Finite Group Databasebeta

← C67×#33001 #30712252

Subgroups — C67×#33001 #30712252

Order 17152.

1242 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3428C211yesmaximal, characteristic
2C67×#3095C211yesmaximal, characteristic, complemented
3C67×#3007C211yesmaximal, characteristic
4C67×#2499C211yesmaximal, characteristic, complemented
5C67×#1882C212yesmaximal, complemented
6C67×#1882C212yesmaximal, complemented
7C67×#1860C212yesmaximal, complemented
8C67×#1859C212yesmaximal, complemented
9C67×#1826C211yesmaximal, characteristic, complemented
10C67×#1822C211yesmaximal, characteristic, complemented
11C67×#1789C211yesmaximal, characteristic
12C22×#41×C67C2211yescharacteristic
13C2×#137×C67C2212yes—
14C2×C22⋊Q8×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#485×C67—22no—
28#484×C67—21no—
29#483×C67—21no—
30#452×C67—22no—
31#451×C67—21no—
32#450×C67—21no—
33#428×C67C2212yescomplemented
34C2×#104×C67C2212yes—
35C2×#104×C67C2212yes—
36#419×C67C2212yescomplemented
37#417×C67C2212yes—
38#407×C67C2211yescharacteristic
39#407×C67C2211yescharacteristic
40#397×C67—22no—
41#391×C67C2211yescharacteristic, complemented
42#391×C67C2211yescharacteristic, complemented
43#387×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×Q8×C67C2311yescharacteristic
59C22×D4×C67—21no—
60C22×D4×C67D411yescharacteristic
61C23×C4×C67—21no—
62C23×C4×C67C2311yescharacteristic
63#137×C67—24no—
64#137×C67—24no—
65C2×Q16×C67—22no—
66C2×SD16×C67—22no—
67C2×M16×C67C2312yes—
68C2×M16×C67C2312yes—
69C22⋊Q8×C67—21no—
70C22⋊Q8×C67—21no—
71C22⋊Q8×C67—21no—
72C22⋊Q8×C67—21no—
73C4×Q8×C67—22no—
74C4×Q8×C67—22no—
75C4×Q8×C67—22no—
76C4×Q8×C67—22no—
77C4×D4×C67—24no—
78C4×D4×C67—24no—
79C4×D4×C67—22no—
80C4×D4×C67—22no—
81C4×D4×C67—22no—
82C4×D4×C67—22no—
83C4×D4×C67—24no—
84C4×D4×C67—24no—
85#117×C67—24no—
86#117×C67—24no—
87#117×C67—21no—
88#117×C67—21no—
89#117×C67—21no—
90#117×C67—21no—
91#117×C67—21no—
92#117×C67—21no—
93#117×C67—22no—
94#117×C67—22no—
95#117×C67D412yes—
96#117×C67C2312yes—
97C2×C4⋊C4×C67—22no—
98C2×C4⋊C4×C67—22no—
99C2×C4⋊C4×C67—21no—
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×C67D412yes—
111C2×C42×C67—24no—
112C2×C42×C67—21no—
113C2×C42×C67D412yes—
114C2×C42×C67C2311yescharacteristic
115C2×C42×C67C2311yescharacteristic
116#107×C67—22no—
117#106×C67—22no—
118#104×C67—24no—
119#104×C67—24no—
120#103×C67—22no—
121#103×C67—22no—
122#103×C67—22no—
123#102×C67—22no—
124C22⋊C8×C67—21no—
125C22⋊C8×C67—21no—
126C22⋊C8×C67—21no—
127C22⋊C8×C67—21no—
128#95×C67—21no—
129#95×C67—22no—
130#95×C67—22no—
131#95×C67—22no—
132C24×C67—21no—
133C24×C67C2×D411yescharacteristic
134#41×C67—24no—
