Finite Group Databasebeta

← C67×#32998 #30712249

Subgroups — C67×#32998 #30712249

Order 17152.

1284 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3428C211yesmaximal, characteristic, complemented
2C67×#3094C211yesmaximal, characteristic, complemented
3C67×#3006C211yesmaximal, characteristic
4C67×#2500C211yesmaximal, characteristic, complemented
5C67×#1883C212yesmaximal, complemented
6C67×#1883C212yesmaximal, complemented
7C67×#1860C212yesmaximal, complemented
8C67×#1859C212yesmaximal, complemented
9C67×#1826C211yesmaximal, characteristic, complemented
10C67×#1822C211yesmaximal, characteristic, complemented
11C67×#1789C211yesmaximal, characteristic, complemented
12C22×#41×C67C2211yescharacteristic
13C2×AGL(1,ℤ/8)×C67C2212yes—
14C2×#121×C67C2211yescharacteristic
15#520×C67—21no—
16#520×C67—21no—
17#520×C67C2212yes—
18#520×C67C2212yes—
19#519×C67—21no—
20#518×C67—21no—
21#518×C67—22no—
22C4×#41×C67—22no—
23C4×#41×C67—22no—
24C2×#117×C67—22no—
25C2×#117×C67C2211yescharacteristic
26C2×#117×C67C2211yescharacteristic
27#482×C67—22no—
28#481×C67—21no—
29#480×C67—21no—
30#449×C67—22no—
31#448×C67—21no—
32#447×C67—21no—
33#428×C67C2212yescomplemented
34C2×#104×C67C2212yes—
35C2×#104×C67C2212yes—
36#418×C67C2212yescomplemented
37#417×C67C2212yes—
38#407×C67C2211yescharacteristic
39#407×C67C2211yescharacteristic
40#394×C67—22no—
41#390×C67C2211yescharacteristic, complemented
42#390×C67C2211yescharacteristic, complemented
43#388×C67—22no—
44#384×C67C2211yescharacteristic, complemented
45#382×C67C2211yescharacteristic, complemented
46C4×C22⋊C4×C67C2211yescharacteristic
47#343×C67C2212yescomplemented
48#343×C67C2212yescomplemented
49#339×C67C2212yescomplemented
50#339×C67C2212yescomplemented
51C2×#41×C67—24no—
52C2×#41×C67—24no—
53C2×#41×C67—21no—
54C2×#41×C67—21no—
55C2×#41×C67D412yes—
56C2×#41×C67C2312yes—
57C22×Q8×C67D411yescharacteristic
58C22×D4×C67—21no—
59C22×D4×C67—21no—
60C22×D4×C67D411yescharacteristic
61C22×D4×C67C2311yescharacteristic
62C23×C4×C67—21no—
63C23×C4×C67C2311yescharacteristic
64AGL(1,ℤ/8)×C67—24no—
65AGL(1,ℤ/8)×C67—24no—
66C2×SD16×C67—22no—
67C2×D8×C67—22no—
68C2×M16×C67C2312yes—
69C2×M16×C67C2312yes—
70#121×C67—21no—
71#121×C67—21no—
72#121×C67—21no—
73#121×C67—21no—
74C4×Q8×C67—22no—
75C4×Q8×C67—22no—
76C4×Q8×C67—22no—
77C4×Q8×C67—22no—
78C4×D4×C67—24no—
79C4×D4×C67—24no—
80C4×D4×C67—22no—
81C4×D4×C67—22no—
82C4×D4×C67—22no—
83C4×D4×C67—22no—
84C4×D4×C67—24no—
85C4×D4×C67—24no—
86#117×C67—24no—
87#117×C67—24no—
88#117×C67—21no—
89#117×C67—21no—
90#117×C67—21no—
91#117×C67—21no—
92#117×C67—21no—
93#117×C67—21no—
94#117×C67—22no—
95#117×C67—22no—
96#117×C67D412yes—
97#117×C67C2312yes—
98C2×C4⋊C4×C67—22no—
99C2×C4⋊C4×C67—22no—
100C2×C4⋊C4×C67—22no—
101C2×C4⋊C4×C67D411yescharacteristic
102C2×C4⋊C4×C67D411yescharacteristic
103C2×C4⋊C4×C67C2311yescharacteristic
104C2×C4⋊C4×C67C2311yescharacteristic
105C2×C4⋊C4×C67C2311yescharacteristic
106C2×C22⋊C4×C67—24no—
107C2×C22⋊C4×C67—21no—
