Finite Group Databasebeta

← C67×#61722 #30740973

Subgroups — C67×#61722 #30740973

Order 17152.

2014 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3496C211yesmaximal, characteristic
2C2×#556×C67C211yesmaximal, characteristic, complemented
3C67×#3353C218yesmaximal, complemented
4C67×#3324C218yesmaximal, complemented
5C67×#3226C212yesmaximal, complemented
6C67×#3193C214yesmaximal, complemented
7C67×#3103C212yesmaximal, complemented
8C67×#3094C214yesmaximal, complemented
9C67×#2975C211yesmaximal, characteristic, complemented
10C2×H(5,𝔽2)×C67—21no—
11#556×C67—24no—
12#556×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×#126×C67—22no—
21C2×#124×C67—21no—
22C2×C22⋊Q8×C67C2212yescomplemented
23C2×#121×C67C2214yes—
24C2×C4×Q8×C67—21no—
25C2×C4×Q8×C67C2211yescharacteristic, complemented
26C2×C4×D4×C67C2211yescharacteristic
27C2×#117×C67C2212yescomplemented
28#500×C67C2214yescomplemented
29#486×C67C2214yescomplemented
30#484×C67C2218yescomplemented
31#482×C67C22116yescomplemented
32#481×C67C2214yescomplemented
33#480×C67C2218yescomplemented
34#475×C67C2214yescomplemented
35#474×C67C2218yescomplemented
36#468×C67C22116yescomplemented
37#460×C67C2214yescomplemented
38#450×C67C2218yescomplemented
39#448×C67C2214yescomplemented
40C4×SD16×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×Q8×C67C2311yescharacteristic
51C22×D4×C67—24no—
52C22×D4×C67—24no—
53C22×D4×C67D411yescharacteristic
54C22×D4×C67C2314yes—
55C23×C4×C67C2312yes—
56C2×SD16×C67C2318yes—
57C2×M16×C67C2318yes—
58C4⋊Q8×C67—22no—
59C4⋊Q8×C67C2314yes—
60C4⋊Q8×C67C2312yes—
61#127×C67C2312yes—
62#126×C67—216no—
63#124×C67—28no—
64#124×C67—22no—
65#123×C67—28no—
66C22⋊Q8×C67—24no—
67C22⋊Q8×C67—24no—
68C22⋊Q8×C67C2318yes—
69#121×C67—24no—
70#121×C67—24no—
71#121×C67—22no—
72#121×C67—24no—
73#121×C67C2314yes—
74#121×C67C2318yes—
75#120×C67—28no—
76C4×Q8×C67—28no—
77C4×Q8×C67—22no—
78C4×Q8×C67C2314yes—
79C4×D4×C67—22no—
80C4×D4×C67D414yes—
81C4×D4×C67C2314yes—
82C4×D4×C67C2314yes—
83#117×C67—24no—
84#117×C67C2318yes—
85C2×C4⋊C4×C67—22no—
86C2×C4⋊C4×C67—21no—
87C2×C4⋊C4×C67—21no—
88C2×C4⋊C4×C67D411yescharacteristic
89C2×C4⋊C4×C67C2312yes—
90C2×C4⋊C4×C67C2311yescharacteristic
91C2×C4⋊C4×C67C2314yes—
92C2×C22⋊C4×C67—22no—
93C2×C22⋊C4×C67—22no—
94C2×C22⋊C4×C67—24no—
95C2×C22⋊C4×C67D412yes—
96C2×C42×C67—22no—
97C2×C42×C67—21no—
98C2×C42×C67C2311yescharacteristic
99#107×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—24no—
114C2×Q8×C67—22no—
115C2×Q8×C67—42no—
116C2×Q8×C67—44no—
117C2×Q8×C67C2412yes—
118C2×D4×C67—28no—
119C2×D4×C67—24no—
120C2×D4×C67—24no—
121C2×D4×C67—22no—
122C2×D4×C67—22no—
123C2×D4×C67—24no—
124C2×D4×C67—48no—
125C2×D4×C67—416no—
126C2×D4×C67—416no—
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—24no—
134C22×C4×C67—44no—
135C22×C4×C67C2×D412yes—
136C22×C4×C67C2×D411yescharacteristic
137C22×C4×C67C2×D414yes—
138C22×C4×C67C2×D411yescharacteristic
