Finite Group Databasebeta

← C67×#11090 #30690341

Subgroups — C67×#11090 #30690341

Order 17152.

852 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#2905C211yesmaximal, characteristic, complemented
2C2×#413×C67C211yesmaximal, characteristic
3C2×#413×C67C211yesmaximal, characteristic
4C2×#410×C67C211yesmaximal, characteristic
5C2×#344×C67C211yesmaximal, characteristic, complemented
6C2×#344×C67C211yesmaximal, characteristic, complemented
7C67×#1742C211yesmaximal, characteristic, complemented
8C22×#41×C67C2211yescharacteristic
9C22×M16×C67C2211yescharacteristic
10C22×M16×C67C2211yescharacteristic
11C22×M16×C67C2211yescharacteristic
12C22×M16×C67C2211yescharacteristic
13C2×#117×C67C2211yescharacteristic
14C2×#117×C67C2211yescharacteristic
15#424×C67—22no—
16#424×C67—22no—
17#423×C67—22no—
18#413×C67—22no—
19#413×C67—22no—
20#413×C67—22no—
21#413×C67—22no—
22C2×#101×C67—21no—
23C2×#101×C67—21no—
24C2×#100×C67—21no—
25C2×#100×C67—21no—
26#410×C67—22no—
27#410×C67—22no—
28C2×#99×C67C414yes—
29#344×C67—22no—
30#344×C67—22no—
31#343×C67—22no—
32C2×#41×C67—21no—
33C2×#41×C67—21no—
34C2×#41×C67—21no—
35C2×#41×C67—21no—
36C2×#41×C67—21no—
37C2×#41×C67—21no—
38C2×#41×C67—21no—
39C2×#41×C67—21no—
40C2×#41×C67—21no—
41C2×#41×C67—21no—
42C2×#41×C67—21no—
43C2×#41×C67—21no—
44C22×Q8×C67D411yescharacteristic
45C22×D4×C67C2×C412yes—
46C22×D4×C67D411yescharacteristic
47C23×C4×C67Q812yes—
48C23×C4×C67C2311yescharacteristic
49C2×M16×C67—21no—
50C2×M16×C67—21no—
51C2×M16×C67—21no—
52C2×M16×C67—21no—
53C2×M16×C67—21no—
54C2×M16×C67—21no—
55C2×M16×C67—21no—
56C2×M16×C67—21no—
57C2×M16×C67—21no—
58C2×M16×C67—21no—
59C2×M16×C67—21no—
60C2×M16×C67—21no—
61C2×M16×C67—21no—
62C2×M16×C67—21no—
63C2×M16×C67—21no—
64C2×M16×C67—21no—
65C2×M16×C67—21no—
66C2×M16×C67—21no—
67C2×M16×C67—21no—
68C2×M16×C67—21no—
69C2×M16×C67D412yes—
70C2×M16×C67D412yes—
71C2×M16×C67D412yes—
72C2×M16×C67D412yes—
73C22×C8×C67—21no—
74C22×C8×C67—21no—
75C22×C8×C67—21no—
76C22×C8×C67—21no—
77#117×C67—21no—
78#117×C67—21no—
79#117×C67—21no—
80#117×C67—21no—
81#117×C67—21no—
82#117×C67—21no—
83#117×C67D412yes—
84#117×C67D412yes—
85C2×C4⋊C4×C67—21no—
86C2×C4⋊C4×C67—21no—
87C2×C22⋊C4×C67C2×C412yes—
88C2×C22⋊C4×C67C2×C412yes—
89C2×C42×C67—21no—
90C2×C42×C67—21no—
91C4⋊C8×C67—28no—
92#101×C67—24no—
93#101×C67—24no—
94#100×C67—24no—
95#100×C67—24no—
96#99×C67—28no—
97C24×C67C4⋊C412yes—
98C24×C67C22×C411yescharacteristic
99#41×C67—41no—
100#41×C67—41no—
101#41×C67—41no—
102#41×C67—41no—
103#41×C67—41no—
104#41×C67—41no—
105#41×C67—41no—
106#41×C67—41no—
107#41×C67—41no—
108#41×C67—41no—
109#41×C67—41no—
110#41×C67—41no—
111#41×C67—41no—
112#41×C67—41no—
113#41×C67—41no—
114#41×C67—41no—
115C2×Q8×C67—21no—
