Finite Group Databasebeta

← C67×#32996 #30712247

Subgroups — C67×#32996 #30712247

Order 17152.

1382 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#520×C67C211yesmaximal, characteristic, complemented
2C67×#3094C211yesmaximal, characteristic, complemented
3C67×#3006C211yesmaximal, characteristic
4C67×#2501C211yesmaximal, characteristic, complemented
5C67×#1883C214yesmaximal, complemented
6C67×#1859C214yesmaximal, complemented
7C67×#1826C211yesmaximal, characteristic, complemented
8C67×#1821C211yesmaximal, characteristic, complemented
9C67×#1789C211yesmaximal, characteristic, complemented
10C22×#41×C67C2211yescharacteristic
11C2×AGL(1,ℤ/8)×C67C2212yes—
12C2×#123×C67—22no—
13C2×#121×C67C2211yescharacteristic
14#520×C67—22no—
15#520×C67—22no—
16#520×C67—22no—
17#520×C67C2214yes—
18C2×C4×Q8×C67—21no—
19C2×C4×D4×C67—21no—
20C2×C4×D4×C67—22no—
21C2×#117×C67—21no—
22C2×#117×C67C2211yescharacteristic
23C22×C4⋊C4×C67—21no—
24C22×C4⋊C4×C67C2211yescharacteristic
25#482×C67—22no—
26#481×C67—21no—
27#480×C67—21no—
28#449×C67—22no—
29#448×C67—21no—
30#447×C67—21no—
31#428×C67C2212yescomplemented
32C2×#104×C67C2214yes—
33#418×C67C2212yescomplemented
34#417×C67C2212yes—
35#407×C67C2211yescharacteristic
36#407×C67C2211yescharacteristic
37#399×C67—22no—
38#390×C67C2212yescomplemented
39#384×C67C2212yescomplemented
40C4×C22⋊C4×C67C2211yescharacteristic
41#343×C67C2214yescomplemented
42#339×C67C2214yescomplemented
43C2×#41×C67—28no—
44C2×#41×C67—21no—
45C2×#41×C67—21no—
46C2×#41×C67D412yes—
47C2×#41×C67C2312yes—
48C22×Q8×C67D411yescharacteristic
49C22×D4×C67—21no—
50C22×D4×C67—21no—
51C22×D4×C67D411yescharacteristic
52C22×D4×C67C2311yescharacteristic
53C23×C4×C67—21no—
54C23×C4×C67—22no—
55C23×C4×C67D412yes—
56C23×C4×C67C2311yescharacteristic
57AGL(1,ℤ/8)×C67—28no—
58C2×SD16×C67—22no—
59C2×D8×C67—22no—
60C2×M16×C67C2312yes—
61C2×M16×C67C2312yes—
62#123×C67—48no—
63#121×C67—21no—
64#121×C67—21no—
65#121×C67—21no—
66#121×C67—21no—
67C4×Q8×C67—24no—
68C4×Q8×C67—24no—
69C4×D4×C67—28no—
70C4×D4×C67—24no—
71C4×D4×C67—24no—
72C4×D4×C67—28no—
73#117×C67—24no—
74#117×C67—21no—
75#117×C67—21no—
76#117×C67—24no—
77#117×C67D412yes—
78#117×C67C2312yes—
79C2×C4⋊C4×C67—22no—
80C2×C4⋊C4×C67—24no—
81C2×C4⋊C4×C67—21no—
82C2×C4⋊C4×C67—21no—
83C2×C4⋊C4×C67—21no—
84C2×C4⋊C4×C67—21no—
85C2×C4⋊C4×C67—24no—
86C2×C4⋊C4×C67—24no—
87C2×C4⋊C4×C67D412yes—
88C2×C4⋊C4×C67D411yescharacteristic
89C2×C4⋊C4×C67D411yescharacteristic
90C2×C4⋊C4×C67C2311yescharacteristic
91C2×C4⋊C4×C67C2312yes—
92C2×C22⋊C4×C67—22no—
93C2×C22⋊C4×C67—21no—
94C2×C22⋊C4×C67—21no—
95C2×C22⋊C4×C67—21no—
96C2×C22⋊C4×C67—24no—
97C2×C42×C67—22no—
98C2×C42×C67—21no—
99C2×C42×C67C2312yes—
100#107×C67—22no—
101#106×C67—22no—
102#104×C67—28no—
103#103×C67—22no—
104#102×C67—22no—
105#102×C67—22no—
106#102×C67—22no—
107C22⋊C8×C67—21no—
108C22⋊C8×C67—21no—
109C22⋊C8×C67—21no—
110C22⋊C8×C67—21no—
111#95×C67—21no—
112C24×C67—21no—
113C24×C67—21no—
114C24×C67C2×D411yescharacteristic
115#41×C67—28no—
116#41×C67—28no—
117#41×C67—24no—
118#41×C67—24no—
119#41×C67—24no—
120#41×C67#4118yes—
121C2×Q8×C67—24no—
122C2×Q8×C67—21no—
123C2×Q8×C67C2×D412yes—
124C2×D4×C67—28no—
125C2×D4×C67—22no—
