Finite Group Databasebeta

← C67×#18865 #30698116

Subgroups — C67×#18865 #30698116

Order 17152.

922 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3506C211yesmaximal, characteristic
2C67×#3254C211yesmaximal, characteristic, complemented
3C67×#3231C211yesmaximal, characteristic, complemented
4C67×#3110C211yesmaximal, characteristic, complemented
5C67×#1654C212yesmaximal, complemented
6C67×#1647C212yesmaximal, complemented
7C67×#1646C212yesmaximal, complemented
8C67×#1646C212yesmaximal, complemented
9C67×#1559C211yesmaximal, characteristic, complemented
10C67×#1496C211yesmaximal, characteristic, complemented
11C67×#1486C211yesmaximal, characteristic, complemented
12#559×C67—22no—
13#556×C67—21no—
14#555×C67C2212yes—
15#553×C67—22no—
16#550×C67—21no—
17#547×C67C2212yes—
18#538×C67—22no—
19#535×C67—21no—
20#533×C67C2211yescharacteristic
21#533×C67C2211yescharacteristic
22#532×C67C2211yescharacteristic
23#525×C67—22no—
24#520×C67—21no—
25C2×C4×D4×C67C2211yescharacteristic
26#500×C67C2211yescharacteristic, complemented
27#498×C67C2211yescharacteristic, complemented
28#487×C67C2211yescharacteristic, complemented
29#487×C67C2211yescharacteristic, complemented
30#485×C67—22no—
31#484×C67—21no—
32#479×C67—22no—
33#477×C67—21no—
34#474×C67C2212yes—
35#474×C67C2212yescomplemented
36#468×C67—22no—
37#461×C67C2212yes—
38#461×C67C2212yescomplemented
39#460×C67—21no—
40#450×C67—21no—
41#440×C67—22no—
42#432×C67C2211yescharacteristic
43#431×C67C2211yescharacteristic
44#423×C67C2211yescharacteristic
45#333×C67C2212yescomplemented
46#332×C67C2212yescomplemented
47D4⋊C8×C67C2212yes—
48D4⋊C8×C67C2212yes—
49D4⋊C8×C67C2212yescomplemented
50D4⋊C8×C67C2212yescomplemented
51C2×#41×C67—22no—
52C2×#41×C67—21no—
53C22×D4×C67D411yescharacteristic
54C23×C4×C67—21no—
55C2×SD16×C67—22no—
56C2×SD16×C67—22no—
57C2×M16×C67—21no—
58C4⋊Q8×C67D411yescharacteristic
59C4⋊Q8×C67C2311yescharacteristic
60C4⋊Q8×C67C2311yescharacteristic
61C4⋊Q8×C67C2311yescharacteristic
62C4⋊Q8×C67C2311yescharacteristic
63#127×C67—22no—
64#127×C67D411yescharacteristic
65#126×C67—24no—
66#125×C67—21no—
67#125×C67—21no—
68#124×C67—22no—
69#124×C67—24no—
70#124×C67—21no—
71#123×C67—24no—
72#123×C67—24no—
73C22⋊Q8×C67—21no—
74C22⋊Q8×C67—22no—
75C22⋊Q8×C67—21no—
76C22⋊Q8×C67—21no—
77C22⋊Q8×C67D412yes—
78#121×C67—24no—
79#121×C67—22no—
80#121×C67—22no—
81#121×C67—22no—
82#121×C67—21no—
83#121×C67D412yes—
84#120×C67—24no—
85C4×Q8×C67—22no—
86C4×Q8×C67—21no—
87C4×Q8×C67—21no—
88C4×Q8×C67—21no—
89C4×Q8×C67D412yes—
90C4×D4×C67—24no—
91C4×D4×C67—22no—
92C4×D4×C67—24no—
93C4×D4×C67—21no—
94C4×D4×C67—21no—
95C4×D4×C67—21no—
96C4×D4×C67D412yes—
97C4×D4×C67C2312yes—
98C4×D4×C67C2312yes—
99#117×C67—21no—
100#117×C67—22no—
101#117×C67—22no—
102#117×C67—21no—
103C2×C4⋊C4×C67—22no—
104C2×C4⋊C4×C67D411yescharacteristic
105C2×C22⋊C4×C67—21no—
106C2×C42×C67C2311yescharacteristic
107#107×C67—22no—
108#107×C67—22no—
109#107×C67—22no—
110#106×C67—22no—
111C4⋊C8×C67—21no—
112C4⋊C8×C67—21no—
113C4⋊C8×C67C2312yes—
114C4⋊C8×C67C2312yes—
115#103×C67—22no—
116#103×C67—22no—
117#103×C67—22no—
118#103×C67—22no—
119#102×C67—22no—
120#102×C67—22no—
121C22⋊C8×C67—21no—
122C22⋊C8×C67—21no—
123#97×C67—21no—
124C4×C8×C67C2311yescharacteristic