108C2×C22⋊C4×C67—21no—
109C2×C22⋊C4×C67—21no—
110C2×C22⋊C4×C67—22no—
111C2×C22⋊C4×C67—22no—
112C2×C22⋊C4×C67D412yes—
113C2×C42×C67—24no—
114C2×C42×C67—21no—
115C2×C42×C67D412yes—
116C2×C42×C67C2311yescharacteristic
117C2×C42×C67C2311yescharacteristic
118#107×C67—22no—
119#106×C67—22no—
120#104×C67—24no—
121#104×C67—24no—
122#103×C67—22no—
123#102×C67—22no—
124#102×C67—22no—
125#102×C67—22no—
126C22⋊C8×C67—21no—
127C22⋊C8×C67—21no—
128C22⋊C8×C67—21no—
129C22⋊C8×C67—21no—
130#95×C67—21no—
131#95×C67—22no—
132C24×C67—21no—
133C24×C67—21no—
134C24×C67C2×D411yescharacteristic
135#41×C67—24no—
136#41×C67—24no—
137#41×C67—28no—
138#41×C67—24no—
139#41×C67—24no—
140#41×C67—22no—
141#41×C67—22no—
142#41×C67#4114yes—
143#41×C67#4114yes—
144C2×Q8×C67—24no—
145C2×Q8×C67—21no—
146C2×Q8×C67C2×D412yes—
147C2×D4×C67—24no—
148C2×D4×C67—24no—
149C2×D4×C67—22no—
150C2×D4×C67—22no—
151C2×D4×C67—24no—
152C2×D4×C67—21no—
153C2×D4×C67—21no—
154C2×D4×C67—21no—
155C2×D4×C67—22no—
156C2×D4×C67—22no—
157C2×D4×C67—41no—
158C2×D4×C67—41no—
159C2×D4×C67—82no—
160C2×D4×C67C2×D412yes—
161C22×C4×C67—28no—
162C22×C4×C67—21no—
163C22×C4×C67—22no—
164C22×C4×C67—22no—
165C22×C4×C67—22no—
166C22×C4×C67—22no—
167C22×C4×C67—21no—
168C22×C4×C67—24no—
169C22×C4×C67—24no—
170C22×C4×C67—21no—
171C22×C4×C67—21no—
172C22×C4×C67—21no—
173C22×C4×C67—21no—
174C22×C4×C67C2×D411yescharacteristic
175C22×C4×C67C2×D411yescharacteristic
176C22×C4×C67C2×D411yescharacteristic
177C22×C4×C67C2×D411yescharacteristic
178C22×C4×C67C2×D411yescharacteristic
179C22×C4×C67C2×D412yes—
180C22×C4×C67C2×D412yes—
181C22×C4×C67C2411yescharacteristic
182SD16×C67—44no—
183D8×C67—44no—
184M16×C67—22no—
185M16×C67—22no—
186M16×C67—22no—
187M16×C67—22no—
188C2×C8×C67—22no—
189C2×C8×C67—22no—
190C4⋊C4×C67—24no—
191C4⋊C4×C67—24no—
192C4⋊C4×C67—21no—
193C4⋊C4×C67—21no—
194C4⋊C4×C67—21no—
195C4⋊C4×C67—21no—
196C4⋊C4×C67—21no—
197C4⋊C4×C67—21no—
198C4⋊C4×C67—21no—
199C4⋊C4×C67—21no—
200C4⋊C4×C67—82no—
201C4⋊C4×C67C2×D412yes—
202C4⋊C4×C67C2×D412yes—
203C22⋊C4×C67—24no—
204C22⋊C4×C67—24no—
205C22⋊C4×C67—22no—
206C22⋊C4×C67—22no—
207C22⋊C4×C67—22no—
208C22⋊C4×C67—22no—
209C22⋊C4×C67—44no—
210C22⋊C4×C67—42no—
211C22⋊C4×C67—41no—
212C22⋊C4×C67—41no—
213C22⋊C4×C67—82no—
214C42×C67—24no—
215C42×C67—24no—
216C42×C67—22no—
217C42×C67—22no—
218C42×C67—22no—
219C42×C67—22no—
220C42×C67—22no—
221C42×C67—22no—
222C23×C67—24no—
223C23×C67—22no—
224C23×C67—21no—
225C23×C67—21no—
226C23×C67—41no—
227C23×C67—41no—
228C23×C67—42no—
229C23×C67—42no—
230C23×C67—41no—
231C23×C67—41no—
232C23×C67—41no—
233C23×C67—81no—
234C23×C67—81no—
235C23×C67#12011yescharacteristic
236C23×C67#12011yescharacteristic
237C23×C67C22×D411yescharacteristic
238Q8×C67—24no—
239Q8×C67—22no—
240Q8×C67#12314yes—
241D4×C67—28no—
242D4×C67—24no—
243D4×C67—24no—
244D4×C67—24no—
245D4×C67—22no—
246D4×C67—42no—
247D4×C67—42no—
248D4×C67—82no—
249D4×C67—82no—
250D4×C67#12314yes—
251C2×C4×C67—24no—
252C2×C4×C67—28no—
