Finite Group Databasebeta

← C67×#61697 #30740948

Subgroups — C67×#61697 #30740948

Order 17152.

2082 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3496C211yesmaximal, characteristic
2C2×#554×C67C211yesmaximal, characteristic, complemented
3C67×#3335C218yesmaximal, complemented
4C67×#3295C218yesmaximal, complemented
5C67×#3227C212yesmaximal, complemented
6C67×#3187C214yesmaximal, complemented
7C67×#3103C212yesmaximal, complemented
8C2×#481×C67C214yesmaximal, complemented
9C67×#2972C211yesmaximal, characteristic, complemented
10C2×H(5,𝔽2)×C67—21no—
11#554×C67—24no—
12#554×C67C2218yescomplemented
13#547×C67C2214yes—
14#546×C67—24no—
15D42×C67C2214yes—
16#537×C67—21no—
17#534×C67—22no—
18#532×C67C2212yes—
19C2×C4⋊Q8×C67C2211yescharacteristic, complemented
20C2×#125×C67—21no—
21C2×C22⋊Q8×C67—22no—
22C2×C22⋊Q8×C67C2212yescomplemented
23C2×#121×C67C2214yes—
24C2×C4×Q8×C67C2211yescharacteristic, complemented
25C2×C4×D4×C67—21no—
26C2×C4×D4×C67C2211yescharacteristic
27C22×C4⋊C4×C67C2212yescomplemented
28#498×C67C2214yescomplemented
29#486×C67C2214yescomplemented
30#483×C67C2218yescomplemented
31#481×C67C22116yescomplemented
32#481×C67C2214yescomplemented
33#481×C67C2218yescomplemented
34#476×C67C2218yescomplemented
35#475×C67C2214yescomplemented
36#465×C67C22116yescomplemented
37#460×C67C2214yescomplemented
38#450×C67C2218yescomplemented
39#448×C67C2214yescomplemented
40C4×SD16×C67C2218yescomplemented
41C8×D4×C67C2218yescomplemented
42#431×C67C2212yescomplemented
43C2×#106×C67C2218yescomplemented
44C2×#102×C67C2218yescomplemented
45C2×C22⋊C8×C67C2214yescomplemented
46H(5,𝔽2)×C67—48no—
47C2×#41×C67—24no—
48C2×#41×C67—22no—
49C22×Q8×C67—21no—
50C22×Q8×C67C2311yescharacteristic
51C22×D4×C67—24no—
52C22×D4×C67—24no—
53C22×D4×C67D411yescharacteristic
54C22×D4×C67C2314yes—
55C23×C4×C67—22no—
56C23×C4×C67C2312yes—
57C2×SD16×C67C2318yes—
58C22×C8×C67C2318yes—
59C4⋊Q8×C67—22no—
60C4⋊Q8×C67C2312yes—
61C4⋊Q8×C67C2314yes—
62#127×C67C2312yes—
63#125×C67—28no—
64#124×C67—22no—
65#123×C67—28no—
66C22⋊Q8×C67—24no—
67C22⋊Q8×C67—24no—
68C22⋊Q8×C67—216no—
69C22⋊Q8×C67C2318yes—
70#121×C67—24no—
71#121×C67—24no—
72#121×C67—22no—
73#121×C67—24no—
74#121×C67C2314yes—
75#121×C67C2318yes—
76#120×C67—28no—
77C4×Q8×C67—22no—
78C4×Q8×C67C2314yes—
79C4×D4×C67—28no—
80C4×D4×C67—22no—
81C4×D4×C67D414yes—
82C4×D4×C67C2314yes—
83C4×D4×C67C2314yes—
84C2×C4⋊C4×C67—22no—
85C2×C4⋊C4×C67—24no—
86C2×C4⋊C4×C67—21no—
87C2×C4⋊C4×C67—21no—
88C2×C4⋊C4×C67—22no—
89C2×C4⋊C4×C67D411yescharacteristic