126C2×D4×C67—22no—
127C2×D4×C67—24no—
128C2×D4×C67—21no—
129C2×D4×C67—21no—
130C2×D4×C67—21no—
131C2×D4×C67—24no—
132C2×D4×C67—44no—
133C2×D4×C67—41no—
134C2×D4×C67—41no—
135C2×D4×C67C2×D412yes—
136C22×C4×C67—28no—
137C22×C4×C67—21no—
138C22×C4×C67—22no—
139C22×C4×C67—22no—
140C22×C4×C67—24no—
141C22×C4×C67—21no—
142C22×C4×C67—22no—
143C22×C4×C67—21no—
144C22×C4×C67—22no—
145C22×C4×C67—28no—
146C22×C4×C67—22no—
147C22×C4×C67—21no—
148C22×C4×C67—44no—
149C22×C4×C67—84no—
150C22×C4×C67C2×D411yescharacteristic
151C22×C4×C67C2×D411yescharacteristic
152C22×C4×C67C2×D411yescharacteristic
153C22×C4×C67C2×D411yescharacteristic
154C22×C4×C67C2×D411yescharacteristic
155C22×C4×C67C2×D414yes—
156C22×C4×C67C2411yescharacteristic
157SD16×C67—44no—
158D8×C67—44no—
159M16×C67—24no—
160M16×C67—24no—
161C2×C8×C67—22no—
162C2×C8×C67—22no—
163C4⋊C4×C67—28no—
164C4⋊C4×C67—22no—
165C4⋊C4×C67—21no—
166C4⋊C4×C67—22no—
167C4⋊C4×C67—21no—
168C4⋊C4×C67—22no—
169C4⋊C4×C67—22no—
170C4⋊C4×C67—21no—
171C4⋊C4×C67—21no—
172C4⋊C4×C67—48no—
173C4⋊C4×C67C2×D412yes—
174C4⋊C4×C67C2×D412yes—
175C22⋊C4×C67—28no—
176C22⋊C4×C67—24no—
177C22⋊C4×C67—22no—
178C22⋊C4×C67—22no—
179C22⋊C4×C67—48no—
180C22⋊C4×C67—41no—
181C22⋊C4×C67—41no—
182C42×C67—28no—
183C42×C67—22no—
184C42×C67—22no—
185C42×C67—24no—
186C23×C67—24no—
187C23×C67—22no—
188C23×C67—22no—
189C23×C67—22no—
190C23×C67—41no—
191C23×C67—42no—
192C23×C67—42no—
193C23×C67—42no—
194C23×C67—42no—
195C23×C67—41no—
196C23×C67—42no—
197C23×C67#12011yescharacteristic
198C23×C67#12011yescharacteristic
199C23×C67C22×D411yescharacteristic
200Q8×C67—24no—
201Q8×C67—22no—
202Q8×C67#12314yescomplemented
203D4×C67—28no—
204D4×C67—24no—
205D4×C67—28no—
206D4×C67—22no—
207D4×C67—42no—
208D4×C67—42no—
209D4×C67—82no—
210D4×C67—82no—
211D4×C67#12314yescomplemented
212C2×C4×C67—24no—
213C2×C4×C67—28no—
214C2×C4×C67—22no—
215C2×C4×C67—22no—
216C2×C4×C67—22no—
217C2×C4×C67—22no—
218C2×C4×C67—22no—
219C2×C4×C67—28no—
220C2×C4×C67—22no—
221C2×C4×C67—24no—
222C2×C4×C67—22no—
223C2×C4×C67—24no—
224C2×C4×C67—22no—
225C2×C4×C67—42no—
226C2×C4×C67—42no—
227C2×C4×C67—42no—
228C2×C4×C67—44no—
229C2×C4×C67—42no—
230C2×C4×C67—44no—
231C2×C4×C67—42no—
232C2×C4×C67—41no—
233C2×C4×C67—82no—
234C2×C4×C67—88no—
235C2×C4×C67#12011yescharacteristic
236C2×C4×C67#12011yescharacteristic
237C2×C4×C67C22×D411yescharacteristic
238C2×C4×C67C22×D411yescharacteristic
239C2×C4×C67C2×#4114yes—
240C8×C67—42no—
241C8×C67—42no—
242C22×C67—21no—
243C22×C67—21no—
244C22×C67—21no—
245C22×C67—21no—
246C22×C67—21no—
247C22×C67—24no—
248C22×C67—21no—
249C22×C67—21no—
250C22×C67—44no—
251C22×C67—48no—
252C22×C67—42no—
253C22×C67—42no—
254C22×C67—84no—
255C22×C67—81no—
256C22×C67—81no—
257C22×C67—81no—
258C22×C67#45312yes—
259C22×C67C2×#12011yescharacteristic
260C22×C67#52512yes—
261C4×C67—21no—
262C4×C67—24no—
263C4×C67—21no—
264C4×C67—44no—
265C4×C67—44no—
266C4×C67—42no—
267C4×C67—42no—
268C4×C67—81no—
269C4×C67C2×#12312yes—
270C4×C67#52512yes—
271#32996C6711yesmaximal, characteristic, direct product factor, Sylow
272C2×C67—22no—
273C2×C67—22no—
274C2×C67—41no—
275C2×C67—44no—
276C2×C67—161no—
277C2×C67#284911yescharacteristic