135#41×C67—24no—
136#41×C67—28no—
137#41×C67—24no—
138#41×C67—24no—
139#41×C67—22no—
140#41×C67—22no—
141#41×C67#4114yes—
142#41×C67#4114yes—
143C2×Q8×C67—24no—
144C2×Q8×C67—21no—
145C2×Q8×C67—21no—
146C2×Q8×C67—21no—
147C2×Q8×C67—22no—
148C2×Q8×C67—22no—
149C2×Q8×C67C2×D412yes—
150C2×D4×C67—24no—
151C2×D4×C67—24no—
152C2×D4×C67—22no—
153C2×D4×C67—22no—
154C2×D4×C67—24no—
155C2×D4×C67—21no—
156C2×D4×C67C2×D412yes—
157C22×C4×C67—28no—
158C22×C4×C67—21no—
159C22×C4×C67—22no—
160C22×C4×C67—22no—
161C22×C4×C67—22no—
162C22×C4×C67—22no—
163C22×C4×C67—21no—
164C22×C4×C67—24no—
165C22×C4×C67—24no—
166C22×C4×C67—21no—
167C22×C4×C67—21no—
168C22×C4×C67—21no—
169C22×C4×C67—21no—
170C22×C4×C67—21no—
171C22×C4×C67C2×D411yescharacteristic
172C22×C4×C67C2×D411yescharacteristic
173C22×C4×C67C2×D411yescharacteristic
174C22×C4×C67C2×D411yescharacteristic
175C22×C4×C67C2×D411yescharacteristic
176C22×C4×C67C2×D412yes—
177C22×C4×C67C2×D412yes—
178C22×C4×C67C2411yescharacteristic
179Q16×C67—44no—
180SD16×C67—44no—
181M16×C67—22no—
182M16×C67—22no—
183M16×C67—22no—
184M16×C67—22no—
185C2×C8×C67—22no—
186C2×C8×C67—22no—
187C4⋊C4×C67—24no—
188C4⋊C4×C67—24no—
189C4⋊C4×C67—21no—
190C4⋊C4×C67—21no—
191C4⋊C4×C67—21no—
192C4⋊C4×C67—21no—
193C4⋊C4×C67—21no—
194C4⋊C4×C67—21no—
195C4⋊C4×C67—21no—
196C4⋊C4×C67—21no—
197C4⋊C4×C67—44no—
198C4⋊C4×C67—41no—
199C4⋊C4×C67—41no—
200C4⋊C4×C67C2×D412yes—
201C4⋊C4×C67C2×D412yes—
202C22⋊C4×C67—24no—
203C22⋊C4×C67—24no—
204C22⋊C4×C67—22no—
205C22⋊C4×C67—22no—
206C22⋊C4×C67—22no—
207C22⋊C4×C67—22no—
208C22⋊C4×C67—42no—
209C22⋊C4×C67—41no—
210C22⋊C4×C67—41no—
211C42×C67—24no—
212C42×C67—24no—
213C42×C67—22no—
214C42×C67—22no—
215C42×C67—22no—
216C42×C67—22no—
217C42×C67—22no—
218C42×C67—22no—
219C23×C67—24no—
220C23×C67—22no—
221C23×C67—21no—
222C23×C67—21no—
223C23×C67—41no—
224C23×C67—41no—
225C23×C67—81no—
226C23×C67—81no—
227C23×C67#12011yescharacteristic
228C23×C67#12011yescharacteristic
229C23×C67C22×D411yescharacteristic
230Q8×C67—24no—
231Q8×C67—22no—
232Q8×C67—42no—
233Q8×C67—42no—
234Q8×C67#12314yes—
235D4×C67—28no—
236D4×C67—24no—
237D4×C67—24no—
238D4×C67—24no—
239D4×C67—22no—
240D4×C67#12314yes—
241C2×C4×C67—24no—
242C2×C4×C67—28no—
243C2×C4×C67—22no—
244C2×C4×C67—21no—
245C2×C4×C67—21no—
246C2×C4×C67—22no—
247C2×C4×C67—22no—
248C2×C4×C67—22no—
249C2×C4×C67—24no—
250C2×C4×C67—24no—
251C2×C4×C67—22no—
252C2×C4×C67—22no—
253C2×C4×C67—22no—
254C2×C4×C67—22no—
255C2×C4×C67—22no—
256C2×C4×C67—22no—
257C2×C4×C67—41no—
258C2×C4×C67—41no—
259C2×C4×C67—42no—
260C2×C4×C67—42no—
261C2×C4×C67—41no—
262C2×C4×C67—42no—
263C2×C4×C67—41no—
264C2×C4×C67—41no—
265C2×C4×C67—41no—
