Finite Group Databasebeta

← C67×#9115 #30688366

Subgroups — C67×#9115 #30688366

Order 17152.

952 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#407×C67C211yesmaximal, characteristic
2C67×#2848C211yesmaximal, characteristic
3C2×#328×C67C212yesmaximal, complemented
4C67×C22⋊#95C211yesmaximal, characteristic
5C67×#1294C212yesmaximal, complemented
6C24×C4×C67C2211yescharacteristic
7C22×M16×C67—21no—
8C2×C22⋊Q8×C67C411yescharacteristic, complemented
9C2×C22⋊Q8×C67C411yescharacteristic, complemented
10C2×C22⋊Q8×C67C2211yescharacteristic
11C22×C22⋊C4×C67—21no—
12#407×C67—24no—
13C2×C22⋊C8×C67—21no—
14C2×C22⋊C8×C67C2212yes—
15#385×C67—21no—
16#385×C67—21no—
17#385×C67—21no—
18#380×C67C411yescharacteristic, complemented
19#380×C67C411yescharacteristic, complemented
20#380×C67C2211yescharacteristic
21#379×C67—21no—
22C2×#95×C67—21no—
23C2×#95×C67C2212yes—
24#328×C67—24no—
25C25×C67D411yescharacteristic
26C22×Q8×C67C2×C411yescharacteristic
27C23×C4×C67—22no—
28C23×C4×C67—22no—
29C23×C4×C67—22no—
30C23×C4×C67—42no—
31C23×C4×C67—42no—
32C23×C4×C67C2×C411yescharacteristic
33C23×C4×C67C2×C411yescharacteristic
34C23×C4×C67D411yescharacteristic
35C23×C4×C67D411yescharacteristic
36C23×C4×C67D411yescharacteristic
37C23×C4×C67C2311yescharacteristic
38C2×M16×C67—24no—
39C2×M16×C67—42no—
40C2×M16×C67—42no—
41C22×C8×C67—22no—
42C22⋊Q8×C67—22no—
43C22⋊Q8×C67—22no—
44C22⋊Q8×C67—42no—
45C22⋊Q8×C67—42no—
46C2×C4⋊C4×C67—21no—
47C2×C4⋊C4×C67—21no—
48C2×C4⋊C4×C67—21no—
49C2×C4⋊C4×C67C2×C411yescharacteristic
50C2×C4⋊C4×C67C2×C411yescharacteristic
51C2×C4⋊C4×C67C2×C411yescharacteristic
52C2×C22⋊C4×C67—24no—
53C2×C22⋊C4×C67—21no—
54C2×C22⋊C4×C67—21no—
55C2×C22⋊C4×C67—21no—
56C2×C22⋊C4×C67—21no—
57C2×C22⋊C4×C67—21no—
58C2×C22⋊C4×C67—21no—
59C2×C22⋊C4×C67—42no—
60C2×C22⋊C4×C67—42no—
61C22⋊C8×C67—24no—
62C22⋊C8×C67—22no—
63C22⋊C8×C67D414yes—
64#95×C67—21no—
65#95×C67—21no—
66#95×C67—21no—
67#95×C67—22no—
68#95×C67—22no—
69#95×C67—42no—
70C24×C67—22no—
71C24×C67—22no—
72C24×C67—21no—
73C24×C67—21no—
74C24×C67—22no—
75C24×C67—22no—
76C24×C67—41no—
77C24×C67—41no—
78C24×C67C22⋊C411yescharacteristic
79C24×C67C22⋊C411yescharacteristic
80C24×C67C2×D411yescharacteristic
81C2×Q8×C67—22no—
82C2×Q8×C67—41no—
83C2×Q8×C67—41no—
84C22×C4×C67—22no—
85C22×C4×C67—22no—
86C22×C4×C67—21no—
87C22×C4×C67—21no—
88C22×C4×C67—22no—
89C22×C4×C67—22no—
90C22×C4×C67—22no—
91C22×C4×C67—22no—
92C22×C4×C67—22no—
93C22×C4×C67—22no—
94C22×C4×C67—21no—
95C22×C4×C67—21no—
96C22×C4×C67—24no—
97C22×C4×C67—21no—
98C22×C4×C67—21no—
99C22×C4×C67—21no—
100C22×C4×C67—21no—
101C22×C4×C67—41no—
102C22×C4×C67—41no—
103C22×C4×C67—41no—
104C22×C4×C67—41no—
105C22×C4×C67—41no—
106C22×C4×C67—41no—
