—341no—
387C23×C4—341no—
388C2×M16—174no—
389C2×M16—174no—
390C2×M16—174no—
391C2×M16—174no—
392C2×M16—342no—
393C2×M16—342no—
394C22×C8—171no—
395C22×C8—171no—
396C22×C8C22×D1711yescharacteristic
397C22×C8C22×D1711yescharacteristic
398#117—174no—
399#117—342no—
400#117—342no—
401#117—342no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—171no—
405C2×C4⋊C4—171no—
406C2×C4⋊C4—174no—
407C2×C4⋊C4—171no—
408C2×C4⋊C4—171no—
409C2×C4⋊C4—342no—
410C2×C4⋊C4—342no—
411C2×C4⋊C4—342no—
412C2×C4⋊C4—341no—
413C2×C4⋊C4—341no—
414C2×C4⋊C4—341no—
415C2×C4⋊C4—341no—
416C2×C4⋊C4—341no—
417C2×C4⋊C4—341no—
418C2×C4⋊C4—341no—
419C2×C4⋊C4—341no—
420C2×C4⋊C4—684no—
421C2×C4⋊C4C22×D1711yescharacteristic
422C2×C4⋊C4C22×D1711yescharacteristic
423C2×C4⋊C4C22×D1711yescharacteristic
424C2×C4⋊C4C22×D1711yescharacteristic
425C2×C22⋊C4—341no—
426C2×C22⋊C4—341no—
427C2×C42—171no—
428C2×C42—171no—
429C2×C42—171no—
430C2×C42—342no—
431C2×C42—342no—
432C2×C42—341no—
433C2×C42—341no—
434C2×C42C22×D1711yescharacteristic
435#107—22no—
436#107—174no—
437#107—174no—
438#107—342no—
439#106—174no—
440#106—342no—
441#106—342no—
442#106C4×D1714yescomplemented
443#97—22no—
444#97—342no—
445C4×C8—342no—
446C4×C8—342no—
447#95—21no—
448#95—21no—
449#95—341no—
450#95—341no—
451#95—341no—
452#95—341no—
453#95—341no—
454#95—341no—
455C17#2115911yescharacteristic, complemented, Sylow
456C24—171no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—171no—
462C22×C4—172no—
463C22×C4—172no—
464C22×C4—172no—
465C22×C4—172no—
466C22×C4—172no—
467C22×C4—172no—
468C22×C4—171no—
469C22×C4—171no—
470C22×C4—342no—
471C22×C4—342no—
472C22×C4—342no—
473C22×C4—342no—
474C22×C4—344no—
475C22×C4—341no—
476C22×C4—341no—
477C22×C4—341no—
478C22×C4—341no—
479C22×C4—681no—
480C22×C4—681no—
481C22×C4—682no—
482C22×C4—682no—
483C22×C4—682no—
484C22×C4—682no—
485C22×C4—682no—
486C22×C4D4×D1711yescharacteristic
487C22×C4D4×D1711yescharacteristic
488C22×C4C23×D1711yescharacteristic
489M16—1716no—
490M16—348no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—172no—
494C2×C8—172no—
495C2×C8—172no—
496C2×C8—172no—
497C2×C8—341no—
498C2×C8—341no—
499C2×C8C2×C4×D1712yes—
500C2×C8D4×D1712yes—
501C2×C8Q8×D1712yes—
502C2×C8Q8×D1712yes—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—172no—
512C4⋊C4—172no—
513C4⋊C4—344no—
514C4⋊C4—341no—
515C4⋊C4—341no—
516C4⋊C4—341no—
517C4⋊C4—341no—
518C4⋊C4—341no—
519C4⋊C4—341no—
520C4⋊C4—682no—
521C4⋊C4—682no—
522C4⋊C4—682no—
523C4⋊C4—682no—
524C4⋊C4—682no—
525C4⋊C4—682no—
526C4⋊C4—682no—
527C4⋊C4C2×C4×D1712yes—
528C4⋊C4C2×C4×D1712yes—
529C22⋊C4—344no—
530C22⋊C4—682no—
531C42—21no—
532C42—21no—
533C42—342no—
534C42—342no—
535C42—344no—
536C42—341no—
537C42—341no—
538C42—682no—
539C23—172no—
540C23—172no—
541C23—172no—
542C23—341no—
543C23—341no—
544C23—341no—
545C23—341no—
