Finite Group Databasebeta

← #17097737

Subgroups — #17097737

Order 4352.

814 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763113C211yesmaximal, characteristic, complemented
2C2×#11762909C211yesmaximal, characteristic
3#14531704C211yesmaximal, characteristic, complemented
4#14530305C211yesmaximal, characteristic
5#14530265C211yesmaximal, characteristic, complemented
6#14530158C211yesmaximal, characteristic
7#14530119C211yesmaximal, characteristic, complemented
8#14529524C211yesmaximal, characteristic, complemented
9#14529350C211yesmaximal, characteristic, complemented
10#14529142C211yesmaximal, characteristic
11#14529142C211yesmaximal, characteristic
12#14529051C211yesmaximal, characteristic, complemented
13#14529051C211yesmaximal, characteristic, complemented
14#14529045C211yesmaximal, characteristic
15C17×#2454C211yesmaximal, characteristic
16C22×#63229C2211yescharacteristic
17C2×C4×Dic34C2211yescharacteristic
18C2×C4×Dic34C2211yescharacteristic
19C2×C4×Dic34C2211yescharacteristic
20#11763113—24no—
21#11762909—24no—
22C2×#10587478C2211yescharacteristic
23C2×#10587478C2211yescharacteristic
24C2×#10587479C2211yescharacteristic
25C2×#10587479C2211yescharacteristic
26C2×#10587479C2211yescharacteristic
27C2×C4⋊#3622C2211yescharacteristic
28C2×#10587481C2211yescharacteristic
29C2×#10587481C2211yescharacteristic
30C2×#10587481C2211yescharacteristic
31C2×#10587503C2211yescharacteristic
32C2×#10587503C2211yescharacteristic
33C2×#10587503C2211yescharacteristic
34C2×#10587474C2211yescharacteristic
35#11762707C2211yescharacteristic
36#11762707C2211yescharacteristic
37#11762704—21no—
38#11762704—21no—
39#11762675C2211yescharacteristic
40#11762673—21no—
41#11762660—21no—
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48#11762564C2211yescharacteristic
49#11762564C2211yescharacteristic
50C2×C17×#125C2211yescharacteristic
51C2×C4×D4×C17C2211yescharacteristic
52C17×#399—21no—
53C17×#396—21no—
54C17×#390C2211yescharacteristic
55C17×#390C2211yescharacteristic
56C17×#389C2211yescharacteristic
57C17×#389C2211yescharacteristic
58C17×#385—21no—
59C17×#380—21no—
60C4×C4⋊C4×C17C2211yescharacteristic
61C22×D4×C17C2311yescharacteristic
62C23×C4×C17—21no—
63C22×Dic34C2311yescharacteristic
64C17×#125—22no—
65C17×#125—41no—
66C4×D4×C17—22no—
67C4×D4×C17—41no—
68C2×C4⋊C4×C17—21no—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17D411yescharacteristic
73C2×C4⋊C4×C17D411yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C22⋊C4×C17—21no—
78C2×C22⋊C4×C17—21no—
79C2×C22⋊C4×C17—21no—
80C2×C22⋊C4×C17—21no—
81C2×C22⋊C4×C17—21no—
82C2×C42×C17C2311yescharacteristic
83C2×C42×C17C2311yescharacteristic
84C2×C42×C17C2311yescharacteristic
85C2×#63229—22no—
86C2×#63229—22no—
87C2×#63229Q814yes—
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×#63225—21no—
92C2×#63225—21no—
93C2×#63225—21no—
94C2×#63225—21no—
95C2×#63225—21no—
96C2×C4×Dic17—21no—
97C2×C4×Dic17—21no—
98C2×C4×Dic17—21no—
99C22×#3622C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101C22×#3622C2311yescharacteristic
102C22×#3622C2311yescharacteristic
103C4×Dic34—22no—
104C4×Dic34—22no—
105C4×Dic34—22no—
106C4×Dic34—22no—
107C4×Dic34—41no—
108C4×Dic34—41no—
109C17×#95—21no—
