Finite Group Databasebeta

← C67×#60238 #30739489

Subgroups — C67×#60238 #30739489

Order 17152.

2090 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3449C211yesmaximal, characteristic
2C67×#2978C214yesmaximal, complemented
3C67×#2977C214yesmaximal, complemented
4C67×#2976C218yesmaximal, complemented
5C67×#2935C214yesmaximal, complemented
6C67×#2932C212yesmaximal, complemented
7C67×#2931C212yesmaximal, complemented
8C67×#2921C211yesmaximal, characteristic, complemented
9C2×#433×C67C211yesmaximal, characteristic, complemented
10C67×#2880C214yesmaximal, complemented
11C22×#41×C67C2212yes—
12C2×#131×C67C2214yescomplemented
13C2×#131×C67C2214yescomplemented
14#537×C67C414yes—
15#536×C67C414yes—
16#535×C67C414yes—
17#534×C67C414yes—
18#532×C67C2212yes—
19#532×C67C2212yes—
20#525×C67C2214yes—
21#525×C67C2214yes—
22C2×#117×C67C2211yescharacteristic
23#436×C67C2214yescomplemented
24#436×C67C2214yescomplemented
25#436×C67C2218yescomplemented
26#435×C67C2218yescomplemented
27#435×C67C2218yescomplemented
28#435×C67C2218yescomplemented
29#435×C67C2218yescomplemented
30C8×D4×C67C2214yescomplemented
31C8×D4×C67C2214yescomplemented
32C8×D4×C67C2218yescomplemented
33#433×C67C2214yescomplemented
34#433×C67C2212yescomplemented
35#433×C67C2212yescomplemented
36#433×C67C2214yescomplemented
37#433×C67C2214yescomplemented
38#423×C67C2212yescomplemented
39C2×C4⋊C8×C67C2212yescomplemented
40#408×C67C2218yes—
41#408×C67C2218yes—
42#408×C67C2214yescomplemented
43#408×C67C2214yescomplemented
44#407×C67C2212yes—
45#407×C67C2218yescomplemented
46C2×C22⋊C8×C67C2212yes—
47C2×C22⋊C8×C67C2218yescomplemented
48C4×M16×C67C2212yescomplemented
49C2×#97×C67C2211yescharacteristic, complemented
50C2×C4×C8×C67C2211yescharacteristic, complemented
51C2×#41×C67—48no—
52C2×#41×C67C2×C414yes—
53C2×#41×C67C2×C414yes—
54C2×#41×C67C2314yes—
55C2×#41×C67C2314yes—
56C22×Q8×C67C2×C412yes—
57C22×D4×C67C2×C414yes—
58C22×D4×C67C2×C412yes—
59C23×C4×C67C2311yescharacteristic
60C23×C4×C67C2314yes—
61#131×C67—48no—
62#131×C67—48no—
63C2×M16×C67C2314yes—
64C2×M16×C67C2318yes—
65C2×M16×C67C2318yes—
66C22×C8×C67C2312yes—
67C22×C8×C67C2312yes—
68C22×C8×C67C2318yes—
69C22×C8×C67C2318yes—
70C4⋊Q8×C67C2×C412yes—
71C4⋊Q8×C67C2×C412yes—
72#127×C67C2×C412yes—
73#127×C67C2×C412yes—
74#124×C67C2×C414yes—
75#124×C67C2×C414yes—
76#123×C67C2×C418yes—
77#123×C67C2×C418yes—
78C22⋊Q8×C67C2×C418yes—
79C22⋊Q8×C67C2×C418yes—
80#121×C67C2×C418yes—
81#121×C67C2×C418yes—
82#121×C67C2×C414yes—
83#121×C67C2×C414yes—
84#121×C67C2×C418yes—
85#120×C67C2×C418yes—
86#120×C67C2×C418yes—
87C4×D4×C67C2314yes—
88C4×D4×C67C2318yes—
89C4×D4×C67C2314yes—
90#117×C67C2×C414yes—
91#117×C67C2312yes—
92#117×C67C2312yes—
93C2×C4⋊C4×C67C2×C412yes—
