Finite Group Databasebeta

← #17052694

Subgroups — #17052694

Order 4352.

982 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762911C211yesmaximal, characteristic
2C2×#11762911C211yesmaximal, characteristic
3C2×C4×#63219C211yesmaximal, characteristic
4C4⋊C4×Dic34C211yesmaximal, characteristic, complemented
5#14531695C211yesmaximal, characteristic
6#14529522C211yesmaximal, characteristic, complemented
7#14529517C211yesmaximal, characteristic, complemented
8#14529055C211yesmaximal, characteristic, complemented
9#14529048C211yesmaximal, characteristic
10#14529041C211yesmaximal, characteristic, complemented
11#14529034C211yesmaximal, characteristic
12#14528952C211yesmaximal, characteristic, complemented
13#14528800C211yesmaximal, characteristic, complemented
14#14528785C211yesmaximal, characteristic
15C17×#1771C211yesmaximal, characteristic, complemented
16C22×#63219C2211yescharacteristic
17C2×C4×Dic34C2211yescharacteristic
18C2×C4×Dic34C2211yescharacteristic
19C2×C4×Dic34C2211yescharacteristic
20C2×C4×D68C2211yescharacteristic
21C2×C4×D68C2211yescharacteristic
22C2×C4×D68C2211yescharacteristic
23#11762911—24no—
24#11762911C418yescomplemented
25C4×#63219—24no—
26C2×#10587489C2211yescharacteristic
27C2×#10587489C2211yescharacteristic
28C2×#10587489C2211yescharacteristic
29C2×#10587488C2211yescharacteristic
30C2×#10587488C2211yescharacteristic
31C2×#10587487C2211yescharacteristic
32C2×#10587487C2211yescharacteristic
33C2×#10587487C2211yescharacteristic
34C2×#10587492C2211yescharacteristic
35C2×#10587492C2211yescharacteristic
36C2×#10587492C2211yescharacteristic
37#11762707C2211yescharacteristic
38#11762707C2211yescharacteristic
39#11762707C2211yescharacteristic
40#11762707C2211yescharacteristic
41#11762706—21no—
42#11762706—21no—
43#11762680—21no—
44#11762680—21no—
45#11762679C2211yescharacteristic
46#11762678—21no—
47#11762675C2211yescharacteristic
48#11762668—21no—
49C4⋊C4×Dic17—21no—
50#95×D17—21no—
51#11762572C2211yescharacteristic
52#11762572C2211yescharacteristic
53#11762572C2211yescharacteristic
54#11762572C2211yescharacteristic
55C2×C17×C4⋊C8C2211yescharacteristic
56C2×C17×#97C2211yescharacteristic
57C2×C17×#97C2211yescharacteristic
58C2×C4×C8×C17C2211yescharacteristic
59C4×C4⋊C4×C17C2211yescharacteristic
60C17×#336C2211yescharacteristic
61C17×#336C2211yescharacteristic
62C22×D68C2311yescharacteristic
63C22×C4×D17—21no—
64C22×C4×D17—21no—
65C22×C4×D17—21no—
66C22×Dic34C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C22×C8×C17C2311yescharacteristic
70C22×C8×C17C2311yescharacteristic
71C2×C4⋊C4×C17D411yescharacteristic
72C2×C4⋊C4×C17D411yescharacteristic
73C2×C42×C17C2311yescharacteristic
74C2×C42×C17C2311yescharacteristic
75C2×C42×C17C2311yescharacteristic
76C2×#63227—21no—
77C2×#63227—21no—
78C2×#63227—21no—
79C2×C4⋊Dic17—21no—
80C2×C4⋊Dic17—21no—
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×#63225—21no—