253C2×C4×C67—22no—
254C2×C4×C67—21no—
255C2×C4×C67—21no—
256C2×C4×C67—22no—
257C2×C4×C67—22no—
258C2×C4×C67—22no—
259C2×C4×C67—24no—
260C2×C4×C67—24no—
261C2×C4×C67—22no—
262C2×C4×C67—22no—
263C2×C4×C67—22no—
264C2×C4×C67—22no—
265C2×C4×C67—22no—
266C2×C4×C67—22no—
267C2×C4×C67—41no—
268C2×C4×C67—41no—
269C2×C4×C67—42no—
270C2×C4×C67—42no—
271C2×C4×C67—41no—
272C2×C4×C67—41no—
273C2×C4×C67—42no—
274C2×C4×C67—41no—
275C2×C4×C67—41no—
276C2×C4×C67—81no—
277C2×C4×C67—81no—
278C2×C4×C67—84no—
279C2×C4×C67—84no—
280C2×C4×C67#12011yescharacteristic
281C2×C4×C67#12011yescharacteristic
282C2×C4×C67C22×D411yescharacteristic
283C2×C4×C67C22×D411yescharacteristic
284C2×C4×C67C2×#4112yes—
285C2×C4×C67C2×#4112yes—
286C8×C67—42no—
287C8×C67—42no—
288C22×C67—21no—
289C22×C67—21no—
290C22×C67—21no—
291C22×C67—21no—
292C22×C67—24no—
293C22×C67—21no—
294C22×C67—21no—
295C22×C67—21no—
296C22×C67—21no—
297C22×C67—42no—
298C22×C67—42no—
299C22×C67—81no—
300C22×C67—81no—
301C22×C67—82no—
302C22×C67—82no—
303C22×C67—84no—
304C22×C67—81no—
305C22×C67—81no—
306C22×C67—81no—
307C22×C67C2×#12011yescharacteristic
308C22×C67#52511yescharacteristic
309C22×C67#52511yescharacteristic
310C4×C67—21no—
311C4×C67—24no—
312C4×C67—21no—
313C4×C67—42no—
314C4×C67—42no—
315C4×C67—42no—
316C4×C67—42no—
317C4×C67—42no—
318C4×C67—81no—
319C4×C67—81no—
320C4×C67C2×#12312yes—
321C4×C67#52512yes—
322#32998C6711yesmaximal, characteristic, direct product factor, Sylow
323C2×C67—21no—
324C2×C67—21no—
325C2×C67—41no—
326C2×C67—41no—
327C2×C67—44no—
328C2×C67—161no—
329C2×C67#284911yescharacteristic
330C2×C67#301811yescharacteristic
331C2×C67#302111yescharacteristic
332#3428C2×C6711yescharacteristic, complemented
333#3094C2×C6711yescharacteristic, complemented
334#3006C2×C6711yescharacteristic
335#2500C2×C6711yescharacteristic, complemented
336#1883C2×C6712yescomplemented
337#1883C2×C6712yescomplemented
338#1860C2×C6712yescomplemented
339#1859C2×C6712yescomplemented
340#1826C2×C6711yescharacteristic, complemented
341#1822C2×C6711yescharacteristic, complemented
342#1789C2×C6711yescharacteristic, complemented
343C67#3299811yescharacteristic, direct product factor, Sylow
344C22×#41C22×C6711yescharacteristic
345C2×AGL(1,ℤ/8)C22×C6712yes—
346C2×#121C22×C6711yescharacteristic
347#520—21no—
348#520—21no—
349#520C22×C6712yes—
350#520C22×C6712yes—
351#519—21no—
352#518—21no—
353#518—22no—
354C4×#41—22no—
355C4×#41—22no—
356C2×#117—22no—
357C2×#117C22×C6711yescharacteristic
358C2×#117C22×C6711yescharacteristic
359#482—22no—
360#481—21no—
361#480—21no—
362#449—22no—
363#448—21no—
364#447—21no—
365#428C22×C6712yescomplemented
366C2×#104C22×C6712yes—
367C2×#104C22×C6712yes—
368#418C22×C6712yescomplemented
369#417C22×C6712yes—
370#407C22×C6711yescharacteristic
371#407C22×C6711yescharacteristic
372#394—22no—
373#390C22×C6711yescharacteristic, complemented
374#390C22×C6711yescharacteristic, complemented
375#388—22no—
376#384C22×C6711yescharacteristic, complemented