94C2×C22⋊C4×C67C2×C412yes—
95C2×C42×C67C2311yescharacteristic
96C2×C42×C67C2311yescharacteristic
97C4⋊C8×C67C2314yes—
98C4⋊C8×C67C2314yes—
99C4⋊C8×C67C2314yes—
100C4⋊C8×C67C2314yes—
101C22⋊C8×C67C2314yes—
102C22⋊C8×C67C2314yes—
103C22⋊C8×C67C2318yes—
104C22⋊C8×C67C2318yes—
105C22⋊C8×C67C2318yes—
106C22⋊C8×C67C2318yes—
107C22⋊C8×C67C2318yes—
108#97×C67C2312yes—
109#97×C67C2312yes—
110#97×C67C2314yes—
111C4×C8×C67C2311yescharacteristic
112C4×C8×C67C2312yes—
113C4×C8×C67C2311yescharacteristic
114C4×C8×C67C2314yes—
115C24×C67C22×C411yescharacteristic
116C24×C67C22×C414yes—
117#41×C67—28no—
118#41×C67—28no—
119#41×C67—48no—
120#41×C67—48no—
121#41×C67—48no—
122C2×Q8×C67—44no—
123C2×Q8×C67C22×C414yes—
124C2×Q8×C67C22×C414yes—
125C2×D4×C67—44no—
126C2×D4×C67—48no—
127C2×D4×C67C22×C418yes—
128C2×D4×C67C22×C414yes—
129C2×D4×C67C22×C414yes—
130C2×D4×C67C22×C418yes—
131C2×D4×C67C22×C414yes—
132C2×D4×C67C22×C418yes—
133C2×D4×C67C22×C414yes—
134C22×C4×C67—24no—
135C22×C4×C67—48no—
136C22×C4×C67C22×C411yescharacteristic
137C22×C4×C67C22×C412yes—
138C22×C4×C67C22×C412yes—
139C22×C4×C67C22×C412yes—
140C22×C4×C67C22×C418yes—
141C22×C4×C67C22×C414yes—
142C22×C4×C67C2412yes—
143C22×C4×C67C2418yes—
144C22×C4×C67C2411yescharacteristic
145M16×C67—28no—
146M16×C67—48no—
147M16×C67—48no—
148C2×C8×C67—24no—
149C2×C8×C67—24no—
150C2×C8×C67—48no—
151C2×C8×C67—48no—
152C2×C8×C67C2412yes—
153C2×C8×C67C2414yes—
154C2×C8×C67C2418yes—
155C2×C8×C67C2412yes—
156C4⋊C4×C67C22×C414yes—
157C4⋊C4×C67C22×C414yes—
158C4⋊C4×C67C22×C414yes—
159C4⋊C4×C67C22×C418yes—
160C4⋊C4×C67C22×C414yes—
161C22⋊C4×C67C22×C414yes—
162C22⋊C4×C67C22×C414yes—
163C22⋊C4×C67C22×C418yes—
164C22⋊C4×C67C22×C418yes—
165C22⋊C4×C67C22×C418yes—
166C22⋊C4×C67C22×C418yes—
167C42×C67C22×C412yes—
168C42×C67C22×C412yes—
169C42×C67C2412yes—
170C42×C67C2411yescharacteristic
171C42×C67C2411yescharacteristic
172C23×C67—22no—
173C23×C67—44no—
174C23×C67—42no—
175C23×C67—48no—
176C23×C67C23×C412yes—
177C23×C67C23×C418yes—
178C23×C67C23×C411yescharacteristic
179Q8×C67—28no—
180Q8×C67—44no—
181D4×C67—216no—
182D4×C67—28no—
183D4×C67—44no—
184D4×C67—48no—
185D4×C67—44no—
186D4×C67—48no—
187D4×C67—44no—
188C2×C4×C67—22no—
189C2×C4×C67—24no—
190C2×C4×C67—28no—
191C2×C4×C67—28no—
192C2×C4×C67—24no—
193C2×C4×C67—44no—
194C2×C4×C67—48no—
195C2×C4×C67—42no—
196C2×C4×C67—42no—
197C2×C4×C67C23×C412yes—
198C2×C4×C67C23×C412yes—
199C2×C4×C67C23×C414yes—
200C2×C4×C67C23×C418yes—
201C2×C4×C67C23×C412yes—
202C2×C4×C67C23×C411yescharacteristic
203C2×C4×C67H(5,𝔽2)14yes—
204C2×C4×C67C2511yescharacteristic
205C8×C67—22no—
206C8×C67—24no—
207C8×C67—22no—
208C8×C67—28no—
209C22×C67—22no—
210C22×C67—22no—
211C22×C67—28no—
212C22×C67—28no—
213C22×C67—28no—
214C22×C67—24no—
215C22×C67—82no—
