Finite Group Databasebeta

← #17097380

Subgroups — #17097380

Order 4352.

820 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×#11763116C211yesmaximal, characteristic, complemented
2C2×C4×#63230C211yesmaximal, characteristic
3#14532656C211yesmaximal, characteristic
4#14530308C211yesmaximal, characteristic
5#14530266C211yesmaximal, characteristic, complemented
6#14530155C211yesmaximal, characteristic
7#14530122C211yesmaximal, characteristic, complemented
8#14529509C211yesmaximal, characteristic, complemented
9#14529343C211yesmaximal, characteristic, complemented
10#14529139C211yesmaximal, characteristic
11#14529139C211yesmaximal, characteristic
12#14529052C211yesmaximal, characteristic, complemented
13#14529038C211yesmaximal, characteristic, complemented
14#14529030C211yesmaximal, characteristic
15C17×#2351C211yesmaximal, characteristic, complemented
16C22×#63230C2211yescharacteristic
17C2×#10587556C2211yescharacteristic
18C2×C4×D68C2211yescharacteristic
19#11763116—24no—
20C4×#63230—24no—
21C2×#10587478C2211yescharacteristic
22C2×#10587478C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×#10587479C2211yescharacteristic
25C2×#10587479C2211yescharacteristic
26C2×C4⋊#3622C2211yescharacteristic
27C2×#10587506C2211yescharacteristic
28C2×#10587506C2211yescharacteristic
29C2×#10587506C2211yescharacteristic
30C2×#10587480C2211yescharacteristic
31C2×#10587480C2211yescharacteristic
32C2×#10587480C2211yescharacteristic
33C2×C4×#3622C2211yescharacteristic
34#11762744—21no—
35#11762733—21no—
36#11762711C2211yescharacteristic
37#11762711C2211yescharacteristic
38#11762707C2211yescharacteristic
39#11762707C2211yescharacteristic
40#11762680—21no—
41#11762668—21no—
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48#11762564C2211yescharacteristic
49#11762564C2211yescharacteristic
50C2×C4×Q8×C17C2211yescharacteristic
51C2×C4×Q8×C17C2211yescharacteristic
52C2×C4×Q8×C17C2211yescharacteristic
53C17×#389—21no—
54C17×#389—21no—
55C17×#389C2211yescharacteristic
56C17×#389C2211yescharacteristic
57C17×#382—21no—
58C17×#382C2211yescharacteristic
59C4×C4⋊C4×C17—21no—
60C4×C4⋊C4×C17C2211yescharacteristic
61C22×Q8×C17C2311yescharacteristic
62C22×D68C2311yescharacteristic
63C22×C4×D17—21no—
64C4×Q8×C17—22no—
65C4×Q8×C17—22no—
66C4×Q8×C17—22no—
67C4×Q8×C17—22no—
68C4×Q8×C17—41no—
69C4×Q8×C17—41no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17D411yescharacteristic
76C2×C4⋊C4×C17D411yescharacteristic
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C4⋊C4×C17C2311yescharacteristic
79C2×C4⋊C4×C17C2311yescharacteristic
80C2×C42×C17—21no—
81C2×C42×C17—21no—
82C2×C42×C17—21no—
83C2×C42×C17C2311yescharacteristic
84C2×C42×C17C2311yescharacteristic
85C2×C42×C17C2311yescharacteristic
86C2×#63230—22no—
87C2×#63230—22no—
88C2×#63230Q814yes—
89C2×#63227—21no—
90C2×#63227—21no—
91C2×#63227—21no—
92C2×#63227—21no—
93C2×#63227—21no—
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96C2×C4⋊Dic17C2311yescharacteristic
97C2×#63225—21no—
98C2×#63225—21no—
99C2×#63225—21no—
