Finite Group Databasebeta

← C67×#22081 #30701332

Subgroups — C67×#22081 #30701332

Order 17152.

968 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#480×C67C211yesmaximal, characteristic
2C4×#121×C67C211yesmaximal, characteristic
3C2×#389×C67C211yesmaximal, characteristic, complemented
4C67×#1943C212yesmaximal, complemented
5C67×#1936C212yesmaximal, complemented
6C67×#1921C212yesmaximal, complemented
7C67×#1874C211yesmaximal, characteristic
8C67×#1864C212yesmaximal, complemented
9C67×#1853C211yesmaximal, characteristic, complemented
10C67×#1823C211yesmaximal, characteristic, complemented
11C67×#1749C211yesmaximal, characteristic, complemented
12C2×#121×C67C2211yescharacteristic
13C2×C4×D4×C67—21no—
14C2×C4×D4×C67C2211yescharacteristic
15C22×C4⋊C4×C67—21no—
16C22×C4⋊C4×C67C2211yescharacteristic
17C22×C4⋊C4×C67C2211yescharacteristic
18C22×C42×C67C2211yescharacteristic
19#480×C67C418yescomplemented
20C2×#107×C67C2212yes—
21C2×#106×C67C2212yes—
22C2×#102×C67C2212yes—
23C2×#102×C67C2212yes—
24C2×C22⋊C8×C67C2211yescharacteristic
25C2×C22⋊C8×C67C2211yescharacteristic
26C2×#97×C67C2212yes—
27#390×C67C2211yescharacteristic
28#389×C67—21no—
29#389×C67—21no—
30#389×C67C2212yes—
31#389×C67C2212yes—
32#389×C67C2211yescharacteristic
33#388×C67—21no—
34#386×C67—21no—
35#380×C67C2211yescharacteristic
36C4×C4⋊C4×C67C2211yescharacteristic
37C4×C22⋊C4×C67—21no—
38C2×#95×C67—21no—
39#340×C67C2212yes—
40#340×C67C2212yes—
41#340×C67C2212yes—
42#340×C67C2212yes—
43#336×C67C2212yes—
44C22×D4×C67—21no—
45C22×D4×C67C2311yescharacteristic
46C23×C4×C67—21no—
47C23×C4×C67—21no—
48C23×C4×C67—21no—
49C23×C4×C67D411yescharacteristic
50C23×C4×C67D411yescharacteristic
51C23×C4×C67C2311yescharacteristic
52C22×C8×C67C2312yes—
53C22×C8×C67C2312yes—
54#121×C67—22no—
55#121×C67C2×C414yes—
56C4×D4×C67—42no—
57C4×D4×C67—44no—
58C2×C4⋊C4×C67—21no—
59C2×C4⋊C4×C67—22no—
60C2×C4⋊C4×C67—22no—
61C2×C4⋊C4×C67—22no—
62C2×C4⋊C4×C67—21no—
63C2×C4⋊C4×C67—21no—
64C2×C4⋊C4×C67—42no—
65C2×C4⋊C4×C67—44no—
66C2×C4⋊C4×C67C2×C414yescomplemented
67C2×C4⋊C4×C67C2312yes—
68C2×C4⋊C4×C67C2311yescharacteristic
69C2×C4⋊C4×C67C2311yescharacteristic
70C2×C4⋊C4×C67C2312yes—
71C2×C4⋊C4×C67C2311yescharacteristic
72C2×C22⋊C4×C67—21no—
73C2×C22⋊C4×C67—21no—
74C2×C22⋊C4×C67—21no—
75C2×C22⋊C4×C67—21no—
76C2×C22⋊C4×C67—21no—
77C2×C42×C67—21no—
78C2×C42×C67—42no—
79C2×C42×C67D411yescharacteristic
80C2×C42×C67D411yescharacteristic
81C2×C42×C67C2311yescharacteristic
82C2×C42×C67C2311yescharacteristic
83#107×C67C2×C418yescomplemented
84#106×C67—42no—
85#102×C67—42no—
86#102×C67C2×C418yescomplemented
87C22⋊C8×C67—22no—
88C22⋊C8×C67C2×C414yescomplemented
89#97×C67—42no—
90#95×C67—22no—
91#95×C67—22no—
92#95×C67—21no—
93#95×C67—21no—
94#95×C67—21no—
95C24×C67—21no—
96C24×C67C2×D411yescharacteristic
97C2×D4×C67—24no—
98C2×D4×C67—21no—
99C2×D4×C67—22no—
100C2×D4×C67—22no—
101C2×D4×C67—42no—
102C2×D4×C67—42no—
103C2×D4×C67C22×C412yes—
104C22×C4×C67—22no—
105C22×C4×C67—22no—
106C22×C4×C67—24no—
107C22×C4×C67—22no—
108C22×C4×C67—22no—
109C22×C4×C67—21no—
110C22×C4×C67—22no—
