Finite Group Databasebeta

← C67×#61720 #30740971

Subgroups — C67×#61720 #30740971

Order 17152.

2078 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3496C211yesmaximal, characteristic, complemented
2C2×#555×C67C211yesmaximal, characteristic, complemented
3C67×#3354C218yesmaximal, complemented
4C67×#3325C218yesmaximal, complemented
5C67×#3227C212yesmaximal, complemented
6C67×#3192C214yesmaximal, complemented
7C67×#3100C212yesmaximal, complemented
8C67×#3094C214yesmaximal, complemented
9C67×#2975C211yesmaximal, characteristic, complemented
10C2×H(5,𝔽2)×C67—21no—
11#555×C67—24no—
12#555×C67C2218yescomplemented
13#547×C67C2214yescomplemented
14#546×C67—24no—
15D42×C67C2214yescomplemented
16#537×C67—21no—
17#534×C67—22no—
18#532×C67C2212yes—
19C2×C4⋊Q8×C67C2211yescharacteristic, complemented
20C2×#124×C67—22no—
21C2×#121×C67—21no—
22C2×#121×C67C2214yes—
23C2×#121×C67C2212yescomplemented
24C2×C4×D4×C67C2211yescharacteristic
25C2×C4×D4×C67C2211yescharacteristic, complemented
26C2×#117×C67—21no—
27C2×#117×C67C2212yescomplemented
28#498×C67C2214yescomplemented
29#486×C67C2214yescomplemented
30#482×C67C22116yescomplemented
31#481×C67C2218yescomplemented
32#481×C67C2218yescomplemented
33#480×C67C2214yescomplemented
34#476×C67C2218yescomplemented
35#474×C67C2214yescomplemented
36#469×C67C22116yescomplemented
37#459×C67C2214yescomplemented
38#447×C67C2218yescomplemented
39#447×C67C2214yescomplemented
40C4×D8×C67C2218yescomplemented
41#436×C67C2218yescomplemented
42#431×C67C2212yescomplemented
43#428×C67C2218yescomplemented
44#418×C67C2218yescomplemented
45#407×C67C2214yescomplemented
46H(5,𝔽2)×C67—48no—
47C2×#41×C67—24no—
48C2×#41×C67—22no—
49C22×Q8×C67—21no—
50C22×D4×C67—24no—
51C22×D4×C67—22no—
52C22×D4×C67—24no—
53C22×D4×C67D411yescharacteristic
54C22×D4×C67C2311yescharacteristic
55C22×D4×C67C2314yes—
56C23×C4×C67—21no—
57C23×C4×C67C2312yes—
58C2×D8×C67C2318yes—
59C2×M16×C67C2318yes—
60C4⋊Q8×C67—22no—
61C4⋊Q8×C67C2312yes—
62C4⋊Q8×C67C2314yes—
63#127×C67C2312yes—
64#124×C67—216no—
65#124×C67—22no—
66#123×C67—28no—
67C22⋊Q8×C67—24no—
68#121×C67—28no—
69#121×C67—24no—
70#121×C67—24no—
71#121×C67—24no—
72#121×C67—22no—
73#121×C67—24no—
74#121×C67C2318yes—
75#121×C67C2318yes—
76#121×C67C2314yes—
77#120×C67—28no—
78C4×D4×C67—22no—
79C4×D4×C67—22no—
80C4×D4×C67D414yes—
81C4×D4×C67C2314yes—
82C4×D4×C67C2314yes—
83C4×D4×C67C2314yes—
84#117×C67—24no—
85#117×C67—28no—
86#117×C67C2318yes—
87C2×C4⋊C4×C67—21no—
88C2×C4⋊C4×C67D411yescharacteristic
89C2×C4⋊C4×C67C2311yescharacteristic
90C2×C4⋊C4×C67C2312yes—
91C2×C4⋊C4×C67C2314yes—
92C2×C22⋊C4×C67—22no—
