Finite Group Databasebeta

← #30855528

Subgroups — #30855528

Order 17152.

1330 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16864162C211yesmaximal, characteristic
2C2×#16863444C211yesmaximal, characteristic, complemented
3C2×#16863436C211yesmaximal, characteristic, complemented
4#22568729C211yesmaximal, characteristic
5C2×#16863395C211yesmaximal, characteristic
6#22560272C214yesmaximal, complemented
7#22560267C214yesmaximal, complemented
8#22559947C211yesmaximal, characteristic, complemented
9C2×#343×C67C211yesmaximal, characteristic, complemented
10C22×#11761628C2211yescharacteristic
11#16864162—24no—
12#16864162C418yescomplemented
13C2×#14524051C2214yes—
14C2×#14524049—22no—
15C22×#11761612C2211yescharacteristic
16C22×#11761612C2211yescharacteristic
17C2×M16×D67—22no—
18C2×#14524002C2211yescharacteristic
19#16863444—24no—
20C2×#14523978C2214yescomplemented
21#16863436—22no—
22#16863436—22no—
23C2×#14523976C2212yescomplemented
24C2×#14523974C2212yescomplemented
25#16863406—21no—
26#16863402—21no—
27#16863398C2212yes—
28#16863398C2212yes—
29#16863395—22no—
30#16863395C2214yes—
31#16863393—22no—
32C2×#14523966—21no—
33C2×#14523963—21no—
34#16863164C2212yescomplemented
35#16863164C2212yescomplemented
36C22×M16×C67C2211yescharacteristic
37C22×M16×C67C2211yescharacteristic
38C22×C4⋊C4×C67C2211yescharacteristic
39#343×C67C2214yescomplemented
40C2×#11761628—21no—
41C2×#11761628—22no—
42C2×#11761628—21no—
43C2×#11761628—22no—
44C2×#11761628C2×C414yes—
45C2×#11761628D412yes—
46C2×#11761628Q814yes—
47C2×#11761628C2312yes—
48C22×Dic134D411yescharacteristic
49C22×D268D411yescharacteristic
50C22×C22⋊D67—21no—
51C22×C4×D67—21no—
52#14524051—28no—
53#14524051C2×C4116yescomplemented
54#14524049—216no—
55C2×#11761612—21no—
56C2×#11761612—22no—
57C2×#11761612—22no—
58C2×#11761612C2×C414yes—
59C2×#11761612D412yes—
60C2×#11761612D412yes—
61C2×#11761612D414yes—
62C2×#11761612C2312yes—
63C2×#11761611—24no—
64M16×D67—216no—
65C22×C67⋊C8—21no—
66C22×C67⋊C8C2312yes—
67C2×C8×D67—24no—
68#14524002—21no—
69#14524002—21no—
70#14524002—21no—
71#14524002—21no—
72C2×C4⋊Dic67C2311yescharacteristic
73C2×C4⋊Dic67C2311yescharacteristic
74C2×C22⋊Dic67—21no—
75C2×C4×Dic67—21no—
76#14523978—28no—
77#14523976—24no—
78#14523974—24no—
79#14523968—22no—
80#14523968—22no—
81#14523966—24no—
82#14523965—22no—
83#14523965—22no—
84#14523963—24no—
85C23×C4×C67—21no—
86C23×C4×C67C2311yescharacteristic
87C2×M16×C67—21no—
88C2×M16×C67—22no—
89C2×M16×C67—21no—
90C2×M16×C67—22no—
91C2×M16×C67—22no—
92C2×M16×C67C2×C414yes—
93C2×M16×C67C2312yes—
94C2×M16×C67C2312yes—
95C22×C8×C67—21no—
96C22×C8×C67—21no—
97C2×C4⋊C4×C67—21no—
98C2×C4⋊C4×C67—21no—
99C2×C4⋊C4×C67—21no—
100C2×C4⋊C4×C67—21no—
101C2×C4⋊C4×C67—21no—
102C2×C4⋊C4×C67C2312yes—
103C23×D67—21no—
104#11761628—24no—
105#11761628—24no—
106#11761628—24no—
107#11761628—24no—
108#11761628—24no—
109#11761628—22no—
110#11761628—22no—
111#11761628C4⋊C418yes—
112#11761628C22×C414yes—
113#11761628C2×Q814yes—
114C2×Dic134—22no—
115C2×Dic134—21no—
116C2×Dic134C4⋊C414yes—
117C2×Dic134C2×D412yes—
118C2×D268—22no—
119C2×D268—21no—
120C2×D268C4⋊C414yes—
121C2×D268C2×D412yes—
122C2×C22⋊D67—24no—
123C2×C22⋊D67—22no—