111C22×C4×C67—22no—
112C22×C4×C67—21no—
113C22×C4×C67—21no—
114C22×C4×C67—22no—
115C22×C4×C67—41no—
116C22×C4×C67—41no—
117C22×C4×C67—41no—
118C22×C4×C67—42no—
119C22×C4×C67—42no—
120C22×C4×C67—42no—
121C22×C4×C67—42no—
122C22×C4×C67—42no—
123C22×C4×C67—42no—
124C22×C4×C67—42no—
125C22×C4×C67—42no—
126C22×C4×C67C22×C412yes—
127C22×C4×C67C2×D411yescharacteristic
128C22×C4×C67C2×D411yescharacteristic
129C22×C4×C67C2×D411yescharacteristic
130C22×C4×C67C2×D411yescharacteristic
131C22×C4×C67C2×D411yescharacteristic
132C22×C4×C67#4112yes—
133C22×C4×C67C2411yescharacteristic
134C2×C8×C67—22no—
135C2×C8×C67—22no—
136C2×C8×C67—22no—
137C2×C8×C67—22no—
138C2×C8×C67—22no—
139C2×C8×C67C22×C414yes—
140C4⋊C4×C67—24no—
141C4⋊C4×C67—22no—
142C4⋊C4×C67—21no—
143C4⋊C4×C67—22no—
144C4⋊C4×C67—22no—
145C4⋊C4×C67—22no—
146C4⋊C4×C67—22no—
147C4⋊C4×C67—42no—
148C4⋊C4×C67—44no—
149C4⋊C4×C67—44no—
150C4⋊C4×C67—42no—
151C4⋊C4×C67C22×C414yes—
152C4⋊C4×C67C22×C412yes—
153C22⋊C4×C67—24no—
154C22⋊C4×C67—42no—
155C22⋊C4×C67—42no—
156C22⋊C4×C67—42no—
157C22⋊C4×C67—42no—
158C42×C67—22no—
159C42×C67—22no—
160C42×C67—41no—
161C42×C67—41no—
162C42×C67—42no—
163C23×C67—21no—
164C23×C67—21no—
165C23×C67—21no—
166C23×C67—21no—
167C23×C67—22no—
168C23×C67—41no—
169C23×C67—41no—
170C23×C67—41no—
171C23×C67—41no—
172C23×C67—41no—
173C23×C67—41no—
174C23×C67C4×D412yes—
175C23×C67AGL(1,ℤ/8)12yes—
176C23×C67AGL(1,ℤ/8)12yes—
177C23×C67C22×D411yescharacteristic
178D4×C67—24no—
179D4×C67—48no—
180D4×C67—42no—
181C2×C4×C67—21no—
182C2×C4×C67—21no—
183C2×C4×C67—21no—
184C2×C4×C67—21no—
185C2×C4×C67—24no—
186C2×C4×C67—24no—
187C2×C4×C67—22no—
188C2×C4×C67—24no—
189C2×C4×C67—21no—
190C2×C4×C67—21no—
191C2×C4×C67—21no—
192C2×C4×C67—21no—
193C2×C4×C67—21no—
194C2×C4×C67—21no—
195C2×C4×C67—21no—
196C2×C4×C67—21no—
197C2×C4×C67—21no—
198C2×C4×C67—44no—
199C2×C4×C67—44no—
200C2×C4×C67—41no—
201C2×C4×C67—42no—
202C2×C4×C67—42no—
203C2×C4×C67—41no—
204C2×C4×C67—48no—
205C2×C4×C67—41no—
206C2×C4×C67—42no—
207C2×C4×C67—41no—
208C2×C4×C67—41no—
209C2×C4×C67—41no—
210C2×C4×C67—42no—
211C2×C4×C67—41no—
212C2×C4×C67—41no—
213C2×C4×C67—42no—
214C2×C4×C67—84no—
215C2×C4×C67—84no—
216C2×C4×C67C4×D412yes—
217C2×C4×C67#13712yes—
218C2×C4×C67C23×C411yescharacteristic
219C2×C4×C67C2×#4111yescharacteristic
220C2×C4×C67H(5,𝔽2)12yes—
221C2×C4×C672−1+412yes—
222C8×C67—24no—
223C8×C67—42no—
224C22×C67—22no—
225C22×C67—22no—
226C22×C67—22no—
227C22×C67—22no—
228C22×C67—22no—
229C22×C67—22no—
230C22×C67—42no—
231C22×C67—42no—
232C22×C67—42no—
233C22×C67—81no—
234C22×C67—81no—
235C22×C67—81no—
236C22×C67—81no—
237C22×C67#44214yes—
238C22×C67C2×C4×D411yescharacteristic
239C22×C67#53411yescharacteristic
240C22×C67#53611yescharacteristic
241C22×C67C2×AGL(1,ℤ/8)11yescharacteristic
242C22×C67C2×AGL(1,ℤ/8)11yescharacteristic
243C22×C67C2×#13711yescharacteristic
244C22×C67#57511yescharacteristic
245C4×C67—22no—
246C4×C67—22no—
247C4×C67—41no—
248C4×C67—41no—
249C4×C67—41no—
250C4×C67—44no—
251C4×C67—42no—
252C4×C67—81no—
253C4×C67—82no—
254C4×C67—81no—
255C4×C67#52012yes—
256C4×C67#53912yes—