266C2×C4×C67—41no—
267C2×C4×C67—41no—
268C2×C4×C67—41no—
269C2×C4×C67—81no—
270C2×C4×C67—81no—
271C2×C4×C67—84no—
272C2×C4×C67—84no—
273C2×C4×C67#12011yescharacteristic
274C2×C4×C67#12011yescharacteristic
275C2×C4×C67C22×D411yescharacteristic
276C2×C4×C67C22×D411yescharacteristic
277C2×C4×C67C2×#4112yes—
278C2×C4×C67C2×#4112yes—
279C8×C67—42no—
280C8×C67—42no—
281C22×C67—21no—
282C22×C67—21no—
283C22×C67—21no—
284C22×C67—21no—
285C22×C67—24no—
286C22×C67—21no—
287C22×C67—21no—
288C22×C67—21no—
289C22×C67—21no—
290C22×C67—42no—
291C22×C67—42no—
292C22×C67—81no—
293C22×C67—81no—
294C22×C67—84no—
295C22×C67C2×#12011yescharacteristic
296C22×C67#52511yescharacteristic
297C22×C67#52511yescharacteristic
298C4×C67—21no—
299C4×C67—24no—
300C4×C67—21no—
301C4×C67—42no—
302C4×C67—42no—
303C4×C67—42no—
304C4×C67—42no—
305C4×C67—42no—
306C4×C67—81no—
307C4×C67—81no—
308C4×C67—81no—
309C4×C67C2×#12312yes—
310C4×C67#52512yes—
311#33001C6711yesmaximal, characteristic, direct product factor, Sylow
312C2×C67—21no—
313C2×C67—21no—
314C2×C67—41no—
315C2×C67—41no—
316C2×C67—44no—
317C2×C67#284911yescharacteristic
318C2×C67#301811yescharacteristic
319C2×C67#302211yescharacteristic
320#3428C2×C6711yescharacteristic
321#3095C2×C6711yescharacteristic, complemented
322#3007C2×C6711yescharacteristic
323#2499C2×C6711yescharacteristic, complemented
324#1882C2×C6712yescomplemented
325#1882C2×C6712yescomplemented
326#1860C2×C6712yescomplemented
327#1859C2×C6712yescomplemented
328#1826C2×C6711yescharacteristic, complemented
329#1822C2×C6711yescharacteristic, complemented
330#1789C2×C6711yescharacteristic
331C67#3300111yescharacteristic, direct product factor, Sylow
332C22×#41C22×C6711yescharacteristic
333C2×#137C22×C6712yes—
334C2×C22⋊Q8C22×C6711yescharacteristic
335#520—21no—
336#520—21no—
337#520C22×C6712yes—
338#520C22×C6712yes—
339#519—21no—
340#518—21no—
341#518—22no—
342C4×#41—22no—
343C4×#41—22no—
344C2×#117—22no—
345C2×#117C22×C6711yescharacteristic
346C2×#117C22×C6711yescharacteristic
347#485—22no—
348#484—21no—
349#483—21no—
350#452—22no—
351#451—21no—
352#450—21no—
353#428C22×C6712yescomplemented
354C2×#104C22×C6712yes—
355C2×#104C22×C6712yes—
356#419C22×C6712yescomplemented
357#417C22×C6712yes—
358#407C22×C6711yescharacteristic
359#407C22×C6711yescharacteristic
360#397—22no—
361#391C22×C6711yescharacteristic, complemented
362#391C22×C6711yescharacteristic, complemented
363#387—22no—
364#384C22×C6711yescharacteristic, complemented
365#382C22×C6711yescharacteristic, complemented
366C4×C22⋊C4C22×C6711yescharacteristic
367#343C22×C6712yescomplemented
368#343C22×C6712yescomplemented
369#339C22×C6712yescomplemented