88C2×#63225—21no—
89C2×#63225—21no—
90C2×#63225—21no—
91C2×#63225—21no—
92C2×#63225—21no—
93C2×#63225—21no—
94C2×C4×Dic17—21no—
95C2×C4×Dic17—21no—
96C2×C4×Dic17—21no—
97C2×#63219C2×C414yes—
98C2×#63219D414yes—
99C2×#63219Q814yes—
100C4×D68—22no—
101C4×D68—22no—
102C4×D68—22no—
103C4×D68—41no—
104C4×D68—41no—
105C4×D68C2×C414yes—
106C4×Dic34—22no—
107C4×Dic34—22no—
108C4×Dic34—22no—
109C4×Dic34—41no—
110C4×Dic34—41no—
111C4×Dic34C2×C414yescomplemented
112C17×C4⋊C8—22no—
113C17×#97—22no—
114C17×#97C2×C414yescomplemented
115C4×C8×C17—22no—
116C17×#95D411yescharacteristic
117C17×#95D411yescharacteristic
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587492—22no—
125#10587492—22no—
126#10587492C2×C414yes—
127#10587489—22no—
128#10587489—41no—
129#10587489C2×C414yescomplemented
130#10587488—22no—
131#10587488—41no—
132#10587487—22no—
133#10587487—22no—
134#10587487C2×C414yescomplemented
135C23×D17—21no—
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2411yescharacteristic
143C22×Dic17—21no—
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17—21no—
148C22×Dic17—21no—
149C22×Dic17—21no—
150C2×D68—21no—
151C2×D68—21no—
152C2×D68—21no—
153C2×D68C22×C412yes—
154C2×D68C2×D412yes—
155C2×D68C2×Q812yes—
156C2×C4×D17—24no—
157C2×C4×D17—42no—
158C2×C4×D17—42no—
159C2×C4×D17—42no—
160C2×C4×D17—42no—
161C2×C4×D17—81no—
162C2×C4×D17—81no—
163C2×C4×D17—81no—
164C2×C4×D17—81no—
165C2×Dic34—21no—
166C2×Dic34—21no—
167C2×Dic34—21no—
168C2×Dic34C22×C412yes—
169C2×Dic34C2×D412yes—
170C2×Dic34C2×Q812yes—
171C2×C8×C17—21no—
172C2×C8×C17—21no—
173C2×C8×C17—21no—
174C2×C8×C17—21no—
175C2×C8×C17—21no—
176C2×C8×C17—21no—
177C2×C8×C17—21no—
178C2×C8×C17—21no—
179C2×C8×C17C22×C412yes—
180C2×C8×C17C22×C412yes—
181C2×C8×C17C2×D412yes—
182C2×C8×C17C2×Q812yes—
183C4⋊C4×C17—22no—
184C4⋊C4×C17—41no—
185C42×C17—21no—
186C42×C17—21no—
187C42×C17—21no—
188C42×C17—21no—
189C42×C17C22×C412yes—
190C42×C17#4112yes—
191#63227—24no—
192#63227—42no—
193#63227—42no—
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199C4⋊Dic17—21no—
200C4⋊Dic17—21no—
201C4⋊Dic17—21no—
202C4⋊Dic17—21no—
203C4⋊Dic17—21no—
204C4⋊Dic17—42no—
205C4⋊Dic17—81no—
206C4⋊Dic17C22×C412yes—
207C4⋊Dic17C22×C412yes—
208#63225—24no—
209#63225—42no—
210#63225—42no—
211#63225—81no—
212#63225—81no—
213#63225—81no—
214#63225—81no—
215C4×Dic17—24no—
216C4×Dic17—42no—
217C4×Dic17—42no—
218#63219C4⋊C4116yescomplemented
219#22678—171nomaximal, Sylow
220C23×C17C22×D411yescharacteristic
221C22×D17—22no—
222C22×D17—22no—
223C22×D17—22no—
224C22×D17—41no—
225C22×D17—41no—
226C22×D17—41no—
227C22×D17—41no—
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—21no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17C4×D412yes—
244C2×C4×C17C4×D412yes—
245C2×C4×C17#13712yes—
246C2×C4×C17C23×C411yescharacteristic
247C2×C4×C17C22×D411yescharacteristic