546C23C2×D4×D1711yescharacteristic
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—172no—
566C2×C4—172no—
567C2×C4—172no—
568C2×C4—174no—
569C2×C4—174no—
570C2×C4—174no—
571C2×C4—174no—
572C2×C4—172no—
573C2×C4—172no—
574C2×C4—341no—
575C2×C4—341no—
576C2×C4—341no—
577C2×C4—341no—
578C2×C4—342no—
579C2×C4—342no—
580C2×C4—684no—
581C2×C4—684no—
582C2×C4—688no—
583C2×C4—681no—
584C2×C4—681no—
585C2×C4—681no—
586C2×C4—681no—
587C2×C4—681no—
588C2×C4—681no—
589C2×C4—681no—
590C2×C4—681no—
591C2×C4—681no—
592C2×C4—681no—
593C2×C4—1364no—
594C2×C4C22×C4×D1711yescharacteristic
595C2×C4C2×D4×D1711yescharacteristic
596C2×C4C2×Q8×D1711yescharacteristic
597C2×C4C2×Q8×D1711yescharacteristic
598C2×C4#41×D1711yescharacteristic
599C2×C4#41×D1711yescharacteristic
600C8—22no—
601C8—174no—
602C8—342no—
603C8C4⋊C4×D1714yescomplemented
604C22—174no—
605C22—342no—
606C22—342no—
607C22—342no—
608C22—342no—
609C22—342no—
610C22—342no—
611C22C4×D4×D1711yescharacteristic
612C22#121×D1711yescharacteristic
613C22C4⋊Q8×D1711yescharacteristic
614C22AGL(1,ℤ/8)×D1711yescharacteristic
615C22#137×D1711yescharacteristic
616C22#1176360511yescharacteristic
617C22#1176360811yescharacteristic
618C4—42no—
619C4—42no—
620C4—41no—
621C4—41no—
622C4—81no—
623C4—81no—
624C4—174no—
625C4—342no—
626C4—342no—
627C4—682no—
628C4—682no—
629C4—1361no—
630C4—1361no—
631C4C2×C4⋊C4×D1711yescharacteristic
632C4C4×Q8×D1711yescharacteristic
633C4C22⋊Q8×D1711yescharacteristic
634C4#125×D1711yescharacteristic
635C2—344no—
636C2D17×#38911yescharacteristic
637C2#1453758911yescharacteristic
638C2#1453760611yescharacteristic
639C2#1453840211yescharacteristic
640C2#1453841311yescharacteristic
641C2#1453932911yescharacteristic
642C2#1453933511yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1944C2
socleC23×C17C22×D4
prime radicalC23×D17C2×D4
cosocleC22×C4×C17C24
hypercenter#1944D17
hypocenterC17#21159
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1944

Complements

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

NH ≅ G/Ntype
6. #14529524C2semidirect
7. #14529522C2semidirect
8. #14529519C2semidirect
9. #14529516C2semidirect
10. #14529515C2semidirect
11. #14529513C2semidirect
12. #14529511C2semidirect
15. C17×#1944C2semidirect
25. #11762848C4semidirect
115. C17×#106C2×C4semidirect
125. #10587489C2×C4semidirect
130. #10587479C2×C4semidirect
132. #10587478C2×C4semidirect
216. #63218C4⋊C4semidirect
280. C8×C17C2×C4⋊C4semidirect
282. #3622C2×C4⋊C4semidirect
289. #1944D17semidirect
442. #106C4×D17semidirect
455. C17#21159semidirect
603. C8C4⋊C4×D17semidirect

Maximal subgroups

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

subgroupindex
C2×#117628502
C2×#117628482
C2×#117627612
#145312652
D17×#3892
#145295242
#145295222
#145295192
#145295162
#145295152
#145295132
#145295112
#145285022
C17×#19442
#2115917