377#382C22×C6711yescharacteristic, complemented
378C4×C22⋊C4C22×C6711yescharacteristic
379#343C22×C6712yescomplemented
380#343C22×C6712yescomplemented
381#339C22×C6712yescomplemented
382#339C22×C6712yescomplemented
383C2×#41—24no—
384C2×#41—24no—
385C2×#41—21no—
386C2×#41—21no—
387C2×#41D4×C6712yes—
388C2×#41C23×C6712yes—
389C22×Q8D4×C6711yescharacteristic
390C22×D4—21no—
391C22×D4—21no—
392C22×D4D4×C6711yescharacteristic
393C22×D4C23×C6711yescharacteristic
394C23×C4—21no—
395C23×C4C23×C6711yescharacteristic
396AGL(1,ℤ/8)—24no—
397AGL(1,ℤ/8)—24no—
398C2×SD16—22no—
399C2×D8—22no—
400C2×M16C23×C6712yes—
401C2×M16C23×C6712yes—
402#121—21no—
403#121—21no—
404#121—21no—
405#121—21no—
406C4×Q8—22no—
407C4×Q8—22no—
408C4×Q8—22no—
409C4×Q8—22no—
410C4×D4—24no—
411C4×D4—24no—
412C4×D4—22no—
413C4×D4—22no—
414C4×D4—22no—
415C4×D4—22no—
416C4×D4—24no—
417C4×D4—24no—
418#117—24no—
419#117—24no—
420#117—21no—
421#117—21no—
422#117—21no—
423#117—21no—
424#117—21no—
425#117—21no—
426#117—22no—
427#117—22no—
428#117D4×C6712yes—
429#117C23×C6712yes—
430C2×C4⋊C4—22no—
431C2×C4⋊C4—22no—
432C2×C4⋊C4—22no—
433C2×C4⋊C4D4×C6711yescharacteristic
434C2×C4⋊C4D4×C6711yescharacteristic
435C2×C4⋊C4C23×C6711yescharacteristic
436C2×C4⋊C4C23×C6711yescharacteristic
437C2×C4⋊C4C23×C6711yescharacteristic
438C2×C22⋊C4—24no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—22no—
443C2×C22⋊C4—22no—
444C2×C22⋊C4D4×C6712yes—
445C2×C42—24no—
446C2×C42—21no—
447C2×C42D4×C6712yes—
448C2×C42C23×C6711yescharacteristic
449C2×C42C23×C6711yescharacteristic
450#107—22no—
451#106—22no—
452#104—24no—
453#104—24no—
454#103—22no—
455#102—22no—
456#102—22no—
457#102—22no—
458C22⋊C8—21no—
459C22⋊C8—21no—
460C22⋊C8—21no—
461C22⋊C8—21no—
462#95—21no—
463#95—22no—
464C24—21no—
465C24—21no—
466C24C2×D4×C6711yescharacteristic
467#41—24no—
468#41—24no—
469#41—28no—
470#41—24no—
471#41—24no—
472#41—22no—
473#41—22no—
474#41#41×C6714yes—
475#41#41×C6714yes—
476C2×Q8—24no—
477C2×Q8—21no—
478C2×Q8C2×D4×C6712yes—
479C2×D4—24no—
480C2×D4—24no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—24no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—41no—
490C2×D4—41no—
491C2×D4—82no—
492C2×D4C2×D4×C6712yes—
493C22×C4—28no—
494C22×C4—21no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—21no—
500C22×C4—24no—
501C22×C4—24no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4C2×D4×C6711yescharacteristic
507C22×C4C2×D4×C6711yescharacteristic
508C22×C4C2×D4×C6711yescharacteristic
509C22×C4C2×D4×C6711yescharacteristic
510C22×C4C2×D4×C6711yescharacteristic
511C22×C4C2×D4×C6712yes—
512C22×C4C2×D4×C6712yes—
513C22×C4C24×C6711yescharacteristic
514SD16—44no—
515D8—44no—
516M16—22no—
517M16—22no—
518M16—22no—
519M16—22no—
520C2×C8—22no—
521C2×C8—22no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—82no—
533C4⋊C4C2×D4×C6712yes—
534C4⋊C4C2×D4×C6712yes—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—44no—
542C22⋊C4—42no—
543C22⋊C4—41no—
