Finite Group Databasebeta

← #30873594

Subgroups — #30873594

Order 17152.

782 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863436C211yesmaximal, characteristic
2#22563100C211yesmaximal, characteristic
3#22560949C211yesmaximal, characteristic, complemented
4#22560934C211yesmaximal, characteristic, complemented
5#22560798C211yesmaximal, characteristic, complemented
6#22560794C211yesmaximal, characteristic, complemented
7#22560787C211yesmaximal, characteristic, complemented
8#22560777C211yesmaximal, characteristic, complemented
9#22560301C211yesmaximal, characteristic
10#22560295C211yesmaximal, characteristic
11#22560240C211yesmaximal, characteristic
12#22560107C211yesmaximal, characteristic, complemented
13C4×#14523976C211yesmaximal, characteristic, complemented
14#22559993C211yesmaximal, characteristic
15C67×#2345C211yesmaximal, characteristic
16C22×#11761612C2211yescharacteristic
17C2×#14524002C2211yescharacteristic
18#16863436C418yescomplemented
19C2×#14523976C2211yescharacteristic
20C2×#14523976C2211yescharacteristic
21C2×#14523976C2211yescharacteristic
22C2×#14523974C2211yescharacteristic
23C2×#14523974C2211yescharacteristic
24C2×#14523974C2211yescharacteristic
25C2×C22⋊(C67⋊C8)C2211yescharacteristic
26C2×#14523954C2211yescharacteristic
27C2×C4×C67⋊C8C2211yescharacteristic
28#16863294C2211yescharacteristic
29#16863294C2211yescharacteristic
30#16863290—21no—
31#16863290—21no—
32#16863263—21no—
33#16863256C2211yescharacteristic
34C4×C4⋊Dic67C2211yescharacteristic
35C4×C4⋊Dic67C2211yescharacteristic
36#16863247—21no—
37#16863158C2211yescharacteristic
38#16863158C2211yescharacteristic
39#16863158C2211yescharacteristic
40#16863158C2211yescharacteristic
41#16863158C2211yescharacteristic
42#16863158C2211yescharacteristic
43#16863158C2211yescharacteristic
44#16863158C2211yescharacteristic
45#16863158C2211yescharacteristic
46#16863158C2211yescharacteristic
47#16863151C2211yescharacteristic
48#16863151C2211yescharacteristic
49C22×C4⋊C4×C67C2211yescharacteristic
50#389×C67C2211yescharacteristic
51#389×C67C2211yescharacteristic
52#387×C67—21no—
53#387×C67—21no—
54#380×C67—21no—
55#380×C67C2211yescharacteristic
56C4×C4⋊C4×C67C2211yescharacteristic
57C4×C4⋊C4×C67C2211yescharacteristic
58C4×C22⋊C4×C67—21no—
59C4×C22⋊C4×C67C2211yescharacteristic
60C2×#11761612—22no—
61C2×#11761612—22no—
62C2×#11761612C2×C414yes—
63C22×C67⋊C8C2311yescharacteristic
64C22×C67⋊C8C2311yescharacteristic
65C22×C67⋊C8C2311yescharacteristic
66C22×C67⋊C8C2311yescharacteristic
67#14524002—22no—
68#14524002C2×C414yes—
69C2×C4⋊Dic67C2311yescharacteristic
70C2×C4⋊Dic67C2311yescharacteristic
71C2×C4⋊Dic67C2311yescharacteristic
72C2×C4⋊Dic67C2311yescharacteristic
73C2×Dic67⋊C4—21no—