116C2×Q8×C67—21no—
117C2×Q8×C67—21no—
118C2×Q8×C67—21no—
119C2×Q8×C67—21no—
120C2×Q8×C67—21no—
121C2×D4×C67—22no—
122C2×D4×C67—22no—
123C2×D4×C67—22no—
124C2×D4×C67—22no—
125C2×D4×C67—22no—
126C2×D4×C67—22no—
127C2×D4×C67—21no—
128C2×D4×C67—21no—
129C2×D4×C67—21no—
130C2×D4×C67—21no—
131C2×D4×C67—21no—
132C2×D4×C67—21no—
133C22×C4×C67—22no—
134C22×C4×C67—22no—
135C22×C4×C67—22no—
136C22×C4×C67—22no—
137C22×C4×C67—22no—
138C22×C4×C67—22no—
139C22×C4×C67—22no—
140C22×C4×C67—22no—
141C22×C4×C67C4⋊C412yes—
142C22×C4×C67C2×D411yescharacteristic
143C22×C4×C67C2×D411yescharacteristic
144C22×C4×C67C2×D411yescharacteristic
145C22×C4×C67C2×D411yescharacteristic
146C22×C4×C67C2×D411yescharacteristic
147C22×C4×C67C2×D411yescharacteristic
148C22×C4×C67C2×D411yescharacteristic
149C22×C4×C67C2×Q811yescharacteristic
150C22×C4×C67#4111yescharacteristic
151C22×C4×C67#4111yescharacteristic
152C22×C4×C67#4111yescharacteristic
153C22×C4×C67#4111yescharacteristic
154C22×C4×C67#4111yescharacteristic
155C22×C4×C67#4111yescharacteristic
156M16×C67—22no—
157M16×C67—22no—
158M16×C67—22no—
159M16×C67—22no—
160M16×C67—22no—
161M16×C67—22no—
162M16×C67—22no—
163M16×C67—22no—
164M16×C67—41no—
165M16×C67—41no—
166M16×C67—41no—
167M16×C67—41no—
168M16×C67—41no—
169M16×C67—41no—
170M16×C67—41no—
171M16×C67—41no—
172C2×C8×C67—22no—
173C2×C8×C67—22no—
174C2×C8×C67—22no—
175C2×C8×C67—22no—
176C2×C8×C67—41no—
177C2×C8×C67—41no—
178C2×C8×C67—41no—
179C2×C8×C67—41no—
180C2×C8×C67—41no—
181C2×C8×C67—41no—
182C2×C8×C67—41no—
183C2×C8×C67—41no—
184C4⋊C4×C67—22no—
185C4⋊C4×C67—22no—
186C4⋊C4×C67—41no—
187C4⋊C4×C67—41no—
188C22⋊C4×C67—22no—
189C22⋊C4×C67—22no—
190C22⋊C4×C67—22no—
191C22⋊C4×C67C22⋊C414yes—
192C42×C67—22no—
193C42×C67—22no—
194C42×C67—41no—
195C42×C67—41no—
196C23×C67—22no—
197C23×C67—22no—
198C23×C67—22no—
199C23×C67—81no—
200C23×C67—82no—
201C23×C67C2×C22⋊C411yescharacteristic
202C23×C67C2×C4⋊C411yescharacteristic
203C23×C67#11711yescharacteristic
204C23×C67C4×D411yescharacteristic
205C23×C67C4×D411yescharacteristic
206C23×C67C4×D411yescharacteristic
207C23×C67C4×D411yescharacteristic
208Q8×C67—41no—
209Q8×C67—41no—
210Q8×C67—41no—
211Q8×C67—41no—
212D4×C67—42no—
213D4×C67—42no—
214D4×C67—42no—
215D4×C67—42no—
216D4×C67—41no—
217D4×C67—41no—
218D4×C67—41no—
219D4×C67—41no—
220C2×C4×C67—22no—
221C2×C4×C67—22no—
222C2×C4×C67—22no—
223C2×C4×C67—24no—
224C2×C4×C67—42no—
225C2×C4×C67—42no—
226C2×C4×C67—42no—
227C2×C4×C67—42no—
228C2×C4×C67—42no—
229C2×C4×C67—82no—
230C2×C4×C67—82no—
231C2×C4×C67#12011yescharacteristic
232C2×C4×C67#12011yescharacteristic
233C2×C4×C67#12011yescharacteristic
234C2×C4×C67#12111yescharacteristic
235C2×C4×C67#12111yescharacteristic
236C2×C4×C67#12111yescharacteristic