107C22×C4×C67—41no—
108C22×C4×C67—48no—
109C22×C4×C67—48no—
110C22×C4×C67—42no—
111C22×C4×C67—42no—
112C22×C4×C67—42no—
113C22×C4×C67—42no—
114C22×C4×C67—42no—
115C22×C4×C67—42no—
116C22×C4×C67C22⋊C411yescharacteristic
117C22×C4×C67C22⋊C411yescharacteristic
118C22×C4×C67C22×C411yescharacteristic
119C22×C4×C67C2×D411yescharacteristic
120C22×C4×C67C2×D412yes—
121C22×C4×C67#4112yes—
122M16×C67—48no—
123C2×C8×C67—24no—
124C2×C8×C67—42no—
125C2×C8×C67—42no—
126C4⋊C4×C67—22no—
127C4⋊C4×C67—41no—
128C4⋊C4×C67—41no—
129C4⋊C4×C67—42no—
130C4⋊C4×C67—42no—
131C4⋊C4×C67—41no—
132C4⋊C4×C67—41no—
133C22⋊C4×C67—42no—
134C22⋊C4×C67—48no—
135C22⋊C4×C67—42no—
136C22⋊C4×C67—42no—
137C22⋊C4×C67—42no—
138C22⋊C4×C67—42no—
139C22⋊C4×C67—41no—
140C22⋊C4×C67—41no—
141C23×C67—21no—
142C23×C67—21no—
143C23×C67—21no—
144C23×C67—21no—
145C23×C67—24no—
146C23×C67—24no—
147C23×C67—21no—
148C23×C67—21no—
149C23×C67—21no—
150C23×C67—21no—
151C23×C67—21no—
152C23×C67—21no—
153C23×C67—21no—
154C23×C67—21no—
155C23×C67—21no—
156C23×C67—21no—
157C23×C67—21no—
158C23×C67—21no—
159C23×C67—44no—
160C23×C67—44no—
161C23×C67—44no—
162C23×C67—44no—
163C23×C67—42no—
164C23×C67—42no—
165C23×C67—42no—
166C23×C67—42no—
167C23×C67#9912yes—
168C23×C67#10412yes—
169C23×C67C2×C22⋊C411yescharacteristic
170C23×C67#12112yes—
171C23×C67AGL(1,ℤ/8)11yescharacteristic
172C23×C67#13711yescharacteristic
173C23×C67#13712yes—
174Q8×C67—82no—
175C2×C4×C67—24no—
176C2×C4×C67—24no—
177C2×C4×C67—21no—
178C2×C4×C67—21no—
179C2×C4×C67—21no—
180C2×C4×C67—21no—
181C2×C4×C67—21no—
182C2×C4×C67—21no—
183C2×C4×C67—21no—
184C2×C4×C67—21no—
185C2×C4×C67—21no—
186C2×C4×C67—42no—
187C2×C4×C67—42no—
188C2×C4×C67—44no—
189C2×C4×C67—44no—
190C2×C4×C67—44no—
191C2×C4×C67—42no—
192C2×C4×C67—42no—
193C2×C4×C67—44no—
194C2×C4×C67—44no—
195C2×C4×C67—44no—
196C2×C4×C67—41no—
197C2×C4×C67—41no—
198C2×C4×C67—41no—
199C2×C4×C67—41no—
200C2×C4×C67—41no—
201C2×C4×C67—41no—
202C2×C4×C67—41no—
203C2×C4×C67—41no—
204C2×C4×C67—41no—
205C2×C4×C67—41no—
206C2×C4×C67—41no—
207C2×C4×C67—41no—
208C2×C4×C67—82no—
209C2×C4×C67—82no—
210C2×C4×C67—82no—
211C2×C4×C67—82no—
212C2×C4×C67#9912yes—
213C2×C4×C67C4×D412yes—
214C2×C4×C67#12011yescharacteristic
215C2×C4×C67#12311yescharacteristic
216C8×C67—44no—
217C22×C67—22no—
218C22×C67—22no—
219C22×C67—21no—
220C22×C67—21no—
221C22×C67—22no—
222C22×C67—22no—
223C22×C67—21no—
224C22×C67—21no—
225C22×C67—22no—
226C22×C67—22no—
227C22×C67—22no—
228C22×C67—22no—
229C22×C67—22no—
230C22×C67—22no—
231C22×C67—48no—
232C22×C67—48no—
233C22×C67—41no—
234C22×C67—41no—
235C22×C67—41no—
236C22×C67—41no—
237C22×C67—41no—
238C22×C67—41no—
239C22×C67—41no—
240C22×C67—41no—