74C2×Dic67⋊C4—21no—
75C2×C22⋊Dic67—21no—
76C2×C22⋊Dic67—21no—
77C2×C4×Dic67—21no—
78C2×C4×Dic67—21no—
79C2×C4×Dic67—21no—
80#14523976—22no—
81#14523976C2×C414yes—
82#14523976C2×C414yes—
83#14523974—22no—
84#14523974C2×C414yes—
85#14523974C2×C414yes—
86C22⋊(C67⋊C8)—22no—
87#14523954—22no—
88C4×C67⋊C8—22no—
89C67⋊#95—21no—
90C67⋊#95—21no—
91C67⋊#95—21no—
92C67⋊#95—21no—
93C67⋊#95—21no—
94C23×C4×C67—21no—
95C23×C4×C67C2311yescharacteristic
96C2×C4⋊C4×C67—21no—
97C2×C4⋊C4×C67—22no—
98C2×C4⋊C4×C67—21no—
99C2×C4⋊C4×C67—21no—
100C2×C4⋊C4×C67C2×C414yes—
101C2×C4⋊C4×C67C2311yescharacteristic
102C2×C4⋊C4×C67C2311yescharacteristic
103C2×C4⋊C4×C67C2311yescharacteristic
104C2×C4⋊C4×C67C2311yescharacteristic
105C2×C22⋊C4×C67—21no—
106C2×C22⋊C4×C67D411yescharacteristic
107C2×C22⋊C4×C67D411yescharacteristic
108C2×C42×C67—21no—
109C2×C42×C67—21no—
110C2×C42×C67C2311yescharacteristic
111C2×C42×C67C2311yescharacteristic
112#95×C67—21no—
113#95×C67—21no—
114#95×C67—21no—
115#95×C67—21no—
116#95×C67—21no—
117#95×C67D411yescharacteristic
118#95×C67D411yescharacteristic
119C22×Dic67—21no—
120C22×Dic67—21no—
121C22×Dic67—21no—
122C22×Dic67—21no—
123#11761612—28no—
124C2×C67⋊C8—21no—
125C2×C67⋊C8—21no—
126C2×C67⋊C8—21no—
127C2×C67⋊C8—21no—
128C2×C67⋊C8—21no—
129C2×C67⋊C8—21no—
130C2×C67⋊C8—21no—
131C2×C67⋊C8—21no—
132C2×C67⋊C8C22×C412yes—
133C2×C67⋊C8C22×C412yes—
134C2×C67⋊C8#4112yes—
135C2×C67⋊C8#4112yes—
136C4⋊Dic67—21no—
137C4⋊Dic67—21no—
138C4⋊Dic67—21no—
139C4⋊Dic67—21no—
140C4⋊Dic67—21no—
141C4⋊Dic67—21no—
142C4⋊Dic67C22×C412yes—
143C4⋊Dic67C22×C412yes—
144Dic67⋊C4—42no—
145Dic67⋊C4—42no—
146C22⋊Dic67—24no—
147C22⋊Dic67—81no—
148C4×Dic67—24no—
149C4×Dic67—81no—
150C4×Dic67—81no—
151C24×C67C2×D411yescharacteristic
152C22×C4×C67—21no—
153C22×C4×C67—21no—
154C22×C4×C67—21no—
155C22×C4×C67—21no—
156C22×C4×C67—21no—
157C22×C4×C67—21no—
158C22×C4×C67—24no—
159C22×C4×C67—21no—
160C22×C4×C67—21no—
161C22×C4×C67—81no—
162C22×C4×C67—81no—
163C22×C4×C67C22×C412yes—
164C22×C4×C67C2×D411yescharacteristic
165C22×C4×C67C2×D411yescharacteristic
166C22×C4×C67C2×D411yescharacteristic
167C22×C4×C67C2×D411yescharacteristic
168C22×C4×C67C2×D411yescharacteristic
169C22×C4×C67#4112yes—
170C22×C4×C67C2411yescharacteristic
171C4⋊C4×C67—24no—
172C4⋊C4×C67—21no—
173C4⋊C4×C67—21no—
174C4⋊C4×C67—21no—
175C4⋊C4×C67—21no—
176C4⋊C4×C67—21no—
177C4⋊C4×C67—21no—
178C4⋊C4×C67—81no—
179C4⋊C4×C67—81no—
180C4⋊C4×C67C22×C412yes—
181C4⋊C4×C67C22×C412yes—
182C22⋊C4×C67—22no—
183C22⋊C4×C67—41no—
184C22⋊C4×C67—42no—
185C42×C67—21no—
186C42×C67—21no—
187C42×C67—21no—
188C42×C67—21no—
189C42×C67—81no—
190C42×C67—81no—
191C23×C67—21no—
192C23×C67—21no—
193C23×C67—21no—
194C23×C67—21no—
195C23×C67—21no—