90C2×C4⋊C4×C67C2311yescharacteristic
91C2×C4⋊C4×C67C2312yes—
92C2×C4⋊C4×C67C2318yes—
93C2×C4⋊C4×C67C2314yes—
94C2×C22⋊C4×C67—22no—
95C2×C22⋊C4×C67—24no—
96C2×C22⋊C4×C67D412yes—
97C2×C42×C67—21no—
98C2×C42×C67C2311yescharacteristic
99#106×C67C23116yes—
100#106×C67C2318yes—
101C4⋊C8×C67C2314yes—
102#103×C67C2318yes—
103#102×C67C23116yes—
104#102×C67C2318yes—
105C22⋊C8×C67C2314yes—
106C22⋊C8×C67C2318yes—
107C4×C8×C67C2314yes—
108C24×C67—24no—
109C24×C67C2×D412yes—
110#41×C67—416no—
111#41×C67—48no—
112C2×Q8×C67—21no—
113C2×Q8×C67—22no—
114C2×Q8×C67—24no—
115C2×Q8×C67—42no—
116C2×Q8×C67—82no—
117C2×Q8×C67C2412yes—
118C2×D4×C67—28no—
119C2×D4×C67—24no—
120C2×D4×C67—22no—
121C2×D4×C67—22no—
122C2×D4×C67—24no—
123C2×D4×C67—48no—
124C2×D4×C67—416no—
125C2×D4×C67—416no—
126C2×D4×C67SD1618yescomplemented
127C2×D4×C67C2×D414yes—
128C2×D4×C67C2414yes—
129C22×C4×C67—21no—
130C22×C4×C67—21no—
131C22×C4×C67—22no—
132C22×C4×C67—24no—
133C22×C4×C67—28no—
134C22×C4×C67—24no—
135C22×C4×C67—44no—
136C22×C4×C67—84no—
137C22×C4×C67C2×D411yescharacteristic
138C22×C4×C67C2×D414yes—
139C22×C4×C67C2×D412yes—
140C22×C4×C67C2×D411yescharacteristic
141C22×C4×C67C2411yescharacteristic
142C22×C4×C67C2414yes—
143SD16×C67—48no—
144C2×C8×C67—48no—
145C2×C8×C67C2418yes—
146C4⋊C4×C67—21no—
147C4⋊C4×C67—28no—
148C4⋊C4×C67—22no—
149C4⋊C4×C67—24no—
150C4⋊C4×C67—24no—
151C4⋊C4×C67—28no—
152C4⋊C4×C67—24no—
153C4⋊C4×C67—22no—
154C4⋊C4×C67C2×D414yes—
155C4⋊C4×C67C2412yes—
156C4⋊C4×C67C2414yes—
157C4⋊C4×C67C2414yes—
158C22⋊C4×C67—28no—
159C22⋊C4×C67—24no—
160C22⋊C4×C67—28no—
161C22⋊C4×C67—24no—
162C22⋊C4×C67C2×D418yes—
163C42×C67—24no—
164C42×C67C2×D412yes—
165C42×C67C2412yes—
166C23×C67—24no—
167C23×C67—42no—
168C23×C67—42no—
169C23×C67—48no—
170C23×C67—84no—
171C23×C67—84no—
172C23×C67C2×SD1618yes—
173C23×C67C22×D414yes—
174C23×C67C22×D411yescharacteristic
175Q8×C67—24no—
176Q8×C67—42no—
177Q8×C67—44no—
178Q8×C67—84no—
179D4×C67—28no—
180D4×C67—416no—
181D4×C67—48no—
182D4×C67—44no—
183D4×C67—82no—
184D4×C67—84no—
185D4×C67C2×SD16116yes—
186C2×C4×C67—28no—
187C2×C4×C67—22no—
188C2×C4×C67—22no—
189C2×C4×C67—22no—
190C2×C4×C67—24no—
191C2×C4×C67—24no—
192C2×C4×C67—44no—
193C2×C4×C67—42no—
194C2×C4×C67—41no—
195C2×C4×C67—42no—
196C2×C4×C67—44no—
197C2×C4×C67—42no—
198C2×C4×C67—48no—
199C2×C4×C67—48no—
200C2×C4×C67—88no—
201C2×C4×C67C2×SD1614yes—
202C2×C4×C67C22×D411yescharacteristic
203C2×C4×C67C22×D412yes—
204C2×C4×C67C22×D414yes—
205C2×C4×C67C22×D412yes—
206C2×C4×C67C2511yescharacteristic
207C8×C67—28no—
208C22×C67—21no—