100C2×#63225—21no—
101C22×#3622C2311yescharacteristic
102C22×#3622C2311yescharacteristic
103C22×#3622C2311yescharacteristic
104C22×#3622C2311yescharacteristic
105C4×D68—22no—
106C4×D68—41no—
107#10587556—22no—
108#10587556—41no—
109C17×#95—21no—
110C17×#95—21no—
111C17×#95—21no—
112C17×#95—21no—
113C17×#95D411yescharacteristic
114C17×#95D411yescharacteristic
115#10587506—22no—
116#10587506—22no—
117#10587506D414yes—
118#10587502—21no—
119#10587502—21no—
120#10587480—22no—
121#10587480—22no—
122#10587480—41no—
123#10587479—22no—
124#10587479—22no—
125#10587479—22no—
126#10587478—22no—
127#10587478—41no—
128C4⋊#3622—22no—
129C4×#3622—22no—
130C23×D17—21no—
131C2×Q8×C17—21no—
132C2×Q8×C17—21no—
133C2×Q8×C17—21no—
134C2×Q8×C17C2×D412yes—
135C2×Q8×C17C2×Q812yes—
136C2×Q8×C17#4112yes—
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2411yescharacteristic
148C22×Dic17—21no—
149C22×Dic17—21no—
150C22×Dic17—21no—
151C2×D68—21no—
152C2×D68—21no—
153C2×D68—21no—
154C2×D68—21no—
155C2×D68—21no—
156C2×D68C2×Q812yes—
157C2×C4×D17—42no—
158C2×C4×D17—42no—
159C2×C4×D17—81no—
160C4⋊C4×C17—24no—
161C4⋊C4×C17—22no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—41no—
168C4⋊C4×C17—42no—
169C4⋊C4×C17—42no—
170C4⋊C4×C17—42no—
171C4⋊C4×C17—42no—
172C4⋊C4×C17#4112yes—
173C42×C17—24no—
174C42×C17—21no—
175C42×C17—21no—
176C42×C17—21no—
177C42×C17—21no—
178C42×C17—21no—
179C42×C17—21no—
180C42×C17—42no—
181C42×C17—42no—
182#63230—28no—
183#63227—42no—
184#63227—42no—
185#63227—81no—
186#63227—81no—
187#63227—81no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—21no—
192C4⋊Dic17—21no—
193C4⋊Dic17C2×D412yes—
194#63225—42no—
195#63225—42no—
196#63225—81no—
197#63225—81no—
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#26375—171nomaximal, Sylow
211C23×C17C22×D411yescharacteristic
212C22×D17—22no—
213C22×D17—41no—
214C22×D17—41no—
215C22×D17—41no—
216C22×D17—41no—
217C22×D17—41no—
218C22×D17—41no—
219Q8×C17—22no—
220Q8×C17—22no—
221Q8×C17—22no—
222Q8×C17C22⋊Q814yes—
223C2×C4×C17—22no—
224C2×C4×C17—21no—
225C2×C4×C17—22no—
226C2×C4×C17—22no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—22no—
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×C17—21no—
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—41no—
251C2×C4×C17—41no—
252C2×C4×C17#13512yes—
253C2×C4×C17C22×D411yescharacteristic
254C2×C4×C17C22×Q811yescharacteristic
255C2×C4×C17C2×#4111yescharacteristic
256C2×C4×C17C2×#4111yescharacteristic
257C2×C4×C17H(5,𝔽2)11yescharacteristic
258C2×C4×C172−1+411yescharacteristic
259C2×Dic17—22no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17—41no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268D68—22no—
269D68—22no—
270D68—41no—
271D68—41no—
272C4×D17—84no—
273#3622—22no—
274#3622—22no—
275#3622—41no—
276#3622—41no—
277C2×#479—171no—
278C2×C4×SD16—171no—
279#2454—171no—
280#2351D1711yescharacteristic, complemented
281#2092—171no—
282#2085—171no—
283#2065—171no—
284#2059—171no—
285#1941—171no—
286#1916—171no—
287#1880—171no—
288#1880—171no—
289#1865—171no—