93C2×C22⋊C4×C67—22no—
94C2×C22⋊C4×C67—21no—
95C2×C22⋊C4×C67—24no—
96C2×C22⋊C4×C67D412yes—
97C2×C42×C67—22no—
98C2×C42×C67C2311yescharacteristic
99#107×C67C2318yes—
100#106×C67C23116yes—
101C4⋊C8×C67C2314yes—
102#102×C67C2318yes—
103#102×C67C2318yes—
104#102×C67C23116yes—
105C22⋊C8×C67C2314yes—
106C22⋊C8×C67C2318yes—
107C4×C8×C67C2314yes—
108C24×C67—21no—
109C24×C67—24no—
110C24×C67C2×D412yes—
111#41×C67—416no—
112#41×C67—48no—
113C2×Q8×C67—22no—
114C2×Q8×C67—24no—
115C2×Q8×C67—82no—
116C2×D4×C67—28no—
117C2×D4×C67—24no—
118C2×D4×C67—28no—
119C2×D4×C67—24no—
120C2×D4×C67—21no—
121C2×D4×C67—22no—
122C2×D4×C67—22no—
123C2×D4×C67—24no—
124C2×D4×C67—42no—
125C2×D4×C67—48no—
126C2×D4×C67—416no—
127C2×D4×C67—416no—
128C2×D4×C67—84no—
129C2×D4×C67C2×D414yes—
130C2×D4×C67C2414yes—
131C2×D4×C67C2412yes—
132C22×C4×C67—21no—
133C22×C4×C67—22no—
134C22×C4×C67—24no—
135C22×C4×C67—24no—
136C22×C4×C67—24no—
137C22×C4×C67—44no—
138C22×C4×C67—44no—
139C22×C4×C67C2×D412yes—
140C22×C4×C67C2×D411yescharacteristic
141C22×C4×C67C2×D414yes—
142C22×C4×C67C2×D411yescharacteristic
143C22×C4×C67C2411yescharacteristic
144C22×C4×C67C2414yes—
145D8×C67—48no—
146M16×C67—48no—
147C2×C8×C67C2418yes—
148C4⋊C4×C67—24no—
149C4⋊C4×C67—21no—
150C4⋊C4×C67—22no—
151C4⋊C4×C67—24no—
152C4⋊C4×C67—22no—
153C4⋊C4×C67C2×D414yes—
154C4⋊C4×C67C2414yes—
155C4⋊C4×C67C2412yes—
156C4⋊C4×C67C2414yes—
157C22⋊C4×C67—28no—
158C22⋊C4×C67—24no—
159C22⋊C4×C67—28no—
160C22⋊C4×C67—24no—
161C22⋊C4×C67—28no—
162C22⋊C4×C67—24no—
163C22⋊C4×C67C2×D418yes—
164C42×C67—28no—
165C42×C67C2×D412yes—
166C42×C67C2412yes—
167C23×C67—24no—
168C23×C67—22no—
169C23×C67—24no—
170C23×C67—42no—
171C23×C67—42no—
172C23×C67—42no—
173C23×C67—42no—
174C23×C67—48no—
175C23×C67—42no—
176C23×C67—84no—
177C23×C67—84no—
178C23×C67C22×D414yes—
179C23×C67C22×D411yescharacteristic
180Q8×C67—44no—
181Q8×C67—84no—
182D4×C67—28no—
183D4×C67—24no—
184D4×C67—28no—
185D4×C67—42no—
186D4×C67—416no—
187D4×C67—48no—
188D4×C67—44no—
189D4×C67—82no—
190D4×C67—84no—
191D4×C67—88no—
192C2×C4×C67—28no—
193C2×C4×C67—22no—
194C2×C4×C67—22no—
195C2×C4×C67—24no—
196C2×C4×C67—24no—
197C2×C4×C67—44no—
198C2×C4×C67—42no—
199C2×C4×C67—41no—
200C2×C4×C67—48no—
201C2×C4×C67—44no—
202C2×C4×C67—42no—
203C2×C4×C67—48no—
204C2×C4×C67—48no—
205C2×C4×C67AGL(1,ℤ/8)14yes—
206C2×C4×C67C22×D414yes—
207C2×C4×C67C22×D411yescharacteristic
208C2×C4×C67C22×D412yes—
209C2×C4×C67C22×D412yes—
210C2×C4×C67C2511yescharacteristic
211C8×C67—28no—
212C22×C67—21no—
213C22×C67—21no—
214C22×C67—24no—
215C22×C67—24no—
216C22×C67—24no—
217C22×C67—42no—
218C22×C67—44no—