237C2×C4×C67#12111yescharacteristic
238C2×C4×C67#12111yescharacteristic
239C2×C4×C67#12111yescharacteristic
240C2×C4×C67#12111yescharacteristic
241C2×C4×C67#12111yescharacteristic
242C2×C4×C67C22⋊Q811yescharacteristic
243C2×C4×C67C22⋊Q811yescharacteristic
244C2×C4×C67C22⋊Q811yescharacteristic
245C2×C4×C67C22⋊Q811yescharacteristic
246C2×C4×C67C22⋊Q811yescharacteristic
247C2×C4×C67C22⋊Q811yescharacteristic
248C2×C4×C67#12311yescharacteristic
249C2×C4×C67#12311yescharacteristic
250C2×C4×C67#12311yescharacteristic
251C2×C4×C67#12311yescharacteristic
252C2×C4×C67#12311yescharacteristic
253C2×C4×C67#12411yescharacteristic
254C2×C4×C67#12411yescharacteristic
255C2×C4×C67#12411yescharacteristic
256C2×C4×C67#12611yescharacteristic
257C2×C4×C67#12611yescharacteristic
258C2×C4×C67#12711yescharacteristic
259C8×C67—42no—
260C8×C67—42no—
261C8×C67—42no—
262C8×C67—42no—
263C22×C67—41no—
264C22×C67—41no—
265C22×C67—41no—
266C22×C67—41no—
267C22×C67—41no—
268C22×C67—41no—
269C22×C67—41no—
270C22×C67—42no—
271C22×C67—82no—
272C22×C67—82no—
273C22×C67—82no—
274C22×C67#38011yescharacteristic
275C22×C67#38511yescharacteristic
276C22×C67#38511yescharacteristic
277C22×C67#38611yescharacteristic
278C22×C67#38811yescharacteristic
279C22×C67#38811yescharacteristic
280C22×C67#39011yescharacteristic
281C4×C67—44no—
282C4×C67—42no—
283C4×C67—82no—
284C4×C67#39211yescharacteristic
285C4×C67#39311yescharacteristic
286C4×C67#39411yescharacteristic
287C4×C67#39411yescharacteristic
288C4×C67#39611yescharacteristic
289C4×C67#39611yescharacteristic
290C4×C67#39711yescharacteristic
291C4×C67#39911yescharacteristic
292#11090C6711yesmaximal, characteristic, direct product factor, Sylow
293C2×C67—21no—
294C2×C67—21no—
295C2×C67—21no—
296C2×C67—21no—
297C2×C67—21no—
298C2×C67—21no—
299C2×C67—21no—
300C2×C67—82no—
301C2×C67#143911yescharacteristic
302#2905C2×C6711yescharacteristic, complemented
303C2×#413C2×C6711yescharacteristic
304C2×#413C2×C6711yescharacteristic
305C2×#410C2×C6711yescharacteristic
306C2×#344C2×C6711yescharacteristic, complemented
307C2×#344C2×C6711yescharacteristic, complemented
308#1742C2×C6711yescharacteristic, complemented
309C67#1109011yescharacteristic, direct product factor, Sylow
310C22×#41C22×C6711yescharacteristic
311C22×M16C22×C6711yescharacteristic
312C22×M16C22×C6711yescharacteristic
313C22×M16C22×C6711yescharacteristic
314C22×M16C22×C6711yescharacteristic
315C2×#117C22×C6711yescharacteristic
316C2×#117C22×C6711yescharacteristic
317#424—22no—
318#424—22no—
319#423—22no—
320#413—22no—
321#413—22no—
322#413—22no—
323#413—22no—
324C2×#101—21no—
325C2×#101—21no—
326C2×#100—21no—
327C2×#100—21no—
328#410—22no—
329#410—22no—
330C2×#99C4×C6714yes—
331#344—22no—