248C2×C4×C17C22×Q811yescharacteristic
249C2×C4×C17C2×#4111yescharacteristic
250C2×C4×C17H(5,𝔽2)11yescharacteristic
251C2×C4×C172−1+411yescharacteristic
252C2×Dic17—22no—
253C2×Dic17—22no—
254C2×Dic17—22no—
255C2×Dic17—22no—
256C2×Dic17—22no—
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×Dic17—41no—
267C2×Dic17—41no—
268C2×Dic17—41no—
269C2×Dic17—41no—
270C2×Dic17—41no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273D68—22no—
274D68—22no—
275D68—22no—
276D68C2×C4⋊C414yes—
277C4×D17—84no—
278C4×D17—84no—
279C4×D17—84no—
280Dic34—22no—
281Dic34—22no—
282Dic34—22no—
283Dic34C2×C4⋊C414yes—
284C8×C17—22no—
285C8×C17—41no—
286C8×C17—41no—
287C8×C17C2×C4⋊C414yes—
288C2×#440—171no—
289C2×#440—171no—
290C2×C4×SD16—171no—
291Q8×C4⋊C4—171no—
292#2349—171no—
293#1943—171no—
294#1942—171no—
295#1865—171no—
296#1864—171no—
297#1863—171no—
298#1862—171no—
299#1847—171no—
300#1805—171no—
301#1802—171no—
302#1771D1711yescharacteristic, complemented
303C22×C17C2×C4×D411yescharacteristic
304C22×C17D4211yescharacteristic
305C22×C17D4×Q811yescharacteristic
306C22×C17C2×#13711yescharacteristic
307C22×C17#57511yescharacteristic
308C22×C17#57611yescharacteristic
309C22×C17#57711yescharacteristic
310C2×D17—24no—
311C2×D17—42no—
312C2×D17—42no—
313C2×D17—42no—
314C2×D17—42no—
315C2×D17—42no—
316C2×D17—42no—
317C4×C17—22no—
318C4×C17—22no—
319C4×C17—22no—
320C4×C17—41no—
321C4×C17—41no—
322C4×C17—41no—
323C4×C17C22×C4⋊C411yescharacteristic
324C4×C17#52011yescharacteristic
325C4×C17#54711yescharacteristic
326C4×C17#55411yescharacteristic
327Dic17—24no—
328Dic17—42no—
329Dic17—42no—
330Dic17—81no—
331Dic17—81no—
332Dic17—81no—
333Dic17—81no—
334C22×SD16—171no—
335C2×C4×Q8—171no—
336C2×C4×Q8—171no—
337C2×C4×Q8—171no—
338C2×C4×D4—171no—
339C2×C4×D4—171no—
340C2×C4×D4—171no—
341#440—178no—
342#440—344no—
343C4×SD16—344no—
344C2×#107—171no—
345C2×#107—171no—
346C2×#107—171no—
347C2×#106—171no—
348C2×#106—171no—
349C2×C4⋊C8C2×D1711yescharacteristic
350C2×#103—171no—
351C2×#103—171no—
352C2×#103—171no—
353C2×#102—171no—
354C2×#102—171no—
355C2×#102—171no—
356C2×#97C2×D1711yescharacteristic
357C2×#97C2×D1711yescharacteristic
358C2×C4×C8C2×D1711yescharacteristic
359#389—171no—
360#389—171no—
361#389—171no—
362#389—171no—
363#389—341no—
364#389—341no—
365#385—341no—
366#385—341no—
367#384—171no—
368#384—341no—
369#382—171no—
370#380—341no—
371C4×C4⋊C4—341no—
372C4×C4⋊C4C2×D1711yescharacteristic
373C2×#95—341no—
374#340—171no—
375#340—171no—
376#340—171no—
377#340—171no—
378#336C2×D1711yescharacteristic
379#336C2×D1711yescharacteristic
380C2×C17D4×C4⋊C411yescharacteristic
381C2×C17C4×#13711yescharacteristic
382C2×C17#294611yescharacteristic
383C2×C17#326611yescharacteristic
384C2×C17#326811yescharacteristic
385C2×C17#338511yescharacteristic
386C2×C17#338811yescharacteristic
387D17—44no—
388C22×Q8—171no—
389C22×D4—171no—