219C22×C67—42no—
220C22×C67—44no—
221C22×C67—42no—
222C22×C67—44no—
223C22×C67—84no—
224C22×C67—84no—
225C22×C67—168no—
226C22×C67C2×AGL(1,ℤ/8)12yes—
227C22×C67C23×D411yescharacteristic
228C4×C67—22no—
229C4×C67—21no—
230C4×C67—24no—
231C4×C67—44no—
232C4×C67—42no—
233C4×C67—44no—
234C4×C67C2×AGL(1,ℤ/8)14yes—
235C4×C67#58512yes—
236#61720C6711yesmaximal, characteristic, direct product factor, Sylow
237C2×C67—22no—
238C2×C67—44no—
239C2×C67—82no—
240C2×C67—84no—
241C2×C67#348511yescharacteristic
242C2×C67C22×AGL(1,ℤ/8)11yescharacteristic
243C2×C67#358511yescharacteristic
244#3496C2×C6711yescharacteristic, complemented
245C2×#555C2×C6711yescharacteristic, complemented
246#3354C2×C6718yescomplemented
247#3325C2×C6718yescomplemented
248#3227C2×C6712yescomplemented
249#3192C2×C6714yescomplemented
250#3100C2×C6712yescomplemented
251#3094C2×C6714yescomplemented
252#2975C2×C6711yescharacteristic, complemented
253C67#6172011yescharacteristic, direct product factor, Sylow
254C2×H(5,𝔽2)—21no—
255#555—24no—
256#555C22×C6718yescomplemented
257#547C22×C6714yescomplemented
258#546—24no—
259D42C22×C6714yescomplemented
260#537—21no—
261#534—22no—
262#532C22×C6712yes—
263C2×C4⋊Q8C22×C6711yescharacteristic, complemented
264C2×#124—22no—
265C2×#121—21no—
266C2×#121C22×C6714yes—
267C2×#121C22×C6712yescomplemented
268C2×C4×D4C22×C6711yescharacteristic
269C2×C4×D4C22×C6711yescharacteristic, complemented
270C2×#117—21no—
271C2×#117C22×C6712yescomplemented
272#498C22×C6714yescomplemented
273#486C22×C6714yescomplemented
274#482C22×C67116yescomplemented
275#481C22×C6718yescomplemented
276#481C22×C6718yescomplemented
277#480C22×C6714yescomplemented
278#476C22×C6718yescomplemented
279#474C22×C6714yescomplemented
280#469C22×C67116yescomplemented
281#459C22×C6714yescomplemented
282#447C22×C6714yescomplemented
283#447C22×C6718yescomplemented
284C4×D8C22×C6718yescomplemented
285#436C22×C6718yescomplemented
286#431C22×C6712yescomplemented
287#428C22×C6718yescomplemented
288#418C22×C6718yescomplemented
289#407C22×C6714yescomplemented
290H(5,𝔽2)—48no—
291C2×#41—24no—
292C2×#41—22no—
293C22×Q8—21no—
294C22×D4—24no—
295C22×D4—22no—
296C22×D4—24no—
297C22×D4D4×C6711yescharacteristic
298C22×D4C23×C6711yescharacteristic
299C22×D4C23×C6714yes—
300C23×C4—21no—
301C23×C4C23×C6712yes—
302C2×D8C23×C6718yes—
303C2×M16C23×C6718yes—
304C4⋊Q8—22no—
305C4⋊Q8C23×C6714yes—
306C4⋊Q8C23×C6712yes—
307#127C23×C6712yes—
308#124—216no—
309#124—22no—
310#123—28no—
311C22⋊Q8—24no—
312#121—28no—
313#121—24no—
314#121—24no—
315#121—24no—
316#121—22no—
317#121—24no—
318#121C23×C6714yes—