216C22×C67—88no—
217C22×C67#51812yes—
218C22×C67C24×C411yescharacteristic
219C4×C67—22no—
220C4×C67—28no—
221C4×C67—22no—
222C4×C67—22no—
223C4×C67—24no—
224C4×C67#51812yes—
225C4×C67C2×H(5,𝔽2)12yes—
226#60238C6711yesmaximal, characteristic, direct product factor, Sylow
227C2×C67—22no—
228C2×C67—48no—
229C2×C67—42no—
230C2×C67C2×#51811yescharacteristic
231C2×C67#357312yes—
232#3449C2×C6711yescharacteristic
233#2978C2×C6714yescomplemented
234#2977C2×C6714yescomplemented
235#2976C2×C6718yescomplemented
236#2935C2×C6714yescomplemented
237#2932C2×C6712yescomplemented
238#2931C2×C6712yescomplemented
239#2921C2×C6711yescharacteristic, complemented
240C2×#433C2×C6711yescharacteristic, complemented
241#2880C2×C6714yescomplemented
242C67#6023811yescharacteristic, direct product factor, Sylow
243C22×#41C22×C6712yes—
244C2×#131C22×C6714yescomplemented
245C2×#131C22×C6714yescomplemented
246#537C4×C6714yes—
247#536C4×C6714yes—
248#535C4×C6714yes—
249#534C4×C6714yes—
250#532C22×C6712yes—
251#532C22×C6712yes—
252#525C22×C6714yes—
253#525C22×C6714yes—
254C2×#117C22×C6711yescharacteristic
255#436C22×C6714yescomplemented
256#436C22×C6714yescomplemented
257#436C22×C6718yescomplemented
258#435C22×C6718yescomplemented
259#435C22×C6718yescomplemented
260#435C22×C6718yescomplemented
261#435C22×C6718yescomplemented
262C8×D4C22×C6714yescomplemented
263C8×D4C22×C6714yescomplemented
264C8×D4C22×C6718yescomplemented
265#433C22×C6714yescomplemented
266#433C22×C6712yescomplemented
267#433C22×C6712yescomplemented
268#433C22×C6714yescomplemented
269#433C22×C6714yescomplemented
270#423C22×C6712yescomplemented
271C2×C4⋊C8C22×C6712yescomplemented
272#408C22×C6718yes—
273#408C22×C6718yes—
274#408C22×C6714yescomplemented
275#408C22×C6714yescomplemented
276#407C22×C6712yes—
277#407C22×C6718yescomplemented
278C2×C22⋊C8C22×C6712yes—
279C2×C22⋊C8C22×C6718yescomplemented
280C4×M16C22×C6712yescomplemented
281C2×#97C22×C6711yescharacteristic, complemented
282C2×C4×C8C22×C6711yescharacteristic, complemented
283C2×#41—48no—
284C2×#41C2×C4×C6714yes—
285C2×#41C2×C4×C6714yes—
286C2×#41C23×C6714yes—
287C2×#41C23×C6714yes—
288C22×Q8C2×C4×C6712yes—
289C22×D4C2×C4×C6714yes—
290C22×D4C2×C4×C6712yes—
291C23×C4C23×C6711yescharacteristic
292C23×C4C23×C6714yes—
293#131—48no—
294#131—48no—
295C2×M16C23×C6714yes—
296C2×M16C23×C6718yes—
297C2×M16C23×C6718yes—
298C22×C8C23×C6712yes—
299C22×C8C23×C6712yes—
300C22×C8C23×C6718yes—
301C22×C8C23×C6718yes—
302C4⋊Q8C2×C4×C6712yes—
303C4⋊Q8C2×C4×C6712yes—
304#127C2×C4×C6712yes—
305#127C2×C4×C6712yes—
306#124C2×C4×C6714yes—
307#124C2×C4×C6714yes—
308#123C2×C4×C6718yes—
309#123C2×C4×C6718yes—
310C22⋊Q8C2×C4×C6718yes—
311C22⋊Q8C2×C4×C6718yes—
312#121C2×C4×C6718yes—
313#121C2×C4×C6718yes—