125C4×C8×C67C2311yescharacteristic
126C24×C67—21no—
127#41×C67—44no—
128#41×C67—82no—
129#41×C67—82no—
130C2×Q8×C67—21no—
131C2×Q8×C67—21no—
132C2×Q8×C67—21no—
133C2×Q8×C67—21no—
134C2×Q8×C67C2×D412yes—
135C2×D4×C67—22no—
136C2×D4×C67—24no—
137C2×D4×C67—24no—
138C2×D4×C67—21no—
139C2×D4×C67—21no—
140C2×D4×C67—21no—
141C2×D4×C67—42no—
142C2×D4×C67C2×D412yes—
143C2×D4×C67C2×D412yes—
144C22×C4×C67—24no—
145C22×C4×C67—21no—
146C22×C4×C67—22no—
147C22×C4×C67—22no—
148C22×C4×C67—22no—
149C22×C4×C67—81no—
150C22×C4×C67—81no—
151C22×C4×C67C2×D411yescharacteristic
152C22×C4×C67C2×D411yescharacteristic
153C22×C4×C67C2×D411yescharacteristic
154SD16×C67—44no—
155SD16×C67—82no—
156M16×C67—81no—
157C2×C8×C67—22no—
158C2×C8×C67—22no—
159C2×C8×C67—21no—
160C2×C8×C67—21no—
161C4⋊C4×C67—24no—
162C4⋊C4×C67—21no—
163C4⋊C4×C67—22no—
164C4⋊C4×C67—21no—
165C4⋊C4×C67—22no—
166C4⋊C4×C67—21no—
167C4⋊C4×C67—21no—
168C4⋊C4×C67—21no—
169C4⋊C4×C67—21no—
170C4⋊C4×C67—21no—
171C4⋊C4×C67—21no—
172C4⋊C4×C67—21no—
173C4⋊C4×C67—41no—
174C4⋊C4×C67—41no—
175C4⋊C4×C67C2×D412yes—
176C4⋊C4×C67C2×D412yes—
177C4⋊C4×C67C2×D412yes—
178C22⋊C4×C67—24no—
179C22⋊C4×C67—24no—
180C22⋊C4×C67—24no—
181C22⋊C4×C67—22no—
182C22⋊C4×C67—22no—
183C22⋊C4×C67—22no—
184C22⋊C4×C67—22no—
185C22⋊C4×C67—22no—
186C22⋊C4×C67—41no—
187C42×C67—24no—
188C42×C67—22no—
189C42×C67—41no—
190C42×C67C2×D411yescharacteristic
191C42×C67C2×D411yescharacteristic
192C42×C67C2×D411yescharacteristic
193C42×C67C2411yescharacteristic
194C23×C67—22no—
195C23×C67—22no—
196C23×C67—42no—
197C23×C67—81no—
198C23×C67—81no—
199C23×C67#12011yescharacteristic
200Q8×C67—42no—
201Q8×C67—42no—
202Q8×C67—41no—
203Q8×C67—41no—
204Q8×C67—42no—
205Q8×C67—41no—
206Q8×C67—41no—
207D4×C67—22no—
208D4×C67—22no—
209D4×C67—22no—
210D4×C67—22no—
211D4×C67—42no—
212D4×C67—42no—
213D4×C67—44no—
214D4×C67—81no—
215D4×C67—84no—
216D4×C67—84no—
217C2×C4×C67—22no—
218C2×C4×C67—21no—
219C2×C4×C67—21no—
220C2×C4×C67—22no—
221C2×C4×C67—22no—
222C2×C4×C67—22no—
223C2×C4×C67—22no—
224C2×C4×C67—41no—
225C2×C4×C67—44no—
226C2×C4×C67—42no—
227C2×C4×C67—42no—
228C2×C4×C67—41no—
229C2×C4×C67—41no—
230C2×C4×C67—82no—
231C2×C4×C67—81no—
232C2×C4×C67—84no—
233C2×C4×C67#12011yescharacteristic
234C2×C4×C67#12011yescharacteristic
235C2×C4×C67#12011yescharacteristic
236C2×C4×C67C22×D411yescharacteristic
237C2×C4×C67C22×D411yescharacteristic
238C2×C4×C67C22×D411yescharacteristic
239C8×C67—41no—
240C8×C67—41no—
241C8×C67—42no—
242C8×C67—42no—
243C22×C67—21no—
244C22×C67—21no—
245C22×C67—21no—
246C22×C67—42no—
247C22×C67—42no—
248C22×C67—42no—
249C22×C67—42no—
250C22×C67—42no—
251C22×C67—42no—
252C22×C67—84no—
253C22×C67C2×#12011yescharacteristic
254C4×C67—21no—
255C4×C67—21no—
256C4×C67—21no—
257C4×C67—21no—
258C4×C67—42no—
259C4×C67—42no—
260C4×C67—42no—
261C4×C67—42no—
262C4×C67—81no—
263C4×C67—81no—
264C4×C67#57512yes—
265C4×C67#57611yescharacteristic
266C4×C67#57711yescharacteristic
267#18865C6711yesmaximal, characteristic, direct product factor, Sylow
268C2×C67—41no—
269C2×C67—82no—
270C2×C67—82no—
271C2×C67#300511yescharacteristic
272C2×C67#301211yescharacteristic
273C2×C67#302211yescharacteristic
274#3506C2×C6711yescharacteristic