370#339C22×C6712yescomplemented
371C2×#41—24no—
372C2×#41—24no—
373C2×#41—21no—
374C2×#41—21no—
375C2×#41D4×C6712yes—
376C2×#41C23×C6712yes—
377C22×Q8D4×C6711yescharacteristic
378C22×Q8C23×C6711yescharacteristic
379C22×D4—21no—
380C22×D4D4×C6711yescharacteristic
381C23×C4—21no—
382C23×C4C23×C6711yescharacteristic
383#137—24no—
384#137—24no—
385C2×Q16—22no—
386C2×SD16—22no—
387C2×M16C23×C6712yes—
388C2×M16C23×C6712yes—
389C22⋊Q8—21no—
390C22⋊Q8—21no—
391C22⋊Q8—21no—
392C22⋊Q8—21no—
393C4×Q8—22no—
394C4×Q8—22no—
395C4×Q8—22no—
396C4×Q8—22no—
397C4×D4—24no—
398C4×D4—24no—
399C4×D4—22no—
400C4×D4—22no—
401C4×D4—22no—
402C4×D4—22no—
403C4×D4—24no—
404C4×D4—24no—
405#117—24no—
406#117—24no—
407#117—21no—
408#117—21no—
409#117—21no—
410#117—21no—
411#117—21no—
412#117—21no—
413#117—22no—
414#117—22no—
415#117D4×C6712yes—
416#117C23×C6712yes—
417C2×C4⋊C4—22no—
418C2×C4⋊C4—22no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—22no—
421C2×C4⋊C4D4×C6711yescharacteristic
422C2×C4⋊C4D4×C6711yescharacteristic
423C2×C4⋊C4C23×C6711yescharacteristic
424C2×C4⋊C4C23×C6711yescharacteristic
425C2×C4⋊C4C23×C6711yescharacteristic
426C2×C22⋊C4—24no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4D4×C6712yes—
431C2×C42—24no—
432C2×C42—21no—
433C2×C42D4×C6712yes—
434C2×C42C23×C6711yescharacteristic
435C2×C42C23×C6711yescharacteristic
436#107—22no—
437#106—22no—
438#104—24no—
439#104—24no—
440#103—22no—
441#103—22no—
442#103—22no—
443#102—22no—
444C22⋊C8—21no—
445C22⋊C8—21no—
446C22⋊C8—21no—
447C22⋊C8—21no—
448#95—21no—
449#95—22no—
450#95—22no—
451#95—22no—
452C24—21no—
453C24C2×D4×C6711yescharacteristic
454#41—24no—
455#41—24no—
456#41—28no—
457#41—24no—
458#41—24no—
459#41—22no—
460#41—22no—
461#41#41×C6714yes—
462#41#41×C6714yes—
463C2×Q8—24no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—22no—
468C2×Q8—22no—
469C2×Q8C2×D4×C6712yes—
470C2×D4—24no—
471C2×D4—24no—
472C2×D4—22no—
473C2×D4—22no—
474C2×D4—24no—
475C2×D4—21no—
476C2×D4C2×D4×C6712yes—
477C22×C4—28no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4C2×D4×C6711yescharacteristic
492C22×C4C2×D4×C6711yescharacteristic
493C22×C4C2×D4×C6711yescharacteristic
494C22×C4C2×D4×C6711yescharacteristic
495C22×C4C2×D4×C6711yescharacteristic
496C22×C4C2×D4×C6712yes—
497C22×C4C2×D4×C6712yes—
498C22×C4C24×C6711yescharacteristic
499Q16—44no—
500SD16—44no—
501M16—22no—
502M16—22no—
503M16—22no—
504M16—22no—
505C2×C8—22no—
506C2×C8—22no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—44no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4C2×D4×C6712yes—
521C4⋊C4C2×D4×C6712yes—
522C22⋊C4—24no—
523C22⋊C4—24no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—42no—