390C23×C4—341no—
391C23×C4—341no—
392C23×C4—341no—
393C2×SD16—174no—
394C2×SD16—174no—
395C2×SD16—174no—
396C22×C8C22×D1711yescharacteristic
397C22×C8C22×D1711yescharacteristic
398C22×C8C22×D1711yescharacteristic
399C22×C8C22×D1711yescharacteristic
400C4×Q8—174no—
401C4×Q8—342no—
402C4×Q8—342no—
403C4×Q8—342no—
404C4×Q8—681no—
405C4×Q8—681no—
406C4×D4—174no—
407C4×D4—342no—
408C4×D4—342no—
409C4×D4—342no—
410C4×D4—681no—
411C4×D4—681no—
412C2×C4⋊C4—171no—
413C2×C4⋊C4—171no—
414C2×C4⋊C4—171no—
415C2×C4⋊C4—171no—
416C2×C4⋊C4—171no—
417C2×C4⋊C4—171no—
418C2×C4⋊C4—341no—
419C2×C4⋊C4—341no—
420C2×C4⋊C4—341no—
421C2×C4⋊C4—341no—
422C2×C4⋊C4—341no—
423C2×C4⋊C4—341no—
424C2×C4⋊C4—341no—
425C2×C4⋊C4—341no—
426C2×C4⋊C4—341no—
427C2×C4⋊C4D6811yescharacteristic
428C2×C4⋊C4D6811yescharacteristic
429C2×C22⋊C4—341no—
430C2×C22⋊C4—341no—
431C2×C22⋊C4—341no—
432C2×C42—341no—
433C2×C42—341no—
434C2×C42—341no—
435C2×C42C22×D1711yescharacteristic
436C2×C42C22×D1711yescharacteristic
437C2×C42C22×D1711yescharacteristic
438#107—174no—
439#107—342no—
440#107—681no—
441#106—342no—
442#106—681no—
443C4⋊C8—22no—
444#103—174no—
445#103—342no—
446#103—342no—
447#102—174no—
448#102—342no—
449#102—342no—
450#97—22no—
451#97C4×D1714yescomplemented
452C4×C8—22no—
453#95—341no—
454#95—341no—
455#95—341no—
456#95—341no—
457#95—341no—
458#95—341no—
459#95D6811yescharacteristic
460#95D6811yescharacteristic
461C17#2267811yescharacteristic, complemented, Sylow
462C24—341no—
463C2×Q8—172no—
464C2×Q8—172no—
465C2×Q8—172no—
466C2×Q8—341no—
467C2×Q8—341no—
468C2×Q8—341no—
469C2×D4—172no—
470C2×D4—172no—
471C2×D4—172no—
472C2×D4—341no—
473C2×D4—341no—
474C2×D4—341no—
475C22×C4—341no—
476C22×C4—341no—
477C22×C4—341no—
478C22×C4—344no—
479C22×C4—341no—
480C22×C4—341no—
481C22×C4—341no—
482C22×C4—341no—
483C22×C4—682no—
484C22×C4—682no—
485C22×C4—682no—
486C22×C4—682no—
487C22×C4—1361no—
488C22×C4—1361no—
489C22×C4—1361no—
490C22×C4—1361no—
491C22×C4C2×D6811yescharacteristic
492C22×C4C2×D6811yescharacteristic
493C22×C4C2×D6811yescharacteristic
494C22×C4C2×D6811yescharacteristic
495C22×C4C2×D6811yescharacteristic
496C22×C4C2×D6811yescharacteristic
497C22×C4C23×D1711yescharacteristic
498SD16—1716no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8C2×C4×D1712yes—
508C2×C8C2×C4×D1712yes—
509C2×C8D4×D1712yes—
510C2×C8Q8×D1712yes—
511C4⋊C4—22no—
512C4⋊C4—41no—
513C4⋊C4—172no—
514C4⋊C4—172no—
515C4⋊C4—341no—
516C4⋊C4—341no—
517C4⋊C4—344no—
518C4⋊C4—341no—
519C4⋊C4—341no—
520C4⋊C4—341no—
521C4⋊C4—341no—
522C4⋊C4—341no—
523C4⋊C4—341no—
524C4⋊C4—341no—
525C4⋊C4—341no—
526C4⋊C4—682no—
527C4⋊C4—682no—
528C4⋊C4—682no—
529C4⋊C4—1361no—
530C4⋊C4—1361no—
531C4⋊C4—1361no—
532C4⋊C4—1361no—
533C4⋊C4—1361no—
534C22⋊C4—344no—
535C22⋊C4—682no—
536C22⋊C4—682no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—344no—
542C42—682no—
543C42—682no—
544C42C2×C4×D1712yes—