290#1863—171no—
291#1862—171no—
292C22×C17#52511yescharacteristic
293C22×C17D4211yescharacteristic
294C22×C17D4×Q811yescharacteristic
295C22×C17C2×#13511yescharacteristic
296C22×C17#57511yescharacteristic
297C22×C17#57711yescharacteristic
298C22×C17#57711yescharacteristic
299C2×D17—42no—
300C2×D17—42no—
301C2×D17—42no—
302C2×D17—81no—
303C2×D17—81no—
304C2×D17—81no—
305C2×D17—81no—
306C4×C17—24no—
307C4×C17—22no—
308C4×C17—41no—
309C4×C17—42no—
310C4×C17—41no—
311C4×C17—41no—
312C4×C17—41no—
313C4×C17—41no—
314C4×C17—81no—
315C4×C17—81no—
316C4×C17C2×C22⋊Q811yescharacteristic
317C4×C17#53911yescharacteristic
318C4×C17#54711yescharacteristic
319C4×C17#55411yescharacteristic
320Dic17—42no—
321Dic17—81no—
322Dic17—81no—
323C22×SD16—171no—
324C2×#125—171no—
325C2×C4×Q8C2×D1711yescharacteristic
326C2×C4×Q8C2×D1711yescharacteristic
327C2×C4×Q8C2×D1711yescharacteristic
328C2×C4×D4—171no—
329#479—344no—
330C4×SD16—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×C4×C8—171no—
344#399—341no—
345#396—341no—
346#390—171no—
347#390—171no—
348#389—21no—
349#389—21no—
350#389—171no—
351#389—171no—
352#389C2×D1711yescharacteristic
353#389C2×D1711yescharacteristic
354#385—341no—
355#382—21no—
356#382C2×D1711yescharacteristic
357#380—341no—
358C4×C4⋊C4—21no—
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#244911yescharacteristic
369C2×C17#310211yescharacteristic
370C2×C17#312611yescharacteristic
371C2×C17#327111yescharacteristic
372C2×C17#328211yescharacteristic
373C2×C17#338111yescharacteristic
374C2×C17#338811yescharacteristic
375D17—82no—
376C22×Q8C22×D1711yescharacteristic
377C22×D4—171no—
378C23×C4—341no—
379C2×SD16—174no—
380C2×SD16—342no—
381C2×SD16—342no—
382C22×C8—171no—
383C22×C8—171no—
384C22×C8—171no—
385C22×C8—171no—
386#125—342no—
387#125—681no—
388C4×Q8—22no—
389C4×Q8—22no—
390C4×Q8—22no—
391C4×Q8—22no—
392C4×Q8—41no—
393C4×Q8—41no—
394C4×D4—342no—
395C4×D4—681no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—171no—
402C2×C4⋊C4—171no—
403C2×C4⋊C4—171no—
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—341no—
414C2×C22⋊C4—341no—
415C2×C22⋊C4—341no—
416C2×C22⋊C4—341no—
417C2×C22⋊C4—341no—
418C2×C42—21no—
419C2×C42—21no—
420C2×C42—21no—
421C2×C42C22×D1711yescharacteristic
422C2×C42C22×D1711yescharacteristic
423C2×C42C22×D1711yescharacteristic
424#107—342no—
425#107—681no—
426#106—342no—
427#106—342no—
428#106—342no—
429C4⋊C8—342no—
430#103—174no—
431#103—342no—
432#103—342no—
433#102—342no—
434#102—342no—
435#102—681no—
436C4×C8—342no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—341no—
442#95—341no—
443#95C22⋊D1711yescharacteristic
444#95C22⋊D1711yescharacteristic
445C17#2637511yescharacteristic, complemented, Sylow
446C24—341no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8#6325212yes—
451C2×Q8D4×D1712yes—
452C2×Q8Q8×D1712yes—
453C2×D4—172no—
454C2×D4—341no—
455C2×D4—341no—
456C2×D4—341no—
457C2×D4—341no—
458C2×D4—341no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—341no—
464C22×C4—341no—
465C22×C4—341no—
466C22×C4—682no—