110C17×#95—21no—
111C17×#95D411yescharacteristic
112C17×#95D411yescharacteristic
113#10587503—22no—
114#10587503—22no—
115#10587503—41no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587481—22no—
121#10587481—22no—
122#10587481D414yes—
123#10587479—22no—
124#10587479—22no—
125#10587479—41no—
126#10587478—22no—
127#10587478—22no—
128C4⋊#3622—22no—
129#10587474—22no—
130C24×C17—21no—
131C2×D4×C17—21no—
132C2×D4×C17—21no—
133C2×D4×C17—21no—
134C2×D4×C17—21no—
135C2×D4×C17—21no—
136C2×D4×C17C2×Q812yes—
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—42no—
141C22×C4×C17—42no—
142C22×C4×C17—81no—
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2411yescharacteristic
150C22×Dic17—21no—
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—21no—
154C2×Dic34—21no—
155C2×Dic34—21no—
156C2×Dic34—21no—
157C2×Dic34C2×D412yes—
158C2×Dic34C2×Q812yes—
159C2×Dic34#4112yes—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—41no—
166C4⋊C4×C17—41no—
167C4⋊C4×C17—42no—
168C4⋊C4×C17—42no—
169C4⋊C4×C17—81no—
170C4⋊C4×C17—81no—
171C4⋊C4×C17C2×D412yes—
172C22⋊C4×C17—42no—
173C22⋊C4×C17—42no—
174C22⋊C4×C17—81no—
175C22⋊C4×C17—81no—
176C22⋊C4×C17—81no—
177C42×C17—21no—
178C42×C17—21no—
179C42×C17—21no—
180C42×C17—21no—
181C42×C17—21no—
182C42×C17—21no—
183#63229—28no—
184C4⋊Dic17—21no—
185C4⋊Dic17—21no—
186C4⋊Dic17—21no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17#4112yes—
190#63225—24no—
191#63225—42no—
192#63225—42no—
193#63225—42no—
194#63225—42no—
195C4×Dic17—24no—
196C4×Dic17—42no—
197C4×Dic17—42no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622—21no—
208C2×#3622C2×D412yes—
209C2×#3622C2×Q812yes—
210#26399—171nomaximal, Sylow
211C23×C17—22no—
212C23×C17—41no—
213C23×C17—41no—
214C23×C17—41no—
215C23×C17—41no—
216C23×C17—41no—
217C23×C17—41no—
218C23×C17C22×D411yescharacteristic
219D4×C17—22no—
220D4×C17—22no—
221D4×C17—41no—
222D4×C17—41no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—22no—
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—41no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17—84no—
251C2×C4×C17C22×D411yescharacteristic
252C2×C4×C17C22×Q811yescharacteristic
253C2×C4×C17C2×#4111yescharacteristic
254C2×C4×C17C2×#4111yescharacteristic
255C2×C4×C17H(5,𝔽2)11yescharacteristic
256C2×C4×C172−1+411yescharacteristic
257C2×Dic17—22no—
258C2×Dic17—22no—
259C2×Dic17—22no—
260C2×Dic17—22no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17—41no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268C2×Dic17—41no—
269Dic34—22no—
270Dic34—22no—
271Dic34—22no—
272Dic34C22⋊Q814yes—
273#3622—22no—
274#3622—22no—
275#3622—41no—
276#3622—41no—
277C2×#479—171no—
278C2×#440—171no—
279#2454D1711yescharacteristic
280#2351—171no—
281#2092—171no—
282#2085—171no—
283#2065—171no—
284#2059—171no—
285#1944—171no—
286#1917—171no—
287#1880—171no—
288#1880—171no—
289#1865—171no—
290#1865—171no—
291#1864—171no—
292C22×C17—42no—
293C22×C17—42no—
294C22×C17—42no—
295C22×C17—81no—
296C22×C17—81no—
297C22×C17—81no—
298C22×C17—81no—
299C22×C17#52511yescharacteristic
300C22×C17D4211yescharacteristic
301C22×C17D4×Q811yescharacteristic
302C22×C17#57511yescharacteristic
303C22×C17#57511yescharacteristic
304C22×C17#57711yescharacteristic