545C42#6325212yes—
546C23—342no—
547C23—342no—
548C23—342no—
549C23—681no—
550C23—681no—
551C23—681no—
552C23—681no—
553C23C22×D6811yescharacteristic
554Q8—174no—
555Q8—342no—
556Q8—342no—
557Q8—342no—
558D4—174no—
559D4—342no—
560D4—342no—
561D4—342no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—342no—
578C2×C4—342no—
579C2×C4—342no—
580C2×C4—342no—
581C2×C4—342no—
582C2×C4—681no—
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—681no—
594C2×C4—681no—
595C2×C4—681no—
596C2×C4—681no—
597C2×C4—681no—
598C2×C4—1364no—
599C2×C4—1364no—
600C2×C4—1364no—
601C2×C4C4×D6812yes—
602C2×C4C4×D6812yes—
603C2×C4#1058759512yes—
604C2×C4C22×C4×D1711yescharacteristic
605C2×C4C2×#6325211yescharacteristic
606C2×C4C2×D4×D1711yescharacteristic
607C2×C4C2×Q8×D1711yescharacteristic
608C2×C4#1058769611yescharacteristic
609C2×C4#1058769711yescharacteristic
610C8—22no—
611C8—41no—
612C8—41no—
613C8C4⋊C4×D1714yes—
614C22—344no—
615C22—682no—
616C22—682no—
617C22—682no—
618C22—682no—
619C22—682no—
620C22—682no—
621C22C2×C4×D6811yescharacteristic
622C22D4×D6811yescharacteristic
623C22Q8×D6811yescharacteristic
624C22C2×#1058759511yescharacteristic
625C22#1176360911yescharacteristic
626C22#1176361411yescharacteristic
627C22#1176362111yescharacteristic
628C4—22no—
629C4—22no—
630C4—22no—
631C4—41no—
632C4—41no—
633C4—41no—
634C4—344no—
635C4—682no—
636C4—682no—
637C4—1361no—
638C4—1361no—
639C4—1361no—
640C4—1361no—
641C4C2×C4⋊C4×D1711yescharacteristic
642C4#1176328911yescharacteristic
643C4#1176346411yescharacteristic
644C4#1176351911yescharacteristic
645C2—684no—
646C2C4⋊C4×D6811yescharacteristic
647C2C4×#1058759511yescharacteristic
648C2#1453761811yescharacteristic
649C2#1454029911yescharacteristic
650C2#1454032911yescharacteristic
651C2#1454169611yescharacteristic
652C2#1454173111yescharacteristic
subgroupG/H
centerC23C22×D68
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1771C2
socleC23×C17C22×D4
prime radicalC22×D68C23
cosocleC22×C4×C17C24
hypercenter#1771D17
hypocenterC17#22678
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1771

Complements

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

NH ≅ G/Ntype
4. C4⋊C4×Dic34C2semidirect
6. #14529522C2semidirect
7. #14529517C2semidirect
8. #14529055C2semidirect
10. #14529041C2semidirect
12. #14528952C2semidirect
13. #14528800C2semidirect
15. C17×#1771C2semidirect
24. #11762911C4semidirect
111. C4×Dic34C2×C4semidirect
114. C17×#97C2×C4semidirect
129. #10587489C2×C4semidirect
134. #10587487C2×C4semidirect
218. #63219C4⋊C4semidirect
302. #1771D17semidirect
451. #97C4×D17semidirect
461. C17#22678semidirect

Maximal subgroups

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

subgroupindex
C2×#117629112
C2×C4×#632192
C4⋊C4×Dic342
#145316952
#145295222
#145295172
#145290552
#145290482
#145290412
#145290342
#145289522
#145288002
#145287852
C17×#17712
#2267817