314#121C2×C4×C6714yes—
315#121C2×C4×C6714yes—
316#121C2×C4×C6718yes—
317#120C2×C4×C6718yes—
318#120C2×C4×C6718yes—
319C4×D4C23×C6714yes—
320C4×D4C23×C6718yes—
321C4×D4C23×C6714yes—
322#117C2×C4×C6714yes—
323#117C23×C6712yes—
324#117C23×C6712yes—
325C2×C4⋊C4C2×C4×C6712yes—
326C2×C22⋊C4C2×C4×C6712yes—
327C2×C42C23×C6711yescharacteristic
328C2×C42C23×C6711yescharacteristic
329C4⋊C8C23×C6714yes—
330C4⋊C8C23×C6714yes—
331C4⋊C8C23×C6714yes—
332C4⋊C8C23×C6714yes—
333C22⋊C8C23×C6714yes—
334C22⋊C8C23×C6714yes—
335C22⋊C8C23×C6718yes—
336C22⋊C8C23×C6718yes—
337C22⋊C8C23×C6718yes—
338C22⋊C8C23×C6718yes—
339C22⋊C8C23×C6718yes—
340#97C23×C6712yes—
341#97C23×C6712yes—
342#97C23×C6714yes—
343C4×C8C23×C6711yescharacteristic
344C4×C8C23×C6712yes—
345C4×C8C23×C6711yescharacteristic
346C4×C8C23×C6714yes—
347C24C22×C4×C6711yescharacteristic
348C24C22×C4×C6714yes—
349#41—28no—
350#41—28no—
351#41—48no—
352#41—48no—
353#41—48no—
354C2×Q8—44no—
355C2×Q8C22×C4×C6714yes—
356C2×Q8C22×C4×C6714yes—
357C2×D4—44no—
358C2×D4—48no—
359C2×D4C22×C4×C6718yes—
360C2×D4C22×C4×C6714yes—
361C2×D4C22×C4×C6714yes—
362C2×D4C22×C4×C6718yes—
363C2×D4C22×C4×C6714yes—
364C2×D4C22×C4×C6718yes—
365C2×D4C22×C4×C6714yes—
366C22×C4—24no—
367C22×C4—48no—
368C22×C4C22×C4×C6711yescharacteristic
369C22×C4C22×C4×C6712yes—
370C22×C4C22×C4×C6712yes—
371C22×C4C22×C4×C6712yes—
372C22×C4C22×C4×C6718yes—
373C22×C4C22×C4×C6714yes—
374C22×C4C24×C6712yes—
375C22×C4C24×C6718yes—
376C22×C4C24×C6711yescharacteristic
377M16—28no—
378M16—48no—
379M16—48no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—48no—
383C2×C8—48no—
384C2×C8C24×C6712yes—
385C2×C8C24×C6712yes—
386C2×C8C24×C6718yes—
387C2×C8C24×C6714yes—
388C4⋊C4C22×C4×C6714yes—
389C4⋊C4C22×C4×C6714yes—
390C4⋊C4C22×C4×C6714yes—
391C4⋊C4C22×C4×C6718yes—
392C4⋊C4C22×C4×C6714yes—
393C22⋊C4C22×C4×C6714yes—
394C22⋊C4C22×C4×C6714yes—
395C22⋊C4C22×C4×C6718yes—
396C22⋊C4C22×C4×C6718yes—
397C22⋊C4C22×C4×C6718yes—
398C22⋊C4C22×C4×C6718yes—
399C42C22×C4×C6712yes—
400C42C22×C4×C6712yes—
401C42C24×C6711yescharacteristic
402C42C24×C6711yescharacteristic
403C42C24×C6712yes—
404C23—22no—
405C23—44no—
406C23—42no—
407C23—48no—
408C23C23×C4×C6712yes—
409C23C23×C4×C6718yes—
410C23C23×C4×C6711yescharacteristic
411Q8—28no—
412Q8—44no—
413D4—216no—
414D4—28no—
415D4—44no—
416D4—48no—
417D4—44no—
418D4—48no—
419D4—44no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—28no—
423C2×C4—28no—
424C2×C4—24no—
425C2×C4—44no—
426C2×C4—48no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4C23×C4×C6712yes—
430C2×C4C23×C4×C6712yes—
431C2×C4C23×C4×C6714yes—
432C2×C4C23×C4×C6718yes—
433C2×C4C23×C4×C6712yes—
434C2×C4C23×C4×C6711yescharacteristic
435C2×C4H(5,𝔽2)×C6714yes—
436C2×C4C25×C6711yescharacteristic
