Finite Group Databasebeta

← C67×#11153 #30690404

Subgroups — C67×#11153 #30690404

Order 17152.

804 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#421×C67C211yesmaximal, characteristic
2C67×#2896C211yesmaximal, characteristic
3C2×#413×C67C211yesmaximal, characteristic
4C2×#344×C67C211yesmaximal, characteristic, complemented
5C2×#344×C67C211yesmaximal, characteristic, complemented
6C67×#1738C211yesmaximal, characteristic, complemented
7C67×#1733C211yesmaximal, characteristic, complemented
8C22×#41×C67C2211yescharacteristic
9C22×M16×C67C2211yescharacteristic
10C22×M16×C67C2211yescharacteristic
11C22×M16×C67C2211yescharacteristic
12C22×M16×C67C2211yescharacteristic
13C2×#117×C67C2211yescharacteristic
14C2×#117×C67C2211yescharacteristic
15#421×C67—22no—
16#421×C67—22no—
17C2×#104×C67—22no—
18#419×C67—21no—
19#418×C67—21no—
20#417×C67—22no—
21#416×C67—22no—
22#413×C67—22no—
23#413×C67C414yescomplemented
24C2×#101×C67—21no—
25C2×#100×C67—21no—
26#344×C67—22no—
27#344×C67—22no—
28#341×C67—22no—
29#339×C67—22no—
30C2×#41×C67—21no—
31C2×#41×C67—21no—
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×C67C2×C412yes—
41C2×#41×C67D412yes—
42C22×Q8×C67D411yescharacteristic
43C22×D4×C67—21no—
44C22×D4×C67D411yescharacteristic
45C23×C4×C67—21no—
46C23×C4×C67C2311yescharacteristic
47C2×M16×C67—21no—
48C2×M16×C67—21no—
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×C67C2×C412yes—
68C2×M16×C67C2×C412yes—
69C2×M16×C67D412yes—
70C2×M16×C67D412yes—
71C22×C8×C67—21no—
72C22×C8×C67—21no—
73C22×C8×C67—21no—
74C22×C8×C67—21no—
75#117×C67—21no—
76#117×C67—21no—
77#117×C67—21no—
78#117×C67—21no—
79#117×C67—21no—
80#117×C67—21no—
81#117×C67D412yes—
82#117×C67D412yes—
83C2×C4⋊C4×C67—21no—
84C2×C4⋊C4×C67—21no—
85C2×C22⋊C4×C67—21no—
86C2×C22⋊C4×C67—21no—
87C2×C42×C67—21no—
88C2×C42×C67—21no—
89#104×C67—28no—
90#103×C67—24no—
91#102×C67—24no—
92#101×C67—24no—
93#100×C67—24no—
94C24×C67—21no—
95C24×C67#4111yescharacteristic
96#41×C67—22no—
97#41×C67—22no—
98#41×C67—22no—
99#41×C67—22no—
100#41×C67—22no—
101#41×C67—22no—
102#41×C67—22no—
103#41×C67—22no—
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—
112C2×Q8×C67—21no—
113C2×Q8×C67—21no—
114C2×Q8×C67—21no—
115C2×Q8×C67—21no—
116C2×Q8×C67—21no—
117C2×Q8×C67C22⋊C412yes—
118C2×D4×C67—22no—
119C2×D4×C67—22no—
120C2×D4×C67—21no—
121C2×D4×C67—21no—
122C2×D4×C67—21no—
123C2×D4×C67—21no—
124C2×D4×C67—21no—
125C2×D4×C67—41no—
126C2×D4×C67—41no—
127C2×D4×C67—41no—
128C2×D4×C67—41no—
129C2×D4×C67C22⋊C412yes—
130C22×C4×C67—21no—
131C22×C4×C67—22no—
132C22×C4×C67—22no—
133C22×C4×C67—41no—
134C22×C4×C67—41no—
135C22×C4×C67—41no—
136C22×C4×C67—41no—
137C22×C4×C67—41no—
138C22×C4×C67—41no—
139C22×C4×C67C22×C411yescharacteristic
140C22×C4×C67C2×D411yescharacteristic
141C22×C4×C67C2×D411yescharacteristic
142C22×C4×C67C2×D411yescharacteristic
143C22×C4×C67C2×D411yescharacteristic
144C22×C4×C67C2×D411yescharacteristic
145C22×C4×C67C2×D411yescharacteristic
146C22×C4×C67#4111yescharacteristic
147C22×C4×C67#4111yescharacteristic
148C22×C4×C67#4111yescharacteristic
149C22×C4×C67#4111yescharacteristic
150C22×C4×C67#4111yescharacteristic