305C22×C17#57711yescharacteristic
306C4×C17—41no—
307C4×C17—41no—
308C4×C17—41no—
309C4×C17—41no—
310C4×C17—41no—
311C4×C17—41no—
312C4×C17—42no—
313C4×C17—81no—
314C4×C17—81no—
315C4×C17C2×C22⋊Q811yescharacteristic
316C4×C17#53911yescharacteristic
317C4×C17#54711yescharacteristic
318C4×C17#55411yescharacteristic
319Dic17—24no—
320Dic17—42no—
321Dic17—81no—
322Dic17—81no—
323C22×SD16—171no—
324C2×#125C2×D1711yescharacteristic
325C2×C4×Q8—171no—
326C2×C4×Q8—171no—
327C2×C4×Q8—171no—
328C2×C4×D4C2×D1711yescharacteristic
329#479—344no—
330#440—344no—
331C2×#107—171no—
332C2×#107—171no—
333C2×#106—171no—
334C2×#106—171no—
335C2×#106—171no—
336C2×C4⋊C8—171no—
337C2×#103—171no—
338C2×#103—171no—
339C2×#103—171no—
340C2×#102—171no—
341C2×#102—171no—
342C2×#102—171no—
343C2×#97—171no—
344#399—21no—
345#396—21no—
346#390C2×D1711yescharacteristic
347#390C2×D1711yescharacteristic
348#389—171no—
349#389—171no—
350#389—341no—
351#389—341no—
352#389C2×D1711yescharacteristic
353#389C2×D1711yescharacteristic
354#385—21no—
355#382—171no—
356#382—341no—
357#380—21no—
358C4×C4⋊C4—341no—
359C4×C4⋊C4C2×D1711yescharacteristic
360#340—171no—
361#340—171no—
362#340—171no—
363#340—171no—
364#336—171no—
365#336—171no—
366#336—171no—
367#336—171no—
368C2×C17—82no—
369C2×C17#244911yescharacteristic
370C2×C17#311111yescharacteristic
371C2×C17#312611yescharacteristic
372C2×C17#327111yescharacteristic
373C2×C17#327511yescharacteristic
374C2×C17#338811yescharacteristic
375C2×C17#338811yescharacteristic
376C22×Q8—171no—
377C22×D4C22×D1711yescharacteristic
378C23×C4—21no—
379C2×SD16—174no—
380C2×SD16—342no—
381C2×SD16—342no—
382C22×C8—171no—
383C22×C8—171no—
384C22×C8—171no—
385C22×C8—171no—
386#125—22no—
387#125—41no—
388C4×Q8—342no—
389C4×Q8—342no—
390C4×Q8—342no—
391C4×Q8—342no—
392C4×Q8—681no—
393C4×Q8—681no—
394C4×D4—22no—
395C4×D4—41no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—171no—
401C2×C4⋊C4—171no—
402C2×C4⋊C4—171no—
403C2×C4⋊C4—341no—
404C2×C4⋊C4—341no—
405C2×C4⋊C4—341no—
406C2×C4⋊C4—341no—
407C2×C4⋊C4—341no—
408C2×C4⋊C4C22⋊D1711yescharacteristic
409C2×C4⋊C4C22⋊D1711yescharacteristic
410C2×C4⋊C4C22×D1711yescharacteristic
411C2×C4⋊C4C22×D1711yescharacteristic
412C2×C4⋊C4C22×D1711yescharacteristic
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C42—341no—
419C2×C42—341no—
420C2×C42—341no—
421C2×C42C22×D1711yescharacteristic
422C2×C42C22×D1711yescharacteristic
423C2×C42C22×D1711yescharacteristic
424#107—342no—
425#107—342no—
426#106—342no—
427#106—342no—
428#106—681no—
429C4⋊C8—342no—
430#103—174no—
431#103—342no—
432#103—342no—
433#102—342no—
434#102—342no—
435#102—681no—
436#97—342no—
437#95—21no—
438#95—21no—
439#95—341no—
440#95—341no—
441#95—341no—
442#95—341no—
443#95C22⋊D1711yescharacteristic
444#95C22⋊D1711yescharacteristic
445C17#2639911yescharacteristic, complemented, Sylow
446C24—21no—
447C2×Q8—172no—
448C2×Q8—172no—
449C2×Q8—172no—
450C2×Q8—341no—
451C2×Q8—341no—
452C2×Q8—341no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4Q8×D1712yes—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—81no—
465C22×C4—341no—
466C22×C4—341no—
467C22×C4—341no—
468C22×C4—341no—
469C22×C4C2×C22⋊D1711yescharacteristic