278C2×C67C2×#45311yescharacteristic
279C2×C67#302011yescharacteristic
280C2×#520C2×C6711yescharacteristic, complemented
281#3094C2×C6711yescharacteristic, complemented
282#3006C2×C6711yescharacteristic
283#2501C2×C6711yescharacteristic, complemented
284#1883C2×C6714yescomplemented
285#1859C2×C6714yescomplemented
286#1826C2×C6711yescharacteristic, complemented
287#1821C2×C6711yescharacteristic, complemented
288#1789C2×C6711yescharacteristic, complemented
289C67#3299611yescharacteristic, direct product factor, Sylow
290C22×#41C22×C6711yescharacteristic
291C2×AGL(1,ℤ/8)C22×C6712yes—
292C2×#123—22no—
293C2×#121C22×C6711yescharacteristic
294#520—22no—
295#520—22no—
296#520—22no—
297#520C22×C6714yes—
298C2×C4×Q8—21no—
299C2×C4×D4—21no—
300C2×C4×D4—22no—
301C2×#117—21no—
302C2×#117C22×C6711yescharacteristic
303C22×C4⋊C4—21no—
304C22×C4⋊C4C22×C6711yescharacteristic
305#482—22no—
306#481—21no—
307#480—21no—
308#449—22no—
309#448—21no—
310#447—21no—
311#428C22×C6712yescomplemented
312C2×#104C22×C6714yes—
313#418C22×C6712yescomplemented
314#417C22×C6712yes—
315#407C22×C6711yescharacteristic
316#407C22×C6711yescharacteristic
317#399—22no—
318#390C22×C6712yescomplemented
319#384C22×C6712yescomplemented
320C4×C22⋊C4C22×C6711yescharacteristic
321#343C22×C6714yescomplemented
322#339C22×C6714yescomplemented
323C2×#41—28no—
324C2×#41—21no—
325C2×#41—21no—
326C2×#41D4×C6712yes—
327C2×#41C23×C6712yes—
328C22×Q8D4×C6711yescharacteristic
329C22×D4—21no—
330C22×D4—21no—
331C22×D4D4×C6711yescharacteristic
332C22×D4C23×C6711yescharacteristic
333C23×C4—21no—
334C23×C4—22no—
335C23×C4D4×C6712yes—
336C23×C4C23×C6711yescharacteristic
337AGL(1,ℤ/8)—28no—
338C2×SD16—22no—
339C2×D8—22no—
340C2×M16C23×C6712yes—
341C2×M16C23×C6712yes—
342#123—48no—
343#121—21no—
344#121—21no—
345#121—21no—
346#121—21no—
347C4×Q8—24no—
348C4×Q8—24no—
349C4×D4—28no—
350C4×D4—24no—
351C4×D4—24no—
352C4×D4—28no—
353#117—24no—
354#117—21no—
355#117—21no—
356#117—24no—
357#117D4×C6712yes—
358#117C23×C6712yes—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—24no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—24no—
366C2×C4⋊C4—24no—
367C2×C4⋊C4D4×C6712yes—
368C2×C4⋊C4D4×C6711yescharacteristic
369C2×C4⋊C4D4×C6711yescharacteristic
370C2×C4⋊C4C23×C6711yescharacteristic
371C2×C4⋊C4C23×C6712yes—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—24no—
377C2×C42—22no—
378C2×C42—21no—
379C2×C42C23×C6712yes—
380#107—22no—
381#106—22no—
382#104—28no—
383#103—22no—
384#102—22no—
385#102—22no—
386#102—22no—
387C22⋊C8—21no—
388C22⋊C8—21no—
389C22⋊C8—21no—
390C22⋊C8—21no—
391#95—21no—
392C24—21no—
393C24—21no—
394C24C2×D4×C6711yescharacteristic
395#41—28no—
396#41—28no—
397#41—24no—
398#41—24no—
399#41—24no—
400#41#41×C6718yes—
401C2×Q8—24no—
402C2×Q8—21no—
403C2×Q8C2×D4×C6712yes—
404C2×D4—28no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—24no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—24no—
412C2×D4—44no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4C2×D4×C6712yes—
416C22×C4—28no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—28no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—44no—
429C22×C4—84no—
430C22×C4C2×D4×C6711yescharacteristic