124C2×C22⋊D67—24no—
125C2×C22⋊D67—22no—
126C22×Dic67—21no—
127C22×Dic67—21no—
128C22×Dic67—21no—
129C2×C4×D67—24no—
130C2×C4×D67—22no—
131C2×C4×D67—24no—
132C2×C4×D67—22no—
133#11761612—22no—
134#11761612—22no—
135#11761612—42no—
136#11761612C22⋊C418yes—
137#11761612C22⋊C418yes—
138#11761611—216no—
139C2×C67⋊C8—22no—
140C2×C67⋊C8—42no—
141C2×C67⋊C8C22×C414yes—
142C2×C67⋊C8C2×D414yes—
143C2×C67⋊C8C2×D414yes—
144C8×D67—216no—
145C4⋊Dic67—21no—
146C4⋊Dic67—21no—
147C4⋊Dic67—21no—
148C4⋊Dic67—21no—
149C22⋊Dic67—41no—
150C22⋊Dic67—41no—
151C4×Dic67—41no—
152C4×Dic67—41no—
153C24×C67#4111yescharacteristic
154C22×C4×C67—22no—
155C22×C4×C67—41no—
156C22×C4×C67—41no—
157C22×C4×C67—82no—
158C22×C4×C67C22×C412yes—
159C22×C4×C67C2×D412yes—
160C22×C4×C67C2×D411yescharacteristic
161C22×C4×C67C2×D411yescharacteristic
162C22×C4×C67C2×D411yescharacteristic
163C22×C4×C67C2×D411yescharacteristic
164C22×C4×C67C2×Q812yes—
165C22×C4×C67#4111yescharacteristic
166C22×C4×C67#4112yes—
167C22×C4×C67C2411yescharacteristic
168M16×C67—24no—
169M16×C67—22no—
170M16×C67—22no—
171M16×C67—22no—
172M16×C67—42no—
173M16×C67C22×C414yes—
174C2×C8×C67—24no—
175C2×C8×C67—22no—
176C2×C8×C67—22no—
177C2×C8×C67—42no—
178C4⋊C4×C67—22no—
179C4⋊C4×C67—22no—
180C4⋊C4×C67—41no—
181C4⋊C4×C67—41no—
182C23×C67—41no—
183C23×C67—41no—
184C23×C67#11712yes—
185C23×C67#12111yescharacteristic
186C23×C67#12111yescharacteristic
187C23×C67C22⋊Q812yes—
188C23×C67C2×#4111yescharacteristic
189C22×D67—22no—
190C22×D67—24no—
191C22×D67—81no—
192C22×D67—81no—
193C2×C4×C67—42no—
194C2×C4×C67—82no—
195C2×C4×C67—82no—
196C2×C4×C67—82no—
197C2×C4×C67C2×C22⋊C412yes—
198C2×C4×C67C2×C22⋊C412yes—
199C2×C4×C67C2×C4⋊C412yes—
200C2×C4×C67C2×C4⋊C412yes—
201C2×C4×C67#11712yes—
202C2×C4×C67#12112yes—
203C2×C4×C67#12112yes—
204C2×C4×C67#12111yescharacteristic
205C2×C4×C67#12111yescharacteristic
206C2×C4×C67C22⋊Q812yes—
207C2×C4×C67C22⋊Q812yes—
208C2×C4×C67C22⋊Q812yes—
209C2×C4×C67C23×C411yescharacteristic
210C2×C4×C67C22×D411yescharacteristic
211C2×C4×C67C22×D411yescharacteristic
212C2×C4×C67C22×D411yescharacteristic
213C2×C4×C67C22×Q811yescharacteristic
214C2×C4×C67C2×#4111yescharacteristic
215C22⋊D67—24no—
216C22⋊D67—24no—
217C22⋊D67—28no—
218C2×Dic67—22no—
219C2×Dic67—24no—
220C2×Dic67—41no—
221C2×Dic67—41no—
222C2×Dic67—81no—
223C2×Dic67—81no—
224D268—22no—
225D268—24no—
226D268C2×C4⋊C414yes—
227C4×D67—24no—
228C4×D67—24no—
229C4×D67—28no—
230Dic134—22no—
231Dic134—24no—
232Dic134C2×C4⋊C414yes—
233C8×C67—24no—
234C8×C67—42no—
235C67⋊C8—42no—
236C67⋊C8C2×C22⋊C418yes—
237C22×C67—21no—
238C22×C67—21no—
239C22×C67—21no—
240C22×C67—21no—
241C22×C67—21no—
242C22×C67—21no—
243C22×C67—42no—
244C22×C67—42no—
245C22×C67#38012yes—
246C22×C67#38012yes—
247C22×C67C2×#11711yescharacteristic
248C22×C67C2×#12111yescharacteristic
249C22×C67C2×C22⋊Q811yescharacteristic
250C2×D67—24no—
251C2×D67—42no—
252C2×D67—42no—
253C2×D67—84no—
254C4×C67—82no—
255C4×C67#38012yes—
256C4×C67#38012yes—
257C4×C67C22×C22⋊C411yescharacteristic
258C4×C67C22×C4⋊C411yescharacteristic
259C4×C67C2×#12111yescharacteristic
260C4×C67C2×C22⋊Q811yescharacteristic
261Dic67—24no—
262Dic67—81no—
263Dic67—81no—