437C8—22no—
438C8—24no—
439C8—22no—
440C8—28no—
441C22—22no—
442C22—28no—
443C22—28no—
444C22—24no—
445C22—22no—
446C22—28no—
447C22—82no—
448C22—88no—
449C22#518×C6712yes—
450C22C24×C4×C6711yescharacteristic
451C4—22no—
452C4—22no—
453C4—24no—
454C4—22no—
455C4—28no—
456C4#518×C6712yes—
457C4C2×H(5,𝔽2)×C6712yes—
458C2—22no—
459C2—42no—
460C2—48no—
461C2C2×#518×C6711yescharacteristic
462C2C67×#357312yes—
subgroupG/H
centerC2×C4×C67C25
commutatorC22C24×C4×C67
Frattini, cosocleC2×C4C25×C67
socleC22×C67C24×C4
prime radicalC67×#3449C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
2. C67×#2978C2semidirect
3. C67×#2977C2semidirect
4. C67×#2976C2semidirect
5. C67×#2935C2semidirect
6. C67×#2932C2semidirect
7. C67×#2931C2semidirect
8. C67×#2921C2semidirect
9. C2×#433×C67C2semidirect
10. C67×#2880C2semidirect
12. C2×#131×C67C22semidirect
13. C2×#131×C67C22semidirect
23. #436×C67C22semidirect
24. #436×C67C22semidirect
25. #436×C67C22semidirect
26. #435×C67C22semidirect
27. #435×C67C22semidirect
28. #435×C67C22semidirect
29. #435×C67C22semidirect
30. C8×D4×C67C22semidirect
31. C8×D4×C67C22semidirect
32. C8×D4×C67C22semidirect
33. #433×C67C22semidirect
34. #433×C67C22semidirect
35. #433×C67C22semidirect
36. #433×C67C22semidirect
37. #433×C67C22semidirect
38. #423×C67C22semidirect
39. C2×C4⋊C8×C67C22semidirect
42. #408×C67C22semidirect
43. #408×C67C22semidirect
45. #407×C67C22semidirect
47. C2×C22⋊C8×C67C22semidirect
48. C4×M16×C67C22semidirect
49. C2×#97×C67C22semidirect
50. C2×C4×C8×C67C22semidirect
226. #60238C67direct (N × H)
233. #2978C2×C67semidirect
234. #2977C2×C67semidirect
235. #2976C2×C67semidirect
236. #2935C2×C67semidirect
237. #2932C2×C67semidirect
238. #2931C2×C67semidirect
239. #2921C2×C67semidirect
240. C2×#433C2×C67semidirect
241. #2880C2×C67semidirect
242. C67#60238direct (N × H)
244. C2×#131C22×C67semidirect
245. C2×#131C22×C67semidirect
255. #436C22×C67semidirect
256. #436C22×C67semidirect
257. #436C22×C67semidirect
258. #435C22×C67semidirect
259. #435C22×C67semidirect
260. #435C22×C67semidirect
261. #435C22×C67semidirect
262. C8×D4C22×C67semidirect
263. C8×D4C22×C67semidirect
264. C8×D4C22×C67semidirect
265. #433C22×C67semidirect
266. #433C22×C67semidirect
267. #433C22×C67semidirect
268. #433C22×C67semidirect
269. #433C22×C67semidirect
270. #423C22×C67semidirect
271. C2×C4⋊C8C22×C67semidirect
274. #408C22×C67semidirect
275. #408C22×C67semidirect
277. #407C22×C67semidirect
279. C2×C22⋊C8C22×C67semidirect
280. C4×M16C22×C67semidirect
281. C2×#97C22×C67semidirect
282. C2×C4×C8C22×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#34492
C67×#29782
C67×#29772
C67×#29762
C67×#29352
C67×#29322
C67×#29312
C67×#29212
C2×#433×C672
C67×#28802
#6023867