139C22×C4×C67C2411yescharacteristic
140C22×C4×C67C2414yes—
141SD16×C67—48no—
142M16×C67—48no—
143C2×C8×C67C2418yes—
144C4⋊C4×C67—28no—
145C4⋊C4×C67—24no—
146C4⋊C4×C67—24no—
147C4⋊C4×C67—21no—
148C4⋊C4×C67—22no—
149C4⋊C4×C67—24no—
150C4⋊C4×C67—22no—
151C4⋊C4×C67C2×D414yes—
152C4⋊C4×C67C2412yes—
153C4⋊C4×C67C2414yes—
154C4⋊C4×C67C2414yes—
155C22⋊C4×C67—28no—
156C22⋊C4×C67—28no—
157C22⋊C4×C67—24no—
158C22⋊C4×C67—28no—
159C22⋊C4×C67—24no—
160C22⋊C4×C67C2×D418yes—
161C42×C67—28no—
162C42×C67—24no—
163C42×C67C2×D412yes—
164C42×C67C2412yes—
165C23×C67—24no—
166C23×C67—24no—
167C23×C67—42no—
168C23×C67—42no—
169C23×C67—48no—
170C23×C67—84no—
171C23×C67—84no—
172C23×C67C22×D414yes—
173C23×C67C22×D411yescharacteristic
174Q8×C67—24no—
175Q8×C67—42no—
176Q8×C67—48no—
177Q8×C67—44no—
178D4×C67—28no—
179D4×C67—28no—
180D4×C67—416no—
181D4×C67—48no—
182D4×C67—44no—
183D4×C67—82no—
184D4×C67—84no—
185C2×C4×C67—28no—
186C2×C4×C67—22no—
187C2×C4×C67—22no—
188C2×C4×C67—22no—
189C2×C4×C67—24no—
190C2×C4×C67—24no—
191C2×C4×C67—44no—
192C2×C4×C67—42no—
193C2×C4×C67—41no—
194C2×C4×C67—42no—
195C2×C4×C67—48no—
196C2×C4×C67—42no—
197C2×C4×C67—44no—
198C2×C4×C67—48no—
199C2×C4×C67AGL(1,ℤ/8)14yes—
200C2×C4×C67C22×D414yes—
201C2×C4×C67C22×D411yescharacteristic
202C2×C4×C67C22×D412yes—
203C2×C4×C67C22×D412yes—
204C2×C4×C67C2511yescharacteristic
205C8×C67—28no—
206C22×C67—21no—
207C22×C67—21no—
208C22×C67—24no—
209C22×C67—24no—
210C22×C67—24no—
211C22×C67—44no—
212C22×C67—44no—
213C22×C67—44no—
214C22×C67—84no—
215C22×C67—168no—
216C22×C67C2×AGL(1,ℤ/8)12yes—
217C22×C67C23×D411yescharacteristic
218C4×C67—22no—
219C4×C67—21no—
220C4×C67—24no—
221C4×C67—42no—
222C4×C67—42no—
223C4×C67—44no—
224C4×C67—44no—
225C4×C67C2×AGL(1,ℤ/8)14yes—
226C4×C67#58512yes—
227#61722C6711yesmaximal, characteristic, direct product factor, Sylow
228C2×C67—22no—
229C2×C67—44no—
230C2×C67—84no—
231C2×C67#348511yescharacteristic
232C2×C67C22×AGL(1,ℤ/8)11yescharacteristic
233C2×C67#358611yescharacteristic
234#3496C2×C6711yescharacteristic
235C2×#556C2×C6711yescharacteristic, complemented
236#3353C2×C6718yescomplemented
237#3324C2×C6718yescomplemented
238#3226C2×C6712yescomplemented
239#3193C2×C6714yescomplemented
240#3103C2×C6712yescomplemented
241#3094C2×C6714yescomplemented
242#2975C2×C6711yescharacteristic, complemented
243C67#6172211yescharacteristic, direct product factor, Sylow
244C2×H(5,𝔽2)—21no—
245#556—24no—
246#556C22×C6718yescomplemented
247#547C22×C6714yes—
248#546—24no—
249D42C22×C6714yes—
250#537—21no—
251#534—22no—
252#532C22×C6712yes—
253C2×C4⋊Q8C22×C6711yescharacteristic, complemented
254C2×#126—22no—
255C2×#124—21no—
256C2×C22⋊Q8C22×C6712yescomplemented
257C2×#121C22×C6714yes—
258C2×C4×Q8—21no—
259C2×C4×Q8C22×C6711yescharacteristic, complemented
260C2×C4×D4C22×C6711yescharacteristic
261C2×#117C22×C6712yescomplemented
262#500C22×C6714yescomplemented