275#3254C2×C6711yescharacteristic, complemented
276#3231C2×C6711yescharacteristic, complemented
277#3110C2×C6711yescharacteristic, complemented
278#1654C2×C6712yescomplemented
279#1647C2×C6712yescomplemented
280#1646C2×C6712yescomplemented
281#1646C2×C6712yescomplemented
282#1559C2×C6711yescharacteristic, complemented
283#1496C2×C6711yescharacteristic, complemented
284#1486C2×C6711yescharacteristic, complemented
285C67#1886511yescharacteristic, direct product factor, Sylow
286#559—22no—
287#556—21no—
288#555C22×C6712yes—
289#553—22no—
290#550—21no—
291#547C22×C6712yes—
292#538—22no—
293#535—21no—
294#533C22×C6711yescharacteristic
295#533C22×C6711yescharacteristic
296#532C22×C6711yescharacteristic
297#525—22no—
298#520—21no—
299C2×C4×D4C22×C6711yescharacteristic
300#500C22×C6711yescharacteristic, complemented
301#498C22×C6711yescharacteristic, complemented
302#487C22×C6711yescharacteristic, complemented
303#487C22×C6711yescharacteristic, complemented
304#485—22no—
305#484—21no—
306#479—22no—
307#477—21no—
308#474C22×C6712yes—
309#474C22×C6712yescomplemented
310#468—22no—
311#461C22×C6712yes—
312#461C22×C6712yescomplemented
313#460—21no—
314#450—21no—
315#440—22no—
316#432C22×C6711yescharacteristic
317#431C22×C6711yescharacteristic
318#423C22×C6711yescharacteristic
319#333C22×C6712yescomplemented
320#332C22×C6712yescomplemented
321D4⋊C8C22×C6712yes—
322D4⋊C8C22×C6712yes—
323D4⋊C8C22×C6712yescomplemented
324D4⋊C8C22×C6712yescomplemented
325C2×#41—22no—
326C2×#41—21no—
327C22×D4D4×C6711yescharacteristic
328C23×C4—21no—
329C2×SD16—22no—
330C2×SD16—22no—
331C2×M16—21no—
332C4⋊Q8D4×C6711yescharacteristic
333C4⋊Q8C23×C6711yescharacteristic
334C4⋊Q8C23×C6711yescharacteristic
335C4⋊Q8C23×C6711yescharacteristic
336C4⋊Q8C23×C6711yescharacteristic
337#127—22no—
338#127D4×C6711yescharacteristic
339#126—24no—
340#125—21no—
341#125—21no—
342#124—22no—
343#124—24no—
344#124—21no—
345#123—24no—
346#123—24no—
347C22⋊Q8—21no—
348C22⋊Q8—22no—
349C22⋊Q8—21no—
350C22⋊Q8—21no—
351C22⋊Q8D4×C6712yes—
352#121—24no—
353#121—22no—
354#121—22no—
355#121—22no—
356#121—21no—
357#121D4×C6712yes—
358#120—24no—
359C4×Q8—22no—
360C4×Q8—21no—
361C4×Q8—21no—
362C4×Q8—21no—
363C4×Q8D4×C6712yes—
364C4×D4—24no—
365C4×D4—22no—
366C4×D4—24no—
367C4×D4—21no—
368C4×D4—21no—
369C4×D4—21no—
370C4×D4D4×C6712yes—
371C4×D4C23×C6712yes—
372C4×D4C23×C6712yes—
373#117—21no—
374#117—22no—
375#117—22no—
376#117—21no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4D4×C6711yescharacteristic
379C2×C22⋊C4—21no—
380C2×C42C23×C6711yescharacteristic
381#107—22no—
382#107—22no—
383#107—22no—
384#106—22no—
385C4⋊C8—21no—
386C4⋊C8—21no—
387C4⋊C8C23×C6712yes—
388C4⋊C8C23×C6712yes—
389#103—22no—
390#103—22no—
391#103—22no—
392#103—22no—
393#102—22no—
394#102—22no—
395C22⋊C8—21no—
396C22⋊C8—21no—
397#97—21no—
398C4×C8C23×C6711yescharacteristic
399C4×C8C23×C6711yescharacteristic
400C24—21no—
401#41—44no—
402#41—82no—
403#41—82no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8C2×D4×C6712yes—
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—42no—
416C2×D4C2×D4×C6712yes—
417C2×D4C2×D4×C6712yes—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—81no—
424C22×C4—81no—
425C22×C4C2×D4×C6711yescharacteristic