467C22×C4—682no—
468C22×C4—1361no—
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—24no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—41no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—172no—
502C4⋊C4—341no—
503C4⋊C4—341no—
504C4⋊C4—341no—
505C4⋊C4—341no—
506C4⋊C4—341no—
507C4⋊C4—682no—
508C4⋊C4—682no—
509C4⋊C4—1361no—
510C4⋊C4—1361no—
511C4⋊C4#6325212yes—
512C22⋊C4—682no—
513C22⋊C4—682no—
514C22⋊C4—1361no—
515C22⋊C4—1361no—
516C22⋊C4—1361no—
517C42—24no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—42no—
525C42—42no—
526C23—342no—
527C23—681no—
528C23—681no—
529C23—681no—
530C23—681no—
531C23—681no—
532C23—681no—
533C23C22×C22⋊D1711yescharacteristic
534Q8—22no—
535Q8—22no—
536Q8—22no—
537Q8#1058758214yes—
538D4—342no—
539D4—342no—
540D4—681no—
541D4—681no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
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—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—342no—
572C2×C4—681no—
573C2×C4—681no—
574C2×C4—681no—
575C2×C4—681no—
576C2×C4—681no—
577C2×C4—681no—
578C2×C4—681no—
579C2×C4—681no—
580C2×C4—1364no—
581C2×C4#1058763612yes—
582C2×C4C2×#6325211yescharacteristic
583C2×C4C2×#6325211yescharacteristic
584C2×C4C2×D4×D1711yescharacteristic
585C2×C4#1058769111yescharacteristic
586C2×C4C2×Q8×D1711yescharacteristic
587C2×C4#1058769411yescharacteristic
588C8—342no—
589C8—342no—
590C8—681no—
591C8—681no—
592C22—682no—
593C22—682no—
594C22—682no—
595C22—1361no—
596C22—1361no—
597C22—1361no—
598C22—1361no—
599C22#1176332011yescharacteristic
600C22D4×C22⋊D1711yescharacteristic
601C22Q8×C22⋊D1711yescharacteristic
602C22C2×#1058763611yescharacteristic
603C22#1176360711yescharacteristic
604C22#1176361711yescharacteristic
605C22#1176361711yescharacteristic
606C4—22no—
607C4—24no—
608C4—41no—
609C4—42no—
610C4—41no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—81no—
615C4—81no—
616C4—682no—
617C4—1361no—
618C4—1361no—
619C4C2×#1058758211yescharacteristic
620C4#1176340611yescharacteristic
621C4#1176346011yescharacteristic
622C4#1176351411yescharacteristic
623C2—1362no—
624C2#1453259311yescharacteristic
625C2#1453880211yescharacteristic
626C2#1453907511yescharacteristic
627C2#1454037011yescharacteristic
628C2#1454049211yescharacteristic
629C2#1454165511yescharacteristic
630C2#1454172311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2351C2
socleC23×C17C22×D4
prime radicalC22×D68C23
hypercenter#2351D17
hypocenterC17#26375
LCSG ▷ C22×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
1. C2×#11763116C2semidirect
5. #14530266C2semidirect
7. #14530122C2semidirect
8. #14529509C2semidirect
9. #14529343C2semidirect
12. #14529052C2semidirect
13. #14529038C2semidirect
15. C17×#2351C2semidirect
280. #2351D17semidirect
445. C17#26375semidirect

Maximal subgroups

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

subgroupindex
C2×#117631162
C2×C4×#632302
#145326562
#145303082
#145302662
#145301552
#145301222
#145295092
#145293432
#145291392
#145290522
#145290382
#145290302
C17×#23512
#2637517