264#20679—671nomaximal, Sylow
265C2×C67—22no—
266C2×C67—21no—
267C2×C67—21no—
268C2×C67—22no—
269C2×C67C2×#38011yescharacteristic
270C2×C67#306811yescharacteristic
271C2×C67#306911yescharacteristic
272D67—44no—
273C2×#568—671no—
274C2×#430—671no—
275C2×#428—671no—
276#2896—671no—
277C2×#417—671no—
278#1816—674no—
279#1815—674no—
280#1742—671no—
281C2×#343D6711yescharacteristic, complemented
282C67#2067911yescharacteristic, complemented, Sylow
283C22×#41—671no—
284#568—678no—
285#568—1344no—
286C2×#131—674no—
287C2×#131—1342no—
288C22×M16—671no—
289C22×M16—671no—
290C22×M16—1342no—
291C22×M16C2×D6711yescharacteristic
292C22×M16C2×D6711yescharacteristic
293C2×#117—671no—
294C22×C4⋊C4C2×D6711yescharacteristic
295#430—1344no—
296C2×#108—674no—
297#428—1342no—
298#428—1342no—
299C2×#107—672no—
300C2×#106—672no—
301#419—1341no—
302#418—1341no—
303#417—672no—
304#417—672no—
305#417—674no—
306#417—1342no—
307#416—1342no—
308C2×#103—1341no—
309C2×#102—1341no—
310#343—672no—
311#343—672no—
312#343C2×D6714yescomplemented
313C2×#41—674no—
314C2×#41—672no—
315C2×#41—674no—
316C2×#41—672no—
317C2×#41—1341no—
318C2×#41—1342no—
319C2×#41—1341no—
320C2×#41—1342no—
321C22×Q8—671no—
322C22×D4—671no—
323C22×D4—1341no—
324C23×C4—21no—
325C23×C4—1341no—
326C23×C4C22×D6711yescharacteristic
327#131—6716no—
328#131—13416no—
329#131—1348no—
330C2×M16—21no—
331C2×M16—22no—
332C2×M16—21no—
333C2×M16—22no—
334C2×M16—22no—
335C2×M16—674no—
336C2×M16—672no—
337C2×M16—672no—
338C2×M16—674no—
339C2×M16—672no—
340C2×M16—1344no—
341C2×M16—1342no—
342C2×M16—1342no—
343C2×M16—13416no—
344C2×M16—1341no—
345C2×M16C4×D6714yes—
346C2×M16C22×D6712yes—
347C2×M16C22×D6712yes—
348C22×C8—21no—
349C22×C8—21no—
350C22×C8—672no—
351C22×C8—1344no—
352C22×C8—1341no—
353#117—1341no—
354#117—1341no—
355#117—1341no—
356#117—1341no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—671no—
363C2×C4⋊C4—671no—
364C2×C4⋊C4C22×D6712yes—
365C2×C22⋊C4—1341no—
366C2×C42—1341no—
367#108—1348no—
368#107—1344no—
369#106—1344no—
370#103—1344no—
371#103—1342no—
372#103—1342no—
373#102—1344no—
374#102—1342no—
375#102—1342no—
376C24—1341no—
377C24#1176162811yescharacteristic
378#41—678no—
379#41—674no—
380#41—674no—
381#41—1344no—
382#41—1344no—
383#41—1344no—
384#41—1344no—
385#41—1344no—
386#41—1342no—
387#41—1342no—
388C2×Q8—674no—
389C2×Q8—672no—
390C2×Q8—1342no—
391C2×Q8—1341no—
392C2×D4—674no—
393C2×D4—672no—
394C2×D4—1344no—
395C2×D4—1342no—
396C2×D4—1344no—
397C2×D4—1342no—
398C2×D4—1341no—
399C2×D4—1342no—
400C22×C4—22no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—82no—
404C22×C4—1341no—
405C22×C4—1341no—
406C22×C4—1341no—
407C22×C4—1344no—
408C22×C4—1342no—
409C22×C4—1344no—
410C22×C4—1342no—
411C22×C4C2×C4×D6712yes—
412C22×C4D4×D6712yes—
413C22×C4D4×D6711yescharacteristic
414C22×C4D4×D6711yescharacteristic
415C22×C4D4×D6711yescharacteristic
416C22×C4D4×D6711yescharacteristic
417C22×C4Q8×D6712yes—
418C22×C4#1176162811yescharacteristic
419C22×C4#1176162812yes—
420C22×C4C23×D6711yescharacteristic
421M16—24no—
422M16—22no—
423M16—22no—
424M16—22no—
425M16—42no—
426M16—2682no—
427M16—678no—
428M16—678no—
429M16—13416no—
430M16—1342no—
431M16—1342no—
432M16C2×C4×D6714yes—