257#22081C6711yesmaximal, characteristic, direct product factor, Sylow
258C2×C67—24no—
259C2×C67—82no—
260C2×C67#232211yescharacteristic
261C2×C67C2×#44211yescharacteristic
262C2×C67C4×#13711yescharacteristic
263C2×C67#319311yescharacteristic
264C2×C67#319411yescharacteristic
265C2×C67#323411yescharacteristic
266C2×C67#323811yescharacteristic
267C2×#480C2×C6711yescharacteristic
268C4×#121C2×C6711yescharacteristic
269C2×#389C2×C6711yescharacteristic, complemented
270#1943C2×C6712yescomplemented
271#1936C2×C6712yescomplemented
272#1921C2×C6712yescomplemented
273#1874C2×C6711yescharacteristic
274#1864C2×C6712yescomplemented
275#1853C2×C6711yescharacteristic, complemented
276#1823C2×C6711yescharacteristic, complemented
277#1749C2×C6711yescharacteristic, complemented
278C67#2208111yescharacteristic, direct product factor, Sylow
279C2×#121C22×C6711yescharacteristic
280C2×C4×D4—21no—
281C2×C4×D4C22×C6711yescharacteristic
282C22×C4⋊C4—21no—
283C22×C4⋊C4C22×C6711yescharacteristic
284C22×C4⋊C4C22×C6711yescharacteristic
285C22×C42C22×C6711yescharacteristic
286#480C4×C6718yescomplemented
287C2×#107C22×C6712yes—
288C2×#106C22×C6712yes—
289C2×#102C22×C6712yes—
290C2×#102C22×C6712yes—
291C2×C22⋊C8C22×C6711yescharacteristic
292C2×C22⋊C8C22×C6711yescharacteristic
293C2×#97C22×C6712yes—
294#390C22×C6711yescharacteristic
295#389—21no—
296#389—21no—
297#389C22×C6712yes—
298#389C22×C6712yes—
299#389C22×C6711yescharacteristic
300#388—21no—
301#386—21no—
302#380C22×C6711yescharacteristic
303C4×C4⋊C4C22×C6711yescharacteristic
304C4×C22⋊C4—21no—
305C2×#95—21no—
306#340C22×C6712yes—
307#340C22×C6712yes—
308#340C22×C6712yes—
309#340C22×C6712yes—
310#336C22×C6712yes—
311C22×D4—21no—
312C22×D4C23×C6711yescharacteristic
313C23×C4—21no—
314C23×C4—21no—
315C23×C4—21no—
316C23×C4D4×C6711yescharacteristic
317C23×C4D4×C6711yescharacteristic
318C23×C4C23×C6711yescharacteristic
319C22×C8C23×C6712yes—
320C22×C8C23×C6712yes—
321#121—22no—
322#121C2×C4×C6714yes—
323C4×D4—42no—
324C4×D4—44no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—42no—
332C2×C4⋊C4—44no—
333C2×C4⋊C4C2×C4×C6714yescomplemented
334C2×C4⋊C4C23×C6712yes—
335C2×C4⋊C4C23×C6711yescharacteristic
336C2×C4⋊C4C23×C6711yescharacteristic
337C2×C4⋊C4C23×C6712yes—
338C2×C4⋊C4C23×C6711yescharacteristic
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C42—21no—
345C2×C42—42no—
346C2×C42D4×C6711yescharacteristic
347C2×C42D4×C6711yescharacteristic
348C2×C42C23×C6711yescharacteristic
349C2×C42C23×C6711yescharacteristic
350#107C2×C4×C6718yescomplemented
351#106—42no—
352#102—42no—
353#102C2×C4×C6718yescomplemented
354C22⋊C8—22no—
355C22⋊C8C2×C4×C6714yescomplemented
356#97—42no—
357#95—22no—
358#95—22no—
359#95—21no—
360#95—21no—
361#95—21no—
362C24—21no—
363C24C2×D4×C6711yescharacteristic
364C2×D4—24no—
365C2×D4—21no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4C22×C4×C6712yes—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4C22×C4×C6712yes—
394C22×C4C2×D4×C6711yescharacteristic
395C22×C4C2×D4×C6711yescharacteristic
396C22×C4C2×D4×C6711yescharacteristic