263#486C22×C6714yescomplemented
264#484C22×C6718yescomplemented
265#482C22×C67116yescomplemented
266#481C22×C6714yescomplemented
267#480C22×C6718yescomplemented
268#475C22×C6714yescomplemented
269#474C22×C6718yescomplemented
270#468C22×C67116yescomplemented
271#460C22×C6714yescomplemented
272#450C22×C6718yescomplemented
273#448C22×C6714yescomplemented
274C4×SD16C22×C6718yescomplemented
275#436C22×C6718yescomplemented
276#431C22×C6712yescomplemented
277#428C22×C6718yescomplemented
278#418C22×C6718yescomplemented
279#407C22×C6714yescomplemented
280H(5,𝔽2)—48no—
281C2×#41—24no—
282C2×#41—22no—
283C22×Q8—21no—
284C22×Q8C23×C6711yescharacteristic
285C22×D4—24no—
286C22×D4—24no—
287C22×D4D4×C6711yescharacteristic
288C22×D4C23×C6714yes—
289C23×C4C23×C6712yes—
290C2×SD16C23×C6718yes—
291C2×M16C23×C6718yes—
292C4⋊Q8—22no—
293C4⋊Q8C23×C6714yes—
294C4⋊Q8C23×C6712yes—
295#127C23×C6712yes—
296#126—216no—
297#124—28no—
298#124—22no—
299#123—28no—
300C22⋊Q8—24no—
301C22⋊Q8—24no—
302C22⋊Q8C23×C6718yes—
303#121—24no—
304#121—24no—
305#121—22no—
306#121—24no—
307#121C23×C6714yes—
308#121C23×C6718yes—
309#120—28no—
310C4×Q8—28no—
311C4×Q8—22no—
312C4×Q8C23×C6714yes—
313C4×D4—22no—
314C4×D4D4×C6714yes—
315C4×D4C23×C6714yes—
316C4×D4C23×C6714yes—
317#117—24no—
318#117C23×C6718yes—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4D4×C6711yescharacteristic
323C2×C4⋊C4C23×C6711yescharacteristic
324C2×C4⋊C4C23×C6712yes—
325C2×C4⋊C4C23×C6714yes—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4D4×C6712yes—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42C23×C6711yescharacteristic
333#107C23×C67116yes—
334#106C23×C6718yes—
335C4⋊C8C23×C6714yes—
336#103C23×C6718yes—
337#102C23×C6718yes—
338#102C23×C67116yes—
339C22⋊C8C23×C6714yes—
340C22⋊C8C23×C6718yes—
341C4×C8C23×C6714yes—
342C24—24no—
343C24C2×D4×C6712yes—
344#41—416no—
345#41—48no—
346C2×Q8—21no—
347C2×Q8—24no—
348C2×Q8—22no—
349C2×Q8—42no—
350C2×Q8—44no—
351C2×Q8C24×C6712yes—
352C2×D4—28no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—24no—
358C2×D4—48no—
359C2×D4—416no—
360C2×D4—416no—
361C2×D4C2×D4×C6714yes—
362C2×D4C24×C6714yes—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—44no—
369C22×C4C2×D4×C6712yes—
370C22×C4C2×D4×C6711yescharacteristic
371C22×C4C2×D4×C6714yes—
372C22×C4C2×D4×C6711yescharacteristic
373C22×C4C24×C6711yescharacteristic
374C22×C4C24×C6714yes—
375SD16—48no—
376M16—48no—
377C2×C8C24×C6718yes—
378C4⋊C4—28no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—22no—
385C4⋊C4C2×D4×C6714yes—
386C4⋊C4C24×C6712yes—
387C4⋊C4C24×C6714yes—
388C4⋊C4C24×C6714yes—
389C22⋊C4—28no—
390C22⋊C4—28no—
391C22⋊C4—24no—
392C22⋊C4—28no—
393C22⋊C4—24no—
394C22⋊C4C2×D4×C6718yes—
395C42—28no—
396C42—24no—
397C42C2×D4×C6712yes—
398C42C24×C6712yes—
399C23—24no—
400C23—24no—
401C23—42no—
402C23—42no—
403C23—48no—
404C23—84no—
405C23—84no—
406C23C22×D4×C6714yes—