151C22×C4×C67#4111yescharacteristic
152C22×C4×C67#4111yescharacteristic
153M16×C67—22no—
154M16×C67—22no—
155M16×C67—22no—
156M16×C67—22no—
157M16×C67—22no—
158M16×C67—22no—
159M16×C67—22no—
160M16×C67—22no—
161M16×C67—41no—
162M16×C67—41no—
163M16×C67—41no—
164M16×C67—41no—
165M16×C67—41no—
166M16×C67—41no—
167M16×C67—41no—
168M16×C67—41no—
169C2×C8×C67—22no—
170C2×C8×C67—22no—
171C2×C8×C67—22no—
172C2×C8×C67—22no—
173C2×C8×C67—41no—
174C2×C8×C67—41no—
175C2×C8×C67—41no—
176C2×C8×C67—41no—
177C2×C8×C67—41no—
178C2×C8×C67—41no—
179C2×C8×C67—41no—
180C2×C8×C67—41no—
181C4⋊C4×C67—22no—
182C4⋊C4×C67—22no—
183C4⋊C4×C67—41no—
184C4⋊C4×C67—41no—
185C22⋊C4×C67—22no—
186C22⋊C4×C67—22no—
187C22⋊C4×C67—41no—
188C22⋊C4×C67—41no—
189C42×C67—22no—
190C42×C67—22no—
191C42×C67—41no—
192C42×C67—41no—
193C23×C67—22no—
194C23×C67—41no—
195C23×C67—41no—
196C23×C67—81no—
197C23×C67—82no—
198C23×C67#11711yescharacteristic
199C23×C67#12111yescharacteristic
200C23×C67#12311yescharacteristic
201C23×C67#12311yescharacteristic
202C23×C67#12311yescharacteristic
203C23×C67#12411yescharacteristic
204C23×C67#12611yescharacteristic
205Q8×C67—22no—
206Q8×C67—22no—
207Q8×C67—41no—
208Q8×C67—41no—
209D4×C67—22no—
210D4×C67—22no—
211D4×C67—42no—
212D4×C67—42no—
213D4×C67—42no—
214D4×C67—42no—
215D4×C67—41no—
216D4×C67—41no—
217C2×C4×C67—22no—
218C2×C4×C67—41no—
219C2×C4×C67—41no—
220C2×C4×C67—42no—
221C2×C4×C67—42no—
222C2×C4×C67—42no—
223C2×C4×C67—42no—
224C2×C4×C67—42no—
225C2×C4×C67—42no—
226C2×C4×C67—82no—
227C2×C4×C67—82no—
228C2×C4×C67C2×C22⋊C411yescharacteristic
229C2×C4×C67#11711yescharacteristic
230C2×C4×C67C4×D411yescharacteristic
231C2×C4×C67C4×D411yescharacteristic
232C2×C4×C67C4×D411yescharacteristic
233C2×C4×C67C4×D411yescharacteristic
234C2×C4×C67#12011yescharacteristic
235C2×C4×C67#12011yescharacteristic
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×C67#12111yescharacteristic
243C2×C4×C67#12311yescharacteristic
244C2×C4×C67#12311yescharacteristic
245C2×C4×C67#12311yescharacteristic
246C2×C4×C67#12311yescharacteristic
247C2×C4×C67#12311yescharacteristic
248C2×C4×C67#12311yescharacteristic
249C2×C4×C67#12311yescharacteristic
250C2×C4×C67#12411yescharacteristic
251C2×C4×C67#12411yescharacteristic
252C2×C4×C67#12411yescharacteristic
253C2×C4×C67#12611yescharacteristic
254C2×C4×C67#12611yescharacteristic
255C2×C4×C67#12611yescharacteristic
256C8×C67—42no—
257C8×C67—42no—
258C8×C67—42no—
259C8×C67—42no—
260C22×C67—41no—
261C22×C67—41no—
262C22×C67—41no—
263C22×C67—41no—
264C22×C67—41no—
265C22×C67—41no—
266C22×C67—41no—
267C22×C67—42no—
268C22×C67—82no—
269C22×C67—82no—
270C22×C67—82no—
271C22×C67#38111yescharacteristic
272C22×C67#38811yescharacteristic
273C22×C67#38811yescharacteristic
274C22×C67#39411yescharacteristic
275C22×C67#39411yescharacteristic
276C22×C67#39711yescharacteristic
277C22×C67#39711yescharacteristic
278C4×C67—42no—
279C4×C67—82no—
280C4×C67—82no—
281C4×C67#38611yescharacteristic
282C4×C67#38611yescharacteristic
283C4×C67#38811yescharacteristic
284C4×C67#38811yescharacteristic
285C4×C67#39411yescharacteristic
286C4×C67#39411yescharacteristic