209C22×C67—24no—
210C22×C67—21no—
211C22×C67—24no—
212C22×C67—48no—
213C22×C67—44no—
214C22×C67—44no—
215C22×C67—44no—
216C22×C67—168no—
217C22×C67C22×SD1618yes—
218C22×C67C22×SD1612yes—
219C22×C67C23×D411yescharacteristic
220C4×C67—22no—
221C4×C67—21no—
222C4×C67—24no—
223C4×C67—42no—
224C4×C67—44no—
225C4×C67—42no—
226C4×C67—44no—
227C4×C67C22×SD1614yes—
228C4×C67#58512yes—
229#61697C6711yesmaximal, characteristic, direct product factor, Sylow
230C2×C67—22no—
231C2×C67—28no—
232C2×C67—84no—
233C2×C67#348511yescharacteristic
234C2×C67C23×SD1611yescharacteristic
235C2×C67#358511yescharacteristic
236#3496C2×C6711yescharacteristic
237C2×#554C2×C6711yescharacteristic, complemented
238#3335C2×C6718yescomplemented
239#3295C2×C6718yescomplemented
240#3227C2×C6712yescomplemented
241#3187C2×C6714yescomplemented
242#3103C2×C6712yescomplemented
243C2×#481C2×C6714yescomplemented
244#2972C2×C6711yescharacteristic, complemented
245C67#6169711yescharacteristic, direct product factor, Sylow
246C2×H(5,𝔽2)—21no—
247#554—24no—
248#554C22×C6718yescomplemented
249#547C22×C6714yes—
250#546—24no—
251D42C22×C6714yes—
252#537—21no—
253#534—22no—
254#532C22×C6712yes—
255C2×C4⋊Q8C22×C6711yescharacteristic, complemented
256C2×#125—21no—
257C2×C22⋊Q8—22no—
258C2×C22⋊Q8C22×C6712yescomplemented
259C2×#121C22×C6714yes—
260C2×C4×Q8C22×C6711yescharacteristic, complemented
261C2×C4×D4—21no—
262C2×C4×D4C22×C6711yescharacteristic
263C22×C4⋊C4C22×C6712yescomplemented
264#498C22×C6714yescomplemented
265#486C22×C6714yescomplemented
266#483C22×C6718yescomplemented
267#481C22×C67116yescomplemented
268#481C22×C6714yescomplemented
269#481C22×C6718yescomplemented
270#476C22×C6718yescomplemented
271#475C22×C6714yescomplemented
272#465C22×C67116yescomplemented
273#460C22×C6714yescomplemented
274#450C22×C6718yescomplemented
275#448C22×C6714yescomplemented
276C4×SD16C22×C6718yescomplemented
277C8×D4C22×C6718yescomplemented
278#431C22×C6712yescomplemented
279C2×#106C22×C6718yescomplemented
280C2×#102C22×C6718yescomplemented
281C2×C22⋊C8C22×C6714yescomplemented
282H(5,𝔽2)—48no—
283C2×#41—24no—
284C2×#41—22no—
285C22×Q8—21no—
286C22×Q8C23×C6711yescharacteristic
287C22×D4—24no—
288C22×D4—24no—
289C22×D4D4×C6711yescharacteristic
290C22×D4C23×C6714yes—
291C23×C4—22no—
292C23×C4C23×C6712yes—
293C2×SD16C23×C6718yes—
294C22×C8C23×C6718yes—
295C4⋊Q8—22no—
296C4⋊Q8C23×C6714yes—
297C4⋊Q8C23×C6712yes—
298#127C23×C6712yes—
299#125—28no—
300#124—22no—
301#123—28no—
302C22⋊Q8—24no—
303C22⋊Q8—24no—
304C22⋊Q8—216no—
305C22⋊Q8C23×C6718yes—