470C22×C4C2×C22⋊D1711yescharacteristic
471C22×C4C2×C22⋊D1711yescharacteristic
472C22×C4C2×C22⋊D1711yescharacteristic
473C22×C4C2×C22⋊D1711yescharacteristic
474C22×C4C2×C22⋊D1711yescharacteristic
475C22×C4C23×D1711yescharacteristic
476SD16—348no—
477C2×C8—172no—
478C2×C8—172no—
479C2×C8—341no—
480C2×C8—341no—
481C2×C8—341no—
482C2×C8—341no—
483C2×C8—341no—
484C2×C8—341no—
485C2×C8—341no—
486C2×C8—341no—
487C2×C8—341no—
488C2×C8—341no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—81no—
499C4⋊C4—81no—
500C4⋊C4—172no—
501C4⋊C4—344no—
502C4⋊C4—341no—
503C4⋊C4—341no—
504C4⋊C4—341no—
505C4⋊C4—341no—
506C4⋊C4—341no—
507C4⋊C4—682no—
508C4⋊C4—682no—
509C4⋊C4—682no—
510C4⋊C4—682no—
511C4⋊C4D4×D1712yes—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—81no—
515C22⋊C4—81no—
516C22⋊C4—81no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—344no—
524C42—682no—
525C42—682no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23C22×C22⋊D1711yescharacteristic
534Q8—174no—
535Q8—342no—
536Q8—342no—
537Q8—342no—
538D4—22no—
539D4—22no—
540D4—41no—
541D4—41no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—84no—
570C2×C4—342no—
571C2×C4—342no—
572C2×C4—342no—
573C2×C4—342no—
574C2×C4—681no—
575C2×C4—681no—
576C2×C4—681no—
577C2×C4—681no—
578C2×C4—681no—
579C2×C4—681no—
580C2×C4—681no—
581C2×C4—681no—
582C2×C4C2×D4×D1711yescharacteristic
583C2×C4#1058769111yescharacteristic
584C2×C4C2×Q8×D1711yescharacteristic
585C2×C4#1058769411yescharacteristic
586C2×C4#41×D1711yescharacteristic
587C2×C4#41×D1711yescharacteristic
588C8—342no—
589C8—342no—
590C8—681no—
591C8—681no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22#1176331611yescharacteristic
600C22D4×C22⋊D1711yescharacteristic
601C22Q8×C22⋊D1711yescharacteristic
602C22#1176360711yescharacteristic
603C22#1176360711yescharacteristic
604C22#1176362011yescharacteristic
605C22#1176362011yescharacteristic
606C4—41no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—41no—
611C4—42no—
612C4—41no—
613C4—81no—
614C4—81no—
615C4—344no—
616C4—682no—
617C4—1361no—
618C4—1361no—
619C4C22⋊Q8×D1711yescharacteristic
620C4#1176340511yescharacteristic
621C4#1176346111yescharacteristic
622C4#1176351311yescharacteristic
623C2—82no—
624C2#1453259011yescharacteristic
625C2#1453888711yescharacteristic
626C2#1453907011yescharacteristic
627C2#1454036511yescharacteristic
628C2#1454041111yescharacteristic
629C2#1454172911yescharacteristic
630C2#1454172911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2454C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
hypercenter#2454D17
hypocenterC17#26399
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2454

Complements

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

NH ≅ G/Ntype
1. C2×#11763113C2semidirect
3. #14531704C2semidirect
5. #14530265C2semidirect
7. #14530119C2semidirect
8. #14529524C2semidirect
9. #14529350C2semidirect
12. #14529051C2semidirect
13. #14529051C2semidirect
445. C17#26399semidirect

Maximal subgroups

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

subgroupindex
C2×#117631132
C2×#117629092
#145317042
#145303052
#145302652
#145301582
#145301192
#145295242
#145293502
#145291422
#145290512
#145290452
C17×#24542
#2639917