319#121C23×C6718yes—
320#121C23×C6718yes—
321#120—28no—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4D4×C6714yes—
325C4×D4C23×C6714yes—
326C4×D4C23×C6714yes—
327C4×D4C23×C6714yes—
328#117—24no—
329#117—28no—
330#117C23×C6718yes—
331C2×C4⋊C4—21no—
332C2×C4⋊C4D4×C6711yescharacteristic
333C2×C4⋊C4C23×C6711yescharacteristic
334C2×C4⋊C4C23×C6712yes—
335C2×C4⋊C4C23×C6714yes—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4D4×C6712yes—
341C2×C42—22no—
342C2×C42C23×C6711yescharacteristic
343#107C23×C6718yes—
344#106C23×C67116yes—
345C4⋊C8C23×C6714yes—
346#102C23×C6718yes—
347#102C23×C67116yes—
348#102C23×C6718yes—
349C22⋊C8C23×C6714yes—
350C22⋊C8C23×C6718yes—
351C4×C8C23×C6714yes—
352C24—21no—
353C24—24no—
354C24C2×D4×C6712yes—
355#41—416no—
356#41—48no—
357C2×Q8—22no—
358C2×Q8—24no—
359C2×Q8—82no—
360C2×D4—28no—
361C2×D4—24no—
362C2×D4—28no—
363C2×D4—24no—
364C2×D4—21no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—24no—
368C2×D4—42no—
369C2×D4—48no—
370C2×D4—416no—
371C2×D4—416no—
372C2×D4—84no—
373C2×D4C2×D4×C6714yes—
374C2×D4C24×C6714yes—
375C2×D4C24×C6712yes—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4C2×D4×C6712yes—
384C22×C4C2×D4×C6711yescharacteristic
385C22×C4C2×D4×C6714yes—
386C22×C4C2×D4×C6711yescharacteristic
387C22×C4C24×C6711yescharacteristic
388C22×C4C24×C6714yes—
389D8—48no—
390M16—48no—
391C2×C8C24×C6718yes—
392C4⋊C4—24no—
393C4⋊C4—21no—
394C4⋊C4—22no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4C2×D4×C6714yes—
398C4⋊C4C24×C6714yes—
399C4⋊C4C24×C6712yes—
400C4⋊C4C24×C6714yes—
401C22⋊C4—28no—
402C22⋊C4—24no—
403C22⋊C4—28no—
404C22⋊C4—24no—
405C22⋊C4—28no—
406C22⋊C4—24no—
407C22⋊C4C2×D4×C6718yes—
408C42—28no—
409C42C2×D4×C6712yes—
410C42C24×C6712yes—
411C23—24no—
412C23—22no—
413C23—24no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—48no—
419C23—42no—
420C23—84no—
421C23—84no—
422C23C22×D4×C6714yes—
423C23C22×D4×C6711yescharacteristic
424Q8—44no—
425Q8—84no—
426D4—28no—
427D4—24no—
428D4—28no—
429D4—42no—
430D4—416no—
431D4—48no—
432D4—44no—
433D4—84no—
434D4—82no—
435D4—88no—
436C2×C4—28no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—48no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—48no—
448C2×C4—48no—
449C2×C4AGL(1,ℤ/8)×C6714yes—
450C2×C4C22×D4×C6714yes—
451C2×C4C22×D4×C6711yescharacteristic
452C2×C4C22×D4×C6712yes—
453C2×C4C22×D4×C6712yes—
454C2×C4C25×C6711yescharacteristic
455C8—28no—
456C22—21no—
457C22—21no—
458C22—24no—
459C22—24no—
460C22—24no—
461C22—42no—
462C22—44no—
463C22—42no—
464C22—44no—
465C22—42no—
466C22—44no—
467C22—84no—
468C22—84no—