332#344—22no—
333#343—22no—
334C2×#41—21no—
335C2×#41—21no—
336C2×#41—21no—
337C2×#41—21no—
338C2×#41—21no—
339C2×#41—21no—
340C2×#41—21no—
341C2×#41—21no—
342C2×#41—21no—
343C2×#41—21no—
344C2×#41—21no—
345C2×#41—21no—
346C22×Q8D4×C6711yescharacteristic
347C22×D4C2×C4×C6712yes—
348C22×D4D4×C6711yescharacteristic
349C23×C4Q8×C6712yes—
350C23×C4C23×C6711yescharacteristic
351C2×M16—21no—
352C2×M16—21no—
353C2×M16—21no—
354C2×M16—21no—
355C2×M16—21no—
356C2×M16—21no—
357C2×M16—21no—
358C2×M16—21no—
359C2×M16—21no—
360C2×M16—21no—
361C2×M16—21no—
362C2×M16—21no—
363C2×M16—21no—
364C2×M16—21no—
365C2×M16—21no—
366C2×M16—21no—
367C2×M16—21no—
368C2×M16—21no—
369C2×M16—21no—
370C2×M16—21no—
371C2×M16D4×C6712yes—
372C2×M16D4×C6712yes—
373C2×M16D4×C6712yes—
374C2×M16D4×C6712yes—
375C22×C8—21no—
376C22×C8—21no—
377C22×C8—21no—
378C22×C8—21no—
379#117—21no—
380#117—21no—
381#117—21no—
382#117—21no—
383#117—21no—
384#117—21no—
385#117D4×C6712yes—
386#117D4×C6712yes—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C22⋊C4C2×C4×C6712yes—
390C2×C22⋊C4C2×C4×C6712yes—
391C2×C42—21no—
392C2×C42—21no—
393C4⋊C8—28no—
394#101—24no—
395#101—24no—
396#100—24no—
397#100—24no—
398#99—28no—
399C24C4⋊C4×C6712yes—
400C24C22×C4×C6711yescharacteristic
401#41—41no—
402#41—41no—
403#41—41no—
404#41—41no—
405#41—41no—
406#41—41no—
407#41—41no—
408#41—41no—
409#41—41no—
410#41—41no—
411#41—41no—
412#41—41no—
413#41—41no—
414#41—41no—
415#41—41no—
416#41—41no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4C4⋊C4×C6712yes—
444C22×C4C2×D4×C6711yescharacteristic
445C22×C4C2×D4×C6711yescharacteristic
446C22×C4C2×D4×C6711yescharacteristic
447C22×C4C2×D4×C6711yescharacteristic
448C22×C4C2×D4×C6711yescharacteristic
449C22×C4C2×D4×C6711yescharacteristic
450C22×C4C2×D4×C6711yescharacteristic
451C22×C4C2×Q8×C6711yescharacteristic
452C22×C4#41×C6711yescharacteristic
453C22×C4#41×C6711yescharacteristic
454C22×C4#41×C6711yescharacteristic
455C22×C4#41×C6711yescharacteristic
456C22×C4#41×C6711yescharacteristic
457C22×C4#41×C6711yescharacteristic
458M16—22no—
459M16—22no—
460M16—22no—
461M16—22no—
462M16—22no—
463M16—22no—
464M16—22no—
465M16—22no—
466M16—41no—
467M16—41no—
468M16—41no—
469M16—41no—
470M16—41no—
471M16—41no—
472M16—41no—
473M16—41no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4C22⋊C4×C6714yes—
494C42—22no—
495C42—22no—
496C42—41no—
497C42—41no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—81no—
502C23—82no—
503C23C2×C22⋊C4×C6711yescharacteristic
504C23C2×C4⋊C4×C6711yescharacteristic
505C23#117×C6711yescharacteristic
506C23C4×D4×C6711yescharacteristic
507C23C4×D4×C6711yescharacteristic
508C23C4×D4×C6711yescharacteristic
509C23C4×D4×C6711yescharacteristic