241C22×C67#38611yescharacteristic
242C22×C67C2×#9911yescharacteristic
243C22×C67C2×#9911yescharacteristic
244C22×C67#41711yescharacteristic
245C22×C67#41911yescharacteristic
246C22×C67C2×#10411yescharacteristic
247C22×C67#42111yescharacteristic
248C22×C67#46812yes—
249C22×C67#47012yes—
250C4×C67—21no—
251C4×C67—22no—
252C4×C67—21no—
253C4×C67—22no—
254C4×C67—41no—
255C4×C67—84no—
256C4×C67—81no—
257C4×C67—81no—
258C4×C67—81no—
259C4×C67—81no—
260#9115C6711yesmaximal, characteristic, direct product factor, Sylow
261C2×C67—22no—
262C2×C67—22no—
263C2×C67—42no—
264C2×C67—42no—
265C2×C67#150411yescharacteristic
266C2×C67#150811yescharacteristic
267C2×C67#151111yescharacteristic
268C2×C67#151311yescharacteristic
269C2×C67#187111yescharacteristic
270C2×C67#189411yescharacteristic
271C2×C67#189811yescharacteristic
272C2×#407C2×C6711yescharacteristic
273#2848C2×C6711yescharacteristic
274C2×#328C2×C6712yescomplemented
275C22⋊#95C2×C6711yescharacteristic
276#1294C2×C6712yescomplemented
277C67#911511yescharacteristic, direct product factor, Sylow
278C24×C4C22×C6711yescharacteristic
279C22×M16—21no—
280C2×C22⋊Q8C4×C6711yescharacteristic, complemented
281C2×C22⋊Q8C4×C6711yescharacteristic, complemented
282C2×C22⋊Q8C22×C6711yescharacteristic
283C22×C22⋊C4—21no—
284#407—24no—
285C2×C22⋊C8—21no—
286C2×C22⋊C8C22×C6712yes—
287#385—21no—
288#385—21no—
289#385—21no—
290#380C4×C6711yescharacteristic, complemented
291#380C4×C6711yescharacteristic, complemented
292#380C22×C6711yescharacteristic
293#379—21no—
294C2×#95—21no—
295C2×#95C22×C6712yes—
296#328—24no—
297C25D4×C6711yescharacteristic
298C22×Q8C2×C4×C6711yescharacteristic
299C23×C4—22no—
300C23×C4—22no—
301C23×C4—22no—
302C23×C4—42no—
303C23×C4—42no—
304C23×C4C2×C4×C6711yescharacteristic
305C23×C4C2×C4×C6711yescharacteristic
306C23×C4D4×C6711yescharacteristic
307C23×C4D4×C6711yescharacteristic
308C23×C4D4×C6711yescharacteristic
309C23×C4C23×C6711yescharacteristic
310C2×M16—24no—
311C2×M16—42no—
312C2×M16—42no—
313C22×C8—22no—
314C22⋊Q8—22no—
315C22⋊Q8—22no—
316C22⋊Q8—42no—
317C22⋊Q8—42no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4C2×C4×C6711yescharacteristic
322C2×C4⋊C4C2×C4×C6711yescharacteristic
323C2×C4⋊C4C2×C4×C6711yescharacteristic
324C2×C22⋊C4—24no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—42no—
332C2×C22⋊C4—42no—
333C22⋊C8—24no—
334C22⋊C8—22no—
335C22⋊C8D4×C6714yes—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—22no—
340#95—22no—
341#95—42no—
342C24—22no—
343C24—21no—
344C24—21no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—41no—
349C24—41no—
350C24C22⋊C4×C6711yescharacteristic
351C24C22⋊C4×C6711yescharacteristic
352C24C2×D4×C6711yescharacteristic