287C4×C67#39711yescharacteristic
288C4×C67#39711yescharacteristic
289#11153C6711yesmaximal, characteristic, direct product factor, Sylow
290C2×C67—21no—
291C2×C67—21no—
292C2×C67—21no—
293C2×C67—21no—
294C2×C67—21no—
295C2×C67—21no—
296C2×C67—21no—
297C2×C67—82no—
298C2×C67#144011yescharacteristic
299C2×#421C2×C6711yescharacteristic
300#2896C2×C6711yescharacteristic
301C2×#413C2×C6711yescharacteristic
302C2×#344C2×C6711yescharacteristic, complemented
303C2×#344C2×C6711yescharacteristic, complemented
304#1738C2×C6711yescharacteristic, complemented
305#1733C2×C6711yescharacteristic, complemented
306C67#1115311yescharacteristic, direct product factor, Sylow
307C22×#41C22×C6711yescharacteristic
308C22×M16C22×C6711yescharacteristic
309C22×M16C22×C6711yescharacteristic
310C22×M16C22×C6711yescharacteristic
311C22×M16C22×C6711yescharacteristic
312C2×#117C22×C6711yescharacteristic
313C2×#117C22×C6711yescharacteristic
314#421—22no—
315#421—22no—
316C2×#104—22no—
317#419—21no—
318#418—21no—
319#417—22no—
320#416—22no—
321#413—22no—
322#413C4×C6714yescomplemented
323C2×#101—21no—
324C2×#100—21no—
325#344—22no—
326#344—22no—
327#341—22no—
328#339—22no—
329C2×#41—21no—
330C2×#41—21no—
331C2×#41—21no—
332C2×#41—21no—
333C2×#41—21no—
334C2×#41—21no—
335C2×#41—21no—
336C2×#41—21no—
337C2×#41—21no—
338C2×#41—21no—
339C2×#41C2×C4×C6712yes—
340C2×#41D4×C6712yes—
341C22×Q8D4×C6711yescharacteristic
342C22×D4—21no—
343C22×D4D4×C6711yescharacteristic
344C23×C4—21no—
345C23×C4C23×C6711yescharacteristic
346C2×M16—21no—
347C2×M16—21no—
348C2×M16—21no—
349C2×M16—21no—
350C2×M16—21no—
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×M16C2×C4×C6712yes—
367C2×M16C2×C4×C6712yes—
368C2×M16D4×C6712yes—
369C2×M16D4×C6712yes—
370C22×C8—21no—
371C22×C8—21no—
372C22×C8—21no—
373C22×C8—21no—
374#117—21no—
375#117—21no—
376#117—21no—
377#117—21no—
378#117—21no—
379#117—21no—
380#117D4×C6712yes—
381#117D4×C6712yes—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C42—21no—
387C2×C42—21no—
388#104—28no—
389#103—24no—
390#102—24no—
391#101—24no—
392#100—24no—
393C24—21no—
394C24#41×C6711yescharacteristic
395#41—22no—
396#41—22no—
397#41—22no—
398#41—22no—
399#41—22no—
400#41—22no—
401#41—22no—
402#41—22no—
403#41—41no—
404#41—41no—
405#41—41no—
406#41—41no—
407#41—41no—
408#41—41no—
409#41—41no—
410#41—41no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8C22⋊C4×C6712yes—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—41no—
425C2×D4—41no—
426C2×D4—41no—
427C2×D4—41no—
428C2×D4C22⋊C4×C6712yes—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4C22×C4×C6711yescharacteristic
439C22×C4C2×D4×C6711yescharacteristic
440C22×C4C2×D4×C6711yescharacteristic
441C22×C4C2×D4×C6711yescharacteristic
442C22×C4C2×D4×C6711yescharacteristic
443C22×C4C2×D4×C6711yescharacteristic
444C22×C4C2×D4×C6711yescharacteristic
445C22×C4#41×C6711yescharacteristic
446C22×C4#41×C6711yescharacteristic
447C22×C4#41×C6711yescharacteristic
448C22×C4#41×C6711yescharacteristic
449C22×C4#41×C6711yescharacteristic
450C22×C4#41×C6711yescharacteristic
451C22×C4#41×C6711yescharacteristic
452M16—22no—
453M16—22no—
454M16—22no—