407C23C22×D4×C6711yescharacteristic
408Q8—24no—
409Q8—42no—
410Q8—48no—
411Q8—44no—
412D4—28no—
413D4—28no—
414D4—416no—
415D4—48no—
416D4—44no—
417D4—84no—
418D4—82no—
419C2×C4—28no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—48no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—48no—
433C2×C4AGL(1,ℤ/8)×C6714yes—
434C2×C4C22×D4×C6714yes—
435C2×C4C22×D4×C6711yescharacteristic
436C2×C4C22×D4×C6712yes—
437C2×C4C22×D4×C6712yes—
438C2×C4C25×C6711yescharacteristic
439C8—28no—
440C22—21no—
441C22—21no—
442C22—24no—
443C22—24no—
444C22—24no—
445C22—44no—
446C22—44no—
447C22—44no—
448C22—84no—
449C22—168no—
450C22C2×AGL(1,ℤ/8)×C6712yes—
451C22C23×D4×C6711yescharacteristic
452C4—22no—
453C4—21no—
454C4—24no—
455C4—44no—
456C4—44no—
457C4—42no—
458C4—42no—
459C4C2×AGL(1,ℤ/8)×C6714yes—
460C4#585×C6712yes—
461C2—22no—
462C2—44no—
463C2—84no—
464C2C67×#348511yescharacteristic
465C2C22×AGL(1,ℤ/8)×C6711yescharacteristic
466C2C67×#358611yescharacteristic
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×#556×C67C2semidirect
3. C67×#3353C2semidirect
4. C67×#3324C2semidirect
5. C67×#3226C2semidirect
6. C67×#3193C2semidirect
7. C67×#3103C2semidirect
8. C67×#3094C2semidirect
9. C67×#2975C2semidirect
12. #556×C67C22semidirect
19. C2×C4⋊Q8×C67C22semidirect
22. C2×C22⋊Q8×C67C22semidirect
25. C2×C4×Q8×C67C22semidirect
27. C2×#117×C67C22semidirect
28. #500×C67C22semidirect
29. #486×C67C22semidirect
30. #484×C67C22semidirect
31. #482×C67C22semidirect
32. #481×C67C22semidirect
33. #480×C67C22semidirect
34. #475×C67C22semidirect
35. #474×C67C22semidirect
36. #468×C67C22semidirect
37. #460×C67C22semidirect
38. #450×C67C22semidirect
39. #448×C67C22semidirect
40. C4×SD16×C67C22semidirect
41. #436×C67C22semidirect
42. #431×C67C22semidirect
43. #428×C67C22semidirect
44. #418×C67C22semidirect
45. #407×C67C22semidirect
227. #61722C67direct (N × H)
235. C2×#556C2×C67semidirect
236. #3353C2×C67semidirect
237. #3324C2×C67semidirect
238. #3226C2×C67semidirect
239. #3193C2×C67semidirect
240. #3103C2×C67semidirect
241. #3094C2×C67semidirect
242. #2975C2×C67semidirect
243. C67#61722direct (N × H)
246. #556C22×C67semidirect
253. C2×C4⋊Q8C22×C67semidirect
256. C2×C22⋊Q8C22×C67semidirect
259. C2×C4×Q8C22×C67semidirect
261. C2×#117C22×C67semidirect
262. #500C22×C67semidirect
263. #486C22×C67semidirect
264. #484C22×C67semidirect
265. #482C22×C67semidirect
266. #481C22×C67semidirect
267. #480C22×C67semidirect
268. #475C22×C67semidirect
269. #474C22×C67semidirect
270. #468C22×C67semidirect
271. #460C22×C67semidirect
272. #450C22×C67semidirect
273. #448C22×C67semidirect
274. C4×SD16C22×C67semidirect
275. #436C22×C67semidirect
276. #431C22×C67semidirect
277. #428C22×C67semidirect
278. #418C22×C67semidirect
279. #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×#556×C672
C67×#33532
C67×#33242
C67×#32262
C67×#31932
C67×#31032
C67×#30942
C67×#29752
#6172267