431C22×C4C2×D4×C6711yescharacteristic
432C22×C4C2×D4×C6711yescharacteristic
433C22×C4C2×D4×C6711yescharacteristic
434C22×C4C2×D4×C6711yescharacteristic
435C22×C4C2×D4×C6714yes—
436C22×C4C24×C6711yescharacteristic
437SD16—44no—
438D8—44no—
439M16—24no—
440M16—24no—
441C2×C8—22no—
442C2×C8—22no—
443C4⋊C4—28no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—48no—
453C4⋊C4C2×D4×C6712yes—
454C4⋊C4C2×D4×C6712yes—
455C22⋊C4—28no—
456C22⋊C4—24no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—48no—
460C22⋊C4—41no—
461C22⋊C4—41no—
462C42—28no—
463C42—22no—
464C42—22no—
465C42—24no—
466C23—24no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—41no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—41no—
476C23—42no—
477C23#120×C6711yescharacteristic
478C23#120×C6711yescharacteristic
479C23C22×D4×C6711yescharacteristic
480Q8—24no—
481Q8—22no—
482Q8#123×C6714yescomplemented
483D4—28no—
484D4—24no—
485D4—28no—
486D4—22no—
487D4—42no—
488D4—42no—
489D4—82no—
490D4—82no—
491D4#123×C6714yescomplemented
492C2×C4—24no—
493C2×C4—28no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—28no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—82no—
514C2×C4—88no—
515C2×C4#120×C6711yescharacteristic
516C2×C4#120×C6711yescharacteristic
517C2×C4C22×D4×C6711yescharacteristic
518C2×C4C22×D4×C6711yescharacteristic
519C2×C4C2×#41×C6714yes—
520C8—42no—
521C8—42no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—24no—
530C22—44no—
531C22—48no—
532C22—42no—
533C22—42no—
534C22—84no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22#453×C6712yes—
539C22C2×#120×C6711yescharacteristic
540C22#525×C6712yes—
541C4—21no—
542C4—24no—
543C4—21no—
544C4—44no—
545C4—44no—
546C4—42no—
547C4—42no—
548C4—81no—
549C4C2×#123×C6712yes—
550C4#525×C6712yes—
551C2—22no—
552C2—22no—
553C2—41no—
554C2—44no—
555C2—161no—
556C2C67×#284911yescharacteristic
557C2C2×#453×C6711yescharacteristic
558C2C67×#302011yescharacteristic
subgroupG/H
center, socleC22×C67C2×#120
commutator, Frattini, cosocleC22×C4C24×C67
prime radicalC67×#3006C2
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
1. C2×#520×C67C2semidirect
2. C67×#3094C2semidirect
4. C67×#2501C2semidirect
5. C67×#1883C2semidirect
6. C67×#1859C2semidirect
7. C67×#1826C2semidirect
8. C67×#1821C2semidirect
9. C67×#1789C2semidirect
31. #428×C67C22semidirect
33. #418×C67C22semidirect
38. #390×C67C22semidirect
39. #384×C67C22semidirect
41. #343×C67C22semidirect
42. #339×C67C22semidirect
202. Q8×C67#123semidirect
211. D4×C67#123semidirect
271. #32996C67direct (N × H)
280. C2×#520C2×C67semidirect
281. #3094C2×C67semidirect
283. #2501C2×C67semidirect
284. #1883C2×C67semidirect
285. #1859C2×C67semidirect
286. #1826C2×C67semidirect
287. #1821C2×C67semidirect
288. #1789C2×C67semidirect
289. C67#32996direct (N × H)
311. #428C22×C67semidirect
313. #418C22×C67semidirect
318. #390C22×C67semidirect
319. #384C22×C67semidirect
321. #343C22×C67semidirect
322. #339C22×C67semidirect
482. Q8#123×C67semidirect
491. D4#123×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#520×C672
C67×#30942
C67×#30062
C67×#25012
C67×#18832
C67×#18592
C67×#18262
C67×#18212
C67×#17892
#3299667