529C22⋊C4—41no—
530C22⋊C4—41no—
531C42—24no—
532C42—24no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C23—24no—
540C23—22no—
541C23—21no—
542C23—21no—
543C23—41no—
544C23—41no—
545C23—81no—
546C23—81no—
547C23#120×C6711yescharacteristic
548C23#120×C6711yescharacteristic
549C23C22×D4×C6711yescharacteristic
550Q8—24no—
551Q8—22no—
552Q8—42no—
553Q8—42no—
554Q8#123×C6714yes—
555D4—28no—
556D4—24no—
557D4—24no—
558D4—24no—
559D4—22no—
560D4#123×C6714yes—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4#120×C6711yescharacteristic
594C2×C4#120×C6711yescharacteristic
595C2×C4C22×D4×C6711yescharacteristic
596C2×C4C22×D4×C6711yescharacteristic
597C2×C4C2×#41×C6712yes—
598C2×C4C2×#41×C6712yes—
599C8—42no—
600C8—42no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—24no—
610C22—42no—
611C22—42no—
612C22—81no—
613C22—81no—
614C22—84no—
615C22C2×#120×C6711yescharacteristic
616C22#525×C6711yescharacteristic
617C22#525×C6711yescharacteristic
618C4—21no—
619C4—24no—
620C4—21no—
621C4—42no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—81no—
627C4—81no—
628C4—81no—
629C4C2×#123×C6712yes—
630C4#525×C6712yes—
631C2—21no—
632C2—21no—
633C2—41no—
634C2—41no—
635C2—44no—
636C2C67×#284911yescharacteristic
637C2C67×#301811yescharacteristic
638C2C67×#302211yescharacteristic
subgroupG/H
center, socleC22×C67C2×#120
commutator, Frattini, cosocleC22×C4C24×C67
prime radicalC22×#41×C67C22
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
2. C67×#3095C2semidirect
4. C67×#2499C2semidirect
5. C67×#1882C2semidirect
6. C67×#1882C2semidirect
7. C67×#1860C2semidirect
8. C67×#1859C2semidirect
9. C67×#1826C2semidirect
10. C67×#1822C2semidirect
33. #428×C67C22semidirect
36. #419×C67C22semidirect
41. #391×C67C22semidirect
42. #391×C67C22semidirect
44. #384×C67C22semidirect
45. #382×C67C22semidirect
47. #343×C67C22semidirect
48. #343×C67C22semidirect
49. #339×C67C22semidirect
50. #339×C67C22semidirect
311. #33001C67direct (N × H)
321. #3095C2×C67semidirect
323. #2499C2×C67semidirect
324. #1882C2×C67semidirect
325. #1882C2×C67semidirect
326. #1860C2×C67semidirect
327. #1859C2×C67semidirect
328. #1826C2×C67semidirect
329. #1822C2×C67semidirect
331. C67#33001direct (N × H)
353. #428C22×C67semidirect
356. #419C22×C67semidirect
361. #391C22×C67semidirect
362. #391C22×C67semidirect
364. #384C22×C67semidirect
365. #382C22×C67semidirect
367. #343C22×C67semidirect
368. #343C22×C67semidirect
369. #339C22×C67semidirect
370. #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×#30952
C67×#30072
C67×#24992
C67×#18822
C67×#18602
C67×#18592
C67×#18262
C67×#18222
C67×#17892
#3300167