196C23×C67—21no—
197C23×C67C4×D412yes—
198C23×C67C22×D411yescharacteristic
199C2×C4×C67—21no—
200C2×C4×C67—21no—
201C2×C4×C67—21no—
202C2×C4×C67—21no—
203C2×C4×C67—22no—
204C2×C4×C67—22no—
205C2×C4×C67—22no—
206C2×C4×C67—22no—
207C2×C4×C67—22no—
208C2×C4×C67—22no—
209C2×C4×C67—21no—
210C2×C4×C67—21no—
211C2×C4×C67—21no—
212C2×C4×C67—21no—
213C2×C4×C67—21no—
214C2×C4×C67—21no—
215C2×C4×C67—21no—
216C2×C4×C67—21no—
217C2×C4×C67—21no—
218C2×C4×C67—41no—
219C2×C4×C67—41no—
220C2×C4×C67—41no—
221C2×C4×C67—41no—
222C2×C4×C67—41no—
223C2×C4×C67—41no—
224C2×C4×C67—41no—
225C2×C4×C67—41no—
226C2×C4×C67—41no—
227C2×C4×C67—41no—
228C2×C4×C67—84no—
229C2×C4×C67C4×D412yes—
230C2×C4×C67#13712yes—
231C2×C4×C67C23×C411yescharacteristic
232C2×C4×C67C2×#4111yescharacteristic
233C2×C4×C67C2×#4111yescharacteristic
234C2×C4×C67C2×#4111yescharacteristic
235C2×C4×C672−1+411yescharacteristic
236C2×C4×C672−1+411yescharacteristic
237C2×Dic67—22no—
238C2×Dic67—22no—
239C2×Dic67—41no—
240C2×Dic67—41no—
241C2×Dic67—41no—
242C2×Dic67—41no—
243C2×Dic67—41no—
244C2×Dic67—41no—
245C2×Dic67—41no—
246C2×Dic67—41no—
247C2×Dic67—41no—
248C2×Dic67—41no—
249C67⋊C8—22no—
250C67⋊C8—41no—
251C67⋊C8—41no—
252C67⋊C8#11714yes—
253C22×C67—22no—
254C22×C67—22no—
255C22×C67—22no—
256C22×C67—41no—
257C22×C67—41no—
258C22×C67—41no—
259C22×C67—41no—
260C22×C67C2×C4×D411yescharacteristic
261C22×C67#54711yescharacteristic
262C22×C67#54711yescharacteristic
263C22×C67C2×#13711yescharacteristic
264C22×C67#57511yescharacteristic
265C22×C67#57611yescharacteristic
266C22×C67#57711yescharacteristic
267C4×C67—22no—
268C4×C67—22no—
269C4×C67—41no—
270C4×C67—41no—
271C4×C67—41no—
272C4×C67—42no—
273C4×C67—42no—
274C4×C67—81no—
275C4×C67—81no—
276C4×C67C2×#11711yescharacteristic
277C4×C67#51911yescharacteristic
278C4×C67#55211yescharacteristic
279C4×C67#55211yescharacteristic
280Dic67—42no—
281Dic67—42no—
282Dic67—81no—
283Dic67—81no—
284#22623—671nomaximal, Sylow
285C2×C67—42no—
286C2×C67#234411yescharacteristic
287C2×C67C4×#13711yescharacteristic
288C2×C67#294611yescharacteristic
289C2×C67#334611yescharacteristic
290C2×C67#335011yescharacteristic
291C2×C67#335511yescharacteristic
292C2×C67#335611yescharacteristic
293C2×#428—671no—
294#2347—671no—
295#2345D6711yescharacteristic
296#1944—671no—
297#1941—671no—
298#1921—671no—
299#1920—671no—
300#1919—671no—
301#1918—671no—
302#1825—671no—
303#1823—671no—
304#1809—671no—
305#1776—671no—
306C4×#107—671no—
307#1753—671no—
308C67#2262311yescharacteristic, complemented, Sylow
309C22×M16—671no—
310C2×#117—671no—
311C22×C4⋊C4C2×D6711yescharacteristic
312#428—678no—
313C2×#107—671no—
314C2×#107—671no—
315C2×#107—671no—
316C2×#106—671no—
317C2×#106—671no—