397C22×C4C2×D4×C6711yescharacteristic
398C22×C4C2×D4×C6711yescharacteristic
399C22×C4#41×C6712yes—
400C22×C4C24×C6711yescharacteristic
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8C22×C4×C6714yes—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—21no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—42no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—42no—
418C4⋊C4C22×C4×C6714yes—
419C4⋊C4C22×C4×C6712yes—
420C22⋊C4—24no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C42—22no—
426C42—22no—
427C42—41no—
428C42—41no—
429C42—42no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—22no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23C4×D4×C6712yes—
442C23AGL(1,ℤ/8)×C6712yes—
443C23AGL(1,ℤ/8)×C6712yes—
444C23C22×D4×C6711yescharacteristic
445D4—24no—
446D4—48no—
447D4—42no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—48no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—84no—
482C2×C4—84no—
483C2×C4C4×D4×C6712yes—
484C2×C4#137×C6712yes—
485C2×C4C23×C4×C6711yescharacteristic
486C2×C4C2×#41×C6711yescharacteristic
487C2×C4H(5,𝔽2)×C6712yes—
488C2×C42−1+4×C6712yes—
489C8—24no—
490C8—42no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22#442×C6714yes—
505C22C2×C4×D4×C6711yescharacteristic
506C22#534×C6711yescharacteristic
507C22#536×C6711yescharacteristic
508C22C2×AGL(1,ℤ/8)×C6711yescharacteristic
509C22C2×AGL(1,ℤ/8)×C6711yescharacteristic
510C22C2×#137×C6711yescharacteristic
511C22#575×C6711yescharacteristic
512C4—22no—
513C4—22no—
514C4—41no—
515C4—41no—
516C4—41no—
517C4—44no—
518C4—42no—
519C4—81no—
520C4—82no—
521C4—81no—
522C4#520×C6712yes—
523C4#539×C6712yes—
524C2—24no—
525C2—82no—
526C2C67×#232211yescharacteristic
527C2C2×#442×C6711yescharacteristic
528C2C4×#137×C6711yescharacteristic
529C2C67×#319311yescharacteristic
530C2C67×#319411yescharacteristic
531C2C67×#323411yescharacteristic
532C2C67×#323811yescharacteristic
subgroupG/H
center, socleC23×C67C22×D4
commutatorC2×C4C23×C4×C67
Frattini, cosocleC22×C4C24×C67
prime radicalC2×#121×C67C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C67 ◁ C22×C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×#389×C67C2semidirect
4. C67×#1943C2semidirect
5. C67×#1936C2semidirect
6. C67×#1921C2semidirect
8. C67×#1864C2semidirect
9. C67×#1853C2semidirect
10. C67×#1823C2semidirect
11. C67×#1749C2semidirect
19. #480×C67C4semidirect
66. C2×C4⋊C4×C67C2×C4semidirect
83. #107×C67C2×C4semidirect
86. #102×C67C2×C4semidirect
88. C22⋊C8×C67C2×C4semidirect
257. #22081C67direct (N × H)
269. C2×#389C2×C67semidirect
270. #1943C2×C67semidirect
271. #1936C2×C67semidirect
272. #1921C2×C67semidirect
274. #1864C2×C67semidirect
275. #1853C2×C67semidirect
276. #1823C2×C67semidirect
277. #1749C2×C67semidirect
278. C67#22081direct (N × H)
286. #480C4×C67semidirect
333. C2×C4⋊C4C2×C4×C67semidirect
350. #107C2×C4×C67semidirect
353. #102C2×C4×C67semidirect
355. C22⋊C8C2×C4×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#480×C672
C4×#121×C672
C2×#389×C672
C67×#19432
C67×#19362
C67×#19212
C67×#18742
C67×#18642
C67×#18532
C67×#18232
C67×#17492
#2208167