426C22×C4C2×D4×C6711yescharacteristic
427C22×C4C2×D4×C6711yescharacteristic
428SD16—44no—
429SD16—82no—
430M16—81no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—22no—
434C2×C8—22no—
435C4⋊C4—24no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4C2×D4×C6712yes—
450C4⋊C4C2×D4×C6712yes—
451C4⋊C4C2×D4×C6712yes—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—41no—
461C42—24no—
462C42—22no—
463C42—41no—
464C42C2×D4×C6711yescharacteristic
465C42C2×D4×C6711yescharacteristic
466C42C2×D4×C6711yescharacteristic
467C42C24×C6711yescharacteristic
468C23—22no—
469C23—22no—
470C23—42no—
471C23—81no—
472C23—81no—
473C23#120×C6711yescharacteristic
474Q8—42no—
475Q8—42no—
476Q8—41no—
477Q8—41no—
478Q8—42no—
479Q8—41no—
480Q8—41no—
481D4—22no—
482D4—22no—
483D4—22no—
484D4—22no—
485D4—42no—
486D4—42no—
487D4—44no—
488D4—81no—
489D4—84no—
490D4—84no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—41no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—82no—
505C2×C4—81no—
506C2×C4—84no—
507C2×C4#120×C6711yescharacteristic
508C2×C4#120×C6711yescharacteristic
509C2×C4#120×C6711yescharacteristic
510C2×C4C22×D4×C6711yescharacteristic
511C2×C4C22×D4×C6711yescharacteristic
512C2×C4C22×D4×C6711yescharacteristic
513C8—41no—
514C8—41no—
515C8—42no—
516C8—42no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—84no—
527C22C2×#120×C6711yescharacteristic
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—42no—
533C4—42no—
534C4—42no—
535C4—42no—
536C4—81no—
537C4—81no—
538C4#575×C6712yes—
539C4#576×C6711yescharacteristic
540C4#577×C6711yescharacteristic
541C2—41no—
542C2—82no—
543C2—82no—
544C2C67×#300511yescharacteristic
545C2C67×#301211yescharacteristic
546C2C67×#302211yescharacteristic
subgroupG/H
center, socleC22×C67C2×#120
commutator, Frattini, cosocleC42C24×C67
prime radicalC67×#3506C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C67 ◁ C2×C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#3254C2semidirect
3. C67×#3231C2semidirect
4. C67×#3110C2semidirect
5. C67×#1654C2semidirect
6. C67×#1647C2semidirect
7. C67×#1646C2semidirect
8. C67×#1646C2semidirect
9. C67×#1559C2semidirect
10. C67×#1496C2semidirect
11. C67×#1486C2semidirect
26. #500×C67C22semidirect
27. #498×C67C22semidirect
28. #487×C67C22semidirect
29. #487×C67C22semidirect
35. #474×C67C22semidirect
38. #461×C67C22semidirect
45. #333×C67C22semidirect
46. #332×C67C22semidirect
49. D4⋊C8×C67C22semidirect
50. D4⋊C8×C67C22semidirect
267. #18865C67direct (N × H)
275. #3254C2×C67semidirect
276. #3231C2×C67semidirect
277. #3110C2×C67semidirect
278. #1654C2×C67semidirect
279. #1647C2×C67semidirect
280. #1646C2×C67semidirect
281. #1646C2×C67semidirect
282. #1559C2×C67semidirect
283. #1496C2×C67semidirect
284. #1486C2×C67semidirect
285. C67#18865direct (N × H)
300. #500C22×C67semidirect
301. #498C22×C67semidirect
302. #487C22×C67semidirect
303. #487C22×C67semidirect
309. #474C22×C67semidirect
312. #461C22×C67semidirect
319. #333C22×C67semidirect
320. #332C22×C67semidirect
323. D4⋊C8C22×C67semidirect
324. D4⋊C8C22×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#35062
C67×#32542
C67×#32312
C67×#31102
C67×#16542
C67×#16472
C67×#16462
C67×#15592
C67×#14962
C67×#14862
#1886567