306#121—24no—
307#121—24no—
308#121—22no—
309#121—24no—
310#121C23×C6714yes—
311#121C23×C6718yes—
312#120—28no—
313C4×Q8—22no—
314C4×Q8C23×C6714yes—
315C4×D4—28no—
316C4×D4—22no—
317C4×D4D4×C6714yes—
318C4×D4C23×C6714yes—
319C4×D4C23×C6714yes—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4D4×C6711yescharacteristic
326C2×C4⋊C4C23×C6711yescharacteristic
327C2×C4⋊C4C23×C6712yes—
328C2×C4⋊C4C23×C6718yes—
329C2×C4⋊C4C23×C6714yes—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4D4×C6712yes—
333C2×C42—21no—
334C2×C42C23×C6711yescharacteristic
335#106C23×C6718yes—
336#106C23×C67116yes—
337C4⋊C8C23×C6714yes—
338#103C23×C6718yes—
339#102C23×C6718yes—
340#102C23×C67116yes—
341C22⋊C8C23×C6714yes—
342C22⋊C8C23×C6718yes—
343C4×C8C23×C6714yes—
344C24—24no—
345C24C2×D4×C6712yes—
346#41—416no—
347#41—48no—
348C2×Q8—21no—
349C2×Q8—22no—
350C2×Q8—24no—
351C2×Q8—42no—
352C2×Q8—82no—
353C2×Q8C24×C6712yes—
354C2×D4—28no—
355C2×D4—24no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—24no—
359C2×D4—48no—
360C2×D4—416no—
361C2×D4—416no—
362C2×D4SD16×C6718yescomplemented
363C2×D4C2×D4×C6714yes—
364C2×D4C24×C6714yes—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—28no—
370C22×C4—24no—
371C22×C4—44no—
372C22×C4—84no—
373C22×C4C2×D4×C6711yescharacteristic
374C22×C4C2×D4×C6714yes—
375C22×C4C2×D4×C6712yes—
376C22×C4C2×D4×C6711yescharacteristic
377C22×C4C24×C6711yescharacteristic
378C22×C4C24×C6714yes—
379SD16—48no—
380C2×C8—48no—
381C2×C8C24×C6718yes—
382C4⋊C4—21no—
383C4⋊C4—28no—
384C4⋊C4—22no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—28no—
388C4⋊C4—24no—
389C4⋊C4—22no—
390C4⋊C4C2×D4×C6714yes—
391C4⋊C4C24×C6712yes—
392C4⋊C4C24×C6714yes—
393C4⋊C4C24×C6714yes—
394C22⋊C4—28no—
395C22⋊C4—24no—
396C22⋊C4—28no—
397C22⋊C4—24no—
398C22⋊C4C2×D4×C6718yes—
399C42—24no—
400C42C2×D4×C6712yes—
401C42C24×C6712yes—
402C23—24no—
403C23—42no—
404C23—42no—
405C23—48no—
406C23—84no—
407C23—84no—
408C23C2×SD16×C6718yes—
409C23C22×D4×C6714yes—
410C23C22×D4×C6711yescharacteristic
411Q8—24no—
412Q8—42no—
413Q8—44no—
414Q8—84no—
415D4—28no—
416D4—416no—
417D4—48no—
418D4—44no—
419D4—84no—
420D4—82no—
421D4C2×SD16×C67116yes—
422C2×C4—28no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—48no—
435C2×C4—48no—
436C2×C4—88no—
437C2×C4C2×SD16×C6714yes—
438C2×C4C22×D4×C6711yescharacteristic
439C2×C4C22×D4×C6712yes—
440C2×C4C22×D4×C6714yes—
441C2×C4C22×D4×C6712yes—
442C2×C4C25×C6711yescharacteristic
443C8—28no—
444C22—21no—
445C22—24no—
446C22—21no—
447C22—24no—
448C22—48no—
449C22—44no—
450C22—44no—