318C2×#106—671no—
319C2×C22⋊C8—671no—
320C2×#97—671no—
321C2×C4×C8—671no—
322#389—671no—
323#389—671no—
324#389C2×D6711yescharacteristic
325#389C2×D6711yescharacteristic
326#388—1341no—
327#388—1341no—
328#387—21no—
329#387—21no—
330#383—1341no—
331#380—21no—
332#380—671no—
333#380C2×D6711yescharacteristic
334C4×C4⋊C4—671no—
335C4×C4⋊C4—671no—
336C4×C4⋊C4—1341no—
337C4×C4⋊C4C2×D6711yescharacteristic
338C4×C4⋊C4C2×D6711yescharacteristic
339C4×C22⋊C4—21no—
340C4×C22⋊C4C2×D6711yescharacteristic
341#340—671no—
342#340—671no—
343#340—671no—
344#340—671no—
345#340—671no—
346#340—671no—
347#340—671no—
348#340—671no—
349#340—671no—
350#340—671no—
351#336—671no—
352#336—671no—
353C23×C4—21no—
354C23×C4C22×D6711yescharacteristic
355C2×M16—674no—
356C2×M16—1342no—
357C2×M16—1342no—
358C22×C8—671no—
359C22×C8—671no—
360C22×C8—671no—
361C22×C8—671no—
362#117—674no—
363#117—1342no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—671no—
369C2×C4⋊C4—671no—
370C2×C4⋊C4—671no—
371C2×C4⋊C4—671no—
372C2×C4⋊C4—1341no—
373C2×C4⋊C4—1341no—
374C2×C4⋊C4C4×D6714yes—
375C2×C4⋊C4C22×D6711yescharacteristic
376C2×C4⋊C4C22×D6711yescharacteristic
377C2×C4⋊C4C22×D6711yescharacteristic
378C2×C4⋊C4C22×D6711yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—1341no—
381C2×C22⋊C4—1341no—
382C2×C22⋊C4C22⋊D6711yescharacteristic
383C2×C22⋊C4C22⋊D6711yescharacteristic
384C2×C42—21no—
385C2×C42—21no—
386C2×C42—1341no—
387C2×C42—1341no—
388C2×C42—1341no—
389C2×C42C22×D6711yescharacteristic
390C2×C42C22×D6711yescharacteristic
391#107—674no—
392#107—674no—
393#107—1342no—
394#106—674no—
395#106—674no—
396#106—1342no—
397C22⋊C8—1342no—
398#97—1342no—
399C4×C8—1342no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—1341no—
406#95—1341no—
407#95—1341no—
408#95—1341no—
409#95—1341no—
410#95C22⋊D6711yescharacteristic
411#95C22⋊D6711yescharacteristic
412C24C2×C22⋊D6711yescharacteristic
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—81no—
423C22×C4—81no—
424C22×C4—1341no—
425C22×C4—1341no—
426C22×C4—1341no—
427C22×C4—1341no—
428C22×C4C2×C4×D6712yes—
429C22×C4C2×C22⋊D6711yescharacteristic
430C22×C4C2×C22⋊D6711yescharacteristic
431C22×C4C2×C22⋊D6711yescharacteristic
432C22×C4C2×C22⋊D6711yescharacteristic
433C22×C4C2×C22⋊D6711yescharacteristic
434C22×C4#1176162812yes—
435C22×C4C23×D6711yescharacteristic
436M16—1348no—
437C2×C8—672no—
438C2×C8—672no—
439C2×C8—672no—
440C2×C8—672no—
441C2×C8—1341no—
442C2×C8—1341no—
443C2×C8—1341no—
444C2×C8—1341no—
445C2×C8—1341no—
446C2×C8—1341no—
447C2×C8—1341no—
448C2×C8—1341no—
449C4⋊C4—24no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—81no—
457C4⋊C4—81no—
458C4⋊C4—2682no—
459C4⋊C4—2682no—
460C4⋊C4—672no—
461C4⋊C4—672no—
462C4⋊C4—1341no—