510Q8—41no—
511Q8—41no—
512Q8—41no—
513Q8—41no—
514D4—42no—
515D4—42no—
516D4—42no—
517D4—42no—
518D4—41no—
519D4—41no—
520D4—41no—
521D4—41no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4#120×C6711yescharacteristic
534C2×C4#120×C6711yescharacteristic
535C2×C4#120×C6711yescharacteristic
536C2×C4#121×C6711yescharacteristic
537C2×C4#121×C6711yescharacteristic
538C2×C4#121×C6711yescharacteristic
539C2×C4#121×C6711yescharacteristic
540C2×C4#121×C6711yescharacteristic
541C2×C4#121×C6711yescharacteristic
542C2×C4#121×C6711yescharacteristic
543C2×C4#121×C6711yescharacteristic
544C2×C4C22⋊Q8×C6711yescharacteristic
545C2×C4C22⋊Q8×C6711yescharacteristic
546C2×C4C22⋊Q8×C6711yescharacteristic
547C2×C4C22⋊Q8×C6711yescharacteristic
548C2×C4C22⋊Q8×C6711yescharacteristic
549C2×C4C22⋊Q8×C6711yescharacteristic
550C2×C4#123×C6711yescharacteristic
551C2×C4#123×C6711yescharacteristic
552C2×C4#123×C6711yescharacteristic
553C2×C4#123×C6711yescharacteristic
554C2×C4#123×C6711yescharacteristic
555C2×C4#124×C6711yescharacteristic
556C2×C4#124×C6711yescharacteristic
557C2×C4#124×C6711yescharacteristic
558C2×C4#126×C6711yescharacteristic
559C2×C4#126×C6711yescharacteristic
560C2×C4#127×C6711yescharacteristic
561C8—42no—
562C8—42no—
563C8—42no—
564C8—42no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—42no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22#380×C6711yescharacteristic
577C22#385×C6711yescharacteristic
578C22#385×C6711yescharacteristic
579C22#386×C6711yescharacteristic
580C22#388×C6711yescharacteristic
581C22#388×C6711yescharacteristic
582C22#390×C6711yescharacteristic
583C4—44no—
584C4—42no—
585C4—82no—
586C4#392×C6711yescharacteristic
587C4#393×C6711yescharacteristic
588C4#394×C6711yescharacteristic
589C4#394×C6711yescharacteristic
590C4#396×C6711yescharacteristic
591C4#396×C6711yescharacteristic
592C4#397×C6711yescharacteristic
593C4#399×C6711yescharacteristic
594C2—21no—
595C2—21no—
596C2—21no—
597C2—21no—
598C2—21no—
599C2—21no—
600C2—21no—
601C2—82no—
602C2C67×#143911yescharacteristic
subgroupG/H
centerC4×C67#393
commutatorC24C22×C4×C67
Frattini, cosocleC23×C4C23×C67
socleC2×C67#1439
prime radicalC22×#41×C67C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C4×C67 ◁ C23×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#2905C2semidirect
5. C2×#344×C67C2semidirect
6. C2×#344×C67C2semidirect
7. C67×#1742C2semidirect
292. #11090C67direct (N × H)
302. #2905C2×C67semidirect
306. C2×#344C2×C67semidirect
307. C2×#344C2×C67semidirect
308. #1742C2×C67semidirect
309. C67#11090direct (N × H)

Maximal subgroups

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

subgroupindex
C67×#29052
C2×#413×C672
C2×#410×C672
C2×#344×C672
C67×#17422
#1109067