353C2×Q8—22no—
354C2×Q8—41no—
355C2×Q8—41no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4C22⋊C4×C6711yescharacteristic
389C22×C4C22⋊C4×C6711yescharacteristic
390C22×C4C22×C4×C6711yescharacteristic
391C22×C4C2×D4×C6711yescharacteristic
392C22×C4C2×D4×C6712yes—
393C22×C4#41×C6712yes—
394M16—48no—
395C2×C8—24no—
396C2×C8—42no—
397C2×C8—42no—
398C4⋊C4—22no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—42no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—42no—
405C22⋊C4—48no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—42no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—24no—
418C23—24no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23#99×C6712yes—
440C23#104×C6712yes—
441C23C2×C22⋊C4×C6711yescharacteristic
442C23#121×C6712yes—
443C23AGL(1,ℤ/8)×C6711yescharacteristic
444C23#137×C6712yes—
445C23#137×C6711yescharacteristic
446Q8—82no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4#99×C6712yes—
485C2×C4C4×D4×C6712yes—
486C2×C4#120×C6711yescharacteristic
487C2×C4#123×C6711yescharacteristic
488C8—44no—
489C22—21no—
490C22—21no—
491C22—22no—
492C22—22no—
493C22—21no—
494C22—21no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—48no—
504C22—48no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22#386×C6711yescharacteristic
514C22C2×#99×C6711yescharacteristic
515C22C2×#99×C6711yescharacteristic
516C22#417×C6711yescharacteristic
517C22#419×C6711yescharacteristic
518C22C2×#104×C6711yescharacteristic
519C22#421×C6711yescharacteristic
520C22#468×C6712yes—
521C22#470×C6712yes—
522C4—21no—
523C4—21no—
524C4—22no—
525C4—22no—
526C4—41no—
527C4—84no—
528C4—81no—
529C4—81no—
530C4—81no—
531C4—81no—
532C2—22no—
533C2—22no—
534C2—42no—
535C2—42no—
536C2C67×#150411yescharacteristic
537C2C67×#150811yescharacteristic
538C2C67×#151111yescharacteristic
539C2C67×#151311yescharacteristic
540C2C67×#187111yescharacteristic
541C2C67×#189411yescharacteristic
542C2C67×#189811yescharacteristic
subgroupG/H
center, socleC23×C67C2×C22⋊C4
commutatorC22×C4C22×C4×C67
Frattini, cosocleC23×C4C23×C67
prime radicalC25×C67D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C67 ◁ C24×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×#328×C67C2semidirect
5. C67×#1294C2semidirect
8. C2×C22⋊Q8×C67C4semidirect
9. C2×C22⋊Q8×C67C4semidirect
18. #380×C67C4semidirect
19. #380×C67C4semidirect
260. #9115C67direct (N × H)
274. C2×#328C2×C67semidirect
276. #1294C2×C67semidirect
277. C67#9115direct (N × H)
280. C2×C22⋊Q8C4×C67semidirect
281. C2×C22⋊Q8C4×C67semidirect
290. #380C4×C67semidirect
291. #380C4×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#407×C672
C67×#28482
C2×#328×C672
C67×C22⋊#952
C67×#12942
#911567