451C22—44no—
452C22—168no—
453C22C22×SD16×C6718yes—
454C22C22×SD16×C6712yes—
455C22C23×D4×C6711yescharacteristic
456C4—22no—
457C4—21no—
458C4—24no—
459C4—44no—
460C4—42no—
461C4—44no—
462C4—42no—
463C4C22×SD16×C6714yes—
464C4#585×C6712yes—
465C2—22no—
466C2—28no—
467C2—84no—
468C2C67×#348511yescharacteristic
469C2C23×SD16×C6711yescharacteristic
470C2C67×#358511yescharacteristic
subgroupG/H
center, socleC22×C67C23×D4
commutator, Frattini, cosocleC2×C4C25×C67
prime radicalC67×#3496C2
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
2. C2×#554×C67C2semidirect
3. C67×#3335C2semidirect
4. C67×#3295C2semidirect
5. C67×#3227C2semidirect
6. C67×#3187C2semidirect
7. C67×#3103C2semidirect
8. C2×#481×C67C2semidirect
9. C67×#2972C2semidirect
12. #554×C67C22semidirect
19. C2×C4⋊Q8×C67C22semidirect
22. C2×C22⋊Q8×C67C22semidirect
24. C2×C4×Q8×C67C22semidirect
27. C22×C4⋊C4×C67C22semidirect
28. #498×C67C22semidirect
29. #486×C67C22semidirect
30. #483×C67C22semidirect
31. #481×C67C22semidirect
32. #481×C67C22semidirect
33. #481×C67C22semidirect
34. #476×C67C22semidirect
35. #475×C67C22semidirect
36. #465×C67C22semidirect
37. #460×C67C22semidirect
38. #450×C67C22semidirect
39. #448×C67C22semidirect
40. C4×SD16×C67C22semidirect
41. C8×D4×C67C22semidirect
42. #431×C67C22semidirect
43. C2×#106×C67C22semidirect
44. C2×#102×C67C22semidirect
45. C2×C22⋊C8×C67C22semidirect
126. C2×D4×C67SD16semidirect
229. #61697C67direct (N × H)
237. C2×#554C2×C67semidirect
238. #3335C2×C67semidirect
239. #3295C2×C67semidirect
240. #3227C2×C67semidirect
241. #3187C2×C67semidirect
242. #3103C2×C67semidirect
243. C2×#481C2×C67semidirect
244. #2972C2×C67semidirect
245. C67#61697direct (N × H)
248. #554C22×C67semidirect
255. C2×C4⋊Q8C22×C67semidirect
258. C2×C22⋊Q8C22×C67semidirect
260. C2×C4×Q8C22×C67semidirect
263. C22×C4⋊C4C22×C67semidirect
264. #498C22×C67semidirect
265. #486C22×C67semidirect
266. #483C22×C67semidirect
267. #481C22×C67semidirect
268. #481C22×C67semidirect
269. #481C22×C67semidirect
270. #476C22×C67semidirect
271. #475C22×C67semidirect
272. #465C22×C67semidirect
273. #460C22×C67semidirect
274. #450C22×C67semidirect
275. #448C22×C67semidirect
276. C4×SD16C22×C67semidirect
277. C8×D4C22×C67semidirect
278. #431C22×C67semidirect
279. C2×#106C22×C67semidirect
280. C2×#102C22×C67semidirect
281. C2×C22⋊C8C22×C67semidirect
362. C2×D4SD16×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#34962
C2×#554×C672
C67×#33352
C67×#32952
C67×#32272
C67×#31872
C67×#31032
C2×#481×C672
C67×#29722
#6169767