455M16—22no—
456M16—22no—
457M16—22no—
458M16—22no—
459M16—22no—
460M16—41no—
461M16—41no—
462M16—41no—
463M16—41no—
464M16—41no—
465M16—41no—
466M16—41no—
467M16—41no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C42—22no—
489C42—22no—
490C42—41no—
491C42—41no—
492C23—22no—
493C23—41no—
494C23—41no—
495C23—81no—
496C23—82no—
497C23#117×C6711yescharacteristic
498C23#121×C6711yescharacteristic
499C23#123×C6711yescharacteristic
500C23#123×C6711yescharacteristic
501C23#123×C6711yescharacteristic
502C23#124×C6711yescharacteristic
503C23#126×C6711yescharacteristic
504Q8—22no—
505Q8—22no—
506Q8—41no—
507Q8—41no—
508D4—22no—
509D4—22no—
510D4—42no—
511D4—42no—
512D4—42no—
513D4—42no—
514D4—41no—
515D4—41no—
516C2×C4—22no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4C2×C22⋊C4×C6711yescharacteristic
528C2×C4#117×C6711yescharacteristic
529C2×C4C4×D4×C6711yescharacteristic
530C2×C4C4×D4×C6711yescharacteristic
531C2×C4C4×D4×C6711yescharacteristic
532C2×C4C4×D4×C6711yescharacteristic
533C2×C4#120×C6711yescharacteristic
534C2×C4#120×C6711yescharacteristic
535C2×C4#121×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#123×C6711yescharacteristic
543C2×C4#123×C6711yescharacteristic
544C2×C4#123×C6711yescharacteristic
545C2×C4#123×C6711yescharacteristic
546C2×C4#123×C6711yescharacteristic
547C2×C4#123×C6711yescharacteristic
548C2×C4#123×C6711yescharacteristic
549C2×C4#124×C6711yescharacteristic
550C2×C4#124×C6711yescharacteristic
551C2×C4#124×C6711yescharacteristic
552C2×C4#126×C6711yescharacteristic
553C2×C4#126×C6711yescharacteristic
554C2×C4#126×C6711yescharacteristic
555C8—42no—
556C8—42no—
557C8—42no—
558C8—42no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—42no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22#381×C6711yescharacteristic
571C22#388×C6711yescharacteristic
572C22#388×C6711yescharacteristic
573C22#394×C6711yescharacteristic
574C22#394×C6711yescharacteristic
575C22#397×C6711yescharacteristic
576C22#397×C6711yescharacteristic
577C4—42no—
578C4—82no—
579C4—82no—
580C4#386×C6711yescharacteristic
581C4#386×C6711yescharacteristic
582C4#388×C6711yescharacteristic
583C4#388×C6711yescharacteristic
584C4#394×C6711yescharacteristic
585C4#394×C6711yescharacteristic
586C4#397×C6711yescharacteristic
587C4#397×C6711yescharacteristic
588C2—21no—
589C2—21no—
590C2—21no—
591C2—21no—
592C2—21no—
593C2—21no—
594C2—21no—
595C2—82no—
596C2C67×#144011yescharacteristic
subgroupG/H
centerC4×C67#394
commutatorC22×C4C22×C4×C67
Frattini, cosocleC23×C4C23×C67
socleC2×C67#1440
prime radicalC22×#41×C67C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C67 ◁ C23×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
4. C2×#344×C67C2semidirect
5. C2×#344×C67C2semidirect
6. C67×#1738C2semidirect
7. C67×#1733C2semidirect
23. #413×C67C4semidirect
289. #11153C67direct (N × H)
302. C2×#344C2×C67semidirect
303. C2×#344C2×C67semidirect
304. #1738C2×C67semidirect
305. #1733C2×C67semidirect
306. C67#11153direct (N × H)
322. #413C4×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#421×C672
C67×#28962
C2×#413×C672
C2×#344×C672
C67×#17382
C67×#17332
#1115367