463C4⋊C4—1341no—
464C4⋊C4—1341no—
465C4⋊C4—1341no—
466C4⋊C4—1341no—
467C4⋊C4—1341no—
468C4⋊C4C2×C4×D6712yes—
469C4⋊C4C2×C4×D6712yes—
470C22⋊C4—22no—
471C22⋊C4—41no—
472C22⋊C4—42no—
473C22⋊C4—5361no—
474C22⋊C4—1344no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—81no—
480C42—81no—
481C42—5361no—
482C42—5361no—
483C42—1344no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23C4×C22⋊D6712yes—
491C23C22×C22⋊D6711yescharacteristic
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—84no—
522C2×C4—2681no—
523C2×C4—2681no—
524C2×C4—2681no—
525C2×C4—2681no—
526C2×C4—2681no—
527C2×C4—2681no—
528C2×C4—2681no—
529C2×C4—2681no—
530C2×C4—2681no—
531C2×C4—2681no—
532C2×C4—1342no—
533C2×C4—1342no—
534C2×C4C4×C22⋊D6712yes—
535C2×C4#1452407412yes—
536C2×C4C22×C4×D6711yescharacteristic
537C2×C4#41×D6711yescharacteristic
538C2×C4#41×D6711yescharacteristic
539C2×C4C2×#1176162811yescharacteristic
540C2×C4#1452409011yescharacteristic
541C2×C4#1452409011yescharacteristic
542C8—2681no—
543C8—2681no—
544C8—674no—
545C8—1342no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22C2×C4×C22⋊D6711yescharacteristic
554C22#1686404511yescharacteristic
555C22#1686404511yescharacteristic
556C22C2×#1452407411yescharacteristic
557C22#1686419411yescharacteristic
558C22#1686419911yescharacteristic
559C22#1686420711yescharacteristic
560C4—22no—
561C4—22no—
562C4—41no—
563C4—41no—
564C4—42no—
565C4—42no—
566C4—41no—
567C4—81no—
568C4—81no—
569C4—2682no—
570C4—2682no—
571C4—5361no—
572C4—5361no—
573C4#117×D6711yescharacteristic
574C4#1686387311yescharacteristic
575C4#1686407911yescharacteristic
576C4#1686407911yescharacteristic
577C2—42no—
578C2#2256307911yescharacteristic
579C2C4×#1452407411yescharacteristic
580C2#2256904111yescharacteristic
581C2#2257268811yescharacteristic
582C2#2257274811yescharacteristic
583C2#2257280911yescharacteristic
584C2#2257282411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC67×#2345C2
socleC23×C67C22×D4
prime radicalC24×C67C2×D4
cosocleC22×C4×C67C24
hypercenter#2345D67
hypocenterC67#22623
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2345

Complements

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

NH ≅ G/Ntype
3. #22560949C2semidirect
4. #22560934C2semidirect
5. #22560798C2semidirect
6. #22560794C2semidirect
7. #22560787C2semidirect
8. #22560777C2semidirect
12. #22560107C2semidirect
13. C4×#14523976C2semidirect
18. #16863436C4semidirect
308. C67#22623semidirect

Maximal subgroups

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

subgroupindex
C2×#168634362
#225631002
#225609492
#225609342
#225607982
#225607942
#225607872
#225607772
#225603012
#225602952
#225602402
#225601072
C4×#145239762
#225599932
C67×#23452
#2262367