469C22—168no—
470C22C2×AGL(1,ℤ/8)×C6712yes—
471C22C23×D4×C6711yescharacteristic
472C4—22no—
473C4—21no—
474C4—24no—
475C4—44no—
476C4—42no—
477C4—44no—
478C4C2×AGL(1,ℤ/8)×C6714yes—
479C4#585×C6712yes—
480C2—22no—
481C2—44no—
482C2—82no—
483C2—84no—
484C2C67×#348511yescharacteristic
485C2C22×AGL(1,ℤ/8)×C6711yescharacteristic
486C2C67×#358511yescharacteristic
subgroupG/H
center, socleC22×C67C23×D4
commutator, Frattini, cosocleC2×C4C25×C67
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C2×C4×D4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. C67×#3496C2semidirect
2. C2×#555×C67C2semidirect
3. C67×#3354C2semidirect
4. C67×#3325C2semidirect
5. C67×#3227C2semidirect
6. C67×#3192C2semidirect
7. C67×#3100C2semidirect
8. C67×#3094C2semidirect
9. C67×#2975C2semidirect
12. #555×C67C22semidirect
13. #547×C67C22semidirect
15. D42×C67C22semidirect
19. C2×C4⋊Q8×C67C22semidirect
23. C2×#121×C67C22semidirect
25. C2×C4×D4×C67C22semidirect
27. C2×#117×C67C22semidirect
28. #498×C67C22semidirect
29. #486×C67C22semidirect
30. #482×C67C22semidirect
31. #481×C67C22semidirect
32. #481×C67C22semidirect
33. #480×C67C22semidirect
34. #476×C67C22semidirect
35. #474×C67C22semidirect
36. #469×C67C22semidirect
37. #459×C67C22semidirect
38. #447×C67C22semidirect
39. #447×C67C22semidirect
40. C4×D8×C67C22semidirect
41. #436×C67C22semidirect
42. #431×C67C22semidirect
43. #428×C67C22semidirect
44. #418×C67C22semidirect
45. #407×C67C22semidirect
236. #61720C67direct (N × H)
244. #3496C2×C67semidirect
245. C2×#555C2×C67semidirect
246. #3354C2×C67semidirect
247. #3325C2×C67semidirect
248. #3227C2×C67semidirect
249. #3192C2×C67semidirect
250. #3100C2×C67semidirect
251. #3094C2×C67semidirect
252. #2975C2×C67semidirect
253. C67#61720direct (N × H)
256. #555C22×C67semidirect
257. #547C22×C67semidirect
259. D42C22×C67semidirect
263. C2×C4⋊Q8C22×C67semidirect
267. C2×#121C22×C67semidirect
269. C2×C4×D4C22×C67semidirect
271. C2×#117C22×C67semidirect
272. #498C22×C67semidirect
273. #486C22×C67semidirect
274. #482C22×C67semidirect
275. #481C22×C67semidirect
276. #481C22×C67semidirect
277. #480C22×C67semidirect
278. #476C22×C67semidirect
279. #474C22×C67semidirect
280. #469C22×C67semidirect
281. #459C22×C67semidirect
282. #447C22×C67semidirect
283. #447C22×C67semidirect
284. C4×D8C22×C67semidirect
285. #436C22×C67semidirect
286. #431C22×C67semidirect
287. #428C22×C67semidirect
288. #418C22×C67semidirect
289. #407C22×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#34962
C2×#555×C672
C67×#33542
C67×#33252
C67×#32272
C67×#31922
C67×#31002
C67×#30942
C67×#29752
#6172067