544C22⋊C4—41no—
545C22⋊C4—82no—
546C42—24no—
547C42—24no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C23—24no—
555C23—22no—
556C23—21no—
557C23—21no—
558C23—41no—
559C23—41no—
560C23—42no—
561C23—42no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—81no—
566C23—81no—
567C23#120×C6711yescharacteristic
568C23#120×C6711yescharacteristic
569C23C22×D4×C6711yescharacteristic
570Q8—24no—
571Q8—22no—
572Q8#123×C6714yes—
573D4—28no—
574D4—24no—
575D4—24no—
576D4—24no—
577D4—22no—
578D4—42no—
579D4—42no—
580D4—82no—
581D4—82no—
582D4#123×C6714yes—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—84no—
611C2×C4—84no—
612C2×C4#120×C6711yescharacteristic
613C2×C4#120×C6711yescharacteristic
614C2×C4C22×D4×C6711yescharacteristic
615C2×C4C22×D4×C6711yescharacteristic
616C2×C4C2×#41×C6712yes—
617C2×C4C2×#41×C6712yes—
618C8—42no—
619C8—42no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—24no—
629C22—42no—
630C22—42no—
631C22—81no—
632C22—81no—
633C22—82no—
634C22—82no—
635C22—84no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22C2×#120×C6711yescharacteristic
640C22#525×C6711yescharacteristic
641C22#525×C6711yescharacteristic
642C4—21no—
643C4—24no—
644C4—21no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—81no—
651C4—81no—
652C4C2×#123×C6712yes—
653C4#525×C6712yes—
654C2—21no—
655C2—21no—
656C2—41no—
657C2—41no—
658C2—44no—
659C2—161no—
660C2C67×#284911yescharacteristic
661C2C67×#301811yescharacteristic
662C2C67×#302111yescharacteristic
subgroupG/H
center, socleC22×C67C2×#120
commutator, Frattini, cosocleC22×C4C24×C67
prime radicalC67×#3006C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C23×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#3428C2semidirect
2. C67×#3094C2semidirect
4. C67×#2500C2semidirect
5. C67×#1883C2semidirect
6. C67×#1883C2semidirect
7. C67×#1860C2semidirect
8. C67×#1859C2semidirect
9. C67×#1826C2semidirect
10. C67×#1822C2semidirect
11. C67×#1789C2semidirect
33. #428×C67C22semidirect
36. #418×C67C22semidirect
41. #390×C67C22semidirect
42. #390×C67C22semidirect
44. #384×C67C22semidirect
45. #382×C67C22semidirect
47. #343×C67C22semidirect
48. #343×C67C22semidirect
49. #339×C67C22semidirect
50. #339×C67C22semidirect
322. #32998C67direct (N × H)
332. #3428C2×C67semidirect
333. #3094C2×C67semidirect
335. #2500C2×C67semidirect
336. #1883C2×C67semidirect
337. #1883C2×C67semidirect
338. #1860C2×C67semidirect
339. #1859C2×C67semidirect
340. #1826C2×C67semidirect
341. #1822C2×C67semidirect
342. #1789C2×C67semidirect
343. C67#32998direct (N × H)
365. #428C22×C67semidirect
368. #418C22×C67semidirect
373. #390C22×C67semidirect
374. #390C22×C67semidirect
376. #384C22×C67semidirect
377. #382C22×C67semidirect
379. #343C22×C67semidirect
380. #343C22×C67semidirect
381. #339C22×C67semidirect
382. #339C22×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#34282
C67×#30942
C67×#30062
C67×#25002
C67×#18832
C67×#18602
C67×#18592
C67×#18262
C67×#18222
C67×#17892
#3299867