433C2×C8—24no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—42no—
437C2×C8—2682no—
438C2×C8—674no—
439C2×C8—674no—
440C2×C8—674no—
441C2×C8—13416no—
442C2×C8—1342no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—1341no—
448C4⋊C4—1341no—
449C4⋊C4—1341no—
450C4⋊C4—1341no—
451C22⋊C4—2681no—
452C22⋊C4—2681no—
453C42—2681no—
454C42—2681no—
455C23—41no—
456C23—41no—
457C23—5361no—
458C23—5361no—
459C23—1342no—
460C23—1344no—
461C23#1452400012yes—
462C23#1452401611yescharacteristic
463C23#1452401611yescharacteristic
464C23#1452402112yes—
465C23C2×#1176162811yescharacteristic
466Q8—674no—
467Q8—1344no—
468Q8—1342no—
469D4—674no—
470D4—1348no—
471D4—1344no—
472D4—1344no—
473D4—1344no—
474D4—1342no—
475C2×C4—42no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—2681no—
480C2×C4—2681no—
481C2×C4—5361no—
482C2×C4—5361no—
483C2×C4—1344no—
484C2×C4—1342no—
485C2×C4—1348no—
486C2×C4—1344no—
487C2×C4—1344no—
488C2×C4C22⋊C4×D6712yes—
489C2×C4C22⋊C4×D6712yes—
490C2×C4C4⋊C4×D6712yes—
491C2×C4C4⋊C4×D6712yes—
492C2×C4#1452400012yes—
493C2×C4#1452401611yescharacteristic
494C2×C4#1452401611yescharacteristic
495C2×C4#1452401612yes—
496C2×C4#1452401612yes—
497C2×C4#1452402112yes—
498C2×C4#1452402112yes—
499C2×C4#1452402112yes—
500C2×C4C22×C4×D6711yescharacteristic
501C2×C4C2×D4×D6711yescharacteristic
502C2×C4C2×D4×D6711yescharacteristic
503C2×C4C2×D4×D6711yescharacteristic
504C2×C4C2×Q8×D6711yescharacteristic
505C2×C4C2×#1176162811yescharacteristic
506C8—24no—
507C8—42no—
508C8—2682no—
509C8—678no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—42no—
517C22—42no—
518C22—2682no—
519C22—2682no—
520C22—5364no—
521C22—1344no—
522C22#1686325512yes—
523C22#1686325512yes—
524C22C2×#1452400011yescharacteristic
525C22C2×#1452401611yescharacteristic
526C22C2×#1452402111yescharacteristic
527C4—82no—
528C4—5361no—
529C4—5361no—
530C4—1344no—
531C4#1686325512yes—
532C4#1686325512yes—
533C4C2×C22⋊C4×D6711yescharacteristic
534C4C2×C4⋊C4×D6711yescharacteristic
535C4C2×#1452401611yescharacteristic
536C4C2×#1452402111yescharacteristic
537C2—22no—
538C2—22no—
539C2—21no—
540C2—21no—
541C2—2684no—
542C2C2×#1686325511yescharacteristic
543C2#2256993611yescharacteristic
544C2#2256994511yescharacteristic
subgroupG/H
centerC22C2×#14524016
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC2×#343×C67C2
socleC22×C67C2×#121
prime radicalC22×#11761628C22
cosocleC22×C4×C67C24
hypercenterC2×#343D67
hypocenterC67#20679
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
2. C2×#16863444C2semidirect
3. C2×#16863436C2semidirect
6. #22560272C2semidirect
7. #22560267C2semidirect
8. #22559947C2semidirect
9. C2×#343×C67C2semidirect
12. #16864162C4semidirect
20. C2×#14523978C22semidirect
23. C2×#14523976C22semidirect
24. C2×#14523974C22semidirect
34. #16863164C22semidirect
35. #16863164C22semidirect
39. #343×C67C22semidirect
53. #14524051C2×C4semidirect
281. C2×#343D67semidirect
282. C67#20679semidirect
312. #343C2×D67semidirect

Maximal subgroups

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

subgroupindex
C2×#168641622
C2×#168634442
C2×#168634362
#225687292
C2×#168633952
#225602722
#225602672
#225599472
C2×#343×C672
#2067967